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DoubleQuantum Light Scattering by MoleculesR. Bersohn, YohHan Pao, and H. L. Frisch

Citation: The Journal of Chemical Physics 45, 3184 (1966); doi: 10.1063/1.1728092 View online: http://dx.doi.org/10.1063/1.1728092 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/45/9?ver=pdfcov Published by the AIP Publishing

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3184 DIELEMAN, DE JONG, AND MEIJER

comparison with metalloid vacancy compensation. The dominant energy term in the latter compensation mechanism is Hv. The value of Hv can be estimated at about 2 eV.19 We see that the balance between these two incorporation mechanisms can shift from more metalloid vacancy compensation in ZnS to more Fe-donor compensation in CdS and ZnSe. What was found actually was that compensation by the Fe donors increased in this direction. However, this is not evidence of a competition with metalloid vacancy compensation, other incorporation mechanisms still being possible at present. Since the evidence for the self-coactivated dissolution of the alkali metals is not yet convincing,

19 F. A. Kroger, The Chemistry of Imperfect Crystals (North-Holland Publ. Co., Amsterdam, 1964).

THE JOURNAL OF CHEMICAL PHYSICS

it is even possible that the solubility of these metals is largely determined by the presence of shallow donors, i.e., they dissolve as monovalent ions, just as has been found for the elements Cu, Ag, and Au. What remains is an expected increase in compensation by Fe donors from ZnS to CdS and ZnSe, because of the decreasing ionization energy of these donors in this direction.

ACKNOWLEDGMENTS We wish to thank Dr. C. Z. van Doorn for helpful

discussions, Dr. W. Kwestroo for the preparation of high-purity ZnS and CdS, and the department of Dr. N. W. H. Addink for the chemical analysis. We are also grateful to Mr. T. S. te Velde for making available to us some of his results.

VOLUME 45, NUMBER 9 1 NOVEMBER 1966

Double-Quantum Light Scattering by Molecules R. BERSOHN

Department of Chemistry, Columbia University, New York, New York AND

YOH-HAN PAO AND H. L. FRISCH Bell Telephone Laboratories, Incorporated, Murray Hill, New J er sey

(Received 19 May 1966)

Double-quantum light scattering by a system of molecules is discussed in this paper. Expressions have been obtained for the scattered light intensity considering both the coherent and incoherent contributions. In that coherent contributions are also considered in this treatment, it goes beyond the scope of previous studies. It is shown that, for molecules of low symmetry, elliptically polarized light must be used in order to determine five independent quadratic forms in the 18 symmetric components (f3i;k+{3.k;). According to the present results, the apparent discrepancy between the observed value of l for the depolarization ratio for eCI. and the value to be expected from theory may be due to the fact that the coherent contribution had been neglected in previous theoretical considerations. In general, orientational correlation is essential if there is to be appreciable contribution from coherent scattering. For macromolecules, this constitutes a major difference between single- and double-quantum scattering, and additional information may be ex-pected if the latter is investigated experimentally.

I. INTRODUCTION

I F a system is irradiated by photons of frequency W and scatters photons of frequency 2w, it is said to exhibit double-quantum scattering. More generally we call two-quantum light scattering the process in which two quanta of frequencies WI, W2 are incident on a system and a single quantum with frequency W3=Wl+W2 is scattered. At low frequencies such scatter-ing is routine and is described in terms of the nonlinear electrical parameters of the system. On the other hand at optical frequencies the phenomenon has only recently been discovered. l The reason for this contrast is simply that for a given energy density there are far more quanta at low frequencies than at optical frequencies.

The possibility of nonlinear optical processes was discussed theoretically in 1931 by Mayer.2 Franken3 has given a review of his first nonlinear optical experi-ments and a phenomenological theory. The latter is based on the observation that double-quantum scatter-ing depends on the third-order tensor (3 in the expan-sion of the electric polarization of the system

1 P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Letters 7, 118 (1961).

(1) or

where a and ~ are second and third order tensors, re-spectively. It can be inferred from this equation that

2 M. Goeppert-Mayer, Ann. Physik 9, 273 (1931). il p, A. Franken and J. F. Ward, Rev. Mod. Phys. 35, 23 (1963).


LIGHT SCATTERING BY MOLECULES 3185

in a system with a center of symmetry the polarization vector must change sign when the electric vector re-verses direction; hence fJ must vanish identically. One would therefore expect that only crystals which lack a center of symmetry would exhibit nonlinear effects. Armstrong et al.4 have analyzed these effects in detail and have shown that coherent double-quantum scatter-ing is to be expected in certain crystals at certain orientations but not, in general, in fluids.

Terhune, Maker, and SavageS succeeded in observing double-quantum scattering in such "fluids" as H20, CC4, CHaCN, and fused quartz. The intensities ob-served were so small (",10-1a of the incident intensities) that they represent not a contradiction of the previous theory but rather an incoherent scattering from single molecules or small clusters of molecules. Future ex-periments following the pioneering work of Terhune et at. should provide information about (1) the fJi;k tensor of the individual molecules and (2) the structure of liquids and dissolved macromolecules.6

To explore these matters, we present in this paper a derivation of expressions for the scattered light in-tensity, considering both the coherent and incoherent contributions. By coherent contributions we here refer to the cooperative scattering of neighboring molecules

whose positions and orientations are correlated. It is understood the scattering from different regions of the liquid (whose size is of the order of the wavelength of the light or larger) is incoherent. In that coherent contributions7 are also considered in this treatment, it goes beyond the theoretical treatments of Li,8 Kielich,9 Cyvin et al.,l0 and Terhune et at.5 However, in contrast to these previous discussions, the present treatment is limited to elastic scattering; that is, the scattered light is exactly at the sum frequency and the Raman effect is neglected.

II. SCATTERING BY AN INDIVIDUAL MOLECULE

A. Tensor fJi;k A derivation is given in Appendix A of the probability

per unit time of the scattering into a solid angle dO of a light quantum of wave vector ka and polarization Aa by a system of individual elements a (molecules or segments of a larger molecule) when given intensities of photons kl' AI, k2, A2 are incident. The derivation uses, for the sake of clarity, a quantized radiation field but yields (as might be expected) the same expression previously derived by semiclassical means, viz.,

+ (JJ.X1") Oq (JJ.xa") qr (JJ.X2") rO + (I-'X2") Oq (I-'XI") qr (JJ.X1") rO (Wl-Woq) ( -W2-WOr) (W2-WOq) ( -WI-WOr)

K=ka-kl-k2, and Ra is a vector from an arbitrary origin to a fixed point in the scattering unit a.

Let us consider a single radiating dipole t' in classical electromagnetic theory. If It is a unit vector from the dipole to point of observation at a distance R, then the field vectors are

S..,= (wNc2R)(t' xlt) xIt, Hsc= (wNc2R) (t' xlt)

(4) (5)

4 J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).

R. M. Terhune, P. D. Maker, and C. M. Savage, Phys. Rev. Letters 14, 681 (1965).

GR. Bersohn, J. Am. Chem. Soc. 86, 3505 (1964).

and the Poynting vector is

(6) The corresponding quantum-mechanical expression

7 A brief summary of preliminary results of this aspect of the present investigation was presented at the Third International Conference on Electromagnetic Scattering, Amherst, Mass., 1965, by H. C. Frisch.

8 L. Y. Yuan, Acta Phys. Sinica 20, 164 (1964). G S. Kielich, Bull. Acad. Polon. Sci. 12, 53 (1964). 10 S. J. Cyvin, J. E. Rauch, and J. C. Decius, J. Chem. Phys.

43,4083 (1965).


3186 BERSOHN, P(O, AND FRISCH

for the energy flux vector scattered by a single molecule ex IS

Wka-klk.}iwkl 0,= 271" (~)4 (~) l(w1)I(w2) R2dfJ R2 me he

X 1 fa (kaAa, k2A2, k1~q) 12hwall. (7) Comparing Eqs. (6) and (7) we see that the com-

ponent of the induced dipole in the direction of polari-zation of the scattered wave is

JJ.A3 = (t)!(efNm2) [OAI (WI) OA2(W2) /Wa(WIW2) I] Xfa (kaA3, k2A2, kIAI) , (8)

where we have substituted

i=I,2. (9) Now, by the definition of the tensor {3, we have

(10) and hence we can establish the correspondence

(3AIA2AI = (t)1/2(eh2/m2) [1/wa(wIW2) 1/2Jfa(k3A3, k2A2, kIAI) (11)

between the matrix element of the {3 tensor and the scattering amplitude fa. The corresponding matrix element for a group of molecules would be a sum over the scattering amplitudes for the individual molecules weighted by phase factors.

B. Theorems on the Tensor {3.;k

Two general theorems can be established from the tather complicated formulas of Eqs. (2) and (3). In rhe dipole approximation which we have adopted the individual scattering units ex are small compared to the wavelength and only the electric dipole operator t' occurs in the matrix elements. If the system ex has a

center of symmetry not all the three quantities f}oq, t'qT, and t'rO can simultaneously be different from zero. Therefore the tensor {3 for the unit ex vanishes and-in the dipole approximation-harmonic scattering is im-possible. The theorem depends on the existence of a center of symmetry in the unit ex and makes no refer-ence to the presence or absence of a center of symmetry in the system as a whole.

Kleinman's theoremll and its limitations are also apparent from inspection of the formulas. If the im-portant intermediate states have energies so much greater than the energy of the three light quanta in-volved that the photon frequencies can be neglected in the denominators, the 3 ! terms of the formula become the 3! permutations on the three indices. The tensor {3i;k becomes symmetric in all its indices and the applied electric fields of the light waves become equivalent to static fields. The theorem appears to be valid for crystals such as crystalline quartz12 and KH2P04 with incident ruby laser light (6943 A~1.8 eV). However, these crystals have little absorption below 7.5 eV. On the other hand, obvious nonlinear resonance effectsl3- I5 are shown in cases where the absorption bands are closer to the harmonic such as in CS2, ZnS, and crystals of benzpyrene and benzanthracene. In the absence of a theoretical evaluation of Eq. (3), the limits of validity of Kleinman's theorem must be determined experi-mentally. This is not assumed in this paper, in contrast to the work of Cyvin, Rauch, and Decius.1o

C. Estimate of the Cross Section for Double-Quantum Scattering

If we try to form the cross section from Eq. (2) ac-cording to Eq. (Al1) , we have a practical difficulty that for a three-particle process the effective cross sections are dependent on beam intensities. As an ex-ample, suppose we consider a single incident beam for which WI=W2=!W3=W. Then

( 12)

A measure of the harmonic component in the scattered radiation is the ratio of the differential scattering cross sections (12) and (AlO) , i.e.,

(dO'/dfJ)harmonic e2 ( h )2 1(w) 1 ~fa(2) exp(iKRo) 12 (dO'/dfJ).ingle hc 471" me'l ~ 1 L:.fa(l) exp(iKRa) 12 . (13)

The ratio SeW) of the squares of the structure factors is dimensionless and should be of the order of mag-nitude of unity except at frequencies w such that 2w is close to an absorption peak. Equation (13) can be rewritten more simply as

0'2/0'1 =ex( 471"ao2) [I (w) /wJs(w) , (14) where ex is the fine-structure constant and ao is the

a

Bohr radius. The physical meaning of this equation is that when the incident flux lew) is so great that on

11 D. A. Kleinman, Phys. Rev. 126, 1977 (1962). 12 R. C. Miller, Phys. Rev. 131,95 (1963). 13 B. Lax, J. G. Mavroides, and D. F. Edwards, Phys. Rev.

Letters 8, 166 (1962). 14 P. D. Maker, R. M. Terhune, and C. M. Savage, Phys.

Rev. Letters 12, 507 (1964). 15 Y. H. Pao and P. M. Rentzepis, J. Chern. Phys. 43, 1281

(1965) .



the average one photon passes through an atomic area in the period of the light wave the harmonic ratio is equal to the fine structure constant. Inserting the values of the physical constants and using 6943 A as a wavelength we have

UdCl1 =2X 10-341 (w) S(w). (15) Equation (15) shows that in order to observe appre-ciable harmonic intensity we must use a wavelength such that Sew) is very large or an exceptionally intense la&er beam. Contemporary technology permits peak pulsed powers outputs in the megawatt to gigawatt range which corresponds to l()2c1Q27 photons/sec. The actual flux 1 (w), the number of photons per square centimeter per second, may be increased considerably by focusing the incident light which permits observable intensities of the harmonic.

III. DEPOLARIZATION OF THE SCATTERED LIGHT

A. Observable Functions of the Components of {3 Consider a small volume of the medium as a source

of dipole radiation. The induced dipole l' is by definition

(16) Assume the initial light beam is incident along the X axis. (These innocent looking a&sumptions restrict

one to phenomena describable in terms of the 18 sym-metric components (3ijk+{3ikj only as is fully explained by Giordmaine.16) The polarization of the light wave will determine the actual quadratic form in the {3's which is observed; but, as we see below, the use of elliptically polarized light is necessary to obtain all the observables.

The inten&ities of single-quantum scattering can be expressed in terms of the principal molecular pol ariz-abilities an, a22, and a33 through the quadratic forms (an2 +a222 +(332) and (ana22+ana3a+a22aa3)' This can be done by measuring the intensities of the two different polarizations of radiation scattered at 90. No matter what polarization properties one uses in the incident light, and no matter at what scattering angle one looks, one can only measure these two quadratic functions of the polarizabilities. For double-quantum scattering one can measure five quadratic forms in the (in general) 18 symmetric components of the polarizability (3ijk.

To prove this theorem assume that the incident light travels in the X direction with an arbitrary polariza-tion given by

t= to{cos~ coswtj+ sin~ cos(wt+15)kl, (17) where ~ and 5 are arbitrary angles. For unpolarized light one must average the final intensities over 5. For circular polarization one sets ~ = t1r and 5 =!1r and for linearly polarized light ~=O. The second-harmonic dipole moment induced by this electric vector is

l'=~: toto = ![{,Bxzz sin2~ cos2(wt+15) +,BXYY COS2~ cos2wt+(,Bxyz+{3xZY) sin~ cos~ cos(2wt+15) Ii + {,Byzz sin2~ cos2 (wt+15) +,Byyy COS2~ cos2wt+ ((3yyz+,ByZY) sin~ cos~ cos (2wt+15) I j

+ {,BZYY COS2~ cos2wt+,Bzzz sin2~ cos2(wt+15) + ((3ZYz+,BzZY) simp co~if; cos (2wt+15) Ik]C;o2

where a static component has been subtracted. An oscillating dipole l' will give rise to a scattered

harmonic wave at a distance R whose electric vector is given by Eq. (4). If we take R to be arbitrary, i.e.,

of the previous vectors the new vectors are

e1 =[I/(I-Az2)1/2]( -Ayi+AXj), e2=[I/(I-A~)1/2][ -AXAZi-AYAzj+(1-AZ2)k],

(18)

(19) ea=R/R=Axi+Ayj+Azk. (21) then performing the cross products we have

(20)

In an experiment one does not measure the magnitude of t along three mutually perpendicular axes but along two axes which are mutually perpendicular and per-pendicular to the direction of observation, R. In terms

Then, in this basis,

(2W)2 1

(1-Az2)1/2 [ -JlXAY+JLYAX]e1

1 + (l-Az2) 1/2 [JlXAXAZ+JLYAYAZ-JlZ(1-AZ2) ]e2

(22)

16 J. A. Giordrnaine, Phys. Rev. 138, A1599 (1965).


3188 BERSOHN, PAO, AND FRISCH

The two independent intensities which can be meas-ured are

and (23)

(24) In these equations the subscript Av refers to a time

average over the period of the incident light wave which is to be carried out first. The brackets refer to an average over a large number of molecules. In this latter average the cross products like J!.xJ!.y which occur have zero average values. This is because the liquid, unlike the individual molecules is isotropic. The aver-aged intensities are then

h = [(2W)4/41TC3J[1/ (1-AZ2) JI (.uX2)A,Ay2+ (.uy2)AVAX2}, (25) (26) 12 = [(2w) 4/41TC3J[1/ (1-Az2) J I (Jl.X2)A,AX2AZ2+ (.uy2)A,Ay2AZ2+ (.uZ2)A, (1-A~) 21,

where

(J!.X2)A,/02=i(j3xzZ2) sin~+i(j3xyy2) cos41/t+i sin21/t cos2!JtfJxyz+fJxzy)2+2fJxzzfJxyy cos2c5), (27a) (.uy2)A,/02=i(j3YZZ2) sin~+t.


TABLE 1. Expectation values of quadratic functions of f3 in space-fixed coordinates expressed as coefficients of quadratic functions of f3 in molecule-fixed coordinates.

a f31 f32 f3. f3. f3. 1'1 1'2 1'3 Il, Il.

2 2 2 1 1 2 1 1

3190 BERSDHN, PAD, AND FRISCH

The sums are carried out for i, j, k =X, Y, Z but i, j, k are never to be equal to each other.

A check on the values of the coefficients in Table I is that if one assumes the molecular tensor is totally symmetric, the first, second, and sixth lines reduce to identical expressions derived by Cyvin, Rauch, and Decius. Also, addition and subtraction of various rows confirms the expressions of the quadratic functions of the space-fixed components of fJ in terms of the a, b, c, d, e terms previously discussed.

C. Depolarization Ratio for Linearly Polarized Light

Inasmuch as we have shown that one does not obtain additional information by looking in any direction but the cla~sic 90 angle, for the case of linearly po-larized light we can evaluate Egs. (25) and (26) with Az=Ax=O, Ay=l and siny,=1. This means the incident light travels in the X direction, is polarized in the Z direction, and the scattered light is observed in the Y direction. Under these conditions

Ix = [(2w) 4/47rc3J(.uX2)Av = [(2w)4/327rc3J(Bxz~), (45) I z = [( 2w) 4/47rc3J(.uZ2)AV = [(2w) 4/327r3J(fJzzZ2), (46) where Ix(z) is the intensity of light polarized in the X (Z) orientation, and the depolarization ratio is

( 47) The general expressions of Eqs. (45) and (46) for

a molecule with no symmetry are contained in the first two lines of Table 1. They may greatly simplified for molecules of rather high symmetry as is shown in Appendix B.

IV. SCATTERING BY A SYSTEM OF MOLECULES; ORIENTATION CORRELATION

In this paper we have derived scattering probabilities for light scattering by isolated molecules in a vacuum. The intensity of light scattered by a real system of many molecules is not just a multiple of the scattering by a single molecule. In the first place internal fields

alter the effective magnitude of the scattering inter-action. Also, there may be coherent scattering in that the partial waves scattered by the various molecules or segments of a given large molecule may interfere. Finally, the observed intensity will depend on the size of the extent of coherence between small volume ele-ments which in turn depends on the degree of mismatch of the indices of refraction at wand 2w.

A. Internal Fields

The effect of the polarization of the neighbors is ::ondensed into the Lorentz factor (e+ 2) /3 which is the ratio of the local electric field to the electric field of the light wave acting in a vacuum; e is of course the dielectric constant at the frequency w. The intensity of single-quantum scattering is increased by the fourth power of [e(w) +2J/3. As pointed out by Armstrong et al.,4 the intensity of double-quantum scattering in which quanta WI and W2 produce the quantum W3 must be multiplied by the square of the product

![e(wl) +2J![e(w2) +2J![e(w3) +2]. The rule is that a Lorentz factor must appear at each vertex of the perturbation diagram to augment the radiation field acting there.

B. Coherent and Incoherent Contributions to the Scattering

The structure factor FJLM(K) of a group of N scattering units can be written as the sum

N FJLM(K) = L!JLMa (W3, W2, WI) exp(iKRa )

a=l

x exp(iKRa). (48) The scattering intensity which is proportional to the square of the structure factor equals

N l(w3) = (167rw34jc5R2)I(wl)I(w2) i{e(wl) +2J2 t[e(w2) +2J2 t{e(w3) +2J2X 1 L fJJLM(W3, W2, WI) exp(iKRa) 12

=KN (BJLM2), where

N

a=1

(49)

(BJLM2) =N-l L I fJJLM(a) 12+N-l LL fJJLM (a) fJJLM (a') exp[iK (Ra, -Ra)] a=1 aa'=l

For a real fluid the sums are equivalent to averaging over an ensemble of randomly oriented configurations. We may characterize the two terms in this equation as incoherent and coherent, respectively. The incoherent terms have already been discussed in Sec. II1.B.

(50)

C. Intermolecular Correlations

In the coherent part of Eq. (50) one needs to average over relative distances and mutual orientations of pairs of molecules. Let g(rl' r2; Ql, Q2) be the two-



particle correlation function, i.e., p2g drldr.ylJhdQ2 is the probability of observing Molecules 1 and 2 in drldr2 at rl, r2 with orientations in dQldQ2 at QI~(p=N IV, the number density). Equation (50) can then be rewritten with r=rl-r2, r=1 rl-r21, and g(rl, r2, QI, Q2) replaced by g(r, QI, Q2)

(flJLM2) = 8~2 1 .BJLM2(1)dQI+P (8~Y ~ 1 dr 1 dQI f dQ2g(r, QI, 02).BJLM(1).BJLM(2) cosKr

One can write a general expansion

g(r, QI, Q2) =! gi(r)hi(QI, Q2), (52) i..()

where the hi(QI, Q2) are suitable orthogonal polynomials in the trigonometric functions of the Euler angles desig-nated by QI, Q2, ho=l and the gi(r) are the expansion coefficients. The normalization of Eq. (52) is given by

(8:2)2111 drdQldOg(r, QI, Q2) -lJ=p 1 dr[go(r) -lJ =pkTK-1, (53)

where K is the compressibility of the fluid. The coherent part of Eq. (51) can be written17 as

the sum of an intensity scattered as a result of the geometry of the surface of the matter plus a scattering from the density fluctuations. The two terms are ob-tained by substituting

in the second term of Eq. (51). Omitting further dis-cussion of the surface scattering we are left with the coherent scattering which is caused by the microscopic inhomogeneities involving correlations in orientation of molecules:

On inserting the expansion of g, Eq. (52) into Eq. (55), we find

100 sinKr 00 27rp drr2 -- L: gi(r) 8 i (J LM), o Kr i==1 (56) where

8i(JLM)

=(8~2r f dQI f dQ2hi(01, Q2).BJLM(1).BJLM(2). (57)

17 J. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954).

(51)

In Eq. (56) the summation excludes the term i=O because there is no contribution to the incoherent scat-tering when g(r, QI, Q2) is replaced by go(r). The co-herent part of the two-photon scattering vanishes if there are no correlations in orientations. This is a sig-nificant difference from the case of ordinary Rayleigh scattering. The reason is just that (averaged over all orientations)

(.BJLM+.BJML) =0, (58) (aJL) = t8JL (ax.,+ayu+a .. ) . (59)

D. Approximate Treatment of geT, QI, Q2) The general expansion of Eq. (52) for g(r, QI, O2)

is rather formidable and an approximate treatment is sufficient at this early stage of consideration of the problem. In the absence of phase transitions (liquid-crystal formation), one expects correlations in orienta-tions to occur only between adjacent molecules in a liquid. The range of all gi(r) is therefore small, say of the order of the position R2 of the next-nearest peak of go(r). In view of the absence of reliable information on g(r, 01, Q2) or its expansion, one can obtain an order of magnitude estimate by replacing the series, Eq. (52) in hi by a single term and approximating the coefficient gi(r) by the values

gi(r) =0 gi(r) =1 (60)

For small molecules we may replace the factor sinKr/Kr in the integrand of Eq. (56) by unity and with the above substitution for gi(r) the integral in (56) becomes

(2/7r)p!-rr(R23_R I3)8 i (JLM). (61) The interpretation is that RI is the distance of closest approach of typical molecules in the liquid and that beyond this distance the orientation of the molecules are correlated ai> far out as the distance R2 The quantity !-rr(R23_RI3)p can be replaced by z, the effective number of nearest neighbors in the fluid.

E. Application to Coo. Molecules

While otherwise a priori arbitrary, the function h;(Ql , Q2) must be invariant under the symmetry opera-



tions of the two molecules. We now consider the dis-tribution function appropriate for molecules of sym-metry Coo" The treatment holds precisely for linear molecules without a center of symmetry and perhaps adequately for any symmetric top. To obtain a pos-sible form of the function hi(Ql, Q2) let us consider the following heuristic argument. Suppose that the orient-ing intermolecular force were the interaction between the static dipoles of the linear ( or symmetric-top) molecules. The interaction energy between two such dipoles is17

In a dilute system where the distribution function is only determined by pair interactions, we have

The form of the potential energy can thus be used as a

guide to the correlation function. Thus we may write hi(Qt,~) =t(COS012-3 COsOt COs(2)

= t (sinOt sin02 cOS!/>t COS!/>2 + sinOt sin02 sin!/>t sin!/>2 - 2 COsOt COs(2)

=E(aXZlaXZ2+aYzlaYz2-2azZlaZZ2)' (64) where E should be ""JJ.2/Rt8kT if the dipole orienting force determined the distribution function.

The direction cosines aZZ1aZZ2 appear with a different coefficient than the sum aXZlaXZ2+aYzlaYz2' Had we assumed that the orientation distribution was a func-tion only of the angle 012 between the symmetry axes, then only

COsOt2 =aXZlaXZ2+aYzlaYz2+aZZlazZ2 (65) would have appeared. It is reasonable therefore to take as a general form

h(Qt, Q2) =tt(axZ1aX .. +aYz1aYz2) +t2azZlaZZ2' (66) One can now write for any molecule, regardless of

its space group, whose liquid has a distribution func-tion of this form

8 t (XZZ) =El [8~ f dQlaXx/Jx .. (1) r = ({3xZ~)ooh, 8 t (ZZZ) =E2 [8~2 f dQlazz1{3 ... (1) r = ({3zzZ2)ooh.

(67)

(68)

Substituting the transformations

(3xzz(l) =axolazxl{3zxx+aXZlaZlIlz{3zuU+aXz13{3zzz+aXXlaZX,aZZl ({3xzx+{3xxz) +aXII,aZlI,aZZ' ((3l1zv+{3w.) + .. " (69) (3ZZZ( 1) = azz,3{3 ... +azZ,azX12({3zn+{3xzx+{3x"z) +az"az1I1z ((3zw+{3YZII+{3wz) +. . . (70)

(omitted terms make no contributions to the integrals) and integrating, one finds 8z(ZZZ) = t{i{3 ... +I"5" ({3zu+{3xz.'+{3xxz+{3zW+{3l1Zll+{3WZ) J2, (71)

(72) 8 t (XZZ) =tl[-h{3 ... +h({3zxx+{3zw) --io({3xxz+{3xzx+{3VlIZ+{3l1ZII) J2. For linear molecules (Coov) the only nonvanishing components16 of {3 are 13m, {3zxx = {3ZYY, and {3xzx+{3xxz = (3l1zlI+{3w;

Inspection of Table I and Eqs. (34)-(44) shows that


nave no 7r*' state to go to except one constructed of 3fJ7r orbitals whose energy would be prohibitively high. Therefore, we expect fJ ... to be very small; fJxx. and its permutations would be determined by a u~*~

u*~u sequence of transitions corresponding to a nitro-gen lone-pair transition to an antibonding 7r state, to an antibonding u state, and then back to the ground state, and therefore fJxxz, fJxzx, and flz"", would all be of substantial size. If we crudely assume fJzzz""'O, 2fJzx::::::' flxzx+fl",xz =fl, then

and the depolarization ratio is

(75)

(76)

(77)

A check on this simplified theory would be possible by studying the depolarization ratio in the limit when the coherent contribution vanishes. Practically this could be done in two ways. The harmonically generating liquid can be mixed with a liquid composed of centro-symmetric molecules, e.g., benzene, CS2, liquid SF6, or liquid Xe. The depolarization ratio in the dilute limit is, of course, the incoherent contribution. Alternatively the pure liquid could be studied as a function of tem-perature; in the high-temperature limit, correlation effects should disappear.

F. Application to Tetrahedral Molecules

For tetrahedral (Td ) molecules the only nonvanish-ing elements16 of the fl tensor are flX1Jz=flxzy=flyzx= flzxy=flzyx=fl. By inspection of Eqs. (43) and (44) and Table I, one sees that the incoherent contributions to (flzzz'l) and (fJxzZ2) are -Hfl2 and :/r;f32, respectively. To calculate the coherent contribution we need to adopt a correlation function hi(flt, Q2)' Because of the nature of the nonvanishing fliik we must use a function which contains each of the coordinates x, y, z at least once and symmetrically. The simplest possible func-tions which fulfill this requirement are

2

Ii II (aXxiaYYiaZ .. +aXYiaYziaZX,+axz,aYx,aZYi i=l

and

2 2 2 E( II aXXiaXlliaX .. + II aYxiaYlliaYzi+ II aZXiaZlliaZ.,).

i=-l i=l i-I

(79)

To obtain the coherent contributions to the scattering,

we consider the transformations

fJzzz(i) =6fJazxiaZy,aZ." flxzz (i) = 2fl (aXXiazlliaZZ, +aXl/iaZXiaZ .. +ax .. azx,azl/') .

(80) From the form of these transformations it is clear that the function, Eg. (78) cannot contribute to coherent scattering and it is therefore not considered. The second function makes the following contributions:

(f3zzZ2)ooh = E{32 ( 6 (adazI/2ad) )2=rh-r;Ef32. (82) The depolarization ratio for tetrahedral molecules

is defined as

(f3XZZ2) (flXZz'l)inooh + (f3xzZ2)ooh (f3zzz'l) (f3ZZz'l)inooh+ (f3zZz'l)coh

:/r;f32+rh-r;E{32 2+rhE -Hf32+n\-r;Ef32 3+-hE'

(83)

This result is, from the experimental point of view, perhaps the most significant one of this paper. The reason is that the absolute intensity of the harmonic scattered radiation is very difficult to measure ac-curately. The ratio of two intensities, the depolariza-tion ratio is much easier to measure. For molecules of most space groups this ratio involves a fairly large number of independent components of fJ; it is therefore difficult to draw much of a conclusion from the result. But in the case of tetrahedral and D3k molecules the ratio is independent of the values of the fJ tensor. Terhune et al.5 observed a depolarization ratio of "'i for CCI4 Comparing this value with Eg. (83) we draw the following conclusions:

(a) In CCl4 E is positive, which means that neigh-boring molecules tend to orient themselves so that they are identically oriented. This is just the configura-tion which would cause immediately adjacent CI atoms to avoid each other and is therefore reasonable.

(b) The orienting potential which determines E is comparable with or greater than kT. Had it been weaker, the depolarization ratio would have been closer to i.

From this single datum it is perhaps unwise to draw very broad conclusions. If the theory just given is generally right however it would be of interest to study the depolarization ratio in the limit when orientational correlation disappears (d. preceding section). The depolarization ratio in a benzene-CCI. mixture should



1.0

0.8

0.6

Q

0.4

0.2

10 0

0 2 1 3 4 U2

FIG. 2. Angular distribution of double-quantum scattering by a macromolecule in random-coil conformation. Abscissa u l is a measure of differences in refractive indices as well as k vectors. Coil dimension also enters as a parameter [Eq. (88) of text].

rather speedily approach j with decreasing mole frac-tion of CCl4

G. Application to Macromolecules

In what way can double-quantum light scattering add to our knowledge of macromolecules? In a previous paper one of us6 discussed this question and arrived at the conclusion that except for the facility of being able to look at zero angle (which is not at K =0 for double-quantum scattering) there is no additional in-formation available from double-quantum scattering over single-quantum scattering. This is true if and only if the scattering elements are oriented the same way throughout the molecule or at least in some specific pattern along the molecule. When this is not the case, the coherent contribution to the scattered light for double-quantum scattering will be different from that of single-quantum scattering.

In solutions sufficiently dilute so that interference occurs only between segments of the same molecule, the function g(r, nl, n2) of Eq. (52) refers to the inter-segmental pair-distribution function. The function g(rih ni , nJ ) describes in part the structure of the polymer, being the joint probability that two segments separated by a distance rij have the orientation ni , nj .

Of particular interest are two types of polymers, one being the synthetic linear polymers consisting of a chain of N repeating units which while chemically identical have different tacticity. By tacticity we mean that each unit can, in principle, exist in two states of orien-tation. These are analogous to a one-dimensional Ising lattice comprised of a set of spins with two possible orientations. And the second type being the poly-nucleotides in which successive bases are stacked par-allel to each other. The function g is a measure of the

range of this stacking. For a double helix of poly-nucleotides a one-dimensional Ising model can be used to describe the states of hydrogen bonding between the bases and thus the over-all structure.

The function which we use for the segments of the macromolecules is

where K is the reciprocal of an orientation correlation length. The explanation of this simple function is as follows: The delta function arises from the fact that nearby configurations are strongly correlated. The exponential function is rigorously obtained for a one-dimensional Ising model in which only nearest-neighbor orienting forces are considered. Furthermore, in general, it appears that the exponential decay of a correlation function reflects the short-range character of the inter-molecular forces.Is

The structure function used for macromolecules is

() 1 "N" sinkr ij P (j =- L...iL...i--N2 ir'i=l krij ,

(85)

where k= (2-n/A) sin(j and P((j) is defined by the equa-tion

J((j) =J(OH(1+ cos2(j)P((j). (86) J((j) is the intensity of scattering of (unpolarized) radiation through an angle (j. For any assumed model for the structure of a polymer the function P((j) can be calculated and compared with experiment. 19 In

1.0 ........ :--------------------,

0.8

0.6

Q

0.4

0.2

10

0 0 2 3 4 5

x

FIG. 3. Angular distribution of double quantum scattering by a thin rod. Abscissa x is a measure of differences in refractive indices as well as k vectors [Eq. (87) of text].

18 M. E. Fisher, in The Equilibrium Theory of Classical Fluids, H. L. Frisch and J. L. Lebowitz, Eds. (W. A. Benjamin, Inc., New York, 1965), p. III-86.

19 B. H. Zimm, R. S. Stein, and P. Doty, Polymer Bull. 1,90 (1945) .



particular, for a thin rod

() 1 j2X sinwdw (sinX)2 PO =- ----X 0 w X'

where X =~kL, and for a random coil

where U=tk2R2.

(87)

(88)

In the present problem we require a modified struc-ture function

1 N sinkr Q(O; K) =2 2:2: --" exp( -Krij). N i>"j krij

For a thin rod

1 12X sinw (VW) Q(k; v) =- -exp -- dw X 0 w X

l e-' sinhv cos2X X 1+ . 2X sm where V=~KL.

For a random coil

2 j1 (l-y) Q(k;v)=-( )1/2 -1-'2- dy

7rU 0 Y

v cotXe-2.] X '

(89)

(90)

xjOO exp [-(~~+~)]{cosz-_v SinZ}dZ, (91) o 4 uy U 1/2 U 1/2

where v=KR/61/2 This integral can be evaluated in terms of error functions of complex arguments but the resulting cumbersome expression is harder to program than the double integral above which was evaluated directly. The numerical results for these two functions are shown in Figs. 2 and 3, respectively.

The three values of v-O, 1, lO-for which the curves were drawn correspond, respectively, to a correlation length l/K which is OC), comparable with the molecular size, and .......,lo of this size. The loss of angular correla-tion with distance leads to a general decrease of the coherent contribution to the scattering. The angular dependence becomes steeper the longer the correlation length.

Experimentally it may be difficult to measure the absolute intensity of the scattered light. The absolute intensity is of interest only in the determination of the absolute values of the /3 tensor. What is far easier to determine is the angular dependence of the light scat-tering. The latter depends for single quantum scatter-ing on the shape and characteristic size of the molecule. For double-quantum scattering it depends on the size and shape and on the orientation correlation length.

kKm IVn Wo -

. k'

k'

1/10

(al (h) FIG. 4. Feynman diagrams for the scattering of a single light

quantum. In (a) the scattered quantum is emitted after the in-cident quantum is absorbed. In (b) the reverse is true.

It is perhaps worth repeating6 the point that for single-quantum scattering at a frequency w

k= (47r/Ao)n(w) sin~O, (92) where Ao is the vacuum wavelength, 27rc/w, but that for double-quantum scattering at a frequency 2w

k=[47r/(2Ao)](n(2w)n(w) sint9 + a[n(w) -n(2w) ]}2)1/2. (93)

In other words, to calculate k at a given scattering angle 0, one must know the indices of refraction of the solvent at both the incident and the scattered wave-lengths.

ACKNOWLEDGMENTS

R. B. would like to acknowledge the support of the U. S. Public Health Service Grant CA-07712-02 as well as the hospitality of the Bell Telephone Laboratories where part of this work was done. We would also like to thank R. L. Kornegay for his able numerical com-putations and Paul Maker for helping to correct a serious error.

APPENDIX A: DERIVATION OF THE TWO-PHOTON SCATTERING AMPLITUDE

1. Matrix Elements of the Matter-Radiation Interaction

Use of a quantized radiation field20 permits a simple picture of the light-scattering phenomena on a mo-lecular level. To make the formalism perfectly clear we first discuss ordinary scattering. To begin with, our system is imagined to consist of a single molecule and an interacting radiation field enclosed within the walls of a large box of volume V. The Hamiltonian can be written

JC =JCmatter+JCradiation +JCinteraction , (A1) where

JCinteraction = t {_.!2.. PrA(rj) +2 e; 2 A2(rj )} j=l mjC mjC

=JCint(l) +JCint (2). (A2) 20 W. Heitler, Quantum Theory of Radiation (Clarendon Press,

Oxford, England, 1954).



The sum over j extends over the N charged particles of the molecule. The zeroth-order eigenfunctions of the system (in the absence of interaction) are of the form

!/ta (fl' "', fN) I nkA, nk'A', nk"X", ). (A3) The occupation numbers nkA are the number of quanta with wave vector k and polarization A present in the box.

The scattering of photons can be described by tran-sitions in which the material system in a state !/to ab-sorbs the incident quantum k, is virtually excited, and

then emits the scattered quantum k' and is virtually de-excited to its final state !/tn. Such a process may be pictured by the Feynman diagram of Fig. 4(a). The emission may precede the absorption in which case the process is represented by Fig. 4(b). The prob-ability or cross section for the scattering transition is measured by the rate at which the amplitude for the final state builds up in the over-all wavefunction of the system. This amplitude is computed from time-de-pendent perturbation theory which in turn necessitates the calculation of the matrix elements of x'int:

(A4a)

(A4b)

x'int(2) which contains two A operators may annihilate two quanta, create two quanta or annihilate one quantum and create another. Here we include only the latter sort of matrix element

(A4c)

All the calculations which depend explicitly on the quantized field operators are condensed in the matrix ele-ments (A4a), (A4b), and (A4c). Suppose that the scattering system is constructed of m units (each of which contains na charged particles) which can be regarded as independent of each other. Within each unit the expo-nentials can be regarded as constants so that the matrix elements can now be written

(ASb)

(ASc)

R.. is the vector from an arbitrary origin to a fixed point in the scattering unit a. The vectors fj (for each a) ex-tend from this fixed point to the instantaneous electron positions.

2. Ordinary Scattering Cross Sections in Terms of Structure Factors of Scattering Units

The form of the matrix elements shows that the scattering transition is accomplished by the use of x'int(1) twice or x'lnt(2) once. Using second-order and first-order perturbation theory, we find that the transition probability per unit time is given by

(A6) m

where p,. is the density of final states n per unit energy per unit volume. The latter are characterized by a distribu-


tiGHT SCATTERiNG BY MOLECULES 3197

tion in the angle 8, cp of the scattered photon of wave vector k' and polarization A' relative to the incident photon characterized by k, A

(A7)

where dn is the solid angle within which the vector k' lies and w' is the frequency of the scattered quantum. Substituting the appropriate matrix elements of the form of Eq. (2) we have

(AS)

where K=k'-k. Our initial conditions correspond to nk'>" =0 and nk>'~O. If we use the abbreviation

(A9)

and substitute the initial conditions, our formula reduces to

The probability of a scattering event occurring per unit time is just the differential scattering cross section times the incident flux. As the incident flux is just nk>.cjV, we can write

du Wn+-O dn dnnn(cjV)' (All)

The resulting expression for the cross section con-tains all ordinary one-photon scattering phenomena, elastic and inelastic, and at all photon energies below relativistic energies. In the next section we derive an analogous formula for the scattering of two quanta.

3. Harmonic-Scattering Cross Sections

In single-quantum scattering there are two (2!) Feynman diagrams corresponding to the two possible ~equences of emission and absorption. The harmonic scattering is described by the annihilation of two in-cident photons of frequency WI, W2 and polarization AI, A2 with the creation of a photon with frequency Wa and polarization As. There are six possible Feynman dia-grams (d. Fig. 5) for this process corresponding to the six different sequences of annihilation of two quanta and creation of a third quantum. To each diagram there corresponds a different set of matrix elements

(AlO)

which means that in the sum _ 271' 1 ~ [Xint(1) ]nq[Xint(1) ]qp[Xint(1) ]pO 12 ~+-O- ~ ~ Ii m,p (Eo-Eq) (Eo-Ep)

(A12) there will be six terms. [The interaction energy JCint(2) does not contribute to the elastic harmonic scattering because its matrix elements are all diagonal in the electronic energy and Xint(l), with which it would have to be combined, has matrix elements which are only off diagonal in the electronic energy. As we omit discussion of the Raman effect we omit further men-tion of these terms.]

~k3 1/10 !flo (/Iq (/Ir - -k I k. k~ 1/10 k 3 (/Iq (/I "2 (110 k2

FIG. 5. FeY1l111an diagrams for elastic double-quantum scatter-ing. Three more diagrams are obtained by permitting Wl and W2.



If we consider only the case of elastic scattering where the material system does not change its state, we set W3=W1+W2 in Eq. (A12), use Eq. (A7) for Pn and obtain

211" I" ('K 'D) {e2 ""[ (p>'.t'")oq(h.")qr(h,")rfJ WkaXl .... k.}.2.kl}.I=~ 7 exp ~ '''~ m1i 7~ (Eo-Eq+fu.J1) (Eo-Er+fu.J1+fu.J2)

+ (h.")Oq(hl") qr(h,")rfJ + (hl")Oq(h,") qr(ht)rfJ + (h.")Oq(h3") qr(hl")rfJ (Eo- E q+fu.J2) (Eo- Er+fu.J1+fu.J2) (Eo- E q+fu.J1) (Eo- Er -fu.J2) (Eo- E q+fu.J2) (Eo- Er -fu.J1)

+ (P).t)Oq(hl") qr(h.a)rfJ + (h,a)Oq(ht) qr(ht)rO ]} [2 (Eo- E q -fu.J1-fu.J2) (Eo- Er -fu.J2) (Eo- Eq-fu.J1-li(2) (Eo- Er -fu.J1)

X (211"IiC)3 (~) nklnk.(nk.+1) d~32V (~)3. (A13) V e4 k1k2k3m Ii (211"c) 3 me2

If we adopt the initial condition nk.=O and replace the bulky expression in the curly brackets by the dimensionless snttering amplitude/a (2) and let (nk;c/V) =1(w.), we find

(A14)

APPENDIX B: INCOHERENT CONTRIBUTIONS FROM MOLECULES OF DIFFERENT POINT GROUPS

The general and complex formulas of Table I and Eqs. (34)-(44) expressing the incoherent contributions to the scattering in terms of components (3ijk in the molecular framework simplify if the molecule is symmetrical. In the text the intensities are given for the groups Ta and Coo . In this appendix, results applicable to linearly po-larized light for D3h , C3., C2., and C3 are given. The components (3ijk have been re-expressed in terms of the com ponents Sij where16

S;j=(3,jj for j = 1, 2, 3, S i4 =(3i23+(3i32, SiS = (3.13+(3i31, S16=(3,12+(3i21 .

Individual Groups D3,.

/R 2)_IS 2 \/-,zzz -""{ 16, /R 2)_2S2 \/-'x .. -TI16

Coo. (ftm2) =-ts332+-Js-S33( S31 +S15) +rh( S31+S15) 2, (ftx .. 2 ) = -Iss3l+As312+-hS152 -rh-S31S15+rh-S31S33 --s\S3SS15 .

Cs. (ftm2) = t S332+-s\S162+rh ( S31 +S15) 2+-Js-S33(S31 +S15) , (3x .. 2 ) = -hs3l+rl-sS162+As312 +rhS31S33 +-IS-S152 --s\S33S15 -mS31S 15 .

C2 (ft ... 2) =tS332+ls-(S31+S15)2+-h(S32+S24) 2+-Is S33 (S31+S15) +-s\S33(S32+S24) +rh(S31+S15) (S32+S24), (ftx .. 2) =-hS332+A (S312+S322) +rhS33 (S31 +S32) +-s\S31SS2+rhS152+S242 -rhS15S24

--hSSS(S24+S15) --h(SSlS15+SS2S24) -rh(S31S24+S32S15).

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