HO RITE 550 HEEELE EN OF THE ROTATION MATRICES IN TERMS OF i/iRODRIGUEZ'S PRAMETERqH ME~qN ANI (ITALY) IST DISTRUTTURA DELLA MATERIA F E RUNESE ET AL JUN 83
UNCASSIFIED AIRCSL-TR-83643 DAJARI8-C09 F/ 111
EEEEEEEs
*25 LA11
MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREW4 OF STANDARDS-1963-A
S CHEMICOLS SYSTEMSL LABOR01TORY US Army Armament Research and Development Command
L Aberdeen Proving Ground, Maryland 21010
TECHNI CAL REPORT ARCSL-TR-83043
O0 THE ELMETS OF THE ROTATION MATRICES INTERMS OF RODRIGUEZ' S PARAMETERS
1.0 by
F. BorgheseP. DentiR. Saija
G. Toscano
I Universitai di Messina4=4 Istituto di Struttura dell& Materia
0. I. Sindoni
Physics BranchResearch Division
June1983DTICJune1983ELECTE
JAN 5 0
D
Approved for public release; distribution unlimited.
84 '
Disclaimer
The findings in this report are not to be construed as an officialDepartment of the Army position unless so designated by other authorizingdocuments.
Disposition
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it to the originator.
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READ INSTRUCTIONSREPORT DOCUENTATION PAGE BEFORE COMPLETING FORMI.REpONT NMSER 12 OVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER
ARCSL-TR-83043 nA /3 ___________
4. TITLE ( a~s S. TYPE OF REPORT A PERIOD COVERED
THE ELEMENTS OF THE ROTATION MATRICES IN TERMS Technical ReportOF RODRIGUEZ'S PARAMETERS_____________
6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR~q) 6. CONTRACT OR GRANT NUMBER()
F. Borghese G. ToscanoP. Denti 0. 1. SindoniR. Saija DAJA3 7-81-C-0895
9. PERFORING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK
Comnander, Chemical Systems Laboratory AE OKUI UBR
ATTN: DRDAR-CLB-PSAberdeen Proving Ground, MD 21010
11I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATECommander, Chemical Systems Laboratory June 1983ATTN: DRDAR-CLJ-IR 13. NUMBER OF PAGES
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Approved for public release; distribution unlim~ited.
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16. SUPPLEMENTARY NOTES
It. KEY WORDS (Cwe n revrse aide it noeesry and Iintify by block mmber)
Rotation matrices Kayley-Kicin parametersRodriguez's parametersDirection cosines
Angle of rotation
ShL AESTUACi (bifio an -~M94 S N -momSA ad 1~0tf~ bY 6!loCk 7"-,er
The elements of the rotation matricesI D M, M for arbitrary L, as veil as those
of the matrix S which rotates the cartesiy coordinates, are expressed in
terms of Rodriguez's parameters, viz., the direction cosines of the axis,4144"dd a. Ifs9do' e'e'
)6 $ , and the angle of rotation,
W I F=79 we no E~or IF ?o MW s soSSLETE UNCLASSIFIEDSECUMNY CLASSIFICATION OF THIS PAGE (Shinm Dot& Entered)
inmu~y pAjiTMag OF TUIS PAGWW- Dklf OIm
BLs~
2 UNCLASSIFIEDSZCUOt*rY CLASSIPICAIO air THIS PAGgruhe. dfte EaemO*
PREFACE
The work described in this report is based on work supported inpart by the US Army European Research Office through contract DAJA37-81-C-0895, and in part by the Centro Nazionale Ricerche (CNR) (National Centerfor Research) through the Gruppo Nazionale Struttura della Hateria (GNSM)(National Group for the Structure of the Hatter). This work was startedin 1978 and is on-going.
The use of trade names in this report does not constitute anofficial endorsement or approval of the use of such commercial hardware orsoftware. This report may not be cited for purposes of advertisement.
Reproduction of this document in whole or in part is prohibitedexcept with permission of the Commander, Chemical Systems Laboratory, ATTN:DRDAR-CLJ-IR, Aberdeen Proving Ground, Maryland 21010. However, the DefenseTechnical Information Center and the National Technical Information Serviceare authorized to reproduce the document for United States Government pur-poses.
This report has been approved for release to the public.
.Accession
For
NTiS &I
DTIC TABU.nnouned 0Justifioatior
K Distribution/
AvailabilitY Codes
IA-ail end/or
Dist I special so@
3
Blank
CONTENTS
Page
1 INTRODUCTION ...................... 7
2. DISCUSSION ......... ... .. .. ...................... 7
3. CONCLUSION ....................... . 10
LITERATURE CITED .................... 11
DISTRIBUTION LIST ...... ................... . 13
'N0
I F " ' .,. .., . ,, ,r . . ... . .. ....., .,, , . ,. ...., ....."5 , . . q ," , " " , " ,, *, " , .. . , ' . , . . .. . .- , • . ' ' ' ", ." " . ' - '
* ~ ~TA~ -L .* * * .-... * ,t -~l ILIC -
Blank
6
- *- ~ ~V~.. .* VVV
THE ELEMENTS OF THE ROTATION MATRICES IN
TERMS OF RODRIGUEZ'S PARAMETERS
1. INTRODUCTION
In quantum mechanics and group theory, the elements of the rotation
matrices D are expressed in terms of the eulerian angles whose mutual in-
dependence offer undoubted advantages. However, there is one application
in which expressing the elements of -Z in terms of Rodriguez s parameters,
viz., the direction cosines of the axis, X, v, v, and the angle of rotation,
w, may be very useful. The application we are thinking of is the calculation2of sy metrized combinations of irreducible spherical tensors, particularly
scalar and vector spherical harmonics. In this case, while their use to describe
the rotations is rather instinctive, the lack of mutual independence of Rodriguez's
parameters does not matter.
2. DISCUSSION
Bassani and Pastori-Parravicini,3 who restrict their analysis to
two-component spinors, have used Rodriguez's parameters to study the rotational
properties of functions. We will obtain general expressions for the elements
of the matrices DJ ,for arbitrary J, and, particularly for the elements of
the matrix S describing the rotation of the cartesian coordinates:
xi = Sijx j (I
Recalling that the matrices D3 form an irreducible representation
of order 2J+1 of the rotation group in three dimensions on a basis of simulta-
neous eigenvectors of J2 and J
D< J, 'e ~ -J,M > 2)
'' 7
%%ee lp is the vector angle of rotation, and the explicit expression forJ 4the elamess %.,1 is gives (e.g., by Sammermesh) in terms of the Kayley-
Klein parmters. a sod b, as
M ~ k (J+N-k)tkI(J-M'-k)I(M'-M+k)t
Xa J-Mk(a*) J-'kb k(-b*)M-k (3)
It is weil known that the Kayley-Kicin parameters are used in classical
mechanics as a tool to integrate gyroscopic problems; but for present pur-
poses, they are better related to the elements of D1
--(:* ~
Therefore, once it is shown that a and b can be expressed in terms of
As p, v, and w, the task will be done. To this end, notice that for
Ja ;, Equation 2 yields:
, I ei 0 J , > jexpi (a + P 2y + V a]~. (4)
where the a 'a are the Pauli spin matrices. Now, let us put
xVp +Vat&i U (then, by substituting into Equation (4), we get
exp (O8
2~ k 2 15
T. -- . . . . . . . . .- -- 9 . . . -. - A
according to the customary definition of the functions of a matrix.
Since it is immdiately verified that
p2 ku1 P2 k + 1
Equation (5) can be rewritten as
D W )2i k (2) 2k + (L..k+l 1 r2k+11
2 2i"sn (6)
Therefore, the expression for D3 becomes
Cos 1E+ i v sin-1 i (A - iii sin7
D"
' i( X+ i) siml cos!E -i V s
.6 and the Kayley-Klein parameters take on the desired form
a w coo!E + iv sin (7I2 2 (a
b - i (OX i Pi ) sin! (7b)
9
When dealing with symmetry properties of sets of spherical
tensors centered at molecular or crystal sites, the site coordinates
have to be rotated in order to get the permutations induced by the
4 group operations. This can be done easily through the use of the.4 .5matrix S of Equation (1), whose expression has been given by Goldstein
Z
in terms of a and b:
I1
! (a2 + a*2 - b2 - b* 2 ) ! (b*2 - b 2 - a2 + a*2 ) - (a b+a* b*)2 2
S " (a2 - a*2 - b2 + b2) -1 (a2 + a*2 + b2+ b2) i(ab-a*b*)Z .2 2
ab*+a* b i (ba*-ab*) aa* bb* (8)
Substituting Equation (6) into Equation (7) then yields
2 2S = (l-cos)+vsinw +(l-jicosw lv (l-cos)-Asinw (9)
v X (1 - cos )- psin w Pv (1-cosw)+Xsinw v 2 + (l-v 2 ) cosW
3. CONCLUS ION
In conclusion, it is pointed out that Equation (9) has been
successfully coded with Equation (3) with a and b given by Equation (7)
as part of a program to get symmetrized combinations of spherical multi-
poles centered at the sites of a molecule. In spite of the occurrence of
complex quantities, we were also able to implement the coding to reduce
the request for complex computer algebra to a minimum.
10
S... . . . .. -.... . ., .
.- ' ,. , ,. I ? .. J'.*. ' . . • .....-.. ....-... ., ". . . - - . - -. . . . - . - -. .. ... •,,, ,m ,l iallla l~w~n ,,= ma,,,. m .,=m.ll ni idnkdnL.lku , _ : - , , A , A __ -. .. -. : .
LITERATURE CITED
1. Thomson, W., and Tait, G. P. Treatise on Natural
Philosophy. Cambridge University Press, Cambridge, England. 1923.
2. Rose, E. M. Elementary Theory of Angular Momentum.
Wiley, Nev York. 1957.
3. Bassani, G. F. and Pastori-Parravicini, G. Electronic
States and Optical Transitions in Solids. Pergamon Press, Oxford, England.
1975.
4. Hammermesh, M. Group Theory. Addison-Wesley, Reading,
Massachusetts. 1962.
5. Goldstein, H. Classical Mechanics. Addison-Wesley, Reading
Massachusetts. 1962.
@.
l'V
€ ' '. , .' o '. .;".".'.. -' "'2.2.'-.'' -. • '-.-'. -°-'. .'" " - "' "--."".".' ,i.","- -. ..
* * - - - - - -. - - * - - - - - - - - - - - - - - - - -----..--
A
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4
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F,
12
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.
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US Might Vision and EO Laboratories Alexandria, VA 22333ATTN: ORSEL-NV-VI (Dr. R. G. Buser) 1
ATTN: DRSEL-NV-VI (Mr. R. Bergemann) I Or. W. Michael Farmer, Assoc Prof, PhysicsATTN: DELNV-VI (Luanne Obert) I University of Tennessee Space Institute I
ATTN: DELNV-L (D. N. Spector) I Tullahoma, TN 37388Fort Belvoir, VA 23651
CommandantAcademy of Health Sciences, US ArmyATTN: HSHA-CDH 1ATTN: HSHA-IPM 2Fort Sam Houston, TX 78234
Science Applications Inc.ATTN: Dr. Frederick G. Gebhardt 1
3 Preston CourtBedford, MA 01730
Science Applications Inc.ATTN: Mr. Robert E. Turner 1
1010 Woodmen Drive, Suite 200
Dayton, OH 45432
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