Synchrotron Radiation: Properties and Production
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Welcome to Chicago
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Discovery of x-rays ~100 years ago
! X-rays were discovered (accidentally) in 1895 by Wilhelm Konrad Roentgen. !
! Roentgen won the first Nobel Prize in 1901 “for the discovery with which his name is linked for all time: the... so-called Roentgen rays, as he himself called them, X-rays…”!
!
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”X-ray Cream Furniture Polish - Guaranteed since 1891” Umm… x-rays weren't discovered until 1895!"
Synchrotron Radiation
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Discovery of Synchrotron Radiation - ~50 years ago!0=#-&2"%2"#'2$+4$?"#'R0PS'L$9'@29%'">9.2,.+'R$--4+.#%$55=S'62"O'$'TG':.U'9=#-&2"%2"#'HVWT;'
Excerpted from Handbook on Synchrotron Radiation, Volume 1a, Ernst-Eckhard Koch, Ed., North Holland, 1983.
On April 24,[1947] Langmuir and I [Herbert Pollack] were running the machine and as usual were trying to push the electron gun and its associated pulse transformer to the limit. Some intermittent sparking had occurred and we asked the technician to observe with a mirror around the protective concrete wall. He immediately signaled to turn off the synchrotron as "he saw an arc in the tube." The vacuum was still excellent, so Langmuir and I came to the end of the wall and observed. At first we thought it might be due to Cherenkov radiation, but it soon became clearer that we were seeing Ivanenko and Pomeranchuk [i.e., synchrotron] radiation.
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Synchrotron Radiation
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From Synchrotrons to Storage Rings
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( ![Å] = 12.4 / E[keV] )!
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Square brackets indicate the units to be used in the calculation.
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Evolution of Synchrotron Radiation Sources
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Synchrotron Radiation Facilities Around the World
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Bending Magnet Source
Undulator Source
Radiation from Moving Charges
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Radiation Patterns From Accelerating Charges
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Radiation from Highly-Relativistic Particles
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Radiation Sources at 3rd Generation Facilities There are two different sources of radiation at 3rd generation sources:!
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! insertion devices (IDsSn'periodic arrays of magnets inserted between the BMs (wigglers or undulators)!
The important parameters to know about each one is:!
! Spectral distribution! !
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Bending Magnet Sources
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Radiation Sources at 3rd Generation Facilities There are two different sources of radiation at 3rd generation sources:!
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! Spectral distribution! !
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Planar Insertion Devices
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Characterizing Insertion Devices
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Wigglers and Undulators
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Wiggler Radiation Sources
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Radiation Sources at 3rd Generation Facilities There are two different sources of radiation at 3rd generation sources:!
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Undulator Radiation
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1 Å X-rays from a 3 cm Period Magnetic Field?
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'r''a%'+".9'#"%'9..'$'9%$?-'O$)#.?-'@.5+'62"O'%&.'1#+15$%"2K'>1%'2$%&.2'$'?O.D,$2=4#)'[email protected]+'$#+'$99"-4$%.+'[email protected]+'R+1.'%"'%&.'2.5$?,49?-'%2$#96"2O$?"#'"6'%&.'O$)#.?-'@.5+'"6'%&.'+.,4-.S;''r''c&.'<.24"+'"6'%&.'N'$#+'h'@.5+'$2.'("2.#%Q'-"#%2$-%.+'9"'%&$%7''!.D62$O.'k'!a8's'% Rb;b'-O'%"'F;b'ÉOS $#+'9"'%&.'.5.-%2"#'"9-455$%.9'R$#+'&.#-.'2$+4$%.9S'L4%&'%&$%'9$O.'<.24"+'+24,.#'>='%&.'N:'@.5+9;'
'O4 V0JJ8)2"D-+WR"'h$-M'4#'%&.'5$>'62$O.7'
&r''81.'%"'%&.'6$-%'%&$%'%&.'.5.-%2"#'49'%2$,.54#)'%"L$2+9'19K'%&.'2$+4$?"#'.O4E.+'>='%&.'.5.-%2"#'49'8"<<5.2'9&4Ñ.+'%"'&4)&.2'62.e1.#-4.9'R9&"2%.2'L$,.5.#)%&9S;''c&.'2.5$?,49?-'8"<<5.2'9&4Ñ')".9'$9'ÖHD#'s'ÖHÄ#') 1/2% K'$#+'9"'%&.'L$,.5.#)%&'">9.2,.+'4#'%&.'5$>'497'
'T4 "?0L<+'&;0'R"" """!5$>')'R !a8s%SR1/2%) = R !a8sF%FS'&
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!ID = 3.3cm" !14000(APS)!x"ray (cm) #0.8x10
"8
#0.08 nm
d#D$_49'R$ kGSK'L.'-$#'L24%.7'
'!#_D2$='k'R !a8sF%F#SRH'Ä'zFsFS''
!#_D2$='K'-$#'>.'$+J19%.+'>='hK'4;.;'>=',$2=4#)'z'Rk'G;GVbW'!a8{-O|'h"{MZ|'S;'c&49'-$#'>.'$-&4.,.+'>=',$2=4#)'%&.'-122.#%'4#'%&.'L4#+4#)9'"6'.5.-%2"O$)#.?-'+.,4-.9'"2'>=',$2=4#)'%&.'9.<$2$?"#'>.%L..#'%&.'1<<.2'$#+'5"L.2'<"5.9'Rw%&.')$<xS'4#'<.2O$#.#%'O$)#.%'+.,4-.9;''["1'-$#`%'2.+1-.'%&.')$<'%""'O1-&'94#-.'="1'L455'-1%'4#%"'%&.'<$2?-5.'>.$O'$#+'546.?O.'L455')"'%"'<"%q'''
Tuning the Peaks of Undulator Radiation
O!X"(L"
C#"T!T"(L"
/?C"N!Y"(L"
D08+$"3:<Z+4["$0\)$"3:<T/'4"
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The energy spread of the interference peak (central cone) is given by:''
''''''''''''''''''''''*NsN'k'*$/$'m'Hs#B'''(like a grating!).!!For a given K-value (gap), the wavelength at angle $ is !H'k'R !a8sF%FSRH'Ä'zFsF'Ä'%2$HFS'''
Undulator Energy Spread and Angular Distribution
The central cone opening angle, $HK !for the odd harmonics is given by:!!$1/ F'k'R$_D2$=sF(SHsF'!
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Undulator Radiation Patterns and Spectra
C'$18&.02">&$+&;0'"3D))"#JJ)'$+K"E4"
!! 1#+15$%"29'+.@#.+'$9'a89'L4%&'&"24Q"#%$5'+.u.-?"#'
$#)5.'m'Hs%'K'4;.;K''z'mH'
! 9<.-%21O'<.$M.+'$%'_D2$='9<.-4@-'_D2$='.#.2)4.9K'>1%'<.$M9'$2.'%1#$>5.'>=',$2=4#)'z'''Rz'k'G;VW'h{c| !a8{-O|S''
! $%'%&.'<.$M9'R&$2O"#4-9S'%&.'&"24Q"#%$5'$#+',.2?-$5'"<.#4#)'$#)5.9'"6'%&.'2$+4$?"#'49')4,.#'>=7'
R!_D2$='s'F(SHsF''{%=<4-$55='$'6.L'O4-2"2$+4$#9|'
! %"').%'%&.'%21.'"<.#4#)'$#)5.K'#..+'%"'-"#94+.2'%&.'"<.#4#)'$#)5.'"6'%&.'.O4o#)'<$2?-5.9'
! I4)&5='54#.$25='<"5$24Q.+'"#'&$2O"#4-;''X$#'19.'9<.-4$54Q.+'+.,4-.9'%"'-"#%2"5'<"5$24Q$?"#'
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3155'<"5$24Q$?"#'-"#%2"5'62"O'9"12-.'
X42-15$2'9L4%-&4#)'2$%.'
'''fHGG'O9'RG;\'IQS'R'fFGG'*S'
(4#.$2'9L4%-&4#)'2$%.'
'''fbG'9'R'fHFGG'*'S'
#$%&'()$"@-0.0'"/012()"CJ,2&$)"3#@/AC4"J20])(."
Electromagnetic Helical Undulator (4-ID-C)
N5.-%2"DO$)#.%9'19.+'%"'O$M.'.5.-%2"#')"'4#'$'&.54-$5'"2>4%;''c&49'<2"+1-.9'-42-15$25='<"5$24Q.+'_D2$=9'R)""+'6"2'O$)#.?9O'._<.24O.#%9S;'
bW'
_'
:$)#.?-'34.5+'$5"#)'+.,4-.'
! /$2?-5.9'$2.')2"1<.+'%").%&.2'>='%&.'$-?"#'"6'%&.'P3'-$,4?.9'4#%"'>1#-&.9'Rb\G':IQS;'''
*%'%&.'*/07'
g #$%12$5'>1#-&'5.#)%&'R4;.;'4#'%&.'54O4%'"6'Q.2"'-122.#%S'b\'<9.-'3^I:'
g %=<4-$55='$>"1%'HGG'<9.-'3^I:''
! HHGW'O'-42-1O6.2.#-.'Rb;Y]'O4-2"9.-"#+'<.24"+n'fFTG'MIQS'
g &$2O"#4-'#1O>.2'HFVY'4;.;'HFVY'.,.#5='9<$-.+'wP3'>1-M.%9x'$2"1#+'%&.'24#)'
g O4#4O1O'9<$-4#)'49'F;]'#9.-'
! '8.%$459'"6'%&.'?O.'9%21-%12.'+.<.#+9'"#'%&.'@55'<$E.2#K'4;.;'L&4-&'>1-M.%9'&$,.'.5.-%2"#9'4#'%&.O;'
!
Time Structure of the Radiation
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! bFW'.e1$55='9<$-.+'>1#-&.9'D'$<<2"_4O$%.5='HH'#9.-'>.%L..#'>1#-&.9'
D $<<2"_4O$%.9'$'-"#?#1"19'9"12-.''
! FW'.e1$55='9<$-.+'>1#-&.9'D'$<<2"_4O$%.5='H\W'#9.-'>.%L..#'>1#-&.9'
D -"O<2"O49.'>.%L..#'-"#?#1"19'9"12-.'$#+'<159.+'9"12-.'
! H'Ä'T_]'R&=>24+'O"+.S''D'$'94#)5.'>1#-&'6"55"L.+'>=']')2"1<9'"6'T'>1#-&.9'
D'?O4#)'._<.24O.#%9'
Typical APS Filling Patterns
0-&.O$?-'"6'*/0'
&=>24+'O"+.''
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Laser Pump Pulse
X-ray Probe Pulse Excited State
Time
Laser pulse pump X-ray pulse probe Intrinsic time resolution: 30 – 100 ps fwhm
Storage Ring
GS
ES1
ES2
h!
FC
Pump-probe Chemistry
! A<'1#?5'#"LK'L.'&$,.'-$5-15$%.+'%&.'2$+4$?"#'<2"<.2?.9'62"O'$'94#)5.'.5.-%2"#K'&"L.,.2'4#'$'9%"2$).'24#)K'%&.'2$+4$?"#'49'.O4E.+'62"O'$#'.#9.O>5.'"6'.5.-%2"#9'L4%&'9"O.'@#4%.'94Q.'$#+'+4,.2).#-.'+49%24>1?"#;''
! h"%&'%&.'%2$#9,.29.'$#+'5"#)4%1+4#$5'<2"<.2?.9'"6'%&.'<$2?-5.'>.$O'4#'$'9%"2$).'24#)'$2.'%&.'.e1454>241O'<2"<.2?.9'"6'%&.'<$2?-5.'>.$OK'>1%'&.2.'L.'$2.'4#%.2.9%.+'4#'%&.'!"#$%&'"%''<2"<.2?.9;'
! c&.'<2"+1-%'"6'%&.'<$2?-5.'>.$O'94Q.'$#+'+4,.2).#-.'49'<2"<"2?"#$5'%"'$'<$2$O.%.2'"6'%&.'>.$O'-$55.+'%&.'.O4E$#-.'R1#4%9'$2.'5.#)%&'_'$#)5.S;'R0..'*<<.#+4_'YS !
'
Transverse Properties of Particle Beams
! The emittance is a constant of the storage
ring, although one can trade off beam size for divergence as long as the (something proportional to the product) remains constant. !
['
Ü'C'
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G-9"$0"5)"'))$".0"7'05"&<01.".-)".2&'D%)2D)"J&2;(8)"<)&L"J20J)2;)D^''*5%&"1)&'%&.'u1_'62"O'h:'$#+'a8'9"12-.9'-$#'>.'+.%.2O4#.+'L4%&"1%'+.%$45.+'M#"L5.+).'"6'%&.'9"12-.'94Q.'$#+'+4,.2).#-.K'"#.',.2='4O<"2%$#%'-&$2$-%.249?-'"6'%&.'>.$OK'#$O.5="<2+,-.')DD['2.e142.9'$'O"2.'4#6"2O$?"#'"#'%&.'<$2?-5.'>.$O`9'94Q.'$#+'+4,.2).#-.;'
'h24)&%#.99'&$9'1#4%9'"67'''''
<&"%"#9s9.-sG;Ht'h^s9"12-.'$2.$s9"12-.'9"54+'$#)5.''
Flux/4'2 +h +v +&`'+,`""
'L&.2.'+4'R+4`S'49'%&.'.y.-?,.'"#.'94)O$',$51.'"6'%&.'9"12-.'94Q.'R+4,.2).#-.S'4#'%&.'4%&'+42.-?"#;''c&.'.y.-?,.'9"12-.'94Q.'$#+'+4,.2).#-.'&$9'-"#%24>1?"#9'62"O'>"%&'%&.'<$2?-5.'>.$O'$#+'%&.'2$+4$?"#'4%9.56;''
Transverse Properties and Photon Beam Brightness
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Stuff a Storage Ring Source can’t do
" G-&."J&2&L).)2D"5018$"1D)2D"8+7)".0"D))"_)'-&'()$`^"g a#-2.$9.+'>24)&%#.99'R4;.;K'5$2).2'-"&.2.#%'62$-?"#'k'3-"&.2.#%'s'3%"%$5S'
g 0&"2%.2'<159.9'R62"O'HGG'<9'%"'HGG'69.-S'
" G)"&2)"&<01."&.".-)"8+L+.D"0="5-&."D.02&,)"2+',D"(&'"$0R!g /$2?-5.'>.$O'.O4E$#-.'R9"12-.'94Q.'_'+4,.2).#-.S7'
r I"24Q"#%$5'.O4E$#-.7' 'b'_'HGDV'OD2$+9'
r *,.2$).'>24)&%#.997' 'HGHVDHGFG'_D2$=9s9.-DG;Ht'>LDOOFDO2$+F'
– Particle beam longitudinal properties:!r h1#-&'5.#)%&7 ' 'FG'OO'RTG'<9.-S''
" _C8;L&.)"/.02&,)">+',D"&'$"I')2,9">)(0%)29"*+'&(D"-&%)".-)"J0.)';&8".0"$)8+%)2"3')&24"=1889"DJ&;&889"(0-)2)'."<)&LD"<1."+="5)"5&'."=)L.0D)(0'$"J18D)D"&'$"=188"(0-)2)'()["1D)".-)"805")L+\&'()"&'$"D-02."J18D)D".-&."(&'"<)",)')2&.)$"<9"8+')&2"&(()8)2&.02D!"
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X-ray Free Electron Lasers (X-FELs) " (4#$-D>$9.+'_D2$='62..'.5.-%2"#'5$9.29'RC3N(9S'
9&"15+'<2",4+.'O$#='"6'%&.'+.942.+'4O<2",.O.#%9;"
g 3155'-"&.2.#-.q'g 3.O%"9.-"#+'<159.'5.#)%&9q'
" *#"KA2&9"aI*'19.9'%&.'&4)&'>24)&%#.99'"6'$#')8)(.20'",1'"-"1<5.+'%"'$#'.O4E$#-.D<2.9.2,4#)'54#$-;'
" c&.')$4#'4#'%&.'5$9.2'49'">%$4#.+'%&2"1)&'$'<2"-.99'-$55.+'/)8=A#LJ8+b)$"/J0'.&')01D"IL+DD+0''-&''/#/I."
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Self-Amplified Spontaneous Emission"
The LCLS produces extraordinarily bright pulses of synchrotron radiation in a process called “self-amplified spontaneous emission” (SASE). In this process, an intense and highly collimated electron beam travels through an undulator magnet. The alternating north and south poles of the magnet force the electron beam to travel on an approximately sinusoidal trajectory, emitting synchrotron radiation as it goes.
N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!
N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N!Electron Bunch
Undulator Magnet
c0-'"d&8&9$&["*?*/"
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Self-Amplified Spontaneous Emission
N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!
N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N!Electron Bunch
Undulator Magnet
N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!
N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N!
The electron beam and its synchrotron radiation are so intense that the electron motion is modified by the electric and magnetic fields of its own emitted synchrotron light. Under the influence of both the undulator magnet and its own synchrotron radiation, the electron beam is forced to form micro-bunches, separated by a distance equal to the wavelength of the emitted radiation.
c0-'"d&8&9$&["*?*/"
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Self-Amplified Spontaneous Emission
N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!
N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N!Electron Bunch
Undulator Magnet
N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!
N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N!
N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!
Coherent Synchrotron Radiation
N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S!N! N!N! N!S! S! S! S! N!
These micro-bunches begin to radiate as if they were single particles with immense charge. The process reaches saturation when The micro-bunching process has gone as far as it can go.
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Free Electron Lasers (FELs)
Start-up
0%$2%D1<'9%$).'External signal or spontaneous radiation interacts with the e-beam resonantly at undulator !&
&Energy modulation # density modulation (microbunching) # coherent radiation at ! # "._<"#.#?$5')2"L%&'R(ZS'
Undulator Hall at the Linac Coherent Light Source (LCLS) at SLAC
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Spectral Properties for X-ray Free Electron Lasers
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'The future looks bright……
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Linac Coherent Light Source at SLAC
Existing 1/3 Linac (1 km) (with modifications)
Near Experiment Hall
Far Experiment Hall
Undulator (130 m)
X-FEL based on last 1-km of existing 3-km linac
New e- Transfer Line (340 m)
1.5-15 Å (14-4.3 GeV)
X-ray Transport Line (200 m)
UCLA
Injector (35º) at 2-km '
Advanced Photon Source Upgrade
! */0'-122.#%5='&$9'$'<2"<"9$5'%"'+"'$'O$J"2'1<)2$+.'"6'%&.'6$-454%='",.2'%&.'#._%'\DT'=.$29;''
! N9?O$%.+'-"9%''49'fáb\G:K'$#+'L455'>24#)'%&.'6$-454%='%"'%&.'9%$%.D"6'%&.D$2%'6"2'$'b2+').#.2$?"#'
9%"2$).'24#)'9"12-.;'
! h..#'$<<2",.+'>='8dN;''X122.#%5='4#'%&.'+.94)#'<&$9.'
! 3429%'1<)2$+.9'L455'9%$2%'4#'%&.'3$55'"6'FGHb'
• B.L'$#+'1<)2$+.+'>.$O54#.9'
• a#-2.$9.'-122.#%'%"'H\G'O*;''• 8.54,.2'HGG'?O.9'>.E.2'%.O<"2$5'2.9"51?"#K'+"L#'%"'H'<4-"9.-"#+;'
'
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*+,$#-.+'
/&"%"#'0"12-.'A<)2$+.'
R*/0DAS'<2"J.-%'
10 Renovated Front Ends 11 New Front Ends
4 Long Straight Sections
5 Planar Undulators
6 Revolver Undulators
3 Superconducting Undulators
SPX SRF Cavities and Cryostats
8 Beamline Upgrades
6 New Beamlines
2 Polarized Undulators
APS-U Project Baseline Scope
The classical formula for the radiated power from an accelerated electron is:!!!!!Where P is the power and , the acceleration. For a circular orbit of radius r, in the non-relativistic case, , is just the centripetal acceleration, v2/r. In the relativistic case:!!!!!!Where - = t/% = proper time, % = 1/√1-#2 = E/moc2 and # = v/c!
Appendix 0: Radiated Power from Charges at Relativistic Velocities
!
P =2e2
3c 3a2
!
a =1mo
dpd"
=1mo
#d#movdt
= # 2dvdt
= # 2v 2
r
!
P =2e2
3c 3" 4v 4
r2=2ce2
3r2E 4
mo4c 8
Boxes with lines like this indicate an important equation.!
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c&.2.'$2.'%L"'<"4#%9'$>"1%'%&49'.e1$?"#'6"2'%"%$5'2$+4$%.+'<"L.27''
'H;''0-$5.9'4#,.29.5='L4%&'%&.'O$99'"6'%&.'<$2?-5.'%"'%&.'W%&'<"L.2'R<2"%"#9'2$+4$%.'-"#94+.2$>5='5.99'%&$#'$#'.D'L4%&'%&.'9$O.'%"%$5'.#.2)=K'N;S'
''F;''0-$5.9'L4%&'%&.'W%&'<"L.2'"6'%&.'<$2?-5.`9'.#.2)=''R$'T'Z.U'9%"2$).'24#)'2$+4$%.9'FWGG'?O.9'O"2.'<"L.2'%&$#'$'H'Z.U'24#)'L4%&'%&.'9$O.'2$+419S'
!
Appendix 0: Dependence on Mass and Energy of Radiated Power
!
P =2e2
3c 3" 4v 4
r2=2ce2
3r2E 4
mo4c 8
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^&.#','ll'-K'R# m GSK'%&.'9&$<.'"6'%&.'2$+4$?"#'<$E.2#'49'$'-5$994-$5'+4<"5.'<$E.2#;''P.-$55'%&$%'9<.-4$5'2.5$?,4%='9$=9'%&$%'$#)5.9'%2$#96"2O'$97'''''''*#+'9"'$9'="1'-2$#M'1<'#K'%&.'2$+4$?"#'<$E.2#'>.)4#9'%"'+.6"2O'R4#'%&.'5$>'62$O.S;'
Appendix 0: Radiation Patterns When V << C
!
tan"lab =sin" '
# cos" ' + $( )At the APS with E = 7 GeV, !! = E/moc2 = 7 GeV/0.511 MeV!
! = 1.4 x 104!
1/! = 73 x 10-6!
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c&.'9<.-%2$5s$#)15$2'+49%24>1?"#'"6'Å9=#-&2"%2"#'2$+4$?"#Å'L$9'L"2M.+'"1%'>='!;'0-&L4#).2'4#'HVWV;'0-&L4#).2'6"1#+'%&.'9<.-%2$5'+49%24>1?"#'62"O'$#'$--.5.2$?#)'<$2?-5.K'1#+.2'%&.'4#u1.#-.'"6'$'-"#9%$#%'O$)#.?-'@.5+K'L$9'$'9O""%&5=',$2=4#)'61#-?"#'"6'<&"%"#'.#.2)='$#+'%&$%'%&.'9<.-%21O'-"15+'>.'<$2$O.%.24Q.+'>='$'-24?-$5'.#.2)=K'N-;'
' ' ''''''''''' ' '''''N-'k'b&-%bsW'2;'
'I.2.'&'49'/5$#-M`9'-"#9%$#%'$#+'. /0 %&.'2$+419'"6'-12,$%12.'"6'%&.'%2$J.-%"2=;''B"%.'%&$%'%&.'-24?-$5'
.#.2)='9-$5.9'$9'%b;''In practical units, the critical energy can be written as:'''
N-{M.U|'k'F;FH]'Nb{Z.U|s'.{O|'k'G;GYY\H'h{MZ|'NF{Z.U|!'
Appendix 1: BM Spectral Distribution
At the APS the bending magnets have a field strength of 5.99 kilogauss and the ring operates at E = 7 GeV . The critical energy of the radiation emitted from the BM is:!!
Ec[keV] = 0.06651 B[kG] E2[GeV]!or!
Ec = 0.06651(5.990)(72) = 19.5 keV or 0.64 Å.(
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! The opening angle of the entire radiation field (i.e., the power) is approximately:'
$x = $y ) 1/%,&
!where $x is the horizontal angle and $y is the vertical angle.!!! The collimation in the horizontal direction is lost and what is observed is
just the vertical opening angle.!
! Flux from a bending magnet is usually quoted as flux per unit horizontal angle (integrated over the vertical angle). There are no "simple" closed-form solutions for the photon flux, F, as a function of wavelength from a bending magnet however there are numerous series approximations that can easily be evaluated with a calculator/computer.!
!
Appendix 1: BM Angular Distribution and Flux
From a bending magnet (B = 5.99 kG) at the APS operating at E = 7GeV and I = 100 mA (at the critical energy, integrated over all vertical angles) we get:!$
$ $$<FO<%P$$#$QRQN$%.-1-#'O'0*$E$RIQS$8T$E$:&"< %)&
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Appendix 2: Where did “K” come from?
!
Fx = max = "m0 ˙ v x = e! v #! B = ecB0 sin 2$z
%ID
&
' (
)
* +
˙ v x =ecB0
"m0
sin 2$z%ID
&
' (
)
* + z = ct
vx = ,ecB0
"m0
%ID
2$ccos 2$ct
%ID
&
' (
)
* + = ,
eB0
"m0
%ID
2$cos 2$ct
%ID
&
' (
)
* +
x =eB0
"m0c%ID
2$-
. / 0
1 2
2
sin 2$ct%ID
&
' (
)
* + =
eB0
m0
%ID
2$c
-
. /
0
1 2
1"%ID
2$-
. / 0
1 2 sin 2$z
%ID
&
' (
)
* + = K 1
"%ID
2$-
. / 0
1 2 sin 2$z
%ID
&
' (
)
* +
xmax = K 1"%ID
2$-
. / 0
1 2 and dx
dz-
. / 0
1 2 max
=K"
where K =eB0
2$%ID
m0c
-
. /
0
1 2
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Ne1$?"#'"6'O"?"#'6"2'$'2.5$?,49?-'-&$2).+'<$2?-5.'
4#'$'O$)#.?-'@.5+'
Spectral Distribution:!Wiggler radiation is the incoherent superposition of radiation from each pole of the wiggler. As with the bending magnet, the spectral distribution of the emitted radiation from a wiggler is smoothly varying as a function of photon energy and is characterized using the critical energy as a parameter. '
Appendix 3: Wiggler Radiation Spectral Distribution
The APS has built a wiggler with a magnetic field, Bo , of 10 kilogauss, a period of 8.5 cm, and a length of 2.4 meters (N=28).!!
K = 7.9, Ec = 32.6 keV, $max = 577 microradians, xmax = 8 microns!
Wiggler Flux:!The flux from a wiggler can calculated by multiplying the bending magnet equations by 2N where N is the number periods (and by using the appropriate critical energy!).!
The APS Wiggler (Bo =10 kG, $ID = 8.5 cm, L = 2.4 meters; N=28, has a flux (at the critical energy integrated over all vertical angles) of:!!
dF/d%x = 4.8 x 1014 photons/sec - 0.1 % BW - mrad %x !!BC0-&""5'FGHH''':4559'
''
Appendix 4: Undulator Flux
To determine the flux from an undulator, we integrate over both the vertical and horizontal angular distributions of the central cone and so the flux will have units of:!!
x-rays/sec-0.1% bw.!!As with bending magnets, there is not a closed form expression for the flux, but it can be approximated by:!!
Fn = 0.72 x 1011 N Qn I[mA] ph/sec-0.1% bw!where:!
Qn(K) ) 1 for K >1.!!
!!For APS Undulator A (N = 72) with the storage ring running at 7 GeV and 100 milliamps, typical flux values for the first harmonic (n=1) are:!!
a"e"TAY"K"NFNE""J-fD)(AF!Ng"<5!"
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Appendix 5: APS Electron Beam Parameters
BC6$&9#'$U(1. &H =$N$P$QREV$:E&"<$"#<$"$*-9%+(#/$2&")-$-=$A0&)*"+$0:(W"#*0$1-$
.-&(;-#1"+$0:(W"#*07$-=$RIVS$X$1.0&0=-&0$$
$&V M$RIRHY$P$QREV$:E&"<I$$$
$$Z.0$%"&)*+0$>0":$'-9&*0$'(;0$"#<$<(A0&/0#*0$"1$1.0$'1&"(/.1$'0*)-#'$"&0[$$
'H$M H\R$:(*&-#'!'H' M QQ$:(*&-&"<("#'!
'V M V$:(*&-#'!'V' M N$:(*&-&"<("#'$
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Appendix 5: Comparison of Radiation & Electron Beam Properties
#@/"C'$18&.02"#"-&D"&"8)',.-"0="O!E"L).)2D!""a02"Nh"2&$+&;0'".-)"'&.12&8"0J)'+',"&',8)"+DR""02' = "[!/2L]"e""E!Y"L+(202&$+&'D!""Z-)"(022)DJ0'$+',"D012()"D+U)"0=".-)"2&$+&;0'"+DR""
02"e"ij!*fk'Ol"e"N!X"L+(20'D!"
"?0LJ&2)".-+D"5+.-".-)"%)2;(&8"D+U)"&'$"$+%)2,)'()"0=".-)"#@/"<)&LR""'V' = *(+,-"."#/,#$%('V =(0(+,-".$%(
! a6'Z$1994$#'+49%24>1?"#9'$2.'$991O.+'6"2'>"%&'%&.'<$2?-5.'>.$O'$#+'%&.'2$+4$?"#'
4%9.56K'2.915%$#%'9"12-.'94Q.'$#+'+4,.2).#-.'49'
%&.'e1$+2$%12.'91O'"6'%&.'%L"'-"O<"#.#%9K'
#$O.5=7'
'
'+i = 1 [0r2 + 0i
2] and +I' = 1 [0r'2 + 0i'2].!
!! ^.'-$#'#"L'-$5-15$%.'%&.'>.$O'>24)&%#.99'
94#-.'
B = Flux/4'2 +H +V +I`'+U`"
!
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! *'O"#"D.#.2).?-K'-&$2).+'<$2?-5.K'1#+.2'%&.'$-?"#'"6'$'54#.$2'6"2-.'91-&'$9'49'6"1#+'4#'$'9%"2$).'24#)K'%2$-.9'$'-"#%"12'4#'<&$9.'9<$-.'%&$%'49'$#'.554<9.;'''
Appendix 6: Phase Space of the Charged Particles
= Arctan(
[!"]
#'
#
[!$]
2%/($&")2' )
1/2
1/2
! 0IKU(s) and 0IKU'(s) $2.'%&.'"#.D94)O$',$51.9'"6'%&.'
%2$#9,.29.'<"94?"#'$#+'+4,.2).#-.K'2.9<.-?,.5=K'$%'
9"O.'<"94?"#K'9K'$2"1#+'%&.'9%"2$).'24#);'
! c&.'$2.$'"6'%&.'.554<9.'49'<2"<"2?"#$5'%"'$'<$2$O.%.2'"6'%&.'>.$O'-$55.+'%&.'.O4E$#-.'R1#4%9'$2.'5.#)%&'_'$#)5.S;'
! (4"1,455.�9'c&."2.O'9%$%.9'%&$%'6"2'$'9=9%.O'91-&'$9'+.9-24>.+'&.2.K'%&.'<&$9.'9<$-.',"51O.'9&"15+'2.O$4#'$'-"#9%$#%;'''a#'"%&.2'L"2+9'%&.'$2.$'"6'%&.'<&$9.'9<$-.'.554<9.'R.O4E$#-.S'49'-"#9%$#%'.,.#'46'%&.'9&$<.'"6'%&.'.554<9.'-&$#).9'R<.24"+4-$55=S'$9'"#.')".9'$2"1#+'%&.'<$2?-5.`9'%2$J.-%"2=; !a#'%&.'$>",.'+4$)2$OK ('49'%&.'.O4E$#-.'$#+',, #, '
$#+'%'+.9-24>.'%&.'9&$<.'"6'%&.'.554<9.'$%'$#='<"4#%'4#'%&.'9%"2$).'24#);',, #, '$#+'%'$2.'9"O.?O.9'-$55.+'%&.'cL499'<$2$O.%.29;'
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Diffraction Limited Source Size and Divergence
! c&.'.y.-?,.'<&$9.'9<$-.'"6'%&.'2$+4$?"#'9"12-.'R+4'$#+'+4`S'&$9'-"#%24>1?"#9'62"O'94Q.'$#+'+4,.2).#-.'"6'%&.'<$2?-5.'>.$O').#.2$?#)'%&.'2$+4$?"#'$#+'%&.'4#%24#94-'9"12-.'94Q.'$#+'+4,.2).#-.'"6'%&.'2$+4$?"#'4%9.56;'BD".-)2)"&2)"8+L+.".0"-05"DL&88".-)")m)(;%)"J-&D)"DJ&()"&2)&"3+!)![")L+\&'()4"(&'"<)^''[.9K'="1'$2.'9?55'>"1#+'>='%&.'I.49.#>.2)'A#-.2%$4#%='/24#-4<5.;''P.-$557'
!
"x"px # ! /2pxpz
=$x or "pxpz
= "$x and pz = !k = !(2% /&)
so :"x"px = "x"$x pz = "x"$x[!(2% /&)] # ! /2"x"$x # & /4%
! 'c&49'49'%&.'9"D-$55.+'+4y2$-?"#'54O4%;'3"2'-.#%2$5'-"#.'"6'%&.'1#+15$%"27'
02' or (*2) = "[!/2L]'$#+'9"''
02'"2'R*x) k'Ö{!(s]'O|'
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Brightness vs. Photon Energy for Various SR Facilities
! c&.'&4)&'>24)&%#.99'>.$O9'
$%'b2+').#.2$?"#'9"12-.9'
R*(0'$#+'*/0K'6"2'._$O<5.S'
2.915%9'4#'$#'_D2$='>.$O'L4%&'
<$2?$5'%2$#9,.29.'R"2'9<$?$5S'
-"&.2.#-.;''
! c&49'>.$O'<2"<.2%='-$#'>.'
$#'4O<"2%$#%'<$2$O.%.2'4#'
9"O.'._<.24O.#%9'91-&'$9'
<&"%"#'-"22.5$?"#'
9<.-%2"9-"<=K'_D2$='
&"5")2$<&=K'4O$)4#)K'.%-;'
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The Linac Coherent Light Source (LCLS) at SLAC
c&.'CD2$='62..'.5.-%2"#'5$9.2'RC3N(S'2.54.9'"#'$'&4)&'.#.2)='54#.$2'$--.5.2$%"2'R(aB*XS'$#+'#"%'"#'$'9=#-&2"%2"#s9%"2$).'24#);"
c&.'(X(0'19.9'%&.'5$9%'M45"O.%.2'"6'$#'._49?#)'b'MO'(aB*X'$%'%&.'0%$#6"2+'(4#.$2'*--.5.2$%"2'X.#%.2'R$%'0%$#6"2+'A#4,.294%=K'X*S
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