+ All Categories
Home > Documents > V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [...

V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [...

Date post: 17-Jun-2020
Category:
Upload: others
View: 3 times
Download: 0 times
Share this document with a friend
199
Transcript
Page 1: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

������������ �������������������������� ���� ����� �����������! "�#������$�%��� ���� "�� ���&������')(��������� *����

"��+�����#���,( -.!�������� �� /���0�����

12436587:9;7<7:=6>@?�9A2@2458B24C24365�BD58E�1DFG24?�5IH42$CKJ�E83@LM7:9;NO7P1DHDB1DE8E8QR9;5IB�EI3@LK7:9;NO7

CKJ�24365)SPH69UTV58F47:9A2WLXCKJ�YW24FO1Z2O3DN@Q[LKBV5JRCKF�24365�BD58\MF4585�CKJBVCVN@24CKF$CKJ�E8369;Q]CV7:CKE83@L

^`_a]bdcfe�ghbji0elk]mUnpo4qGrtsu�v ghmUw6x�yzy|{

Page 2: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

m

} ~ ����� ���V�%�%� �����*�

� sZ��r�q v _ g�m��zs�b�q���bdsZm��Pb�sZ���hmU�P�W��wUqz�|�z�Pb�q�b�sZ��e�c|bdsZq�gPc|�ZnZ��gPbdsZ�.bd��g��P�<q���bdsZ�����|m�bd��n� mU�Z��nZqz�%� q v _ ghmU�zs�b u r[bd�¡e��£¢Rcfe�w�m�¤f��n�� _ �j�Zm�k���gh��m�b _ q��&aRbdgde�bdsZr�w _ nZ�,¥¦���zcZwUe�bdm�qz��§|¨ª©�«|¨i�cZ�£e�rt¬R�Zq®­¦w���nZ�z���<����b��"c|�hbje�w¯­°e _ �¦�W�£�<e�nZ��q��Dbds|�¡c|���£q��Ke�� _ ��e�bd��ghmAe�wIr�q��Rbde�mU�|��npmU�O±qzg&nZ��g�m¯kz��np�²g�qz�p±Gb�sZmU�°b�sZ���hmU��¨

Page 3: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

mUm

³µ´V¶�·®¸:¹fº�·� sZm��jb�sZ���hmU�j������¬R��bdq v g�q®k]mUn|�»� v ��r�bdghqz��r�q v m�r»nZmUe��z�Zq��hbdm�r�bdq]qzwU�&b�sZg�qzc|�zspb�sZ�»e�n|k�e��Zr����P����bq��Me�bdqz�PmUr��<q]nZ��w�wUm��Z�»b���rtsZ�Zm�¢RcZ����¨°¼½���hcZ����q��Mq v e�r�m¯b _�¾ e��|n�m¯bd�&r�q��Z��m��hbd����b�b�g��le�bd�P����b¦�²g�qz��IqzcZ�Zn�b�q��²gh�����¿bte�b����dÀ[±M�ÂÁ|r�m�b���n��¿bte�b�� v q v cZwUe�bdm�qz���<q]nZ��w�wUm��Z�.e��Zn�� v ��r�b�gde�wÃ�²�le�b�cZg��<�z���]Ä��g�e�bdm�qz�Åe�gh��nZ��e�w�b»­¦m�b�sO¨ � sZ�µ����r�e v �µ�Æe�r�bdq�g»bd��rtsZ�Zm�¢]c|�ÇmU�/c|����nÅb�q�e�r�r�qzcZ��b"�²qzg»q v e�r�m�b _mU�����P��gh�z����b£ÈfcGÁ)e��Zn�e�b�qz�PmUr v q v cZwUe�bdm�qz��rle�wUr�c|wAe�b�mUqz�Z��¨ u b�sZqzghqzcZ�zs�bdgh�le�b��<���Rb»q��`wUq®­����g�m����j�<���"�W��g¦q v e�r�m�b _ mU�¦�zm¯kz���#mU�.bds|�»r�qGnZ� u i u a�xG{l©Z±f­¦sZm�rts#�PqGn|m�¤f����e�n|�Æy�©,¤fwU���¦�²q�g��c|�Z����¢]c|����b v q v cZwUe�bdm�qz� v g�q]r����h��m��Z��kGmUe u i u a�r�q�wUwUm���m�qz�fe�wÉĽgde�nZmAe�b�m¯k��Êg�qzcGbdmU�|����¨ � sZ�.���ËÄrle v ���Æe�r[bdqzg&�ÂÁ v g�������m�qz�Z��e�g���e��fe�w _ b�mUr�e�wUw _ �ÂÁ]bd���|nZ��np�²g�qz�Ì�Wq�cZ�ZnÇbdqÊ�²g����0�hbde�bd����±8e�w�wUq®­¦mU�Z�q v e�r�m�b _ bdq��I�ÇnZ�le�w�b"­¦m¯bdsÍ���PqGq�bdsZw _ e�r�g�q����ÊbdsZ�Çm�qz�Zm��de�b�mUqz��b�sZg�����sZq�wUnO¨ � sZ�.r�qz��bdm��RcZm�b _e vZv g�q�e�rts#q��Kb�sZmU�¦bdsZ����m���mU�¦r�qz��bdgde��hbd��n#­¦m�b�s�b�sZ�/cZ�hcfe�w4bd��rtsZ�Zm�¢]c|�"q��Îm��Rb�g�q]nZcZr�mU�Z�Pe���mUqz�GÄmU�hmU�Z� v wAe��h��e"�PmUr�g�q�¤Z��wUn<­¦sZmUrts.�zg�e�nZcfe�w�w _.Ï nZmU�h��qzw¯kz����ÐZbdsZ�0s|mU�zsZw _ �ÂÁGr�m�b���n.¢Rcfe���bdc|�Ñ��sZ��wUw��mU��b�q�bds|�µr�qz��b�mU�RcZcZ�p¨ u r�qz� v g���sZ���|��m�k�� v wUe���m��/qzc|b�wUm��Z��n�­¦sZmUrts�­¦m�wUwÒe�w�wUq®­Ór�q�wUwUm���m�qz�fe�wÉÄgde�nZmAe�b�m�k����<q]nZ��wUwUm��Z�¡q��Me�g��Zm¯bdgde�g _ sZ�lelk _ � v ��r�mU����¨ � sZmU�&�hrtsZ���P�0­ qzgh¬R�jm��pe"�Z��­�r�qzc v wUmU�|�v m�r�bdc|g���±8�²qzg¦­¦sZmUrts#�Z��­ÕÔ£e�cZ��b��Æe�r�bdq�g���sZe®k��/�I������nZ��k���wUq v ��nO¨ � s|�/�ZcZ�|nZwU��nGÄ ¾½Ö®× À¿Ø@ÙÒe��Zn�ZcZ�|nZwU��nGÄ ¾½Ö®× À¿Øp�Zw�qGrt¬ v g�m��<m¯bdm�k����Îq��Ibds|mU�Ã�<q]nZ��wZsfelkz���W�����PnZ��k���wUq v ��nO±|e��`sfe��`e0bd���hb`�<q]nGÄcZw��`�²qzgKe���m��Z�zwU�Ò�GÄÚ��sZ��wUw|q��fbdsZ�`�ZcZ�ZnZw���nGÄ ¾ËÖ®× ÀËØOÙÉÛ£�ZwUq]rt¬ v g�m��<m¯bdm�k���¨ u �hrtsZ���P�°�²qzgKe�c|bdq���e�b�mUrnZ�[bd��gh�<m��fe�b�mUqz�"q��@�Zc|�ZnZwUm��Z�0r�cGbhĽq�Ü@�Îsfe��`e�wU��q0�I�����<n|���Pqz�Z�hb�gde�b���nO¨ � sZ�¦�Z��­Ý�Pq]nZ��wGmU�ÎnZ�ÂÄ��m��z�Z��n<m��<e0�PqGn|cZwAe�gK��e��|�Z��gMb�q/e�w�wUq®­Í�²c|bdc|g��¦nZ��k���w�q v �P����bd�Ãbdq¡�W� v ��g¿�²qzg��P��n,mU�Êb����hbde��Zw���hb�� v �,e��Zn�bdq�bte�¬��Çe�n|k�e���bte��z��q��¦�Pq]nZ��gh��sZmU��s v ��gh�²q�g��<e��Zr��<r�qz� v c|b���gh��¨ � sZ� v q v c|wAe�Äbdm�qz�O±Rq v e�r�m�b _ e��Zn�r�qz��bdmU�RcZm¯b _ �<q]nZ��w�wUm��Z�£e�g��¦bds|���µ�ZghqzcZ�zs�bÒbdqz����bds|��g m��Pb�sZ�������Z��gde�b�mUqz�Pq��e.� _ ��bdsZ�[bdmUrP� v ��r[bdghcZ���²qzg0����g�mU���»w�mU�Pm�b�­°elkz��wU���Z��bds|�/q�� s _ nZghqz�z���ZmUr,� _ �¿bd���P��¨ u � v ��r�m¯¤frr�q]nZ�¦mU�έ¦ghm�b�b����,bdq¡e�w�wUq®­ÝÞzß � nZm�k���ghb�qzg ^ e�w��<��gÎ����ghmU���Òw�mU�Pm�bKqz�Z����ghk�e�b�mUqz�Z�Îb�q¡�I��e��fe�w _ �h��nO±g��[bdcZgh�ZmU�|�»nZmUe��z�Zq��hbdm�r°mU�G�²qzg��<e�bdm�qz�"qz� v wUe����<e���w���r�b�g�qz�PnZ���Z��m¯b _ ±Rbd��� v ��gde�bdcZgh��e��Zn<gh��r�qz�,Ä�Zm��fe�bdm�qz�,�hbde�bd��¨ � sZ��r�qGn|��m��Ò�hsZq®­¦��bdq v ghq:k]m�nZ����qGq]nP¤|bd�Ãbdq£b _ v mUrle�wIn|m�kz��ghb�qzg`� v ��r�b�gde/e��Zn­¦mUw�wW�I�0nZ�[kz��w�q v ��npmU��bdq,e�� u i u aÇ� v ��r�b�gde�w@¤|b�bdm��Z� v ghqGr���nZcZgh��¨ � sZ�0�²c|b�cZg��0nZm�g���r�bdm�qz�Z� �²q�ge vZv wUmUr�e�bdm�qz�µq��Dbds|�£­`qzgh¬.e�g��0qzcGbdwUm��Z��n@¨

Page 4: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

m�mUm

³Çº:à4áDâ]ã<äÆå]æ6çIå]è�å]áW·®¶¼jsfelkz�,�W�������²qzg¿bdcZ�fe�bd�ÊnZcZghmU�Z�µb�sZ�Êr�q�cZg��h�Êq��`b�sZmU��bdsZ����mU��b�qÇsfelkz�Êbds|�ÊsZ��w v e��Zn���c vZv qzg¿bq�����e�� _ v ��q v w���¨Å¼ÚbPmU�"­¦m¯bds v wU��e���c|g��µbdsZe�bP¼¡b�sfe��Z¬�� _ ��c v ��g¿k]mU��q�g<éÎghq��˨Îê�c|�zsa]cZ�,Ä�P��g����²q�g�bdsZ�Ê���Rb�sRcZ��mUe����p±De�b�b�����bdm�qz��bdq.nZ��bde�mUwë± v e�bdmU���Zr��Pe��|n�­`�le�w�bds�q��`mUn|�le���­¦sZm�rts�sZ���s|q:­`��nµe�bÃ��k���g _ �¿bte��z�¦q��Ibds|��éÎsfi"¨�¼D­ q�cZwUn<e�wU�hq£wUmU¬��°bdq0bdsZe��Z¬"� _jì qzmU��bÃ��c v ��ghk]mU�hqzgÎ�²g�qz�¥ u oα@i�g�¨I¥jm�rtsfe�g�n�ê�e�ghg�mU�hqz���²q�g�sZm����hc vZv qzghb�e��Zn����Zr�qzc|gde��z���<����bl¨ � sZe��Z¬R�£e�wU�hq<b�qµi�g�¨í m��z��w ^ e�nZ�Z��wUwK�²q�g�bdsZ�Pe�n|k]m�r��Êbdsfe�b£sZ�<sZe��0�zm�k����)e��Zn��²qzg0sZm���k�e�wUcfe��|wU�"sZ��w v mU��ghcZ�Z�Zm��Z�u cGbdqz�¿bdg�c|r�bdc|g���¨P¼�e��%�zg�e�bd�[�²cZwMbdqpéÎg�q��˨ � c|ghb ^ ��s|g�mU�|�z��g��²qzg£e�wUwUq®­¦m��Z�µ�P�<e�r�r������£bdq.sZm��q v e�r�m�b _ r�qGn|�µe��Zn��²qzg£bdsZ�<sZ��w v e��Zn�e�n|k]mUr��<b�sfe�b»sZ�<�ze®k��µ�P�Pn|cZg�m��Z�#m¯bd�¡r�qz��kz��g��hmUqz��b�qu i u a�xG{l©|¨<¼�e���mU�Zn|���|b���n�bdq�i�g�¨DÞ�mU�%o6e��Z�.­¦sZq#���Z��cZgh��n�b�sfe�b»e�w�wMmU��bd��gde�r[bdm�qz�Z��­¦m�b�s¥ u o$g�e������PqGq�bdsZw _ e��Zn�­¦sZq�sZe���bte�¬����)e���mU��bd��g����hb£mU��b�sZmU��­`qzgh¬#�²g�qz��bdsZ�Ê�I����mU�Z�Zm��Z�Z¨¼°­`qzcZwUn#e�wU�hq�w�mU¬��£bdqPbdsfe��Z¬pi�g�¨IéV�[bd��g0î�r�ïðsZm�ghb���g��²q�gjb�sZ�/cZ�h���²cZwDnZm���r�c|����m�qz�Z��e�b�k�e�ghmUqzcZ��hbde��z���µq��0bds|mU�<­`qzgh¬I¨ � sfe��Z¬R�.e�w���q)b�q�bds|��¥jcGbdsZ��gh�²qzghn u�vZv wU��b�qz�Ýo6e��Iqzgde�bdqzg _ �²q�g<bds|��z���Z��ghqzcZ� v g�q®k]mU�hmUqz�Çq��MeÊrle��h�»el­ e�g�n�nZcZghmU�Z�"b�sZ�£r�qzcZgh���¡q��V� _ �hb�cZnZm�����¨î _ r�qzwUw����zcZ���&mU�Çbds|�»�zghqzc v sfelkz�»e�wU�hq,�W������eÊ�hqzcZg�r��»q��M���|r�qzcZg�e��z���<���Rbjbds|g�qzcZ��sZqzc|b¦bds|�­`qzg�¬8¨K¼W­ qzc|wUn/w�mU¬��Ãb�qjb�sfe��Z¬¡Ô0qzghnZqz�£�²qzg6sZmU�4�²g�m����Zn|��sZm v ±�s|qz�Z���¿b _ e��Zn¡b�sZ�Ò��e�� _ m��Z��m��zs�b��²cZwr�q��<�P����bd�Vbdsfe�bKsZ�&sfe��K��e�n|��¨K¼De��ñ�zg�e�bd�[�²cZwGb�q£i�g�¨�� qz�Z�Zq�g ^ e�w�wAe��Zr��Ò�²q�gKbdsZ� cZ���&q��8�hqz�P�q��8sZm��K�RcZ�P��ghmUrle�wGghqzc|bdm��Z���Me��Zn/�²qzgKe�wUw�q®­¦mU�Z�j�<�Òbdq�bde�¬��&q�c|bÎe�� _ v s _ ��mUr��Îg���wAe�b���nÊe���g����h��m�qz�qz�Êe���¢Rcfe��hsÊ�fe�wUw]mU�|�hbd��e�n"q��Zqz�"� _ r�q�� v cGbd��g�¨ � sfe��Z¬R�Ãe�wU�hq��²qzgVmU��bdghqGnZc|r�mU�|���P�`b�q���¬RmUm��Z�Z¨� sfe��|¬]��bdq � e�b�sZwU�������²q�g�gh�le�nZm��Z�<b�sZg�q�cZ�zs��"c|rts�q��Mb�sZmU�¦bdsZ����m���±Oe�� _ g�����e�mU�Zm��Z�,� v ��wUw�mU�Z��PmU�hbde�¬����Çe�gh��� _ g���� v q��Z��m��ZmUw�m�b _ ¨ � sfe��Z¬R�µbdqÍi�g�¨ ê�e�ghk�� _ u �ZnZ��g��hqz�Ý�²qzgÇe�w�wjb�sZ��sfe�ghn­`qzg�¬�bdsfe�b"sZ�ÇnZmUnÅq�� u i u aŧ|{z{�Ä£m¯b"�<e�nZ��b�sZ�ÇnZ�[kz��w�q v �P����b,q��¦bds|�Çr�q]nZ�µmU�ÅrtsZe v bd��g©.e v qz���hmU�Zm�wUm�b _ ¨0¼j­`qzcZw�n�e�wU�hqÇwUm�¬��»b�qÇbdsfe��|¬]�£i�g�¨4î#e�ghbdm���ò"Ðóî�c|wUwAe��Z�»�²qzg�sZ��w v mU�Z���<�Êe�bk�e�g�m�qzcZ�°�¿bte������&nZ��� v m¯bd�0�I��mU�|�"��q,�ZcZ� _ ­¦m¯bdsÇsZm��&q:­¦�Ç­`qzg�¬8¨ � sfe��|¬]�¦e�wU��q"b�q u w�wAe��P�²qzg°sZm���²g�m����Zn|��sZm v e��Znp�²q�g¦e�wUwIbdsZ�£r�q�� v cGbd��gje�nGkGm�r��£sZ�£sfe��¦�zm¯kz���O¨� q�b�sZ�p�²q�wU¬�e�bÊbd�le ¾ i0elk]mUn@±°Ô£e�mUwë±Òê����|n _ ±¦Ô0qzghnZqz�O± u w�wAe��Íe��|nÞ�qzs|�8À[±Òbdsfe��|¬]�,�²qzgÊbds|�wAe�cZ�zsZ�Êe��Zn��²qzg"�Zq�bÊ�zm�k]m��Z�#�P�Çe�g���nÍrle�ghnO¨)a v ��r�mUe�w`b�sfe��Z¬R�"bdq�� _$Ï �²��w�wUq®­Ìqzw _ � v mAe��|��Ðî#e�ghrle�q|±Zê����Zn _ e��Zn.a]�le��O¨Î¼½� v e�ghb�mUr�c|wAe�gÎbdsZe��Z¬R� bdq"ê����|n _ �²q�g°e�w�w8b�sZ���ze��<��� q��O��¢Rcfe��hsO±k]mU�hm�bd�°b�qÊbdsZ�0­`��mU��sRb���ghqGq��Ñe��ZnÇ�²qzg°b�sZ�£�Z��­�­ qzghnZ�¦bdsfe�b _ qzc.bte�cZ��sRbj�P��¨� q0� _ �²ghmU���ZnZ�K�²ghqz�ô� sRcZghrts ¾ Þ�mU�p±�i0elk]mUn,i"±�éÎsZmUw�m v ±�i0elkGm�n,î)±za]bd�[kz�°õʱza]bd� v sZ���@±Gi0el­¦�O±a]b���kz�ÊÔ,±Wi����Zm�����±@o4����± � e�b�sZwU�����8À°bdsfe��Z¬R���²qzg�e�wUw4bds|�/�zgh�le�b�bdmU�P���&­`�"sfelkz�/sfe�nO±@e��Zn��²q�g

Page 5: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

m�k

�Zq�b°e���¬Rm��Z�"�P��bdq/�hbdq v bte�w�¬]m��Z�£­¦sZ���.¼M­`����b¦qz�Çe��Iqzc|b � _ bdsZ����m���¨ u w���qZ±Rbdsfe��Z¬R� bdq u k]g�mUw�²qzg b�sZ�£e�w�wIb�sZ�0���Zr�qzcZg�e��zmU�|�"���<e�mUw��`b�sfe�b&�hsZ�0�����Rb�e��|nµbdq � e�g����O±8�°e�ghw _ ±ZöKm�qz�fe"e��|n � qz��²qzg°b�sZ��m�g&sZ��w v q®k���g¦bdsZ�£wAe��hb&�²��­ _ ��e�g���¨î _ �Æe��<m�w _ sfelkz�£�W�����#e,�hqzcZghr��¡q��K���Zr�q�cZgde������P����b�e��Znp��c vZv qzg¿bjb�sZg�q�cZ�zsZqzcGbje�wUwO� _ wUm¯�²��±�Zq�b ì cZ�¿b¦bdsZ�¡wAe��¿b&�²��­ _ �le�g����²q�g°bdsZm��¦b�sZ���hmU��¨°aGqP� v ��r�mAe�wOb�sfe��Z¬R�j�zq,bdq,� _ �Pq�bds|��g�±Z�Æe�bdsZ��ge��ZnP�Zghq�bdsZ��g���±Göfg�e�����g`e��Zn<¼Ëe�mU�O¨ � sfe��Z¬R�`e�wU�hq£b�q/o6e�ghmU�ZnZe0e��Zn í e�b�sfe��fe���wZ�²qzgMbds|�j�|wU���h��m��Z�bdsZe�b£bdsZ� _ sfelkz�<�W�����O¨P¼�­`qzcZw�n�w�mU¬��,b�qÇbdsfe��|¬�� _ �j�Zr�w��<i0elk]mUn��²qzg£���Zr�qzc|gde��zm��Z�.�<�"b�q�hbde _ qz�<�²qzg`b�sZ�0éÎsfi"¨ � sfe��Z¬R�¦e�gh��nZcZ��b�q"� _ Ô0gde��.îpcZ�Zg�q»�²qzg sZ��g �Æe�m¯bds|�²cZw8�hc vZv qzghb¦e��Znr�q��Zr���g���e��Zn�¼�e���m��ZnZ���|bd��n�b�qp� _ éKe v e�o4q]rts��²q�g�bdsZ�,�[Á|e�� v w��Êq�� s|mU�0wUm¯�²�Êe��Zn)e�w�wKsZm��­`qzg�nZ�jq��D­¦mU�hnZqz�p¨

Page 6: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

~ �%÷ø�)ùô÷ø��ú

û»âIáW·®å]áW·:¶ üýÒþ ¶�·Pâ8ÿ�� þ ç��6¸:åG¶ þ��ýÒþ ¶�·Pâ8ÿ�� ¹Z´VäÆå]¶ �6þëþÆþ� � áI·®¸:â4æ�Dº�· þ âIá �{z¨¯{ u m��<�&e��Zn�q�� ì ��r[bdm�k���� ¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {{z¨ x ^ e�rt¬R�zghqzcZ�Zn.bdsZ��qzg _ ¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{

{z¨ xG¨�{ ßÎ�hrle v �¡�Æe�r�bdq�g&�fe���m�r0nZ��¤Z�Zm�b�mUqz�Z��¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{{z¨ xG¨óx ßÎ�hrle v � v g�qz�fe��ZmUw�m�b _ e��Znp���P��gh�z����b¦ÈfcGÁô¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {:x{z¨ xG¨ § u �Z��q�g v b�mUqz�Ç�Æe�r[bdqzg¦e��|n v q v cZwUe�bdm�qz�Z� ¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {�{z¨ xG¨ © � q��Rb�mU�RcZcZ��q v e�r�m�b _ ¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {��{z¨ xG¨ o4m��Z�0�Zg�qze�nZ���|mU�Z��¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®«{z¨ xG¨�� éDq v cZwUe�bdm�qz�Ç�PqGnZ��wUw�mU�Z�ͨ¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ x��

{z¨ § ßKÁZe�� v w����Vq��fq v e�r�m�b _ e��|nÊ� v ��r�b�gde�wG����g�mU���Km��Êe��hb�g�q v s _ �hmUrle�wZe��Zn/wAe��Iqzgde�bdqzg _v wAe����<e�� ¨¡¨»¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ §��{z¨ §|¨�{ � s|���hqzwAe�gÃe�bd�Pqz� v s|��g����V�Zgde��|rtsZmU�Z�¡wUm��Z�°g�e�bdm�qz�Ãe��`e£n|mAe��z�|qz�hb�mUr°�²qzg

n|��nZcZr�mU�Z�Pq v b�mUr�e�w@nZ� v bds%¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ §��{z¨ §|¨óx o4q®­�s _ nZghqz�z��������g�mU���¦�<���"�W��g&q v e�r�m�b _ �`öfcZ�hmUqz�Çn|m�kz��ghb�qzg v wAe��h��e ©Z{{z¨ §|¨ § êjmU�zsÍs _ nZg�q��z���$����ghmU���P�P���/�W��gÊq v e�r�m�b _ �paGqzwUe�g v g�q��<m��Z���Zr��.o _ Ä

�<e��.wUm��Z����±Zo _ ��e���r�qz��bdmU�RcZc|� e��Zn ^ e�w��<��g°wUm��Z��gde�nZmAe�b�mUqz�$¨»¨¡¨¡¨ ©�{z¨ §|¨ © êjmU�zs.�h��g�m����¦qz�Z�h��g¿k�e�bdmUq��Z���Òö8c|��mUq��pn|m�kz��ghb�qzg v wAe����<e ¨£¨»¨¡¨»¨¡¨¡¨ ©�«

k

Page 7: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

k]m

� ��� ¹fº þ ·��ðè)â4æDå]äÆä þ á6ç�â8ÿ0äÆâ]ã ¶�åR¸ þ å]¶µè�å]è�´ÎåR¸�¶<ÿÚâ8¸ � â � �DäƹG· þ âIáð¹fáDæ�å]è�åR¸��ç8åGáI·���� � ºz¹fäƺ �Däƹ|· þ âIáD¶"!#�%$6å,åG¶�ºz¹ � å � ¸:â8´V¹Z´ þ ä þ ·��#¹fáDæ�¹Z´K¶�âI¸ � · þ âWá�ÿÚ¹fº�·lâI¸¹ �&� ¸:â8¹fº'$ ( �xG¨¯{ ¼½��bdg�q]nZcZr[bdm�qz��¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ G{xG¨ x ^ e��hmUr � sZ��qzg _ ¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ zxxG¨ § a v e�b�mAe�wIk�e�g�mUe�bdm�qz�µmU�Çb�sZ�£���PmU���hmUqz�.r�qG�*)<r�mU���Rb)¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��xG¨ª© éÎg�q]q��Vb�sfe�b¦�z���Z��ghmUr�+ ¾-, À°e��Zn/.ñk]��q v bdm�rle�wWnZ� v b�s#r�cZghk�����e�g��0k�e�wUm�n ¨¡¨¡¨ �«xG¨� éÎg�q�¤fw�� � sZ��q�g _ ¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��xxG¨0� ßMk�e�wUcfe�b�mUqz��q���+ ¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��xxG¨�� � qz� v c|bte�bdmUq��Çq�� v q v c|wAe�b�mUqz�Z� cZ��m��Z�Pe��Z��q�g v b�mUqz�Ç�Æe�r[bdqzgh� ¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ �z§xG¨01 ���hmU�Z�Êbds|�¡r�qGn|�»e���eÊw�mU�Z��g�e�bdm�qÊnZmAe��z�Zqz�¿bdmUr ¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��©xG¨ « ßKÁZe�� v w���q��&b�sZ�ÇcZ�h�µq��¦bdsZ��r�q]nZ���²qzgÊêj�µwUm��Z�PmUnZ���Rb�m�¤fr�e�bdm�qz��cZ�hmU�Z���°i£a

¢RcZmU�[b¦��cZ��qz�Z����ghk�e�b�mUqz�Z� ¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ���xG¨¯{®yÝöfcZghb�sZ��g¦bdsZ��qzg _ e��|n#r�q]nZ�¡nZ��k���w�q v �P����b2�°¼½�|r�wUc|��mUq���q��Îe<�fe�rt¬R�zghqzcZ�Zn#gde�Ä

nZmAe�bdmUq���¤f��w�n ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��§xG¨¯{®y|¨¯{ � s|��qzg _ ¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��§

xG¨¯{z{�� qz�Zr�wUcZ�hmUqz�Z� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��3 û»âWáI· þ á4� þ ·��ÍâIÿ�$ þ ç�$¶�å]¸ þ å]¶,â � ¹fº þ ·*� 576§|¨¯{ ¼½��bdg�q]nZcZr[bdm�qz��¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ �8�§|¨ x u �fe�w _ bdm�r�r�qz��b�mU�RcZm�b _ ¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ �8�

§|¨ xG¨�{ � q��Rb�mU�RcZm¯b _ q��V���PmU���hmUqz�Çr�qG�*)Pr�mU����b ¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ �81§|¨ xG¨óx � q��Rb�mU�RcZm¯b _ q��Ke��Z��q�g v b�mUqz�.r�q]�9)Pr�mU����b̨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ 1�©

§|¨ § � qz��bdg�e��hb¦­¦m�b�sÇbdsZ�£q]r�r�c v e�bdm�qz� v g�q��fe��Zm�wUm�b _ �²qzgh��e�w�mU�h� ¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ 1��§|¨ª© � sZ�£cZ���¡q��Vr�qz��b�mU�RcZqzcZ�¦����rle v �»e��Zn�e��Z�hqzg v b�mUqz�Ç�ÂÁ v g�������m�qz�Z� ¨£¨»¨¡¨»¨¡¨¡¨ 1�1

§|¨ª©Z¨�{ a]qz�<�0m�wUw�cZ�hb�gde�b�mUqz�Z�»¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ 1z«§|¨� � qz�Zr�wUcZ�hmUqz�Z� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ «z§

: ; â � �Dä²¹|· þ âWá$è�â4æ6å]äëä þ á6ç�¹fáDæXºzâIè � �4·®¹|· þ âIá < :©Z¨¯{ u �pq®kz��g¿k]mU��­�q��Vb�sZ�£�<q]nZ��w ¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ «�©

©Z¨¯{z¨�{ i��[¤f�Zm¯bdmUq��Çq��K�|wUq]rt¬��hb�g�cZr[bdcZgh���j�²qzg¦e�n|k�e��Zr���npr�qz� v c|bde�bdm�qz�.¨»¨¡¨¡¨ « �©Z¨ x Ô¡e�cZ��b&�Æe�r�b�qzg��¦g��[k]mU��m¯bd��nø¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ «z«

Page 8: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

k]mUm

©Z¨ xG¨�{ ^ e�rt¬R�zg�qzc|�Zn ¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ «z«©Z¨ xG¨óx Ô£e�cZ��b&�Æe�r[bdqzg&�[k�e�wUcfe�bd��n.�²qzg ì Ä ì r�qzc v w�mU�Z� ¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ «z«©Z¨ xG¨ § � ghqz���¦r�qzc v w�mU�Z�<Ô£e�cZ��b&�Æe�r�b�qzg��¦�[k�e�wUcfe�b���n ¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®y�y

©Z¨ § � sZ�£�Z��­�r�qzw�wUmU�hmUqz�Ze�w¯ÄÚgde�nZmUe�bdmÉkz�¦r�qGnZ�$¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®yzx©Z¨ §|¨�{ Ø@ÙMe��ZnpØ�g�����qzw�c|bdm�qz�%¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®yzx©Z¨ §|¨óx Ø@ÙÉÛ�g�����qzw�c|bdm�qz� ¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®y�§

©Z¨ª© u c|b�qz�<e�bdm�r�nZ��b���r[bdmUq��pq��Kbds|�¡�|cZ�ZnZw�mU�Z�Êr�c|bhÄÚq�Ü@��¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{�{©Z¨ª©Z¨�{ ßÎ�Z��g�� _ wU��k���w6mU��k����hb�mU��e�bdmUq�� ¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{�{©Z¨ª©Z¨óx � q�wUwUm���m�qz�fe�w8g���nZmU�¿bdghmU�Zc|b�mUqz�Ç�[k�e�wUcfe�bd��n ¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {:x�x

©Z¨� � qz�Zr�wUcZ�hmUqz�Z� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§zx( = � å]º þ ¹fä ÿÚåR¹|·>�D¸:å�¶9�4áW·�$6åG¶ þ ¶ÊÿÚâ8¸ÇºzâIáW· þ á?�&�Vè å]æ6ç8å�¶ � å]º�·®¸:¹fä ¹fá6¹fä��6¶ þ ¶ �@3A3

G¨¯{ aG��ghmU���jw�mU�Pm�b`� v ��r[bdgde�w4�<q]nZ��w�wUm��Z�/�fe�rt¬]��g�qzcZ�|n ¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§�§G¨ x Ô£���Z��g�e�wOr�qz� v qz�Z����bd�jq��Vb�sZ�£����g�mU���¦wUmU�Pm�bÒ�²�le�bdcZgh�ø¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§�©

G¨ xG¨�{ u b�qz�PmUr v q v cZwUe�bdm�qz�Z�µ¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§�©G¨ xG¨óx ò v e�r�m�b _ ¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§�G¨ xG¨ § o4m��Z� v g�q�¤fwU�0�[k�e�wUcfe�b�mUqz� ¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§�«G¨ xG¨ © ^ qzcZ�ZnGÄ �²g�����±f�Zgh���P���hb�gde�s|wUcZ�Z�Pe��|n�m�� v c|g�m�b _ r�q��Rb�g�m��Zc|bdm�qz�Z��¨»¨¡¨¡¨ {®§�«

G¨ § u � v ��r�m�¤fr¡r�q]nZ���²qzg¦�PqGnZ��wUw�mU�Z�¡bdsZ� ^ e�w��<��g°���PmU�h��m�qz�ÇmU�Çbds|�/Þzß � nZm�k���ghb�qzg�¨,{l©|{G¨ §|¨�{ i��[bte�m�wU�°q��Vb�sZ� v s _ �hmUr�e�w v g�q��ZwU��� ¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l©|{G¨ §|¨óx îpqGnZ��wUw�mU�Z�/­`qzgh¬ ¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l©�xG¨ §|¨ § öVm�b�b�mU�Z�»bdsZ�»Þzß � nfe�bde,e��Zn.nZmAe��z�Zqz�¿bdmUr0gh���hcZw�b���¨�¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l© �

G¨ª© � qz�Zr�wUcZ�hmUqz�Z� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {��x6 BÇå]á6å]¸:¹fä&ºzâIáDº�äC�D¶ þ âIáV¶"¹fáDæÝÿD�4·��D¸:åpã"â8¸:à � ( :

�|¨¯{ Ô£���Z��g�e�wOr�qz�Zr�w�cZ��m�qz�Z�je��Znp��c|�<�<e�g _ ¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {��©�|¨ x u gh�le��&�²qzg°�²c|b�cZg��0­`qzg�¬�¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {�8

�|¨ xG¨�{ ßKÁ]bd���Z��m�qz�Z�¦bdq u i u apxG{l©ð¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {�E��|¨ xG¨óx öfc|b�cZg��0­`qzg�¬<�²qzg°bdsZ�£�h��g�m����¦wUm��<m¯b r�qGn|� ¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {�E��|¨ xG¨ § éDq v cZwUe�bdm�qz�Ç�[Á]b����Z�hmUqz�.�²qzg&sZ��e®k _ � v ��r�mU��� ¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {�E1

FÊþ ´Kä þ â8ç8¸:¹ � $G� � 6 �

Page 9: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

kGm�mUm

³ BǹH�DáW·<ÿÚ¹fº�·®â8¸µå�ü|¹fäC�6¹|· þ âIá � 6AIu ¨¯{ ^ e��hmUrMghcZwU���O�²qzgO�zg�e v sZm�rle�w��[Á v g������hmUqz�»e��Zn£�<e��Zm v cZwAe�b�mUqz�jq��|� wU���Z��rts]Ä¿Ô0qzg�nfe��

r�q]�9)Pr�m�����bd�<¨»¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {>��«u ¨ x u�vZv wUmUr�e�bdm�qz�Êq��IbdsZ�j�zg�e v sZm�rle�wZ�<e��Zm v cZwUe�bdm�qz�"�P��bds|qGnÊ�²qzg`� wU���Z��rts]Ä¿Ô0qzg�nfe��

r�q]�9)Pr�m�����bd�<¨»¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {���§u ¨ xG¨¯{ Ô£e�cZ��b&�Æe�r[bdqzgh�&�²qzg ì Ä ì r�qzc v wU��npm��Zm�b�mAe�wWe��Zn.¤f�fe�w4�hbte�bd��� ¨»¨¡¨»¨¡¨¡¨ {���§u ¨ xG¨ x Ô£e�cZ��b&�Æe�r[bdqzgh�&�²qzg&r�ghqz���ËĽr�q�c v wUm��Z�ñ¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {>1�y

Page 10: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

J �Mú�� �LK M �V�ON ��ù ú

{z¨¯{ ßμ � ê��¡§zy�©QPu qz�|����g¿k�e�bdm�qz�.q��Vb�sZ�£��qzwUe�g°rtsZghqz�Pqz� v s|��g�� ¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ x{z¨ x u �pmU�<e��z��q��Vb�sZ�£mU�Z�hmUnZ�0q��Vb�sZ�»Þzß � bdqz¬�e��<e�¬Ê¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ §{z¨ § i�mAe���gde�� q��DbdsZ�£w�mU�Z�0q��V��m��zs�b¦cZ����n#m��.b�sZ�£���hrle v � v g�qz�Ze��ZmUw�m�b _ �²qzg��<e�w�mU��� {®§{z¨ª© ¥�e��Z�z�0q��Dk�e�wUm�nZm�b _ q��V�Zg�q�e�nZ���Zm��Z�Pe vZv ghqlÁ|m���e�bdmUq��Z� �²qzgSRUT*¨¡¨£¨»¨¡¨»¨¡¨¡¨ xzx{z¨� ¼½qz�GÄÚ�<m�r�g�q�¤f��w�n v wUq�b�bd��nÇ�²qzg°k�e�g�m�qzcZ� v wUe����<e v e�g�e��<�[bd��g��&¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ x�©{z¨0� ßKÁZe�� v w��&q��IbdsZ�&s _ nZg�q��z����«�Ä¿x ^ e�w��<��gK�h��g�m����Ò� v ��r[bdgde�wfwUmU�|� v g�q�¤fwU�¦e��Î�[k�e�w¯Ä

cfe�b���np� _ bdsZ�¡éÃéÃér�q]nZ�#¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ x8�{z¨�� ¼½wUw�cZ�hb�gde�b�mUqz�µq��Me��.q v e�r�m¯b _ �<q]nZm¯¤f��nÇw�mU�Z� v ghq�¤fw�� ¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ x81{z¨01 a v ��r�b�g�cZ��q��Vb�sZ�»� 9Æ9 �"c|w�bdm v wU�[b°cZ�h��n�mU�ÇbdsZ�£�Zg�e��ZrtsZm��Z�,gde�b�mUq,�hb�cZn _ ¨¡¨¡¨ § �{z¨ « éÎwUq�b&q��Vb�sZ�»� 9ë9 w�mU�Z��gde�b�mUqÊcZ�h��n#m��ÇbdsZ�£nZ��nZcZr�b�mUqz�pq��Vq v b�mUrle�w@nZ� v bdsp¨»¨¡¨¡¨ §z«{z¨¯{®yðaGrtsZ����e�b�mUr0q��MÞzß � nZm¯kz��g¿bdqzg&�hsZq®­¦mU�Z�,bds|�£bde�g�����b v wUe�bd���¦e��Znpr�g _ q v cZ� v ¨ ©Rx{z¨¯{z{ u �Z�hqzg v bdmUq��.�Æe�r�bdq�g&g����hcZw�b��&�²g�qz� u i u apxG{l©Ê�²qzg&nZm¯kz��g¿bdqzg&r�qz�ZnZm¯bdm�qz�Z�#¨¡¨¡¨ ©�§{z¨¯{:xÝò v e�r�m¯b _ �<q]nZm�¤Z��n�w�mU�Z�¡gde�b�mUqµgh���hcZw�b����²qzg�V�W X8RZYµ�²g�qz� u i u a�x]{l©��²qzg�nZm¯Ä

kz��ghbdq�gjr�qz�ZnZm¯bdm�qz�Z� ¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ©z©{z¨¯{®§ðaGqzwUe�g v g�qz�PmU�Z���Zr��%RUTÍqz�Z����ghk�e�b�mUqz�#bde�¬������ _ b�sZ� ^ m�� ^ ��e�g0aGq�wAe�gjqz�Z�h��g¿Ä

k�e�bdqzg _ m���{l« ��y�¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ © �{z¨¯{l©ðo _ �<e����h��ghmU���0w�mU�Pm�b¦qz�|����g¿k�e�bdm�qz��q��Òeµ��q�wAe�g v g�qz�PmU�|���Zr���±Wbde�¬������²g�qz��bdsZ�

nfe�bde,m���aGrtsZ�PmU��n|��g&��b�e�wë¨ ¾ {l«z«z«��8Àµ¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ © �{z¨¯{� ^ e�w��<��g°����g�mU���¦wUmU�Pm�b`qz�Z�h��g¿k�e�bdmUq��pq��Vb�sZ�»Þzß � nZm�k���ghb�qzg&g����zmUq�� ¨»¨¡¨»¨¡¨¡¨ �yxG¨¯{ ßKÁZe�� v w��0q��DbdsZ� v ghqGr������hmU�Z�Pq v b�mUqz�|�&��r�gh�����#mU� u i u apx]{l© ¨¡¨£¨»¨¡¨»¨¡¨¡¨ �§xG¨ x aGrtsZ����e�b�mUr��jq��Vb�sZ�£nZmU�h¬��z��qz�<�[bdg _ r�qz�|��mUn|��g���n�� _ u i u a�xG{�©"¨£¨»¨¡¨»¨¡¨¡¨ �§xG¨ § aGrtsZ����e�b�mUr��jq��Vb�sZ�£r _ wUm��ZnZg�m�rle�wW�z��q��<�[bdg _ r�qz�|��mUn|��g���n#� _ u i u a�xG{�©�¨¡¨¡¨ �©

m¯Á

Page 11: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

Á

xG¨ª© i�mAe���gde��P�`q��Db�sZ�£� v e�bdmAe�w@nZm��hbdghmU�ZcGbdmUq��µq��6bdsZ�0���<m����hmUqz�Çr�qG�*)<r�mU���Rbje��&r�qz�GÄ��m�nZ��gh��np� _ u i u apxG{l©�¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ 8�

xG¨� ßKÁZe�� v w��0q��DbdsZ�£�z���Z��ghmUr".Õe��|n[+Ýr�cZghk���� v g�q]nZcZr���n#� _ u i u a�x]{l© ¨¡¨¡¨ �zyxG¨0� ßKÁZe�� v w��0q��Dk�e�g�mUq�cZ�&s _ n|g�qz�z��� v g�q�¤fw����je��¦rle�wUr�c|wAe�b���nÇ� _ bdsZ� u i u a�xG{�© �z§xG¨�� ßKÁZe�� v w��/q��Îb�sZ�"���<��g��z���Rb�ÈfcGÁ#wUmU�|�/gde�bdmUq<��k�e�w�cfe�bdm�qz�.kG�0r�qzwUc|�<��nZ���Z�hm�b _

�²g�qz� u i u apxG{l©�¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ � xG¨01 ßKÁZe�� v w���q��6bds|�¡q��Z����ghkz��n��°i£aµ� v ��r[bdghcZ�ÑcZ�h��npmU�µbds|�0wUmU�|��mUnZ����bdm�¤Zrle�b�mUqz�

­ q�g�¬�¨¡¨»¨¡¨»¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ � �xG¨ « \ ¾�]4^ À`�²cZ�Zr[bdm�qz�Ç�²qzg&sZ��w�mUcZ� ��k�e�w�cfe�bd��n.cZ��m��Z� u i u aÇ©Z{:x*¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ �z«xG¨¯{®yÝßÎ��rle v �¦�Æe�r�b�qzg v wUq�b`�����Z��gde�b���n�c|��mU�|� u i u aPxG{l©£�²qzgÎbds|��sZ��wUm�cZ�ôwUm��Z�&mUn|���GÄ

bdm¯¤frle�b�mUqz�µ­`qzgh¬�¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��yxG¨¯{z{Ío4mU�Z�0g�e�bdm�q v wUq�b&�z���|��gde�bd��n�cZ��m��Z� u i u a.xG{l©Ê�²qzg`bdsZ�£sZ��wUmUc|� w�mU�Z��m�nZ����b�m�¤|Ä

rle�b�mUqz�µ­`qzgh¬ ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ �G{§|¨¯{ i�mAe���gde��Óm�wUwUc|�hbdg�e�bdm��Z�£bdsZ��r�qz��bdghmU�Zc|b�mUqz��q��Dq:k���ghwAe vZv mU�Z� v g�q�¤fw���� b�q"bdsZ��mU�GÄ

bd���zgde�b�mUqz�Çm��.��¢R� ¾ §|¨¯{:xzÀø¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��«§|¨ x ßÎ�<m�����m�qz�Çr�q]�9)Pr�mU����b¦�²qzg°b�sZ� ^ e�wU�P��g ����ghmU��� v w�q�b�b���npe�r�g�qz�h�¦b�sZ�£����g�mU���¦wUmU�Pm�b_1z§§|¨ § u �Z�hqzg v bdmUq��Êr�q]�9)Pr�m�����bÃ�²qzgKbds|� ^ e�wU�P��gK����ghmU��� v wUq�bhbd��n<e�r�g�q����Mbds|�j�h��ghmU���Îw�mU�Pm�bU1��§|¨ª© � qz��bdm��]c|qzcZ�Ã���hrle v � v g�qz�fe��ZmUw�m�b _ v wUq�bhbd��n,�²q�gÃb�sZ�¦rle��h��q��Wb�sZ� ^ e�wU�P��gÎ�h��ghmU���

mU�µb�sZ�¡�²c|��mUq��pn|m�kz��ghb�qzg𨻨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ 1z«§|¨� � qz��bdm��]c|qzcZ�Ò����r�e v � v ghqz�fe��Zm�wUm¯b _ v w�q�b�b���n<�²q�gÃb�sZ�jrle��h��q��Ob�sZ��o _ �<e����h��ghmU���

mU�Ç�hqzwAe�g v g�qz�PmU�Z���Zr��0r�qz�ZnZm¯bdm�qz�Z�p¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ «zy§|¨0� u �Z�hqzg v bdmUq��.�Æe�r�bdq�g v wUq�bhbd��n.�²q�g°bdsZ�¡o _ ��e��.�h��g�m����¦wUm��<m¯bʨ¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ «�x©Z¨¯{ � sZ�£nZ��bde�mUw��&q��DbdsZ�£�ZcZ�|nZwUm��Z�Ê��rtsZ���<� v ghq v q�����n.�²qzg°b�sZ�£�Z��­ �Pq]nZ��w�¨¡¨¡¨ « ©Z¨ x éVq v cZwAe�b�mUqz�Çgh���hcZw�b��¦�²ghqz� b�sZ�£s _ n|g�qz�z���#�PqGn|cZwU��q��Vb�sZ�£�Z��­ �ÒĽ¥�r�q]nZ�,¨¡¨ {®y�©©Z¨ § ,a`�b c gh����c|w�bd�<�²qzg��Ad0§�e��Çe)�²cZ�Zr[bdm�qz�q��£nZ���Z��m¯b _ �²qzgfe�ghd*§���e��|n ]4^ d

{�i y 2j�{®y8kml ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®y�«©Z¨ª© éÎwUq�bV��sZq®­¦m��Z��bdsZ�`¤f�Z�°�¿bdg�c|r�bdc|g��°�Zw ì � v w�m�bhbdmU�|��m��/bdsZ� Ønd�§"o xjb�gde��Z�hm�b�mUqz�

�²qzgSe�gpd�x81|±f§8�rq ©z©Ê¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{ly©Z¨� s#tCu8vxw t ��� v e�gde�bdmUq��.�²ghqz�ys�tzu>{|wzv r�����bdghqzmUnÇÄ}eE~ nZ� v ���Zn|���Zr�� ¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{�©©Z¨0� s t u8vxw t ��� v e�gde�bdmUq��.�²ghqz�ys t u>{|wzv r�����bdghqzmUnÇÄÒØ�nZ� v ���|nZ���Zr��𨻨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{�©

Page 12: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

ÁGm

©Z¨�� s `�b c ��� v e�gde�b�mUqz�Ç�²ghqz�ys `�b r�����bdghqzmUnµÄ}eE~ nZ� v ���|nZ���Zr�� ¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{��©Z¨01 s `�b c ��� v e�gde�b�mUqz�Ç�²ghqz�ys `�b r�����bdghqzmUnµÄÒ�pnZ� v ���|nZ���Zr�� ¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{��©Z¨ « s `�b c ��� v e�gde�b�mUqz�Ç�²ghqz�ys `�b r�����bdghqzmUnÇ�²qzg���� ~C� � v e�g����Rb&ÄÒw4nZ� v ���ZnZ���Zr��µ¨¡¨¡¨ {z{>�©Z¨¯{®y�s `�b c ��� v e�gde�b�mUqz�Ç�²ghqz�ys `�b r�����bdghqzmUnÇ�²qzg����&��� � v e�g����Rb&ÄÒw4nZ� v ���ZnZ���Zr��µ¨¡¨¡¨ {z{>�©Z¨¯{z{�s `�b c�� ��� v e�gde�bdmUq��.�²ghqz��s `�b c r�����bdg�q�mUnÇ�²qzg���� ~C� � v e�gh����b&Ä}eE~&nZ� v ���ZnZ���Zr���¨ {z{�1©Z¨¯{:x�s `�b c�� ��� v e�gde�b�mUqz�.�²g�q���s `�b c r�����b�g�qzm�n��²qzg����&��� � v e�gh����b Û�d ¾ Ù4�Ý{>XzxzÀ`Ä}eE~

nZ� v ���|nZ���Zr�� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{�1©Z¨¯{®§�s `�b c�� ��� v e�gde�b�mUqz�.�²g�q���s `�b c r�����bdghqzmUn.�²qzgS������ � v e�gh����b°Û�d ¾ ÙH�X{>XzxzÀ`Ä}eE~

nZ� v ���|nZ���Zr�� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{�1©Z¨¯{l©�s `�b c�� ��� v e�gde�bdmUq��.�²ghqz��s `�b c r�����bdg�q�mUnÇ�²qzg � ��~C� � v e�gh����b&ÄÒ��n|� v ���ZnZ���Zr�� ¨ {z{l«©Z¨¯{��s `�b c�� ��� v e�g�e�bdm�qz���²g�qz��s `�b c r�����bdg�q�mUn��²qzg��m� ��� � v e�g�����b ì d0wÉÄt{���xP�hbde�bd�/Ä&�nZ� v ���|nZ���Zr�� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{l«©Z¨¯{>��s `�b c�� ��� v e�gde�bdmUq��.�²ghqz��s `�b c r�����bdghqzmUn.�²qzg��m� ��� � v e�gh����b ì d�w��"{��zx»�¿bte�b���ÄÒ�nZ� v ���|nZ���Zr�� ¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {z{l«©Z¨¯{���s `�b c�� ��� v e�gde�bdmUq��.�²ghqz��s `�b c r�����bdg�q�mUnÇ�²qzg���� ~C� � v e�gh����b&ÄÒw4nZ� v ���|nZ���Zr��µ¨¡¨ {:x�y©Z¨¯{>1�s `�b c�� ��� v e�gde�bdmUq��.�²ghqz��s `�b c r�����bdg�q�mUnÇ�²qzg������� � v e�gh����b&ÄÒw4nZ� v ���|nZ���Zr��µ¨¡¨ {:x�y©Z¨¯{®«���s `�b c�� X8��s `�b c �²qzg � � ~C� � v e�gh����bl± ì d�wÉÄt{��zxñ¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {:x]{©Z¨ x�y���s `�b c�� X8��s `�b c �²qzg������� � v e�gh����bl± ì d�wÉÄt{��zxñ¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {:x]{©Z¨ xG{���s `�b c�� X8��s `�b c �²qzg���� ~C� � v e�gh����bl± ì d�w��Ê{>�zx ¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {:x]{©Z¨ xzx���s `�b c�� X8��s `�b c �²qzg � ���� � v e�gh����bl± ì d�w��Ê{>�zx ¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {:x]{©Z¨ x�§ u Ä k�e�wUcZ���£�²q�g¡b�sZ�<g�e��Z�z�Pq�� v s _ �hmUr�e�w v e�gde��P��b���gh�¡b�sfe�b¡bdsZ�<e�c|b�qz�hb�g�cZr[bdcZgh�

r�q]nZ�0­°e��jghcZ�O±fØ�e��ZnneE~°nZ� v ���Zn|���Zr�� ¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {:xE�©Z¨ x�©Ý¼½qz�.¢Rcfe�nZghc v q�wU��r�qzwUw�mU��m�qz�fe�w8g�e�bd���¦e��¦e"�²cZ�Zr�b�mUqz�Çq��Db���� v ��gde�b�cZg��£e��Zn.�GĽ�hsZ��w�w�{:x8�©Z¨ x��¼½qz�.¢Rcfe�nZghc v q�wU�£r�qzw�wUm���mUq��fe�w8gde�b����je��je"�²cZ�Zr[bdmUq��pq��K�GÄÚ��sZ��wUw6e��|n/eE~ ¨»¨¡¨¡¨ {:x8�©Z¨ x8�ð� g�m¯bdmUr�e�w0nZ���Z��m¯b _ e�b#­¦sZm�rts�bdsZ�)mUqz�X¢Rcfe�nZghc v qzwU�)r�qzw�wUmU�hmUqz�|�Ç�fe�wAe��Zr���bdsZ�

� v qz��bte��|��qzcZ�j���PmU�h��m�qz�Çgde�b����&�²g�q�� bds|� ì d�wÉÄt{��zx¡����b�q��VwU�[kz��w��*¨£¨»¨¡¨»¨¡¨¡¨ {:xE1©Z¨ x��Ý� g�m¯bdmUr�e�w��]Ľ��s|��wUw v wUq�b4�hsZq®­¦mU�Z�¦bds|�`r�g�m�b�mUr�e�w�k�e�w�cZ�Ãe�b6­¦sZmUrts£bds|�Òm�qz�£r�qzw�wUm���mUq��Z�

�fe�wUe��Zr���b�sZ�£� v qz��bte��|��qzcZ�j���PmU�h��m�qz�Çgde�b����&�²g�q�� bds|� ì d�wÉÄt{��zx/�h��bjq��VwU�[kz��w�� {:xE1©Z¨ x81���� ��� �[k�e�wUcfe�b���n,�²qzgÃbdsZ���ÒĽw�mU¬��¦� _ �¿bd���Ì�²qzgÎb�sZ�»{>�>�[x8�>�tx*�n��Ø?�"���[b q��4wU�[kz��w���{®§�y©Z¨ x�«���������� ^z���� `�b���¡¢`�b0£ ~ �9¤ ¾|] ��� ^¥���*  À�À ��� �[k�e�wUcfe�b���nÅ�²qzg"b�sZ���ÒĽwUm�¬��Ç� _ �¿bd��� �²qzg"bdsZ�

{����Âx����Âx*�n��Ø4�����[b�q��Kw���kz��wU�¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§�y

Page 13: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

ÁGmUm

©Z¨ §zy���� ��� X ¾ ����� � � ^¥�*�� `�b���¡�`�b�£ ~ �9¤ ¾�] � � ^z����  À �"� ÀM��k�e�w�cfe�b���nÊ�²qzgMbds|���ÒÄÚwUm�¬��&� _ �¿bd����²qzg°bds|�P{>�>�tx����Âx*�n��Ø?���h��bjq��VwU�[kz��w�� ¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§G{

©Z¨ §|{���� ��� X ¾ ����� � � ^¥�*�� ��¡#�9¤ ¾|] ��� ^z����  À ����¦�§©¨¥` À6�[k�e�wUcfe�b���n¡�²qzg4bdsZ�°�ÒĽwUm�¬��Ã� _ �¿bd����²qzg°bds|�P{>�>�tx����Âx*�n��Ø?���h��bjq��VwU�[kz��w�� ¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§G{

G¨¯{ ßKÁZe�� v w��Mq��Rbds|� v q v cZwAe�b�mUqz��nfe�bte�¤fwU� v g�q]nZcZr���n»� _ bdsZ�Îs _ nZg�q��z���¡r�qzw�wUm���mUq��fe�w¯Ägde�n|mAe�b�m�kz��r�q]nZ��¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§8�

G¨ x � qz��bdm��]c|cZ� e��Z�hqzg v bdmUq��.�Æe�r�bdq�g��&�²qzg°bds|�»o _ �<e��.����ghmU��� ¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§81G¨ § � qz��bdm��]c|cZ�Óelkz��gde��z��nf+Å�²qzgMb�sZ�jo _ �<e��Pr�qz��bdm��RcZcZ� v w�q�b�b���nPe��Òe0�²cZ�Zr�b�mUqz�

q�� v wAe��h��e"w����Z��bds ¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®§81G¨ª© u r�qz� v e�ghmU�hqz�.q��MéÃéÃé v ghq�¤fw����&­¦m�b�s#Ô0g�m����pÐ �&�ÂÁ v g�������m�qz� ¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l©zyG¨� ¼½wUw�cZ�hb�gde�b�mUqz�pq��Mb�sZ�"wUm��Z���jq��Ã�hmU�zs�b�q��ÎbdsZ� �£� §µm��Z�hb�g�cZ�P����b�mU��b�sZ� ^ e�wU�P��g

����g�mU���¦wUmU�Pm�b`qz�Z�h��g¿k�e�bdmUq��Z� ¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l©z§G¨0� éVq v cZwAe�b�mUqz�Çgh���hcZw�b��¦�²q�g°b _ v mUrle�w4nZm�k���ghb�qzg&r�q��ZnZm�b�mUqz�|�"¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l©�©G¨�� u �fe�w _ bdm�rle�w�w _ r�qz��b�mU�RcZqzcZ������rle v � v ghqz�fe��Zm�wUm¯bdmU����e��|n e��Z��qzg v bdm�qz� �Æe�r�b�qzg��

��k�e�w�cfe�bd��nÇ�²qzg°b _ v mUr�e�w4nZm�k���g¿bdqzg&r�qz�ZnZm¯bdmUq��Z� ¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l© G¨01 ò v bdm�rle�w�w _ bdsZm�rt¬Çe��Zn.bds|mU� v q v cZwAe�bdmUq��Z� �²qzg°b _ v m�rle�wOn|m�kz��ghb�qzg¦r�qz�Zn|m�bdm�qz�Z� ¨ {l©��G¨ « � qz��bdghmU�Zc|b�mUqz�|�£b�q�b�sZ�����<m�����m�qz��r�q]�9)Pr�m�����b/�²ghqz��bds|��nZm¯Ü@��g�����b»s _ n|g�qz�z���

r�qz� v qz�Z����b���¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l©�1G¨¯{®yð� qz��bdghmU�Zc|b�mUqz�|�Db�q�bdsZ�`bdq�bde�wRmU��bd���Z��m¯b _ �²ghqz� e�wUw�bds|� s _ nZg�qz�����,e��ZnÊm�� v cZg�m¯b _

���Pm�bhbd��gh�%¨¡¨»¨¡¨¡¨»¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {l©�1G¨¯{z{Íi�mAe����Zqz�hb�mUrj¤Zb¦bdq v cZw����£©�§ ��§�G± ^ e�wU�P��g°�h��g�m����jwUm��<m¯b � v ��r�bdghcZ� ¨»¨¡¨»¨¡¨¡¨ {l©z«G¨¯{:xi�mAe����Zqz�hb�mUrj¤Zb¦bdq v cZw����£©�§ ��§81|± ^ e�wU�P��g°�h��g�m����jwUm��<m¯b � v ��r�bdghcZ� ¨»¨¡¨»¨¡¨¡¨ {��yu ¨¯{ Ô£g�e v sZm�rle�wWg�� v g���������bde�bdm�qz��q��Kbds|�/� w����Z�hrtsGÄ¿Ô0qzghnfe��.r�q]�9)Pr�m�����bø¨»¨¡¨»¨¡¨¡¨ {���yu ¨ x*Ô£g�e v sZm�rle�wWg�� v g���������bde�bdm�qz��q��Kbds|�£ïðm��z�Z��gS��Ä ì r�q]�9)Pr�mU����b�¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {��]{u ¨ § � sfe��Z���¡m��.r _ r�w�mUr0qzghnZ��g¦q��DïðmU�z�|��g¦§�Ä ì � _ �/�Wqzw�¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {��]{u ¨ª© � qz�"�Zm��ZmU�|�"ïðmU�z�Z��g¦§�Ä ì � _ �/�Wqzw�� ¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {���x

Page 14: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

J �Mú�� �LK ª «­¬ ®Vù ú

{z¨¯{ aGcZ�P��e�g _ q��Vb�sZ�»nZe�bte"�²qzg&bds|�/� 9ë9 x����Âx*�[�m���� � ��x��:x*�H�S�*¯&~C� � b�gde��Z�hm�b�mUqz��²qzg&�le�rts�g�e��hb���g¦��r�e�� v qz�hm�bdm�qz�*¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ©�y{z¨ x aGcZ�P��e�g _ bte��Zw��0q��Db _ v mUrle�w v g�qz�PmU�|���Zr�� v s _ �hmUr�e�w v g�q v ��g¿bdm����ñ¨£¨»¨¡¨»¨¡¨¡¨ ©��{z¨ § aGcZ�P��e�g _ bte��ZwU�<q��&b _ v mUr�e�w`�P�le����²g���� v e�b�sZ�"q��¦s _ nZg�q��z��� v s|q�bdqz�|�"mU��e

��qzwUe�g v ghqz�PmU�Z���Zr�� ¨£¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ © 1xG¨¯{ ò��Z����ghkz��n�e��|nÇrle�w�r�cZwUe�bd��nÇ��qzwUe�g&e�b��<qz� v sZ��g��0sZ��wUmUc|� w�mU�Z��gde�b�mUqz� ­¦m¯bds.e��|n

­¦m�b�sZqzc|b&b�sZ�£��ÜW��r�b���q��Kq v e�r�m¯b _ ¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ ��§©Z¨¯{ ¾ Û°��ÛGÀÊr�q�c v wUm��Z��bdq ¾ Û�±?�$Û�±ÉÀÊr�qzc v wUmU�|�³²�e��|n¥�k�e�wUc|���P��k�e�wUcZe�bd��n$�²qzg

k�e�g�m�qzcZ�&wU�[kz��wU�&q��Vg�����qzw�c|bdm�qz� ¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {®yG{G¨¯{ � sZ�£nZmAe��z�Zqz�¿bdmUr�k�e�wUcZ���jgh��b�cZg��Z��n��²ghqz� b�sZ�0¤Zb¦��sZq®­¦��mU�Ǥf�zcZgh�´G¨�{�{ ¨¡¨¡¨ {��yG¨ x i�mAe����Zqz����nÇ��w���r�b�g�qz�pnZ���Z��m¯bdmU���&�²g�qz�ÌbdsZ�¡öVïXê�î e��Ze�w _ ��m��&q��Mî���m��z� ¨¡¨¡¨ {��yG¨ § � sZ�£nZmAe��z�Zqz�¿bdmUr�k�e�wUcZ���jgh��b�cZg��Z��n��²ghqz� b�sZ�0¤Zb¦��sZq®­¦��mU�Ǥf�zcZgh�´G¨�{®x ¨¡¨¡¨ {�]{G¨ª© i�mAe����Zqz����nÇ��w���r�b�g�qz�pnZ���Z��m¯bdmU���&�²g�qz�ÌbdsZ�¡öVïXê�î e��Ze�w _ ��m��&q��Mî���m��z� ¨¡¨¡¨ {�]{u ¨¯{ ¾ Û°��ÛGÀÊr�q�c v wUm��Z��bdq ¾ Û�±?�$Û�±ÉÀÊr�qzc v wUmU�|�³²�e��|n¥�k�e�wUc|���P��k�e�wUcZe�bd��n$�²qzg

k�e�g�m�qzcZ�&wU�[kz��wU�&q��Vg�����qzw�c|bdm�qz� ¨¡¨¡¨»¨¡¨»¨¡¨£¨»¨¡¨¡¨»¨¡¨»¨¡¨¡¨»¨¡¨£¨»¨¡¨»¨¡¨¡¨ {���«

ÁGmUm�m

Page 15: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� � ��� ����

� � ������� � ������� �

����� ���! "$#&%(' )+*-,/.10324�65+.1"

798;:=<?>@8BAC<D:=<E:=<GFIHKJCL;<MACNOHQPSRT>@HQUV:=J9UVHKN;ACW=WX:=P;YZFIH\[GN;:]RQYQP;H\<M>D:XJ^RQP_RQWa`K<D:X<bHQFdc;W]RQ<MUeRT<G>D8;[DH\LBY\8H\f;<MAC[hgQRi>@:XHQP(HQFb>@8;A6:X[Z<McjA6J6>@[DRBkmln>o:X<Z[MACJCHQY\P;:X<MACN(>@8;RT>1<DcpACJq>@[MH\<DJ6H\cr`(HQFG8;:XYQ8 >@ACU+cpAC[@Ri>@L;[MAc;WXRQ<DUVRQ<Oc;[MH�gs:=N;AC<tc;[MH\f_RQfBW=`u>@8;AvU+H\<M>S<MACP;<M:=>D:=g\AwRQPBNx[MA6gQA!RQW=:XP;Y�>y`KcpAwHTFzU+A!RQ<ML;[DA6UVA6Pr>{ 8;:=J|8}J!RTP}fpA~UVRQN;AoH\P}RT>@H\U+<Ck��sL;J|8 <h>@L;NB:XAC<^RQ[MAzRQcBc;WX:=J!RQf;W=A { 8;A6>D8;AC[�>@8;Amc;WXRQ<DUVR�:X<3[DA��U+HQ>@A^RQ<G:XPORT<M>@[MH\c;8r`K<D:=JC<�H\[GP;ACRQ[Dfr`V:XP�>@8;A^W]RQfpH\[@Ri>@H\[h`\k�798BA�ACU+:=>M>@:=P;YmRT>@H\U+<?RQ[DA�:XPVUeRQPr`JC:=[DJ6L;UV<h>|RQPBJCAC<b[MA6g\ACRQWX:=P;YoHQFp>@8;A3ACPrgK:X[MH\P;U+ACPr>G:XP { 8;:=J|8+>D8;A6`�W=:XAERQP;N+:=>�:=<4>@8;A6<DA�<D:=>DL_RT>D:XH\P;<{ 8;:=J|8zc;[MH�gK:=N;A�>@8;AGN;:]RTY\P;H\<h>@:XJ�HQc;cjHQ[M>@LBP;:=>D:XAC<6k��EH\P;<D:=N;AC[/>D8;A?<MH\W]RT[/J|8;[DH\U+H\<Mc;8;AC[M:XJ!�TJCH\[MH\P_RQWc;WXRQ<DUVR}RQPBN�>@8;AVW]RQfpH\[DRT>@H\[h` FIL;<M:XH\P�cBW]RQ<MUeRBkt798;Aq`�8_R!gQAOUeRT[D�QA6N�<D:=UV:=W]RQ[M:=>D:XAC<^FI[DH\U�>@8BAcpH\:XPr>�HTF�gK:=A { HQF�<DcpACJ6>D[@RQW�ACU+:X<D<M:XH\P��BRQWa>@8;HQL;Y\8O>@8BAC:X[?ACP;A6[DYQ`O:XP;c;Ls>9UVA6J|8_RQP;:=<DU+<9RQ[MAZ��L;:=>DAN;:a��A6[DACPr>Ck3�EHQ>@8�RQ[MA~8;:=Y\8t>@ACU+cpAC[@Ri>@L;[MA~c;WXRQ<DUVRQ< { :=>D8�>D8;A~<MH\W]RT[�LBc;cjA6[�Ri>@U+H\<Dc;8BAC[DAmACW=ACJ��>@[MH\P�>DACU+cjA6[@RT>DL;[DA-[@RTP;Y\:XPBYvFI[MH\U �����x���T��� >@H����}�x�!�Q�|� :=P�>@8BA���L;:=A6>V<DL;P�RTP;NL;cu>@H�� �S� ���\¡@� :=Pu¢_RQ[DA6<Ck�798;A�UVHKN;A6[DP£FIL;<D:=H\P�cBW]RQ<MUeRB�G<DLBJ|8�RQ<�>@8;A}¤QH\:XPr>V¥�L;[DHi�cpA!RQP�7�H\[DLB<t¦§¤\¥�7�¨m>DH\�iRQUeRT�p��:X<~HQF3<DH\U+A { 8;RT>�8;:=Y\8;AC[m>@A6UVcpAC[DRT>@L;[MAORT>��ª©t����� ¡ � >DH�«�������Q¬@� ¦­©\®B¯�°±>@HV�Q�r®B¯�°~¨�:XPS>@8;AoJ6H\[DAT�_f;LB> { :=>D8S>@ACU+cpAC[@Ri>@L;[MAC<3:XPS>@8BAm�TA6`tN;:ag\A6[M>@HQ[c;WXRQ<DUVR+gQAC[M`}<D:=UV:=W]RQ[E>DHe>@8BA&<DH\WXRQ[�J!RQ<MAQk1798;A&J6H\[DAmFIL;<D:=H\P c;WXRQ<DUVRV8;RQ<ZACW=ACJq>@[DHQP N;ACPB<D:=>D:XA6<HQF9�$©t�²�!�K³µ´9¶·���²����³ �q¸q¹}º ´ { 8;:XW=A�>D8;A+<DH\WXRQ[oJCH\[MH\P_RQWbc;WXRQ<DUVR-:=<oHQF�W=H { AC[mN;A6P;<D:a>y`�Ri>� �e����� ¬ ¸�¹ º ´ f;Ls>V[M:X<M:XP;Y(<h>@A!RTN;:XWa`�RQ<�H\P;AtU+H�g\AC<+N;H { P�>D8;[DHQL;Y\8£>@8;AtJ|8;[MH\UVHQ<Dc;8;A6[DATk�EHQ>D8e>y`KcjA6<�HQF/cBW]RQ<MUeRoN;:X<Mc;W]R!`�UVRQ[D�TACNOYQ[@RQN;:=ACPr>@<?:XP�>@A6UVcpAC[DRT>@L;[MAZRQPBNON;ACPB<D:=>y`}¦IJCH\U+c_RQ[MA

Page 16: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

»�:=Y\L;[DA��Qk=�\¼b½�Am©\�Q¾«¿À H\f;<MAC[MgTRT>D:XH\P-HQF�>@8;Ao<MH\W]RQ[EJ|8;[DH\U+H\<Mc;8;AC[MAzRi>9Á�ÂG�·¾r�BÃ@�\�\�\���K>|RQ�TACPL;<M:XP;Y�>@8BA~¥�ly7·:=P;<M>D[DL;U+ACPr>3H\P�fpHrRQ[MN->D8;A~�;Ä1½ZÄ<Mc_RQJ6ACJC[DRTFÅ>!k

>@8BA~<MH\W]RQ[�>D[@RQPB<D:=>D:XH\Pt[MACY\:=H\P}RTP;N�8B:XY\8-<D8;ACRQ[�W]R!`\A6[D<^RQP;N ½9�yU+HsNBA1cjA6N;AC<h>|RQW=<^:=P->D8;A1FIL;<D:=H\PJ!RT<DA�¨��KLB:=>@A?RQc_RT[M>�FI[MH\U«WXHKJ!RTWKNB:X<M>DL;[Df;RQP;JCA6<4RQPBN&:XP;<h>|RQfB:XWX:a>@:=AC<Ck/798BAC<DA?>@ACU+cpAC[@Ri>@L;[MAC<�A6P;<DLB[DA>@8;RT>Z>@8BA&c;W]RT<DUVRV:X<^8;:XYQ8;W=`-:XHQP;:X<MACN(RQPBN >@8;A&[MACWXRT>@:ag\ACWa`-WXH { N;ACPB<D:=>D:XA6<ZUVACRQP}>D8_RT>Z>D8;A&:XHQPACPrgK:X[MH\P;U+ACPr>&:=<mFµRT[oFI[MH\UÆWXHKJ!RQW4>D8;AC[MUVHKNB`KP_RQU+:XJ�A6�KLB:XWX:=f;[D:=L;U-k À W=<DH;�Ç>@8BAeN;:=UVA6P;<D:=H\P;<oHQF>@8BAVc;WXRQ<DUVRQ<1A6P;<DL;[MA+>D8_RT>mH\P;A+8_RQ<o:=P�Y\A6P;AC[DRQW�A!RT<M`vAC<MJ!RQcpAQ��>@8_Ri>~:=<oH\cB>@:=J!RQW�>@8;:=P;P;AC<M<C��HQF>@8BAmA6UV:a>D>DACNt[@RQNB:]RT>D:XH\Pdk4»ÈHQ[�>@8;A6<DAm[DA!RT<DH\P;<6�_>@8;Am<DcpACJq>@[@RTWÇRQP_RQWa`s<M:X<�FIH\[9>D8;AC<MAm> { H+>y`KcpAC<�HQFc;WXRQ<DUVR+8;RQ<�N;Aqg\A6WXH\cpACN(<M:XU+:XW]RT[DW=`V:XP}ReU+HsN;A6WXW=:XP;Y�c;:=J6>DL;[DA { 8;:XJ|8 Aqg\A6Pw:=P�>D8;A~FIL;<M:XH\P�J!RQ<MA:X<9HTFÅ>@ACP-J!RTWXWXA6N ÉÊJ6H\[DHQP_RQW§ËakÌ :=>D8²>@8BA N;A6gQACW=H\c;U+ACPr>tHQF1Í^�n[@R!`u>@ACW=AC<MJCH\cpAC<tRQPBN�P;A { <@Ri>@ACW=WX:a>@AC<eRQP;N�<Mc_RQJCA6JC[DRTFÅ> { :=>D8<DJ6:XA6P�>D:=Î_J�c;R�`KW=HrRQN;<+¦IAQkÏY;k�>@8BA�½^L;f;fBWXA+�sc_RTJCA�7�ACWXA6<DJ6H\cjAT���;Ä1½ZÄ&�d73Ð À ��¥3����½ ÀZÑ^Ò Ð À ¨>@8BAC[DA-RQ[MAOFIA { <DcpACJq>@[@RTW9[DACYQ:XH\P;<zWXA6FÅ>z>@8_RT>�JCRQP;P;HT>�fjASU+A!RQ<ML;[DA6Ndk�798;:=<�UVACRQP;<&>D8_RT>�>@8BA<DcpACJq>@[DR�[DA6<DL;Wa>@:XPBYwFI[MH\U RQWXU+H\<h>+RQPr`�cpH\<M<D:XfBWXA�RT>@H\U+:XJO>D[@RQPB<D:=>D:XH\PuJ!RTP²fpA}H\f;<MAC[hg\ACN RTP;NN;:XRQY\P;H\<h>@:=JZ:XPBFIHQ[DUVRT>@:=H\POH\fB>@RQ:XPBACNdk

Page 17: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©

»�:=Y\L;[DA��QkÓ�s¼ À P�:=UeRTY\AZHQF�>@8;Ao:=P;<D:=N;A1HQF�>@8;A~¤\¥�7Ô>DH\�iRQUeRT�pk

�scpACJq>@[DHQ<DJCHQc�`OJ!RQPSfpA1N;:=gK:=N;ACNO:=Pr>@H&>@8BAZf;[DH\RQNSJ!RT>DACY\HQ[D:XA6<�HQFbÉ { R!g\A6WXA6P;YQ>@8�Ë_RQP;N ÉÕ:XPr>@A6P;<D:a>y`pËf_RQ<MACNv<DcpACJq>@[MH\<DJ6H\cr`\k~Ì�R!g\A6WXACPBYQ>@8v<DcpACJq>@[MH\<DJ6H\cr` >|RQ�TAC< { R�gQACW=ACP;YQ>D8v:XPBFIH\[MUeRi>@:XHQP�<ML;J|8�RT<WX:=P;A3JCA6P�>D[DH\:=N;<?RQP;NOW=:XP;A { :=NB>@8;<ERQP;NV[DA6>DL;[DPB<E:XPsFIH\[DUVRT>@:=H\P�H\P+>@8;A�WXHKJ!RQW_cBW]RQ<MUeRoJCH\PBN;:=>D:XH\P;<<DLBJ|8VRT<�¢_H { g\A6WXHKJC:=>D:XA6<C�i:XH\Pm>@ACU+cpAC[@Ri>@L;[MAC<�RQPBN&:XP~JCAC[h>|RQ:=P&JC:=[DJ6L;UV<h>|RQPBJCAC<�:XH\PzRQP;N&A6WXACJq>@[MH\PN;A6P;<D:a>@:XA6<Ck�ln>mN;HKAC<Z>D8;:X<oc;[M:XUVRQ[D:=W=`SgK:]RO��P;H { WXA6N;Y\A+HQFG>@8;A�c;WXRQ<DUVReN;:=<M>@[M:Xf;Ls>@:XHQP FIL;P;J6>D:XH\PB<RQP;Nu[DA6W]RT>DAC<�>@8BA { R!g\ACW=ACP;YT>@8²:=PBFIH\[MUeRT>D:XH\P�>@Hv>@8;A�RQc;c;[MH\c;[M:]RT>DASc;W]RQ<MUeR(c_RT[@RQU+A6>DAC[D<6�GFIHQ[AqÖBRQU+c;W=AZ[DA6W]RT>D:XP;Yz>D8;Ao<D8;:aFÅ>9:XPtR&W=:XP;A�JCACPr>D[DH\:=Nt>DH�R�N;H\c;c;W=AC[9<M8;:=FÅ>9:=PO>@8;A1A6UV:a>D>D:XP;Y�Ri>@H\U+<CklyPeYQACP;A6[@RQWÈ:=>G:=<GPBHQ>EPBACJCA6<D<DRQ[M`+>@H~�KPBH { N;A6>@RQ:XW=ACN+:XPBFIH\[MUeRi>@:XHQP�H\P+>D8;AZRi>@H\U+:XJ9cpH\c;L;WXRT>@:=H\P;<FIH\[�<DL;J|8�>@A6J|8;P;:=�KLBAC<Ck�798B:X<�:=<�U+H\<M>DW=`Q�Qf;LB>�P;HQ>4RQW { R!`s<6�\>@8;AEJ!RQ<MAQk�»_H\[�AqÖBRQU+c;WXAT�\N;:XRQY\P;H\<M:XP;Yc;WXRQ<DUVReACW=ACJq>@[DHQPwNBACP;<M:=>@:=AC<ZFI[MH\UªH\fB<DAC[hg\A6N�JCHQWXWX:=<D:=H\P_RQW=W=`OfB[DHrRQNBACP;A6N�W=:XP;A { :XNB>D8;<1[DA6�KLB:X[DA6<:XPsFIH\[DUVRT>@:=H\P1H\Pm>@8;AGcjHQc;L;W]Ri>@:XHQP;</FIH\[/>@8;A?WX:XPBA�f;[DHrRTN;ACP;:=P;Y�JCRQWXJ6L;W]Ri>@:XHQP;<Ckǽ�H { A6gQAC[6�KfB[DHrRQNBW=`<DcpA!RT�K:=P;Y;�B>D8;AC<MAZ>@A6J|8;P;:X��L;A6<�RT[DAZ:=P;N;ACcpACPBN;ACPr>�HQFÇ>@8;A�cjH\cBL;W]RT>D:XH\PeN;:X<h>@[D:=f;LB>D:XH\P { :=>@8B:XPO>@8BAACU+:=>M>@:=P;Y(RT>DH\UV<6k�798;AtH\f;<MAC[hgQRi>@:XHQP;<�HQF^<DL;J|8�<DcpACJ6>D[@RvN;A6gQACW=H\cjA6N²RTWXH\P;YQ<D:XNBAe>@8;AtU+HsNBACWX<{ 8;:=J|8�L;<MA { R!g\A6WXACPBYQ>@8�<DcpACJ6>D[DH\<MJCH\cr`SFIH\[9>D8;AC:=[9:XPr>@A6[Dc;[MA6>|Ri>@:XHQPdk798;:=<E>@8;A6<D:=<�FIHsJ6L;<DA6<9W]RQ[MY\ACWa`�H\PSL;<D:=P;Y�:XPr>DACP;<M:=>y`O<McjA6J6>D[DH\<MJCH\cr`tRQ<�R&UVACRQP;<�HQF�:=P�>DAC[Mc;[DAq>@:XPBYH\f;<MAC[hgQRi>@:XHQP_RQW�N;RT>|RBk�lyP-:XPr>DACP;<M:=>y`O<McjA6J6>D[DH\<MJCH\cr`-:XPBFIHQ[DUVRT>@:=H\Pe<ML;J|8}RT<9WX:XPBAZ8;AC:=Y\8r>@<3H\[�W=:XP;A

Page 18: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾

[@Ri>@:XHQ<VRT[DA-L;<DA6Ndk 798;A}8;A6:XY\8r>VHQFZ<McjA6J6>D[@RQW�WX:=P;AC<�:X<+<M>D[DH\PBY\W=`�NBACcpACP;N;A6Pr>SH\P�>D8;A Ri>@H\U+:XJcpH\c;L;WXRT>@:=H\Pt<M>@[ML;J6>DL;[DA~HQF�>D8;AzA6UV:a>D>D:XP;Y�RT>DH\U+<C�_>@8�L;<^UVHKN;ACW=<3N;AC<M:XY\P;A6N RQ<3>@HKH\W=<3FIHQ[�:=Pr>@ACPK�<D:a>y`£<DcpACJq>@[DHQ<DJCHQc�`�RQ[DAtÎ_[MUVWa`�Y\[DHQL;P;N;A6N²:XP²RT>@HQUV:=JScjH\cBL;W]RT>D:XH\P£U+HsNBACWXW=:XP;YBk À YrRQ:XP£>@8BAN;:XRQY\P;H\<h>@:=J�UVHKN;ACW=<18_R!g\A+>@[@RTJ|�QACN�>@8;A+N;A6gQACW=H\c;U+ACPr>zHTF�H\f;<MAC[hgQRi>@:XHQP;<mRQP;N�>D8;A+:XP;J6[DA!RT<D:XPBY[DA6<DH\W=LB>@:=H\PO>@H { 8;:=J|8�<McjA6J6>@[DR+JCRQP-fjAoH\f;<MAC[hg\ACN�klyP;:a>@:]RTWXW=`+:a> { RQ<�UVRQ:XP;Wa`V>D8;Am<M>@[MH\P;Y+[DA6<DH\P_RTP;JCAmWX:=P;AC< { 8;:=J|8�J6H\L;WXN-fpAm[MAC<MH\W=gQACN RTP;N�:=Pr>@ACPK�<D:a>y`�<McjA6J6>@[MH\<DJ6H\cr` { RQ<�L;<DA6Nu>DH([DACWXRT>@AV>@8;AO:=Pr>@ACPB<D:=>D:XA6<&HQF�<DL;J|8uWX:=P;AC<~>@H >@8BAScjH\cBL;W]RT>D:XH\P<M>D[DL;Jq>@L;[MAQ� { 8;:XJ|8+JCHQL;WXN�>@8BACP+fjA3[DA6W]RT>DACN�>@H1>D8;A9c;W]RT<DUVRZACU+:=>D>DAC[�H\[�A6WXA6J6>@[MH\P�N;ACPB<D:=>D:XA6<?RTP;N>@A6UVcpAC[DRT>@LB[DAC<�:aF�:XPBFIH\[MUeRi>@:XHQP { RQ<+R�gTRQ:=W]RQfBWXAeHQP�RTWXWEHTF�>D8;AtN;H\U+:XP_RTP�>�RT>@H\U+:XJVc;[DHKJCA6<D<MAC<{ 8;:=J|8VN;Aq>@A6[DU+:XP;A�>D8;A3<M`K<M>DACU-ËÕ<GcjHQc;L;W]Ri>@:XHQP&N;:X<h>@[M:Xf;LB>D:XH\P�k�798;A3[MAC<DHQP_RQP;J6A^W=:XP;A6<�HQFp:XPr>@A6[DAC<h>>@A6P;N;A6N�>@HvJ6H\UVAOFI[MH\U�cBW]RQ<MUeRQ<zHQF^>D8;AtJCH\[MH\P_RQW?>y`scpAQ�ERQP;N�RQ<�<DLBJ|8²JCHQL;WXN£fpA-U+HKN;ACW=WXACN[DA6W]RT>D:=gQACW=`}<D:=UVc;Wa`\k+798;AC[MA { AC[MAeP;H-H\cB>D:XJCRQW4>@8;:=J|��P;AC<M<zA6�jACJq>@<~RQP;Nw>@8;A+N;H\U+:XP_RTP�>mRi>@H\U+:XJc;[MHsJ6AC<D<MAC<9>@8;RT>98_RQNO>DH�fjA1J6H\P;<M:XN;A6[DACN { AC[MA1W]RQ[MY\ACWa`+>D8;A1<M>D[DH\PBY�N;:XcpH\WXA��×RQW=WXH { ACN+JCH\W=WX:X<M:XH\P;RQWRQP;N�<DcpH\Pr>|RQPBACH\L;<z[@RT>DAC< { 8B:XJ|8�NBA6>@A6[DU+:XP;A6N�>@8BAVcpH\c;L;WXRT>@:=H\P;<oHQF�>D8;A+c;[D:=P;JC:=c_RQWb��L_RQPr>@L;U<D8BACWXW=<Ck�Ì :=>@8m>@8;A?f;[DACRQNB>D8�:XP~>D8;A { R!g\A6WXACPBYQ>@8�[@RTP;Y\AGHQFÈ<DcpACJq>@[DHQUVAq>@AC[M<C�\HQf;<DA6[MgTRT>@:=H\P;<�JCH\L;W=NfpAVUVRQN;A+HQF9U+H\<M>m[MACY\:=H\P;<mHQF�:=P�>DAC[MAC<M>&RQP;N�<DH U+H\<M>mHQFE>@8;A+UeRT:XPw[DAC<MH\P_RQPBJCAeW=:XP;A6<oJCH\L;W=NfpAeH\f;<MAC[hg\ACN�k Ñ HT>@AV>D8_RT>m>@8;AVY\ACPBAC[@RTW?U+A6>D8;HsN�:XPwN;A6gQACW=H\c;:XPBY}>@8;A+N;:XRQY\P;H\<h>@:=J+>DACJ|8;P;:=��L;A{ RQ<9>@H�JCHQP;<M>D[DL;Jq>�>D8;AoRT>DH\UV:=J�U+HsNBACW§� { :=>D8tRQWXWp>@8;AoRTc;c;[DHQc;[D:XRT>@A�>D[@RQPB<D:=>D:XH\Pe[@RT>DAC<C�s>@H&>@8BACPFIH\W=WXH { >@8;:=<&>@8;[MH\L;Y\8£>@H�R(<M`KPr>@8;Aq>@:XJ-<DcpACJq>@[ML;U��?H\[+RT>�WXACRQ<M>+R(>@8BACH\[MA6>@:=J!RQW9:=Pr>@ACPB<D:=>y`Qk À<M`KPr>@8BA6>@:=JO<DcpACJq>@[ML;UØYQACP;A6[@RT>DACN�FIH\[~:XPr>@A6P;<D:a>y`w<DcpACJ6>D[DH\<MJCH\cr`�RQP_RQWa`K<D:X<m:X<~[DAqFIAC[D[MACN�>DH RQ<zRÉÕ<DcpACJ6>D[@RQW�FIA!RT>DL;[DATË=ko798;A�H\cB>D:XU+:X<DRT>@:=H\PtHQF?ReÎ;>1HQF�>D8;:X<�FIACRT>@LB[DA&>@HORQP(H\f;<MAC[MgQACN�<DcpACJ6>D[DL;U[DAq>@L;[MP;<^R�NB:]RQY\PBH\<M>D:XJZHQF�c;WXRQ<DUVR&c_RQ[DRQUVAq>@A6[D<3<DL;J|8}RQ<9>@A6UVcpAC[DRT>@L;[MAQ�;NBACP;<M:=>y`\�ÈAq>@JQkÀ <�c;WXRQ<DUVRQ<3HQF�Y\[MA!RT>DAC[3N;A6P;<D:a>y`�¦IH\[�<DH\U+A6>D:XU+AC<9HQF�YQ[DA!Ri>@AC[�N;:=UVA6P;<D:=H\P;<3:XPt>@8;A~RQ<h>@[DHQc;8r`s<×�:XJCRQWEJCRQ<DA!¨ { A6[DAtH\f;<MAC[MgQACNd�?H\cB>D:XJ!RTW?>@8B:XJ|��P;AC<M<VAq��A6J6>D<+fpACJCRQU+AS<D:XYQP;:=Î_JCRQPr>!k�»_H\[&AqÖBRQU+c;W=A:XPrgQAC<M>D:XYrRi>@:XHQP;<o:XPr>@H-<DH\WXRQ[1c;[MH\UV:=P;ACPBJCAC<mWXA6Nw>DH->D8;A+JCH\P;J6WXL;<M:XH\PwP;HQ>~H\P;Wa` >@8;RT>o>@8;Aq` { AC[MAAqÖK>D[DACU+ACWa`vH\cB>D:XJ!RTW4>@8;:=J|�(:XPv>@8;AVÙd`sUVRQP�<DAC[M:XA6<O¦µ½^:X[DR!`rRQUVRB�G��Ú\Û\©r¨��Çf;LB>m>@8_RT>zRQf;<MH\[Dcs>@:XHQPHQF�>@8;AtÙd`KUVRQP£JCH\Pr>@:=P�L;L;U { RQ<�H\P;ASHQF3>D8;AtUeRQ:=P�<DHQL;[DJ6AC<�FIH\[&c;[MH\UV:=P;ACPBJCASACPBAC[DYT`�YrRQ:=P¦­Ü�H\W]RTP;N�RTP;N À P;ÝCA6[C�G��ÚrÞs��¨qk�798;A+P;ACA6N�>@H-:XPr>DAC[DcB[DA6>mH\cs>@:XJCRQWXWa`�>D8;:XJ|�}<DcpACJ6>D[@R-WXACNv>@Ht>@8BAN;Aqg\ACW=H\c;U+ACPr>1HQFbgTRQ[M:XH\L;<�>@A6J|8;P;:X��L;A6<Z>@HSN;ACRQW { :a>@8 HQc_RQJC:a>y` RTNißML;<M>DUVA6Pr>@<�>DHOcjHQc;L;W]Ri>@:XHQP;<C�WX:=P;A�<D8_RTcjA6<&RQP;N�ACU+AC[MY\ACPr>z¢;LsÖsAC<Ck-798;A6<DAV>DACJ|8;PB:X��L;AC<zJ!RQP�fpA+f;[DHrRTN;W=`vJCRT>@A6Y\H\[D:=<DA6N�:XPr>DHÉÕ[@RQN;:XRT>@:ag\AO>D[@RQP;<hFIAC[6ËbRQP;NÔÉÕAC<MJ!RQcpAtFµRQJ6>DH\[CË4U+A6>D8;HKN;<+RQP;N { :XWXWEfjAtN;:=<DJCLB<D<DA6N�W]Ri>@AC[�:=P�>@8;:=<

Page 19: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

à

J|8_RQcs>@AC[6�pRQP;Nt:XPtJ|8_RQcs>@AC[��skÀ <�<DcpACJ6>D[DH\<MJCH\cB:XJ&>@A6J|8;P;:=�KLBAC<oRQNBgTRQPBJCACN(:a>ZfpACJCRQUVA~cjHQ<D<D:=f;WXAm>@HO[MAC<MH\W=gQA&Î_P;ACWa`-<Dc;W=:=>ZUzL;Wa�>@:=c;WXAq>@<�HTF���P;H { P-RT>DH\UV:=J�<h`s<h>@A6UV<6k?798;:=<9RQW=WXH { A6NO>@8;A1[MACN;:=<M>D[D:XfBLB>@:ag\AZ[DRT>@A6<9fjAq> { A6ACP�>D8;AC<MAÎ_P;A6W=`}<Dc;W=:=>1W=A6g\A6WX<1>@HSfpA�AqÖBRQU+:XP;A6NwRQP;NwRT>@HQUV:=JzU+HsN;A6WX< { AC[MAVNBA6g\A6WXH\cpACNv>@H-RQJ6JCH\LBP�>1FIHQ[<DLBJ|8�>@[@RTP;<D:a>@:XHQP�[DRT>@A6<Ckt798;AOAq��A6J6>@<zHQF9[DA��yN;:=<M>D[D:XfBLB>@:ag\A�ACW=ACJ6>D[DH\P�RQP;N�:=H\P�J6H\WXW=:X<M:XH\P;< { AC[MA:XPBJCWXLBN;ACN�:=Pz>@8;A3Ri>@H\U+:XJGUVHKN;A6WXWX:=P;Y3<DL;J|8�>@8;RT>�P;H\PK�y<M>@RT>@:=<M>D:XJ!RTW§�i[DAC<MH\W=gQACN�cpH\c;L;WXRT>@:=H\P;<�JCH\L;W=NfpAmAqgTRQWXL_Ri>@ACN�k�798;AC<MA~U+HsNBACWX<9J6H\L;W=Nt>D8;ACP-fpAmA6UVcBWXH�`\A6N->DH+:=Pr>@AC[Mc;[DAq>3>@8;Ao:XPr>DACP;<M:=>y`OHTF�>@8BAH\f;<MAC[hg\ACN(<McjA6J6>D[@RQWÇU&L;Wa>@:=c;WXAq>9:XPt>@A6[DU+<^HQF�>@8;AmN;H\U+:XP;RQPr>^cBW]RQ<MUeR�c_RQ[DRQU+A6>@A6[D<�:XPrg\HQW=g\A6N}:=P>@8BAe[DA��yN;:=<M>D[D:XfBLB>@:ag\A�>@[@RTP;<D:a>@:XHQP;<V¦µ<DLBJ|8£RQ<~>D8;Ae:=H\P�RTP;N�ACW=ACJq>@[DHQP�N;A6P;<D:a>@:XA6<|¨�k�lyP�>D8;:X< { R�`U+H\[DA�NBA6>|RT:XWXA6N�c;W]RQ<MUeR^:XPBFIH\[MUeRi>@:XHQP~>@8;RQP { RQ<bcpH\<D<M:Xf;W=A�L;<D:=P;Yo[DA6<DH\P_RTP;JCA�WX:=P;AC<�RTWXH\P;AEJCH\L;W=NfpAmNB:]RQY\PBH\<DA6NdkÀ <eUVH\[MA}A�ÖBHT>@:XJ�c;W]RT<DUVRQ< { AC[MA(Y\ACP;A6[@RT>DACN�H\P²A!RQ[h>@8 RQP;N²<DcpACJ6>D[@RQW�H\f;<DA6[MgTRT>D:XH\P;<VHQFmRQ<×�>@[MH\c;8r`K<D:=J!RQW�H\fKßMA6J6>@<�:XU+c;[DH�g\A6Nd��:a>�fpACJCRQU+A~cpH\<D<M:Xf;W=A~>DHeH\f;<MAC[hg\A&>@8BAzJCHQUVc_RT[@RT>D:=g\A6W=` { A!RQ�<DcpACJq>@[DRQWÇWX:XPBAC<9H\[M:XY\:=P_RT>@:=P;YzFI[MH\Uá8;:=Y\8t�KL;RQPr>@L;U$<D8BACWXW=<CkG798;AC<MA~HQf;<DA6[Mg\A6N(8;:XYQ8-<MAC[M:XAC<�WX:=P;AC<RQ[MAeHQFÅ>DACP�f;W=ACP;NBACN { :=>D8�P;A6:XY\8�fpH\L;[M:XP;Y�UVA6U&fjA6[D<zRQP;N�RQ[MAV>y`Kc;:=J!RQW=W=`vH\f;<MAC[hg\ACN�FI[MH\U�WXH {>@A6UVcpAC[DRT>@LB[DAQ�j8;:XYQ8 N;ACPB<D:=>y`�c;WXRQ<DUVRQ<Ck�798;AC<MA�<DcpACJ6>D[@R { AC[MA&Î_[D<h>1HQFGRQW=W�:=Pr>@AC[Mc;[DAq>@A6N(L;<D:=P;Y{ R�gQACW=ACP;YQ>D8O<DcpACJq>@[MH\<DJ6H\cr`(¦Å>@8;A { R�gQACW=ACP;YQ>D8OHQFj>@8;A�W]RQ<h>GH\f;<DA6[MgTRQf;W=A3N;:X<MJC[DAq>@A�WX:=P;A { RT<EL;<MACN>@HzN;A6>DAC[DU+:XPBA^>@8BA�c;WXRQ<DUVR~A6WXACJq>@[MH\PON;A6P;<D:a>y`\�;<MACAZlyP;YQWX:X<?â«7�A6WXW=AC[C�p��Ú\©QÚr¨qk47�ACJ|8BP;:X��L;A6< { AC[MA>@8BACP�NBA6g\A6WXH\cpACN�<DL;J|8�>D8_RT>m:XPr>DACP;<M:=>y`v<DcpACJq>@[MH\<DJ6H\cr`�J6H\L;WXNwfpAVcpAC[hFIH\[DU+ACNwH\Pw<DAC[M:XA6<~W=:XU+:=>H\f;<MAC[hgQRi>@:XHQP;<&¦µÜ�:XYrRQ[MH�gtA6>�RQW§ka�Ç��Ú\Ú\ãr¨�k�lyPr>@A6[Dc;[MA6>@:=P;YV>D8;AC<MA&<DcpACJq>@[DRV[DA6��L;:X[MAC<���P;H { W=ACN;Y\A~HQF>@8BAeRT>DH\U+:XJ�c;[MHsJ6AC<D<MAC<z>DH}RTP;N�FI[MH\Uä>@8;A+8;:=Y\8���L_RTP�>DL;UÆ<D8;A6WXW=<C� { :=>@8�R�PBACACN�>@H�RQJCJ6H\L;Pr>c;[MH\cjA6[DWa` FIH\[1>@8;A�>@[DRQP;<M:=>@:=H\P;<ZFI[MH\U�8;:XYQ8w<MAC[M:XAC<~f;WXA6P;N;A6N�WX:=P;AC<oRTP;N�>D8;AV<MUVHKHQ>D8�U+AC[MY\:XP;YHQF^<DL;J|8u8;:=Y\8�Ps�y<M8;ACW=W�fpH\L;P;NucjHQc;L;W]Ri>@:XHQP;<~>D8;[DH\LBY\8�>@H(FI[DA6AS<M>|Ri>@AC<6k À J6JCH\L;Pr>VU&L;<h>+RQWX<MHfpAe>|RQ�TACP£HQF�c;WXRQ<DUVR�U+:XJ6[DHQÎ_A6WXN�Aq��A6J6>D<�H\P�>@8;AS8B:XY\8�Wa`K:XP;Y�cjH\cBL;W]RT>D:XH\PB<&RQP;N£RT<D<DHKJC:XRT>@A6N>@[DRQP;<M:=>@:=H\P;<�JCH\PB<D:X<h>@A6P�>DW=` { :=>D8;:XP}>D8;AzJ6H\WXW=:X<M:XH\P_RTWa�y[DRQN;:XRT>@:åg\A�FI[DRQU+A { H\[D�Èko�sLBJ|8(UVHKN;ACW=<�JCRQP[DACRQcvReFIL;[h>@8;A6[ZWXAqg\ACW�HQFGN;:]RQYQP;H\<M>D:XJ~:XPBFIH\[MUeRi>@:XHQPt>D8_RQP { RQ<1cjHQ<D<D:=f;WXAmFI[DH\Uæ[DA6<DH\P;RQP;JCA�W=:XP;AH\[�U&LBW=>@:=c;WXAq>�RQP;RQW=`K<D:=<�RQW=H\P;AQk�798;A { H\[M��HQF�>@8;:=<&>@8;A6<D:=<+:=<�c_RQ[h>�HQF^>D8;ASN;Aqg\ACW=H\c;U+ACPr>+HQF>@8B:X<�W]RT>DAC<h>V<McjA6J6>D[DH\<MJCH\c;:=J->DACJ|8;PB:X��L;A RTP;N�<MACA6�K<V>@HwUeRT>DJ|8²RTf;<DH\W=LB>@At:=P�>DACP;<M:=>D:XAC<�HQF�8;:XYQ8<DA6[D:=AC<ZH\fB<DAC[hgTRT>@:=H\P;< { :a>@8(R�Î;>D>DACN >D8;ACH\[MA6>D:XJ!RTW�<McjA6J6>@[ML;U-k À Y\RQ:XP->@8;A&f;RQ<D:=Jzc;[M:XP;J6:Xc;W=A~8_RT<fpACACPtL;<MACN { 8;AC[MA1>@8;A1J6H\P;<M>D[DL;Jq>@:=H\PtHQF�RQP-RT>@H\U+:XJ^UVHKN;A6W�WXACRQN;<E>DH+R&c;[DA6N;:XJq>@ACNt<McjA6J6>@[ML;U-�RQW=WXH { :XP;YmRoÎ;>D>D:XP;YmHQFd>D8;:X<b<M`KPr>@8BA6>@:=J�FIA!Ri>@L;[MA�>DH~>@8BA�H\f;<MAC[hgQRi>@:XHQP;<G>DHzfpA^UVRQN;A^RQP;NVc;W]RT<DUVR

Page 20: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

Û

c_RQ[DRQU+A6>@A6[D<3N;:XRQY\P;H\<MACNdk798;A�P;A�Ös>oHQf;<DA6[MgTRT>@:=H\P_RQW�<M>|RTY\A�:XP FIL;<M:XH\P >DH\�iRQUVRQ��<Z:X<Z>DHSWXHKH\�}Ri>Z>@8;A�<McjA6J6>@[DRSc;[DHKN;L;J6ACNfr` 8;ACR!gK` <DcpACJC:=AC<1:=P�>D[DHKN;L;J6ACN(>@HO>D8;A�c;W]RT<DUVRBk1798;:X<�:X<1RQ<oRO[MAC<ML;W=>1HTFG>D8;A�c;WXRQP;<�>DHSW]R!`\A6[8;ACR�g�`�<McjA6JC:=AC<�:XPr>DH JCA6[M>|RT:XP { RQWXWå�n>@:=WXA6<mHTF�>D8;ASly73¥bÐ�[DACRQJ6>DH\[Ckv798;A6[DAS:=<�R N;L_RQW?c;L;[McjH\<MA:XP�>D8;:X<6��Î_[D<h>@W=`w>@8BAC<DAt8;ACR!gK`�<DcpACJC:=AC<+RQ[MASUVH\[MAe>@A6UVcpAC[DRT>@LB[DAO[MAC<D:=<M>@RQPr>�>@8_RT>�>D8;ASJ!RT[DfpH\Pf_RQ<MACNe>D:XW=AC<�>@8_Ri>�RQ[MA^LB<DACNe:XP+AqÖs:X<h>@:XPBY~[MA!RQJq>@H\[M<CkG�sA6JCH\P;NBW=`�>@8;A6<DA^8;A!R!g�`e<McjA6JC:=AC<EWXR!`\AC[M< { :=WXWc;[MH�gK:=N;A1RzU+A!RQPB<EHQFÇN;Aq>@A6J6>@:=P;Y�8;H { AC[MHsN;A6NO>@8;A { RQW=WX<?RQ[DAZRT>9RQPr`VcjHQ:XPr>!k4798BAZ:XN;ACRz:X<G>@8_Ri>{ 8;A6P£>D8;AeFIL;<M:XH\P�c;W]RQ<MUeR-AqÖscpH\<DA6<&>@8;AO8BA!R!g�`�<DcpACJ6:XA6<�W]R!`\A6[D<C�4>D8;A6` { :XW=WGfpAOAyßMACJ6>DACN£:XPr>DH>@8BAe[DACRQJ6>DH\[Ck�798;AC<MAS<DcpACJ6:XAC< { :XWXW�>D8;ACP�A6UV:a>zJ|8;RQ[@RQJq>@A6[D:X<h>@:=Je<DcpACJq>@[DR { 8;:XJ|8 { :=WXWbfpAeL;<MACN>@HVRQW=AC[M>9>D8;AmH\cjA6[@RT>DH\[D<3HQF�>@8BA~N;AqgK:XJCAo>@H�>@8BAmcB[DH\Y\[MAC<M<�HQF�>@8;A { RQWXW/A6[DH\<M:XH\Pdk?Ù/:XY\8r>3<McjA6JC:XA6<JCHQL;WXNtP;HQ>3fpAoL;<DA6N-FIHQ[�>@8;:=<9fjA6J!RQL;<MAm>D8;A6` { H\LBWXNtfjA1FIL;W=W=`V:XH\PB:X<DA6NS:XPS>@8BAmJ6H\[DAmRQPBN { H\L;W=NACU+:=>?P;H�<McjA6J6>@[DRQW�W=:XP;A6<Ck�lyPO>D8;AZf;L;:=WXNVL;cO>@H&>D8;AZly73¥bÐÔ[MA!RQJq>@H\[E>D8;AC[MA { :XW=WjfpAZc;[MACc_RT[@RT>DH\[M`H\f;<MAC[hgQRi>@:XHQP;<4HQFÈ8;ACR!gK`z<DcpACJ6:XAC<�8B:XY\8&<MAC[M:XAC<�WX:XU+:=>D<�RQ< { ACWXWBRQ<�>D8;A�AqÖK>D[DACU+ACWa`zNBA6>|RT:XWXA6N&Y\[DRQ<D<×�WX:=�QA1<McjA6J6>@[DRVH\fB<DAC[hg\A6N}FI[DH\Uç>@8;AmJCH\U+c;W=AqÖS<MA6>^HQF4A6P;AC[MYQ`tWXAqg\A6WX<3cjHQ<D<D:=f;WXA1:=P�<ML;J|8}<h`K<M>@A6UV<6k»_H\[9U+A!RQP;:=P;YQFIL;Wj:XPr>@A6P;<D:a>y`S<DcpACJq>@[MH\<DJ6H\cr`->DH�fjAocpAC[MFIHQ[DU+ACNtH\PS>@8;A6<DAmP;A { H\f;<MAC[MgTRT>D:XH\PB<C�_:=>:X<3P;A6JCA6<D<@RT[M`}>DHVNBA6g\A6WXH\c RVP;A { RT>@H\U+:XJ1U+HKN;ACW­kE798;A~W]RT>DAC[ { HQ[D�Sc;[MAC<DA6Pr>@ACNv:=P->D8;:X<9>@8BAC<D:=<<DA6AC��<�>DHeN;He>D8;:X<6k9798;Az8B:XY\8�<DAC[M:XA6<�8r`KN;[DH\YQACP;:=J~cpH\c;L;WXRT>@:=H\P { H\[M�tN;A6gQACWXHQcjA6N 8;AC[MAzJCRQP fpAL;<MACN�>DHwRQJCJ6H\L;Pr>�FIH\[&>@8;At8;:=Y\8£W=`K:XPBY(cjH\cBL;W]RT>D:XH\PB<&:XP�>@8BA-8BA!R!g�`£<DcpACJC:=AC<+RQP;N£>@8BA-f;RQ<D:=Jc;WXRQP;<�RT[DAZWXRQ:XNVH\LB> { 8B:XJ|8 { :XW=W�RTWXWXH { >D8;AZW=H { AC[EWXA6gQACW=<?>@H�fpA�NBA!RQWa> { :=>D8SJCH\PB<D:X<h>@A6P�>DW=`Qk�798;Af_RQ<M:XJmUVAq>@8;HKN;H\W=H\YQ`SHTFb:XPr>@A6P;<D:a>y`�<McjA6J6>D[DH\<MJCH\cr`�:X<ZRTN;8;AC[MACN}>DH;�È>@8_RT>�:X<3>@8;AzRT>@HQUV:=JmU+HKN;ACWN;Aqg\ACW=H\cpACN { :=WXW�fjA�J!RT[D[D:=ACNv>@8;[MH\L;Y\8v>@H-RSÎ_P_RQW�<McjA6J6>D[DL;U { 8;:=J|8�JCRQP�fpA�JCH\U+c_RQ[MACN { :=>D8H\f;<MAC[hgQRi>@:XHQP;<z>@H([MA6>DL;[DP£N;:XRQY\P;H\<h>@:=JV:=PBFIH\[DUVRT>D:XH\PwH\P�>@8BAS8;A!R!g�`�<McjA6JC:XA6<�ACU+:=>D>DAC[M<Ckv�EHQ>D8>@8BA�8;:XY\8(<MAC[M:XAC<mRQPBNw8BA!R!g�`vRT>@HQUèU+HsNBACWXW=:XP;YV:XPr>@[MHsNBL;JCA�<MH\UVAzN;:=é+JCLBW=>@:=AC< { 8B:XJ|8vP;ACA6Nw>DHfpA~N;ACRQW=> { :a>@8dk9½�H { Aqg\AC[6��>@8BAC[DAm:X<^R�JCH\[M[DAC<McjHQP;N;:XPBYe[DAq>@L;[MP}:=P->D8;AzJ6AC[M>@RQ:XPr>y`-RQPBN�gTRQ[D:=A6>y`HQF�N;:]RTY\P;H\<MACNtc;W]RT<DUVR�c;RQ[@RQU+A6>DAC[M<9L;<D:=P;Y�>@8;A6<DAmP;A { U+HKN;ACW=<Ck798;A6[DA { AC[MA9R^gTRQ[D:=A6>y`mHQF_cB8�`K<M:XJ!RTWBc;[MHsJ6AC<D<MAC<4>D8_RT>�[DAC��L;:=[DA6N�RT>D>DACPr>@:=H\Pz:=P~>@8B:X<�>D8;AC<M:X<6k�798;A6<DA:XPBJCWXLBN;ACN(H\c;RQJC:a>y`\��J6H\Pr>@:=PKLB:=>y`�HQFG[@RT>DAC<1RQPBN(cjH\cBL;W]RT>D:XH\PB<^FI[DHQUæfpH\L;PBNv>DHeFI[DA6A�<M>@RT>@A6<C�/H\fs�>|RT:XP;:=P;Yz8;:=Y\8O��L_RQW=:=>y`+WX:=P;A�c;[DHQÎ;WXAC<?FIH\[E>@[DRQP;<M:=>@:=H\P;<GP;A!RT[E>@8;A�<DAC[M:XA6<9WX:XU+:=>GRTP;NSN;A!RTWX:XPBY { :=>D8>@8BAS[@RT>DAOAqÖsc;[MAC<D<M:XH\PB<+FIH\[z8;:XY\8��KL;RQPr>@L;Uê<M8;ACW=W�cpH\c;L;WXRT>@:=H\P;<~>DH(Y\[DACRT>@A6[�[DA6<DH\W=LB>@:=H\P�>@8;RQP8_RQ<�fpACACP�N;H\P;AmfpA6FIH\[MAQk9�EAqFIH\[DAoc;[MHsJ6ACA6N;:XP;Y { :=>D8}R�N;AC<MJC[M:XcB>D:XH\P-HQF�>D8;Amf_RQ<D:=Jm>D8;ACHQ[M`tL;<MACN

Page 21: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

Þ

>@8B[DH\L;YQ8;H\LB>�>D8;:X<&>D8;AC<M:X<�:=>�:=< { HQ[M>@8 { 8;:=WXAt<M>DH\c;c;:=P;Y(>@HvH�g\A6[MgK:XA { A!RTJ|8²HQF�>@8;A6<DAtN;:aéVJ6L;Wa�>@:=AC<z>D8_RT>zU&L;<M>&fjAOH�gQAC[DJ6H\U+AQ�bN;AC<MJC[M:Xf;:=P;Y(:XP�fB[DHrRQN�>DAC[MUV<m>@8;ASc;8r`K<D:=J!RQW?P_RT>DL;[DASHQF3ACRQJ|8c;[MH\f;WXA6U�kÄZc_RQJ6:=>y`Z[MA6FIA6[D<Ç>@H9>@8;Abc;[MHsJ6AC<M< { 8BAC[DAGc;8;HT>@H\P;</>D[@R!g\A6WXWX:=P;Y�>@8B[DH\L;YQ8~R9c;WXRQ<DUVR�N;H3P;HQ>�A6<DJ!RTcjAT�f;LB>�RQ[DA�[MAq�yRQf;<DHQ[DfpACN { :=>D8;:XP~>@8;A�cBW]RQ<MUeR3g\H\W=L;UVATk�798;:=<48_RQ<�>@8;A9N;L;RQWBAq��A6J6>�HQFjRQWa>@A6[D:XPBY^>@8BAACU+AC[MY\ACPr>�¢;LsÖtRQPBNtRQNißML;<h>@:XPBYz>@8;A1cpH\c;LBW]RT>D:XH\P+N;:X<h>@[M:Xf;LB>D:XH\P { :=>D8;:XPe>D8;A1RQfB<DH\[Mf;:XP;Y�Ri>@H\U+<CkÀ Wa>@8;HQL;Y\8 >@8BA+A6UVA6[DY\A6Pr>1¢_LsÖv:=<ZL;<DL;RQWXWa`}[MACN;L;J6ACNd��:=>Z:X<ZcpH\<D<M:Xf;W=A&FIH\[ZH\c;RQJC:a>y`}>@HS:=P;JC[MA!RQ<MA>@8BAO¢_LsÖ£:=P£RvYQ:=g\A6P£WX:XPBAO:=F3c;8BHQ>@HT�yRQf;<MH\[DcB>D:XH\PB<�:XP£RQP;HT>@8;A6[�WX:XPBAe:XPBN;:X[MACJ6>DW=`�:XP;J6[DACRQ<DAS>@8BAcpH\c;L;WXRT>@:=H\P�N;A6P;<D:a>y`�HQF9>D8;ASL;c;cpAC[zWXA6gQACW�HTF�>D8;Ae>D[@RQP;<M:=>D:XH\P�:XP���L;AC<h>@:=H\PdkvlnF3>@8;AOcB8;HQ>@HQP;<AC<MJ!RQcpA { :=>D8;H\LB>�[MAq�×RTf;<DH\[McB>@:=H\P�HKJCJCLB[D[D:=P;Y >@8BASc;W]RQ<MUeR :=<&<@RT:XN�>@H(fpA ÉÊHQcB>@:=J!RQW=W=`v>@8;:=PdË4:=P>@8;RT>~>D[@RQPB<D:=>D:XH\P�kSlnF9<DL;J|8�[MAq�yRQf;<DHQ[DcB>D:XH\P;<mN;H HKJCJ6L;[z:a>m:X<m[DA6FIA6[D[MACN�>DH(RQ<mfjA6:XP;YwÉÊH\cs>@:XJCRQWXWa`>@8B:XJ|�ÈË=k�ÄZc_RQJ6:=>y`t:XP�c;[D:=P;JC:=c;WXAoJ6H\L;c;W=AC<�Aqg\A6[M` cpH\:=P�>^:XPt>@8BAzc;WXRQ<DUVR�>DH\Y\Aq>@8;A6[ZRQP;N { A&8;R�gQAR�PBH\Ps�yW=:XP;ACRQ[oRQP;N�P;H\Ps�nWXHKJ!RQW�cB[DH\f;W=ACUä>DH}<DHQW=g\ATkt798;AeUVRQ:XPwRQcBc;[DHrRTJ|8;AC<z>D8_RT>~8_R!g\AVfjA6ACPN;Aqg\ACW=H\cpACN²>DH�U+HsNBACW9H\c_RTJC:=>y`uRT[DA(ÉÊ[DRQN;:XRT>@:ag\AS>@[DRQP;<MFIA6[CËGRQP;NxÉÊA6<DJCRQcjA-FµRQJ6>DH\[CË4>DACJ|8;PB:X��L;AC<6k798;A�>DACJ|8;P;:=��L;A9HQFp[@RQNB:]RT>D:=g\AG>@[DRQP;<MFIA6[4:X<4R�8;A!R!gK:XWa`zJ6H\U+c;LB>|Ri>@:XHQP_RQWsRQc;c;[MHrRQJ|8&>@H1<MH\W=gK:=P;Y�>@8BAJCHQL;c;WXA6N~A6��L_RT>@:=H\P;</HQFB[@RQNB:]RT>D:=g\A�>@[@RTP;<MFIA6[�RTP;N~<h>|RT>D:X<h>@:XJCRQWQf;RQW]RQPBJCAQk�ln>Ç8_RQ<ÇfpACA6PzL;<MACNo>@H�N;A!RQW{ :=>D8�>D8;A~Aq��A6J6>D<ZHQF4HQc_RQJC:a>y`tH\P}cpH\c;LBW]RT>D:XH\Pt<M>D[DL;Jq>@L;[MAQ�jWX:XPBAmcB[DHQÎ_W=AC<^RQP;N�ACU+AC[DYQACPr>^¢;LsÖsAC<FIH\[ÇgTRQ[D:=H\L;<�c;W]RQ<MUeRT<�<ML;J|8+RQ<�<h>@ACW=W]RQ[�RT>DUVHQ<Dc;8;A6[DA6<3¦§��RQ[MWX<MH\Pd�B�!Ú\ã\Ûr¨ÇRQP;N&<MH\W]RT[�c;[MH\U+:XP;A6P;JCA6<¦§ë1H\LB>M>@A6f;[DH\Ý6A&A6>1RQW§ka���!Ú\Ú\©r¨�kZ798;A&A6<DJ!RTcjA&FµRQJq>@H\[�RQc;c;[MHrRQJ|8(:=<ZReU+HsNBAC[@Ri>@AzHQcB>@:=J!RQW�N;ACcs>@8U+A6>@8BHsN { 8;A6[DAz:a>3:X<3RQcBc;[DH\cB[D:]Ri>@AZ>@H�>@[MA!RT>3>D8;AoAC��L_RT>D:XH\P-HQF�[@RQN;:XRT>@:ag\AZ>D[@RQP;<hFIAC[�:XP-R�WX:XPBA!RQ[RQc;cB[DH!Ös:XUVRT>@:=H\Pdk 798;:X<eRQc;c;[MHrRQJ|8�8;RQ<e>@8BA RQNBgTRQPr>@RQY\A�>D8_RT>e:=>V:X<eACRQ<D:=W=`£:=P�>DACY\[DRT>@A6N�:XPr>DHHQ>D8;AC[�UVHKN;A6WX<C�pRQPBN}8_RQ<�fpACACP}L;<DA6N}>@HVN;:]RQYQP;H\<DAoH\cs>@:XJCRQWÇN;ACcs>@8;<^:XPt>@8;A~<DH\WXRQ[�RT>@U+H\<DcB8;AC[MA¦ Ò H�`KWXA�â·ì�JqÌ 8;:X[h>@AC[6�È��Ú\ã\�B�\�E[DHKH\��<�A6>?RQW­k=���T�\�\�r¨��Q>@Ho:XPsFIAC[4:=PBFIH\[DUVRT>D:XH\P~H\P�>@8;A3<M>D[DL;Jq>@L;[MAHQF4>D8;Az<MH\W]RT[�RT>DUVHQ<Dc;8;A6[DAO¦­»�:=<DJ|8�f_RQJ|8;A6[C�jA6>1RTW§ka�d�Q�Q�\�r¨9RQP;N}>DHeN;:XRQY\P;HQ<DAzcBW]RQ<MUeR�N;ACPB<D:=>D:XA6<:XP(FIL;<M:XH\PvN;:ag\A6[M>@HQ[D<V¦­�EA68;[D:=P;Y\A6[C�b��Ú\Ú\ã\¨qk��sA6AVly[MH\P;<V¦h�!ÚrÞQÚr¨�FIHQ[oRQP�H�gQAC[MgK:=A { HQFG>@8BA+f;RQ<D:=JAC<MJ!RQcpAoFµRQJ6>DH\[9AqÖsc;[MAC<D<M:XH\PB<�RQPBN�N;A6[D:agQRi>@:XHQP;<CklyPv>@8;:=<�>@8;A6<D:=<mHQc_RQJC:a>y` :=<oRQJCJ6H\L;Pr>@A6NwFIH\[1LB<D:XPBYt>D8;A�AC<MJ!RQcpA�FµRQJ6>DH\[1RQc;cB[DHrRQJ|8�k À < { :XW=W�fpA<DA6ACPd�;>D8;A1RQcBc;[DHrRTJ|8S:X<EACRQ<D:=W=`+:XPr>DACY\[DRT>@A6NS:XPr>@H&J6H\WXW=:X<M:XH\P_RTWa�y[DRQN;:XRT>@:åg\A�U+HKN;ACW=<�RQP;NSA6UVA6[DY\A6Pr>¢_LsÖwJCRQWXJ6L;W]Ri>@:XHQP;<Ck�¥�<MJ!RQcpA+FµRTJ6>@HQ[D<mRQ[DA�L;<ML_RQW=W=`(RTc;c;WX:=ACN >DH�N;:=<DJ6[DA6>DAQ��WXH { <DAC[M:XA6<~U+ACU&fpAC[<DcpACJq>@[DRQWjW=:XP;A6<^¦I>@8BH\L;Y\8VP;HQ>ERQW { R!`s<6�B<MACA Ò [DR { :XP(¦×��ÚrÞQ�\¨4FIHQ[?RTPeAqÖBRQU+c;W=A^HQFdAC<DJCRQcpA^FµRQJq>@H\[M<L;<MACN}>DHORQJ6JCH\L;Pr>^FIH\[^J6H\Pr>@:=PKLBL;U$H\c_RQJ6:=>y`B¨qk9798;:=<�>D8;AC<M:X<^AqÖK>@A6P;N;<�>D8;:X<�WXH { W=A6g\A6W�RTc;c;[DH\RQJ|8

Page 22: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

ã

>@HmU+HsN;A6W_H\c_RQJ6:=>y`&>D8;[DHQL;Y\8+8;:XY\8+<DA6[D:=AC<GUVA6U&fjA6[D<b:XPr>DHm>D8;A3JCH\Pr>@:=P�L;L;U-k�Ì�A�<D8BH { >@8_RT>G>@8BAA6�jACJq>@<&HTF9H\c_RQJ6:=>y`vJ6H\Pr>@:XP�L;AV<DU+HsHT>@8;Wa` FI[DH\U�fpH\L;PBN�>@H-FI[DA6Ae<M>@RT>@A6<&:XP�R�<M:XU+:XWXRQ[ZUVRQP;P;A6[>@HOëmRTL;Pr>�FµRQJq>@H\[M<ZRQP;N(�BRQ8_Ri�y�EH\Wa>@ÝCUVRQP;P-N;Aqgs:XRT>@:=H\PtFµRQJ6>DH\[D<&¦µfs�§FµRQJ6>DH\[D<@¨qk�798;:X<�RQWXW=H { <3>@8BAL;<ML_RQW_WXH { <DA6[D:=AC<bUVA6U&fpAC[4A6<DJ!RTcjA9FµRQJq>@H\[4>DACJ|8;P;:=��L;AC<b>@HofpA9AqÖK>@A6P;N;ACN+>@H1<MUVHKHQ>@8BW=`zRQJCJ6H\L;Pr>FIH\[9H\c;RQJC:a>y`O:XPt8;:=Y\8t<DA6[D:XA6<3UVA6U&fjA6[D<9>@8B[DH\L;YQ8t>DH�>@8;AoJCHQP�>D:XP�L;L;U-klyP�H\[MN;AC[O>DH�JCH\[M[DA6J6>@Wa`�c;[MACN;:=J6>O>D8;A 8;:=Y\8�<MAC[D:=AC<S<McjA6J6>@[ML;U { AvRQWX<MH�P;A6ACN�>@H£RQJ6JCL;[DRT>@A6W=`U+HsN;A6WZRQW=W^>D8;A(cjHQ<D<D:=f;WXA >D[@RQPB<D:=>D:XH\P²[@RT>DAC<O>@8;RT>tJ!RQP :=PB¢_L;A6P;JCA(>@8BA(8;:XY\8 ��L_RQPr>DL;U <D8BACWXWcpH\c;L;WXRT>@:=H\P;<Ckí798;RT>-:X<C�^H\P;AvU&L;<h>�RQJCJ6H\L;Pr>�FIHQ[tRQWXW�>D8;AvcjHQ<D<D:=f;WXA >D[@RQPB<D:=>D:XH\P�[DRT>@A6<�¦µ:§kÏAQk<DcpH\Pr>|RTP;ACH\LB<C��<h>@:=U&L;W]Ri>@ACN�RQP;N JCH\W=WX:=<D:XHQP_RQW9[@Ri>@AC<@¨V>@8_Ri>S8_R!g\AwRQP�:=PB¢_L;A6P;JCAvH\P�>D8;A(8;:XYQ8W=`K:=P;Y�Ri>@H\U+:XJS<h>|RT>DAC<Ck 798;A�UVHKN;A6WXWX:=P;Y(HQFoRi>@H\U+:XJScpH\c;L;WXRT>@:=H\P£N;:X<h>@[M:Xf;LB>D:XH\PB<+8;RQ<eRwWXH\PBY8;:=<M>@HQ[M`(RTP;N�:=<1P;H { RT>mRTP�RQNsgQRTP;JCA6N�<h>|RQY\ATk�798;A+UVH\<h>o<DL;J6JCA6<D<MFILBWG>DACJ|8;P;:=��L;AV:=PvN;A!RQW=:XP;Y{ :=>D8(cjHQc;L;W]Ri>@:XHQP}N;:=<M>@[M:Xf;Ls>@:XHQP;<�8_RT<1fjA6ACP(>@8_Ri>1HQFGJCH\W=WX:=<D:XHQP_RQWa�n[@RQNB:]RT>D:agQAZ>@8;A6H\[M`�¦­��RT>DAC<1Aq>RQW­k=�;�!Ú\Ûr�\¨?¦µ<DA6A3<DACJq>@:=H\Pt�\kÓ�KkÊÛ�FIH\[4RZUVH\[MA�JCHQUVc;W=A6>DA�N;AC<MJC[M:XcB>D:XH\P�HQFÈ>D8;A9JCH\W=WX:=<D:XHQP_RQWa�n[@RQNB:]RT>D:agQARQc;cB[DHrRQJ|8_¨qk�lyP&>D8;:X<�>@8;A6<D:=<�>@8;A�RQNBgTRQP;J6ACU+ACPr>@<�>DHZAqÖs:=<M>@:=P;YZJ6H\WXW=:X<D:=H\P_RQWå�y[DRQN;:]Ri>@:åg\A4U+HsNBACWX<�RQ[MAcpAC[MFIHQ[DU+ACN { :=>@8B:XP�>@8;A3FI[@RQU+A { H\[M��HQF�>D8;A À >DH\U+:XJ Ò RT>@R~RQPBN À P;RQW=`K<D:=<?�K>@[ML;J6>DL;[DA&¦ ÀZÒ3À �B¨<DLB:=>@A^HQFÇJCHQUVc;Ls>@AC[?JCHKN;AC<1¦§�sL;U+U+AC[D<6�;�T�\�B��¨�k ÀZÒ3À �VJ6H\P;<M:X<M>D<�HQFÇRTPO:XPr>@A6[DJ6H\P;P;A6J6>@A6Nt<DA6>EHQFJCHQUVc;Ls>@AC[�JCHKN;A6<1RQP;N}RT>@H\U+:XJoN_Ri>|RBk9798;AzN_RT>|R+[DA6c;[DA6<DACPr>D<�>@8;A~8;:XYQ8;AC<h>Z��L_RQW=:=>y`tR!gTRQ:XWXRQf;W=AQ�JCH�gQAC[D:=P;Y&U+H\<M>G>y`KcjA6<�HQF�N_RT>|R~P;ACA6N;ACNOFIH\[bA�ÖB:=<M>D:XP;YzRQ<M>D[DH\c;8r`K<D:=J!RQWjRQP;NVFIL;<D:=H\PeJCRQWXJ6L;W]RT>D:XH\PB<Ck�K>@RQP;N_RQ[MN ÀZÒ3À � N_RT>@ReFIH\[DUVRT>@<�¦µ[DAqFIAC[M[DACN(>DH�RT<&ÉÓRQNsFCËjN_RT>|RiÎ_WXA6<|¨ZRQ[MA�N;AC<M:XY\P;A6Nv<DL;J|8v>@8_Ri>>@8BA-J6HsNBAC<+RQ[DAtY\A6P;AC[DRQW9c;L;[McjHQ<DA-RQP;NuRQc;cBWX:XJCRQf;W=Ae>@HwR { :XN;At[DRQP;Y\ASHQF�RT>DH\UV:=JSRQP;N£:=H\P;:XJ<M`K<h>@ACU+<Ck�798;A�J6HsN;A6<mRQ[DA�<Mc;WX:a>1:XPr>@HtA6:XY\8r>o<DA6[D:=AC< { :=>D8vA!RQJ|8w<DA6[D:XA6<oJCH\P;<M:X<h>@:XPBY-HTF�RO<MA6>mHQFJCHKN;A6<CkS»_H\[1A�Ö;RTUVc;W=AQ�Ç<DA6[D:=AC<o>@8;[MACA+JCH\Pr>|RT:XP;<o>@8;A�J6HsN;A6<~FIH\[1J|8;RQ[DY\A+AqÖsJ|8_RTP;Y\AeU+HKN;ACW=WX:XPBY;kÀZÒ3À �o©B�\�b>@8BACPm[DACcB[DAC<MACPr>@<�JCHKN;AbACWXAqg\A6P&HQFs<DAC[M:XA6<�>D8;[DA6AQk�798BA?J6HsN;A6<�RT[DAbL;<DA6N~>DH^U+HsNBACW\>@8BA[@RTN;:]RT>D:=gQAzc;[MH\cjA6[M>D:XAC<^FIH\[ { :=N;A&[@RTP;Y\A&HQFGc;WXRQ<DUVReJ6H\P;N;:a>@:XHQP;<CkZ798BA�[DH\LB>D:XP;A6<1RQ[MA&N;AC<M:XY\PBACN<DLBJ|8 >@8_RT>^>@8;Aq`}JCRQP <MH\W=gQAzFIH\[3>@8;A&cpH\c;LBW]RT>D:XH\P;<3:=P ReP�L;U&fpAC[�HQF�ÉÕc;:XJq>@L;[MAC<CËa� { 8;Aq>@8;A6[Z:XP}Rfs�§FµRQJ6>DH\[C��fs�nFµRTJ6>@HQ[ { :a>@8�AqÖscjHQP;ACPr>@:XRQW�>@A6[DUÆ:XP;J6WXL;NBACNwH\[~J��nFµRQJq>@H\[oFIH\[MU�kV798;:X<mACW=:XU+:XP_RT>DAC<WX:=�QA6W=`SP�L;U+AC[M:XJ!RTW/:XP_RQJ6JCL;[DRQJC:=AC<^RQP;N->@8;A~JCH\U+c;LB>@RT>@:=H\P_RQW�>DACJ|8;PB:X��L;AC<�8_R!g\AzfpACA6P c;[MH�gQACP}>DHfpA~fpHQ>@8�<M>@RQf;WXA~RQP;N-FµRQ<M>Ck9798;AzJ6HsNBAC<ZRT[DA~RQWX<MHeRQfBWXA1>@HV:XPr>@A6[Dc;[MA6>^<DcpACJ6>D[@RQW�U+A!RQ<ML;[DA6UVA6P�>D<RQP;N�� { :a>@8;:=P(>@8;A�<DA6[D:=AC<m<D:aÖ <DAq>zHQF?JCHKN;A6<C�Ç>@H-Y\ACP;A6[@RT>DA�<M`KPr>@8;Aq>@:XJ+<DcpACJq>@[DR�¦µ[MA6FIA6[D[DA6Nw>DH}RT<ÉÕ<DcpACJC:XRQWEFIACRT>@L;[MAC<6ËÓ¨�kwÌ�AOÎ_P;N£:=>&JCH\Prg\A6P;:XA6Pr>+>DH(N;A6Î;P;A-R 8;AC:=[@RQ[MJ|8�`�HQF�<DcpACJq>@[@RTWEFIA!RT>DL;[DA6<JCW=H\<DA6W=`uWX:=P;�QA6N�>@H�cpH\c;L;WXRT>@:=H\Pu<M>@[ML;J6>DL;[DATkîlnF~R�<MA6>tHQFoW=:XP;A6<e:=P�>@8;A <McjA6JC:XRQW^FIACRT>@L;[MA(RQ[MA

Page 23: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

Ú

JCHQP;P;ACJq>@A6NegK:]Roc;L;[DA^f;[@RTP;J|8;:XPBY~W=:XP;A3[@Ri>@:XHQ<4>@8;Aq`eRT[DA�[DA6FIA6[D[MACNV>DH&RQ<3ÉÏFIA!RT>DL;[DA�c;[D:=UV:a>@:ag\AC<6Ë=ïr:aF>@8BA�JCHQP;P;ACJq>@:=H\Pe:=<GgK:]RoAqÖsJ6:=>@A6NOWXAqg\A6WjcpH\c;L;WXRT>@:=H\P�f_RQWXRQP;JCA3>@8BA6`VRQ[DA�J!RTWXWXA6N-ÉÏFIA!Ri>@L;[MAC<CËaïB:aFd>@8BAJCHQP;P;ACJq>@:=H\P}:=<3gK:XR+:=H\P;:X<DRT>@:=H\PSf_RQWXRQP;J6Ao>D8;A6`�RQ[MAoJ!RQW=WXACNvÉÕ<DL;cpAC[hFIA!RT>DL;[DA6<CËak9ln>�:=<�<M8;H { P W]Ri>@AC[8;H { <D:=UV:=W]RQ[�8;AC:=[@RQ[MJ|8;:XJCRQWÈ:XN;ACRQ<1¦I<DACJq>@:=H\Pe¾;ka�\ka��¨�:XPecpH\c;LBW]RT>D:XH\P+<M>D[DL;Jq>@L;[MA�WXACRQNV>DH&JCHQ8;AC[MACPr>H\[MYrRQP;:=<@RT>D:XH\POHTF�J6H\UVcBWXAqÖSRT>@HQUáUVHKN;A6WXWX:=P;Y;k798;A À�Ò�À ��JCHKN;A6<1RQ[DA~RQJ6JCAC<M<DA6Nw>D8;[DH\LBY\8 RQP}l Ò Ù£YQ[@RQc;8B:XJ!RTW�LB<DAC[�:XPr>@A6[MFµRQJ6AQk�798;:X<�RQWXW=H { <>@8BA+LB<DAC[o>@Ht<McjA6JC:=FÅ` { 8;:XJ|8�RT>@HQUV:=J&<M`K<M>DACU�:=<Z>@H-fjA�:=P�gQAC<h>@:XY\RT>@A6Nd� { 8_RT>~RT>@HQUV:=J&N_RT>@RS>DHL;<MAQ� { 8_Ri>9c;W]RT<DUVR~c_RQ[DRQU+A6>@A6[D<E>D8;AZJ6HsN;A�8_RQ<E>DH�fjAZ[ML;POFIH\[ERQP;N { 8_Ri>�FIH\[DUVRT>?>@8;AZHQLB>@c;Ls>N_RT>@RZRQP;NzY\[@RQcB8;<�RQ[MA?>@H�fjA?Y\:=gQACP&:=Pdk�798;:=<4RQW=WXH { <Ç>@8;A�LB<DAC[�>@HZ:=UVU+ACN;:XRT>@A6W=`�<DACA?>@8;A�[MAC<ML;W=>D<HQF9>D8;AeJCRQWXJ6L;W]RT>D:XH\P���UVRQ��:XP;Y�:=>~cjHQ<D<D:=f;WXA�>@H}AqÖsc;WXHQ[DAV>D8;AOc_RT[@RQU+A6>DAC[m<Dc_RQJ6AOFIH\[o>@8;AOYQ:=g\A6P<M`K<h>@ACU-k�lnF3>D8;AC[MAS:X<&P;HvRi>@H\U+:XJVN_RT>|R cB[DAC<MACPr>�FIH\[�R Y\:ag\A6P�J!RQW=JCL;WXRT>@:=H\P�:a>�:X<&cpH\<M<D:XfBWXAV>DHY\A6P;AC[DRT>@Am>@8;AmP;ACJ6AC<M<@RQ[h` N_RT>@R ¦IHQFÅ>@A6P�>@HeRVW=H { AC[���L_RQW=:=>y`B¨�L;<D:=P;YVgTRQ[M:XH\L;< ÀZÒ3À �t[DH\Ls>@:XPBAC<Ck»_H\[EA�ÖBRQUVcBWXA À�Ò�À �OÞQ�B��[DL;P;<?>@8;A1RTLB>@H\<h>@[ML;J6>DL;[DAZJ6HsNBA�¦­��RTN;P;ACW=W§�p��Ú\ã\Û\¨q�KN;A6gQACWXHQcjA6NOFI[DH\U>@8BA&<DL;cpAC[M<M>D[DL;Jq>@L;[MA+J6HsNBAt¦­¥�:X<M<DP;A6[ZA6>1RQW§ka���!ÚrÞT¾�¨�ï ÀZÒ3À � ã\�s�z[DLBP;<Z>@8;A��EH { RQPv<M>D[DL;Jq>@L;[MAJCHKN;Aw¦­�EH { RQPd�^��Ú\ãB��¨m>|RQ:=WXH\[MACN�>DHvc;[DHKN;L;J6AtÎ;WXAC<�HTF À�Ò�À �£N;RT>|R}FIH\[MUeRi>�fr`uÄ&ËÓì�LBWXW]RTP;AQïÀZÒ3À �V�s�\��Y\ACPBAC[@Ri>@AC<G[@RTN;:]RT>D:=gQA�[MACJ6H\U&f;:=P_RT>@:=H\P�N_RT>@RmL;<M:XP;Yo>@8BAZëoRQL;Pr>bFµRQJ6>DH\[D<GN;Aqg\ACW=H\cpACN:XP}¦­�?L;[DY\A6<D<Eâð�sL;U+UVA6[D<C�È��ÚQãrÞ\¨qk ÀZÒ3À �+RQWX<MH~J6H\Pr>|RQ:=P;<ERmWX:=f;[@RQ[h`&HQF��QAq`e<ML;f;[DHQLB>@:=P;AC<b>@8_Ri>J!RTP�fpAoJ!RQW=WXACNOFI[MH\Uç>D8;AoL;<DA6[CËÏ<^H { P�JCHKN;AC<9>@H+RQ<M<D:=<M>�:XPt<DcpACJ6:=Î_JoJCRQWXJ6L;W]RT>D:XH\PB<Ck798;:=<V>@8BAC<D:=<eJ6H\Pr>|RQ:=P;<OR�J6H\U+c;WXAq>@A-N;AC<MJC[D:=cB>@:=H\P�HQF1R�UVHKN;A6W�:=P²c;[MH\Y\[MAC<D<V>@8_RT> { :XW=W�FIH\[DUc_RQ[h>ZHQF�>@8;A À�Ò�À ��<DLB:=>@A~HQFbJCHKN;A6<Ck�798B:X<�U+HKN;ACW { :XW=WÇRQJCJ6L;[@RT>DACWa`}c;[MACN;:=J6>^>@8;A~cjH\cBL;W]RT>D:XH\P<M>D[DL;Jq>@L;[MA HQFoRT[Df;:a>@[@RT[M`£8;ACR�g�`�<DcpACJ6:XAC<ORQP;N²JCRQP�fpA}[ML;P�:XP�:=>D<Vc;[MAC<MACPr>S<M>@RT>@A->@H�U+HKN;ACW8;:=Y\8tWXA6gQACW/8r`KN;[DHQY\ACP;:=J~cpH\c;LBW]RT>D:XH\P;<6k À UVHKN;L;W=AZHQF�>@8;:=<9JCHKN;Ao:X<3L;<MACN�>DH�c;[DA6N;:XJq>�>D8;Ao8;:XYQ8��L_RQPr>@LBUñ<D8;A6WXW/cpH\c;L;WXRT>@:=H\P;<�P;A6ACN;A6N�FIH\[�>D8;Ao<DcpACJq>@[@RTWdFIA!RT>DL;[DA { H\[D�OHTF4J|8_RQcs>@AC[ à k798;AOHT>@8;A6[~UVRQ:XPw:X<M<DL;AORQN;N;[MAC<D<MACN�:=P�>@8B:X<~>D8;AC<M:X<~:X<o>@8;AeRQJCJ6L;[@RT>DAeU+HsNBACWXW=:XP;YtHQF�8;:XY\8�<DA��[D:=AC<~WX:XPBAVc;[MHQÎ_W=AC<Ck}½^:=<M>@HQ[D:XJCRQWXWa`\�ÇWX:=P;A+c;[DHQÎ;WXAC<~8_R!g\ASfpACA6P�L;<DA6N�>@H c;[MH�gs:=N;AOR { :XN;AV[@RQP;YQAHQF�cBW]RQ<MUeR�N;:]RTY\P;H\<h>@:XJo:=PBFIH\[DUVRT>D:XH\Pdk À Pr`�f;[DHrRTN;ACP;A6N <DcpACJq>@[DRQW�W=:XP;AmJCHQP�>@RQ:XPB<^:=PBFIH\[DUVRT>D:XH\PH\P�>D8;ASc;8r`K<D:XJCRQW�c;[MHsJ6AC<M<�>D8_RT>�J6H\Pr>@[M:Xf;LB>DACN�>DH >@8;ASf;[MHrRQN;A6P;:XPBY;kw»ÈHQ[zAqÖBRQU+c;W=AQ��N;H\cBc;WXA6[8_RQWaF]� { :=NB>@8;<GJ!RTPOfjA^L;<DA6NO>@H&NB:]RQY\PBH\<DA^Rzc;WXRQ<DUVRm:=H\PV>DACU+cjA6[@RT>DL;[DA�RQP;NOcB[DHQÎ_W=A�<M8_RQcpAC<�JCRQPfpA1L;<DA6N->DH+:=PBFIAC[?>@8;A1P_Ri>@L;[MAmHTF�>D8;AoJCH\W=WX:=<D:XHQP_RQW_f;[MHrRQN;A6P;:XP;Y�>D8_RT>9c;[MHsNBL;JCA6N->D8;AoWX:XPBAC<D8;RQcjA¦µ:­kÕATk { 8;A6>D8;AC[^ACW=ACJ6>D[DH\P�H\[3:XH\P-f;[MHrRQN;A6P;:XP;Y+N;H\U+:XP;RT>@A6<|¨qk?798;AmJ!RQW=JCL;WXRT>@:=H\PSHQF�W=:XP;A1c;[MHQÎ_W=AC<C�:XPSc;RQ[M>D:XJCLBW]RQ[�J6H\WXW=:X<M:XH\P_RTWXW=`zf;[DH\RQN;ACPBACN-WX:XPBAZc;[DHTÎ_WXA6<C�;:=<3R&P;H\Ps�§>@[D:agK:]RQWpc;[DHQf;WXA6U { :=>D8tUeRQPr`

Page 24: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

���

c;8r`K<D:=J!RQWdcB[DHKJCAC<M<DA6<�cpHQ>DACPr>@:XRQWXWa`�JCH\Pr>@[M:Xf;Ls>@:XPBYz>@H&>D8;AZf;[MHrRQN;A6P;:XP;Y�HTFÇ<DL;J|8-WX:=P;AC<6k4798;A1<D:a>@Ls�RT>D:XH\P+:X<�FIL;[h>@8;A6[?J6H\UVcBWX:XJCRT>@A6N�FIH\[b8;:XY\8+<DA6[D:=AC<EW=:XP;A6<bN;L;A3>@H~>D8;A�[DACWXRT>@:ag\A6W=`�<MUeRQW=W_RQU+H\L;Pr>GHQF8;:=Y\8e<MAC[D:=AC<�HQf;<DA6[MgTRT>@:=H\P_RQWÈN_Ri>|RzL;cpH\P { 8B:XJ|8O>@H~>@AC<h>E>D8;AZWX:=P;A�c;[DHQÎ;WXA�J!RQW=JCL;WXRT>@:=H\P;<Ck Ñ Aqg\AC[×�>@8BACWXA6<D<6�B8;:=Y\8V��L_RTWX:=>y`�W=:XP;A3c;[MHQÎ_W=A�J6HsNBAC<E<ML;J|8tRQ<b>@8;A�ÜbÜGÜ�J6HsN;A&¦µ7�RQWX:=P�A6>9RTW§ka�p�!Ú\Ú à ¨��K8;R�gQAfpACACPtN;Aqg\A6WXH\cpACN�RQP;Nt8_R!g\Aoc;[MH�g\ACPt>@8BACU+<DACWag\A6<3>DH�fjA1[MACW=:]RQf;W=A�RTP;NSgQRTWX:XNVFIH\[9R { :XNBA1[@RQP;YQAHQF�c;W]RT<DUVR&JCH\P;NB:=>@:=H\P;<6kÄZP;JCAoRTWXW�HTF�>D8;AC<MA1:X<M<DL;A6<�8;R�gQAofjA6ACPtN;A!RTW=> { :a>@8S:=>�:X<�cpH\<D<M:Xf;W=A^>DH+J6H\U&f;:=P;A�>D8;A18;:XYQ8O<DA6[D:XA6<WX:=P;A�c;[DHTÎ_WXA6< { 8;:=J|8�JCHQP�>D:XP�L;AV<DU+HsHQ>D8;W=`}>@8;[MH\L;Y\8w>@H}J6H\Pr>@:=PKLBL;U { R!g\A6WXACPBYQ>@8;< { :=>@8�8;:XYQ8WXAqg\A6W�cpH\c;LBW]RT>D:XH\P�[MAC<DLBW=>@<�>DH�Y\A6P;AC[DRT>@A-RvJCH\U+c;W=A6>@AS<h`KP�>D8;A6>D:XJ�<DcpACJ6>D[@RQW�FIACRT>@LB[DA�RT>�<DA6[D:XA6<WX:=UV:a> { R!g\A6WXACPBYQ>@8;<�FIH\[3R { :XN;A1[DRQP;Y\AoHQF�c;WXRQ<DUVR&JCH\PBN;:=>D:XH\P;<6k798;Av[DA6<M>-HQFm>@8;:=<SJ|8_RQcB>DAC[-JCH\PB<D:X<h>@<tHQFzR�f;[D:=A6F~<M>@A6cÔf;RQJ|��FI[DH\U >@8;:=<OÎ_P_RQW�c;:XJq>@L;[MA�RTP;N[DAqgK:XA { <+>@8;A�8;:X<h>@H\[h`�HQF1HQL;[eRTc;c;[DH\RQJ|8�>DH�JCHQP;<D:=N;AC[M:XP;Yw8;:XYQ8�<MAC[D:=AC<V<DcpACJ6>D[@RQW�c;[MACN;:=J6>@:=H\Pdkln>V:X<�8;H\cpACN²>D8_RT>�>@8B:X< { :XWXWEc;[DH�gK:XN;A�R�LB<DA6FILBW^H�g\A6[MgK:XA { HQFoRQW=W�>@8;A-c;8r`K<D:=J!RQW�A6�jACJq>@<ORTP;N>@8BACH\[MA6>@:=J!RQW�:X<M<DL;A6<Z>@8_RT>o8;R�gQA&>@HSfpA�RQJCJ6H\L;Pr>@A6N(FIH\[1RQ< { ACW=W4RQ<1<M8;H { :XP;YO>D8;A�H\f;<MAC[MgTRT>D:XH\PB<>@8;RT>�WXACNS>DH+>D8;AoJCH\P;<h>@[ML;J6>D:XH\P-HQF�>@8;A1Î_P_RTW/UVHKN;ACW­k�798�L;<^>D8;AoP;AqÖK>^<DACJq>@:=H\P�cB[DH�gK:XN;A6<�<MH\U+AHQFÇ>@8BAZf_RQJ|��Y\[MH\L;P;Nt>@8;A6H\[M`V>@8_Ri> { :XW=WjfpAZ[DA6��L;:X[MACNd�;<MH\U+AZHQFÇ>@8;AZ>D8;ACHQ[M`V:X<E>D8;ACPtN;Aqg\ACW=H\cpACN:XP�<DA6J6>@:=H\P��\kÕ©}:=P�>@8BAOJCH\Pr>DAqÖK>�HQF�gTRQ[M:XH\LB<z<DcpACJq>@[DRQWEH\fB<DA6gTRT>D:XH\PB<C��FI[MH\U�<DH\WXRQ[o>@8;[MH\L;Y\8�>DHFIL;<M:XH\Ptc;WXRQ<DUVRQ<Ck�E8_RQcs>@AC[9�m>@8;A6PSY\HKAC<�:=P�>DHzN;Aq>|RQ:=WjHQPe>@8;A�N;:X<MJC[DAq>@AZW=H { <DAC[M:XA6<�<DcpACJq>@[@RTW�WX:=P;A�H\c_RQJ6:=>y`�>@8;A6H\[M`RQP;N >D8;A&J6HsN;Az�KPBH { PwRQ< ÀZÒ3À �(�s�!¾V>@8_RT>�:X<ZLB<DACN(>DHOA6gTRQW=L_RT>@A~H\c_RQJ6:=>y`�UVHKN;:aÎ_J!RT>D:XH\PB<3>DHfpHQ>@8(>@8BA+A6UVA6[DY\A6Pr>m¢;LsÖwRQP;N�RT>DH\UV:=J&cpH\c;L;WXRT>@:=H\P <M>D[DL;Jq>@L;[MAVHTFEH\cs>@:XJCRQWXWa`t>@8;:=J|� c;W]RT<DUVRQ<Ck�E8_RQcs>@AC[&© N;AC<MJC[D:=fjA6<z>@8BAOAqÖK>@A6P;<D:=H\P£HQF9>@8B:X<~>D8;ACHQ[M`w>@H 8;:=Y\8�<MAC[D:=AC<�RTP;N�JCH\Pr>D:XP�L;L;UÆFI[DA����L;ACPBJC:XA6<CkO�E8_RQcB>DAC[m¾-YQHsA6<m:=Pr>@Ht>@8;A�N;Aq>|RQ:=WX<1HQFG>@8;A+8;:=Y\8vPs�y<M8;ACW=W�cjH\cBL;W]RT>D:XH\P(U+HKN;ACW�RTP;NH\L;[&FILs>@L;[MA-cBW]RQP;<�FIH\[&>D8;:X<�J6H\WXW=:X<M:XH\P_RTWa�y[DRQN;:XRT>@:åg\A { H\[D�Èk��E8_RQcB>DAC[ à >@8;A6P²f;[M:XP;YQ<&>@8;:=<�RQWXW>@HQY\A6>D8;AC[?RQP;N+N;ACU+H\P;<h>@[DRT>@A6<?>@8BA�YQACP;A6[@RT>D:XH\P+HQF/RTPORQ[Mf;:=>D[@RQ[h`&8;:=Y\8V<MAC[M:XAC<G<DcpACJq>@[DRQWpFIACRT>@LB[DAQk»�:=P_RQWXWa`+J|8_RQcB>DAC[^Û�<ML;U+UeRQ[M:X<MAC<3H\L;[9UVRQ:XPSJ6H\P;J6WXL;<M:XH\P;<�RQPBN�H\Ls>@WX:=P;AC<�>D8;A1FILB>@LB[DAoN;:X[MACJq>@:XHQPHQF�>@8;:=< { HQ[D�Èk

Page 25: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�\�

����ò óç#z0EôOõ+ö�)+÷(%(' 24ø .Z)+ö4ù

úGûyü�ûMú ý}þsÿ�����������ÿ��� �����þ��hÿ�����������������Zþln>3:X<�Î;[D<M>�L;<MA6FIL;Wd>DH�:XPr>@[MHsN;LBJCA1<DHQUVA1f_RT<D:XJ1>D8;ACH\[h` { 8;:=J|8�JCRQPt>@8;A6P�fpA1L;<DA6N RQ<�>D8;Aof;L;:=WXNs�:XPBYufBWXHKJ|�K<SFIH\[O>D8;AvN;A6gQACW=H\c;U+ACPr>@<-c;[DA6<DA6P�>DACN·:=P W]RT>DAC[SJ|8_RQcs>@AC[M<Ck 798;Av>@AC[MUV:=P;H\W=H\YQ`uHQF»�:=<DJ|8�f_RQJ|8;A6[ZA6>ZRQW§k�¦­�Q�\�\�\¨q�pRQP;N->@8;A~P;HQ>|Ri>@:XHQP�HQF4ly[DH\P;<&¦h��Ú\ÞQÚr¨3RQ[DAmL;<DA6N >@8;[MH\L;Y\8BH\LB>�>@8;:=<>@8BAC<D:=<Ckb798�L;<3>@8;A1FIH\W=WXH { :XP;YzN;A6Î_PB:=>@:=H\P;<98;HQWXNd¼� 7�[@RQPB<DU+:X<D<M:XH\P+FµRQJ6>DH\[�� 7 �È��798;A1R!gQAC[@RTY\AZc;[MH\f_RQf;:=WX:a>y`&>@8_Ri>9Rzc;8BHQ>@H\P { :=>@8B:XPV>D8;A�W=:XP;A

c;[MHQÎ_W=A { :XWXWjc;[DHQc_RQYrRT>DAZFI[DH\U�H\P;AoY\:=gQACPtcpH\:XPr>9>@H+RQPBHQ>@8;A6[Ck� ¥�<DJCRQcpAeÜ�[DHQf_RQf;:=WX:=>y`!�#"��4� Ò AqÎ_P;ACN�:XN;A6Pr>@:XJCRQWXWa`�>DHS>@8;A�>@[DRQP;<DU+:X<M<D:=H\P(FµRQJ6>DH\[1AqÖsJCA6cB>>D8_RT>9>@8BAm<MACJ6H\P;N-cjH\:=Pr>9WX:=AC<9H\LB>D<D:=N;AZ>@8;Aoc;WXRQ<DUVRBk�798_RT>�:X<6�$":=<�>@8;AoR!g\A6[@RQYQA~c;[MH\f_Ri�f;:=WX:a>y`oHQFjRZc;8;HQ>DH\P�ACU+:=>D>DACN { :a>@8;:=Pz>@8BA9WX:XPBAEc;[MHQÎ_WXAEAC<DJCRQc;:=P;Y1FI[DH\Uð>@8;AEc;W]RQ<MUeR�RQWXH\PBYR�c_RQ[h>@:=JCL;WXRQ[�WX:=P;AZHQF�<D:=Y\8r>!k� ¥�<DJCRQcpAO»ÈRQJ6>DH\[Z�Z798BAeU+A!RQPwc;[DHQf_RQf;:=WX:=>y`�>@8_Ri>&R-WX:XPBA+cB8;HQ>@HQP�ACU+:=>M>@A6N�<MH\UVA { 8;AC[MA:=P�>@8BAzc;WXRQ<DUVR { :=WXW/A6<DJ!RTcjAzFI[MH\Uá>@8BAzc;WXRQ<DUVRVRTWXH\P;YVRQPr`-WX:=P;AoHQF�<D:=Y\8r>!k�798;AzcB8;[@RQ<MAÉÕAC<MJ!RQcpAZFµRQJ6>DH\[?>DACJ|8;PB:X��L;AQËB:X<�RTWX<DHzL;<DA6N-:=Pe>@8B:X<?>D8;AC<M:X<E>DH�N;AC<MJC[M:XfpA�>@8;A�Y\ACP;A6[@RQWp>@A6J|8s�P;:=��L;A&HQFbW=:XP;ACRQ[D:=<D:XPBY+>D8;A&AC��L_Ri>@:XHQP(HQFb[@RQNB:]RT>D:=g\Am>@[DRQP;<MFIA6[�FIH\[�>@8;A&AqgTRQWXL_Ri>@:XHQP}HQFb>@8BAA�ÖBcB[DAC<M<D:=H\P;<�N;A6Î_PBACN}8BAC[DATk� À f;<DH\[McB>@:=H\P�FµRQJq>@H\[%�'&��G�oH\P;ASU+:XP�L;<~>@8;ASc;[MH\f_RQfB:XWX:a>y`v>@8_RT>�R P;A6>+RQfB<DH\[McB>@:=H\P�HQF^Rc;8BHQ>@H\P { :=WXWjHsJ6JCL;[3RT>9R&c_RQ[M>D:XJ6L;W]RQ[?cjHQ:XPr>9:XPtR&cBW]RQ<MUeRBk�798;:X<�:=<9<DH\U+A6>D:XU+AC<?[DA6FIA6[D[MACN>DHVRT<3>D8;A�ÉÓ�?:XfpAC[DUVRQPs�y½^H\W=<M>DAC:XPeJCHKA6é+JC:=ACPr>!Ë_H\[ZÉÕP;Aq>^[DRQN;:XRT>@:ag\A1f;[DRQJ|�QAq>!Ëak

Ñ HT>@A�>D8_RT>+>@8;A�H\P;Wa`£N;:=�jAC[MACP;J6A}fpA6> { ACACP�>D8;AC<MA N;A6Î;P;:=>D:XH\P;<VRQP;N�>@8;H\<MA}HQFZly[DH\P;<V:X<�>@8_Ri>8;A�N;A6Î_PBAC<V>D8;A}A6<DJCRQcjA�c;[DHQf_RQf;:=WX:=>y`�>@H�fjAt>@8;A�UVACRQPuc;[DH\f;RQf;:XW=:=>y`�HQFoR�cB8;HQ>@HQP²ACU+:=>M>@A6NRT>eR�cpH\:XPr>V:XP�R�c;WXRQ<DUVRwA6<DJCRQc;:XPBY�RQW=H\P;Y�RQPr`�W=:XP;AtHQFZ<D:XYQ8�>Ck 798;:X<V:X<�>@8;A�<@RQU+A}RT<V>@8BAAC<MJ!RQcpAmc;[DH\f;RQf;:XW=:=>y`VN;A6Î;P;ACN(RT<^RTfjH�g\A~R!g\A6[@RQY\A6N}H�g\A6[^<MH\WX:=N-RQP;Y\W=AQkbln>�:X<9FIACWa>9>@8_RT>�N;AqÎ_P;:=P;Y>@8BA-A6<DJCRQcjA-c;[MH\f_RQf;:=WX:a>y`�RQW=H\P;Y(H\PBA-YQ:=g\A6P�W=:XP;AeHQF�<M:XY\8r>+:X<�RTP�A�ÖBcB[DAC<M<D:=H\P²U+H\[DASNB:X[DA6J6>DW=`RQc;cBWX:XJCRQf;W=A�>@H+RQJq>@L_RQW/HQf;<DA6[MgTRT>@:=H\P;<6k

Page 26: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

���

úGûyü�ûyü ý}þsÿ������(�� ��)�������+*,��.-/�����0�213�4 �5��2� 76�8:9�EH\P;<M:XN;A6[D:=P;Y�>@8;A1H\fB<DAC[hg\A6N�¢_LsÖSAC<DJCRQc;:=P;Y�FI[DH\U�R�cBW]RQ<MUeR~>D8;ACPt>@8BAmA6��L_RT>@:=H\PSHQF�[@RQNB:]RT>D:=g\A>@[DRQP;<hFIAC[b:X<bL;<MACNV>DHoUVHKN;ACWs>@8;A3gTRQ[D:XRT>@:=H\P&HQFj>@8;A3[@RTN;:]RT>D:XH\P�:=Pr>@ACPB<D:=>y`t¦<;s¨4RQWXHQP;YmRoYQ:=g\A6PVW=:XP;AHQF�<D:=Y\8r>!�;N;A6P;HQ>DACN}fr`O>D8;AocjHQ:XPr>@<�RQW=H\P;Y>=yk? ;j¦A@KÃB=µ¨? = CED ¦A@sÃB=µ¨�¶GFE¦A@sÃH=­¨�;j¦A@sÃB=µ¨ ¦×�\k=��¨{ 8;A6[DA� D ¦A@KÃB=µ¨�:X<Z>@8BAVA6UV:=<D<D:=H\P(JCHKA6é+J6:XACPr>mN;AqÎ_P;ACN�RQ<1>@8;A�P�L;U&fpAC[oHQF?c;8;HQ>DH\P;<oACU+:=>M>@A6N�Ri>FI[MAC��L;ACPBJ6`I@tcpAC[9LBP;:=>9>D:XU+AQ�;cpAC[9L;PB:=>9g\HQWXL;U+AQ�;cpAC[3L;PB:=>3<DHQWX:XNtRTP;Y\WXAoRi>3cjH\<M:=>D:XH\P%=yk� FE¦A@sÃB=µ¨Ç:X<�>D8;A9RQf;<MH\[Dcs>@:XHQP&JCHKA6é+JC:=ACPr>!�rNBA6Î_P;A6NV<ML;J|8�>@8_Ri>�F�¦J@sÃB=µ¨+;j¦A@KÃB=µ¨�:X<�>@8;A9P�L;UzfjA6[

HQF3cB8;HQ>@HQP;<&RQfB<DH\[MfjA6N£RT>zFI[MAC��L;ACPBJ6`K@�cpAC[~L;P;:a>z>@:=UVAT��cpAC[~L;P;:=>~g\H\W=L;U+AQ��cpAC[zL;P;:=><MH\WX:=NtRQP;Y\W=AoRT>3cjHQ<D:=>D:XH\P%=yk� ;j¦A@KÃB=µ¨�:X<�>D8;A�:=P�>DACP;<M:=>y`zHQFÈ>@8;A9[DRQN;:]Ri>@:XHQP~Î_A6WXN�RT>4>D8;A�cpH\:XPr>�HTFj:=Pr>@AC[MAC<h>�¦J=µ¨q�rNBA6Î_P;A6NeRT<>D8;AZP�L;U&fpAC[9HQFÇcB8;HQ>@HQP;<�JC[MH\<D<M:XP;Y�R&LBP;:=>�RQ[DACRB�BcpAC[9L;PB:=>E>D:XU+AQ�KcjA6[�L;P;:a>�<MH\WX:=NSRQP;Y\W=AQ�cpAC[3L;P;:a>9FI[DA6�KLBACP;Jq`\k

À W=<DH-N;A6Î;P;Ae>@8BAVH\cs>@:XJCRQW�N;A6cB>@8 ¦MLȨ1HQF�>D8;AVcBW]RQ<MUeRS>@H�fjA�>@8;AVRQf;<MH\[Dcs>@:XHQP�JCHKA6é+J6:XACPr>~:XPs�>@A6Y\[@Ri>@ACN�RQW=H\P;Y(>@8;A-WX:=P;ASHQF�<D:XYQ8�>Ck�798�L;<+>@8;AtH\cs>@:XJCRQW9N;A6cB>@8�:X<�Y\:ag\ACP�fr`�>@8;AtFIH\WXW=H { :XP;YAqÖsc;[MAC<M<D:XHQPd� { :a>@8�>@8;A�<D:=UVcBWX:=Î;ACN�[DA6<DL;Wa>?FIHQ[bRmJ6H\P;<M>@RQPr>9RQfB<DH\[McB>@:=H\P�JCHKA6é+JC:=ACPr>�RTWX<DHm<D8BH { P�k

L/¦A@;¨ C N>OP FE¦A@sÃH=­¨ ? =C FE¦A@;¨+Q ¦×�\kÓ�Q¨798;:=<9Y\:=gQAC<�RQPt:XP;NB:XJ!Ri>@:XHQPOHQF�8;H { FµRQ[9HQP;A1J!RQPvÉÕ<DACATË_:XPr>@H&>D8;Aoc;W]RT<DUVRBk4ln>3<D8BH\L;WXNtfpAoP;HQ>@A6N>@8;RT>�A6��L_RT>@:=H\Pe¦h�\ka��¨/:X<ÇP;H\Ps�nWX:=P;A!RQ[/fpACJCRQL;<MA?>@8BA?A6UV:=<D<D:=H\PmJCHKA6é+JC:=ACPr>9¦ D ¨�RT>�cpH\:XPr>R=;N;A6cjA6P;N;<L;cpH\P(>@8;A�L;cBcjA6[mW=A6gQACW�cpH\c;L;WXRT>@:=H\P(N;ACPB<D:=>y` HQF?>D8;A�>@[DRQP;<D:a>@:=H\P :XPv��L;A6<M>@:=H\Pdk�798;:=<1:XP(>@L;[MPN;A6cjA6P;N;<�LBcjH\P->@8;A~P�L;U&fpAC[�HQF4c;8BHQ>@HT�yRQf;<MH\[DcB>D:XH\PB<3>D8_RT>^J!RQP�AqÖsJC:a>@AzRQP�A6WXA6J6>@[MH\P}:=Pr>@H�>@8;:=<WXAqg\A6W§� { 8B:XJ|8tN;ACcpACPBN;<3L;cjHQPt>D8;A1[@RTN;:]RT>D:XH\P+Î_ACW=NS;ORT>�cjH\:=Pr>T=y�B>@8�L;<3UVRQ��:XP;Y D :=POc;[D:=P;JC:=c;WXARzFIL;P;Jq>@:=H\POHQF4;VRQP;NO>D8KLB<9AC��L_RT>D:XH\Pv¦×�\ka��¨?P;H\PK�yWX:=P;A!RT[Ck Ñ HQ>@A^>@8_RT>?>@8;A1A6UV:=<D<M:XH\PVJCHKA6é+JC:=ACPr>

Page 27: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

��©

U W W C f W C �ÜbWXRQ<DUVR

V V V VV V

V V V VV V

V V V VV V

V V V VV V

»�:=Y\L;[DA��QkÊ©B¼XW2Y[Z]\_^AZ�`Ga]b�cJdfeXghYjikeXa]b�lJYh\�dmc�nkl<epoqYjiqcJdkeXe,lAr+Z]steusf^Aa�vkZ�vfYjghYjcxw bya�^A` Z�gjY[lJ`qzRT>{=�:=P�c;[D:=P;JC:=c;WXAVN;ACcpACPBN;<�L;cpH\P D RT>�RQW=Wb>D8;ASHQ>D8;AC[&cpH\:=P�>D<z:=P�>@8;Aec;W]RQ<MUeR�¦µN;L;AV>@H >@8BAN;A6cjA6P;N;ACPBJCAoHQF/>@8BAZcjHQc;L;W]Ri>@:XHQP;<GRT>T=ÇL;cpH\Pe>@8BAZWXHKJ!RQWÈ[DRQN;:]Ri>@:XHQPVÎ_A6WXNSRQPBNO8;ACPBJCA1L;cpH\PO>@8BAACU+:X<M<D:=H\P�FI[MH\U$ACWX<MA { 8;A6[DA�:=P}>@8;Azc;W]RQ<MUeR\¨9UVRQ��:XPBYV>@8BA&AC��L_RT>D:XH\P}P;H\Ps�nWXHKJ!RQW�RQ< { ACW=W§k^798;AU+A6>@8BHsNOA6UVcBWXH�`\A6Ne>@8;[MH\L;Y\8BH\LB>�>D8;:X<G>@8;A6<D:X<E:X<?>DH&<DHQW=g\A�>@8;A�AC��L_RT>D:XH\POHTFÇ[@RQN;:XRT>@:ag\A�>@[@RTP;<MFIA6[fr`(RQ<M<DL;U+:XPBYe>@8_Ri>Z>@8;A�A6UV:=<D<M:XH\P J6HsAqé+JC:XA6Pr>o:X<�:=P;N;A6cjA6P;N;ACPr>mHQF�>D8;A�:XPr>@A6P;<D:a>y`�>DAC[MU���>D8�L;<UVRQ��:XP;Y�AC��L_Ri>@:XHQP�¦×�\ka��¨�WX:XPBA!RQ[9RTP;N}RTWXWXH { :=P;Y�:=>9>DHVfpAo<DH\Wag\ACN-L;<M:XP;YVRQP-:XPr>@A6Y\[@RT>D:XP;Y&FµRTJ6>@HQ[RQc;cB[DHrRQJ|8�kE798;:=<�RQ<M<DL;U+cB>D:XH\P-HQF4W=:XP;ACRQ[D:a>y` { :=WXWdfpAm[DA!RT<DH\P_RTf;WXA1FIH\[3c;WXRQ<DUVRQ< { 8;:XJ|8}RQ[DAoP;HT><D:=Y\P;:aÎ_J!RQPr>DW=`VH\cB>@:=J!RQW=W=`+>@8;:=J|�SRQP;NS>D8KLB<^N;H�P;HT>^8;R�gQAzR�WXRQ[DY\A1NBACY\[MACAoHQF�<Dc_RT>D:]RQW�JCH\L;cBWX:XPBY;k�EH\P;<M:XN;A6[tR�WX:=P;A}HQFo<M:XY\8r>O>D8;[DHQL;Y\8xR�c;WXRQ<DUVR { :=>D8�>@8;A(HQ[D:XYQ:XP�Ri>O>@8;A}f_RQJ|�²FµRQJ6A(RQ<S:=PÎ_Y\LB[DA&�\kÕ©Bkb�EHQP;<D:=N;AC[EÎ;[D<M>�HQF�RQWXWpR&c;LB[DACWa`VRQf;<DHQ[Df;:=P;Y&c;WXRQ<DUVR { :a>@8SRzJ6H\P;<h>|RQPr>�RQf;<MH\[Dcs>@:XHQPJCHKA6é+J6:XACPr>?RQP;N { :=>D8�:XP;J6:XN;A6P�>b:=Pr>@ACPB<D:=>y`�;j¦A@sÃ|�\¨q�\>@8BA�:=Pr>@ACPB<D:=>y`&H\f;<DA6[MgQACNORi>�Q { :XW=WBfpA�YQ:=g\A6Pfr`

;j¦A@KÃBQq¨ C ;j¦A@sÃ|�\¨M¯ º}|�~���t�h�B����� ¦×�\kÊ©\¨lnF�F�:X<�RQW=WXH { ACN�>DH+gTRQ[h` { :=>D8�[MAC<DcpACJq>�>DH�=�RQP;N->@8;Amc;WXRQ<DUVR�J!RQP}RQWX<MHVA6UV:a>�RTWXH\P;Y�>D8;A~W=:XP;AHQF�<D:=Y\8r>9>@8;AoACU+AC[MY\ACPr>�:XPr>DACP;<M:=>y` { :=WXWdfpA;j¦A@sÃHQ6¨ C N�OP D ¦A@sÃB=µ¨h¯ ºX|�~� �t�[�p� �����y�#�j���#� ? =��K;�¦J@sÃ|�r¨h¯ º}|$~���t�[�p� �j�y�+� ¦×�\kÕ¾r¨798;AmACU+:X<M<D:XHQPtRQP;N-RQf;<MH\[DcB>D:XH\PtJ6HsAqéVJ6:XA6P�>D<�RT[DAoY\:=gQACPtfr`

Page 28: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�!¾

D ¦A@KÃB=µ¨ C �¾t�T��� �h� �.��� � ¦J=µ¨,�B¦A@sÃH=­¨ ¦×�\k à ¨

RQP;N

FE¦A@sÃH=­¨ C �:� � � � ��� ¦<=­¨+�;¦J@sÃB=µ¨C ��� ¦J=µ¨ ¸��ã����� ��{¡ � �@ �� � � � ��� �[� � �B¦A@sÃB=µ¨ ¦×�\kÊÛ\¨

{ :=>D8 � ��� �h� ��C >@8;Am¥�:XPB<M>@A6:XPs� À gTRQWXL;AZFIHQ[�>@8;Ao<DcpH\Pr>|RTP;ACH\LB<^A6UV:=<D<D:=H\P � ��¢ � �j£ º ³¤�� ��� � C À >@H\U+:XJ1cpH\c;L;WXRT>@:=H\POHQF�WXAqg\A6W � � � ¸�¹ º ´ �� ���¥C À >@HQUV:=JZcjHQc;L;W]Ri>@:XHQPSHQF�W=A6gQACW � � ¸�¹ º ´ �� �B¦A@;¨ C A6UV:=<D<M:XH\Ptc;[MHQÎ_WXA { 8B:XJ|8-:X<3RQ<M<DL;U+ACN-AC��L_RQW/>DH�>D8;AmRQf;<MH\[DcB>D:XH\Ptc;[MHQÎ_W=A�� <+�� �:� � � � C ¥b:=P;<M>DAC:=Ps�×��gTRQWXLBAZFIH\[9c;8;HQ>DHT�×RTf;<DH\[McB>@:=H\P � ¢ � � � ¸�¹ � £ º ³,£§¦ º ³¤�7�HV<MACA~>D8;AmN;A6Î_PB:=>@:=H\PtHQF�>D8;AmAC<DJCRQcpA~c;[MH\f_RQfB:XWX:a>y`e>D8;A~W=:XP;A1:=Pr>@ACYQ[@RT>DACN-ACU+AC[DYQACPr>^¢;LsÖ�RT>�f:X< { [M:=>M>@ACP�RQ<�FIH\W=WXH { <C¼

;j¦JQq¨ C N �#¨ � Â:© N�OP D ¦A@sÃB=µ¨h¯ ºX|�~� �t�[�p� � � �y��� � ? =��ª;j¦J@sÃ|�r¨h¯ ºu|�~���t�[�p� �j�����[« ? @C | ��¨ �  | OP D ¦J@sÃB=µ¨M¯ ºu|�~� �t�h�p� �#�[������� ? = ? @�� | ��¨ �  ;�¦J@sÃ|�r¨h¯ º}|�~���t�h�p� �h�y��� ? @| ��¨ �  | OP D ¦J@;¨ ? =��K;j¦A@sÃ|�\¨ ? @� N �#¨ �  N>OP D ¦J@;¨ ? =��ª;j¦A@sÃ@�r¨ ? @C " � � � � N �#¨ �  © N OP D ¦A@;¨ ? =��ª;j¦A@sÃ@�r¨ « ? @C " � � � � © �¾t� �:� � � � N>OP ��� � ¦J=µ¨ ? =�� N �#¨ �  ;j¦J@sÃ|�r¨ ? @ « ¦×�\kÓÞQ¨

Page 29: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� à

798�L;<�>D8;AEA6<DJ!RTcjA�cB[DH\f_RTf;:XW=:=>y`\�t"&�Q:=<�N;AqÎ_P;A6N�RQ<�>D8;AE[DRT>@:=H^HQF_>D8;AEHQc_RQJC:a>y`oUVHKN;:aÎ_ACN~AC<DJCRQc;:=P;Y¢_LsÖt>DH+>D8;AoH\cB>D:XJ!RTWXW=`+>@8B:XP-AC<DJCRQc;:=P;Y+¢;LsÖj� { :a>@8-fjHT>@8t¢_LsÖsAC<^A6gTRQWXL;RT>@A6N}RQ<3W=:XP;A1:=P�>DACY\[DRT>@A6NgTRQWXLBAC<Ck�798;RT>�:X<C�

" � � � � C | ��¨ �  | OP D ¦J@sÃB=µ¨M¯ º | ~� �t�h�p� � � ����� � ? = ? @�� | ��¨ �  ;j¦A@sÃ|�\¨M¯ º | ~���t�h�p� �h�y��� ? @| ��¨ �  | OP D ¦J@;¨ ? =��K;j¦A@sÃ|�\¨ ? @ ¦×�\kÊã\¨798;:=<+:=<�H\L;[�Y\A6P;AC[DRQW9AC<MJ!RQcpA�cB[DH\f_RTf;:XW=:=>y`�AqÖsc;[DA6<D<M:XH\PuFI[DH\U { 8B:XJ|8 { A-N;AC[M:=gQA}RQW=WEFIL;[h>@8;A6[[DA6<DL;Wa>@<6kelyP�J|8;RQcB>@A6[&�O>@8B:X<oAqÖsc;[MAC<D<M:XH\P�:X<oN;A6gQACW=H\cjA6N�RQPBN�A6gTRQW=L_RT>DACNvFIH\[1gQRT[D:XHQL;<1c;W]RT<DUVRY\A6H\UVAq>@[M:XAC<VRQP;N²WX:XPBA-cB[DHQÎ_W=AC<eRQP;N²:=P²J|8_RTcB>@A6[S©�:a>e:=<VA�Ös>DACP;NBACN�>@H�:XP;J6WXL;N;A-JCHQP�>D:XP�L;L;UFI[DA6��L;ACP;J6:XA6<Ck798;A&A6<DJCRQcjAzc;[DH\f;RQf;:XW=:=>y`tJCRQP fpAzL;<MACNwRQ<ZRVc_RQ[@RTUVAq>@[D:=JzRTNißML;<M>DUVA6Pr>ZH\P}>D8;A&<McjH\Pr>@RQP;ACHQL;<ACU+:X<M<D:=H\PvJCHKA6é+JC:=ACPr>~RQWXW=H { :=P;YO>@8BA+W=:XP;A�:=Pr>@ACYQ[@RT>DACNvACU+AC[MY\ACPr>m¢_LsÖvHQF�R-NB:X<DJ6[DAq>@A�<DcpACJq>@[@RTWWX:=P;A�>D[@RQP;<M:=>D:XH\Pd� � ¢ � � �;>@H�fpA { [M:=>M>@ACP-RQ<; C �

¾t� " � � � � © �:� � � � N¬OP ��� � ¦<=­¨ ? =�� N ��¨ �  ;j¦A@sÃ|�\¨ ? @ « ¦×�\kÊÚ\¨

úGûyü�û�­ ®��Zþ¯�� $�:����)�°�]��ÿ��� ±�����²������8�*��}����)�Zþ�EH\P;<M:XN;A6[�P;H { >@8BA-HQc_RQJC:a>y`�Aq��A6J6>@<�FI[DHQUØ>@8BA-cpAC[M<DcpACJq>@:=gQA�HTFZRT>DH\U+:XJecpH\c;LBW]RT>D:XH\P�U+HKN;:a�Î_JCRT>@:=H\PdkOÌ�[D:=>D:XP;YtH\Ls>~>D8;Ae<McjH\Pr>@RQP;ACHQL;<zA6UV:=<D<D:=H\Pw[@RT>DA+FI[MH\U�WXAqg\ACW �³� >@H � ��RQP;N�c;8BHQ>@HT�RQf;<MH\[Dcs>@:XHQP;<�>@H � � FI[DH\U � { A�Y\A6>Z>D8;A&FIH\W=WXH { :XP;Y+AqÖsc;[MAC<M<D:XHQPwFIH\[�>@8;A�PBA6>oc;8;HT>@HT�nACU+:X<D<M:XH\P[@Ri>@AoJCH\Pr>D[D:XfBLB>@:=H\PO>@H�>@8BAm<h>|RT>D:X<h>@:XJCRQWdf_RTW]RQP;J6A1AC��L_RT>D:XH\PB<

Nf´:N �#¨ �  Df� �j� � ¦A@sÃB=µ¨ ? @ ?¶µ ¶ N§´�N ��¨ �  F � � � � ¦A@sÃH=­¨�·�¦A@sÃB=µ¨ ? @ ?¶µ ¦h�\ka���r¨{ 8;A6[DAtH\P;AOJ6H\P;<M:XN;A6[D<�ACU+:X<M<D:=H\P�FI[DHQU�R}gQH\WXL;U+AOA6WXACU+ACPr>�RTP;N�c;8;HQ>DHT�×RTf;<DH\[McB>@:=H\P�>DH >@8;:=<ACW=ACU+ACPr>!��P;ACY\W=ACJq>@:XPBYu<h>@:=U&L;W]Ri>@ACN A6UV:=<D<D:=H\Pd�^RQP;N�·�¦J@sÃB=µ¨e:X<S>D8;Av[@RQN;:XRT>@:=H\P�N;A6P;<D:a>y`xRT>¸=×kÀ <M<DL;U+:XPBYz>@8_Ri>E>@8BAZACU+:X<M<D:XHQPVFI[DHQU >D8;A�gQH\WXL;U+A�A6WXA6UVA6P�>E:X<?L;P;:=FIHQ[DUí:XPORQWXWÈN;:=[DACJq>@:=H\P;<�RTP;N:XPr>DACY\[DRT>@:=P;Y~H�gQAC[E>D8;A�P;HQ[DUVRQWX:=<DACN+ACU+:X<M<D:=H\Pec;[MHQÎ_W=AQ��>@8;A^P;A6>E<DcpH\Pr>|RQP;A6H\L;<�A6UV:=<D<M:XH\PV[@RT>DA^:=<

N§´�N ��¨ �  Dk� � � � ¦A@sÃH=­¨ ? @ ?¹µ ¶ Nº´:N �#¨ �  F � � � � ¦J@sÃB=µ¨+·�¦A@KÃB=µ¨ ? @ ?¹µ ¦h�\ka�\��¨

Page 30: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

��Û

C N§´�N ��¨ �  Dk� �[� � ¦A@sÃH=­¨ ? @ ?¹µ © ��¶ | ´ | �#¨ �  F � � � � ¦J@sÃB=µ¨+·�¦A@KÃB=µ¨ ? @ ?¹µ| ´ | ��¨ �  Df� � � � ¦J@sÃB=µ¨ ? @ ?¹µ « ¦h�\ka���\¨

C & � � � � ¦J=µ¨ N§´�N ��¨ �  Df� �h� � ¦A@KÃB=µ¨ ? @ ?¹µ ¦h�\ka��©r¨798_RT>�:=<

& � � � � ¦J=µ¨ C �3¶ | ´ | ��¨ �  F � � � � ¦A@sÃB=µ¨�·�¦J@sÃB=µ¨ ? @ ?¹µ| ´ | �#¨ �  Df� �j� � ¦A@sÃB=µ¨ ? @ ?¶µ ¦h�\ka�!¾�¨& � � � � :X<4[MA6FIAC[M[DA6N+>DH~RT<4>@8;A3RQfB<DH\[McB>@:=H\P&FµRQJ6>DH\[C�TFIH\WXW=H { :=P;Y�>D8;A�>@A6[DU+:XP;H\W=H\YQ`oHQF�»�:=<DJ|8�f_RQJ|8;A6[A6>VRQW­k9¦­�Q�\�\�r¨�k�ln>�J!RQPufpA-<MACA6P�>@8;RT>>& � � � � :=<�H\P;AtUV:=P�L;<&>@8;AS>DHQ>|RQW9P�L;UzfjA6[�HQFZc;8BHQ>@HT�RQf;<MH\[Dcs>@:XHQP;<�RT>»¦ N;:=gK:=N;ACNtfr`S>@8;A1>DHQ>|RQW�PKLBU&fjA6[3HQF4<McjH\Pr>@RQP;ACHQL;<^A6UV:=<D<M:XH\P;<3Ri>»¦ kb798_RT>�:=<& � � � � ¦J=µ¨1[DA6c;[DA6<DACPr>D<~>@8BAVc;[MH\f_RQfB:XWX:a>y`}HTF9Rtc;8;HQ>DH\P�PBHQ>zfpAC:=P;Y-RQf;<MH\[DfpACN�RT>mcjH\<M:=>D:XH\P¼=�:=P>@8BAmcBW]RQ<MUeRBkln>4:=<�:=P�>DAC[MAC<M>D:XP;Y�>@HZJ6H\P;<M:XN;A6[�>@8BA�WX:=UV:a>@:XPBY�J!RQ<MA�HQFpR�fBW]RQJ|�ofpHsNB`Qk�lyPz>D8;:X<�J!RQ<MAE>@8BAE[@RTN;:]RT>D:XH\PN;A6P;<D:a>y`¸·�¦J@;¨3:=<^YQ:=g\A6P fr`t>@8;A~<DH\L;[MJCAmFIL;P;J6>D:XH\P�AC��L;:agQRTWXACPr>�>@H D ¦J@sÃB=µ¨p½�F�¦A@KÃB=µ¨qk À WX<DHB�p:aF4>@8BAc;WXRQ<DUVRZ:X<GR1f;W]RTJ|�&fjHKNB`&>D8;[DHQL;Y\8;H\Ls>?:a>@<4gQH\WXL;U+A9>@8;A6P¾FE¦A@sÃH=­¨�RQPBN D ¦A@KÃB=µ¨4RQ[DA3:XPBN;ACcpACP;NBACPr>HQF�cjHQ<D:=>D:XH\P�k4798_RT>�:X<

& � � � � ¦<=­¨ C �3¶ | ´ | ��¨ �  F � � � � ¦A@;¨¹¿ �h�B��t�[�H� ? @ ?¹µ| ´ | �#¨ �  Df� � � � ¦A@;¨ ? @ ?¶µC � ¦h�\ka� à ¨798�L;< { A�8_R!g\A?JCH\U+c;W=A6>@AG[DA��×RQfB<DH\[McB>@:=H\PzHQF_RQWXWrA6UV:a>D>@A6Nzc;8BHQ>@H\PB<4RT>�RQPr`mcjH\:=Pr>u=_RQ<�AqÖscpACJ6>DACNFI[DHQU�N;Aq>|RQ:=WXACNtf_RTW]RQP;J6A1L;P;N;A6[�>D8;AC[MUVHKNB`KP_RQU+:XJ�AC��L;:XW=:Xf;[M:XL;U J6H\P;N;:a>@:XHQP;<CkÑ HT>@A^>D8_RT>?FIH\[?>D8;AZRQfB<DH\[McB>@:=H\PeFµRQJq>@H\[?RQ[DYQL;UVA6Pr>�8;A6[DA { A1RQ[MA�JCHQP;<D:=N;AC[M:XP;YzRQf;<DHQ[DcB>D:XH\PSRTP;NACU+:X<M<D:=H\P(RT>oROcpH\:=P�>�:XP}>D8;A�c;WXRQ<DUVRBk1ln>1:=<ZP;HSWXHQP;Y\AC[�>@8;A�A6<DJCRQc;:XPBYO¢_LsÖ >D8_RT>1:=<ZHQFG:XPr>DAC[h�AC<h>1f;LB>^>@8;AzcjH\cBL;W]RT>D:XH\P-U+HsN;:aÎ_J!Ri>@:XHQP�Ri>�>@8BA&cjHQ:XPr>�=ykZ798BA&c;8;HQ>DHT�×RTf;<DH\[McB>@:=H\P�>D[@RQPB<D:=>D:XH\P[@Ri>@AtJ!RQP£fpA-RQf;<DHQ[DfpACNu:XPr>@H(>@8BA-<McjHQP�>@RQP;A6H\L;<+ACU+:X<D<M:XH\P�[DRT>@Atfr`�:=P;JCW=L;N;:XPBY >@8;A-RQf;<MH\[DcK�>@:=H\P(FµRQJ6>DH\[oRQ<~RSc_RQ[@RTUVAq>@[D:=J�RQNißML;<h>@U+ACPr>!k+798_RT>m:=<C�Çc;8;HQ>DHT�×RTf;<DH\[McB>@:=H\P;<oRQ[MA�:XP;J6WXL;N;A6N�RT<P;A6YrRT>@:ag\A~<DcpH\Pr>|RQP;A6H\L;<ZA6UV:=<D<M:XH\P;<6k3lyP H\L;[^JCH\W=WX:X<M:XH\P;RQWa�n[@RQN;:XRT>@:åg\A^J!RQW=JCL;WXRT>@:=H\P;<9>@8B:X<�:X<�A6gTRQWa�L_RT>DACN�fr`�U&L;W=>D:Xc;Wa`K:XP;Yt>@8;AO¥b:=P;<M>DAC:=P À �§gQRTWXL;A�FIH\[zACRQJ|8�>@[DRQP;<D:a>@:=H\P�fr`�:=>@<~JCH\[M[DA6<DcpH\P;N;:=P;YRQf;<MH\[Dcs>@:XHQP�FµRQJ6>DH\[Ckw½^A6P;JCA { 8BACPuH\P;AO<MH\W=gQAC<&>@8BAS<M>|Ri>@:X<h>@:=J!RQWGf_RQWXRQP;JCAeAC��L_RT>D:XH\P;<z:XP;J6WXL;Ns�:XPBYt>D8;AC<MAeRTf;<DH\[McB>@:=H\P(FµRQJ6>DH\[D<1>D8;A�H\c_RQJ6:=>y`(U+HsNB:=Î_A6NwcpH\c;LBW]RT>D:XH\P <h>@[ML;J6>DL;[DAV:X<1c;[MHsN;LBJCACN�k

Page 31: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

��Þ

798;A�RQfB<DH\[McB>@:=H\P FµRQJ6>DH\[�A�Ösc;[DA6<D<D:=H\PwHQF�¦h�\ka�!¾�¨3:X<ZH\LB[Z<M>@RQ[M>D:XP;YOA�Ösc;[DA6<D<D:=H\PvFIH\[^>D8;A�H\c_RQJ6:=>y`U+HsN;:aÎ_J!Ri>@:XHQP;<�>@HmcpH\c;L;WXRT>@:=H\P�<M>@[ML;J6>DL;[DATk�ln>E:=<bN;A6gQACWXHQcjA6Ne:XP+J|8_RQcs>@AC[E�ZFIH\[�gQRT[D:XHQL;<bY\ACH\U+Aq�>@[M:XA6<&RQP;N�WX:XPBA+cB[DHQÎ_W=AC<6k �E8_RTcB>@A6[�©->D8;ACP�c;[MH�gs:=N;AC<m>@8;AeAqÖK>@A6P;<D:=H\P�HQF9>D8;:X<mAqÖsc;[MAC<D<M:XH\P�>DHJCHQP�>D:XP�L;L;UçFI[DAC��L;A6P;JC:=AC<6k798�L;< { A&8_R!g\Am> { HSf;RQ<D:=J~A�ÖBcB[DAC<M<D:=H\P;<C��¦h�\kÕãr¨3RQP;N�¦×�\ka�!¾�¨q� { 8;:=J|8 J!RQP}fjAmL;<DA6N >@HeRQJCJ6H\L;Pr>FIH\[�>D8;AoA6�jACJ6>D<^HTF�H\c;RQJC:a>y`O:XP-RQP-H\cB>D:XJCRQWXWa`+>D8;:XJ|�ec;W]RQ<MUeRsk

úGûyü�û<À Á!���)��+�»8�8�1 ������ÿ4��.-798;A6[DA�:X<1HQF?JCHQL;[D<MAVRSJ6H\Pr>@:XP�L;LBUäAC��L;:agQRTWXACPr>1>DHt>D8;A�N;:X<MJC[MA6>@A�W=:XP;AzH\c_RQJ6:=>y`}>D8_RT> { AV8;R�gQAWXHKH\�TACNVRT>!k�½�AC[MA { A^J6H\P;<M:XN;A6[?c;8BHQ>@HT�n:XH\PB:X<@Ri>@:XHQP;<�RQP;N+[DA6JCH\Uzf;:XP_Ri>@:XHQP;<b[@RT>D8;AC[�>@8;RQPec;8BHQ>@HT�RQf;<MH\[Dcs>@:XHQP;<�RQPBNO<DcpH\Pr>|RQPBACH\L;<9A6UV:=<D<M:XH\P;<6k��sA6A Ò [@R { :=Pv¦h��Ú\ÞQ�r¨4FIH\[ERTPOAqÖBRQU+c;WXA^HQFÇJCHQP�>D:XPs�L;L;UèRQf;<MH\[Dcs>@:XHQP�FµRQJq>@H\[M<�N;A6[D:=gQACNvRQPBN L;<DA6N(:XP�c;[@RQJq>@:=JCAS¦I:XP�8;:X<^J!RQ<MA~>@HeA6gTRQWXL;RT>@Amc;W]RT<DUVR[DA6W]RiÖBRT>D:XH\P�>@:=UVA6<DJCRQWXA6<|¨qk�798;AZJCHQP�>D:XP�L;L;U AC<MJ!RQcpA1c;[MH\f_RQf;:=WX:a>y`\� { 8;:=J|8 { :XWXWpRQW=<DH~fjA^[DA6FIA6[D[MACN>@HORQ<�>@8BAzfpH\L;P;Ns�§FI[DA6A&AC<MJ!RQcpA&c;[MH\f_RQf;:=WX:a>y`\�È:X<^N;AC[M:=g\A6N(fjA6WXH { k Ñ HQ>@Az>D8;Az<M:XU+:XWXRQ[D:a>@:XA6< { :=>D8>@8BAmfpH\L;PBNs�yfpH\L;P;N-AqÖsc;[MAC<M<D:XHQPdk798;AmJCH\Pr>D:XP�L;L;Uá:=Pr>@ACYQ[@RT>DACNS¢_LsÖ�RT>3f-:X< { [M:=>M>@ACP-RQ<

;j¦JQq¨ C N§Â<à �mÄ�Å © N OP D OxÆ ¦A@sÃB=µ¨h¯ º | ~� � ~ÈÇ �h�p� �#�[������� ? =��ª;j¦A@sÃ@�r¨M¯ º | ~��� ~ÉÇ �[�p� �j�y�j��� « ? @C | Âxà �mÄ�Å | OP D OxÆ ¦J@sÃB=µ¨M¯ º | ~� � ~ÉÇ �[�p� �����y���#� ? = ? @�� | Âxà �mÄ�Å ;�¦J@sÃ|�r¨h¯ º | ~��� ~ÈÇ �h�p� �h�y�#�+� ? @| Âxà �mÄ�Å | OP D OxÆ ¦J@;¨ ? =��K;j¦A@sÃ|�\¨ ? @� N§Âxà �mÄ�Å N OP D OxÆ ¦A@;¨ ? =��ª;j¦A@sÃ@�r¨ ? @C " ��N§Âxà �mÄ�Å N>OP D OxÆ ¦J@;¨ ? =��ª;j¦A@sÃ@�r¨ ? @C " � © �¾t� �qÊ � � N>OP ��Ê ¦J=µ¨ ? =�� N§Âxà �mÄ�Å ;j¦J@sÃ|�r¨ ? @ « ¦h�\ka��Ûr¨

Ñ HT>@A1>@8_Ri>�Ë-[DACcB[DAC<MACPr>@<�>@8;AoA�Ös>DACP;<M:XH\P�HQF4fpH\L;PBN�<h>|RT>DAC<C� � � �_:XPr>DH�>D8;AmJCH\Pr>@:=P�L;L;U�gK:]R�>@8BA>@[DRQP;<hFIH\[DUVRT>@:=H\P � ��¢ Ì Ëdkx798;A fpH\L;P;NK�nFI[DA6A}A6UV:=<D<D:=H\P�RQP;N�RQfB<DH\[McB>@:=H\P²J6HsAqéVJ6:XA6P�>D<SRQ[MA>|RT�QACP-FI[DH\Uáì�ACPBÝCACW�â Ü�AC�TAC[D:=<&¦×��Ú\© à ¨qk4798_Ri>^:=<

Page 32: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

��ã

D OxÆ ¦A@;¨ ? @ C �¾t� � � ¨<Í ¦<Îs¨,Q Ê ? ÎÏ�¹� � Â+ιРÊ]� �Ñ D OxÆ ¦J@;¨ C �¾t� � � ¨<Í ¦<Îs¨,Q Ê ? ÎÏ�RÒ � Â,ιРÊB� �? ËRÓ ? Ë? @ ¦h�\ka��Þ\¨

RQP;N

F OxÆ ¦A@B¨ C���� �ǯ �¹t¸ Í ��mÊ ? Ë? @RÔ¶Õ�Õ�§Ê ¦h�\ka��ãr¨{ 8;A6[DA Í ¦xÎs¨�:=<G>D8;A1ì�RiÖ { A6WXWX:XRQP�g\ACW=HsJ6:=>y`�N;:=<M>D[D:XfBLB>@:=H\P�FIH\[GFI[MACA^ACWXA6J6>D[DH\P;<6�¯Ð ÊB� :X<b>@8;A3>|RT[DY\Aq>RQ[MA!R&FIH\[�FI[DA6AoACWXA6J6>D[DH\P-J!RQcs>@L;[MAmYQ:=g\A6P�fr`

Ð ÊB�¥C � �©¶Ö © ¯ �]×¹ � ¸ ´ �Ø ³�§Ù � ³Ê�Ù�Ú Ë �� ´ Ô¶Õ�ÕÊB� ¦h�\ka��Úr¨

Í ��mÊ :=<�>@8;AofpH\L;P;NK�nFI[DA6AoH\<DJ6:XWXWXRT>@HQ[E<M>D[DA6P;YQ>@8�RQP;N Ô Õ�Õ�§Ê :X<�>@8BAmfpH\L;PBNs�nFI[MACA~ëmRTL;Pr>9FµRQJ6>DH\[Ck798�L;<�>D8;A�AC<MJ!RQcpA�c;[MH\f_RQfB:XWX:a>y`�" � :X<�NBA6Î_P;A6N�RQ<�>D8;AE[DRT>@:=H�HQFB>@8;A?H\c_RQJ6:=>y`mUVHKN;:aÎ_ACN~AC<DJCRQc;:=P;Y¢_LsÖ�>@H�>@8BA(H\cB>@:=J!RQW=W=`£>@8B:XP�AC<MJ!RQcB:XP;Y�¢_LsÖj� { :=>@8�fpHQ>D8�¢_LsÖsAC<-A6gTRQW=L_RT>@A6N RQ<SJCHQP�>D:XP�L;L;U:XPr>DACY\[DRT>@A6Nt¢;LsÖsAC<Ck?ln>9FIH\WXW=H { <E>@8_Ri>

" �¥C | Âxà �_Ä�Å | OP D OxÆ ¦A@sÃB=µ¨h¯ º | ~� � ~ÈÇ �h�p� � � ����� � ? = ? @�� | Âxà �mÄ�Å ;j¦A@sÃ|�\¨M¯ º | ~� � ~ÉÇ �[�p� �j����� ? @| OP | Âxà �_Ä�Å D OxÆ ¦A@;¨ ? @¥�K;�¦J@sÃ|�r¨ ? = ¦h�\kÊ�Q�r¨

Ñ HT>@A^>D8_RT>?>@8;A6[DAZ:=<EP;H&<ML;f;<MJC[D:=cB>�ËOH\P%" � :XP+>@8;AZNBA6Î_P;:a>@:=H\Pe<M8;H { PS8;AC[MAQkb798;:X<?:X<?fjA6J!RQL;<MA>@8BAeAC<MJ!RQcpAecB[DH\f_RTf;:XW=:=>y`(:=<~:=P�>DACY\[DRT>@A6N�H�g\A6[z>@8BA { 8;H\W=AV<MA6>zHQF9JCHQP�>D:XP�L;L;UÆ<h>|RT>DAC<>Ëd��>D8�L;<" � [DA6c;[DA6<DACPr>D<^>@8BA~c;[MH\f_RQfB:XWX:a>y`eHTF4R�[DA6JCH\U&fB:XP_RT>D:XH\PScB8;HQ>@HQPd�_c;[DHKN;L;J6ACN-FI[DH\UñRTP�A6WXACJq>@[MH\PRQPr` { 8;AC[MA^:=P&>@8;A9J6H\Pr>@:=PKLBL;U±[MACJCHQU&f;:XPB:XP;YZ>DHZ>@8;A9<h>|RT>DA � ��AC<MJ!RQc;:=P;YoRQW=H\P;YZ>@8BA9J|8;H\<DA6PVW=:XP;AHQF�<D:XYQ8�>Ck À U+H\[DAoY\A6P;AC[DRQWÇAqÖsc;[MAC<D<M:XH\P->@8_Ri>^:=<3P;HQ>�:XPr>@A6Y\[@Ri>@ACN-H�g\A6[�>D8;A { 8BH\WXAoJCHQP�>D:XP�L;L;U:X<9cB[DAC<MACPr>@A6N :XPtJ|8_RTcB>@A6[^©sklyP(R+<D:XU+:XWXRQ[�UVRQP;P;A6[�>D8;A&RTf;<DH\[McB>@:=H\PtFµRQJ6>DH\[�FIHQ[^fpH\L;P;NK�nFI[DA6Am>D[@RQPB<D:=>D:XH\PB<�UVR!`-fjA { [D:a>D>@A6P:XPt:a>@<9Y\A6P;AC[DRQWdFIH\[DU-k4798;AmP;Aq>^:=H\P;:=<@RT>D:XH\PO[DRT>@A1:=<9Y\:=gQACP-fr`

Nº´:NºÂxà �mÄ�Å D�Ê � � ¦A@sÃH=­¨ ? @ ?¹µ ¶ Nº´:N§Â<à �mÄ�Å F � � Ê ¦J@sÃB=µ¨+·�¦A@KÃB=µ¨ ? @ ?¹µ ¦h�\kÊ�s��¨

Page 33: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

��Ú

C Nº´:NºÂxà �mÄ�Å D�Ê � � ¦A@sÃH=­¨ ? @ ?¹µ © ��¶ | ´ | Â<à �mÄ�Å F � � Ê ¦J@sÃB=µ¨+·�¦A@KÃB=µ¨ ? @ ?¹µ| ´ | Âxà �_Ä�Å DkÊ � � ¦J@sÃB=µ¨ ? @ ?¹µ « ¦h�\kÊ�\�\¨

C & � � Ê ¦J=µ¨ N§´�N§Âxà �mÄ�Å DkÊ � � ¦A@KÃB=µ¨ ? @ ?¹µ ¦h�\kÊ�Q©r¨

{ 8;A6[DAS>@8;A-RQfB<DH\[McB>@:=H\P£RQP;N£ACU+:X<M<D:=H\P�JCHKA6é+JC:=ACPr>@<�8_R!gQA->D8;AS<@RQU+ASN;AqÎ_P;:=>D:XH\PB<�RQ<�:XP�>@8BAfpH\L;P;Ns�§FI[DA6A+A6<DJCRQcjA�FµRQJq>@H\[1JCRQ<DATke798_RT>m:=<C�/>@8;A+RQfB<DH\[McB>@:=H\P(FµRQJ6>DH\[ZFIH\[ofjHQL;P;Ns�§FI[DACA�c;8BHQ>@HT�RQf;<MH\[Dcs>@:XHQP-:=<9Y\:=gQACP-fr`

& � � Ê ¦<=­¨ C �3¶ | ´ | Âxà �_Ä�Å F � � Ê ¦A@KÃB=µ¨+·�¦A@sÃH=­¨ ? @ ?¹µ| ´ | Âxà �_Ä�Å DkÊ � � ¦J@sÃB=µ¨ ? @ ?¹µ ¦h�\kÊ�T¾�¨798;AO<M:XU+:XWXRQ[D:a>y`}fpA6> { ACA6PuA6��L_RT>@:=H\P;<O¦h�\kÊ�Q�r¨��9¦h�QkÓ�T¾r¨mRQPBN�>@8;AefjH\LBP;Ns�yfpH\L;PBN�JCH\LBP�>DAC[Mc_RQ[M>D<¦h�QkÊãr¨��d¦×�\ka�!¾�¨4:X<GJCW=A!RQ[6k4798;:X<?<D:=UV:=W]RQ[M:=>y`m:X<G<D8;H { POFIH\[MUeRTWXW=`z:XP+J|8_RQcB>DAC[9© { 8;AC[MA^>D8;A�fpH\L;P;NK�fpH\L;P;N�AqÖsc;[DA6<D<M:XH\P;<�RQ[DAm<D8BH { P�>@HVRQP_RQWa`K>D:XJCRQWXWa`eJ6H\Pr>@:XP�L;A~:XPr>@H�>D8;AzfpH\L;P;NK�nFI[DA6AoH\P;AC<^L;cpH\PRQc;cBWX:XJCRT>@:=H\POHQF�>@8BAo>D[@RQPB<MFIH\[MUeRT>D:XH\P � ��¢ Ì Ë/kúGûyü�û¤Û Ü��,���(�� ����X���R���+��5lyP c;[DRQJ6>D:XJ6Az:=>^:X<3FIH\L;PBN}>@8_Ri>�>@8BAzAC<MJ!RQcpA�c;[MH\f_RQf;:=WX:a>y`SRQP;NvRTf;<DH\[McB>@:=H\P�FµRTJ6>@HQ[^A�ÖBcB[DAC<M<D:=H\P;<J!RTP+fpA98;:=Y\8;W=`m<MACP;<M:=>@:ag\A�>DHo>D8;A9ACU+:=>M>@:=P;YZRQPBNVRTf;<DH\[Mf;:XPBYZWX:=P;A�c;[MHQÎ_WXA6<C�r<MACA�FIH\[4A�ÖBRQUVcBWXA�>@8BA:XW=WXL;<h>@[DRT>@:=H\P;<4:=Pe�EA68;[D:=P;Y\A6[Z¦×��Ú\Ú\ãr¨�k4»_H\[�>@8B:X<b[DACRQ<DH\P��;RQP;NV:XP+H\[DNBAC[b>@H~RQJCJ6L;[@RT>DACWa`�JCH\U+c_RQ[MARQPr`w>@8;A6H\[DAq>@:XJCRQWG<DcpACJ6>D[DL;U ¦ { 8;Aq>@8;A6[&H\cB>D:XJCRQWXWa` >@8;:=P�H\[m>@8;:=J|�B¨ { :=>D8�>@8_Ri>&H\f;<MAC[hg\ACN£:a>z:=<P;A6JCAC<M<@RQ[h`t>@H�UVHKN;A6WdACU+:X<M<D:XHQP�RTP;N-RQf;<DHQ[DcB>D:XH\PtWX:=P;A1c;[MHQÎ_WXA6< { :=>D8}R�8;:=Y\8tN;ACY\[MACAoHQF�c;[DA6JC:a�<D:=H\Pdk4798B:X<E:=<EA6<DcpACJC:XRQWXWa`�>@[ML;A^FIH\[b>@8BAZ8;:XYQ8e<DA6[D:=AC< { H\[M�Vc;[MAC<MACPr>@A6N�:=PeJ|8_RQcs>@AC[ à { 8;A6[DAZ>@8BA>@8BACH\[MA6>@:=J!RQW?ACW=ACJ6>D[DH\P�f;[DHrRTN;ACP;A6N£WX:XPBA { :XNs>@8;<�RT[DAOL;<MACNuRQ<�R�N;ACP;<M:=>y`�N;:]RQYQP;H\<M>D:XJTk 798;AC[MARQ[MA1UeRQPr`e<M>|RTP;N_RQ[MN->DAqÖK>@fpHsHQ�K< { 8;:=J|8tY\:=gQA~RTP-H�gQAC[MgK:=A { HTF�>D8;A1f_RQ<M:XJZ>@8BACH\[h`SfjA68;:XPBN-W=:XP;Af;[MHrRQN;A6P;:XP;Y²¦µ<MACA ëo[M:XACU-���!ÚrÞT¾;��ëo[M:XACU-����Ú\ÚrÞs���sH\fpACW=UeRQP£Aq>ORQW­k=�^��Ú\ãB��RQP;N�J|8_RQcB>DAC[���©HQF���RT>@A6<C�?��Ú\ÛQ¾�¨�kSì�HQ<M> { H\[M�(:XPv>@8;:=<zRQ[MA!RtJCH\PBJCACPr>D[@RT>DAC<&H\Pw>@8;A+A6gTRQW=L_RT>D:XH\PvHQFEc;[DA6<D<DLB[DAf;[MHrRQN;A6P;:XP;YeRQ<Z:a>�:=<�>D8;A&U+H\<M>^JCH\U+c;W=AqÖ}RQP;N HTFÅ>@ACP�>@8;A~UVH\<h>Z:=UVcpH\[M>@RQPr>�WX:XPBA~f;[MHrRQN;A6P;:XPBYU+ACJ|8_RQPB:X<DU-k-798;AOP;A�Ös>zFIA { <DA6J6>D:XH\P;< { :XWXWbY\:ag\AVR}f;[M:XA6F�H�gQAC[hgs:=A { HQF�>D8;AOUVRQ:XPwf;[DH\RQN;ACPK�:XPBY�AqÖsc;[DA6<D<M:XH\P;<6�ÈJCH\P;J6ACPr>@[DRT>@:=P;Y�H\PtR�N;AC<MJC[M:XcB>D:XH\PSHQF�>@8BA1c;8r`s<M:XJCRQWdc;[MHsJ6AC<D<MAC<�fjA68;:XPBN-ACRQJ|8U+ACJ|8_RQPB:X<DU-ko798;A�FIH\WXW=H { :XP;YV<DACJq>@:=H\P;< { :XW=W�WXR!`}H\Ls>Z>@8;A�f;RQJ|��Y\[DH\LBP;N(>@8;A6H\[M`}P;ACJ6AC<D<DRQ[M`(>DHc;LB>9>D8;A1WX:=P;AZc;[MHQÎ_W=AZJ!RQW=JCL;WXRT>@:=H\P;<?HQF�J|8_RQcB>DAC[D<��s�B©B�;¾+RQP;N à :XPr>DH�JCH\Pr>@A�ÖK>!kG798;AC[MAmA�Ös:X<M>D<3R

Page 34: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�Q�

{ :XNBAm[DRQP;Y\AmHQF�f;[DHrRTN;ACP;:=P;Y�UVA6J|8_RQP;:=<DU+<3:XP;J6WXL;N;:=P;Y Ò H\c;c;W=AC[9f;[MHrRQN;A6P;:XPBY;�_P_RT>DL;[@RQWÇf;[MHrRQNs�ACPB:XP;Y;��c;[DA6<D<ML;[DAOfB[DHrRQNBACP;:=P;Y(RQ< { A6WXW?RQ<z:=P;<M>D[DL;U+ACPr>@RQWbA6�jACJ6>D< { 8;:=J|8�RQW=>DAC[o>@8;AVH\f;<MAC[MgQACNc;[MHQÎ_WXATk�798;Aof_RQ<M:XJZW=:XP;AZcB[DHQÎ_W=AZAqÖsc;[MAC<M<D:XHQP;<�FIHQ[�A!RQJ|8-HQFÇ>@8BAC<DAoU+ACJ|8_RTP;:X<MUV<3RQ[MA1c;[DA6<DA6P�>DACNRQW=H\P;Y { :=>D8-RQPr`-RQc;c;[MH�Ös:=UeRT>DAZAqÖsc;[MAC<M<D:XHQP;<^>D8_RT>�RQ[MA1L;<DA6N}:XPS>D8;AoW]RT>DAC[9J|8_RQcs>@AC[M<CklyPzcB[@RQJq>@:XJ6AbH\P;AGL;<DL_RTWXW=`1RQ<D<ML;UVA6<�>D8_RT>�>@8;AGACU+:X<M<D:=H\PmRQP;N&RTf;<DH\[McB>@:=H\Poc;[DHTÎ_WXA6<�RQ[MA?:=N;ACPr>D:XJ!RTW§k798_RT>e:X<6�9:XP²>D8;A�>D:XU+A-FI[MH\U c;8;HQ>DHT�×RQfB<DH\[McB>@:=H\P�>DH�<DL;fB<DAC��L;A6Pr>�[MAq�nACU+:X<D<M:XH\P��E>@8;A6[DA 8;R�gQAfpACACP�<DLBé+J6:XACPr>�JCH\W=WX:X<M:XH\PB<E<DLBJ|8}>@8_Ri>�>D8;AmN;:X[MACJ6>D:XH\PtHQF�>@8;AmACU+:=>M>@A6N-cB8;HQ>@HQP�:=<�PBHQ>�[DACWXRT>@A6N>@H+>@8_Ri>�FI[MH\U { 8;:=J|8}:=> { RQ<ZRTf;<DH\[MfjA6Ndk9798�L;<�>D8;A~A6UV:a>D>DACN-c;8;HQ>DH\P}8_RT<�fpACACP->@8;A6[DUVRQWX:=<DA6NRQP;N£>@8BA-[MAC<ML;W=>@RQPr>�ACU+:X<D<M:XH\P£c;[MHQÎ_W=AS:X<�Y\H�g\A6[DP;A6N²fr`�>D8;At<@RQU+Atc;W]RQ<MUeR cB[DH\cpAC[h>@:XA6<&>@8_Ri>WXACRQN >@H£>@8;A�RQf;<DHQ[DcB>D:XH\Pxc;[DHTÎ_WXATk�798;:=<}RQ<M<DL;U+cB>D:XH\Px:X<tHQFÅ>DACP [MA6FIA6[D[DA6NÔ>DH�RQ<-JCH\U+c;W=A6>@AFI[DA6��L;ACP;Jq`w[DACNB:X<M>D[D:=f;LB>@:=H\Pu¦§��Ð Ò ¨1RQP;Nw:X<1UVRQN;A�>@8B[DH\L;YQ8;H\LB>1>D8;AV[MAC<h>~HTFE>@8BA�>D8;AC<M:X< { :=>D8>@8BA�cpH\<D<M:Xf;:=WX:=>y`zHQF/N;:a��A6[DA6P�>?ACU+:X<M<D:=H\PeRTP;NORQfB<DH\[McB>@:=H\Pec;[MHQÎ_W=AC<GfjA6:XP;Y~:XP;J6WXL;N;A6NORT>ERmFILs>@L;[MAN_RT>DAo:=F�P;ACJ6AC<M<@RQ[h`\k

Ý)Þ¶ßáàXÞ¹âfã�äfå4âfÞIÞÏæ�Þ¶çtäfè798;A fB[DHrRQNBACP;:=P;Y�N;LBA}>@Hw>@8;A�:XP;8;A6[DA6P�>e>@8;A6[DUVRQW3UVHQ>D:XH\PuHQFZ>D8;A A6UV:a>D>@:=P;YvH\[ORTf;<DH\[Mf;:XPBYRT>DH\UV<6�Bc;[MH�gs:=N;ACNS>D8;AC:=[?g\A6WXHKJC:a>y`eN;:=<M>D[D:XfBLB>@:=H\PO:X<9ì}RiÖ { A6WXW=:]RQP { :a>@8SJ|8_RT[@RQJq>@AC[M:X<h>@:XJ^>@ACU+cpAC[h�RT>DL;[DAo71�_[DA6<DL;Wa>@<3:XP-R�WX:=P;AZc;[MHQÎ_W=A1Y\:=gQACP-f�`e>@8;AoP;H\[MUeRTWX:X<MACN Ò H\cBc;WXA6[9AqÖsc;[DA6<D<M:XH\P

�;¦J@;¨ C �Ö �4é�@�ê ¯ º}ëtì+íºì �î ìxïTð Ù ¦h�\kÊ� à ¨{ 8;A6[DA

é�@�ê C �@ P�ñ �\®KÁ¹ ¦h�\kÊ�QÛr¨:X<�>D8;A Ò H\c;c;W=AC[98_RTW=F { :XNB>D8tHQF�>D8;AoYrRQL;<M<D:]RTP-W=:XP;A�c;[DHQÎ;WXAQkò ã¯äfå2âfã�ó�ôRâfõ$ã�ö4Þ¯÷2ø<÷2ù798;A6[DAEA�Ös:X<M>D<�R3U+ACJ|8_RQPB:X<DU«LB<DL_RQW=W=`1[MA6FIA6[D[DA6N~>DH�RQ<�P_RT>DL;[@RTWrf;[DHrRTN;ACP;:=P;Y;�i[DA6<DL;Wa>@:XPBY3FI[MH\U·>@8BAÎ_P;:a>@A�W=:=FIAq>@:XU+AbHQF_>@8BA�ACU+:=>M>@:XPBYZRT>DH\Uî:=P&:=>D<�A�ÖBJ6:=>DACN�<M>@RT>@Ao¦µRQ<4c;[MACN;:=J6>DACN�fr`z>@8BA9½^A6:X<DA6P�fjA6[DYL;P;J6AC[h>|RQ:=P�>y`tcB[D:XPBJC:XcBWXAQ��é>ú�é{û�ü × ½\�k�Ǩqk

Page 35: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�s�

Ñ Ri>@L;[DRQW�f;[DH\RQN;ACPB:XP;Y-AqÖs:X<h>@<m:XP;NBACcpACP;N;A6Pr>@W=`vHTF9RQPr`vAqÖK>DAC[DP;RQWb:XPB¢_LBACP;J6AC<mH\P�>D8;A�ACU+:=>D>D:XP;YRT>DH\U�kîln>t[DA6<DL;Wa>@<S:XPxR�c;[MHQÎ_W=A(�KPBH { PÔRQ<-Ù/H\[DA6Pr>@ÝC:XRQP { :=>D8 R { :=NB>@8 J|8_RT[@RQJq>@AC[M:X<MACNxf�`é�@ký C���� � � � ½\�f��k798;Amc;[DHTÎ_WXA1A�ÖBcB[DAC<M<D:=H\P�:=<3HQF�>D8;A1FIH\[DU

�B¦A@;¨ C ¦Jé�@ký;¨ �¦A@�¶ª@ P ¨ � �£¾È¦Jé�@kýB¨ � ¦h�\kÊ�\Þ\¨

798;:=<Of;[DH\RQN;ACPB:XP;Y�U+ACJ|8;RQP;:X<MU :=<eP;A6Y\WX:=Y\:XfBWXA { 8;ACP J6H\UVc;RQ[DA6N²>@H�c;[MAC<M<DL;[MAvRQP;N�N;H\cBc;WXA6[f;[MHrRQN;A6P;:XP;Y�FIH\[+>D8;A JCRQ<DA6<O>@8_RT>ORQ[DA�L;<DL;RQWXWa`£ACP;J6H\L;Pr>@A6[DACNx:XPuFIL;<D:=H\P�RQPBN�RQ<M>D[DH\cB8�`K<M:XJ!RTWc;WXRQ<DUVRQ<Ckþ Þ¶÷Rè§øMä]ÿªÞ¹æ�Þ¶çtäfè»_H\[&N;A6P;<DAtc;WXRQ<DUVR JCHQP;N;:=>D:XH\PB<&:=>�:=<&WX:=�QA6W=`w>@8_RT>�fpHQ>D8�P_RT>@LB[@RQW�fB[DHrRQNBACP;:=P;YvRQP;N£N;H\cBc;WXA6[f;[MHrRQN;A6P;:XP;Y { :=WXWGfjASPBACY\W=:XY\:=f;WXA+JCH\U+c_RQ[MACN { :a>@8�>@8BASJCH\W=WX:X<M:XH\P;RQWbf;[DHrRTN;ACP;:=P;Y(:XPBN;L;JCA6N£f�`>@8BA-J|8;RQ[DY\A6N²c_RT[M>@:=JCW=AC<�HQF�>@8;A-c;WXRQ<DUVRB�b:XPr>@A6[@RQJq>@:=P;Y >@8;[MH\L;Y\8£>@8BA}��>|RQ[M��A6�jACJq>!k Ñ ACLB>D[@RQWf;[MHrRQN;A6P;:XP;Y~RQP;N+>@8;A^WXH\PBYm[DRQP;Y\A��GRQPVN;AC[bÌ�RQRQWX<bf;[MHrRQN;A6P;:XP;Y { :XW=W;fjA�P;A6Y\WXA6J6>@A6N { :=>D8VHQP;W=`f;[MHrRQN;A6P;:XP;Y1NBL;A9>@HoACW=ACJ6>D[DH\PB<bRQP;N�:XH\PB<�fjA6:XP;Y1J6H\P;<D:=N;AC[MACN+:XP&>D8;:X< { H\[D�Èk4798BA�U+H\<h>bY\ACP;A6[@RQWAqÖsc;[MAC<M<D:XHQP�FIH\[3R+�K>@RQ[D�OfB[DHrRQNBACP;A6N}WX:=P;AZc;[MHQÎ_W=A1:X<

�;¦��E¨ C ���� ¯�� N��P ¯��Ǧ Ì �)£�¨� e¦J£i¨ ? £�� ¦h�\kÊ�Qãr¨{ 8;A6[DA� V¦A£�¨�:=<�[DAqFIAC[M[DACN >DHSRQ<�>D8;A�RQLs>@HKJCH\[M[DACWXRT>@:=H\PtFIL;P;J6>D:XH\P}HQF�>@8;AzWX:XYQ8�>^RQU+c;W=:=>@LBN;AzRTP;N:X<9AqgTRQWXL_Ri>@ACNOFI[MH\U���L_RTP�>DL;UñU+ACJ|8_RQPB:XJ!RTWd��P;H { WXACNBY\AoHQFÇ>@8;AocpAC[M>DL;[DfpAC[�RQPBN-A6UV:a>D>DAC[Ck�798;AJCHQUVc;W=A6>DAm>D[DA!Ri>@U+ACPr>ZHQFb<ML;J|8(f;[MHrRQN;A6P;:XP;Ye:X<�JCH\U+c;WX:=J!RT>DACN�f�`->@8BA~FµRQJq>�>@8;RT>Z<MWXH { UVH�gK:XPBYcpAC[M>DL;[DfpAC[M<9:XPr>@A6[@RQJq> { :=>D8O>@8;A1A6UV:a>D>DAC[E:=PSR&N;:a��A6[DACPr> { R!`O>@H&>D8;A�FµRQ<h>@A6[9cjA6[M>@LB[DfpAC[D<6k��^<DLK�RQW=W=`VH\P;AoHQF�> { HVWX:XU+:=>D:XP;Y~RQc;cB[DH!Ös:XUVRT>@:=H\P;<�:=<9L;<DA6N�>@HVRQJ6JCH\L;Pr>3FIH\[�>D8;Aof;[DHrRTN;ACP;:=P;Y+A6�jACJ6>D<HQFj>@8;A^J|8_RQ[DYQACNecpAC[h>@L;[MfjA6[D<6k�lnFd>@8;A�cpAC[M>DL;[DfpAC[?:X<�FµRQ<h>�RQW=WB>@8BA�f;[MHrRQN;A6P;:XP;YmJCHQWXWX:=<D:=H\P;< { :=WXW_fpA<M>@RT>@:=<M>D:XJ!RTWXW=`m:=P;N;ACcpACPBN;ACPr>bFI[DH\UîA!RQJ|8VHQ>@8BAC[bRQP;N+J!RTP+fpA3:XP;NB:=gK:XN;L;RQWXWa`mRQJCJ6H\L;Pr>@A6NVFIH\[6k�798;:=<:X<~��P;H { P�RT<z>@8;A�ÉÊ:=UVc_RTJ6>�RTc;c;[DH!Ös:XUVRT>D:XH\PdË?¦µÙÇHQ[DACPr>DÝQ�9��Ú\�QÛr¨~RQPBN�WXA!RTN;<~>DHvR Ù/H\[DA6P�>DÝC:XRQPc;[MHQÎ_WXATk?lnF�>@8;AmcpAC[M>DL;[DfpAC[3:=<3gQAC[h`t<DWXH { >D8;ACP-:=>�JCRQP�fpAoJCH\PB<D:XNBAC[DA6N�>DHVfpA~<h>|RT>D:XH\P;RQ[M`VFI[DH\U>@8BAZACU+:=>M>@AC[6ËÕ<?cjHQ:XPr>�HQF/gK:=A { RQP;Ne>D8;AZf;[MHrRQN;A6P;:XPBY&A6�jACJq>@<9HQF/>D8;AZ[DA6<DL;Wa>|RQPr>�Î;ACWXNSNBL;A�>@H&RQWXW

Page 36: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�\�

0

1

2

3

4

5

6

10 12 14 16 18 20 22

Log (

T)

/ T

in K

Log (Ne) / Ne in cm^-3

Impact ions

Impact electrons

Static Ions

Static electrons

»�:=Y\L;[DAu�\kϾ;¼�Ð�RQP;Y\A�HQF&gTRQW=:XN;:a>y`�HQF�f;[MHrRQN;A6P;:XPBY�RQc;cB[DH!Ös:XUVRT>@:=H\P;<SFIH\[�� F?�Z>@RQ�QA6P·FI[DH\U�K>@RQUVUçA6>^RQW­k/¦h�!Ú\Ú\ãr¨�k

>@8BA+ÉÕ<M>@RT>@:=H\P_RQ[h`jË_cpAC[h>@L;[MfjA6[D<�AC<h>@:XUVRT>DACNdï_>@8B:X<3:X<3[DAqFIAC[M[DACN->@HVRQ<3>@8;A�ÉÊ��L_RT<D:a�n<M>@RT>@:=J~RTc;c;[DH!Ös:a�UVRT>@:=H\PdË/¦µ½^H\Wa>@<MUeRQ[M�È����ÚB�!Úr¨qk�EHQ>D8�:=H\P;<�RQP;N�ACW=ACJ6>D[DH\PB<^f;[MHrRQN;A6P->D8;AmWX:XPBAC<9>@8;[MH\L;Y\8t>@8BA&�EH\LBWXH\U&f-:XPr>DAC[@RTJ6>@:=H\PtfjAq> { A6ACP>@8BACU+<DACWag\A6<3RQP;NO>@8BAZACW=ACJ6>D[DH\PB<�:XP+>@8;AZA6UV:a>D>D:XP;YzRT>@H\U-k�798�L;<�>D8;AZH\PBW=`VN;:=�jAC[MACP;J6AZfjAq> { A6ACP>@8BA�fB[DHrRQNBACP;:=P;YoA6�jACJ6>D<?HTFj>D8;A3:XH\P;<bRTP;NVA6WXA6J6>@[MH\P;<bJCHQUVA6<bFI[MH\Uî>D8;AC:=[�N;:=�jAC[M:XP;Y1<McjA6ACNd� { :=>D8>@8BA~A6WXACJq>@[MH\P;<9>y`Kc;:XJCRQWXWa`O8_R!gK:XP;YeR+UzL;J|8�8B:XY\8;A6[�R�gQAC[DRQY\Ao>@8;A6[DUVRQWdg\A6WXHKJC:=>y`e>@8_RQP->@8;Ao:=H\P;<Ck798�L;<?>y`Kc;:=J!RQW=W=`&>D8;A�ACWXA6J6>D[DH\P;<?RQ[MA�>D[DA!Ri>@ACNV:XP+>@8;A�:XU+c_RQJq>GRQP;NV>D8;A�:XH\P;<G:XP�>@8BA^��L_RQ<M:a�n<M>|Ri>@:XJRQc;cB[DH!Ös:XUVRT>@:=H\Pdk�ln>+<D8BH\L;WXNufpA-PBHQ>@A6Nu>D8_RT>+[DACRQWX:a>y`�N;HKAC<�P;HQ>VRQW { R!`K<�FµRQWXWE:XPr>@HvAC:a>@8;A6[�HQF>@8BAC<DA~> { HOW=:XU+:=>D:XP;Y�RTc;c;[DH!Ös:XUVRT>D:XH\P;<6�_f;LB>�:XP�>D8;Az:=Pr>@AC[MUVA6N;:]Ri>@Ao[DA6Y\:XU+A~fpA6> { ACA6P >@8;Am> { HBklyP�Î_YQL;[DA-�\kϾ RQP�:=WXW=L;<M>D[@RT>D:XH\P(HQF?>@8;AVfjHQL;P;N_RQ[M:XA6<~HQFE>@8;A6<DAe> { H(RQc;cB[DH!Ös:XUVRT>@:=H\P;<1FIH\[o:XH\PB<RQP;N�ACW=ACJ6>D[DH\PB<~:=<m<M8;H { P�FIH\[o<DHQUVA�c;WXRQ<DUVRSJCH\P;NB:=>@:=H\P;<oWX:=�QA6W=` >DH�HKJCJ6L;[~:=Pvc;[@RQJq>@:=JCAQke798;ARQ[M[DH { <?:=P;N;:=J!RT>DA�>@8;A3[DA6Y\:XU+AC<4FIHQ[ { 8B:XJ|8�>@8;A�:XU+c_RQJq>bRQP;NV�KL;RQ<D:å�y<M>@RT>@:=J3RQc;c;[MH!ÖB:=UeRi>@:XHQP;<�RQ[MAgTRQWX:=Ndk Ñ HT>@A~>D8_RT>�>@8;AzY\[@RQcB8}:X<�H\P;Wa`SgTRQWX:=N-FIH\[�� F?�Èf;Ls>ZJCHQL;WXN�:XP}cB[D:XPBJC:XcBWXAofjA~A6gTRQW=L_RT>@A6NFIH\[3RQPr`SW=:XP;ATk

Page 37: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�Q©

Ý��4Þ ø<ßáà2ã�çtä>ã�à2àRâfõ���øJßáã¯äfø<õ ÷À <o<h>|RT>DACNvc;[DAqgK:XH\L;<MW=`Q�d>@8;A�:XU+c_RQJq>1RQcBc;[DH!Ös:XUVRT>@:=H\P(RQ<M<DL;U+AC<1>D8_RT>1J6H\WXW=:X<D:=H\P;<�RQ[DA�<h>|RT>D:X<M>D:a�J!RTWXW=`V:XPBN;ACcpACP;NBACPr>^RQP;Nt>@8�L;<9>@8;AZ>DHQ>|RTWdf;[DH\RQN;ACPB:XP;Y�Aq��A6J6>�HQF�>@8;AoJCHQWXWX:=<D:=H\P;<GJ!RQP-fjA1AqgTRQWXLs�RT>DACNtFI[DH\U >@8;Ao<DLBUñHQF�ACRQJ|8-f;[DHrRTN;ACP;:=P;Y�JCH\W=WX:=<D:XHQPdklyP�c;[DRQJ6>D:XJ6A+>D8;Ae:=UVc_RTJ6>~RQc;c;[MH!ÖB:=UeRi>@:XHQPw:=<~LB<DL_RQW=W=`(gTRQW=:XN(FIH\[o>@8;A+>@[DACRT>@U+ACPr>~HQF9A6WXACJq>@[MH\PJCHQWXWX:=<D:=H\P_RQWjf;[DHrRTN;ACP;:=P;Y+N;L;Ao>@HV>D8;AC:=[9FµRQ<M>�>@8;A6[DUVRQW/U+HQ>@:=H\P-RQP;N�>D8�L;<^>D8;A~<M8;H\[h>^>D:XU+AC<DJCRQWXAHQF?>@8;A6:X[oJCH\W=WX:=<D:XHQP;< { :=>D8�>D8;AVA6UV:a>D>D:XP;YtRT>DH\U-ke798;:=<~f;[MHrRQN;A6P;:XPBY�:=P�:=<DH\WXRT>@:=H\P c;[MHsN;LBJCAC<zRÙ/H\[DA6Pr>@ÝC:XRQP+c;[DHQÎ;WXA { :=>D8VJ|8_RT[@RQJq>@AC[M:X<h>@:XJ38_RTW=F { :XNB>D8;< { 8;:XJ|8eRQ[DA3c;[M:XUVRQ[D:=W=`mFIL;P;J6>D:XH\PB<GHTF�>@8BAcpAC[M>DL;[DfpAC[3N;A6P;<D:a>y`\kEëo[M:XA6U�¦h�!Ú\Û\�r¨EN;A6[D:ag\AC<^R�<D:=UVc;W=:=Î_A6NSAqÖsc;[MAC<D<M:XH\P-FIH\[�R&>@[@RTP;<D:a>@:XHQPOFI[DH\U� ��¢ � { :=>@8-RQP-ACW=ACJq>@[DHQP-fB[DHrRQNBACP;A6N}8_RQWaF { :XNB>D8�Y\:ag\A6P-fr`� C à�� Û����!� ºK� � ³��n´ � º ³ Á º ³�� � © ="! Ô ³ P � ¾+����� � Á �� � � ³�� � ¡ ¶£� � ��� à « � �$# � � #� � � ¶ � � ¦h�\kÊ�QÚr¨Ñ HT>@A�>@8_RT>~:=F Ò H\c;c;W=AC[of;[MHrRQN;A6P;:XP;Y�:X<~RQWX<MH�<M:XY\PB:=Î_JCRQPr>~>D8;ACPw>@8;AV[DAC<ML;W=>@RQPr>zcB[DHQÎ_W=A { H\L;W=NfpAoR�J6H\Prg\H\W=LB>@:=H\PSHQF�R�ÙÇHQ[DACPr>DÝC:]RTP�RTP;N�ëmRQLB<D<D:XRQPSc;[MHQÎ_WXA { 8B:XJ|8t:X<9[MA6FIAC[M[DA6N->DH�RQ<�R��bH\:XYQ>c;[MHQÎ_WXATkG�sL;J|8�c;[DHQÎ;WXAC<3RT[DAoHQFÅ>@A6P-R�gTRQ:=W]RQfBWXA1RQ<�>@RQf;W=AC<C�_ATkÕYBk À WXW=ACPw¦×��Ú\ÚB��¨�k

Ý��4Þ&%�å4ã�è§ø�'Hè_äºã�äfø<ç%ã�àRà2âfõ��4ø<ßáã�äºø<õ ÷À >�>D8;A1HQ>@8BAC[�A�Ös>D[DA6UVAT�;:=>EUeR!`VfjA�>D8;A1J!RT<DAZ>@8;RT>E>D8;A1J|8_RQ[MY\ACNtcpAC[M>DL;[DfpAC[M<98_R!g\A1P;A6Y\WX:=Y\:XfBWXAU+HQ>@:=H\P�NBL;[D:=P;Y�>D8;AOWX:aFIA6>D:XU+A�HQF9>@8;AVACU+:=>M>@:XPBY�L;cBcjA6[zWXAqg\A6W§k(��P;N;A6[z>@8;A6<DAOJ6:X[MJCL;U+<M>@RQP;JCA6<>@8BAZA6�jACJq>@<9HQF/>D8;A�cpAC[h>@L;[MfjA6[D<9J!RTPSfjA^fjA6<M>9U+HsNBACWXW=ACNeRQ<�RzÎ_A6WXNOfB[DHrRQNBACP;:=P;Yz>@8BAZWX:=P;AQk�798;A��L_RQ<M:a�y<h>|RT>D:XJ�RQc;c;[MH�Ös:=UeRT>D:XH\P+:X<?L;<DL;RQWXWa`+RQc;c;W=:XACNORQ<?FIH\W=WXH { <Ck�798;AZcpAC[M>DL;[DfpAC[M<3RQ[DA�RQ<D<ML;U+ACN>@H~fjA�<M>@RT>@:=H\P_RQ[h`�RQP;NVH\P;A^J!RQW=JCL;WXRT>@A6<b>D8;A�Aq��A6J6>EHQF�>D8;A�J6H\P;<DA6��L;ACPr>9<h>|RT>D:XJ�ACWXA6J6>D[D:XJ9Î_A6WXNVH\P>@8BA�WX:=P;AC<�gK:]Ro>@8;A��K>@RQ[D��A6�jACJq>!kGÄZP;A�>@8;A6PORQJ6JCH\L;Pr>D<EFIH\[b>@8;A�<M>@RT>@:=<M>@:=J!RQW_NB:X<M>D[D:=f;LB>@:=H\P+HQFd>@8BAÎ_A6WXN+f�`+AqgQRTWXL_RT>D:XP;Y1>D8;A^cB[DH\f_RTf;:XW=:=>y`~N;:X<h>@[D:=f;LB>D:XH\P�HQFj>@8;A�ACW=ACJ6>D[D:=J3Î;ACWXN�k4798;A��K>|RQ[M��c;[DHTÎ_WXA6<RQ[MAo>D8;ACP�R!g\A6[@RQY\A6N}H�g\A6[9>@8;:=<9N;:X<h>@[D:=f;LB>D:XH\Pdk798;Awc;[DHQf_RQf;:=WX:=>y`�N;:=<M>@[M:Xf;Ls>@:XHQP�HQFz>D8;Av:XH\P ACW=ACJq>@[D:=J(Î_ACW=NÔ:=<SL;<DL_RTWXW=`�N;ACP;HT>@ACNÔfr`*) ¦"+z¨{ :=>D8,+ C ú{½tú P RQP;N ú P C ¯f½�¦ �P � { 8BAC[DA¸¦ P :X<&>@8BA�U+A!RQP£N;:=<M>@RQP;JCA-fpA6> { ACACP²c_RQ[M>D:XJ6WXAC<:XP >D8;A�c;W]RT<DUVRBkz798;:=<1[DA6c;[DA6<DACPr>D<~>@8BA+cB[DH\f_RTf;:XW=:=>y`t>@8_Ri>mRtJ6AC[h>|RQ:=P(ACWXA6J6>D[D:XJ&Î;ACWXN�¦<ú&¨q�/NBL;A>@H->@8;AVc;[DA6<DA6P;JCASHQF�<DL;[M[DH\L;PBN;:XP;Y�:XH\PB<C� { :=WXW�fpAeA�ÖBcpAC[M:XA6P;JCA6N�f�`w>@8BAVA6UV:a>D>@:=P;Y�RT>@H\UÆRTP;N:X<�L;<ML_RQWXWa`t[DAqFIAC[D[MACN}>DHORQ<�>@8;A+ÉÊ<h>|RT>D:X<h>@:XJCRQW/U+:XJC[MHQÎ_A6WXN-N;:X<h>@[D:=f;LB>D:XH\PtFIL;P;Jq>@:XHQPdË=k9798BAC<DA�8;R�gQA

Page 38: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�T¾

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20 25 30

W(F

)-

E/E0

»�:XY\L;[MA&�\k à ¼�lyH\Ps�nUV:=JC[MHQÎ_ACW=NSc;WXHT>D>@A6NSFIH\[9c;W]RT<DUVR&c_RQ[@RTUVAq>@AC[M<�RQ<9N;AC<MJC[M:XfpACN}:=PO>@8;Ao>@A�ÖK>!k

fpACACP { :XNBACW=`OA6gTRQWXL;RT>@A6N-:=P { H\[M�O<DLBJ|8 RQ<9>@8;RT>3HQF�½�HsHQcjA6[m¦h��Ú\ÛQÛr¨ERQP;N�½^HKH\cpAC[m¦×��Ú\Û\ãr¨�k�lyP>@8BA~:=N;A!RQW/Y\RQ<^W=:XU+:=>?>@8;Aq`t[DACNBL;JCAm>@H�>@8;A~½^H\Wa>@<MUeRQ[M�VÎ_A6WXN�¦­½^HQW=>@<MUeRT[D�È�d��ÚB�!Úr¨qkE»�:XY\LB[DA��\k à<D8BH { <mReU+:XJ6[DHQÎ;ACWXN}N;:X<h>@[D:=f;LB>D:XH\P-FIL;P;J6>D:XH\P�FIH\[1ROc;WXRQ<DUVRVJ6H\P;<D:=<M>D:XP;YSHQFGÚQÚ�.0/21�RQP;N��3./ P RT>^RQPtACW=ACJ6>D[DH\Pt>@A6UVcpAC[DRT>@LB[DAoHQF?��¯�° RTP;N}RTP�A6WXACJq>@[MH\PtN;ACP;<M:=>y`tHQFE�m�����K³ ��¸�¹}º ´�kb798;AÀ Üb¥4ÍîJCHKN;AV¦µlyY\W=AC<D:XRQ<C�ÈÙ/A6fjH { :=>DÝmâíì�J�ë1H { RTPd�Ç��Ú\ã\©r¨E:=<3L;<DA6N�>@8B[DH\L;YQ8;H\LB>3>@8B:X<9>@8;A6<D:=<^:=PRQPr`tU+:XJC[MHQÎ_A6WXNtJ!RQW=JCL;WXRT>@:=H\P;<6k?798;:=<^[MH\LB>D:XP;Ao:X<3[MACYrRQ[MN;ACN}RQ<�fjA6:XP;Y+8;:XYQ8;W=`ORQJCJ6L;[@RT>DA&RQPBN�:=<{ :XNBACW=`eL;<DA6NdïÈ<DA6A~J|8_RTcB>@A6[^©�HQF4ëo[M:XA6Uä¦×��Ú\ÚrÞ\¨?FIHQ[9UVH\[MA1N;A6>@RQ:XW=<�H\Pt>@8;AoL;<MAoHQF�>D8;:X<3J6HsN;ATkln>Ç:X<�P;HQ>ÇP;H\[MUeRTWXW=`�RTc;c;[DHQc;[D:XRT>@A�>DH3UVHKN;ACWi>@8;A�Aq��A6J6>D<�HQF�>D8;A�ACW=ACJ6>D[DH\PB</L;<D:=P;Y9>@8;:=<�>@A6J|8;P;:X��L;ATk798;AZ:=H\POJ6H\WXW=:X<D:=H\P;<�H\PO>D8;AZHQ>D8;AC[?8_RQP;NtRT[DA�HTFÅ>@ACPeUVHKN;ACW=WXA6Ne:XP+>@8;:=<�RQcBc;[DH!Ös:XUVRT>@:=H\Pdk�798;AC:=[<DW=H { A6[ogQACW=HsJ6:=>@:=AC<ZU+A!RTP;<Z>@8_Ri>1>@8;Aq` HQFÅ>DACPvFµRQWXW�:XPr>@HS>D8;A�[DA6Y\:XU+A { 8;A6[DA�>@8;A6:X[1:=P�>DAC[DRQJ6>D:XH\P;<{ :=>D8�>D8;AzA6UV:a>D>@:=P;Y+RT>@HQU$RQ[DA~H�gQAC[^WXH\P;YVcjA6[D:=HsN;<3HQF�>@:XU+AQk9lyP}Y\ACP;A6[@RQW�RV[MA!RQ<MH\P_RQfBWXAmY\L;:XNBA:XP-N;Aq>@AC[MUV:=P;:XPBY+>D8;A~f;RQW]RQPBJCAmfjAq> { A6ACP >@8BA~> { HSRQc;c;[MHrRQJ|8;A6<�:=<�>D8_RT>3>@8_Ri>^>D8;Az��L_RQ<M:a�n<M>|Ri>@:XJRQc;cB[DH!Ös:XUVRT>@:=H\P�:X<bgQRTWX:XN+:=Fj>@8;A^WX:=P;A { :=NB>@8+:X<GW]RQ[MY\A3JCH\U+c_RQ[MACN { :a>@8+>@8;A^:XPrg\A6[D<DA�HQFÇRm>y`Kc;:=J!RQWJCHQWXWX:=<D:=H\P²>@:=UVAT��RTP;Nx>D8;Av:XU+c_RQJq>-RQc;c;[MH�Ös:=UeRT>D:XH\P²:X<SgTRQWX:=N�:=Fo>@8;:=<SgQRTWXL;A(:=<S<DUVRQWXW { 8BACPJCHQUVc_RT[DACNS>DH+>D8;Ao:XPrg\A6[D<DAmHQF�R&>y`scB:XJ!RTW/JCH\W=WX:X<M:XH\P+>@:=UVATkÑ HT>@AZ>@8;RT>9:XPScB[@RQJq>@:XJ6AQ�sH\P;A1N;HKAC<9P;HT>98_R!g\AoH\P;Wa`eHQP;A1f;[DH\RQN;ACPB:XP;Y�U+ACJ|8_RTP;:X<MUç:=PB¢_L;A6P;JC:=P;Y

Page 39: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� à

>@8BAEWX:=P;AGc;[DHQÎ;WXAQk�lyP�N;A6P;<DA�cBW]RQ<MUeRQ<�H\P;A?:X<�WX:=�QACWa`1>@HZYQA6>4:=PB¢_L;A6P;JCA6<�FI[DHQUfpHQ>D8~>@8BA�ACW=ACJ6>D[DH\PB<RQP;Nt>@8BAm:=H\P;< { 8;:=J|8�UzL;<M>�fpA1<D:XUzL;W=>@RQP;ACHQL;<DWa`SRQJ6JCH\L;Pr>DACN�FIHQ[Ck4 çÏç.õ åR÷ äføJ÷4ù65�õ$â�ôXõ�ä3� øxõ³÷ ã�÷Rö Þ¶óxÞ¶çtäºâfõ³÷7ç.õ³ó<ó<øJè§øxõ ÷2ã�óuôRâfõ$ã�ö4Þ¯÷2ø<÷2ùlyPoc;[@RTJ6>@:=JCA�:a>/:X</J6H\UVU+H\P�>DH9>|RQ�TAb>D8;A4FIH\W=WXH { :XP;YERQc;c;[MHrRQJ|8dk�798;A�ACU+:=>M>@:=P;Y9RT>DH\U :=</RQ<D<ML;U+ACN>@HVWX:=Am:=P ReJ6H\P;<h>|RQPr>ZA6WXA6J6>@[M:XJoÎ_ACW=N c;[MHsN;LBJCACN}f�`->@8BAVÉÕ<M>|Ri>@:XHQP_RQ[M`pËÈ:XHQP;< { 8B:XJ|8 c;[MHsNBL;JCA6<1R�K>@RQ[D�uf;[DH\RQN;ACPBACN�<DAq>SHQFoW=:XP;A6<Ck·798;A6<DA(W=:XP;A6<ORQ[MA�>@8BACP�FIL;[h>@8;A6[Of;[DH\RQN;ACPBACN fr`²A6WXACJq>@[MH\PJCHQWXWX:=<D:=H\P;< { :=>D8;:XP+>@8;AZ:=UVc;RQJ6>?RQc;cB[DH!Ös:XUVRT>@:=H\Pdk4»�:XP_RQW=W=`&>D8;A�<h>|RT>D:X<h>@:XJCRQWÈN;:X<h>@[M:Xf;LB>D:XH\PVHQF/>@8BA:XHQP;:XJbÎ_ACW=N&:X<�RQJ6JCH\L;Pr>DACN�FIH\[�fr`zR!g\AC[DRQY\:=P;Y�>@8BAEf;[MHrRQN;A6P;ACN�<MA6>�HQFÈW=:XP;A6< { :a>@8z>D8;A�c;[MH\f_RQfB:XWX:a>y`N;:=<M>@[M:Xf;Ls>@:XHQPtFILBP;J6>D:XH\PtHQF�>@8;Am:XH\PtÎ_A6WXNdk?798KLB<^>@8BAmA6WXA6J6>@[MH\P;<^RQ[DA~RQJ6JCH\L;Pr>DACN}FIH\[9LB<D:XPBYV>@8BA:XU+c_RQJq>bRQc;cB[DH!Ös:XUVRT>@:=H\P�RQP;N�>D8;A3:XH\P;<4gK:XR1>D8;A9��L_RQ<M:a�y<h>|RT>D:XJ3RQcBc;[DH!Ös:XUVRT>@:=H\PdkÇ798;A���L_RQPr>@L;UU+ACJ|8_RQPB:XJ!RTWsFIH\[MUHTFp>D8;:X<�:X<4<M8;H { P�fjA6WXH { k�ÄZP;A { H\L;W=N&>@8;A6P�JCH\Prg\HQW=g\A�>D8;A9[DA6<DL;Wa>@:=P;Y�c;[MHQÎ_W=A{ :=>D8S>@8;A Ò H\c;cBWXAC[9cB[DHQÎ_W=AoRQP;N-:XP;<h>@[ML;UVA6Pr>|RQW�FILBP;J6>D:XH\Pt:=F�RQcBc;[DH\cB[D:]Ri>@AQk798;:=<�c;[DHKJCA6<D<�:X<�AqgTRQWXL_Ri>@ACN�P�L;U+AC[M:XJCRQWXWa`o:XP&>@8BA3Üb:=U±Ü�RQUíÜ�H\L;Uñ¦µÜbÜGÜG¨�J6HsN;Am¦I7�RTWX:XPzA6>GRQW­k=���Ú\Ú à �r7�RTWX:XP�Aq>�RQW­k=�_�!Ú\ÚrÞ\¨�k�798;:=<GJ6HsNBA^:=< { ACW=W;>@A6<M>@A6NSRQP;N+c;[MH�gK:=N;AC<ERQPeRTJCJCLB[@RT>DA^AqgQRTWXL_RT>D:XH\PHQF^�K>@RQ[D�wf;[MHrRQN;A6P;ACN£c;[MHQÎ_WXA6<zN;LBAO>@H :=H\P�RQP;N�A6WXA6J6>@[MH\P�cpAC[h>@L;[MfjA6[D<Ckv798;ASJ6HsNBAe>|RQ�TAC<�:=PRT>DH\UV:=J1N_RT>@R+J6H\Pr>|RQ:=P;:XP;Y�A6P;AC[MYQ`tWXAqg\ACWÇ:=PBFIH\[DUVRT>D:XH\PtRQPBN�<McjH\Pr>@RQP;ACHQL;<�A6UV:=<D<D:=H\PSgTRQWXL;A6<�RT<{ ACWXW^RQ<�>@8;A�<DAq>OHQF1cBW]RQ<MUeRvc_RQ[DRQUVAq>@A6[D<VFIHQ[ { 8B:XJ|8²>D8;A}c;[MHQÎ_W=A�:=<+>DH�fpA}A6gTRQW=L_RT>DACN�RTP;N[DAq>@L;[MP;<&>@8BASACU+:X<D<M:XH\P�JCHKA6é+JC:=ACPr>&FIH\[z>D8;AO>@[DRQP;<M:=>@:=H\P�HTF^:=Pr>@AC[MAC<h>-¦µ:­kÕATk ³�87 ��� �h� �.��� � �;¦J@;¨D¨�k798;Aoc;[MHQÎ_WXA�J!RQP-fjA1A6:=>D8;AC[9P;HQ[DUVRQWX:=<DACNeH\[9P;HQ>�RQPBNt>D8;A1cpH\c;L;WXRT>@:=H\PON;A6P;<D:a>y`SJ!RQPtfpAo<DA6>9>DH>@8BAz�sRQ8_Ri�y�EH\W=>DÝCUVRQP;POgTRQW=L;AC<9H\[�R�c;[MAq�yJCRQWXJ6L;W]Ri>@ACNtcpH\c;L;WXRT>@:=H\PON;A6P;<D:a>y`tN;:X<h>@[M:Xf;LB>D:XH\PSLB<DACN�k798�L;<Z>@8BA�ÜbÜGÜxJCHKN;AQ��JCH\L;cBWXACN { :a>@8}>D8;A À�Ò�À �}JCH\W=WX:X<M:XH\P;RQWa�n[@RQN;:XRT>@:åg\A�JCHKN;AC<1JCRQP(c;[DH�gK:XNBARQJ6JCL;[DRT>@AoWX:=P;AZc;[MHQÎ_W=AC< { :=>D8-RQf;<DHQWXLB>DA1:XPr>@A6P;<D:a>@:XA6<EFIH\[3R { :XNBA1[@RQP;YQA1HQF�cBW]RQ<MUeR&J6H\P;N;:a>@:XHQP;<CkÀ P�AqÖBRQU+c;WXA+HQF�R�c;[MHQÎ_WXA+A6gTRQW=L_RT>DACNwL;<D:=P;Y�>D8;AOÜbÜbÜ«JCHKN;AV:=<o<D8;H { P�:XPvÎ_Y\LB[DA}�QkÊÛBke798;AÚ�¶w�~�ERQWXU+AC[G<DA6[D:XA6<EW=:XP;A�:X<G<D8;H { POFIH\[G>@8;A^J!RQ<MA { 8;A6[DAZH\PBW=`�>@8BAZACW=ACJ6>D[DH\Pef;[DHrRTN;ACP;:=P;Y&RTP;N>@8BACP�>@8;At:=H\P�c;WXLB<&ACW=ACJ6>D[DH\P£f;[MHrRQN;A6P;:XPBY(:X<�JCHQP;<D:=N;AC[MACNdk�ln>�JCRQP£fjAS<MACA6Pu>D8_RT>&FIH\[&>D8;AC<MAc;WXRQ<DUVR-J6H\P;N;:a>@:=H\P;<C�Ç>@8BAeACW=ACJq>@[DHQP�f;[DH\RQN;ACPB:XP;Y�N;H\U+:XP_RT>DAC<1>@8BAeWX:=P;A�<D8_RTcjAT��RQW=>D8;H\L;YQ8�>@8BA:XHQP-fB[DHrRQNBACP;:=P;Y�:X<9<M>D:XW=W�<D:=Y\P;:aÎ_J!RQPr>CklyP�>@8BAO>@8;A6<D:X< { H\[M��c;[MAC<DA6Pr>@ACNu8;A6[DAtR U+H\[DAeRQc;c;[MH�Ös:=UeRT>DAeU+A6>D8;HsN { 8B:XJ|8�J!RTcB>@L;[MAC<�>@8BAUVRQ:XP~fjA68_R!gs:=H\L;[�HQFpRQWXW�>@8BA�f;[DH\RQN;ACPB:XP;YZU+ACJ|8;RQP;:X<MUV<�:X<�RQWX<MHZL;<DA6Ndk4798;:=<�FµRQ<h>!�rRQcBc;[DH!Ös:XUVRT>@AU+A6>@8BHsNv:X<oHQF?L;<MAVc_RT[M>@:=JCL;WXRQ[DWa`}:=PvJ|8_RQcB>DAC[ à { 8;A6[DAVR-WXRQ[DYQA&P�L;U&fpAC[mHQFE8r`KN;[MH\Y\ACP�<DA6[D:XA6<

Page 40: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�QÛ

0

0.02

0.04

0.06

0.08

0.1

0.12

3815 3820 3825 3830 3835 3840 3845 3850

Em

issiv

ity

wavelength / A

electron broadening onlyion + electron broadening

»�:=Y\L;[DAe�\kÊÛs¼1¥�Ö;RTUVc;W=A&HQFb>@8BA+8r`KN;[MH\Y\ACP�Úi�×�O��RQWXU+AC[^<DA6[D:XA6<1<DcpACJq>@[@RTW�WX:XPBA&c;[DHTÎ_WXA�RT<1A6gTRQWa�L_RT>DACNxfr`�>@8;AvÜbÜbÜáJCHKN;A FIH\[tRTPxA6WXA6J6>@[MH\P N;ACPB<D:=>y`�HQFe�-� �!�K³ �q¸q¹}º ´�RQP;NxRQP A6WXACJq>@[MH\P>@A6UVcpAC[DRT>@LB[DA1HQFE�~����� � ��k

c;[MHQÎ_WXA6<3P;ACA6N->DHVfpAoA6gTRQWXL;RT>@A6N�RTP;N->D8;AmRQJCJ6L;[@RTJ6`tHQF�>@8;A1gQAC[M`-8;:=Y\8S<DA6[D:=AC<�UVA6U&fjA6[9:X<9P;HT>HQF^c;[D:=UeRT[M`�:XU+cjH\[h>|RQPBJCAQkuë1[D:=ACU ¦h��ÚQÛ\�r¨zN;A6[D:=gQAC<VR(<D:XU+c;W=:=Î_A6N£RQWXY\A6f;[@RT:XJOA�ÖBcB[DAC<M<D:=H\PuFIHQ[A6gTRQW=L_RT>D:XP;Y :=H\PuRQP;N£ACW=ACJq>@[DHQPufB[DHrRQNBACP;A6N�c;[MHQÎ_W=AC<�FIH\[&8r`KN;[DHQY\ACPd� { :=>D8£>D8;At:XH\PB<&>@[DACRT>@A6N:XPv>@8BAV��L_RQ<M:a�n<M>|Ri>@:XJ�RTP;N�>D8;AVA6WXA6J6>@[MH\P;<m:XP(>@8BAV:=UVc_RTJ6>mRQc;cB[DH!Ös:XUVRT>@:=H\Pdk�798;Ae<DRQU+A+f;RQ<D:=JAqÖsc;[MAC<M<D:XHQPO:X<bRQW=<DHmY\:=gQACP+:XPe�sH\fpACW=UeRQP�Aq>ERTW§kȦ×��Ú\ãB��¨�k�798;A^AqÖsc;[MAC<D<M:XH\PB<�RQ[MA�NBAC<D:=Y\P;A6NV>@HmfpA��L;:XJ|��>@HtRQc;cBW=`\��L;<D:=P;YSWXHKH\�\�yL;c >@RQf;W=AC< { 8;:=J|8vJCH�g\A6[zRO[MA!RQ<MH\P_RQf;W=A&[@RTP;Y\A�HQFGACW=ACJq>@[DHQP�RTP;N:XHQP-fB[DHrRQNBACP;:=P;Y�8_RQW=F { :=NB>@8;<6kb798;Aof_RQ<D:=JoAqÖsc;[DA6<D<M:XH\P�L;<DA6N�:=<; C �+ P �

� �_� � Áz¦"9bà � ¨ ¦h�\kÕ©\�r¨Áz¦"9bà � ¨�:X<9FIH\L;PBN�FI[DHQUá>@8;AmWXHKH\�\�yL;ct>|RTf;WXA6<C� � ¦I>D8;A~8;RQW=F { :XNB>D8}N;L;Am>@H+ACW=ACJ6>D[DH\P�f;[DH\RQN;ACPK�:XPBY�¨�:=<�A6gTRQW=L_RT>DACN}FI[MH\UèA6��L_RT>@:=H\P£¦×�\kÓ�TÚr¨q�:+ P C � � Ûs��¯ � Ù; :=<�>@8;A&½^H\Wa>@<DUVRQ[M�SP;H\[DUVRQWÇÎ_ACW=N<M>D[DA6P;YQ>@8��È� �m� � C à��Êà ����� º # � º # � �m� �[�=<� � Ù º � Ù �>9 C@? ½Q� �_� � { :a>@8 ? C é�A ½B+ P klyP�AqgQRTWXL_RT>D:XP;Yt>@8B:X<&RTc;c;[DH!Ös:XUVRT>DAVA�ÖBcB[DAC<M<D:=H\P£RQP�A�Ös>DACP;<M:XH\P { RQ<&UVRQN;A�>@H}UeRQ�TAe:a>zU+H\[DA[DHQf;L;<M>Ck À <^H\P;Amc;[DHQY\[DA6<D<DA6<�L;c}R+YQ:=g\A6P}<MAC[D:=AC<6�jA6WXACJq>@[MH\P�fB[DHrRQNBACP;:=P;YeN;HQUV:=P_RT>@A6<C�ÈRQPBN�>@8BAc;[MHQÎ_WXA6<V>DACP;N�>DH { RQ[MN;<tÙ/H\[DA6P�>DÝC:XRQP�c;[MHQÎ_WXA6< { :a>@8²>D8;AC:=[e8_RQWaF]� { :=NB>@8;<+N;Aq>@AC[MUV:=P;ACN²c;L;[DA6W=`

Page 41: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�\Þ

fr`e>D8;AZACW=ACJq>@[DHQPSf;[DHrRTN;ACP;:=P;Y�AqÖsc;[MAC<M<D:XHQPdkb798KLB<�FIH\[EgTRQWXLBAC<�HQFC9v>@8;RT>�FµRQW=WpHQLB>@<M:XN;Aoë1[D:=ACU-ËÕ<>|RTf;WXAT�i:a> { RQ<�RT<D<DLBUVA6Nm>D8_RT>�ACW=ACJ6>D[DH\Pof;[MHrRQN;A6P;:XPBY�NBH\UV:=P_RT>DACNoRQP;N~R3Ù/H\[DA6P�>DÝC:XRQP~A�Ösc;[DA6<D<D:=H\P{ RQ<EL;<MACN { :=>@8eRm8;RQW=F]� { :XNB>D8+YQ:=g\A6Pefr` � k�798;:X<G:X<bP;HT>ERTW { R�`K<ERogTRQW=:XN+RQc;c;[MH!ÖB:=UeRi>@:XHQPdk�»_H\[AqÖBRQU+c;W=A&:=F4>D8;A�:XHQP f;[DH\RQN;ACPB:XP;YO:=<Z<M>D:XW=W�<M:XY\P;:aÎ_J!RTP�>^>@8;A6P(>@8;A&JCRQWXJ6L;W]Ri>@ACN}WX:XPBA { :XW=W�PBHQ>1fpARQ<1f;[MHrRQNwRQ<o:=> { H\LBWXNv:XPv[DACRQWX:a>y`\k À <o:X<1<M8;H { P�:=PwJ|8;RQcB>@A6[ à ��ëo[M:XA6U�ËÏ<ZAqÖsc;[MAC<M<D:XHQP�c;[MH�gQA>@H�fpA�gTRQW=:XNVFIHQ[�UeRTP�`VHQFÇ>@8BAZc;W]RT<DUVRzJCH\PBN;:=>D:XH\P;<G>@8_Ri>3RQ[DA�:XPrg\A6<M>@:=YrRT>DACNS:XPO>D8;:X<?>@8;A6<D:X<6ï;<DA6A<DA6J6>D:XH\P à kÊ�skÊ©&FIH\[9U+H\[MA1N;A6>@RQ:XW=<Ck798�L;<o>@8;A�c;[MA6FIA6[D[DA6N�U+A6>D8;HsN FIHQ[1c;[DHQÎ;WXA&JCRQWXJ6L;W]RT>D:XH\P}:XP(>@8;:=<Z>@8;A6<D:=<18_RQ<ofjA6ACP�RQ<ZFIH\W=WXH { <Ck»_H\[1<D:=UVcBWX:=Î;ACN(JCHQP;N;:=>D:XH\PB<C� { 8;AC[MA Ò HQc;c;WXA6[1H\[o<D:=UVc;W=A�Ù/H\[DA6Pr>@ÝC:XRQPwc;[DHQÎ;WXAC<mRQ[MA�gQRTWX:XN��/P;HJCHQUVc;W=AqÖ-c;[MHQÎ_W=AzJ!RTWXJCLBW]RT>D:XH\P;<�RQ[MAzN;HQP;AQk�»ÈHQ[^J6H\P;N;:a>@:XHQP;< { 8BAC[DAz<D:XYQP;:=Î_JCRQPr>1��>|RQ[M��f;[MHrRQNs�ACPB:XP;Y~HsJ6JCL;[M<E:XP+fpHQ>@8+>@8;A^:XH\P;<GRQPBNOACW=ACJ6>D[DH\PB<C��>@8;A^ÜbÜGÜ£JCHKN;A�:X<GL;<DA6Nd�BA6<DcpACJ6:]RQW=W=`&FIH\[GWX:=P;AC<>@8;RT>oRQ[MA&>@HSfpA�L;<DA6NvFIHQ[ZN;:]RTY\P;H\<h>@:XJzc;L;[DcpH\<MAC<Ck~lyPvJ|8_RQcB>DAC[ à g\A6[M`(8;:=Y\8(8r`KN;[DH\YQACPv<DA6[D:XA6<U+ACU&fpAC[�c;[DHTÎ_WXA6<^U&LB<M>^fjA~A6gTRQWXL;RT>@A6Ndk9798;A&RTc;c;[DH!Ös:XUVRT>DA~A�ÖBcB[DAC<M<D:=H\P :X<�L;<MACN L;PBN;AC[^<DL;J|8JC:=[DJ6L;UV<h>|RQPBJCAC<9>DHV<ML;c;c;W=ACU+ACPr>3>@8;AmÜbÜbܲ[DA6<DL;Wa>@<9FIH\[�>@8BAmW=H { <MAC[D:=AC<�UVA6U&fpAC[9WX:=P;AC<6kD à4äfø<ç.ã�ó�ä3�2ø<çFE�÷4Þ¶è§è ÞÏæ�Þ¯çtäºèln>O:=<ORQW=<DH�cjH\<M<D:=f;WXA-FIH\[VHQc_RQJC:a>y`�>DH�U+HsNB:=FÅ`uR�Y\:ag\A6P�H\f;<MAC[MgQACN WX:=P;A-c;[DHQÎ;WXAQk·ÄZP²>@[DR�gQACWå�WX:=P;Ye>D8;[DH\LBY\8vROc;W]RT<DUVRe:=>1:=<ZU+H\[DA~WX:X�TACWa`�>D8_RT>1W=:XP;A~JCACPr>D[DA�c;8;HQ>DH\P;< { :=WXW�fjA�RQfB<DH\[MfjA6N�RT<JCHQUVc_RT[DACNw>@H->@8;H\<MAe:=Pw>D8;AeW=:XP;A { :XPBY\<CkS798;:=<zJCRQP�fpAe<MACACP�FI[DH\U�A6�KL;RT>@:=H\P�¦×�\kÊã\¨1FIH\[o>@8BARQf;<MH\[Dcs>@:XHQPeFµRQJ6>DH\[C� { 8;AC[MA�F�:X<E<D:XYQP;:=Î_JCRQPr>@Wa`V8B:XY\8;A6[�RT>�WX:XPBA^J6ACPr>@[MA~RT<�JCH\U+c_RQ[MACN { :a>@8O>@8BAWX:=P;A { :XP;YQ<Ck�798KLB<Z>@8;A~c;[DHQÎ;WXAo>@8_RT>^Î_P_RQW=W=`SA6UVA6[DY\A6<^FI[MH\Uñ>@8;A~fjHKNB`-HQF4>D8;Azc;WXRQ<DUVR�JCH\L;W=N8_R!g\Am[DA6N;L;J6ACN}:=Pr>@ACPB<D:=>y`SRT>9>@8;AoW=:XP;AZJ6ACPr>@[MAQ�ÈWXACRQN;:=P;Y&AC:a>@8;A6[9>@H+R�WX:XPBA�¢ÈRT>M>@ACPB:XP;Y�H\[9Aqg\ACP}RWX:=P;Am:XPrg\A6[D<D:=H\Pdk��sACA~Î_Y\L;[MAe�\kÊÞ�FIH\[�AqÖBRQU+c;W=AC<�HTF�>@8;:=<^AqgQRTWXL_RT>DACN LB<D:XPBY ÀZÒ3À �}�s�!¾VFIHQ[^>@8BAJ!RT<DAoHQF�R�c_RT[@RQfpH\WX:=J�N;A6P;<D:a>y`tN;:X<h>@[M:Xf;LB>D:XH\P { :=>@8B:XP-R�J6`KWX:=P;N;[D:=J!RQWjc;W]RT<DUVRBk

úGûyü�ûHG II����8�*+�}����)�²1 �����2*+*,�+��5798;A~<D:XU+c;W=:=FÅ`K:XP;Y+RQ<M<DL;U+cB>D:XH\P;<�JCHQUVU+H\P;Wa`SUeRTN;A { 8BACP(U+HsN;A6WXW=:XP;Y�c;WXRQ<DUVR+RT>@H\U+:XJ1cpH\c;LK�W]Ri>@:XHQP;<EJCRQPtfjA6<M>3fpA1L;P;N;A6[D<M>DHsHKN-L;cjHQPSJCH\P;<M:XN;A6[@RT>D:XH\PSHTF�>D8;A1gTRQ[D:=H\L;<9RT>DH\U+:XJZRTP;NSc;W]RT<DUVRWX:aFIA6>D:XU+AC<CkS798BAC<DAeWX:=FIAq>@:=UVA6<~RTWXWXH { H\P;AV>DH}JCRQWXJ6L;W]RT>DAeRTP²ÉÊA6��L;:XW=:Xf;[M:XL;Uª>D:XU+A�JCH\P;<h>|RQPr>CË�N;A��<DJ6[D:=f;:XP;Y�>D8;Am>D:XU+A�>|RT�QACP�FIH\[9A!RQJ|8�c;W]RT<DUVR�J6H\P;<h>@:=>DL;ACPr>�>@H�[DA6W]RiÖO>@H+:=>@<9A6��L;:XW=:Xf;[M:XL;U <M>@RT>@ATk

Page 42: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�Qã

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 5 10 15 20 25 30

line

heig

htJ

Wavelength from line center / pm

optically thin profilemoderately thick profile

thick profile

»�:=Y\L;[DAm�\kÊÞs¼LKNMfZ]` sfghe,l³a]b¯a�skZ_r�Yjcxw�` aºoºYPOke,o�gjYhife}sf^Ja Okghe,l³e�QBZ]ghnkZBcAe,o�nkl<Yhif\SR2WCRUTWVYX[Zkz>\�dfeRcJa�s:sf^Aa]OkgheY[l)cJdfe�a_sºcJY[r+Z�gjghw cJdkYji�a�ike�z^\�dfe�ife_M§c»ghaa`³e,^�a_ife�YhlTcJdfe�` aºoºe+^�ZBcAe+ghw cJdfY[rHb�r,Z�lJe c ika]cJe�cAdkZBcTYjcAl cAa�s¾Yhld Z]c<cJe,ife,o¹ze\�dfeXgjaa`e,l<c�r+Z�lJeuYhl�cJdfeXa_sºcJY[r+Z�gjghwTcJdkYhrHb�r,Z�lJe cºifa�cJe}cJd�ZBc4YjcAl4r+e+imcJ^AeuYhl�YhifQ�e,^<cAe,oÏz798;Ao[DA6W]RT>D:=gQA�gTRQWXL;A6<9HQF�>@8;A6<DA1>@:=UVA^JCH\P;<h>|RQPr>D<^N;Aq>@A6[DU+:XP;A6<�>@8;A1H\cs>@:XUVRQWjRT>@HQUV:=J�U+HsN;A6WXW=:XP;YRQc;cB[DHrRQJ|8�k Ò A6Î_PBA)Lhg �jihk Å i <DL;J|8�>D8_RT>�:a>4N;A6<DJC[M:XfpAC<4>D8;A�NB`KP_RQU+:XJCRQW�>@:XU+AC<MJ!RQW=AEHQF_>D8;A�c;WXRQ<DUVRB�>@8;RT>9:X<?>@8;A�>D:XU+AC<MJ!RQW=A�L;cpH\P { 8;:=J|8t>D8;AZ>y`Kc;:XJCRQW�cBW]RQ<MUeRzc;RQ[@RQU+A6>DAC[M<EgTRQ[M` ¦µH\[EHQFÅ>@A6PS:XPO>@8BAJ!RT<DA&HQFbWXRQfpH\[@RT>DH\[M`Oc;W]RQ<MUeRT<C�È>@8;Az>D:XU+AC<MJ!RQW=AC<3FIH\[ { 8;:=J|8 >@8;A~c;W]RT<DUVRVAqÖs:=<M>@<@¨qkqL[g �Pi[k Å i RQWX<MHN;A6<DJC[M:XfpAC<9>@8BAo>y`Kc;:=J!RQWj>@:XU+AC<MJ!RQW=A�FIH\[�cBW]RQ<MUeR&c_RT[M>@:=JCW=AC<E>DH+J6[DH\<M<9>@ACU+cpAC[@Ri>@L;[MAoRQP;NtN;ACPB<D:=>y`<DJCRQWXAEWXACPBYQ>@8;<6kb�EH\U+c_RQ[M:XP;Y�>D8;AE>D:XU+AEJCHQP;<M>@RQPr>GFIHQ[�A!RTJ|8Vc;WXRQ<DUVR�c;[MHsJ6AC<M< { :=>@8{L[g �Pi[k Å i Y\:ag\A6<RQPS:=P;N;:XJCRT>@:=H\P+HQF { 8;A6>D8;AC[�>D8_RT>Ec;[DHKJCA6<D<98_RQ<?>@:XU+A^>DH&AC��L;:=WX:=f;[D:XRT>@A3fpA6FIH\[MA�>@8;A�c;W]RQ<MUeRmJCH\PK�N;:a>@:XHQP;<^gTRQ[h`\k Ò L;Az>DHO>y`scB:XJ!RTW4c;W]RT<DUVRVJCHQPBÎ_P;A6UVA6P�>1>D:XU+AC<qL[g �Pi[k Å i :=<�HQFb>D8;A�H\[DNBAC[ZHQFGH\PBA<DA6JCH\PBN}FIH\[3>@8;AmJCH\[MA~cBW]RQ<MUeR�HQF�R�FIL;<D:=H\Pt>@H\�iRQUVRQ�SRQPBN RQ<�WXH { RQ<m��� ¹ £mFIH\[3>@8BA~N;:ag\A6[M>@HQ[c;WXRQ<DUVRBk�Ä1PBA�J!RQP+RQW=<DHZNBA6Î_P;A9J6AC[h>|RQ:=POÉÊA�Ös>D[D:=P;<D:=JQËT>D:XU+AEJ6H\P;<M>@RQPr>@<6�K<ML;J|8+RQ<uL6 º Âq�\Y\:=gK:=P;Y�>@8BA>@:=UVAbFIH\[�ACW=ACJ6>D[DH\PK�yACW=ACJq>@[DHQP&JCH\W=WX:=<D:XHQP;</>@H^>D8;AC[MUeRTWX:X<MAG>D8;AEFI[MACAEA6WXA6J6>@[MH\P;<6�tL ¨ º ¨ N;AC<MJC[M:Xf;:=P;Y�>@8BAAC��L;:agTRQWXA6P�>o:=H\Ps�n:XH\P�AC��L;:Xc;RQ[M>D:=>@:=H\P(RQPBN¼L ¨ º Â�FIHQ[1:XH\Ps�nACW=ACJ6>D[DH\P}AC��L;:XW=:Xf;[DRT>@:=H\P�¦I:§kÏAQk�>D8;A&>@:=UVAFIH\[9>@8BAzACW=ACJq>@[DHQP RQP;N�:XH\P-N;:=<M>@[M:Xf;Ls>@:XHQP;<�>@HV[DA!RTJ|8}>@8;Am<@RTUVAmJ|8_RQ[DRQJ6>DAC[D:=<M>D:XJo>@ACU+cpAC[@Ri>@L;[MA�¨qkÒ A6Î;P;A�L ¨ à � <ML;J|8(>@8_Ri>1:=>^N;AC<MJC[M:XfpAC<Z>@8BA&:XH\PB:X<@Ri>@:XHQP�[MACW]R�Ö;Ri>@:XHQP�>@:=UVA6<DJCRQWXAoFIH\[^>D8;A&c;WXRQ<DUVRBk798;A6[DAoRQW=<DH&A�ÖB:=<M>3J6AC[M>@RQ:XP ÉÕ:XPr>D[D:XPB<D:XJTËr>@:XU+A�J6H\P;<h>|RQPr>@<3N;A6<DJ6[D:XfB:XP;Y&<h>|RT>DAC< { :a>@8;:=Pe>@8;AoRi>@H\U+:XJ<M`K<h>@ACU�LBP;N;AC[1J6H\P;<M:XN;A6[@RT>D:XH\Pdk�L Å Y\:=gQAC<Z>D8;A�[DA6W]RiÖBRT>D:XH\P}>D:XU+AzFIH\[�>D8;A�U+A6>|RT<M>|RTf;WXA&<h>|RT>DAC<6�L à FIHQ[�>@8;AoH\[MN;:XP_RT[M`OA�ÖsJC:=>DACN-<M>@RT>@A6<�RQPBN L i FIH\[�>D8;AmRQLB>@Hi�y:XHQP;:X<M:XP;Y&<h>|RT>DAC<6k�EAqFIH\[DAVWXHKH\��:XP;Y-RT>m>y`Kc;:XJCRQWGgTRQW=L;AC<oFIH\[mA!RQJ|8�HTF�>@8;A6<DA+>@:XU+A+JCH\P;<h>|RQPr>D<C�4J6H\P;<D:=N;AC[m>@8;AVJ!RQ<MA

Page 43: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�QÚ

{ 8;A6[DA~RQWXWdHTF�>@8BAm:=Pr>@[D:=P;<D:=JoRQP;N-AqÖK>@[M:XP;<M:XJ1>@:=UVA�JCH\P;<h>|RQPr>D<ZRQ[MAmUzL;J|8�W=AC<M<�>D8_RT>TL[g �ji[k Å i �_>@8BAc;WXRQ<DUVRm:=<b:XPe:=<DH\WXRT>@:=H\P�RQP;N+>@8;A6[DA^:X<GP;HzA6<DJ!RTc;:XP;Y~[@RQNB:]RT>D:XH\P�Î_A6WXNdk4lyPV>@8;:=<?JCRQ<DA�>@8;A�c;W]RT<DUVR:X<1<DRQ:XN(>DH-fpA�:XP�ÉÏ>@8;A6[DU+HsNs`sP;RQUV:=J&AC��L;:=WX:=f;[D:=L;U�Ë�¦I73¥?¨1RQPBN�ACRQJ|8�cB[DHKJCAC<M<~:=<1f_RQWXRQP;JCA6Nvf�`:=>D<9:XPrg\A6[D<MAe¦I[DAqFIAC[D[MACNt>@H+RQ<9>@8BA&ÉÊc;[M:XP;J6:Xc;W=AZHQF�N;A6>@RQ:XW=ACNSf_RTW]RQP;J6AQËʨqkl��P;N;AC[3<ML;J|8}J6H\P;N;:a>@:XHQP;<C�>@8BAmFI[MACAzcBW]RQ<MUeR�c_RQ[h>@:=JCWXA6<&¦IACW=ACJ6>D[DH\PB<C�È:XH\PB<^Aq>@J�¨98_R!g\A�ì}RiÖ { A6WXW=:]RQPtg\A6WXHKJC:=>y`ON;:X<h>@[D:=f;LB>D:XH\P;<J|8_RQ[DRQJ6>DAC[M:X<DA6N}fr`O>D8;Ao<@RQU+AZ>@A6UVcpAC[DRT>@L;[MAQ�_N;A6<DJ6[D:XfpACN-fr`

Í ¦nm�¨ C ¾t�po ¹�k��®KÁrq ; Ù ¯ º � Åts Ù � � � �8u�v � m � ¦h�\kÕ©B��¨

798;A&cBW]RQ<MUeR+:XPr>@A6[DP_RQW�[@RTN;:]RT>D:XH\PtÎ_A6WXN :=<^>@8;RT>ZHQF�Ref;W]RTJ|��fpHsNs`}RTP;N Y\:ag\ACP(fr`->@8;AzÜ�W]RQP;J|�FIL;P;Jq>@:=H\Pd¼

·�¦J@;¨ C ã�� × @¸ ´ ë ¯fw � � u[v ¶�� ð ¦h�\kÕ©r�\¨

RQP;NS>D8;A1fjHQL;P;Nt<M>@RT>@A6<9HQFÇ>@8;AmRT>DH\UV<ERQP;Nt:XHQP;<�8_R!g\Am�EHQW=>@Ý6UeRTP;PSN;:X<h>@[M:Xf;LB>D:XH\PB<C�K<DL;J|8t>@8_Ri>>@8BAm[DRT>@:=H�HQF�cjH\cBL;W]RT>D:XH\PSNBACP;<M:=>@:=AC<9HQF�fpH\L;P;N-<M>@RT>@A6< Ì RQPBN2xe:=<9Y\:=gQACPtfr`

� ¨�zy C � ¨� y ¯ � Õ|{ º Õ~} � � � u[v � ¦h�\kÕ©\©r¨{ 8;A6[DAz� ¨ RQPBN�� y RT[DA�>D8;A�<h>|RT>D:X<M>D:XJCRQW { A6:XY\8r>9HQFÇW=A6gQACWX< Ì RQPBN�xÈk�798;AoRQf;<MH\WXLB>DA�cpH\c;L;WXRT>@:=H\P;<HQF�WXAqg\ACW Ì RQ[MAoY\:=gQACPtfr`S>@8;A~�BRQ8;Ri�×�EHQW=>@Ý6UeRTP;PSAC��L_RT>D:XH\P� ¨ C�� 1 �  � × �

�k� ¹ ®KÁ ¡; Ù � ¨�3� 1 ¯ � Õ|{ � � � u[v � ¦h�\kÕ©Q¾�¨

{ 8;A6[DA�� 1 RQP;N � 1 RT[DAo>@8;Amc_RQ[MACPr>�<h>|RT>D:X<h>@:XJCRQW { AC:XYQ8�>D<�RTP;N}cpH\c;L;WXRT>@:=H\P;<9N;A6P;<D:a>@:=AC<�HQF4>@8BAc_RQ[MACPr>9<h`s<h>@A6U�kb798;A173¥uRQ<D<ML;U+cB>@:=H\POACP;RQf;WXA6NS<DH\U+A�NBACY\[MACAZHQF�RQP;RQW=`K<D:=<G>DH�fjA^cjA6[MFIH\[MUVA6NH\P�JCA6[M>@RQ:XP�c;WXRQ<DUVRQ<6k4»_H\[4:=P;<M>@RQP;JCA9>D8;A�FµRQJ6>�>D8_RT>�>D8;A9<DL;P�ËÕ<bc;8;HT>@H\<Mc;8;AC[MA^:=<�RQc;cB[DH!Ös:XUVRT>@A6W=`f;WXRQJ|�zfpHsNs`&RQPBN+A6UV:a>@<�RoJ|8;RQ[@RQJq>@A6[D:X<h>@:=J9Ü�W]RQP;J|��:XRQP�<DcpACJq>@[DLBU { RT<�L;<DA6N+>DH1N;A6>DAC[DU+:XPBAE>@8_Ri>>@8BAmcB8;HQ>@HQ<Dc;8;A6[DAm8_RQ<�R&>@A6UVcpAC[DRT>@L;[MAoHQF��·ÛsÃ|�\�\�\��kì�H\<h>1c;WXRQ<DUVRQ<�8;R�gQA�RQP(AC<MJ!RQcB:XP;YS[DRQN;:]Ri>@:XHQP-Î_A6WXN { 8;:=J|8vRQJ6>D<�>@HSU+H�g\Az>D8;A&c;WXRQ<DUVROR { R�`FI[DHQU J6H\P;N;:a>@:=H\P;<GHQFÇ73¥9kBlnF�RQW=Wp:=Pr>@[D:=P;<D:=J�RTP;NOAqÖK>D[D:XPB<D:XJ�>@:=UVA�JCH\PB<M>|RTP�>D<3RQ[DA^U&L;J|8SW=AC<D<G>@8_Ri>L[g �ji[k Å i >@8;ASJ6H\WXW=:X<M:XH\P;<~RQ[MASA6é+JC:=ACPr>+ACP;H\LBY\8u>DH(H�g\AC[MJCH\U+AS>@8;AScpH\c;LBW]RT>D:XH\P�W=H\<D<MAC<�N;L;AO>DH[@RTN;:]RT>D:=gQAeN;A6J!R!`\�b>@8;:=<�:X<&[MA6FIA6[D[DA6N�>@HvRQ<OÉÕWXHKJ!RQWb>@8BAC[DU+HKNB`KP_RQU+:XJeA6��L;:XW=:Xf;[M:XL;U-Ë?¦­Ùs73¥G¨�k4lyP

Page 44: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©\�

>@8B:X<1J!RT<DA�>@8;A�73¥ A6��L_RT>@:=H\P;<o8;H\W=N FIH\[Z>@8BA�FI[MACA+RQP;NvfpH\L;P;Nwc_RQ[h>@:=JCWXA6<C�/f;Ls>1>@8;A�[@RTN;:]RT>D:XH\PÎ_A6WXN�:=<1>@8_RT>~HQF�R�N;:=WXLB>DACNwf;W]RQJ|�(fpHKNB`\k À WXW�<h`s<h>@A6UV< { :XWXW�8;R�gQAOR-JC[D:a>@:=J!RQW4NBACP;<M:=>y`�RQfpH�gQA{ 8;:=J|8uJ6H\WXW=:X<M:XH\P;<~N;H\U+:XP_Ri>@AeH�gQAC[�<DcpH\Pr>|RQPBACH\L;<�A6UV:=<D<D:=H\P�WXHQ<D<DA6<Ck��scjHQP�>@RQP;A6H\L;<�ACU+:X<M<D:XHQPJCHKA6é+J6:XACPr>D<3>y`Kc;:=J!RQW=W=`V<DJCRQWXAoRQ<��±�º½ � � # ¦ � � fpAC:XPBY&>@8;A1L;cBcjA6[���L_RTP�>DL;Uá<M8;ACW=WŨq� { 8;:XW=AZACW=ACJ��>@[MH\PtJCH\W=WX:=<D:XHQP_RQWÈJC[MH\<D<9<MACJ6>D:XH\P-<DJCRQWXAoRT<��±�f½ � � � kb798�L;<�RQ<9H\P;AoU+H�gQAC<9>@H�8B:XY\8;A6[9��L_RQPr>@L;U<D8BACWXW=<C�_ÙK73¥²JCHQP;N;:=>D:XH\PB<9fjA6JCH\U+AC<3U+H\[DAZgTRQW=:XNdklnFjJCH\W=WX:X<M:XH\PB<4RQ[MA9P;HQ>GRQf;W=AE>@HoJ6H\UVcpACPB<@RT>DA3FIHQ[�[@RQN;:XRT>@:=H\PzW=H\<D<MACN�>@8;A3c;WXRQ<DUVR�:=<�:XPOÉÕP;H\Ps�nWXHKJ!RQW>@8BAC[DU+HKNB`KP_RQU+:XJ�A6��L;:XW=:Xf;[M:XL;U-ËT¦ Ñ ÙK73¥G¨qk�»_H\[dU+H\<M>dc;W]RQ<MUeRT<LC º Â�� L à � L Å RQP;N1FIH\[�UeRQPr`c;WXRQ<DUVROJCH\PBN;:=>D:XH\P;<+¦I:XP;J6WXL;NB:XP;YOFIL;<M:XH\P(>DH\�iRQUeRT�K<@¨:Lhg �jihk Å i �3L ¨ à � �3L Å k+lyPv>@8BAC<DA+JCHQP;N;:a�>@:=H\P;<�>D8;A&FI[DA6A&ACW=ACJq>@[DHQP;<18_R!g\A�>@:=UVAm>@HSJ6H\WXW=:X<D:=H\P_RQW=W=`V>@8;A6[DUVRQWX:=<DA~RQP;N(FIH\[DUèRtì}RiÖ { ACW=WX:]RTPg\A6WXHKJC:a>y` N;:=<M>@[M:Xf;Ls>@:XHQPdko798;A�AqÖsJC:a>@ACNv<h>|RT>DAC<1gTRQ[M`}H\PvRSU&L;J|8v��L;:XJ|�TAC[1>@:=UVA6<DJ!RTWXAz>D8_RT>1>@8BAU+A6>|RT<M>|RTf;WXA6<^RTP;N�JCRQP}fpAoJCH\PB<D:XNBAC[DA6N�>DHVfpA~:=Pt�KL;RQ<D:å�y<M>@RT>@:=JoAC��L;:XW=:Xf;[M:XL;U { :=>D8->D8;ACU-kG798;AU+A6>|RT<M>|RTf;WXA6<�>D8;AC[MUeRTWX:X<MAtH\P�<M:XU+:XWXRQ[z>@:=UVA6<DJCRQWXA6<�>DHw>D8;A-c;W]RQ<MUeRvNB`KP_RQU+:XJ6<C�ERQ<VNBHsA6<V>@8BA:XHQP;:X<DRT>@:=H\P(f_RQWXRQP;JCATkS�sH-fpHQ>@8v>@8;A+U+A6>|RT<M>|RTf;WXA6<~RTP;N�:=H\P;:X<DRT>@:=H\P(f_RQWXRQP;J6A+UzL;<M>~fjA�<MH\W=gQACNFIH\[GNB`KP_RQU+:XJCRQWXWa` { :=>D8;:XP+>@8;A^JCH\Pr>DAqÖK>9HQF/cBW]RQ<MUeR1>@[DRQP;<DcpH\[h>�RQPBNO:XH\PB:X<@Ri>@:XHQP+f;RQW]RQPBJCA�JCHKN;AC<6k798;A~AqÖsJC:a>@ACN(<h>|RT>DAzcpH\c;L;WXRT>@:=H\P;<3J!RTP�>@8BACP(fpA~AqgQRTWXL_RT>DACN { :a>@8�[DAC<McjA6J6>Z>DHV>D8;AzU+A6>@RQ<M>@RQf;WXAcpH\c;L;WXRT>@:=H\P;<Ck�lyPO<DHQW=gK:XP;YoFIH\[b>@8;A^AqÖsJC:a>@A6NS<M>|Ri>@A�cpH\c;LBW]RT>D:XH\P;<bH\PBA�U&L;<h>�RTJCJCHQL;Pr>�FIH\[?RQWXW_>@8BAcpH\<D<M:Xf;W=A�>@[@RTP;<D:a>@:XHQP;<�fjAq> { A6ACPOA�ÖsJC:=>DACN+<M>@RT>@A6<ERQ< { ACW=WÈRQ<�>@[@RTP;<D:a>@:XHQP;<4>@Ho>@8;A3U+A6>@RQ<M>@RQf;WXA6<�HQF>@8BA^<DRQUVA^RQP;NVRQNißDRTJCACPr>E:XH\P;:=<@RT>D:XH\P�<h>|RQYQAC<Ck�798B:X<b:X<G[DAqFIAC[D[MACN+>@H~RQ<G>D8;AoÉÊc;[M:XP;J6:Xc;W=A3HQFd<h>|RT>D:X<h�>@:=J!RQW�f_RQWXRQP;J6AQËak À WXWÇ>@8BA�[@RT>DAC<ZJCRQP(fjA&RQ<D<MACU&f;W=ACN(:=P�>DHSRO<M`K<h>@ACUªHQFbAC��L_RT>D:XH\PB<ZN;AC<MJC[M:Xf;:=P;Y>@8BA { R!`K<b:XP { 8B:XJ|8�A!RTJ|8VRi>@H\U+:XJG<M>@RT>@A�JCRQP�fpA�cjHQc;L;W]Ri>@ACN&RQP;N�N;A6cjH\LBW]RT>DACNdk�798;:X<�RTc;c;[DH\RQJ|8:X<�[DAqFIAC[M[DACN->@HeRQ<�>D8;AVÉÕJCHQWXWX:=<D:=H\P_RQWå�y[@RTN;:]RT>D:agQAQËj¦§�G�yÐ�¨3RQc;cB[DHrRQJ|8��jN;Aqg\A6WXH\cpACN fr`���RT>DAC<^A6>�RQW§k¦h�!Ú\Ûr�\¨ERTP;N�YQACP;A6[@RQW=:X<DA6Ntf�`��KL;UVU+AC[M<3â±½�HsH\cpAC[~¦×��Ú\ã\©r¨b>@H+RQJCJ6H\L;Pr>�FIH\[9UVAq>|RQ<h>|RQfBWXAC<6k�EH\P;<M:XN;A6[}RTP·RT>DH\UV:=J(ACP;A6[DYQ`xW=A6g\A6W Ì RQPBN·RQ<M<DACUzf;WXA�RQWXWZ>D8;AwJCH\W=WX:X<M:XH\P;RQW§�3[@RTN;:]RT>D:=gQA�RTP;N:XPBN;L;JCA6N([@RT>DAC<�>@8_RT>�J!RQP(cpH\c;LBW]RT>DA~H\[^N;ACcpH\c;LBW]RT>DAz:=>Ck^798BA�<DL;UæHQFbRQW=W�>D8;AC<MA&[@Ri>@AC<�AC��L_RQW=<>@8BA+[DRT>@A�HQFEJ|8_RQP;YQAVHQF?>@8;A�cpH\c;L;WXRT>@:=H\P(N;A6P;<D:a>y`v:XP(>@8_Ri>~W=A6gQACW§k+798;A+Y\ACPBAC[@RTW4FIH\[DUäHTFE>@8BA<M>@RT>@:=<M>D:XJ!RTW�f_RQWXRQP;J6AmA6��L_RT>@:=H\P;<3:X<3RT<�FIH\WXW=H { <

�¨ �~� ¨ Ò � ¨ �[� ¨ � N �#¨ �  ·�¦A@B¨ � ¨ �h� ¨ ? @�� � Â[� �  �¨ ��� ¨ � � Â[� � g �¨ ��� ¨ Ó}� ¨ �

Page 45: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©B�

� �¨ � ��� ¨ Ò N �#¨ �  ·�¦A@B¨ � ¨ � �[� ¨ ? @�� � Â�� �  �¨ � ��� ¨ � � Â[� � g �¨ � �[� ¨ ÓR� ¨ � �� �  � 1 F �����¨ � � � � 1 F � ´ �¨ � �  � 1 N ·�¦A@;¨ �qÊ � ¨ ? ËC © �¨ ��� ¨ Ò N ��¨ �  ·�¦A@;¨ � ¨ � ¨ � ? @�� � Â�� �  �¨ � ¨ � � � Â�� � g �¨ � ¨ � Ó� �¨ � ��� ¨ Ò � ¨ � ¨ � � � N ��¨ �  ·�¦A@;¨ � ¨ � ¨ � � ? @�� � Â[� �  �¨ � ¨ � � � � Â�� � g �¨ � ¨ � � Ó� N ·�¦A@;¨ � ¨ � Ê ? Ë¥� � Â[� �  �¨ � ¿ � � Â�� � g �¨ � ¿ « � ¨ � ? � ¨? û ¦h�\kÕ© à ¨

Ì 8;A6[DA+k=kak=kak� � ¨ �[� ¨ �G<DcpH\Pr>|RTP;ACH\LB<^A6UV:=<D<D:=H\POFI[DHQU ÌJ��¢ Ì� ·�¦A@B¨ � ¨ �h� ¨ �G<M>@:=U&L;WXRT>@A6NSACU+:X<D<M:XH\POFI[MH\U Ì<��¢ Ì� ·�¦A@B¨ � ¨ � ¨ � �Gc;8;HQ>DHT�×RQfB<DH\[McB>@:=H\POFI[DH\U Ì � ¢ Ì� � Â[� �  �¨ ��� ¨ �GACW=ACJq>@[DHQP-J6H\WXW=:X<MH\P_RQWjN;Aq�nAqÖsJC:a>|RT>D:XH\POFI[MH\U Ì � ¢ Ì� � Â[� � g �¨ ��� ¨ �Gc;[MHQ>@H\PtJ6H\WXW=:X<DHQP_RQWpN;Aq�nAqÖsJC:a>|RT>D:XH\PSFI[DHQU Ì � ¢ Ì� � Â[� �  �¨ � ¨ � �GACW=ACJq>@[DHQP-J6H\WXW=:X<M:XH\P_RTWjA�ÖBJ6:=>@RT>@:=H\POFI[DH\U Ì2¢ ÌJ�� � Â[� � g �¨ � ¨ � �Gc;[MHQ>@H\PtJ6H\WXW=:X<D:=H\P_RQWpAqÖsJ6:=>|Ri>@:XHQPOFI[DH\U Ì}¢ Ì �� | ·�¦A@;¨ � ¨ � Ê ? Ët�Gc;8;HT>@HT�n:XH\P;:=<@RT>D:XH\P+FI[DH\U Ì� � Â[� �  �¨ � ¿ �GACW=ACJ6>D[DH\P-JCHQWXWX:=<D:=H\P_RQWÈ:XHQP;:X<DRT>@:=H\PeFI[DHQU Ì� � Â[� � g �¨ � ¿ �Gc;[DHT>@H\PtJCHQWXWX:=<D:=H\P_RQWp:XH\PB:X<@Ri>@:XHQPeFI[DH\U Ì� �  � 1 F �P���¨ �G[DRQN;:]Ri>@:=gQA1[DA6JCH\U&fB:XP_RT>D:XH\PV>@H Ì� � � � 1 F � ´ �¨ �b>D8;[DA6AmfpHKNB`O[MACJCHQU&f;:XP;RT>@:=H\PO>@H Ì� �  � 1 | ·�¦A@;¨ �qÊ � ¨ ? ËS�G<h>@:=U&L;W]Ri>@ACNt[MACJCHQU&f;:XP;RT>@:=H\Pe>@H Ì

Page 46: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©r�

À <MA6>1HTFb<DL;J|8vAC��L_Ri>@:XHQP;<�JCRQP fpA�RQ<D<MACU&fBWXACN { :a>@8�H\P;A&A6�KL;RT>@:=H\P}FIH\[�A!RQJ|8wRT>DH\U+:XJoACP;A6[DYQ`WXAqg\A6W§� Ì �È>@8_Ri>�A�ÖB:=<M>D<Ck3ln>Z:=<3>D8;Az<M:XU&LBW=>|RTP;ACH\LB<�<MH\WXLs>@:XHQP�HQF�>@8;:=<^<MA6>^HQF�AC��L_Ri>@:XHQP;<^>D8_RT>^c;[DHT�N;L;J6AC<Z>D8;A�RT>DH\UV:=J~cpH\c;LBW]RT>D:XH\P�<M>@[ML;J6>DL;[DATkm�EACJCRQL;<DA�>D8;A&AqÖsJ6:=>@A6N(<M>@RT>@A&cpH\c;LBW]RT>D:XH\P-gQRTWXL;A6<RQ[MA~:=P�A6�KLB:XWX:=f;[D:=L;U { :=>@8t>@8BA~U+A6>@RQ<M>@RQf;WXAocpH\c;L;WXRT>@:=H\P;<�>D8;AC:=[ ? � ¨ ½ ? û�>DAC[DU+<3J!RQP�fjAm<DAq>^>DHÝCA6[DH;kÄZP;AeJCRQP { [M:=>DAV>@8B:X<m<DA6>zHQF9A6�KL;RT>@:=H\P;<m:XP�UVRT>D[D:aÖ(P;HQ>|Ri>@:XHQP�:=P;N;AqÖsA6N�fr`�:GRQP;Ntß6k-ì}RT>@[M:aÖACW=ACU+ACPr>@<?<DL;J|8-RQ<S ¨ y RQ[DA�UeRQNBA^L;cOFI[MH\UíAC��L_Ri>@:XHQPv¦h�\kÕ© à ¨GRQP;NONBAC<DJ6[D:=fjA�RQWXW_>@8BAZc;[DHKJCA6<D<MAC<fr` { 8;:=J|8}WXAqg\A6W Ì JCRQPtfjAocpH\c;L;WXRT>@A6NSFI[DH\UçWXA6gQACWBß6� { [M:=>D>DACPtFIH\[DUVRQW=W=`+RQ<

y ¨ CE� ¨ � y � � Â�� �  �¨ � y � � gY� � g �¨ � y � �����~� ¦h�\kÕ©\Ûr¨798;AmN;:]RTY\H\P_RQWjACW=ACU+ACPr>@<6�_<DL;J|8}RQ<� ¨[¨ �ÈRT[DA1>@8;A1>DHQ>|RQW�WXH\<M<9[@RT>DAC<9FI[DH\UçWXAqg\ACW Ì �;Y\:=gQACP-fr`

¶� ¨h¨ C �yf�� ¨ y ¨ � � Â[� �  �¨ � ¿ ¦h�\kÕ©rÞ\¨Ò ACPBHQ>@:=P;Y^U+A6>@RQ<M>@RQf;W=A?W=A6g\A6WX< { :=>D8zY\[MACAC�m<DLBf;<DJ6[D:Xcs>@<C�QL;<D:=P;Y^>D8;AEL;<ML_RQWK<DL;U+UVRT>@:=H\PoP;HQ>|Ri>@:XHQPH\P;Ao8_RT<3>D8;A1FIH\WXW=H { :XP;Y&[DRT>@A1A6��L_RT>@:=H\PtFIH\[9WXAqg\ACW��VHQF�:XH\P;:=<@RT>D:XH\Pe<M>|RTY\A+¦��r¨? � ���,��? û C ��� � �$�+�� �� � y�� ���+�y �� ���]� � �$� 1j³ ��]� �� ��� í � ��� º ³ �� í ¦h�\kÕ©\ãr¨RQP;NtFIH\[9A�ÖBJ6:=>DACN-<M>|Ri>@A Ì H\P;Ao8_RQ<

� C ¨ � � �$�+�� �� ¨ y]� ���,�y �� ¨ �]� � ��� 1j³ ��]� �� ¨ � í � �$� º ³ �� í ¦h�\kÕ©\Úr¨{ 8;A6[DA�Ð º RTP;N%Ð 1 NBACP;HQ>DAoUVAq>|RQ<h>|RQf;W=AC<9HQF�RQN�ßDRQJCA6P�>�:=H\P;:X<DRT>@:=H\Pe<h>|RQYQAC<Ckb798;Ao[MAC<DLBW=>@:=P;Y�<DAq>HQF�AC��L_Ri>@:XHQP;<3RQ[DAZRQ<D<MACU&f;W=ACNt:XPr>DH&>@8;A1J6H\WXW=:X<D:=H\P_RQWå�y[DRQN;:]Ri>@:åg\A�UVRT>@[M:aÖeRQP;Nt<MH\W=gQACNtfr`OUVRT>@[M:aÖ:XPrgQAC[D<M:XH\P�FIH\[�>@8BA^A�ÖBJ6:=>DACNe<h>|RT>DA^cpH\c;LBW]RT>D:XH\P;< { :a>@8+[DA6<DcpACJ6>G>@Ho>@8BA^U+A6>@RQ<M>@RQf;WXA6<^¦ { 8;:=J|8ORQ[MARQ<M<DL;U+ACN���P;H { P�FI[DHQU�Ns`sP;RQUV:=J!RQWb:=H\P;:X<DRT>@:=H\Pvf_RQWXRQP;JCA!¨qk�»_H\[zJ6H\UVcBLB>|RT>D:XH\P;RQW�<D:=UVc;W=:=Î_JCRi�>@:=H\Pe>D8;Ao�G�×ÐxUeRi>@[D:åÖ+AC��L_RT>D:XH\PS:=<EJ6H\Prg\ACPr>D:XH\P_RTWXW=`+<MH\W=gQACNtL;<D:=P;Y�<DJCRQWXA6NO>@A6UVcpAC[DRT>@L;[MAC<9RTP;NN;A6P;<D:a>@:XA6<

û× C ®�Á�Â;f� �� �P ¦h�\kϾr�r¨

Page 47: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©\©

û � C ®�Á �;� �� �P ¦h�\kϾ;��¨

�r C � #f� � © � ? ´PF ´ � Â� ¡P ¦h�\kϾ��\¨{ :=>D8t<D:XU+:XWXRQ[GN;A6Î_PB:=>@:=H\P;<EFIHQ[�>@8;A1:=H\PS>@ACU+cpAC[@Ri>@L;[MAC<3RQP;NtN;A6P;<D:a>@:=AC<Ckb798;AocpH\c;L;WXRT>@:=H\PO<MH\WXLs�>@:=H\P�:X<�AqÖsc;[DA6<D<MACNORT<�N;A6gK:]Ri>@:XHQPzFµRQJ6>DH\[D<�¦I[DAqFIAC[D[MACN�>@HoRT<�ÉÊfs�§FµRQJ6>DH\[D<6ËÓ¨�FI[MH\U±�BRQ8_Ri�y�EH\Wa>@ÝCUVRQP;PgTRQWXLBAC<C�_NBA6Î_P;A6N�gK:]R

� ¨ C�� 1 �  � ×���k� ¹ ®KÁ ¡

; Ù � ¨�3� 1 ¯ Õ"{ � u[v Q ¨ ¦h�\kϾr©r¨�sH\WagK:XP;Yv>@8;A�AC��L_RT>D:XH\PB<VFIH\[+>@8;A�fs�nFµRQJq>@H\[M<VcB[DH�gK:XN;A6<SR�<MH\WXLB>D:XH\P£>@8;RT>O:=<VU+H\[DA-A!RT<D:XWa`£:XPs�>@A6[Dc;[MA6>DACNí¦I:§kÏAQk9:=PÉÊÙs73¥��y<Mc_RQJ6AQËʨ-RQPBN WXACRQN;<S>@HuJ!RTP;JCA6WXW]Ri>@:XHQP;< { 8;:=J|8 UVRQ�QA(>@8BA�<h`s<h>@A6UP�L;U+AC[D:=J!RQW=W=`(U+H\[MA+<h>|RQfBWXAQk-798;AVAC��L_RT>D:XH\P;<~J!RQP�RQWX<MH}fpAe<MH\W=gQACN�FIH\[mJq�§FµRQJ6>DH\[D<e¦ C Q�¶·��¨q�H\[�¯ Õ"{ � u[v�� Q ¨ FµRQJ6>DH\[D<6k�798;A�J|8;HQ:XJCA~HQF�>@8BAeÉÊcB:XJ6>DL;[DATËjHTF�>@8;A&<MH\WXLs>@:XHQP}:X<^N;A6>DAC[MUV:=P;ACN fr`->@8_Ri>{ 8;:=J|8ec;[MH�gK:=N;AC<G>@8;A^UVH\<h>GPKLBUVA6[D:XJCRQWXWa`&<h>|RQf;W=A^<MH\WXLs>@:XHQPdk4»_H\[bAqÖBRQU+c;W=A3FIHQ[GgQAC[M`+W=H { Á�Â�>@8BAQ ¨ gTRQWXLBAC<EJCRQPOfpA�g\A6[M`VJCW=H\<DA�>DHzÝ6AC[DH&FIHQ[?>@8BA�AqÖsJ6:=>@A6NS<M>@RT>@A6<9RQP;NOW=A!RQNV>@H&P�L;U+AC[M:XJ!RTWpA6[D[MH\[D<6�{ 8;A6[DA!RT<�R�<DHQWXLB>D:XH\PS:=Pt>DAC[MUV<3HQF�¯ Õ|{ � u�v� Q ¨ :=<9U&L;J|8-UVHQ[DA1<M>@RQf;W=AQk� õ å2àRóJø<÷4ùáè§çF�4Þ¯ß Þ¯èlyP >@8B:X< { H\[D�È�ÇROY\A6P;AC[DRQW4cpH\c;L;WXRT>@:=H\P RTc;c;[DH\RQJ|8(:X<ZNBA6g\A6WXH\cpACN { 8;:=J|8v:X<^gTRQW=:XN}FIHQ[ZRQW=W�:=H\P;<C�>@8;RT>m:X<C�ÇHQFEfjHT>@8wWX:XYQ8�>mRTP;N�8BA!R!g�`�<McjA6JC:=AC<m:XP�RQ[Df;:a>@[DRQ[D:=W=`-8;:XYQ8;W=`}:XH\P;:=<DA6Nw<h`K<M>@A6UV<6kO798�L;<ACPBAC[DYT`vWXA6gQACW�<h>@[DLBJ6>@LB[DAC<~8_R!g\A�>@H-fjA�N;ACRQW=> { :a>@8 { 8;:XJ|8w[@RTP;Y\A�FI[DH\UäÙd�×�}>@AC[MUV<V¦µÐ�L;<M<DA6WXWa��BRQLBP;N;AC[M<^J6H\L;c;W=:XP;Yr¨?FIH\[9WX:=Y\8r>3RQP;N�P;A!RQ[3P;A6LB>@[DRQW/<M`K<h>@ACU+<3>@H+:XPr>@A6[DU+ACN;:XRT>@A1J6H\L;c;W=ACN-WXA6gQACW=<FIH\[�U+ACN;:=L;U { AC:=Y\8r>ORQPBNuFµRQ:=[DWa`�:XHQP;:X<MACN£<M`K<M>DACU+<(¦"���æ¾r�r¨~>DHSß×�aß�J6H\L;c;W=ACNu8;A!R!g�`\�EgQAC[h`8;:=Y\8;W=`V:XHQP;:X<MACNt<M`K<M>DACU+<CkÀ Wa>@8;HQL;Y\8&>@8BA�AqÖs:X<h>@:=P;Y1WX:=Y\8r>4ACW=ACU+ACPr>�J6H\L;c;W=:XP;YZ<MJ|8;ACU+AQ� { 8;A6[DA3RZf;LBP;N;WXA6Ns�yP_��HQ[4f;L;P;NBWXACNK�P;W]�BÙwA6WXA6J6>@[MH\Pt<D:a>@<9H\PS>@HQc-HTF4RQP-Ù��h�OJ6H\L;c;W=ACNtJCH\[MAQ�;cB[DH�gK:XN;A6<3>D8;AoP;ACJ6AC<M<@RQ[h`tc;[DA6JC:X<M:XH\POFIHQ[<DcpACJq>@[MH\<DJ6H\c;:XJ�RTP_RQW=`K<M:X<�HTF�>y`scB:XJ!RTW�WX:=Y\8r>�A6WXA6UVA6P�>Z<DcpACJ6>D[@RB��:=>ZA6P;JCHQL;Pr>@AC[M<oc;[DH\fBWXACU+< { 8BACPAqÖK>DACP;N;A6Nx>DH�>@8BA <M`�>@A6UV<V>@8_RT>SRQ[DA�N;A!RQWa> { :a>@8�:=P²>@8;:=<V>D8;AC<M:X<Ckð»_H\[V8;A!R!g�`�<McjA6JC:=AC<O>@8BA[DA6W]RT>D:=gK:X<h>@:=JO<Dc;W=:=>M>@:XPBY HQF�>DAC[MUV<�:XPr>DHvÎ;P;At<M>@[ML;J6>DL;[DA�WXAqg\ACW=<�fjA6JCH\U+AC<�<D:=Y\P;:aÎ_J!RQPr>+RQPBN�>@8BA

Page 48: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©Q¾

[DA6<DL;Wa>@:=P;Y~fB[DA!RT�KNBH { PS:=POÙ��h��JCH\LBc;WX:=P;Y;�KRQWXHQP;Y { :a>@8V<D:XYQP;:=Î_JCRQPr>GJCH\PBÎ;Y\L;[@Ri>@:XHQPe:XPr>DAC[@RTJ6>@:=H\Pd�U+A!RQP;<Z>D8_RT>Z>D8;A&Î_P;A�<h>@[DLBJ6>@LB[DA�WXH { W=`K:XPBYOWXAqg\ACW�cjH\cBL;W]RT>D:XH\PB<�J!RTPwPBH-W=H\P;Y\A6[�fpA�RQ<D<ML;U+ACN>@H�fpA1<M>@RT>@:=<M>D:XJ!RTW�RQPBN-UzL;<M>3fpAZ>@[MA!RT>DACNt:XP;NB:=gK:XN;L;RQWXWa`\k�798;:X<�:=<�[DAq¢_ACJq>@A6N}<DcpACJq>@[MH\<DJ6H\c;:XJCRQWXWa`\�{ 8;A6[DA~RV¤i�n[DA6<DH\Wag\ACNtA6P;AC[MYQ`tWXAqg\ACW/cB:XJ6>DL;[DA1:=<9L;<DL;RQWXWa`eL;<MACN-FIH\[3RQP_RQWa`K<D:X<EHQF�U+ACNB:XL;Ue�T8;ACR�g�`<DcpACJ6:XA6<Ck À W=<DH;�3>@8BAwcB:XJ6>DL;[DA(HTF&R�f;LBP;N;WXA6Ns�yP_� A6WXACJq>@[MH\P f;L;:XWa>OH\P�>DH\c HQF&R�U+A6>@RQ<M>@RQf;WXAc_RQ[MACPr>3J!RTPSP;H�WXH\PBY\AC[?fjA1<ML;<M>@RQ:XP;A6NSFIH\[E8;:=Y\8;Wa`VA�ÖsJC:=>DACNtcpH\c;L;WXRT>@:=H\P;<Ck�»_H\[�<DLBJ|8-A6WXA6J6>@[MH\P;<6�[DA6W]RT>D:=gK:X<h>@:=J�:XPr>@A6[@RQJq>@:=H\P�N;H\U+:XP_RT>DAC<�H�g\A6[?A6WXA6J6>@[MH\<M>@RT>@:=J3WXA!RTN;:XP;YZ>DH~<Mc;:XP+<M`K<h>@ACU f;[MA!RQ��N;H { PRQP;N�:a>e:=<eU+H\[DA}RQc;c;[MH\c;[M:]RT>DA�>@H�UVHKN;ACW^R { A!RQ��Wa`�JCHQL;c;WXA6N�P;W^ACWXA6J6>D[DH\P�f;LB:XW=>VH\P�R�Ùd�×�JCHQL;c;WXA6N�HQ[�:=Pr>@AC[MUVA6N;:]Ri>@A1JCHQL;c;WX:=P;Y+¤+J6H\[DATkE798;:=<3<M`K<M>DACU$:X<3<DHQUVAq>@:XU+AC<9[MA6FIA6[D[DA6N�>DHVRT<oÉ ß×�yWJCHQL;c;WXA6NdËakSln>z<M8;H\L;W=N�RQW=<DH}fpA+[DACJ6H\Y\P;:=<DA6N�>@8_Ri>m>D8;AC[MAe:=<~P;HQ[DUVRQWXWa`(RtJ|8_RQP;Y\AVHQF�>D8;AeU+H\<M>RQc;cB[DH\c;[M:]RT>DAVJ6H\L;c;W=:XP;Y-<DJ|8BACU+AtRQ<zH\PBAec_RQ<M<DA6<zFI[DH\UÆW=H { WXA6gQACW=<~>@H}8;:XY\8BW=`(A�ÖBJ6:=>DACN�WXAqg\A6WX<Ck798�L;<bFIH\[4>@8;A3U+HsNBACWBcB[DAC<MACPr>@A6NO:XP�J|8;RQcB>@A6[?¾1>D8;A9FIH\WXW=H { :=P;Y�fBL;P;N;W=:XP;Y1[MACY\:=UVA6<bRQ[DA9N;AqÎ_P;A6Nd�{ :=>D8�A!RQJ|8+U+AC[DYQ:XP;YZ<MUVHKHQ>D8;W=`m:XPr>DHZ>@8;A9P;A�ÖK>GRQ<4>@8;A9W=A6gQACWX<�fpACJ6H\U+A�<M>@RT>@:=<M>@:=J!RQW=W=`mcpH\c;L;WXRT>@A6NRQP;N-J!RTP�fpA1f;L;P;NBWXACN-H�g\A6[Ck

Page 49: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

© à

� � � P ¼ � �P� 7Y� �� P � �6�±� ³ ¼ � �P�8� 7 � � �p� � � y �� ³ � �6�±� � ¼ � �P� � 7 � � � � � � y� � � �6�±� ´ ¼ � �P�8� 7 � � �p� � �� ´ � �6�±� � ¼ � �P�8� 7 � � �p� � ¦h�\kϾ\¾�¨»_H\[ � � � P ��>@8;A�gTRQWXA6P;JCAVACW=ACJ6>D[DH\P�:X<m:XPw>@8;AVJCH\[MAeJCHQPBÎ_Y\L;[DRT>@:=H\P�RTP;N�>D8;AeU&LBW=>@:=ACW=ACJ6>D[DH\P<M>@RT>@A6<ORQ[MA�:=Pu>D8;A�FIHQ[DU � �L��� { 8;AC[MA � :=<+>D8;A�J6H\PBÎ_YQL;[@RT>D:XH\P���� >D8;A}c;RQ[D:a>y`£RQP;N@� >@8BA>@HT>|RQWÈRQPBY\L;W]RT[�UVH\U+ACPr>DL;U�k�»_H\[�>@8;A�[DA6<M>?HQF�>D8;A^fBL;P;N;W=:XP;Ym[DA6Y\:XU+AC<bH\P;A38_RT<ERQP � =dA6WXACJq>@[MH\Pf;L;:=W=>^L;cpH\P(Rt¤i�yJ6H\L;c;W=ACN c;RQ[DA6P�>Z<M>|Ri>@At¦ � g_� g¡�Yg�¨&¦I8;ACP;J6A6FIH\[h>@8([MA6FIAC[M[DA6Nv>DHSRQ<�>@8;AeÉÕf;L;PBN;WXA��¦_�fgi¨ � =yË4U+HKN;ACW]¨qkv»_H\[ � P � �¢� � ³ >D8;A-¤vW=A6gQACWX<&RQ[DAO>D[DA!Ri>@ACN£:=P;N;:=gK:=N;L_RQW=W=`\��RT> � ³ >D8;A6`RQ[MA�A6�jACJ6>D:=gQACW=`}N;ACYQACP;A6[@RT>DAVRTP;Nv<M>|Ri>@:X<h>@:=J!RQW=W=`tcpH\c;L;WXRT>@A6Nw[MACWXRT>@:ag\Az>DHSA!RQJ|8wHQ>@8BAC[oRQP;NvJCRQPfpA&f;L;P;NBWXACN >DH\Y\Aq>@8;A6[Ck~�s:XU+:XWXRQ[DWa`SRT> � � >@8BA�ß×�y<h>|RT>DAC<1RQ[MA&A6�jACJq>@:ag\ACWa`}NBACY\A6P;AC[DRT>@A�RQPBN(<M>|R��>@:=<M>D:XJ!RTWXW=`�[DA6W]RT>D:=gQACW=`�cjHQc;L;W]Ri>@ACN�RQP;N�J!RQP�fjA-f;L;PBN;WXA6N²H�g\A6[eRQPBN�RT> � ´ >@8;A-Wa�n<M>|Ri>@AC<�JCRQPfpA&f;L;P;NBWXACN}H�gQAC[6k Ò A6>@A6[DU+:XP;:=P;Y�>@8;A&cpH\:=P�>ZRT> { 8B:XJ|8 >@8BAzJCHQWXWX:=<D:=H\P_RQW/[MACN;:=<M>@[M:Xf;Ls>@:=gQAz[DRT>@A6<H�g\A6[DJCHQUVAz>D8;A&[DRQN;:]Ri>@:=gQAzN;A6J!R!` [DRT>@A6<1RQW=WXH { <�>D8;A&f;LBP;N;WX:=P;YeJ6LB>M�nHQ�j<�¦µ:­kÕATk � P ¢ � � ¨�>DHOfpAA6gTRQW=L_RT>DACNdk4Ì�H\[D�+:X<?c;[DA6<DA6P�>DACN-:XPSJ|8_RTcB>@A6[9¾&H\LB>DWX:XPB:XP;Y~R&<DJ|8;A6UVA1fr` { 8;:XJ|8S>@8BAC<DA1J6LB>M�nHQ�j<J!RTP�fjA�NBA6>@A6J6>DACNVRQLB>@HQUeRT>D:XJCRQWXWa`oRQP;N�L;<MACN�:XP~>@8;A9RT<D<DA6U&f;Wa`&HQF_>D8;A3RQc;c;[MH\c;[M:]RT>DAEJCHQWXWX:=<D:=H\P_RQWå�[@RTN;:]RT>D:=gQA^UVRT>@[M:aÖ&��P;HQ>DA^>D8_RT>E>D8;A�:=P;JC[MA!RQ<MA�:=Pe<D:=ÝCA^HQFd>@8BA { HQ[D��:XP;Y~UeRi>@[D:=JCA6<?P;A6JCA6<D<@RT[M`eFIHQ[>@8B:X<1<DJ|8BACU+A+:=<1<M>D:XWXW { :=>D8;:XP >D8;A�J!RQc;RQJC:a>y` HQF?UVHKN;A6[DP { HQ[D��<M>@RT>@:=H\P;<Ckz798;A�<DA6>mHQF?JCHQL;c;WXA6NU+HsN;A6WX<4<M8;H { PV8BAC[DA9UVRQ[M�K<bRZ<M:XY\PB:=Î_JCRQPr>b:XU+c;[DH�g\A6UVA6Pr>�H�g\AC[�>D8;A9AqÖs:X<h>@:=P;Y1<DJ|8;A6UVA9FIH\[�W=:XY\8r>ACW=ACU+ACPr>@<6kÌ :=>D8�>D8;Az[DRQP;Y\AmHQF�J6H\L;c;W=:XP;Y+<DJ|8;A6UVA6<ZL;<MACN(:XPt>@8B:X<�P;A { U+HsN;A6WÇ:=>�:X<�P;ACJ6AC<M<@RQ[h`}>@HVN;A6Î;P;ARQc;cB[DH\c;[M:]RT>DA4>@[DRQP;<MFIHQ[DUVRT>@:=H\P;<dfpA6> { ACA6P&<M`K<M>DACU+<Ck4798�L;<�[@Ri>@AC<�U&L;<h>�fjAbA6gTRQW=L_RT>DACN1FIH\[/JC[MH\<D<×�JCHQL;c;WX:=P;Y�>D[@RQP;<M:=>D:XH\P;<4RQ< { A6WXW_RT<4FIH\[�>@[DRQP;<M:=>@:=H\P;< { :a>@8;:=Pz>@8BA9P;A { f;L;P;N;W=Aq��¦_�Yg�¨ � =pUVHKN;ACW­k À <U+H\<M>z[@RT>DAeAqÖsc;[MAC<M<D:XHQP;<�JCH\Pr>@RQ:XP�>D8;AC:=[zRTP;Y\L;WXRQ[~J6H\U+cjH\PBACPr>@< { :a>@8;:=P�ëoRQL;Pr>zFµRTJ6>@HQ[D<C��>@8;:=<WXACRQN;<1>@H�RtP;ACA6N�>@H-N;A6[D:=gQAOëoRQL;Pr>oFµRQJq>@H\[M< { 8;:=J|8�NBA!RQW { :=>D8�RTWXW�>D8;AVcpH\<M<D:XfBWXA�:XPB:=>@:XRQW4RTP;NÎ_P_RTW�<M>|Ri>@AC<mHQFG>@8;:=<oP;A { <MJ|8;ACU+AQkV798;:X<o8_RT<mfpACA6P�N;HQP;A+RQP;Nv>@8;A�[DA6<DL;Wa>@<m<DL;U+UVRQ[D:=<DACNv:=P>|RTf;WXA1¾;ka�\� { :a>@8S>@8BAmNBA6>|RT:XWX<9YQ:=g\A6Pt:XP-RQc;cpACP;NB:aÖ À k

Page 50: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©\Û

À cB[DH\Y\[MAC<M<D:XHQP(HQFbJCH\LBc;WX:=P;Ye<MJ|8;ACU+AC<1:=<�L;<MACN >@8B[DH\L;YQ8;H\LB>Z>D8;:X<�>@8;A6<D:=<Cko�E8_RTcB>@A6[D<���RQPBNv�AqÖBRQU+:XPBAVH\c;RQJC:a>y`v:XPwWX:XYQ8�>mA6WXACU+ACPr>D<O¦­�¤£�£?:=P�<MACJ6>D:XH\P��\kÕ©Bk=�+>@8;A6P�Y\A6P;AC[M:XJ+WX:XYQ8�>mA6WXACU+ACPr>D<FIH\[o>@8;AVH\c_RQJ6:=>y` { H\[D�vHQF9J|8;RQcB>@A6[��\¨1L;<D:=P;Y�>D8;AeA�Ös:X<M>D:XP;Y ÀZÒ3À ��W=:XY\8r>mACW=ACU+ACPr>~JCH\L;cBWX:XPBY<DJ|8BACU+AQk4Ì :=>D8+>D8;A�AqÖK>@A6P;<D:=H\PeHQFj>@8;A3H\c;RQJC:a>y` { HQ[D�&>@Hm8;:=Y\8VWa`K:XP;Yo8r`KN;[DH\YQACP;:=J^cpH\c;L;WXRT>@:=H\P;<:XPSJ|8;RQcB>@A6[D<�©�RQP;N à �_R&J6H\L;c;W=:XP;Y�<MJ|8;ACU+A { 8;AC[MAmRQP � =�A6WXA6J6>@[MH\PS<D:a>@<9H\P-R&f_RT[DAoP�L;JCW=ACL;<9:=<L;<MACNdk^798;:X<^WXA!RTN;<�H\P->@He>D8;Az8B:XY\8�WXA6gQACW�8;A!R!g�` <McjA6JC:=AC< { H\[M��HQF4J|8_RQcB>DAC[Z¾ { 8BAC[DA~>@8;AmFIL;WXWf;L;PBN;WXA��q¦_�fg�¨ � =�UVHKN;ACW�:=<oL;<DA6Ndk Ñ HQ>DA+>D8_RT>o>@8BAVJ6HsN;A+N;Aqg\ACW=H\cpACN�:=P�J|8;RQcB>@A6[z¾-:X<oP;HQ>m>@8BAÎ_P_RTW_8;A!R!g�`�<DcpACJ6:XA6<?H\PBAQ�rf;LB>b>D8;A38�`KN;[MH\Y\A6P;:XJ3U+HsN;LBWXA { 8B:XJ|8 { :XWXWKFIH\[DU>@8;A38;:=Y\8���L_RQPr>@L;U<D8BACWXW3J6H\UVcpH\P;A6Pr>VHTF�>@8BA-Î_P;RQW3JCHKN;AQk�798;A}J6H\P;JCA6cB>VHQF1RQP�AqÖsJC:a>@A6N²8r`KN;[DH\YQACP;:=J�A6WXACJq>@[MH\PJCHQL;c;WXA6N�>DHeRV¤i�y[MAC<DHQW=g\A6N}J6H\[DAo:X<�R�gTRQWX:=N�RTc;c;[DH!Ös:XUVRT>D:XH\POFIH\[9>D8;A~8B:XY\8;Wa`OA�ÖBJ6:=>DACN}<h>|RT>DAC<�HQF8;ACR�g�`�<DcpACJ6:XAC<6�bRQP;N�cB[DH�gK:XN;A6<&R}U+HsNBL;WXA�>@8;RT> { AOJCRQP�>@A6<M>�RTYrRQ:XPB<M>mAqÖs:X<h>@:=P;Y}8r`KN;[DHQY\ACP;:=JWX:=Y\8r>9ACW=ACU+ACPr>3JCHKN;A6<Ck

���a¥ ¦@§-#z ¨ª©C.1"ª)�««)�¨ #&0��62dù #z%(' "L¨�.1092�ö�#¬©�"�.1öb�C.1"ä�!% #z"t­24ö�)�¨(øOù-"4�C0�#�©u#z%(' ©C#&*�)Vö�#m24)+ö4ù ¨ª©C#&"4 #z"

úGû�­9ûMú ®°¯���þ���*��X G�}�1 �?þ��±¯��4 ��³² �� ����Zÿ´¯��+��5 *+�+���E ��}����?þ!��þ � �����X5¶µ���?þ¹��×ÿ��]�� ±���4��8Zÿ4�+��5 ���:��hÿ���*����R�:>¯¥�<DJCRQcjAzc;[DH\f;RQf;:XW=:=>D:XAC<98;R�gQA&HQFÅ>DACP}fpACA6P(L;<DA6N}>@HeN;:]RQYQP;H\<DAmH\cB>D:XJCRQWÇN;ACcB>D8;<�¦­¤QH\[DN_RTPd���!Ú\ÛrÞsïÒ H�`KWXA~â ì�J6Ì 8;:=[M>DAC[C����Ú\ã\�sï:·1ACA6P_RQP(⤷1:=P;Y\<M>DH\Pd����Ú\ã\Ûs︷oRQ<h>@P;A6[C����Ú\Ú\Úr¨�k À <�ReN;ACU+H\Ps�<M>D[@RT>D:XH\PtHQF�>@8;:=<9>@ACJ|8BP;:X��L;AmJCH\PB<D:XNBAC[^RQP�A�ÖBRQUVcBWXA1>|RQ�TACP-FI[DH\U�>D8;Ao<M>DL;NB`-HQF��E[DHKH\��<^Aq>^RQW§k¦§�T�\�\�r¨�k�ÄZf;<DA6[MgTRT>D:XH\P;< { AC[MA-JCRQ[D[M:XA6NuHQLB>�H\P��sACcs>@ACUzfjA6[eÞ�>@8d�3��Ú\ÚQÛvLB<D:XPBYv>D8;A-����ì}¥bЦIÌ :=WX8;A6WXUæA6>mRQW­k=���!Ú\Ú à ¨^:=P;<M>D[DL;U+ACPr>mH\P�fpHrRQ[DN(>D8;AV�;Ä1½ZĪ¦­�;ÄZW]RQ[ZRQP;N�½^ACW=:XH\<Mc;8;A6[D:XJ�Ä1fK�<DA6[MgTRT>DH\[M`B¨1<Dc_RQJ6ACJ6[@RTFÅ>Cke798;A+H\f;<MAC[MgTRT>D:XH\PB<mJ6H\P;<M:X<M>DACN�HQF��!ã-J6[DH\<M<h�yW=:XU&fv<MJ!RQP;<e¦µ:=P£�\kÊÚ-RQ[MJ<DA6JV<h>@A6c;<|¨1HQFG>@8;A { R!gQACWXA6P;YQ>D8�[DA6Y\:XHQPw<ML;[D[MH\L;P;NB:XP;YtgTRQ[D:=H\L;<1U&L;Wa>@:=c;WXAq>@<Ck~Ì�A�c;[MAC<DA6Pr>z>@8BA[DA6<DL;Wa>@<GFIH\[?>D8;A1�¹£"£Ç�.£ � � � �]º �b�.£�� � �)� » ¦×���\©\Û ¿À ¨bU&L;W=>D:Xc;W=A6>C�r<DACA^Î_Y\L;[MA&�\kÕãBk4798;AZHQfKßMACJ��>@:ag\A { RQ<Z>@HtN;:XRQY\P;HQ<DA&H\c;RQJC:a>@:XA6<�FI[DHQUèf;[DRQP;J|8;:=P;YtWX:=P;Az[DRT>@:=HSH\f;<DA6[MgTRT>D:XH\P;<^FI[DH\Uæ>@8;A�<MH\W]RQ[WX:=U&fdk�798;:X<4FIHQWXWXH { <4HQP�FI[MH\U Ò H�`KWXA�â·ì�JqÌ 8;:X[h>@AC[^¦×��Ú\ã\�r¨ { 8;HocjA6[MFIH\[MUVA6NVR1<M:XU+:XW]RT[�<h>@L;NB`H\PS>@8BAz�*£�£"£4�.£�� �-´ º �E� � � ´ º ¦h�Q��Þ à ¿� ¨EU&L;W=>D:Xc;W=A6>Ck

Page 51: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©rÞ

»�:=Y\L;[DAo�\kÕãB¼UTºs�epr¤cJ^Anf`!a]b.cAdfeL¼¸½�½�`�nfgjcJYhsfgje+cn�l<epo YjiTcAdfe4vf^�Z]i�r�dfYjik\X^AZ]cJYha�lxcAnkoºw_z>\�dke4ghZ�^J\_e�gjYhifeRa�i cJdkeghe�b�c4dkZ�iko�l<Y[oºeuYhl2Z)l<cJ^Aa�ik\¿¾tMºw§\�e,i:ghYjikelyP <MJ!RQP;PB:XP;YeRQJC[MH\<D<�>D8;AzW=:XU&ftHQF�>@8;A~<DL;Pd�p>@8;A~Y\ACHQUVAq>@[D:=J!RQWd>D8;:XJ|��P;A6<D<1HQF�>@8;A~WX:XPBAmHTF�<D:=Y\8r>:XPBJC[DACRQ<DA6<Ck�ÄZc_RTJC:=>y`}A6�jACJq>@<m:XP >@8B:XJ|�}W=:XP;A6<ZU+A!RQP;<Z>D8_RT>1>D8;AC:=[�¢_LsÖsAC< { :=WXW�gTRQ[h` N;:a��A6[DA6P�>DW=`FI[DHQU >D8;H\<MA�HQFd>D8;A�>D8;:XPVWX:XPBAC<Ck�798;A�[DRT>@:=HzHQF/<ML;J|8SWX:=P;AC<?RQW=WXH { <G>D8;A�Aq��A6J6>D<�HQFÇH\c_RTJC:=>y`�>DH&fpA:X<MH\W]Ri>@ACN�k�^<M:XP;Y�A6��L_RT>@:=H\Pî¦h�\kÕÚr¨ { A(<DA6A(>@8_RT>S>D8;A([@Ri>@:XH�HQFo> { H£<DcpACJq>@[DRQWZWX:=P;A :=P�>DACP;<M:=>D:XAC<eJ!RQP fpA{ [D:a>D>DACP-RQ< ; � � Ù � � Ù; � � À � � À C " � � Ù � � Ù" � � À � � À � � � Ù � � Ù� � � À � � À | OP � � � Ù ? =| OP � � � À ? = ¦h�\kϾ à ¨

{ 8;A6[DA � ³ RTP;N � � ³ [MACc;[MAC<MACPr>�>@8;AOW=H { A6[&RQP;N�LBc;cjA6[&WXAqg\A6WE:XPBN;:XJ6AC<z[MAC<McjA6J6>@:ag\A6W=`�HTF3WX:XPBA��\k�s:=UV:=W]RQ[MW=` � � RQP;N � � � [MACc;[MAC<MACPr>3>@8;A1W=H { AC[�RQP;NSL;c;cpAC[9W=A6gQACW�:=P;N;:=JCAC<E[DAC<McjA6J6>D:=g\A6W=`eHQFÇWX:=P;A1�sklnF�>@8;AoWX:=P;AC<9HQ[D:XYQ:XP_RT>DA^FI[MH\Uç>D8;Ao<@RQU+AoL;c;cpAC[3WXAqg\ACWd>D8;ACP �³� ³ C � �� C �³� >@8�L;< � � � À C�� � � Ù C��� � kG798;RT>�:X<C� ; � � Ù � � Ù; � � À � � À C " � �h� � Ù" � �h� � À ��� �[� � Ù��� �[� � À ¦h�\kϾrÛr¨798�L;<}HQf;<DA6[MgTRT>@:=H\P;<-HQFz>D8;A�[DRT>@:=H£HQFz> { H�WX:=P;AC<tA6UV:a>D>@A6N�FI[DH\U RuJCH\U+UVHQPxLBc;cjA6[�W=A6gQACW§�

Page 52: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©\ã

�@£�£��.£�� � �T� » ÀÙ :XPOHQL;[�J!RT<DAQ�_A6WX:=UV:=P_RT>@A6<G>D8;A1N;A6cjA6P;N;ACPBJCAmH\PO>@8;A1LBc;cjA6[9WXAqg\ACW�cjH\cBL;W]RT>D:XH\PN;A6P;<D:a>@:XA6<1RQP;N(:=<DH\WXRT>@A6<�>D8;A�H\c_RTJC:=>y`-:=PB¢_L;A6P;JCA�H\P�>@8;A�W=:XP;Am[@RT>D:XH;k1�s:XP;J6A~>D8;A À �§gQRTWXL;A6<ZRQ[MA��P;H { Pd�GRQP�H\f;<MAC[hgQRi>@:XHQP�HQF9>@8;AS:=Pr>@ACPB<D:=>y`�[@Ri>@:XH�WXACRQN;<z>DH(R N;ACNBL;JCA6N�A6<DJ!RTcjASc;[MH\f_RQfB:XWX:a>y`[@Ri>@:XHB� { 8;:XJ|8�:XP(>@L;[MP�W=A!RQNB<o>DH}RTP�H\cs>@:XJCRQW�N;A6cB>@8�N;A6>DAC[DU+:XP;RT>@:=H\Pdk�798;A+H\cB>D:XJ!RTW4N;ACcs>@8�Ri>WX:=P;AZJ6ACPr>@[MAQ�_>|RQ�TACP-FI[DH\Uáì�:=>DJ|8;ACW=WÇâÂÁ�ACUVRQP;<M�r`w¦h��ÚQÛB��¨q�B:X<9Y\:ag\ACPtfr`

L P C FE¦A@ P ¨,QC � � ��Û����!� ºK� ñ à Á ¨ A P Í � �h� �t��� Q ¦h�\kϾ�Þ\¨

{ 8;A6[DAÁ ¨ :=<�>@8;Ao:=H\PS>@ACU+cpAC[@Ri>@L;[MA��j·��­kà :X<�>@8BA~Ri>@H\U+:XJZUVRQ<M<�P�L;UzfjA6[�HTF�>D8;AmRQf;<DHQ[Df;:=P;Y+RT>@H\U+<�� RBkÏU�kÏLdk��§k@ P :X<�>D8;AoJCA6P�>D[@RQW/FI[MAC��L;ACPBJ6`tHQF�>@8;A1>D[@RQP;<M:=>D:XH\PSHQF�:XPr>@A6[DA6<M>��Ͻ�Ý_�­kA P :=<�>@8;AoJCA6Pr>@[@RTW { R!g\A6WXA6P;YQ>@8�HQF�>D8;A1>@[DRQP;<D:a>@:=H\POHQF�:XPr>@A6[DAC<h>�� J6U��­k��� :=<�>@8;AoP�L;U&fpAC[3N;A6P;<D:a>y`SHQF�>@8;AoW=H { AC[3W=A6g\A6W/HQF�>@8;A1>@[DRQP;<M:=>@:=H\P � ¸q¹ º ´ �­kÍ � �[� � :=<�>@8;AmRQfB<DH\[McB>@:=H\PtH\<DJ6:XWXWXRT>@HQ[E<M>D[DA6P;YQ>@8á� N;:XU+ACP;<M:XH\PBWXAC<M<,�­kQ�:=<�>@8;Aoc;8r`K<D:=J!RQW/>@8B:XJ|��P;AC<M<^HTF�>D8;Aoc;W]RQ<MUeR�RTWXH\P;Y&>D8;AoWX:=P;A1HQF�<D:XYQ8�>¥� JCU>�­k798�L;<?>D8;A3[@RT>D:XH1HQFj> { H~H\cB>D:XJ!RTW;N;ACcs>@8;<GJCH\[M[DA6<DcpH\P;N;:=P;Ym>DH~W=:XP;A6< { 8;:XJ|8VRQ[M:X<DA9FI[MH\U±>@8BA�<DRQU+AL;c;cpAC[3W=A6g\A6W§� � � �_:X<9YQ:=g\A6Ptf�` L P � � À � � �L P � � Ù � � � C A P � � À � � �A P � � Ù � � � Í P � � À � � �Í P � � Ù � � � ��� À��� Ù ¦h�\kϾrãr¨À <Ç>@8BAE�Ä£�£�WX:XPBAC<ÇRQ[MA?J6WXH\<MAb>DH\Y\Aq>@8;A6[C��>@8;AG[@RT>D:XH9HQFs>@8;AbJ6ACPr>@[DRQW { R!g\A6WXA6P;YQ>@8B<�:X<ÇJ6WXH\<MAb>DH^LBP;:=>y`RQP;N�>D8;A¾L�[DRT>@:=H}:=<zJCHQP�>D[DH\W=WXA6N�fr`�>D8;AO[DRT>@:=H}HQF9>D8;Ae> { HwHQ<DJC:=WXWXRT>@H\[1<h>@[MACP;YQ>D8;<�RQP;N�W=H { A6[WXAqg\A6W^cpH\c;L;WXRT>@:=H\P;<6� { 8;:XJ|8�RQ[DA�<M>|Ri>@:X<h>@:=J!RQW=W=`�cpH\c;LBW]RT>DACNdk À P�HQcB>@:=J!RQW=W=`�>D8;:XP�cjH\cBL;W]RT>D:XH\PJ!RTWXJCLBW]RT>D:XH\PSL;<M:XP;Y ÀZÒ3À � { RQ<^NBH\P;Ao>@H�g\A6[D:=FÅ`S>D8;:X<6k?ln> { RQ<3FIH\L;P;Nt>@8_Ri>�>D8;AmWXH { A6[3Y\[DH\LBP;N>@A6[DU [MACW]Ri>@:=gQA(cjHQc;L;W]Ri>@:XHQP;< { A6[DAwg\AC[h`xJCWXHQ<DA(>@Hu<M>@RT>@:=<M>@:=J!RQWZRQP;N :=P;<DA6P;<D:a>@:ag\A(>@Huc;W]RT<DUVRJCHQP;N;:=>D:XH\PB<1H\[1Î_P;:=>DA�H\cB>@:=J!RQW�N;ACcB>D8;<Cke798;A+FIHQWXWXH { :=P;YSW]R!`\A6[mR�gQAC[DRQY\ACN�AC<DJCRQcpAVc;[MH\f_RQfB:XWX:a>y`AqÖsc;[MAC<M<D:XHQP£¦µ[MA6FIAC[M[DA6N}>@HeRQ<mÉnÅÔ ¦xL P ¨qËʨ { RQ<^N;A6gQACW=H\cjA6N}FIH\[�RQP�:XPBÎ_PB:=>@Aoc;WXRQP;Amc_RQ[DRQWXW=ACWd<MW]RQf-HQF8;H\U+H\Y\A6P;ACHQL;<3N;ACP;<M:=>y`

Page 53: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©\Ú

»�:=Y\L;[DA��\kÕÚB¼�Æ$gha]c2a�b�cAdfeS¼ÈÇ~Ç>ÉËÊ[ÌÍlÎ6ÏÍYÐ�ÑaÉBÊhÏÍ�Î6ÏÍÐ4gjYhife�^�ZBcAYja�zC\�dfe�l<a_gjY[o�gjYhife�lJdfaa`4l�cAdfe�a_sºcJY[r+Z�gjghw»cAdfYhid nYMq^AZ]cJYhakzN\�dfeue+^A^Aa�^�vkZ]^�l^Je d e,r�c�cJdfeXnkikr�e,^<c�Z]Yhi_cxwTYhiqcJdfeLOkc<cJYhif\Ta�b¯cAdfeuc|`³a d n3M§epl+zÅÔ ¦ML P ¨ C �Ö � N��º �

ÒÓ �3¶�¯ º � �8Ô:Õ Â íYÖ ÙF P é¬ ×Ø ? · ¦h�\kϾrÚr¨798�L;<~>D8;A�UVAq>@8;HKNwHQF9RQP;RQW=`K<D:=< { RT<zRQ<oFIH\W=WXH { <C¼Ù��<D:XPBY�A6�KL;RT>@:=H\P�¦×�\kϾrÛr¨^>@8BAV[DRT>@:=H-HTFE>@8BA> { HZAC<MJ!RQcpA�c;[MH\f_RQf;:=WX:a>@:XA6< { RT<4N;ACNBL;JCA6N�FI[MH\U·>@8BA�:XPr>@A6P;<D:a>y`m[@RT>D:XH�H\f;<DA6[MgTRT>D:XH\Pdk�798;A6P�L;<D:=P;YAqÖsc;[MAC<M<D:XHQP-¦h�\kϾrÚr¨�>@8BAEJCHQ[D[DA6<DcpH\P;N;:=P;Y�HQcB>@:=J!RQW�N;A6cB>@8t¦<F P érj¨Ç:XP~H\P;AGHQF;>@8BA?>@[DRQP;<M:=>@:=H\P;< { RQ<A6gTRQW=L_RT>DACNSP�L;U+AC[D:=J!RQW=W=`Qk4¥���L_RT>@:=H\Pw¦h�QkÕ¾rã\¨ { RQ<�>@8;A6PtL;<DA6Nt>DH&N;A6N;L;JCA1>D8;A1H\cB>D:XJ!RTWjN;A6cB>@8t:=P>@8BA&HQ>@8BAC[�W=:XP;A~RQP;N�>@8;:=< { RQ<1AqÖK>@[DRQcpH\W]RT>DACN->@HO:=PBFIAC[�>@8;A&HQcB>@:=J!RQWÇN;A6cB>@8;<1HTFbRQWXWÇHQ>D8;AC[1�¢£"£>@[DRQP;<M:=>@:=H\P;<6klyPtH\[DNBAC[�>@H�U+A!RT<DL;[MAZ>@8;A1W=:XP;A^:XPr>@A6Y\[@RT>DACNS<McjA6J6>@[DRQWd¢_LsÖO:XPSACRQJ|8tHQF�>D8;Am��£�£/WX:=P;AC< { AoL;<MACN>@8BAeÎ;>D>D:XP;Y�c;[DHKJCA6N;L;[MA ÀZÒ3À ��Û\�\��¦§�sLBUVU+AC[M<C�b�Q�Q�B��¨qk�798;:X<~Î;>D>D:XP;Y�[DH\LB>D:XP;AVcjA6[MFIH\[MUV<zRUVRiÖs:XU&L;UèWX:=�QACW=:X8;HKHKN}Î;>mHQF�ëmRQLB<D<D:XRQPwc;[DHQÎ;WXAC<1>DH�RTP�H\fB<DAC[hg\A6N�<DcpACJq>@[DLBUØRTP;N�cB[DHKN;L;JCA6<:XPr>DACY\[DRT>@A6N ¢_LsÖsAC<6�/JCACPr>D[@RQW { R!g\A6WXACPBYQ>@8wcjHQ<D:=>D:XH\PB< { :a>@8vRQ<M<DHKJC:]Ri>@ACN(A6[D[MH\[ZAC<h>@:=UeRT>DAC<6k1798;A[DA6<DL;Wa>|RQPr>4¢;LsÖ&[DRT>@:=H�c;W=HQ>�:X<�<M8;H { P�:XPzÎ_YQL;[DAZ�QkÊÚ��dP;HQ>DA?>@8;RT>4>@8BA�<M>D[@RQ:=Y\8r>4WX:=P;AG:XP~>D8;:X<�Î_Y\L;[MA<D8BH { <G>@8;A3>@8BACH\[MA6>@:=J!RQWÈH\cs>@:XJCRQWXWa`~>D8;:XP+[@Ri>@:XHB�rY\:=gQACPefr`�>D8;A�[DRT>@:=HmHTF�>@8BA�> { H À �ngTRQWXLBAC<Ck�798;AN;:=<Dc;WXR�`QACN A6[D[MH\[D<^RQ[DAmY\:ag\ACP�fr`->D8;Az<M��L_RQ[DA~[DHKHQ>�HQF4>D8;Az<ML;U$HQF�>D8;Az<M�KL;RQ[DA6<Z:XPt>@8;A~AC[M[DH\[M<:XPS>D8;A1¢_LsÖsAC<3FIH\[9A!RTJ|8}WX:=P;AQk798;A^[@RT>D:XH�;j¦ ´� ¶ ³� ¨p½�;j¦ ³� ¶ ³� ¨�N;Aqgs:XRT>@A6<?<M:XY\PB:=Î_JCRQPr>@W=`zFI[DH\U :a>@<bH\cB>D:XJCRQWXWa`z>@8B:XP�gQRTWXL;A3H\PVN;:X<MJRQP;N}RT>�>@8;A~WX:XUzf�¦Jü Ú à ÚVRT[DJz<MACJ!¨q��[MA6>DL;[DP;:=P;YV>DHV:a>@<3>@8;:=P�gTRQW=L;AzRTfjH�g\Am>@8;AmWX:=U&fdk Ñ AC:a>@8;A6[

Page 54: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾r�

Ü�HQ<CkX¦"Ú�¨ ÅÔ ´_� � º ³�� � L P � ´_� � º ³�� � Ü�H\<CkX¦"Úr¨ ÅÔ ´_� � º ³�� � L P � ´_� � º ³�� �ÚQ¾r©skÊ�\Û �BkÊ©s� à kÕ©\� Ú à ÚBkÕÚQ¾ �Bka��ã ���BkÊ�ÚQ¾\¾BkÊÚQ¾ �BkÊ©s� à kÊ� à ÚQÛB�\kÕã\ã �BkÊ�Qã ÛBkÊ�Q�ÚQ¾rÛskÊãB� �BkÊ©\� ¾;kÕÚ à ÚQÛ\©BkÊÞ à �BkϾrÛ �skÊã\�ÚQ¾rãskÊÛ\Ú �BkÓ�TÚ à kÊÞQ� ÚQÛ à kÕÛ\© �BkÕã\ã �BkÊ©QÛÚ à �sk à Û �BkÓ�QÞ ÛBkÊ�Q© ÚQÛrÞsk à � �BkÕã\ã �BkÊ©QÛÚ à �KkÕ¾\¾ �BkÓ�Tã ÛBkÕ�\� ÚQÛ\ÚBkÕ©\ã � �Ú à ¾BkÊ©B� �BkÓ�TÛ ÛBkÊÞQ� Ú\Þs�\kÊ� à � �Ú à Ûsk=��Ú �BkÓ�TÚ à kÕã\� Ú\ÞQ©Bka��© � �Ú à ãskÊ�\Û �BkÓ�QÞ ÛBkÊ�Q� Ú\Þ à kÕ�\� � �

7�RQf;WXA+�\ka�\¼ÛTºnf` ` Z�^Jwqa]b�cJdke�ofZBc�Z bya�^4cJdfeS¼ÜÇ~Ç�V�Ý Í V8Þ Íhß ÌHà Í Î�V ÝáV8Þ Í¶Í[â Ï�à Í cJ^�Z]i�l<YjcJYha�iqbya�^Re,Z_r�dq^�Z�l<cJe,^lAr+Z]iqsta_lJY#cAYja_i¹zWX:=P;A�<D8;H { <�U&LBJ|8�WX:=U&f�fB[D:XYQ8�>DACP;:=P;Y;��:XP;NB:XJ!Ri>@:XPBYt>D8_RT>~>@8;Aq`�RT[DAefpHQ>D8�H\cB>D:XJ!RTWXW=`�>@8;:=J|�vH\PN;:=<DJQk�798;At¦­©.½\�Z¶��f½\�\¨3WX:=P;A~:=<�A�ÖBcpACJq>@A6Nv>DHOfpA&UVHQ[DAm<DACPB<D:=>D:=gQAz>@HOJ|8;RQP;Y\A6<1:XP H\c;RQJC:a>y`}RT<:=>�8_RQ<oROY\[MA!RT>DAC[ZHQ<DJC:=WXWXRT>@H\[�<M>D[DA6P;YQ>@8wRQPBN(<M>|Ri>@:X<h>@:=J!RQW { A6:XY\8r>mRQPBN >@8;A6[DA6FIHQ[DA�RQP(:=P;JC[MA!RQ<MACNH\cB>D:XJCRQW?NBACcB>D8dk�Ä1P�NB:X<DJORQP;N£RT>&>D8;AOW=:XU&fw>@8;:=<zWX:=P;AeA�ÖscjA6[D:XA6P;JCA6<�R f;:XYQY\AC[~[DACNBL;J6>D:XH\P�:=P:XPr>DACP;<M:=>y`+RQP;N�>@8BA^:=P�>DACP;<M:=>y`�[DRT>@:=Hm:=<bWXH { A6[�>@8_RQPV:=>D<GHQcB>@:=J!RQW=W=`m>@8;:=P+gTRQW=L;AQk À fjH�gQA�>D8;A^W=:XU&f>@8BA+NBACP;<M:=>y`(HQFG>@8BA+cBW]RQ<MUeReFµRQW=WX<ZHT�£RQP;Nv>@8;A�¢_LKÖBA6<O¦µRQP;Nw8;ACPBJCA�>@8;A�W=:XP;A�[DRT>@:=H\<|¨^<M8;H\L;W=N[DAq>@L;[MP�>DH->D8;AC:=[mHQcB>@:=J!RQW=W=`->@8;:=PwgTRQW=L;AC<6kV798;:=<zRTc;cjACRQ[D<o>@H-fpA�>D8;A+J!RQ<MAV8;A6[DAT��f;LB>mFIL;[h>@8;A6[<M>DL;NB`�:XP;NB:XJ!Ri>@AC<o>@8;RT>z<MJ!RT>M>@A6[DACN�W=:XY\8r>C��H\[mL;P;[MAC<MH\W=gQACN�Î_P;Ae<M>@[ML;J6>DL;[DA6<&fpA6`\HQP;N�>@8;AVWX:=U&fd�U+HsN;:aÎ_AC<+>@8;:=<O<D:=UVcBWXA}J6H\P;J6WXL;<M:XH\P𦵻�:X<MJ|8�f_RQJ|8;A6[C�o�Q�Q�B��¨qkã��<D:=P;Y�>@8;A(U+A6>D8;HKN�N;AC<MJC[M:XfpACNRQfpH�g\AZ>@8BAZH\cB>D:XJ!RTWjN;A6cB>@8B<9:XPV>D8;A+¦­©.½\��¶��f½\�\¨bWX:=P;A { AC[MA1A6gTRQW=L_RT>@A6NSRT>9ACRQJ|8S<DJCRQPtcjHQ<D:=>D:XH\PRQP;N&RQ[DA?<D8;H { P�:XPm>|RQf;W=A^�Qk=�\k�lyP��E[DHKH\��<�Aq>�RQW­ks¦§�Q�\�Q�r¨/>D8;A�H\cs>@:XJCRQW�N;ACcs>@8;<�RT[DAEA�ÖK>@[@RTcjH\WXRT>@A6N>@H�J6W]RQ<M<D:aFÅ`V>D8;AZ>@[DRQP;<D:a>@:=H\P;<�:=PSN;:=�jAC[MACPr>^�*£"£/WX:XPBAC<�RTP;N-R&<D:XU+:XWXRQ[G<M>@LBNB`S:X<�RQWX<MH&cjA6[MFIH\[MUVA6NH\PtHQ>D8;AC[3J!RT[DfpH\P�UzL;W=>D:Xc;W=A6>@<6k798;:=<~>@A6J|8;P;:=�KLBASHQF9N;Aq>@AC[MUV:=P;:XPBY->D8;AOH\cB>D:XJCRQWbN;ACcB>D8;<&:=P�RQW=Wb>D8;AOW=:XP;A6<~HQF�H\P;AO:=H\P�FI[MH\UÆR<D:=P;Y\W=A&f;[@RTP;J|8;:XPBYSWX:XPBA&[@RT>D:XHO8;RQ<1c;[DH�gK:XNBACNvL;< { :=>D8vRtcpH { A6[MFIL;W�J|8;RQ[@RQJq>@A6[D:X<DRT>@:=H\P HQF?WX:=P;AC<{ 8;:=J|8VRT[DA9H\cB>D:XJCRQWXWa`m>D8;:XP��rRQP;N�8;A6P;JCA3<DLB:=>|RTf;WXAEFIHQ[ Ò :=�jAC[MACPr>@:XRQWB¥�U+:X<M<D:XHQP+ì�A!RQ<ML;[DA~¦ Ò ¥?ì}¨RQP_RTW=`K<D:=<Ck À W=>DAC[DP;RT>@:ag\ACWa`\�TAC<DJCRQcpA�cB[DH\f_RTf;:XW=:=>@:=AC<ÇJ!RTP+fpA�L;<MACN�>@HoRQN�ßML;<M>�HQf;<DA6[Mg\A6NV:=Pr>@ACPB<D:=>D:XA6<HQF�H\cB>D:XJ!RTWXW=`�>D8;:XJ|�OW=:XP;A6<C�_RTWXWXH { :=P;Yz>@8BACU�>DHVfpA1L;<DA6N}:=P-<ML;J|8 R�<h>@L;NB`Qk?798�L;<�:=>3:=<9cjHQ<D<D:=f;WXA

Page 55: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾;�

>@HoN;:XRQY\P;HQ<DA9H\cB>D:XJCRQW;N;A6cB>@8;<b:=PVR1U+HsNBAC[@Ri>@ACWa`zHQcB>@:=J!RQW=W=`o>@8;:=J|�&c;WXRQ<DUVR�FI[DH\Uîf;[@RTP;J|8;:XPBYmW=:XP;A[@Ri>@:XH�HQf;<DA6[MgTRT>@:=H\P;<3L;<M:XP;Y�AC<MJ!RQcpAmc;[DH\f;RQf;:XW=:=>y`�>@A6J|8;P;:=�KLBAC<Ck

úGû�­9ûyü Ü>�åä ¯T-��� ���5��2�áþ¯�2 $���ÇþG1 �R1 ���4 G�)����ÿ4��.-æ²(ç�8Zþ��+��� ���8è)�4 ���� ��*+��þ�13��EH\P;<M:XN;A6[�P;AqÖK>�>D8;ASJ!RQ<MASHQF�8r`sNB[DH\Y\A6P�HQc_RQJC:a>y`�:=P�>@8BAOFIL;<D:=H\P�>@H\�iRQUVRQ�vN;:ag\AC[h>@H\[�[MACY\:=H\Pdk798;A~N;:=gQAC[M>DH\[�¦I<DA6A~Î_Y\LB[DAe�Qk=���\¨3:=<^<M:=>@L;RT>@A6N fjA6WXH { R+PKLBWXWÇcjHQ:XPr>�:=P�>D8;A~>DH\�iRQUeRT�OcjHQWXH\:=N_RQWUVRQY\P;Aq>@:XJ�Î_A6WXN�<DL;J|8�>D8_RT>~c;W]RQ<MUeRt:XU+c;L;[M:=>D:XAC<1A6Pr>@AC[M:XP;Y�>@8;AV<DJC[DRQcpAq�yHT��WXR!`\AC[m¢_H { L;P;:=U��cpACN;A6N�>DH�>@8BA�>|RQ[MY\A6>�c;W]RT>DAC<�RTP;N�RQ[DA?Î_W=>DAC[DA6N�H\LB>�FI[DH\Uð>@8;AEUeRQ:=P&fpHsNB`mHQFÈ>D8;A�c;WXRQ<DUVRBk�798;Ac;WXRQ<DUVR�:XU+c;L;[D:a>@:=AC<9RQPBN}<DcpACPr>�FIL;ACWÇA�ÖBcpAC[M:XA6P;JCA~JCH\W=WX:X<M:XH\PB<�RQP;N�cjHQ<D<D:=f;W=`O[@RQN;:XRT>@:ag\A1J6HsH\W=:XP;YL;cpH\P}A6Pr>@[M`�:XPr>DH+>D8;AzN;:ag\A6[M>@HQ[^[MACN;L;J6:XP;YV>@8;A6:X[�ACP;A6[DY\:=AC<ZRTP;N}U+H\U+ACPr>|RBk9798�L;< { 8;ACP}>D8;A6`ACPBJCH\L;Pr>DAC[Z>@8BA�N;:=gQAC[M>DH\[^>@RQ[DYQA6>1c;WXRT>@A~<M>D[D:X�TA�cjHQ:XPr>@<�>@8;Aq`(RQ[MA&WXA6<D<ZAq��A6J6>@:ag\A�:=P <Dc;Ls>D>@A6[D:=P;YUVRT>@A6[D:]RTW�f_RTJ|�(:XPr>@HO>D8;A�c;W]RQ<MUeRVJCH\[MAQk�798;A�[DAqg\A6[D<DA+¢_H { :=P >@8;A�N;:ag\AC[h>@H\[oRT<D<D:=<M>D<m:=P(JCH\PK�<M>D[@RQ:=P;:XPBY~RTP�`+AyßMACJ6>DACNOUVRT>DAC[D:XRQWÈRQP;NV[DAq>@L;[MP;:XP;Ym:=>b>DH { RQ[MN;<?>D8;A�N;:ag\A6[M>@HQ[C�sRQWXW=H { :=P;Y1:=>b>DHzfpAAqÖK>D[@RQJq>@ACN-fr`O>@8BAmJ6[M`QH\c;L;U+cdkb798;AoN;:=gQAC[h>@H\[9[MACY\:=H\Pt:X<9R�WXH { >@A6UVcpAC[DRT>@L;[MA+¦Aü±�o���!� � �v¨q�8;:=Y\8 N;A6P;<D:a>y`u¦Aüá�&�£��� ³µ´ ¸�¹ º ´ ¨�c;WXRQ<DUVR { 8BAC[DA�>D8;A&ACW=ACJq>@[DHQP;<oRQP;N(:=H\P;<ZUVR!`}[MACJCHQU&f;:XPBARQP;NtR~8;:XY\8eJCH\P;J6ACPr>@[DRT>@:=H\PSHQFÇP;A6LB>@[DRQWX<ERQ[DA�c;[DA6<DACPr>CkG798BA1P;ACLB>D[@RQWjc;[DA6<DA6P;JCAo:X<?UeRT:XPr>|RQ:=P;ACNfr`e>D8;A1[DA6J6`KJCW=:XP;Y�HQF�8r`KN;[DH\YQACPtH\PS>@8;A { RQW=WX<�RTP;Ntf�`eN;:X[MACJq>3YrRQ<�c;LséVPBY+:=Pr>@H&>@8BA1N;:=gQAC[M>DH\[Ck�^PBN;AC[G>@8;A6<DAZJ6H\P;N;:a>@:XHQP;<ERTP;NOA6�jACJq>@:ag\AZc;WXRQ<DUVRoN;:XU+ACP;<M:XH\PB<C��H\P;A^UV:=Y\8r>?A�ÖscjA6J6>�>D8;A�W=H { AC<h>8r`KN;[DH\YQACPuÙd`sUVRQP�W=:XP;A6<z>@H(fpACJ6H\UVAeH\cB>@:=J!RQW=W=`(>D8;:XJ|�È�GRQW=>D8;H\L;YQ8�>@8;At�ERQWXU+AC[~WX:XPBAC< { H\L;W=NJCA6[M>@RQ:XP;Wa`S[DA6UeRT:XPS>@8;:=PdkEÄ1c;RQJC:a>y`S:XPt>@8;AêéLë+WX:XPBA18_RQ<�fjA6ACP}HQf;<DA6[Mg\A6N :XP À � Ò ¥�Í��yL;cBY\[@RQNBA¦­�?AC8;[M:XP;Y\A6[C�d�!Ú\ÚrÞ\¨ERTP;N�:=PO>@8;A~¤\¥�7 NB:=g\A6[M>DH\[m¦­ÙÇRQJ|8;:=Pd�d��ÚQÚrÞ\¨qk�lyP�¤\¥�7 :a> { RT<3FIHQL;P;NS>@8_Ri>>@8BAC[DA { RT<&<D:=Y\P;:=Î;J!RQPr>~WX:XPBAVRTf;<DH\[McB>@:=H\P�:=PÄé � f;Ls>z>@8;RT>z:a>mN;:XN�P;HQ>�Ri��A6J6>z>D8;Ae:=H\P;:=<@RT>D:XH\Pf_RQWXRQP;J6A+HQ[1cjH { AC[mWXH\<M<Ck�Ì�RQPwA6>~RQW9¦h��Ú\Ú à ¨�L;<DA6N�RQPwAC<MJ!RQcpA�FµRQJ6>DH\[mRQc;c;[MHrRQJ|8v>@H-c;[DA6N;:XJq>>@8BA�A6�jACJ6>D<oHQFÛé � RQfB<DH\[McB>@:=H\P(:XP(c;[MHißMACJq>@A6N�ly73¥bбJ6H\P;N;:a>@:XHQP;<C��Î;P;N;:XPBYO>@8_RT>oHQc_RQJC:a>y` { :=WXW8_R!g\Ao>@H�fpAo:XP;J6WXL;N;A6Nt:XPt:XH\PB:X<@Ri>@:XHQPOf_RQWXRQP;JCAoRTP;N�cpH { A6[3WXH\<M<�JCRQWXJ6L;W]Ri>@:XHQP;<Ck798;A�H\cB>D:XJCRQWXWa`�>D8;:XPvf;[DRQP;J|8;:=P;YtWX:XPBA&[@RT>D:XHSHQFSéLë¹½B� � FI[DHQUäAC��L_RT>D:XH\P�¦×�\kÕ¾\Ûr¨�:X<o�BkÊÞQÚBk�Ä1fK�<DA6[MgTRT>D:XH\P;<~>|RQ�TACP£FI[DH\UØ�EAC8B[D:XPBY\AC[t¦×��Ú\Ú\ã\¨mH\P�>D8;A À � Ò ¥4Í��yL;c;YQ[@RQN;ASU+A!RT<DL;[MACN�:=>z>@H(fpARQfpH\LB>1¾BkÊ�B�/:=P;N;:=J!RT>D:XP;Y2é¶ëOH\c_RQJ6:=>y`\k À PvAC<MJ!RQcpA�c;[DHQf_RQf;:=WX:=>y`�RQPBNvRQf;<DHQ[DcB>D:XH\P FµRQJq>@H\[1J6HsNBA8_RQ<�fpACA6P�N;A6gQACWXHQcjA6N+fr`&�EAC8;[M:XP;YQAC[�RQPBN�>D[@RQPB<DW]Ri>@ACNz:XPr>@HZ>D8;A ÀZÒ3À �zJ6HsN;AT� ÀZÒ3À ���s�C¾;k��sA6A

Page 56: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾��

»�:=Y\L;[DA~�\ka���B¼¶Tºr�dfe,` Z]cJY[r2a�b�ìK´\¾cAa bBZ]`TZ]b»ofYPQ_e+^JcJa�^³^Je,\�Yha�i�lJdfaa`�Yhif\�cJdfeRcAZ�^J\_e�c�sfghZ]cJeplZ�iko�r+^Jw_a�sfnk`)s¶z>@8BA&JCHKN;A&RTP;N}>@8BACH\[h`}N;A6<DJ6[D:Xcs>@:XHQP(RQ<�YQ:=g\A6P(:XP}�EAC8;[M:XP;YQAC[�¦×��Ú\Ú\ã\¨�RQPBN}>@8;AzN;AC<MJC[D:=cB>@:=H\P :=PJ|8_RQcs>@AC[Z�sk9798;AmFIH\WXW=H { :=P;Y�AqÖBRQU+c;WXAm:X<9>@RQ�QA6P FI[DH\Uá>D8;Az�?AC8;[M:XP;Y\A6[CËÏ<�[MACcpH\[M>�RQP;N�c;[DH�gK:XN;A6<ReL;<MA6FIL;W�:XW=WXL;<h>@[@Ri>@:XHQP-HTF�>@8;A~Y\ACP;A6[D:=JS¦µ:­kÕATkj<MJ!RQWXRQf;W=A�¨3RQc;c;W=:XJ!Ri>@:XHQP�HQFGRTf;<DH\[McB>@:=H\P�FµRTJ6>@HQ[D<^>DHRQJ6JCH\L;Pr>�FIH\[9H\c_RTJC:=>y`OUVHKN;:=Î;J!RT>D:XH\P;<?>@HVRT>@H\U+:XJ�cjH\cBL;W]RT>D:XH\PS<h>@[ML;J6>DL;[DATkE798;AoJCHKN;A { RT<^[ML;PFIH\[^8�`KN;[MH\Y\A6P(ACU+:X<D<M:XH\P�:XP->@8;A~N;:=gQAC[M>DH\[�[MACY\:=H\P { :a>@8�>D8;AzFIH\W=WXH { :XPBY+HQcB>@:=H\P;<C¼3R+g\H\:=YQ>�W=:XP;Ac;[MHQÎ_WXA { RT<4J|8;H\<MACP { :=>@8~>@8;A9Ù/H\[MACPr>@ÝE8_RQW=F { :XNB>D8&fjA6:XP;YZ�Bk=�C¾�>@:=UVA6<�>@8BA�N;H\c;cBWXAC[�8_RQWaF { :XNB>D8¦µ<MACAV�EAC8B[D:XPBY\AC[V¦h�!Ú\Ú\ãr¨�FIHQ[�ßMLB<M>@:aÎ_J!Ri>@:XHQP(HQFE>D8;:X<1J|8;HQ:XJCA!¨qk À YrRQ<Z>@A6UVcpAC[DRT>@LB[DA�HQF��Q©BÃ|�B�!�· RQP;N(RTP(ACWXA6J6>D[DH\P}>DACU+cjA6[@RT>DL;[DA~HQF3�\�\Ã@Û\� à · { AC[MA&<DA6WXACJq>@A6Ndk À J6`KW=:XP;N;[M:XJCRQWÇc;W]RQ<MUeR�Y\A��H\U+A6>D[M` { RQ<3J|8;H\<MACP { :a>@8-RQP�RQ<DcpACJq>^[DRT>@:=H�HQF à kG798BA~Aq��A6J6>@:ag\AmWXA6P;YQ>@8tHQF�>@8BA~c;WXRQ<DUVR { RQ<J|8;H\<MACPS>@HzfjAoÞKk à J6U�RTP;NO>@8;A^JCH\W=L;U+PON;ACPB<D:=>y` ¦ C W=ACP;YQ>D8���cBW]RQ<MUeRmN;ACP;<M:=>y`B¨GgTRQ[M:XACNVFI[DH\U�O�²����³ ��¸�¹}º � >@H�� � �O�²���K³ # ¸q¹}º � k À c_RQ[DRQfpH\WX:=J&N;ACJ6[DACRQ<DAO:=Pw>D8;A+L;c;cpAC[zW=A6gQACWbN;ACPB<D:=>y`N;:=<M>@[M:Xf;Ls>@:XHQP { RQ<~<DACW=ACJq>@ACN�RQP;Nw>@8;AVWX:XPBA�[@RT>D:XH\<oc;[MACN;:=J6>@A6NdkO798;AV[DA6<DL;Wa>@<mHQF�>@8BAVJ6HsN;AVRQ[MA<D8BH { P�:XPSÎ_Y\LB[DAC<o�\ka�\�mRQP;Nw�\ka���sklyP�Î;Y\L;[DA��\ka�\�o>@8;A~RQf;<MH\[Dcs>@:XHQPtFµRTJ6>@HQ[D<3A6gTRQW=L_RT>@A6N}Ri>�R�JCHQWXL;U+PtN;ACP;<M:=>y`-HQF?�z���!� ³ � ¸�¹ º �RQ[MA+<M8;H { Pdk+798;A+`r�yRiÖs:X<oY\:=gQAC<1>@8BAVRTf;<DH\[McB>@:=H\PvFµRQJ6>DH\[ZgTRQWXL;A { 8B:XWXA&>D8;A&Ö��×RiÖs:X<o<M8;H { <~>@8BAH\cB>D:XJCRQWÇN;ACcB>D8 J!RTWXJCLBW]RT>DACN�FIHQ[�>D8;A~gTRQ[M:XH\L;<3>@[DRQP;<M:=>@:=H\P;<6k Ñ HQ>DA~>@8;RT>^>D8;A�RQfB<DH\[McB>@:=H\P-FµRTJ6>@HQ[

Page 57: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾r©

»�:=Y\L;[DA3�\ka�\�\¼�R4vklJa�^AsºcJYha�iub�Z_r¤cAa�^¶^Ae,lJnfgjcAl¶bya�^¯dmwºoº^Aa�\_e+iuby^Aa�`(íe+df^AYjik\�e+^:Ê|Xhî î ïÐ.^Ae+sf^Aaºoºnkr�epounklJYjif\åR2WCRUTVYX[Zkz>\�dfe4Yhikoºe�M i§nf`�v�e,^Al³g[Z]vte+ghgjYhif\XcAdfetQ_e+^JcJY[r+Z]g�gjYhifepl³r+a�^A^Jepl<sta�iko�cAa�cJdke2sf^AYji�r�YhskZ]gBð_n�Z]imcJnf`GlJdfe+ghgta]bÏcJdkenfsfste+^RZ]iko:gjaa`e+^�l<cAZ]cJe,l,z

Page 58: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾\¾

»�:XY\LB[DA��\ka���s¼�¾RskZ�r+Y#cxw�`)aºoºYPOkepo:ghYhifeu^AZ]cJYha)^Ae,lJnfgjcAlby^Aa�`�RRWCRåT±V3X�Z�bya�^4oºY�Q�e,^<cAa�^�r�a_ikoºYjcJYha�ikl,z

Page 59: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾ à

JCLB[Mg\Am:X<3[MACc;[MAC<DA6Pr>@ACN}f�`S>D8;Am<DH\W=:XNSWX:=P;AoRQP;Nt>@8BAmA6<DJCRQcjAmc;[MH\f_RQf;:=WX:a>y`eJ6L;[MgQA~fr`S>@8BA~N;HT>D>@A6NWX:=P;AQk À gQAC[M>D:XJCRQWEW=:XP;A+:X<~N;[@R { P�FIH\[m>@8;A+Î_[D<h>&FIA { Ùd`KUeRQP�RQP;N£��RQW=UVA6[m>D[@RQPB<D:=>D:XH\PB<C�4RTP;N{ 8;A6[DA}>D8;AC<MA WX:=P;AC<�JCLs>O>@8;A}RQf;<MH\[DcB>D:XH\P£FµRQJq>@H\[VJCL;[hg\A H\PBA}J!RTP�[DACRQN²HQ�x>@8BA RQf;<MH\[Dcs>@:XHQPFµRQJq>@H\[GFIH\[G>D8_RT>?>@[@RTP;<D:a>@:XHQPdk À WXW_>@8;AZ��RQWXU+AC[bW=:XP;A6<?8_R!gQA1P;ACY\W=:XY\:=f;WXA3RTf;<DH\[McB>@:=H\PVFµRTJ6>@HQ[D<�RTP;NAC<MJ!RQcpAmc;[DH\f;RQf;:XW=:=>D:XAC<6� { 8B:XWXA^>@8;AmÙd`KUeRQP-WX:=P;AC<�RQ[DAoU+HsN;A6[@RT>DACWa`V>D8;:XJ|�Èk»�:=Y\L;[DAo�\ka���o<D8;H { <E>D8;A�c;[MACN;:=J6>DACNO:=P�>DACP;<M:=>y`�W=:XP;A3[@Ri>@:XHmHQFCéLë.½B� � RQ<?RmFIL;PBJ6>@:=H\PeHTF/JCH\W=L;U+PN;A6P;<D:a>y`\kbÄZcB>D:XJ!RTWXW=`1>D8;:XJ|�~cjHQc;L;W]Ri>@:XHQPzN;A6P;<D:a>@:XA6<�RQ[MA�L;<DA6NV:=Pz>@8B:X<4JCRQWXJ6L;W]RT>D:XH\Pmfr`�U+HsN;:aFÅ`K:XP;Y>@8BA1<DcpH\Pr>|RQP;A6H\L;<9A6UV:=<D<D:=H\POJ6HsAqéVJ6:XA6P�>D<3:XPe>D8;A1JCH\W=WX:=<D:XHQP_RQWa�n[@RQNB:]RT>D:agQA�<DL;fB[DH\LB>D:XP;A1HTF À�Ò�À ��s�!¾}f�`�>@8;ASRTc;c;[DHQc;[D:XRT>@ASRTf;<DH\[McB>@:=H\P�FµRQJq>@H\[M<Ckv798;AtAC<MJ!RQcB:XP;Y ¢_LKÖ�FIH\[&ACRQJ|8£WX:=P;Ae:=<~>@8BACPU+HsN;:aÎ_ACNSfr`O>D8_RT>�WX:=P;AQËÏ<�A6<DJ!RTcjAmc;[DHQf_RQf;:=WX:=>y`VRQPBN->D8;AoH\cB>@:=J!RQW=W=`�>@8B:XJ|�eW=:XP;A�[@RT>D:XH&FIH\L;PBNdkbln>J!RTP-fpAo<DA6ACPt>@8_RT>9>D8;AoJCH\W=L;U+PSN;ACP;<M:=>y`SJ6H\[D[MAC<McjH\PBN;:XP;Y�>DH&>@8;AoH\f;<MAC[hg\ACNÜéLë¹½B� � WX:=P;A�[DRT>@:=HHQF�¾;kÕ�&:X<3RQc;cB[DH!Ös:XUVRT>@A6W=`OÞ �Óà �����K³ �^¸q¹}º � � { 8;:XJ|8-:XU+c;WX:=AC<?R�P;ACLB>D[@RQW/8r`KN;[MH\Y\ACP-N;A6P;<D:a>y`SHQF� � �������K³ ��¸q¹}º ´��Bf;[DHrRTN;W=`eJCH\P;<M:X<h>@ACPr> { :=>D8-NB:=g\A6[M>DH\[3JCH\P;NB:=>@:=H\P;<6k�^<M:XP;YZRQf;<DHQ[DcB>D:XH\PzFµRTJ6>@HQ[�>@A6J|8;P;:=�KLBAC<b<DL;J|8+RQ<�>@8;A�HQP;AEH\Ls>@WX:=P;ACNz8;AC[MAQ�rH\c_RTJC:=>y`~RQN�ßML;<M>DUVA6P�>D<>@H�U+HsNBAC[@Ri>@ACWa`OH\cs>@:XJCRQWXWa`�>@8;:=J|�Oc;W]RT<DUVR&cjH\cBL;W]RT>D:XH\PeN;ACP;<M:=>D:XAC<3JCRQP�fpA1FIH\L;P;N�k

úGû�­9û�­ ñ��+5Û¯ ¯ -��� ���5��2� þ¯�2 $���Çþ 1 �21 ���4 °������ÿ4��.-ê² ò��)*��X 0�� ���1 �óµ���2�Zÿ��±Ü>-:1 ��� *+�,���ÇþÈô:Ü>-:1 ����ÿ���� ��+��8�8�1 �����öõK��*+13�4 K*+�,��� ��X���+�}����)��scpACJq>@[@RTW�H\f;<DA6[MgTRT>D:XH\P;<oHQFE<DH\WXRQ[1c;[MH\UV:=P;ACPBJCAC<o<M8;H { >@8_Ri>1>@8;Aq`wRQ[DA�H\cB>D:XJCRQWXWa`�>D8;:XJ|� :=PwRQWXWÙd`KUeRQP�WX:XPBAC<o>@8;[MH\L;Y\8�>DH�>D8;ASÙd`KUeRQP�JCH\Pr>D:XP�L;L;U-k »�:=Y\L;[MA��\ka��©-Y\:=gQAC<&RTP÷� F :XUVRQY\A�HQFRvc;[DH\U+:XPBACP;J6AtRQP;N£Î_Y\L;[MA��\ka�!¾wRQPuAqÖBRQU+c;WXASHTF1RQP£H\cB>D:XJCRQWXWa`�>D8;:XJ|��Ùd`KUVRQP�<MAC[M:XAC<�W=:XU+:=><DcpACJq>@[ML;U�k�sH\WXRQ[4c;[MH\U+:XP;A6P;JCA6<bRQ[DA9WXRQ[DY\AT�\N;A6P;<DAT�KJ6HsHQW;<M>D[DL;Jq>@L;[MAC<b:XP&>D8;A3<DH\WXRQ[�RT>DUVH\<Mc;8;A6[DAQ�r:=UVU+AC[M<DA6N:XP�Rtf_RQJ|��Y\[DHQL;P;N�[DRQN;:]Ri>@:XHQPvÎ;ACWXNvFI[DHQUä>@8;A+<DH\WXRQ[1c;8;HT>@H\<Mc;8;AC[MASRQP;NwJ|8;[MH\UVHQ<Dc;8;A6[DATktÐ�A��gK:XA { <ZJ!RQP}fjAmFIH\L;P;N :=P ½�:X[@R!`\RQUeR}¦h��ÚQã à ¨q�¸Á�:=[D�QA6[�¦×��Ú\ã\ãr¨��pÜb[MAC:=<M>�¦×��Ú\ã\Ú\¨�RQPBN 7�RTP;N;fpAC[DYi�½�RTP;<D<MACPí¦×��Ú\Ú à ¨qkí�KACA(>|RQfBWXA��\kÊ��FIH\[O>y`Kc;:XJCRQWZc;8r`K<D:=J!RQWZgTRQW=L;AC<6k 798;Aq`xRQ[DA(<h>|RQf;W=A(H\PxR>@:=UVA6<DJCRQWXA^HQF { A6AC��<3L;Pr>@:XWp>@8;Aq`OHQFÅ>DACPSA6[DL;cB>9FI[MH\U >@8;AZ<MH\W]RT[9RT>@U+H\<Mc;8;AC[MAQk�798;Aq`SRQ[MA1<DH\U+Aq�>@:=UVA6<eRT<D<DHKJC:XRT>@A6N { :a>@8��EHQ[DH\P_RTW�ì�RQ<D<S¥jßMA6J6>@:=H\P;<}¦§�9ì}¥9ËÏ<|¨���>D8;H\L;Y\8u>@8;A�P_RT>DL;[DA�HQFZ>@8BARQ<M<DHKJC:]Ri>@:XHQP(:X<1L;P;J6WXACRQ[Ck�798;A6[DA+:X<1U&L;J|8w:XPr>DAC[DA6<M>m:XPv<MH\W]RQ[1cB[DH\U+:XP;A6P;JCA6<oRT>o>@8;A�U+H\U+ACPr>!�

Page 60: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾rÛ

»�:=Y\L;[DA��\ka��©B¼�Tºa�g[Z]^4sf^Aa�` Yhife+i�r�eløÙù¬a�vklJe+^óQBZBcAYja_i�c�Z]b�e,i�v§w�cAdfe�ú<Z]ikY#cAa�^2a�b�cJdfe�íYh\Wíe,Z�^åT§a�g[Z]^2a�vklJe+^�ûQBZBcAa�^AwTYji�X[îfüaýfz>þ4a�cJe}cJdkeUQBZ�^JYha�nkl�r�dkZ�^AZ_r¤cAe+^AYhl<cJY[r4sf^Aa�` Yhife+i�r�eXr�a_`)sta�ike+imcAl[c§YÉz e�zfZ)g[Z]^A\�e}v�aºoºw»a�b¯sfg[Z�lJ`TZlJnkl<ste+i�oºe,o�Yji�cAdfe r�a�^Aa�i�Z�`�Y#cAd¥ZTbye[` bya§a]cJsta�YhimcAlR^Ja§a�cJe,o�Yji�cAdfe)sfdfa]cAa_lJsfdfe,^Je_zL¾Ri�r+gjaml<e�YhiklJs�epr¤cAYja_i�cJdkeQ�e,^<cAYhr,Z]gBOkghghZ�` e+imcAZ�^Jw»YhimcJe,^JikZ�g¹lxcA^Jn�r¤cJnk^Jeua]b¯cJdfeXsk^Ja_`)Yhife,ikr�eXr+Z�i�v�e�lJe+e,i¹z

»�:=Y\L;[DA��\ka�!¾;¼ ÿ¹w§`TZ]i l<e,^JYhe,l�gjYh` Y#c�a�vklJe+^óQBZBcJYha�iIa]bTZ%lJa�g[Z]^�sf^Aa�` Yhife+i�r�e cuc�Z]b_e+iIby^Ja_`3cAdfe�ofZ]cAZ�YhiTºr�df` Yhe,oºe,^»e�c�Z�gÉzåÊ|X[î�î î�vËФz þ2a]cAe�cJdfe�` e,^J\_Yjif\{a�b}cAdfe�dfYh\�d%lJe+^AYjepl gjYhifeplTYjimcAa{cJdfe�r+a�imcJYhi§nfnf`�z(¼³a_`SûskZ�^Je2cJdfeRdfe+Yh\�dmcAl³a]b¶cJdfeRghZ_lxc³bye�`%a_vkl<e,^�QBZ�vfgje2gjYhife,l:`�YjcJdTcAdfe}\_Z]Yhi»Yhi�ghYjike2dfe,Yj\_d_c�l³Z_l a_ife2sk^Ja_\�^Ae,lAl<epl$oºaa`�icJdke:lJe+^AYjepl+zÙ\�dfe:l<`TZ]ghg Yhikr+^JepZ�lJe»Yji�oºYhr,ZBcAe,lXcAdkZBc�cJdfe:gjYhifepl�Z]^Ae»ifa�c�^Ae d epr¤cJYhif\�a_sºcJY[r+Z�gjghw¥cJdfYhi¬r+a�ikoºYjcJYha�i�l`�YjcJd±ía�gjc � `TZ�ifi:nksfs�e,^4ghe�Q�e,g¹oºe+i�l<YjcJYhe,l[cfvfnºc2Z�^Jeuvte+Yhif\Tl<Yh\�ikYjO�r,Z]imcJghw»^Ae,Z�vkl<a_^Jvte,o¹z

Page 61: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾�Þ

� L;RQPr>@:=>y` >y`Kc;:XJCRQWdgTRQWXLBAC<¥�WXA6J6>@[MH\PtN;ACPB<D:=>y` à �����K³ P ¶��m������³y³ ¸q¹}º ´¥�W=ACJ6>D[DH\Ptc;[MAC<M<DL;[MA � � � ?�� � ¸�¹ º �7�ACU+cjA6[@RT>DL;[DA ÞsÃ@�\�\�\�ëoRQ<3c;[DA6<D<DLB[DA ©1¶�Û ?�� � ¸�¹}º �½3`KN;[DH\YQACP}NBACP;<M:=>y` ©������K³y³ ¸q¹}º ´

�s:=ÝCA ���\�BÃ@�\�\�Z¶�Û\�Q�BÃ|�\�\�\® ¹ WXH\P;Yà Ã|�\�\�Z¶����sÃ|�\�\�r® ¹ >D8;:XJ|�

à �BÃ|�\�Q�r® ¹ 8;:XYQ8lyH\PB:X<@Ri>@:XHQPSN;ACY\[MACAoHQF�8r`sNB[DH\Y\A6P �BkÊ�\Þ

7�RQf;W=A+�QkÓ�s¼lT§nf` `TZ]^AwTcAZ�vfgjeua�b¯cxw§sfY[r+Z�gÏsf^Ja_` Yjife,ikr�eusfd§wºlJYhr,Z]gÏsf^Aa�ste+^JcJYhe,l,z:XPtc_RT[M>@:=JCL;WXRQ[E:=PtAqÖsc;W]RT:XP;:=P;Yz>@8BAC:X[9<ML;c;cpH\[M>3U+ACJ|8;RQP;:X<MUá:XPS>D8;A1<DH\WXRQ[3RT>DUVH\<Mc;8;A6[DAoRQP;Nt>@8BA>@[M:XY\YQAC[/FIH\[Ç>@8;A6:X[�RQf;[ML;cB>4A6[DL;cs>@:XHQPd�Q<MACA9�KJ|8;UV:=ACN;A6[�Aq>�RQW­kK¦h��ÚQÚ\ãr¨q�Q�sJ|8;U+:XA6N;AC[�A6>4RQW§kK¦×��Ú\Ú\ÚQRr¨q��sJ|8;U+:XA6N;AC[3Aq>�RTW§k/¦×��Ú\Ú\ÚTfȨq�È�KJ|8;UV:=ACN;A6[3A6>�RQW­kǦ§�Q�Q�\�r¨q�_�K>@A6WXWXUVRQJ|8;A6[9âÌ :=AC8;[~¦×��Ú\ÚQ¾�¨�k798;A�H\cs>@:XJCRQWÇ>@8;:=J|�KPBAC<D<oHQF�>D8;A�Ùd`KUeRQP}<DAC[M:XA6<1:X<�ACRQ<D:=W=`-<DA6ACPvFI[DH\UæRQP(A6<M>D:XUVRT>@A~HQFGc;8;HQ>DH\PU+A!RQP}FI[MACA&c;RT>@8;<�¦µY\:ag\ACP}f�`�>@8;Az[DACJ6:Xc;[MHsJCRQW�HQF�>@8;A&RQf;<DHQ[DcB>D:XH\P J6HsAqé+JC:XA6Pr>�¨q�jJCH\U+c_RQ[MAz>@8BAU+A!RQPtFI[DA6A~c_Ri>@8;<^HQF�>@RQf;WXA��\kÕ© { :=>@8->y`Kc;:XJCRQWÇc;[DH\U+:XPBACP;J6AmNB:XU+ACP;<M:XH\P;<6kE798;AzÙd`sUVRQP-<DA6[D:XA6<c;8;HT>@H\P;<3RT[DA18;ACR�gK:=W=`eRQf;<DHQ[DfpACNd� { :a>@8tR�U+A!RQPOFI[MACA1c_RT>D8SHQF�WXA6<D<�>@8;RQP-R&JCACPr>D:XU+A6>@A6[C� { 8B:XWXA>@8BA&��RQW=UVA6[3<DAC[M:XA6<�c;8;HT>@H\P;<�8_R!g\A&U+A!RQP-FI[DA6Azc_RT>D8;<�HTF4>@8;A~<@RQU+AzHQ[DN;A6[�RQ<�>@8;A~N;:XU+ACPB<D:XHQPHQF�>@8;Atc;[MH\U+:XP;A6P;JCATk À JCH\P;<MAC��L;A6P;JCA�HQF^>D8;:X<�8BA!R!g�`uRQf;<DHQ[DcB>D:XH\P£:X<~>@8;AtL;P�L;<ML_RQW3ACP;A6[DYQ`f_RQWXRQP;J6A3FIHQ[?c;[MH\U+:XP;A6P;JCA6<C�rÎ_[D<h>EFIH\LBP;Nefr`+Ü�HQW]RQP;NeRQP;N À P;Ý6AC[Z¦h��ÚrÞK��¨qk�ln>E:=<GW=:X�TACW=`z>@8_RT>G>@8BAUVRQ:XP(A6P;AC[MYQ`(YrRQ:=P(:X<1N;LBA�>DH-Ù/`KUVRQPvJCH\Pr>D:XP�L;L;U�RTf;<DH\[McB>@:=H\Pd�ÇRQPBNv>D8_RT>1>D8;A�UeRQ:=P WXHQ<D<DA6<RQ[MAZN;L;A�>DH�½qFwACU+:X<D<M:XH\P�k�Ü�8;HQ>@Hi�×RQf;<MH\[Dcs>@:XHQPOHQF�Ùd`KUeRQPSW=:XP;A6<E:=<?>@8BAZN;H\U+:XP_RTP�>Ec;[DHKJCA6<D<9:=PcpH\c;L;WXRT>@:=P;Y9>@8;AGAqÖsJC:a>@A6Nz<M>@RT>@A6<�HQFs>@8;A?c;[DHQUV:=P;ACP;J6Ab8�`KN;[MH\Y\A6P�RT>@HQUV<6k4�s:=UV:=W]RQ[MW=`^RQf;<MH\[Dcs>@:XHQPHQF4>D8;A&Ùd`KUVRQP}J6H\Pr>@:XP�L;LBUæNB[D:=gQAC<�>@8;Am:XH\PB:X<@Ri>@:XHQP-f;RQW]RQPBJCAmHQF�fpHQ>@8�8�`KN;[MH\Y\A6P(RQP;N 8BACWX:=L;URT>DH\UV<9:=PS>@8;Aoc;WXRQ<DUVRBk798;A?c;[DA6<DACPBJCA�HQF;H\c_RQJ6:=>y`1:=P&RQW=W�Ù/`KUVRQPmWX:XPBAC</>@8;[MH\L;Y\8m>@H�RQP;Nz:=P;JCW=L;N;:=P;Y9>@8;A?JCH\Pr>@:=P�L;L;U«WXA6N>@H�>@8BA~N;Aqg\A6WXH\c;U+ACPr>^HQF�>D8;A~RQP_RQWa`K>D:XJCRQWXWa`VJ6H\Pr>@:=PKLBH\L;<�AC<DJCRQcpAzc;[MH\f_RQf;:=WX:a>y`ORQPBN RQf;<MH\[Dcs>@:XHQPFµRQJq>@H\[oAqÖsc;[MAC<M<D:XHQP;<&<D8BH { P�:XPwJ|8_RQcB>DAC[&©ske798;A6<DAOA�ÖBcB[DAC<M<D:=H\P;<&RTWXWXH { H\c_RQJ6:=>y` >DH}fpAVN;ACRQW=>{ :=>D8�RQ<�<DA6[D:=AC<�WX:=UV:a>�WX:XPBAC<4U+AC[MY\AE>@HQY\A6>D8;AC[bRQPBN�f;WXA6P;N�:XPr>@HZ>D8;A9JCH\Pr>D:XP�L;L;U-�Q>@8�L;<bc;[MH�gs:=N;:XPBY

Page 62: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾rã

7�[DRQP;<M:=>@:=H\P UVACRQPSFI[DACAoc_Ri>@8�¦µUe¨���@� � � ¾B�~����� º � ¹©i�@� ã � ã\������� º �@¹¾T�@� � � � à ����� P ¹à �@� � �Êà ¾+����� P ¹©i�×� Ú � ¾�������� �@¹¾T�×� Û � ãQ¾+����� # ¹à �×� � � �Q¾+����� �D¹¾T�y© � �Êà Û������ # ¹à �y© � � �\Ú������ �D¹à �n¾ � � Ûr������� # ¹

7�RQf;W=A��\kÊ©s¼lTºnf` ` Z�^Jw»cAZ]vkgjeXa�b¶cxw§sfY[r+Z]gÏ` epZ]iqby^Je,euskZBcAdkl�a]b�d§w§of^Ja_\�e+i:sfdfa�cJa�i�l�Yji�Z l<a_ghZ�^�sf^Ja_` Yjife,ikr�e_zRoUVACRQP;<�HQFdRQP_RQWa`s<M:XP;Y1<McjA6J6>D[@R~<ML;J|8ORT<b>D8;A�H\P;A3<D8;H { PV:XP�Î_Y\L;[MAo�\k=�C¾;k�½^H { A6gQAC[C�K>@8;A3WXRQ[DY\AH\cB>D:XJCRQW9N;ACcs>@8;<OHTFZc;[MH\UV:=P;ACPBJCAC<+c;LB>D<V>D8;ACU :XPr>DH�Rw[DACYQ:XU+A { 8BAC[DA}R�[DRQN;:]Ri>@:=gQAS>@[@RTP;<MFIA6[RQc;cB[DHrRQJ|8u:X<�U+H\[MAtRQc;c;[MH\c;[D:XRT>@ATk�798;A6[DA�RQWX[MA!RQNB`�AqÖs:X<h>�WXH { <DAC[M:XA6<+U+ACUzfjA6[�c;[DH\U+:XPBACP;J6A[@RTN;:]RT>D:=gQA�>@[DRQP;<hFIAC[SJCHKN;A6<w¦­½^ACRQ<DW=A6`�â�ì�:=8_RQWXRQ<C�o��ÚrÞTÛ�RQP;NÔë1H\LB>D>DACf;[MH\ÝCA}A6>-RQW­k=�o��Ú\ÚQ©r¨qkln>38_RQ<3fpACA6P�N;A6UVHQP;<M>D[@RT>DACNt>@8_RT>�R�[DRQN;:]Ri>@:=gQA�>@[DRQP;<hFIAC[�RQc;c;[MHrRQJ|8 { 8;:=J|8}RQW=WXH { <EFIH\[9c;RQ[M>D:]RQWFI[DA6��L;ACP;Jq`�[DA6N;:X<h>@[M:Xf;LB>D:XH\PS¦­ÜbÐ Ò ¨�:X<4P;A6JCA6<D<@RT[M`&>DH1UeRT>DJ|8�>@8;A�HQf;<DA6[Mg\A6NeW=H { <MAC[D:=AC<�U+ACU&fpAC[WX:=P;A�c;[DHQÎ;WXAC<�RQP;NO>@Hzc;[DH\cpAC[MW=`VRQJ6JCH\L;Pr>9FIH\[?cjHQc;L;W]Ri>@:XHQPVU+HsNB:=Î_JCRT>@:=H\P ¦­ëoHQLB>D>DACf;[MH\ÝCAZAq>3RQW­k=���Ú\ÚQ©r¨qk�ln>ZUVR!`�fpA&cpH\<D<M:Xf;W=Am>DHOJCHQP;<M>D[DL;Jq>oRVgTRQWX:=N�8B:XY\8�<DAC[M:XA6<ZAC<MJ!RQcpAzFµRQJq>@H\[^JCHKN;AQ��RQ<�>@8BA8;:=Y\8;AC[GcpH\c;L;WXRT>@:=H\P;< { :XW=W_fjA�JCW=H\<DA3>@Hz�EH\Wa>@ÝCUVRQP;P�gTRQW=L;AC<G<DHz>D8;A�JCH\L;cBWX:XPBY~HQFj>@8;A^AC��L_RT>D:XH\PB<¦h�Qk=��¨�RQPBN(¦h�QkÊ© à ¨�:=<bP;HQ>?RQP+:X<D<ML;AQk À WX<MH;��c_RQ[M>D:]RQWsFI[DAC��L;A6P;J6`V[DA6N;:X<h>@[M:Xf;LB>D:XH\P�� { 8;:XJ|8VU&L;<M>?fpARQJ6JCH\L;Pr>DACN�FIH\[o:=Pw>D8;AVW=H { <DA6[D:=AC<mUVA6U&fjA6[D<6��fpACJCHQUVA6<~W=AC<D<m<D:=Y\P;:aÎ_J!RQPr>~RQ<oH\P;A+c;[MH\Y\[DA6<D<MAC<L;cuR(<DA6[D:XA6<+RQP;N£JCH\U+c;W=A6>@AOFI[MAC��L;A6P;J6`£[MACN;:=<M>@[M:Xf;Ls>@:XHQP ¦§��Ð Ò ¨zJ!RQPufpAtRQ<D<ML;UVA6Ndk�ln> { RQ<N;A6JC:XNBACNe>D8_RT>?P;H&N;:XRQY\P;H\<h>@:=J^RTNBgTRQPr>|RQY\A { H\L;WXNefjA^RQJ|8;:=A6g\A6NOFI[DH\UíNBA6g\A6WXH\cB:XP;Y&<ML;J|8SRzJ6HsNBARQP;N(>D8_RT>1>@8BA�JCHKN;A�HQFGJ|8_RQcB>DAC[m©S<D8;HQL;WXN(fpA�[DAq>|RQ:=P;ACN(FIH\[ZcBW]RQ<MUeRQ< { :=>@8vU+H\[DAzUVHKN;AC[DRT>@AH\cB>D:XJCRQWdN;A6cB>@8;<6k798;A�c;[DH\U+:XPBACP;J6A-HQf;<DA6[MgTRT>@:=H\P;<+[@RQ:=<DAtH\P;AtÎ_P_RTW�:=<D<ML;AQk�798_Ri>e:=<C�?>D8;AC[MA RQ[MA�cBW]RQ<MUeRQ<&FIHQ[{ 8;:=J|8²>@8;A�:XPB¢;L;ACP;J6A HQFoRwf_RQJ|��Y\[MH\L;P;N�[DRQN;:]Ri>@:XHQP£Î;ACWXN�H\P�H\LB[VA6<DJ!RTcjA(RTP;N�RQf;<MH\[Dcs>@:XHQPFµRQJq>@H\[�A�Ösc;[DA6<D<D:=H\P;<�<D8;HQL;WXNSfpA1:XP;J6WXL;NBACNdk Ñ H\[DUVRQWXWa`�>@8;A1f;RQJ|��Y\[DH\LBP;N�[DRQN;:XRT>@:=H\PVÎ_A6WXNO>DAC[DU

Page 63: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¾rÚ

:XPVAC��L_RT>D:XH\P(¦h�\kÕãr¨?RTP;Ne>@8BA�f_RQJ|��Y\[MH\L;P;Nt[DRQN;:]Ri>@:XHQP�Î;ACWXNeJCH\Pr>@[M:Xf;Ls>@:XHQPV>@H�·�¦J@;¨G:=PeA6�KL;RT>@:=H\P¦h�Qk=�!¾r¨^RQ[MAz<DAq>Z>@HeÝCAC[MH;k1ln>ZcB[DH�g\A6<ZFµRQ:X[MW=`t<M>D[@RQ:=Y\8r>DFIH\[ { RT[DN}>DHO:XP;J6WXL;NBAz<DLBJ|8�RVf_RQJ|��Y\[DHQL;P;NÎ_A6WXN :=P}>@8;Am>@8;A6H\[M`�N;Aqg\ACW=H\c;U+ACPr>ZFIH\[�>@8;AzAC<DJCRQcpA&c;[DHQf_RQf;:=WX:=>y`-RTP;N(RQf;<MH\[Dcs>@:XHQP}FµRQJ6>DH\[�A�Ö��c;[MAC<D<M:XH\PB<Ck�798B:X< { RQ<�N;H\P;AtRTP;N�>@8;AS:=P;:=>D:]RQW�<h>@ACcB<+:=P�N;A6gQACW=H\c;:XPBYvR JCH\[M[DAC<McjHQP;N;:XPBYwJ6HsNBA{ AC[DAmcpAC[MFIHQ[DU+ACNd�;<MACAm<DA6J6>@:=H\P��ska���Bk

úGû�­9û<À ñ��+5Û¯äþ��4 $���Çþ �)�Zþ¯�4 >èX�}������Zþ ²6ç�8Zþ����)� ���8è��2 ���� ���*���þ�1 �À Wa>@8;HQL;Y\8(<DA6[D:=AC<oWX:=UV:a>^cB[DH\U+:XP;A6P;JCA�HQf;<DA6[MgTRT>@:=H\P;<1J6H\L;WXN(PBHQ>ofjA�U+HKN;ACW=WXACN��j>@8BAC[DA�A�ÖB:=<M>DACNR~PBACACNO>DH&:XPr>DAC[DcB[DA6>?<D:=UV:=W]RQ[4FILB<D:XHQPeN;:ag\AC[h>@H\[?<DcpACJq>@[@Rsk�798;AC<MA�>y`KcjA6<9HQF/<DA6[D:=AC<EW=:XU+:=>�<McjA6J6>D[@R{ AC[DA3Î_[M<M>ERQP_RQWa`s<MACNVf�`�lyPBY\WX:=<�â 7�ACWXW=AC[^¦h�!Ú\©\Úr¨�� { 8;AC[MA�>D8;A^<MUVHKHQ>D8+U+AC[MY\:XPBY1HQFd8;:=Y\8�<DA6[D:XA6<WX:=P;AC<�:=Pr>@H >@8BA-J6H\Pr>@:=PKLBL;U { RQ<�<D8;H { P�RTP;N�>@8;AtcpH\:XPr>+RT> { 8;:=J|8u>D8;ASU+AC[DYQ:XP;Y(HKJCJ6L;[D[MACNL;<MACN�RT<~RSN;A6P;<D:a>y`(N;:XRQY\P;H\<h>@:=JQk�Ð3ACJCA6Pr>@W=` UVH\[MA�RQNBgTRQP;J6ACNwUVHKN;A6WX<18_R!g\A+fpACACPwN;Aqg\ACW=H\cpACNRQW=WXH { :XP;Y1<MAC[M:XAC<GWX:=UV:a>@<�>@HmfjA3:XPBJCWXLBN;ACN+:XP�<DHQW]RQ[GRT>DUVH\<Mc;8;A6[DA3JCHKN;AC<�¦ Ò RTc;cjA6Pd��A6>ERQW§ka�_��Ú\ã\Þ\¨q�:XPeRT>@HQUV:=J^JCRQWXJ6L;W]RT>D:XH\PB<bFIHQ[?>@8BAZl×Ð9Ä Ñ c;[MHißMACJq>m¦§�KA!RT>DH\Pd���!Ú\Ú\�r¨bRQP;NO>DHz:XPr>DAC[DcB[DA6>E>@H\�iRQUVRQ�N;:ag\AC[h>@H\[m<DcpACJq>@[DR�¦­Ü�:XY\RQ[DH�gvA6>&RQW§ka�G��Ú\Ú\ãr¨�kt»�:XYQL;[DAt�\ka� à <M8;H { <�RQP�AqÖBRQU+c;WXA+HQF3R���RQWXU+AC[<DA6[D:=AC<3WX:=UV:a>EH\fB<DAC[hg\A6N H\PS>@8;A~¤\¥�7·[MA!RQJq>@H\[6�B>|RQ�TACP-FI[DH\Uáì�AC:=Y\<3A6>^RQW§kd¦h��ÚQÚ\ãr¨qkÀ <�H\P;A&c;[MH\Y\[MAC<D<MAC<oL;cvRe<MAC[M:XAC<6��>@8BA�<DcpACJ6>D[@RQW�WX:=P;AC<^f;WXA6P;N >@H\YQA6>@8BAC[1RQPBN(RQ<D<M:XY\PBUVA6P�>ZHQFGRc;8;HT>@H\P >DH�ROc;RQ[M>D:XJCLBW]RQ[^L;c;cpAC[1WXAqg\A6W4fjA6JCH\U+AC<oRTU&f;:XYQL;H\L;<6km798BA�UVA6[DY\A6NwW=:XP;A6<�<MUVHKHQ>@8BW=`AqÖK>DACP;Nx:XPr>@H�>D8;A(JCH\Pr>D:XP�L;L;U-�ZRQPBNx:a>SfpACJCHQUVA6<SN;:=é+JCLBW=>O>DHuNBA6Î_P;AwR�<D8;RQ[Dc JCHQP�>D:XP�L;L;UACNBY\AQk�798;:=<oRQN;N;<oL;PBJCAC[h>|RQ:=Pr>y` >@HtACW=ACJq>@[DHQPv>DACU+cjA6[@RT>DL;[DA+RQPBNwNBACP;<M:=>y`(N;:XRQY\P;HQ<M>@:=JC< { 8B:XJ|8N;A6cjA6P;NvL;cjHQP >@8;A&J6H\Pr>@:=PKLBL;UªA6N;Y\A-¦µ<ML;J|8wRQ<�>D8;H\<DA�NBAC<DJ6[D:=fjA6Nw:=P J|8_RQcs>@AC[M<����SRQP;N��Q�&HQFë1[D:XA6U ¦h��ÚQÚrÞ\¨D¨�k À JCHKN;A { RQ<~N;A6gQACW=H\cjA6N�>@H�c;[DA6N;:XJq>�R}<h`sPr>D8;A6>D:XJO<McjA6J6>@[ML;U-�4[DAqFIAC[D[MACN�>DHRQ<eRuÉÕ<DcpACJ6>D[@RQW3FIA!Ri>@L;[MAQËa��HQF1<MLBé+JC:XA6Pr>S��L_RQWX:a>y`�>@8;RT>e>D8;A-ÎB>D>@:=P;Y�HTFZ>@8;:=<�FIA!RT>DL;[DA->@H�¤\¥�7N;:ag\AC[h>@H\[E<DcpACJ6>D[@R�JCHQL;WXNSfpAZL;<MACNt>@H&N;:XRQY\P;HQ<DAZc;WXRQ<DUVRzA6WXACJq>@[MH\PO>@A6UVcpAC[DRT>@L;[MAQ�;NBACP;<M:=>y`SRTP;N[DA6JCH\Uzf;:XP_Ri>@:XHQPS<M>|Ri>@AQk�EH\U+c;[DA68;ACPB<D:=gQA�U+HsNBACWXW=:XP;Y^HQF�>D8;:X<b[MACY\:=H\P�[DAC��L;:=[DA6<?8;:=Y\8+�KL;RQWX:a>y`&WX:=P;A9c;[MHQÎ_WXA6<�FIH\[�>@8;A�8;:XYQ8<DA6[D:=AC<ÇUVA6U&fpAC[D<6��>@8;A�cpH\c;L;WXRT>@:=H\PZU+HsNBACWXW=:XP;Y?U&L;<h>ÇfjAGRQc;cB[DH\c;[M:]RT>DA�FIH\[/8;:=Y\81��L_RQPr>@LBU·<D8;A6WXW=<RQP;N->@8;Ao<MUVHKHQ>@8-U+AC[DYQ:XP;Y�HQF�8;:XY\8t<MAC[D:=AC<^WX:XPBAC<9:XPr>DH+>D8;A~J6H\Pr>@:=PKLBL;Uñ8_RQ<3>@H+fjA~RQJ6JCH\L;Pr>DACNFIH\[6k±798;AwJCH\PrgQACPr>@:=H\P_RQW1U+A6>D8;HsN�HTFzN;ACRQWX:=P;Y { :a>@8�>@8;A(<MAC[M:XAC<tU+AC[MY\:XP;Y�:=<e>@H£:=P�>D[DHKN;L;J6ARv<M>|Ri>@:X<h>@:=J!RQW�U+:XJ6[DHQÎ;ACWXN { 8B:XJ|8²cB[DH\Y\[MAC<M<D:=gQACWa`£<D8;:aFÅ>@<�>@8;A-8;:=Y\8£�KL;RQPr>@L;U�<M8;ACW=WX<�:XPr>@Hv>@8BA

Page 64: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

à �

»�:=Y\L;[DA+�\ka� à ¼�í�Z]gh` e+^RlJe+^AYhe,l4ghYh`)Yjcua�vklJe+^óQBZBcJYha�i�a�b�cJdfeSì�K¸\ oºYPQ_e+^JcJa_^2^Ae+\�Yha�i cfcAZ b�e,i�by^Ja_`���e+Yh\_l2e�c}Z�gÉzÊ|Xhî î�ï�ФzJCHQP�>D:XP�L;L;U >@8;[MH\L;Y\8�Î;ACWXN�:=H\P;:=<@RT>D:XH\Pdk 798_RT>S:=<C�3:=>S:=<ORQ<M<DL;U+ACN�>D8_RT>SH\PBJCAvRQP�A6WXACJq>@[MH\P[DACRQJ|8;A6<^R�J6AC[M>@RQ:XPSPK�y<D8BACWXW�:=>�:=<3RQLB>DH\UeRi>@:XJCRQWXWa`�:XHQP;:X<MACNtRQP;NS>D8KLB<9fjA68_R!g\AC<^RQ<9:aF�:a>�:=<9:XPO>@8BAJCHQP�>D:XP�L;L;U-kG�sA6AzÜ�:XY\RQ[DH�gOAq>�RQW§k/¦h��Ú\ÚQãr¨GFIH\[3RQP-AqÖBRQU+c;W=A1HQF�>D8;:X<9U+A6>D8;HKN-LB<DACN-:XPtc;[DRQJ6>D:XJ6A>@H�N;A!RQW { :=>D8²<MAC[D:=AC<OU+AC[MY\:XPBY�:=P�>@8BA}N;:ag\AC[h>@H\[VHQFZ>D8;A À W=J!RT>DH\[e�G�hì�HKN�>DH\�iRQUeRT�pkx798;:=<>@8BAC<D:=<?>@RQ�QA6<�RQPeRQW=>DAC[DP;RT>@:ag\A�RQc;c;[MHrRQJ|8d�sc;[DH�gK:XPBYm>D8_RT>?fjH\LBP;Ns�yfpH\L;PBNec;8;HT>@HT�n[@RT>DAC<GJCH\Pr>@:=P�L;ARQP_RTW=`�>@:=J!RQW=W=`^:XPr>@H3>@8;AbFI[DA6A?<h>|RT>DAC<6� { :=>D8~PBH�P;A6ACN~>DH~ÉÕN;:X<M<DH\Wag\ATËi8B:XY\8m��L_RQPr>@L;U«<M8;ACW=WX<�:XPr>@H3>@8BAJCHQP�>D:XP�L;L;U-k�798;AbUV:=JC[DHTÎ_ACW=N1A6�jACJ6>D<�RT[DAbRQJ6JCH\L;Pr>DACN~FIHQ[/:XPZ>@8BAGW=:XP;A�f;[DHrRTN;ACP;:=P;Y3J!RQW=JCL;WXRT>@:=H\P;<RQP;N£:a>�:X<&:=Pr>@ACPBN;ACN£>@8_Ri>&>@8;Aq` { :XW=WEfpAS:XP;J6WXL;NBACNx¦­RT<�RQP£AqÖK>@[DRv[DRT>@AT�?PBHQ>�RQ<+RQP�:=H\P;:X<M:XP;YÎ_A6WXNȨo:XP�FILs>@L;[MAeJCHQWXWX:=<D:=H\P_RQWå�y[@RTN;:]RT>D:agQA { H\[D�uk-798;AON;Aq>|RQ:=WX<mHQF�>@8B:X<~RQW=>DAC[DP;RT>@:ag\AORTc;c;[DH\RQJ|8RQ[MA}c;[MAC<DA6Pr>@ACN :=P�J|8_RQcB>DAC[S©�RQP;N�R�J6H\Pr>@[DRQ<M> { :a>@8u>@8;A J6H\Prg\A6P�>D:XH\P;RQW�U+A6>D8;HsN�Y\:=gQACP�:=P<DA6J6>D:XH\P(©skÊ©Bk�ln> { :=WXWÇfpAz<D8BH { P(:=P <DA6J6>D:XH\P à kÊ©BkÕ©+>D8_RT>^>@8;A&¤\¥b7ðN;:ag\A6[M>@HQ[�JCHKN;A~J!RQP(:=P;N;ACA6NÎ;>�H\f;<MAC[hgQRi>@:XHQP_RQWdN_Ri>|R�RQP;NtN;:XRQY\P;H\<MAoWXHKJ!RQWjc;W]RQ<MUeR&c;RQ[@RQU+A6>DAC[M<Ck

Page 65: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� � ��� ����

� � ���� ���� � ��� ������ �� � ��� ����� � �!�" ��!�

� ��� # ��$� �%��� � �&� ' �(�)�* ��+� �,� �

��� ��-�.�/�&� 0 '�1 �2�,�3�4' �(�)�* ��+� � 5 6 � �

�!���7��� � � �-�&# ��# ��� ���� �,� � ��# � ���*� �* ��+�

�%�8�9���&� ��� � �-�&���9�

:<;>= ?A@�B3CEDGF H IJB�KLDM@

NJOQPSRUTVOXWZY\[^]L_a`Q]LRbTc_dPfeg]cR%[^OQ]h[^OQ]Lij_dk<lmin_poqir`Q]csSsfPft\uvijYXWnTLPS[wkGPStxsSizy4Rb]L_bPf]LRaoq]coveg]c_dR%{QRbPStQu[^O\]|]cRdT}WnYg]~l�WZTL[^in_(WnY\YQ_bi�WZTVO��jWnt\`�PS[bR(WZYQYQsfPST}WZ[bPfijt�WZR�[^O\]~TLi�`Q]~���%�������}�Q��N�y|ia�r{QWnt�[^PS[bPf]LRWn_d]�]��nWZsf{XWZ[b]c`��U[^OQ]�]cRdT}WnYg]�YQ_dijeXWne\PfsfP�[wk����h��lmij_G]LoM]L_buj]Lt�[x�X{� 2Wn`z¡d{QRd[boM]Lt�[bR¢Wnt\`2[^O\]WneQRdij_bY�[^Pfintxl�WnT�[^ij_���£��Jlmin_UYgijYQ{QsfWZ[^PSijtGoMir`QP�¤XT}WZ[bPfijt\Rc�QRd]c]v]L�r{QWZ[^PSijtQR��¥�j�§¦��%WntQ`*�¨�j���}����inlTVOXWnY�[^]c_h�j����%�������c�pPSR�`Q]L_bPS�ZWZ[bPS�n]�inl�[bOQ]3y~ij_b©�iZlQª|]LOQ_bPStQuj]L_�y%OQiU`\]L�j]LsfijYg]c`�WJTcir`Q]3lmij_�[bOQ]�YQ_^WZTL[^PST}Wns]L�ZWnsS{XWZ[bPfijthinl��¨�j�«¦j��WntQ`<�¥�j���}���¬��NJOQPSR(TVOXWZY\[^]L_(ij{\[bsfPStQ]cRA[^OQ]~[^OQ]Lij_dkheg]cOQPStQ`�[^O\]|tQ]�y­�a�p���Tcir`Q]Z�ZOQPSujOQsSPfujO�[^PStQu~[bOQ]E�nWZ_bPfin{QR�WnYQY\_bi} �PfoMWZ[^PSijtQR®y%OQPSTVOvWn_d]�o�WZ`Q]n��NJOQ]¯ij_bPSujPftXWZsnª|]LOQ_bPStQuj]L_Tcir`Q]ay|WnR�{\Rb]c`<PStM[bOQ]�Rd[b{Q`\kMinl�Wnt<WZPf_¥°±sfPS©n]%YQs²WZRboMWJ¡d]L[��´³n]Lt�[bRbTVOQ©Z]n�A�>µjµj¶j����[^OQ]LtGWnoq]ctQ`Q]L`WntQ`h]� r[b]ctQ`Q]L`�y%PS[bO�WJsfWZ[^]L_��j]c_dRbPSijt�eg]cPStQu%{QRb]L`�[^ipWnTLTcij{\t�[�lmij_AijYXWZTcPS[wkaPfthsfi>y�[^]coqYg]c_^W·[^{Q_d]

¶��

Page 66: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶j�

`QP��j]c_¥[^ij_�YQsfWnRboMWnR|��ª|]LOQ_bPStQuj]c_L�\�>µnµ�¸j����NJOQ]~ij_bPSujPftXWZsjTcir`Q]~y|WnR�y%_bPS[d[^]Lt�Pft�ª|���r¹¥º�lmij_(��»��¼�WntQ`xs²WZ[b]c_J[^_bWntQRds²WZ[b]c`G[^i¾½�PfRd{XWns�ª�WnRdPfT�lmij_J¿À¹¨Áa��»J¿2�µ�¶Ãe�kĪ|]cOQ_dPftQun]c_c�|NJOQ]�TLi�`Q]�y|WnRTcint��n]c_¥[^]c`$[^i _d{Qt$WnR¾Wnt*�a�p����Tcir`Q]n�E_d]Llm]c_d_b]L`$[biÅWnRv�a�p���*���}�"��Rd]c]Ä��{\oMoq]c_dRc�¯�nÆjÆ��WntQ`Ä[^OQ]�ij_bPSujPftQWns®_d]cYgij_d[UinlEª|]LOQ_bPStQuj]L_c�A�>µjµj¦j�����a�p���Â���}�qTcintQRbPSRd[^RUinl�WZt¹¨��Ç-uj_bWnYQOQPST}WnsPft�[b]c_dl�WZTc]Jy%OQPfTVOqRbYQW>y%t\R�W�lmij_d[b_^WntM¸j¸�Tcir`Q]%[^i�]L�ZWnsS{XWZ[b]%[bOQ]%£"WntQ`��7]� �YQ_d]cRdRbPfintQRc�¯È(PSuj{Q_b]�����GRbOQi>y%RÃ[^OQ]<Y\_birTc]cRdRbPStQu�inY\[^PSijtQRvRdTc_b]L]ct$y%OQ]c_d]Ä[bOQ]<YQsfWnRboMWÂYXWn_bWnoq]L[^]L_bR¾Wn_d]<]ct�[^]L_b]L`��NJOQ]ÄTcir`Q]ÄWnsfRdi _b]L�r{\Pf_b]LR�WZt­PStQYQ{\[<É«Wn`\l�Æn�\Ê(¤Xsf]<TLijt�[VWnPStQPft\u�WZsfs~[bOQ]G�p°´�nWZsf{Q]LRc�¯TcijsSsfPfRdPfijtQWns_^W·[^]cR%]�[^T�lmij_�[^O\]hPSijtG{QtQ`Q]L_%TcijtQRdPf`Q]L_^WZ[bPfijt®�NJOQ]�TLi�`Q]aij{\[^Y\{\[^RJWZt<ijYXWnTLPS[wkMoMir`QPS¤Q]c`M�n]c_dRbPfintÄiZl�[^OQPSR~¤XsS]n��y%P�[^O�[bOQ]��p°´�ZWnsf{Q]LR�Wn`z¡d{QRd[b]c`e�kÅ[^OQ]MWnYQYQ_dijYQ_dP²WZ[b]MWZeQRbij_dY\[^PSijtÅl�WnTL[bij_bRL��NJOQPfR�¤XsS]qTcWnt�eg]q{QRb]L`-Pft�TcijsSsfPSRbPfintXWns�°±_^Wn`\P²WZ[bP��n]T}WZsfTc{\s²WZ[bPfijtQRA[^iaYQ_bir`Q{QTL]%WntÃijY\[bPfT}WZsfsSka[^OQPSTV©�YgijYQ{QsfWZ[^PSijt�`\PfRd[b_bPSeQ{\[^PSijt������%���Ã���c��OXWnR([wy~ioMWnPftÃuj_bWnYQOQPST}Wns\ij{\[^Y\{\[^RL�rP�[ERbOQi>y%RE[^OQ]puj]LtQ]c_dPfTJ£"WntQ`��7TL{Q_d�n]cR~y%P�[^O¾Rb]Lsf]LTL[^]L`M[b_^WntQRdPS[bPfijtQRoMWn_b©Z]c`���WnR%Pft<¤Xun{Q_b]q�j���j�z��WnRJy~]csfsAWZRpY\sfin[d[^Pft\uÃsfPft\]�_^WZ[bPfijR��n]c_bRd]cRUTcijsS{QoMtG`Q]LtQRbP�[wk���WnR%PSt¤Xuj{\_b]¾�j���z�j���

:<;¬: ËÍÌ�Î3K}I Ï)ТÑ�DqC�Ò

¹wt<[^OQ]�]��ZWnsf{XW·[^Pfint�inl(��WntQ`G£Ó]� �YQ_d]cRdRbPfintQRc�ÔTcintQRbPS`Q]c_bWneQsf]aRbPfoqYQsSPS¤XTcWZ[^PSijtMPSR�YgijRbRdPfeQsS]Uy%PS[bO[^O\]ÃWnRbRd{QoqY\[^PSijt inlETc]L_d[^WnPft uj]Lijoq]L[^_dPf]LR�WntQ`ÅRdYXWZ[^PfWns�`QPfR¥[^_bPSeQ{\[bPfijtxinlE[^OQ]v]LoMPSRbRdPfijt¢TLi�]�ÕðTcPS]ct�[}�ENJOQ]�uj]Lijoq]L[^_dPf]LRJTcijtQRdPf`Q]L_b]L`Âe�k�[^O\]hTLi�`\]�WZ_b]nÖ

× ¹wRdin[^_dijYQPfT�RdYQOQ]L_b]hinl(_bWn`QPS{QR%Ø>�×9Ù s²WZtQ]�YXWn_bWnsfsS]cs®`\PfRb©�inl�[bOQPfTV©�tQ]LRbRa�nØ>�× º~krsfPStQ`Q]L_�y%PS[^OG_bWn`QPf{\RUØaWntQ`Gsf]LtQun[^Ox�nÚX�

��]L]�¤Xuj{Q_d]cR����Û�qWntQ`Å�r�«Ü¾lmij_p[bOQ]v`Q]�[VWnPSsfR%inl3[bOQ]cRd]vuj]LijoM]�[^_dPf]cRL�%NJOQ]v]cRdT}WnYg]vYQ_dijeXWne\PfsfP�[wk�PSR]L�ZWnsS{XWZ[b]c`�int<W�sSPftQ]%inl�RdPfujO�[¯[^OQ_dij{QujOM[^OQ]pTc]Lt�[b_b]�iZl�[^OQ]URbYQO\]c_b]Z�XWnsSijtQu�[^OQ]UTc]ct�[b_^Wns¼W· �PfR¯inl[^O\]h`\PfRb©�WntQ`GYg]c_bYg]ct\`QPfTL{Qs²Wn_~[^iÃ[^OQ]�TL]ct�[^_bWnsAW· �PfRJinlA[bOQ]�TLkrsfPStQ`Q]c_L��NJOQ]hWneQRdij_bY\[bPfijt�l�WZTL[^in_PfR�]L�ZWnsS{XWZ[^]L`$WZ[�[bOQ]<Tc]Lt�[^_b]<iZlJ[^OQ]<RdYQOQ]c_d]GWntQ`$WZ[�[^OQ]<oqPf`\Y¼ijPSt�[�iZlp[bOQ]�TL]ct�[^_bWnsJW· �PfR�inlegin[^O�[bOQ]a`\PfRb©qWnt\`�[^OQ]UTLkrsfPStQ`Q]L_���t\in[^]p[^OXWZ[~[^OQ]aWneQRdij_bY\[bPfijtql�WnTL[bij_|]��ZWnsf{XW·[^]c`MPftq[^OQ]�TLi�`\]PfRJintQsSkM�nWZsfPf`GW·[J[^OQPSRJY¼ijPSt�[¬���

Page 67: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶nÜ

È(PSuj{Q_b]������jÖvÝXÞz߬à�ázâ«ãJä¬å�æ´çzã%ázè´ä>é¨ãdê´ê�ë§ìzí�äLá>æ�ë«ä�ìZêEê�é¥è�ãdã¥ìvë«ì¾î�ïAî(ð¾ñ>òdó·ôUõ�ä¬æ´ã�æ�çzãUè´ß�ìzí�ã�ä¬å�ázâÛß�ê�àJßá·ß�è´ß�à|ã¥æ�ãdè´ê�ö�çzë«é±çhìzã¥ãb÷�æ�äUøZã|ã¥ìcæ�ãdè�ãb÷�ùnë§ì�á·ß¬è�æ�ëÛé¨úzâÛ߬è�æ�çzã|û®ä�â§æ�ü¥àJß�ìzìh÷>ãdýcëÛßVæ´ë§äLì�å�ß�é¨æ�äLèAæ�çZßVæ3ëþæ3ú·ê�ãd÷�ë«ìæ�ç·ã�ë§ìzë§æ�ëÛ߬âráZäLázúzâÛßVæ�ë«ä�ìaãdê�æ�ë«àJßVæ´ë§äLì�ô

È(PSuj{Q_b]v���«��Ö�ð>é±çzãdà�߬æ�ëÛé¥ê(ä¬ågæ�çzã�÷>ëÛê�ÿ�íLã¥ä�à�ã¥æ�è���é¥ä�ì·ê�ëÛ÷>ã¥è´ãd÷�ø���î�ï�î�ð�ñ>ò¥óZùnæ´ß¬ÿLã¥ì�å è´ä�àÓæ´çzã|è�ãdáZäLè�æ(ä¬åû®ã¥çzè´ë«ìzí�ãdè���ò������¨ô

Page 68: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶Z�

È(PSuj{Q_b]����§Ü\Ö�ðzé±çzã¥àJßVæ´ë«édêQä¬åzæ´çzã�é��}â«ë§ì·÷zè�ëÛé¥ß�âcíLã¥äLà|ã¥æ�è��Eé¨äLì·ê�ëÛ÷>ãdè�ãb÷Eø��Eî�ïAî(ð�ñ>ò¥óZùbæ´ß¬ÿLã¥ì¯å è�äLà æ�çzã�è´ã¥ánä�è�æä¬åXû®ã¥çzè´ë«ìzí�ãdè���ò������¨ôº|ijtQRdPf`Q]L_J¤X_bR¥[p[bOQ]�]cRdT}WnYg]�Y\_bijeXWZeQPfsSPS[wkq]� �YQ_d]cRbRdPfijtÄlm_bijo�TVOXWnY\[b]c_%ijtQ]ZÖ

� �������������������� � �"!$#���%'& �)(+*,.-0/21�3 �54�687���4+9 # 9 �;: �<�������"= � �"!VÆ��>%'& ��(+*? -0/�1�3 �@6A�@7��B9 ������������C����D� ���E!$#�� 9 # 9 � �����S�>��aRdRb{Qoq]�[bOXWZ[p[^OQ]L_b]�PfRptQi�eXWZTV©run_bij{Qt\`¢_^Wn`\P²WZ[bPfijt<¤X]Lsf`*� = ���"!VÆj�F��Æ��pWZtQ`x[bOXWZ[HG"PSRUPStQ`Q]�°Yg]ctQ`Q]Lt�[ainl�RbYQWZ[^PfWns�YgijRbP�[^Pfint�y%PS[^O\Pft�[^O\]�YQsfWnRboMWÃWntQ`G]L�r{QWns�[^iJIG�� �Q���ENJOQ]hWnRdRb{QoqY\[^PSijtGinlTcintQRd[^Wnt�[�Wne\Rbij_dY\[^PSijt¾Tcir]LÕqTcPS]ct�[¯_b]cY\_b]cRd]ct�[^R~[^OQ]%oMWnPft¾WnY\YQ_bi} �PfoMWZ[^PSijtvPSt¾[bOQ]pTcir`Q]n��]c��{QP���°WnsS]ct�[�[^ixWnRbRd{QoMPStQuÄWÄTLijtQR¥[VWnt�[�sfi>y|]L_�sf]L�n]csEYgijYQ{QsfWZ[^PSijt `Q]ct\RbPS[wk [bOQ_bin{QujOQij{�[h[bOQ]¾YQs²WZRboMW�jinsf{Qoq]n�-NJO\PfRÃRbO\ij{Qsf`$eg]ÄW¢_b]}WZRbijtXWZeQsf]ÄWnYQY\_bi} �PfoMWZ[^PSijt�lmij_vlm{\RbPfint"`\PS�j]L_d[bij_ÃTcijt\`QPS[bPfijtQRWntQ` lmij_�oMWnt�k¢sSi>y[b]coqY¼]L_^WZ[b{Q_b]ÃY\s²WnRdo�WnRL�û�YXWnTLPS[wkÂ]�K¼]cTL[bRhWn_b]ÃTLijoMoqijtQs�kÂPSt��j]cR¥[^PSu�WZ[^]L`Pft*[b_^WntQRdPS[bPfijtQRvy%P�[^O"]cP�[^OQ]L_¾[bOQ]Guj_bin{QtQ`"ij_¾oM]�[VWnR¥[VWneQsS]cR¾WnRÃ[bOQ]cPS_qsSi>y|]L_qsS]L�n]cs´�"ª|]LT}Wn{\Rb]inl%[^O\]ÄsfWn_bun]�YgijYQ{QsfWZ[^PSijtQRvinl%[bOQ]cRd]ÄsS]L�n]csfRÃP�[ÃRbO\ij{Qsf`$eg]��ZWnsfPS`�[^iÅWnRdRb{Qoq]<[^OXW·[v[^OQ]�k$Wn_d]RbYQWZ[^PfWnsfs�kMPSt��nWZ_bP²WZt�[c��NJOQ]�]cRdT}WnYg]�Y\_bijeXWZeQPfsSPS[wkqPfR�[bOQ]ctxujPS�n]ctGe�k

� � � �����L� � ���� � � �"!$#���% & � (+*,NM-0/21 6A7��54 9 # 9 �� ������� � ���� � ���E!$#�� 9 # 9 �� � �����L� �C���� � � �"!$#���% / � & ��� 6 M-O/21 6 9 # 9 �� �5�2��� �C���� � ���"!<#�� 9 # 9 � ������n�

��]L¤QtQ] �'P � #���Rb{\TVOx[^OQWZ[%]coqPfRdRbPfint<Tcir]LÕqTcPS]ct�[pT}WntÄe¼]�y%_dPS[b[b]ctÄWnR

Page 69: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶j¶

� � �"!$#��Q� ��0RTS � 42U �OV;� 4 � #���W\���Q�

� �XP �Y#���WQ� �Q� �����«Üj�WntQ`ZIG P Rb{QTVOÄ[^OXW·[

IG|� �Q�[� V\�^] �¦'R _a`cb`ed � �� �� 42U � S � 4�U � WQ� �Q�� IG P WQ� �Q� �����§���

�aRdRb{QoqPft\u¾[bOXWZ[J[bOQ]�]coqPfRdRbPfintGWntQ`ÄWneQRdij_bY\[bPfijtGsSPftQ]UYQ_bin¤Qsf]cRpWn_d]�PftQ`\]cYg]ctQ`Q]Lt�[UiZl(YgijRbP�[^PSijt��[^O\]h]LRbTcWnY¼]hYQ_dijeXWneQPSsfP�[wkMTcWntÄe¼]�y%_bP�[b[b]ctÄWnR

� ���C������'P �Y#�� ���5���L� WQ� �Q��% / � & ��� 6 M-gfXh�/�1 6 9 � 9 #�C������XP 9 # �����Û¶n�º|ijtQRdPf`Q]L_%tQ]� r[%[^OQ]hWne\Rbij_dY\[^PSijt<l�WnTL[bij_J]� �YQ_d]cRbRdPfijt®�gR¥[VWn_¥[^PStQuÃy%PS[^OG]L�r{QWZ[^PSijt-�¨�j���}�����

£�� #��i��kj ��lm��������� G����E!$#��>n����"!<#�� 9 � 9po� l � �5�2��� � ���E!$#�� 9 � 9po �����2qj��aujWnPft��Ôijt\]vWnRdRb{Qoq]cRa[bOXWZ[p[^OQ]�WneQRdij_bY\[bPfijtxTcir]LÕqTcPS]ct�[�PSRUPStQ`Q]LY¼]LtQ`Q]ct�[�iZli#w�aÈXij_%[^OQ]�YQ{Q_¨°YgijRb]LR�inl~[^OQ]qTcir`Q]q£!PSRh]��ZWnsf{XW·[^]c`�WZ[h[^OQ]qTc]Lt�[^_b]Minl�in{Q_�Y\s²WnRdo�W<uj]LijoM]�[^_¥kj��`Q]ctQiZ[^]c`�e�ksr��e��­¹±[�PSRqWnRbRd{QoM]L`9[^OQWZ[q]coqPfRbRdPfijt$lm_bino [^OQPSRMYgijPSt�[¾PSRM{\tQPSlmij_do�Pft­WnsSs�`QPf_d]cT�[^PfintQRx��P´�þ]n���lm��������� � � �"!$#�� 9 � 9po �7�0R �XP ��NJO�{QR

£���Ø��s���kj IG P � l � �����L� WQ� �Q�>n�� �"!VØ�� 9 � 9po�0R �XP �����Û¸n�

NJOQ]h_^Wn`\P²WZ[bPfijt�¤Q]csf`<[b]c_bo�PfRJujP��j]ctÄe�k

n����Q�i� t �u / l"6 �'P � #��>WQ���\��% / � & � 6 M-gf'h�/21 6 9 # �����«¦j�

Page 70: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶0q

Homogeneous distribution Linear distribution

Parabolic distributoin

l=0 l=2bl=b

l=0 l=b l=2b

l=2bl=0 l=b

(b)

(b)

ε

ε

ε

È(PSuj{Q_b]p�r�§�QÖ(ï�ëÛ߬í�è±ß¬àJêQä�ånæ´çzã�ê�á·ß¬æ�ëÛ߬â·÷>ëÛê�æ�è´ë§øzúzæ�ë«ä�ì%ä�ånæ´çzãAãdà|ëÛê´ê�ë«ä�ì%é¥äcã>v%é¨ë«ã¥ìcæ¼ß�êgé¨ä�ìZê�ëÛ÷>ã¥è´ãd÷|ø���î�ïAî(ðñ>òdó·ôy%PS[bOxwA� o ��`Q]cRdTc_dPfeQPStQuGW�Y¼inPft�[�ijt [^O\]ÃYQs²WnRdo�W�Rd{Q_dl�WZTc]n��NJOr{\R�[bOQ]Ã_^WZ`QP²WZ[bPfijtx¤X]csS` PfR�[^O\]_b]LRb{Qs�[VWnt�[E_bWn`QP²W·[^Pfint��ZPStQTcPS`Q]ct�[�lm_dijo�WZsfs\YgijRdRbPfe\sf]�sSPftQ]LR(inl¼RbPSujO�[�Pft�[^i�[bOQ]JTc]Lt�[^_b]Z�rWnTcTLij{Qt�[^PStQulmij_%WZ[d[^]Ltr{QWZ[^PSijtÄWnsfijt\uÃ]}WnTVOxsfPft\]�inl�RbPfunO�[c�ÁUiZ[^]v[^OQWZ[�]c��{XW·[^Pfint9�����§¦��UPfRaRbPfoqPfsfWn_%[^i�[^O\]Ãtr{\oM]L_^WZ[bij_�inl¯[^OQ]Ã]L��{XWZ[^PSijt"�´�r�Û¶j�¬��WnsSsfi>y%PftQuijtQ]Ã[biÄRdPfujt\PS¤XTcWnt�[^sSkÂRbPSoMYQsSPSl kG[^O\]qt�{Qoq]c_dPfTLR�inl~[bOQ]¾T}WnsSTc{QsfWZ[^PSijt��v�UR�y%Pfsfs�e¼]¾Rd]c]ct�Pft [^O\]tQ]¬ �[%Rd]cT�[^Pfint��\[^O\]�]coq]c_bun]ct�[J�X{� <Wnsfijt\uÃWvujPS�n]ctGsfPStQ]aiZl�RdPfujO�[JTcWnt<eg]�]� �YQ_b]LRbRd]c`ÂY\{Q_b]LsSk<WZRW�lm{QtQTL[bPfijtqinl®ijY\[^PST}WnsX`\]cY\[bO���NJO�{QRFn����\��TcWnt�eg]p]L�ZWnsf{QWZ[^]L`qlmij_¯W�_^Wnt\uj]UiZl�ijY\[bPfTcWnsX`Q]cY�[^OQRL�{QRd]c`ÅPfoqoM]L`QP²W·[^]cs�k�[^i<]��ZWnsf{XW·[^]Ã� WntQ` TcWnsfsS]c`¢_d]cYg]}WZ[b]c`Qs�kÂPSt¢[^OQ]�RbijsSPf` Wnt\ujsf]�Pft�[^]Luj_^WZ[bPfijtlmij_~£a�3NJO�{QRJP�[|PSR|ijtQs�kqtQ]cTL]cRbRbWn_dkM[^i�y~ij_d©Mij{�[�[^OQ]Uuj]ct\]c_bPSTHn����\�¯�rRFy�TL{Q_d�n]�ijtQTc]�WntQ`�[^O\]_b]LRd[pinl�[^OQ]�TcWnsfTL{Qs²WZ[bPfijt\R�T}WntÄeg]hYg]c_¥lmij_boq]c`<lm_dijo�PSt�[b]c_dY¼ijsfWZ[^PSijt�inl�[^O\]cRb]h_b]LRb{Qs�[^RL�

Page 71: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶j¸

:<;$z {}|¢ÌhB�KcÌ�~��hÌvCEKcÌhB�KcDq@ K}@ B3Ð¢Ñ Ñ�� KcÎ3Î3KcDq@ I%D<ÑH� IpKcÑ�@�B

NJOQ]%Tcir`Q]JWZsfsfi>y%R(lmin_(�ZWn_bPSij{QR3`QPSRd[b_bPfe\{\[^PSijtQR(PStv[^OQ]J]LoMPSRbRdPfijtvTLi�]�ÕqTcPf]Lt�[�y%PS[^O\Pftv[^O\]��jijsS{QoM]inl�[^OQ]�YQsfWnRboMW\�3NJO\]cRb]�ijY\[bPfijtQR�Wn_b]× OQinoMijun]ctQ]Lij{QR%`QPfR¥[^_dPfeQ{\[bPfijtM[^OQ_dij{QujOQin{\[%[^OQ]�YQsfWnRboMW\�× sSPftQ]cWn_�`Q]LTc_b]cWnRb]�lm_bino�YQsfWnRboMWÃTc]ct�[b_b]�[^i¾]c`Qun]n�× YXWZ_^WnegijsfPSTa`Q]LTc_d]}WnRd]�lm_bino�YQsfWnRboMWvTc]Lt�[b_b]�[^i¾]c`\uj]n�

¿�_bPS[bPftQuÃin{\[J[^OQ]�PSt�[b]ctQRdPS[wk<]¬ �YQ_b]LRbRbPSijt¢WZtQ`G_b]cYg]}W·[^]c`\sSk<PSt�[b]cuj_bWZ[^PStQuÃe�kGYXWn_¥[^R%_b]LRb{Qs�[^R%Pft

= � �"!V��Q� t �� � ���"!<#��>% / � & � 6 M-�/21 6 9 #� � ���"!V��IG����\� j � b ���"!V��IG�� �Q� � : � b b ���E!V��IG�� �Q�>� jN�B�8�B�

jH% & M-O/21 6 �^� � � �"!VÆ��IG����\� j � b ���"!^Æ��IG~���Q� � : � b b ���"!^Æ��IG~���Q� � j��8�B�8��� �����«µj�NJOQ]%�ZWn_bPfWZ[^PSijt¾PftÃ[^OQ]UWnsSsfi>y|]L`M]LoMPSRbRdPfijt¾Tcir]LÕqTLPf]ct�[|`QPfR¥[^_bPSeQ{\[bPfijtQR�PSR¯WZ[¯oMijR¥[¯Rb]cTLijtQ`�in_b`Q]L_oMWn©�PftQu|[bOQPfR�[b]cTVOQt\Pf��{Q]¯Wnt�]LÕqTcPS]ct�[Ay�W}k�iZl�]��ZWnsf{XW·[^Pft\u|[^OQ]�sSPftQ](inlrRbPfunO�[APSt�[^]cun_^WZ[bPfijt���¿ÀO\]ct[^O\PfR�]¬ �YQ_b]LRbRbPSijt"PfR�]��ZWnsf{XW·[^]c`�lmij_v]}WZTVO*]coqPfRbRdPfijt�Tcir]LÕqTcPS]ct�[Ã`QPSRd[^_dPfeQ{�[^Pfint���[bOQ]�lmijsfsSi>y%PftQu]� �YQ_d]cRdRbPfintQR�Wn_d]ÄWnTVOQPf]��j]L` ��_d]cRd{QsS[bRqy%P�[^O$[^OQ]GPft�[b]cuj_bWZ[^PSijt$]� r[^]LtQ`QPft\u�lm_dijo ÆJ� �nØ�WZtQ`lm_bino)Æ�� ØJWn_d]URdOQi>y%t�RbPStQTc]%[^OQ]�kqWn_b]p{QRd]c`�PStq[bOQ]UTcWnsfTL{Qs²W·[^PfintÃinlA��Wnt\`�£$_b]LRbYg]cTL[bPS�n]csSk\�¬ÖÈXij_�[^O\]hO\ijoMinuj]ctQ]Lij{QRJ`QPSRd[b_bPfe\{\[^PSijt<ijtQ]�OXWnR

t �� � ���"!$#��>% / � & � 6 M-O/21 6 9 #�� � ���\�IG�� �Q��� �kjJ% M-O/�1 6 ���t ���� � ���"!$#��>% / � & ��� 6 M-O/21 6 9 #�� � ���\�IG�� �Q� � �kjJ% � M-O/�1 6 ��� �´�����>Æ��ÈXij_�[^O\]hsSPftQ]cWn_�`QPSRd[b_bPfe\{\[^PSijt�[^O\]cRb]heg]cTcinoM]

t �� � ���E!$#���% / � & � 6 M-O/�1 6 9 #�� � ���E!VØ��IG�� �Q� � �kj �IG~���Q�¥Ø : �IG|���Q�¥Ø % & M-0/21 6 � �t ���� � � �"!$#���% / � & ��� 6 M-�/�1 6 9 #�� � ���E!VØ��IG�� �Q� �IG|���Q�¥Ø � �kj"�0% & M-0/21 6 � :�% & � M-0/21 6 ��� �´�����j�z�

Page 72: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶n¦

WntQ`

t �� � � �"!$#���% / � & � 6 M-O/�1 6 9 #�� � � �"!VØ��IG~���Q� � �kj �IG~���Q� � Ø � : _ �IG|���Q�¥Ø : �IG����\� � Ø � d % & M-O/21 6 � �t ���� � � �"!$#���% / � & ��� 6 M-�/�1 6 9 #�� � � �"!VØ��IG~���Q� �IG|���Q�¥Ø � ��j �IG|���Q�¥Ø : _ �k: �IG����\�dØ d % & � M-O/21 6 � �g�´�����z�j�lmij_�[^O\]~YXWZ_^WnegijsfPST�`QPfR¥[^_dPfeQ{\[bPfijt®��NJOQPfR�_b]LRb{Qs�[^R�Pfth]cRbTcWnYg]|Wnt\`vWneQRdij_bY�[^Pfinthl�WnT�[^ij_A]¬ \Y\_b]cRdRbPSijtQRinl

� � t �5���L� W\���Q��� �^j�% & � M-0/�1 6 �����IG~���\�dØ 9 �� t �5���L� W\���Q��� �^j�% & ��� /21 6 ��XyA� �Q� 9 �

£���Ø��Q� ��0R t l t ������� WQ� �Q��% & M-O/21 6 � / lE6 9 � 9po

� ��0R t l t ������� WQ� �Q��% & � /21�3 lE6 9 � 9po �´�����>Ü��

lmij_�[bOQ]�OQijoqijuj]ct\]cij{QR%`QPSRd[b_bPfe\{\[^PSijt��

� � � t ������� �IG����\�dØ WQ���\� �IG�� �Q�dØ � �kj"�0% & M-0/21 6 � :�% & � M-0/21 6 � � 9 �� �it ������� �yA���\� WQ� �Q� �y����Q� � �kj"�0% & � /21 6 :�% & ��� /21 6 � 9 �

£���Ø��Q� ��'R t l t �5���L� � �IG P Ø·� o � j �IG P Ø·� o � % & M-0/21 6 � / lE6 � 9 � 9Eo

� ��'R t l t �5���L� � W\���Q�yA���E! o � j W\���Q�yA���E! o � % & � /21�3 lE6 � 9 � 9po �´�����}���

lmij_�[bOQ]�sfPft\]}Wn_�`\PfRd[b_bPSeQ{\[^PSijtGWntQ`

� � Ü� t �����L� � �IG~���Q�¥Ø W\���Q� �IG�� �Q�dØ � ��j �IG|� �Q�dØ : � �k: �IG~���Q�¥Ø �F% & � M-O/21 6 � �O� 9 �

Page 73: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¶nµ

� � t �����L� � �y����Q� W\���Q� �yA� �Q� � ��j �yA� �Q� : � �k: � IyA���\� �F% & � M� /21 6 �'� 9 �

£���Ø��[� ��XR t l t ������� � �

�gIG|���Q�¥Ø·� o �b�L�gIG3Ø·� o �b� j � �IG P Ø·� o � : ��gIG~���Q�¥Ø·� o �b�L�TIG P Ø��c� o � �% & M-0/�1 6 � / l"6 � 9 � 9Eo

� ��XR t l t ������� � �0WQ� �Q�

��y����"! o �b� � j � �'WQ���E! o �yA� �"! o � : �0W\���Q���y����"! o �b� � �F% & � /�1�3 lE6 � 9 � 9po �´�����z¶j�

lmij_�[^OQ]�YQWn_^WnegijsSPfTE`QPfR¥[^_dPfeQ{\[bPfijt®���EWnTVOÃiZlX[^OQ]LRb]J]� �YQ_d]cRdRbPfintQREWn_b]¯lm{QtQT�[^PfintQR3inlÔijY�[^PfTcWns�`Q]cY�[^O�gIG P ØL�3WZtQ`¾sfPft\]|YQ_din¤Xsf]Z��WnsfsSi>y%PftQuUuj]ctQ]L_bPST%�2Wnt\`q£���R�y¾TL{Q_d�n]cRE[^i�eg]J]L�ZWnsS{XWZ[^]L`���Rb]L]J¤Xuj{Q_d]���«¶��

:<;>� ��C�D<D �ÓB�ТÌhB�¡¾Ñ�@¢Ñ�CEK}I£¢�¤¦¥¨§9Ì�@ F © �ÄÎ8Dª|ÂB�K}I%Ìe~$F¢Ñ�|ÂB3ÐIpH C��qÑ�Î,ÌvC�Ñ �hÌe~cK}F

NJOQ]%Tcir`Q]J]��nWZsf{XWZ[b]cR([bOQ]%]cRdT}WnYg]pWntQ`¾WneQRdij_bY�[^Pfintvl�WnTL[bij_�]� �YQ_d]cRbRdPfijtMWnR�uj]LtQ]c_dPfT�TL{Q_d�n]cRE[^OXW·[Wn_d]�ijtQs�k�lm{\tQTL[bPfijtQR�inlpijY\[bPfT}WZs¯`\]cY\[bO��ÅNJOQ]<ijY\[bPfT}WZs¯`\]cY\[bO$lmin_�]}WZTVO"YgijRdRbPfe\sf]M[b_^Wnt\RbPS[bPfijtPfR<[bOQ]ctÀy|in_b©n]L`�ij{\[xWnt\`2P�[^RGTcij_d_b]LRbYgijtQ`QPStQu­]LRbTcWnY¼]�WntQ`2WneQRdij_bY�[^PfintÀl�WZTL[^in_Glmin{QtQ`7e�kPft�[b]c_bYgijsfWZ[^PSijt<inl([bOQ]�uj]ct\]c_bPST�TL{Q_d�n]n��NJO�{QRp[^OQ]huj]LtQ]c_dPfT�Tc{Q_¥�j]�PfRpWnRbRd{Qoq]c`x[biMeg]��nWZsfPf`�lmin_]L�n]c_¥kÂYgijRbRdPfeQsS]h[b_^Wnt\RbPS[bPfijt<y%P�[^OQPStx[^O\]�ijt\]�Pfint���¹±[�PSR%tQin[�PSoMoq]c`\P²WZ[b]csSkMije��rPfij{\Rp[bOXWZ[p[^OQPSRWnYQY\_bi�WnTVO�PfR��ZWnsfPS`��qNJOQ]c_d]�Wn_d]¾[wy~i¢PSRbRb{\]cR�[^iÂeg]¾`Q]}WZsS[�y%PS[bO���¤Q_bRd[bsSkÅTcWnt�[bOQ]M]¬ �YQ_b]LRbRbPSijtWns�y�W}krR�e¼]vy%_dPS[d[^]ctÅPSt¢[^]c_doMR�iZl~inY\[^PST}Wns�`Q]cY\[bO�WZtQ` Rb]LTcijtQ`\sSk OQi>y�`Qi�]LR�ijtQ]¾WnTLTcij{\t�[�lmin_[^O\]�l�WZTL[J[^OQWZ[p`QPBK¼]c_d]ct�[%[^_^WZtQRbP�[^PfintQR�T}WZtxOXW}�n]�`QP8K®]L_b]Lt�[%sSPftQ]�Y\_bin¤XsS]cR<«N�ihRdOQi>y9[bOXWZ[�[^O\]J]� �YQ_b]LRbRdPfijtQR¯T}WZtMWZsSy|W>krREeg]�y%_bPS[d[^]LtqPStv[^]L_boqR�inl®inY\[^PST}Wns�`Q]cY\[bOMTLijtQRdPf`Q]L_[^O\]�lmijsfsSi>y%PftQuMWn_bun{QoM]Lt�[}ÖUª~kÂPStQRbYg]cT�[^PSijtÂlm_bino&]L�r{QWZ[^PSijtQR¾�����S�}Æ��J°��´�����z¶j�%PS[�TcWnt eg]vRb]L]ct[^OQWZ[��ÓWZtQ`�£�T}Wnt�e¼]¯y%_bP�[b[^]Lt�PSth[b]c_boqR�inlXijY\[bPfTcWns�`Q]cY�[^OÃWnt\`vsfPStQ]�YQ_biZ¤Xsf]Z�(¹±[�T}Wnt�e¼]~RbOQi>y%t[^OQWZ[�[^OQPSR�PfR�uj]LtQ]c_bWnsfs�kÂ[bOQ]¾T}WnRd]qP�l¯ijtQ]Ãy%_bP�[^]cR�[bOQ]Ã]coqPfRdRbPfint�TLi�]�ÕqTcPf]Lt�[hy%P�[^O [^O\]qRdYXWZ[bP²Wns�ZWn_bPfWZ[^PSijtÂ]¬ r[^_^WZTL[^]L`���[^OXW·[�PfR � � �"!$#��m� � ���E!VØ���¬3�Y#��%y%OQ]c_d]­¬(� #��U`Q]�¤XtQ]cR�[bOQ]v�ZWn_bPfWZ[^PSijtÂinl �WnsSijtQu®#w�h¹±[�OXWnR�eg]c]ctÅRdOQi>y%t�PSt ]c��{XWZ[bPfijtQRÃ�´�����>Æ��%°h�´�����z¶j�p[bOXWZ[�[bOQ]ÃsfPStQ]�PSt�[^]cun_^WZ[b]c`¢�X{� T}WZt-e¼]qy%_bP�[b[^]Lt-WnR°¯ /21�3 � 6M-0/21 6�± ��y����Q�²!$WQ� �Q�b��y%OQ]c_d]�u¢PSRh[bOQ]�WZYQYQ_binYQ_bPfWZ[^]¾lm{QtQT�[^Pfint�lm_bino [bOQ]cRd]]c��{XW·[^PfintQRc�ENJOQ]�]cRdT}WnYg]�WntQ`xWneQRbin_bY\[bPfijt<l�WnT�[^ij_dRJT}WntG[^OQ]Ltxeg]�y%_bP�[b[^]Lt��

Page 74: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

qjÆ

È(PSuj{Q_b]M���«¶�Ö�ÝÔÞz߬à�ázâ«ã|ä�å¼æ´çzãJí�ãdìzã¥è´ëÛéT³"߬ìZ÷�´�é¨úzè´ý�ãbê3ázè�ä>÷>úZé¨ãd÷�ø���î�ïAî�ðÃñzò¥ó·ô%õ�ä¬æ´ã�æ�ç·ß¬æ¯ß�ê3ä�ì·ãázè´ä�íLè�ãbê�ê�ãdêQúzáUß�í�ë«ý�ãdì�ê�ã¥è´ë«ãdêXæ´çzã�³xß�ì·÷T´¾ýV߬â«úzãbêgë«ì·é¨è´ãdßLê�ãLô"µ�çz㶳x߬ì·÷T´MýV߬â«úzãdê¼ß¬âÛê�ä�ë«ì·é¨è´ãdßLê�ã�å è´ä�à�ä�ì·ãê�ã¥è´ë§ãbê®æ�ä|æ�çzã�ì·ã¨Þ}æ

Page 75: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

q\�

� � � �� � ������� � ���"!$#��>% / � & � 6 M-0/21 6 9 � 9 #� �� � ������� � � �"!$#�� 9 � 9 #� �C�� ¬3�Y#�� ���5���L� � ���E!VØ���% / � & � 6 M-0/�1 6 9 � 9 #�C�� ¬3�Y#�� 9 # �������L� � ���"!^ØL� 9 � 9 #� � �� ¬3�Y#�� � �5���L� ¯ /�1�3 � 6M-O/�1 6 ± ��yA� �Q�!<WQ���\�b� 9 � 9 #

� �� ¬3�Y#�� 9 # � �����L� �XP ��Ø���WQ� �Q� 9 � 9 #� �XP ��Ø�� ���� ¬3� #�� �������L� h�/�1 6M-0/�1 6 ± ��y����Q�²!$WQ� �Q�b� 9 � 9 #�XP ��Ø�� �C�� ¬3� #�� 9 # ���5�2��� W\���Q� 9 � 9 #� �C�� ¬3�Y#�� ���5���L�¸·M-gf ± ��y����Q�²!$WQ� �Q�b� 9 � 9 #� �� ¬(� #�� 9 # �´�����¦q��

£���ØL�Q� �kj¹��lm��������� IG�� �Q�>n����E!VØ�� 9 � 9po� l � ������� � � �"!VØ�� 9 � 9po� �kj ��lm��������� IG�� �Q� ¯ /�1�3 � 6M-0/�1 6 ± ��yA� �"! o �²!$WQ� �Q�b� 9 � 9Eo��lm��������� �'P ��Ø��>WQ���\� 9 � 9po� �kj �XP ��Ø�� � l � �5�2��� W\���Q� ± ��yA� �"! o �²!$WQ� �Q�b� 9 � 9Eo�'P ��Ø��¨�0R� ��lH�������L� WQ� �Q� ± ��y����"! o �!$W\���Q�d� 9 � 9po

�0R �´�����z¸j�NJO�{QRc��lmin_MWÅujP��j]ct"sfPStQ]GYQ_biZ¤Xsf]Z�~ijt\]xTcWnt9uj]LtQ]c_bWZ[^]Äuj]LtQ]c_dPfTx� WntQ`"£)��RMijY\[bPfT}WZs�`Q]cY�[^OTc{\_d�j]LRc��¹±[�RdOQij{QsS`Ãe¼]|tQin[^]L`v[^OXW·[(lmij_�e¼in[bOq� WntQ`ã-W�t\ij_boMWnsfPSRb]L`�sfPStQ]¯YQ_bin¤Qsf]|PSR({QRd]c`Ãy%PS[bO[^O\]vWnT�[^{XWns�sfPStQ]�OQ]cPSujO�[�Tcint�[^WnPft\]c`¢PStx[^O\] �XP [b]c_boÄ� Ù _di>��PS`Q]c`¢[bOXWZ[p[^OQ]�R^Wnoq]�tQin_boMWnsfPSRb]c`YQ_din¤Xsf]�OQijsf`\Rvlmij_ÃWnsSs~[^_bWntQRdPS[^PSijtQRL�3[^OQ]<uj]LtQ]c_dPfTG��WntQ`$£��rRºy$Tc{Q_¥�j]<y%PfsSs|OQinsf`-lmij_v]cWnTVO[^_bWntQRdPS[^PSijt��~NJOQPfRpPfR%sfPS©n]LsSk�[biMeg]h[bOQ]hT}WnRd]vWnRpPS[pPfR%oMWnPft\sSk<WZ`·¡bWnTc]Lt�[aRd]c_bPS]cRUoM]Love¼]L_bRJy%PS[bO]cP�[^OQ]L_�[^O\]<uj_bij{\tQ`$ij_ÃW oM]�[VWnR¥[VWne\sf]�WZRv[^OQ]<sSizy~]c_ÃsS]L�n]cs|[bOXWZ[¾Wn_d]<ijY\[bPfT}WZsfsSk [bOQPfTV©Ô��NJOQ]R^WZoM]<eQ_di�Wn`Q]LtQPft\u�oq]cTVOQWntQPfRdoMRvWn_b]GsfPS©n]LsSk�[^iÅWnYQY\sSk�[biÅRb{QTVO­sfPft\]cRvujP��rPftQuÂ[bOQ]co RbPfoqPfsfWn_YQ_din¤Xsf]LRc�3NJO\]UOQPSujOQ]L_¯Rd]c_dPf]cRE[^_bWntQRdPS[^PSijtQRL�jy%OQPfTVO�Wn_b]%oqij_b]%sfPS©n]LsSk�[bi�OQW>�n]U`\PBK¼]c_b]Lt�[|YQ_din¤XsS]cRc�

Page 76: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

q��Wn_d]Minl [b]ct�ijY\[^PST}WnsSsSkÄ[^OQPSt���eQ{\[�]L�j]Lt*PSl¯[^OQ]�k�Wn_b]qtQin[h[^OQPSt�[bOQ]Lk�OXW}�n]�sfP�[b[bsf]v]K®]LTL[vint�[^O\]YgijYQ{QsfWZ[^PSijt�_b]LRb{Qs�[^R%inl�[^OQ]�Tcir`Q]Z�

:<;²» ��C�Dº¼x~cÑ Ï Ð¢ÑaDMC3Ò��]}WZsfPft\uqt\]� r[ay%P�[^Ox[bOQ]��ZWn_dPfij{\RpsSPftQ]�YQ_din¤XsS]�ijY�[^PfintQRc�X[^O\]c_b]�Wn_b]�[^O\_b]c]�eXWnRdPfThYQ_bin¤Qsf]cRL�®WZtQ`[^O\_b]c]�TLijt��jijs��j]c`xYQ_biZ¤Xsf]LRUWZsfsfi>y~]c`<lmij_JPStG[bOQ]�Tcir`Q]n�XtQWnoM]LsSk

× ��ijY\YQsf]L_× Ç�ij_d]ct�[<½LP²Wnt×N¾ ijsS[bRboMWn_b©× ½EijPSun[× ��ijY\YQsf]L_¥° ¾ ijsS[bRboMWn_b©× ��ij{\eQsf]���ijY\YQsf]L_

ÁUin_boMWnsfPSRb]c`ÅYQ_din¤XsS]cRhWn_b]quj]LtQ]c_bWZ[^]L`�t�{Qoq]c_bPST}WnsSsSkÂPft [^OQ]qTcir`Q]¾lm_bijo+]� �YQ_b]LRbRdPfijtQR�Rb{QTVO*WZR[^O\ijRb]¯PfthRb]LTL[^PSijtq�j�Û�r�Û¶���NJOQ]|��ijYQYQsS]c_�OXWns�l²°±y%PS`\[^OQRAWZ_b]E`Q]L[b]c_doMPStQ]c`hlm_bijo�]¬ �YQ_b]LRbRbPSijtG�¨�j�Û�'q����[^O\]ÃÇ�ij_b]Lt�[�½cPfWntÅWntQ` ¾ ijsS[bRboMWn_b©<y%Pf`�[^OQR�WZ_b]vRd]L[�[bi<OXWnsSl([^OQ]¾��ijY\YQsf]L_Uy%Pf`�[^O Wnt\`¢lmij_a[^O\]Tcint��nijsS�n]c`xYQ_bin¤Qsf]cRL�Q[^OQ]�tQint�°¨��ijY\YQsf]L_|y%Pf`\[bOGPfRJ]ct�[b]c_b]L`¢e�k�[^O\]h{\Rb]c_L�NJOQ]�PSt�[^]cun_^WZ[bPfijthi>�j]c_�[^OQ]¯y�W}�j]Lsf]LtQun[^OÃPSR�Yg]c_¥lmij_boq]c`�t�{QoM]L_bPST}WnsSsSkj�ZWZ[�[^OQPSR(YgijPft�[�[^OQ]~]cRdT}WnYg]YQ_dijeXWneQPSsfP�[wk�TcWntGe¼]�lmij{Qt\`Älm_dijo�]¬ \Y\_b]cRdRbPSijt*�����Û¶n���

:<;$¿ ÀD�hÌe~cH¢Ì�B�KLDM@ D ��¢N�i�]L�ZWnsS{XWZ[b]Ä[bOQ]ÂWZeQRbij_dY\[^PSijt­l�WZTL[^in_c�|P�[qPfRqtQ]cTL]cRbRbWn_dk"[^i�Pft�[^]Luj_^W·[^]Äiz�n]c_MRbijsSPf`­WntQujsS]n�|Rdi©�tQi>y%sf]L`Quj]ÂiZl�[^OQ]xYQs²WZRboMW�uj]Lijoq]L[^_¥k$PfRq_b]L�r{\Pf_b]L`��7NJOQPfRqPft�[^]Luj_^W·[^Pfint­PSRMYg]c_¥lmij_boq]c`­tr{r°oq]c_bPST}WnsSsSkn��ÈXij_%[^OQ]�RbYQOQ]L_bPST}Wns�uj]cinoM]�[^_dkG£7T}WZt¢e¼]�_b]�[^{Q_dtQ]c`¢PSoMoq]c`\P²WZ[b]csSkqy%PS[bOx[bOQ]�RdijsfPS`WntQunsf]%Pft�[^]Luj_^W·[^PfintqPSt�[^_bir`Q{QTLPftQu�W�l�WnTL[bij_Einl��0R��3ÈÔin_E[bOQ]pTLkrsfPStQ`Q]c_¯Wnt\`�`QPSRb©Ãuj]LijoM]�[^_dPf]cRE[^O\]

Page 77: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

qjÜ

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 20 40 60 80 100

Delta lambda / pm

Doppler profileLorentzian profile

Voigt profile

È(PSuj{Q_b]����@q\Ö�ÝXÞz߬à�ázâ«ã¼ä�å·ýV߬è´ë§äLú·ê\ç��}÷zè�äLí�ã¥ì¯ázè�äÁ·â«ãdêQß�êQé¥ß�â«é¥úzâÛßVæ�ãb÷¯ø��Eî�ïAî�ð�ñ>òdó·ôrõ�ä¬æ´ã¼æ´ç·ßVæÔß�â§â>ázè´ä²ÁZâ§ãbê߬è´ã¯ì·ä�è´à�ß�â§ëÛê�ãd÷�ß�ì·÷�ã¥ýVß�â§ú·ß¬æ�ãb÷�å ä�è�æ�ç·ã�ê´ß¬à�ã|ê�ã¥æ(ä�ågázâÛß�ê�à�ßpá·ß¬è±ß¬à�ã¥æ�ã¥è±êdôµ�çzãÄÃ�äLè�ãdìcæ�ü¥ëÛ߬ìhö�ë«÷}æ´çhëÛê3ê�ã¥ææ�ä�ç·ß�âþå\æ�ç·ãE÷>ä�ázá·â§ãdè�ö�ë«÷}æ´ç�ß�ê�ëÛê�æ�çzãiÃrä�è´ã¥ìcæ�üdë«ß�ìUé¥ä�à�áZäLìzã¥ìcæ�ä¬å\æ�ç·ã�ý�ä�ë«í¬æ�ázè´ä²Á·â«ã�ôRdkroqoM]�[^_¥kMiZl�[^OQ]pYQ_dijeQsf]Lo PfR¯]� �YQsfinPS[^]L`MPStM[bOXWZ[~PS[~PfR¯ijtQsSk¾t\]cTc]LRbRbWn_dkq[^i�]L�ZWnsf{QWZ[^]%[^OQ]UTcijtr°[^_dPfeQ{�[^PfintMlm_bino intQ]�Rd[b_bPSY����at<]Lsf]coq]ct�[|inlARb{Q_¥l�WnTc]�WZ_b]}W�PfR|TLijtQRdPf`Q]L_b]c`ÄWnsSijtQu�WvujPS�n]ct�R¥[^_bPSYWntQ`�PS[^R¯Tcint�[b_bPSeQ{\[^PSijt¾[^i�[bOQ]a_bWn`QPfWZ[^PSijtäX]csS`<WZ[|Ø%]��ZWnsf{XW·[^]c`®�3NJOQ]aTLijt�[^_dPfeQ{\[bPfijtqlm_bijo)]cWnTVO]csS]coq]ct�[hWnsSijtQu�[bOQ]¾Rd[^_dPfY PSR�[^OQ]LtÅy~ij_d©n]c`�ij{\[c��y%PS[^OÅWZt�]LtQ`�Tcij_d_b]cT�[^PSijt {QRb]L`�[biGWnTcTLij{Qt�[lmij_�[bOQ]��ÔWZ[%]ct\`QRUiZl(egin[^OÄuj]LijoM]�[^_dPf]cRL�

:<;Å Æ Dª� | HÂB3ÌhB�KcDq@ D �Ç|�Dª| HÈ~cÌ�B3KcDq@ Î�H Î3K}@.¡ Ì�É Î(DqCs|ÂB3KcDq@�nÌvIJB3DqCEÎ

NJOQ]%oMWnPftvY\{Q_bYgijRb]%inlg[^O\]JTcir`Q]%PfR3[^i�WZsfsfi>y"ijYXWZTcPS[wk�oMir`QP�¤X]c`ÃYgijYQ{QsfWZ[^PSijtv`QPSRd[b_bPfe\{\[^PSijtQR�[bieg]v]L�ZWnsS{XWZ[^]L`¢y%PS[bOQPft¢�a�p���¢TcijsSsfPfRdPfijtQWns�°±_^Wn`QPfWZ[^P �j]aTcir`Q]cRL��NJOQ]LRb]¾Tcir`Q]LR�WnRbRd]coveQsS]ÃWnsSs�[^O\]TcinsfsfPSRbPSijtXWns��n_^WZ`QP²WZ[bPS�n]UWZtQ`�RdY¼ijt�[^WntQ]cin{QR|_bWZ[^]LR|eg]L[wy~]c]ctGW·[^ijoqPfTJsS]L�j]LsfRL�\Wnt\`M[^O\]ct�RdijsS�n]U[^O\]RdkrR¥[^]co [biGYQ_b]L`QPfT�[hWGYgijYQ{\s²WZ[bPfijt¢R¥[^_d{QTL[b{Q_b]Z�ÃNJOQ]¾PftQ`QP��rPf`Q{XWZs(_^W·[^]cR�WZ_b]Ã]cP�[^OQ]L_�]L�ZWnsS{XWZ[^]L`lm_bino.Wnsfuj]LeQ_^WZPfT�]� �YQ_d]cRbRdPfijt\R�ij_U[^Wn©n]LtÂlm_bino8[^OQ]�e¼]LRd[�W}�ZWnPfsfWneQsS]�ijeQRb]L_d�ZWZ[bPfijtXWZsrÊZTcWnsfTL{Qs²WZ[b]c`

Page 78: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

qn�`XWZ[^W\�(»�t\]~[wkrY¼]~inlQPStQYQ{\[�`XWZ[^WZ¤XsS]¯PSR�©�tQi>y%tÃWZR3Wnt�Wn`\l�Æn�%¤Xsf]Z�Z[bOQ]cRd]|ijY\[bPfTcWnsfs�ka[^O\Pfth`XWZ[VW·¤Xsf]LRT}WZt¢e¼]hTcint��n]c_¥[^]c` [bi�WnTLTcij{\t�[almij_%ijYXWnTLPS[wkÄ]�K¼]cT�[^Ra{\RbPft\u����%���Â���}�Q�JNJOQPSRUPSRU`\ijtQ]vWZ{\[^iZ°oMWZ[^PST}WnsSsSkÄPft [^O\]MTLi�`Q]Z�A]}WnTVO�[^_^WZtQRbP�[^Pfint�eg]cPStQuG_b]cWn`�PSt�lm_dijo&[bOQ]qPftQYQ{�[�¤Xsf]Z��P�[^R��p°´�ZWnsf{Q]oqi�`QP�¤X]c`Äe�kG[^OQ]�WnYQYQ_dijYQ_bPfWZ[^]h£2WntQ`xy%_dPS[d[^]ctÄ[^iq[^OQ]�ij{\[^Y\{\[�ɧijY\[bPfT}WZsfsSkM[^OQPSTV©ÔÊ®Wn`�l�Æn�q¤Qsf]n��asSsEinlJ[^OQ]M]� �PfR¥[^PStQuÂ�a�p����Tcir`Q]cRvWnRbRd{QoM]M[^OXWZ[h[^O\]�YQs²WZRboMWÄPSR�ijY\[bPfTcWnsfs�kÂ[^O\Pft��([bO�{QR�[^O\]WneQRdij_bY�[^Pfint¾l�WZTL[^in_E[b]cTVOQt\Pf��{Q]pinl�oqi�`\PSl krPftQu�[bOQ]^��PftQR¥[^]cPStM�%°±�ZWnsS{Q]cR3y%P�[^Oq£$l�WZTL[^in_bREPfR�PS`Q]}WnsPft�[^OXWZ[3PS[�WZsfsfi>y%R�[^OQ]aɧijY\[bPfTcWnsfs�k�[^OQPSt�ÊZTLijsfsSPfRdPfijtXWZs�°w_bWn`QPfWZ[^P �j]3TLi�`\]cR3[^i�Y\_birTc]cRdR�ijYXWnTLPS[wk�oMir`�°PS¤Q]c`ÄWn`\l�Æn�äXsS]cRpWntQ`Äuj]LtQ]c_bWZ[^]hWv[^OQPSTV©�YgijYQ{QsfWZ[^PSijt�Rd[b_b{QT�[^{Q_d]n�

:<;$Ë Ì Î3K}@.¡ B3Ð¢Ñ I%DGF¢Ñ Ì�Î&Ì ~}K}@¢Ñ C�ÌhB�KcD F KcÌ�¡M@¢DqÎ�B3K}I���%�������}�vOXWZRJWnt�]¬ �[b]ctQRdPfijt�lm_dijo)[^OQ]Uij_bPSujPftQWnsXTcir`Q]ay%OQPSTVOGWnsfsSi>y%R¯[bOQ]�{QRd]c_|[bivRbYg]cTLPSl kMW_^WZtQuj]|iZlXTcijsS{Qoqt�`Q]LtQRbP�[^Pf]LR�lmij_Ay%OQPSTVOv[^ia_d{Qt�[bOQ]JWneQRdij_bY�[^Pfint�l�WnT�[^ij_�T}WnsSTc{QsfWZ[^PSijt��ANJOQ]�TLi�`\][^O\]ctÅ]L�ZWnsS{XWZ[^]LR�WneQRdij_bY\[bPfijt¢l�WZTL[^in_bR�lmij_�]cWnTVOÅTcijsS{QoMt `\]ctQRdPS[wkj�AYQ_di�`\{QTcPStQu<Tcij_d_b]LRbYgijtQ`QPStQuWn`\l�ÆZ��¤Xsf]LR~WZtQ`¾ijYXWnTLPS[wkvoqir`QPS¤X]L`ÃY¼inYQ{Qs²W·[^Pfintv`QPfR¥[^_dPfeQ{\[bPfijt\RJ�m�rPfW��a�p���¾���}�QÊþR�i>y%tMR¥[VWZ[bPfR¥°[^PST}WnsgeXWnsfWntQTL]�Rb{Qe\_bij{\[bPftQ]}����¹±[JPSR~[bOQ]ctÄW�Rdo�WnsSs¼oMWZ[d[^]c_¯[^iÃsSi�ij©Ã{\YG[bOQ]�]LRbT}WZY¼]�YQ_dijeXWne\PfsfP�[wk]� �YQ_d]cRdRbPfintxW·[J]}WnTVO<TLijsf{\oMt�`\]ctQRdPS[wk�lmin_~[wy~iÃTVOQijRb]LtÄ[b_^Wnt\RbPS[bPfijt\R|Wnt\`<{QRb]�]L�r{QWZ[^PSijt��¥�j�þ��q��[^ipy|in_b©hij{\[�[^OQ]~ijYXWnTLPS[wk�oqi�`QP�¤X]c`�sfPStQ]E_^WZ[bPfiQ�ANJO�{QR3ijt\]|T}WZtv]L�ZWnsS{XWZ[^]~Wnt�khujPS�n]ctv]LoM]L_buj]Lt�[�X{� <sfPft\]�_^WZ[bPfi¾WnR%WÃlm{QtQT�[^PfintÄiZl(TLijsf{Qoqt<`Q]LtQRbP�[wkj�|��]c]�¤Xuj{\_b]����«¸vlmin_%Wntx]¬ \WnoMY\sf]�inl�[^OQPSRc�NJOQ]MoMir`QPS¤Q]c`ÅY¼inYQ{Qs²W·[^PfintQR�y%OQPfTVO�Wn_b]q]L�ZWnsS{XWZ[b]c`�Wn_b]¾lmij_�ijtQ]qY¼inPft�[hPftÅ[^OQ]qYQs²WZRboMWÄWZtQ`PS[aPfRaWnRbRd{QoM]L`¢[^OXW·[a[^O\]Lk¢_d]coMWnPftxTcijt\Rd[VWZt�[hWZsfijtQuq[^OQ]�sfPft\]�inlERdPfujO�[c��NJOQPfR�PSRatQiZ[�]ct�[bPf_b]LsSkTcintQRbPSRd[^]Lt�[�y%PS[bO9[^OQ]xRb]Lsf]cT�[^PSijtÓinlhW�oqir`QPS¤X]L`­{\YQY¼]L_�sf]��j]Lsa`Q]LtQRbP�[wk9`QPSRd[b_bPfe\{\[^PSijt­PSt9[^O\]WneQRdij_bY�[^Pfint­l�WnT�[^ij_MT}WnsSTc{QsfWZ[^PSijt�� NJOXWZ[�PfRL��[^OQ]x{QYQYg]c_�sS]L�j]LsUYgijYQ{\s²WZ[bPfijt"oMir`QP�¤XT}WZ[bPfijt$PSR[VWZ©n]ct9Pft�[^i�WnTcTLij{Qt�[�PSt­[^O\]x]��nWZsf{XWZ[bPfijt"inl�£ÍWntQ`Ó�v�JeQ{\[MtQin[MPft­[bOQ] V;� 4 {QRd]c` PSt­[^O\]]coq]c_duj]ct�[E�X{� qTcWnsfTL{Qs²W·[^Pfint���¿ÀP�[^OQPStÃ[^OQPSR�sfPfoqPS[^WZ[^PSijt�[bOQ]%sfPStQ]J_^WZ[bPfi�ijY�[^PfintqiZl®[^O\]pTLi�`\]UTcWntYQ_diz�rPS`Q]vWntx]cR¥[^PfoMWZ[b]hiZlEWnt�kÄujPS�n]ctÂ_bWZ[^PSiqlmin_UPSoMoq]c`QPfWZ[^]�TLijoqYXWn_bPSRbijtGy%PS[bOÂije\Rb]c_¥�ZWZ[^PSijtXWns`XWZ[^W\���UsS[b]c_btQWZ[^P��j]cs�kÓ[^O\]�{QRd]c_xT}Wnt2]L�ZWnsf{QWZ[^] [^O\]�oqi�`\PS¤X]L`7YgijYQ{\s²WZ[bPfijtÀ`Q]LtQRbP�[wk7Wnsfijt\u[^O\] sfPft\]¢inlhRbPSujO�[�[^OQ]LoMRd]cs��j]cRÄWnt\`7Yg]c_¥lmij_bo [^OQ] PSt�[^]cun_^WZ[b]c` �X{�  WntQ`ÀOQ]ct\Tc]ÅsfPStQ]Â_bWZ[^PSiT}WZsfTc{\s²WZ[bPfijt�[bOQ]coqRb]LsS�n]cRc�

Page 79: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

q�¶

È(PSuj{Q_b]U���«¸�Ö�ÝÔÞz߬à�ázâ«ãAä¬åjæ�ç·ã�ã¥à�ãdè�íLã¥ìcæ)Í·ú>ÞJâ«ë§ì·ãAè´ß¬æ�ë«äEã¥ýVß�â§ú·ß¬æ�ë«ä�ìJý>êgé¥ä�â«úzà�ì%÷>ã¥ìZê�ë§æ��|å è�äLà�î�ïAî�ð%ñzò¥óZôµ�çzã�è±ßVæ´ë§äcê�߬è´ã(ãdýV߬â«ú·ßVæ´ãd÷Uå ä�è�ç��>÷>è´ä�í�ãdìpö�ë§æ�ç�æ´çzã�ë§ìZ÷>ã¨Þ>ãdê�é¨ä�è´è´ãdê�áZäLì·÷>ë«ìzí¯æ´äJúzázánã¥è�ß�ì·÷aâ§äVö®ã¥è�á·è�ë«ì·é¨ë«á·ß�âÎ ú·ß¬ìcæ´úzà*ì}úzà¯ønã¥è±ê¥ô�µ�çzã(ázâÛß�ê�àJßEá·ß�è´ß�à|ã¥æ�ãdè´ê¼ß¬è´ãAæ�ç·äLê�ãAå è´ä�à-æ�ç·ã�å ú·ê�ë«ä�ìU÷zë§ýLã¥è�æ�ä�è®êmæ´ú·÷¦�Jä�å�ê�ãbéwæ´ë§äLì�ò�ô Ïzô ñ

Page 80: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

qOq:<;$Ð À�ÑÄÌ�� |È~cÑ)D ��B�ТÑ)H Î(Ñ�Dª��B3ТÑ)I%DGF¢ÑÒ�jDqC Ó ÑÔ~}Kc@¢ÑÍKcF¢Ñ�@ÖÕ

B3K�¼�I%Ì�B�KLDM@ H Î3K}@.¡ Æ�× { Ø<H KLÑaB Î(H @ DªÉ Î(Ñ�C��hÌ�B3KcDq@ Î�+{\Rb]Llm{\saPfsSsf{QR¥[^_bWZ[^PSijt$inl����%���9���}�-PSRqlmij{\tQ`ÓPft9YXWn_d[Minl�[bOQ]ÂRdY¼]LTL[^_bWnsasSPftQ]ÄPf`\]ct�[^P�¤XT}W·°[^PSijtÂy~ij_b©xinl¯ª|_di�ij©�R�]�[hWns´���¨�>µjµjµ��¬��¹wt¢[^OQPSR�Rd[b{Q`\kx[^OQ]qº|ij_dijtXWns���P²WZujtQijR¥[^PfT���Yg]cT�[^_binoM]¬°[^]L_¾�´º����\�pPft\Rd[^_d{Qoq]ct�[�ijt�egi�Wn_d`Â[^OQ]Ã�Q» ¾ »/RbYXWZTc]cTL_^WZl [�y|WnR�{\Rb]c`¢[bi�ije\[^WnPft¢W�TcijoqYQsS]L[^]RbYg]cT�[^_d{Qo8inl�[^OQ]���{QPS]L[%Rd{Qtx{\RbPft\u¾[bOQ]�Pft\Rd[^_d{Qoq]ct�[}ÊþR�lm{Qsfs¼y�W}�j]Lsf]LtQun[^OÄ_^WZtQuj]q��ÜnÆj¦Hj-Ün¦\�FÙS �¶��>Ü�j­qjÜjÜ ÙS ���j[bOQ]%WnPfoeg]cPft\u�[bi�Pf`Q]Lt�[bPSl k�WnR�oMWnt�kvRbYg]cT�[^_bWnsXsfPStQ]cREWnR�YgijRdRbPfe\sf]n���at¾]� \WnoqYQsS]inl%WZt�ijeQRd]c_¥�j]c`*RdY¼]LTL[b_^Wns¯_b]LujPfijt�PfRhRbO\izy%t�Pft�¤Quj{Q_b]����§¦Äy%P�[^O�Rdijoq]Minl|[^OQ]qsfPStQ]¾Pf`Q]Lt�[bPS¤\°T}W·[^PfintQRvoMWn_b©Z]c`����r[b_bPfT�[vTL_bPS[b]c_dP²Wxy~]c_d]<{QRb]L`­PSt�[^OQ]�Pf`Q]Lt�[bPS¤XTcWZ[^PSijt-inl%[bOQ]�sfPStQ]cRL�ÅÈ(Pf_dRd[bsSky|W>�n]csS]ctQun[bOGTcijPStQTcPS`Q]ctQTL]Uy%P�[^O<WnTLTc]LY\[^]L`G�ZWnsS{Q]cR¯y�WZR�Tcijt\RbPf`\]c_b]L`��XWnt\`�lmij_¯sfPft\]cR~y%O\]c_b]U[^OQPSRTcinPftQTLPf`Q]LtQTc]�y�WZRaujiri�`®��É«YgijRbP�[^PSijtxYQWZ[b[b]c_bt\RcÊÔTcij{\sf`Âeg]h{QRb]L`Â[^iMTcijt\¤Q_bo�WntÄPf`Q]Lt�[bPS¤XTcWZ[^PSijt��NJOQ]LRb]%Y¼inRbPS[bPfijt�YXWZ[d[^]c_dtQR�_b]�lm]c_([bi�Wa[^]LTVOQtQPf��{Q]�¤Q_bRd[EPSt�[^_bir`Q{QTL]c`ÃPSt�[^OQ]%WZtXWnsSkrRdPfR(iZl¼`QWZ[VWUlmin_[^O\]v��YQWnTc]Ls²WneÂ�¾º ¾ ���"�À]¬ �Y¼]L_bPfoq]ct�[pPftÄÇAWntQu\�ÛÚÂWnRbint.ÜÒÚxT�¿ÀOQPf_¥[^]L_��¨�>µjµjÆj��WntQ`Â]Ls²WnegiZ°_^W·[^]c`¾PStê|_birij©�R�]L[¯WZs´�\�¥�>µjµnµ����(��Yg]cTL[b_^W�y~]c_b]J[^Wn©n]Ltqi>�n]c_¯W�_^Wnt\uj]JinlgRbYXW·[^P²WZs\Wnt\`¾[b]coqY¼in_^WnsYgijPft�[^R�WntQ`ÅWnRdRbPfuntQ]c`ÅW�{\tQPf��{Q]ÃYgijRbP�[^PSijtÂPStQ`Q]� ¼��NJO�{QR�[^O\]Ã`QPBK¼]c_d]ct�[�ijeQRd]c_¥�nW·[^PfintQR�_b]��X]cT�[[wkrYQPfTcWnsÔ��{QPf]�[|Rd{Qt��ZWn_bPfWneQPSsfPS[bPf]LR(Pftq`Q]LtQRbP�[wkj��[b]coqY¼]L_^WZ[b{Q_b]Z��Rd[^_d{QTL[b{Q_b]Z�s�¯WZTVO�RbYg]cT�[^_^WZs¼sSPftQ]JPSR[^O\]ct<_b]L`Q{QTL]c`<[^iÃW�RbPft\ujsf]%�ZWnsf{Q]Ue�kM[^Wn©�PftQuvP�[^R¯Pft�[^]LtQRbP�[wk�PSt�Tcin{Qt�[^R�`QP��rPf`Q]L`�e�k�P�[^R�W}�n]c_^WZuj]i>�j]L_�WnsSs�Y¼ijRdPS[bPfijtQRL��NJOQ]LRb]vYgijPft�[bRaWn_d]h[bOQ]ctxYQsfiZ[b[^]L`ÂsSiju�Wn_dPS[^O\oMPST}WnsSsSkqWnu�WZPftQR¥[UYgijRdPS[^PSijtGPft�°`Q]¬ ®�U¿ÀOQ]ct¢[bOQ]�YgijRbP�[^PSijtÂYXW·[b[^]L_btQRplmij_aPS`Q]ct�[bPS¤X]L`¢sfPStQ]cRUWn_b]�Wn_b_bWntQuj]L`¢Pftxij_b`\]c_�inl3RbPfoqPfsfWn_YXWZ[d[^]L_btQRL��WZtQ`Ås²Wneg]csSsf]c`xy%PS[^O [bOQ]¾PfijtQR�[b]coqY¼]L_^WZ[b{Q_b]Ä� lm_bijo.[^O\]qYg]}WZ©¢inl¯PS[bR;Ý��rÞ � �b�¬��PS[�PSRlmij{Qt\`Â[^OXW·[U[bOQ]c_d]vWn_d]�RbPSoMPSs²Wn_|YXWZ[b[b]c_dtQRUlmij_psfPStQ]cRp]coqPS[d[^]L`xe�kÄ[^OQ]hR^Wnoq]�PSijtxWntQ`Âlmij_psfPStQ]cRlm_bino!PSijtQR(lmij_doM]L`ÃWZ[EWZYQYQ_bi} �PfoMWZ[b]csSk�[bOQ]�R^WZoM]|[b]coqY¼]L_^WZ[b{Q_b]Z�3NJO�{QR3[^O\]JY¼ijRdPS[bPfijt�YXWZ[b[b]c_dtQRinl¯{QtQPS`Q]ct�[bPS¤X]L`�sSPftQ]LR�T}Wnt�WnT�[hWnR�W�¤XtQun]c_bY\_bPft�[�PStQ`QPfTcWZ[^PStQu�[bOQ]ÃYQ_bineXWneQsS]v[^]coqYg]c_^W·[^{Q_d]vinllmij_do�WZ[bPfijtxinlE[^OQ]ÃsSPftQ]Z�¼[bOr{\R�WZPf`QPStQu�PSt PS[^RaPf`Q]Lt�[bPS¤XTcWZ[^PSijt���¹±[�y|WnR�Rbijoq]L[bPfoq]cRUY¼ijRdRbPSeQsf]�[bi{QRd]�sfPft\]aPft�[b]ctQRdPS[wkM_^WZ[bPfijR¯[^iU¡d{Q`Qun]�y%OQ]L[bOQ]c_Jij_|tQin[JsSPftQ]LR|ij_dPfujPStXWZ[^]L`Mlm_dijo [bOQ]�R^WZoM]�]LoMP�[d°[^PStQuvPSijt<WntQ`<PSt�[^O\PfR|y|W}k�lm{Q_¥[^OQ]L_�Tcijt�¤X_boÍTc]c_¥[VWnPSt<Pf`Q]Lt�[^PS¤QT}WZ[bPfijtQRL�3��eQ_dPf]�lA`Q]cRdTc_bPSY\[^PSijt�PSRYQ_d]cRb]Lt�[^]c`�eg]csfi>y�ijtx[^OQ]v{\Rb]vinlE�a�p��� ���}��PSt¢WnPS`QPft\u�PftÄ[^OQ]�Pf`Q]Lt�[bPS¤XTcWZ[^PSijtÂiZl�lmij{Q_ ¾ ]v¹sfPStQ]cRL�ÈX_bijoÀy|W>�n]csS]ctQun[bO�TLijPft\TcPf`\]ctQTL]EWntQ`�[^OQ]�RdPfoqPfsfWn_bP�[^Pf]LRÔinl�[^OQ]LPf_�YgijRbP�[^PSijt�YQsSin[^RL�·P�[®y|WnRARd[b_bijtQunsSkRb{\RbYg]cTL[b]c`Â[bOXWZ[Jlmij{\_JijeQRb]L_d�n]c`¢sSPftQ]LR�y|]L_b]

Page 81: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

q�¸

È(PSuj{Q_b]����§¦\ÖÃÝXÞzß�à|á·â§ã%ä�å�æ�çzã%äLø·ê�ã¥è´ý�ãd÷ ß®ï3ðvê�ánãdé¨æ�è´úzà4ú·ê�ãb÷vë§ì¾æ�çzã%â«ë«ìzã%ëÛ÷>ã¥ìcæ�ë@ÁZédßVæ�ë«ä�ìÃö®ä�è´ÿjùræ±ß¬ÿLã¥ìå è´ä�à"û®è´äcäLÿ>ê¼ã¥æ�߬âfôp�mò����Lwô

Page 82: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

qj¦×N¾ ]�à��gá � ·�â � j �¦á>��ã ·�ä · ¶j�j�"�«�j¶j¸ ÙS×N¾ ]�à��gá � ·�â � j �¦ázÜã ·�ä · ¶nÜ�¸"�§Æjµ\� ÙS×N¾ ]�à��gá � · â � j �¦á·�²ã · ä · ¶n¦n�C�§ÜOqjÆ ÙS×N¾ ]�à��gá � · â � j �¦á·�²ã � ä · ¶nµ\�O�þ�Q�j� ÙS

NJOQ]�¶j�n����¶ZÜ�¸��pWntQ`7¶n¦Z�*sfPft\]cRMy~]c_d]�WnsfspRd[b_bijtQu"Wnt\`Ó[^OQ]L_b]¢y|WnRGtQi*`Qij{Qe�[ÄWne¼in{\[�[bOQ]cPS_Pf`\]ct�[^P�¤XT}WZ[bPfijt¢��Rd]c]�¤Xuj{\_b]��r�«¦v{QY\Y¼]L_�YQsfiZ[~lmij_¯[^OQ]�¶n¦n�vsSPftQ]}����Pft�l�WZTL[|[bOQ]cRd]�sfPStQ]cR¯y|]L_b]�{QRd]c`WnR3TcWnsfPSeQ_^W·[^Pfint�sfPStQ]cR�Pftv[bOQ]�Rd[b{Q`\kn��NJOQ]p¶nµ\��PSt�[^]c_dTcijo�eQPftXW·[^Pfint�sfPStQ]|intv[^OQ]Jin[bOQ]c_(OXWntQ`Ãy|WnR�j]L_dk<y~]}Wn©��mRb]L]h¤Xun{Q_b]����§¦Ãsfi>y|]L_%YQsfin[V��WntQ`Ä]L�rPf`\]ctQTL]�inl�[^O\]hTLijtQRdPfRd[b]ctQT�kÄinl�[bOQ]�ijeQRd]c_d�n]c`Pft�[b]ctQRdPS[^PS]cR�y|WnRptQ]c]L`Q]c`Älmij_%WÃ`Q]L¤Xt\PS[^]�PS`Q]ct�[^P�¤XT}W·[^Pfint��3NJOQ]hijeQRd]c_d�n]c`¢sSPftQ]aPft�[^]LtQRbP�[wk<_^WZ[bPfijRinlg]}WZTVOMsSPftQ]¯[^i�[^OQ]p¶Z¦n��sfPStQ]|PSt�[b]ctQRdPS[wkvWn_b]JRdOQi>y%tMPStÃTcijsS{QoMtv[wy~i�inlÔ[VWne\sf]%�����j���asfRdi�RbOQi>y%tPfR�[bOQ]¾YQ_b]L`QPfT�[^]L`�ijY\[bPfTcWnsfs�kÄ[bOQPftÅPSt�[b]ctQRdPS[wk _bWZ[^PSi���Tcinsf{Qoqt¢[^OQ_d]c]}��y%O\]c_b]¾[^O\]MYgijYQ{QsfWZ[^PSijtQRy~]c_b]�]L�ZWnsS{XWZ[b]c`Àlm_bijo ���%���¼�J¹±[ÄT}WZt2eg]ÅRb]c]Lt2[bOXWZ[<[^O\]�ije\Rb]c_¥�j]L` = �´¶nÜj¸j��å = ��¶n¦n����WZtQ`= �´¶j�n�j��å = ��¶n¦n���J_^WZ[bPfijRasfPf]�TcsSijRb]v[bi�[^OQ]LPf_�ijY�[^PfTcWnsfs�kG[^OQPSt¢�nWZsf{Q]LRc�®y%OQPSsf]�[bOQ] = �´¶nµ��z��å = ��¶n¦n���_^W·[^PfiMRbPSujtQPS¤QT}Wnt�[^s�kG`Q]L�rPfWZ[^]LRalm_bijo.P�[}�p¹±[�y|in{Qsf`¢eg]�]¬ �Y¼]LTL[^]L` [^OXWZ[U[^OQ]�Rd[^_dijtQu�sSPftQ]LRÃ��¶n¦n�Q�¶nÜ�¸ÃWnt\`¶n�j�j�~y~ij{Qsf`Geg]�ijY\[^PST}WnsSsSk¾[^O\PfTV©<Wnt\`xRdiqOQW>�n]�WÃ_d]c`Q{QTL]c`x]coq]c_bun]ct�[%�X{� <lm_dijo�[^O\]Rb{\t��U¹±l�[bOQ]LkÄOXW}�n]vRbPSoMPSs²Wn_�inYXWnTcP�[^PS]cR%[^OQ]�kxoMW}kG]� �Yg]c_bPS]ctQTL]vRbPSoMPSs²Wn_JPSt�[^]ct\RbPS[wkG_d]c`Q{\TL[^PSijtQRL�oqiz�rPStQuv[^OQ]LPf_J_bWZ[^PSijRJeXWnTV©�[bi¾[bOQ]�]� �Yg]cTL[b]c`Ä[^OQPStG�ZWnsS{Q]cRL�N�i�Rd]c]ÄPSlpijYXWZTcPS[wk�Tcij{\sf`$]� �YQsfWnPft*[^O\]<`Q]L�rP²W·[^Pfint"iZlp[bOQ] = �´¶nµ\�>��å = �´¶Z¦n���h_bWZ[^PSi¢lm_bijo [^OXW·[ijeQRd]c_¥�j]c`®�J���%���9�r�}��y|WnRM{QRb]L`­[^i�_b]�[^{Q_dtÓijYXWZTcPS[wk*WntQ`9sSPftQ]<_bWZ[^PSiÅ]cRd[bPfoMWZ[^]LRc�­�&`QPSRb©uj]LijoM]�[^_¥k-y�WZRMTVOQinRb]ct"y%PS[bO9W s²WZ_buj]GWnRdY¼]LTL[M_^WZ[bPfi¢[bi�RdPfov{QsfWZ[^]�W�tQ]cWn_¾Pft\¤QtQPS[b]<Rbs²WZe$lmin_[^O\]�Rdijs²Wn_pWZ[boMinRbYQOQ]L_b]Z�%��ijYQYQsS]c_%YQ_biZ¤Xsf]LRpy~]c_d]v{QRb]L` WnsSijtQu¾y%PS[^OxWMO\ijoMinuj]ctQ]Lij{QRp`Q]ct\RbPS[wk`QPSRd[^_dPfeQ{�[^Pfint��|y%PS[bO9[^O\]Â[^]LoMYg]c_bWZ[^{Q_d]¢eg]cPft\u�[VWn©Z]ctÀWnR<�n¦�!VÆjÆnÆOæ lm_bino [bOQ]¢Yg]}Wn©"inl�[^O\]ÝM��Þ � �~lm{Qt\TL[^PSijt��QRd]c]�¤Xuj{Q_d]����«µ����O\]csfPS{Qo `Q]ctQRdPS[wkGinl~�\ç��>Æ · � ]è & � y�WZRp[^Wn©n]LtÄlm_dijo�[^O\]Tcin_b_b]LRbYgijtQ`QPStQu��ZWnsf{\]�PftÄ[^OQ]�½E]c_btQW0½L½cWG]�[�Wns|�¥�>µn¦\�z�JRbijsfWn_�WZ[boMinRbYQOQ]L_b]voqir`Q]cs��®WnRdRb{QoqPft\u[^OQWZ[Ã[^OQ]<O\]csfPS{Qo `\]ctQRdPS[wk-[bi e¼]<[^]Lt$Y¼]L_bTc]Lt�[�inl%[bOQ]GO�kr`Q_dijuj]Lt­`Q]LtQRbP�[wkj�*NJOQ]GsS]ctQun[bO*inl[^O\]�Rds²WneÀy�WZRád{Q`\uj]c`4WnYQYQ_di> �PSo�WZ[b]cs�k9lm_bino [^OQ]�OQ]LPfujO�[Glm_bijo [^OQ]�YQOQin[bijRbY\OQ]c_d]�[bi­[^O\][^_bWntQRdPS[^PSijt-_d]cujPSijt��EWnu�WZPft-[VWZ©n]ct*lm_dijo [bOQ]�½E]c_dtXW0½L½cW oMir`Q]cs���NJO\]<PftQYQ{�[qWn`\l�Æn�¢¤XsS]�y|WnRuj]LtQ]c_bWZ[^]L`¢{QRbPStQuM[bOQ]�Wn{\[^inRd[^_d{QTL[b{Q_b]�Tcir`Q]��rPfW¾[bOQ]��a�p���¢¸nÆ\�hPft�[b]c_dl�WZTc]n�J�UtÂ�%°±�ZWnsf{\]�lmin_[^O\]hPSt�[^]c_dTcijo�eQPftXW·[^Pfint�sfPStQ]�inl(Æ��2qOá & · y|WnR%[VWn©Z]ctGlm_bijo8��{QoqoM]L_bR��¥�>µj¸j¸j���

Page 83: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

qjµ

È(PSuj{Q_b]U���§µ\Ö�éT�8êpë�Qå úzìZéwæ�ë«ä�ì%å ä�èÔçzãdâ§ë«úzà-ã¥ýVß�â§ú·ß¬æ�ãb÷�ú·ê�ë«ìzí¯î�ïAî�ð%ó·ò^ñ¦ì�ì·ä¬æ�ãAæ�ç·ß¬ægæ�ç·ã�ánãd߬ÿ|ä¬åjæ´çzã�éT�8ê�ë>â«ë§ãbê�ßVæ�߬ønä�úzæAñ�gí�îîî�ï¾ô

Page 84: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¸nÆ

È(PSuj{Q_b]U�����>Æ\Ö�ݼê�éd߬ánã�å�ß�éwæ´ä�èÔá·â§ä�æ¼í�ãdìzã¥è±ßVæ´ãd÷|ú·ê�ë§ì·í¯î�ïAî(ð%ñ>òdó3å ä�èXæ´çzã�ç·ã¥â«ë§úzà-â«ë§ì·ãAë«÷zã¥ìcæ�ë@ÁZé¥ß¬æ�ë«ä�ìJö®ä�è´ÿjô

Page 85: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¸��

È(PSuj{Q_d]h���S�n�jÖ¶Ãrë§ìzã3è±ßVæ´ë§ä�ázâ«ä¬æ�í�ãdìzã¥è±ßVæ´ãd÷Uú·ê�ë§ìzíJî�ïAî�ð�ñzò¥ó~å ä�è�æ�ç·ã�çzã¥â«ë§ú·à"â«ë§ìzã�ëÛ÷>ãdìLæ´ë5Áné¥ßVæ´ë§äLìUö®ä�è´ÿjô

Page 86: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¸j�

ðURdPftQu�[bOQ]cRd]p�ZWnsS{Q]cREWntq]cRdT}WnYg]%l�WnTL[bij_�T}WnsSTc{QsfWZ[^PSijt�y|WnREY¼]L_dlmij_doM]L`��3NJOQ]%_d]cRb{\sS[^R¯Wn_d]%RbOQi>y%tPft¢¤Quj{Q_b]LR����S�}Æ<WntQ`������j�j��NJO\]�¶Z¦n�Q��¶nÜ�¸�WZtQ`�¶j�j��sfPStQ]cR�Wn_b]ÃTLsf]}WZ_bsSkx]� r[b_b]coq]cs�k¢ijY�[^PfTcWnsfs�k[^O\PfTV©Ô�pÁUin[b]h[bOXWZ[a[bOQ]LkÄOXW}�n]�RbPSoMPSs²Wn_|ijYXWnTLPS[^PS]cRL�Ô[bO�{QRa[^O\]cPf_%_bWZ[^PSijRURdOQij{QsS`Âeg]�TLsfijRd]�[biM[^O\]ijY\[bPfTcWnsfs�k<[^OQPSt¢�nWZsf{Q]LRq��P´�þ]n�g[^OQ]LPf_UPft�[^]LtQRbP�[^Pf]LR�RbTcWnsf]ÃRdPfoqPfsfWn_bs�k�y%PS[bO ijY\[^PST}Wns�`Q]LY\[^O�Rbi�[bOQ]cPS__^W·[^PfinR�TcWnt*Rd[bPfsfsE_d]coMWnPft�TcsfinRb]M[biÂ[^O\]�ijY\[bPfT}WZsfsSkÂ[^OQPSt-_^WZ[bPfiQ��]L�n]ct-[^O\ij{QujO-[bOQ]�Pft\`QPS�rPf`\{XWnssfPStQ]cR�Wn_b]�ijY�[^PfTcWnsfs�k¾[^OQPSTV©\�¬��NJOQ]�¶nµ\��sSPftQ]�intG[bOQ]�in[bOQ]c_JOXWZtQ`GT}WntGeg]�Rb]c]Ltx[biÃe¼]�ijY�[^PfTcWnsfs�k[^O\Pft��¼WntQ`xRbi�[bOQ]�ije\Rb]c_¥�j]L`�`\]L�rP²WZ[bPfijtGlm_bino.[bOQ]�[bOQPftxsfPStQ]�PfR%[^i�eg]�]¬ \Yg]cT�[^]L`�����i�WM��{QPfTV©WnRdRb]cRdRboq]ct�[qinlJ[^O\]<]cRbTcWnYg]<YQ_bijeQWneQPfsSPS[bPf]cR�{QRdPftQuÂ���%���*���}�ÂOXWnRv��{XWnsSPS[^WZ[^P��j]cs�kÅ]� �YQsfWnPftQ]L`[^O\]hineQRb]L_d�j]L`¢_^W·[^PfinRc�NJOQ]ÃPSt�[b]ctQRdPS[wkÂ_^WZ[bPfijRpy|]L_b]v[^O\]ctÅ]cR¥[^PfoMWZ[b]c`°Jt\in[^]v[bOXWZ[�W�_bWZ[^PSi�WntXWZsSkrRbPSR�sfPS©n]�[bOQPfRaPfR�tQiZ[[^O\]¢o�WZPft9Y\{Q_bYgijRb]Âinl�[^OQ]xTcir`Q]n�pRbPStQTc]Ä[^OQ]Â]L�ZWnsf{QWZ[^]L` WneQRdij_bY\[bPfijt­l�WnTL[bij_�PSR�ijtQs�k$�nWZsfPf`WZ[�[^O\]qTL]ct�[^_d]�inl~[^OQ]¾uj]Lijoq]L[^_¥k {QtQ`\]c_hPft��j]LRd[^PSu�WZ[bPfijt�WntQ`Å[^O\]M_bWZ[^PSiÄ]��ZWnsf{XW·[^PfintÅWnRbRd{QoM]LR[^OQWZ[�P�[UPSRU�ZWnsSPf`Glmij_p[bOQ]�y%O\ijsf]�sfPft\]hiZlERbPfunO�[c�a��]LRbYQP�[^]�[^OQPSRaP�[py|WnR�{\Rb]Llm{\s�[^i�]cRd[bPfoMWZ[^]�[^O\]_^W·[^PfinR¯y%P�[^OQPSt�[^OQ]Ue¼in{QtQ`QR�iZl�[^OQ]�TLi�`\]a[^ivun]L[JWZt�PftQ`\PfT}W·[^PfintMinl�[bOQ]cPS_|eg]cOXW}�rPfin{Q_c���2_^WntQun]inl~TcijsS{Qoqt `Q]ctQRdPS[bPf]cR�Rd{Q_b_dij{QtQ`QPStQu<[^OQ]Ã�ZWnsS{Q]vlmij_�y%OQPSTVO�¤Xun{Q_b]M�����>Æ�y�WZRhun]ctQ]L_^WZ[b]c`�y~]c_d]TVOQijRd]ct<WZtQ`M[bOQ]a_bWZ[^PSijR�YQ_bir`Q{QTL]c`��ENJOQ]p_b]LRb{Qs�[^R|Wn_b]%RbO\izy%t<PStq¤Quj{Q_b]a�����j�j��Ápin[^]%[^OQWZ[¯e¼in[bO[^O\] = ��¶nÜ�¸j��å = �´¶n¦n����WntQ` = �´¶j�j�n��å = �´¶Z¦n����TL_bijRdRq[bOQ]cPS_qineQRb]L_d�j]L`9�ZWnsf{\]cRqWZ[qWnYQY\_bi} �PfoMWZ[^]LsSk[^O\]�RbWnoM]hTcinsf{Qoqtx`\]ctQRdPS[wk���ñ)�O�§Üòç-�>Æ ·Yó ]è & � �¬�JNJOQ] = ��¶nµ\�z��å = �´¶n¦n���~ijtx[bOQ]�iZ[^OQ]L_UOXWZtQ``Qir]cR�t\in[�_d]}WnTVO P�[^RMijeQRb]L_d�n]c` �ZWnsf{Q]Z�%WnsSy|W}krR�sSkrPStQu�eg]csSizy.P�[}�7NJOQPfRMPfRMtQin[�{\tQ]� �Yg]cTL[b]c`WnRvy~]ÄWn_b]<_d{QtQtQPStQu¢[^OQ]<TLi�`\]ÄPSt-[^OQ]<_d]cujPSoM]�e¼]�kjijtQ`$y%OQPSTVO"P�[ÃPfR�tQij_boMWnsSsSkÅ�ZWnsfPS`��ÅNJOQ]YQ_d]c`QPSTL[^]L` = ��¶nµ\�z��å = �´¶n¦Z���p_bWZ[^PSi�PfR�[bi�i<sSizy��®PftQ`QPST}WZ[bPftQuM[^OXWZ[ay|]¾OXW}�j]¾i>�j]L_b]LRd[^PSo�W·[^]c`Å[^O\]¹��´¶Z¦n����Pft�[^]LtQRbP�[wkj�(�UR3Wp_b]cRd{QsS[(inlQ[bOQPfR�WntXWns�krRbPfR�PS[�y|WnR(`Q]LTcPS`Q]c`�[bi�_b{\t�[^OQ]ÄÚôðaÇrNJ¹�_^Wn`\P²WZ[bPS�j][^_bWntQR¥lm]c_ÄRd[^]LsfsfWn_GWZ[^oqijRdYQOQ]c_d]ÅTcir`Q]Ó��º�Wn_bsSRbijt®�v�>µj¦Oqj��[bi­Y\_b]c`\PfTL[G[bOQ]�sSPftQ] _bWZ[^PSijRc� NJOQPSRWnYQY\_bi�WnTVO<PSR|oqij_b]UsfPS©n]cs�kÃ[^ivunPS�j]aW�eg]L[b[b]c_�]LRd[bPfoMWZ[^]Uinl�[^OQ]asfPft\]U_bWZ[^PSijR¯[bOXWnt��a�p���<���}�ÃWZRPS[({QRb]LR¯W�Rdijs²WZ_3WZ[boMijRdYQOQ]L_b]Joqi�`Q]Ls\eXWZRb]c`¾uj]Lijoq]L[^_¥kÃWnt\`Ã`Q]ctQRdPS[wkÃ`\PfRd[b_bPSeQ{\[^PSijt<�mPftv[bOQ]JT}WnRd]OQ]L_b]�[^OQ]�oqi�`\]cs�inl�½E]c_dtXW0½�½}W¾]L[UWns´�����>µn¦\�z���¶ÚôðaÇrNJ¹|y�WZRp_d{QtxWntQ`GsfPStQ]�_^WZ[bPfijR|YQ_bir`Q{QTL]c`<°Rb]L]vTcinsf{Qoqtxlmij{\_Uinl([VWneQsS]v�r�S�j�p�aRaT}Wnt¢eg]�Rd]c]Lt¢lm_bijo8[bOQ]�[VWZeQsf]�[^OQ]�YQ_b]L`QPfT�[^]L`¢_^WZ[bPfijRUWn_d]ujiri�`¾lmij_Eegin[^O¾[^OQ] = �´¶nÜj¸j��å = ��¶n¦n���3Wnt\` = �´¶j�n�j��å = ��¶n¦n����_bWZ[^PSijRc��NJOQ]ainY\[^PST}WnsSsSk�[bOQPfTV©Ãoqir`Q]csoMWZ[^TVOQ]LR�[^OQ] = ��¶nµ\�z��å = �´¶n¦n���¯_^W·[^Pfi�e¼]�[b[^]L_�[^OXWZtG[bOQ]�[^O\Pft<oqi�`Q]Ls´��[^OQin{QujOGPS[JPSR�Rd[bPfsSs®WvsSPS[b[bsf]sfi>y���aR¯W�_b]LRb{Qs�[Einlg[^OQPSREWntXWns�k�RdPfR([bOQ]%sfPStQ]|y|WnREY¼ijRdPS[bPS�j]LsSk�Pf`Q]Lt�[bPS¤X]L`qWnRiõ.% = �¦á � ·�â � jÂ�¦á·�²ã � ä ·

Page 87: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¸nÜ

öUW·[^Pfi Úx]cWnRb{Q_d]c` Ù _b]L`QPfT�[^]L`.öpWZ[^PSi»�Y\[}�3[bOQPftG_bWZ[^PSi ÚôðUÇ�NJ¹|_bWZ[^PSi

¹��´¶nÜj¸j��ÊZ¹���¶n¦n��� �¨�O�Û�k÷ÓÆ��§Ü��Fç-�}Æ & · ��Æ"�2q�j �O�§Ü��Äç��}Æ & · �¥�0�Û¶j�øç-�>Æ & ·¹��´¶j�n�j��ÊZ¹���¶n¦n��� ��Ü��2q^÷4�O�§Æ��Fç-�}Æ & � �¥�0�«¦�j*�C�«�j�Äç��}Æ & � �´¶E�«µ��øç-�>Æ & �¹��´¶nµ��z��ÊZ¹���¶n¦n��� ��µ��«¦^÷ÓÜ��«�j�Fç-�}Æ &pù �¥�0�«Æ�j �O�§¦��Äç��}Æ &pù �����«Ü��øç-�>Æ &pùN�WneQsf]Â�����jÖôú�øZê�ãdè�ýLãd÷qß�ì·÷�é¥ß¬âÛé¨ú·â«ß¬æ�ãd÷�ê�ä�âÛ߬èJßVæ´à|äcê�á·çzã¥è´ãUç·ã¥â«ë§úzà!â§ë«ìzã�è±ßVæ�ë«äLê|ö�ë§æ�ç<߬ìZ÷qö�ë§æ�çzäLú>æJæ�ç·ãã�ûjãdé¨æ´ê�ä�å�äLá·ß�é¥ëþæ��Lô¶nµ\�0�§�Q�n� ÙS �ÃN�iÄ[bOQ]qWn{\[bOQij_cÊþR�©�tQi>y%sf]c`\uj]q[bOQPfR�sSPftQ]ÃOQWnR�tQin[�eg]c]ct�ijeQRd]c_d�n]c`-eg]Llmin_b]¾Pft [^O\]Rb{\t��|NJOQ]hRbO\ij_d[pPfsSsf{QR¥[^_bWZ[^PSijt�RbO\izy%txOQ]c_d]�Y\_bi>�rPf`Q]LRaW¾ujiri�`G]� \WnoqYQsS]�inl([bOQ]�{QRb]hinl3�a�p������}�hlmij_EoMin_b]pTcijoqYQsf]¬ qYQ_bineQsf]LoMR�[^OQWnt�PS[bR¯WnRbRd{QoMY�[^PfintQR|oqPfujO�[¯WnsSsfi>y��(NJO\]�Tcir`Q]pYQ_bi>�rPf`Q]LRW*oM]cWntQRGinlh��{QPfTV©�s�k ]cR¥[^PfoMWZ[bPftQu�[bOQ]ÅijYXWnTLPS[wk ]K®]LTL[bRÄint7RdY¼]LTL[b_^Wns�sfPStQ]cR�inl�oqir`Q]c_bWZ[^]LsSkijY\[bPfTcWnsfs�k�[^O\PfTV©¾YQs²WZRboMWnRc�rWntQ`MPSl�[bOQ]psfPft\]cR~Wn_b]%lmij{QtQ`M[^i�l�WZsfsXij{\[bRbPS`Q]UiZl�[^OQ]UijY\[bPfT}WZsg`\]cY\[bOQRlmij_Ay%OQPfTVO�[^OQ]~Tcir`Q]|PSR��ZWnsSPf`��zPS[�WnsSsfi>y%RA[^OQ]~Pft��j]LRd[^PSu�WZ[bij_�[bia`Q]LTcPS`Q]~y%OQ]�[^OQ]L_(P�[(PSR�y~ij_d[bO�y%OQPfsS]`QijPStQu�W�oMin_b]JTcinoMYQsS]� vijYXWnTLPS[wkvR¥[^{Q`�kÃijtÃ[^OQ]%sSPftQ]LR�PftÃ��{Q]cR¥[^PSijt��rWnR�y�WZR¯`\ijtQ]JPftÃ[bOQPfR�TcWnRb]Z�

:<;>=�ü ýGH C3B�ТÑ�C B�ТÑ�DqC3Ò Ì�@ F I%DGF¢Ñ F¢Ñm�MÑ�~cDª|È� Ñ�@�B þ ?A@ÖÕI^~}H Î3KLDM@ Dª�2Ì É¢Ì�IFÿ ¡qC�DqH @ F C�ÌvF KLÌ�B�KLDM@ ¼ÅÑ�~}F

���������� ��� ������»�tQ]Ginla[bOQ]<¤X_bR¥[MWnRbRd{QoMY�[^PfintQR¾o�Wn`\]<Pft$]L�ZWnsS{XWZ[^PStQu¢[^O\]Ä]LRbTcWnY¼]xWntQ`­WneQRdij_bY�[^Pfint$l�WZTL[^in_]� �YQ_d]cRdRbPfint4y�WZRÄ[bOQ]�]¬ �Tcsf{\RbPfint2inlvW"eQWnTV©�uj_bij{\tQ`4_bWn`QP²W·[^PfintÓ¤X]csS`����( �YQ_b]LRbRdPfijtQR-�´�����>Æ��[^O\_bij{QunO�[^iÄ�´�r�S�}���~T}Wnt<eg]�_b]¬°w]��nWZsf{XWZ[b]c`�[biÃPftQTLsf{Q`Q]�WvTcijt�[^_dPfeQ{�[^Pfint�lm_bijo [^O\]�eXWnTV©�uj_bin{QtQ`_^WZ`QP²WZ[bPfijt¾¤X]Lsf`��EÁpin[^]a[bOXWZ[�[bOQ]�YXWZ_^WnegijsfPST%]� �YQ_b]LRbRdPfijtGPfR|tQin[�PStQTcsS{Q`Q]L`<Pft�[bOQ]ay~ij_b©MRbOQi>y%tOQ]L_b]n�Q[bOQij{QunOxP�[%PfRJRd[b_^WnPSujO�[blmij_¥y�WZ_b`<[^i¾]� r[b]ctQ`Ä[^OQ]hWnsSuj]ceQ_bW�[^iMWnsfsSi>y7lmin_JPS[}�ÈXij_�[^O\]hO\ijoMinuj]ctQ]Lij{QRJ`QPSRd[b_bPfe\{\[^PSijt�[^OQ]�sSPftQ]�PSt�[^]cun_^WZ[b]c`G�X{� �]cRpe¼]LTcijoq]

t �� � � �"!$#���% / � & � 6 M-O/21 6 9 #�� � ���Q�IG����\� � �kjJ% M-O/�1 6 ��� : = ���"!^���% & M-'/21 6 �

Page 88: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¸Z�

t ���� � � �"!$#���% / � & ��� 6 M-O/21 6 9 #�� � ���Q�IG����\� � �kjJ% � M-O/�1 6 � � : = ���"!^Æ���% & M-0/21 6 ��� �´�����>¦��WntQ`

t �� � ���"!<#��>% / � & � 6 M-�/21 6 9 #�� � ���"!^ØL�IG�� �Q� � �Tj �IG�� �Q�dØ : �IG|� �Q�dØ % & M-O/�1 6 � ��: = � �"!VÆ��>% & M-0/�1 6 �t ���� � ���"!<#��>% / � & ��� 6 M-�/21 6 9 #�� � ���"!^ØL�IG�� �Q� �IG|� �Q�dØ � �kj$�0% & M-O/�1 6 � :�% & � M-O/21 6 � � : = � �"!VÆ��>% & M-0/�1 6 ����´�����>µ��

lmij_�[bOQ]�sfPft\]}Wn_�`\PfRd[b_bPSeQ{\[^PSijt��NJOQ]LRb]h_b]cRd{QsS[%PStG]cRbTcWnYg]�WZtQ`ÄWneQRbin_bY\[bPfijt<l�WnT�[^ij_dRJ]� �YQ_b]LRbRdPfijtQR%lmij_%WÃOQijoqijuj]LtQ]cin{QRJ`Q]ct\RbPS[wk`QPSRd[^_dPfeQ{�[^Pfint<inl

� � ��nØ �'P : ��������� n����E� �Q� 9 � t �5���L� � �'PIG P � �kj�% & � M-O/21 6 ��� :�n����¯���\��% & � M-O/�1 6 ��� 9 �

£��Y#��[� ��0R t l t��� � WQ���\��% & M-O/�1 6 � / lE6 j IG����\��XP n����E���Q�>% & M-�/21 6 � / lE6 � 9 � 9Eo �´���«�nÆ��

WntQ`

� � �Ø �XP :9� � ������� n����E� �Q� 9 �ç � �it ������� � � �"!VØ��IG|���Q� �IG����\�dØò� �kj$�'% & M-0/�1 6 � :�% & � M-0/�1 6 ��� :�n����¯���Q�>% & � M-�/21 6 � � 9 �

£�� #��Q� ��0R t l t��� � _ �IG P Ø·� o � j �IG P Ø·� o � % & M-gf � / lE6 d jIG�� �Q��XP ��Ø�� n����~� �Q��% & M-O/21 6 � / lE6 � 9 � 9po �´���«���z�

lmij_aW�sfPft\]}Wn_U`Q]ctQRdPS[wkx`QPfR¥[^_dPfeQ{\[bPfijt®�gy%OQ]L_b] n����E���Q�aPSRU[^O\]ÃPftQTLPf`Q]Lt�[aeXWnTV©�uj_dij{QtQ`Å_^WZ`QP²WZ[bPfijt`Q]LtQRbP�[wk�WZ[�lm_b]c��{Q]LtQTLk¸�¼��� oqi�`\PS¤X]L`Å�n]c_dRbPfint�inl|���%�����r�}�<y�WnR�YQ_bir`Q{QTL]c`�[^ixWnTcTLij{Qt�[lmij_URb{QTVOÅW�eQWnTV©�uj_bij{\tQ` ¤X]csS`��a�UsS[^O\ij{QujOÄ[^OQ]v_d]cRd{QsS[^Wnt�[�]cRdT}WnYg]ÃYQ_dijeXWneQPSsfP�[^Pf]LR�y|]L_b]vlmij{\tQ`

Page 89: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¸j¶

[^iheg]py~]csSsgeg]cOQW>�n]c`���PS[¯YQ_diz�n]c`�[bi�eg]p`QPSÕqTc{\sS[�[^i�]L�ZWnsS{XWZ[^]%oq]}Wnt\PftQunlm{\sXWneQRdij_bY\[bPfijt¾l�WnT�[^ij_dRlmij_GWnsSs�YgijRbRdPfeQsS]�inY\[^PST}Wns�`\]cY\[bOQRc��ÈÔin_<]� \WnoqYQsf]Z��Tcijt\RbPf`\]c_<[^OQ]ÅTcWnRb]�inlÃW$YQsfWnRboMW�[^OXW·[RbO\izy%R�ijtQs�k oMir`Q]L_^WZ[b]ÃijYXWnTLPS[wkÅWnt\`�OXWZR�Wnt�Pft\TcPf`\]ct�[heXWnTV©�uj_bin{QtQ`�_bWn`QPfWZ[^PSijt¢¤X]csS`�y%O\PfTVOPfR�TLsfijRd]v[^iGeQsfWnTV©¢egi�`\kn�¾��{Q]v[^i<[bOQ]¾sfi>y�inYXWnTcP�[wk¢inl¯[^O\]qY\s²WnRdo�W\�¼[^OQ]¾_^WZ`QP²WZ[bPfijtx¤X]csS`�W·[[^O\]�Tc]Lt�[b]c_vy%PSsfsEe¼]�TcsfinRb]M[bi eQs²WZTV©�egir`\k �moMinRd[^s�kÅ`Q{Q]�[biÂ[^OQ]�eXWnTV©�uj_bin{QtQ`*¤X]csS`Ô���¢ÁUin[b]lm_bino�[bOQ]¾WneQRdij_bY\[bPfijtxl�WnTL[bij_�]� �YQ_d]cRdRbPfintQRM�´���«�nÆ��pWntQ`"�´�r�Û���>�p[^OXW·[�[^OQ]�ov{QsS[bPfYQsSPf]L_pintÈn����PfR GÂå � ��¹±lU[^OQ]<eXWZTV©run_bij{Qt\`9_^WZ`QP²WZ[bPfijt�`Q]ctQRdPS[wk*PSRÃeQs²WZTV©�egi�`\kn�~intQ]<ijtQs�k-uj]�[^R�� � ÆÅP�lG9WZtQ` � OXW}�j]¾ª|ijs�[<½Lo�Wnt\tÂYgijYQ{QsfWZ[^PSijtQRc� ¾ i>y~]L�n]c_c��[bOQ]vTcir`Q]ÃWZRbRb{\oM]LR�Wnt Pft\PS[^PfWns�]¬ \TLPS[b]c`Rd[^WZ[^]�YgijYQ{QsfWZ[^PSijt�R¥[^_b{\TL[^{\_b]�y%PS[^OG`Q]�l�Wn{Qs�[pª|ijsS[�½coMWntQt<`Q]��rP²WZ[bPfijt�l�WnT�[^ij_dR%inl�Æ\�S���my%P�[^O<[^O\]ijY\[bPfijt­lmij_M[bOQ]¢{QRd]c_�[^i*]Lt�[^]c_GW�`QPBK¼]c_d]ct�[<`Q]���PfWZ[^PSijt9l�WnT�[^ij_^��� NJOQPSR�Rb]L_d�n]cR<[^i-PStQTc_d]}WnRd]G�å � �3WnsfsSizy%PStQux[bOQ]�eXWZTV©run_bij{Qt\`$¤Q]csf`*TLijt�[^_dPfeQ{\[bPfijt�[^i��$[bi¢eg]cTcinoM]�sS]cRdRv[^OXWnt*t\]cu�WZ[bPS�n]ijtQ]Z�3_b]LRb{Qs�[^PStQuxPSt�tQ]Lu�WZ[bPS�j]�WneQRdij_bY\[bPfijtÅl�WnT�[^ij_dRc�űl�[^OQ]MRdkrRd[b]co y|]L_b]<WnsSsfi>y~]c`Å[^i¢P�[^]c_bWZ[^][^O\]ct<[^OQ]�sfWn_bun]a_^WZ`QP²WZ[bPfijtq¤X]csS`�y|in{Qsf`<PStQTc_d]}WnRd]�[^OQ]�YgijYQ{QsfWZ[^PSijt�iZl�[bOQ]�]� �TLPS[^]L`GRd[VW·[^]cR%WZtQ`©n]L]cYG[^OQ]hWne\Rbij_dY\[^PSijt�l�WnTL[bij_bR|Y¼ijRdPS[bPS�j]Z�¯��Pft\Tc]�[^O\]�`QPf_d]cTL[bPfijt<iZl�[bOQ]�YQ_d]cRb]Lt�[Uy~ij_b©MRb]L]c©�Rp[bi]� �YQsSijPS[%[bOQ]�sSPftQ]cWn_%tXWZ[b{Q_b]hinl([bOQ]�]LRbTcWnY¼]hl�WnTL[bij_pWnYQYQ_di�WnTVO¢t\iMlm{Q_¥[^OQ]L_py~ij_b©<y|WnRa`\ijtQ]hijt[^O\PfR%WnRdY¼]LTL[}�

:<;>=¾= Æ Dq@ Im~cH Î3KcDq@ Î�À`Q]cRdTc_dPfY\[bPfijt¾inl¼[^OQ]%���%���q���}��ijYQWnTcP�[wkÃTcir`Q]%OXWnREeg]c]ctMujPS�n]ctMWntQ`qPS[ETcWntMeg]%Rb]L]ctM[bOXWZ[¯PS[Pft�[b]cuj_bWZ[^]LR3y|]Lsfs\y%P�[^OÃ[^OQ]%]¬ \PSRd[bPftQu��a�p���ÃTLijsfsSPfRdPfijtXWZs�°w_bWn`QPfWZ[^P��j]�Tcir`Q]LRc�3¹±[EYQ_di>��PS`Q]cR¯W���{QPfTV©WntQ`x]�K¼]cT�[^PS�n]hy|W}kÄinl3]��nWZsf{XWZ[bPftQu¾ijYQWnTcP�[wkGWn`·¡d{QR¥[^oq]ct�[^Rp[^iqe¼iZ[^OxWZ[^ijoqPfT�YgijYQ{\s²WZ[bPfijtQR%WZtQ`]coq]c_duj]ct�[%�X{� �lm_bijo8oqir`Q]c_bWZ[^]LsSkMijY\[^PST}WnsSsSk¾[^O\PfTV©�YQsfWnRboMWnRL�

Page 90: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� � ��� ���� � � ��������� ����� ��� � ��� � ���������� ��� � �!�����

"$#&% ')(�*,+.-$/�0�12*�34-5(627985:9;<84=<8?>A@CBD7FEG:H@I84;KJMLNB?O9=P8?=QLR>�BSEUTVB?O9TWEG@IXV>9Y$LR:FEUB4XV@[Z\B?LU>9=<XV=P@I84>A@ITVZ\XV>]E^=PXW>9YUTW8C=<:_8?B`@I;IEGT=<84;<XV8?=aJM;PLRb TVL&cdTVZNXW>9Y]e9XW=PB?;<8`@I8fbg8?bihj84;�TWXV>98?=g@I79;PLRO9YR7k@<Ll@I798^B?LR>A@IXV>mO9O9b cnXo@I79XV>k@I7H88?=PBSEU:_8�JpEUB`@ILR;iJM;<EUbg84cqLR;<rtsu6279XW=iXW=Qe9LU>98�hAZ�EU>9EUTVZm@IXVBSEUTVTVZ�8wvN@I8?>He9XW>9Y�@<798�O9=POFEUTxh_LRO9>9eNyh_LRO9>9eue98`zF>9XV@<XWLR>H={LUJ.@I798|84=<B?EU:j8|:H;<LRhFEGh9XWTVXV@IXV8?=}EU>9e\EUh9=PLR;<:~@IXWLU>^JpEUB4@<LR;<=�XV>A@IL�h_LRO9>9e~y�JM;<848;<84YRXWbg8?=4s|��@�XW=�=<7HL�cn>\@I7FE�@KhHTW8?>He98?e\79XWYU7\=P8?;PXW8?=�TVXW>984=�b584;<YR8QcnXo@I7�E$hH;<LAEUeH8?>984e]B?LU>m@<XW>~yO9O9b�=Pb5LNLU@I7HTVZCEUB4;<LR=P=�@<798�=P8?;PXW8?=qTWXVb5Xo@�cnXV@<7a>9LK>98?84ea@ILKXW>A@I;PL~eHO9B?8}EU>aXVLR>9XV=<XW>HY�bgXWB4;<LUz98?TWecn79XVBD7�84�j8?B4@<XV�U8?TVZ���e9XW=P=<LRTo�R84=?�,79XVYR7]�mOFEG>m@<O9b�=<7984TWTW=KXW>A@IL�@I798aB?LR>A@IXV>mO9O9b�s^62798�B4LR>A@IXV>NONyLRO9=�84=<BSEG:j8C:9;PLRhFEUh9XVTWXo@IXW84={EU;P8C84�GEUTWO9EG@I84e\XV>uE�>984c�B4L~eH85EG>9euO9=<84e�@IL�EUe��PO9=P@�79XVYR7�=<84;<XW84=EU>9e�h_LRO9>9e~y�JM;<848�B4LR>A@IXV>NOHO9b 8?bg8?;PYR8?>A@}XW>A@I84>9=<Xo@IXV8?=xJMLR;2LR:9EUB?Xo@[ZRsq�~XWbgXWTWEU;<ToZ|@I798�B4LR>A@IXW>mO9LUO9=EUh9=PLR;<:~@IXWLU>aJpEUB4@<LR;<=xEU;<8}O9=<84e$@<L|b5LNe9XoJ�ZC79XWYR7a=<8?;PXW84=x=<:_LR>A@DEU>H8?LRO9=284b5XV=<=PXWLR>aEU>9e�JM;<848{8?TV8?Bwy@I;PLR>�;<8?B4LRbQh9XV>FEG@<XWLR>$;IE�@I8?=2JMLR;n:_LR:9O9TWEG@IXVLR>$BSEUTVB?O9TWEG@IXVLR>�XV>^EU>�EU>FEGTWLRYRLUO9=2baEG>9>98?;2@<Lg@<798?XV;h_LRO9>9e~y�hjLUO9>9e�B4LRO9>A@I84;<:FEU;�@I=4s�627HXW=�BD79EU:H@I84;�B4LR>9=<XVe98?;P=xLR>9ToZC7AZ~eH;<LRYR84>9XWB�LR:FEUB4XV@[ZR�mcnXV@<7a@I7H8;<84B?LRYU>9XV@<XWLR>�@<7FEG@2@I7H8�@<798?LU;PZ$BSEG>�h_8�8`vN@I84>9e984e^@ILC>9LR>Ny[7AZNe9;<LRYU8?>9XVB�84TW84b584>m@<=?s"$#w� ��(f�Q `¡�*�3S1 1¢-g(�*�3?(�0�3`*£¡¤�v~:9;<84=<=<XVLR>9=g¥§¦Rs©¨Rª}EU>9e�¥§¦Rso¦S«mª}EG;<8K@<798Q=�@DEU;�@IXW>HY�:jLUXW>A@I=¢JMLR;�e984b5LU>9=P@<;IEG@<XW>9YaEU>FEUToZN@<XWB�B?LR>Ny@IXV>mO9XV@[ZUsK�[>�@I7H8QJMLRTWTVL�cnXV>9Y5EUTWYR84h9;IE~�£hjLUO9>9e~y�hjLROH>9e��mOFEU>A@IXo@IXW84=�EU;<8ie98?>9LG@I8?e\8?Xo@I7984;{hAZf@I7H8

¬U­

Page 91: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¬R¬=<OHh9=<B4;<XW:~@I=n�©>9>®�NLU;n��h9h£�o�~EU>9eC@I798¢hjLUO9>9e~y�JM;<8?8n�mOFEU>A@<XV@IXV8?=�cnXV@<75=PO9h9=PB?;<XV:H@I=n��hHJ?�mLR;2��¯�°£�RcnXV@<7>9LCBD7FEU>HYR8KXV>$b58?EU>9XW>HY9s�±�8�=P8?84r�@ILC=<79L&c²@<7FEG@

³q´�´ ¥pµ ª.¶¦n·¹¸}º ´�´ ¥¼»~½Iµ ª�¾j¥�»~½<µ ª�¿N»¸}¸PÀ¢Á ´�´ ¥¼»~½Iµ ªP¿mÂ2¿N» ¥pÃHsV¦&ªB?LU>m@<XW>mO984=�EU>9EUTVZm@IXVBSEUTVTVZCXV>m@<L

³q´MÄ ¥pµ ª.¶¦n· ¸}º ´MÄ ¥�»N½Iµ ª§¾j¥�»~½Iµ ª�¿~»¸}¸ À Á ´MÄ ¥�»N½Iµ ªP¿mÂ2¿N» ¥pÃHsÆÅUªEU>9e$@I79EG@

Ç ´�´ ¶ ¸<È®É)¸ ´Ê Á ´�´ ¥¼»~½DËpªPÌGÍ ¸9ÎϣРÎoÎ[Ñ É<Ò ÓÕÔ×ÖÙاÓÚÔ ¿mËp¿N»¸ 飃 ¸ ´Ê Á ´�´ ¥¼»~½DËpªP¿mËM¿N» ¥pÃHs©ÃRªB?LU>m@<XW>mO984=}XV>m@<L

Ç ´MÄ ¶�¸ 飃 ¸ ´Ê Á ´MÄ ¥�»N½DËpªPÌ Í ¸9ÎϣРÎVÛSÑ É<Ò Ó Ô ÖÙØ§Ó Ô ¿mËp¿N»¸ 飃 ¸ ´Ê Á ´MÄ ¥�»N½DËpªP¿mËM¿~» ¥pÃHs�«Aª62798�hNy[h�EU>Helh~y�Jn84b5XV=<=PXWLR>]EU>9e�EUhH=<LR;P:H@IXVLR>]B4L~8`Ü5B4XW84>m@<=CEU;<85e984z9>98?e�XV>lBD7FEU:H@<8?;�¦Us�±�8=<848?rC@ILQ:H;<L&�R8G�9O9:_LR>a@I;<EU>9=�JMLR;<b5EG@IXVLR>CLUJ)¯jÝHÞàß[°£�~@<7FEG@ Á ´�´ Þ Á ´MÄ EG>9e º ´�´ Þ º ´MÄ �~TW8SEGe9XW>9Y@ILCB4LR>A@IXW>mO9Xo@[Z�XV> Ç EU>9e ³ s.átLU;2B?TâEG;<XV@[ZU�~@<798 Á EU>He º 8wv~:9;<84=<=<XVLR>9=}EU;<8�;P8?:9;PL~eHO9B?84e�h_8?TVL�c�s

Á ´�´ ¥¼»9ª.¶ ¦«Uãnä�å Ôoæ å Ô×ç åUètå Ô å ¥�»9ª ¥pÃHsÆéUªcnXWTVT®B4LR>A@IXW>mO98�XV>m@<L

Á ´MÄ ¥�»9ª�¿~» ¶ ¦«Uã�ê ä�ëpì ¥pí~ªPîDï&¿AíAð ê äKñ ímòFï å ðó Á ´MÄ ¥¼»9ªô¶ ¦«Uã ê ä�ëpì ¥pí~ªPîDï&¿AíAð�õ ä�ñ ímòFï å ¦¿~°�ö ¿N°¿N» ¥pÃHs ­ ªcnXV@<7 ì ¥MíHª�hj84XW>9Yi@<798�÷^E�v~cq8?TVTWXWEU>a�R84TWLNB?Xo@[Zae9XV=P@<;<XWhHOH@IXVLR>gJMLU;�JM;<848{8?TV8?B4@<;<LR>H=2EU>9e�òFï å XV=�@I7H8@DEG;<YR8`@¢EU;<8?ECJMLR;xJM;P8?8�8?TV8?B`@I;<LU>�B?EU:H@<O9;<8G�FYRXV�U8?>�hmZ

òFï å ¶ Å�øÃNù à ÌSúwûü úwýwþ ¦ÿ��å�� � �ï � � °9ú¯ þ����ï å ¥pÃHs ¬ ª

Page 92: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¬ ¨�~XVb5XVTâEU;PTVZ

º�å?å Ô ¥�»9ª.¶ ä�å ã)ÌSúü ý ì Ýå?å Ô èFå?å Ô ¥¼»9ª ��å?å Ô ¥pÃHs©¨RªcnXWTVT®B4LR>A@IXW>mO98�XV>m@<L

º)å ïm¥�»9ª.¶ ä�å ã�Ì úü ý ì Ýå ï ¿~°¿N» ���å ï ¥pÃHs��Rª }LG@I8a@<7FEG@|@I7H8�:9;<LNLUJM=Q=P79L&cn> 7H8?;<8�7HLRTWelJMLR;� ��N6n¤�:_LR:9O9TWEG@IXVLR>le9XV=P@I;PXWh9O~@IXWLU>9=KcnXo@I7l@I7H8XW>HB?TWOH=<XWLU> LUJ�hNy�JpEUB`@ILR;P=?�nî å �2JMLR;gh_LRO9>9e =�@DEG@<8?=acn7HXWBD7 B?LU>m@<XW>mO98�XW>A@IL�îDïfJpEUB`@ILR;P=5JMLR;gJM;<848=P@IEG@I84={¥M;<84:9;<84=<8?>A@<XW>9Y�e984�NXWEG@IXVLR>9=,JM;<LUb�E�÷^E�vNc�84TWTVXâEU>|JM;<848`y�8?TW84B4@<;<LR>|�R84TWLNB?Xo@[Z|e9XW=�@I;PXWh9OH@<XWLR>Fª`s��������� �������! "�$#% ��'&(�*),+.-/ 0102 ���435�$+56738 "+9���62798$;<84=<O9To@DEU>A@g8?bgXW=P=<XVLR>�;IEG@<8�@ILuTW8`�R84T2>�JMLR;CE�YRXo�R84>�=P8?;PXW8?=4��LR>HB?8�@I798$:9;PLUzFTV8?=|=P@IEU;P@|@<Lh9TV8?>9eu@ILRYU84@I7H8?;�EG@�79XWYU7\=P8?;PXW8?=�bg8?bQh_8?;P=?�)XW=�=<XVb5:HTVZ�@I798g=<O9b LGJq@<7985hNy[h\8?bgXW=<=PXWLR>u;<EG@I84=L&�R84;�EUTVTj@<;IEU>9=PXV@<XWLR>9=2XV>�@I798�=P8?;<XV8?=4sq627FE�@nXW=?�

Á�å ¥¼»9ª.¶ ¦«Uã;: å Ô ä�å ÔVæ å Ô©ç åAèFå Ô å ¥¼»9ª ¥¼ÃHso¦=<Aª> =�@<798C=<8?;PXW84=�TVXWbgXV@}XW=�EU:9:H;<LAEUBD7H8?e£�)LR>9ToZf>98?EU;<hAZu:9;PLUzFTV8?=�B?LR>A@<;<XWhHOH@I8G�)@<798|�GEUTWO984=�LUJ ä�å ÔEU>9e æ å Ô©ç å eHL]>9LU@5�GEU;PZ�=<XVYR>9XozFBSEU>A@<TVZ�EU>9ekBSEU> h_8^;P8?:9TWEUB?84e hAZk;P8?:9;P8?=P8?>A@DEG@<XV�U8f�UEGTWO984=ä�?å Ô EU>9e æ ?å Ô×ç å EU>9eu@DEUrG8?>\LROH@�LUJ�@I798C=PO9b5b5EG@<XWLR>£sK62798C8?bgXW=P=<XWLU>�;IEG@<8C@<L$TW84�U8?T�>�@I7H8?>h_8?B?LUb584=Á�å ¥¼»9ª.¶ ¦«Uã ä@?å Ô æ ?å ÔÆç å : å Ô èFå Ô å ¥¼»9ª ¥¼ÃHso¦R¦�ª

áFLR;�7HXWYR7Q=P8?;PXW8?=�bg8?bihj84;<=?�G@I798x:H;<LUzFTV8?=,hH;<LAEUeH8?>gEU>9eQ@<8?>9eQ@<L{@I798x=<EUbg8xhFEU=PXWBxe9XV=P@<;<XWhHOH@IXVLR>£�LR>9ToZ =P79XVJ�@<8?e =<TVXWYR7A@IToZ�XW> JM;<84�NOH8?>9B`Z e9O98�@IL�@I7H8�XW>9B4;<8SEG=<8?e 8?>H8?;<YGZ LUJ�@I798uO9:9:_8?;�TV84�U8?T�s±�8QBSEUTVB?O9TWEG@I8�@I7H8|ES�R8?;<EUYR8Q84b5XV=<=PXWLR>f;<EG@I8KL&�R8?;{E�=<b5EUTWT�JM;<84�mO98?>9B`ZfXV>m@<8?;��UEGT��_8?�mOFEUT @IL�@I7H8=<84:FEU;<EG@IXVLR>$hj8`@[c�848?>fTVXW>98{B48?>A@I;P8?=BAg»Á�å ¥�»9ª.¶ ¦«Uã ä ?å Ô æ ?å ÔÆç å ¦Ag»DC 飃 : å Ô ètå Ô å ¥�»9ª�¿~» ¥¼ÃHso¦�ÅRª

Page 93: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¬ �

0

5e-15

1e-14

1.5e-14

2e-14

2.5e-14

3e-14

3.5e-14

4e-14

7.2e+14 7.4e+14 7.6e+14 7.8e+14 8e+14 8.2e+14 8.4e+14

Frequency / Hz

∆ν

á�XVYRO9;<8{ÃHsV¦'E*F9GIHKJMLNHPOQGSR�RSTVUXWNLNHKWYG�Z[J�WY\[]_^"`PZaWYLNGSbcT=WYG�`PZ�`Kd1`feP]"LNRIHKg[g[G�Z[J�g[LY`PhcR�]�U�WY`�WN\[]iGSZaWN]"JPLjHfWNGS`MZkG�Zl]�maZnpocqSrtsPuq$v `MZcUYG�w=]�LxWY\[]$^"`PZaWYLNG�b[T=WYG�`PZydzLN`PO{]|HP^j\yg[LY`PhcR�]�WN`lWY\[]�LN]"JMGS`MZ}`Pd�G�ZaWY]"JMLNHKWYG�`PZy~5G�WN\[GSZ�WN\[]$w=`PW�WY]|wR�GSZ[]|U q%� d8`MZ[]�WNLNHM^]�U_`PT=W_WN\[]B^"`PZaWYLNG�b[T=WYG�`PZVUxdzLY`MO�]|HP^j\�g[LN`KhVRS]�G�Z�WYTcLYZ n U�]�]�WY\c]$bV`MR�w�U�]�JPO�]"ZaWNU_`Kd5WY\c]g[LN`KhcR�]�U u G�W8GIU�]�maT[G�efHKR�]"ZaW5WY`�G�ZaWY]"JMLNHKWYG�Z[J�`feP]�L�`MZ[]iZ[`PLNO�HKR�G�UY]�wkg[LN`KhcR�] qáFLR;�79XWYR75=<84;<XW84=�bg8?bQh_8?;P= èFå Ô å ¥¼»9ª�XV=�EG:9:9;<LSv~XWb5EG@<8?TVZi@I798}=IEUbg8}JMLR;qEUe��<EGB?8?>A@�:9;<LUz9TW8?=4�~LU>9TVZ=<7HXVJ�@I84eaXV>gJM;P8?�mO98?>HB4Z5hmZ�Ag»js > =qB?EU>ah_8}=P8?84>aJM;<LRb z9YRO9;<8}ÃHso¦R�A@I798}=<O9bgb5EG@IXVLR>|XW>9=PXWe98n@I7H8XW>A@<8?YR;<EUT�=PXWbg:9TVZ�=<84;P�U8?=�@IL$=<84TW8?B`@�EUe��<EUB48?>A@�=<84YRbg8?>A@I=�JM;<LUb�@<798|>9LR;PbaEUTVXW=P8?ef:9;PLUzFTV8?=?�)EG>9eXW>�@I798QB?EU=<8iLUJ�zFYRO9;P8|ÃHso¦icq8|7FES�R8|EU>�XV>A@I8?YU;IEUT L&�R8?;�LR>98QB4LRb5:HTW84@<8i:9;PLUzFTV8Us{��J�@<798Q;<84YRXWLU>Ag»5XW=.@I7H8?>aXV>9B?;P8SEU=P8?ea@<LiB4L��U8?;�A��¯ Ý :9;PLUzFTV8?=?�m@I7H8�XW>A@<8?YR;<EUT9c�LUO9TWe5hj8}L��U8?;�A��¯ Ý >9LU;<b5EUTWXV=<8?e:9;PLUzFTW84=?s$627mO9=K@<798aXV>A@I8?YU;IEUT.L&�R84;�Ag»�;<84e9O9B48?=i@<L�A �¯ Ý ¥pX�sÚ8Us�@<798a>mO9bQh_8?;�LUJ2TVXW>98CB48?>A@I;P8?=cnXV@<79XW>�@<798�Ag»�JM;<8?�mO984>9B4Z�;P8?YRXVLR>tª`s� ¢LU@I8$@I7FE�@aEG=gLUO9;|JM;<8?�mO984>9B4Z 8?TV8?bg8?>A@5EU:9:9;PLAEUBD7984=@I7H8}=P8?;<XV8?=qTWXWbgXV@?�Uh9;PLAEUe984>98?ea:9;<LUz9TW8?=�JM;PLRb @I798¢B?LU>m@<XW>mO9O9b�JMLR;Pb�:FEU;�@qLGJ£@I79XV=.B?LR>A@I;PXWh9O~@IXWLU>@IL�@<798DAg»u;P8?YRXVLR>�¥�@I79XV={XW=�bgLR=P@{B?TW8?EU;<ToZ�=<8?84>�XV>f@I798|>mO9bg8?;PXWB?EUT�=PXWbQO9TWEG@IXVLR>^TWEG@I84;�XV>f@I79XV==<84B4@<XWLR>tªws �[>�@I7H8�TVXWbgXV@KLUJ%Ag»�h_8?B4LRbgXW>9Y��R84;PZ�=PbaEUTVT���@I7H8�ES�U8?;IEGYR8�L&�U8?;C@I798$JM;<84�NOH8?>9B`Z;<84YRXWLU>u@<8?>9eH=�@<L�¿ �¯ Ýp� ¿N»js|627mO9=�@I798C>mO9bQh_8?;�LUJq=<:_LR>A@DEG>98?LROH=K84b5XV=<=<XVLR>9={L&�U8?;�@I798C;<84YRXWLU>¿N»$XV=2YRXV�U8?>$hAZ

Á�å ¥¼»9ª.¶ ¦«Uã2ä@?å Ô æ ?å ÔÆç å ¿$�¯ Ý¿N» ¥¼ÃHso¦&ÃAª

Page 94: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨�<6279XV=2XW=�@<798?>�EU>FEGTVZm@IXVBSEUTVTVZC8`vN@<8?>9e984e�@<L|B?LR>A@<XW>mO9O9b =�@DEG@<8?=2O9=PXW>9YC8wvH:H;<8?=P=<XVLR>9=2JMLR; ä@?å Ô EG>9eæ ?å ÔÆç å @<;IEU>H=PJMLR;Pb584e�XV>m@<LQ@I798�B4LR>A@IXW>mO9OHb�s.627FEG@¢XV=?�

ä@?å Ô ¶ ä�ñ<ä�ëx� û úÅ�ã ü,��� ñ��8���� ?å ÔÅ 8`v~: õ ¾ ?å Ô�� ñ ö î ?å Ô ¥¼ÃHso¦S«mªXW=28wvN@I8?>He98?efXV>A@ILQJM;<848�=P@DE�@I8?=¢hAZ�TW8`@<@<XW>9Y �¯ Ý Þàß[°^EG>9e$O9=<XV>9Y

� ï{¶ ¨Gã ü þDímúD¿Aíä�ñ û þ ¥¼ÃHso¦�éRªJMLR;Q@<798$=P@DE�@IXW=�@IXVBSEUT�c�84XWYR7A@CLUJ�EU>�8?TW84B4@<;<LR>�XW>�=�@DEG@<8�°²¥�÷�84>2�?84Tk�(��84rU84;<XW=4��¦=�UÃAéRª`s]62798B?LU>m@<XW>mO9O9b =P@IEG@IXV=P@IXVBSEUTtcq8?XWYU7m@nXV=xe98`zF>98?e�L&�R84;nEQb5LRbg8?>A@<O9b�EG>9e��RLRTVO9b58�8?TW84b584>A@nEU>9e�XV=e98`zF>98?eC=PO9BD7|@I7FEG@,XV@,XV=�B4LR>9=<XV=P@<8?>A@�cnXV@<7 � å �U@<7FEG@�XV=�@<798?;P82XW=�>HL �x� @I8?;Pb�JMLR; @I798�=P@DE�@IXW=�@IXVBSEUTcq8?XWYU7m@¢LUJ @I7H8�XVLR>£s > TW=<LCO9=P8

¾ ?å Ô ¶��%� úÊ�¯ Ý ú ¥¼ÃHso¦ ­ ªJMLR;x@<798�XWLR>9XV=IEG@<XWLR>a:jLU@<8?>A@IXWEUT£LUJ�TW8`�R84T��¯ Ý s�627mO9=

ä�?å Ô å Þ äKñPä�ëx� û9úÅ�ã ü,�� ñ � � � ¦Å ¨Gã ü þDímúD¿Aíä�ñ û þ 8wv~: � · �%� úÊ° ú �� ñ � îDï¶ ä�ë «Gã�í ú õ ¦Å�ã �� ñ ö � � 8`v~: � · �%� úÊ° ú �� ñ�� îwï ü þü � � ¿Aí¶ ä�ë «Gã�í ú õ üÅ�ã �� ñ ö � � 8`v~: � ·��%� úÊ° ú �� ñ�� îwï�¿Aí ¥¼ÃHso¦ ¬ ª62798i÷^E�vNcq8?TWTVXâEU>��U8?TVL~B4XV@[Z�eHXW=P@<;<XVh9OH@IXVLR>£� ì ¥pí~ª�XW=2YUXV�R84>�hAZ

ì ¥Mí~ªP¿Aí|¶ «Uã�í ú õ üÅ�ã ��� ñ ö � � 8`v~: � · ü ímúÅ �� ñ=� ¿Aí ¥¼ÃHso¦&¨AªEU>9e$@I7H8�84>98?;PYUZ\¥��¢ï�ªxLUJ @I798�JM;P8?8�8?TV8?B4@<;<LR>$XV=

�¢ï{¶ ¦Å ü í ú ¶ �%� úÊ° ú ¥¼ÃHso¦=�Aª627mO9=¢TW84@P@IXV>9Y|¯ Ý Þ ß�°�XV>�84�mOFEG@IXVLR>]¥¼ÃHso¦ ¬ ªqYUXV�R84=

Page 95: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨H¦

ä�å Ô å Þ ä�ëMì ¥Mí~ªPîDï�¿Aí ¥¼ÃHs©Å'<Aª6 Lg@<;IEU>9=�JMLR;<b @I798 > y��GEUTWOH8?=?�H8`v~:9;<84=<= > XV>�@I8?;Pb5=2LGJ�LR=PB?XVTWTâE�@ILR;q=P@I;P8?>9YG@I79=4�_X¼s�8Gs

æ ?å ÔÆç å ¶ � å� ?å Ô ¨�ã ú Ì úü ý þ » ú ?å Ô å ì Ý ?å Ô å �� ?å Ô å ¥¼ÃHs©Å~¦�ªEU>9e�O9=P8

» ?å Ô å ¶ �%� úÊû õ ¦¯ ú · ¦�¯ Ý ú ö ¥¼ÃHs©ÅRÅRªEU>9e

ì Ý ?å Ô å ¶ ÅV�à ù ÃGã ¦� å ¦ÿ��å=� · �?å Ô � � þ ¦¯ þ �¯ Ý þ ¥¼ÃHs©ÅUÃAª@ILCLRh~@DEUXV>

æ ?å Ô©ç å ¶ � å� ?å Ô ¨Gã�úwÌSúü ýwþ¡  �%� úÊû(¢ ú õ ¦¯ ú · ¦�¯ Ý ú ö ú Å �ÃNù ÃGã ¦� å ¦ÿ �å=� · �?å Ô � � þ ¦¯ þ �¯ Ý þ �£� ?å Ô å¶ ¦� ?å Ô ¨Gã ú Ì úü ý þ   �%� úÊû ¢ ú ÅV�à ù Ã�ã ¦ÿ �å=� · �?å Ô � � ¦¯ þ ¯ Ý þ � � ?å Ô å ¥¼ÃHs©ÅG«mª��84@P@IXW>HY�¯�Þ ß[°\EU>9euO9=<XV>9Y$8?�m>k¥¼ÃHso¦�éRª¢JMLU;{@I798|=�@DEG@<XW=P@<XWB?EUT�cq8?XVYR7A@KLGJ�EG>\84TW84B4@I;PLR>uXW>�@I7H8B?LU>m@<XW>mO9O9b�YRXV�U8?=æ ?å Ô©ç å Þ · ä�ñ û9þ¨Gã ü þ í ú ¿Aí ¨Gã�úIÌSúü ý`þ � ú � øÊû ú Å �ÃNù ÃGã ¦ÿ��å�� � �ï � � ¦¯ þ ° þ����ï å¶ ä�ñ ûí ú ¿Aí ÌSúü ø ý þ � ú � øÊ Å �à ù à ¦ÿ �å�� � �ï � � ¦¯ þ ° þ'� ��ï å ¥¼ÃHs©ÅRéRª

627984>^O9=PXW>9Y¦Å ü í ú ¶ �%� úÊ° ú ¥¼ÃHs©Å ­ ª

Page 96: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨AÅ@ILCB4LR>A�R8?;�@}@<798�¿AíaXW>A@<L5¿N°��~XWE

¿míQ¶�· ¦ü í Å �%� úÊ° þ ¿N° ¥¼ÃHs©Å ¬ ª62798�@I;<EU>9=PJMLU;<bg8?e > y��UEGTWO98{XV=æ ?å Ô©ç å Þ � · ä�ñ û ¦í ú ¦¿Aí ÌSúü øwý`þ � ú � øÊ �¥¤¦ Å �à ù à ¦ÿ �å=� � �ï � � ¦¯ þ ° þ��£��ï å�§¨¶ · äKñ û � ü °9úÅ �%� úÊ � � · ü í~°9þÅ �%� úÊ ¿N° � ÌSúü ø ý þ � ú � øÊ ¤¦ Å �à ù à ¦ÿ �å=� � �ï � � ¦¯ þ ° þ�� ��ï å §¨¶ ¤¦ ä�ñ í Å�øÃNù à ÌSú`ûü úDýwþ ¦ÿ��å � � �ï � � °9ú¯ þ ���ï å ¦¿N° §¨¶ ä�ñ ímòFï å ¦¿~° ¥¼ÃHs©ÅU¨Aª

cn79XVBD7�XV=x@I798�h_LRO9>9e~y�JM;<848�84b5XV=<=PXWLR>�;<EG@I8Gs627mO9=i@<798ah_LRO9>9eNy[h_LRO9>9el84b5XV=<=PXWLR>\B?LN84ÜgB?XV8?>A@|B4LR>A@IXW>mO984=i=Pb5LNLU@I7HTVZuXW>A@<L�@I7H8ah_LRO9>9e~y�JM;<8488`v~:9;P8?=P=<XWLU>fEU= �¯ Ý Þàß[°£s.627FEG@¢XW=Á ?å Ô©ç å ¥¼»9ªx·®Þ ¦«Uã ê ä�ëpì ¥pí~ªP¿AíAð õ äKñ íNòFï å ¦¿N° ö ¿N°¿N» ¥¼ÃHs©Å'�Aª

EU>9e�EU>FEGTVZm@IXVBQB?LR>A@<XW>mO9XV@[Zfhj8`@[c�848?>l84�NO9EG@IXVLR>9=5¥¼ÃHs©éRª{EU>He ¥pÃHs ­ ª�79EU=�hj848?>\=<79L&cn>£s > B4L~eH8c�EU=�cn;<Xo@<@<8?>a@<LQe98?bgLR>9=�@I;<EG@I8¢@I798{B4LR>A@IXV>NOHXV@[ZgLUJ�@I7H8�;<84=<O9To@DEU>A@284b5XV=<=PXWLR>gB?LN84ÜgB?XV8?>A@nEUB4;<LR=P=@I7H8^=<84;<XV8?=�TVXWbgXV@Ssª©�LU@I7«��LR;<84>m@|�?XWEU> :9;PLUzFTW8�8`v~:9;P8?=P=<XWLU>9=$EU>9ek@<798^8wv~:9;<84=<=<XVLR> LUJy¬�;<XV8?b¥�¦�� ­ <Aª�B?LROHTWe]h_8aOH=<8?e�XV>\@<798�B?EUTWB4O9TâEG@<XWLR>®��EG>9e]8wvHXV=P@<XW>9Y >l­$> ��=PO9h9;<LUOH@IXV>98?=icq8?;P8�O9=P8?e@IL�84�GEUTVOFEG@<8u@<798u>98?B48?=P=IEU;�Z®¬�EUO9>A@$JpEUB4@<LR;<=4� > y��GEUTWO984=�EU>9e :jLR:HO9TâEG@<XWLR> h~y�JpEUB4@<LR;<=4s�áFLR;@I7H8ae984b5LR>H=P@I;<EG@IXVLR>\=<79L&cn>l7984;<8G�i��LR;<84>m@|�?XWEU>]:9;PLUzFTV8?=QEG;<8gO9=<84e]@I79;PLRO9YR7HLROH@Ss, ¢LU@I8C@I7FE�@@I7H8�:9;PL~LGJ�LRO~@ITWXV>98?e�7984;<8�cqLR;<rm=aXV>9e984:j84>9e98?>A@<TVZ�LUJ¢@I798�BD79LRXVB?8$LUJ�TWXW>H8�:9;<LUz9TW8U�.@I7H8�LU>9TVZ>984B?8?=P=<Xo@[Zu7FEU=�h_8?8?>�@<7FEG@{@<798|TWXV>98Q:9;PLUzFTW84={bQO9=�@�hj8|>HLR;<b5EUTWXV=<84e£s��[>\:H;IEUB`@IXWB48QLR>98QcqLRO9TVeO9=P82EU:9:9;PLR:9;PXâEG@<8�TWXV>98.:9;<LGzFTW8q8`v~:9;<84=<=PXWLR>9=.EG>9eicnXWe~@I79=�JMLU;�@I7H8�:9TWEU=<b5E}O9>He98?;,XV>A�R8?=�@IXVYAEG@IXVLR>£s�~848�BD79EU:H@I84;{éiJMLR;nEG>�LR:9EUB?Xo@[Z�BSEGTWB?OHTâEG@<XWLR>�OH=<XW>HYg@<798�:9;<LUz9TW8�8`v~:9;P8?=P=<XWLU>^LUJx¬�;PXW84b�s62798�B?LNe98.@DEUrG8?=�@<798�=IEUbg8�EU:9:9;PLAEUBD7iXV>�@I;<8?EG@IXV>9Yn@I798�B?LR>A@<XW>mO9XV@[Z�EU= @<798�:9;<LNLUJ~=<79L&cn>|EUh_L&�R8Gs627FEG@iXW=4��E$cxES�R84TW84>9YU@I7�84TW84b584>m@KXW=�B?LU>9=<XVe98?;P8?elXV>�@I7H8a=<:_8?B`@I;PO9b�EU>9e]@<798aB4LR>A@I;<XVh9OH@<XWLR>9=

Page 97: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨RÃ

0

5e+14

1e+15

1.5e+15

2e+15

2.5e+15

3e+15

3.5e+15

3550 3600 3650 3700 3750

Em

issi

on

co

eff

icie

nt¯

Wavelength / A

resultant emission coefficientbound-free component

bound-bound component

á�XVYRO9;<85ÃHs©Å£E�°±O�GIUYUYG�`PZ�^`�]²$^"GS]�ZMW*dz`PL_WN\[]k³�HKR�O�]"L*UY]"LNG�]�Uig[R�`KWYWY]|wDHP^"LY`aUYU´WN\[]kUY]"LNGS]|UiRSG�O�G�W q¶µ \[]k^"`�w[]~�HPU5LNT[Zldz`PL.·�¸9¹ s�º�r|»a¼X½"¾�¿yÀ'ÁKÂVà ¸9¹ r�ÄfÅ�q5Æ `PWY]iWN\cHfW9WY\[]*^"`PZaWYG�Z�T[T[Oª^`MZMWNLYG�b[T=WNGS`MZ%G�U9O�HPw=]iT[g}`Kd!H^`MZaWYG�ZaTc`PTcU5UY]W5`Kd2b'`PT[Zcw=ÇÈdzLN]"]xR�GSZ[]igcLY`PhcRS]|U qJM;<LUb EUTVT9TWXV>982EU>9eCB4LR>A@IXW>mO9OHb :9;<LGzFTW84=.@<L�@I79XV=�8?TW84b584>A@�EU;<8n84�GEUTVOFEG@<8?e£s�6279XW=.;<84YRXWLU>|XW=,@I7H8?>=<B?EU>9>984ekEGB?;<LU=<=|@I798�=P8?;PXW8?=|TVXWbgXV@KEU>9e�@<798�;<84=<O9To@I=|:9TVLU@<@<8?e£s\�[>�@I798�h9TV8?>9eH8?e�;<84YRXWLR>�EG>9eB?LU>m@<XW>mO9O9b @I79XV=�c�E&�U8?TV8?>9YU@<7$=<8?YUb584>m@nB4LR;<;P8?=P:jLR>He9=x@ILCEU>f��8?TV8?bg8?>A@S�~LUJ :jLU:9O9TâE�@IXWLU>£�mbaEUeH8O9:�JM;<LRb�@I798�B4LR>A@IXV>NOHLRO9=|JM;<848�=�@DEG@<8?=CLR;Q@<798�:9=<84O9e9LGy�B?LR>A@<XW>mO9LRO9=g79XVYR79TVZ�8`v~B4XV@I84e =�@DEG@<8?=4s> @�TWL&c =P8?;PXW8?=�c�ES�R8?TV8?>9YG@I79=�@I798gbaEGXW>�B?LU>m@<;<XVh9OH@IXVLR>�@IL�@I7H8gc�ES�R8?TV8?>9YG@I7]84TW84b584>m@�XV={JM;<LRb@I7H8�=PO9bàLUJ�EGTWTxh_LRO9>9eNy[h_LRO9>9e�:9;<LGzFTW84=?s > =5LU>98�EU:9:9;PLAEUBD7984=5@I7H8�=P8?;<XV8?=gTWXVb5Xo@S��EU>He @I7H8=P@IEG@I84=nhj84B?LRbg8�XW>9e9XV=P@<XW>9YROHXW=<79EUh9TW8}EU>9e�XV@xXW=2bgLR;<8�EU:H:9;<LR:H;<XâE�@I8�@<LgbgLNe98?T®@<79XW=2=PO9bgbaEG@<XWLR>EU=nEU>�XV>A@I8?YU;IEUTjLUJ)@I798�B4LR>A@I;<XVh9OH@<XW>9YQ:H;<LUzFTV8?=xL&�U8?;n8?>984;<YUZUs.62798�B?LR>A@<XW>mO9O9b XW=xbgL~eH8?TWTV8?e�EG=E�B?LU>m@<XW>mO9LROH=�=<84@�LUJ�TVXW>98K:9;<LUz9TW8?=|¥¼EU=P=<O9bg8?eÉ��LR;<84>m@|�?XWEU>fcnXV@<7�B?LR>9=�@DEU>A@�cnXWe~@I7tªw��EU>9e�@I7H8B?LU>m@<;<XVh9OH@IXVLR>$@IL5@<798�cxES�R84TW84>9YU@I7^8?TW84b584>A@}JMLROH>9e�JM;PLRb @I798KXW>A@I84YR;IEGT�L��U8?;}8?>984;<YUZ�LUJ,@<798?=P8h_LRO9>9e~y�JM;<8485:9;PLUzFTV8?=?sg627985;<8?=PO9TV@IEU>A@�8?bgXW=<=PXWLR>uB4L~8`Ü5B4XW84>m@KXW>u@I7H8Cc�ES�R84TW8?>HYU@I7l84TW84b584>m@�XV=@I7H8�=<O9b LUJ)@I798{h_LRO9>He~y[h_LRO9>9e�EU>He�hjLROH>9e~y�JM;P8?8{B4LR>A@I;PXWh9OH@<XWLR>H=?s�62798�;P8?=<OHTV@I=nEG;<8{=P79L&cn>�XV>zFYROH;<8�ÃHs©Å~s }LG@I8Q@I79EG@�@I798|h_LRO9>He~y[h_LRO9>9eu:9;PLUzFTV8?=�b5EUrU8CE�B?LU>m@<;<XVh9OH@IXVLR>uEG@�B4LR>A@IXV>NOHO9b c�E&�U8?TV8?>9YU@<79=EU>9e�=<XVb5XVTâEU;PTVZ2@<798qhH;<LAEUeH8?>984eKB4LR>A@IXW>mO9OHb B?LR>A@I;PXWh9O~@I8?=�EG@)h_LRO9>9eNy[h_LRO9>9e�cxES�U8?TW84>9YU@<79=?s,��@ XV=

Page 98: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨U«@I7H8¢B4LRbQh9XV>FEG@<XWLR>|LUJ_@I7H8?=<8¢B?LU>m@<;<XVh9OH@IXVLR>9=,cn79XVBD75TV8SEUeH=.@<L�@I798¢=<bgL~LG@I7|B?LU>m@<XW>mOFEG@<XWLR>gEUB4;<LR=P=@I7H8x=<84;<XW84=,TWXVb5Xo@ LUJF@I798�8?bgXW=<=PXWLR>KB?LN84ÜgB?XV8?>A@Ss�h~y�JpEUB`@ILR;P=�cq8?;<82B?EUTWB4O9TâE�@I8?e�JMLR;,EGTWTm@I;IEG>9=<Xo@IXWLU>9=?�@I7HLRO9YR7�JMLR;{@<798|79XWYU798?;{@<;IEU>9=PXV@<XWLR>9=}@I798`ZuEU;<8|84�mOFEUT,@IL�LU>98Us > b5E�vNc�84TWTVXâEU>^�U8?TVL~B4XV@[ZfeHXW=�y@I;PXWh9O~@IXWLU>$c�EU=�EU=<=PO9b584e�¥MX�sÚ8UstîDï�¶ ¦�ª`s�Ê}L&c�8`�R84;?�j@<798�:9;<LNLUJ,LGJ,B?LU>m@<XW>mO9Xo@[Z$XW=2JMLR;n8?bgXW=P=<XWLU>JM;<LUb�EG>f8?TV8?bg8?>A@{LUJ.:_LR:9OHTâEG@<XWLR>�XW>^@<798iB4LR>A@IXV>NOHO9b��_cn79XWBD7�XV=}@<798?>l��=<B?EU>9>984e£�£e9L&cn>�XW>A@<Lh_LRO9>9e�=�@DEG@<8?=?s�627mO9=Q@<798�8wvH:_8?B`@I84e�84b5XV=<=PXWLR>l;P8?=PO9TV@<XW>9Y�JM;<LRb�Ef>9LR>Ny�÷^E�vNcq8?TVTWXâEG>]e9XV=P@I;PXoyh9OH@<XWLR>�LUJ,JM;P8?8K8?TW84B4@<;<LR>9=¢BSEG>fhj8�bgL~e984TWTV8?e�cnXV@<7�@I7H8KXV>m@<;<LNe9O9B`@IXWLU>�LUJ�@I798KEU:9:9;PLR:9;PXâEG@<8iîDï�GEUTWOH8?=?sx }LU@<8x@I7FEG@.@<79828?bgXW=P=<XWLU>gB4L~8`ÜgB?XW84>A@�cq8?TVT9XW>A@<L�@<798nB?LR>A@<XW>mO9O9b XV=�baEGe982O9:CJM;<LRb EUTWT@I7H82B?LR>A@I;PXWh9O~@IXW>HY�hjLROH>9e~y�JM;P8?8n:9;PLUzFTW84=.EU>9eCXW=�8?�mOFEUT9@<L�@<7982O9=POFEUTtEU>FEGTVZm@IXVB�8`v~:9;P8?=P=<XWLU>aO9=P8?e@ILab5LNe98?T @I798iB?LR>A@IXV>mO9O9b�:9;<LUz9TW8$¥M8?�mOFEG@<XWLR>�¥pÃHs ­ ª<ªws�62798|EU:9:H;<LAEUBD7�LGJqB4LR>A@IXV>NOHXV@[Z�=<79L&cn>7984;<8�XW=C=<XVYR>9XVz9BSEU>A@IToZle9Xo�®84;<84>m@CJM;PLRb�@I7H8�B4LR>A�R84>m@<XWLR>9EUT�EG:9:9;<LREUBD7£�qcn7H8?;<84hAZkEG> XVLR>9XW=PXW>9YbgXWB?;PLUzF84TWe�XW=2XV>A@I;<LNe9O9B48?e£sq�~848�=<8?B`@IXVLR>^Ã~s©ÃQJMLR;nE|e9XV=<B?OH=<=<XVLR>�LR>�@<79XW=2:_LRXV>m@?s��������� �������! "�$#% ��'&(�*),Ë_Ìl01�*ͱÎB�! ���Ï35�$+56738 "+9����[>CE{=<XVb5XVTâEU;�b5EU>9>H8?;�@<798xh_LRO9>9e~y�hjLUO9>9eCEUh9=PLR;<:~@IXWLU>QB?LN84ÜgB?XV8?>A@S�A84�mOFEG@IXVLR>$¥pÃHs©¨Rª`�RB4LR>A@IXW>mO984=XW>A@<L}Xo@I=�B?LU;<;<84=<:_LR>9e9XV>9Y�h_LRO9>He~y�JM;P8?8.8`v~:9;P8?=<=PXWLR>®�R8?�mOFEG@<XWLR>a¥¼Ã~s��Aªws��N@DEU;�@IXV>9Y¢cnXo@I7�@I7H8qh_LRO9>9eNyh_LRO9>9e EUhH=<LR;P:H@IXVLR> B?LN84ÜgB4XW8?>A@�8wv~:9;<84=<=<XVLR>£�nJMLR;�EGh9=<LR;P:H@IXVLR>k@<L�79XWYR7 >~y[=P798?TVTW=g@I798fOH:9:j84;TW8`�R84TW=2h_8?B?LUb58�:9;PLRYR;P8?=<=PXV�U8?TVZ�B?TWLU=<8?;x@<LRYR8`@I7984;2O9>A@IXWT£EUe��<EUB48?>A@}EUh9=<LU;<:H@<XWLR>�:9;PLUzFTV8?=2=P@IEU;P@2@<LL&�R84;<TâEG:£s�62798{>mO9bQh_8?;�LUJ EUh9=PLR;<:H@<XWLR>H=xEG@qJM;<8?�mO984>9B4Z�»ae9O98�@<LQEUTVT_@<798�L&�U8?;<TWEU:9:9XV>9YQTVXW>984=qXV=

º�å ¥¼»9ª.¶ ä�å ã)ÌSúü ý : å Ô ì Ýå4å Ô �£�å4å Ô ètå4å Ô ¥�»Hª ¥¼ÃHs�Ã�<Aª62798KES�R8?;<EUYR8�LUJ @I79XV=2�mOFEU>A@IXo@[Z$L��U8?;¢EC=<b5EUTWT_JM;<8?�mO984>9B4Z$;P8?YRXVLR>,Ag»$XV=

º)å ¥�»Hª�¶ ä�å ã)ÌSúü ý ¦Ag»�C 飃 : å Ô ì Ýå?å Ô ��å?å Ô èFå?å Ô ¥¼»9ªP¿N» ¥¼ÃHs�ÃH¦�ªÐ >]EG:9:9;<LREUBD79XW>HY$@<798C=<84;<XW84=�TWXWbgXV@¢@I7H8g:H;<LUzFTV8?=4� èFå?å Ô ��@<7FEG@�B?LU>m@<;<XVh9OH@I8|@<L$@<798�A5»�;<84YRXWLU>7FES�R8gEU:H:9;<LSv~XWb5EG@I84TVZ�@<798|=IEUbg8Q=P7FEU:_8CEU>9euEU;P8Q=<XVb5:9ToZ�=<79XoJ�@I8?euEGTWLR>9YaXW>fJM;<84�mO98?>9B`ZRsK62798LR=PB?XWTVTâEG@<LR;�=P@I;P8?>9YG@I7 EU>9e�¬�EUO9>A@|JpEUB4@<LR;i�GEUTVO98?=QJMLR;K@I798�B4LR>A@I;<XVh9OH@<XW>9Yf@<;IEU>9=PXV@<XWLR>9=iBSEU>�h_8;<84:9TâEGB?8?e�hAZ�;<84:9;<84=<84>m@IEG@IXo�R8��UEGTWO984=�¥�¶ ì Ýå ?å Ô EU>9e � �å ?å Ô ª`s�627HXW=n:9;<LNe9O9B48?=

Page 99: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨Aé

º)å ¶ ä�å ã)Ì úü ý ì Ýå ?å Ô ��å ?å Ô ¦Ag»�C 騃 : å Ô èFå?å Ô ¥¼»9ªP¿N» ¥¼ÃHs�ÃAÅRª> YREUXW> B?LU>9=<XVe98?;PXW>9Y�Ag» 84�NO9EUT�@IL�@I798u=<84:FEU;<EG@IXVLR> h_84@[cq8?84>!@I7H8�TVXW>98f:H;<LUzFTV8?=�E�=<XWbgXWTWEU;EU;PYRO9bg8?>A@�@IL�@<798g8?bgXW=<=PXWLR>uB?EU=<8g79LRTVe9=?s�627H858`�®84B4@QLGJ�@I7H85=PO9b5b5EG@<XWLR>�XV=�@<L^B4LR>A@I;<XVh9OH@<8EUe��<EGB?8?>A@|B4LRbg:jLR>H8?>A@I=|XV>�@I7H8�TWXW>H85:9;PLUzFTV85@IL�@I798�XV>m@<8?YR;<EUT�=<O9BD7�@I79EG@i@<798�XW>A@<8?YR;<EG@IXVLR>]XV=84�j8?B`@IXo�R8?ToZ|L&�R84;�@<798n>NOHbQhj84;qLGJ®TVXW>98xB48?>A@I;P8?=qXW>�Ag»js_©�8?B?EUO9=P8¢@<798n:9;<LUz9TW8?=�EU;P8n>9LR;<b5EUTVXW=<84e£�@I7H8xXW>A@I84YR;IEGT~;P8?e9O9B48?=�@ILlA �¯ Ý EU>9eCEU=´Ag»|h_8?B4LRbg8?=,XV>HzF>9Xo@I8?=PXWb5EUTWToZ{=PbaEUTVTN@<798xXV>m@<8?YR;<EUTm@I8?>He9=@ILg¿ �¯ Ý s.62798�EUh9=<LU;<:H@<XWLR>$B?LN84ÜgB4XW8?>A@¢@I7984>�h_8?B4LRbg8?=º å ?å Ô ¥�»9ª.¶ ä�å ã�ÌSúü ý ì Ýå ?å Ôz�£�å ?å Ô ¿ �¯ Ý¿N» ¥¼ÃHs�ÃRÃAª� �å?å Ô Þ � ��å ï EU=�=<79L&cn>uXW>Ñ©qO9;<YR84=<=¶� �~O9bgbg8?;<=Q¥�¦=�U¨ ¬ ª}EG>9e ì Ýå ?å Ô Þ · ì Ýå ï EU={=P8?84>\XV>^@I7H8e98`zF>9XV@<XWLR>H=xJM;<LRb ÷�8?>2�48?T.�Ò� 8?rU84;<XV=Q¥§¦=�RÃAéRªws.áF;<LRb 8?�mOFEG@<XWLR>�¥¼Ã~sÆÅRÅUª

¿ �¯ Ý¿N» Þ · ¿N°¿N» ¥¼ÃHs�ÃU«mª��@gXW=C>9L&c B4TW8SEG;C@<7FEG@58?�mOFEG@<XWLR>!¥¼Ã~s©ÃRÃRªiB?LR>A@IXV>mO98?=5XW>A@ILu8?�mOFE�@IXWLU> ¥¼Ã~s��AªiLU> @I;<EU>9=�JMLR;<bgXW>9Y¯ Ý Þàß[°£s.627FEG@¢XW=4�

º�å?å Ô ¥¼»9ªôÞ ä�å ã)Ì úü ý ���å ï ¥§· ì Ýå ï ª   · ¿N°¿~» ¢¶ ä�å ã)ÌSúü ý � ��å ï ì Ýå ï ¿N°¿N»¶ º�å ïm¥¼»9ª ¥¼ÃHs�ÃAéRª62798nB?LU>m@<XW>mO9LROH=.EUh9=<LU;<:H@<XWLR>|B4L~8`Ü5B4XW84>m@�c�EU=.EUTV=<L�8`�UEGTWOFEG@<8?e|>mO9bg8?;PXWBSEGTWTVZUs´Ó¢=<XV>9Y{@I7982=<EUbg8@I84BD79>9XV�NOH8aEU=�JMLR;{@<798C8?bgXW=P=<XWLU>�B?LN84ÜgB?XV8?>A@S�)@I798CB4LR>A@I;<XVh9OH@<XWLR>9={@<L�E�c�ES�R84TW8?>HYU@I7�8?TW84b584>A@cq8?;<8$8`�UEGTWOFEG@<8?e�EU>Hel@I79XV=icxES�U8?TW84>9YU@<7 84TW8?bg8?>A@���=<B?EU>9>984e£��EUB?;PLR=<=Q@<798�=P8?;<XV8?=CTWXVb5Xo@Ss�62798B?LU>m@<XW>mO9O9b B?LR>A@I;PXWh9O~@IXWLU>|XW=�bgL~e984TWTV8?eCEU=,@<798nXW>A@I84YR;IEUTHL��U8?;�JM;<848n8?>984;<YRXV8?=.LUJ_@I798nB?LU>m@<;<XVh9OH@PyXW>HY�B?LR>A@IXV>mO9O9b :9;PLUzFTV8?=�¥¼EU=P=<O9bg8?e���LR;<84>A@t�?XWEU> XW> @<798uBSEUTVB?O9TWEG@IXVLR> =<7HL�cn> 798?;P8&ªws�62798h_LRO9>9e~y�hjLUO9>9egB?LR>A@I;PXWh9O~@IXWLU>|XW=�84�GEUTVOFEG@<8?eCJM;<LRb@I798n=<OHb LUJ�EGTWTH=P8?;PXW8?=.:9;PLUzFTV8?=�[email protected]&c =<84;<XW84=c�E&�U8?TV8?>9YU@<79=?�HEU>9e5EU=.EU>gXW>A@I84YR;IEGT9L&�R8?;�h_LRO9>9eg8?>984;<YRXV8?=�EGJ�@<8?;�E�B?8?;�@DEUXV>gc�ES�R84TW8?>HYU@I75B?OH@<LU��s�[>|:9;<EUB4@<XWB?8�XV@�c�EU=,JMLROH>9eQ@I7FE�@,@I798�=<O9b�L��U8?;.EUTVT~h_LRO9>He~y[h_LRO9>9e|:H;<LUzFTV8?=,B4LRO9TVe|hj8x8wvN@I8?>He98?e

Page 100: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨ ­

0

5e-07

1e-06

1.5e-06

2e-06

2.5e-06

3e-06

3.5e-06

4e-06

3550 3600 3650 3700 3750

Ab

sorp

tion

co

eff

icie

nt

Wavelength / A

resultant absorption coefficientbound-free component

bound-bound component

á�XVYRO9;<8KÃHs�Ã1E�Ô8bcUY`PLNg=WNGS`MZ¶^`�]²$^G�]"ZaW.dz`PL8WN\[]�³�HPRSO�]"L.U�]�LYG�]�U9g[RS`PW�WN]�w%HP^"LY`aUYU5WY\[]�U�]�LYG�]�U5R�G�O�GSW q´µ \[]*^"`�w[]~�HPU�LNT[Zkdz`ML8·�¸5¹ s�º�r�»�¼Õ½�¾�¿}À�ÁKÂ[à ¸�¹ r|ÄfÅ�q@ILC=POHÜgB?XW84>A@ITVZ�79XWYR7�:H;<LUzFTV8?=2@ILCB?EG@IBD7�EUTWT_@I798�e98`@DEUXVTW=xLGJ�@<798�XW>A@I84YR;IE�@IXWLU>£s,62798�;<84=<O9To@I=¢EU;P8=<7HL�cn>�XW>�zFYROH;<8�ÃHs�ÃHs627mO9=¢EU>FEUToZm@IXWB¢B?LU>m@<XW>mO9Xo@[ZaXV>�8SEUBD7$�mOFEU>A@<XV@[Z�XV>a@I7H8�EUh9=<LU;<:H@<XWLR>aJpEGB4@ILU;xEU>9e�84=<B?EU:j8�:9;PLRhFE�yh9XVTWXV@[ZC84�mOFEG@IXVLR>9=27FEG=xhj848?>$=<7HL�cn>�EU>9e�@<798�h_LRO9>9e~y�hjLUO9>9e$EUh9=PLR;<:~@IXWLU>aJpEUB4@<LR;<=xEU>9e$8?=PBSEU:_8:9;PLRhFEUh9XVTWXo@IXW84=xB?LU>m@<XW>mO98�=<bgL~LG@I79ToZ�XW>A@ILC@I7H8KB4LR;<;P8?=<:_LR>9eHXW>9YghjLUO9>9e~y�JM;<8?8�8`v~:9;P8?=P=<XWLU>9=?sn6279XV=7FEU=nh_8?84>�8`�UEGTWOFEG@<8?e$JMLR;x@[cqL5:9;<EUB4@<XWB?EUT£8`vHEUbg:9TV8?=2XW>$=P8?B4@<XWLR>�ÃHsÚ«9s"$#D" Ö -g(�*�+���×�*(Ø 34*xÙ *xÙ�Ú -$1¢1¢0ÑÛf�K*�34-5( Û�+�-�Üf��Ü�3S S3`*£¡ ÝR-g+´Þß �i S3�× ß627984;<82XW=,EU>CEUTV@<8?;P>FEG@IXo�R82EG:9:9;<LREUBD7|@IL{e98?EUTHcnXo@I7Q@I7H82e984@IEUXWTV8?ei@<;<8SE�@Ibg8?>A@�LUJt@I7H82TWXW>H8qh9TV8?>9eNyXW>HYi>98?EU;�@I7H8{=<84;<XW84=xTWXVb5Xo@Ss)62798�cqLR;Pr5=�@I8?bg=�JM;PLRb�@I798{LU;<XWYUXW>FEUTtEG>FEUTVZN=PXW=�LGJ)�[>9YRTWXV=�� 6 8?TVTW8?;¥�¦��RÃ��Aª{cn79L�e984;<XV�U8?e�EU>]8?�mOFEG@<XWLR>l;P8?TWEG@IXV>9Y�@<798�TWEU=P@KLRh9=P8?;P�GEUhHTW8�=P8?;<XV8?=QTVXW>98C@IL�@I7985TWLNBSEUT8?TV8?B`@I;<LU>�e98?>9=PXV@[ZUs > =�Ea;PLRO9YR7�84=P@<XWb5EG@I8�@I798QB4LR>A@IXV>NOHO9b�XV=�LUJ�@<8?>�B4LR>9=<XVe98?;P8?ef@<L�EUeH�GEU>9B48e9L&cn>u@IL�@I7HXW=Q��=<:9O9;PXWLROH=�B?LR>A@IXV>mO9O9b 84e9YR8U�os > bgLR;<8QeH84@DEGXWTW84ef@I;P8SEG@<b584>m@�XV=�@IL$XV>m@<;<LNe9O9B48

Page 101: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨ ¬E�:9TâEG=<b5EubgXWB4;<LUz98?TWe ¥p=<8`@aOH: hAZl@<798áàFO9B4@<OFEG@<XW>9YuXWLU>9=CXW>l@I7H8$�NXVB?XW>HXV@[ZlLGJ}@<798$8?bgXV@<@<XW>9YEG@<LRb LR;�XWLR>tª{@IL^EUBD79XV84�R85=<bgL~LG@I7]bg8?;PYRXW>HY�LUJxh9TW84>9e98?e]TWXW>H8?=�XW>A@IL�@I7985B?LR>A@<XW>mO9O9b�s$62798EU;PYRO9bg8?>A@�XW=,@I79EG@.@I79XV=,zF8?TVe5XV>m@<;<LNe9O9B48?=qE�:_LU@<8?>A@IXWEUTHcn79XVBD7aBSEG>5XVLR>9XW=P8279XWYU79TVZi8`v~B?Xo@I8?eg8?TV8?Bwy@I;PLR>9=4�t@<7mO9=¢@<798?;P8K8wv~XW=P@<=�EC�mOFEU>A@IOHb�TV84�U8?T EUh_L��U8�cn7HXWBD7�8?TW84B4@<;<LR>9=ncnXWTVT£h_8�XVLR>9XV=<8?e�h_84JMLR;P8@I7H84Zf79E&�U8i@I7H8|BD7FEU>9B48Q@IL�=P:jLR>A@IEU>98?LUO9=<ToZ^e984BSESZ^@<L$EaTWL&cq8?;{TV84�U8?T,EG>9e�8?TV8?B4@<;<LR>H={XW>^@<798?=P88`v~B4XV@I84el=P@IEG@I84=�cnXVTWT�h_8?7FES�U8�EU=�XVJ�@I798`Z\cq8?;P8�84�j8?B`@IXo�R8?ToZ�JM;<848584TW8?B`@I;PLR>9=?s > >k��L~B4B?O9:FE�@IXWLU>:9;PLRhFEUh9XVTWXo@[Zj��B?EU>lh_8a8`�UEGTWOFEG@<8?e£��YRXV�NXW>HY�@<798�:9;PLRhFEUh9XVTWXo@[Zf@I7FE�@QEU>]8`v~B?Xo@I8?e�84TW84B4@I;PLR>lXV=K8`JâyJM8?B`@IXo�R8?ToZ��©e9XV=<=PLRTV�U8?e£� XW>A@IL$@I7H8aB?LU>m@<XW>mO9O9b�s$6279XW=KEU:9:H;<LAEUBD7]7FEU=ih_8?84>]cnXWeH8?TVZuOH=<8?e]hjLU@<7XW>ubgL~eH8?TWTVXW>9Y5=<84;<XW84=�8?bgXW=<=PXWLR>�:H;<LUzFTV8?=�EU>9euXV>\B4LRTWTVXW=PXWLR>FEGToy[;<EUe9XWEG@IXÙ�R8�BSEGTWB?OHTâEG@<XWLR>9=4si��@�c�EU=XW>A@<;<LNe9O9B?84e�hAZá¬�O9;<L&�NXWBD7â�¹¤�>9YR84T��Ú=<7A@�¥§¦=� ¬R¬ ª�@ILQLUhH@DEUXV>�EQ>9LU>~y[e9Xo�R84;<YR84>m@x:FEU;P@<XV@<XWLR>CJMO9>9Bwy@IXVLR>|EU>HeghAZ��.XVYAEU;PL��K84@�EUT�sH¥�¦=�'�R¨Aª�XV>Q79XW=�b5LNe98?TVTWXV>9Y}LGJ®E�@ILRbgXWB�:jLU:9O9TâE�@IXWLU>9=,EU>Heg84b584;<YR84>A@àFO~v~84=¢JMLR;$©�EUTWbg8?;x=P8?;PXW8?=¢TWXVb5Xo@�B?EUTWB4O9TâE�@IXWLU>9=?sÊ}OHh9>984Z 8`@fEUT¼s�¥�¦����U«mª�XW>A@<;<LNe9O9B?8\EU> L~B4B?O9:9EG@IXVLR> :H;<LRhFEGh9XWTVXV@[ZR�¢e98`�R8?TVLR:_8?e²XV>ãÊ¢O9b5bg8?;� ÷�XW7FEUTWEU=�¥§¦=�R¨R¨Rª`�¢XW> @I7H8?XW;a=P@I84TWTWEU;�EG@<b5LU=<:97984;<84=�B4L~eH8f@IL�XW>HB?TWOHe98f@I798�8`�®84B4@<=�LGJKTVXW>98h9TV8?>9e9XV>9Y^>H8SEU;�@I798�=P8?;PXW8?=iTWXWbgXV@?s���@QXV=iXV>9=P@<;<O9B`@IXV�U85@<Lf8`vHEUbgXW>98C@I7H8?XW;K8`v~:9;P8?=<=PXWLR>lJMLU;K@I7H8@ILG@DEUT�EGh9=<LR;P:H@IXVLR>$B?LN84ÜgB?XV8?>A@S�FeH8?;<Xo�R84e�JM;<LUb�8?�mOFE�@IXWLU>]¥¼Å~sÚ«mª�LUJ´Ê}O9h9>H84Z�84@¢EUT¼s)¥�¦����U«mªKEº)å ¶ ä�å ¤¦ :å Ôpä å � å Ô ã�Ì úü ý ì�å Ô å�åqå Ô å ¥�»Hª � æç è;é å Ôpä å ¥�¦n· � å Ô ª=ê ñ �ë.ì Ø ÄNí Ô íØ§É »,î » å?å Ôò å ï »,ï » å?å Ô §¨ ¥¼ÃHs�à ­ ª

cn7984;<8�ò å ïuXV=�8?�mO9Xo�UEGTW8?>A@�@<L�LRO9;�h_LRO9>He~y�JM;P8?8�EUhH=<LR;P:H@IXVLR> B?LN84ÜgB4XW8?>A@?s�62798�=PXWbgXWTWEU;<Xo@IXW84=cnXV@<7�8?�mOFE�@IXWLU>9=�¥pÃHs©Ã'<AªqEU>Heu¥¼ÃHs�ÃRÃAª�EU;<8}84�NXWeH8?>A@Ss�62798�b5EUXV>5e9Xo�®84;<84>9B?8{XV=q@<798�XV>m@<;<LNe9O9B`@IXWLU>LUJ2@<798aJpEUB`@ILR; � å cn79XWBD7l@<7984Z];<8`JM8?;i@<L�EU=i@<798f�©eHXW=<=PLRTWO~@IXWLU>�JpEUB`@ILR;4�Vsf627H8?XW;ih9TW84>9e98?e�TVXW>98B?LU>m@<;<XVh9OH@IXVLR>�XV=�YR;<EUe9OFEUTVTVZg;<84e9O9B48?e$hAZa@I7HXW=�JpEUB`@ILR;qcn79XWTV8}@I7H8{B?LR>A@<XW>mOFEG@<XWLR>�LGJ @<798{h_LRO9>9eNyJM;<848CEUh9=PLR;<:H@<XWLR>�B4L~8`ÜgB?XW84>A@�@<L�hjLUO9>9e~y�hjLROH>9e�cxES�R84TW84>9YU@I7H=�XV=�=<XVbQO9TV@IEU>984LRO9=<ToZ�XW>9B4;<8SEG=<8?e®s627mO9=5EU=CLR>98�EU:9:9;PLAEUBD7984=5@<798$=<8?;PXW84=gTVXWbgXV@K@I798�EUh9=PLR;<:~@IXWLU>�B4L~8`ÜgB?XW84>A@5XV=Ce9;<Xo�R8?>�@ILu@I7H88`vN@<8?>9e984e�B4LR>A@IXV>NOHO9b��GEUTVO98ihAZ�@I798ib5XVB?;<LGzF8?TVe£sl ¢LU@I8�@I7FEG@�@I798QB4LR>A@IXV>NOHO9b�XV=}@<;<8SE�@I8?euEG=E|:9;<LGzFTW8{cnXo@I7�EC=P7FEU;P:�84e9YR8U�F>HLU@}EU=nEg=PO9b LUJ�h_LRO9>9e~y�JM;<848�:H;<LUzFTV8?=4s6279XV=qEG:9:9;<LREUBD75eHXV�j8?;<=,JM;PLRb LUO9;<=.XW>|@<7FEG@�cq8¢79E&�U8}=P79L&cn>5@<7FEG@�@I7H8¢hHTW8?>He98?eg:9;<LGzFTW84=.EUTWLR>HYcnXV@<75@<798{h9;PLAEUe984>98?e�B4LR>A@IXV>NOHO9b B4LR>A@I;PXWh9OH@<XWLR>5YRXV�U8}84b5XV=<=PXWLR>aEU>9e�EUhH=<LR;P:H@IXVLR>aB4L~8`Ü5B4XW84>m@<=cn79XVBD7^bg8?;<YU8K=Pb5LNLU@I7HTVZa@<79;<LUO9YR7�@<Lg@<798�B?LR>A@IXV>mO9O9b��tcnXo@I79LUOH@{EG>mZ$>H8?8?e�@ILgXW>A@I;PL~e9OHB?8KEU>XWLU>9XW=PXW>9Y{bgXWB4;<LUz98?TWe®s�62798n@I84BD79>9XV�NOH8}LUJjbgL~e9XoJ�ZNXW>9Y�@<798n=P@IEG@IXV=P@IXVBSEUT~hFEUTWEU>9B?8n84�NO9EG@IXVLR>9=,cnXV@<7

Page 102: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨R¨E�TW8`�R84T,e9XW=P=<LRTVOH@IXVLR>^JpEUB`@ILR;�XW={XV>9B?LR;P;<84B4@�LR>�@<798|YR;<LUO9>9e9=�LUJ�e98`@DEUXVTW84efhFEUTWEU>9B?8QEU>9eu8?>984;<YUZB?LU>9=<84;P�GEG@IXVLR>£s Ð J�B?LUO9;<=P8�@I7H8^bgXWB?;PLUzF84TWe�84�j8?B4@<=abQOH=P@�h_8�XW>9B4TWO9e984ekXV> @I7H8^B?LUTWTWXV=<XVLR>FEUTÙy;IEGe9XâEG@<XV�U8}84�NO9EG@IXVLR>9=2EU>He�@I798{TVXW>98¢h9;PLAEUe984>9XW>9Yi8`v~:9;<84=<=PXWLR>9=x@<L|EUTVTWL&c @I7H8{:jLU:9O9TâE�@IXWLU>9=�EG>9e:9;PLUzFTW84=5@<L]h_8fEUB4B?O9;<EG@I84TVZ�84�GEUTVOFEG@<8?e£s 62798fbgXWB4;<LUz98?TWe�84�j8?B`@I=�LU> @I7H8^TWXV>98$:9;<LGzFTW84=�EU;P8EUTV;<8SEGeHZ�XW>HB?TWOHe98?e�XW>�@I7HXW=|c�LU;<r�cnXV@<7�@<798�OH=<8�LUJ�@<798��_�_� h9;PLAEUe984>9XW>9Y\B?LNe98Gsk62798�XW>~yB?TVO9=<XVLR>uLUJ�@<798Cb5XVB?;PLUzF8?TVefcnXV@<79XW>�@I7H85B4LRTWTVXW=<XVLR>FEUTÙy[;<EUe9XâE�@IXÙ�R8�JM;IEUbg84cqLR;Pr�EU=�E�;<EG@I8|cnXo@I7\EB?LU;<;<84=<:_LR>9e9XV>9YiXV>A�R8?;P=<8{:H;<LNB?8?=P=�XV=qTV84J�@�JMLR;�JMO~@IO9;P8¢cqLR;<rts.62798}XWe98?E�XV=�@<7FEG@�@I798}b5XVB?;<LGzF8?TVe£��=84�j8?B`@nLR>�@I7H8�:jLR:HO9TâEG@<XWLR>H=xEU;P8{:9;<XVbaEG;<XWToZ|e9O98{@<L|XW>9eHXV�NXWe9O9EUTjXVLR>9=xbgL&�NXW>9YQB4TWLR=P8?;�@<7FEU>�@I7H8>98?EU;<84=P@2>984XWYR7mh_LRO9;qe9XW=�@DEU>9B48is�627984=<8{XVLR>9=qBSEU>�;P8?bgL&�R8�EU>a8?TW84B4@<;<LR>�:_LR=P=<XWhHTVZ|�NXâEQE�BD7FEU;PYR88`v~BD7FEG>9YR8|bg8?BD7FEU>HXW=<b�s¢6279XW=�EUTWTVL�cn=¢XW>HB?LR;P:jLR;<EG@IXVLR>�LUJ�@I798Kb5XVB?;PLUzF8?TVe^EG=�E5:9;PL~B48?=P=�=IEG@<XW=�yJ�ZNXW>9Y|bgXWB4;<LR;P84�U8?;<=PXWh9XVTWXo@[ZCB?LR>9=PXW=�@I8?>A@<TVZacnXo@I79XV>a@I7H8�B?LRTVTWXW=PXWLR>9EUToy�;IEUe9XWEG@IXÙ�R8qJM;IEUbg84cqLR;<raEU>9e�XV=:9TWEU>9>984eQJMLR;�@I7H82>98`vN@�=P@IEUYR8xLUJ >�­k> �|eH84�R84TWLR:Hb584>m@?s,627mO9=�EUTV@<79LRO9YU7Q@I798xzF>9EUTHB4LR>A@IXW>mO9LUO9=B?LN84ÜgB4XW8?>A@<=,XW>�@I79XV=�BD7FEG:H@I84;,EU;P8�=<XVb5XVTâEU;j@IL�@<79LR=<8q:9;<LNe9O9B48?eQhAZ�@I798qL~B4B?O9:9EG@IXVLR>i:9;PLRhFEUhHXWTWXo@[Zbg84@I7HL~e£�~@I798�:979XVTWLR=PLR:979XV8?=2LUJ @I7H8�@[cqL5EU:9:9;PLAEUBD7984=�EU;P8��mO9XV@<8�eHXV�j8?;<84>A@Ss6 Lf@I798$EUO~@I79LR;4��=irN>HL�cnTV8?e9YU8�@I798�YR84>98?;PXWB�:H;<LNLUJnLUJ¢B?LR>A@<XW>mO9XV@[Z�=<7HL�cn>�7984;<8�cn79XVBD7�79LRTWeH=JMLR;aEU>AZk:97AZN=<XVBSEUT}=<8`@�LUJ�TWXV>98�:9;<LUz9TW8?=57FEU=�>HLU@�h_8?84> e98?bgLR>9=�@I;IE�@I8?e h_84JMLU;<8U�2cnXo@I7 bgLR=P@EUOH@<79LR;P=QLR:H@<XW>9Y�JMLR;�@<798�LNB?B4O9:FEG@<XWLR>l:H;<LRhFEGh9XWTVXV@[Z�bg84@<79L~e®s^62798�B4TWLR=P8?=�@|8`vHEUbg:9TW85LUJ2@I7H8:9;PL~LUJ{LUJ{@I7HXW=aBD79EU:H@I84;aXV=5JMLRO9>He XW> �~8SEG@<LR>¹¥§¦=����<Aªws{Ê}8^B4LR>9=PXWe984;<=5@<798fB?EU=<8�LUJ�>9L]TVXW>98e9XV=<=<LUTWOH@<XWLR> ¥pX¼s�8Gs)>9L^bgXWB4;<LUzF84TWeu84�j8?B`@I=IªKEU>9e�=P@IEG@I84=K@I79EG@K@<798g=<O9bôLUJ2b5EU>AZuL&�R84;<TâEG:9:9XW>HY79XVYR7a=P8?;<XV8?=qTWXW>H8¢:H;<LUzFTV8?=�¥pB4LR>9=<XVe98?;P8?ea@<Lih_8�=�@I8?:5JMO9>9B4@<XWLR>H=Dª�XV>g@<798�7HXWYR75=<8?;PXW84=xEUh9=PLR;<:~@IXWLU>B?LN84ÜgB4XW8?>A@,@<8?>9eH=�@IL&c�EU;<e9=�@I798q:979LU@<LGy[XVLR>9XW=<EG@IXVLR>�;IEG@<8Us´Ê}L&c�8`�R84;�798�@I7H8?>QYRLN8?=�LR>i@<L{O9=<8xEU>LNB?B?OH:FEG@IXVLR>�:9;PLRhFEUh9XVTWXo@[ZaEG:9:9;<LREUBD7�@<LCe98SEUTjcnXV@I7$=P8?;<XV8?=2TVXWbgXV@I=qXW>�79XV=2b5LNe984TWTWXV>9Y�LUJ,E�@ILRbgXWBeFEG@IEQJMLR;27AZ~eH;<LRYR84>�JMLR;x@<798��[;<LR>�:9;PL��P8?B`@Ss"$#"ð ñòÙ�Ú 0Ñ×.Ú -DÝ 1n-g(�*�3S($0f-g0Ñ×óÚ%×,1n��Û@Ú �Q(�/ ��ÜÑ×�-g+*Û^*,3?-g(Úkô;Û�+_Úl×x×,3?-g(Ñ×62798¢EUh9=PLR;<:~@IXWLU>QJpEUB4@<LR;.EU>9eC8?=PBSEU:_8n:9;<LUhFEUh9XVTWXV@[Z�8wvH:H;<8?=P=<XVLR>9=.BSEU>g>9L&c hj828`�UEGTWOFEG@<8?e£s��~O9BD7�mOFEU>A@IXo@IXV8?=gBSEU>kEUTW;P8SEUeHZ�h_8^8`�GEUTWOFE�@I8?e�JMLR;Ce9XW=PB?;P84@I8�=<:_8?B`@I;IEGT}TVXW>984=^¥M=<848fBD7FEU:H@<8?;�ÅRªgEG>9eJMLR;CB?LU>m@<XW>mO9O9b�;IEUe9XWEG@IXVLR> ¥ ­ ;<EScnXW>£�¢¦=� ¬ <Rª`s�Ó}=PXW>9Yu@I798�B?LU>m@<XW>mO9Xo@[Zl@I7984LR;PZ�=P79L&cn>k7984;<8G�

Page 103: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨��

0.999986

0.999988

0.99999

0.999992

0.999994

0.999996

0.999998

1

3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200

Esc

ape

prob

abili

tyõWavelength / A

Escape probability

á�XVYRO9;<8KÃHsÚ«2E v `PZaWNGSZ�T[`MTcU8]|UY^�HKg']*g[LN`PbcHPb[G�RSGSWXökg[R�`KWYWY]|w%dz`PL9WY\[]�^"HMU�]_`Pd!WY\[]�³�HKR�O�]�L.UY]"LNGS]|U8G�Z%WY\c]_dzTcUYGS`MZw=G�eP]�L�WN`PL q · ¸ ¹ r÷º´r�»�¼Õ½¾¿}À�ÁKÂfà ¸ ¹ r�ÄfÅ� R�]"Z[JPWY\'¹xøP^O  HPUNU�TcO�G�Z[J9ù`PLN]"ZaWYú�G�HPZxg[LY`PhcR�]�UHPZcw_H8û}Hfü=~�]�RSR�GIHKZeP]�RS`=^"G�WXö*w[G�U�WYLNGSbcT=WYG�`PZ�dz`PL2WN\[]�dzLN]"]5]"R�]�^WYLN`PZcU q£Æ `KWN]�WY\[]8UYO�`a`PWY\�O�]"LNJPG�Z[J´`Kd'WY\[]5]|UY^�HKg']�gcLY`MbcHKb[G�R�G�WXö*HM^LN`MUNUWY\c]_U�]�LYG�]�U�R�GSO�GSW q@I7HXW=nBSEU>�hj8�8`vN@<8?>9e984e^@IL5XW>9B4TWO9eH8�84TW84b584>m@<=���cnXo@I79XV>£�9@I798�h9TV8?>9e984e^79XVYR7�=<8?;PXW84=};P8?YRXVLR>^EG>9eB?LU>m@<XW>mO9O9b�s¹62798�B?LU>m@<XW>mO9LROH=�8?bgXW=P=<XWLU> EU>9e EUhH=<LR;P:H@IXVLR> B?LN84ÜgB4XW8?>A@<=�EG;<8fB4LRbQh9XV>98?ek@<L:9;PL~e9OHB?8�@I798�>984B?84=<=IEG;PZ�LR:FEUB4XV@[Z�8wv~:9;<84=<=<XVLR>�EU=¢ECJMO9>HB4@IXVLR>$LUJ�:9TâEU=PbaEQ:9EU;IEUbg84@<8?;P=?s����ýB��� þk��-ÿ+ ��� #l0�1Í!Ë´�! ���l0������� ������������������������! ����#"$���%�'&Ó}=PXW>9YC@I7H8i;<84=<O9To@DEU>A@ Á EU>9e º 8wv~:9;<84=<=<XVLR>9=}XV@¢XW=¢:jLU=<=<XVh9TW8�@<L58`�UEGTWOFEG@<8 Ç EUB?;PLR=<=¢@I798�=<84;<XW84=TWXVb5Xo@Ss�(�LR>H=<XWeH8?;.E{79LRbgLRYR84>98?LUO9=�:9TâEG=<b5E�=<O9BD7C@I79EG@ Á EU>9e º EU;P82XW>9e984:j84>9e984>m@�LUJ_:_LR=<Xo@IXVLR>£s¤��mOFEG@IXVLR>�¥¼ÃHs�ÃAª�;P8?e9O9B48?=¢@IL

Ç ¶ ¸IÈ£É�) Ñ ÉDÖÐ Ñ ÉDÖ ¥§¦¢·�Ì Í Ð Ñ ÉDÖ´ ªP¿N»î ¸ 飃 Á ¥�»9ª�¿~» ¥¼ÃHs�à ¬ ª

(�LR>9=PXWe984;}@<798�©xEGTWbg8?;n=<84;<XW84=�TVXWbgXV@2XV>�JMO9=PXWLR>�e9XV�U8?;P@<LR;�:HTâEU=PbaEU=4sn6279XW=¢;<84YRXWLR>�XW=¢8`v~:_8?B4@<8?e@IL|h_8{LR:~@IXWB?EUTWToZ|@I79XV>£�~cnXo@I7a@<798�TWL&cq8?;��£ZNb5EU>�b584bQh_8?;<=�hj84XW>9Yi@<798{LR>HTVZg@I79XVBDr5TVXW>984=?s_ }8`�Ay8?;�@I7984TW84=<=¢XV@n:9;PL��NXVe98?=}EgOH=<84JMOHT�BD7984BDr$LR>�LUO9;nb58`@I79LNe£s�62798K84=<B?EU:j8�:9;PLRhFEUh9XVTWXo@[ZaB4L~e98�c�EU=;<OH>�JMLR; ä�ñ ¶¦+*l¦=< � ø ý ü Í þ&� � ñ ¶¦SÌ�,5�tî2¶!é ý ü �9@<798�;<8?=PO9TV@<=¢EU;<8�=<7HL�cn>�XW>�zFYUO9;<8�ÃHsÚ«9s��@�BSEG>�hj8|=P8?8?>u@I79EG@{@I7H8|8?=<B?EU:_8QJpEUB4@<LR;�JMLR;�@<798D©xEUTVb584;�TVXW>984=�@I79;PLRO9YR7f@<L$EU>9euXW>HB?TWOHe9XW>9Y

Page 104: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

��<

0.0001

0.001

0.01

0.1

1

890 900 910 920 930 940 950 960

Esc

ape

prob

abili

tyõWavelength / A

Escape probability

á�XVYRO9;<8}ÃHs©é£E v `PZaWYG�Z�T[`PTcU�]�UN^"HPgV]9g[LY`MbcHKbcGSR�G�WXö�g[R�`KWYWY]|w�dz`PL�WY\[]x^"HMU�]9`Kd÷WY\c]´ùöaO�HPZkU�]�LYG�]�U GSZBU�`MR�HPL g[LN`PO�GSÇZ[]�Zc^]|U q · ¸ ¹ r�º�r|»a¼�-�¾�¿yÀ'ÁP£à ¸ ¹ r|ÄtÅ� R�]"ZcJKWY\�¹ r�º�r�»/."¾�¿yq10 Z[R�ö}]�RS]|^�WYLN`PZ�b[LN`MHMw=]"Zc]�w�ù�`PLN]"ZaWNú"GIHKZg[LN`KhcR�]�U�~�]"LN]´TcU�]|w�GSZ�WY\cG�U8^�HKRI^T[RIHfWNGS`MZ q!µ \[]iù�ö�O�HKZlU�]�LYG�]�U�R�G�O�GSW8`=^"^"T[LNU5HKW�243 rPr65Ô q@I7H8�B?LR>A@<XW>mO9O9b�XW=4��JMLR;QEGTWT.:9;IEGB4@IXVBSEUT�:9O9;P:jLU=<8?=4�.LR:H@IXVBSEUTVTVZf@<79XW>®s� }LG@I85@<7FEG@ Ç B4LR>A@IXW>mO984==<bgLNLU@I79ToZ�EGB?;<LU=<=n@I798�=<84;<XV8?=nTWXVb5Xo@�¥#7²Ã ­ «mé98> ªws }8wvN@KB4LR>9=PXWe984;�@<798|BSEG=<8CLUJ�@I798��£ZNbaEU>u=P8?;<XV8?=�TWXVb5Xo@¢XW>u=<LRTWEU;{:9;PLRbgXW>984>9B?84=?s|�[>�@I79XV=�BSEU=P8LR>98�8`v~:j84B4@<=.@<798��£ZNbaEG>|TWXW>H8?=,EG>9e|B?LR>A@<XW>mO9O9b¹@IL�h_8x8`vN@<;<8?bg8?ToZiLU:H@IXVBSEUTVTVZ�@I7HXWBDrts���@�=P79LRO9TVeh_8g>HLU@I84e\@<7FEG@�@I7H85E�@ILRbgXWBQ:_LR:9OHTâEG@<XWLR>9=�XW>uE$=<LRTWEU;{:9;PLRb5XV>98?>HB?8CEU;P8g=�@I;PLR>9YRToZfB?LU>m@<;<LRTVTW84ehAZ�:979LG@ILGy[EUh9=<LU;<:H@<XWLR>�LUJ�XW>HB?XWeH8?>A@n;IEUeHXâEG@<XWLR>�EG>9e�@<7FEG@¢EQJMO9TVT£;IEUeHXâEG@<XV�R8}@I;IEG>9=PJM84;}EG:9:9;<LREUBD7XW=};<84�NOHXW;<84ef@IL�EGB?B?OH;IEG@<8?TVZ�bgL~eH8?T)@I798Q:_LR:9OHTâEG@<XWLR>9=4�!àFO~v~8?=�EU>He�8`vHEUB4@{TWXW>H8K:9;PLUzFTV8i=<79EU:j84=¥p=P8?8@¬�LUOH@<@<8?h9;PL��?8u84@^EUT¼sV��¦=���RÃRª`s�±]8uO9=P8\@<798\=<LRTWEU;�:9;<LUb5XV>98?>9B48�BSEU=P8]EG=�E�BD7984BDr @<L=<848$XVJn@I798�B4LR>A@IXV>NOHO9b 8`v~:9;P8?=P=<XWLU>9=5eH84�R84TWLR:_8?e�7984;<8$BSEG>�:H;<8?eHXWB4@C;P8SEU=PLR>FEUhHTW8�B?LU>m@<XW>mO9O9bLR:FEGB?XV@<XW84=�cn79XWBD7�=<bgLNLU@I79ToZabg8?;PYR8�XW>A@<Lg7HXWYR7$=<84;<XV8?=nLR:FEUB4XV@<XW8?=4s,627mO9= Ç c�EU=n84�GEUTWO9EG@I84e$JMLU;äKñ ¶d¦:*k¦=< � Ê ý ü Í þ � � ñ ¶�¦&Ì�,5��î|¶d¦:*k¦=<<; ý ü �)@I798g;<84=<O9To@I=KEU;<8C=<7HL�cn>]XW>�zFYRO9;P8aÃ~sÆé~s }LG@I8}@<7FEG@�@I7984;<8}XW=�XW>9e9848?e�=PXWYR>HXVzFB?EU>A@�LR:9EUB?Xo@[ZaE�@�B4LR>A@IXW>mO9OHb�c�ES�R84TW8?>HYU@I79=nEG>9e�EK=Pb5LNLU@<7bg8?;<YUXW>9Y|LUJ Ç XV>A@ILChjLUO9>9e~y�hjLROH>9e�LU:FEUB?Xo@[Z�EUB4;<LR=P=¢@<798�=<84;<XW84=nTWXWbgXV@{¥=7 �~¦R¦ 8> ª`s> ���?���@$�����BADCE�F�<�%���@�G�! ��F��"��H�%�'&62798¢YR8?>H8?;<XVB}8wv~:9;<84=<=<XVLR>5JMLU;qEG>aEUhH=<LR;P:H@IXVLR>gJpEGB4@ILU;�XV>g@<8?;Pb5=.LUJj@I798¢B?LR>A@<XW>mO9LRO9=qEUh9=PLR;<:~@IXWLU>EU>9e�8?bgXW=P=<XVLR>�B?LN84ÜgB?XV8?>A@I=¢XW=

Page 105: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�H¦³ ¶�¦2·¹¸ 飃 º)å ¥�»N½Iµ ª�¾j¥�»~½<µ ª�¿N»¸PÀ�¸<ȮɣÁ�å ¥�»N½Iµ ªP¿N»H¿m ¥¼ÃHs�ÃR¨Aª

�[>�@<798g84�GEUTWO9EG@IXVLR>�LGJ ³ �)@I7H85XV>m@<8?>9=PXV@[Zu@I84;<b XV=�8`�GEUTWOFE�@I8?e�hAZ�XV>A@I8?YU;IEG@<XW>9Y�@I7H8584b5XV=<=<XVLR>£�EG@P@I8?>mOFE�@I8?euhAZ�:979LU@<LGy§EUhH=<LR;P:H@IXVLR>£�_L��U8?;{@<798i�RLUTWO9bg8iLUJ�@<798|:9TWEU=<b5Eg@<L�YR84@�@I798Q;P8?=PO9TV@IEU>A@XW>A@<8?>9=PXV@[Z�E�@�@I7H8|:jLRXV>A@{LUJ�XW>A@<8?;<84=P@?siáFLR;�@<798QB?EU=<8|LUJ ³ E�@{@I798|B48?>A@I;P8|LUJqE�79LRbgLRYR84>98?LROH=e984>9=<Xo@[Z$:9TâEU=PbaEQLGJ�=P:9798?;PXWB?EUT�YR8?LUb58`@I;PZ�¥pLUJ�;IEGe9XWO9=2hFªx@<798�EUh9=PLR;<:H@<XWLR>�JpEUB`@ILR;2;P8?e9OHB?8?=n@IL³ ¶ ¸PȣɣÁ ¥�»9ª�Ì Í ´ Ð Ñ ÉDÖ ¿N»¸<ȣɣÁ ¥¼»9ªP¿N» ¥¼ÃHs�Ã��Aª

á�XVYRO9;<8{ÃHs ­ =P79L&cn=2@I798�;<84=<O9To@I=xJMLR;q@I798¶�£ZNbaEG>�=P8?;PXW8?=2TVXWbgXV@qEG@n8wvH:_8?B`@I84e�=PLRTâEU;�:9;<LRbgXW>H8?>9B48B?LU>9e9XV@<XWLR>H=?s ³ XV=�84�GEUTVOFEG@I84egEU=�E�JMOH>9B4@<XWLR>CLUJ_JM;<84�NOH8?>9B`Zg@<L�:9;<LNe9O9B48n@<798¢EUh9=PLR;<:H@<XWLR>QJpEGB4@ILU;EUB4;<LR=P=�@I798�=<8?;PXW84=�TWXWbgXV@?s5Ê}L&cq84�U8?;?�mJMLR;�EUe��PO9=�@Ibg8?>A@I=�@<L�EG@<LRb5XVBq:_LR:9OHTâEG@<XWLR>KLR;�JM;P8?8x84TW8?B`@I;PLR>e9XV=P@I;PXWh9O~@IXWLU>9=�XV@�XW=�>98?B48?=P=IEU;�Z�@<L^e98`zF>98�E�:jLR:HO9TâEG@<XWLR>u84TW8?bg8?>A@�¥¼EU>He]7984>9B?8�E$JM;<84�NOH8?>9B`Z8?TV8?bg8?>A@wªqJMLU;xcn79XWBD7$@ILgBSEUTVB?O9TWEG@I8}@I798 ³ JpEUB`@ILR;P=?s­ 84z9>9XW>9Y�Ag»^EUB4B?LR;Pe9XW>9YUTVZ�YRXo�R84=}@I7H8K8wvH:H;<8?=P=<XVLR>fbgLR=P@�EU:9:9;PLR:9;PXâEG@<8�@<LaLROH;}e984=<XV;<8?e^EU:9:9TVXoyBSE�@IXWLU>£s�± Xo@I7QAg»]B?L&�U8?;<XV>9Y�LR>98aB?LRbg:9TW8`@I85=<:_8?B4@<;IEUTqTWXV>985Xo@QXV=i:_LR=<=PXWh9TV8g@<LfYR8?>H8?;IE�@I8�@I7H8O9=POFEUT�eHXW=<B4;<8`@I8�TVXW>985EUh9=PLR;<:H@<XWLR>�JpEUB4@<LR;<=4s^± XV@<7QAg»]BD79LR=<84>�@ILfh_8�E^=PbaEUTVT.8?TW84b584>A@S�,@I7H8EUh9=PLR;<:~@IXWLU>�JpEUB`@ILR;nB4LR;<;P8?=<:_LR>9eHXW>9Yg@IL5Eg=PbaEUTVT�:jLU:9O9TâE�@IXWLU>�8?TW84b584>A@�h_8?TVL�c @I798�B?LU>m@<XW>mO9O9b8?eHYR8U�£LU;�XW>�@I798QB4LR>A@IXV>NOHO9b�Xo@I=P8?TVJ,BSEU>�h_8KJMLROH>9e£s���J*Ag»^XW=}BD79LR=<84>�@IL�B?L&�U8?;�°^¶JI Þ <¥pX¼s�8Gst»g¶!» ë Þ I!�~cnXV@<7�» ë @I798{JM;P8?�mO984>9B4Z�EG@2@<798�B?LR>A@<XW>mO9O9b 8?e9YR8Sª`�9cq8�E&�U8?;<EUYR8�L&�R8?;n@I7H8cn79LRTV8�B?LR>A@<XW>mO9O9bdEU>He�YU84@n@I798�=<EUb58�8wv~:9;<84=<=<XVLR>fEG= ­ ;<EScnXW>�¥�¦=� ¬ <Aª`�~X�sÚ8Us³ ë ¶ ¦ � ¦¯LKD¯ ñ � ë � ã)ÌSúü4M ý ¯NK Í � Ò ë C OÉQPSR

¥¼»~½Iµ ª ¿ ì¿N» ¿N» �¶ ¦2· ¦¯NKD¯ ñ � ë � ã�Ì úü4M ý ¯NK Í � Ò ëcC OÉQPSR¥�»~½<µ ª ì Ýå ï ��"�å ï ût° þÅ �%� úÊ ¿N» �cn7984;<8a¥p¯NKD¯ ñ � ë ªxXW=x@<798�;<8?B4LRbQh9XV>FEG@<XWLR>�;<EG@I8{JM;PLRb @<798�B?LR>A@IXV>mO9O9b @ILCTV84�R84T�XÊ}L&cq84�U8?;?��XoJ�Ag» XW=�BD79LU=<8?> @IL�h_8\=PbaEGTWTW84;5@<7FEU> @I79XV=�c�8uBSEG> 8wv9EGb5XV>98�;<84B?LRbQhHXW>FEG@<XWLR>JM;<LUb�e9Xo�®84;<8?>A@n:FEG;P@I=2LGJ @<798�÷^E�vNcq8?TVTWXâEG>�e9XW=�@I;<XVh9OH@<XWLR>�LGJ JM;P8?8�84TW8?B`@I;PLR>9=?s.6279XV=2B?LROHTWe�@I7H8?>h_8�OH=<8?e�@ILuTWLNLRr]EG@|@I798$:_LR:9O9TWEG@IXVLR>�e9XV=P@I;PXWh9O~@IXWLU> �©XV>9=<XVe98U��@<798$B?LR>A@<XW>mO9O9b EU>He�@<7NOH=C@<LXW>A�U8?=P@<XWYAE�@I8�:979LU@<LGy[XVLR>9XV=IEG@<XWLR>5e9;<Xo�R8?>�>9LR>Ny�÷^E�vNcq8?TVTWXâEG>�JM;<8?8�84TW8?B`@I;PLR>�eHXW=P@<;<XVh9OH@IXVLR>9=4s

Page 106: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�AÅ

1e-90

1e-80

1e-70

1e-60

1e-50

1e-40

1e-30

1e-20

1e-10

1

890 895 900 905 910 915 920 925 930 935

Esc

ape

prob

abili

tyõWavelength / A

Absorption factorEscape probability

á�XVYRO9;<8�ÃHs ­ E5Ô8bcUY`PLNg=WNGS`MZidpHP^WY`PL£g[R�`KW�WN]�wxdz`MLWN\[]�ùö�O�HPZ*UY]"LNGS]|U�R�GSO�GSWdz`PL£WXö�g[GI^"HKRMg[LN`PO�GSZc]"Zc^"] gcHPLNHPO�]"WY]�LNU qµ \[]_^"`�w[]´~�HPU�LNT[ZBdz`PL8· ¸ ¹ riºyr|»a¼�-�¾�¿yÀ'ÁP RS]�Z[JKWN\}¹ riº�r�»/."¾�¿yÂcà ¸ ¹ r�ÄfÅ }LG@I8{@I79EG@2JMLR;�@<798�=<XV@<OFEG@<XWLR>�=P79L&cn>£�F@I798�EGh9=<LR;P:H@IXVLR>�JpEUB`@ILR;2XV=x=<XVYR>9XVz9BSEU>A@IToZaTV8?=P=x@I7FEU>$@I7H88?=PBSEU:_8KJpEGB4@ILU;}E�@}@<798K=<EUbg8�c�ES�R8?TV8?>9YG@I7£s26279XV=}XV>9e9XVBSEG@<8?=n@I798�BSEU=P8Kcn7984;<8iEg:97HLU@ILR>�7FEU=}EBD7FEU>HB?8�LUJH84=<B?EU:9XW>HY�EUTVLR>9YnE2YRXo�R8?>�TWXV>98,LUJ~=<XVYR7A@S��h9OH@�@<798.:9;<LRh9EUh9XWTVXV@[Z}LUJHEn:H79LU@IL�y§EUh9=PLR;<:~@IXWLU>EG@{@<798|B?84>A@I;<8CLUJ�@I7H8|:9TâEU=PbaE5XW=��U8?;PZfTâEU;PYR8Qe9O98Q@<L�@I798CTWEU;<YR8K;IEUeHXâEG@<XWLR>^z98?TWe�XV>9B?XVe98?>A@�E�@@I7H82B?8?>A@<;<8i¥pYR84>98?;<EG@I84e5hAZi@<798n;<[email protected]_@I7982:HTâEU=PbaE��ULRTWO9bg8&ªws��[>|@I7H8?=<8nB4XW;<B4O9bg=P@DEG>9B?84=.@<798?;P8XW=nE|=�@I;<LU>9YR8?;nXV>1àFO984>9B?8�LR>�@I798�LR:~@IXWB?EUTWToZ5@<79XWBDr5:jLU:9O9TâE�@IXWLU>9=�@I79EU>�LU>$@<798�8?bg8?;PYR8?>A@$àFO~vjs }LG@I8Q@I79EG@�c�8C7FES�U85;P8?:9TWEUB?84eu@<798|>9LU@<XWLR>�LGJxE�79XVYR7�=<84;<XW84=�TW84�U8?T,cnXo@I7uE�=<b5EUTWT JM;<84�NOH8?>9B`Z8?TV8?bg8?>A@�B?LR;P;<84=<:_LR>9e9XV>9Y�@<LiEG>g84TW84b584>m@�LUJ®:jLU:9O9TâE�@IXWLU>£s ±�82c�LUO9TWeC@I7984>a>9848?eg@IL�b5LNe98?T9EUTWT@I7H82:jLR=P=<XVh9TW8x;<EG@I84=,@IL�EU>9eQJM;PLRb�@<798?=P8n8?TW84b584>A@I=?s > b58?EU>9=�LUJ_84>9B?LR;P:jLU;IEG@<XW>9Y�E{B4LR>A@IXW>mO9LUO9=³ XW>A@IL�@<7985:H;<8?=P8?>A@QB4LRTWTVXW=PXWLR>FEGToy[;<EUe9XWEG@IXÙ�R8�BSEGTWB?OHTâEG@<XWLR>fcqLRO9TVe�h_8|@IL�=P@IXVTWT,B4LR>9=PXWe984;K:97HLU@ILGyEUh9=PLR;<:~@IXWLU>uEU>9eu8?bgXW=P=<XWLU>^JM;<LRb E�e9XW=PB?;P84@I8CTV84�R84T��£hHOH@�@IL$84�GEUTVOFEG@<8i@I798|XV>A@I8?>H=<XV@[Z�@I84;<b XV>8?�mOFE�@IXWLU>^¥pÃHs�ÃR¨Aª JM;PLRb�@<7982L&�R8?;PTâEU:H:9XW>9Y�:9;PLUzFTV8?=?s�627NOH=.cq8n7FES�R8n@I7982XV>A@I8?>H=<XV@[Z|EG@,@I7H8¢:_LRXV>m@LUJqXW>A@I84;<84=P@iEU=�@I798C;<84=<O9To@DEU>A@�XW>A@<8?>9=PXV@[Z�e9OH8C@<L^EGTWT,L&�R84;<TWEU:9:9XV>9Y$:9;<LUz9TW8?=4��cn79XWBD7\BSEU>u@I7H8?>h_8�EUh9=<LU;<h_8?e$hAZ$EU>�XW>He9XV�NXWeHOFEUT®:H;<LUzFTV8U�~LR;x8?bgXV@P@I84e�JM;<LRb @I7FEG@x=IEUbg8�:9;<LGzFTW8Gs�627FEG@nXW=4�HLROH;EUh9=PLR;<:~@IXWLU>$JpEGB4@ILU;28`v~:9;<84=<=PXWLR>�hj84B?LRbg8?=4s³ ¶�¦n· ¸<È£É)º Ø ëUT ìWV ñ�XÙñ ¥�»9ª§¾ ´ Ó ñ¼å Ø ¥¼»9ªP¿N»¸ À ¸ 騃 Á Ø ëUT ìWV ñ#XÙñ ¥�»HªP¿N»H¿m ¥¼ÃHsÚ«÷<Aª

cn7984;<8Q@I7H8QJM;<8?�mO984>9B4ZfXW>A@I84YR;IEG@<XWLR>^XW={L&�R84;{@I798|LU>98Q79XVYR7f=<84;<XV8?=�=<:_8?B`@I;<EUT,TWXV>98i:H;<LUzFTV8U��EG>9e@I7H8�84b5XV=<=PXWLR>�EU>9e EUh9=PLR;<:~@IXWLU>�B4L~8`ÜgB?XW84>A@I=gB?LR;P;<8?=P:jLU>9e @<L\@<798��UEGTWO984=CJMLU;C@<7FEG@Ce9XV=<B4;<84@<8

Page 107: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�RÃTWXV>98U��X¼s�8Gs£>9L�TWXV>98Qh9TV8?>9e9XV>9Y$XW=�EUB4B?LROH>m@<8?e]JMLU;?s|��@�XW=�LR>HTVZfXV>�@I798gXW>A@I84>9=<Xo@[Zf@I84;<b @<7FEG@�@I7H8JMO9TVTVZ5h9TW84>9e98?e�8?bgXW=P=<XVLR>�EU>9e�EUh9=PLR;<:H@<XWLR>$B4L~8`Ü5B4XW84>m@<=�EG;<8�B?LU>9=<XVe98?;P8?e£s6279XV=5EG:9:9;<LREUBD7 79EU=g@<798�EUeH�GEU>A@DEGYR8�@I7FEG@CXo@C;<84�NOHXW;<84=5bgXW>HXWb5EUTqBD79EU>9YR84=g@<Lu@<798�8`v~XW=�@IXW>HYB?LUTWTWXV=<XVLR>FEUTÙy[;IEGe9XâEG@<Xo�U8�JM;IEUbg84cqLR;<rtsZY¢EG@I84=KEGTW;<8?EUeHZ^8`v~XW=�@�JMLU;{@I7984=<8g79XWYR7�>Ny[=<7H8?TWTV=?� EU>9eu@I7H8³ =�8`�GEUTWOFE�@I8?eu7984;<8CBSEG>�h_8CEUe9e984e�eHXW;<84B4@<TVZR��=PO9BD7lE�@<8?BD79>9XV�mO98C7FEU=�hj848?>\O9=<84e�XV>u=<8?B`@IXVLR>é~s©Å~s©Å~sÐ >98�LUJ~@<798�TWLR>9Y2@<8?;<b²EU:9:9TVXWBSE�@IXWLU>9=£LUJ~@I79XV=)cqLR;Pr�XW=£@I7H8qXV>m�U8?=�@IXWYREG@IXVLR>�XW>A@ILn>9LR>Ny�÷^E�vNcq8?TVTWXâEG>JM;<848Q8?TV8?B`@I;<LU>�e9XW=�@I;<XVh9OH@<XWLR>9=4s�áFLR;}@<79XW=}@I798Q=P:j84B?XozFBSEG@<XWLR>�LGJ.:jLR:HO9TâEG@<XWLR>\�©84TW84b584>m@<=?�jXW>f@I7H8B?LU>m@<XW>mO9O9b EU>9e EG@a7HXWYR7 �NO9EU>A@IO9b =<7984TWTW=CcqLRO9TWekhj8�>98?B48?=<=<EU;PZU��EG>9ek@I7H8fb5LU;<8�YR8?>984;<XVBEUh9=PLR;<:~@IXWLU>$JpEGB4@ILU;28`v~:9;<84=<=PXWLR>�XW=x@I7H8KEG:9:9;<LU:9;<XWEG@I8�LR>H8{@ILgO9=<8Gs"$#�[ Ö -g(�1} ?0Ñ×,3?-g(Ñ×> >[email protected]?LR>A@IXV>mO9XV@[Z�LGJFEUh9=PLR;<:~@IXWLU>�JpEUB`@ILR;P=�EG>9eK84=<B?EU:j8q:9;<LUhFEUh9XVTWXV@<XW84=®JM;<LRb!hjLROH>9eK@<79;<LROHYR7@IL�JM;<848}=�@DEG@<8?=�79EU=qh_8?84>�=<7HL�cn>®s�6279XW=qEUTWTVL&cn=�LU:FEUB?Xo@[Z|@IL�hj8}b5LNe984TWTW84e|XW>5e9XW=PB?;<8`@I8¢=<:_8?B`@I;IEGTTWXV>98?=4�nh9TW84>9e98?e 79XVYR7 =<84;<XV8?=�TWXV>98?=�EU>9e XW> @<798�B?LR>A@<XW>mO9O9b�s�627H8uEU:9:9;PLAEUBD7 =<79L&cn> 798?;P8XW=5B?LR>A@<;IEU=�@I8?e cnXo@I7 @I7H8fL~B4B?O9:9EG@IXVLR> :9;PLRhFEUh9XVTWXo@[Z�JMLR;<b5EUTWXV=<bàEG>9e =<79L&cn> @ILlh_8�ElbgLR;<8bg8SEU>9XV>9YUJMO9Tjb58`@I79LNe$LUJ�8`vN@I84>9e9XW>HY5:97HLU@ILGy�;IEG@<8?=¢EUB4;<LR=P=n@<798�XWLR>HXW=IE�@IXWLU>a@I79;P8?=P79LRTWe®s��@|=P79LRO9TVelhj8a>9LU@I84el@I7FE�@QcnXV@<7]@I798ae984�U8?TWLU:9b584>A@CLUJ2@I798�EU>FEUToZN@<XWB?EUTWToZ�B?LR>A@<XW>mO9LRO9=i8?bgXW=�y=<XVLR>�B?LN84ÜgB?XV8?>A@¢XW>$@I79XV=2BD7FEU:H@<8?;�Xo@nXW=2XVb5bg8?e9XWEG@I84TVZg:_LR=<=PXWh9TV8�@<LC:9;<84e9XWB`@{EU>�8?bg8?;PYR8?>A@$àFO~v@I79EG@iB4LR>A@IXV>NOH8?=Q=Pb5LNLU@I7HTVZfJM;PLRbôhjLROH>9e~y[h_LRO9>He�@IL^hjLROH>9e~y�JM;P8?8CcxES�U8?TW84>9YU@<79=QJMLR;KEU>]LR:~@IXoyBSEGTWTVZ�@<79XW>^:9TâEU=PbaE~s}6279XV=�7FEG={hj848?>�e9LU>98iJMLR;n@I798iBSEU=P8QLUJ�@I7H8�©xEUTVb584;¢=P8?;PXW8?=�TWXWbgXV@nXV>^@I7H8JMO9=PXWLR>�e9Xo�R84;P@ILU;2EU>9e�XV=xe98?=PB?;PXWh_8?e$XW>�=P8?B`@IXWLU>^éNs©ÃHs�62798�O9=<8�LUJ)@<798{8?=PBSEU:_8�:9;<LUhFEUh9XVTWXV@[ZgEG>9eEUh9=PLR;<:~@IXWLU>�JpEUB`@ILR;�8`v~:9;P8?=<=PXWLR>H=|e984�U8?TWLU:j84elXW>\@I79XV=iBD7FEU:~@I8?;iEUTWTVL&cn=�@<79XW=�@[ZN:j8�LGJ2=<:_8?B`@I;IEGTJM8SE�@IO9;P8�YU8?>984;IEG@<XWLR>�@ILCh_8�8`vN@I84>9e98?e�@ILgLR:H@<XWBSEGTWTVZC@<79XWBDr�B4LR>9e9Xo@IXVLR>9=�¥M=<8?8�=<84B4@<XWLR>^é~s©Å~sÆÅUª`s

Page 108: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� � ��� ����

� � � ������� ��� � ��� ������� � � � � �

� ��� � �������� ���

���! #"%$'&)(*",+.-/ #0213+4�!576.02186/$'0:9;$<&)-%1=02&=02�> #0:?�5.@A+B&=(*"%02CD0E@F5G1)",+.�H?%I<$'�%JK L"%0M+.&L&=02CONHI'P!+4�%?&L5.I<QH L$<5G�E5.@R(:5GI<I'$<&L$'5G�,+.ITS�1L+.?%$U+V #$T6G0�CK+4 #1=$<(20:&W&LQH$' #0:?D L53+.I<IX@FQH #QH1L0Y(25GI'I<$'&L$<5.�,+.IZS[1#+.?H$U+4 L$Z6.0W9\5G1=]_^���`-,+41= #$'(2Q%I<+.1a$' a$<&b?%02&=$<JG�%0:?c #5d&LQ%-%-R5G18 )&= LQ%?%$'02&)5G�c"%0e+e6fPc&=-g0:(2$'02&h+.&b9\0:I<Ii+.&j(:5G�%&L$'&= L02�> &= LQ%?%$'02&\$'�>6G5GIZ6/$'�%JkNg5GQH�%?B+.�H?K@F1L0:0j&8 *+4 L02&2^����K #"%$'&l(*"%+.-H #0:1m+k"%$<JG"KnfQ,+.�> #Q%Co&="%02I'IRCD5X?%Q%I<0$<&a?%0p6G02I'5G-R02?c9Y"%$'(*"c0p6.+4I<Q,+4 L02&a">P/?H1L5GJG0:�%$<(q-g5.-%Q%IU+V #$<5.�%&2r% L"%$<&s9Y$<I'ItNg0kQ%&=02?c L5d-%1=02?%$'(: j+.I<I #"H0E-g5G-HQ%IU+4 L$<5G�c?%$<&8 #1L$'N%QH L$<5G�%&j�%02(:02&L&[email protected] #"%0M">PX?%1L5.JG02�u&L0:1L$<0:&hI<$'CK$Z j&=-g0:(: #1L+.IW@F0e+V #Q%1=0(25X?%0E5.@A(*",+.-/ #021qv/^w���u&=02(p #$<5.�uxH^'yE #"H0EJG02�%0:1#+.I�-H$<(: LQ%[email protected]) #"%0E�%0p9�CD5/?H02I�$'&h5GQH LI<$<�H02?z^{ 0:(: #$'5G�|x%^~}O?%0:&L(:1L$<NR02&b+M�%0:9�&=0: a5.@��k+.QH�f s@�+4(: #5.1L&s@F5G1YQ%&L0k$<�� L"%03CK5X?%02I�^��s"%0k?%0: #+.$<I'&m5.@ #"H0a"H$<JG"E�/S[&L"%0:I<I%(:5GI<I'$<&=$<5G�,+4IZS�1L+.?%$<+4 #$T6G0�CD5/?%QHI<0\$'�E #"%0s(:5GCK-HQH *+4 L$<5G�%+.IH$'CK-%I'02CD02�> *+V #$<5.�O+.1=0 #"H02�!-%1=02&L0:�> #02?u$<��&L0:(: #$'5G�!x%^��B+.�%?c$<�`&=02(: L$<5G�cx%^�xB+.�!+.�,+4I'PX&L$'&j$<&b?%5G�%0w$<�> #5D #"%0w64+.I'$<?%$Z �P+.�%?�+.QH L5GCK+4 #$'5G�B5.@� #"%03N%QH�%?%I<$'�%JM(2QH 8S�5.�`&L(*"%0:CK0k-%1L5.-g5G&=02?|$<�B L"%03�%0:9�CD5X?%02I�^

�`��� ��� �3�D�h�W�|�2�j� �M�����`� � ���`�h��s"%0:1L0l+.I<1=0e+.?/P)0p /$'&= ¡(25.I<I<$'&L$'5G�,+.ITS�1#+4?%$U+4 L$Z6.0t(25X?%02&[email protected]<$<J."f ¢02I'02CD02�> #&z9Y$Z #"%$'�) #"%0�&8 #1LQH(: #QH1L0�5.@ #"H0s£)¤Y£ { &LQ%$Z #0m54@g(:5/?H02&b¥ { QHCKCD021=&2rf}.¦.¦Hy7§p^i�s"%02&=0Y(25X?%02&�",+e6.0aNR020:�D9A02I'I/ L02&= L02?K+.�%?D+.1=0-%1=576.02�E #5hNR0m0p�¢0:(: #$Z6G0s$'�O?%0e+4I<$<�HJb9Y$' L"OI<$'JG"> W0:I<02CD02�> L&2^�£b&W",+.&�NR020:�E?%02&=(21=$<NR02?M$<�O(*"%+.-H #0:1yGrX$' l$<&�$'�,+.-%-%1=5G-%1=$U+4 L0Y L5O+.-H-%I'PO L"%02&=0)(25X?%0:&A #5k"%02+�6fPK&L-R02(:$<02&\+.�%?d+.�B+.IZ #0:1L�,+4 L$'6.0b&=(*"%02CD0

¨ x

Page 109: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨ v

©_ªW«¥ ©V¬2ªf¬G«�¬ §®­z¯T° «¥ ©V¬eªf¬G«�¬ §8­¢¯T°¥ ©V¬eªf¬G«�¬ §®­z¯¥ ©V¬2ªf¬G«�¬ §®­

­¢±­�²­¢³­¢´­gµ

¶�$'JGQ%1L0jx%^'y.·�¸z¹Vº�»�º®¼[½¿¾ÁÀÁÂzÃpÄ/¼Å¹7º�Æ7Ç7ÈV»7À�¾ÁÈ7ÉYÂÅÊ[¹7º8ËmºWÌ7ÍÅÃpÌ.Ã:Â�º=»sÄTÃ:Ít¼�¹Vº�È7º8Î�Ë\Ã�»�º=À'ÏCOQ%&8 BNR0�?H0:6G0:I<5G-R02?¢^ж�$'JGQ%1=0�x%^Zyc&="%5�9Y&d #"H0c�%0p9ÑJG02�%0:1#+.Ia-%$'(: #QH1L0�9Y$' #"Ò #"%0�&=9Y$Z #(*"%$'�%J-%1=$<�%(:$<-,+.IHnXQ%+.�> #Q%C�&="%02I'I<&�?%0:Ó%�%$<�%J3 #"H0a&=Q%NR?%$'6X$<&=$<5G�H&2^W���M #"%$'&��%0:9ÔCD5X?%02I%5.�%0Y(25G�%&=$<?%0:1L&A+@FQ%I'I'PE1=02&L5.I'6G0:?B-%$'(: #QH1L0Y@F5G1l L"%0jI'5�9ÔI<0p6G02I'&2rf9Y$' L"K #"H0a64+.I'02�%(:0j02I'02(p #1L5.�d$<�M #"H0j(25.1L0w^WÕb021=0 ©1L0:-%1L0:&L0:�f L&j #"H0q(:5G�HÓ,JGQH1#+4 L$<5G�zr ª #"H0q-,+41L$' �P�+.�%? « L"%0k L5. *+4It+.�%JGQHIU+.1YCD5GCK0:�> #Q%CÖ54@� #"H0&=PX&8 #02C|^��[@i L"%0h64+.I<0:�%(20h0:I<02(p #1=5G��$<&s$<��+ECD5G1L0h"H$<JG"%IZPD0p /(2$Z #02?|-%1=$<�%(:$<-,+.IgnfQ,+.�> #Q%C×&="%02I'Iz$' $<&�-R5G&=&L$<NHI<0\ L536X$'0:9Ò #"%0Y02I'02(p #1L5.�K+4&mØÁN%QH$<I' 2Ù.5G�D+)-,+41L02�> l&= L1LQ%(p #Q%1=0.^�Õa021L0 ©V¬ 1L0:-%1L0:&L02�> L&A #"H0-,+.1=02�> qI<0:6.02I�Ù�&k(25G�/Ó,JGQ%1L+4 #$'5G�zr ªf¬ $Z #&3-,+.1=$' �PÚ+.�H? «�¬ $' L&3 L5. *+4I�+.�%JGQ%I<+.1hCD5GCK0:�> #Q%C|^M�s"%0?%0:(21L02+.&L$'�%JkI<0:6.02I'&�5.@R1L02&=5GI<Q/ #$<5.�ENg0:(25GCD0m64+.I<$'?E+.&� #"%0sI'0:6.02I<&�NR02(:5GCD0sCK5G1=0s?%02J.02�%0:1#+4 L0b+4�%?I<0p6G0:I,-g5G-HQ%IU+4 L$<5G�H&�NR02(25.CK0Y&= #+4 #$'&= L$<(e+4I<I'P3?H$<&= L1L$'N%QH #0:?z^�Û)�%0s$'&W #"%0:1L0:@F5.1L0a+.N%I'0\ #53N%Q%�H?%I<0s #"H0-R5G-%Q%I<+4 #$'5G�%&m-%1=5GJG1=02&L&=$'6.02I'PB$<�> #5E #"H02&L0kI<0:&L&Y1L0:&L5GIZ6G0:?�-%$'(: LQ%1L0:&2^ÜÝ$' L"| L"%0k&LQ%NR?%$'6X$'&L$<5.�%&Y$<�� #"%0kI<0p6G02I¡&=(*"%02CD02&)&="%5�9Y�`$<��Ó,J.Q%1L0kx%^Zyw$' Y$'&Y-g5G&=&L$'N%I<0h #5KCK5X?/SQ%I<+.1L$'&L0E #"%0K+.&=&L02CwN%I'Pu5.@A #"%0K(25GI'I<$<&=$<5G�%+.IZS[1#+.?%$<+4 #$T6G0kCK+4 #1=$Z gr¡$<�u-,+.1= L$<(:Q%IU+.1h@F5G1h L1#+.�%&=$' L$<5G�%&9Y"%0:1L0O #"H0O-,+.1=02�> 3$<&b$<�>64+.1L$<+.�> e^3Þ\5G�H&L$<?H021)N%1=$<0pß%Pc"H579 #"H0O?%$'�g021=02�> hI'0:6G0:I<&)54@�1L02&=5GI<Q/ #$<5.� #"%+4 D&LQ%(*"à+ÚCD5X?%02Im-R5G&=&L02&=&L0:&B&L"%5.Q%I<?áNR0|?H0e+.IZ E9Y$' #"¢^â¶,5G1O L1#+.�H&L$' L$<5G�H&w9Y$' L"%$<�� #"H0� �9A5I<5�9A02&8 O&L(*"H02CD02&�¥F$Å^ã0.^¡­Ðä�­¡²L§pr�5.�%0d",+4&O+|@FQHI<I'PÚåVS�1L0:&L5GIZ6G0:?æ&=PX&= L02C�9Y$' #"�(25G1=1L0:&L-R5G�%?%$'�%J #1L+.�%&=$' #$'5G�%&qNg0p �9\0:02�â+.I<I� #"%0d-g5G&=&L$'N%I<0BI'0:6.02I<&:^Ú�s"%0�1#+4 L02&[email protected] #"%0:&L0�1=02JG$'CK0:&wCOQ%&8 ENR0B5.@"%$'JG"ænfQ,+.I'$' �Pu?%Q%0d L5` #"%0d?%0: #+.$<I'02?�&=-g0:(: L1L5G&=(25G->Pæ L",+4 E$'&w0p /-R02(p #02?á #5cNg0B-R0218@F5G1LCD02?�5G� #"H0j1L0:&LQ%IZ #$'�%Jw&=-g0:(: #1L+H^��s"%02&=0)1#+V #02&m+.1L0j+.&L&=02CON%I'02?d@F1=5GC� #"%0bCK5.&= A(25GCD-%1L0:"%02�H&L$'6.0h(e+.I'(2Q/SIU+V #02?u+.�H?uCD0e+4&LQ%1=02?u?,+4 *+�+e64+.$<I<+.N%I'0.^k�s"%0D L1L02+4 #CD02�> 35.@l #"%0:&L0MI<5�9A0213I<0p6G0:I<&)@�+.I'I<&j9Y$Z #"%$'�

Page 110: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨Gç

#"H0M+.-%-%1=5>+.(*"Ú5.@� #"H0E£)¤Y£ { }.¦Gèd&8 *+4 L$<&8 #$<(2+.I�N%+.IU+.�H(20O(:5/?%04r¢ #"H$<&)(:5/?%0EQH&L02&h@FQ%�H?,+.CD02�> *+.I02�H021LJ4PcI'0:6G0:IW+.�H?`1#+4 L0w?,+V *+K@F1=5GCÑ-%1L0¿S®+.&=&L02CwN%I<0:?!£)¤Y£ { +.?/@�Ó,I<0:&2^)�s"%0w$<�Hß,QH02�%(:0O5.@� #"H0"%$'JG"%021h-R5G-%QHIU+4 L$<5G�%&)+.1L0E-%1=5Vé=02(p #0:?�5G�> #5� L"%0MI<5�9\0:1hI<0p6G02I�1=02&=Q%I' L&2^M�s"%0E #1=0e+4 LCK0:�f k5.@l #"H0I<5�9ÝI'0:6G0:I<&A$<&m+4I<1L02+.?HPM9A02I<IRQ%�%?H021L&8 #5X5/?B9Y$Z #"%$'�K L"%0)(25.�f L0p X s5.@¡0¿ /$<&= L$<�%Jw£)¤Y£ { (:5/?%0:&2r,+4�%?NR02(25.CK0:&s+.�B$'&L&LQH0)5.@¡J>+4 L"%021=$<�%Jq5G1\(2+.I<(:Q%IU+V #$<�HJw+.I'Ig54@z #"%0j�%02(:02&L&L+.1=P�+4 #5GCD$<(Y?,+V *+H^��s"%$'&\$'&+d&=$<JG�%$ZÓ,(e+4�f b *+.&=]�@F5G1a #"%0w"%0e+e6X$<0:1)&L-R02(:$<02&j?%Q%0O #5K #"%0OI<+.1LJ.0w&L0p #&)54@AI'0:6.02I<&O¥�+.1=$<&=$<�%JD@F1L5GC #"H0wåM1L0:&L5GI'QH #$'5G�_§l9Y"%$'(*"�CwQ%&= aNR0k?H0e+.IZ m9Y$' L"z^�s"%$'&\(*",+.-/ #021A(25G�H(202�> L1#+4 L02&s5G�K?%0:6.02I<5.-%$<�%JO+qJG0:�%021L+.IgCD5X?%02IR+.�%?d(25X?%0a L",+4 \(e+.��+4&L&L0:CON%I'0 #"H0E(25GI'I<$<&=$<5G�%+.IZS[1#+.?%$<+4 #$T6G0hCK+4 L1L$Z cN%I'5/(*]f&)@F5G1j L"%0E1L0:Cd+4$<�%$'�%Jd #"H1L020E?H02JG1=0202&k5.@l1L0:&L5GI'QH #$'5G�z^�s",+4 a$'&2rH L"%0h@F5GI<I'579Y$'�%JON%I'5/(*]K5.@� #"%03(25.I<I<$'&L$'5G�,+.ITS�1#+4?%$U+4 L$'6.0sCd+4 L1L$T d9Y$'I<IzNR03+.&L&=02CONHI<02?¢^

­z¯T° ­¢¯Ö­­z¯T° êêêAëíì'ìïîðëíì'ìïîðëTìZìïî êêê­¢¯ êêêAëíì'ìïîðëíì'ìïîðëTìZìïî êêê­ êêêAëíì'ìïîðëíì'ìïîðëTìZìïî êêê

¥Fx%^'y�§

ÜÝ"%0:1L0m02+.(*"BØÁ0:I<0:CK0:�f 2Ù4$'�q L"%$<&iCd+4 L1L$T k$<&�$<�w@�+.(p W+bCd+4 L1L$T k-,+.1= L$' L$<5G�k5.@,$Z #&�5�9Y�zrG+.(2(:5GQ%�> #$'�%J@F5G1) L1#+.�%&=$' L$<5G�%&)9Y$Z #"%$'�u+.�%?uNg0p �9\0:02�� L"%0E6.+41L$<5.Q%&h1L0:JG$<CD02&35.@A1L0:&L5GI'QH #$'5G�z^E�s"%0DN%Q%�%?HI<$<�HJ&L(*"H02CD0c #5�Ng0�Q%&L0:?ñ@F5G1D L1#+.�H&L$' L$<5G�H&D9Y$Z #"%$'�Ò L"%0c-,+41= #$Z #$'5G�â&8 #1=Q%(: LQ%1L0`54@D¥�x%^Zy7§E$<&K&L"%5�9Y�CD5G1L0l@F5G1LCK+.I<IZP3$<��¥Fx%^Á}G§p^�òb5. L0\ #",+V W #"%0sNHQ%�%?%I'$<�%J%rV9Y"%02�E64+.I'$<?zr4$<CD-%I<$'02&�&= *+V #$<&8 #$'(e+.IX1L0:IU+4 L$'6G0-R5G-%Q%I<+4 #$'5G�%&m5.@� #"H0k(:5GCD-g5G�H02�> aCK0:CONg0:1L&s5.@� #"H0kNHQ%�%?%I'0.^

­�²mäà­ÚóÔ­¢³ ·õôMöã÷�ø#ù#ø*ú�øLûíüeý þYÿ�� ú ôMöã÷®øLù#ø*ú�øLûTü2ý þLúô öã÷�ø#ù#ø*ú�øLûíüeý þ#ú ÿ ¥Å} «�� y7§¥Å} «�¬�� y7§:¥Å}2° � y7§ ô öã÷®øLù#ø#ú�ø#ûTü2ý þ­¢³bäà­ÚóÔ­¢´ ·õôMöã÷�ø#ù#ø*ú�øLûíüeý_ÿ�� þ ôEö�÷�ø*ù#ø#ú�ø#ûíüeý þôMöã÷�ø#ù#ø*ú�øLûíüeý þYÿ ¥Å}2° � y7§}%¥�}.¯ � y7§ ôEö�÷�øLù*ø*ú�øLûíüeý­¢´bäà­ÚóÔ­gµ ·õôMöã÷�ø#ù#ø*ú�øLûíüOÿ�� ý ôEö�÷�øLù#ø#ú�ø#ûTü2ý

Page 111: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨��ô öã÷�ø#ù#ø*ú�øLûíüeý ÿ ¥�}.¯ � y7§­ ³ ô ö�÷�øLù#ø#ú�ø#ûTü ¥Fx%^~}.§

�s"%0s1L+4 #0:&�@F5G1i L"%02&=0s"%$<J."%021i&= *+V #02&i9Y$<I<IfJG0:�%021L+.I<IZPkNg0m">PX?%1=5GJG02�H$<(m0¿ H-H1L02&=&L$'5G�%&a¥�5.1WnfQ,+.�> #Q%C?%0p@F02(: ¡ L"%025G18PhN,+.&=02?k0p X #0:�%&L$'5G�%&¡5.@/">P/?H1L5GJG0:�%$<(�0¿ /-%1L0:&L&L$'5G�%&#§t@F1=5GC +s1#+4?%$U+.I4Cd+V #1L$T b02I<0:CK0:�> -R5G$<�> A5.@z6X$<0p93r/ #"H0jCK+.$<�d+.I' L021#+V #$<5.�%&�Ng0:$<�%Jq$<�K L"%0)+4�%JGQ%I<+.1A-%+.1= L&\1L0:nfQ%$<1=02?d L5E+.(2(:5GQ%�> \@F5.1 #"H0)�%0:9�?%0:JG1L0:02&s5.@¡1=02&=5GI<QH L$<5G�d+.�%?B L"%0)�%0p9 (:5GQ%-%I'$<�%Jq-%$<(p #Q%1=02&\ L",+4 A9Y$<I<IRNR0jQ%&=02?z^��s"%0:&L0+.�%J.Q%IU+.1D-,+.18 #&D9Y$<I<IYNR0c$'�%(25G1=-g5.1#+4 L02?à$'�à #"H0!�3+.Q%�> d@�+.(p #5G1=&2^ �s"XQH&B�%0:9��3+.Q%�> d@�+4(: #5.10p /-%1=02&=&L$<5.�%&BCOQ%&8 BNR0c?%0:1L$Z6G02?â9Y"%$<(*"Ý+.I'I<5�9�@F5.1D L1#+.�%&=$' L$<5G�%&MNR0: �9A020:� +4I<Is #"%0��%02(:02&=&#+.18PI<0p6G0:I<&Y5.@i1L02&=5GI<Q/ #$<5.�u¥�&L0:0q&=02(p #$<5.�|xH^~}/r%9Y$Z #"�1L0:&LQ%IZ #&Y&LQHCKCK+.1L$'&L0:?|$'�B *+.NHI<0hx%^Zy7§p^� �� � ����������������� ������ !�#"%$ &'�)(%*+",�)*+(��-&.�/��(1032+4305�+"��-2 "���687�*��90;:�������<\0p@F5G1L0O-H1L5X(2020:?%$<�HJd9Y$' L"c #"%0w?%0: #+.$<I'&b5.@� L"%0w�H0:9YI'P�-%1=5G-g5.&L02?!(:5/?H0O$' b$<&bQ%&L0p@FQ%Ii L5B1L0p6/$'0:9+.�%?u?%0pÓ,�%0DCK5G1=0O(e+41L0:@FQHI<I'P� #"%0MN,+4&L$<(MN%QH$<I<?H$<�%JBNHI<5X(*]X&h L",+4 h9Y$<I'IWNR0EQ%&L0:?æ+.�%?u"%5�9; L"%0:P(e+41L1=Pd #"%1=5GQ%JG"� #5E #"H03Ó%�,+.It(25.I<I<$'&L$'5G�,+.ITS�1#+4?%$U+4 L$Z6.0sCd+V #1L$T g^=?>A@�BDCFEHGDIKJLIAMNIPORQTS£j&A(e+.�dNg0b&L020:�[email protected]�¥Fx%^Zy7§� L"%0jCD5G&8 \N,+4&L$<(aCK+4 #1=$Z MN%QH$<I<?H$<�%J3N%I'5/(*]M$<&� #"%+4 [email protected] L1#+.�%&=$' L$<5G�%&NR0: �9A0202�Ô �9A5�I<0p6G02I'&B5.@h1=02&=5GI<QH L$<5G�â5G1K9Y$Z #"%$'�Ô+�&=$<�%JGI'0�&=(*"%02CD0.ra&LQH(*"Ý+.&B­z¯T°VU ­¢¯�ra5.1­z¯T°WU ­z¯T°,^)�s"%0:&L0EN%QH$<I<?H$<�%JDN%I<5X(*]f&b9Y$'I<I¡Ng0w1L0:@F0:1L1=02?` #5B+.&qØ�N%I<5X(*]|-%1L$'CK$Z #$'6.02&:Ù_+.�,+.I'5GJG5GQ%& #5� #"%0�Øã@F0e+V #Q%1=0c-%1=$<CD$' #$Z6G0:&2Ù�Q%&L0:?Ô$<�à L"%0`&=-g0:(: #1L+.Ib@F02+4 #QH1L0cJ.02�%0:1#+4 L$<5G�ñ(:5/?H02&B5.@3£j¤a£ {ç ¦>}�S ç ¦.xc¥ { Q%CKCD021=&2rg}.¦G¦/y7§prR+.�%?c$<�� L"%0k"%$<JG"�&L0:1L$<0:&j&=-g0:(: #1L+.I¡@F0e+4 LQ%1L0qJG02�%0:1#+4 L$<5G�|9\5.1L]�5.@(*",+.-/ #021Kv/^��s"%0�N%I<5X(*]�-%1=$<CD$' L$'6G0K1L0:-%1L0:&L02�> L&K #"H0�@F5.Q%�%?,+4 L$<5G��QH-g5G��9Y"%$'(*"à+4I<IA L"%0|ÞlS�XCK+4 #1=$<(20:&b9Y$'I<I¡Ng0wN%Q%$<IZ e^b�s"%0DÞlS�Xõ(25X?%0O?%0p6G0:I<5G-R02?Ú$<�� #"%$'&j(*",+.-/ #021h$'&jCD5/?HQ%IU+.1=$<&=02?|&LQ%(*" #"%+4 Y0e+.(*"�CK5X?%Q%I'0)1L0p #Q%1=�%&b+MNHI<5X(*]d-%1=$<CD$' #[email protected]+.&L&=02CON%IZP�$<�> L5EIU+.1=JG021aÞlSYXÐ&= L1LQ%(p #Q%1=02&2^=?>A@�BDCZSÜ�0u1=0:@F0:1� L5� #"%0u+.&=&L02CwN%I<0:?Ð&= L1LQ%(p #Q%1=0Ú5.@ON%I'5/(*]â-%1L$'CK$Z #$Z6G02&B+.&!ØÁN%I'5/(*]f&:Ù'rh+4�,+.I<5.JG5GQ%&d L5&L-R02(p #1L+.IOØ�@F02+4 #QH1L02&:Ù'^��s"fQ%&�£j¤a£ { �Hy.y!1LQ%�Ý$<�Ý$Z #&B">PX?%1L5.JG02��&L0p #Q%-Ð+.I<1=0e+.?HPâ1L0p #Q%1=�%&|+N%I'5/(*]�@F5.1O #1#+4�%&L$Z #$<5.�%&ENg0p �9\0:02�ñ­¢¯\[5];+.�%?Ò­�[ &8 *+4 L02&M9Y"%$<(*"Ò(25G�> #+.$<�%&MN%I'5/(*]�-%1L$'CK$Z #$'6.02&

Page 112: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨ è

@F5G13­z¯\[3]^U ­`_T¯a_b[c_b]3_<ri­¢¯\[5]�U ­`_b[d_Ur�­�[�U ­`_T¯a_b[c_e]c_W+.�H?�­�[�U ­R_f[d_U^dòb5. L0D L",+4 k$' 3$'&-R5G&L&=$<N%I'0M L5c+4&L&L0:CON%I'0D L"%02&=0dN%I'5/(*]f&3 #5|5GNH *+4$<�ÚIU+.1=JG0213CD5G1L0MJG0:�%021L+.I�(25GI'I<$'&L$<5.�,+.IZS[1#+.?H$U+4 L$Z6.0CK+4 #1=$<(20:&m9Y"%$<(*"�+.I'I<5�9Ý@F5G1s(25GCD-%I'0p /$' L$<02&\&LQ%(*"c+.&Y-,+.1L0:�> b(*"%+.�%JG$'�%JE #1#+4�%&L$Z #$<5.�%&2^gZh E3Q'GNid>A@�BNC�S�s"%$'&�+4&L&L0:CON%IZPÔ5.@qN%I<5X(*]f&|$'&�1L0:@F0:1L1=02?Ý #5â+.&|+ÔØ�&LQH-g0:1LN%I'5/(*]_ÙZrh+.�%?�$<&|+.�,+4I<5GJG5.Q%&d #5á #"H0&L-R02(p #1L+.ImØÁ&=Q%-R021=@F02+4 #QH1L0.ÙZ^E£�&LQH-g0:1LN%I'5/(*]�CK$'JG"> ) �PX-%$<(2+.I<IZPc(:5G�> *+.$'�!5G�Ú$' #&j?%$U+.J.5G�,+.I�r¢N%I'5/(*]-,+.18 #$Z #$<5.�%&A$'�K9Y"%$'(*"B #"%0j-,+.1L0:�> m(25.�HÓ,JGQ%1L+4 #$'5G�B?%5X02&\�%5. m(*"%+.�%JG04^��s"%0)5.�_S�?%$<+.JG5G�%+.I_-,+.1= L$ZS #$'5G�%&l9\5GQHI<?B #"H02��(25.�f #+.$<�d-,+.1L0:�> Y(*",+.�%JG$'�%JON%I'5/(*]f&:^��s"%0h&LQ%-R021=N%I<5X(*]DCd+4 L1L$T K9A5GQ%I'?d #"H02�NR0b&=5GI'6.02?d L5q5GN/ *+.$'�d+3(25GCD-%1=02"%0:�%&L$Z6G0)-R5G-%Q%I<+4 #$'5G�E&L5.I<QH L$<5G�M@F5G1� #"%0b+4 L5GCK$'(s&=PX&= L02C Q%�%?%0:1(25.�%&L$'?%021L+4 #$'5G�z^jWQTSkBlGDI\E-MNIA@`mn@%o#MqpHQrEHs)GqMNI\B h >As)Gtid>A@�BNCuSvMqG h BwM h GNQxo�@%GyMNp-Qrm-Q'z{JL@�|-Q}>�s"%0`Ó,�%+.Ih+.$'C 5.@3 L"%0!�%0:9ðCD5/?%0:Ij$'&d #"%0Ú+4&L&L0:CON%IZPñ5.@k&LQH(*"Ð+æ&=Q%-g0:1LN%I'5/(*]_^ �s"%0`Ó,1L&8 ?%$<+.JG5G�,+4ItN%I<5X(*]B9Y$<I'It(25G�> *+4$<��+.I<It #"H0k L1#+.�%&=$' L$<5G�%&s9Y$Z #"%$'�| L"%0�¥ ©Rª�« §s&L(*"%0:CK04rg9Y"%0:1L0q L"%021=0$<&h�H5|?H0:Ó,�%0:?�-,+41L02�> 2^E�s"%0K�H0p X k&L0: k5.@A?%$<+.JG5G�,+4I�N%I'5/(*]f&3?%02&=(21=$<NR0D�H5G�/S�-%+.1L0:�f 3(*"%+.�%JG$'�%J #1L+.�%&=$' #$'5G�%&Y9Y$' L"%$<�| #"%0�¥ ©V¬eªf¬G«�¬ §8­¢¯T° « &L(*"%0:CK04r¢9Y$' L"c #"%0w5.�RS[?%$U+4JG5G�,+.ItNHI<5X(*]X&)+.(2(:5GQ%�> #$'�%J@F5G1D-,+.1=02�> B(*",+4�%JG$<�HJ� #1L+.�%&=$' #$'5G�á9Y$' #"H$<�â L"%$<&D&L(*"H02CD0!+.�%?à L1#+.�H&L$' L$<5G�H&D?H579Y�ñ L5Ô¥ ©Rª�« §p^£j&�",+4&|NR020:� �H5. #0:?zrb L"%02&=0�N%I'5/(*]f&�9Y$<I'IjNR0u+.&L&=02CON%I'02?ÝQ%&=$<�%Já #"%0!0¿ /$<&= L$<�%Já£j¤a£ { }.¦.è+.-%-H1L5>+.(*"¢^ �s"%0`"%$'JG"%0:1d?%$<+.JG5G�,+4Ia0:I<0:CK0:�f L&B+.1=0c #"H02�ñN%I'5/(*]f&[email protected]�%5G�XS�-,+.1=02�> B(*"%+.�%JG$'�%J #1L+.�%&=$' #$'5G�%&\9Y$Z #"%$'�� #"%031L0:Cd+4$<�%$'�%Jw #"H1L02031=02&L5.I<QH L$<5G��&L(*"H02CD02&2r, L",+4 a$<&sN%I'5/(*]f&Y5.@� #"%0h@F5G1=C~���� � ö��_ûTü2ý þ �jöe�_ûíüeý þ � ö��_ûTü2ý þ �)ö��_ûíüeý � ö��_ûíüeý þ��jöe�_ûíü� ö��_ûTü2ý��jö��_ûTü2ý þ � öe�_ûíüeý��jöe�_ûíüeý � ö��_ûTü2ý��jö��_ûTü� öe�_ûíüq�jö��_ûTü2ý þ � öe�_ûíüq�jö��_ûTü2ý � ö��_ûTük�)ö��_ûíü

������ ¥Fx%^Á�G§

�s"%0a�%02$'JG"fNg5.Q%1L$'�%Jq54�RS[?%$U+.J.5G�,+.I/N%I<5X(*]f&\+.1=0s@F5G1�-,+.1L0:�> \(*",+4�%JG$<�HJk L1#+.�%&=$' L$<5G�%&W9Y$Z #"%$'�D L"%02&=0 #"H1L020b&L(*"%0:CK0:&2^��s"%0pPB+.1=0b�H02JGI'$<JG$'N%I<0m(:5G�> #1=$<N%QH L5G1L&l #53 #"%0b-g5.-%Q%IU+V #$<5.�D&8 #1=Q%(: LQ%1L04r%+.I'I<5�9Y$<�%J #"H02C� #5qNg0a&L0p \ #5��20:1L5%^��s"H021L0b5G�%IZPM1L02CK+.$'�%&� #"%0aÓ,1L&8 A@F0p9Ý"%5G1L$��25G�> #+.I_+.�%?d6.0218 #$<(2+.I_&= L1L$<-H&+.(:(25GQ%�> L$<�%JM@F5G1m #1L+.�%&=$' #$'5G�%&\@F1=5GC� L"%03Q%-%-R021s #"%1=0203&L(*"H02CD02&b L5E #"%03I<5�9A021s �9A5%^��s"%02&=0O+.1=0

Page 113: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

¨G¨

Ó,I'I<02?�$'��Q%&L$'�%JD L"%03-%1L57é=02(: L$<5G��&=(*"%02CD0.rR+.&a+.I'1L0e+4?HPBQ%&=02?|$<��£)¤Y£ { }.¦Gè/^

�`��� �����!�B�Ñ�l���Y���D���ð���j�|�k�W�:���h��Y�� � �u03"R$#��(���*��+2¶,5G1s #"%$'&s9\5G1=]��%0:9õ�k+.QH�f a@�+.(p #5G1=&Y9A021=0w?%0p6G0:I<5G-R02?c+.(2(:5GQ%�> #$'�%[email protected]+4I<Iz #"H0k-R5G&=&L$<NHI<03(25GQXS-%I'$<�%JD&L(*"%0:CK0:&)@F5G1b L"%0w$'�%$' L$U+.I¡+4�%?cÓ,�,+4Ii&8 *+4 L02&2^h�k+.QH�f b@�+.(: L5G1L&j+.1L0q02&=&L02�> L$U+.I'I'P|nfQ,+.�> #Q%CCD02(*",+.�H$<(e+4IR(:5G1L1=02(p #$<5.�%&\5G�K(2I<+.&L&=$<(e+4I<I'Pw?%021=$'6.02?B5G&=(2$'I<IU+V #5G1�&8 #1L0:�%J. L"%&2^��s"%0b5G1L$'JG$<�,+4I_�k+4Q%�> @�+.(p #5G1=&[email protected]�Nzr,N/SÅ@�+.�H?|@USÅ@i L1#+.�%&=$' L$<5G�%&m9A021=0w-%Q/ a L5GJG0p #"%0:1b$'���|02�)�:02I;���i0:].021=$<&O¥8y ¨ �GvG§p^ c¡D(2$<0:�> sCK0p #"%5X?%&A",+e6G0hNR0202��?H0:6G0:I<[email protected] #"H02$<1A?%0: L021=CK$'�,+4 #$'5G�c¥K<\Q%1=JG02&=&#� { Q%CDCK0:1L&2ry ¨ è � §p^W�[ Y$'&mQ%&=Q,+.I¢ L5E0:64+.I<Q%+4 #0j L"%0h1#+4?%$U+.Ig+.�%?|+.�HJGQ%IU+41\(25.CK-R5G�%0:�f L&s5.@¡ #"%0k�3+.Q%�> m@�+4(: #5.1&L0:-,+.1L+4 #0:I'PG^��s"%03NR5GQ%�%?XS�NR5GQ%�%?c�k+4Q%�> s@�+.(: L5G1s$<&s?%0:&L(:1L$<NR02?�N>P

¢T£¤¦¥ ¤e§ ¥©¨Tª�¨ _ §�ÿ¬« �ª } µy­�®�¯u® _° ³²²±k³ ´ µ yµ ÷R¶ ¤¦¥ ¤e§�· £¤¦¥ ¤e§ ¥©¨Tª�¨ _ § ¥Fx%^�x>§9Y"%0:1L0 µ ÷ $'&m #"%0h&8 *+4 L$<&8 #$<(2+.Ig9A02$'JG"> a5.@¡ #"%03-,+.1=02�> Y&= #+4 #04r · $'&m #"%0h1L+.?%$<+.I¢$'�f L02JG1L+.I¢(:5GCD-g54S�%0:�f b+.�H? ¶ (:5G�> *+.$'�%&s #"%0k+.�%J.Q%IU+.1A@�+.(: L5G1L&:^ ¶ +.�%? · (2+.�c+4I<&L0hNR030p /-%1=02&L&=02?Ú+.&

¶ ¤¦¥ ¤e§�· £¤¦¥ ¤e§ ¥©¨Tª�¨ _ §�ÿ ä ¥©[ � ] � « �-¸¹� §®­z¯T°�º « ¸1» ¼ » ¥K[ _� ] _ � « _� ¸ _� §8­ _ ¯ _ ° _ º _ « _ ¸ _,½ ¥Fx%^~v.§�s"%0O0p64+.I<Q,+V #$<5.��5.@� L"%$<&b0p /-%1=02&L&=$<5G�Ú+4I<I<5�9Y&a$<CDCK0:?%$U+V #03(e+.I'(2Q%I<+4 #$'5G�|5.@W #"H0O(25G1=1L0:&L-R5G�%?%$'�%J�3+.Q%�> M@�+.(: L5G1E6/$<+!02nfQ,+4 L$<5G�Ð¥�x%^ãxf§¿^â�s"fQ%&M #"%0|0:64+.I'Q,+4 L$<5G��5.@j02nfQ,+4 L$<5G�Ð¥�xH^~vG§w1=02-%1=02&=02�> #& #"H0BCd+4$<��NR5X?HP�54@s9\5G1=]�$'�f6.5GI'6.02?�$'�� #"%0d(e+.I'(2Q%I<+4 #$'5G��54@a L"%0B�%0p9��k+.QH�f w@�+.(: L5G1L&:^!�s"%0NR5GQ%�%?/SÅ@F1L0:0Y+.�%?E@F1L0:0pS[@F1=020a�3+.Q%�> �@�+.(p #5G1=&�(e+.�MNR0a0p64+.I<Q,+V #02?EN>PM+4�,+.I'Pf L$<(e+4I<I'P30¿ X #02�H?%$<�%J3 #"H0NR5GQ%�%?/S[Ng5.Q%�%?|0p /-%1L0:&L&=$<5G�� #5M@F1L0:0k&8 *+4 L02&:r_+.&Y$<�� #"%03(:5G�> #$<�fQ%$Z �Pd9A5G1L]�54@�(*",+4-H #0:1j�H^�Y��Y� ¾�05*�������03",�)��(¿��4305 !*+0d�)�-2À����(ÂÁl:!Áu"���*�7� !�!�+�Þ\5G�%&=$<?%0:1¡+m #1#+4�%&L$Z #$<5.�)Ng0p �9\0:02�k �9\5s@FQ%I'I'PjåVS�1=02&L5.I'6G0:?qI'0:6.02I<&:r�9\0�0p6.+4I<Q,+4 L0W #"%0 ¶ ¤¦¥ ¤ § · £¤�¥ ¤ § ¥K¨}ª�¨T_í§(25.CK-R5G�%0:�f K5.@b #"%0c�k+4Q%�> K@�+.(p #5G1D+.�H?à&L-HI<$' E L"%0|-g5G&=$' L$<5G��6G0:(: L5G1 ¼ $'�f L5�$Z #&M1#+4?%$U+.IY+4�%?&L-H"%021=$<(e+4It",+.1LCD5G�%$'()(25.CK-R5G�%0:�f L&2^��s"%030p /-%1=02&=&L$<5.�� #"%+4 Y�%020:?%&Y #5MNg0h0p64+.I<Q,+V #02?�NR02(:5GCK0:&

Page 114: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦.¦

à ¥©[ ¬ ] ¬G«�¬ §®­z¯T° « ¸1» ¼ » ¥©[ _¬ ] _¬ « _¬ §®­ _ ¯ _ ° _ « _ ¸ _ÅÄ ÿà ¥©[ ¬ ] ¬G«�¬ §®­z¯T° « ¸1»Æ¼ÈÇ ²AÉ » ¥©[ _¬ ] _¬ « _¬ §®­ _ ¯ _ ° _ « _ ¸ _ Ä ¥Fx%^ ç §���u0:64+.I<Q%+4 #$'�%J� #"%$'&h02nfQ,+4 L$<5G�zr�+�&L$'CK$'IU+.1a #02(*"H�%$<nfQ%0M #5� #"%0M5G�H0DQH&L02?�$<�! #"%0tÊ¢S { (25.Q%-%I<0:?�3+.Q%�> m@�+.(p #5G1l9\5G1=]K54@H<\Q%1LJ.02&L&#� { Q%CDCD021L&)¥8y ¨ è � §l$'&\Q%&=02?z^l¶,1L5.C +4-%-g0:�%?%$T XS[£Ð #"H0jÓ,�,+.I1L0:&LQ%IZ a$<&

» à ¥K[ � ] � « � §8­z¯T° « »�» ¼ÈÇ ² »�» ¥©[ _� ] _ � « _� §8­ _ ¯ _ ° _ « _bÄ » ³ÿ ¥Å} «�� y7§2¥�}2° � y7§2¥�}2° _ � y7§:¥Å} « _ � y7§!ËH¥K[ � ] � « �dÌ [ _� ] _ � « _� §Í » äñ­¢¯ » ¼%» ­ _ ¯ _ ½ » ³ » äñ¯ »¦» Ç ² »¦» ¯ _ ½ » ³ÏÎÐ Ñ ° « « �

« _ ° _ y Ò ÓÔ³ ÎÐ Ñ °l_ ¯P_ yNÕ.}

¯ ° y Ò ÓÔ³¥Fx%^ � §

�i+.N%I<0Yx%^ZyY&L"%5�9Y&l #"%0Y1L0:&LQ%IZ #&�5.@g #"%$'&�0p /-%1L0:&L&=$<5G�D@F5G1�6.+41L$<5.Q%&�I<0p6G02I'&�5.@¢NHQ%�%?%I'$<�%J31L0:&L5GI'QH #$'5G�z^�s"%0j #+.N%Q%I<+4 #0:?K64+.I'Q%02&l(e+.�K #"%0:�BNg0bQ%&L0:?B #5q0:64+.I<Q%+4 #0 ¶ ¤�¥ ¤e§a· £¤¦¥ ¤ § ¥©¨}ª²¨T_Z§prf #"fQ%&m�k+4Q%�> \@�+.(p #5G1=&@F5G1ié®SZés(:5GQ%-%I'02?� #5)é®S'é\(25.Q%-%I<0:?�I'0:6.02I<&Y(2+.��Ng03(e+4I<(2QHIU+4 L02?z^�Y���Ö ×Ø(���&9&Ù"���*�7� !�!�+�Ú¾�05*����Û��03",�9��(�&��-4c05 *+0d�9�-2Þ\5G�%&=$<?%0:1\+3 #1L+.�%&L$Z #$'5G�M$<�M9Y"%$<(*"�+mé®S'é�(25GQ%-HI<02?KI<0p6G02I_JG5X02&� #5w+.�ÜÊ¢S { (:5GQ%-%I'02?dI'0:6.02IR5G1�6/$'(20¿S6G0:1L&L+H^�£j&3 #"%0M6G0:1=PuI<5�9�I'0:6.02I� #1=0e+4 LCK0:�f k@F5G1h #"H0D@FQH LQ%1L0D"%02+�6fPu$<5.�u9A5G1L]!CK+ePÚNg0M?%5.�%0$<�|+.�xÊ¢S { (25.Q%-%I<$'�%JO@F1#+4CK0p9\5G1=]B$' Y$'&s�%02(:02&=&#+.18P| L5K?H0:6G0:I<5G-c�3+.Q%�> Y@�+.(: L5G1L&s #"%+4 a+.I<I'5�9�@F5.1 #"H02&L0�(21=5G&L&®S�(25.Q%-%I<$'�%J! #1L+.�%&L$Z #$'5G�%&2^Ò�s"%0|CK0p #"%5X?Ò5.@)&=5GI<QH L$<5G��@F5GI'I<5�9Y&E(2I'5G&L0:I'Pæ9Y$Z #"á #",+V @F5G1k #"%0ké®SZéq(25GQH-%I<$'�%Jc #5Mé®S'éq(25.Q%-%I<$'�%Jc-H$<(: LQ%1L0K@F1L5GC #"H0�-%1L0p6/$'5GQ%&O&=02(p #$<5.�zrW #"%$'&q L$<CD0K #"H00p /-%1=02&=&L$<5.�� #5MNR030:64+.I<Q%+4 #0:?|$'&2·

» ä ¥©[ � ] � §®­z¯\[3] « »¦» ¼wÇ ² »¦» ¥©[ _� ] _ � « _� §8­ _ ¯ _ ° _ « _ ½ » ³ ¥Fx%^ÁèG§¶,1L5GC�+.-%-R02�%?H$Z �£ L"%$<&Y1L0:?%Q%(:02&Y #5

ÿ ¥�} «�� y7§:¥Å} « _ � y7§:¥Å}l[ � y�§2¥Å}È] � y�§2¥Å}:° _ � y7§ ³ ¥Å} « _� � y7§

Page 115: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦/y

Ý Þeßà ßá ßâÆãä åä æä áä

Ý Þ ßà ßá ßâ ãä åä æä

Ý Þeßà ßá ßâÆãä åä

Ý Þ ßà ßá ßâ ãä

Ý Þ\çà çá ßâ ãå æáÝ èá éêâÝ èáä éêâÝ èæ éêâÝ èæä éêâÝ èá éêâÝ èæä éêâ

ë ìí îïð ë ìñ îïð

ë ìí îïð ë ìñ îïð

ò æáá ß áä æäêóì

ò æä åäêô èå æêóì

ò æä åäêô èå æêóì

õ öåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõì õöåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõì õöåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõìú ñûõ öåõõ ÷ïõõ åä øõì õ öãåõÆùõ ãä åä øõì

Ý Þ çà çá ßâÆãå æÝ èáä éêâÝ èæ éêâÝ èæä éêâÝ èá ßéêâÝ èæ éêâë ìí üîïðë ìý îïð ë ìñ îïð

ë ìí üîïðë ìý îïð ë ìñ îïð

ò æä åäêô èå æêóì

ò æä åäêô èå æêóì

õ öåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõì õöåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõì õöåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõìú ñûõ öåõõ ÷ïõõ åä øõì õ öãåõÆùõ ãä åä øõì

Ý Þ\çà çá ßâ ãåë ìíû îïð ë ìñû îïð

ë ìýû îïðë ìí üîïð ë ìñû îïð

èÝ èá ßéêâ

èÝ èá ßéêâ

õ öåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõì õöåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõì õöåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõìú ñûõ öåõõ ÷ïõõ åä øõì õ öãåõÆùõ ãä åä øõì

Ý ÞKçà çá ßâ ãë ìíû îïð ë ìñû îïð

ë ìýû îïðë ìí üîïð ë ìñû îïð

Ý èá ßéêâ

Ý èá ßéêâ

ú ñõ öåõõ ÷ïõõ åä øõì õ öãåõÆùõ ãä åä øõìú ñõ öåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõìú ñõ öåõõ ÷ ïõõ åä øõì õ ö ãåõÆùõ ãä åä øõìú ññûõ öåõõ ÷ ïõõ åä øõì õ öãåõÆùõ ãä åä øõì

�i+.N%I<0Kx%^ZyG·Üþ�ÿ���ÿ���Ê8ÃpÇ7ÌVÀ�¾ÁÈ7É)¼�Ã�þ�ÿ�����ÿ�����Ê®Ã:Ç7Ì7ÀÁ¾�ÈVÉ�á½pÈV»� �*½¿ÀÁÇ7º=Âlº��*½¿ÀÁÇV½¿¼�º=»qÄTÃ:Í��*½¿Íž�Ã:ÇVÂ�ÀÁº��pº=ÀÁÂ�ÿÄÍź=Â�ÃpÀÁÇ�¼�¾ÁÃpÈfÏ

Page 116: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦G}

Í���� ¥Å}�� � y�§2¥ ¯ y7§ ´ � ÎÐ Ñ ] ¯ ] �� [ « Ò ÓÔ

³ ÎÐ Ñ y ¯a_ ¯� « « _ Ò ÓÔ

³

Í ÎÐ Ñ « _� � yNÕG}¯P_ °l_ « _ Ò ÓÔ

³ ÎÐ Ñ « _� yNÕG} �[ ] � [ � Ò ÓÔ³

Í » äà­z¯ » ¼R» ­ _ ¯ _ ½ » ³ » äÔ¯ »¦» Ç ² »�» ¯ _ ½ » ³ ¥Fx%^ ¨ §£j&YNR0:@F5G1=0h #"%$'&s(e+.�|Ng0k+e6G0:1#+.J.02?� #5K+.(2(:5GQ%�> Y@F5G1Y+.�>P�?%0:&L$<1=02?|?%02J.1L020k5.@i1L0:&L5GI'QH #$'5G�z^

�`��� ���`� �`�j� �a�D�e�2�k�W�2�D�r�O���R�3�O�!� �k���:�D� �a���`�Ü�0!�%5�9ð&=020:]à #5�-%1=5/?HQ%(20Ú+æ&=Q%N%1L5.QH #$'�%0` #",+V �5GQH L-%QH #&B+áÞlS�XðCK+4 L1L$Z Ò@F5G1d�%5G�/S[-,+.1L0:�> (*",+.�HJG$<�%J! L1#+.�%&=$' L$<5G�%&O L",+4 M(2+.�àNR0�(e+.I'I<0:?Ò1=02(:Q%1L&=$'6G0:I'P� L5�+4&L&L0:CON%I'0� L"%0�&LQ%-R021=N%I<5X(*]�?%0¿S&L(:1L$'Ng0:?d0e+41LI<$'021:^i�s"%$'&A$'&�?%5G�%0Y@F5G1�­z¯t+.�H?K­|I<0p6G02I'&�5.@z1=02&L5.I<QH L$<5G�K+.�%?M #"%0aCD0: #"H5/?D$<&� #0:&= L02?@F5G1s�%I ésI'0:6.02I<&:^��Ö � ���t05�+2��Ú(��-&}�� !*��������¶,5G1s #"%0k"%$'JG"%I'Pd0p /(2$Z #0:?c&= #+4 #0:&b5.�%0q"%+.&j+4�c�|5G1Y�%I¡02I<0:(: L1L5G�|5G�� L5G-|5.@�+KåVS�1=02&L5.I'6G0:?c(:5G1L04^�s"%$'&j$<&a&L$'CK$'IU+.1m L5K L"%0O0p /$'&= #$'�%Jd">PX?%1=5GJG02�!CD5X?%0q54@�£)¤Y£ { �/yGy�¥ { Q%CDCK0:1L&2rz}.¦G¦Hy�§m9Y"H$<(*"+.&=&L02CwN%I<0:&|1L+4 #0:&�@F5G1�+.�Ý�%Ih5G1���02I'02(: L1L5G�Ý5G�Ý #5.-�54@O+�N,+.1=0Ú�XQH(2I<0:Q%&2rj&L5GIZ6/$'�%J�@F5G1B #"H0-R5G-%Q%I<+4 #$'5G�%&2^��5O+.?,+4-H A L"%0a(25X?%0b@F5G1�5GQH1A-%QH1L-R5G&L0:&\ #"%0a-,+.1=02�> m&8 *+4 L$<&8 #$<(2+.I%9A02$<J."f A0p /-%1=02&=&L$<5.�B9m+.&ACK5X?/S$'Ó%02?B #5M+4(2(25.Q%�> s@F5G1m+MåVS[1L02&=5GI'6.02?�(25.1L0.^��s"H0h(25X?%0)9m+4&sCK5X?%Q%I<+.1L$'&L02?K&LQ%(*"� #"%+4 s$' \1L0p #Q%1=�%& #"H0sN%I<5X(*]q-H1L$<CD$' L$'6.02&�$<�%?%$Z6X$<?%Q,+4I<I'P.^�Û)�%0s(25GCD-%I'$<(e+V #$<5.�k9\+.&W #"%+4 �£)¤Y£ { �/yGym0:64+.I'Q,+4 L02&W #"H0�%02+.1LIZPd?%0:JG02�H021#+V #0h #1L+.�%&L$Z #$'5G�d1L+4 #0)CK+4 L1L$Z K0:I<02CD02�> L&s5GQH #&=$<?%0b #"%0h1=579 +.�H?|(:5GI<QHCK�BI'5/5.-%&2^�[ s�%0:02?%&s L5E?%5O #"%$'&mNR02(e+4Q%&L03$' \Q%&L0:&Y1#+4 L02&\@F1=5GC 0:I<&=0:9Y"%0:1L03$<�d L"%0hCK+4 #1=$Z D #5E0p6.+4I<Q,+4 L0j #"H0I<$Z@F0: L$<CD02&h5.@A #"%0KI<0:6.02I'&w$<�>6.5GI'6.02?�$'�� L"%0K?H02JG0:�%021L+4 #0B(2+.I<(:Q%IU+V #$<5.�%&2^K���� #"%0K�%0:9o(:5/?H0K #"H0�%02+.1LIZPd?%0:JG02�H021#+V #031#+4 L02&Y+.1=0)0:64+.I'Q,+4 L02?B9Y$' L"%$<�K #"%0h1=579Ð+.�%?�(:5GI<Q%CD�BI'5/5G-H&2r/ L"%0)I'$'@F0: L$<CD02&NR02$<�HJh02&= L$<CK+4 #0:?O@F1L5GCõ #"H0sI<0:6.02I%1L+.?%$<+4 #$Z6G0\1L+4 #0:&�0:64+.I'Q,+4 #0:?EQ%&L$'�%Jh">PX?%1L5GJ.02�%$'(s0p /-%1L0:&L&=$<5G�%&:^ÜÝ"%0:�� #"H0K(:5/?%0D$<&k+.-H-%I<$'02?! #5�"%0e+e6fPu&L-R02(2$'02&3 #"%0DI<$Z@F0: L$<CD02&)(2+.��NR0M02&= L$<CK+4 #0:?u@F1=5GC #"H0$<�H$' #$<+.Ig+.QH #5.&= #1=Q%(: LQ%1L031=Q%�|9Y"H$<(*"|&L0: L&bQH-| L"%0302�%0:1LJ.P�I'0:6G0:It&= L1LQ%(p #Q%1=0.^

Page 117: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦.�

�s"%0�(25GCD-%I<0p #0:?â">PX?%1L5.JG02�ñCD5X?%Q%I<0�54@) #"%0|(25X?%0|(e+.�âNg0|1LQH�â@F5G1D+�QH&L021K?%0:Ó%�%02?ñ&=0: K5.@02I'02(p #1L5.�à #0:CK-R021L+4 #Q%1=02&K+.�%?ñ?H02�%&=$' #$'02&:^ { 5.CK0|$<I'I<Q%&8 #1#+V #$'6.0�1L0:&LQ%IZ #&B+41L0c&="%5�9Y�Ô$<�ÒÓ,JGQ%1=0x%^Á}/^��[ m9\+.&m@F5GQ%�H?� L",+4 m L"%0)�%0p9ÐCD5X?%Q%IU+41A6G0:1L&=$<5G��5.@¡ L"%0h(25X?%0h+.J.1L020:?�9Y$' #"B L"%0)-%1=0:6X$<5.Q%&">PX?%1L5GJ.02��1L0:&LQ%IZ #&\@F1=5GCo£)¤Y£ { �Hy.yGrH+4�%?d #"%+4 \ L"%0)-R5G-%Q%I<+4 #$'5G�K1=02&=Q%I' L&\64+.1L$'02?�+.&A0p /-R02(: L02?9Y"%0:�Ô�%5G�/S��2021=5 «�¬ 64+.I'Q%02&M9A021L0cQ%&L0:?z^Ð�s",+V B$<&:rm #"%0�N/S[@�+4(: #5.1L&K+.1L0�Q%�,+.IZ #021=02?Ô+4�%?à #"H0+.(p #Q,+.It-R5G-%QHIU+4 L$<5G�d64+.I'Q%02&a+.1=0h1L0:?%Q%(20:?cN>P|+O@�+.(: L5G1s5.@m¥Å} «e¬�� y7§p^��Ö Y� ������(��-&}�� !*��������£á #02&8 �CD5/?%QHI<0W9\+.&W(25.�%&= L1LQ%(p #[email protected]� L"%0\�%I é¡-,+41= i5.@% #"H0\�%0p9á(25X?%0.^WÞ\5G�%&L$'?%021� #"%0AN%Q%�%?HI<02?XS�%I_+.�H?KN%QH�%?%I<0:?/S��DCD5/?%QHI<0s5.@R #"%0Y(:5/?%0b+.�%?D&L-HI<$' W5G�%0Y&L$'�%JGI<0s�DI<0p6G0:IR$'�> #53$' L&��%IïéW&LQ%NH&= *+V #02&:^�s"%0!-R5G-%Q%I<+4 #$'5G�à54@k L"%02&=0!�%Iïéd&=Q%N%&= #+4 #0:&|+.1L0c L"%02�Ý&=5GI'6.02?Ô$<�H(2I<QH?%$<�%Jæ+.I<IY #"%0` L1#+.�H&L$' L$<5G�1#+V #02&�NR0: �9A020:� L"%0!�%IïéK&=Q%N%&8 *+4 L02&�+.�%?ñ #"H5G&L0! #5á+4�%?ñ@F1L5GC #"%0!&=Q%1L1=5GQ%�%?H$<�%J��%I)+.�%?Ý�&= #+4 #0:&2^��s"%0Y">PX?%1L5.JG02�KCK5X?%Q%I'0m5.@R #"%0Y(25X?%0Y$<&�QH&L02?K$<�D+h-%1L$'5G1�(e+4I<(2QHIU+4 L$<5G�w L530:64+.I<Q%+4 #0s #"H0&LQH1L1L5.Q%�%?%$'�%J!-g5.-%Q%IU+V #$<5.�%&w+4�%?��%0:9 1#+V #0d0¿ /-%1L0:&L&L$'5G�%&MQ%&=02?� #5`0p64+.I<Q,+V #0d L"%0B(25.�%�%02(p #$'�%J1#+V #02&� #5k+.�[email protected]; #"%0Y�%I é�&=Q%N%&8 *+4 L02&2^����D #"%$'&�9\+ePE #"%0a-R5G-%Q%I<+4 #$'5G�E?%$<&8 #1=$<N%QH L$<5G�O9Y$Z #"%$'�D+.��%I éa&L0p 35.@�I<0p6G0:I<&)(2+.�!Ng0q$<&=5GIU+4 L02?c+.�%?` #"H0w�%I éb1L02&=5GI'6.02?!1L+4 #0:&)0p H+.CD$<�%0:?z^)�s"%0O+.?H64+.�> *+.J.05.@¢JG02�%0:1#+4 L$<�%Jk&LQH(*"�+3 L02&8 \CD5/?HQ%I<0sNR0:@F5G1=0b?H0:6G0:I<5G-H$<�%Jq+k�H0:9ÝN%I'5/(*]O-H1L$<CD$' L$'6.0m(25X?%0a$<&� #",+V 5G�%03?%5X02&Y�%54 b�H0202?| #5E9\+.$' aQ%�> #$'I¢ #"H039Y"H5GI<03N%I'5/(*]B-H1L$<CD$' L$'6.0jCD5/?%QHI<0)$'&a(:5GCD-%I<0p #0hNR0:@F5G1=0NR02$<�HJE+.N%I'0j #5O L02&= Y$Z e^W£b&m@FQH #QH1L0)CD5/?HQ%I<0:&m+.1=03+.?%?%0:?�5G�> #5E #"%0j(25GI'I<$<&=$<5G�%+.IZS[1#+.?%$<+4 #$T6G0s(:5/?H0$' \9Y$<I<IRNR02(25.CK0h-%1=5GJG1=02&L&=$'6.02I'PdCK5.1L0)?%$Å¡D(2Q%IZ m #5E0:1L1L5.1s(*"%02(*]d L"%03(25G�> L1L$<NHQH #$'5G�%&\@F1=5GC 02+.(*"N%I'5/(*]q-%1L$'CK$Z #$'6.0.^¡�s"%0:1L0Y$<&�+)@FQ%18 #"%0:1�-g5G$'�> W@F1L5GC #"%0m L02&8 ACD5X?%Q%I<04r. #"H0s�%Iïé�-R5G-%Q%I<+4 #$'5G�%&[email protected]+h&L$'�%JGI<0\�K&="%02I'I%(e+.�DNR0aQH&L02?M #53JG0:�%021L+4 #0a+h&L-R02(p #1L+.I,02CD$<&=&L$<5.�E-%1L5.Ó%I<0m9Y"%$'(*"D9Y$'I<IH&="%5�9ñ #"H0?%$'&= #5.1= #$'5G�B-%1=5/?HQ%(20:?|N>PK #"%0)1=02I<+4 #$Z6G0)-R5G-%Q%I<+4 #$'5G�%&l5.@¡ #"%0j�%Iïé\&LQHN%I<0p6G02I'&Y+.�%?B L"%02$'1m1L0:IU+4 L$'6G002�H021LJ4Pæ&L"H$'@ #&:^Ú�s"%$<&O$'&E+.��$'CK-R5G1= #+.�> w(:5G�%�%0:(: L$<5G�æ9Y$Z #"�5GQ%1wI<$<�H0d-%1=5.Ó,I<0K&= LQ%?%$<0:&M+.�%?�$'&?%0:CK5G�H&= #1L+4 #0:?�I<+4 #0:1s$<�B L"%$<&Y&L0:(: L$<5G�z^��Q%1L�%$'�%J` #5� #"%0�?%0p *+.$'I<&k5.@Y #"%0K #02&8 M(25X?%0d@F5G1O+�9Y"%5GI'0BN%I<5X(*]u-%1L$'CK$Z #$'6.0.r�+`-g0:1= #QH1LN,+4 L$'6.0 �PX-g0K5.@s+4-%-%1L5G+.(*"�$'&q?%5.�%0K$'�u9Y"H$<(*"�@F1=5GC� #"H0dJG0:�%021L+.IAÞlS�X�CK+4 #1=$Z ÚN%I'5/(*]` L"%0D­z¯mN%I'5/(*]-%1=$<CD$' #$Z6G0a9Y"%$<(*"|$<&m L5KNR0hIU+4 L021s0¿ H-%+.�%?%0:?c$<�> L5D­¢¯T°d&LQ%NHI<0:6.02I'&b$'&s&L02I'02(p #02?¢^\Û)�%03",+4&

Page 118: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦4x

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120

b-f

act

or

Principal quantum shell

Te=2.32e4 K

Ne=1.e4 cm^-3Ne=1.e8 cm^-3

Ne=1.e12 cm^-3Ne=1.e16 cm^-3

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120

b-f

act

or

Principal quantum shell

Te=2.32e7 K

Ne=1.e4 cm^-3Ne=1.e8 cm^-3

Ne=1.e12 cm^-3Ne=1.e16 cm^-3

¶i$<JGQ%1=0hx%^Á}/·"!%Ã:Ì7Ç7À~½*¼�¾ÁÃpÈbÍ�ºLÂ�ÇVÀã¼[¢ÄTÍ�Ã:Ë�¼�¹7º�¹�#�»�ÍÅÃpÉ:º8ÈaËmÃ�»�Ç7ÀÁº�ÿÄH¼�¹Vº�È7º8Î%$'&(`Ê®Ã�»�º:Ï

Page 119: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦Gv

~��������������

¯*)�+%,¿¯*)�-�. U U/ -�01)3254e­ U

­z¯6,¿¯*)�-�./ / ­7,p¯*)�-�.

� ��������������

~��������������

ôM²ì'ìì'ìì'ìì'ìô78:9 �

� ��������������ÿ

~��������������

·<; [ìZììZììZììZììZì

� ��������������¥�x%^Zy�¦>§

�s"%0Ú0p /$'&= #$'�%Já(25X?%0Ú&L5GIZ6G0:&� L"%0ÚCd+V #1L$T ñ0:nfQ,+4 #$'5G�Ý$<��+�N/S[@�+4(: #5.1�-%$<(p #Q%1=0`9Y"%021=0u64+.1=$<5GQ%&+4 L5GCK$'(b(:5G�%&8 *+.�> #&\",+e6G0hNR0202�B #+.].0:��5GQH \@F1=5GCoNg54 #"d L"%0)1#+V #0j0¿ /-%1L0:&L&L$'5G�%&Y+.�H?B #"%0h+V #5GCD$<(-R5G-%Q%I<+4 #$'5G�%&2r/JG$'6X$'�%J

-=)7­>2�? Í ~���� � ¤¦¥ @�§ � ¤�¥ ü § ý § � ¤¦¥BAü §� üeý ¥ @�§ � ü2ý ¥ ü § ý § � ü2ý ¥BAü §� Aü ¥ @�§ � Aü ¥ ü § ý § � Aü ¥CAü §� ����~���� , @�§, ü § ý §, Aü §

� ���� ÿ~���� ·<; [ @·<; [ ü2ý·<; [ Aü

� ���� ¥�x%^ZyGy7§�s"%03 #0:&= a(25X?%035GNH #+.$<�%&m L"%03&L5GI'QH #$'5G�B@F5G1m L"%03N/S[@�+.(p #5G1=&s5.@� #"%$'&s02nfQ,+4 L$<5G�z^��s",+V Y$<&2r~���� , @�§

, ü § ý §, Aü §

� ���� ÿ y-D)V­E2�? Í ~���� � ¤¦¥ @ § � ¤¦¥ ü § ý § � ¤¦¥CAü §� ü2ý ¥ @ § � üeý ¥ ü § ý § � üeý ¥CAü §� Aü ¥ @�§ � Aü ¥ ü § ý § � Aü ¥CAü §

������F ² ~���� ·<; [ @·<; [ ü2ý·<; [ Aü

������ ¥�x%^Zy7}G§

Û)�%0h(e+4�� #+.].0h0:nfQ,+4 #$'5G�u¥�xH^'yGy�§\+.�%?�1=0pS®+41L1#+4�%JG0)$Z m #5E$'&L5GI<+4 #0a #"%0)&=02I'02(: L02?|�%IzN%I'5/(*]D-%1=$<CD$ZS #$Z6G04^l�s"H$<&s$<&s-R0218@F5G1LCD02?�+.&m@F5GI'I<5�9Y&2·

ë � üeý ¥ @�§ îXë , @ § î � ë � ü2ý ¥ ü § ý § îXë , ü § ý § î � ë � üeý ¥CAü § îXë , Aü § î ÿ y-D)V­E2�? ë ·<; [ üeý î ¥�x%^Zy��>§

ë � üeý ¥ ü § ý § îXë , ü § ý § î ÿ y-D)V­E2�? ë ·<; [ üeý î ¯ ë � ü2ý ¥ @�§ îfë , @�§ î ¯ ë � üeý ¥CAü § îXë , Aü § î

ÿ ë ­E4�+ ·<; [ ü2ý î ¥�x%^Zyexf§Ü�0j #1=0e+4 A #"%$'&s+.&m+O�H0:9�&L0p Y5.@¡02nfQ,+4 L$<5G�H&A #5E&=5GI'6.0j@F5G1l #"%0<, ü § ý § 5.�B #"%0)I'0:@ A",+.�%?�&=$<?%04rH #"H01L$'JG"> )",+4�%?!&L$'?%0ONR02$<�HJB]f�%5�9Y�Ú@F1=5GCÑ #"%0O-H1L0:6X$'5GQ%&)&=5GI<QH L$<5G�¢^)�s"%0E1L0:5G1LJG+.�%$<&L+4 #$'5G�c(2+.�!NR0(*"%0:(*].02?!N>P�&="%5�9Y$<�%JM #"%+4 s #"%0G, ü § ý § 1L02&=Q%I' L&a+.JG1L0:0h9Y$' #"� L"%02$'1s-%1L$'5G1\64+.I<QH02&2^

Page 120: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦ ç

�s"%0D�%0p X w&= #0:-�$'&3 L5�5.-g0:��Q%-u #"%0D�%I�N%I<5X(*]`-%1=$<CD$' L$'6G0w$<�> #5|$' L&k�HIïé)&LQ%N%I'0:6.02I<&:^��s"fQ%&w5G�H0&L-HI<$' L&h0e+.(*" ·<; [ üeý $<�> #5 ·<; [ ü2ýÅý F ²�H[³O+.�%? ·<; [ ü2ý[ýJI ²�H[³2^ { $'CK$'IU+.1=I'P|0e+4(*" , üeý $'&3&L-%I'$' 3$<�> L5, üeý[ý F ²�H[³ +.�%?K, ü2ý[ýCI ²�H[³ ^ ~�������

� üeý þ ¥ ü § ý § þ § ì'ì�ìZìoì'ìì'ì ì'ì�ìZìoì'ìì'ì ì'ì�ìZìoì'ìì'ì ì'ì�ìZìoì'ì

� �������~�������, üeý þìZììZììZì

� ������� ÿ~�������·<; [ üeý þ

ìZììZììZì

� ������� ¥�x%^Zy7vG§

��5D0¿ /-,+.�%?� #"%0�XaÕ { @F5G1m�HIïé\1L0:&L5GI'QH #$'5G��$' s$'&s+.&=&LQ%CD02?� #",+V s #"%0:&L035�6G021L+.I<I¢1#+4 L02&s$<�>6.5GI'6X$<�HJ #"H0k54 #"%0:1a�|+.�H?��%It&8 *+4 L02&Y(25.�f L1L$'N%QH #0h L5M L"%03�%Iïém-R5G-%Q%I<+4 #$'5G�%&m&8 *+4 L$<&= L$<(2+.I<IZPG^��s"%+4 a$<&

·<; [ ü,ý�ý F ²�H[³mÿ ­ ¯}.¯ � y ´ ·<; [ ü,ý ¥�x%^Zy ç §

·<; [ ü,ý�ýJI ²�H[³mÿ ­ ¯ � y}.¯ � y ´ ·<; [ ü,ý ¥�x%^Zy � §£ &L$<CD$<I<+.1w&=-%I<$Z L L$<�%JÚ$<&D+.&=&LQ%CD02? ¥�+.&K+!Ó,1L&8 B&= L02-_§O@F5G1E #"H0�(:5GI<I'$<&L$'5G�,+.IA1#+4 L02&D5G�â L"%0|I<0:@ ",+.�H?�&L$<?H0.^��s"%$'&\$<&té=Q%&= L$'Ó,0:?�+.&\@F5GI'I<5�9Y&2·i�s"%0hI<0p@ \"%+.�%?�&L$'?%0)$'&\CK+.?%0jQH-�5.@¡N/S[@�+4(: #5.1L&m+4�%?1#+V #02&�&LQ%(*"D+.&W&=-g5.�f #+.�%0:5GQ%&�02CD$<&=&L$'5G�w1L+4 #0:&2r>(25.I<I<$'&L$'5G�,+.IG1L+4 #0:&W0: L(.^W�s"H0s(25GCONH$<�,+4 L$<5G�k5.@_ #"H0&= #+4 #$'&= L$<(e+4IX9A02$'JG"> �$<�O #"H0sN/S[@�+.(p #5G1=&�+.�%?M$'�O #"%0s1L+4 #0m0¿ /-%1L0:&L&L$'5G�DCK02+.�%&W #"%+4 W #"H0s(25GI'I<$<&=$<5G�%+.I1#+V #02&q5G�� L"%0dI'0:@ k",+.�H?�&L$'?%0K64+.18PÚ$<�u #"%0D&#+.CD0M9m+eP�+.&k L"%0dI'$<�%0M&8 #1L0:�%J. L"%&w9A5GQ%I<?¢^��s"%0(25.1L1L0:&L-R5G�%?%$'�%Jh9\0:$<JG"> � L021LCD&W",+e6.0b+4I<1L02+.?HPqNg0:02�M9\5.1L].0:?K5.QH �$<�w L"%0Y-%1L0p6X$<5GQ%&=I'Pq0:64+.I'Q,+4 #0:?�3+.Q%�> s@�+.(: L5G1s9\5.1L]_^l¶_1=5GC L"%0q�k+4Q%�> s@�+.(: L5G1s *+.N%I'02&:rH #"%0h9A02$'JG"> #$<�HJM L021=CÖ$<&

y}%¥�} «�¬�� y�§ ¥�}2° _ � y7§2¥�} «�¬�� y�§2¥Å}:° � y7§ ÎÐ Ñ °l_;¯a_ yNÕG}¯ ° y Ò ÓÔ

³

ÿ y} ¥Å}2° � y7§:¥Å}2° _ � y�§ ÎÐ Ñ °�_ ¯a_ yNÕG}¯ ° y Ò ÓÔ

³¥�x%^Zy�è>§

£m E #"H$<&O-R5G$<�> q #"%0�0:nXQ%+4 #$'5G�%&E(e+4�á+.J>+.$'�æNg0�&=5GI'6.02?Ò+.�%?æ L"%0�1L0:&LQ%IZ #&E&="%5GQ%I'?á+.JG1L0:0d9Y$' L", ü § ý § ý § F ²�H[³hÿL, ü § ý § ý § I ²�H[³hÿL, ü § ý § ^h£b@ #0:1b L"%$<&a@FQ%1= L"%021)(*"H02(*]_rt9\0O(2+.�!-%1=5/(:0202?` L5�$<�%(:I<Q%?H0q #"H0-%1=5G-g0:1h1#+4 L0E0p /-%1=02&L&=$<5G�H&q5G�! L"%0MI<0:@ )",+.�%?Ú&L$'?%0D+.�%?Ú0:64+.I'Q,+4 L0O #"%0EN/SÅ64+.I<Q%0:&)@F5G1j L"%0M�%0:9

Page 121: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦ ��%I é�CD5/?%QHI<0q¥�+.JG+.$<�K+.&L&=Q%CD$<�%Jk]f�%5�9YI<02?HJG0j5.@¢ #"%0a1L$'JG"> \",+4�%?d&=$<?%0M, ü +.�H?N, üeý 64+.I<QH02&*§¿^W�s"%$'& #0:&= ECD5X?%Q%I<0M(2+.��Ng0KQ%&L0:?æ #5`0¿ H+.CK$'�%0K L"%0d1L+4 #0K0p /-%1=02&L&=$<5G�H&M+.�%?æ(*"%0:(*]æI<$'CK$Z #$'�%JB(e+4&L02&:^���|-,+.1= L$<(:Q%IU+.1m L"%03(25G�H?%02�%&L+4 #$'5G�|5.@� L"%03CK5X?%Q%I'0hNfPB(*"%5G&L$'�%Jd1=02-%1=02&=02�> *+4 L$'6.0qI'0:6G0:I<&a(e+.�|NR0$<�>6.02&= L$<J>+V #02?¢^¶,5G1b L"%0q L02&8 3(25X?%0q9Y1=$' L L02�!"%0:1L0O5.�%I'P|5G�%0q&L$'�%JGI<0q­`_t$'&j(:5G�%&L$'?%021=02?!&=Q%(*"! #",+4 )+d?%$'&= #5.1= #0:?­`_9U ­!&L-R02(p #1#+4Iz #1#+4�%&L$Z #$<5.�B(e+.��NR0hCK5X?%0:I<I<0:?z^��s"H$<&s(25X?%0h$'&m?%02&=$<JG�H02?� #5ENR03+O-%1L5X5.@¡ #",+V #"H0)1L0:?%$<&8 #1L$'N%QH L$'6G0j1#+4 L02&m(2+.��Ng0)QH&L02?�$'�d #"%0h�H0:9 �%IïéAN%I'5/(*]D-%1=$<CD$' #$Z6G04^POb-R5G�B6.021L$ZÓ,(e+V #$<5.�5.@� #"%$'&s&LCK+.I<I'021\ L02&8 bCD5X?%Q%I<0b #"%03IU+41LJG0:1sN%I<5X(*]d-%1=$<CD$' L$'6G0a #0:&= b(25X?%0)9Y$<I'IzNg0h9Y1=$' L L02�z^{ 5�-%1=5/(:0202?H$<�%JB9Y$Z #"! #"%0M+4I<JG0:N%1#[email protected]) #"%0E&=Cd+.I'I<0:1a L02&= kCD5/?HQ%I<0q5G�%0E1L0p #Q%1=�%&h #5�0:nXQ%+4 #$'5G�¥�xH^'yex>§i+.�%?q9Y1L$' L02&W$Z �5.QH �CD5G1L0l@FQ%I<IZP3$<�q L021=CK&W54@R+.I<I> #"H0m-g5.&L&L$'N%I<0A�O64+.I<Q%0:&i L",+4 W(2+.�M+.-%-R0e+41$<�� L"%03�%It-,+.18 #$' L$<5G�¢^��s",+4 a$'&2r

~�������� ü�Q[ýRQ ¥ ü�Q[ýRQ � ü�Q[ýRQ ¥ ü�QSI ² ý�T ìZì ìZì� ü Q I ² ý T ¥ ü Q ý Q � ü Q I ² ý T ¥ ü Q I ² ý T ìZì ìZì

ìZì ì'ì � ü�U�ýRU ¥ ü�U�ýVU ìZììZì ì'ì ìZì ìZì

� �������~�������, ü�Q�ýRQ, ü Q I ² ý T, ü=UÅýVUì'ì

� ������� ÿ~�������­W4�+ ·<; [

ìZììZììZì

� ������� ¥�x%^Zy ¨ §

ÜÝ"%0:1L0h­WX®¯YXl1=02-%1=02&=02�> #&s #"H0h�/S�&="%02I'Iz+.�%?�(:5G1L1=02&=-g5G�H?%$<�%JEIg&= #+4 #0:&m #",+4 m+.1L0b #5ENR0)0p /-,+.�H?%02?Q%-R5G�|$<�> #5E�%I és&=Q%N/S[&= *+V #02&:^A��&=5GIU+V #$<�HJO #"%031L0:?%$<&8 #1=$<N%QH L$'6.0h1#+4 L02&Y5G�%03$<&sI'0:@ m9Y$Z #"z·

ë � ü�U�ýVU ¥ ü=UÅýVU îXë , ü�U�ýRU î ÿ ë ·<; [ ü=UÅýVU î ¯ ë � ü�U�ýRU ¥ ü Q ý Q îXë , ü Q ý Q î ¯ ë � ü�U�ýRU ¥ ü Q I ² ý T îfë , ü Q I ² ý T î ¯ ìZì'ìÿ ë ­E4�+ ·<; [ ü�U�ýVU î�s"%0K-%1L5X(20:?%Q%1L0d(25G�> #$'�fQ%02&E+4&q?%0:&L(:1L$<NR02?æ@F5.1w+�9Y"%5GI<0M #0:&= wN%I<5X(*]Ú-%1L$'CK$Z #$'6.0.^WXY02&=Q%I' L&[email protected]­WXwÿ��!+.1=0B&L"H579Y�á"%0:1L04^�Þ\5G�%&=$<?%0:1w #"H0�1L02?H$<&= L1L$'N%QH #$Z6G0B1L+4 #0:&2^�Û)�%I'P�?%$<-R5GI'0D L1#+.�%&=$' L$<5G�%&9Y"%$'(*"w1L0:?%$<&8 #1=$<N%QH L0� #"%0�02I'02(p #1L5.�%&i$'�sé¡+.1L0�&=$<JG�%$ZÓ,(e+4�f 2^i�s"H021L0A+.1=0A5.@/(25.Q%1L&=0\$<5.�knfQ,+4?%1LQ%-R5GI'0 #1L+.�%&=$' #$'5G�w L",+4 �-%1=576X$'?%0aåk(*",+4�%JG$<�HJ) #1#+4�%&L$Z #$<5.�ON%QH A+.&W L"%02&=0b+41L0m9Y$' L"%$<�M+hJG$Z6G0:�D�HIïé�JG1L5.Q%- #"H0:P�?%5M�%54 Y+.I' L021\ L"%03$<�%?H$'6X$<?%Q%+.I>ém-R5G-%Q%I<+4 #$'5G�%&2^i�s"fQ%&a L"%0hCK+.$<�BCD02(*"%+.�%$<&=C�5.@i$<�H?%$<1=02(: « 1L0:?%$<&8 #1L$'N%QH L$<5G�á+.�H?â"%0:�%(20OéM1=02?%$'&= L1L$<NHQH #$'5G��$<&MN>Pæ #1L+.�%&8@F021M@F1L5GC ­z¯T° « L5u­¢¯[Z�y#°�_ « _ ^òb54 #0Y #",+4 ANR02(e+4Q%&L0b5.@z #"H0b�%02+.1l?%02JG0:�%021L+.(:PK$<�M #"H0j1L0:IU+4 L$'6X$<&8 #$'(m-%$<(p #Q%1=0b54@t­z¯8¥�¯ � yNÕG}.§�+4�%?­z¯ � yf¥�¯ ¯ yNÕG}G§¿r>$' W$'&�5G�HI'Pk #"%0m(:5GI<I'$<&=$<5G�,+4IG1#+4 L02&�NR0: �9A020:�D­¢¯8¥�¯ � yqÕG}G§ « +4�%?O­z¯ � yf¥�¯ ¯ yNÕG}G§ « _

Page 122: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦.è

9Y"%$'(*"uCK+4 = #021:^w�s"%0:&L0K+.1L0M+4I<I<5�9A02?!?%$'-g5.I<0q L1#+.�H&L$' L$<5G�H&)$<�%?HQ%(20:?uN>P!-R5G&=$' #$Z6G0O$'5G�!$<CD-,+.(p e^£Ñ(25X?%0�N,+.&=02?Ò5G�Ò L"%0� #"%0:5G1=Pæ5.@+��02�%J.02I<IZPu� { 0e+4 L5G� ¥®y ¨Gç x>§w(e+.�ÒNg0�QH&L02?Ò #5Ú0:64+.I'Q,+4 L0 #"H0`$<5G�à(25GI'I<$<&=$<5G�%+.I\ #1L+.�%&=$' #$'5G�à1L+4 #0:&BNR0: �9A0202��+.?7éL+.(20:�f �Ia&= #+4 #0:&2^¶,5G1K+.�Ô02�%0:1LJ.PâI<0p6G0:I&L0:-,+.1L+4 #$'5G�ñJG1=0e+4 L021d L",+.�ñ #"%+4 B@F5G1d9Y"H$<(*"ñ #"H0 �i02�HJG02I'I'P]\ { 02+4 #5G�Ý+4-%-%1L5G+.(*"Ô$<&D6.+4I<$<?¢rY+.�$<CD-,+.(p w-%+.1#+.CD0: L021q(25X?%0!¥\<\Q%1LJ.02&L& � { QHCKCD021=&2rmy ¨l�.ç §k(2+.�æNR0BQ%&=02?z^!�s"%0:&L0�(:5/?%0:&E+.1=0+.I'&L5MQ%&L0:?�$'��&L02(p #$'5G��xH^�xE #5M0p6.+4I<Q,+4 L0hN%Q%�%?HI<$<�HJD(:QH =S[5.�g&2^�s"%0^, üeý þ @�+.(p #5G1=&k0p64+.I<Q,+V #02?!@F5G1 ° ±wÿ�� ç +.�H?`_bahÿÖy ì ¦>v Í y�¦6c�d ¥�1=02?%Q%(:02?u #02CD-R021#+V #Q%1=0_ba�ÿ y Í y�¦6e�dÚ§M+.1=0�&="%5�9Y�Ý+.&d+Ú@FQ%�%(p #$<5.�à5.@j?%02�%&=$' �P ¥�ôGaK+4&L&LQHCK0:?ñ02nfQ,+.IY #5�ô ¤ §M$'�Ó,JGQH1L0Mx%^��H^Oòa5. #0E #"%+4 k+4 3I<5�9?%0:�%&L$Z #$<0:&3(25G1=5G�,+.IW(:5G�%?%$Z #$'5G�%&h+.-H-%I'P`+.�%?! #"%0M-R5G-%Q%I<+4 #$'5G�%&+.1=0O?%0: L021=CK$'�%02?`N>Pc&L-R5G�> *+.�%0:5GQ%&h0:CK$'&L&L$'5G�`+4�%?!(25GI'I<$'&L$<5.�,+.It0p /(:$' *+V #$<5.��@F1L5.C L"%0OJG1=5GQ%�%?z^£m w+.Ng5.QH Dy Í y�¦ ³�³ -�º F ´ L"%0hé®S[1L0:?%$<&8 #1L$'N%QH L$'6G0E1L+4 #0:&qNR02(:5GCD0K&=$<JG�%$ZÓ,(e+4�f q+.�H?�(25.Q%-%I<0M #"H0�%02+.1LIZPB?%0:JG02�%0:1#+4 L0j°wÿЯfZ�yNÕG}OCK0:CONR021L&s54@W+.?7éL+.(20:�f aI'&m #5GJG0p #"%0:12^lòb54 #0) #"%+4 Y #"%0h1=$<&L03$'� #"H0hN/S[@�+.(p #5G1=&m$<&\?%Q%0h #5E-R5G-%Q%I<+4 #$'5G�K@F1=5GC L"%0)�_ÿ3}O&L"H02I<I¢Ng0:(25GCD$<�%Jw&L$<J.�%$'Ó,(2+.�> e^W�s"H$<&m9\+.&(25.�HÓ,1LCD02?!QH&L$<�HJB£)¤Y£ { }/y ç ^)£j&b #"%0O?H02�%&=$' �P`$'�%(21=0e+.&=02&j@FQH1= #"H021)5G�H0w9\5.Q%I<?!"%5.-g0w L5B&L0:0 #"H0g, üeý SÅ@�+.(: L5G1L&�(25.�f6.021=JG$<�%Jw+.�%?M #"%0:�K L"%0g, ü S[@�+.(p #5G1=&l(:5G�>6G0:1LJG$'�%J%^�¶,5G1� #"H0b0¿ %+4CK-%I'0Y&L"%5�9Y�$<��Ó,JGQ%1=0Ox%^��K #"%0wNg0:",+e6/$'5GQ%1h+V )"%$'JG"%021b?%02�H&L$' L$<0:&)$<&b(25G�> #1=5GI<I'02?`N>P| #"%0w�/S�&="%02I'I�NR02",+e6X$<5GQH1+.�%?�$Z A9\+.&s�%54 m-R5G&L&=$<N%I'0b #5E$'&L5GI<+4 #0Y #"H0)Ig(:5G�>6G0:1LJG0:�%(204^mòb0p6G0:1= #"H02I<0:&L&s #"H0)$<�Hß%Q%02�%(:0)5.@¡ #"H0é®S�1=02?%$'&= L1L$<NHQH #$Z6G0)1L+4 #0:&a$'&a+.&s0p /-R02(: L02?z^j IKS/Mq@%GNMkQ}|nE;GN@1hd>\QTSÜÝ$' L"� L"%0d�%I ék�%5G�XS�&= #+4 #$'&= L$<(e+4I�-g5G-HQ%IU+4 L$<5G�Ú1=02&LQHI' #&w+c?%$'&= #5.1= #0:?�-%1=5.Ó,I'0K(2+.�æNR0d-%1=5/?%QH(202?¢^¶�$'1L&= LI'P`+�&= L$<(*]!-%I'5. 35.@m L"%0i, ü2ý þ @�+4(: #5.1L&k$<&3JG02�H021#+V #02?¢^B�s"%$<&k$<&h L"%02��N%1L5>+4?%02�%0:?æQ%&=$<�%J�+&= #+.�%?,+41L?M¤)5G-%-HI<021i-%1L54Ó,I<0A0p /-%1L0:&L&=$<5G�E #5j-%1L5X?%Q%(:0Y+j?%$<&8 #1=5. #0:?O-%1L54Ó,I<0\@F5.1W+jJG$Z6G0:�E­bj ¬®¬ alk�U­ ýBmon a�kW L1#+.�%&=$' L$<5G�z^iòb54 #0m #"%+4 �$'@, L"%0a+.N%&=5GI<QH L0m-R5G-%Q%I<+4 #$'5G�%&W+41L0sQ%&=02?D+.I<5.�%J)9Y$' L"E(e+.I'(2Q%I<+4 #0:?£aSÅ64+.I<Q%0:&)+.�H?`N%1L5G+.?%02�H02?Ú-%1L54Ó,I<0:&2rz+4�`0p /-R02(: L02?Ú02CD$<&=&L$'5G�`-%1=5.Ó,I<0k(e+4�!Ng0O-H1L5X?%Q%(20:?z^)Ü�0&L"H579Ô L"%0aN/S[@�+.(p #5G1�1=02&LQHI' #&l"%0:1L0Y #5k$<&L5.IU+4 L0Y L"%0a-g5G-HQ%IU+4 L$<5G�M+4�%?d0:�%021=J.PD&L(e+4I<$<�HJkNR02"%+�6X$'5GQ%12^¶�$'JGQ%1L0bx%^ãxD&="%5�9Y&a+O&L0: Y54@i(2+.I<(:Q%IU+V #$<5.�%&A@F5G1A #"%0h­�ÿ � U }q #1#+4�%&L$Z #$<5.�z^W�s"%0h-%I'5. #&[email protected]+3&L(e+4I<02?D?%0:�%&L$Z �PM5.@Wy Í y�¦ ²Fµ -�º F ´ +.�H?d+3&L(2+.I<0:?K #0:CK-R021L+4 #QH1L0Y5.@iy Í y�¦6p�dæ^�òb5. L0a L",+4 � #"H002�H021LJ4P!&L-%I'$' = #$<�HJd$<�H(21L02+.&L0:&h9Y$' #" ° ±ert1=02&=Q%I' L$<�%J�$'�Ú+�?H$<&= L5G1= L$<5G�c5.@� #"%0E&=-g0:(: #1L+.I�02�>6.02I<5.-g04^�s"%0D1L0:&LQ%IZ *+.�> q-%1L5.Ó%I<0M$<&h�%5.1LCK+.I<$'&L02?! L5�5G�H0.^d�s"H0K"%$'JG"%0:13 L02CD-g0:1#+4 LQ%1L0E L",+.�� L",+4 kQ%&=02?@F5G1mÓ,J.Q%1L03x%^��E-%1L5X?%Q%(:02&Y #"%03$<�H(21L02+.&L0:?�NXS[@�+.(p #5G1m64+.I'Q%02&:^

Page 123: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y�¦ ¨

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1e+14 1e+16 1e+18 1e+20 1e+22 1e+24 1e+26 1e+28 1e+30

b_

nlj f

acto

r

q

Density / cm^-3

3 s 1/2

3 p 1/2

3 p 3/2

3 d 3/2

3 d 5/2

¶�$'JGQ%1=0hx%^Á�/·�rts�u v�Í�ºLÂ�Ç7À�¼Å¢ÄTÃ:ÍtÈ6w:xm½pÂt½AÄTÇ7ÈVÊ®¼�¾ÁÃpÈ)ÿÄ,»�º=ÈVÂ�¾�¼*#sÄTÃ:Ízy|{}w~xD�\½¿È4»M���Ew%�=� ���������D�t�DÏ

Page 124: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGye¦

0

12

34

56

6200 6250 6300 6350 6400 6450 6500

b-fa

cto

r

�z0=44

0

12

34

56

6300 6350 6400 6450 6500 6550

b-fa

cto

r

�z0=36

0123456

6380 6400 6420 6440 6460 6480 6500 6520 6540 6560 6580

b-fa

cto

r

Wavelength (A) * z0 * z0

z0=28

¶�$'JGQ%1L0bx%^ãx%·P!_ÀÁÿ¼zÂ�¹VÃ*Ît¾�È7Él¼�¹7º}�VÈ7ºWÂ�¼�ÍÅÇVÊ�¼ÅÇ7Í�ºiÈ7À �¢Â�Ì7À�¾�¼�¼Å¾�ÈVÉA¾ÁÈY¼Å¹7º}�7w~x����W¼ÅÍŽpÈVÂ�¾�¼�¾ÁÃpÈXÏ%¸z¹Vº[��& ½���¾ÁÂR¾Á¼�¹VºlÎz½��pº8ÀÁº8ÈVÉ¿¼�¹���y��{ ½¿ÀÁÀ�Ã*Ît¾ÁÈ7Ém¼Å¹7º�Íź=Â�Ç7Àã¼[ÂtÄTÃ:Í[�*½¿Íž�Ã:ÇVÂzy { ¡¼�ÃYÆ.ºlÌVÀ�Ãp¼�¼�ºL»)ÃpÈ)¼Å¹7ºAÂŽ¿ËmºlÂÅÊ8½pÀ�º:Ïz¸z¹Vº�Ì7ÀÁÿ¼Å½¿ÍźzÄTÃpÍWy�{}w~�D���=x=�����D�3�:½�Íź=»�Ç4Ê®º=»\¼Åº8ËmÌ.º8Í[½*¼�ÇVÍ�º¡ÃpÄ��E���|�=�S�ݽ¿È4»s½�Í�ºL»�ÇVÊ8º=»Y»�º=ÈVÂ�¾ã¼*#\ÿÄ1�E������ *¡�¢S£�¤6¥¿Ï

Page 125: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGy.y

�`�8� ���c�W�D� �k���k� �`�j�����s���2�D� �D�����`� ¦Â�!�!�!�e�e�¨§ � �c�}�,�ª©F��[ h$'&)+.?H64+.�> #+.JG025.Q%&)$'@� #"%0O�H0:9(25GI'I<$'&L$<5.�,+.IZS[1#+.?H$U+4 L$'6G0b(25X?%0O(2+.�Ú+.QH L5GCd+V #$<(2+.I<IZPB?%0p #0:(: h #"H0N%Q%�H?%I<$'�%JÚ(2QH 8S�5.�g&D+.�%?Ò&L0: D #"H02C +.-H-%1L5G-H1L$U+V #02IZPG^����Ò #"%$'&E&L02(p #$'5G�Ò L"%0|(21L$Z #0:[email protected]&LQ%(*"(2Q/ =S�54�¢&�@F5G1� #"H0b�%0p9Ô(25.I<I<$'&L$'5G�,+.ITS�1#+4?%$U+4 L$Z6.0ACD5X?%02I,+41L0a?%0: L021LCD$<�H02?z^W�s"H0b-%1=5GN%I'02C�$'&�Ó%1L&= LI'P0p H+.CD$<�H02?�@F1L5GCÖ+.��02�%0:1LJ.P�I'0:6G0:It-g0:1L&=-g0:(: #$Z6G0q+.�%?� #"%0:�|@F1=5GCÖ+E1L+4 #03-R5G$<�> Y5.@�6X$<0:93^�s"%0�Ó,1=&= ¡&= L02-k$<�) L"%0�"%0e+e6fPh&L-R02(:$<0:&�(:5/?H0l$'&t+.�k+.QH L5G&= L1LQ%(p #Q%1=0�1LQ%�h #5Y?%0p #0:1LCD$<�%0� L"%0�02�%0:1LJ.PI<0p6G0:I,&= #1=Q%(: LQ%1L04^��s"%$<&�&=02(p #$<5.�K&="%5�9Y&A"%5�9â �P/-H$<(e+4I,+.QH L5G&= L1LQ%(p #Q%1=0a?%+4 *+h(e+4�KNR0sQ%&L0:?d$'�O #"%$'&JG0:�%021L+.I_&L(*"%0:CK0j L5q?H0: #0:1LCD$<�%0Y #"H0b(:1L$' L$<(2+.I,�/S�&="%02I'I<&�NR0: �9A0202��NHQ%�%?%I'$<�%Jk1L0:JG$<CD02&:^�Þ\5G�%&=$<?%0:15G�%(:0�CK5G1=0B #"%0�N%QH�%?%I<$'�%J!&L(*"H02CD0�5.@b02nfQ,+V #$<5.� ¥Fx%^~}.§p^æ�[ DCOQ%&8 DNg0�?%0p #0:1LCD$<�%0:?â$Z@a L"%021=0$<&k$<�H?%020:?�64+.I<$'?Ú-g5G-HQ%IU+4 L$<5G�Ú&=02-,+.1L+4 #$'5G�u$<��­¢¯m&L-,+4(20d54@m #"%0D$<�> L021=64+.I'&3­¢±Bä�­�óo­�²k+4�%?­�²Oä�­�ó ­g³q@F5G1kN%Q%�H?%I<$'�%J� #5cNR0K+.-%-%1=5G-%1L$<+4 #04^�Þ\5G�H&L$<?H021q L"%$<&hÓ,1=&= LI'[email protected] #"%0D02�%0:1LJ.PI<0p6G0:It-g5G$'�> Y5.@i6X$'0:93^�\� � «r�+�-(���¬ ���4��H Ü�!��4��-&'������0d�)������s"%0K1L02I<+4 #$Z6X$<&= L$<(O&=02-,+41#+4 L$<5G�%&35.@A #"%0M64+.I<0:�%(20D02I'02(p #1L5.�uÓ,�H0d&= L1LQ%(p #Q%1=0dI'0:6G0:I<&[email protected] #"%0D­z¯(20:�> #1L5.$<?zr% #+.].0:�|@F1=5GCÑÞ\5G�%?H5G��� { "%5G18 #I'0:Pu¥®y ¨ �>vG§¿r%+.1=03JG$'6.02��N>P

­ ® üeýR®R¯ TY°l± ÿ ° µ²D² ³ ·´³­ µ ­ ­¥�¯ � y�§ ¯ ­¥�¯ � yNÕG}G§ ´­ ® üeý ®Jµ TY°l± ÿ ° µ² ² ³ ·´³­ µ ­ ­ ¯ ¯ ­¥�¯ � yqÕG}G§ ´ ¥�x%^Á}.¦>§

9Y"%0:1L0 ° ²�$<&W L"%0a$<5G�M(*",+.1=JG0 � y.r ² $'&W #"%0YÓ,�%0Y&8 #1LQH(: #QH1L0j(:5G�%&8 *+.�> \+.�%? ·´³ $'&W #"%0�XYPX?%NR021=J02�H021LJ4PdQ%�H$' e^c<APD(25G�> #1L+.&= 2r/ L"%0 « &L02-%+.1#+4 L$<5G�B+.1L$'&L02&l@F1L5GC� L"%0)0:I<02(p #1=5G&= #+4 #$'(b$'�f L021L+.(: L$<5G�D5.@ #"H0q64+.I<0:�%(20q02I'02(: L1L5G��9Y$' #"� #"%0q-,+.1L0:�> h(25G1=0.^b�s"%0O$<�> L021#+4(: #$'5G�c$'&b?H$U+.JG5.�,+.I¡$<� « 9Y$' L"c #"H0I<0p6G0:It&L02-%+.1#+4 L$<5G�H&Y@F1=5GC L"%0K¥ «�¬ §®­z¯T°K(:02�> #1=5G$<?|0p /-%1=02&=&L$<NHI<0k$<�B #"[email protected]

­ ® üeý ¶Lú ÿ � ¤ ¥�·V³ ¤(¸ ³ ¤ �u¢ ³ ¤º¹ ³ ¤ §ÿ ° ² ·g³­ ´ � ¤ ¥�·V³ ¤�»¸ ³ ¤ �u¢ ³ ¤�»¹ ³ ¤ § ¥�x%^Á}/y7§

Page 126: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGy�}

9Y"%0:1L0h #"%0~¼8N,+.1L1=02?f½EnfQ,+4�f L$' L$<02&m+.1L0)9A0e+.]fIZPB?%0:-g0:�%?%02�> b5G� ° ²\+.�%?�­Ú+.�%?B L"%0G·g&:r »¸ &Y+4�%?»¹ &b+41L035.@i5G1L?%0:1sQ%�%$Z �PG^��s"%0 »¸ &a+.�H? »¹ &a+.1L0) L"%03Q%&LQ,+4I�&=IU+4 L021s$'�f L02JG1L+.I<&:^Þ\5G�%&=$<?%0:1O+|(21L$Z #$'(e+.IW0:�%021=J.P�&L-%I'$' L L$<�%J ­ ® X)9Y"%$<(*"�$<&h #"%0KCK$'�%$<CwQ%C�I'0:6G0:IA&=02-,+41#+4 L$<5G�u1L0¿SnfQ%$<1=02?�@F5G1O �9A5�I'0:6.02I<&O L5ÚNg0�1=02&L5.I'6G0:?z^��s"%$'&D(:1L$Z #$<(2+.I\0:�%021=J.P�&=02-,+.1L+4 #$'5G��$<&E?H0: #0:1LCD$<�%0:[email protected]ð #"H0D$'5G�!(25.I<I<$'&L$'5G�c1L+4 #0:&q+4�%?�$'&j #"H0DCD$<CD$<CwQ%C×02�%0:1LJ.PÚ&=02-,+41#+4 L$<5G�!1=02nfQ%$<1=02?Ú@F5G1h$<5.�(25.I<I<$'&L$'5G�%&s #5B&8 *+4 L$<&8 #$<(2+.I<IZP�-R5G-%Q%I<+4 #0k #"%0O0:�%021=J.P`I'0:6.02I<&b@F5G1b L"%0w #0:CK-R021L+4 #QH1L0O+4�%?!?%02�H&L$' �P5.@h$'�> #021=02&8 e^��s"XQH&�+�(21=$' L$<(e+4Is�/S�&="%02I'Ib­g³d9Y$<I'Is5/(:(2Q%1d+.Ng5�6.0c9Y"%$<(*"ÚéDI<0p6G02I'&B+.1=0c&8 *+4 L$<&= L$ZS(e+4I<I'Pd-g5.-%Q%IU+V #02?¢^ { $'CK$'IU+.1=I'PM #"H021L039Y$<I'IzNR0q+M(21=$' L$<(e+4Iz�/S�&="%02I'I¡­�²Y+4Ng5�6G039Y"%$'(*"� #"H0wåDI<0:6.02I'&+.1=0w&= #+4 #$'&= L$<(e+4I<I'PB-g5G-HQ%IU+4 L02?z^)¶_5.1a L"%0O-%1=5G-g5.&L02?!NHQ%�%?%I'$<�%JD #5BNR0E+.-%-H1L5G-%1=$U+4 L0k9A0E1L0:nfQ%$<1=0­¢³ ½ ­¡²p^��i+.]f$<�%Jh #"H0a1L+4 #$'5)5.@g02nfQ,+4 L$<5G��¥Fx%^~}4¦>§� #5K¥Fx%^~}Xy7§�5G�%0s(2+.�M&L020Y #"%+4 � #"%$'&W&L0:-,+.1L+4 #$'5G�5.@iN%Q%�%?HI<$<�HJD$'�> #021864+.I<&m­g±jäâ­uóÔ­¡²s+4�%?�­¡²mäà­uóÔ­g³b$'&Y+.-%-%1=5G-%1=$U+4 L0h$'@

­g³­¡²7¾ ° ² ² ³tH[´ ½ y ¥�x%^Á}G}G§¿�m-Q'G�À1ÁÙSkBls�>\I\m[À¹Q�ÂZs�JLE;>\QÄÃ�Å�s9GDi3@RmEò>\IACRQ�IKSv@ZQT>\QTBwMqGN@Rm;I\BÜS QÇÆ h QTm;BlQ£j�w$<I<I'Q%&= L1#+4 L$<5G�h54@H #"H0\02�H021LJ4PwI'0:6G0:I/&=(e+.I'$<�%JY@F5G1t #"H0sÞm+.1=Ng5G�XS�I<$'].0�&L0:nfQ%02�%(:0s$<&�&L"%5�9Y�ENR02I'5793^�s"%0E&=Q%-g0:1L&8 #1LQH(: #QH1L0D(25X?%0|¥K �$<&=&L�%0:1h0: k+.I�^'riy ¨�� x>§Y9\+.&3Q%&L0:?Ú L5�JG02�H021#+V #0D+.�!02�H021LJ4P`I<0p6G0:I&= L1LQ%(p #Q%1=0d@F5G13 #"%0BÞlS�I'$<].0D&L0:nfQ%02�%(:0B9Y$' L"æ+|64+.I'02�%(:0d0:I<02(p #1=5G�� #"%+4 w(:5GQ%I'?�NR0d-%I<+.(20:?�$<��+1#+4�%JG0d54@Y�á+.�%?æIl&L"%0:I<I'&2^`�s"%0d(25G1=0B(25G�/Ó,JGQ%1L+4 #$'5G��9\+.&O"%0:I<?æ(:5G�%&8 *+.�> D+4 dy�2 ³ }È2 ³ }DÉá+4�%? #"H0+XsP/?HNg0:1LJw0:I<0:(: #1=5G�B+4?%?%02?d5G�K L5G-d5.@¢ #"%$'&A(:5G�HÓ,JGQH1#+4 L$<5G�z^i�s"%0j(:5G1L0b(25G�/Ó,JGQ%1L+4 #$'5G�D L"fQ%&JG0:�%021L+4 #0:?� �9A5c-R5G&L&=$<N%I'0E-,+.1L0:�> w&8 *+4 L02&2r��,+.CD02IZP ³�Ê ²�H[³O+.�%? ³�Ê ´tH[³�^K�s"%0D02�H021LJ4PÚ&L-%I'$' L L$<�%JNR0: �9A0202�Ô L"%02&=0!&= *+V #02&d$<&D0p /-g0:(: L02?Ô #5æNR0`&=$<JG�%$ZÓ,(e+4�f LI'Pá&8 #1L5.�%JG021D #",+4�à #"%0B°á5.1 « Ó%�%0&= L1LQ%(p #Q%1=0w&=-%I<$Z L #$'�%JM+.�%?|&L"H5GQ%I<?|&L(2+.I<0q+.& ° µ² Õ4­ ´ ^m¶,5G1s02+.(*"`-,+.1=02�> b(25G�HÓ%JGQ%1#+V #$<5.��9A0w"%+�6.0 #"H0k@FQ%18 #"%0:1b&=-%I<$Z L L$<�%JE$'�f L5D �9A5d-R5G&=&L$<NHI<0a°B&= #+4 #0:&a9Y$Z #"B°Mÿ ¯ � yNÕG}E5G1l°Dÿ ¯ ¯ yNÕG}/r_&=$<�%(:0 #"H021L0k$<&Y5G�HI'P�5G�H0k5.QH #0:1b0:I<02(p #1=5G�| [email protected]� L"%0qI¡+4�%?c&YnfQ,+.�> #QHC �fQ%CwNg0:1L&Y #5DJG$'6.0qQ%&Y #"H01L0:&LQ%IZ *+.�> iéA@F5G1l #"%0)64+.I'02�%(:0)02I'02(p #1L5.�z^��s"%$<&\02�%0:1LJ.PdI<0:6.02Iz&=-%I<$Z L L$<�%Jq$<&\0¿ /-g0:(: #0:?| L5E&L(2+.I<0h+4&$<��0:nfQ,+4 #$'5G��¥�x%^Á}.¦>§¿^W�s"%0:1L030p /$<&8 #&a+w@FQ%18 #"%0:1s-g5G&=&L$'N%I<0)0:�%021=J.P�&=-%I<$Z L #$'�%JO?%QH0) #5O #"H03 L5. *+4ItånfQ,+.�> #QHC �fQ%CwNg0:1m+.�%?D #"%$'&A&="%5GQ%I'?d&=(e+.I'0j+.&l$<�D02nfQ,+V #$<5.�!¥�xH^~}/y�§p^�¶,5G1�5GQ%1\ÞlS�I'$<]40a&=02nfQ%0:�%(209A0q",+e6.0k L"%0h@F5GI<I'579Y$'�%Jw-R5G&=&L$<NHI<0) #54 *+.I¡åMnfQ,+.�> #Q%C×�fQ%CwNg0:1L&2^

Page 127: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGye�

Ë+s)GNQTm`M ­ ¯ ° «³ Ê ²�H[³ ­ ¯ ¯ ¯ yNÕ.} ¯ ¯ ¯ y³ Ê ²�H[³ ­ ¯ ¯ � yqÕG} ¯ � y ¯³ Ê ´tH[³ ­ ¯ ¯ ¯ yNÕ.} ¯ � y ¯ ¯ ¯ y ¯ ¯ }³ Ê ´tH[³ ­ ¯ ¯ � yqÕG} ¯ � } ¯ � y ¯ ¯ ¯ y�[ A$<&�0p /-R02(: L02?d L",+4 l #"%0aIU+41LJG0:&= l02�%0:1LJ.PD&=-%I<$Z L #$'�%J)9Y$<I'I%NR0bNR0: �9A020:��-,+.1L0:�> \&8 *+4 L02&2rX@F5GI<I'5�9\0:?N>P� #"%0méY&=-%I<$Z L L$<�%JM+.�H?� #"H02�� L"%0wåD&L0:-,+.1L+4 #$'5G�z^\�[@�0:6.021=Pf L"%$<�%JK&L(2+.I<0:&j+.&a0p /-R02(p #02?� #"%0:�c$' 9Y$<I'I,Ng0a-R5G&L&=$<N%I'0Y L5w-%1=02?%$'(: l #"%0aN%Q%�H?%I<$'�%Jq(:QH =S[5.�g&\+.�H?B+.&=&L02CwN%I<0Y #"%0b(25GI'I<$'&L$<5.�,+.IZS[1#+.?H$U+4 L$Z6.0CK+4 #1=$Z d02I<0:CK0:�> #&a+.&Y$<�> #0:�%?%0:?z^�s"%1=020w&=Q%-R021L&8 #1=Q%(: LQ%1L0E1=Q%�%&b9A021=0w-R0218@F5G1LCD02?�$<�|5G1L?%0:1b L5D6.021L$Z@ P� #"%0q02�%0:1LJ.P|&L(2+.I<$'�%J%^s¶,5G1+.I'IW1LQ%�H&k L"%0K5.-H #$'5G�!9m+.&kQ%&=02?� L5�&=-g0:(2$Z@ Pu+|(:5G1L0M(25.�HÓ,JGQ%1L+4 #$'5G��+.�%?�+.?%?u5G�æ+ XYPX?%NR021=J02I'02(p #1L5.�z^��s"H0B(25X?%0d1=0: LQ%1L�%0:?æ #"%0K02�%0:1LJ.P�I<0p6G0:IA64+.I<QH02&k$<�ØXsPX?%NR021LJ!QH�%$' L&k9Y$Z #"æ1=02&L-R02(p #5E L"%03I<5�9\0:&= aI<0p6G0:It?%0:Ó,�H02?z^AÛ)�%I'PB+E&L$<�HJGI<0h(:5G1L0h(:5G�HÓ,JGQH1#+4 L$<5G��9\+.&Y&L-R02(2$ZÓ,02?|$<��5G1=?%021m L5$<&=5GIU+V #0Y #"%0j64+.1=$<5GQH&\02�H021LJ4Pd&=-%I<$Z L #$'�%JG&:^��s"%0)Ó,1=&= s1=Q%�B9\+.&[email protected]?BQ%&=$<�%JO+MvV@HXYPX?%NR021=J02I'02(p #1L5.��+.�%? ° ²39\+.&w64+.1=$<0:?�@F1L5.C yK #5!è>vX^c¶,5G13 #"%0d&L02(:5G�%?�1=Q%� ° ²39\+.&O"%0:I<?æ(:5G�%&= #+.�> +4 ç�� L5M+.I<I'579�&=Q9¡D(2$<0:�> b1=02&=5GI<QH L$<5G��54@i L"%03&L-%I'$' L L$<�%Jq #5MNg0hJ.02�%0:1#+4 L02?�+.�%?��|64+.1=$<0:?B@F1L5GCx` #5áy ç ^LÊ¡+.&8 #I'P�9Y$' L" ° ²Dÿ � }!+.�H?�­ ÿ ye�Hrl¯a9\+.&O64+.1L$'02?æ@F1L5.C }� #"H1L5GQ%J."á L5�y�¦H^��[ 9\+.&�-g5G&=&L$'N%I<0`@F1=5GC #"H02&L0Ú1=Q%�%&� #5�6G0:1L$'@ Pà #"%0 ° ²¿r)­õ+4�%?ЯqNg0:",+e6/$'5GQ%1�5.@k #"H0u1=02&=5GI'6.02?02�H021LJ4PcI'0:6G0:I�&=-%I<$Z L #$'�%JG&b+.&j?H0:Ó,�%0:?Ú$<�`0:nXQ%+4 #$'5G�%&E¥Fx%^Á}.¦>§b+4�%?á¥�xH^~}/y�§p^)òa5. #0q #",+4 j$' b$<&a #"%$'&&L(2+.I<$'�%JwNR02"%+�6X$'5GQ%1A L",+4 \$<&\&=5GQ%JG"> s"H021L04rH�%54 \ #"H0)0:64+.I'Q,+4 #$'5G�D5.@¡0p H+.(p s02�%0:1LJ.PKI<0p6G0:Ig64+.I<QH02&2r"%0:�%(20D #"%0:1L0d$'&3�%5c�%0:02?�@F5.1w+�CK5.1L0M(25GCD-%I'0p �+4 L5GCK$'(E&= L1LQ%(p #Q%1=0K #5�Ng0D02�> L021L0:?æ$<�> L5� #"H0&LQH-g0:1L&= L1LQ%(p #Q%1=0w(:5/?%04^Þ\5G�%&=$<?%0:1L$'�%JOÓ,1L&8 s #"%0j-,+.1L0:�> Y02�%0:1LJ.P�I'0:6.02Iz&=-%I<$Z L #$'�%J%^�����Ó,JGQ%1=02&sx%^ÁvkS�x%^ ç 9\0h&="%5�9Ð L"%0 ° ²+.�%?�­`NR02"%+�6X$'5GQ%1s5.@¡ #"H0h-,+.1L0:�> Y&L-%I'$' L L$<�%JH^��s"%0 ° ²\&L(2+.I<$'�%JO$<&m+.&s0p /-R02(: L02?|$Å^ã0.^ ° µ² @F5.1m"%$<J."° ²p^�£b&�@F5G1W #"%0s­_S�?%0:-g0:�%?%0:�%(204r/ #"H0a-%+.1L0:�f l0:�%021=J.PE&L-HI<$' = #$'�%J)$<�H$' #$<+.I<IZP)@�+.I'I<&�54��&L",+41L-%IZPw9Y$' L"�u+.�%?Ú #"%0:�Ú1L02CK+.$'�%&)(25.�%&= #+.�> e^M�[ 3&L"%5.Q%I<?ÚNR0E�%5. #0:?u L",+4 h L"%0ECd+4JG�%$' LQ%?%0O54@A L"%0E-,+.1=02�> &L-HI<$' = #$'�%JO$<&\&L$<J.�%$'Ó,(2+.�> #IZPKJ.1L0e+V #021m L",+4 Y0:$' #"H021\ L"%0Aés5.1YåM&=-%I<$Z L L$<�%J�¥F(25GCD-,+.1=0j #"H03+.N%&L5.I<QH L002�H021LJ4P�?%$'�g021=02�%(:0364+.I'Q%02&s54@iÓ,J.Q%1L03x%^ÁvE9Y$Z #"� #"H5G&L035.@�Ó,JGQ%1=02&ax%^ � SlxH^'y�èG§p^�s"%0|Ng0:",+e6X$<5GQ%1M5.@b #"%0 ® üeý þ Ó,�%0�&= L1LQ%(p #Q%1=0cI'0:6G0:Im@F1L5GC #"%0 ® ü2ý (20:�> #1L5.$<?zrA$Å^ã0.^W0:nXQ%+4 #$'5G�¥�xH^~}.¦G§K9m+.&� #"H02��$<�>6G0:&= #$'J>+4 L02?z^ ��+4]X$'�%J� #"%0u1L0:&LQ%IZ #&�@F1L5.C #"%0 ° ²d1LQ%� $' �(2+.�ÐNR0Ú&L0:02�@F1L5.C Ó,JGQ%1=0�x%^ � #"%+4 M #"%0|&L-%I'$' = #$<�HJ`6.+41L$<0:&M+.& ° µ² @F5G1D+.I'Im-R5G&L&=$<N%I'0qéE64+.I<QH02&2^Ò£j&M@F5G1E #"H0

Page 128: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGy2x

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1000

1 10 100

Ene

rgy

diffe

renc

e

Z1

Parent energy separation(4.*10**(-6))*x**4

¶�$'JGQ%1L0\x%^~vX·}ÌzÍoÎ�ÏÑÐ Í Â�º=ÌV½¿Í[½*¼Å¾�Ã:È�ÄTÍÅÃpË`ÌzÍoÎDÒºÐ*ÏgÊ®º=È2¼�ÍÅÃp¾~»�&Çy   »�º=Ì4º=ÈV»�º8È4Ê®ºpÏ1Ó�ÿ¼�º¢¼Å¹V½*¼,¼Å¹7ºtº8È7º=Í�É=#lÂ�Ì7À�¾�¼�¼Å¾�ÈVÉ�*½¿Íž�ºLÂt½pÂ>y ¡  ÄTÃpÍt¹7¾ÁÉp¹´y   ½pÂzºt��Ì.º=Ê®¼�º=»fÏ

78.0296

78.0297

78.0298

78.0299

78.03

78.0301

78.0302

4 6 8 10 12 14 16

Ene

rgy

diffe

renc

e

Ô

n of Rydberg electron

Parent energy separation

¶�$'JGQ%1=0)x%^ ç ·"ÌzÍoÎ ÏÑÐ Í Â�º8ÌV½pÍŽ¿¼�¾ÁÃpÈYÄTÍÅÃpËÕÌzÍ�Î Ò(ÐlÏ Ê8º8È2¼�ÍÅÃp¾~»�&��w»�º=Ì4º=ÈV»�º=ÈVÊ®º:Ï

Page 129: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGy�v

�/S[?%02-R02�%?H02�%(:0�$' k(e+.��Ng0K&L020:��$<�uÓ,JGQ%1=0dx%^�è| L",+4 [email protected]. #"�-,+.1=02�> #&w L"%0d&=-%I<$Z L L$<�%J�&L(2+.I<0:&+.&ByqÕ4­ ´ ^Ú�s"%0�ITS�?%0:-g0:�%?%02�H(20�$<&w&L"%5�9Y�á$<�æÓ,J.Q%1L0:&Ex%^ ¨ S)x%^Zy�¦H^Ú�[ E(e+4�ÒNR0B&=0202�á #"%+4 O #"H0&L-HI<$' = #$'�%JM?%5/0:&b$'�%?%0:02?�64+.18P�0p H+.(p #I'P�+.&a-%1=02?%$'(: #0:?c$'��0:nXQ%+4 #$'5G��¥Fx%^~}4¦>§pr,�%54 #0w$'��-%+.1= L$<(2QHIU+.1 #"%+4 s #"%03(:5G�%&= #+.�> j54@�-%1=5G-R5G1= L$<5G�,+4I<$' �PD$'&Y+.I<&=5M+.&Y0p /-g0:(: L02?z^Þ\5G�%&=$<?%0:1d�%0¿ X d #"H0 ® üeý þ#ú &L-%I'$' L L$<�%J!@F1=5GC� #"%0 ® üeý þ (:02�> #1=5G$<?à+.&D-%1L0:?%$<(p #02?àNfPÒ0:nXQ%+4 #$'5G�¥�xH^~}/y�§p^k��+.]f$'�%J�Ó,1L&8 h #"%0 ° ²�S[?%02-R02�%?H02�%(:0D5.�%0E(e+.�u&L0:0O@F1L5GC Ó,JGQ%1=02&3x%^ZyGywSYx%^Zy��d L",+4 h #"H002�H021LJ4P�?%$Z�¢0:1L0:�%(20:&O+.1=0d$<�H?%020:?æI<$'�%0e+.1h9Y$Z #" ° ²p^|�s"%0K�/S�?%0:-g0:�%?%0:�%(20�$'&q&="%5�9Y��$<��Ó%JGQ%1L0:&x%^ZyexOSlx%^Zy ç +.�%?|$<&s&L0:02�| #5M&L(2+.I<03+.&3yNÕV­ ´ ^�s"%0qI�NR02"%+�6X$'5GQ%1a$<&Y�%5. b&L5K&= #1L+.$<J."f =@F5G1=9\+.1=?zr, #"%0k02�%0:1LJ.P|?%$Z�¢0:1L0:�%(20q&L-%I'$' #&Y$'�f L5D �9A5BCK+.$<�JG1=5GQ%-%&q?%0:Ó,�H02?�N>PÚ #"%0w°ÚÿÖ¯ ¯ yNÕG}�+.�%?`°ÚÿÖ¯ � yqÕG}|&8 *+4 L02&�¥F&L0:0dÓ,JGQ%1=02&qx%^'y � Sjx%^'yeè>§p^¶,5G1\I'5�9 IR #"%0:1L0h$'&m+O"%$'JG"B?%0:JG1L0:0h5.@¡6.+41L$U+4�%(20b$<�B #"H0 ­ ® ü2ý þLú Ng0:",+e6/$'5GQ%1:r%+.&m0¿ /-g0:(: #0:?�?HQ%0 #5| #"%0d-g0:�%0: L1#+4 L$<5G��5.@m #"H02&L0d5G1LN%$Z *+.I'&q$'�> #5� L"%0d(:5G1L04^c£b&w5G�%0K$<�%(:1L02+.&L0:&w #5�"%$<JG"H021wI� #"H0NR02",+e6X$<5GQH1wNR02(25.CK0:&w&LCD5X5. #"zri&L(2+.I<$'�%J`+.-H-%1L5e /$<CK+4 #0:I'PÚ+4&dyNÕ.¯ ´ @F5.1q #"H0d(e+4&L0B5.@\ #"%0 ³ Ê ´tH[³-,+.1=02�> e^����|Ng54 #"�(e+.&=02&Y #"%03&=-%I<$Z L #$'�%JE(2I'0e+.1=I'PD@�+.I<I'&m5.�`9Y$Z #"�$<�H(21L02+.&L$'�%JMIÅ^¶,1L5GC� L"%$<&�9A5G1=].02?M0¿ %+4CK-%I'0m$' i(e+4�ENg0m&=020:�D L",+4 W0¿ /-%1L0:&L&L$'5G�%&b¥�xH^~}.¦G§i+.�%?|¥Fx%^Á}/y7§i+41L0m$'�%?%020:?64+.I<$'?z^��s"XQH&l$Z �$<&W+.I<&=5h-g5G&=&L$'N%I<0\ L53+.&L&=02&� #"H0s­¢³/Õ4­¡²�1L+4 #$'5)&L"H579Y�D$<�E0:nfQ,+4 #$'5G�|¥�xH^~}G}.§�@F5G1W #"H0ÞlS�I'$<]40w(e+4&L0.^q¶�$'JGQ%1L0:&jx%^Zy ¨ Ssx%^~}.}K&L"%5�9 #"%0w1#+4 L$<5B5.@ ­ ® üeý þ#ú L5 ­ ® ü2ý þ ^3¶,1L5GC 02nfQ,+4 L$<5G�H&¥�xH^~}.¦G§j+.�%?à¥�x%^Á}/y7§a$' h(2+.��NR0E&L0:02�Ú #",+V h #"%$'&h1#+4 L$<5�$<&j02nfQ%$'64+.I'02�> h #5�­g³/Õ4­�²¿^w£j&30p /-R02(: [email protected]:nfQ,+4 #$'5G�|¥�xH^~}G}.§pr4 L"%0m1L+4 #$'5jNR02(25.CK0:&WJG1L02+4 #0:1� #"%+.�E5G�%0s+41L5GQ%�H? ° ²lÿ }.¦H^��s"%+4 �$<&:r. #"H0(*"%5G$'(20k5.@WI'0:6.02I¡N%Q%�%?HI<$<�HJD$'�> #5M #"%0k&L-R02(:$'Ó,0:?c1L0:JG$<5.�%&Y$<&s64+.I<$'?�@F5.1b+4-%-%1L5e /$<CK+4 L02I'P ° ² ½ }.¦H^�s"%$'&m6.+4I<Q%0)64+.1=$<02&A@F5G1m #"%0h?H$'�g021L0:�> b-R5G&=&L$<NHI<0s°M6.+4I<Q%0:&Y+.&s(e+.�|NR0h&L020:�|@F1=5GC Ó%JGQ%1L0:&Yx%^'y ¨ Sx%^Á}G}/^¶,1L5GC #"%0dÞlS�I'$<]40E&LQ%-R021=&= #1=Q%(: LQ%1L0K?,+4 #+|$Z h9m+.&q+.I'&L5�-g5.&L&L$'N%I<0O L5�?H02(2$'?%0M5G�Ú #"%0M(21=$' L$<(e+4IW�64+.I<QH02&O@F5G1O02+.(*"ÒN%Q%�%?%I'$<�%J!1=02JG$'5G�z^��[@j5G�H0�&L-R02(2$ZÓ,02&D+!(21=$' L$<(e+4IA0:�%021=J.P�?%$Z�¢0:1L02�H(20�¥ ­ ® X8§+.NR5�6G0)9Y"%$'(*"�N%Q%�%?%I'$<�%Jw$<&\�H5EI<5G�%J.021m+4-%-%1L5.-%1L$<+4 #04r/­¡²m+4�%?�­¢³s(2+.��Ng0j$<�H@F0:1L1L0:?B@F1L5GC 02nfQ,+7S #$'5G�Ô¥Fx%^Á}G}G§p^`¶_5.1w0p H+.CD-%I'0KQ%&=$<�%J` L"%0B0:�%021=J.P�&=-%I<$Z L L$<�%J�64+.I<Q%0:&O5.@s L"%0 ® ± �5Ö °�± üVö~ý F ²�H[³ û +4�%?® ± � Ö °�± üVö~ý F ²�H[³ û'ú�× µ I'0:6.02I<&j+.�%?Ú+ ­ ® Xmÿ¦ ì ¦Hy ·´Ø #"%0wN%Q%�%?%I'$<�%JK(2QH 8S�5.�g&h+.1=0O+4 j­�²sÿx�+4�%?­¢³\ÿ y7vX^

Page 130: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGy ç

1e-05

0.0001

0.001

0.01

0.1

1

10

1 10 100

Ene

rgy

diffe

renc

e / R

y

Ù

Z1

2P1/2 parent j=l-1/22P1/2 parent j=l+1/22P3/2 parent j=l-1/22P3/2 parent j=l+1/2

¶�$'JGQ%1L0�x%^ � ·>Ì s�u v Â�º8ÌV½pÍŽ¿¼�¾ÁÃpÈiÄTÍ�Ã:Ë�Ì s�u Ê®º8È2¼ÅÍ�Ã:¾Á»}&3y   »�º=Ì4º=ÈV»�º=ÈVÊ®º:Ï]Ó�ÿ¼�º_¼�¹V½¿¼f¼�¹7ºRº8È7º=Í�É=#iÂ�Ì7À�¾�¼�¼Å¾�ÈVÉz�*½¿Íž�ºL½pÂ>y=¡  ÄTÃpÍt¹V¾�É:¹´y   ½:ÂzÌ7Í�ºL»�¾~Ê�¼�ºL»j¾�Èhº�Ú:Ç4½*¼�¾ÁÃpÈ�þÛ�4Ï �D�=�®Ï�Ü¡Èhº8ÀÁº=Ê®¼�ÍÅÃpÈ7¾~ÊWÊ®Ã:È5�VÉpÇVÍŽ¿¼�¾ÁÃpÈaÃpÄW��Ý��|�DÝ����tÞ#þß��à  �áo� ���DâÎz½pÂzÇVÂ�º=»a¼�¹7ÍÅÃpÇVÉp¹7Ã:Ç�¼z¼�¹7º�Â�Ç7Ì.º8Í[ÂF¼ÅÍ�ÇVÊ®¼�Ç7ÍźWÍ�ÇVÈfÏ

0.001

0.01

0.1

1

1 10 100

Ene

rgy

diffe

renc

e / R

y

Ù

n of Rydberg electron

2P1/2 parent j=l-1/22P1/2 parent j=l+1/22P3/2 parent j=l-1/22P3/2 parent j=l+1/2

¶�$'JGQ%1L0Yx%^�èH·PÌ s�u v Â�º8ÌV½pÍŽ¿¼�¾ÁÃpÈ\ÄTÍÅÃpËãÌ s�u Ê8º8È2¼�ÍÅÃp¾~»ä&HÈb»�º8Ì.º8ÈV»7º8ÈVÊ8ºpÏ,¸z¹7º�º=È7º8ÍÅÉD#mÂ�Ì7À�¾�¼�¼Å¾�ÈVÉ��*½¿Íž�ºLÂR½:Âz��å���¥½pÂ�º��eÌ.º=Ê®¼�ºL»3ÄTÍÅÃpËݺ�Ú2ÇV½*¼Å¾�Ã:È�þÛ�4Ï �D�=�®Ï�Ü¡ÈwÀ1�*½¿ÀÁÇ7ºmÿÄExYÄTÃ:Íi¼Å¹7ºsÃpÇ7¼�º8ÍWº=À�ºLÊ�¼�ÍÅÃpÈq΢½:Â�ÇVÂ�ºL»3¼Å¹7ÍÅÃpÇ7É:¹7ÃpÇ�¼�¼�¹VºÈq»7º8Ì.º8ÈV»�º=ÈVÊ®º\ÍÅÇ7ÈfÏPy   ΢½:Â�¹Vº8À~»kÊ®Ã:ÈVÂ�¼Å½¿È2¼i½¿¼�æD�\¼�Ãjº8È4Â�Ç7Íź�¼Å¹V½*¼�¼�¹7º=Í�ºA΢½:ÂiÂ�Ç5çYÊ®¾Áº8È2¼�À�º|�pº8À%Â�Ì7À�¾�¼�¼Å¾�ÈVÉj½¿¼¹7¾ÁÉp¹jÈj¼�ÃsÆ.º�Í�ºLÂ�Ã:ÀC�:º=»>Ï

Page 131: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGy �

0.001

0.01

0.1

1 10

En

erg

y d

iffe

ren

ce / R

y

è

l

j=l-1/2j=l+1/2

(0.4887)/(2*x**(2)+x)(0.4887)/(2.*x**(2)+3.*x+1)

¶�$'JGQ%1L0lx%^ ¨ ·zÌ s�u v Â�º8ÌV½pÍŽ¿¼�¾ÁÃpÈWÄTÍÅÃpËéÌ s�u Ê®º=È:¼ÅÍ�Ã:¾Á»�ÄTÃpÍ1��à  láo� ÌV½pÍ�º=È:¼ê&>Àe»7º8Ì.º8ÈV»�º=ÈVÊ®º:ÏH¸z¹Vºgº=È7º8ÍÅÉD#�Â�Ì7À�¾�¼�¼Å¾�ÈVÉ�*½¿Íž�ºLÂ�º��7½pÊ�¼ÅÀC#3½:Â�Ì7Í�ºL»�¾~Ê�¼�ºL»hÄTÍÅÃpËÔº�Ú:Ç4½*¼�¾ÁÃpÈyþÛ�4Ï �D�=��Ï�Ü�ÈqÈ�*½¿ÀÁÇ7º\ÿÄ[��xaÎz½pÂiÇVÂ�º=»kÄTÃpÍi¼�¹7ºs½¿Ç7¼�Ã:Â�¼�ÍÅÇVÊ®¼�Ç7ÍźÍÅÇ7Èq½pÈV»hÀf�*½¿Íž�ºL»)ÄTÍ�Ã:Ëë�s¼Å¹7Í�Ã:Ç7Ép¹3¼�ÃG���7Ï"y   Îz½pÂ�½¿É2½¿¾ÁÈ3¹7º8À~»kÊ8ÃpÈVÂ�¼Å½pÈ2¼i½*¼�æ��s¼�Ãaº=ÈVÂ�Ç7Í�ºA¼�¹4½*¼�¼�¹7º=Í�ºA΢½:ÂÂ�Ç5çYÊ®¾Áº8È2¼¡À�º|�pº=ÀfÂ�Ì7ÀÁ¾�¼�¼�¾ÁÈ7Ém¼ÅÃ\Æ.º�Íź=Â�ÃpÀJ�pº=»fÏ

0.001

0.01

0.1

1 10

En

erg

y d

iffe

ren

ce / R

y

è

l

j=l-1/2j=l+1/2

(0.4887)/(2*x**(2)+x)(0.4887)/(2.*x**(2)+3.*x+1)

¶�$'JGQ%1L0ax%^Zy�¦H·PÌ s�u v Â�º8Ì4½¿Í[½*¼�¾ÁÃpÈmÄTÍÅÃpËìÌ s�u Ê®º=È2¼�ÍÅÃp¾~»mÄTÃpÍE�|à ¥íá�� ÌV½pÍ�º=È:¼b&HÀ>»�º=Ì4º=ÈV»�º=ÈVÊ®º:Ï_¸z¹7ºiº=È7º8ÍÅÉD#mÂ�Ì7À�¾�¼l&¼�¾ÁÈ7É:�*½¿Íž�ºLÂ/º��7½pÊ�¼ÅÀC#�½:Â/ÌVÍ�ºL»�¾ÁÊ®¼�ºL»�ÄTÍÅÃpË!º|Ú2ÇV½¿¼�¾ÁÃpÈ þÛ�VÏV������Ï]Ü¡È\È��*½¿ÀÁÇ7º¢Ã¿Äê��x¡Îz½pÂ/ÇVÂ�º=»lÄTÃpÍ/¼�¹7ºt½¿Ç7¼�Ã:Â�¼�ÍÅÇVÊ®¼�Ç7ÍźÍÅÇ7Èq½pÈV»hÀf�*½¿Íž�ºL»)ÄTÍ�Ã:Ëë�s¼Å¹7Í�Ã:Ç7Ép¹3¼�ÃG���7Ï"y   Îz½pÂ�½¿É2½¿¾ÁÈ3¹7º8À~»kÊ8ÃpÈVÂ�¼Å½pÈ2¼i½*¼�æ��s¼�Ãaº=ÈVÂ�Ç7Í�ºA¼�¹4½*¼�¼�¹7º=Í�ºA΢½:ÂÂ�Ç5çYÊ®¾Áº8È2¼¡À�º|�pº=ÀfÂ�Ì7ÀÁ¾�¼�¼�¾ÁÈ7Ém¼ÅÃ\Æ.º�Íź=Â�ÃpÀJ�pº=»fÏ

Page 132: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGyeè

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40 50 60 70 80 90

Ene

rgy

diffe

renc

e / R

y

Ù

Z1

j=l-1/2 J=2j=l-1/2 J=3j=l+1/2 J=3j=l+1/2 J=4

¶�$<JGQH1L0hx%^ZyGyG·"Ì s�u voî Â�º8Ì4½¿Í[½*¼�¾ÁÃpÈaÄTÍ�Ã:ËÕÌ s�u v Ê8º8È2¼�ÍÅÃp¾~»aÄTÃpÍz��à  �á�� Ì4½¿Íź8È2¼E&by   »7º8Ì.º8ÈV»�º=ÈVÊ®º:Ï

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 10 20 30 40 50 60 70 80 90

Ene

rgy

diffe

renc

e / R

y

Ù

Z1

j=l-1/2 J=1j=l-1/2 J=2j=l-1/2 J=3j=l-1/2 J=4

¶�$'JGQ%1L0hxH^'y7}X·"Ì s�u voî Â�º8ÌV½pÍŽ¿¼�¾ÁÃpÈYÄTÍÅÃpËÕÌ s�u v Ê®º=È2¼�ÍÅÃp¾~»YÄTÃ:Íz��à ¥oáo� ÌV½pÍ�º=È:¼Rÿäw þYïÈ�i��å��D��&by   »�º8Ì.º8È4»�º8ÈVÊ8ºpÏ

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 10 20 30 40 50 60 70 80 90

Ene

rgy

diffe

renc

e / R

y

Ù

Z1

j=1+1/2 J=2j=1+1/2 J=3j=1+1/2 J=4j=1+1/2 J=5

¶�$'JGQ%1L0hxH^'y��/·"Ì s�u voî Â�º8ÌV½pÍŽ¿¼�¾ÁÃpÈYÄTÍÅÃpËÕÌ s�u v Ê®º=È2¼�ÍÅÃp¾~»YÄTÃ:Íz��à ¥oáo� ÌV½pÍ�º=È:¼Rÿäw þYï�ðñ��å��D��&by   »�º8Ì.º8È4»�º8ÈVÊ8ºpÏ

Page 133: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

yGy ¨

0.0001

0.001

0.01

0.1

1

1 10 100

Ene

rgy

diffe

renc

e / R

y

Ù

n

j=l-1/2 J=2j=l-1/2 J=3j=l+1/2 J=3j=l+1/2 J=4

(0.95*10**(1))*x**(-3)

¶�$<JGQH1L0hx%^Zyex%·"Ì s�u voî Â�º8Ì4½¿Í[½*¼�¾ÁÃpÈaÄTÍ�Ã:ËòÌ s�uVv Ê8º8È2¼�ÍÅÃp¾~»aÄTÃpÍz��à  �á�� Ì4½¿Íź8È2¼E&_È)»7º8Ì.º8ÈV»�º=ÈVÊ®º:Ï

1e-05

0.0001

0.001

0.01

0.1

1

1 10 100

Ene

rgy

diffe

renc

e / R

y

Ù

n

j=l-1/2 J=1j=l-1/2 J=2j=l-1/2 J=3j=l-1/2 J=4

(4.*10**(1))*x**(-3)

¶�$'JGQ%1=0hx%^'y�v/·"Ì s�u voî Â�º=ÌV½¿Í[½*¼Å¾�Ã:ÈYÄTÍÅÃpËòÌ s�u v Ê®º=È:¼ÅÍ�Ã:¾Á»YÄTÃpÍ[�|à ¥íáo� ÌV½¿Íź8È2¼f�owiÀC&o��ó��AÂF¼[½*¼�º:&RÈ)»�º=Ì4º=ÈV»�º8È4Ê®ºpÏ

0.0001

0.001

0.01

0.1

1

1 10 100

Ene

rgy

diffe

renc

e / R

y

Ù

n

j=l+1/2 J=2j=l+1/2 J=3j=l+1/2 J=4j=l+1/2 J=5

(4.*10**(1))*x**(-3)

¶�$<J.Q%1L0)xH^'y ç ·"Ì s�u voî Â�º8ÌV½pÍŽ¿¼�¾ÁÃpÈaÄTÍ�Ã:ËòÌ s�u v Ê®º=È2¼�ÍÅÃp¾~»YÄTÃ:Íz��à ¥oáo� ÌV½pÍ�º=È2¼ê�owiÀÛð���óD�lÂ�¼Å½¿¼�ºP&_Èh»�º8Ì.º8È4»�º8ÈVÊ8ºpÏ

Page 134: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7}4¦

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1 10

Ene

rgy

Diff

eren

ce /

Ry

Ù

l

j=l-1/2 J=l-1j=l-1/2 J=lj=l+1/2 J=l

j=l+1/2 J=l+1

¶�$'JGQ%1=0)x%^Zy � ·"Ìzs�uVvoîhÂ�º=ÌV½¿Í[½*¼Å¾�Ã:ÈYÄTÍÅÃpËòÌ[s�u v�Ê®º8È2¼ÅÍ�Ã:¾Á»aÄTÃpÍz�|à  �áo� ÌV½¿Íź8È2¼E&_ÀX»�º=Ì4º=ÈV»�º=ÈVÊ®º:Ï

0.0001

0.001

0.01

0.1

1 10

Ene

rgy

Diff

eren

ce /

Ry

Ù

l

j=l-1/2 J=l-2j=l-1/2 J=l-1

j=l-1/2 J=lj=l-1/2 J=l+1j=l+1/2 J=l-1

j=l+1/2 J=lj=l+1/2 J=l+1j=l+1/2 J=l+2

¶�$'JGQ%1=0)x%^Zy�èH·"Ì s�uVvoî Â�º=ÌV½¿Í[½*¼Å¾�Ã:ÈYÄTÍÅÃpËòÌ s�u v Ê®º8È2¼ÅÍ�Ã:¾Á»aÄTÃpÍ � à ¥íáo� ÌV½¿Íź8È2¼E&_ÀX»�º=Ì4º=ÈV»�º=ÈVÊ®º:Ï

Page 135: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7}Xy

0.0 20.0ô 40.0ô 60.0ôZõ

1

0.0

1.0

2.0

∆E

nlj /

ö E nljJ

∆÷

E5f(l−1/2)/∆E5f(l−1/2)J=2

∆÷

E5f(l−1/2)/∆E5f(l−1/2)J=3

¶�$'JGQ%1L0Mx%^Zy ¨ ·ùøäÌzs�u voî6åDøäÌzs�u vsÄTÃpÍ���à  láo� ÌV½pÍl&º8È2¼����owiÀB&í��óD��Ï

0.0 20.0ô 40.0ô 60.0ôZ1

0.0

1.0

2.0

∆E

nlj/∆

En

ljJ

∆÷

E5f(l−1/2)/∆E5f(l−1/2)J=1

∆÷

E5f(l−1/2)/∆E5f(l−1/2)J=3

∆÷

E5f(l−1/2)/∆E5f(l−1/2)J=2

∆÷

E5f(l−1/2)/∆E5f(l−1/2)J=4

¶�$'JGQ%1L0Mx%^Á}.¦H·ùøäÌzs�u voî6åDøäÌzs�u vsÄTÃpÍ���à ¥oáo� ÌV½pÍl&º8È2¼����owiÀB&í��óD��Ï

0.0 20.0ô 40.0ôZ1

0.0

1.0

2.0

∆E

nlj/∆

En

ljJ

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=3∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=3

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=3

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=3∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=3

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=3

¶�$<J.Q%1L0Dx%^Á}/yG·ùø�Ì[s�u voîÈå�øäÌzs�u vsÄTÃ:Í���à  láo� Ì4½¿Í�&º=È:¼����owiÀÛð���ó��7Ï

0.0 20.0ô 40.0ô 60.0ôz1

0.0

1.0

2.0

∆E

nlj/∆

En

ljJ

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=4

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=2

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=3

∆÷

E5f(l+1/2)/∆E5f(l+1/2)J=5

¶�$<J.Q%1L0Dx%^Á}G}/·ùø�Ì[s�u voîÈå�øäÌzs�u vsÄTÃ:Í���à ¥oáo� Ì4½¿Í�&º=È:¼����owiÀÛð���ó��7Ï

Page 136: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7}.}

�\�Y� ×Ø�� ! ��&)�����+05 �(��H2 ��&T�9(%�!��*��������{��4305 !*+0d�)�-2�[ s",+4&sNg0:02��&L0:02�� #",+V m #"%0j02�%0:1LJ.Pd&L-%I'$' L L$<�%Jq5.@i+wJG$Z6G02��+V #5GCD$<(b&=PX&= L02CÖ(2+.��Ng0j-%1L0:?%$<(p #02?¢^Ûj@%(:5GQ%1L&=0l L"%0l+.(: LQ,+.I>&=5GQ%1L(:0A54@/ L"%0AI'0:6.02I<&¡NR02(:5GCK$'�%Js&= #+4 #$'&= L$<(e+4I<I'Pb-R5G-%Q%I<+4 #0:?d¥ 9Y$' L"q1=02&L-R02(p #5c0e+.(*"á5. L"%021#§k$<&k #",+V E$<5G��(25GI'I<$'&L$<5.�%&31L02?H$<&= L1L$'N%QH #0K #"%0B0:I<0:(: #1=5G�%&w9Y$Z #"%$'�æ+cJ.$'6G0:��&L0p E5.@I<0p6G0:I<&m@�+.&8 #021s L",+.�� #"%0pP�(e+4��?%0:(e+ePB L"%1L5GQHJG"�&=-g5.�f #+.�%0:5GQ%&Y02CD$<&=&L$<5.�z^W�s"%0:&L0k$<5G��(25.I<I<$'&L$'5G�%&+.1=0K&8 #1L5.�%JGI'P`?%02-R02�%?H02�> EQ%-R5G�Ú #"%0D02�%0:1LJ.PuI'0:6G0:I�&L-%I'$' L L$<�%JHr¡+.�%?� L"fQ%&w$Z 3$<&k$<CD-g5G18 *+.�> ) L5NR0d+.NHI<0M #5� L1#+.(*]! L"%0d&=-%I<$Z L #$'�%J�@F5G1q+.�>P�&8PX&= #0:C NR02$'�%J|$<�>6G0:&= #$'J>+4 L02?zr�+4&q$'�� #"H0K-%1=0:6X$<5.Q%&&L0:(: L$<5G�z^c�i5!0:�%&LQ%1=0d #"%+4 O #"H0dI<0p6G0:I<&E+.1=0d&8 *+4 L$<&= L$<(2+.I<IZP!-g5.-%Q%IU+V #02?�5G�%0BCOQH&= OI'5/5.]Ú+4 O #"H0+.(p #Q,+.IR(25.I<I<$'&L$'5G�,+.I/1#+4 L02&2^��s"H021L0)+.1L0Y �9\5ECK+.$'�K(:5GI<I'$<&L$'5G�,+.I/1#+4 L02&l #",+4 A(e+4�B1L0:?%$<&8 #1L$'N%QH L0b #"H002I'02(p #1L5.�%&j+4CK5G�HJG&= Y #"H0 « I<0:6.02I'&b5.@W+KJG$Z6G0:��¥ « � §8­¢¯T°BJG1L5GQH-� #"%+4 jCwQ%&= bNg0k(25.�%&L$'?%021=02?�@F5.1 #"H0O­�²b(2Q/ =S�54�� L5�Ng0O?H$U+.JG�H5G&L0:?z^h�s"%0OÓ,1=&= 3-%1=5/(:02&L&h$'&j(:5GI<I'$<&L$'5G�,+.I'I'Pd$<�%?%QH(202?!nfQ,+4?%1LQ%-R5GI'0 #1L+.�%&=$' #$'5G�%&zNR0: �9A020:�wCD02CONR021=&t5.@X #"%0a¥ « � §8­¢¯T°bJG1=5GQ%-zr4+.�%?h #"H0�&L02(:5G�%?q$<&t?%$'-g5GI'0i L1#+.�%&=$' L$<5G�%&@F1L5.CÖ+.?VéL+4(202�> bIz&= #+4 #0:&2^úth s�|-G h E3@R>\QxMqGNs�m;S IAMNIA@Rm;S�z�IPMNpHIKmuMNpHQx@RmHQᥠ« � §®­z¯T°KÀ%GN@ h EÞ\5G�%&=$<?%0:1j+M&L0p j54@ « I<0p6G02I'&s9Y$' #"H$<�|+DJ.$'6G0:�c-,+41L02�> b+.�%?d°,^A�s"%0k(21=$' L$<(e+4Iz?%02�H&L$' �P�+4 s9Y"H$<(*" #"H0wåVS[I<0p6G02I'&m9Y$<I'IzNg0h1=02I<+4 #$Z6G02IZPB&8 *+4 L$<&= L$<(2+.I<IZPK-R5G-%QHIU+4 L02?�9Y$<I'I¢NR0kJ.$'6G0:��N>P

ô ö�û aoüDü ûXíT ÿOäþý ú ½ ÕMä3ä ÿ öJû aoü�ü ûúk�aú § ¥*_ öJû aíü�ü û § ½ ½ ¥�x%^Á}.�>§9Y"%0:1L03ä ý ú ½ ?%02�H5. #0:&� L"%0Y+e6G0:1#+.JG0s5�6.021 « 5.@_ #"H0sI<0:6.02I%1=02(:$<-%1=5/(2+.IH1L+.?%$U+V #$'6.0\I'$'@F0: L$<CD0\+4�%?ä3äþÿ ö�û aíü�ü ûúk�aú § ½ ½ ?%0:�%5. #0:&b L"%031#+4 L0k(:5/0/¡D(2$<0:�> j&=Q%CKCD02?|5�6G0:1aÓ,�%+.It&= #+4 #0:& « _z+.�H?c+e6G0:1#+.J.02?5�6G0:1w$<�H$' #$<+.I�&= *+V #02& « ^dòb5. L0D L",+4 k #"%$'&k$'&h@F5G1h #1#+4�%&L$Z #$<5.�%&hNg0p �9\0:02�Òå�-g5G-HQ%IU+4 L$<5G�H&h5.@m #"H0&#+4CK0D¥ « � §8­¢¯T°dJG1L5.Q%-z^�[ W9\+.&�-R5G&L&=$<N%I'0A #5h0p64+.I<Q,+V #0\ L"%0m$<5.�O(25GI'I<$'&L$<5.�,+.I>1L0:?%$<&8 #1=$<N%QH L$<5G�q1#+4 L0mQ%&=$<�%J3+)(:5/?%0\N,+.&L0:?M5G� #"H0a L"%025G18PMJG$'6.02�K$<�D£jI<?H021�0: \+.IÅ^,¥8y ¨ v ç §p^i�s"%0j(:5/?%0Y9\+.&l #+.].0:�[email protected]; #"%0a£)¤Y£ { ?,+4 #+.N,+.&=05.@�(25X?%02&j+4 jål ��õ+4�%?�9\+.&j5.1L$<J.$<�,+.I'I'PD?%0p6G02I'5G-R02?cN>P { Q%CDCD021L&:^m�s"%0q(25X?%0k9\+.&b5G1L$'JG$<�,+4I<I'P?%0:&L$<J.�%02?� #5!CD5X?%02IA-%1L5. L5G��(25.I<I<$'&L$'5G�,+.I�1L+4 #0:&w@F5G1E+.�>P�JG$'6.02�� *+.1=JG0: 2^Ú�[ O9m+4&D-R5G&=&L$<NHI<0K L5&L(2+.I<0a #"%0b-%1L54 #5G�B(:5GI<I'$<&=$<5G�,+4IH(21=5G&L&\&L02(p #$'5G�%&\ L5O+.-%-HI'PM #5w+.�>Pd(:5GI<I'$<?%$'�%J3$<5G�z^i�s"%$<&A$<&A&L"%5�9Y�N%1=$<0:ßHP�Ng0:I<5�93^

Page 137: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7}4�

Å GN@`S S�S QTBwMNIA@Rm�SkBls)>KI\m[À¹o�@%GÜp-Q's'OWÁÛIA@Rm;S�s"%0D-%1L54 #5G�u(25GI'I<$'&L$<5.�,+.I�(:1L5G&=&q&=02(p #$<5.�%&q&=(e+.I'0M$<��+|&8 #1#+4$<JG"> [email protected]=9\+.1L?uCd+4�%�%021h L5| L"%0K(:1L5G&=&&L0:(: L$<5G�%&D@F5G1D$<5G�â(25GI'I<$<&=$<5G�H&2^ñ�s"%0c$<5G�â(25GI'I<$<&=$<5G�%+.I\1L+4 #04r ÿ%¥º_ ¤ §prs$<&M@F5GQ%�H?â@F1L5.C L"%0c(:1L5G&=&&L0:(: L$<5G� � ¤ ¥��H§¿^�£j&=&LQ%CD$<�%JÒ+n�c+V X9A02I'I<$U+4�Ý6.02I'5/(:$' �Pñ?H$<&= L1L$'N%QH #$'5G�ñ@F5G1d L"%0Ú(25GI'I<$'?%$<�%J�$<5G�H&ÿH¥*_ ¤ §m$<&sJG$Z6G0:�|N>P

ÿ%¥º? ¤ § ÿ � ³± x ª�� º ¤} ª .Ç_ ¤�� ´tH[³ � ³ 4�� ±±� ���� � � ¤ ¥��H§����ÿ � ³� Q è ª�� º ¤} ª .Ç_ ¤ � ´tH[³ y} º ¤ � ³ º F T±¤ � ¤ ¥��H§�� ® ¥�x%^Á}4xf§

�s"%0h$'5G��(25GI'I<$'&L$<5.�,+.I,(21=5G&L&s&=02(p #$<5.�%&Y+.1L0h0:nfQ,+.I¢ L5 ° ¤ ³ ¥F L"%0h0:�g02(p #$'6.0h(*",+.1=JG0h5.@¡ #"%0h(:5GI<I'$<?%0:1*§ #$'CK0:&) #"%0D-%1=5. #5G�u(25.I<I<$'&L$'5G�,+.I�(21=5G&L&k&L0:(: #$'5G�u0:64+.I<Q%+4 #0:?�+4 3 L"%0K&L+.CD0M6.02I'5/(:$' �PG^D�s",+4 k$'&k$Z@® ¬ $'&m #"%03-%1=5. #5G�|02�H021LJ4P�+.�%? ® ¤ $'&m #"%03$<5.�|0:�%021=J.PGr% L"%02�® ¤ ÿ y} º ¤ � ³

ÿ y} º ¤ ® ¬²³ º ¬ÿ º ¤º ¬ ® ¬ ¥�x%^Á}GvG§

�s"fQ%&� ¤ ¥ ® ¤ §5U ° ³¤ � ¬ ¥ ® ¬ § ¥�x%^Á} ç §

�s"%0:1L0:@F5.1L0ÿH¥*_ ¤ § ÿ � ³� Q è ªº T±¤ � y} ª .�_ ¤�� ´tH[³ ® ¤ 4�� � �� � ° ³¤ � ¬ ¥ ® ¬ §�� ® ¤

ÿ � ³� Q è ªº T±¤ � y} ª .�_ ¤ � ´tH[³ º ¤º ¬ ® ¬ 4 � ø � ø �� � ° ³¤ � ¬ ¥ ® ¬ § º ¤º ¬ � ® ¬ÿ � ³� Q è ªº T±¬��� y

} ª .�� 8 ø8 � _ ¤� "!#´tH[³ ® ¬ 4 � ø ¥ ø � � � § ° ³¤ � ¬ ¥ ® ¬ §�� ® ¬

ÿ ° ³¤ ÿ ¬ ¥*_ ¬ § ¥�x%^Á} � §

Page 138: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7}Vx

9Y"%0:1L0_ ¬ ÿ º ¬º ¤ _ ¤ ¥�x%^Á}.è>§

$<&O L"%0|&L(e+4I<02?Ò-%1=5. #5G�� L02CD-g0:1#+4 LQ%1L04^Ò�s"fQ%&K$<�á5G1=?%021E #5Ú1=Q%�Ò L"%0|(25X?%0�@F5G1M+.�>Pá-g5.&L&L$'N%I<0(25.I<I<$'?%021:r_5G�%0w"%+.&b L5B0:64+.I'Q,+4 #03 #"H0O-%1L54 #5G�c L02CD-g0:1#+4 LQ%1L0q+4 b #"%$'&j&L(2+.I<0:?c #0:CK-R021L+4 #QH1L0O+4�%?COQ%IZ #$'-%I'PM #"H0k(:1L5G&=&b&=02(p #$<5.�|N>P ° ³¤ ^�s"%0BÞlS�I<$'].0D&L0:nXQH02�%(:0B9m+.&qQ%&L0:?æ #5�$<I<I'Q%&= L1#+4 L0O #"%0KNg0:",+e6X$<5GQ%1k5.@Yô ö�û aoüDü ûXíT @F5G13 #"%0K02I<0:(: L1L$<(nfQ,+.?%1=Q%-R5GI<0B L1#+.�%&=$' L$<5G�%&:^��s"H0|&=Q%-R021L&8 #1=Q%(: LQ%1L0|(25X?%0u¥K �$'&L&L�H021M0: D+.I�^'rsy ¨�� xf§k9\+.&M1LQH�á L5JG0:�%021L+4 #0w L"%0w@FQ%�H?,+.CD02�> *+.Ii+4 L5GCK$'(3?,+4 *+K1L02nfQ%$'1L0:?` #5B0p6.+4I<Q,+4 L0k L"%02&=0E1#+4 L02&:^)�s"%0O&8P/&8 #0:C9\+.&b1=02&8 #1L$'(: L02?� #5E �9\5K+.I'I<5�9\0:?�(25G�HÓ%JGQ%1#+V #$<5.�%&2r%�%+.CK0:I'P

$ y�2 ³ }62 ³ }DÉ ³$ y�2 ³ }62 ³ }DÉc·.­%�

9Y"%0:1L0)­"�M1=02-%1=02&=02�> #&m L"%0+XsPX?%Ng0:1LJE0:I<02(p #1=5G�d #"%+4 \(:5GQ%I'?BNg0b-%IU+4(202?d$<�B+k1#+.�HJG0j5.@z�/S�&="%02I'I<&2^�s"%0k-g5.&L&L$'N%I<0håVS[64+.I<QH02&s #",+V s #"%03&=PX&8 #02C×(2+.�� *+.]40w+.1=0h&L"%5�9Y��$<�� *+.NHI<0hx%^ãx%^Á}/^

Þ\5.�HÓ,JGQ%1L+4 #$'5G� �i+.1L0:�> é -g5G&=&L$'N%I<0kåy52 ³ }È2 ³ }DÉ ³ ³ Ê ²�H[³ ��ÕG} }Xr'yy52 ³ }È2 ³ }DÉ ³ ³ Ê ²�H[³ yNÕG} y.rÁ¦y52 ³ }È2 ³ }DÉ ³ ³�Ê ´tH[³ ��ÕG} �Hr~}Xr'yGr�¦y52 ³ }È2 ³ }DÉ ³ ³ Ê ´tH[³ yNÕG} }Xr'y

y�2 ³ }È2 ³ }DÉ|·>­%� ³ Ê ²�H[³ vlÕG} �/r~}y�2 ³ }È2 ³ }DÉ|·>­%� ³ Ê ²�H[³ ��ÕG} }Xr'yy�2 ³ }È2 ³ }DÉ|·>­%� ³ Ê ´tH[³ ��ÕG} �Hr~}Xr'yGr�¦y�2 ³ }È2 ³ }DÉ|·>­%� ³ Ê ´tH[³ vlÕG} x%rÁ�/r~}/rZy

¶,5G1m #"H0k-HQ%1L-R5G&L0:&b54@�0p6.+4I<Q,+4 L$<�%JO L"%0k$<5G��(:5GI<I'$<&L$'5G�,+.IR1#+V #02&:r,$' s9\+.&a?%02(:$<?%0:?| L5KI'5/5G]K+4 Y #"H0y�2 ³ }È2 ³ }�É�¥ ³ Ê ´tH[³p§pr2°Eÿ ¯ ¯ yNÕG}b&=0: W54@,I<0p6G02I'&� L5j&=020m+V i9Y"%$'(*"O-R5G$<�> � #"%0l$<5G�q(25GI'I<$'&L$<5.�%&zN,+.I<+.�%(:0 #"H0B&L-R5G�> *+.�H025GQ%&E0:CK$'&L&=$<5G�æ1L+4 #04^cÜÝ$' L"� #"H0�(25G�HÓ%JGQ%1#+V #$<5.��(*"%5G&=02�Ò$' q$<&O-R5G&=&L$<NHI<0K L5`JG0p 02I'02(p #1L$'(hnfQ,+.?%1=Q%-g5.I<0) #1L+.�%&=$' #$'5G�%&mNR0: �9A020:�c #"H0 « I'0:6.02I<&Y5.@� #"H$<&sJG1=5GQ%-|+.�%?|?%$<-R5GI'0h+.I<I'5�9\0:?&L-R5G�> *+4�%025GQH&A0:CK$'&L&L$'5G�E #5q+.�>PE5.@g #"%0YJG1=5GQ%�%?D(25G�/Ó,JGQ%1L+4 #$'5G�E #021=CK&l+.I'I<5�9A02?E@F5G1�9Y$' L"%$<�E #"H0

Page 139: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7}.v

­ « ó�yb&=02I<0:(: L$<5G�D1LQ%I'0q¥�+.&� #"%0:1L0Y9Y$<I'I,+.I'9\+ePENR0j+3(*",+.�%J.0b5.@¢-,+.1=$' �PENR0: �9A020:�B #"%0�XYPX?%NR021=J02I'02(p #1L5.��(:5G�HÓ,JGQH1#+4 L$<5G�u+.�%?Ú #"H0DJ.1L5GQ%�H?_§p^D¶,5G1) #"H0K"%$'JG"%IZPc0¿ H(:$' L02?�0:I<0:(: #1=5G�%&:r� L"%021=0M9Y$'I<I+.I'&L5D0p /$<&8 )CK+.�>P�54 #"%0:1b&=-g5G�> #+.�%025.Q%&)02CD$<&=&L$'5G��?H02(e+eP�1L5GQ/ #02&a #",+V )+.1=0q�H5. h+.(:(25GQH�f [email protected]$' L"%$<�� #"%0E&=$<CD-%I<0w+4 #5GCD$<(q&= L1LQ%(p #Q%1=0E(*"%5G&L0:�[email protected] #"%0E&=Q%-R021L&8 #1=Q%(: LQ%1L0D1LQ%�H&2^w�s"fQ%&3$Z j9\+.&?%0:(2$<?H02?c L5B0:64+.I'Q,+4 #0k">PX?%1L5GJ.02�%$'(w£aSÅ64+.I<Q%0:&b@F5G1a0e+4(*"æy�2 ³ }È2 ³ }DÉ�·s­%�B&L0p h5.@�I'0:6.02I<&:^b���� #"%$'&9\+�P|+ECD5G1L03+4(2(2QH1#+4 L0kI'$'@F0p #$<CD0j(:5GQ%I'?|NR03(e+.I'(2Q%I<+4 #0:?B@F5G1sQ%&=0q$'�|0:nfQ,+4 #$'5G��¥�x%^Á}.�>§¿^�s"%0�$<5G��(25GI'I<$'&L$<5.�Ô(25X?%0��%0:02?%&| #5â]X�H579 L"%0�0:�%021=J.PÝ64+.I<Q%0:&2r302I'02(p #1L$'(ÚnXQ%+.?%1LQH-g5GI'0�I'$<�%0&= L1L0:�%J. #"H&A+4�%?E #"%0Y&= #+4 #$'&= L$<(e+4IX9A02$'JG"> #&�@F5G1�0e+4(*"KI'0:6G0:I%$<�>6.5GI'6.02?D$<�O #"H0m #1#+4�%&L$Z #$<5.�E+.&�9\0:I<I,+4& #"H0q0p�¢0:(: #$Z6G0q(*",+.1LJ.0w5.@i #"%0k-%1L57é=02(: L$<I'0w+.�H?c$<5.�� L02CD-g0:1#+4 LQ%1L0k5.@W L"%0k-%IU+.&=Cd+/^\�s"%0q&LQ%-R021®S&= L1LQ%(p #Q%1=0)(25X?%0a9m+.&\1LQ%�d L5OJG0p \ #"H0j02�H021LJ4PdI'0:6G0:IR64+.I'Q%02&\+.�%?d L"%0j0:I<0:(: #1=$<(bnXQ%+.?%1LQH-g5GI'0bI'$<�%0&= L1L0:�%J. #"H&2^D�s"%$<&h9\+.&q?H5G�%0E@F5G13+�1#+.�HJG0M5.@\�XS�&L"H02I<I'&2r ° ²pÙã&q+4�%?�$'5G�` #0:CK-R021L+4 #QH1L02&:^K£bI<-%"%+-,+.18 #$'(2I<0:&m9\0:1L0qQ%&L0:?!+.&s #"%0k(25.I<I<$'?%021=&\ #5D$<I'I<Q%&8 #1L+4 #0j L"%0k1L02&=Q%I' L&b$'�� #"%0k(25G�> L0p X )5.@�+E@FQ%&L$'5G� #5.]4+4Cd+.]_^��5c0:64+.I'Q,+4 L0K0:nXQ%+4 #$'5G�Ô¥Fx%^~}4�>§3$Z 39m+.&q #"%0:�æ�%02(:02&=&#+.18P� L5�@F5G1=C #"%0D6.+41L$<5.Q%&w+e6.021#+4JG02&:^�äý ú ½ 9\+.&a(e+.I'(2Q%I<+4 #0:?�N>P�Ó,�%?%$'�%JE #"%0k+e6G0:1#+.JG035�6G0:1)åE5.@� #"%0h L5. *+.I¢&L-R5G�> *+.�%0:5GQ%&a02CD$<&=&L$<5.�1#+V #02&K@F1L5GC 0e+4(*"�I'0:6.02IÅ^ { $<CD$<I<+.1LIZPñä3ä ÿ ö�û aíü�ü ûúk�aú § ¥*_ ö�û aoüDü û § ½ ½ 9\+.&B@F5GQ%�%?àNfPâ *+.]f$'�%J� #"H0+e6G0:1#+.JG0|5�6G0:1 « 5.@b #"%0� L5. *+.I\$<5G�á(25.I<I<$'&L$'5G�,+.I�1#+V #0�¥�0¿ H(:$' #+4 #$'5G�Ò+.�%?á?%0pS[0p /(2$Z *+4 L$<5G�,§q@F1L5GC0e+4(*"æåB #5�+.I'Ii L"%0E5. #"H021h-R5G&L&=$<N%I'0Då�I'0:6.02I<&h$'�! #"%0bé#ÿ)IZS#y�\G}M&L0: k5.@lI'0:6G0:I<&:^O�s"%0M1L0:&LQ%IZ #&3+.1=0&L"H579Y��$<��Ó,JGQH1L02&Yx%^Á}.�wSlx%^Á} ç ^�[ m(2+.�BNR0j&=0202�� L",+4 l #"%0h+e6.021#+4JG0j£YS[64+.I<QH02&\$'�%(21=0e+.&=0b9Y$Z #" ° ²A+.�H?d@�+.I<I'&�5.��9Y$' #"d$<�%(:1L0e+4&L$<�HJ­�^D�s"%$'&k$'&3+.&30p /-R02(: L02?æ?HQ%0E #5� #"%0M0p /(:$' #0:?�0:I<02(p #1=5G��NR02$'�%J�"%02I'?�(:I<5G&=021) L5| L"%0M([email protected]"%$'JG" ° ²prR #"fQ%&h+.I'I<5�9Y$<�HJD@F5G1a&= L1L5G�HJG021)?H02(e+eP| #5d L"%0OI'579A021b&= #+4 #0:&2^ { $'CK$'IU+.1=I'PD@F5G1j"H$<JG"%0:1j�zr #"H0k0:I<0:(: #1=5G�%&a+.1=03@FQH1= #"[email protected]� #"%03(:5G1L0k+.�%?�&=5KI'02&L&YI'$<].0:I'PK #5M?%02(2+ePG^òb54 #0E #",+V h #"%0M(21=$' L$<(e+4I�?%0:�%&L$Z �P!&L(2+.I<$'�%J�$<&3+.&30¿ H-R02(p #0:?�9Y$' L"uô öJû aoü�ü ûX T $'�%(21=0e+.&=$<�%JB9Y$Z #" ° ²+.�%?�?%0:(21=0e+.&=$<�%JK+.&Y5G�%0h-%1=5GJG1=02&L&=02&aQ%-� #"%0h64+.1L$'5GQ%&m�XS�&L"H02I<I'&2^��[ Y(2+.�c+4I<&L5ENR03&L0:02�� #",+4 [email protected]+OI<$'JG"> \0:I<0:CK0:�f m5.@ ° ²'&;yh+4�%?�­(&�y�v/rX L"%0h(21=$' L$<(e+4IR?H02�%&=$' �[email protected]&= #+4 #$'&= #$'(e+.IR1L0:?%$<&8 #1=$<N%QH L$<5G�$<��åE5/(:(2Q%1=&j+V a+M1=0e+.&=5G�,+.NHI<0h64+.I<Q%0D¥�ô�X T &;y Í y�¦ ²�± -�º F ´ §p^���|5G1L?%0:1s #5M0:64+.I'Q,+4 #0h L"%03(21L$Z #$'(e+.Ig�/S�&="%02I'I¡+.Ng5�6.039Y"H$<(*"� #"%0kI<0p6G02I'&Y+.1L03&8 *+4 L$<&= L$<(2+.I<IZPK-R5G-%QXSIU+V #02?|9\0q-%I<54 qäëý ú ½ ÕH¥8ä3äëÿ öJû aíü�ü ûúk�aú § ¥º_ öJû aíü�ü û § ½ ½ ô ¤ mÅü §prR&L0:0qÓ,JGQH1L0wxH^~} � @F5G1Y #"%0k1L0:&LQ%IZ #&:^�s"%0h(:1L$' L$<(2+.I_�/S�&="%02I'Ig$'&A #"H0)5G�%0h+V \9Y"%$'(*"B #"%0h(:Q%1=6.02&s(21=5G&L&hy.^W�s"%0h-%I'5. \&="%5�9Y&s #"%0)1=02&=Q%I' L&@F5G1q+�1L+.�%JG0K5.@s0:�g02(p #$Z6G0d�fQ%(:I<02+.1w(*",+41LJG0:&2^c�s"H0d1L0:&LQ%IZ #&w+.1L0d+.&q5.�%0K9A5GQ%I'?æ0p /-R02(: 2r�9Y$' L"

Page 140: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7} ç

Hydrogenic <A> values

46

810

1214

16n 0

510

1520

2530

35

z1

1e+07

1e+08

1e+09

1e+10

1e+11

1e+12

1e+13

1e+14

<A>

¶�$'JGQ%1L0kx%^Á}.�H· Ü>&(�*½¿ÀÁÇ7º=¡ÄTÃpÍ¡¼Å¹7ºlÍ[½¿ÈVÉpº�ÃpÄ%ÌV¹�#�Â�¾ÁÊ=½¿À/ÌV½¿Í[½¿Ëmº®¼Åº8Í[Âz¼�¹4½*¼�¼Å¹7ºA½pÇ�¼�Ã2ÂF¼ÅÍ�Ç4Ê�¼�ÇVÍ�ºlÊ8Ã�»�º�Îz½p¡ÍÅÇ7È���O½¿ÈV»gy   »7º8Ì.º8ÈV»�º=ÈVÊ®º:Ï

z1=2z1=7

z1=13

1001000

10000100000

1e+061e+07

1e+08Tion / K 4

68

1012

1416

n

1e-451e-401e-351e-301e-251e-201e-151e-101e-05

1

<<q>>

¶�$'JGQ%1=0hx%^~}Vx%·*) Ã:ÈgÚ2ÇV½:»�Í�ÇVÌ4Ã:À�º�Ê®Ã:À�ÀÁ¾~Â�¾ÁÃpÈV½pÀGÍŽ¿¼�º=Ât½:Ât½AÄTÇ7ÈVÊ�¼Å¾�Ã:È)ÿÄX¼Åº8ËmÌ.º8Í[½*¼�ÇVÍ�º�½pÈV»aÈ�&UÂ�¹7º8ÀÁÀ<Ï

Page 141: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7} �Tion=1E4Tion=1E5Tion=1E6

46

810

1214

16n 0

510

1520

2530

35

z1

1e-241e-221e-201e-181e-161e-141e-121e-101e-081e-06

0.0001

<<q>>

¶�$'JGQ%1=0hx%^~}.v/·+) ÃpÈ<Ú2ÇV½:»�Í�ÇVÌ4Ã:À�º�Ê®Ã:À�ÀÁ¾~Â�¾ÁÃpÈV½pÀGÍŽ¿¼�º=Âz½:Ât½\ÄTÇ7ÈVÊ�¼Å¾�Ã:È)ÿÄ/È�&UÂ�¹7º8ÀÁÀX½pÈV»My   Ï­,& ¨ @F5G1 ° ²lÿ }D+.�H?|­,&;y7vw@F5G1 ° ²�ÿ � ^j I\E3@R>AQxs)>K>A@}z�Q}|uMqGNs�mHSkIAMqI\@RmHS�i3Q'M�z�QTQTmÛs�|.- s�BlQTm`MÜ>�SvMks9MqQTS�s"%0:1L0�$<&q+c&=02(25.�%?áCK0:(*",+.�%$'&LC 5.@Y$<�H?%$<1=02(: « 1=02?%$'&= #1=$<N%Q/ #$<5.��N>P� L1#+.�H&=@F021k@F1L5GC ­¢¯Z° « L5­z¯zZ;y#°�_ « _ ^�òb54 #0B #"%+4 ENR02(e+4Q%&L0�5.@Y #"H0��%0e+.1E?H02JG0:�%021L+.(:P�$'�æ #"%0�1=02I<+4 #$Z6/$'&= L$<(D-%$<(p #Q%1=0B5.@­z¯ ýJI ²�H[³l+.�%?E­¢¯ � y ý F ²�H[³�rG$Z W$<&i5G�%IZPq L"%0m(25.I<I<$'&L$'5G�,+.IG1L+4 #0:&�Ng0p �9\0:02�M­z¯ ýJI ²�H[³ « +4�%?E­z¯ � y ý F ²�H[³ « _9Y"%$'(*"BCd+V L #0:12^W�s"H02&L03+41L0)+4I<I<5�9A02?K?%$<-R5GI<0s L1#+.�%&=$' L$<5G�%&lN%QH m+.J>+.$'�K$'�%?%Q%(:02?�N>PK-g5G&=$' L$'6G0a$<5.�$<CD-,+.(p e^W�s"H0q(:1L$' L$<(2+.I¢?%0:�%&L$Z �PB@F5G1s1=02?%$'&= #1=$<N%Q/ #$<5.��NfP� L"%$<&s-%+4 #">9\+�P�$'&

ô öJû aoü�ü ûX T ÿOäÕý ú ½ ÕOäkä ÿ öJû aíü�ü ûüeýÁúk�müeý0/ ² ú § ¥*_ öJû aoü�ü û § ½ ½ ¥�x%^Á} ¨ §�s"%0m(:1L$Z #$<(2+.I>?%0:�%&L$Z �Pq@F5G1 « 1=02?%$'&= #1=$<N%Q/ #$<5.�q$'&i #"H0\&LCK+.I'I<021¡5.@H #"%0sôGXíTz64+.I<QH02&i@F1=5GCõ02nfQ,+4 L$<5G�H&¥�xH^~}.�G§�+.�H?�¥�xH^~} ¨ §p^�£bI' L021L�%+4 #$Z6G02IZPGr7@F5G1�&L-R02(2$ZÓ,02?Oô öJû aoü�ü û $<�k #"%0A-%IU+4&LCK+HrV9A0\&=020:]k L"%0\I'579A02&8 64+.I<QH0h5.@�­â¥�ÿ�­�²L§[email protected]¯ ½ yh@F5G1m9Y"%$'(*"| L"%0302nfQ,+.I'$' #$'02&Y+.1=03+.(2(:02-H #+.N%IZP�&#+4 L$<&8Ó,02?z^��1#+.�%&=$' L$<5G�%&b@F1L5GC ­z¯T° « #5�­¢¯bZ�y*°l_ « _¡I<0p6G02I'&)9\0:1L0M0:64+.I'Q,+4 L02?`@F5G1j L"%0MÞlS[I<$<]40O&L0:nfQ%02�%(:0D L5$<I'I<Q%&8 #1L+4 #0d L"%0|(21L$Z #$'(e+.Im?H02�%&=$' �Pâ+.�%?Ò�/S�&="%02I'IsNg0:",+e6X$<5GQ%1:^ñ¶,5G1E #"H0�1L+4 #0:&2rA L"%0|&#+.CD0�(:5/?H0

Page 142: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7}4è

Tion=1.E4 KTion=1.E5 KTion=1.E6 K

46

810

1214

16n 0

510

1520

2530

35

z1

1e+10

1e+15

1e+20

1e+25

1e+30

1e+35

1e+40

ncrit / cm^-3

¶�$'JGQ%1L0cx%^~} ç ·ò$gÍžã¼Å¾ÁÊ=½¿À�»�º8ÈVÂ�¾�¼*#B½¿¼bÎt¹7¾~Ê[¹�¼Å¹7ºq¾ÁÃpÈKÚ:Ç4½p»�ÍÅÇ7Ì.ÃpÀÁºkÊ®Ã:À�ÀÁ¾ÁÂ�¾ÁÃpÈVÂaÆV½¿À~½¿ÈVÊ8º3¼Å¹7ºqÂ�Ì4Ã:È2¼Å½¿ÈVº8ÃpÇ4º8Ëm¾~Â�Â�¾ÁÃpÈjÍŽ¿¼�º=¢ÄTÍÅÃpËá¼�¹7ºE�owiÀC&o��ó��AÂ�º8¼tÿÄ%À�º|�pº8À~Â=Ï

1e-05

1

100000

1e+10

1e+15

1e+20

1e+25

4 6 8 10 12 14 16

<A

>/(

<<

q>

>N

ion

)1

n

Z1=2Z1=7

Z1=13Z1=31

¶�$'JGQ%1L0bx%^Á} � ·�$gÍ�¾�¼�¾~Ê8½pÀ4È�&UÂ�¹7º8ÀÁÀGÌ7ÀÁÿ¼tÂ�¹7Ã*Ît¾ÁÈ7Él¼�¹7ºWʮ͞ã¼Å¾ÁÊ=½¿ÀÈ�*½¿ÀÁÇ7ºW½*¼gÎt¹V¾ÁÊ[¹a¼�¹7ºW¾ÁÃpÈbÊ8ÃpÀÁÀ�¾~Â�¾�Ã:ÈVÂ_ÆV½pÀÁ½pÈVÊ®ºi¼�¹VºÂ�Ì4Ã:È:¼[½¿È7º=ÃpÇVÂiº8Ëm¾~Â�Â�¾ÁÃpÈkÍ[½*¼Åº=Â�ÄTÍ�Ã:Ëñ¼�¹7º:�owiÀC&o��ó��sÂ�º®¼�ÃpÄRÀ�º|�pº=ÀÁÂ=Ï�ÜæÊ®ÃpÀÁÀÁ¾Á»�º=Í�»7º8ÈVÂ�¾ã¼*#hÃpÄ}���ª�|�� º¡�¢t£G¤È¥A½pÈV»¼�º=ËmÌ4º=ÍŽ¿¼�Ç7ÍźWÿÄE�"������¡��Ð΢º8Íź�Ç4Â�ºL»>Ï

Page 143: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y7} ¨

+.&w L",+4 q@F5G1q L"%0B­z¯T°` #0:&= E(:5/?H0d9m+4&OQ%&L0:?z^Ú�s"%0B&=Q%-g0:1L&8 #1LQH(: #QH1L0�(25X?%0K9\+.&M+.JG+.$<��1LQ%�� L5JG0:�%021L+4 #0s #"H0b+V #5GCD$<(A?,+4 *+3�%0:(202&=&#+.18PE@F5G1W #"H0a(:5/?H0.^W£b�d+4 L5GCD$<(\&8 #1=Q%(: LQ%1L0a5.@iy�2 ³ }È2 ³ }DÉ�·>­z¯9\+.&wQ%&=02?zri #"%$'&k L$<CD0M L"%0ÜXsPX?%NR021LJc02I<0:(: L1L5G�u9m+.&w+.I<I'5�9\0:?u L5c5X(2(2QH-fP�02$Z #"%0:1k L"%0d-¢r�?�5.1@q&= *+V #02&|+.�H? CK+.].0! L1#+.�%&=$' L$<5G�%&dNg0p �9\0:02� L"%02C 5G1|+.I' L021L�%+4 #$Z6G02IZPà?H579Y�Ý #5� L"%0ÚJG1L5GQH�%?y�2 ³ }È2 ³ }�É ³ (:5G�HÓ,J.Q%1#+4 L$<5G�¢^�XY02&=Q%I' L&h9\0:1L0DJG02�%0:1#+4 L02?u@F5G1k+�1L+.�%JG0M5.@\�/S�&="%02I'I<&3+.�H?��fQ%(:I<0e+41(*",+.1=JG02&:^�s"%0��i0:�%JG0:I<I'P]\ { 0e+4 L5G�q(:5/?H0�1L02nfQ%$'1L0:&i0:�%021=J.PhI<0p6G0:IG6.+4I<Q%0:&2rV+.N%&=5G1L-H L$<5G�35G&=(2$'I<IU+V #5G1g&= #1=02�%J4 #"%&:r&= #+4 #$'&= L$<(e+4IG9\0:$<JG"> #&:rG+.I'I>5.@%9Y"%$'(*"q9A021=0\0p X L1#+.(p #02?q@F1L5GC L"%0A&=Q%-R021L&8 #1=Q%(: LQ%1L0m1=Q%�%&:^W�s"%0\(:5/?H0+.I'&L5k1L02nfQ%$'1L0:&m$<�%$Z #$<+.I%+.�H?dÓ,�,+.IR&= #+4 #0aI<$Z@F0: L$<CD02&2^i¤)Q%0a #5q L"%0j&=$<CD-%I<0Y+V #5GCD$<(Y&= L1LQ%(p #Q%1=0)Q%&=02?$<�q #"%0s&=Q%-g0:1L&8 #1LQH(: #QH1L0Y1LQ%�H&2rG #"H021L0m9A021=0sCd+4�fPq&L-R5G�> *+4�%025GQH&�?%02(2+�PO1L5GQH L02&�9Y"H$<(*"E9\0:1L0s�%54 +.(:(25GQ%�> L02?æ@F5G1:^���5!+.I'I<5�9�@F5G1k1L02+.I<$'&= #$'(MI<$'@F0p #$'CK0:&) #5`NR0d-%1=5/?HQ%(20:?zr�">P/?H1L5GJG0:�%$<(K£aSÅ6.+4I<Q%0:&9A021L0B+.J>+.$'��JG0:�%021L+4 #0:?�@F5G1k0e+.(*"�I'0:6.02Il5.@s$<�> L021L0:&= 2r�+.(2(:5GQ%�> #$'�%J�@F5.1q?%0:(e+ePu #5`+4I<I�-g5.&L&L$'N%I<0I<5�9A021D&= #+4 #0:&2^Ô���à L"%$<&E9\+�Pá L"%0�­"� ³ Ê ²�H[³ &L0: d5.@)I'0:6.02I<&M9A021L0c$<�>6G0:&= #$'J>+4 L02?zrm(:5G�%&=$<?%0:1L$<�HJ+.I'IW #"%0K-g5G&=&L$'N%I<0M?%$'-g5GI'0M(25GI'I<$<&=$<5G�%+.Ii L1#+.�%&=$' L$<5G�%&h L5c+.�H?�@F1L5GC L"%0K+.?VéL+.(:02�> OI�&= #+4 #0:&2r�+4�%?&L-R5G�> *+4�%025GQH&b?%0:(e+eP� #5D+.I'It+.I'I<5�9\0:?�I<5�9A021Y&= #+4 #0:&2^�s"%0Mä ý ú ½ +.�%?Úä3äþÿ ½ ½ 6.+4I<Q%0:&s+.1L0h&="%5�9Y��$'�BÓ,JGQ%1=02&sx%^Á}.èM+.�%?�xH^~} ¨ r,+.�%?�NR5. #"�&=(e+.I'0+.&k5G�%0M9\5.Q%I<?�0p /-g0:(: 2^|�s"H0d(21=$' L$<(e+4IW?%02�H&L$' �P�+4 k9Y"%$'(*"�&= #+4 #$'&= L$<(e+4I�1L02?H$<&= L1L$'N%QH #$'5G�Ú5/(:(2Q%1=&9\+.&Y-%I<5. = #0:?B@F5G1Y+E1#+.�%J.0h5.@�­�+4�%? ° ²\64+.I'Q%02&:r%&L0:03Ó%JGQ%1L0hxH^Á�G¦/^�£j&Y(e+.��NR0h&L0:02�zr, #"H021L03$'&s+&= L1#+.$'JG"> L@F5G189m+41L?�$<�H(21L02+.&L0j5.@¡ #"%0)(:1L$Z #$<(2+.IR?%02�%&=$' �PK9Y$' #" ° ²pr%+.�%?�+O?H02(21=0e+.&=0h9Y$' #"�­i^��s"%021=0$<&i+.I'&L5)+a&LI'$<JG"> �$<�%(:1L02+.&L0l9Y$' #"q(25GI'I<$'?%021t #0:CK-R021L+4 #QH1L0.rV9Y$' L"q #"H0\$<5.�q(:5GI<I'$<&L$'5G�%&tNR02(:5GCK$'�%JjI'02&=&0:�g02(p #$Z6G0K+4 3"%$<J."%021) L02CD-g0:1#+4 LQ%1L0:&2^E�s"%0D(21=$' #$'(e+.Ii�/S�&="%02I'I�+4 h9Y"%$'(*"�1=02?%$'&= L1L$<NHQH #$'5G�!5/(:(2Q%1=&9\+.&\+4I<&L53-%I'5. L L02?E@F5G1l+31#+.�%J.0a54@ ° ²pÙã&2rX&L0:0aÓ,J.Q%1L0Yx%^��HyG^i£j&lNg0p@F5G1L04rf L"%0a1L02&=Q%I' L&l+.1L0Y$<�> LQ%$' L$'6G0+.�%?�6.021L$Z@ PB #"H0k-H1L5G-R5G&L0:?�N%Q%�%?%I'$<�%JMCD0: L"%5/?B9Y$Z #"�&= #+4 #$'&= L$<(e+4I¢1L0:?%$<&8 #1=$<N%QH L$<5G��5X(2(:Q%1L1=$<�%JM+V ­,&�vw@F5G1 ° ²lÿ }D+.�H?|­,&;y7}w@F5G1 ° ²�ÿ � ^�s"fQ%&M5G�%0B(2+.��&L0:0� #",+4 O$' O$'&w-R5G&L&=$<N%I'0B$<��-%1L$'�%(2$'-,+.I� #5!0:64+.I'Q,+4 L0d #"%0d­�²q(2QH 8S�5.�â@F5G1O+4�fPCD02?%$'Q%CN\V"%0e+e6fPB0:I<02CD02�> 2^

Page 144: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y��.¦

<A>

5 6 7 8 9 10 11 12 13 14n 0

510

1520

2530

35

Z1

1e+07

1e+08

1e+09

1e+10

1e+11

1e+12

1e+13

1e+14

<A>

¶�$'JGQ%1L0hxH^~}.è/·3254}î76�º��*½¿ÀÁÇV½*¼Åº=»bÄTÃpÍz¼Å¹7º�$'&UÀ�¾98pºWÂl#�Â�¼�º=ËáÄTÃpÍz¼Å¹7º ��Ý����=Ý��|�SÞ;:D�=<sÂ�º8¼¡Ã¿Ä%À�º|�pº8À~Â=Ï

<q>

5 6 7 8 9 10 11 12 13 14n 0

510

1520

2530

35

Z1

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

<q>

¶�$'JGQ%1L03x%^Á} ¨ ·>2?2A@CB9D �FEGEGHs�uCîJI"s�u0K   îML þß� B0D ��E�EGH �l�N6?6�º|�#½pÀ�Ç4½*¼�ºL»jÄTÃpÍ�¼�¹7ºä$'&UÀÁ¾08:ºlÂl#�Â�¼�º=ËÒÄTÃ:Ít¼�¹7ºM��Ý��|�=Ý����SÞO:��=<Â�º®¼¡Ã¿ÄHÀ�º|�pº=ÀÁ¢ÄTÃ:Í¡½\Ê8ÃpÀÁÀ�¾~»�º=Íz¼�º8ËmÌ.º8Í[½*¼ÅÇ7Í�ºWÃpÄW�"��|�D¡��MÏ

Page 145: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y��/y

Te=1.d4 KTe=1.d5 KTe=1.d6 K

5 6 7 8 9 10 11 12 13 14n 0

510

1520

2530

35

Z1

1e+08

1e+10

1e+12

1e+14

1e+16

1e+18

1e+20

1e+22

Ncrit / cm^-3

¶�$'JGQ%1L0dx%^Á�.¦H·P2Q4:î,6 åqþF2?2Q@ B0D ��E�EGHs�uJîMI"s�u9K   î L þÛ� B0D ��E�EGH �>6?6[��º|�#½pÀ�Ç4½*¼�ºL»EÄTÃ:Íl¼Å¹7ºG$'&<ÀÁ¾08:ºbÂ�#�ÂF¼Åº8ËÐÄTÃpÍ\¼�¹Vº��Ý����DÝ����SÞ;:D�=<YÂ�º8¼tÿÄ%À�º|�pº8À~Â=Ï

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

5 6 7 8 9 10 11 12 13 14

<A

>/(

<<

q>

>N

ion

)R

n

Critical n-shell plot for C-like system

Z1=2Z1=4Z1=5Z1=7

Z1=10

¶�$'JGQ%1L0Kx%^��HyG·A2S4:îT6ÕåNþ�2?2U@ B0D ��E�EGHîJI�î L þÛ� B0D ��E�EGH �V6?6SW�X9Y�s���º|�*½¿ÀÁÇV½*¼Åº=»OÄTÃ:Í�¼�¹Vº<$'&UÀ�¾98pºbÂl#�Â�¼�º8Ë ÄTÃpÍA¼�¹Vº��Ý����DÝ����SÞ�:3�=<)Â�º8¼�ÃpÄ_À�º|�pº8À~Â=ÏPÜæÊ8ÃpÀÁÀ�¾~»�º8ÍW»�º=ÈVÂ�¾�¼*#)ÿÄ[���ª�|�� º¡|¢S£G¤È¥A½pÈV»h¼Åº8ËmÌ.º8Í[½*¼�ÇVÍ�ºAÿÄ[�����|��¡��õ΢º8ÍźÇVÂ�º=»bÄTÃpÍ¡½pÀ�ÀG¼Å¹7º�Íź=Â�Ç7Àã¼[ÂtÂ�¹VÃ*ÎtÈfÏ

Page 146: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

y��G}

�`�[Z \ �D�Â�b� �Â�W�2�D�Â����� #"%$'&O(*",+.-/ #021ECwQ%(*"á5.@Y #"%0�JG1=5GQ%�%?æ9A5G1=]�@F5G1w L"%0��%0p9�(:5GI<I'$<&L$'5G�,+.ITS�1L+.?%$U+V #$T6G0E(25X?%[email protected]+.-%-HI<$<(2+4 #$'5G�h #5a"%0e+e6fPq02I<0:CK0:�> #&�"%+.&iNR020:�OIU+4$<?3?%5�9Y�z^W�s"%0A�%0:9áNHQ%�%?%I'$<�%Ja&L(*"%0:CK0\",+.&�Ng0:02�?%0:&L(21=$<NR02?K+.�%?K+h(25X?%0s?%0:6.02I'5G-g0:?K@F5G1i #"%0YN%Q%�H?%I<0:?/S��%I_+4�%?MN%Q%�%?%I'02?/S[�KNHI<5X(*]w-%1=$<CD$' #$Z6G0:&2^�£ #0:&= \CK5X?%Q%I'0Y L5w0p6.+4I<Q,+4 L0s #"%0bN%Q%�%?%I'02?/S[�%Iïé�N%I<5X(*]E-%1=$<CD$' #$Z6G0:&l"%+.&ANR020:�B?%0:CK5G�H&= #1L+4 #0:?zr%+4�%? #"H0KQ%&=0:@FQ%I'�%02&=&w5.@A #"%$'&k+.-%-%1=5>+.(*"�$'�! #"%0D?%0:6.02I'5G-%CD02�> w54@\@FQH LQ%1L0DCK5X?%Q%I'02&3$<&h(:I<02+.12^K�s"%0�%0:(202&=&#+.18Pu�3+.Q%�> j@�+.(p #5G1=&b@F5G1a #"%0w�%0:9(25X?%0O"%+�6.0ENg0:02�!?%0p6G0:I<5G-R02?Ú+.�H?Ú+.�`$<�>6.02&= L$<J>+V #$<5.�$<�> L5� L"%0D+.QH L5GCd+V #$<5.�`5.@A?%0: L02(: L$<5G�Ú5.@l #"H0DNHQ%�%?%I'$<�%J�(2Q/ =S�54�¢&h-R0218@F5G1LCD02?ÚQ%&=$<�%J|&= #+.�%?,+.1=?$<5.�|(:5GI<I'$<&=$<5G�,+4IR1L+4 #0h0¿ /-%1L0:&L&L$'5G�%&:^�s"%0E #"H02&L$'&h9\5G1=]`-%1=02&L0:�> #02?�"%021=0D$'&3+�&= #0:-%-%$'�%J�&= #5.�%0M$<�> #5� #"H0O@FQ%I<IW?H0:6G0:I<5G-HCK0:�f k5.@l #"H0�%0p9 "H0e+e6fPæ&L-R02(:$<0:&O(25X?%0.^��s"%0BI'579A021qI<0p6G02I'&q9Y$'I<I�Ng0K?%0e+4I' w9Y$Z #"æ$'�� L"%0d0¿ H$'&= L$<�%J`£j¤a£ {}.¦Gèq+.-%-%1=5>+.(*"zr/1L0:nXQH$<1L$'�%JE+k-%",+.&=0)5.@¡$<�> L02�%&=$'6G0b@FQ%�%?,+.CD02�> #+.I¢+V #5GCD$<(Y?,+4 #+w-%1=5/?%QH(: #$'5G�[email protected] #"H0|"H0e+e6fP�&L-R02(:$<02&E L5uNR0�$<�>6G0:&= L$<J>+4 L02?z^�£ (:5/?H0�",+.&E #5!NR0�?%0:6.02I<5.-g0:?Ò L5u0p64+.I<Q,+V #0B #"H0�%0p9Ýé®SZéb(25GQ%-HI<02?Ú+4�%?!(21=5G&L&®S�(25.Q%-%I<0:?u�k+4Q%�> )@�+.(: L5G1L&:^3£jI<&=5%rg L"%0ON%QH�%?%I<$'�%JB(:QH =S[5.�æ&=(*"%02CD0",+.&q #5!NR0K@FQ%I'I'P�+.QH L5GCd+V #02?¢^`�s"%$<&q9\5G1=]�$'&w-%I<+.�%�%0:?æ@F5G1q L"%0B�%0¿ / M-H",+.&L0d5.@Y #"%0B£j¤a£ {(25.I<I<$'&L$'5G�,+.ITS�1#+4?%$U+4 L$Z6.0j?%0p6G0:I<5G-%CD02�> 2^j¶,5G1b+dCD5G1L0k$<�%?H02-H L"c?%$'&L(2QH&L&L$'5G�`5G�� #"%0k@FQH #Q%1=0q9A5G1L]�%0:(202&=&#+.18P� #5M(e+.1=1=PB L"%0h9\5.1L]�"%0:1L0h #"%1=5GQ%JG"� L5K(:5GCK-HI<0: L$<5G�d&L020k(*",+4 L021 ç ^

Page 147: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� � ��� ����

��� �� ������ �������� ��� ��� ����� ������� �� !� " ��#���$��� � % ��& '(� �)� �� *�+���,� �,� ���-� �����

.0/21 304658794;:#<=7?> 79@A:�B+4;C�@$5$DE<F> GIHJ4;<?<?7?KJL MJDNCPOQLR5$GTSUKUHV�W9XZYT[]\_^9`a^6ba[c^=Wd`fehgiYkjlXZePba^=ehm_^9b�\_m_YRmn`$oPp?qZ^9\_^=r]st`fuib�`auieaXZg]sZu0`fXN`aui^;W=XZrv`am_rwgigiYkW?ptr0riX2ox ^Rst^=ri^9efpy`a^=z�pybNp0jlgiriWd`fm_Xtr{Xyj x ptbhm_W|[i\_ptbaYTp0[}ptefpyYT^d`f^=ehb=~R��uim_b�b���rv`aui^9`am_W�bh[�^9W9`aeagiYAmnbea^djl^=ehea^=zE`aX6ptb�p���ba[c^=W9`aefpt\�jl^?py`agiea^y�n~���X�sZ^=ri^9efpy`a^PbhgiW�uRp�ba[c^=Wd`feapt\�jl^=py`fgieh^Pm�`�mnb�ri^9W=^9babfpyeh�N`aXx ehm_ris�`fXZsZ^d`fui^9e�`aui^P[cXZ[igi\_py`fmnXZr���\_mnri^�[ieaXy�}\_^y�tXZ[}pyW=mn`���ptr]zEW=XZrv`am_rwgimn`��;o�XZeh��[ieh^=bh^=rv`f^9z|`auwgibj�pte9�Z`fui^�m_r]zimnq�m_z]g}pt\�W=XZYR[�Xtri^=rv`fb�Xtjc`fu]m_b�jl^=py`fgieh^ x ^9m_ris�ea^djl^=ehea^=z|`fX�pyb���jl^?p�`fgieh^�[]eam_YRmn`amnqt^=b=��~��uimnb�mnb�ptr}pt\nXZsZXZg]b6`fX0`fui^RW=Xt\_\_mnbamnXZr}pt\���efpyzim py`am�qt^ x \_X�W���[iehm_YRmn`amnqZ^9b�ptriz x \nX�W��wb�XtjPW�uipt[]`f^9e¡ ~¢��uim_b�W�u}pt[]`a^=eEXZg�`f\_mnri^=b�`fu]^�YR^9`fu]X�z�Xyj�uim_stu�ba^9eam_^9bNbh[�^9W9`feapt\�jl^=py`fg]ea^�st^=ri^9efpy`am_XZr�pyrizzi^9YTXZr]bh`feapy`f^9b�`fui^�gibh^�Xtj�p£W=X�zi^;`�ptmn\_XZeh^=zQjlXZe�¤Z¥$�¦zimnqt^=eh`aXZe�W=XZr]zimn`am_XZrib9~V6§�V;¨0W=Xtrw`fptm_r]b�qyptehm_XZgib�z]m ptsZr]XZbh`am_W;W=X�zi^9b�`au}py`�[�^9ehjlXZehY,pRYTp�©�m_YNgiY�\nm_�t^9\_mnuiX�X�z��i`�Xtj�pba[c^=Wd`feapt\ijl^?py`agiea^ª`aX;X x bh^=ehqypy`am_XZript\]z}p�`�p]~�V6§�V;¨£«Z¬v­��i`ab�svptgibhbam pyr£[ieaXt�i\_^=b�`aX�ptr|X x bh^=ehqt^=zba[c^=Wd`fehgiYAptrizUoPptb�giba^9z)m_rJ`fu]^£\_m_r]^Em_zi^9rv`fmn�iW?py`am_XZr�o�XZeh��YR^=rv`fmnXZri^=z)mnrUba^=Wd`fmnXZr�­�~¯®0pyriz° ehX�XZ�wb�^9`Ppt\±~³²�´2®t®Z®vµd�wpyriz�mnr|`aui^�ziXZ[i[i\n^=e�bau]mnj¶`�zim_ptsZriXZb�`fmnWªoPXZeh�|Xtj-¨�[}ptz}ptehX�^d`ªpy\·~³²±­t¬Z¬Z¬vµo�uimnW�uT\n^=z£`fX;`fu]^��ieabh`8X x ba^=e�qypy`fmnXZrib$Xtj�bamn[iuiXZr|¸}X2o�b�m_r£`fui^�bhXZ\ pte�py`fYRXZba[]ui^=eh^t~�V�§�V6¨R«Z¬t¹�i`abRº»^=^=YTptr�bh[�^9W9`feapt\Pjl^=py`fgieh^=b£`fX)pyr�X x ba^9ehqt^=z¼bh[�^9W9`aeagiY�`auwgib£zim pysZriXZbhm_risJ[iptefptYR^9`a^=ehbbag]W�u�pyb�`fu]^E[i\ pybaYTpTYTptsZri^d`fm_W��}^9\_z�~�¨�^=^|¨�g]YTYR^=ehb£²·­y¬Z¬]´�µ�jlXZe�pTzi^=bhW=eamn[]`fmnXZrJXtj x Xt`auJXtj`fu]^=ba^|W9X�zi^9b=~N½�m�qZ^9r)`aui^No�XZeh`au)Xtj8bagiW�u)�i`a`am_ris�W=X�zi^=b�ptb6[]\ ptbhY�p�z]m ptsZr]XZbh`am_W=b9�»`fu]^Eba^9eam_^9b

´2¹Z¹

Page 148: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´2¹ ¡

\_mnYTm�`�jl^=py`fg]ea^;oPXtea��jlXZehYTb�p£r}p�`fgieapt\¾ptzizim�`fmnXZrQ`fX£`fu]^�V�§�V6¨Ibhgimn`a^;Xtj�W=X�zi^=b9~¿ XZribhm_zi^9e�`aui^�jlXZ\n\_X2o�m_r]s;^d©�[iea^9babhm_XZrQjlXZe�`fu]^�XZ[�`fm_W=pt\_\��E`fu]m_W��£^=YR^=ehsZ^=rv`�¸}g�©RjleaXZYÀp�[i\ pybaYTpXtj�\_^9rist`fuIXtj�\_mnri^;Xtj�bamnsZuv`�Á��io�m�`fuIriXRm_riW9m_zi^9rv` x ptW��wsZeaXtgiriz�eaptzim_py`fmnXZr��}^9\_z�Â

à ²·Äcµ$Å�ÆRÇ颃 ²·Ä-ÊfË µaÌ|²·ÄcµhÍwÎ ²±Ï�~n´2µ��ui^Iptript\n�w`fmnW?pt\n\n�UW9XZrv`fmnr�g]XZgib£^=YRm_bhbamnXZr�W=X�^9ÐRW9m_^=rv`Rptr]z�^=baW=pt[c^0[ieaX x p x m_\nmn`��)^Ñ©][]ea^=bhbamnXZribjleaXtYÒW�uipt[]`f^9eE¹Jpyea^�g]ba^=z»~JÓ}XZe�pt[i[i\nm_W?p�`fm_Xtrib�bagiW�uÔptb�`fuiXtba^To�^0pteh^�W9XZribamnzi^=ehm_ris¢m_r{`fuimnbW�u}pt[�`f^=e�mn`�m_b�pt\_bhXTr]^=W=^9babapteh�¢`fX�mnriW=\ngizi^�Xt`fu]^=e�W=XZrv`fehm x g�`fm_Xtrib�`aXT`aui^N^9YTmnbabamnXZrÕW=X�^9ÐRW=mn^=rv`bag]W�uUptb�`auiXZbh^�jlehXZY,Xt`fu]^=e�uw��ziehXZsZ^9rJW=XZrv`am_rwg}p]�³uv��zieaXZst^=r x eh^=YRbab�`fefpyui\_gir]s�^=YRm_bhbamnXZr�pyriz^=YRm_bhbamnXZr�jleaXtYk[i\_ptbaYTpNmnYT[ig]eamn`am_^9b=~$¨�^=W9`am_XZrÕÏ�~Ö­Ezi^=pt\_bªmnrQ`fgiehrQo�mn`auIpt\_\c`fui^;W9XZrv`feam x g]`am_XZrib`fX�^9×�gipy`fmnXZrJ²±Ï�~�´�µd~�¨�^=Wd`fm_XtrQÏ�~�¹;zi^=bhW=ehm x ^=b�p�W9X�zi^�`au}py`8m_b x ^=mnris�zi^9qt^=\_Xt[�^9zQptb8p;zim pysZriXZb�`fm_W`fX�XZ\�jlXZe�gibh^NmnrÕ`aui^�z]mnqZ^9eh`aXZe�ea^=stm_XZrÕXtj�¤Z¥$�;�³o�mn`au¢bh^=W9`am_XZrJÏ�~�¹]~¯¹|sZm�q�mnris£`�oPXT^d©]ptYR[i\_^;�i`ab`fX|X x ba^=e�qypy`fmnXZrib�o�mn`au¢p£W9XZeaeh^=ba[cXZriz]m_risRzim pysZriXZbhm_bªXtj�[i\_ptbaYTpE[}pteaptYT^d`f^9eab=~.0/ÑØ ÙÒ4;KJ4;5$DN<UC�GR> B+GRKJ4;KQ@$:ÚG|Û�@$ÜJ4Ý:�4;5�7=4;:�<?7=> 79@ÞÛZ46D�@�SU5$4ߪà�áªàhâ ãÔä]åPæÞçéèëêNåPê;ì;íéî�äiç�å�ï6ð¨�mnriW=^;m�`�m_b�YTptm_r]\n�Ruim_stu0ba^=ehm_^9b�uv��ziehXZsZ^=r���\_mn�t^;b���b�`f^9YTb�`fu}p�`�ptea^ x ^=mnris£W=XZr]bam_z]^=ea^9z�p�`�`fuimnbbh`fptsZ^y��`fui^Qu]m_sZu�r���bau]^=\_\8YRX�zig]\_^RXtj�`fui^�[�XZ[]gi\ py`am_XZr{W=X�zi^Qz]^=baW9eam x ^9zñmnr+W�u}py[]`f^9e ¡ m_bNptrpt[i[]eaXZ[iehm py`a^0YT^=ptrib|Xtj�^9qypt\_gipy`fmnrisJ`fui^Õ[cXZ[igi\_py`fmnXZr�zim_b�`fehm x g]`am_XZr]b=~���ui^¢W=X�zi^I^9qypt\_gipy`f^9bx �·j�ptW9`aXZeabªjlXZe�`fui^;[cXZ[igi\_py`fmnXZrib�gi[0`fXRpte x mn`feapteh�TuimnsZu0r���bhui^=\n\_bPjlXteªzi^9�iri^=zÕ[i\ pybaYTpE[}ptefpyY|�^9`a^=ehb=~�V,bfpyYT[i\n^QXZg]`a[ig]`|z}p�`�py�}\n^0m_bEbhuiX2o�r¼mnr��}sZgieh^ÕÏ�~n´t~���ui^ ° ²�V ¿ ��ò�V6óôµEW=XZ\ngiYRrW=Xtrw`fptm_r]bE`fui^ x �õj�ptWd`fXZehbEjlXZeN`fu]^0[�XZ[]gi\ py`am_XZr]b=~U��uimnb£m_bEbh[i\_m�`Egi[Ôm_rv`aXJqtpyeam_XtgibEW=Xtrw`aeam x g]`h�m_r]sQ[}pte�`fb=�»r}ptYR^=\n�I`au}py`6jlehXZY!`aui^EsZehXZgirizUb�`�py`a^Õ²�YR^9`fptbh`fp x \_^�W=XZrv`aeam x g]`fmnXZrib�W=ptr{pt\_bhX x ^m_r]W=\_g]zi^=z{m_r{ptr�^d©�`f^9ribamnXZr+Xtj8`fu]^TW9X�zi^?µ�mnrU`fui^TÓP´|W=XZ\ngiYRr��ôptriz)`fui^RW=XZrv`aeam x g]`fmnXZrUjleaXZYjlea^9^d�õ^=\_^9W9`aeaXZr)ea^9W=XZY x m_r}p�`fm_XtrJm_rJ`aui^RÓ$­�W=XZ\ngiYRr�~N��ui^9ba^RW=XZrv`aeam x g]`fmnXZrib;pteh^E`fui^9r�eh^=W9XZY|�x mnri^=zQmnrizi^9[�^9rizi^=rv`a\n�R`fXEsZm�qZ^�`aui^ x �õj�ptWd`fXZe8qtpy\_gi^ ° ² ¿ªö ¥ ¿ª÷ µÑ�iptb�p�W�ui^=W��R`fXE^9ribagieh^6`fu}p�``fu]^�bh^=[}pyefpy`am_XZrI[iehX�W9^=zigieh^�o�XZeh�t^=z¢W=XZehea^9W9`f\��Qo�mn`auiXZg]`�W=ptriW9^=\_\_py`fmnXZrQ^9eaeaXte=~��ui^�[iehX�W9^=zigieh^�m_b�gibh^=zTm_rE`fu]^�ba[c^=W9`aefpt\ijl^=py`fg]ea^�W=eh^?py`am_XZr|mnb$ptb�jlXZ\n\_X2o�b=¾`fui^�W9X�z]^�mnb�eag]r|jlXte`fu]^�eh^=×wgimnea^=zR[i\_ptbaYTp6[}pteaptYT^d`f^9eab�ptrizTp;z}py`fpy�}\_^�pyb8mnr£�}sZgieh^�Ï�~n´�[iehX�zig]W=^=z»~���ui^ x �õj�ptWd`fXZe

Page 149: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´2¹ZÏ

qypt\_g]^=bTptea^Õ`fui^9r*ea^=ptz¼jleaXtYø`fuimnbR�}\n^ x �Ô`fu]^�ba[c^=Wd`feapt\�jl^=py`fgieh^¢W9X�z]^t~ù¨]ptu}p�� ° XZ\�`fú=YTptrir[cXZ[igi\_py`fmnXZribEpyea^0`fui^9rÔYTX�zimn�i^=z x ��`fu]^=ba^ x �·qypt\_gi^9bE`fX)sZm�qZ^Q`fu]^0ea^=bhgi\n`fptrv`|[�Xt[igi\ p�`fm_Xtrib=~��uwgibª^=ptW�uQzim�û»^9ea^=rv`�zi^=r]bamn`��Tptr]zT`f^9YT[c^=eapy`fg]ea^�[}ptefpyYT^d`f^=e8ba^d`ªeh^=×wgim_eh^=b�p�bh^=[}pteapy`f^�eagirTXtj`fu]^�[cXZ[igi\_py`fmnXZr|W=X�zi^t~�V�\n`auiXZgisZuR`fuimnb�mnriW=eh^?ptbh^=b8`fui^�eagir��õ`fmnYT^�jlXZe$`aui^�jl^=py`fgieh^�sZ^9ri^=eapy`fmnXZr��mn`6m_b;riXt`�p0YTp�ühXZe;mnbabag]^EjlXZe�pQ�}r}pt\��i`h`fmnris0W=X�zi^t~£V�jlg]`agiea^£V�§�V6¨U�]`a`fmnris0W=X�zi^£o�m_\n\�gibh^p£[iea^Ñ��W=pt\_W9gi\ py`a^=z0bh^9`�Xtj�bh��rv`fui^d`fm_W�ba[c^=Wd`feapRXZe�[cXZbhbam x \n�Qbh[�^9W9`aefpt\ ýy[�Xt[igi\ p�`fm_Xtr�W=XtYT[cXZri^9rw`ab²�W=pt\_\n^=zñ��jl^=py`fgieh^Q�}\_^9b=�¯µ�o�uimnW�uÔW=X2qZ^9e|p¢o�m_zi^�efptrist^QXtj�[i\_ptbaYTp¢[}pyefptYR^9`a^=eab9~U��u}py`£mnb=��`aXXZ[]`am_YRm_bh^N`fui^|bh[�^9^=z+XyjP`fu]^|�]`?�¾`fui^£jl^?p�`fgieh^Rst^=ri^9efpy`am_XZr)W=X�zi^|o�m_\n\ x ^Rbh^=[}pyefpy`a^RjlehXZY�`fu]^�i`h`fm_r]sRm�`fbh^=\njé~ߪà�áªà�á þ¼ê6î$è¾ç�ätÿ� eaX2q�m_zi^9zN`fu]^�[i\_ptbaYTp�ziX�^=b�r]Xt`�j�py\_\ZXZg�`fbamnzi^�`fui^8YTX�zi^9efpy`a^�XZ[�`fm_W=pt\vzi^9[]`fu�ea^=stm_YR^�jlXte¾o�u]m_W�u`fu]^6^=bhW?pt[c^6j�ptW9`aXZe�`a^=W�uir]m_×wgi^;m_b�qtpy\_m_z»��Xtri^6W?pyr0giba^6`aui^6W=Xtrw`am_rwgiXZg]bª^=bhW?pt[c^6j�ptW9`aXZePW9X�z]^6XtjW�u}pt[�`f^=e�¹|`aXQpt\n\_X2o¦XZ[]`fmnW?pt\n\n�R`fuimnW��0ba^9eam_^9b�\nm_YRmn`ªbh[�^9W9`aefpR`aX x ^�YRX�zi^=\n\_^=z»~���uimnb�[]eaX�zigiW=^9bptrÕ^=bhW?pt[c^�[]eaX x p x m_\nmn`��Tptb�p|jlg]riW9`am_XZrIXtjôoªp?qt^=\_^9rist`au�`fu}p�`�W?ptr x ^�gibh^=z¢m_rI^=×wg}py`am_XZr�²±Ï�~n´2µptb�p£YTX�zim��}W?py`am_XZrT`fX£`fui^�^9baW?py[im_ris|¸}g�©�~��ui^|XZ[}pyW=mn`��Jptz�ühgib�`fYR^=rv`fb;`aXQ`fui^|uimnsZuU\n��m_r]s0[�XZ[]gi\ py`am_XZr]b�W?pyr x ^|ziXZri^£mnr)p0r�g]Y x ^9e;XtjoPp2��b9~���r�`aui^;p x baXZeh[]`fmnXZr�j�ptWd`fXZePpt[i[iehXvptW�u���`fui^��ñqypt\_gi^9bªpteh^6gibh^=zIptbª[iptefptYR^9`aeamnW�ptz�ühgibh`��YR^=rv`fb�XZrU`fu]^|¥�mnribh`a^=mnr{V��õqypt\ngi^=b;mnrJ`fui^£W9XZ\_\nm_bamnXZr}pt\���eaptzim p�`fmnqt^;Y�p�`feam�©�~���uimnb;m_b;b�`fefpym_sZuv`h�jlXZe�oªptehzUjlXZe;\_X2o\n��m_r]s0zim_bhW=ea^d`f^£\nm_ri^9b�o�ui^=eh^R`aui^=eh^£m_b;XZr]^£girim_×wgi^��ùjlXte;^?ptW�uU`feaptribhmn`fmnXZr�~Ó}XZeP`aui^�uim_sZu]^=e�ba^9eamn^=b�\_mnri^=b9�]`fu]^=ea^�mnbªbaXZYR^�ptY x m_sZgim�`��Qpybª`fX£o�ui^9`aui^=e�ptrI^=YRmn`h`f^9z0[iuiXt`aXZrXZehm_sZmnr}py`f^9z¢jlehXZY#Xtri^Egi[i[c^=e�\_^9qt^=\�Xte�jleaXZY(m�`fb6ri^9m_sZu x XZgie9~��;r]^£pt[i[iehXvptW�u)m_b�`fX0bh[i\_m�`�`fu]^uimnsZu�\_^dqZ^=\P[�XZ[]gi\ py`am_XZr]b�mnrw`aXU[�Xt[igi\ p�`fm_Xtr+^=\n^=YR^=rv`fb9�$o�mn`fu�W9XZeaeh^=bh[�XZr]zim_risJjleh^=×wgi^9riW9�Ô^9\_^d�YR^=rv`fb�m_r)`fui^|^dqtpy\_g}py`am_XZr)Xtj���~T��ui^;ühg]bh`fm��}W?p�`fm_Xtr)jlXte�`auim_b�mnb�`au}py`=��ptb�`fu]^T\n^9qZ^9\_b�YR^=east^`fXtsZ^9`aui^=e9�]mn`�m_b8riXE\nXZrisZ^9e�YR^?pyrim_risyjlgi\i`fXN`fpt\_�|p x XZg�`ªpNz]m_baW9ea^d`f^�[cXZ[ig]\ py`am_XZrTptriz0pye x m�`fefpyeh�^=\n^=YR^=rv`fb�Xtj-[cXZ[igi\_py`fmnXZrRW?ptr x ^�zi^9�}r]^=z�~��6ri^�W=ptr�`aui^=r0^dqtpy\_g}py`a^�eapy`f^�^d©�[ieh^=bhbam_XtribªjlXZe�pt\_\`fu]^=ba^£��^=\n^=YR^=rv`fb9�}ptriz0mnriba^9eh`�`fu]^=Y�mnrv`fXE`fu]^;ri^9o¦W=XZ\n\_mnbam_Xtr}pt\��õefptz]m py`amnqZ^PY�py`aeam�©T`aX£[ieaX�zigiW9^pT[cXZ[ig]\ py`am_XZrÕbh`aeagiWd`fgieh^Egibamnris��b�^dqtpy\_g}py`a^=z�jlXte�^=ptW�uJXtj$`aui^=bh^Njlea^9×�g]^=riWd�J^=\n^=YR^=rv`fb9~�Ó}XZejlg]`agiea^�pt[][i\_mnW?py`am_XZr|`fX£riXtr����p�©�o�^=\n\_m_ptrQzimnbh`fehm x g�`fm_Xtrib8`auim_bPm_bP`aui^;pt[i[]eaXZ[iehm py`a^6YR^9`fu]X�z�`aXptziXt[]`?~

Page 150: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´2¹t«

�� � ������������������������������������ ��!���"��# $���%�����'&��������(��&�!�"�)���!�������*�+�,���"�"#&���&���"�!���������+-���#��,�����*�!�".&�"�!�+�*�!/ ,�0�1��$��2 3 /547698:/�/ ��; /�/ 3 6<4%698:/�/ ��; /�/����!�1 4�698:/�/ ��; /�='> ��� 47698:/�/ ��; /�?'> ��& 4%698:/�/ ��; /�?.>@ 4#/A8:/�/ ��; /�/ ��� 47698:/�/ ��; 6�? ��*$B�C ��& 4%698:/�/ ��; 6�? ��*$B�C��, 45?D8:/�/ ��; /�? ���$E�!�*�� ��, 47698:/�/ ��; /�F ��*$B�C ��,$E���� 4%698:/�/ ��B /�F �"���G 4#HA8:F�F ��; 6JI ��*$B H +�����B 6

/ ����"�"��J+(������!�" 1�����"��������$�����J�#��!�����+��2 � !�"� �� � ������������-�������J+�!�������� !���1.���$��+�+'������&�"�����2'��!�����+6 / /6 6 HA8:H�=�K�H�H�I�H ��B 6JHL698M?�H�F�/�I�K�I ��B 6�?

/ "�������"�+.�� '*���"�����&�"��J�(����0 H/ ��� � " "�� 6 H �ONP��,���� >�Q �ONR!�������!�" Q ����ESNR����T���; Q6 6 / / HA8:U�K(6JU C I�K ��; / C /A8:/�/�/�/�/�/�/ ��; /�/VHA8:U�K(6JU C I�K ��; / C HA8:U�K(6JU C I�K ��; / C IA8:K�I�K�F�F�=�F ��B /�HH H / / HA8P6JK�H�?�K�/�I ��; /�/WHA8:U C F�K�K�K�F ��B /�HVHA8:H(6J=�=�F�/�I ��; /�/XHA8:H(6J=�=�F�/�I ��; /�/Y698:F(6JI�=�U�U C���B /�UC C / / ?D8 C F�F�F�F�?�/ ��B /(6ZIA8:I�I C�C�C = C���B /(6[KA8:K C 6�6�6JH�? ��B /(6\KA8:K C 6�6�6JH�? ��B /(6[?D8 C /�=(6J=�I�/ ��B /�F? ? / / =A8 C 6J/�/�=�U�/ ��B /�HWKA8:/�I�?�H�/�I�U ��B /(6[KA8:=�=�I�H(6�?�H ��B /(6\KA8:=�=�I�H(6�?�H ��B /(6 C 8:I�I�I�/(6JF�F ��B /�FI I / / HA8:/�U�/�I�H�H�I ��B /�HWKA8:F�I�U�/�H�?�I ��B /(6[KA8:K�U�H�/�F�U�= ��B /(6\KA8:K�U�H�/�F�U�= ��B /(6 C 8:=�K C = C I�? ��B /�FU U / / UA8:I�/ C /�F�=�= ��B / C KA8:K�H(6JI�/�K�= ��B /(6[KA8:K�=�U�I�?�/�I ��B /(6\KA8:K�=�U�I�?�/�I ��B /(6[?D8:U�H C /�F�K�U ��B /�FF F / / HA8M?�I C F�/�=�? ��B / C KA8:K�F�/�/�I�? C���B /(6[KA8:K�K�?�I�K(6�? ��B /(6\KA8:K�K�?�I�K(6�? ��B /(6\IA8:U�/(6 C H�F�I ��B /�F= = / / 698:/�?�H�?�=�? C���B / C KA8:K�=�F(6J=(6JK ��B /(6[KA8:K�K�F�U�/�U�= ��B /(6\KA8:K�K�F�U�/�U�= ��B /(6\UA8:F�= C =�I(6JF ��B /�FK K / / ?D8:F�/(6JF�I�/�F ��B /�?]KA8:K�K�?�6J=�H�U ��B /(6[KA8:K�K�=�=�=�? C���B /(6\KA8:K�K�=�=�=�? C���B /(6\=A8P6JI�H�I�F�I�K ��B /�F6J/ 6J/ / / HA8:/�=�F�=�F C ? ��B /�?]KA8:K�K�F C K�U�K ��B /(6[KA8:K�K�K�?�=�?�= ��B /(6\KA8:K�K�K�?�=�?�= ��B /(6\KA8:U�K�=�K�U C�C���B /�F6�6 6JH / / C 8:/�/�? C H�U�= ��B /�IWKA8:K�K�K�U�/�K�U ��B /(6[KA8:K�K�K�K(6J/�/ ��B /(6\KA8:K�K�K�K(6J/�/ ��B /(6X698 C�C /�=�F�/�/ ��B /�U6JH 6JI / / ?D8:/ C I�?�H�/�? ��B /�UWKA8:K�K�K�K�?�?�H ��B /(6[KA8:K�K�K�K�=�?�U ��B /(6\KA8:K�K�K�K�=�?�U ��B /(6X698:K�K�K�/�/�/�= ��B /�U6 C H�/ / / C 8P6J=�F�?�U(6�? ��B /�FWKA8:K�K�K�K�K�I(6 ��B /(6[KA8:K�K�K�K�K�= C���B /(6\KA8:K�K�K�K�K�= C���B /(6 C 8M?�?�U C�C�C / ��B /�U6�? C / / / HA8:U�?�=�H�/ C / ��B /�=WKA8:K�K�K�K�K�K�U ��B /(6[KA8:K�K�K�K�K�K�= ��B /(6\KA8:K�K�K�K�K�K�= ��B /(6\FA8:I�=�U�/�I�U�/ ��B /�U6JI ?�/ / / IA8M?�/�U�U�F�I�H ��B /�KWKA8:K�K�K�K�K�K�K ��B /(6V698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/Y698 C�C = C H�/�= ��B /�I6JU I�/ / / 698:U�I�=�U�F�U�I ��B /�K^698:/�/�/�/�/�/�/ ��; /�/L698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/XHA8:/�= C F(6J/�U ��B /�I6JF U�/ / / UA8:U�K�U�=(6JU�? ��B 6J/^698:/�/�/�/�/�/�/ ��; /�/L698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/XHA8:K�K�?�F�I�=�I ��B /�I6J= F�/ / / C 8M?�6JF�U C H�= ��B 6J/^698:/�/�/�/�/�/�/ ��; /�/L698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/V?D8:/�F(6�?�I�=�= ��B /�I6JK =�/ / / HA8P6JI(6JF�H�H(6 ��B 6J/^698:/�/�/�/�/�/�/ ��; /�/L698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/XIA8 C 6 C =�/�K(6 ��B /�IH�/ K�/ / / 698:U(6�6JU�I�/�= ��B 6J/^698:/�/�/�/�/�/�/ ��; /�/L698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/XUA8:F�H(6J=�/�=�? ��B /�IH(6\6J/�/ / / 698 C F(6JH�=�U�? ��B 6J/^698:/�/�/�/�/�/�/ ��; /�/L698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/X=A8:H�K�I�?�I�U(6 ��B /�IH�HX6�6J/ / / 698:H�F�U�?�?�I�U ��B 6J/^698:/�/�/�/�/�/�/ ��; /�/L698:/�/�/�/�/�/�/ ��; /�/Y698:/�/�/�/�/�/�/ ��; /�/Y698:/�/ C ?�F�I�H ��B /�?��� 4 6 TSNR� 6 E���; Q ;5 H� 6_4 &���&���"�!��������7�� '2��$������1 +���!������� 7�������; 4 &���&���"�!��������7�� '2��$������1 +���!������� #����G��%�������J+�!���������+���!�2����� 4 &���&���"�!��������7�� '&��������(��&�!�"�)���!�������*�+�,���"�"'�'�� %�������� 4 +�!�,�!�B�����"���3�*�!���� �!��������' $���'&��������(��&�!�"7)���!�������* +�,���"�".���, 4 ����������!�"#,�0�1��$��2���� ����!�*'��������2�0@ 4 ��!�1���!��������.1���"����������# �!��������3 /54 ������"���!�� ��,�!���2��3 6_4 �����'��,�!���2���; 6

����& 4`/ ������1 4 C ��&���+ 4\/ ��"���@ 4Z/ ��������& 4Z/����������& 4`/ ��"�&���+ 4Z/ ����1��J+�& 4X6 3��� � 4W698:/ ��+ 4W698:/�/ 1�; /�=@ 4`/A8:/�/ 1�; /�/ �(����� 4\/A8:/ ��&�0 4Z/A8:/@ 6 4`/A8:/�/ 1�; /�/ 3(��*�& 4\/A8:/ N /A8:/�/ 1�; /�/(QÓ�mnsZgiea^EÏ�~�´ZÂ.acb9dDeDfMg�h�ijbOk%lDg�h�g�mnfpo_dnq�bAlAenr�o$l7sut h�vno�vutAlAq�b9w9o$k%r�bOfjfpipx�ipb9kng9fzy{q�gOlAiMg�h�ip|9o}r�bulDo9~}��o$x�eDfjh�x�gSq�ox�vDb���k_��bOq������������$���������-���9�O�����������(�S�$��~���vno�sAyR� gOr�h�b9q�x¡gSq�o¢wOij|Oo�k�ijk#r�b9fpeD£�k5¤�¥R¦¨§¢��©¨¦«ª­¬��O��izh�vh�vno�r�bOk�h�q�ijsDeDh�ipb9knx®��q�bO£¯h�vDo_wOq�bOeDknl7xJh�gSh�o_ipk�r�b9fpeD£�k�°¢�_gSknl7��q�bO£±q�o$r�b9£�sDijk²g�h�ipb9k�ipk�°¡³A~���vDo_dDfMg9x�£5gdng9q�g9£�o�h�o$q�x�gSq�o�w9ip|9o�k�g�h�h�vDo´h�bOd.b9�­h�vDo<lDgSh�g�m²fjoO~

Page 151: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´2¹¶µ

V�rIpy\n`f^9ear}p�`fmnqt^6pt[][ieaXvpyW�u0m_bP`aXEW=XZr]bam_z]^=eªzimnbaW9ea^9`a^;\_^dqZ^=\�[�Xt[igi\ p�`fm_Xtrib=�wo�mn`auQW=XZehea^9ba[cXZrizimnrisefp�`f^=bIjlXZeI^?pyW�u�\_^dqZ^=\±~·�6ri^)`fui^9rùW=Xtribamnzi^=ehb¢`fu]^+p x baXZeh[]`fmnXZr"jlehXZY `aui^+eh^=bag]\n`�pyrw`I`aXt`�pt\m_rv`a^=ribhmn`��|zigi^�`fXNptrv�|X2qZ^=eh\ pt[][im_ris�[iehXt�}\_^�py`�`fu]^�ehm_sZuv`�jlea^9×�g]^=riWd�T`aX x ^�p x baXZe x ^9z x �£`fu}p�`\_mnri^t~���u}py`�m_b9�iXZgie�p x bhXZea[�`fm_XtrIj�pyW9`fXte�^d©�[iea^9babhm_XZr x ^9W=XZYR^=b9Â

�FÅ´#¸\¹(º�»�¼®½�¾�¿�À�Á�ÂJÃ� ²�Äiµ à Ç�Å ÂJÆu½�Â�½ ²�ÄiµhÍÇĹ$È ¹ º¢» É ½�¾�¿JÀ�Á�Â�Ã� ²�Ä]µhÍÇÄ]Í�ɲ±Ï�~Ö­tµ

o�ui^9ea^E`fu]^Ejlea^=×wgi^9riW9�Jm_rv`f^9sZefpy`am_XZr�m_b6X2qZ^9e6`fui^£Xtri^EuimnsZuJba^9eamn^=b;ba[c^=Wd`feapt\�\_mnri^N[]eaXt�}\n^t�-pyriz`fu]^Õ^9YTmnbabhm_XZrñptrizFp x bhXZea[�`fm_XtrñW9X�^dÐRW=m_^9rv`fbRW=XZehea^=bh[�XtrizF`aX{`aui^0qtpy\_gi^9b|jlXte|`au}py`|zimnbaW9ea^9`a^\_mnri^t�-m±~�^y~�riXÕ\_mnri^ x \n^=rizimnrisIm_b�ptW9W=XZg]rw`a^=z+jlXte=~Ê�õ`�m_b;XZr]\n�JmnrU`fui^Rm_rv`f^9ribam�`��J`f^9eaY�`au}py`�`fu]^jlgi\n\n� x \_^9rizi^=zÕ^=YRm_bhbamnXZrÕptrizÕp x bhXZea[]`am_XZrIW9X�^dÐTW9m_^9rw`ab�pyea^;W=Xtribamnzi^=eh^=z�~��uimnbTpy[i[ieaXZptW�uFuiptbR`aui^Õptz]qyptrv`�pysZ^0`fu}py`|m�`|ea^9×�g]m_ea^9bTYRm_r]m_YTpt\�W�uiptrisZ^9bR`aX)`aui^Õ^d©�m_b�`fm_r]s¿ ��Ë�W9X�zi^|jlefpyYT^doPXZeh�³~ÌË�p�`f^=b�pt\nea^?pyz]�)^d©�m_b�`�jlXte�`aui^=bh^�uimnsZu{r���bhui^=\n\_b=��ptriz{`fui^Í�Úqtpy\_gi^9b^9qypt\ng}py`a^=z�u]^=ea^�W?pyr x ^�mnriW=\ngizi^=z¢zim_eh^=Wd`f\n�t~ªÓ�mnsZgieh^NÏ�~Ö­£bauiX2o�b6bhXZYR^�eh^=bhgi\n`ab�jlXZe�`fu]m_b�`f^9W�u��rimn×�g]^t�}sZ^9ri^=eapy`f^9z¢gibhm_ris£`aui^�uim_sZu0×wg}pyrw`agiY,bau]^=\_\�� W9X�z]^6jleaXZY�W�uipt[]`f^9e�¹E�8r]Xt`f^6`au}py`?�³pyb^d©�[c^=Wd`f^=z»��`fu]^�p x baXZeh[]`fmnXZrUj�ptW9`aXZeab�m_r]W=ea^=ptba^Tptb�XZri^|[iehXZsZeh^=babh^=b�gi[{`aui^Rba^=ehm_^9b=~��õ`�bhuiXZgi\nzx ^0riXt`f^9z�`fu}p�`EjlXZeN`aui^U´ ¡ ¸"­¢`fefpyribam�`fm_Xtr�ptrizÔp x X2qZ^y�8`aui^0\_mnri^=bNptea^ x \_^=r]zim_risJ`aXZsZ^d`fui^9ebamnsZrim��}W?ptrv`a\n�Z�8ptr]z*py`|uimnsZui^9eRXt[]`fmnW?pt\ªz]^=[]`auibTXtri^¢bh^=^9b�p)ba\nm_sZuv`|zi^9W=ea^=ptba^Õm_rFp x bhXZea[�`fm_Xtrj�ptWd`fXZe|jlXZe|`fui^9ba^�`aefptribhmn`am_XZrib9~ ��uim_bRm_bRzigi^¢`fX�`aui^¢j�pyW9`R`fu}py`R`fu]^�efpyzim py`am_XZr��}^9\_z*W=ptgibé�m_r]s�`fui^�[iuiXt`aXy�ép x baXZeh[]`fmnXZribNmnbNzig]^Q`fXUpt\n\8`aui^QX2qZ^9ea\_pt[i[imnrisJ[iehXt�}\_^9b£ptriz�`auwgib£m_b�sZea^=py`f^9e`fuiptr�`au}py`E[]eaX�zigiW=^9z�mnj�`fui^T^=YRmn`h`fm_r]sÕ\nm_ri^£oPptbNmnbaXZ\_py`f^9z�~0��uimnbNr}py`agiefpy\_\n��\_^=ptzib�`aX�YRXZea^[iuiXy`fXy��p x baXtea[]`am_XZrib$mnr£`fui^�\_mnri^ªgir]zi^=e�m_rvqt^=bh`am_svp�`fm_XtrTpyriz|ui^=riW9^�p�ea^9zigiW9^=zQp x baXZeh[]`fmnXZr£j�ptWd�`fXte=~���uimnb£^9û�^=Wd`Tuiptb x ^9^=rFpt\_bhX x ^9^=rñriXt`f^9zÔm_r�`fui^Q`aui^=bhm_b|Xtj�Ó�m_bhW�u x ptW�ui^9e�²±­t¬Z¬]´2µd~+V�b`fu]^6zi^=r]bamn`���m_riW9ea^=ptba^9bª`fui^�p x baXZeh[]`fmnXZr�j�ptWd`fXZehbªzi^=W9ea^=ptba^y�³ptbªXZri^6o�XZgi\nz0^d©�[c^=W9`�²lea^9W?pt\n\�`fu]^zi^d�}rimn`am_XZrIXtjô`fu]^�p x baXZeh[]`fmnXZr0j�ptW9`aXZe�m_r0^9×wg}py`fmnXZr+²�´Z~�´ ¡ µhµd~�õ`ªmnb�^=ptbamn^=e8`fXNz]^?pt\co�mn`fuR`fui^�W=XZrv`am_rwgigiYÞXZ[}ptW9mn`��RYRX�zim��}W?p�`fm_Xtrib�`fXN`aui^�[cXZ[ig]\ py`am_XZrRbh`aeagiWÑ�`fg]ea^t~���ui^�efpy`a^=b�`fu}p�`6pteh^�g]ba^=zUmnrÕ`aui^NW9XZ\_\nm_bhm_XZr}py\���eaptzim_py`fm�qZ^�W=X�zi^=b�ptea^�pt\n\¨�p�©�oP^9\_\�p?qZ^=eé�ptsZ^9z�Xtri^=b9��`fuwgib�mn`�mnb;YTXZb�`�pt[i[iehXZ[ieam_py`f^£`aXÕ^dqypt\_g}p�`f^Rptr�p x baXZeh[]`fmnXZrUj�ptW9`aXZe;`fu}p�`�mnb�pt\_bhXp?qZ^9efptsZ^9z�X2qZ^9e�`aui^To�uiXt\_^|W=XZrv`am_rwgigiY ptr]z+W?pyr x ^Tgibh^=z�pyb�[iptefptYR^9`aeamnWTpyz�ühgibh`aYT^9rv`fb�XZr`fu]^�bh[�Xtrw`fptri^9XZgib�ea^=W9XZY x mnr}py`am_XZr0eapy`f^9b=~�Vë[ieaX�W=^9zigiea^�u}ptb x ^=^9rÕo�ehmn`a`a^=rI`fX|^9qypt\ng}py`a^6`fuimnb²�bh^=^£bh^=W9`am_XZrJ¹]~ ¡ ~n´�µÑ��ptr]z�`fui^Nea^=bhgi\n`ab;pteh^EbauiX2o�r)m_rÕ�}sZgieh^|Ï�~�¹TjlXZe�ptrU^=\n^=W9`aeaXZr�zi^=ribhmn`��¢Xtj´�Î�´2¬ÇÏ�ÐSÑSÒÔÓÇÕ�ptriz¢ptrI^=\_^9W9`aeaXZrI`f^9YT[c^=eapy`fgieh^;Xtj�´AÖD×T~

Page 152: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´2¹�Ø

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10 12 14 16 18 20

Ab

sorp

tion

fa

cto

Upper level quantum nuber

Lyman series

plasma length=1.e4 cm

plasma temperature=1 eV

dens=1.e13 cm^-3dens=5.e13 cm^-3dens=1.e14 cm^-3

Ó�mnsZgiea^RÏ�~Ö­�Â�§Úb9k�h�ipkueDeD£Ûg9snxJbOq�dDh�ipb9k%� gOr�h�b9q�x�gOx´g'��enknr�h�ijbOkÜb9�ÚeDdnd²o$q}fjo$|9o�f¡Ý�eng9kOh�eD£ÞkueD£�s²o$q´��b9q´h�vnoª�tu£5gSk#x�o�q�ipo$xc��bOqÚg}dDfMg9x�£5g}��ijh�v#� � �����'�(� � ���Oß à(á¶âuãJä#�����'�(� � ���¯gSk²l5g}q�g9kDw9o�b9�ådDfpgOxJ£5g<lDo�knx�izh�ijo(x�~

1e-05

0.0001

0.001

0.01

0.1

1

100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09

Ab

rosp

tion

fa

cto

r

Ù

length / cmÓ�mnsZgiea^ÕÏ�~¯¹]ÂÔ§Úb9k�h�ijkueDeD£`g�|9o�q�gSwOo$lÊæç��b9q�h�vDo7ª�tu£5gSkr�b9k�h�ijkueDeD£ZdDfjb9hJh�o$lg9x�g-��eDknr�h�ijbOkÍb9�«dDfMg9x�£5gfpo�kDw9h�v���izh�v7�����è�}�Ü�$� ��� ��� �¶� �D�����è�(à�é5~���o$x�eDfjh�x�gSq�o´w9o$kDo�q�g�h�o$l#e²xJipkDw�h�vno<r�bAlAo´bS�¡r�vngSdDh�o�q�ê

Page 153: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´2¹t®

ߪà�áªà�ë ìRçhï�íê�î}å<ï�í�í\í�ð$î�í�ì6î�ä}çéåPï¨�[c^=Wd`fefpy\]\_mnri^P[]eaXt�}\n^=b$pyea^P^dqypt\_g}p�`f^=z|pyb�zi^=bhW=ehm x ^=z|m_rEW�uipt[]`f^9e�¹]~���u]^ª^d©�[ieh^=babhm_XZr|Xtj»½;eamn^=Y²�´?®Z«Z¬vµªm_b�gibh^=zUptb�`fu]^ x ptbhm_W�[ieaXy�}\_^;jlXZe�pt\_\-\n^9qt^=\_b9� x g]`�mnb�bhg x b�`fmn`ag]`f^9zUptb�pt[i[iehXZ[iehm py`a^�jlXte�t^d�UX x bh^=e�qZ^=z{\_mnri^=b x ��m_YR[ieaX2qt^=z{pt[i[iehX?©]mnY�p�`fm_Xtrib=~ ö ^=ea^£bhgiW�u{uim_sZuU×wg}py\_mn`��¢[iehXt�}\_^9b;pteh^W?py\_W=g]\ py`a^=z x �Ô`aui^ �<�<� W=X�zi^�²l��pt\nm_rÔ^9`Qpt\·~���´?®Z®vÏ�����pt\nm_rñ^9`�pt\·~���´?®Z®åµZµÑ~ �_�<� m_bRgibh^=zjlXZeNpt[i[iehX2©�mnY�py`a^=\��{`aui^��}ehbh`E�]j¶`f^=^9rÔba^=ehm_^9bEYT^9Y x ^=eab9~ ¿ pt\_W9gi\ p�`fm_Xtr+Xtj�`aui^QuimnsZui^=e�ba^9eam_^9bYR^=Y x ^=ehbTmnbRbhXZYR^9o�u}py`TW=XtYT[ig�`�py`am_XZr}py\_\n�{zi^9Y�pyrizim_r]si� x g�`�mnbTpy\_baX{Xtj�\_^=bhbTmnYT[cXZe�`�ptriW9^t�pt\n\_X2o�m_ris|`fu]^|½�ehm_^9Y,[ieaXt�i\_^=b�`fX x ^Ngibh^=z�~;��u]^=ba^£\nm_ri^�[ieaXy�}\_^�W?pt\nW=gi\_py`fmnXZrib�ptea^�W=XZY x m_ri^9zo�mn`auE`fui^�[cXZ[igi\_py`fmnXZrNeh^=bhgi\n`ab�`aX;^9qypt\ng}py`f^ª`aui^ x XZgiriz�� x Xtgiriz|W=XZrv`aeam x g]`fmnXZrN`fX;`aui^�^=YRm_bhbam_XtrW=X�^9ÐRW9m_^=rv`=~�õ`|m_b£mnrv`f^=eh^=b�`fm_r]sJ`fX)W=XZYR[}pteh^�`fu]^Õ\nm_ri^T[ieaXy�}\_^0^Ñ©][]ea^=bhbamnXZribTptb£^9qypt\ng}py`f^9z x ��`fui^¢½;eamn^=Y^=×wg}p�`fm_XtrTo�mn`auT`auiXZba^�^9qypt\ng}py`a^=z�g]bam_r]s�`fu]^ �<�<� W9X�z]^t~���ui^�[ieaXy�}\_^9bPpteh^�[iehm_YTpteamn\n��zi^9[�^9r��zi^9rw`�Xtr�`fui^�^=\_^9W9`aeaXZr0zi^9ribam�`��Z� x g]`�pt\nbaXNqypte��To�m�`fu�`aui^6^=\n^=Wd`feaXtrQ`f^=YR[c^=efp�`fgieh^t~�Ó�mnsZgiea^�Ï�~ ¡bau]X�o�b6`aui^|½;eamn^=Y,[iehXt�}\_^9b6X2qZ^9ea[i\nXt`a`a^=zUXZr�`aui^ �<�<� eh^=bhgi\n`ab�jlXZe6pTefptrist^NXtj�zi^9ribam�`fmn^=b=~ �õ`oPptb8jlXtgiriz£`fu}p�`8`aui^�½�ehm_^9Y [iehXt�}\_^9b�[ieaX�zigiW9^�[]eaXt�}\n^=b�^d©�`aea^=YR^=\��£W=\nXZba^ª`aX�`aui^ �<�_� eh^=bhgi\n`abgi[�`aXJzi^=r]bamn`am_^9bEXtj.ñ�­ÌÎ*´2¬ Ï�Ð Ñ9Ò ÓÇÕ jlXZe�`���[im_W=pt\�z]mnqZ^9eh`aXZeN`f^9YT[c^=eapy`fg]ea^=b0²�òÒ´uÖD×Eµd~UÓ}XZezi^9ribam�`fm_^9bEp x X2qZ^�`auim_b9��`aui^0½;eamn^=YA[iehXt�}\_^9bNgir]zi^=eh^=bh`am_YTpy`f^R`fui^T^=\n^=W9`aeaXZr x eaXvpyzi^=rimnrisJpyriz`fuwgib�[ieaX�zigiW9^�[]eaXt�}\n^=b�o�uim_W�uJptea^;`fX�XRr}pteaehX2o�~'�;g]e�[iehm_YTpteh��m_rv`f^9ea^9bh`�jlXte�`auim_b�o�XZea�0mnb�mnr`fu]^�zim�qZ^=e�`fXZe8ea^9sZm_Xtr�Xtj�`aXZ��ptYTpt�wb8o�m�`fu�z]^=ribhmn`fmn^=b8jleaXZY!²�¬�ón´}¸JÏZµ�ÎÕ´2¬ÇÏ�ÐSÑSÒÔÓÇÕ2��`auwgib�o�^;pteh^m_r)`fu]^Teh^=sZmnYT^Eo�ui^9ea^R`fui^�½�ehm_^9YÝ^d©�[ieh^=babhm_XZr+m_b�bh`fpteh`am_ris0`fX x ea^=pt�UziX2o�r�~Ìô�^9qt^=eh`aui^=\n^=bhb=�mn`�o�mn\_\�b�`fm_\n\�[iehX�zig]W=^�stX�X�z)[ieaXt�i\_^=b�jlXZe�YTptrv�)XtjP`aui^Tz]^=ribhmn`fmn^=b�`fu}p�`�Xtri^Ro�XZgi\nz�oPptrv`�`aXm_rvqt^=bh`am_svp�`f^t�³pyriz0jlXZe�zi^=r]bamn`am_^9b x ^9�tXZriz¢mn`fb�eh^=sZmnXZr0Xtj�qtpy\_m_z]mn`��|`fui^ �<�<� [ieaXy�}\_^9b�ptea^�gibh^=z�~ߪà�áªàJõ öÔåPì;ï�÷.ø²ùSî¡í®í_ú}û�î¡í�æÀð]ðwäcî}î<ü;í�ì;ï�ýñî�ï�÷ëçhæÞê;ì�î³ç�ätÿùè-å�ïPäcî³ç�û;ì�äiç�å�ï6ð��ui^;uimnsZui^9e x XZgiriz��·jlea^9^6uv��z]eaXZsZ^9rIW=XZrv`fmnrwgigiY�W9XZrv`fehm x g]`am_XZr]bªpteh^6^9qypt\ng}py`f^9z�jleaXtYk^Ñ©][]ea^=bé�bamnXZrib$bhgiW�uQptb$^9×wg}py`fmnXZr¢²±¹]~�«vµÑ~���r£`fui^�bh[�^9W9`feapt\ijl^?p�`fgieh^�W=X�zi^t��p6eaXtg]`fmnri^ª�}eab�`�zi^9qt^=\nXZ[�^9zRjlXteptr}py\n��bamnb�Xyj�`fui^ x ea^9YTbhbh`aefptui\ngirisR^=YRm_bhbam_Xtr¢mnr¢`fu]^E¤Z¥$��`fXt�ypyY�pt�U² ö Xteh`fXtr���­y¬Z¬]´�µPjlXZehYT^9z`fu]^ x ptbamnb�jlXZe�^9qypt\_gipy`fmnXZrJXtj8pt\n\ô`aui^£uv��zieaXZst^=rimnW x Xtgiriz��·jlea^=^£W9XZrv`fehm x g]`am_XZr]b=~;��ui^ ö XZeh`aXZr[iehX�W9^=zigieh^=b�ptsZeh^=^9z¢^Ñ©ipyW9`f\���o�m�`fuÕptr0m_r]zi^=[c^=riz]^=rv`�ehXZg]`am_ri^6z]^9qZ^9\_XZ[c^=zÕui^9ea^t~8Ó}XZeªW9XZYR[}py`h�m x m_\_m�`��¢o�m�`fu{V6§�V;¨)ptriz�Xt`fui^9e�pt[i[i\nm_W?p�`fm_Xtrib=�-m�`;oªptb�zi^=W9m_zi^9z�`aX x pyba^|`fui^|oPXtea�Uui^9ea^RXZr

Page 154: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ ¬

01e+102e+103e+104e+105e+106e+107e+108e+109e+101e+11

3600 3650 3700 3750 3800 3850 3900

Em

issi

on c

oeffi

cien

t

þ

Wavelength

Electron density=2.e13 cm^-3

Griem resultsPPP results

01e+112e+113e+114e+115e+116e+117e+118e+119e+11

3600 3650 3700 3750 3800 3850 3900

Em

issi

on c

oeffi

cien

t

þ

Wavelength

Electron density=1.e14 cm^-3

Griem resultsPPP results

0

1e+12

2e+12

3e+12

4e+12

5e+12

6e+12

7e+12

3600 3650 3700 3750 3800 3850 3900

Em

issi

on c

oeffi

cien

t

þ

Wavelength

Electron density=5.e14 cm^-3

Griem resultsPPP results

1e+06

1e+07

1e+08

1e+09

1e+10

1e+11

3600 3650 3700 3750 3800 3850 3900

Em

issi

on c

oeffi

cien

t

þ

Wavelength

Electron density=2.e13 cm^-3

Griem resultsPPP results

1e+08

1e+09

1e+10

1e+11

1e+12

3600 3650 3700 3750 3800 3850 3900

Em

issi

on c

oeffi

cien

t

þ

Wavelength

Electron density=1.e14 cm^-3

Griem resultsPPP results

1e+10

1e+11

1e+12

1e+13

3600 3650 3700 3750 3800 3850 3900

Em

issi

on c

oeffi

cien

t

þ

Wavelength

Electron density=5.e14 cm^-3

Griem resultsPPP results

Ó�mnsZgiea^PÏ�~ ¡ ®��vDo¢q�o$x�eDfzh�gSk�h­s�b9enknluy{s²bOeDknl<o$£�ipx�xJipb9k�r�buo�ÿ#r�ijo$kOhco�|�g9fjengSh�o(l}enx�ijkDw«h�vno¢r�b9fpfjiMxJipb9k²gSfpfjt´sDq�bOgOluyo�kno$l�dDq�bSmnfpo$x®bS����q�ipo�£ ¥������9��¬�r�bO£�d²gSq�o$l ��izh�v�h�vDb�xJo<��q�bO£ h�vno_a�aÚa fjipkDo<sDq�bOgOlAo�knijkDw5r�bulDo9~���vDo<fpb��Úo$qx�o�h´bS��dDfpbSh�x«x�vDb���h�vDo�q�o(xJeDfjh�x«b9kÜg#fpb9w�gSq�izh�vD£�ipr_tOy�g��AiMx�~´��vDo�mnq�x�h�mD�:h�o$o�k-dDq�b9mnfpo$x«bS��h�vDo�¤�g9fj£�o�q´x�o�q�ijo(xgSq�o´o�|�gSfpengSh�o$l�~�®b9h�o}h�vng�h¨h�vDo<lAiMx�r�q�o�dng9knr�t g�h®fMgSq�w9o}lAo�k²xJijh�ipo$x�iMx�fMgSq�w9o$fjt'lDeDo}h�b�h�vDo_ªÇb9q�o�k�h�$ipg9k.dDq�bSm²fjo(xkDb9h®s²o$ijkDw�wOb�bAl.o��uh�o$knx�ijbOknx¢��q�bO£ ��q�ijo$£�� x�dDq�bSm²fjo(x���bOq�h�vDo(xJo}dnfpgOxJ£5g�dngSq�gS£�o�h�o�q�x$~

Page 155: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ ´

`fu]^ ö XZeh`aXZrÕ[ieaX�W=^9zigiea^9b=~8��ui^�uw��ziehXZsZ^9r x eh^=YRbabh`aefptu]\_girisRW=Xtrw`aeam x g]`fmnXZrQjleaXtY�`fui^ ö XZeh`aXZrW=X�zi^;mnbP^9qypt\ng}py`a^=zQjleaXtY�ptript\n�w`fmnW�uv��zieaXtsZ^=rimnW;^d©�[ieh^=babhm_XZr]b�stmnqZ^9r x � ° giehsZ^=bhb�� ¨�g]YTYR^=ehb²�´?®¶ØåµZµÑ~ ö XZeh`aXZr���b�W9X�z]^Ept\nbaX�^dqypt\_g}p�`f^�pt\_\-`fu]^NW=Xtrw`am_rwg}p|jleaXZY(m_YR[igiehmn`��0bh[�^9W=mn^=b=~�Ó}XZe�`fuimnbmn`�eh^=×wgimnea^=b�m_r�jlXZeaYTpy`fmnXZrQXZrÕ`fui^�mnXZrim_bapy`fmnXZrIptrizÕea^9W=XZY x m_r}p�`fm_XtrÕW9X�^dÐRW=m_^9rv`fb�jlXZe�^?ptW�u¢m_XZr��m_bapy`fmnXZrñbh`�pysZ^JXtj6`aui^JmnYT[ig]eamn`���m_r¼×wgi^9bh`am_XZr�~ù��uim_bTm_r]jlXZehY�p�`fm_XtrñmnbTX x `�ptmnri^=z¼gibamnris+`fu]^V6§�V;¨Ubag x eaXZg�`fm_r]^Rz}­yz}py`fp]~�jlXte6o�uim_W�u{m_rU`fg]ear�ptW=W9^=babh^=bNpy`fXZYRm_WNz}py`�pQjlehXZYÝV6§�V;¨)ptz]jé´t´�}\n^=b=~_ô�Xt`f^6`au}py`�pt\_\�m_XZr]m_bfp�`fm_XtrQbh`fptsZ^=b�Xtj�ptrÕm_YR[igiehmn`���ptea^�m_r]W=\_g]zi^=zV�\n\]`fu]^�W9XZrv`fmnr�gip�W9XZrv`feam x g]`am_XZrib�jlehXZY�pt\n\i[cXZbabhm x \n^�m_YR[igieam�`��Nbh[�^9W=mn^=b�ptea^�sZ^9ri^=eapy`f^9zTo�m�`fuimnr`fu]^�bh[�^9W9`feapt\}jl^?py`agiea^�W?py\_W=g]\ py`am_XZr x �£`fuimnb�YT^d`fuiX�z�~���ui^�m_YR[igiehmn`��E^9YTm�`a`a^=eab�YTp?�£XZeamnsZm_ripy`f^jleaXtY p{zim�û»^9ea^9rw`Tea^9sZm_XtrFXtj�`fui^Õ\_mnri^IXtj6bhm_sZuv`R`fu}ptrñ`fui^Õuv��zieaXZst^=r*^9YTm�`a`f^9eab9~*��uwgib�`fu]^W=X�zi^�pt\_\nX�o�b$jlXZe8zim�û»^9ea^=rv`�`f^=YR[c^=efp�`fgieh^=b=��zi^9ribam�`fm_^9bªptr]zQ\_^9rist`fuTXtj-^9YTm�`a`fmnris�ea^9sZm_XZr]b�mnrT`fu]^m_YR[igiehmn`��TW?pt\nW=gi\_py`fmnXZr�~.0/�� � :�B+4;C�7��{CÞC�GIHJ4,ÛtGT5> GIHJ4;<?<=7?KJL @�ÜJ4��kDE<?> 465�4;> 7?:��

:�7=GRK 7?K @�ÜJ4 ����� HU7��R4;5�@�GT5�/ߪà�ë�àhâ í�ä]î�çhíéð å�ùQäcü�íê7ü�ÿ�ð]ç�è-î�íTê�î}å�û;í�íôæ! mn`auQ`fui^;[cXZbhbam x m_\_m�`��|Xtj�sZ^=ri^9efpy`am_risEp|W9XZYR[i\_^d`f^6bh[�^9W9`feapt\»jl^=py`fgieh^;m_riW9XZea[cXZeapy`fmnris�`aui^;W=XZr��`fmnrwgigiY ^=zist^t��p�YEg]W�u�YTXZeh^TW9XZYR[i\_^d`f^�W9XZYR[}pteamnbaXZr{o�mn`au+ba[c^=Wd`feapt\PX x ba^=e�qypy`fmnXZribEmnb�riX2o[cXZbabhm x \n^t~���ui^ÕY�pym_r�bh`agiz]�Ôui^9ea^Õm_b|ptbRpU[cXt`f^9rw`am pt\�z]m ptsZr]XZbamnbN`fX�XZ\PjlXZeE`aui^¢¤Z¥$��zimnqt^=eé�`fXte=~¼V!zi^9baW=ehm_[]`am_XZrñXtj;z]mnqZ^9eh`aXZe|W=XZriz]mn`fmnXZribRptrizñ`���[im_W=pt\�uim_sZuÔbh^=eamn^=bTba[c^=W9`aefp{u}ptb x ^9^=rsZm�qZ^=rÔmnr�W�u}pt[]`a^=e0´Z~ç��rÔ`auim_bNea^=stm_XZr���XZr]^Qba^9^=bTpJbh`fehXZris ° pt\nYT^9eNbh^=eamn^=b£\nm_YRmn`�ba[c^=W9`aeagiYo�ui^9rÔ`aui^Q[i\_ptbaYTp¢z]^9`�pyW�ui^=bEjlehXZYÒ`aui^Qzim�qZ^9eh`fXte�oPpt\_\±~Jò�r]zi^=eE`aui^=bh^0W=m_ehW=giYRbh`fptriW9^=bEm�`NmnbriX|\_XtrisZ^=eP[�XZbhbam x \_^�`aX|ea^=\��QXZrI\_ptrisZYEg]m_e8[ieaX x ^6`f^9YT[c^=eapy`fg]ea^�ptrizIzi^9ribam�`��0YR^?ptbhgiea^9YT^9rw`ab²�bhgiW�uJ[iehX x ^=b6pyea^�^=Y x ^=ziz]^=zJmnrÕ`aui^�z]mnqZ^9eh`aXZe�oPpt\n\_b�µ�ptr]zJpt\�`f^=ehr}py`amnqZ^�[ig]ea^Nbh[�^9W9`aeaXZbhW=XZ[imnW`f^9W�uirimn×�g]^=b�bagiW�u�ptb;`fuimnb;uim_stuUba^=ehm_^9b�Xtri^ x ^9W=XZYR^£Xtj8qypt\_gi^y~N��ui^|[i\_ptbaYTpQ[}pteaptYR^9`f^9eab;mnr`fu]m_bNW=ptba^�ptea^T^d©�`feaptW9`a^=z x �)�i`a`am_ris�p�bh��rv`fui^d`fmnWQba[c^=Wd`feag]Y `fXÕ`fui^�W=XZYR[i\_^d`f^Rba^9eam_^9bN\nm_YRmn`ptrizUW9XZrv`fmnr�g]giY#^9zisZ^EX x ba^9ehqypy`fmnXZrib9~6��uim_b�u}ptb�mnr�j�ptWd` x ^=^=r)ziXtri^No�mn`auJbaXtYT^�bagiW9W=^=bhb x �� m_svpyeaX2qQ^d`�py\·~-²é´2®Z®¶ØZµd~�¨�^9^�bh^=Wd`fm_Xtr¢¹�~¯¹EjlXZe�p£zi^9baW=ehm_[]`am_XZrIXtjô`fu]^�z]mnû�^=ea^9riW=^9b x ^9`�o�^=^9r�`fu]^pt[i[]eaXvptW�uT[iea^9ba^9rw`a^=z0ui^9ea^�ptrizT`aui^�XZri^�giba^9z x � � mnsvpteaX2qc~���ui^�[i\_ptbaYTp�[iptefptYR^9`a^=ehb�o�u]m_W�u

Page 156: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ ­

W?pyr x ^�zi^9zigiW=^9z0m_rR[ieamnriW=mn[i\_^�pyea^�`fui^�^=\_^9W9`aeaXZrR`f^=YR[c^=efp�`fgieh^�jlehXZY `fui^�jlea^9^d� x XZg]rizQW=Xtrw`am_r��gigiY!ba\nXZ[c^£ptrizJmnrv`f^=r]bamn`��¢efp�`fm_X�Xtj$`fui^Nuim_sZu�ba^=ehm_^9b6YT^9Y x ^=eab9�»`fu]^E^=\n^=W9`aeaXZrUz]^=ribhmn`��ÕjleaXZY`fu]^�o�mnz]`fuÕXtj�`aui^�uim_stu¢bh^=eamn^=b�\_mnri^=b9�]`fui^�ea^9W=XZY x m_r}p�`fm_Xtr0bh`�p�`f^�jleaXZY�`aui^�ba^=ehm_^9b�zi^9W=eh^=YR^=rv`ptriz»�8m�j�`fui^�m_YR[igieam�`��{W=XZrv`fehm x g�`fm_Xtr+`fXJ`aui^Q^=YRm_bhbamnXZr�m_bNbam_strimn�}W=ptrv`?��`aui^Iºc��^dû»^9W9`amnqZ^Ipyrizm_YR[igiehmn`��TW=XZr]W=^=rv`aefpy`am_XZrib�jlehXZY�`fu]^�g]rizi^=eh\n��m_r]sRW9XZrv`fmnr�g]giY�\_^dqZ^9\·~��ui^J¤Z¥�� ÷ ��¹+bh[�^9W9`aeaXZYR^9`a^=e|oªptb�giba^9z¼`fX�YT^=ptbagieh^Õ`aui^�u]m_sZuFba^=ehm_^9b ° py\_YR^=e|ba[c^=Wd`fefp�~÷ ��¹]��b�\nm_ri^£XyjPbhm_sZuv`£��ba^=^9b=��p x XZg]`-µt¬#"!Xtj�`aui^|XZg]`a^=e�zimnqt^=e�`fXZe6`fpteasZ^d`�ptb�bauiX2o�r+mnrU�}sZgieh^Ï�~¯Ï�~'�õ`6m_b�^=b�`fmnY�py`a^=z�jlehXZY�`fu]^£¤Z¥$�ë[i\ ptbhY�p|`feaptriba[cXZe�`�W9X�zi^9b�`fuipy`�`aui^ ° py\_YR^=e�^=YRm_bhbam_XtrXZehm_sZmnr}py`f^9b£jleaXZY pt[i[]eaX?©�m_YTpy`f^9\n�ñ­���¹�W9Y Xtj�`fui^¢zimnqt^=eh`aXZe�eh^=sZmnXZrFjlXteR`aui^�qZ^9eh`am_W?py\�[i\_py`f^zimnbaW�u}ptehsZ^�ptriz£¹�� ¡ W9Y�jlXZe¾`fui^8uiXZehm_ú=Xtrw`fpt\w[i\ p�`f^�zim_bhW�u}ptehsZ^t��`fuiXZg]sZu�`aui^=bh^Pqypt\_gi^9b�uip2qt^P\_pteasZ^giriW9^=e�`�ptmnrw`am_^9b=~�V�b�XZri^;W=ptrÕba^=^;jlehXZY��}sZg]ea^�Ï�~¯ÏN`fui^�\_mnri^6Xtj�bamnsZuv`�Xtj ÷ ��¹|[}ptbhba^9b�`auieaXZg]sZu`fu]^�W9^=rv`feh^�Xtjc`fu]^�`aXZ��ptYTpt�NW9eaXZbhb�bh^=W9`am_XZrRbaX�`fu]^�`aXt`�py\}X x ba^9ehqt^=z�mnrv`f^=r]bamn`��£o�mn\_\�[�Xt`a^=rv`fm_pt\_\��W=Xtrw`fptm_rUW9XZrv`fehm x g]`am_XZr]b�jleaXtY#XZg�`�o�mn`fu¢`fui^£zim�qZ^9eh`fXte6ea^9sZm_XZr»~7��rJ`fui^E^Ñ©�[�^9eam_YR^=rv`;`aui^ ÷ ��¹m_r]bh`fehgiYR^=rv`�`�py�t^=b8p�bh^=×wgi^9riW=^�Xtj³`f^9YT[cXZefpy\�bhuiXt`fb$X2qt^=e8p�bamnrisZ\n^�¤Z¥$�F[ig]\_ba^y~���r]jlXteaYTpy`fmnXZrNmnbsZ^9ri^=eapy`f^9z�jleaXZYÀ^=ptW�u�`feaptW���²�m·~�^t~wbh[}py`fm_pt\³[�Xtbamn`am_XZr}µ�p�`P^=ptW�u�`fmnYT^�ba\nm_W9^t~ ÷ ��¹�uiptbªpNbh[}py`am pt\ea^9baXZ\ng]`fmnXZr�Xtj�´2¹yYTY ptrizÔ¬]~�´?riY ba[c^=Wd`fefpy\ªea^9baXZ\ng]`fmnXZr�~UV�rÔ^Ñ©][cXZbhgiea^0`fmnYT^TXtj�´2¬Z¬JYRba^9WoPptb;giba^9zJjlXZe�`aui^EuimnsZu��õr ° pt\nYT^9e�X x ba^=e�qypy`fmnXZrib=~%ô�Xt`f^�`fu}py`�`fui^Nba[c^=W9`aeaXZYR^9`a^=e6mnb6p x \_^�`aXYR^?ptbhgiea^�`aui^ � pybaW�ui^9rFW9XZrv`fm_rwgig]Y ^9zisZ^Iptriz�`aui^Qbh[�^9W9`feapt\Pjl^=py`fgieh^QW=X�zi^0mnbEpt\nbaXUp x \_^T`aXYRX�zi^9\�`fuimnb�bh[�^9W9`aefpt\ôea^9sZm_XZr»~PV�rJptr}pt\���bhm_b�Xtj � ptbhW�ui^=r)ba[c^=Wd`feapTu}ptb��Z^d`�`fX x ^�[c^=e�jlXZeaYR^=z�~Ó}eaXZY`¢^9m_sZb�^9`�pt\±~i²é´2®Z®¶ØZµ¾`���[im_W=pt\}¤Z¥$�Ôzim�qZ^=e�`fXZe�W=XZriz]mn`fmnXZribôoPXZg]\_z x ^�ptrE^9\_^=Wd`fehXZr£zi^=r]bamn`��Xtj�ñ ´�Î0´2¬ Ï�Ð ÑSÒ ÓÇÕ �wri^=g]`aefpt\³zi^9g]`f^9eamngiYÀzi^=ribhmn`��|Xtj�ñ ´�Î0´2¬ ÏJÕ ÑSÒ ÓÇÕ ptr]z�^=\n^=Wd`feaXtr�ptr]z�m_Xtr`f^9YT[c^=eapy`fg]ea^=b�Xtj´ñ ´�Î�´2¬ Ð%$ ~ߪà�ë�à�á & å#÷7íôíhí�ç�ï�ý('+å�î*)+-,/.10124365874,�9 :8;=<>012?58<

Ó�mnsZgiea^6Ï�~�«NbauiX2o�b�`fu]^ x �õqypt\ngi^�eh^=bhgi\n`abªXtj¾`fui^�[�XZ[]gi\ py`am_XZrTW=X�zi^�jlXZePptr0^=\n^=Wd`feaXtr0zi^=ribhmn`��TXtj´%Î�´?¬ Ï�Ð ÑSÒ ÓÇÕ �}ptrizI^=\n^=Wd`feaXtrI`a^=YR[�^9efpy`agiea^;Xyj�´%Î{´2¬ Ð $ ~}�õ`�W=ptr x ^;bh^=^9r¢`fuipy`ª`fu]^�p�`fXZYRm_W[cXZ[igi\_py`fmnXZribNeh^?ptW�uF¨]pyu}p�� ° XZ\n`aú=YTptrir�qypt\ngi^=b x ��p x Xtg]`A@¦ÅÝÏ�~)��ui^ x �õqypt\_g]^=bEmnr��}sZgieh^Ï�~�«EW=XZehea^9ba[cXZriz0`fXE`aui^;z}py`fpy�}\_^�bauiX2o�rÕm_rQ�}stgiea^�Ï�~n´Z~�¥�©]ptYRm_r}p�`fm_Xtr�Xtj¾`fu]^;z}py`�p��}\_^�bauiX2o�b`fuipy`�`aui^£^d©�W=m�`f^9zU\_^9qt^=\�[�XZ[]gi\ py`am_XZr]b�pteh^E\ pyeasZ^9\n�I[cXZ[igi\_py`f^9z�`fuiehXZgisZuUeh^=W9XZY x mnr}py`fmnXZr+²�m·~�^t~

Page 157: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ ¹

Ó�mnsZgiea^¢Ï�~¯Ï�ÂCB�fpfje²x�h�q�gSh�ipb9k�b9�¨h�vDo7fjipkDo(x�bS��x�ipw9v�h5bS�¨h�vDoED}��ê�ipknxJh�q�eD£�o�k�h5ijkh�vDo7¤¢g9fj£�o$q�x�o�q�ipo$x�fjip£�izhb9s²xJo$q�|�g�h�ijbOknx$~`fu]^;W=XZrv`fehm x g�`fm_Xtribª`fX|Á Æ W=XtYT^;\_pteasZ^9\n�|jleaXtYk`fu]^�Ó$­EW=XZ\ngiYTr}µd��ptb�XZri^6o�XZgi\_zI^Ñ©][c^=Wd`�jleaXZYp£bh`aeaXZrist\n�Qeh^=W9XZY x mnrim_ris|[i\_ptbaYTp]~��uwgib=�ôjlXZe6`fu]^Ru]m_sZui^9e ° pt\nYT^9e;ba^=ehm_^9b�`aefptr]bamn`am_XZr]b6o�uim_W�u�ptea^|Xtj�m_rv`a^=ea^9bh`�ui^9ea^t�¾`fu]^Rg][i[�^9e\_^dqZ^9\�[�Xt[igi\ p�`fm_XtribPpyea^6W9\_XZbh^�`fXN`aui^=mneP`fu]^=eaYRX�z]��r}ptYRm_W�^=×wgim_\nm x ehm_giY qypt\ngi^=b9~$V�\nbaXi��XZri^6W=ptru}p?qZ^£W9XZr]�}z]^=riW9^|`au}py`�`fui^Ejleh^=^E^9\_^=Wd`fehXZrib�o�mn\_\ôu}p?qZ^|pÌ�p�©�o�^=\n\_m_ptr�qZ^9\_X�W=m�`���zimnbh`aeam x g]`fmnXZr��YTpt�wm_risR`fui^�uw��ziehXZsZ^9rim_W�W=XZrv`fmnrwg}p�^d©�[ieh^=bhbam_Xtrib6jlXZe;Ì pyriz É ²lo�uimnW�u)ptbabhgiYR^Ê�p�©�o�^=\n\_m pyrzimnbh`fehm x g�`fm_Xtrib�µ�qypt\_mnz�~�§6^9ba[im�`f^�`aui^�j�ptWd`ª`fu}p�`ª��¥�qtpy\_gi^9bPoPXtgi\_z x ^6bag�ÐTW9m_^9rw`=�]`fu]^6ba[c^=Wd`fefpy\jl^?p�`fgieh^�W=X�zi^ª`�pt�y^=b$`fu]^�^9qypt\_gipy`f^9z x �õqypt\_g]^=b�mnrw`aX�ptW=W9XZgirv`�`fX�pt\n\_X2o�jlXZe�^Ñ©�`f^=r]bam_XtrR`aX%ô�ó���¥W=Xtrizimn`am_XZr]b�mnjôri^=W9^=bhbfpte��Z~F .G3*H#7?5�IJ:8;=<>0G2458<

��ui^IXZ[}pyW=mn`��+W=X�zi^0Xtj�W�u}pt[]`a^=eR¹�oPptb|eagirÔjlXZeN`aui^Iba^=\n^=Wd`f^=zñzim�qZ^=e�`fXZe£W9XZrizim�`fmnXZrib£ptr]zFpyb^d©�[c^=Wd`f^=z{`fui^£[]\ ptbhY�p�mnb6XZ[]`am_W=pt\_\��Õ`auim_r x Xt`au)jlehXZY#ptr)^=YR^=east^=rv`;¸}g�©{pyriz{pQ[�XZ[]gi\ py`am_XZr[cXZm_rv`8Xtj»q�mn^9o*jlXZe$`aui^�bh[�^9W9`aefpt\³ea^9sZm_XtrJ²lba^=^��}sZgieh^6Ï�~MµZµÑ~���uwgib�Ì"W?ptr x ^�ba^d`�`aXNXZri^�jlXZe�`fu]^

Page 158: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡t¡

0.1

1

10

100

1000

10000

0 5 10 15 20 25 30 35 40

b-va

lueK

n-shell

Electron density = 1.e14 cm^-3

Electron temperature = 1.e4 cm^-3

Ó�m_sZgieh^�Ï�~�«]Â}acb9dneDfpgSh�ipb9k sAyP� g9r�h�b9q�x¢|ux�dnq�ipknr�ipdng9f�ÝOe²gSk�h�eD£±xJvno�fpf¶��b9q¨lAip|9o$qJh�b9q�r�bOknlAijh�ipb9knx$~^=YR^=ehsZ^=rv`8¸}g�©RW=pt\_W9gi\ py`am_XZr|pyrizTr]X�[cXZ[igi\_py`fmnXZrRptz�ühgib�`fYR^=rv`fb�pteh^�r]^=W=^9babapteh�RjlXZe$`fui^�uimnsZui^9eba^9eamn^=b�`feaptribam�`fmnXZrib=~V�bTu}ptb�pt\_eh^?ptz]� x ^=^9r*bau]X�o�r¼m_rF�isZgiea^�´Z~�´Z´Z��`aui^�\nX2oP^9eQó���Y�ptrF\_mnri^=bRpteh^�\_mn�t^9\n��`aX x ^XZ[]`am_W=pt\_\��T`auim_W��³~ªV�r�XZ[]`am_W=pt\_\��T`auim_W��0[cXZ[igi\_py`fmnXZrIW?pt\nW=gi\_py`fmnXZrQoPptb�[c^=e�jlXZeaYR^=zI`fXR^9qypt\ng}py`a^`fu]^�^9û�^=W9`8Xtj�ó���Y�pyr£ba^=ehm_^9b$[iuiXt`aXy�ép x baXZeh[]`fmnXZr£XZrE`fu]^�^Ñ©�W=mn`a^=z|bh`fpy`f^�[cXZ[ig]\ py`am_XZrEb�`fehgiW9`agiea^y~V6§�V;¨)­�´ ¡ oªpyb�g]ba^=zU`aXÕst^=ri^9efpy`a^E`fui^£XZ[�`fm_W=pt\_\��0`fuimnW��Jptz�j�¬ ¡ jlXZe6[cXZ[ig]\ py`am_XZrJ[]eaX�W=^=bhbamnrisq�m pRV6§�V;¨�­y¬vÏ��³ba^9^��}sZg]ea^£Ï�~pØ]~5�õ`�W?pyr x ^�ba^9^=rJ`au}py`�`fui^�r³Å�­��c¹L� ¡ bhui^=\n\_b�pt\_\-uip2qt^E^=r��u}ptr]W=^=z)[cXZ[igi\_py`fmnXZrib9~6ò�[cXZrJW9\_XZbh^=e6mnrvqZ^=b�`fmnsvpy`fmnXZr�m�`�oPptb�jlXZg]rizñ²ljlXZe�ptrJ^9\_^=Wd`fehXZrUzi^=r]bamn`��Xtj$´#ÎÕ´2¬ Ï�Ð ÑSÒ ÓÇÕ µ�`au}py`8`fui^�m_riW9ea^?pyba^�mnrR`aui^�r}Å�­�[�XZ[]gi\ py`am_XZr|mnb�zig]^�`fX�[iuiXt`aXy�ép x baXZeh[]`fmnXZr��o�uimn\_^�`fu]^8mnriW=eh^?ptbh^�m_r�`fui^�r³Å;¹�� ¡ [�XZ[]gi\ py`am_XZr]b�m_b-YTptm_r]\n��zigi^$`fX�^=\n^=W9`aeaXZr�^Ñ©]W9mn`fpy`fmnXZr;jleaXZY`fu]^T^9riu}ptriW9^=z�r³Å�­Ibaui^9\_\·�¾o�mn`au�`aui^T^dû»^9W9` x ^9W=XZYRm_risÕ\_^9bab�[ieaXtriXZgiriW9^=z�pybNXZri^Rm_r]W=ea^=ptba^9bm_r�r���bhui^=\n\·~���uwgib�`fui^9ea^£m_b�ptrJmnrizim_eh^=Wd`�mnr]¸}gi^9riW=^£Xtj�Xt[}ptW=m�`��ÕXZr�`fu]^E[�Xt[igi\ p�`fm_Xtrib�jlXte�rNM¹]~|V�b;oPptb�ba^=^9r�jlehXZY��}sZg]ea^TÏ�~�´Z�¾ea^=W9XZY x mnr}py`am_XZrUziXtYTmnr}py`f^9b;X2qZ^=e�W9XZ\_\nm_bamnXZr}pt\ô^d©�W9mn`�p�`fm_XtrjlXZe�`aui^EuimnsZui\���^Ñ©]W9mn`a^=z)bh`fpy`f^9b=�¾ptrizJ`aui^=bh^|XZ[}ptW9mn`��¢^9û�^=W9`ab�ptea^£r]Xt`;\_mn�t^9\n�I`fX0W�u}pyrisZ^E`fu]m_b=~ö X2o�^9qt^=e=��`fui^�[iuiXt`aXy�ép x baXZeh[]`fmnXZrR^=riu}pyriW=^9zQ[�Xt[igi\ p�`fm_XtrRXyj�`fui^�r³Å�­�bhui^=\n\cptriz�bhg x bh^=×wgi^=rv`W=Xt\_\_mnbamnXZr}pt\n\n�;^9riu}ptriW9^=zQ[cXZ[ig]\ py`am_XZrib�Xtj»`aui^�uim_sZu]^=e�baui^9\_\nb8YNgibh` x ^�mnriW=\ngizi^9zRjlXZe$W9XZYR[i\_^d`f^d�ri^9bab�mnr0`fui^��}r}pt\-V�§�V6¨IW9X�z]^�`aX�pt\n\_X2o"\_X2o�^=e�ba^=ehm_^9b�YR^=Y x ^9eab�`fX x ^�YRX�zi^9\_\n^=z0W=Xteaea^9W9`a\n�Z~��uwgib�jlXZeP`aui^;ba^9eam_^9b�\_mnYTm�`8o�XZeh��[ieh^=ba^9rv`f^=z¢m_r�`auim_b�W�uipt[]`f^9e�XZ[}ptW9mn`��Tm_bªW9XZribhm_zi^9ea^=zI`fX£uip2qt^p£ri^=st\_m_stm x \n^�^9û�^=Wd`�XZrÕ`aui^�uim_stui\n�T^d©�W=m�`f^9z¢[cXZ[igi\_py`fmnXZrib9~

Page 159: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ Ï

0.999955

0.99996

0.999965

0.99997

0.999975

0.99998

0.999985

0.99999

0.999995

1

3600 3620 3640 3660 3680 3700 3720 3740 3760 3780 3800

Wavelength / A

Escape probabilityAbsorption factor

0.988

0.99

0.992

0.994

0.996

0.998

1

3600 3800 4000 4200 4400 4600 4800 5000

Wavelength / A

Escape probabilityAbsorption factor

Ó�mnsZgiea^ÕÏ�~Mµ� ¦®k²gSfptOh�ipr$gSfpfjt�r�b9k�h�ipkueDb9e²x�o(x�r$gSd�o.dnq�bOsngSsDipfpizh�ijo(x�g9knl�g9snx�b9q�dAh�ipb9k�� g9r�h�b9q�x_|Ax$~¡�¢g�|Oo�fpo�kDw9h�vo�|�g9fjengSh�o(l5��b9qÚh�tudDiMr�gSfålAip|9o$qJh�b9q�r�bOknlAijh�ipb9knx$~¡��vDo«endDd²o$q�dDfjb9h�x�vDb��®x�h�vDo´q�o$x�eDfjh�xÚkDo$g9qÚh�vDo´xJo$q�ipo$xÚfjip£�izh®g9knlh�vno}fjb��¢o�q�dDfpbSh�h�vDo<q�o$x�eDfjh�x�ipknr�fpenlAipkDw5fpb��Úo$q®xJo$q�ipo$x�£�o�£�s²o$q�x$~��¨bSh�o´h�vng�h®bOk�s²b9h�v7x�r$gSfpo$x$�Ah�vno´� g9r�h�b9q�x�gSq�okDo$w9fpijwOijsDfpo9~

Page 160: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ «

Ó�mnsZgiea^JÏ�~jØ]Â�©¨dDd�o�q'dnfjb9h.xJvnb��®x�h�vDo%bOdAh�iMr�g9fjfpt�h�vDipk o��Dr�ijh�o(ld²bOdDeDfMg�h�ipb9k lDo�knx�izh�ijo(x5��bOq5h�vDo%mnq�xJh5mn|OokAy�xJvno�fpfpx¢��bOq�h�t�dnipr$gSf�lAip|9o$qJh�b9q�r�bOknlAijh�ipb9knx$~���vDo´fjb��¢o�q�dDfpbSh¨x�vDb��®x¢h�vDo}b9dAh�ipr$gSfpfjt#h�vDiMrPO#q�o$x�eDfzh�x�~

Page 161: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ µ

QRITSG:8,/U/;V9XWY,0G9GSTZ�WY,�019GS\[�WY,�0G91S]Z_^`:a;b;/[TWG:a;=cN<d<�5e:83*fG2�0G9GUg:8;V<d012?58<�3*9GSN7hci.10G:a7?5�IHj,�95e:a74Wk0T5874,9G<

��ui^T^=YRm_babhm_XZr{W=X�^9ÐRW9m_^=rv`Nptb�^9qypt\_gipy`f^9z+m_r)`fu]^TW9X�zi^Tm_b�YTptzi^Rgi[+XtjPY�pyrw�)W9XZYR[�XZr]^=rv`fb9~Ó�mnsZgiea^6Ï�~�®EbauiX2o�b�`fui^;uv��ziehXZsZ^=r x XZgiriz�� x Xtgiriz�� x XZgir]z��õjleh^=^;ptr]z x ea^=YRbab�`feaptui\_g]ris£W=XZrv`aeam��x g]`am_XZr]b�jlXZe�`fui^EW=ptba^£Xyj8ptrJ^=\n^=Wd`feaXtrJzi^=r]bamn`���Xtj�´ Î�´2¬�Ï�Ð9Ñ9ÒÔÓÇÕ?�»^=\_^9W9`aeaXZr�`a^=YR[�^9efpy`agiea^�Xtj´AÖD×(ptriz+pÕ\nm_ri^Ñ��Xtj �õbam_stuw`6\_^=r]st`fu{Xtj6´?¬¶Ñ9ÒU~ô�Xt`a^R`au}py`�`fu]^TXt`aui^=e�uv��zieaXZst^=r�W9XZrv`fm_rwg}p���mnr[}pte�`fmnW=gi\_pte6`fui^ � ptbhW�ui^=r�W9XZrv`fmnr�g]giY Y�pt�y^=bNp¢bamnsZrim��}W?ptrv`�W=Xtrw`aeam x g]`fmnXZr)`fX¢`aui^�eh^=bhgi\n`fptrv`ba[c^=Wd`fehgiY(py`�`fu]^�YRm_r]m_YTp x ^d`�oP^9^=r¢zim_bhW=ea^d`f^�ba^9eam_^9b�\_m_r]^=b=~$��u]^ x ea^=YRbab�`feaptui\_g]risTW9XZrv`fehm x g��`fmnXZrIm_bª`aui^�\_^?pybh`�bamnsZrim��}W?ptrv`=� x g]`�YEgib�`�bh`am_\_\ x ^�ptW9W=XZg]rw`a^=zÕjlXZe=~��r XZehzi^=eR`fX�^dqtpy\_g}py`a^¢`aui^Jm_YR[igiehmn`��ÔW9XZrv`feam x g]`am_XZrib|`fX�`fui^J^9YTmnbabhm_XZrFmn`Qmnb�ri^9W=^9babfpyeh�¼`aX�wriX2o `fui^�m_YR[igieam�`��£W=XtriW=^9rw`aefpy`am_XZr]bPm_rR`fui^�zimnqt^=e�`fXZe�[i\ ptbhY�p�~�����[]m_W?py\�qypt\_g]^=b�jlXZeª¤Z¥$�"pteh^sZm�qZ^=r�m_rç¢W ¿ efptW��y^=rÔ^9`�pt\±~8²�´2®Z®�Øvµd~TÓ�m_stgiea^RÏ�~�´2¬IbauiX2o�b�`fui^RW=XZrv`aeam x g]`fmnXZrib;`fXI`fui^£`aXt`�pt\m_rv`a^=ribhmn`��QXyj�`aui^;bh��rv`fui^d`fmnW�bh[�^9W9`fehgiY�jlehXZY�W=pte x XZr��iX?©���sZ^=r¢ptriz x ^=e���\n\_mngiYkmnYT[]gieam�`fm_^9bPp�``fu]^PYR^=zim_ptr�W9XZriW9^=rv`feapy`fmnXZribôXtj�¢W ¿ efptW��y^=r¢²�m±~�^y~�mnYT[]gieam�`��6W=XtriW=^9rw`aefpy`am_XZr]b�Xtj-´l"Q�i´l"ëpyriz¬]~¯­m"�XtjP`fui^Tzi^=g]`a^=ehm_giYAzi^9ribam�`��+eh^=bh[�^9W9`fm�qZ^9\n�]µd~I��ui^Tm_YR[igieam�`��UW=Xtrw`aeam x g]`fmnXZr���pt\n`auiXZgistu[ieh^=ba^9rv`?�³ziX�^=b�riXt`ªbam_strimn�}W=ptrv`f\��Qpt\�`f^9eP`fui^;b���rw`aui^9`am_W�ba[c^=Wd`fehgiY¢~8Ó}XZe8`fui^�ea^9bagi\�`fb�bhuiX2o�r¢mnr�}sZg]ea^£Ï�~¯®Tptriz)Ï�~�´2¬£`fui^�uv��z]eaXZsZ^9r x � x � x �·j�ptriz¢m_YR[igiehmn`��0^9YTm�`a`f^9eab�oP^9ea^EstmnqZ^9rJm_z]^=rv`fmnW?pt\\_mnri^6Xtj�bamnsZuv`�\_^=r]st`fuib9�i^=\n^=Wd`feaXtr¢zi^9ribam�`fmn^=b�ptriz0`f^9YT[c^=eapy`fg]ea^=b9~ô�Xy`f^$`fu}p�`ô`aui^8m_riW9\_gibhm_XZr�Xtj�`fui^$qypteamnXZgib¾W=Xtrw`am_rwgigiYÚW9XZrv`fehm x g]`am_XZr]bP²l^9qt^=rEp�`ô`aui^�´2¬�""\_^9qt^=\ µbamnsZrim��}W?ptrv`a\n�Õpt\�`f^=ehb�`aui^Nz]m ptsZr]XZbh`am_W;qypt\_gi^9b�eh^9`fg]eari^9z x �¢pT�]`�`aX�`fui^�X x bh^=ehqt^=zUbh[�^9W9`fehgiYÕ~�õj�`fui^9ba^ÕW=XZrv`fehm x g�`fm_Xtrib|ptea^0^Ñ©�W=\_g]zi^=zÔ`fui^9r¼pUuimnsZui^9e£^=\_^9W9`aeaXZrÔzi^9ribam�`��Ôm_b£m_r�jl^=eaeh^=z���`fu]^�i`|[iehX�W9^=zig]ea^0m_r]W=ea^=ptba^9b£`fui^0YRX�z]^=\Pz]^=ribhmn`���`aX x eaXvpyzi^=r�`fu]^0\_m_r]^=bEbhX)`au}py`E`aui^0YRm_rimnY�px ^9`�o�^=^=r�zim_bhW=ea^d`f^�\_mnri^=b�W=ptr x ^�YTpy`fW�u]^=z�~ߪà�ë�à�ë nIç�ä�äiçhï�ýÀä¡ü�íporqms ÷;î�äiîÀî�ï�÷ ÷�çéî}ý�ï6å�ðwä}çéè\î¡íôð]ì;í�ä]ð�àÓ�m�`fbªo�^=ea^�[c^=e�jlXZeaYR^=zIXZrÕz}py`fp£jlehXZYk`�o�XRzimnû�^=eh^=rv`�zi^=r]bamn`��Iehgirib�o�ui^9ea^�`fui^;`fXt�ypyY�pt�|oPptbW=Xtr]�}sZgieh^=zUbhgiW�uJ`fu}p�`�`aui^Nb�`feamn�t^E[cXZmnrw`�oPptb6XZr¢`fui^�qZ^=e�`fmnW?pt\�ptriz�`aui^=r�XZr¢`aui^EuiXZehm_ú9XZrv`�pt\[i\_py`f^9b�Xtj}`aui^Pz]mnqZ^9eh`aXZeª²±¤Z¥$�ñ[]gi\_bh^PrwgiY x ^=ehb ¡ ¹åµy¹vÏ�pyriz ¡ ¹åµy¹¶Øvµd~�� x ba^9ehqypy`am_XZr}py\�zipy`�p�jleaXZY÷ ��¹|oPptb�p?qtpym_\ p x \_^�jlXZe�`fui^9ba^�`�oPXT[igi\_bh^=b�jlXZe�p£efptr]sZ^�Xtj�`am_YR^6ba\nm_W=^9b�ptrizI`fefpyW��0r�g]Y x ^9eab9~Ó}XZe�W=XZribhm_b�`f^=r]W9�Jm�`�oPptb�zi^=W9m_zi^9zJ`fXQ^Ñ©ipyYTmnri^�`aui^Neh^=bhgi\n`ab�jleaXZY�`aefptW��ÕrwgiY x ^=e�«TjlXZe x Xt`au

Page 162: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ Ø

1e+08

1e+09

1e+10

1e+11

1e+12

1e+13

3600 3650 3700 3750 3800

Em

issi

on c

oeffi

cien

t

t

Wavelength \ A

Total emission coeffHydrogen paschen continuum

Hydrogen bremsstrahlung

Ó�mnsZgiea^¢Ï�~�®]Â�§Úb9k�h�q�ijsneAh�ipb9knx�h�b h�vDo7o$£�iMx�xJipb9k r�buo�ÿ#r�ijo$kOh���q�b9£[h�vDo%lAivu�o$q�o$k�h#v�tAlAq�b9wOo�kr�b9£�d�b9kDo$k�h�x$~��vDo_r�bAlAo}�¢gOx�q�eDk ��b9q¨� � �ç�}�Ü�(� ��� ��� ��� �n� � � �$à�é��no$£�izh�h�ipkDw#fpo�kDw9h�v��xw8y w9���-~¢��vDo<sDq�o�£5x�x�h�q�g9vDfpeDkDwo�£�iMx�x�ipb9kÊiMx�gOx�x�eD£�o$lÜh�b-r�bO£�o#��q�b9£ h�vDo x�g9£�o.o�£�ijhJh�ipkDw-q�o$w9ipb9k�g9x<h�vDo.s²bOeDknluy{s�b9eDknlÍgSknl s�b9enknluyP��q�o$or�bO£�d²bOkDo�k�h�x$~

1e+09

1e+10

1e+11

1e+12

1e+13

1e+14

3600 3650 3700 3750 3800 3850 3900

Inte

nsityz

Wavelength

Total intensityHydrogen b-b and b-f contribution

Hydrogen Bremsstrahlung contributionimpurity contribution

Ó�mnsZgiea^QÏ�~�´2¬]ÂͧÚbOkOh�q�ipsDeAh�ijbOknx}h�b7h�vno#h�b9h�g9f�ipkOh�o�knx�ijh�t-��q�b9£ gSfpf�h�vDo'vutAlAq�b9wOo�kÊgSknl�ij£�dDeDq�ijh�t-o$£�ijhJh�o�q�x�~��vDo�ipkDdDeAh<dng9q�g9£�o�h�o$q�x��¢o�q�o�gOx´ijk-mnwOeDq�o�wD~ �n~���vDo5xJh�o$dÜipk-h�vDo5ip£�dDeDq�izh�t�r�b9k�h�q�ipsDeAh�ipb9kÜiMx´s²o(r�gSe²xJo�h�vnoip£�dneDq�ijh�t.r�bAlAo}g9x�xJeD£�o(x�v�tAlAq�b9wOo�kDiMr�r�b9k�h�ijkueDeD£ o$lAwOo$x¢��b9q�o(g9r�v ijbOkDiMx�gSh�ipb9k.xJh�g9w9oO~

Page 163: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´ ¡ ®

-5e+12

0

5e+12

1e+13

1.5e+13

2e+13

2.5e+13

3e+13

3740 3760 3780 3800 3820 3840 3860 3880 3900

Inte

nsi

ty{

Wavelength / A

high density caseSynthetic feature

Ó�mnsZgiea^�Ï�~n´Z´tÂY|«iMgSw9knbOxJh�iMr¨mnh�h�b�dDeDfMxJoY}�êd~Sê�wA�u¤�gSfp£�o�q®x�o�q�ijo(xÚfpip£�ijh«x�d²o(r�h�q�eD£��uh�q�gOrPO��D�uh������D~ ��w9x[igi\nba^9b=~�V�r£ptr}pt\���bam_b�Xtj}`fu]^=ba^P[igi\_bh^=b�uiptb$pt\nea^?pyz]� x ^9^=r£[c^=ehjlXteaYR^=zI²�¢^=mnsZb�^d`�pt\·~���´2®Z®¶Ø6pyriz¢^9m_sZb�^9`;py\·~���­t¬t¬Z¬vµªo�ui^=eh^E`fui^NÓ ! ö �Xtj�`fui^£®���­��ô´2¬��é­Tptriz+´Z´9��­|\_mnri^=b�oP^9ea^EYR^?pybagieh^=zptriz|`fu]^�^9\_^9W9`fehXZr|zi^=r]bamn`��|mnr]jl^=ehea^9z�~�V�rR^=\n^=Wd`feaXtrRz]^=ribhmn`��|m_b$z]^=zigiW9^=z x �ÍÕ^=m_stb�jlXte$^?ptW�uRXtj`fu]^=^�`fuieh^=^�\_mnri^=b�m_rizi^9[�^9rizi^9rw`a\n�£g]bam_r]s�p�¨�`fptea��[ieaXy�}\_^�^Ñ©][]ea^=bhbamnXZrRjlehXZY ° ea^=st^=e=�³´2®t®¶Ø]~�Ó}XZepQYR^?ptrimnristjlgi\�W=XZYR[}ptehm_bhXZr�`fX x ^EsZm�qZ^=r»�-mn`6oPptb;zi^9W=m_z]^=z)`fX0YRX�zi^9\�^=ptW�u�[]gi\_bh^|py`�p�`fmnYT^ba\nm_W9^ªjlXZe$o�uim_W�u�Ó ! ö ,z}py`fp�ptr]zTui^9riW=^�zim pysZriXZbh^=zR^=\_^9W9`aeaXZrRzi^=r]bamn`��|uiptz x ^=^9r�X x `�ptmnri^=z�~Ó�mnsZgiea^6Ï�~�´Z´;bauiX2o�b�XZgieªXt[]`fmnYTmnba^=zT�i`ª`fX£`aui^;z}py`fpEXtjô[igi\_bh^ ¡ ¹åµy¹vÏ�~}ô�Xt`f^6`au}py`ª`aui^6�i`�oPptbXZ[]`am_YRm_bh^=zQm�`f^=eapy`fm�qZ^9\n� x �Qu}ptr]z�~���ui^�ptg]`aXZY�p�`fm_W�\_^=ptbh`�bh×wg}ptea^9b�XZ[]`fmnYTmnbamnris�Xyjô`aui^6jl^?p�`fgieh^`fX�`fu]^�zipy`�p�m_b�m_r|zi^dqZ^9\_XZ[iYR^=rv`Pptb8p�jlg�`fgieh^�V�§�V6¨RW9X�zi^y~���ui^�zim pysZriXZb�`fm_Wªqypt\ngi^=b$jlehXZY `fu]^�i`�ptea^�bau]X�o�rUmnrÕ`fp x \n^EÏ�~�´�o�mn`fuÕ`fui^�ea^9bagi\�`fb�jleaXZY�`aui^EÓ ! ö ptript\n��bamnb�Xyj<¢^9m_sZb�bauiX2o�rm_r0`fp x \_^�Ï�~Ö­�~��ui^;bh^=W=Xtriz0�i`�[c^=ehjlXteaYR^=z�oPptbª`fX�`fui^6eh^=bhgi\n`ab�Xtj¾[igi\nba^ ¡ ¹¶µt¹¶Ø]~$Ó�m_sZgieh^;Ï�~�´�­�bhuiX2o�b�XZgieP�]``fXQ`auim_b�\_X2o�^=e;zi^9ribam�`��Jz}py`fp]~;��ui^£zi^9zigiW9^=z{[}ptefpyYT^d`f^=ehb;pteh^£bauiX2o�r)m_r�`fp x \n^|Ï�~�¹QptrizU`fu]^Ó ! ö eh^=bag]\n`fb�mnrI`fp x \n^�Ï�~ ¡ ~

Page 164: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´�Ïy¬

� uv��bamnW?pt\¾×wg}ptrv`fm�`�� �8pt\ngi^6jlXZe�b���rv`aui^9`am_W�jl^?py`agiea^¥�\n^=Wd`feaXtr¢zi^9ribam�`�� ¹�Î+´2¬ Ï�Ð Ñ9Ò ÓÇÕ��XtrÕz]^=ribhmn`�� ¹�Î+´2¬ Ï�Ð Ñ9Ò ÓÇÕ¥$\_^=Wd`fehXZrI��^9YT[c^=eapy`fgieh^ ´AÖD×ó-^=r]st`fuÕXtj�\_m_r]^6Xtj�bam_stuw` ¹�Ñ9Ò��YT[ig]eamn`am_^9b ¿ pte x XZr�����©���sZ^=r»� ° ^=e���\n\_mngiY

��YR[igiehmn`��0p x giriz}pyriW=^9b ´8"QÊ?´e"0Êf¬�óÖ­�"��p x \n^�Ï�~n´tÂ}��vno´lDipg9w9kDb�x�h�ipr�|�gSfpeDo$x�q�o�h�eDq�kDo$l'��q�bO£ h�vno´mDh«x�vDb���k'ipk mnw9eDq�o�wA~p�9�

° py\_YR^=eª\nm_ri^ Ó ! ö �� §6m ptsZr]XZba^9zÕz]^=ribhmn`�� ÑSÒ ÓÇÕ®��é­ ¡ ~¯«åµy®v­t®v­Z­ ¡ ón´?®�Î�´2¬ Ï�Ð ÑSÒ ÓÇÕ´2¬��é­ Ï�~pØ]´2Ït¬v­�´2« ¡ óÖ­y®�Î�´2¬ Ï�Ð ÑSÒ ÓÇÕ´Z´9�é­ µ�~ÖÏZÏy®¶ØZ¹Z«vÏ ¡ óÖ­y®�Î�´2¬ÇÏ�ÐSÑSÒÔÓÇÕ��p x \_^�Ï�~Ö­�ÂY|¨iMgSwOkDbOx�o$l.o�fpo$r�h�q�b9k�lAo�knx�ijh�ipo$x¢��q�b9£ h�vDo<°6�C�G�[gSk²gSfptux�iMxÚb9���%o�ipwOx$~

0

1e+12

2e+12

3e+12

4e+12

5e+12

6e+12

7e+12

8e+12

9e+12

1e+13

3660 3680 3700 3720 3740 3760 3780 3800

Inte

nsi

ty{

Wavelength / A

lower density caseSynthetic feature

Ó�m_sZg]ea^�Ï�~�´�­�ÂY|¨iMgSwOkDbOxJh�iMr«mDh�h�b�dDeDfMx�oY}Oêd~Sê��n��¤�gSfp£�o�q®x�o�q�ijo(x�fjip£�izh¨x�d�o$r�h�q�en£ �Dh�q�g9rPO��D�uh������n~ }>w

Page 165: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´�Ï�´

� uv��bamnW?pt\¾×wg}ptrv`fm�`�� �8pt\ngi^6jlXZe�b���rv`aui^9`am_W�jl^?py`agiea^¥�\n^=Wd`feaXtr¢zi^9ribam�`�� ¬�ójØvÏÊÎ�´2¬ Ï�Ð ÑSÒ ÓÇÕ��XtrÕz]^=ribhmn`�� ¬�ójØvÏÊÎ�´2¬ Ï�Ð ÑSÒ ÓÇÕ¥$\_^=Wd`fehXZrI��^9YT[c^=eapy`fgieh^ ´AÖD×ó-^=r]st`fuÕXtj�\_m_r]^6Xtj�bam_stuw` ´?«¶Ñ9Ò��YT[ig]eamn`am_^9b ¿ pte x XZr�����©���sZ^=r»� ° ^=e���\n\_mngiY

��YR[igiehmn`��0p x giriz}pyriW=^9b ´8"QÊ?´e"0Êf¬�óÖ­�"��p x \n^�Ï�~¯¹�Â}��vno´lDipg9w9kDb�x�h�ipr�|�gSfpeDo$x�q�o�h�eDq�kDo$l'��q�bO£ h�vno´mDh«x�vDb���k'ipk mnw9eDq�o�wA~p�(³

° py\_YR^=eª\nm_ri^ Ó ! ö �� §6m ptsZr]XZba^9zÕz]^=ribhmn`�� ÑSÒ ÓÇÕ®��é­ ­�~Mµ ¡ ´2®Z«Z¹åµ ´¶ópØ�Ø�Î�´2¬ Ï�Ð ÑSÒ ÓÇÕ´2¬��é­ ¹�~¯«]´?¹v­ZÏZ­t¹ ­­ón´?¬�Î�´2¬ Ï�Ð ÑSÒ ÓÇÕ´Z´9�é­ ¡ ~ÖÏZ­y¹Z®vÏ�´2® ­­ón´?¬�Î�´2¬ÇÏ�ÐSÑSÒÔÓÇÕ��p x \_^�Ï�~ ¡ ÂY|¨iMgSwOkDbOx�o$l.o�fpo$r�h�q�b9k�lAo�knx�ijh�ipo$x¢��q�b9£ h�vDo<°6�C�G�[gSk²gSfptux�iMxÚb9���%o�ipwOx$~

�C7�<dH#01<><d7h,�9�,/^r58f�;m�15

V�b�XZri^�W?ptr¢ba^=^�jleaXZY�`aui^NsZeapt[iuib�ptrizÕ`fui^�XZ[]`fmnYTmnba^9z¢mnri[ig]`�[}pteaptYT^d`f^9eab=�cea^=ptbaXtr}p x \n^��i`abu}p?qZ^ x ^=^=r+ptW�uimn^9qZ^9z�~���u]^T[ieh^=bh^=rv`��]`fb�`fX0`fui^Rba[c^=Wd`feapI��m_^9\_z)bh��b�`f^=YTpy`am_W?py\_\n��\_X2o�^=e�zi^=r��bam�`fmn^=b�`au}py`E`auiXZbh^0ea^9`agiear]^=z�jleaXZY `fu]^0Ó ! ö pyr}pt\n��bhm_b=~U��u]m_b£m_b x ^9W?ptg]ba^Q`fu]^0W=XZrv`fmnrwg}pW=Xtrw`aeam x g]`fmnXZr¢ptea^�riXt`�W=XZribhm_zi^9ea^9zJm_rI`fui^NÓ ! ö ptr}pt\���bhm_b9~�V�b�oPptb�m_rizimnW?py`a^=zÕm_r¢ba^=Wd`fmnXZrÏ�~�¹]~¯­��wri^=sZ\n^=Wd`fm_r]s�`aui^�W9XZYR[i\_^d`f^�ba^d`PXtj»W=XZrv`am_rwgigiYÀW=Xtrw`aeam x g]`fmnXZrib�W?pyr�\_^=ptzib$`fXNptrRm_r]jl^9eaea^9z^=\n^=Wd`feaXtr�zi^9ribam�`��0`fu}py`�mnb�bh[igieamnXZgibh\n�0uimnsZu�~PÓcpym_\_g]ea^�`aXQptW9W=XZgirv`�jlXZe�pt\_\-W9XZrv`fmnr�g]giY,W=XZrv`aeam��x g]`am_XZr]b�Y�p?�Ipt\nbaX£^Ñ©][]\ ptmnr¢p|bamnYTmn\ pte8zimnbaW=eh^=[}pyriW9�IriXy`f^=z¢ptb�[cXZm_rv`�²lm_m_m µ8Xtjô`fu]^�bh^=ehm_^=b�\nm_YRmn`ptr}py\n��bamnb�Xtj�b�`f^9\_\ pyeP¸³pteh^=b�m_r¢º�pteheaX�� º»mneam_r{²é´2®¶ØvÏtµd~�õ`�bau]XZgi\_z x ^�riXt`f^9z0`fu}py`�X x `�ptmnrim_r]s£ptrIXZ[]`am_YTpt\��i`�pyb�bauiX2o�rJp x X2qZ^�mnbªriXt`�ühgib�`�p|YTpy`h`f^=eXtj�Y�p�`fW�uimnris�`fu]^R\nm_ri^�o�m_z]`auib;XZrU`fui^£u]m_sZu)ba^9eamn^=b;\_mnri^=b9~N��ui^|W9X�zi^|eh^9`fg]earib;`fu]^|p x baXt\_g]`a^m_rv`a^=ribhmn`���Xtj�`fui^�ba[c^=W9`aeagiY�ptriz�m�`Em_b�ri^=W9^=bhbfpte���jlXZe�`auim_b�`fXUYTpy`fW�u�`fui^QX x ba^9ehqypy`fmnXZrib£pybo�^=\_\±~���u�g]b�`fu]^�\nm_ri^6o�mnz]`fu]b=�}zi^9sZea^9^�Xtj�\_mnri^ x \_^9rizimnrisRptriz¢p x baXt\_g]`a^�mnrv`f^=r]bamn`��0pt\_\�uip2qt^�`aXx ^NW=Xteaea^9W9`�jlXZe�pRsZX�X�zÕ�i`�`aX x ^EptW�uimn^9qt^=z�~���uim_b�m_rv`f^9ear}py\�W9XZribamnbh`a^=riWd�Jptz]zib�qypt\ngi^�`aX�`fu]^zim_ptsZriXZb�`fmnW6[}pteaptYT^d`f^9eab�ea^d`fgiehri^=zIjleaXZYk`fui^;�i`=~

Page 166: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´�Ït­

¨�XZYR^�zi^9qt^=\nXZ[iYR^=rv`�Xyj�`fu]^�W9X�z]^�mnb�eh^=×wgimnea^=z x ^9jlXZeh^�m�`�W?ptr x ^�gibh^=zJpyb�p|zim pysZriXZb�`fm_W6`aX�Xt\·~��ui^��ieabh`P`fuimnris�`aXEriXt`a^6m_b8`fu}py`�`fui^9ea^;pyea^�YTptrv�|qypteam_p x \_^9b8`au}py`ªpyû»^9W9`ª`aui^��]`?~$V�r0g]riW=XZr��bh`aefptmnri^=z{YEgi\�`fm��õqypteam_p x \n^�XZ[�`fm_YRm_bh^=zU�i`�o�XZgi\_z{W=^9eh`fptm_ri\��Uea^dqZ^=pt\�Y�ptrv�)\nX�W=pt\�YRm_rimnY�pQpyrizptr�ptY x mnsZgiXZgib�bh^9`6XyjP��XZ[]`fmnYTmnba^9z��}[}pteaptYT^d`f^9eab=~�VrwgiY x ^=e�Xtj�`aui^�qyptehm p x \_^=b9�}bagiW�u)pyb�mnY|�[igiehmn`���W=XZr]W=^=rv`aefpy`am_XZrib�pyea^ªXZr]\n��oP^=pt�w\n�EXteh`fu]XZsZXZr}py\�jlehXZY`aui^ª[cXZm_rv`�Xtj³q�mn^9oFXtj³W=Xtrw`am_rwgigiY^=YRm_bhbamnXZr�~���ui^|r�g]Y x ^9e;Xtj8ba^=pteaW�u{[}pteaptYT^d`f^9eab;YEgib�` x ^£eh^=zig]W=^=z»��XZe�baXZYR^Q�¯st\_X x py\�XZ[�`fm��YRm_bfp�`fm_Xtr���Xyj�pQo�uiXZ\n^ReaptrisZ^£Xyjªzim_ptsZriXZb�`fmnWEYT^=ptbag]ea^=YR^=rv`ab�mnriW=\ngizim_r]s�`fuimnb�[iea^9ba^=rv`�ba^9eam_^9b\_mnYTm�`ªXZr]^�YEg]bh` x ^�ptziXZ[�`f^=z»~�Ó³Xte�`aui^�zi^=YRXZrib�`fefp�`fm_Xtr¢ui^9ea^y�co�^Nu}p?qt^�jl^=\�`�p x \_^;`aX�ba[c^=W9mnj¶�pNrwgiY x ^=ePXtj¾[}pteaptYT^d`f^9eab�`aX£pNeh^?ptbhXZr}p x \n^�zi^9sZea^9^;Xtj¾ptW9W=gieaptW9��m_riz]^=[c^=rizi^9rv`f\n�J²�^9ba[c^=W9m pt\n\n�m_YR[igiehmn`���p x gir]z}ptriW9^Tpyriz)\_^=r]st`fu)Xtj8^=YRmn`h`fmnrisQea^9sZm_Xtr³µ�`fXÕpt\n\_X2oë`fui^£eh^=YTptm_r]m_risQ[iptefptYR^d�`f^9eab�`fX x ^Tqyptehm_^9z+ptr]z�p¢YR^?ptr]m_ristjlg]\��i`EpyW�uim_^dqZ^=z»~ ! ^�riXt`a^R`au}py`�`fui^R`fui^T\_mnri^d�õXtj ��bhm_sZuv`\_^9rist`au�giba^9zñmnr�`fui^Q�]`£Xtj�[igi\nba^ ¡ ¹åµy¹vÏ¢m_bN\_XZrist^=eN`au}ptr�XZri^0YRm_stuw`N^d©�[c^=W9`=~�Ó}XZeN`���[imnW?pt\zim�qZ^=e�`fXZe�YRX�z]^=\_b9��`aui^�\_^=r]st`fuIXtj�^=YRmn`h`fm_r]s£ea^=stm_XZr0mnb'ñë¹7¸ÔϲÑ9ÒU~<�õ`�m_b�[iehX x p x \_^6`fuipy`�W=XZr��`fehm x g�`fm_Xtrib6`fXI`fui^RYR^?ptbhgiea^9z+m_rv`a^=ribhmn`��{pt\nbaXÕW?ptYR^£jleaXZY�`aui^R[}pteh`�Xtj ÷ ��¹���b�\_mnri^d�õXtj ��bhm_sZuv`[}ptbhbamnrisN`fu]eaXZgistuQ`fui^6W9XZea^�Xtj-`fu]^6[i\ pybaYTp]~�V�[cXZbhbam x \n�Rm_YR[ieaX2qt^=YR^=rv`P`aXEXZgie�YT^9uw`aX�z�m_b�`aXgibh^�`aui^ ÷ ¨ ¡ mnribh`aeagiYR^=rv`�ptb�p|ba^=W9XZriz¢ea^9jl^9ea^9riW=^�ba[c^=Wd`feaXtYT^d`f^=e9~��õ`�mnb�bam_YRm_\_pte�`fX|`fui^ ÷ ��¹m_r]bh`fehgiYR^=rv`?��^Ñ©�W=^=[�`�`au}py`;m�`fb6\nm_ri^Ñ��Xtj�bam_stuw`6^d©�W=\ngizi^=b;`aui^Ezim�qZ^=e�`fXZe;eh^=sZmnXZr�~;��uimnb;W=XZg]\_z x ^gibh^=zI`fX£zi^d`f^=ehYTmnri^6`fu]^;W=XZrv`fehm x g�`fm_XtrQ`fX£`fui^;mnrv`f^=r]bamn`��TjleaXZYkea^9sZm_XZr]bªXZg]`abam_z]^6`fui^;zim�qZ^9eh`fXteptrizÔui^9riW=^ x ^0giba^9zñ`aX{ptz�ühgib�`£`fui^ ÷ ��¹UX x bh^=ehqypy`am_XZr]b=~�¥�qZ^9rño�m�`fu�`fu]m_bEm�`£m_bE[]eaX x p x \n�ri^9W=^=bhbfpte��0`fX|YRX�zi^9\�`aui^�[iea^9zim_Wd`f^=zImnrw`a^=ribhmn`��Ipyb�XZeamnsZm_r}p�`fm_r]s�jleaXtYÞ`�o�XTz]mnû�^=ea^9rv`�^=YRmn`a`am_risea^9sZm_Xtrib�o�mn`auim_r£`fu]^�z]mnqZ^9eh`aXZe8ptb�m_b$ziXZr]^�mnr � mnsvpteaX2qN^9`�pt\·~}²�´2®Z®�Øvµd~�¢XZeh^�mnr]jlXZeaYTpy`am_XZr�jleaXZY¤Z¥$�ù[i\_ptbaYTp£YTX�zi^9\_\_mnrisNW9X�z]^=b�m_b�ri^9^=zi^9z¢`aXTsZg]m_zi^;bhgiW�uJptrÕpt[][ieaXvpyW�u�~.0/�� �ÞGRKUC�<=SU:�7=GRKU:��ui^�sZ^9ri^=eapy`fmnXZrIXtj�W9XZYR[i\_^d`f^;ba^9eamn^=b�\_mnYTm�`Pbh[�^9W9`feapt\�jl^?p�`fgieh^=b�u}ptb x ^=^=rÕzi^9YTXZr]bh`feapy`f^9z�pyriz`fu]^{�i`h`fmnrisÔXtjEbhgiW�uùjl^?p�`fgieh^=b¢`aX*¤Z¥$�Ýz]mnqZ^9eh`aXZeIX x ba^9ehqypy`am_XZribÕbau]X�o�r»~(½�X�X�z"�i`abÕo�^=eh^ptW�uimn^9qt^=z�jlXZe x Xt`fu�p+�¯\nX�o��ª²��  ñ�¬�ójØvÏÌÎ ´2¬ Ï�Ð ÑSÒ ÓÇÕ µNptrizÔp{�¯uimnsZu���zi^=ribhmn`�� ²��  ñݹ δ2¬ÇÏ�Ð�Ñ9ÒÔÓÇÕdµªX x bh^=e�qtp�`fm_Xtr�~ª��ui^�ri^d©�`�bh`�pysZ^�o�mn\_\ x ^�`fXT`fptm_\nXZeP`aui^�W9X�z]^�`aX x ^�YRXZeh^�bh[�^9W=m��}W`fXT`fui^|¤Z¥$�zimnqt^=e�`fXZe�ba^9`agi[���jlXZe�^d©]ptYR[i\_^�`�pt�wmnrisQm_rv`aX0ptW=W9XZgirv`�pt\n\�Xyj$`fui^E^9YTmnbabhm_XZrÕjleaXZY`fu]^ª\_mnri^�Xyj³bam_stuw`=�triXy`�ühgibh`�`fuiXtba^PjlehXZYÚ`fui^ªz]mnqZ^9eh`aXZe=�vpyrizEzim_ptsZriXZbhm_ris�`fui^Pm_YR[igieam�`���W=XZrv`a^=rv`

Page 167: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

´�Ïy¹

jlXZeª`aui^�¤Z¥��¦[igi\nba^=b�`fuipy`�oP^9ea^�ptr}pt\���ba^=z»~��uimnb�o�XZeh�Rmnb�[i\ ptr]ri^=z|jlXZe$`fui^�mnYTYR^=zim_py`f^�jlg]`fgieh^t�w`fui^�ptmnY x ^=m_r]s�`aX�uip2qt^6p�W=X�zi^�`fu}py`8W=ptr[c^=ehjlXteaY�ptrEXZ[]`am_YRm_bh^=z��]`�`fX6`aui^�¤Z¥$�Ôuim_sZuNba^=ehm_^9b ° pt\nYT^9e�X x bh^=e�qtp�`fm_Xtrib�jlehXZY`aui^ªzim�qZ^9eh`fXteea^9sZm_Xtr�ptriz+ea^9`agiear+zim ptstriXZbh`am_WRm_r]jlXteaYTpy`fmnXZr)XZr+`fu]^�\_X�W?py\8[]\ ptbhY�pI^=\n^=W9`aeaXZr�`f^=YR[c^=efp�`fgieh^ptriz)^9\_^=Wd`fehXZr)zi^=ribhmn`��t~E��uim_b;o�mn\_\ x ^£mnriW=XZeh[�Xtefpy`a^=zUm_rv`aXI`aui^£^d©�mnbh`fmnrisÕ¤Z¥$� z}p�`�pIptr}pt\���bam_bbaXyj¶`�oªpteh^t~���ui^E[iehXZ[cXZba^9zJW=X�zi^�o�m_\_\-[c^=e�jlXZeaY!pR\_^?pybh`�ba×wg}pteh^=b��i`=�»pyriz�o�m_\n\-W9��W=\n^�`auieaXZg]sZuYEgi\�`fmn[i\_^�ba[c^=W9`aefpRXZriW=^;`aui^�m_rim�`fm py\c�i`h`fmnris|[}ptefpyYT^d`f^=ehb�u}p?qZ^ x ^9^=rÕba^9`=~���rÕzi^dqZ^=\nXZ[imnris£`fuimnbW=X�zi^y�³ptz]qyptrv`�ptst^�o�mn\_\ x ^;`�py�t^=rÕXtjô`fui^�b�`feag]W9`fg]ea^�Xtj�`aui^�^d©�m_b�`fmnris|V6§�V;¨Q�i`a`am_ris|eaXtg]`fmnri^=b9~

Page 168: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� � ��� ����

� � � ������ ����� ����� ������� � ��� � ��� ��� �!�

" �#��$

%'&)( *,+.-/+.0�13254768-94�2;:9<>=?68-9<@13-9A <B:9C C 1D0BE

FHGJILK�MNGPO?KQILK>GSRTK>UJVWO?KQOYXZMNO?[8R]\^O?XJOYVNR^_`IaXZbcOYKWMQIa\ZRTMQIadcXeIaXZMQd]KQUfO?ghMNVQR^_PKWO?VQILO?K�O?i8IaKQKWIadcX3R^XJ[DUfdcUkjl _aRTMNILdcX.imd`[JOY_a_aILXJ\nbTR^_LIa[.opdcVrqsdTMNGtR^KWMQVQdcUJGZu`KQILg;R^_`R^XP[3_vRTqsdcVQRTMNd^VWu]o l KQIad^XtUJ_aR^KQimR^KYw>xyMzR^V{MNIaXP\| ILMQG}R^X}d)bcOYVWb`IaO | dTosqJR^KQILgnO?KWg;R^UfO~o�R^ghMNdcV�MNO?gzGPXJIa� l OYK?�yMNGJOYIaV l KQOYo l _aXJOYKQK�IaX3imd`[JO?_L_aILXJ\7dcUSR^g�jILM�u3OY�sO?ghMNK�dcX�UsdcU l _vRTMQIadcX3KWMNV l gYM l VQO�R^XJ[�OYimOYVQ\cOYXyM�� l`� | RTK�KWGJd | X | I�MNGJILXDMNGJO7g?d^XyMQO � M�d^oopd l V�KQOhMNKBd^o�d^qJKQOYVWbTRTMNILdcXJKYw>��qPVNR^XJgzGPIaXJ\�_LIaXJO�VNR�MNIad�KWM l [Putd^oSMNGJO�KQdc_aR^V�RTMNi8dcKQUPGJO?VWO~[JO?i8dcXkjKWMQVNRTMQO?[9MNGPO l KQO�d^o�O?KWg;R^UfO�UJVQd^qSR^qJIL_aILMQIaOYK]R^K�RTX�d^UPMNILg;R^_�[JO?UPMQG�[JIvR^\^XJdcKWMQIagTwD��qJKWO?VWbTRTMQIadcXPKd^o�MQGJOD�B�y�T����VNRTMQIad�opVQdci�MNGPOD[JILb^O?V{MNdcV�VQO?\^IadcX�d^o�MNGJOt�.xP�.�B� l UJ\cVNRT[JO3KWGJd | OY[�MQGJO l KWOd^o�R^qPKQdcVWUPMNILdcX o�R^ghMNdcVWK�IaX/g?R^_ag l _vR�MNIaXP\mdcUkMNIag?R^_a_�u¡MNGPIagz¢£UsdcU l _vRTMQIadcX�[JO?XPKQILMQIaOYK?w�xkOYVQILO?K�_LIai8ILMdcqJKWO?V{b^R�MNIad^XJKnd^orKWdc_vRTVnUJVWdcimILXJO?XPg?O?KnR^XJ[¡MQGJO�¤c��F�[JILb^O?VWMQdcV~GJIL\cGJ_LIa\cGZMNOY[8MQGJO�UfdcKQKWIaqJIL_aI�M�u}d^o[JOhbcO?_LdcUJILXJ\�R KQUfO?ghMNVQR^_nopO;R�M l VWO¡g?d`[JO'R^K}RTX�O?_aOYgYMQVQdcX![PO?XJKWILM�uc��MQO?i8UsOYVNRTM l VQO¥RTXJ[�UJ_vRTKQimRVQOYg?dci3qJIaXSR�MNIad^X¥KWMzR�MNO�[JIvRT\cXJdcK{MNIagTw��MQGJdcVQd l \cGDMNVWO;RTMQimOYXZM�d^o�_ad | KWO?VQILO?K�i8O?iDqfO?V�dcUSR^gYILM�u | RTKBMNGJOYX8[POYbcOY_adcUfO?[}ILX�R^KQKWdkgYIvRTMQIadcX| ILMQGDMNGJO7gYdcXZbcO?VWKQILdcXmdTofMQGJOHg?d`[JOHd^os¦nOYGJVQILXJ\cOYV�IaXZMNd.�]�§��x8¨`©;ªJw>FHGPO l KQOHd^ofMNGJOHXPO | gYdk[POIaX�MNGJOD«~��x _aIaXPOtIL[JO?XZMNI�¬Sg;R�MNIad^X | dcVQ¢ | RTK�UJVQOYKQOYXyMQO?[�R^XJ[9RTX9O � MNOYXJKQILdcX9d^o�MNGPODimd`[JOY_rMQdIaXPg?_ l [JO§RtqSR^gz¢y\cVWd l XJ[�VNR^[JIaRTMNILdcX}¬SO?_L[�\cI�bcOYX­wBFHGJO�imd`[JIL¬JO?[mR^[Po�®^ª�¬S_aOYK�UPVQd`[ l g?OY[¡qZu�MNGJILKg?d`[JOeR^VWOeILimi8O?[PIvRTMQO?_Lu8RTUJUJ_aILg;R^qP_aO�ILX£O � IaKWMQIaXJ\��]�§��x'gYdc_a_LIaKQILdcXSR^_¯j�VQR^[JIvR�MNI¯bcO7VQd l MNIaXPO?K?��RTXJ[

©±°Tª

Page 169: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©±°^°

g;RTX3qfO l KWO?[eMNd§IaXZbcOYKWMNIL\ZRTMQO�MQGJO�Oh�²OYgYMQK>d^oPimd`[JO?VQRTMNO�dcUJR^g?I�M�u.dcXeO � g?I�MNO?[3KWMzR�MNO�Usd^U l _vR�MNIad^XJK?wFHGJO8XJO?OY[�MQd'imd`[JO?_>KQO?VWIaOYKe_LIai8ILM�KQUfO?ghMNVNR£_aOY[�MQd¥MNGJO�UJVWdkd^opK.dTo~R^XJR^_LuyMNILg}g?dcXZMQIaX l ILM�u g?dcX`jMzRTIaXJOY[ IaX�gzGSR^UPMQO?V.³kw§FHGPIaK�RT_a_ad | K§dcUSR^gYILM�u£R^[±´ l KWMQimOYXyMQK]MNd8MNGJOe� lk� R^XJ[ UsdcU l _vRTMQIadcXPK7MQdqfOD[JO;RT_LM | I�MNG/KWimd`d^MNGP_Lu�R^g?VWdcKQK�MQGJO}Iad^XJIaKQRTMNILdcX£MQGJVQOYKQGJd^_a[­w�FHGJODMQVQO;R�MNi8O?XZM.d^o�MNGPODKQOYVQIaOYK_aILimI�M�MNR^¢^OYXµILX�MQGJIaK�MQGJO?KWIaKeIaK.gYdcXZMNVNRTKWMNOY[ | I�MNG9MNGJO l K l R^_BMNOYgzGJXJIL� l Omd^o~R^KQK l i8IaXJ\¥MNGJRTMeRKWMNRTMNILKWMQIag;RT_²i8Iag?VWd^¬SOY_a[/¶·[JIaKWKQdc_�bcO?KY¸�qsd l XJ[�O � g?ILMQO?[£KWMNRTMNOYK§ILXZMNd}MNGJO�gYdcXZMNIaX lJl i�wFHGJILK§_LO?[¡MQd8MQGJO.Usd^U l _vR�MNIad^X | dcVW¢¡d^o�gzGSR^UPMQO?V7ª | GJO?VWO3R}GJIL\cG¥� l R^XZM l i�KQGJOY_a_¹g?d^_a_aILKQILdcXSR^_¯jVNRT[JIvRTMQILb^O¡imd`[JOY_HR^XJ[ºg?d`[JO | OYVQO£[JOYb^O?_LdcUsOY[­w»FHGJO£imd`[JO?_�opdcVQi8K}USR^V{M�d^o�R9_aR^VQ\cOYV}UJ_vRTXMNd'i8dk[PO?_>MQGJO�UsdcU l _vRTMQIadcXPK�d^o�GJO;R;byu�KQUfO?gYIaOYK?� | ILMNG9MQGJO�g?d`[JO�[JOYb^O?_ad^UsOY[�ILX�MQGJIaK.gzGJR^UPMNOYVUJVWd±b`IL[JIaXJ\3RtXSR�M l VQR^_si8d`[ l _aO | GJILgzG | Ia_L_�qsO]R^[J[JOY[�MNd}Re_ad | _LOYbcOY_²gYdk[JO�d^o¹KWIai8Ia_vRTVBM�ukUfO�MQd���7�.x}¨T®c¼Pw��]X}d l MQ_aILXJOnd^oSMQGJO~GJO?R;b`u3KQUfO?gYIaO?K�i8d`[JO?_ | RTK�\cILb^O?X­� | I�MNG�R][PO?KQgYVQILUPMNILdcXDd^o�MNGPOR^UJUPVQdZR^gzG�MNd£qfO�MNR^¢^OYXµILX�MQGJOmR^KQKWO?iDqJ_�u9d^o�O;R^gzG�g?dc_L_aILKQIad^XSR^_�j½VNR^[PIvRTMQI�b^Oei8d`[ l _aOTw��]K�USR^V{Md^o�MNGJILKeUP_vR^X­��XJO |¿¾ R l XZMto�R^ghMNdcVWK | OYVQO�[POYbcOY_adcUfO?[²w¥�ÀKWgzGJO?i8O8MQd/R l MNdcimRTMNILg;R^_L_Lu£[JOhMNO?ghMMNGPO¡g?VWILMNILg;R^_�Xkj½KQGJOY_a_aKtqsOhM | OYO?Xºq l XJ[P_aIaXP\/VQOY\cIai8O?K | R^K�RT_aKQd KQGJd | X­w�FHGJO¡GJIL\cG�� l R^XZM l iKQGPO?_a_rgYdk[JO�d^o�MNGJILK7gzGSR^UPMQO?V§OYbTR^_ l RTMNOYK§MQGJOmÁpÂ�Ã;ÄyÃcÅ)ñÆÈǵR^XP[!ÁÉÂTÃYÄyÃcÅ)Ã�Æ{DzÊBqJ_ad`gz¢¡UJVWIai8ILMNI�bcOYK~d^oMNGPOmXJO | KQgzGJOYimOTw¡�ËMNO?K{M3i8d`[ l _aODopd^V�MQGJO ÁpÂ�Ã;ÄyÃcÅ)ñÆÈǭʯÌ/g?dci8UfdcXJO?XZM | RTKt[JOhbcOY_adcUfO?[�RTXJ[g;RTX¡qsO]O � MNOYXJ[JO?[¥MQdDOYbTR^_ l RTMNO]RtgYdcimUP_aOYMQO]DzÊ�Ì�qJ_Ldkgz¢�UJVWIai8ILMNI�bcOTw�FHGJO]g?d`[JO�d^o¹gzGJR^UPMNOYVHª3ILKIai8i8O?[JIaRTMNOY_Lu�R^UJUJ_LIag;RTqJ_aO.MNd8MNGJOeGZu`[JVQdc\^O?X9KQOYVQILO?K]_LIai8ILMnKQUfO?gYMQVNR^_ropO;R�M l VWO | dcVQ¢'d^o�gzGJR^UPMNOYV°kwFHGJO§R^KWKQO?i3qJ_Lu}dTo²K l gzG�KWUsOYgYMNVQR^_JopO;R�M l VWO?K | RTK�MQGJO?X8[JOYKQg?VWIaqfO?[­�yqJVWIaXJ\^IaXJ\�MQdc\cOYMQGJO?VBMQGJO7g?dcX`jMNILX l ILM�u^��d^USR^g?I�M�u»R^XJ[»UfdcU l _aRTMNILdcX | dcVQ¢!dTo�MQGJO'UJVQOhbkILd l K�gzGSR^UkMNO?VWK�RTK | OY_a_7R^K�IaXPg?_ l [JIaXJ\O � UfO?ghMNO?[�gYdcXZMNVQILq l MQIadcX�d^o�Iai8U l VWILM�u'KQUfO?gYIaOYK]MNd�MQGJODqSRTgz¢`\^VQd l XP[9g?dcXZMNILX lJl i£w�� [JOYKQgYVQIaU`jMNILdcX8d^orRtgYdk[PO][JOYKQIa\^XJO?[¥RTKnRe[JIvR^\^XJdcKWMQIagHMNd`dc_SopdcV�MNGJO�¤c��FÍ[JILb^O?VWMQdcV�IaK�MNGJOYX¡\cILb^O?X­w�νM | R^KKQGPd | X£MNGSR�M§MQGJO�g?d`[JOeg;R^X'¬JM§qsdTMNG'GJIa\^G�RTXJ[�_Ld | [JOYXJKQI�M�u'dcqJKQOYVWbTRTMQIadcXJK§R^XP[�VWOYM l VQX£_ad`g;R^_UJ_aR^KQimRDUSR^VQR^imOhMNOYVQK?w

%'&�Ï Ð,0�+�1D<ÒÑc680�ÑZ: ÓB:90�+ÕÔ�6m0�Ö

FHGJO�KWM l [JIaOYK�ILXeMQGJIaK¹MNGPO?KQILK�ILXJ[JILg;RTMQO�KWdci8O�RTVQO;RTK�ILX | GJILgzGtMQGJO l KQOYo l _aXJOYKQKBd^oPO � IaK{MNILXJ\��]�§��xUJVWdkgYO?[ l VWO?K§GSR^XJ[P_aIaXP\8d^USR^g?I�M�u¥g;RTX�qfO�Iai8UJVQd)bcOY[­w�FHGJOeUJVQILXJg?ILUSR^_­d^o�MNGJOYKQOeIaK~MQdmILXyMQVQd`[ l gYOKQd^imO�KQUfO?gYIvR^_`g;R^UJR^g?I�M�utILXt�]�§��xD¨k©;ª§opdcV�R�VNRT[JIvRTMQIadcX.¬SOY_a[tg?dcXJKWIaK{MNO?XZM | I�MNGeMNGJO�UsdcU l _vRTMQIadcX

Page 170: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©±°T×

KWMQV l ghM l VWO^��i8dcVQO8[JOhMzR^IL_aKtR^VQOm\cILb^O?X!ILX!KQOYgYMNILdcX�×Pwبkw�©cw£FHGJO?VWO¥R^VQO8M | d9imR±´WdcVeg?d^XJg?OYUPM l RT_IaKWK l OYK.opVQdci@MNGJILK�MNGJOYKQIaK | GPIagzG�R^VQO}O � UfO?ghMNOY[�MNd¥ILXP� l OYXJg?O�K l qJKWMNR^XZMNIaR^_a_�u d l V.VQO?KWO;R^VWgzGµILXMNGPO�ILimi8O?[JIaRTMNO�o l M l VWO^w�FHGJOYKQO�UP_vR^XJXPO?[8[JOYb^O?_ad^UJimOYXZMNKHR^VWO7d l MN_LIaXJOY[mILXmKWO?ghMNIad^XJK~×PwبkwبeRTXJ[×Pwبkw·³Pw

ÙHÚ�Û~ÚWÜ Ý£Þ7ßPà�á�âPã{ä�á�â5ßPä�å�æÍå�ç�Û~Ü�è�~M�MNGJO7i8dci8O?XZM����7�.x8¨k©;ª.K{MzR^V{MNK | ILMQG8R�i8d`[JIL¬SOY[�¦ndc_�MNé?imR^XJX}UfdcU l _vRTMQIadcX}[JILKWMQVQIaq l MNILdcX­�OYbTR^_ l RTMQO?K3MQGJO¡VQOYK l _�MzR^XZM}VQR^[JIaRTMNILdcX�¬SOY_a[»R^XP[ l KQOYK�MQGJIaKeMNd9g?R^_ag l _vRTMQO¡R^qPKQdcVWUPMNILdcXµo�R^ghMNdcVWKR^XJ['GJOYXJg?OeR3MQGJIagz¢mUsdcU l _vRTMQIadcXmKWMQV l ghM l VWO^w�νo�MQGJO?KWO�UsdcU l _vRTMQIadcXPK~R^VWO l KQOY[ RTK~MNGJO.K{MzR^V{MNIaXP\UfdcIaXZM�d^ofo l V{MNGJOYV�ILMNOYVNRTMQIadcXPK?�ZR.UfdcU l _aRTMNILdcXDKWMQV l ghM l VWO | GPIagzG�IaK�gYdcXJKQILKWMQO?XZM | ILMNG}MQGJOHUJ_vRTKQimRVNRT[JIvRTMQIadcX¥¬SOY_a[�i�R;u'qfODR^gzGJILOYb^O?[­�fMNGJOtKWu`KWMQO?i@qsOYIaXJ\mILMNOYVNRTMQO?[ l XZMNIa_¹MQGJO3UfdcU l _aRTMNILdcX'g?dcX`jbcOYVQ\cOY[­w

ÙHÚ�Û~Ú�Û ê.ë�ßJë�ìSà�í!ä�ìSî#ï�ä�ì�ßSð�àñâPàrì�ãÈàrâÍòWã{ó¿ã�ßõô¹ä§ö.àFHGJO}o l M l VQO}KWO?VQILO?K._LIai8ILM | d^VQ¢ [PILb`Ia[JOYK.IaXZMNd¡M | d£R^UJUJVWdZR^gzGJOYK?w}÷�IaVQK{MN_�u�d^XJO}g;R^X�gYdcXJKWMQV l ghMKQUfO?ghMNVQR^_sopO?RTM l VWO]¬JMWMNIaXP\3UJVWdkgYO?[ l VWO?K | GJILgzG l KQO]UJ_vR^KWi�ReIaXPopd^VQimRTMNILdcX}opVQdciËd^MQGJO?V�[JIvR^\^XJdcK{jMNILg?K§dcV§UJ_vR^KWi�R�g?d`[JOYK]MNdmg?dcXJK{MNVQR^IaX�g?O?V{MzR^ILX�¬JMWMNILXJ\mUJR^VNR^i8OYMQO?VWK?w§�]X d^UPMNILimILKQO?['¬JM]d^oBMNGPOdcqJKWO?V{bcO?[�KQOYVQILO?K._aILimI�M§IaK�MQGJO?X�[Jd^XJO�R^K�ILX/MNGJODO � IaKWMQIaXJ\¥�]�§��x�¬JMQMQIaXJ\¥gYdk[PO?K?weFHGJILK]MNOYgzGkjXJIL� l ODg;R^X�qfODIai8i8O?[JIaRTMNOY_Lu¥U l VQK l O?[�R^XJ[�R�o l V{MNGJOYV�imOYiDqfO?V�d^o�MQGJOD���7�.x/KWO?VQILO?K.× | IL_a_qfOeUPVQd)b`Ia[JOY[­wFHGJO£KQOYg?dcXJ[�R^UJUJVWdZR^gzG5IaK�imd^VQO¥_ad^XJ\9MNO?VWi�wºFHGJO'¬JMQMQIaXJ\�gYdk[JO'g?R^X�qfO'i�R^[PO£UJR^VWM8d^o]R\c_LdcqSR^_SdcUkMNIai8IaKQRTMNILdcX}UJVWdkgYO?[ l VWO^w�FHG l K?�PIaXJK{MNO;RT[�d^o¹UfO?V{opdcVQi8IaXP\3R�¬JM~qSRTKQO?[ l Usd^X¡O � MNOYVQXSRT_qfd l XJ[JILXJ\£[SRTMzR'opVQdciødTMNGJOYV3g?d`[JOYKDR^XJ[�[JIvRT\cXJdcK{MNIagYK?�rMNGJO�¬JMWMNIaXP\ g?d`[JO8IaKedcXJO8USR^V{M3d^oHR^XIaXZMQO?\cVQRTMNOY[�¬JM§MNd8MNGJOeg?dci8UJ_LOYMNOeKQOhM]dToBMNdc¢�R^imR^¢¡UJR^VNR^i8OYMQO?VWK?wHFHGJILK�GSRTK]USR^V{MNILg l _aR^V7R^UJUJ_LI�jg;R�MNIad^X MNd¡o l KWIadcX²� | GJO?VWO}MNGJO}KWO?VQILO?K._LIai8ILM§g?d`[JODg?R^X�qsO}ILXyMQO?\cVQRTMNOY[ | ILMQG9O � IaK{MNIaXP\¡UJ_vRTKQimRMNVQR^XJKWUsdcV{M�gYdk[PO?K?wÍÎ�X5MNGJILK�R^UPUJVQdZRTgzG5MNGPO | GJd^_aO£KQOYM�d^o.gYdk[JOYK8opd^VQi8K8MQGJO£¬PMQMNILXJ\�UJVQd`g?O�j[ l VWO^�fR^XJ[£g?d^XJKQILKWMNOYXJgYu£RTimdcXP\cKWM§R^_L_²¬JMWMNO?[�USR^VQR^imOhMNOYVQKHi l K{M§qsOtR^gzGJILOYbcOY[�opdcV7R}bTR^_aIL[¡¬JM;wFHGJILK�UJVQOYKQOYXyMQK�R¡KQIL\cXJIL¬Jg;R^XZM]MNR^KQ¢��sqsd^MQG9IaX MQGJO}R^KWKQOYiDqJ_Lu£d^o�MQGJODKWOYM�d^o�gYdk[JOYK.R^XJ[9ILX MNGPOg?d^imU l MzRTMQIadcXSRT_²dcUPMQIai8IaKWIaXJ\td^o�MQGJO.¬JM;w

Page 171: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©±°^ù

¦nd^MQG�R^UJUJVWdZR^gzGJOYK�VQO?� l ILVQOH[JOhbcOY_adcUJi8O?XZM�d^ofKQOYVQILO?K�_aILimI�M�UJVQd`g?OY[ l VQOYK | GJIagzG8R^VWOHVQdcq l KWM;�ZOhú}jg?ILO?XZM>R^XJ[tMNR^Ia_LdcVQOY[.MNd7MNGJO�KQUfO?gYIL¬Sg�O � UfO?VQILimOYXZMzR^_yKQOhM l U3d^oPMQGJO�UP_vR^KWi�R l XJ[JOYV�ILXZbcO?K{MNIL\ZRTMNILdcX­wÎ�XJI�MNIvRT_a_Lu}MQGJO?VWO3RTVQO�M | dmM�u`UsOYK�dTo>O � UfO?VWIai8O?XZMzRT_­KQUfO?ghMNVNR | GJIagzG | Ia_L_²qfO�imd`[JO?_L_aOY[­w

û£ü¹ý�þ±ÿ����������`þ���� ����������cÿ¹ý����ILMQGõi8IaXJdcV�i8dk[PIL¬Sg?RTMNILdcXJK}MNGJO [JILb^O?VWMQdcV�KWO?VWIaO?K�_aIai8ILM�gYdk[PO/d^o.gzGSRTUPMNOYV£°�g;R^X�qsO/ILXZMNOhj\cVQRTMNOY[�IaXZMNd£MNGJO'¤c��F¿[JILb^O?V{MNdcVt[JIvR^\^XJdcKWMQIagmKQd^oÉM | R^VQOTw�«ndcXJK{MNVNRTIaXZMNK}i l KWMDqfO¡U l M}dcX!MNGPOIaXPILMNIaR^_�¬JMQMQIaXJ\�USR^VQR^i8OYMNOYVQKY�SqSR^KQOY[ l UfdcX£IaXPopd^VQimRTMNILdcX�opVQd^i�d^MQGJO?V7UJ_aR^KQimR}[JIvR^\^XJdcKWMQIagYK?w�Î�XUSR^V{MNILg l _aR^V?��imd^VQO�[JOYMNR^Ia_LO?[3¢yXJd | _LO?[J\cOnd^oJMNGPOng?d^VQO~ILimU l VQILM�u�g?dcXZMNVWIaq l MNIad^XtMNd§MNGJOnKQUfO?gYMQV l ig;RTX�qfO]i8O;R^K l VQOY[ l KWIaXJ\eMNGJO��tx`ª3KQUfO?ghMNVWdcimOhMNOYV?w�FHGJO�g?d`[JO�dTo¹gzGSR^UPMQO?V~° | Ia_L_JqfO]g;RT_a_aOY[�RTKUSR^V{M�d^o�R�_aO?R^KWM]KQ� l R^VWO?[/¬JMWMNIaXP\�UJVWdkgYO?[ l VWO^�­ILX�MQGJODKNRTimOti�R^XPXJO?V]R^K�MQGJO3O � ILKWMQIaXJ\¡�]�§��x¬JMWMNIaXP\}g?d`[JO?KYw�� l KQOYo l _¹g;R^USRTqJIa_LILM�u8d^orMNGJOYKQO�g?d`[JOYK�RTVQO�MNGJRTM~MNGPOYu'R^_L_ad | Iai8U l VWILM�um_aILXJO?KHRTK| O?_a_�RTK�MQGJO�\cO?XPO?VNR�MNO?[�KQUfO?ghMNVQR^_BopO;RTM l VQO}MQd�qfO}IaXJgYdcVQUfdcVQRTMNOY[�ILX9MNGJO}¬JM�ÁpMQGJO�UJVQOYKQOYXJg?O8d^oIai8U l VWILM�u�_LIaXJOYKnIaXmMNGJOe¤c��F [SRTMzRtg;R^X¥qfO�KQOYO?X¥opVQd^i¿¬S\ l VQOD©^wL©±°^Æhw�νMHILKnGJdcUfO?[¡MQGSRTMHK l gzG£R¬JMWMNIaXP\¥g?d`[JO | Ia_L_>VWO?KQd^_LbcODMQGJO}[JILKQg?VWO?USRTXJgYu/opd l XJ[ | GJOYX�MQGJO�÷������ÕR^XJR^_Lu`KQILK�IaK l KQOY[�MQd[JIaR^\cXJdcKWO]OY_aO?ghMNVWdcX¥[JO?XPKQILMQIaOYK�opVQd^i¿bTR^VWIad l K�KQOYVQIaOYK~_aILimI�M�_aILXJO?K | GJO?VWO�R^\cVQOYO?i8O?XZM | R^K~opd l XJ[l KWIaXJ\eMNGJOe©c©YjȨeR^XP[¡¼�j�³e_LIaXJOYK�q l MnXPd^M l KQIaXP\tMNGPO�ù±j�³e_LIaXJOtÁ���O?IL\cK�OhM~RT_ w��P¨T®c®c®ZÆ�w>FHGJILK | dcVQ¢IaKHUP_vR^XJXPO?['opdcV~MNGJO�ILimi8O?[JIaRTMNO§o l M l VQO^w

û�������� � �!�yþ����"�����#��$�cÿ¹ý �FHGJO�UPVQIaXPg?IaUP_aO?K�d^orgYdcXZMNIaX l I�M�umR^XJ[�opO?RTM l VWO�\cOYXJO?VQRTMNILdcX�g?R^X¥qsO�O � MNOYXJ[JOY['MQdDi8dk[JOY_sGPO?_aI l iKQOYVQILO?Ke_aIai8ILMQK?w ��OYVQOmd^o~gYd l VQKWO�qfd^MNG�KQILXJ\c_LOYMtR^XJ[�MNVWIaUJ_LOYMeKQOYVQIaOYKt_LIai8ILMQK | Ia_L_BqsO8UJVWdk[ l g?O?[²wFHGJOYVQOeGSR;bcOeqsOYO?X�M | d8VQO?gYO?XZM�dcqJKQOYVWbTRTMQIadcXJK7d^o�MNGPOeGPO?_aI l i&%­u`imR^X�KWO?VWIaO?K7_LIai8ILM;w�FHGJO.¬SVQK{M| R^K�IaXºR9GJIa\cG�[PO?XJKWILM�u�_aR^qsd^VNRTMQdcVWu�UJ_vRTKQimR9R^XJ[ºR^XSR^_�u`KQO?[ l KWIaXJ\�R9g?dc_L_aILKQIad^XSR^_�j½VNR^[PIvRTMQI�b^Oi8dk[JOY_�Á�'�R^iDqSR»OYM�R^_ wL�]¨^®c®^®ZÆ | GJIagzG IaXJgY_ l [PO?[ R^X O?KQg?R^UfO�UJVWdcqSR^qPIa_aI�M�u5MQd�RTg?g?d l XZM£opd^VdcUSRTg?ILM�u^w7FHGJIaK]R^_a_Ld | O?[£MNGJOtdcqJKWO?VWb^O?[9UJ_aR^KQimR^K7MNdmqsOtg?_vRTKQKQI�¬SO?[�IaXZMNdmIadcXPIaKQILXJ\8R^XJ[/VWO?gYdci�jqJILXJIaXJ\§M�u`UsOTwBFHGJO(%­u`i�RTXDKQOYVQIaOYKB_aILimI�M | RTK�R^_aKWd�dcqJKWO?V{bcO?[�ILX3MNGJOn��ÎQÎWÎ jÈ�5MQdc¢�R^i�RT¢�Á)��GZu`MQOOYMtR^_ wL��¨T®c®c®ZÆ�R^XP[�MQGJO�VQO?gYdciDqJILXSRTMQIadcX9KWUsOYgYMNV l i l KWO?[�MQd�[PIvR^\cXPdcKQODMQGJOmUP_vR^KWi�R¡OY_aO?ghMNVWdcXMNOYimUfO?VQRTM l VQO�R^XJ[£OY�sO?ghMNI�bcOeIadcXJILg�gzGSR^VW\cO�[ l VWIaXJ\�UJ_aR^KQimRD[JIaKWV l UkMNIad^X­w��GJIL_aODMNGPO*%­u`imR^X�KQO?VWIaOYKeILK.d^o~ILXZMNO?VWO?K{M;�¹MNGJO�imR^IaX9d^UJUsd^VWM l XJILM�u/opd^V�o l M l VWO�VQO?KWO;R^VWgzGµ_LIaO?K

Page 172: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©±°T¼

IaX'i8dk[PO?_a_LIaXJ\eMNGJO.MQVQIaUP_aOYM~KQO?VWIaOYK?w�÷SdcV~GJIL\cGJO?VH� l RTXyM l i�KQGPO?_a_LK+%²j{x¡g?d l UJ_aILXJ\}qJVQO?R^¢yK�[Pd | X²�R^XJ[mMNGJO�O � g?ILMQO?[¥OY_aO?ghMNVWdcXJK~g?R^X¥qsO§MNGJd l \cGZMHd^o�R^KnR3i8I � M l VWO]d^o­KQIaXP\c_aOhMHR^XJ[mMNVQILUJ_aOhMnK{MzRTMQO?KYwFHG l K3OY_aOYgYMNVWdcXµd^X�MQGJO8MQVQILUJ_aOhMtKWIa[JO�Oh�²OYgYMNI�bcOY_Lu�KQUfO?XJ[�KQdci8O�MNIai8O}IaX�R'KQIaXP\c_aOhMtK{MzRTMQO�RTXJ[R^VWO�XJdt_adcXP\cO?V�i�RTIaXJ_�uDg?d^XP¬SXJOY[�MNdt[JO?g?R)u`ILXJ\3[Jd | X�MQdtMQGJO3©-,±¨.,�/10µKWGJO?_L_ wBFHG l K~d^XJO | d l _L[O � UfO?ghM�MQGSRTM7MNGJOeVQOYK l _�MNILXJ\}MNVQILUJ_aOhMHKQUfO?ghMNV l i | d l _a['g?dcXPKQIaK{M]dTo>[JILKQgYVQOYMQOt_Ld | OYV§KWO?VWIaO?K§_aILXJO?KR^XJ[9MQGSRTM�R^K.d^XJODUJVWdc\cVQOYKQKWO?K l U9MNGJO}KWO?VWIaO?K�R¡UJR^VWMQIvR^_�� l O?XJgzGJILXJ\'d^o�MNGJO}KWUsOYgYMQV l i | d l _L[qfO]dcqJKWO?V{bcO?[�RTM�MNGJO]UsdcILXZM | GJOYVQO�MQGJO]KWUJIaX¡K{u`KWMNOYi�qJVWO;R^¢y[Jd | X£R^_L_ad | K�MQVQILUJ_aOhMnOY_aO?ghMNVWdcXJK�MQd[JOYg;R;u/[Jd | X9MNd¡MQGJO}\cVWd l XJ[9dcX/MQGJODKWIaXJ\c_LOYM]KQIa[PO^wtx l gzG�R�� l O?XPgzGJIaXJ\£GSR^K]qsOYO?X�dcqJKWO?VWb^O?[Á ��GJXPd!OYM¡R^_ w���©32.2^¼ZÆh�nMQGJd l \^GõILM | RTK¡IaXZMNOYVQUJVWOYMQO?[�R^KmOYb`Ia[JOYXJg?O d^o�imd^_aO?g l _vRTVmRTgYMNI�bTRTMNOY[VQOYg?dci3qJIaXSR�MNIad^X­w«ndc_L_aIaKWIadcXJR^_­VQOY[JIaK{MNVWIaq l MQIadcX£IaK�R^X/OYKQKWO?XZMNIaR^_>ILXJ\cVQOY[JIaOYXZM�opdcV�K l gzG9� l OYXJgzGJILXJ\¡MNd8MzR^¢TODUJ_vRTg?O^wFHGJO8g?dc_L_aILKQIad^XSR^_�j½VNR^[PIvRTMQI�b^Oti8dk[PO?_a_LIaXJ\md^onMQGJO�GJIa\^GJ_Lu O � g?I�MNO?[�KWMNRTMNOYKtd^onGJO?_LI l i#IaKeR^X�R^VQO?RVQOY� l IaVWIaXJ\]RTMQMQO?XZMNILdcX­�cRTXJ[}R�MQO?gzGJXJIL� l O | GJIagzGDR^_a_Ld | KrMNGPO~XSRTM l VQO�d^oSMNGJOnKQUJILXDKWu`KWMQO?i qJVQO?R^¢cj[Jd | XµMQd qfOmILXyb^O?K{MNIa\cRTMNOY[ | d l _a[�qsO l KWOYo l _ w£Î�i8UJVWd±b^O?i8O?XZMNKtIaX�MNGJOmg?dc_L_aILKQIad^XSR^_�j½VNR^[PIvRTMQI�b^Oi8dk[JOY_ | d l _a[�MQGJO?XºVQO?R^Uºo l VWMNGPO?V}qfO?XJOh¬JMNK8IaXºR^_L_nGJOY_aI l i qSRTKQO?[5[JIvRT\cXJdcK{MNIagYK?w!νM�KWGJd l _L[R^_LKQd}qfO�XJd^MQO?[¥MNGSR�MHMNGJO�XPO | KWgzGJO?i8O�UJVQd^UsdcKWO?[¥opdcVnMQGJO.GJO?R)byu¥KWUsOYg?ILO?K | dcVQ¢¡[PdkOYKHXJd^M l KWOMNGPO3g?d^Xyb^O?XZMNILdcXSR^_4%²j{x�gYd l UJ_LIaXJ\�opdcV7MNGJODO � g?ILMQO?[/K{MzRTMQO?K.RTXJ[/KQd | Ia_a_¹qfODRTqJ_aO�MNd�MQVNR^gz¢'MNGPOqfO?GSR;b`Iad l V}d^o7MNGJOYKQO'O � g?I�MNOY[ºK{MzRTMQO?KY�nR^XP[»MQGJO¥O � UfO?ghMNO?[ºKQOYVQILO?K�� l O?XJgzGPIaXJ\Jw�FHGJO'[JO?XPKQILM�uKQOYXJKQI�MNI�bkI�M�u¡d^o�MNGJO�KWUJIaX'KWu`K{MNO?iÒqJVWO;R^¢y[Jd | X�g?d l _L[£R^_aKWd8qfO l KWO?[/RTK�R}[JIaR^\cXJd^KWMNILg]MNd`dc_ wÎ�X'RDVQOYg?OYXyM7O � UsOYVQIai8O?XZM7R�M7¤c��F KWdci8O]UJVWO?_aILimILXSR^V{u}dcqJKWO?VWbTRTMQIadcXPK | OYVQO�MNR^¢^OYX'MQdDOYbTR^_ l RTMQO| GJOhMNGJOYVnR^XZu�GJOY_aI l i MQVQIaUP_aOYMBKQO?VWIaOYK�gYd l _a[�qfO7dcqJKQOYVWb^O?[�Á���O?IL\cK?�k¨^®c®k©±Æhw�Î�X8MQGJO7VQO?\^IadcX8³c×^ªc®5�76 ³Z°^¨^® 5� i8O?i3qsOYVQK�d^ofMNGJO7MNVWIaUJ_LOYM�KWO?VQILO?K]ÁpMQdm©-,�¨98:/<;tÆ | OYVQO�dcqJKQOYVWb^O?[­�PR^XJ[8� l O?XJgzGJILXJ\OY�sO?ghMNK8[JOYMQO?ghMNO?[²wºFHG l K8ILM}R^UJUfO;R^VWK�MQGSRTM}o l V{MNGJOYV | dcVW¢!IaXZMQd�KWO?VWIaO?K�_LIai8ILMtimd`[JO?_L_aILXJ\£opd^VGJOY_aI l iÀMNVWIaUJ_LOYMHKWO?VWIaO?K7ILK�´ l KWMQIL¬SOY[­w

ÙHÚ�Û~Ú>= ? äA@.ë.òCB>ßJãÈänáøà­Þ7ßJà�á�âJãÈä�á ï�ä�ì�ð�à�B�D+E#âF@3àrôrãÈàrâFHGJO¡O � MNO?XPKQIad^X!d^o~MQGJO�UfdcU l _aRTMNILdcX | dcVW¢�d^o7gzGSR^UPMQO?Vtª MNd�[JO;R^_�g?d^XJKQILKWMNOYXZMN_Lu | I�MNGµGPO;R;byuKQUfO?gYIaOYK�UfdcU l _vRTMQIadcXJK�UJVQd)b`Ia[PO?K�R�b^RTVQIaOhM�u3dToso l M l VQO7RTUJUJ_aILg;RTMQIadcXPK?w�FHGJOHUJVQILi�RTVWutimd^MQILbTRTMQIadcXopdcV�MNGJILK | dcVW¢µRTMeMNGPO�UJVWO?KQOYXZM}MNIai8O�IaK�MNGJOmXJO?OY[!MNd�qsO�R^qJ_aO�MNd [JOYMQO?gYM3GJO;R;byu!KWUsOYg?ILO?KDR�MM�u`UJIag?R^_�Î�F7�HG MQO?i8UsOYVNRTM l VQOYK3RTXJ[µ[JOYXJKQI�MNILO?K?w�νM3GSRTKtqfO?OYXµUJVWdcUsd^KQO?[�MNGSR�M3GJO?R;b`u�KWUsOYg?IaOYK

Page 173: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©±°I2

qfO�O?iDqfO?[P[JO?[£RTMn[JIL�sO?VWO?XZMH[JOYUPMNGJK7ILX¡g?OYVWMzRTIaX¡d^o¹MQGJO.g;RTVQqfdcX�¬SqfO?V~gYdci8UsdcKWILMQO | R^_a_fMNIa_LO?K�d^oÎ�F7�HG»R^K�OYVQdcKWIadcXeimR^VQ¢TO?VQKYw>��K�MQGJO�MQIa_LO?KrR^VWO�OYVQd`[JO?[eMNGJO�GJO;R;byueKQUfO?gYIaO?K�_vR;u^O?V�IaKrOÈ´WOYgYMNOY[DIaXZMQdMNGPO~UJ_vRTKQimR�g?d^VQOHR^XP[}ILMNK�KQUfO?ghMNVNRT_JKQIL\cXSRTM l VQO�[JOYMQO?gYMQO?[­w�νM | Ia_a_`R^_LKQd�qfO~XJOYg?OYKQKNRTVWuDopdcV�Î�F7�HGMNd9MNR^¢^O£R^[PbTR^XZMNR^\cO¥d^o7MNGPO£\^dkd`[!MNGJOYVQimR^_~VWO?KQILKWMNR^XJg?O'd^o§GJO?R;b`u»KWUsOYg?ILO?K�IaX!MQGJO¥MNdc¢�R^imR^¢[JOYKQIa\^X­wñ«ndcXJKWIa[JOYV�MNGPO/KQUfO?ghMNV l i opVQd^i RµKWUsOYg?ILO?K¡K l gzG R^K�M l XJ\cKWMQO?X­w��XJO dcqJKQOYVWb^O?K£Rg?d^imUJ_LO � \cVNR^KWK{j½_aIa¢TO~KQUfO?ghMNV l iËR^VQILKQILXJ\]opVQd^i R.g?dci3qJIaXSR�MNIad^XDd^ofMNGJOHimR^XZu}Iad^XJIaKQRTMNILdcX3K{MzR^\cOYKd^oBMQGJO3KWUsOYg?ILO?K�RTXJ[ MNGPOtgYdcimUP_aO � KWOYM]d^o�O?XPO?VQ\Tu _LOYbcOY_aK§UJVQOYKQOYXyM?w]FHGJOt\cVNRTKQK{j½_aIL¢^OeKQUfO?gYMQV l i[Jd`O?K�GJd | OYb^O?V�GSR;bcO§R�VQOYg?dc\^XJIaKQR^qJ_aO§¶ØKWUsOYgYMQVNR^_JOYXZbcO?_LdcUfO^¸yR^VWIaKQILXJ\�opVWdciIai8UsOYVWopO?ghMN_�utVWO?KWdc_Lb^O?[¶ MNVNRTXJKQI�MNIad^X RTVQVNR;u`K?¸�w�FHGJILK�OYXZbcO?_LdcUfO�KWGJd | K/R»KWMNVWdcXJ\ºMNO?i8UfO?VNR�M l VWO�[PO?UfO?XJ[JOYXJg?OTwÒFHGJOimR^VQgzGJILXJ\]dToSMNVNRTXJKQI�MNIad^XDR^VWVNR;u`K�R^KBdcXPO~UJVQd^\cVQOYKQKQOYK�IaKWdTj�X l gY_aO;RTVQ_LueRTXJ[DIaKWdTj�OY_aOYgYMNVWdcXJILg;R^_L_Lu.GSRTKqfO?O?XeKQGPd | X�MNGPO?dcVWOYMNILg;R^_L_LuDÁ �D¸J� l _a_aR^XJOLK�x l i8imOYVQKY�`©M2.2.2ZÆ�w�FHGPIaK | Ia_L_cR^_L_ad | OYb^O?XtR^[�´QR^gYO?XZMGJO?R)byu£KQUfO?gYIaOYK§MQd8qfOeVQO?KWdc_Lb^O?[­w�FHGJOeGJIa\cGPO?VH_aOhbcOY_aK§qsOYg?dci8O�Gyu`[JVWdc\cOYXJIag�IaX£qfO?GSR;b`Iad l V�RTXJ[MNGPO?IaV3KQO?VWIaOYK}_aIai8ILMtKQUfO?ghMNVQR�i8O?VW\cO | I�MNG�MNGJO¥\^O?XJOYVNR^_~gYdcXZMNILX lPl i�j½_aIL¢^O¡RTUJUsO?R^VNRTXJg?O¥d^o§MNGPOKQUfO?ghMNVQR^_¹O?XZbcOY_adcUfO?KYwN�I � O?_]K l i8K¥d)bcO?V'KWUsOYg?IL¬JO?[ | R;b^O?_aOYXJ\^MQG ILXyMQO?V{b^RT_aK | IL_a_§qsO9UfO?V{opdcVQi8O?[ÍdcXÍMQGJO9dcqJKWO?VWb^O?[O?XZb^O?_ad^UsOYK l KQILXJ\!¶ØgYdci8U l MzR�MNIad^XSR^_�¬S_LMQO?VWK?¸�w9FHGJIaK | IL_a_�qfO l KQOY[�MNd9O � MQVNR^ghM�R/KQOhM}d^o§X l i�jqfO?VQK Áȶ·KQIL\cXSRTM l VQO'X l iDqfO?VWK?¸ØÆ�MQd�qfO l KQO?[�R^K�MNGJO£USR^VQR^imOhMNOYVQK}MNd�qsO£imRTMNgzGJOY[�ILX�R�GJIa\^GMNGPVQd l \^GJU l M�¬JMWMNIaXP\£UPVQd`g?O?[ l VQOTw¡¦nd^MQG�XPO l VQR^_�XJOYM | dcVW¢`K3R^XJ[�\cO?XPOYMNILg8R^_a\cd^VQILMQGJi8K | Ia_L_BqfOIaXZb^O?KWMQIa\ZR�MNO?['RTK~UsdTMNO?XZMQIvR^_fUSRTMWMNO?VWX¡VQOYg?dc\^XJILMQIadcX¡i8OYMQGJd`[JK?wBνM | Ia_a_�qfO�UfdcKQKWIaqJ_LO�MNd3\cO?XJOYVNRTMQOMNVQR^IaXPIaXJ\�[SRTMNR8MQdmMNOYKWM�MNGJOe¬JMQMQIaXJ\8UJVQd`g?OY[ l VWO?K?w§�~M�UPVQO?KWO?XZM.O � UfO?VWIai8O?XZMzRT_�KWUsOYgYMQVNRmopdcV7MNGPOGJO?R)byu»KWUsOYg?IaOYK8RTM}O � UfO?ghMNO?[5Î�F7�HG�g?dcXJ[PILMNILdcXJKDg?R^XJXJdTM�qsO¥\cOYXJO?VQRTMNOY[­�~RTXJ[ºR^KDMNGJO¥MQVNR^XkjKQI�MNILdcX�R^VQVQR;u9R^VWO8imR^VW¢^O?[ku/MNOYimUfO?VQRTM l VWO}KQOYXJKQI�MNILb^O^��_ad | O?V�MNOYimUfO?VQRTM l VQO}[SRTMNR | Ia_a_>XJd^M�qfOK l úmgYIaOYXyM?w/FHG l K}g?d^imU l MzRTMQIadcXSRT_�imd`[JO?_L_aILXJ\ | Ia_L_�qfO�VWO?_LIaO?[ l UfdcXµMNd \cO?XJOYVNRTMQO¡KWu`XZMNGPOYMNILgKQIL\cXSRTM l VQO3X l iDqsOYVQKeR^XJ[�UJVWd)bkIL[JODMNGPO}XJO?gYO?KQKQR^VWu9MQVNR^ILXJIaXP\¡[SRTMNR^KQOhMNK?w3F�d¡MQO?KWM�MQGJO}VQdcq l KWM{jXJOYKQK3dTo~MNGJO�¬JMWMNIaXP\�UJVWdkgYO?[ l VQO?KeMNGJO8KWu`XZMNGPOYMNILg�OYXyb^O?_LdcUsOYK | IL_a_�qfOmgYdciDqJILXJO?[ | ILMQG!R^ghM l R^_XJdcILKQO8opVQdci,R^X!O � UsOYVQIai8O?XZMNR^_�[POYb`IagYO¡K l gzG»RTK�¤c��F.w>FHGJILK | Ia_L_�UJVQd`[ l gYO�MNOYKWMD[JRTMzR^KWOYMQKDd^o¶·XJdcIaK{uf¸SKQIa\^XSRTM l VQO.X l iDqfO?VWK7MQdmqfO�¬JMWMNO?[�qZu¡MNGJO�gYdk[JOTwFHGJO | dcVW¢8VWO?� l ILVQO?[mopdcV�MQGJO]gYdci8UJ_aOhMNIad^Xmd^o²MNGJO�GJO;R;byu�KWUsOYg?ILO?K~gYdc_a_LIaKQILdcXSR^_¯j�VQR^[JIvR�MNI¯bcOni8d`[JO?_IaKnKQIa\^XJIL¬Sg?R^XZM;�kq l M§R^gzGJILOYbTR^qJ_LO^w�FHGJO.XPO | gYd l UJ_LIaXJ\3KQgzGJOYimO�GSR^KHqfO?OYX'd l MN_LIaXJOY[¡IaX¥gzGJR^UPMNOYVªJw§FHGJO'Á�Å)ñÆ{DzÊ�RTXJ[»ÁÈÅ;ñÆ{Ç�VQO?KWdc_Lb^O?[9qJ_Ldkgz¢'UJVWIai8ILMQILbcOYK7g;R^X/qfO3OhbTR^_ l R�MNO?[ RTM�UPVQO?KWO?XZM;w]FHGJOKQILXJ\c_LO�Xkj½KQGJOY_a_HMNOYKWM¥gYdk[PO�opdcV�MNGJO�ÁÈÅ)ñÆÈǭʯÌ�VQOYKQdc_ l MNILdcX | IL_a_HqfO O � MQO?XJ[JOY[ÍMNd!OYXJg?dcVWUsd^VNRTMQO

Page 174: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©)×^®

R^X�O?XZMQIaVQO3qJ_ad`gz¢ UJVWIai8ILMQILbcOTw�FHGJIaK | IL_a_rMNGJOYX�qfO l KWO?[�MNd¥MNOYKWM�g?d^XJ[JO?XPKNRTMQIadcX�KQgzGPO?i8O?K?�rMNGPOOY�sO?ghMNK8d^o§MNGJO¥VWO?[JILKWMNVWIaq l MNILb^O¥VNRTMQO?K8R^XJ[ºi8OYMNGPdk[JK}dTo]UfdcU l _aRTMNILdcX!KWdc_ l MQIadcX²wµ�]X ÁÈÅ;Ã�Æ{ǭʯÌqJ_Ldkgz¢mUJVQILimI�MNILb^O�g?d`[JO | Ia_L_sMQGJO?X£qsO | VWILMQMQO?X­wFHGJOtIaXJI�MNIvRT_²qSR^KWO?_LIaXJOe[SRTMNR�opdcV7MNGPOt_Ld | _aOYb^O?_�qJ_ad`gz¢yK | IL_a_¹qfOt\^O?XJOYVNRTMQO?[ l KWIaXJ\8���7�.x�¼c®P©c�R^XJ[ | IL_a_yqfO~UJVQd`[ l gYO?[}ILX}R^[Po�®^ª�opdcVQimRTM;w�FHGJO~[JRTMzR | IL_a_`qsOng?dci8UJ_aOhMNO�opdcV>MQGJO~KQUfO?gYIaOYK l XJ[JOYVIaXZb^O?KWMQIa\ZR�MNIad^X­�kq l MHd^o¹o�R^ILVQ_�u�_ad | � l R^_aI�M�ucwBFHGJO._Ld | _aOYb^O?_s[SRTMzR | IL_a_SMNGPO?X'qsO]K l UJUP_aO?i8O?XZMQO?[| ILMQG/GJIL\cGJO?V�� l R^_aI�M�u'[SRTMNRmopVQdciÒgYdk[JOYK�K l gzG�R^KO��PQ%�%r�§« R^XP[RGHj½i�R�MNVQI � w§FHGPO}_ad | _aOhbcOY_VNR�MNO?K | IL_a_§qsO�RTKQKQOYiDqJ_LO?[�IaXõMQGJO9���7�.x�¨^®c¼µR^UJUJVWdZR^gzG­w FHGJO�IaXP� l O?XJgYO9d^o�MNGJO9GJIL\cGJOYVUfdcU l _aRTMNILdcXJK¡g?R^X MQGJO?X qfO�UPVQd�´WOYgYMNOY[ dcXZMQdºMQGJO?KWO�_Ld | _Lu`ILXJ\�VQOYK l _�MNK£R^XJ[ R�UsdcU l _vRTMQIadcXKQd^_ l MQIadcX¥dcqkMzR^ILXJO?[­wFHGJO7i8dk[ l _vR^VWIaKNR�MNIad^X3d^ofMNGJO7gYdk[JOHd l MN_LIaXJOY[8GPO?VQO§R^_L_ad | K�R^[kb^RTXyMNR^\cOHMNd�qfO~MzR^¢TO?Xmd^o²i8d`[JO?VWXGJIL\cG�UsOYVWopdcVWi�R^XPg?O�USR^VQR^_a_LO?_BgYdcimU l MNOYVQK?wm��R^gzG!qJ_Ldkgz¢ UJVQILimI�MNILb^ODg;RTX�qfO�g;R^_Lg l _aRTMNOY[�ILXJ[JOhjUfO?XJ[JOYXZMN_Lu3dcX�R�[JI��²OYVQO?XZM�UJVWdkgYO?KWKQIaXP\.XJdk[PO^wBFHGJOHqP_ad`gz¢tUJVWIai8ILMQILbcOYK>g?R^XDMNGJOYX�qsOHRTKQKQOYiDqJ_LO?[IaXZMQd'MQGJO�qJ_ad`gz¢yKeR^XJ[�MNGJO}¬SXSRT_�K l UfO?VWqJ_ad`gz¢9i�R�MNVQI � w8x l gzGµUJR^VNR^_L_aOY_�MNOYgzGJXJIL� l O?KtR^VQO�_aIL¢^OY_LuMNd¥qfODVWO?� l IaVWO?[/opdcV�MNGJO}RTKQKQOYiDqJ_�u9R^XJ[9KWdc_ l MNIad^X d^o�MNGPOD_vR^VW\cO3imRTMQVQIagYO?K�XJO?OY[JO?[�opdcV]GPO;R;byuKQUfO?gYIaOYK�i8dk[PO?_a_LIaXJ\PwFHGJO7g?d^_a_aILKQILdcXSR^_¯j�VNRT[JIvRTMQILb^O�g?d`[JO | Ia_a_kqsO7gYdciDqJILXJO?[ | ILMQG8R^X�IadcXJILKNRTMQIadcX3qSR^_vRTXJg?OHg?R^_ag l _vRTMQIadcXMNd3UJVQOY[JIaghMHRtK{u`XyMQGJOYMQIag]opO?RTM l VQO^w�FHGPO]opO;RTM l VQO | Ia_L_�MQGJO?X¥qfO]ILKQdc_aRTMNOY[mILXZMNd3gYdci8UsdcXPO?XZMNKHRTXJ[¬JMWMNO?['MNdDMQGJO�dcqJKWO?VWb^O?['¬S_LMQO?VWO?['O?XZbcOY_adcUfO�KQOY\cimOYXZMNK?w�FHGJO.¬SXSRT_­imd`[JOY_ | Ia_L_²qfO._aILXJ¢^OY['MQd�RUJ_aR^KQimRDMNVNRTXJKQUfdcVWM7gYdk[PO3R^XP[ l KWO?[�IaX£R8\^_adcqSRT_­dcUPMQIai8IaKNR�MNIad^X�VQd l MNILXJO | GJO?VWO?IaX'MNGPO?dcVWOYMNILg;R^_KQUfO?ghMNVQR^_�opO;RTM l VQOYK?��OY_aOYimOYXyMNR^_.g?d^XJg?OYXyMQVNRTMQIadcXPK?��[JOYXJKQI�MNILO?K'R^XJ[ KQUJRTMNIaR^_�[JILKWMNVWIaq l MNIad^XJK¥R^VWOKQILi l _�MzR^XJOYd l KW_Lu8¬JMQMQO?['MNd8dcqJKQOYVWb^O?[ UP_vR^KWi�RDUSRTVNR^i8OYMQO?VQKYw

Page 175: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� � � ��� � ��� �

�� �� � �� ���� ��� ���� ������

���! #"%$'&!&)(+*!,- /.0,!(+12 #3'(+"%(+4657$98: <;'*!">=@?!ACB'DE$'*!,F=@?!AHGIDKJML@;N1E5!"O(P <*Q8O�!(SRT$VU<R+5-UW$V8O <;N*X;'LY8:�-(Z $'5!*I8YL[$'R\8O;N1O"]L@;'1^8:�!(_*!(K`aR\;N5!&!U# <*!bc"2Rd�!(+ef('Afgh;'5!&!U< #*!b%;'Li8:�!(_$V*!bN5!U<$'1]ef;NeE(\*I8:5!ej;'L$c3V$'U<(\*!R+(_(+U#(+RK8:1O;'*k8O;l8O�!(_R+;N12(_R+;N*-m!bN5!1:$98: <;'*>RT$'*kno(_R\;N*I3N(+*- <(+*I8OU#pq(K.r&!12(+"2"O(+,�`s #8:�q8:�-( <*I8O1O;t,!5!R\8O <;N*k;'L]uCghU<(\n!"ORd�tv Z ;N1O,7$V*xw�R\;r(KyfR+ <(\*I8:"%=@"O;Nef(\8O <ef(+"z1O(KL@(+121O(+,�8:;S$V"Eu{3N(+RK8:;N1|R+;N5tv&!U# <*!b!woR+;t(\yfR+ #(+*I8:"}$'*!,k(+465! /3'$VU<(+*I8OU#pS(~.r&!1O(\"O"O(\,�$'"�uH�� #bN*!(\1|��v#�\wxR\;r(KyER\ <(\*68O"dD~Algh;N*!"2 <,!(\18:�-(�`M$T3N(KL@5!*!RK8: <;'*��_=#�V���T���6��D^"25!Rd�q8:�7$V8��� <"�8:�!(�12(+"O5-U#8d$V*68�$V*!bN5!U<$'1^ef;Nef(+*I8:5!ej,!5!(�8O;�V�|$'*!,P�T��J�`s #8O��R\;N1O12(+"O&);N*!,- <*!bSeE$'bN*-(\8: #R_4t5!$'*I8:5!e�*65!e�n)(+1��>A����!(lghU#(+n!"2Rd�rv Z ;N1O,!$'*R+;t(\yfR\ <(+*I8�=������/���W�� � � � � D]R+$'*�n)(�5!"2(+,�8O;l5!*!R+;'5!&!U<(E�_=#�V�@�T� �6��D] #*68O;c8:�!(_R\;N*!"28O #8O5!(+*I8�`M$T3N(~vL@5!*!RK8: #;N*!"�L@;N1¡�V�M$'*!,E�T�TAi� ���¢���W�� � � � � $'*!,l #"�,!(\m7*-(+,S"25!Rd��8O�7$V8�_=#�V�[�T� �I��D¡£ ¤� � � � � ���/���¥�� � � � �M¦ =#�V���T�d�P�§�%�KD =[¨|A#©�D

`s�!(\1O( ¦ =#�V���T�d�P�§�%�KDP£ª�_=#�V���P�2D2�_=#�T�d�%�KDKA«¨^*!b'5!UW$'1EeE;'eE(\*68O5!e &!1O;'n!U<(\eE"E1O(+,-5!R+(P8O;$�"O(K8P;'L�8:�-(+"O(�ghU<(\n!"ORd�tv Z ;N1O,7$V*«R\;r(KyER\ <(\*68O"�`s�- <Rd�¬RT$'*¬no(SL@5-128:�-(+1l1O(\,!5!R+(\,­8:;X$�"O(K8;'L�uH�� #bN*!(+1�®9v/�\w¯$'*!,��! <bN�-(+1zR+;t(\yfR+ #(+*I8:"|5!"O #*!bl3'$V1O <;'5!"Y8:1:$V*!"2L@;N12e�$98: <;'*!"z$'*!,�"2pteEef(\8O12p&!12;N&o(\128O <(+"\A����!(\1O(l(K.r #"28:"_$V*X(\U<(\bI$'*I8�8:(\Rd�!*! #4t5-('J¡m71O"�8_ <*I8:12;r,!5-R+(+,�nIp�°�(\3t #*!"O;N*«= ©�GI±N±NDL@;N1�8:�-(�eE$'*! #&!5!U<$V8: #;N*�;'L]$'*!bN5!U<$'1�ef;NeE(\*I8:5!e²R+;t(\yfR+ #(+*I8:"\Jh`s�!(+12(+nIp0;N*!(lR+;N*I3'(+128O"�8:�-(R+;t(\yfR\ <(+*I8O"� <*I8:;�b'1:$'&!�- <RT$VU)L@;N12e³$'*!,%eE$'*! <&-5!UW$V8O(+"¡8:�!(^1O(+"25!U#8:$'*I8�,! <$'bN1:$VeE"\A����!(]3V$'12 <;N5!"eE$V8:�!(\e�$98: <R+$'UN8:1O$'*!"�L@;N1OeE$V8: #;N*!"´;'L!8O�!(sghU<(\n!"ORd�tv Z ;N1O,7$V*�R+;t(\yfR+ #(+*I8:"�$VU<U6�7$T3N(�R\;N1O12(+"O&);N*!,tv <*-bPbN(\;NeE(K8:12 <RE8O1:$'*-"2L@;N12e�$V8O <;N*-"�;'*�8:�!(c,- W$'bN1O$'e�A����-(%bN1:$'&-�! <R+$'U¡eE(K8:�!;t,� <"|,!(+"2R+12 <n)(+, <*�"28:$'*!,7$V1O,S4657$'*I8O5!ej$'*!bN5!U<$'1seE;'eE(\*68O5!e�8:(~.t8]n);r;Nµ6"�"O5!Rd�S$'"]¶s;N"O(�=�©TGN·N®ID~Jo¸�,!ef;N*!,-"©�®N·

Page 176: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�®'G=�©TGIBV?6D~J�`s #8O��8:�!(�eE;N"�8�R\;NeE&-U<(\8O(�,!(\"OR\1O <&r8: <;'*�no(\ <*!b�L@;N5!*!,� #*�Rd�7$'&-8O(+1fB�;VL�¹h1O #*!µ�$V*!,º $V8ORd�!U<(\1f=�©�G'®N·IDKA|»7;N1]R\;NeE&-U<(\8O(+*!(\"O"\Jx8:�!(_µ'(Kp>,!(K8d$' #U<"Y$'12(�1O(\&!1O;t,!5!R\(+,k�!(+12(_ <*q$'&!&)(+*-,! /.¨}A/©NA����!(�*!(K` Z $'5!*I8sL[$'R\8O;N1O"�$'12(]8:�!(\*�,!(\1O /3N(+,�5!"O #*!b�8:�!(\"O(�1O5!U#(+"� #*�$V&!&o(\*!,! ¢.l¨}A½¼rA¾ ¿�À ÁÃÂ}ÄÆÅTÇ È¡É>Ê+ËYÄ Ì'ÍEÈ ÎfÈ�Â�Ï>Ð>Å+Ç�Â}ÊaË�Ñ�Ï>È�ËYÄÆÄÆÅ+ÍfÒ Â�Ò>Ó Ô ÂzÕ

Ò>ÅTÏ>É>Ê\Â|Ö�Å\ÍEÒ Í_ÌF×ØÊ\ËYÙ>ÄÆÇhÐ�Õ�Ú²ÍfÈ¡ÓSÂ�Ò Ç�Í%Ë�Û Ç�Å\ËYÒcÖ�Ä���!(cghU<(+n-"ORd�rv Z ;N12,7$'*�R\;r(KyfR+ <(\*I8:"�R+$'*�n)(E8O1:$'*!"�L@;N1Oef(+,� <*I8:;��� <bN*-(+1|�9v#�YR+;t(\yfR+ #(+*I8:"�$V""O�-;�`s*� <*l(+4657$98: <;'*�¨|AC¼tA

�zÜ�Ý¥ÞßTàKá £�=�â�©�D�ÜdãtÝ�ã á =�¼'äÆ嬩�D ��æ��|çèêé ë äì í î

ïð =[¨|AC¼'D`s�!(\1O( çèêé ë ä

ì í îïð #"�8:�!(%`s <b'*!(+1_�9v/��R\;r(KyfR+ <(\*I8TAX���! <"_ <"�(~.r&!1O(\"O"O(\,êbN1:$V&!�! <R+$'U<U/pkn6p

$'"2"O;tR+ W$98: <*-b|$z"28O1:$' #bN�I8hU# <*!(M`s /8:�f(T$'Rd��;'Lñ8O�!(^$V*!bN5!U<$'1�eE;'eE(\*68O5!ea3N(\R\8O;N1O"h$-J6nc$'*!,ER^$V*!,�2;N #*! <*!b�8:�!(YU# <*!(\"�$V8s$�"2 <*!b'U<(^&);N #*68M;N1^uH*!;r,-('w/A����!(�ghU<(\n!"ORd�rv Z ;'1O,7$'*l"�pre}no;NUñ <"M,!1O$T`s*�$V" <*Sm!bN5!1O(_¨|A#©f$'*!,>8O�!(��� <bN*!(\1z®9v#�]$'"Y #*>m7bN5!12(�¨}A½¼rA}vs*!;'8O(�8:�7$V8z(+$'Rd�k*!;r,-(��7$'"�$c"O #bN*$V828d$'Rd�!(\,�8:;� #8+Ac¹òpqR\;N*I3N(\*68O <;N*�8:�!(fe�$VbN*!(\8O <R_4657$'*I8:5-e�*65!e}no(\1}$'"2"O;tR+ <$V8:(\,�`s #8O��(+$'Rd�$'*!b'5!UW$'1ief;NeE(\*I8:5!eó3N(\R\8O;N1M <"ò;Nef #8O8O(+,fL@1O;Neó8:�-(],! W$VbN1:$'e�A�¨�U<"2;}*!;'8O(^8:�!$V8M"2;NeE(�3N(\R\8:;'1O"�7$T3N(�,! #1O(+RK8: #;N*7$'Uñ$'1O12;�`s"�$V828d$'Rd�-(+,�8O;�8:�!(\e�A���!(�1O5!U#(+"ML@;N1�b'1:$'&!�- <RT$VUxe�$V*! <&!5-UW$V8O <;N*E;'L�ghU#(+n!"2Rd�rv Z ;N12,7$'*�R+;t(\yfR+ #(+*I8:"�$'1O(z$V"sL@;'U<U<;�`s"\ô©NAs¨�*6p�bN(+;'eE(K8:1O #R^,!(KL@;N1OeE$V8O <;N*f8:�7$V8M&!1O(\"O(+1�3N(\"�8O�!(Y;N1O,-(+1M;'L�8O�!(z$'121O;�`s"s$'*!,%"2 <bN*!"h;N*(+$'Rd�S*!;t,!(� <"�$'U<U#;�`h(\,xA�õ�L�8O�!(}RKprR\U< #R|;N12,!(+1�$V8�$'*Ip�*!;t,!(| #"�Rd�!$'*!bN(\,P8:�!(\1O(|e�5!"28�n)($�R+;'1O1O(\"O&);N*!,! #*!bfRd�7$'*!bN(z #*l8O�!(z"O #bN*l;'L�8O�!(z*!;t,!('A¼rAs¨�Rd�!$'*!bN(Y #*c8:�!(YRKprR\U< #R^;N12,!(+1�;'L¯8O�!(Y�9v/�M"�pte�no;'Uñ1O(\"O5!U/8:"� #*%$�Rd�7$'*!b'(Y <*c8:�-(Y"O <b'*c;'L8O�!(z*!;r,-(|$V"�"O�!;�`s*P #*%m7bN5!12(�¨|A½�rA�-As�M`ò;^U# <*!(\"´12(+&!12(+"2(+*I8: #*!b�8O�!(ò"O$'ef(i8O;'8d$'UN$'*!bN5!U<$'1¯eE;'eE(\*68O5!e�R+$'*|n)(x�2;N <*!(\,�8O;NbN(K8:�!(\1$'"s #*cm7bN5!12(�¨|AH??!As¨êU# <*!(h`s /8:��8§`ò;�;'&!&o;'"O #8O(+U/p�,! #1O(+RK8:(\,E$V1O1O;�`s"¡ <"�(+465! #3V$'U#(+*I8�8:;�$YU< #*!(h`s /8:��*!;�$'121O;�`s"

Page 177: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�BVö

0.0 0.5 1.0

+

a

b

c

Cαβγ

abc=(−1)

a−b−γ(2c+1)

1/2 a b cα β γ

=(−1)a−b−γ

(2c+1)1/2

=(−1)2a

(2c+1)1/2

( )

a

b

c

»� <b'5!1O(]¨|A#©'ôò÷�ø�ùKú�û�üCý�ù~þNø�ÿ�ú�ø�ÿ���ÿ�����ù���ü������ ��û�ÿ��oþ½ÿ�����ý�û��@÷��Kø���ù��ý��Tÿ���ý�üHÿ����±rAs¨FU< #*!(|R\;N1O12(+"2&o;N*-,! <*!bf8:;�$'*S$'*-bN5!UW$V1�ef;Nef(+*I8:5!e é `s /8:��8§`h;%$V1O1O;�`s"] #*�8O�!(}"O$'ef(,! #1O(\R\8: #;N*% <"�(\465! #3V$'U#(+*I8s8:;f$�U< <*-(]`s #8:�l*!;f$'121O;�`s"�8: #eE(\"s$�L[$VR\8:;'1�=�â�©�D � Ü A®-Asõ�L�$'*�$'121O;�`­;'*l$_U# <*!(�`s #8O�l$'*!bN5!U<$'1Mef;Nef(+*I8:5!e é #"M1O(K3N(\1O"O(\,xJ78:�!(Yb'1:$'&!�le�5-"28sn)(e�5-U#8: #&!U< #(+,En6p�$�L[$'RK8:;N1�= â|©�D � Ü ABrAs���!12(+(�$V1O1O;�`s"�e�$Tp�no(|$',!,-(+,S8:;�$�*!;t,!('J);N*!(�8:;E(T$'Rd�>U# <*!(VJ `s /8:�!;'5-8^Rd�!$'*!bN #*!b�8:�-(3V$'U#5!(];'L�8O�!(YbN1:$V&!�xA����!(z$'121O;�`s"�e�5!"�8s$'U#Uñn)(],! <12(+RK8:(+,l(\ #8:�-(+1M #*68O;�;N1M;'5-8ML@1O;'eÃ8:�-(*!;t,!(VA

���!(>u{�� <bN*!(\1�v§¸¡Rdµ9$'1�8Tw´8:�!(\;N1O(\e RT$V*Xn)(%5!"2(+,�8:;>1O(\,!5!R\(l8O�!(%eE$V8:12 /.��ó�I��� ���� �� �"!H�#!$�J`s�!(\1O(%���� | #"h$V*� <121O(\,!5!R+ #n!U<(�8O(+*!"2;N1h;N&)(+1O$V8:;'1]=@"O5!Rd�l$'"h$|"2&!�!(+12 <R+$'Uo�!$'1Oef;N*! #R�DKJN8O;}(~.r8O1:$'RK88:�-(�,-(+&)(+*!,!(\*!R+(|;N*%8:�!(zeE$'bN*!(K8: #RY4t5!$'*I8:5!e�*65!e�n)(+12"+A¡���!(z8:�!(\;N1O(\e�"28:$V8:(\"�8:�7$V8���I��� �&�� '� � ! � ! $�£ ���(�)� �' '�)� �*!'$+ ¼\�]嬩 � ��, � �� , � =[¨|A½�ND-�;V8:(|8O�7$V8�8:�!(|e�$VbN*!(\8O <Rz4657$'*I8:5-e�*65!e�n)(+1� <"�*!;�`FR+;N*I8d$V <*!(\,P <*P$cghU#(+n!"2Rd�rv Z ;N1O,!$'*SR+;9v(\yfR\ <(+*I8+A

Page 178: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�Bt©

+. +.

+.

+.e/

c0 a1f2

d3

b4

{5

}6a b e1

d c f3 = 7 Σ

8(−1)a+e+c−α−ε−γ ( (()

9 )9

)9a f c1

c d e0 e b a/α φ −γ: ε β −α; β δ φ

< γ δ −ε ( )

9b d f4

= 7

»� #bN5!1O(E¨}A½¼rô%÷�ø�ùKú�û�üCý�ùKþ)ø�ÿ�ú�ø�ÿ���ÿ�����ù���ü���=���>��û�ÿ%?%üA@B��ÿ�øDC���E¡ý��Tÿ��sý ü½ÿ����GFIHxû�ÿ%�KJMLNL�ù���ü���_üO�P��Q\ÿ�øiùKþHþL�ù�@B��ÿ���ü½ýSR�J9ù����TJULV�WJUL��Vÿ2øT��Fa

b

c

a

b

c

+ −=

c

»� <bN5-1O(Y¨}AH�-ôP�oû9ù��M@KÿÆüA�Yý�XTý�þHüCý��Kø���ÿ�ø(�� ?%ü�@��ÿ2øZY���E&�[XWLP�\�Kþ

Page 179: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�B'¼

+ −

+ −

a

c

b b

c

a’

a a’

c

b

=

»� <bN5!12(Y¨}A{?!ôP���L���ü���üA�M@N?%ü�@�9ÿ�øZY���E>�KXWLP�\�KþO�

Page 180: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�BV�¾ ¿�] ¾�Ï>Ï>ÊTÅTÇsÂ|ÖÆÅ+ÍfÒ Í_ÌcÖ�ÐSË ÎfÈ¡Â}Ï>Ð>ÅTÇsÂ�ÊfÔ Â�Ò>ÅTÏ>É>Ê+Â�Ö�Å+ÍfÒ Ô Ë^ÖÆÐSÍlÓ

ÌNÍfÈ ×ÃÊ+ËYÙ>ÄÆÇòÐ�Õ¯Ú�ÍEÈ¡ÓSÂ}Ò ÇsÍ%Ë^Û Ç�Å+ËYÒcÖÆÄ���!(�(\3V$'U#57$V8: #;N*c;'L Z $'5!*I8sL[$'R\8O;N1O"\J7"O5!Rd�P$'"M8:�!(zn);N5!*!,tv§n);N5!*!,�(K.r&!12(+"O"2 <;N*

^`_acb a , =Td`e�d ! D¡£ + �f ¼Mg h>i â i !j ���kWl m g ©n á'o acb a ,)p _acb a , =Td`e�d ! D =[¨|AH?ID1O(\,!5!R\(+"h8O;�$|*!(\(+,�8O;}(K3'$VU<57$V8O(rq $'*-,c¶zJt`s�!(\1O( p <"¡8:�!(�1:$',! <$'U7 <*I8O(+bN1O$'U R+;Nef&o;'*!(+*I8�$V*!,o R\;N*I8d$' #*!"ò8O�!(Y$'*-bN5!UW$V1iL[$'RK8:;N12"+A�õ§*c8O�!(]RT$V"O(Y;'L´$}�- <bN�!U/p_(K.rR+ /8:(\,%U<(\3'(+Uxn-5! <U/8h;N*c$}&7$'12(+*I8U<(K3N(\U�Jsq�$V*!,�¶�RT$'*�no(Y12(\`s12 #8O8O(+*

o a)b a , p _acb a , =Tdte�d ! D¡£V��=Tu&v>wxv�y*v>z{v¯D�|(}¢�I�~yZz�� � � =Tu !v w ! v y !v z !v D�| ! } ! � ! � ! y ! z ! $ =[¨|AC±'D¨M8s8:�! #"�&o;' <*I8�8:�!(�¶�� º <"s"2 <ef&!U< /m7(+,c5-"O <*-b�ghU#(+n!"2Rd�rv Z ;N12,7$'*�R+;t(\yfR+ #(+*I8:"\A�����D��� �������N�����x�&� ���������P�~�*�����>����������� �������������¡���r�£¢¤�r���¥�`� ���¦�Z�gh;N*!"2 <,!(\1¯$M8:1:$V*!"O /8: <;'*]no(K8§`h(\(+*�8§`h;�L@5!U#U#p%§9v§12(+"O;'U#3N(\,|U#(\3'(+U<"\J�`h(�(K3'$VU<57$V8O(Æ8:�!( o acb a ,)p _a)b a , =[dte�d`! DR+;'eE&);N*!(\*68E;'L�8:�!( Z $V5!*I8EL[$'RK8:;N1f$'*-,Q"O&-U< #8�8O�!(�&o;N"2 #8O <;N*�3N(\R\8O;N1¨� #*68O;� /8:"_1:$V,! W$'Us$V*!,"O&-�!(+12 <RT$VU��7$'1Oef;N*! #R]R+;'eE&);N*!(\*68O"+A¡���!(�(K.r&!12(+"O"2 <;N*�8:�7$98se�5!"28�n)(Y(\3V$'U#57$V8:(\,�n)(+R\;Nef(+"

© =Tu ª�w>ª*y�ª9D�|(}¢�¦yZz�� � � =Tu !ª w !ª y !ª D�| ! } ! � ! y ! z !¬« £© =Tu ªUw>ª*y�ª�D�|(}¢�¦yZz��­��®x�¯ '� =Tu !ª w !ª y !ª D�| ! } ! � ! y ! z ! « =[¨|A½®ND°�"2 <*!b�8O�!(��� #bN*!(\1�v ¸�Rdµ9$'1�8�8:�!(\1O(+;'e�=[(+4657$98: <;'*�=@¨}AH�IDODh8O;f(~.t8:1:$VR\8�8:�!(z,!(\&o(\*!,!(+*-R+(};VL�8:�-(eE$V8:12 /.�;N*~z $'*!,±z²!@A© =[u ªUw>ª*y�ª9D�|&}/�¦y>z��­��®��¯ s�¢=[u !v w ! v y !v D�| ! } ! � ! y ! z ! «

£ �I³ , � ³´ , ´+ ¼µyE嬩 © =Tu ªUw>ª*y�ª9D�|&}¢��y��c� �\®x�W�)�¢=Tu !v w ! v y !v D�| ! } ! � ! y !¶« =[¨|ACB'D·�5!U/8: #&!U#pEno;V8:�l"O #,!(+"snIp±¸ ´ , �¹³ , � ³ ,´ , ´ $'*!,�5!"2(

Page 181: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�B9?

¤´ , � ³ , � ³´ , ´ � ³ , � ³ ,´ , ´ , £»º ³�³ , º ´�´ , =[¨|A½·ND8:;_bN(K8

¤´ , © =[u ªUwZª*yWª9D�|(}¢�¦yZz¼�­�\®x�¯ s�¢=[u !ª w !ª y !ª D�| ! } ! � ! y ! z !)« � ³ , � ³´ , ´ =�¼`yEå�©�D ��æ��£ º ³�³ , º ´�´ , © =Tu&v>wxv�y"vxD�|&}/�¦y��c� ��®x���)� =Tu !v w ! v y !v D�| ! } ! � ! y !¶« =[¨|A½GND-�;�`³"O5-e n);'8:��"2 <,!(\"�;�3N(+1½zFJi*-;'8:(�8O�7$V8�8O�!(%¶�� º #*�8:�!(l$Vno;�3N(�(+4657$V8O <;N*��7$V"�*!;¾z,!(\&o(\*!,!(+*-R+('Jo$'*!,l8:�65!"�8O�!(z"O5!e�;�3N(\1z <*I8O1O;t,!5!R+(\"�$�L[$VR\8:;'1�=�¼`yEå�©�Dh;N*l8:�!(�¶�� º A© =Tu&v(w�v¿y"v�D�|(}¢�¦yN�À� �\®x�W�)�¢=Tu !v w ! v y !v D�| ! } ! � ! y ! «£ =�¼`yEå�©�Ddã ��æ�� ¤´ , ´ � ³ , � ³´ , ´ © =[u ª�w>ª*y�ª9D�|&}¢��y>z��­��®��¯ s� =Tu !ª w !ª y !ª D�| ! } ! � ! y ! z !¶« =[¨|A#©�öND

���!(\*>5!"O(}8O�!(�ghU#(+n!"2Rd�rv Z ;N12,7$'*>R+;t(\yfR\ <(+*I8O"_=[(\4t5!$V8: #;N*�=@¨}A/©�DOD�8:;c12(+,!5!R\(}8:�-(�e�$98:1O ¢.l,!(~v&)(+*!,!(\*!R+(z8:;|�7J!�>J6�*!o$'*!,��#!WAw�ÁÂu £ =�¼`yEå�©�Ddã ��æ�� ¤Ã¿ÄAÅÃ�ÆMÇà ÄÆ Ç Ä

� ³ , � ³´ , ´ � ³�È��ɳ´ È � ´ � ³ ,Ê � , ³ ,´ ,Ê � , ´ ,Ë © =[u ªUw>ª*y�ªUz±ªVD�|&}¢�6��� �\®x�¯ s�¢=[u !ª w !ª y !ª z !ª D�| ! } ! � ! � !¶« =[¨|A#©N©�D

º #*!R+(Y8:�-(+1O(z #"�*!;ERd�!$'*!bN(z <*%8O�!(z&7$'1O(\*I8�R\;N*-m7b'5!1:$V8O <;N*%8O�! <"�R+$'*�n)(z(K.r&!12(+"O"2(+,>$V"+AwPÁ~u £ =�¼µyE嬩�D:ã ��æ�� º-=[u&v>wxv�y"vZz¾vSe�u !ª w !ª y !ª z !ª D ¤Ã�Ä�ÅÃ&ÆMÇà ÄÆ Ç Ä� ³ , � ³´ , ´ � ³ Ê �G³´ È � ´ � ³ Ê � , ³´ È � , ´ ,

Ë © |(}¢�I��� ��®��¯ '� | ! } ! � ! � !Ì«£ =�¼µyE嬩�D ã �� º-=[u ª�w>ª*y�ª"e�u !ª w !ª y !ª D ¤Ã Ä ÅÃ Æ ÇÇ Ä� ³B, � ³´ , ´ � ³ È �ɳ´ È � ´ � ³ È � , ³ ,´ È � , ´ ,

Ë �Í|&}É� �s� | ! } ! $Î��}¢�I���­®x�¯ '�­} ! � ! � ! $ =[¨|A#©�¼'D

Page 182: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�B'±���!(\*P5!"2(}(+4657$98: <;'*�=@¨}A/©�Dh8:;f5!*!R+;'5!&!U<(��7J7� $'*!,E�*!¥J7�#!o <*I8:;¡}§J7�ÐÏ�J'ÑNJ �ÓÒ�$'*!,Ô}Õ!<J �#!Ï J�Ñ�!WJ�#!Ò A

w�ÁÂu £ =�¼`yEå�©�D ã �� º-=[u ª�w>ª*y�ª"e�u !ª w !ª y !ª D ¤Ã Ä Å�ÃsÖÇ¦Ç Ä Ç�×Ç�ØKǦÄ× Ç¦ÄØ� ³B, � ³´ , ´ � ³ È �ɳ´ È � ´ � ³ È ��,O³B,´ È � , ´ , � Ï �� ��¿ÙC�¿Ú��

Ë � Ï , �� ��,� , Ù � , Ú � , �Í|(}É� �'� | ! } ! $Î��}¥�ÐÏ#�ÓÒ��­®x�¯ s�­} ! � ! Ï � ! Ò $ =[¨|A#©��ND�ÓÒMe�5!"�8s(+4657$'U¯� ! Ò "2;

w�ÁÂu £ =�¼`yE嬩�Ddã �� ºr=Tu ªUw>ª*y�ªT�ÓÒ�e�u !ª w !ª y !ª � ! Ò D ¤Ã Ä Å�à ÖÇ¦Ç Ä Ç ×Ç Ø Ç Ä × Ç Ä Ø� ³ , � ³´ , ´ � ³�È��ɳ´ È � ´ � ³�È�� , ³ ,´ È � , ´ ,

Ë � Ï �� ��¿ÙC��Ú�� � Ï , �� � ,� , Ù � , Ú � , ��|(}É� �s� | ! } ! $¤�Í}¥�ÐÏ#�ÓÒ��­®x�¯ s�­} ! � ! Ï � ! Ò $ =[¨|A#©T?ID°�"2 <*!b�8O�!(Y�� <bN*!(\1�v§¸¡Rdµ9$'1�8s8:�!(\;N1O(\e�;N*!R\(|$VbI$' <*c`ò(�bN(\8w�ÁÂu £ =�¼`yE嬩�Ddã �� ºr=Tu ªUw>ª*y�ª*e�u !ª w !ª y !ª D ¤Ã Ä Å�ÃsÖÇ¦Ç Ä Ç�×Ç Ø Ç Ä ×

� ³B, � ³´ , ´ � ³ È �ɳ´ È � ´ � ³ È � , ³ ,´ È � , ´ ,Ë � Ï �� �� Ù ��Ú[� � Ï , �� � ,� , Ù �¿Ú�� , � Ï , �KÏ� , Ù �¿Ù=�¼�}!嬩�D �� ��|(}É� �s� | ! } ! $Î��}É�c� ®����)�­} ! $

£ =�¼`yE嬩�D ã ��=�¼\}-嬩�D �� ºr=Tu ªUw>ª*y�ª*e�u !ª w !ª y !ª DD�Í|(}É� �'� | ! } ! $Î��}É�c� ®����)�­} ! $Ë ¤Ã Ä Å�ÃsÖÇ¦Ç Ä Ç¦×Ç Ø Ç Ä ×

� ³ , � ³´ , ´ � ³ È �G³´ È � ´ � ³ È � , ³ ,´ È � , ´ , � Ï �� �� Ù � Ú � � Ï , �� ��,� , Ù ��Ú�� , � Ï , �KÏ� , Ù �¿Ù =[¨|A#©�±'D

���!(MghU<(+n-"ORd�rv Z ;N12,7$'*�R\;r(KyfR+ <(\*I8:"Æ$V1O(ò8O�!(+*}1O(+&-1O(+"2(+*I8:(\,_bN1:$'&-�! <R+$'U<U/pY$'*!,|(+4657$V8O <;N*c=[¨|A#©�±'D1O(\,!5!R\(+,�8O;��� <bN*!(\1s®9v#�M"2pte�n);NU#"s$'"�L@;NU<U#;�`s"

Page 183: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�BV®

l’Û

=7

J’Ü1

p

p

J’Ü

j’Ý l

ÛjÝ

1/2

’ j’Ý

1/2

1

(−1)Þ 2l’

(2l+1)1/2

− − − − −

(−1)Þ 2Jp(2J+1)

1/2(−1)Þ 2Jp(2J’+1)

1/2(−1)Þ 2l

(2j+1)1/2 (−1)

Þ 2l’(2j’+1)

1/2

− +ß−

+ß+ß

p

1/2

j’á

1

l’â l

â(−1)Þ 2J’

(−1)2Jp(−1)

2Jp(−1)2l(−1)

2l’(−1)

2l’(2J+)

1/2(2J+1)

1/2(2J’+1)

1/2(2j+1)

1/2(2j’+1)

1/2(2l+1)

1/2

(−1)Þ 2J’

(2J+1)1/2

− +

+

+ 1/2

jã1j’

ãJ’ä

JäJ

äp

l’ l

(−1)2J’

(−1)2l[(2J+1)

2(2J’+1)(2j+1)(2j’+1)(2l+1)]

1/2

=

(−1)2J’

(−1)2l

Due to reversal of arrow in diagram

− +

+

+ 1/2

jã1j’

ãJ’ä

JäJ

äp

l’ l

[(2J+1)2(2J’+1)(2l+1)(2j+1)(2j’+1)]

1/2

=

Cut−off

Page 184: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�B'B

j’ã J’ä

+

1

1

P

j’ã

l’ l

1/2

+ +

+

[(2J+1)2(2J’+1)(2l+1)(2j+1)(2j’+1)]

1/2

Note the change in sign at the node(−1)

Jp+J+j(−1)

j+l+1/2

j’ã J’ä −

− +

+

1

1

P

j’ã

l’ l

1/2

− +

+

[(2J+1)2(2J’+1)(2l+1)(2j+1)(2j’+1)]

1/2(−1)Jp+J+j

(−1)j+l+1/2

j’ã J’ä −

− +

+

1

1

P

j’ã

l’ l

1/2

− +

(−1)2j’

(−1)2Jp

(−1)Jp+J+j

(−1)2j’

(−1)2j (−1)

j+l+1/2

+

[(2J+1)2(2J’+1)(2l+1)(2j+1)(2j’+1)]

1/2

Page 185: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�BV·

£ åæ ç � y y*vy ! � ! © è éê åæ ç �*!ë}À!³©UìN¼} � © è éê =�â�©�D � ³ Ê =�â�©�D ³ Êµí ³ =�â�©�D Ï í ��æ��Ë%î =�¼µyE嬩�D � =�¼`y ! 嬩�D\=�¼+��嬩�D\=�¼+� ! å�©�D+=�¼\}!å�©�D�ï ��æ�� =[¨|A#©�®ND���65!"s`h(��7$T3N(

w�ÁÂu £ =�¼`yEå�©�D ã ��æ��=�¼\}-嬩�D ��æ�� ºr=Tu&v>wxv�y*v�ð�u !v w ! v y !v DD�Í|&}�� �'� | ! } ! $ë� � �`��}É�c� ®��W�c�­} ! $ñ� �Ë = â|©�D�ò ³ Ê í ³ í Ï í ��æ�� åæ ç � y y"vy ! � ! © è éê åæ ç �"!ë}À!éUìN¼} � © è éêË%î =�¼`y�å�©�D � =�¼`y ! å�©�D+=�¼+�^å�©�D+=�¼+� ! 嬩�D\=�¼\}-嬩�D�ï ��æ�� =[¨|A#©�B'D

U<(+$',! #*!b�8:;�8:�-(zm!*7$'U�1O(\"O5!U/8� © =[u&v>wxv�y"v¯D�|(}¢�¦yN�c� �\®x�W�)�¢=[u !v w ! v y !v D�| ! } ! � ! y !¬« � �£ =�¼`yE嬩�D+=�¼\�]嬩�D+=�¼\� ! 嬩�D\=�¼`y ! 嬩�D�º-=[u&v>wxv�y"v�ð�u !v w ! v y !v DË �`�Í|&}�� �'� | ! } ! $ë� � �`��}É�c� ®��W�c�­} ! $ñ� � åæ ç � y y"vy&!«�*! © è éê

� åæ ç �*!ë}À!Ø©�ìN¼} � © è éê� =[¨|A#©�·ND

õ�8� <"¯*!;�`�&o;'"O"O #n!U<(�8O;^"O5-eF(\4657$V8: #;N*�¨}A/©�·�8O;^$'R\R+;N5-*68�L@;N1´$'U<UI&);N"2"O <n-U<(¡,!(+b'1O(+(\"Æ;'L-1O(+"2;NU<5r8: <;'* <*%8O�!(Y8:1:$V*!"O /8: <;'*!"hL@;N1M8O�!(z*!(\` R\;N5!&!U# <*!b�"2Rd�!(+ef('Ai���!(z1O(\"O5!U/8:"�$'12(�"2�!;�`s*P #*l8:$'n!U#(]¨}A/©NA

Page 186: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�BVG

ó ôOõö õ÷ õø­ùú ûú üú ÷ú

ó ô õö õ÷ õø ùú ûú üú

ó ôOõö õ÷ õø­ùú ûú

ó ô õö õ÷ õø ùú

ó ôKýö ý÷ õø ùû ü÷ó þ÷ ÿ�øó þ÷ú ÿ�øó þü ÿ�øó þüú ÿ�øó þ÷ ÿ�øó þüú ÿ�ø

� �� ���

� �����

� �� ���

� �����

�ü÷÷ õ ÷ú üú�

� üú ûú� þû ü�

� üú ûú� þû ü�

��û�� ��� ûú��� �� ùû���� ùú ûú���

��û�� ��� ûú��� �� ùû���� ùú ûú���

��û�� ��� ûú��� �� ùû���� ùú ûú���

� ����û�� ��� ûú��� �� ùû���� ùú ûú���

ó ô ýö ý÷ õø­ùû üó þ÷ú ÿ�øó þü ÿ�øó þüú ÿ�øó þ÷ õÿ�øó þü ÿ�ø� �� ����� �����

� �����

� �� ����� �����

� �����

� üú ûú� þû ü�

� üú ûú� þû ü�

��û�� ��� ûú��� �� ùû���� ùú ûú���

��û�� ��� ûú��� �� ùû���� ùú ûú���

��û�� ��� ûú��� �� ùû���� ùú ûú���

� ����û�� ��� ûú��� �� ùû���� ùú ûú���

ó ôKýö ý÷ õø ùû� ������

� ��� ���

� ��� ������ ����

� ��� ���

þó þ÷ õÿ�ø

þó þ÷ õÿ�ø

��û�� ��� ûú��� �� ùû���� ùú ûú���

��û�� ��� ûú��� �� ùû���� ùú ûú���

��û�� ��� ûú��� �� ùû���� ùú ûú���

� ����û�� ��� ûú��� �� ùû���� ùú ûú���

ó ô[ýö ý÷ õø ù� ������

� ��� ���

� ��� ������ ����

� ��� ���

ó þ÷ õÿ�ø

ó þ÷ õÿ�ø

� ���û�� ��� ûú��� �� ùû���� ùú ûú���

� ���û�� ��� ûú��� �� ùû���� ùú ûú���

� ���û�� ��� ûú��� �� ùû���� ùú ûú���

� �����û�� ��� ûú��� �� ùû���� ùú ûú���

��$'n!U<(_¨|A#©Nô����������Æý��BJ�ú�þ½üA�U@%�T�����! "���! ��Æý��J�ú9þHü��U@$#�ù�M��%¾Qdù~þ�J�ÿG��ÿ�Qdù~þ�J9ù���ÿ��¤�¶�KøxQdùKø�ü��JM��þ½ÿ�Q\ÿ�þO�S���ø�ÿ��[�Kþ�J���ü���µF

Page 187: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�·'ö�����D��� �������N�����x�&� ���������P� �>�����t�µ���>�����������'&���!(|$'*!bN5-UW$'1�$VU<bN(\n!1:$�"2;NU<5-8O <;N*cL@;'1M8:�!(zR+12;N"O" v§R\;N5!&!U# <*!b Z $'5!*I8�L[$'R\8O;N1s&!1O;tR+(\(+,!"�$'"�L@;NU#U<;�`zô

� =[u&v>wxv¯D�|&}Ku�wDy��c� ��®x���c� =Tu !v w ! v y !v D�| ! } ! � ! y ! $£ =�¼µyE嬩�D ��æ�� ¤´ , � ³ , � ³´ , ´ � =[u&v>wxv¯D�|&}[uxwDyZz¼� ��®��¯ '�¢=[u !v w ! v y !v D�| ! } ! � ! y ! z ! $£ =�¼µyE嬩�D:ã ��æ�� ¤Ã¿ÄAÅGÃÃ)(�Ã+*Wà ÄÆÇ Ä

� ³ , � ³´ , ´ �$,.- ³´0/W´21U´ � ³ ,Ê � , ³ ,´ ,Ê � , ´ ,Ë � =[u&v(wxv¯D�|&}Ku�wPz - z , � ��®��¯ s� =Tu !v w ! v y !v z !v D�| ! } ! � ! � ! $-�;V8:(Y8:�7$98^ #*l8O�! <"sU<$'"28�U< #*!(]8:�-(}°Z� º <"s #*!,!(+&)(+*I8O(+*I8];'LS·²$V*!,�8O�!(+12(\L@;N12(|"25!eEef <*-b�no;'8O�"O #,!(+"ò;�3N(+1D·j #*68O1O;t,!5!R\(+"M$ �

3 � ³ í �54 ;N*f8:�!(�¶� º J6�!(+*-R+(^8O�!(�Rd�!$'*!bN(� <*�"2 <bN*f;'Lx8O�!(�=�¼`y]å�©�D8:(\1Oe�Aw�ÁÂu £ =�¼`yE嬩�D ã ��æ�� ¤Ã Ä ÅGÃ6(à *à ÄÆ Ç ÄÃ)( Æ Ç¦Ø[Ã+* ÆÇ�×OÇ Ä × Ç Ä Øà Ä* Æ Ã Ä( Æ

� ³ , � ³´ , ´ � ,7- ³´2/�´21U´ � ³ ,Ê � , ³ ,´ ,Ê � , ´ , � , Ê �� ,´ / Ê �¿Ú ´ / � - Ê Ï -´ 1 Ê ��Ù ´ 1 � Ï , �� � ,� , Ù � , Ú � ,Ë � - , Ê , ,Ê ³ ,Ê´ 1 , Ê ´ / ,Ê ´ Ê , ��=Tu&v>wxvZz - Ê z , Ê D�|(}¥�ÐÏ#�ÓÒ�� ��®x�¯ '� =Tu !v w ! v z - , Ê z , ,Ê D�| ! } ! � ! Ï � ! Ò $

£ =�¼`yE嬩�Ddã ��æ�� ºr=Tu ªUw>ªMz - Ê z , Ê �ÓÒ�ð�u !ª w !ª z !- Ê z !

, Ê � ! Ò DN��|&}¥�ÐÏ�� �\®x�¯ �� | ! } ! � ! Ò $Ë ¤Ã Ä Å�Ã)(à * à ÄÆ Ç Äà ( Æ Ç Ø Ã * ÆÇ × Ç Ä × Ç Ä Øà Ä* Æ Ã Ä( Æ

� ³ , � ³´ , ´ � ,.- ³´2/W´01U´ � ³ ,Ê � , ³ ,´ ,Ê � , ´ ,Ë � , Ê �� ,´2/ Ê ��Ú ´2/ � - Ê Ï -´01 Ê �¿Ù ´01 � Ï , �� � ,� , Ù � , Ú � , � - , Ê , ,Ê ³ ,Ê´ 1 , Ê ´ / ,Ê ´ Ê ,

Page 188: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�·r©

£ =�¼`yE嬩�D ã ��æ��=�¼\}-嬩�D ��æ�� ¤Ã Ä ÅGÃ6(à * à ÄÆ Ç Äà Ä( Æ Ç Ø Ã * ÆÇ × Ç Ä ×� ³ , � ³´ , ´ � ,7- ³´2/�´21U´ � ³ ,Ê � , ³ ,´ ,Ê � , ´ ,

Ë � , Ê �� ,´2/ Ê � Ú ´2/ � - Ê Ï -´01 Ê � Ù ´01 � Ï , �� � ,� , Ù � Ú � , � - Ê , Ê ³ ,Ê´01 Ê ´0/ Ê ´ Ê , � Ï , �KÏ´ Ù , ´ ÙË ��|(}É� �s� | ! } ! $¤�Í}É�c�­®x���)�­} ! $ =[¨|A#©TGID

8 *!(�R+$'*�8:�-(+*�5!"2(�8:�!(EbN1:$V&!�! <R+$'U¡eE(K8:�!;t,!"}$'"�n)(\L@;N12(E8:;P"O <ef&!U# #L¥pP8O�!(lghU#(+n!"2Rd�rv Z ;N1O,!$'*R+;t(\yfR\ <(+*I8O"+A

Page 189: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�·N¼+

+ +

− −

+−

l l’

Jä J’

äL

S

LP

P’

SP

j’ã

1/2

(2J+1)−1/2

(2l+1)1/2

(−1)2J’

(2J+1)1/2

(−1)2S

(2J+1)1/2

(−1)2Jp’

(2J’+1)1/2

(−1)2Sp

(2S+1)1/2

(−1)2Lp

(2L+1)1/2

(−1)2l’

(2j’+1)1/2

(−2)2Lp

(2Jp’+1)1/2

(−1)2l’

(2l+1)1/2

+

+ +

− −

+−

l l’

Jä J’

äL

S

LP

P’

SP

j’ã

1/2

(−1)2J’

(2J+1)1/2

(−1)2S

(−1)2Jp’

(2J’+1)1/2

(−1)2Sp

(2S+1)1/2

(2L+1)1/2

(2j’+1)1/2

(2JP+1)1/2

(−1)2J’

(−1)2Sp

cut−off

=

+

+ + − +

+ +

− −

− −SP

ll’

Jä J’

äL j’

ãLP S J

äP’ 1/2

x y

LP s Jä

P’

1/2

(−1)2S

(−1)2Jp’

(2J+1)1/2

(2J’+1)1/2

(2S+1)1/2

(2L+1)1/2

(2j’+1)1/2

(2JP’+1)1/2

Σ(2x+1)(2y+1)

=

Page 190: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�·'�+

+ + − +

+ +

=

x

− −x

ll’

Jä J’

äL j’

ãLP S J

äP’ 1/2

LP S 1/2Jä

P’

SP− −

(−1)2S

(−1)2Jp’

[(2J+1)(2J’+1)(2S+1)(2L+1)(2j’+1)(2JP’+1)]1/2Σ(2x+1)

− −

+

+

+

+

− −

−−

=LP

S

L x

1

J’ä

l l’

1/2

P’

j’ã

xS J

äP’LP

1/2

SP

Σ(2x+1)(−1)2S

(−1)2Jp’

[(2J+1)(2J’+1)(2S+1)(2L+1)(2j’+1)(2JP’+1)]1/2

−−

+

+

+ + +

− −

− −

−−

LP

S JälL

l

l’

x

J’ä

1/2

P’

j’ã

1 LP 1/2

x

SP

(−1)2S

(−1)2Jp’

[(2J+1)(2J’+1)(2S+1)(2L+1)(2j’+1)(2JP+1)]1/2Σ(2x+1)

cut−off=

SJä

P

x

Page 191: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�·V?−

− −

− −

−−

+9

+9

+9

+9 +9

+9

LP

l:

S;

J<

J<

J’<

l:

l’:

x= x=J’<

l’: 1/2

J<

P’

x=

S;

P

LP1/2

S;

J<

P’

=> Σ(2x+1)

(−1)2S

(−1)2Jp’

[(2J+1)(2J’+1)(2S+1)(2L+1)(2j’+1)(2JP’+1)]1/2

x= L

1

j’?

− −

+9 −

+9

− −

+9 +9

− +9

LPl:

1/2 x=

S;

l:

J<

l:

x=

J< 1J’

<

x= l’:

l’:

x=

j’? J’

<J<

P

S;

1/2

J<

P’LP

S;

P

=> Σ(2x+1)

(−1)2S

(−1)2Jp’

[(2J+1)(2J’+1)(2S+1)(2L+1)(2j’+1)(2Jp’+1)]1/2

− −

+9 −

− −

+9 +9

+9

− −

− +9

+9 +9

LPl:

1/2 x=

S;

l:

J<

l:

x=

J< 1J’

<

x= l’:

l’:

x=

j’? J’

<J<

P

S;

1/2

J<

P’LP

S;

P

=> Σ(2x+1)

(−1)2S

(−1)2Jp’

[(2J+1)(2J’+1)(2S+1)(2L+1)(2j’+1)(2Jp’+1)]1/2

− +9

− +9

(−1)2S

(−1)2J

(−1)x+J’+l’

(−1)l+l’+1

(−1)2Jp’

(−1)l’+J’+x

(−1)j’+J’+Jp’ (−1)

2Lp(−1)

1/2+S+Sp(−1)

x+Lp+S

Page 192: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

©�·N±

£ @Y=�¼7A�嬩�D¡åæ ç w } w�vA u y è éê åæ ç © }Õ! }

A y y ! è éê åæ ç y�!v A ©UìN¼} ! � ! y ! è éê åæ ç y�!v ©UìN¼ Au wxv u&v è éêË = â|©�D � ³ =�â�©�D�òCBr= â|©�D�ò ³ , =�â�©�D�ò Ï , =�â�©�D Ï =�â�©�D.D � =�â�©�D � , = â|©�D ³ ,Ê = â|©�D�òE- Ê = â|©�D � ,ñ= â|©�DF, ÊË%î =�¼`y�å�©�D+=�¼`y ! å�©�D+=�¼*u�å�©�D+=�¼\wSå�©�D+=�¼+� ! 嬩�D\=�¼`y*v , 嬩�D�ï ��£ =�â�©�D � ³ í ò ³ , í ò Ï , í Ï í ò æ�� í � , í ³ ,Ê í òE- ʵí � , í , ÊË =�¼`y�嬩�D ��æ�� =�¼`y ! 嬩�D ��æ�� =�¼*u�嬩�D ��æ�� =�¼\wP嬩�D ��æ�� =�¼+� ! å�©�D+=�¼`y !v 嬩�D ��æ��Ë @ B =�¼7A�å�©�D+= â|©�D�òCB åæ ç w } wxv

A u y è éê åæ ç © }À! }A y y ! è éêË åæ ç y !v A ©UìN¼}À! �"! y&! è éê åæ ç y !v ©Uì'¼ Au wxv u&v è éê =[¨|AC¼'öND

���!(�1O(\"O5!U/8:"�L@;N1M8:�-(z3V$'12 <;N5-"MU<(K3N(+U#"s;'L�12(+"2;NU<5-8O <;N*lR+$'*%8:�!(\*Pn)(Y(\3V$'U<5!$V8:(\,P$'"�n)(\L@;'1O('A

Page 193: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

� ��� ������� ��� ���

�������������! #"$�% #&('�'�)! +*,�� +-/.10203�4�!)5.�6# #7��% 98;:�6<03���! #�=�� %�?><@1AB #C��!DE:4�GF�)�.1HI-J.B���!�56LK�M<NE)5:�O�)PRQTSVU<W<X �

X �������4�!6+ ZY��[8\�� ]�?>1>^�� #_`��)a03�2.�b�M<NE)5:4O�c1�]de'Zc16<03:%03:4�!)�_% Z�f03M���.�6��W �(g=��Ch�R7fci��6������j 9�?>�k�A^ Zl1.�'��56�c1�m.1H�KVM<NE)5:�O�)n7� ]oEp SqW k X1rU �(7fci05��)! ^st�uCh�% B"v:�6�w�)a03.�6+ e�=�yxz�% Z-/OG8;M�:4�a03���! ^Cv�[8\�yK{�� 9�?>�A X EK��5.EO1�BC�.|N�c1�9}B.EO1�B~].16���.�6+ PR�#�LSqX > r

@B�(7fci05��)! Rst�ZCh�m�����#���� {��>�A U ��y�f05.���:�Ovc16�����.��4�!O�'��4c1�fb��2.BO!��)5)2��)!_� ���O�c1������:4OvK��2��)5)A^�(7��!M��5:�6�w����! �"$�� ��?>�> r I�y*zM�����6^�Z'��!6�O���.1Hn��bZciO�:�0�NT.�6�&�NE���5.�w1��6�~m:46���x���:�)5)5:�.�6�ci6����.�6�:�)3ci05:4.�6�7�c1��c16^O���:�6�&n:�w�M;s���6�)2:�0�N�s�:�D��!�205.���K���ci)5��c1)�_% �-/c��e��K���c16^O¡ <�¢��6�)205:�05'^03�£H¤.1�K���c1)2��c$K�M<NE)5:4O!)�C���b�.��a0� Z��KVK¥�?¦<§1@

r �(7��!M��5:�6�w����! {"¨�% q�?>�>�k^ ��yx�)5O�cib9��©�ciO!03.1�5),H¤.��v~m:46���x���:4)5)2:4.�6Tc16^��K�.1b�'���c�03:4.16ªYfci�4O�'^��c��03:�.�6�)�_% ���KVK���C���b�.��20� �-£c��JK��4c16�O¡ ���6�)a03:%03'^0�H¤'���K���ci)5��c1b�M<NE)5:4 R �pZ« SVU<W >

k^�(7��!�5w��!�5.�6+ ^K��% B8¬��)2����c1���­ ^©��% �©Z.�6<03c1:46��i �®$�% 9�?>1>^���% B�n)203�2.�b�M<NE)5:�O�c1�+l1.�'��56�c1�j ^¯ �#�/SqX @ W>^�(7��G03M��i t&=�`�°��± }^ci�4b��!03�!�� =xz�yxz�% ��?> r�r ¨��²�'Zc16<03'^� -/�!O¡MZc16�:�O�)³.1H���6��G��c16��\*fF�.��x��4��OG03�2.�6��f03.���)�_% RK��4��6e'��´KV'^µ+�9Y�.��� Rg(�!F·¶q.��5 

�?¦^��7IMZci03:4c^ Z�°�R"$�R±¸"vc1)a036��!�� 9}9�9�t�� m��>�> X ��n)205�5.�b�M<NE)5:�O�c1��l1.�'��26Zc1�{}B'^b�b��4�!���!6<0�}E���5:���)! � p S � W >

������7I�5��w1���� RK{�� m��>�>�k^ ZK��2:�Dici03�hY�.1���t'^6�:4O�ci03:�.�6

�?A^�

Page 194: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�?A X

� X ��7I�5:46^ � #st��-T��}9�#±¹}^ci03O¡M^�4���! ]®$�yCv�� {�?>�A1k^ {�u��6^w�'���ci����.����!6<03'��J_� #�n�EH¤.��5�Lº(6�:�D1���2)5:�0�NK��5��)2)

� W ��7I�5.E.� e)� Rst�u&=�Z�!0(ci�j�% +�?>�>�>B ���)a03�5.16�.��°N�ci6�����)205�5.�b^MeNE)2:4O�)! +¯�» �LSIX�r1r� U ��7I�5.E.� e)� Rst�u&=�Z�!0(ci�j�% X ¦�¦�¦B ���)a03�5.16�.��°N�ci6�����)205�5.�b^MeNE)2:4O�)! +¯R¼ �LS A�> r�|@B��7I'��5w��!)5)! ��=�R±¸}B'^�������2)� R&=�uK��% m�?> r A^ R-/.�6#�Rgn.10��ZCh�Z�n)205����}B.EO1�% mo � » SVW1U @�?A^��7I'��5w��!)5)! ��=�R±¸}B'^�������2)� R&=�uK��% m�?>�k r R-/.�6#�Rgn.10��ZCh�Z�n)205����}B.EO1�% PRPR�¬SqX @ r� r �(Yfci�5�4)2)5.�6# #-T�� ]�?>�k1A^ +�y�½Y�.���b�'^05���(K��5.�w��5c1��H¤.��n}B.1��DE:46�w�-/'^��03:¾�����!D��!�mgn.16B��~E*�x�Cnc����:�c�03:�D1��*{�3c16^)2H¤���¿KV�2.�µ�������)�:46$-/.?DB:�6�w�.���}E0¡ci05:4OI�f03��.�)5b�M^���5�!)�_% eºnb�b^)3c1�4c�C(�!b9.1�20VÀJÁ W�W ºnb�b^)3c1�4c¨�n)205�5.�6�.1��:�O�c1�+�,µ^)5���aDici03.��aN

�?k^�(Y�.16���.�6+ Bxz�yºh�E±Â}BM�.��a03���!N� B®$�y&h�� R�?> W @B ��u*zM��(05M���.1�2N¨.1H#ci03.1��:�Oz)5b���O!05�3c^_% ^Yfc1�tµ��2:4��w��ºn6�:%D��!�5)5:%0�N�K��5�!)5)

�?>^�(Y�.?F�c16+ �Ch�ust�% ¿�?>�k^�1 ��y*zM^�¨05M���.1�2NL.1HIci05.���:4O°)a03�2'�O!05'��5��c16��³)2b9�!O!03�5c^_% �º(6�:�D1���2)5:�0�N£.1HYfc1�4:%H¤.��56^:�c°K��2��)5)! #7I���5 e���!N

X ¦^��s,cib�b9�!6+ �8\�% Z��6����!�5)2.�6+ �~��� R-J:4MZc1�4c1)� �st�� +��>�k r ��n)205�5.�b�M<NE)5:�O�c1��l1.�'^�56Zc1�­ Z¯moEp S �?><@X ����s��5c1 1�i �®¨�[8\�u©��R±½s,c1��w<c1�26�.� E�°�% m�?>�A�>^ ^��)205�5.�b^MeNE)2:4O�ci��l1.1'��56Zci�j +oE¼ �³SVU @1>X�X ��s��5c�F�:46+ Z&h�`8\�% m�?> r ¦^ Rl^��²�'Zc16<0���}Eb9�!O!03�2.�)5Oi��Cnc1�^:�ci0��^*{�3c16�)aH¤���! ]oE« SVW�WX1W ��s�.?NE�4�hl^�y®$�R±¸-/OG8;M�:4�a03���! RCh�[8\�uK��m�?>�k1¦^ R-/.�6#�Rgn.10��ZCh�Z�n)205����}B.EO1�% ]oEp�¯ S > UerXiU ��x�����.�6��^)� {�°�]Cv�� V�?> r @B ¿�y�n6�w�'��4c1����.�����6<03'��Ã:46Lde'Zc16<03'^�Ä���!O¡MZc16�:�O�)�_% �KV�2:46�O!�!05.�6ºn6�:%D��!�5)5:%0�N�K��5�!)5)

X @B��x�:�)5)56^���� ]8\�% ¿l1.�6^��)� �-¬�% �g('�)5)2µZc1'������! �&h�� V�?> riU {Y�.���b�'^05���hK�MeNE)2:4O�)tY�.����t'�6�:�O�c��03:�.�6�)zbZc1b����nYfKqYfk^ X�r ¦

X A^��©Z�!6�)205���5��c1O¡M��!�� B-¬�1x��i�!0qc1�­�� B�?>�>�>B �K���c1)2��c�KVM<NE)��EY�.16e05�5.��­�1©Z'�)2:4.�6# ��?>�>�>B 1»{o S � W1U @X�r ��©�:4)5O¡MeµZciO¡M����! v®¨�`�°�% n~].EO¡MÅ}9�ust�z±Æ}B'^�������2)�&=�uK��% X ¦1¦�¦^ ���)a03�5.16�.��°N�c16��;��)a03�5.��b�M<NE)5:4O!)� 9¯R¼ �³Sqr A r

Page 195: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�?A W

X k^��©�:4)5O¡MeµZciO¡M����! q®¨�`�°�% X ¦�¦B�� q�ys�:�ciw�6�.�)2:4)hc16�����.E�������4:46^w�.1Hz.�b^05:4O�c1�4�%N£03M^:4O¡ ³)205�5'�OG03'��2��):46�03M^�h��.?F��!�z)5.��4c1��ci03��.�)2b�M����2�1_% RK�MZsÇ05M���)2:4)� Rº(6�:�D1���2)5:�0�N�.1H�}e03�3c�03M�O!��NE���

X >^�(®,.�'^0203�!µ��5.�È!�1 �K{�� �&(��:46^È����­ �K��I±ÃÉ�:4c1�nl^�[��Y��� ,�?>1> W f�n)203�2.�6�.��=NÊc16����n)203�2.�b�M<NE)5:�O�)}B'�b+��}E�����% �pRp S @B� W

W ¦^�(®,�5:����J �&h�RCh�% +�?>�A1¦^ ���)a03�2.�b�M<NE)5:4O�c1��l1.�'��26Zc1�j +oB¯ P³S k1k WW ���(®,�5:����J �&h�RCh�% +�?>�A U #�yKV�4c1)5��c�}Bb���O!05�5.�)2O�.�b<N�_% #-JO?®,�3c�F¥��&(:4���W<X �(®,�5:����J �&h�RCh�+ m��> riU #��}Bb���OG03�3ci�]~]:�6��v7��2.<c1���!6�:46�w¨µ<N�K���ci)5��c1)�_% ���O�c1���!��:�OvKV�2��)2)W�W �(®,�5:����J R&=��Cv�m ��?>1> r +�yKV�2:46�O!:4b�����)�.1H¿KV�4c1)5��c�}Bb9�!O!05�5.�)2O�.�b<N�_� +Yfc1�tµ��2:4��w1�=º(6�:�D1���2)5:�0�NK��5��)2)

W1U �(®,'��2.|DE:�O¡M+ �Ét�Z*,�R±ËxV6^w����­_u)2Me0� ZÉt��}9�% +�?> r1r R}B.?D���KVM<NE)��9l�x�*�K� R»9¼ SIX1W�XW @B��&(��c1)2�4�GN� 9l^�ygh�R±¸-/:�MZc1�4c1)� ^st�� m�?> r A^ ^�n)205�5.�b�M<NE)5:�O�c1��l1.�'^�56Zc1�­ P «�¼ SIX�r1WW A^��&(:4�5c?N�c1��c^ e*,�� 9�?>�A W EK��5.EO��!����:�6�w�)I.1H+03M^����)a03�2.�6�.���:4O�c1��}B.EO�:4�G0�N�.1H{l�c1bZci6+ £oE¼ S � X1XW<r ��&(:4�5c?N�c1��c^ �*,�% m�?>�k<@B Z}B.���ci�zK�MeNE)2:4O�)! {oE«R« SVU �?@W k^��&(.���05)5��c1�5 R Rl^�% m�?>^��>^ ^�n6�6+��K�MeNE)!����~m��:4b^È�:4w<�G /¼ QTS @ r�rW >^��&(.B.1b9�!�� �Y��u©��� m��>�A�A^ ^KVM<NE)5:�O�c1�{C��!DE:4�GFv �o<»9p SIr�rU ¦^��&(.B.1b9�!�� �Y��u©��� m��>�A�k^ ^KVM<NE)5:�O�c1�{C��!DE:4�GFv �o � ¼ SIX �|@U ����&(.��205.�6+ R~��% X ¦�¦^�1 ZK��5:�Dici05�,O�.����t'�6^:4O�c�03:4.16UeX ��&('�µ�6��GN� +�¡�% +&('������!�� �st��®$�% #~]c16�Èi #*,�% ��?>1> U R��)a03�5.16�.��°N/c16��£��)205�5.�b^MeNE)2:4O�)! PRQRP S�|@B�

U<W ��&('������!�� Zst�y®¨ �±¸-/:�MZc1��ci)� Zst�% +�?>�k1k^ ���)a03�2.�b�M<NE)5:4O�c1�{l1.�'��26Zc1�­ R¯R¯]o Sqr > UU�U �z��w1�4��)2:�c1)! RY��`�°�% 9~m�!µ9.?F�:�05È1 Rl^�R±¸-/O?®,.?Ffc16# Rst�� +��>�k W ZK�M<NE)��9C(�GD��Z�° PRQTS �?A�A rU @B�z��6^w��4:�)� Zst�uCh��±Ë*{���4�����! Zxz�� m�?> W >^ ��n)203�2.�b�M<NE)5:�O�c1��l1.�'��26Zc1�­ ZpZ« S�U<W >

Page 196: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�?A U

U A^�z���2.�6�)! 9©I�yxz�+ m�?> r >^ Rl^�R²�'Zci6e0���}Bb���O!05�5.�)2O1��C(c1��:�c�0��^*{�3ci6�)2H¤�!�� PRP¬S �Uer �(l1.1�5�Zc16£Y��% m�?>�A r Z}B.���ci�zK�MeNE)2:4O�)! PmSVU�U �U k^��&h��l1�!6<03)5O¡M^ 1�1 $}Bb9�! <03�5.1)5 1.1b�:4)2O¡M���ºn6<03�!�5)2'�O¡Me'�6�w·��:46^��)£��'^HÌ0¡ciM�6��4:�O¡M���6ÅK���c1)2��c�H¤�5��:�)a�03�5c1M��4)! �*zM��!)5:4)! R��6�)205:�03'B0zH¤'��(KV�4c1)5��ciH¤.��2)5O¡Me'�6�w^ Zºn6�:%D��!�5)5:%0¡ci0n}e03'^0203w<c1�a0� m�?>1><@

U >^��"vc1)a036��!�� 9}9�R�t�u±¸7�MZc�03:�cB ^�°�R"$�% +�?>�> X ���)a03�2.�b�M<NE)5:4O�c1��l1.�'��26Zc1�j ^»9«]o SqU �?A@1¦^��"vc1)a036��!�� 9}9�R�t�R±Ë7�MZci05:�c^ ��=�Z"$�� m��>�>�k^ R-J.�6+�Rgn.i0��ZCv�Z��)a03����}E.BOi�� P p Q¬Sqr A W@B����"vc1)a036��!�n}9���t�% m�?>�>�>B ZK�MeNE)2:4O�c�}BO��2:4b^03c^ � « SqW kB�@ X ��",�!��6Zc16£Ch�uK���±¸",:46�w�)a03.�6��=�y7��% m�?>�k1A^ Z-/.16+�Rgn.10��RCh�^��)205����}B.EO1�% PRP « S�U > W@ W ��",�5c1)5M��!6�:46^6�:4 i.|D��G0(c1�j�% +�?>1> r ZK�M<NE)���K���c1)2��c1)! /» S �?A W k@ U ��~]c1O¡M^:46+ �*,�% $�?>�> r =�u*zM��ª:�6<D���)a03:�w<ci03:�.�6;.1H=.�bZc1O!:�0�N;:46�05M���l�x�*Ã05.� �c1��c1 Í��:%D��!�203.1��5�!w�:4.�6#_� ZK�MZsÂ03M^��)5:�)� Rº(6�:�D1���5)2:�0�N/Y�.����4�!w��,~m.�6���.16

@�@B��~]c16^w�l^�% R-£c1)2.�6�&h�yxz�R±½-/O!8;M�:��205���(Ch�[8\�uK��% m�?>�>�¦^ Z}B.���ci�zK�MeNE)2:4O�)! �o P p SqW �@1A^��~m�GDB:�6�)5.16+ 9Y��R±½©Z.��5�# Z"$�`8\�% m�?><@�@E ZK�MeNE)!�9C��!D��% �pRp Sqr > X@ r ��~m.EO¡M+ ]}9�ust�9±�}B'^�������2)� +&h�yK{�� ��?>�>�>^ 9K��5.EO1��H¤�5.��Î03M^�$ki03MT}^�v&��½F�.��2 e)5M�.�b# ]K�c1�2:4)� b U�Uer

@1k^��~m.��2��6<03È=&=�^�°�% RKV�2.BOi�RCh�^��O�ci�+�9}BO�:­�]���n��)a03���2�Zc1���G Q³S @1>^�@1>^�(-JO?Y��3ciO¡ 1��6# m®$��-¬�Z�G0(c1�j�% +�?>1>�k^ Rl�x�*���K(��>�k<� X �A�¦^�(-J��:4w1)� Z�°�Z�G0nc1�j�% #�?>1>�k^ Rl�x�*��aYfK(��>�k<� X kA^���(-J��:4w1)� ��=�� �©Z'�6��Zc1����6^)5 e:j q8\�� fl1'�b9�!6+ zY� V~]c1�2)5�!6+ f�°�% f~m.EO¡M+ z}��� f�$_y-/'�����c16^�1 I-¬�% }B'������!�5)! B&h�R X ¦1¦�¦$ X�r 03M�xqKq}�O!.�6^H¤�!�5��6^O���.16�O�.16e05�5.����4�!��H¤'�)2:4.�6�c16���b���ci)5��cvb^MeNE)2:4O�)! 7�'��Zcib9�!)20� m� X �¡�?A�03M/l1'�6�� X ¦�¦1¦

A X �(-J��:4w1)� Z�°�% X ¦�¦B�� ZK��5:%D1c�03�,O�.1���t'^6�:4O�ci03:�.�6

Page 197: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�?A�@

A W �(-J��6�È!���j �st�u&=�% +�?> W�r ���)a03�2.�b�M<NE)5:4O�c1��l1.�'��26Zc1�j Q ¼ SqW�W ¦A U �(-J��6�È!���j �st�u&=�Z±½7fc1 1�!�� �l^�y®¨�% m�?> W�r ^�n)203�2.�b�M<NE)5:�O�c1��l1.�'��26Zc1�­ QR�¬Sqr ¦A<@B�(-J��6�È!���j �st�u&=�Z±½K��� i���2:4)� 9Y��u~��� m�?> W @B R-/.16+�Rgn.10��ZCv�Z��)205����}B.EO1�% £p �³SIr�rA�A^�(-J:�03O¡M^���4�­ ��=��Y��y®$��±�Ï#����c16�)2 <N� �-T�`8\�% f�?>�AB�� �_uC(�!)5.�6�c16�O���Cnc1�^:�ci05:4.�6Tc16��ªx��^O!:�05����f03.���)�_% yYfc1�tµ��2:4��w��vº(6�:�D1���2)5:�0�N�KV�2��)2)

A r ��gnc1�tµZcB m}9�% m®,.103.^ +-¬�% #*z)2'�µ�.�:j #&=�% m���Zc^ �*,�+±¹}^ci05.� +"$�% X ¦�¦�¦^ #l1.�'��26Zc1�{.1Hqc1b�b���:4�!�b�M<NE)5:4O!)� QRQ¬SVW �?k X

A�k^�(�t_y-/'�����c16��i ¿-T��®$�]±}B'������!�5)! �&h�yK{�� I�?>�>�>B V�yKV©{Y½��.�b�:�6�w�F�:�03M³03�5c1O���:4��b�'��2:�05:4��)(�)5b���OG03�5.1)5O�.1b�:4O�)a03'��^:4��)=.1H�)5�!6�)5:%03:%DB:%03:���)h.1Hz)5'^�2D��GN�:�6�)205�5'�����6<03)!_� ¿º�",��x��ΩZ'�)2:4.�6�§1xVº*�ci)5 �g(.��Rs�É r �(�¢* U<W k¨C���b�.��20

A�>^�(��M�6^.� qg=�% Ix�È!'���:­ Vg=�% V",�3c1)2M���6�:�6�6�:� 1.?D� f}9�`�¡�% I±�K�:4w<ci�5.?D� ��°�{¶q'#�� ��?>�>1k^ VK�MeNE)2:4O�ci�C(�GDE:4�!FÐ~m�!0505���2)� Q o S k^�?k

r ¦^��K{��6�w1���4�%N� RCv��-T�R±¸}B��ci03.�6# �-¬��l^�% m�?>�A U R-/.�6#�Rgn.10��ZCh�Z�n)205����}B.EO1�% ]o P#��S �?A<@r ����K�:�w<c1�5.?D� ��=�E¶q'+���G0(c1�j�% ��?>�>�k^ ^K���c1)2��c1)fKVM<NE)5:�O�)(c16��JY�.16e05�5.����4�!��©R'^)5:4.16� �»9« SqX ¦<@1@r�X ��K{.���ci6��+ Z�°��±½��6�È!���! Rº=�% m�?> r �� Z}B.���ci�fb�M<NE)5:4O!)� oEp S�U ¦^�r1W ��K��2��:4)a0� (xz�yCv�% h��>�k�>^ ��ys�NE6Zc1��:4O!)�c16^��)205�5'�OG03'��2�L.1Hh²�'�:4�!)5O!��6<0/}B.��4c1��KV�2.���:46��!6�O��!)�_% ",�4'<F��!����O�ci�+�RK�'�µ��j�% Zs�.��2���5�!O¡M<0

riU ��C�.�)2�1 e-¬��xz�% ^�?><@ r ¾��x��4�!���!6<0¡c1�aNv*zM���.��aNh.1HZ��6^w�'���ci��-J.����!6<03'��J_� ign�GF�¶q.��5  S 8;:4���!N~m.�6���.�6 S Y�MZc1b���c16/c16��J&�c1���

r @B�(}EO¡MZc1�5�����! �®¨�u7��% R±¸Yfc1�2�4)2)5.�6+ �-T�� m�?>1k<@B Rl^��Y�.���b�'^0��ZK�M<NB)!�� #¼Rp S @1Ar A^�(}EO¡M���:������!�� m7n�� #&n��:�6�È��!�j #K��% +",'�O��!�3c^ #*,�+±¸É�:4c1�j 9l^�yY��� ¿��>�>�k^ #}B.��4c1��K�M<NE)5:4O!)� ;o Q o SW ¦�>

r�r �(}EO¡M���:������!�� 97��% 9&(��:�6�È��!�j �K��% �É�:�c1�­ Rl^�yY��#±ÑC�'��Zc�FzN� 9K��% {�?>�>�>^ �}E.���c1�(K�M<NE)5:4O!)� Òo Q p S�?¦�>

Page 198: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�?A1A

r k^�(}EO¡M���:������!�� q7��% V",'�O��!�3c^ V*,�`�°�% I&(��:�6�È��!�j VK��% ¿É�:4c1�j �l^��Y��% qK�.��4c16��+ ��=�� z�?>1>�>1µ+ VK��5.EO1�H¤�5.��¹03M��hki05M�}^�v&��\FI.��5 e)2M�.�b+ �K�ci�5:4)! ZbZA�¦<@

r >^�(}EO¡M���:������!�� �7��� ns����4c16��!�1 �Y��� �¶q.�6�w� nst�I¶$�% �É�:�ci�j nl^��Y��f±Æ-£c1�|Ó5c1�5)2 icB �-¬�% X ¦�¦�¦^ ��)a03�5.16�.��°N�ci6�����)205�5.�b^MeNE)2:4O�)! ³¯R¼ Q¬Sqr�X k

k�¦^�(}E��ci05.�6+ �-¬�yl^�� ]�?>1>�¦^ Zl^�RK�M<NE)���7 S �f0���-/.��j�Z�,bB0��RK�M<NB)!�� P ¯ SqW<X @�@k^���(}E.�µ9�!�4��c16+ i�¡�u�¡�% <Éqc1:46�)2M<03��:�6+ <~��`�°��ci6��°¶q'� 1.?D� Exz�u�=�� Z�?>1k^�� �_ux��^O!:�03ci03:�.�6t.1H��f03.1��)�ci6��7��5.�c1����6^:46�w¨.1H¿}Bb���O!05�3c1�]~m:�6���)!_� �}Bb^�5:46^w������¢ÉV���2��c1w

k X �(}EbZc1�Zc1�2.� Rst���G0�ci�j�% X ¦�¦1¦^ ���)a03�2.�6�.��°N�c16��J��)205�5.�b^MeNE)2:4O�)! 9¯�¼Rp Sqr �?Ak W �(}e0¡c1���/ �Ch�q���nc1�­�� X1W � X1r 03MÍ-£c1�2O¡MÂ�?>�>�k^ �K��5.EO����!��:46^w�)�.1H�03M��/�������5:�O�c16ÍK�MeNE)2:4O�ci�}B.EO�:4�G0�N S �f05.���:4O,K��5.EO��!)5)2��)�:�6JKV�4c1)5��c1)! ^��'^µ�'��56¬ yº�}B�

k U �(}e03�����4��c1O¡M��!�� R®$�Z±Ç8;:���M��! Rxz�� Z�n)203�2.�6�.��=N�c16�����)a03�2.�b�M<NE)5:4O!)� P pR« S A<@�@k<@B�(}E'��������2)� R&h�yK{�� m�?> r�r R-/.16+�Rgn.10��ZCv����)2�20��9}B.EO1�% mo ��Q¬S �?¦^�k�A^�(}E'��������2)� R&h�yK{�� m�?>1> U Rl�x�*Âl1.1:46<0(ºn6��^���203c1 e:46�w�:46<05���56�c1�+�5�!b9.1�20� Zl�xV*��¢��C°��> U �a¦�Ak r �(}E'��������2)� &=�uK��% X ¦1¦^�� *zM�� ��s��,} º()5�!� -£c16e'Zc1�­ D1���5)2:4.�6 X � X M<0503b S §�§ic1�Zc1)!�ub�M<NE)��`)205�3ci05M+�uc1O1�`'� B§1c1�Zci)¡§B�

k�k^�(}E'��������2)� Z&h�yK{��±Ç&n.E.�b����� �-¬�u7��� #�?>�k W BK���c1)2��c°KVM<NE)���Y�.16e05�5.��­�^©Z'�)2:4.�6#�� P ¼ S � W �1�k�>^�z*�c16��^µ9�!�5wi��&�c16�)2)5�!6+ �xz�% V�?>�><@B ¿�u*zM���g�c�03'��2�$.1Hz}E.���c1��KV�2.���:46��!6�O��!)�_% �",��'eFI���,�nO�c1�#�K�'�µ��j�% Zs�.��2���5�!O¡M<0

>�¦^�z*�c1��:46+ Z7n���!0nc1�­�� +��>�><@B ^KVM<NE)��9C��!D��Z�° �¼mo S �?>B�?k>^���z*�c1��:46+ Z7n���!0nc1�­�� +��>�> r �l^��²,'�c16<0���}Bb���OG03�5.1)5O1��C(c1��:4ci0��B*��5c16�)2H¤�!�� �¼ Q¬S ><@ W> X �fÉV�!�56ZciÈ�È�cB Rl^�yxz�% ^�(DB�2�!0� Zxz�y&h�� ^~].E��)2���! �Cv�� 9�?>�k^�� e�n)203�2.�b�M<NE)5:�O�c1�]l1.1'��56Zci�+}B'^b�b��4�!���!6<0)5�!�5:4�!)� Z»9¼ S A W @

> W �f8ªc16+ Z�=��}��Z�!0nc1�j�% +�?>1><@B Zl^�Rg('�O��­�R-£ci05���!�� ªo�oe« P¬SqX�X ¦

Page 199: V - Physics Strathclydem Z m zs b ²cZw r q < P bd Vbdsfe bKsZ &sfe K e n| ¨K¼De ñ zg e bd [ ²cZwGb q£ig ¨ qz Z Zq g e w wAe Zr Ò ²q gKbdsZ cZ &q 8 hqz P

�?A r

> U �f8¬��6^È����­ Rº=�Z�G0�ci�j�+ +��>�>�>^ ZK��4c1)5��c$K�MeNE)!�+Y�.�6<05�5.��­�Z©Z'�)5:�.�6+ To ��PTS �?k�>><@B�f8;M<Ne03�i Vst��®$�% �&('���b^M��5�GNB)! �st�`�°�{±Î*�c�NE�4.��! �K{�y~��% X ¦�¦1¦^ �K�M<NE)5:�O�)t.1Hfb���c1��)3ci)� �ÊSU ¦<@ X

>�A^�f8;:��4M��!�4�J ^"$�Z�G0�ci�j�# m��>�><@B �}B.���ci��KVM<NE)5:�O�)! �»{o S k�¦^�> r �(Ï+ci�5�5.^ Zst��-T�R±¸Ï#:4�2:46+ ^&=�% +�?>�k�@B ���)a03�2.�6�.��°N�c16��J��)205�5.�b^MeNE)2:4O�)! {oe» Q¬SqX�U ¦>�k^�(Ï#:��5 i���� �l^�u7��% m�?>�k�kB Z}B.��4c1��K�M<NB)2:4O!)� {oRoBp SVW1U �


Recommended