JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Light Scattering in Terms of Oscillator Strengths and Refractive Indices

CARL M. PENNEYGeneral Electric Research and Development Center, Schenectady, New York 12301

(Received 20 April 1967; revision received 14 September 1968)

Quantum-mechanicaL expressions for nonresonance-dipole light-scattering cross sections are related tocorresponding expressions for oscillator strengths and refractive indices. In particular, Rayleigh and Ramancross sections for atomns are expressed in terms of oscillator strengths and vector-coupling coefficients.Several of our results, including the relationship between Rayleigh scattering and refractive index, differin general from the corresponding results of classical dispersion theorv. We show that these differencesarise from antisymmetric components C 1 a =(Ckj-CjL)/2 of the polarizability tensor, which have beenneglected in the classical analyses. Calculated Rayleigh cross sections for cesium and aluminum atoms,and Raman cross sections for aluminum atoms are presented to illustrate our results. The antisymmetriccontribution is found to be substantial in all of these cross sections; its most obvious effect is to cause thedepolarization (for linearly polarized incident light) to exceed 3 over extended wavelength ranges awayfrom resonance. On the other hand, for atoms initially in states of zero angular momentum, and moleculesunder conditions which allow the use of Placzek's polarizability approximation, our results agree with thoseof classical dispersion theory.

INDEX HEADINGS: Scattering; Cesium; Aluminum; Refractive index; Raman effect; Dispersion.

IN 1899 Lord Rayleigh' disclosed a relationshipbetween the light scattering and refractive index

for an ideal gas of isotropic particles small enough to betreated in the dipole approximation. Later Born2

extended this relationship to apply to particles withanisotropic polarizability tensors. These results havebeen used with expressions for the refractive index interms of oscillator strengths to determine the Rayleighscattering from atoms in terms of oscillator strengths,3

and vice versa.4 However, in Born's derivation of therelationship between Rayleigh scattering and refractiveindex, and in apparently all other derivations of thisand equivalent relationships published heretofore,5 con-tributions to the scattering arising from the antisym-metric components of the polarizability tensor Cmka= (Cjk-CkJ)/2 are neglected, usually through anassumption that the tensor is diagonal in a coordinatesystem appropriately oriented with respect to theparticle. We will show that the resulting expression forthe Rayleigh scattering in terms of a refractive indexor oscillator strengths can be significantly in error evenfar from resonance for atoms initially in states withnonzero electronic angular momentum.

The primary purpose of this paper is, then, to presentexpressions for complete nonresonance Rayleigh andRaman light-scattering cross sections for atoms interms of oscillator strengths, which we will derivedirectly from a quantum-mechanical expression for the

Lord Rayleigh, Phil. Mag. 47, 375 (1899).2 M. Born, Optik (Julius Springer-Verlag, Berlin, 1933).

A. Dalgarno, J. Opt. Soc. Am. 53, 1223 (1963); Douglas W. 0.Heddle, 54, 264 (1964). Both Dalgarno and Heddle based theircalculations on Rayleigh's result for isotropic particles.

I D. W. 0. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt.Soc. Am. 53, 840 (1963); P. Gill and D. WV. 0. Heddle, 53, 847(1963).

1 See for example S. Bhagavantam, Scattering of Light and tireRanman Effect (Chemical Publishing Co., Inc., Brooklyn, N. Y.,1942); H. C. van de Hulst, Light Scattering by Smzall Particles(John Wiley & Sons, Inc., New York, 1957).

cross section. Placzek5 has provided an initial step inthis direction. During the course of his work, he sepa-rated the cross section into three parts, arising fromtrace, symmetric and antisymmetric components of thepolarizability tensor, respectively, and presented thetrace contribution to the Rayleigh scattering in termsof oscillator strengths. However, apparently no onehas presented the symmetric and antisymmetric contri-butions, or composite cross sections in this form.

In Sec. I of this paper, we derive expressions whichdescribe explicitly the angle dependence of the compos-ite light scattering. Besides providing a convenientbasis for the rest of the paper, this derivation facilitatesa simple generalization of previous results regardingthe angle dependence. In Sec. II, we express the matrixelements involved in a result of Sec. I in terms of re-duced matrix elements and vector-coupling coefficients,in order to obtain Rayleigh and Raman cross sectionsfor atoms in terms of oscillator strengths. The contribu-tion of the antisymmetric components of the polariz-ability tensor is included implicitly in these cross sec-tions. On the other hand, it appears explicitly in a newexpression for the Rayleigh cross section in terms of therefractive index which we derive in Sec. III. In Sec.IRT, we discuss the significance of the antisymmetriccontribution and suggest a possible classical explanationfor its presence.

I. ANGLE DEPENDENCE OF THECROSS SECTION

The quantum-mechanical, dipole-approximation,nonresonance light-scattering cross section7 associatedwith a particle transition from an initial state I i) to a

6 G. Placzek, Ilaudbuch der Radiologie, VI (AkademischeVerlagsgesellschaft, Leipzig, 1934), Vol. 2, p. 209.

7 P. A. M. Dirac, QuantmonA Mechanics (Oxford University Press,London, 1958).

34

VOLEUME 59, NUMBER I JANUARY 1969

January1969 SCATTERING IN TERMS OF OSCILLATOR STRENGTHS

final state f f) can be put in the form

(012)i-f= | (Ckj)if ljC2k*

where

1 [(f ID, Ig)(g IDj Ii)(Ckj) i f = -E5

h g cogi-w

+(f IDi Ig)(g D, Ii).

+~ +c.

(1)

(2)

The cross section in Eq. (1) is defined so that(012)i fdQ2 gives the average power of light with(electric field) polarization e2 and propagation directionin the solid angle dQ2 about Q2 scattered by a particlein transitions i- -f from an incident beam of unitpower, polarization el and angular frequency co. Thefrequency cofi is defined by

,ofi= (Ef-E&A, (3)

where Ei and Ef are the energies of initial and finalparticle states. The subscripts j and k refer to com-ponents in a rectilinear laboratory coordinate system.The quantity (Ckj)i.f is the polarizability tensor,expressed in Eq. (2) in terms of matrix elements ofcomponents of the dipole moment for the particle inquestion. The sum over intermediate states in Eq. (2),and thereafter, includes, by implication, an integralover the continuum of positive energy states, whichmay contribute significantly to the scattering far fromresonance.

It is convenient for our purpose and appropriate forfreely oriented particles to employ states which aresimultaneously eigenfunctions of particle angular mo-mentum. We will designate these states by I TJM)where J is the total angular-momentum quantumnumber, M is the magnetic quantum number designat-ing the z component of angular momentum, and Trepresents all other quantum numbers necessary tospecify a particular state. Because the states of freelyoriented particles are degenerate with respect to orien-tation, corresponding observable cross sections involvea sum over final magnetic quantum numbers M' andan average over initial magnetic quantum numbers M.We denote such a cross section by

1(al12)TJ AT'J' ~ Y (a l2)TJ31-7`J J1' M 4

2J+ 1 MM'

Confining subsequent attention to plane polarizations,we choose a coordinate system such that the z axis isparallel to el and e2 lies in the xz plane. Then

Elj= ajz (5)and

E2k==kz cos,0+bkx singk, (6)

where

bkl= 1, k=l,

= 0, otherwise,

and ,6 is the angle between el and e2. Substituting theseexpressions into Eq. (1) and employing Eqs. (2) and(4) along with the magnetic-quantum-number selectionrules, 8

(TJM |Dz~ | YJM')= O. M'-7, M1

(TJMID., T'J'M')=O, M'#MI1,

we obtain directly

(0l2)TJ*T'J' = (o-ZZ)TJ-T'J' COS2t+(Oz.)TJjT'J' Sin2

i', (7)

where

((a XT' J',TJ)4

c4 (2J+ 1)

X E I (Ckj)TJM -T'J'MI' 2Mlf M'

(8)

Equation (7) describes contributions to Rayleigh orRaman scattering, depending on whether or notWTI' JTJ =0. If transitions involving more than onepair of levels (TJ,T'J') contribute to a particularobserved line or band, the corresponding cross sectionmay be obtained by summing (o12)TJ-T' J over appro-priate final levels, and averaging over initial levels.Weighting functions may be incorporated into theseoperations to account for the spectral distribution ofthe incident beam, and spectral response of the detector.The result will be denoted by

T12=Uzz cos2~ti+o-,. sin24,, (9)

where the averaging and summing operations areimplied by the absence of subscripts indicating transi-tions.

It is convenient to rewrite Eq. (9) in terms of thedepolarization p of the observed light scattered at rightangles to the directions of electric field and propagationof a plane-polarized incident beam.9 The depolarizationof the line or band is given by

p = 0rzX/o-zZ. (10)

We note that p can be a sensitive function of the spectraldistribution of the incident beam and spectral responseof the detector. From Eqs. (9) and (10),

O12T,,zz[(l-P) cos2 ip+p]. (11)

8E. U. Condon and G. H. Shortley, The Theory of AtomicSpectra (Cambridge University Press, London, 1957).

9 Note that p differs from the depolarization p' of light scatteredthrough 90° from an unpolarized incident beam. The latterquantity, which is employed frequently in the literature but isperhaps less convenient in an age of laser sources, is related to pby p'= 2p/(I+P).

35

CARL M. PENNEY

Equation (11) provides a concise description of theangular dependence of light scattered from atoms ormolecules small enough to be treated in the dipoleapproximation. It is consistent with correspondingprevious results which have been obtained for the Ray-leigh scattering, particular Raman lines, and rotationalRaman bands,2' 5 6'', 0 but we have shown that it appliesalso to any combination of Rayleigh and Raman lineswhich are involved in the observed scattered light.

Up to this point, we have considered the scatteringfrom an isolated particle. Under certain conditions,which are discussed elsewhere,5 the irradiance of lightscattered from a system of particles, integrated overthe Doppler-Brillouin structure introduced by particlemotions, is given by the sum of individual contributions

12 = I1 (o1 2 /R2

)

where I is the irradiance on the particle, and R is thedistance from the particle to the point of observation.We note that these conditions are usually satisfied in agas away from a critical point, up to pressures of manyatmospheres.

II. CROSS SECTION FOR ATOMS IN TERMSOF OSCILLATOR STRENGTHS

The initial and final states which are significant inRayleigh scattering from atoms often may be character-ized by a single set of quantum numbers T and J."1 Inthis case the Rayleigh cross section is determined by(07zz)TJ-TJ and (aZx)TJOFTJ. Using the selection rules onmagnetic quantum numbers, we obtain from Eqs. (2)and (8)

4 o 4 j 72'J",TJ

#264(2J+ ) l1 T"J" W2T"J",TJ -

X I (TJM I D, I TV "M) |12 | (12)

andcoJ4

(OZ) T JTJ "J=f1

2 c4 (2J+ 1)

X 57 (TJM'j Dx I T"J"M)(T"J"M I D, TJM)

1113frtT"J" WT"1J",TJ Ad

(TJM' I DI I T"PIM')(T"PIM' I Dx I TJM) I+ T"J" ,TJ+(13)

10 S. P. S. Porto, J. Opt. Soc. Am. 56, 1585 (1966).11 Here we propose to ignore any nuclear angular momentum.

A treatment including the nuclear angular momentum revealsthat the scattering integrated over any fine details that arise fromhyperfine splitting of initial and final states does not dependsignificantly on nuclear spin at separations from resonance thatare large compared to the hyperfine splitting of the (initial, inter-mediate, and final) states involved. The coupling of nuclearangular momentum to the scattering nearer resonance is discussedin Ref. 6, Ch. 12. See also A. Ellett, Phys. Rev. 35, 588 (1930).

The oscillator strength for a transition from level TJto level T"J" is

2œn WT" J" ,TJ

fTJ,T'# Jew =-e2J " (2J 1)

X I I (T"J"M I D, I TJM) 1 2. (14)

For f= 0 it is possible to obtain immediately arelationship between the Rayleigh cross section, andoscillator strengths or refractive index, because in thiscase the sums over magnetic quantum numbers inEqs. (12)-(14) reduce to a single term for whichM=M'=O. The magnetic-quantum-number selectionrules may be used to show that (0zx)TOTO=0; there-for, PTO7'o0=0 and, from Eqs. (11), (12), and (14)

/e2 \2

(0a12)TO+7T= W'(MC2

fTO TOT IJ 2X E t- Cos 2Q.

TomJET (,2 T1 J#,TO7'0 a(15)

The corresponding refractive index n is given in termsof oscillator strengths by

n2

-1 e2

fTO,T"J"4 =- E -417rN M TEEJew W2 T,{Jaw ,0-,W2

(16)

where N is the particle-number density. From Eqs.(15) and (16), we obtain

W,4 n 2-1 2

(0r12) TO-TO"=-c - CO21-.C4 47rN/

(17)

This result is equivalent to that obtained by Rayleighfor isotropic particles.

For JO0, we cannot substitute directly from Eq.(14) for the matrix elements in Eqs. (12) and (13)because the functions of the matrix elements summedover the magnetic quantum numbers M and M' aredifferent. However, the dependence of the dipolematrix elements on magnetic quantum numbers isknown, and can be expressed concisely using theWigner-Eckart theorem, which yields, for example,

(T"J"M' I D1 I TJM)

X , 1 M= (-1)J`- K"(T"J"IIDjI TJ)( , (18)

and

(T"J"M" ID, I TJM)

= (- 1)J`"1'"+\2u/2(T"J"IIDjI TJ)

- M" 1 M -M"-_1 M

Vol. 59

January1969 SCATTERING IN TERMS OF OSCILLATOR STRENGTHS

Here (T"J'"jDfl TJ) is a reduced matrix element, equivalent Clebsch-Gordon coefficients, and have beenwhich does not depend on magnetic quantum numbers, tabulated extensively.' Our notation is similar to thatand of Messiah,' 3 who provides a convenient discussion of

l I" 1 ) Athe Wigner-Eckart theorem and properties of the 3-4V-M" 0 M symbols. Employing the symmetry and orthogonality

properties, we obtain from Eqs. (12)-(14), (18), andis a 3-J symbol. The 3-J symbols are proportional to (19)

1

0

4X2 { OT"J" ,TJ 2i (T"lJ"'IDnTJ)12(

112C

4 (2J+ 1) iMl T"J" @2T" J" ,TJ 2 -M

"c4(oyzx,) TJ.TJ = ~ -- i7 I (T"J")ID11TJ) 1

h'2C"(2J±1) M TJ

J)2 2

MJ J(20)

F( il 1 J i/

I -M 1 M-1A--ML WT-J'JTJ-C)

1

0

J J.,

M} --M+1

1 J J/"

0 M-1A-M+1

WT"JI1,TJ+C0

-1 MJI

IJ 'and

fTJ,T"1J"= |MT111T (TIU1111DII TJ) 2.

3et f(2J+ 1)

Substituting from Eq. (22) for the reduced matrix elements in Eqs. (20) and (21), we obtain

(21)

(22)

(0a,) TJ-T'J=9(2J+ 1)Cw(-) EmcI2 M

and

(;2)2 M T J" fPjIrPjI(Oz) TJ -TJ = (2J +1 j@4( E E TJT J

Mc 2 Af (T"J" WT"1J" TJ

fTJT FJI ill

{2 T"J ,TJ-W 2 -M

1 J J"

1 M-1, -M

1

0 M (M+1

1 J

0 M-1/\-M+11 J)\l 12

-1 _

ii.Thus also for JF 0 the Rayleigh cross sections can

be expressed directly in terms of oscillator strengths.In Fig. 1 we show cross sections for ground-state cesiumatoms calculated using Eqs. (23) and (24)."4 The corre-sponding depolarization, plotted in Fig. 2, is in agree-ment with Placzek, who presented an equation for thedepolarization from alkali-metal atoms at light fre-quencies sufficiently near resonance so that only onepair (J"= 4,4!) of excited levels contributes signifi-cantly to the scattering. However, our results do

1" See, for example, M. Rotenberg, R. Bivins, N. Metropolis,and J. K. Woolen, Jr., The 3-f and 6-J Symbols (MIT Press,Cambridge, Mass., 1959).

"3 Albert Messiah, Quantum Mechanics (North-Holland Publish-ing Co., Amsterdam, 1962) Vol. 2, Appendix C.

14 P. M. Stone, Phys. Rev. 127, 1151 (1962).

not agree with those of Born. The disagreement ismost evident from the fact that the relationship betweenRayleigh cross section and refractive index which hederived yields a meaningless (negative) cross sectionfor depolarizations greater than 4; in fact, Bornobtained a limiting value of 3 for the nonresonancedepolarization of Rayleigh scattering. On the otherhand, we find that the depolarization for Rayleighscattering from cesium atoms is greater than 1, andeven exceeds 4 over a considerable wavelength rangeaway from resonance. In Sec. IV, we show that thisand other points of disagreement arise from omissionof the contribution of the antisymmetric componentsof the polarizability tensor in Born's results.

Placzek established originally that the presence ofdepolarization in Rayleigh scattering from atoms re-

X

1

0 M ) 2 2(23)

-MX I- _ 1-

WT"J" ,TJC0 WT"I J",TJ+W(24)

37

CARL M. PENNEY

10F21

6Pix

a .a.

6P312 2

627 8521A 8944A

6000 7000 8000 9000

(8), (18), (19), and (22)

/e2 \2

(0az) TJ -TI"J = '(2J' +1)(\C2/

X

o0.000 andWAVELENGTH OF INCIDENT LIGHT IN ANGSTROMS

FIG. 1. Rayleigh-scattering cross sections for cesium atoms inground states calculated from Eqs. (23) and (24) using semi-empirical oscillator strengths given by Stone,14 viz: f8944=0.394,fs521 = 0.814.

quires nonzero angular momentum in initial states. Inthe case of ground-state alkali-metal atoms, the activeangular momentum is provided by electron spin,"which is coupled to the scattering by the spin-orbitinteraction. Indeed, if this interaction is neglected,paired terms (J"=1, ) in the sum over intermediatestates in Eq. (24) are found to be equal in magnitudefor all incident-light frequencies, but opposite in sign,such that they cancel, and a,, vanishes. The spin-orbitinteraction alters this situation in two ways. First, itsplits the excited levels, causing the frequency depend-ence of opposing terms to differ significantly over arange comparable to the splitting. Consequently theyno longer completely cancel, and oz., shows a strongresonance structure over this range. Coupling of nuclearspin to the scattering arises in a similar fashion atseparations from resonance comparable to the hyperfinesplitting.

A second effect of the spin-orbit interaction is tochange the ratio of oscillator strengths associated withopposing contributions. This effect, which is importantin the higher excited levels of alkali vapor atoms,prevents general cancellation even far from resonance.However, it contributes only slightly to the depolariza-tion over the wavelength range shown in Figs. 1 and 2.

Atoms initially in states with nonzero orbital angularmomentum do not require a strong spin-orbit interac-tion to produce significant depolarization away fromresonance.'" For example, in Fig. 3 we show Rayleighcross sections for aluminum atoms in two initial states.The predicted depolarization for both transitions isunity over the wavelength region considered.

It is also possible to express Raman cross sections interms of oscillator strengths. Under conditions such thatonly one intermediate higher level T"J" contributessignificantly to the scattering, we obtain from Eqs. (2),

15 N. P. Penkin and L. N. Shabanova, Opt. Spectry. (USSR)14, . (1964).

/e 2 \2(0_z.)TJ-T = 91 (2J'+ 1)-)

Mc2/

X

X fTJ,T"J"fT'J',T"J"

1 '/ -M 1 M-1 -M

(26)I J)2

O M/.

Calculated Raman cross sections for two transitionsinvolving aluminum atoms are shown in Fig. 4. Thosefor the f= -= transition are larger than thecorresponding Rayleigh cross sections.

III. RELATIONSHIP BETWEEN RAYLEIGH-SCATTERING CROSS SECTIONS AND

REFRACTIVE INDICES

An expression for the refractive index at angularfrequency co away from resonance in a gas is

n2

-1 2

4,.jNT h ig

to

60( 0 7000

(27)Wg 2 2pf o2 I- gI . I

8000 9000 0,OOOWAVELENGTH OF INCIDENT LIGHT IN ANGSTROMS

FIG. 2. Depolarization of the Rayleigh scattering from cesiumatoms in ground states, calculated from the cross sections inFig. 1.

0

U)U)t0

U.

(.-cT'J' ,TJ)4

XTl>J" ,TJCrT'J',T"J" ((T79J" ,TJ °)

X fTJ,T-J'JfT'J' ,T"1J"

I Jf2 J/,

O M/ -M

1

0

J)2

M)(25)

a3.I

I x IN [ I

/116Ps,2 6P/,2

8521A 8944A

1 I I I I I I II I I I-

4A ---. . . . . . . .I . . . . . . . . . .

Vol. 59

ill

X 57m CM

I

(C')-C7,,JTJ)4

1C0

to"

I . I I I .

January1969 SCATTERING IN TERMS OF

where Pi represents the fraction of particles in statei i). For a system of atoms that are initially in a single

level TJ, Eq. (27) can be put in the form

n2 - 1 2 1 COT"IJ",TJ

47rN 3k 2J+1 T"J SW2

T9J;,TJv Cd

X I(T"J"flDJl TJ) %2 (28)

In order to develop a general relationship betweenthis expression and that for the Rayleigh-scatteringcross section, it is convenient to break the scatteringcross section into the trace, quadrupole and magnetic-dipole contributions introduced by Placzek. Thesecontributions arise from the trace, symnmetric, and anti-symmetric components of the scattering tensor, respec-tively, which are defined by

(C")TJM-T J1MS= 3 Z (Cijj)TJiI T'TJ'M', (29)7,=Y.2

(Ck j)TJJI.T'JSAX' = 42E(Ck;)TJ .JT'J'M'

+ (Cjk)TJM-f.T'J'MP]-(CO)TJMAIT' J'f'6

,kj (30)

and

(Ck a)TJ1I+T'J/M'= 2E(Ckj)TJM.VT'JIM'

- (Cjk)TJJ Tr'J'31I']. (31)

It is apparent that

(Ck j)TJMrT' JAM' = (C0)TJM.+T'J'M18Ak

+ (Ckj) TJM-T' J'M'+ (CAja)TJM..bT/J'J1I'- (32)

Placzek showed that cross products between these com-ponents, which arise in the absolute square of the scat-tering tensor, vanish in the sum over magnetic quantumnumbers M and M'. Consequently, the cross sectioninvolves only the three contributions mentioned pre-

OSCILLATOR STRENGTHS

,0f2l

WAVELENGTH OF INCIDENT LIGHT IN ANGSTROMS

FIG. 4. Raman-light-scattering cross sections for aluminumatoms for the transitions 3p'P1,2 - 3P2P3/2 and 3p2P3/2 -3P2P/2

calculated from Eqs. (25) and (26) using the oscillator strengthsmentioned in the caption to Fig. 3.

viously; i.e.,

(GOjk)TJ IT'J'= (yjh-O)TJ -T'J'

+ (ojjk8)TJ-T' J+ (,-, Ta-TJItJ (33)

where, for example,

(CO-CAT Jo,T J)4

(I(jka)TJTI TI)C4 (2J-f-1)

X Z I (Ck j)TJM-TJ'M' | 2 -M.ll'

(34)

These contributions are expressed in terms of reducedmatrix elements and vector-coupling coefficients inAppendix A.

It is evident from Eq. (31) that the antisyrmetriccontribution must vanish for j= k. Furthermore, thetrace contribution vanishes for j]k because it containsthe factor 5

kj. Therefore,

(Czz)TJIT'IJ = (azz,0)TJ.T' .Jr+ (u 2zz)TJ jT'J', (35)

and

(az.)TJI-T'Jr = (az.')TJI-T' J+( (c:.)TJI-T' J- (36)

Substituting these results into Eq. (11), we obtain

(a12)TJIT'JI = [r-(ZZ TJ-TT'J'+ (aZAT'J)I-T' 1J

Xw(t-PTJr tTJe) CdSi +PTJo iTgJi,

where the depolarization is given by

(37)

3pP Ir2 // \ \ 3P' P3/2

I C723-X

-24

3800 3900 4000 4100

WAVELENGTH OF INCIDENT LIGHT IN ANGSTROMS

FIG. 3. Rayleigh-scattering cross sections for aluminum atomsin 3p2P1 /2 and 3p'P31 , states, calculated from Eq. (23) and (24)using experimental oscillator strengths given by Penkin andShabanova,

6viz: f3944=0.15, f3952=0.15.

PTJ-T'J'`(ZZxs)rJTT1.Jo+ (YZXa)TJOT'J,

(0rzz0) TJ ITS J' + (azz') TrJ -T J'

(38)

Placzek also showed that the depolarization of thesymmetric scattering is 4; i.e.,

(a0'8)TJI-T'J'J= 4 (b'zz ) TI-T' J (39)

39

I

III

�2I

CARL M. PENNEY

This result is rederived from relationships betweenvector-coupling coefficients in Appendix A. Using Eqs.(38) and (39) we can express the symmetric crosssections in terms of the depolarization and antisym-metric cross section. As a result Eq. (37) can be re-written in the form

(0f12)T.TT'J' =[( Z)TJ, T1J-4( 23)TJ- TJI]

X (3/3-4PT J-TW JW)

X:(1-PTJ- T'J') cos2+ ,pTJTJ,]. (40)

Comparison of Eqs. (28) and (Al) reveals that

(a'ZZo)TJrTJ= (o/C)40(12- 1)/4-7rV] 2 . (41)

Substituting from Eq. (41) into Eq. (40), we obtainfor the Rayleigh cross section

F/X\ 4 /It2 - 1 2(Ui2)TJ-TJ= H) V 1-3 (ZTa)TJ-TJ

Lc/ 4-,rN/-

X3/(3 4PTJ-PJ)E(1-PTJ-.TJ) CoS21P+PTJ-TJ]. (42)

IV. DISCUSSION

The relationship between Rayleigh scattering andrefractive index obtained by Born2 can be put into theform

/( )4 n?2- 1\2/ 30'12= (-) 4 (n )E((I - P) cos2V+P]. (43)

\c \ ZrlY \3-4p

Our result, as presented in Eq. (42), differs in that itcontains an antisymmetric contribution in the firstterm. Equations (38) and (39) can be used to showthat the antisymmetric contribution also accounts fordepolarization exceeding 4.

The antisymmetric contribution to Rayleigh scatter-ing from ground-state cesium atoms is particularlyevident because the symmetric contribution vanishesfor those atoms."6 Consequently oz2 arises entirelyfrom the antisymmetric contribution, whose range ofsignificance is indicated thereby in Fig. 1. In order todemonstrate the effect of the antisymmetric contribu-tion on the relationship between Rayleigh scatteringand refractive index for this case, we note that wheneverthe symmetric contribution vanishes, Eq. (42) can berewritten using Eqs. (37) and (41) to read

/W 4 it2- 12

(a912) 7J-TJ - [

c\ 4-irN

X[(1-PTJ-TJ) CoS2'P+pTJ-TJ], (44)

which will differ significantly from Eq. (43), whereverthe depolarization is significant.

15 The symmetric scattering vanishes for transitions J =2JI = L. This is one of the selection rules derived bv Placzek inRef. 6.

Both symmetric and antisymmetric scattering con-tribute to the Rayleigh cross sections for aluminumatoms shown in Fig. 3. The fact that the predicteddepolarization is greater than 4 throughout the rangeof frequencies included in Fig. 3 indicates that the anti-symmetric contribution is significant throughout thatrange.

In contrast to these cases, Placzek showed originallythat the antisymmetric contribution to Rayleigh scatter-ing is negligible away from resonance for atoms in SOstates," and for molecules under conditions for whichhis polarizability theory is valid; i.e., (1) The electronicground state is not degenerate through a mechanismcoupled to the scattering. (2) The incident-light fre-quency and its separation from resonance are largecompared to the splitting and shift of the correspondingelectronic levels due to nuclear motions. In thesesituations, our result for the relationship betweenRayleigh scattering and refractive index becomes equiv-alent to the earlier results of Born, which have hadconsiderable experimental verification. 5 In Appendix Bwe show that under the polarizability theory Eq. (43)should describe the scattering in a band which includesnot only the Rayleigh scattering, but also an arbitraryamount of the adjacent rotational Raman scattering.

It is interesting to speculate about a possible classicalexplanation for the antisymmetric contribution. Bornfound from arguments based on energy conservationthat the classical polarizability tensor should be her-metian away from resonance; i.e., Cki= Cjk*. The usualclassical analysis implies that it should be real awayfrom resonance. Under these two conditions the anti-symmetric part must vanish. However, there is a basicdifference between the usual classical analysis and thatsuggested by the quantum-mechanical results. In theformer, a particle (or, in the case of a molecule whoserotation is taken into account, its angular momentum)is treated as if it maintained constant orientation,whereas in the latter, depolarizing transitions involvea change of the orientation of particle angular momen-tum that is consistent with the selection rule AM= ±-1.It seems likely that an account of this reorientationwithin a classical model would introduce imaginary,and therefore antisymmetric components in the non-resonance polarizability tensor.' 7 We expect that thesecomponents would be most important for asymmetricparticles with very small moments of inertia-typicallyatoms with nonzero angular momentum.

V. CONCLUSION

We have obtained expressions for light-scatteringcross sections for atoms in terms of oscillator strengthsand vector-coupling coefficients and derived a newrelationship between the Rayleigh-scattering cross sec-

17 Born presents a discussion which relates to this point inRef. 2, Art. 73.

Vol. 59

January1969 SCATTERING IN TERMS OF OSCILLATOR STRENGTHS

tion and refractive index. Calculated cross sections forcesium and aluminum atoms are presented. These resultsinclude previously neglected contributions of the anti-symmetric part of the polarizability tensor, which weshow may be significant over wide wavelength rangesfor atoms with nonzero electronic angular momentum.

ACKNOWLEDGMENTS

This paper is drawn in part from a doctoral disserta-tion submitted to the University of Michigan (1965).The author hereby expresses his appreciation to manyassociates there, in particular to R. K. Osborn, forencouragement and numerous clarifying discussions.

(acr J,TJ)4 1 2

(czz0) TJ -T' J' = J JJ' E (Qi+Ri),

9h2

C4 (2J+ 1)2 TJ

(Ca (OT'J',TJ)4 Jl

(aGzzS)TJ-.TJ, = i,h2C4(2J+1) M= J Ti1

(I-_WT'J',TJ)4

(NAzI, 3)TJ -STY' J= ~~E E (Ql+Rl)4h2C4 (2J+ 1) 3f TiJi

(Q,+Rl) M

The work at the University of Michigan was begununder support of a National Aeronautics and SpaceAdministration Traineeship and was completed throughsupport from the Advanced Research Project Agency(Project DEFENDER), monitored by the U. S. ArmyResearch Office, Durham, under Contract No. DA-31-124-ARO-D-403.

APPENDIX A: CROSS-SECTION COMPONENTSIN TERMS OF REDUCED

MATRIX ELEMENTS

Using the Wigner-Eckart theorem and properties ofthe 3-f symbols, we obtain

1

0M- JM

M/\-M

1

0

JM 3(f, +1 21

M 3 (2J+ 1) I,

(Al)

(A2)

CCJf 1 Jf J'

-M 1 M-1 -M

and

(Wo-C0Tt J' TJ)4

(Ozra) TJ-TJo = E4h12 c4 (2J+1) M

X J[XI

L- M

1

1

T E (Q,-Ri)TiJi

MCI 1)(iM

1

0

1

0

1

0

1

-1IM-1M+

1

0

1

-1I

I2 (A3)

JM)1 2

M )i (A4)

(T1Jf1IDII T'J')*(TiJf|IDII TJ)1=

and

(A6)

through the relationships

(A5)

1a M 2

ma k 1 m2R, (T1J1|D|T'J')*(T1J+C JDL' TJ)

W)T1J1,TJ+C0

In order to derive Eq. (39) it is convenient to expressEqs. (A2) and (A3) in terms of 6-J symbols. Writingout the absolute squares in these equations, we obtainproducts of four 3-J symbols, summed over M and M'.Introducing the 6-J symbols

j I j2 j3

1/1 12 13 }

SM

23) 11 12 j3\

3 1n fl n2 -w3

(-1) j3+S+nl+nl(2S+ 1)

11 j2 S(

nt ens M Ml

where

i2

12

•3}IJi11

12

ln2

(A7)

41

JI

J J,

M _M+1

J J,

M _M+1

J f J,

M-1 _H+1

4 CARL M. PENNEY

and employing the orthogonality properties of 3-Jsvmbols, we can show that

(W-WT'J' TJ)4

(Sz)7J>2,,= ' (-)J'-J12 c4 (2J+ 1)

2

X E E (Qi+Ri)(Q2 *+R2 *) E (2S±1)TiJi T2J2 S=O

x{1 1 J' IX t

1 J2 S I

1

J2

§1 [(A A 5)2]' (A8)

and(w-WT' J' .TJ)

4

(az s) T J aTr JR =- 'T(J - i)J'-J+1

2h2c4 (2J+ 1)

2

x E E_ (Qi+Rl)(Q2 *+R 2 *) E (2S+1)TiJi T 2 J2 S=O

1

J 2

J'1 iJ2 1 }r1( 1 I )(1 1A 5s I J2 Sl - 1ooo w

( 1 i 1 1 )] (A9)

0 1 -1 I 0 -1

From tabulated values for 3-f symbols,

11 1 5S I 1 S\ 1 1 S I1 1 S\

\-1 1 oJ0 0 0 \1I 0 -1J o 1 -1J(-2 DA A =-C[A 5)(A 2 1)]- {( )] (AlO)

for S=O, 1, 2. Substituting this result into Eq. (A9)and comparing with Eq. (A8), we obtain Eq. (39),

(ez TJ-T, J, = 4 ( )TJ-TIJI (All)

APPENDIX B: RELATIONSHIP BETWEEN THERAYLEIGH CROSS SECTION AND REFRAC-

TIVE INDEX FOR MOLECULES

Equation (42) has been derived for atoms in initialstates characterized by a single value of J; therefore

it is not directly applicable to molecules in a systemwhere they are excited to states of various rotationalangular momenta. However, using Eqs. (35) and (36)to express (iz,)TJ.-T2J' and (0fZx)TJ.T'J' in Eq. (7), andEq. (39) to relate (ozzs)2,,J, and (0Zxs)TJ-TJ., uponsumming over final states which correspond to a bandof scattered light and averaging over initial states, weobtain

O.12= [O2-Oz3a] [3/(3-4p)]

X[(1-p) cos 24±p] (Bi)in the simplified notation of Eq. (9).

The refractive index for a gas of molecules involvesan average of

1 7,"J", TJ

1 ______ _ (T"J"'JD 1J TJ) 2 (B2)2J+ I TJ 27,, C T ,JTJ-W

over initial levels TJ. On the other hand, under thepolarizability theory the trace cross section for a bandincluding the Rayleigh scattering and perhaps alsosome adjacent rotational Raman scattering (to whichthe trace cross section will not contribute because of itsselection rule AJ= 0) involves an average of the squareof this quantity. Approximating the average of thesquare by the square of the average, we obtain

(B3)

This approximation ought to be excellent under condi-tions allowing use of the polarizability theory, becauseit predicts that the quantity to be averaged is inde-pendent of J.6 Moreover, the antisymmetric contribu-tion vanishes according to the polarizability theory.6

Consequently, we obtain for a band that includesRayleigh scattering and may also include some adjacentrotational Raman scattering,

@\ 4 12 1 /3

0-12= c [N 34 (1-p) cos2.p+pl, (B4)

which is equivalent to the classical result of Born. Thecontribution of rotational Raman scattering to thiscross section is contained within its effect on the depolar-ization.

-- U .

John Gregory (Barnes Engr.), Frank Twitchard (Bendix Res.Labs), and Richard Hoover (NASA) at Washington meeting.

42 Vol. 59

G'ZZ 1= (WIC)4E(112- 1)/4rN]I.