TEMPERATURE DEPENDENCE OF DEPOLARIZED SCATTERED LIGHT NEAR THE CRITICAL POINT

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TEMPERATURE DEPENDENCE OF DEPOLARIZED SCATTERED LIGHT NEAR THE CRITICAL POINT

R. Mountain

To cite this version:

R. Mountain. TEMPERATURE DEPENDENCE OF DEPOLARIZED SCATTERED LIGHT NEAR THE CRITICAL POINT. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-265-C1-268.

�10.1051/jphyscol:1972145�. �jpa-00214935�

JOURNAL DE PHYSIQUE

Colloque C l , suppl6mer~t uu

no

2-3, Tome 33, Fkvrier-Mars 1972, page C1-265

TEMPERATURE DEPENDENCE OF DEPOLARIZED SCATTERED LIGHT NEAR THE CRITICAL POINT

R. D. MOUNTAIN

National Bureau of Standards Institute for Basic Standards Washington, D. C. 20234

R6surn6.

-

Le formalisme de double diffusion de Frish et McKenna est utilise pour etudier en fonction de la temperature la lumikre depolarisee au voisinage du point critique d'un fluide

a

un constituant composk de molkules spheriques. Les termes decrivant les correlations entre deux particules sont estimQ en utilisant la fonction de correlation d'ornstein-Zernike. On trouve que le rapport de I'intensite de la diffusion

H-H a

celle de la diffusion

V-V

diverge comme

k5

In

(llkc).

La possibilite d'utiliser la diffusion depolarisee comme moyen d'exploration des corr6lations entre trois et quatre particules pres du point critique est discutee.

Abstract. - The double scattering formalism of Frish and McKenna is used to investigate the temperature dependence of the depolarized scattering in the vicinity of the critical point of a one component fluid composed of spherical molecules. The terms involving two particle

correlations

are estimated using the Ornstein-Zernike correlation function. The ratio of the intensity of the

H-H

to

^V-V

scattering is found to diverge as

^{k t}

In

( l l k t ) .

The possibility of using depolarized scattering as a probe of three and four particle correlations near the critical point is discussed.

Introduction.

- The two particle correlation func- tion introduced by Ornstein and Zernike [l] to charac- terize critical opalescence has proven to be a very useful concept in the study of the fluid state. The sta- tistical mechanical theory of the pair correlation func- tion has lead to the beginnings of a reasonably success- ful theory of liquids [2]. In this theory one relates the pair function to the interatomic potential and to the function describing the correlation among three particles. In turn the thermodynamic properties are simply related to these microscopic quantities. The two particle function can be directly probed by means of light, X-ray and neutron scattering and much work has been done. It would be of value to have available a means of experimentally studying the three particle correlation function since this important quantity is not at all well characterized.

It is in principle possible to investigate three and four particle correlations in fluids composed of spherically symmetric particles by studying the depo- larized part of light scattered by density fluctuations in the fluid. This is seen by a brief examination of the formalism of double scattering [3-81. It is not, however, immediately evident whether such an inves- tigation can be usefully carried out.

The purpose of this paper is to explore the possibility of learning something about the three and/or four body correlations in a fluid in the vicinity of the critical point by measuring the intensity of the depolarized scattering. In the analysis of this topic we use an exten- sion of the double scattering formalism of Frisch and

McKenna [6] in which the terms involving two, three and four body correlations are explicitly exhibi- ted. The approach we use is to estimate the dominant terms involving pair correlations by employing the Ornstein-Zernike form for the pair correlation function.

If any excess scattering should be observed, it would be attributed to the three and four body correlation func- tion terms.

The extended version of the formalism of Frisch and McKenna is described in the next section. The configurations which lead to depolarized scattering are indicated by diagrams which represent the struc- ture of the terms. The term involving two body correlations which dominates near the critical point is investigated for H-H scattering (the polarization vectors of the incident and scattered light are in the same plane) in the third section. The intensity of H-H scattering is found to be proportional to kt;ln(l/kc) where k is the wave-vector of the light used in the experiment, and 5 is the prnstein-Zernike correlation length. Thus, if this way of studying higher order correlation functions is to be useful in the critical region, the intensity of H-H scattering really should increase more rapidly than ktln(llk5).

Formal Considerations.

- The physical process of double scattering is easily visualized. The light incident on a sample of fluid polarizes molecules.

The dipole field of the polarized molecules then pola- rizes other molecules. I t is the radiation from the

u

other molecules

>>

which is referred to as doubly

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972145

C1-266 R. D. MOUNTAIN

scattered light. Constructive interference requires

some sort of correlation between the ((other mole- cules

⁾⁾

since the fluid is uniform on the average.

The required correlation can involve sets of two, three or four particles as has been shown by numerous authors who have studied the problem of double scattering. In this paper we follow the formal develop- ment of Frisch and McKenna. These authors solve Maxwell's equations recursively, following a procedure described by Landau and Lifshitz [9].

In their analysis, Frisch and McKenna made use of two, three and four body distribution functions.

Distribution functions are not, as those authors were careful to mention, the appropriate functions for doing explicit calculations. Rather, the distribution functions should be expressed in terms of UrselI functions since these functions go to zero when any pair of the particles in question become widely separated [lo]. This property yields finite results when the integrals involved are evaluated, whereas the distribution function forms yield apparently (but not actually) divergent results. Put another way, the Ursell function formulation automatically cancels what appear to be divergences.

The resulting integrals may be conveniently repre- sented by diagrams of the type shown in figure 1.

For a scattering angle of 900, the intensity of the

HH scattered light is expressed in terms of the fully

polarized scattered light as

Here

a

is the scalar polarizitbility of the particles.

The term in the denominator is proportional to the ordinary structure factor for single scattering, namely

The radial distribution function is g2(r). The number density is

p,

and the volume of the fluid is V. The quantity < 1 TI, j 2 > is more complicated but can effectively be expressed as

where F is indicated diagranlmatically in figure 2.

For HH scattering the dipole field term

(-)

has the form

FIG. 2.

-

The quantity F i n eq. (3) in terms of diagrams. The ones with a single bond involve the two particle correlation function. The triangle and rectangle represent three and four

particle correlations.

The explicit form of F is obtained by reexpressing the formal results of Frisch and McKenna in terms

FIG. 1. - ~ ~diagrams which represent theintegrals ~ i ~ ~ l occur-

of Ursell functions rather than distribution functions.

ing in the expression for the intensity of the depolarized light.

Several of the resulting terms vanish identically due to symmetry considerations. Th,e surviving terms are

A dot (0)

represents the particle coordinates which indicated in figure

2.

are integrated. A wiggly line

^(a)

represents the dipole field induced by the applied field

(-+ ^0).

The scat- tered field is represented as an outgoing arrow

(.+).

The two, three and four body Ursell functions are represented by solid lines, triangles and rectangles respectively.

The Critical Region. -

The two particle terms are the familiar results discussetl by Buckingham and Stephen [3]. There is no anomaly arising from these terms in the IH,/Ivv ratio. The second diagram represents the integral

The coordinate system is defined in terms of the k4 A exp(- r / a incident light which is taken to be linearly polarized. f dr eiq-'p ( x ~ ) ~

r

(5)

The light is polarized along the x-axis and the incident

light propagates along the positive z-axis. In what where the Ornstein-Zernike form for the two particle follows we consider the HH scattered light, that is the Ursell function has been introduced and only the scattered light which propagates along the x-axis leading term in D(r), the dipole field, has been retained.

and is polarized along the z-axis. Tbe change in the wave-Vector of the light upon

TEMPERATURE DEPENDENCE OF DEPOLARIZED SCATTERED LIGHT NEAR THE CRITICAL POINT C1-267

scattering is q. As the critical point is approached,

5, the two particle correlation length, increases as

(

T -

T,

I-' where

v

- 0.64. On dimensional grounds this integral is proportional t o t o . On the other hand the denominator < I To 1' > is proportional to

t2. Thus no singularity occurs in the ratio IHH/Ivv from the p2 terms in F.

Next we consider the three and four particle terms involving the pair correlation function. Dimensional analysis indicates a (k5l3 divergence for the three particle term and a t(kc)4 divergence for the four particle term. Dimensional analysis is not sufficient as we shall see, to determine how the three particle term behaves. Also, near the critical point we cannot use the long wavelength limit since kc is typically on the order of 0.01 to 0.1. More detailed analysis reveals that the three particle term is the important one in this range of k t .

The analysis of the three particle integral can be performed by expanding the integrand in terms of spherical Bessel functions and spherical harmonics [12].

In tlus way the angular integrations are straight- forward. The price of this simplicity is an infinite series of considerable complexity. However, for H-H scattering it is possible to extract the significant dependence on the correlation length. Similar results are found for V H scattering. The analysis of the first term in the series is outlined in the appendix. The result is that for kc < 0.1 one finds

=

C k t In- 1

Ivv

k t -

When this expression is plotted on log-log paper and the temperature dependence of 5 is taken into account, one finds, to a good approximation

where a

w

1. This is in very good agreement with some preliminary measurements made by Swinney on xenon [13]. The coefficient C i s to be determined by summing the infinite series. Work is in progress on the summation problem. Until this is completed we

cannot determine whether or not the term involving the two particle correlation function accounts for the observed intensity of the depolarized scattering.

Summary. -

The possibility of using the intensity of the depolarized component of light scattered by density fluctuations to probe the three particle corre- lation function of a fluid in the critical region was studied theoretically.

A formal expression for the

depolarized scattering involving correlations among two, three and four particles was developed and the terms involving the two particle correlation function were analyzed. It was found, considering only two particle correlations, that the ratio of the intensity of the depolarized to polarized scattered light goes as ktln ( l / k < ) in the region which is readily accessible experimentally.

The temperature dependence of this ratio is consis- tent with the preliminary measurements of Swinney.

However, since the numerical coefficient of the divergent term has not been determined, no firm conclusion on the usefulness of this method for probing the three particle correlation function in the critical region can be reached.

Numerous discussions with Martin J. Cooper on this topic have proven to be quite valuable.

Appendix. -

The three body term in F which involves the pair correlation function is the integral

where k is the wave vector of the scattered light, g2(r) is the radial distribution function and D(r) is given by eq. (4). For critical scattering we use the Ornstein-Zernike form

The integrand of F3 is expanded in terms of spherical Bessel functions and spherical harmonics. The first

term in the series, f3, is

Here we have introduced X

=

k t and scaled r , , first kind respectively and the last is a spherical Bessel and r13 by 5. The functions k2(x), i2(x) and

j,(x)

function of the first kind. The terminology of ref. [I 11 are spherical Bessel functions. The first two are isused here.

modified spherical Bessel functions of the third and The evaluation of f3 is straightforward but tedious

18

Cl-268 R. D. MOUNTAIN

and the full expression is not particularly transparent. So far, no general form for the proportionality The dominant part of ^f3 for k t

< 0.1

is coefficient has been found. As. a result the series has

16

z2 ~ k ( k 5 ) ~

In @ / k t )

not been summed to obtain the coefficient of the

f 3 X

5

.

(A.4) (kt)3 In (Ilk<) term in

F,.

When

F,

is divided

by

the

c2

^{factor in}

^I,,

^obtains

I t is apparent from the structure of the integrands

of the other terms in the series representation of

F3

_{i ~ . k(1n-.}

1

that each term will have a part proportional t o f3. Ivv kc ( A . 5 )

References

[I] ORNSTEIN (L. S.) and ZERNIKE (F.), Proc. Acad. Sci., Amsterdam, 1914, 17,793.

[2] M ~ S T E R (A.), Statistical Thermodynamics. Vol. 1.

(Springer-Verlag, Berlin, 1969), chap. 5.

[3] BUCK~NGHAM (A. D.) and STEPHEN (M. J.), Trans.

Faraday Soc., 1957,53,884.

[41 THIBEAU (M.), OKSENGORN (B.) and VODAR (B.), J. Physique, Paris, 1968,29,287.

[5] K~ELICH (S.), Acta Phys. Polon., 1960, 19, 711.

[6] FRISCH (H. L.) and MCKENNA (J.), Phys. Rev., 1965, 139, A68.

[7] THEIMER (0.) and PAUL (R.), J. Chem. Phys., 1965, 42,2508.

[8] TANAKA (M.), Prog. Theoret. Phys., 1968, 40, 975.

[9] LANDAU (L. D.) and LIFSHITZ (E. M.), Electrodyna- mics of Continuous Media (Pergamon Press, Inc., New York, 1960), pp. 377-383.

[lo] LEBOWITZ (J. L.) and PER.CUS (J. K.), Phys. Rev., 1961,122,1675.

1111

and book

of Mathematical Functions AMS 55, ed Abramowitz (M.) and Stegun (I. A.) (National Bureau of Standards Washington, D. C. 1964), chap. 10.

[12] EDMONDS (A. R.), Angular Momentum in Quantum Mechanics (Princeton Universitv Press. Prin- ceton, N. ~.,'1960, chap. 2.

[13] SWINNEY (H. L.), private communication.