CRITICAL POINTSTHE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT

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CRITICAL POINTSTHE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT

H. Swinney, Dominique Henry, H. Cummins

To cite this version:

H. Swinney, Dominique Henry, H. Cummins. CRITICAL POINTSTHE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT. Journal de Physique Colloques, 1972, 33 (C1), pp.C1- 81-C1-90. �10.1051/jphyscol:1972116�. �jpa-00214906�

CRITICAL POINTS

THE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT (*) H. L. SWINNEY, D, L. HENRY (**) and H. Z. CUMMINS (+)

Physics Department, The Johns Hopkins University, Baltimore, Maryland 21218, U. S . A.

R&um6. - La largeur de la raie Rayleigh du Xenon a kt6 rkinterpretee en utilisant de rkcentes mesures thermodynamiques permettant de calculer la partie non singuliere de la largeur de la raie.

Pour la contribution critique, les valeurs ainsi deduites sont en excellent accord avec Yequation de Kawasaki. De plus l'exposant @, qui dkrit la dkpendance asymptotique de la diffusion thermique ,y

par rapport B la temperature rkduite, a une valeur de 0,64. Cette valeur est en bon accord avec celle prevue par Kadanoff-Swift-Kawasaki x

-

Des modifications possibles de l'bquation de Kawasaki, obtenues en utilisant pour la fonction de corrdation une forme differente de celle de Ornstein-Zernike, sont dlscutees ; on montre qu'elles sont faibles.

Abstract. - Rayleigh linewidth measurements of xenon are reinterpreted with the help of recent thermodynamic measurements which permit the background contributions to the linewidth to be computed. The resultant values for the critical contribution to the linewidth are shown to be in excellent agreement with the Kawasaki equation. Furthermore, the exponent @ describing the asymptotic dependence of the thermal diffusivity x on the reduced temperature is found to have a value of 0.64 in good agreement with the Kadanoff-Swift-Kawasaki prediction x t-1 where 5 is the range of the density-density correlation function. This result is in marked contrast to the previous result ^{@ =} 0.75 obtained from the same data without background corrections.

Possible modifications to the Kawasaki equation due to the functional form of the correlation function differing from the Ornstein-Zernike form are discussed and are shown to be small.

I. Historical background. - The first measure- ments of the Rayleigh linewidth for fluids near the critical point were reported six years ago by Alpert and coworkers [I], who measured the linewidth for the binary liquid critical mixture aniline-cyclohexane, and by Ford and Benedek [2], who reported linewidth measurements on sulfur hexafluoride. Those results and subsequent results for the linewidth for several other pure fluids and mixtures were interpreted in terms of the Landau-Placzek theory, in which the spectrum of the light scattered by a fluid is obtained from a solution of the linearized equations of hydro- dynamics. The theory predicts that the spectrum will contain a component with the Lorentzian lineshape, centered at the frequency of the incident light, and with a half-width at a half-maximum given by

tude of the scattering vector (q = 2 nKo sin 3 0 ; n, KO, and 8 are the refractive index, the magnitude of the wavevector of the incident light in vacuum, and the scattering angle, respectively). The thermal djffusivity

x

is given by

x

The Rayleigh linewidth for the different simple fluids and mixtures was found to approach zero as the critical point was approached. Since many of the static properties of systems in the critical region were known to exhibit simple asymptotic power law dependences on the reduced difference temperature,

r

r

where

x

diffusion coefficient, respectively, and q is the magni- (2)

describes the behavior of the thermal diffusivity as the

(*) This research was supported ^by the National Science critical point is approached along the critical isochore

Foundation. or along the coexistence curve (with different para-

(**) Now at the Department of Chemical Engineering, Rice

University, Houston, Texas 77001, U. S . A. meters for the different thermodynamic paths). The

(+) Alfred P. Sloan Research Fellow ; now at the Department linewidth data obtained in mmerous experiments of Physics, New York University, New York, N. Y. 10003. performed in the past six years have been found to be

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972116

C1-82 H. L. SWINNEY, D. L. HENRY AND H. Z. CUMMINS described fairly accurately by the simple exponential

law, with Q, = 0.59 to 0.68 for four binary mixtures (aniline-cyclohexane, Berg6 et al. [4] ; isobutyric acid- water, Chu et al. [5] ; n hexane-nitrobenzene, Chen and Polonsky-Ostrowsky [6] ; and phenol-water, Pusey and Goldberg [7]), @ = 0.73 for CO, (Swinney and Cummins [8]), Q, = 0.75 for xenon (Henry, Swinney, and Cummins [9]), and 4, = 1.26 for SF, (Benedek et al. [lo] ; see also Braun et al. [ll], who recently reported Q, = 0.89 for SF,).

In 1968 Kadanoff and Swift [12] developed a theory for the critical behavior of the transport coefficients and predicted that

x -

-

The analysis of Rayleigh linewidth data in terms of simple exponential laws has been refined during the last few years in two fundamental respects. First, the Landau-Placzek expression (1) for the linewidth has been supplanted by a more general dynamical theory.

Following a concept introduced by Fixman [13], Kada- noff and Swift [12] developed the mode-mode coupling theory which was subsequently extended by Kawasaki.

Kawasaki [14] has derived an expression for the line- width which is applicable over the entire domain from the hydrodynamic regi0.n to the critical point.

Secondly, it has become clear that the simple expo- nential laws are in any case only applicable to the singular part of thermodynamic properties, and that

the exponent for 4, is reduced from 0.73 to 0.62.

Subsequently Benedek et al. [I81 showed that for SF, the divergent part of the thermal conductivity is apparently almost negligible in the temperature region of their experiments, and they calculated [19] that for xenon the asymptotic term in the linewidth accounts for only one-half of the linewidth at T - T, ⁼ 1 OC.

The importance of the large background contribu- tions to the transport coefficit:nts of other critical systems has also just recently become fully appreciated (see the discussion by Kawasaki [20]). For example, from neutron scattering experiments on iron and nickel it appeared that the critical behavior of the spin diffusion coefficient was very different for these two magnets ; a recent reanalysis of the data indicates that the apparent difference in behavior was a consequence of the neglect of the background. The inclusion of the background yields (for both systems) an exponent for the spin diffusion coefficient which is in reasonable agreement with the prediction of the mode-mode coupling theory. As another example, ultrasonic measurements in liquid helium near the lambda point indicated that the exponents characterizing the atte- nuation of first sound exhibited an apparent asymmetry above and below the transition temperature TA. Again it was recently realized that the apparent contradiction to the theory is resolved when the background terms are taken into account.

In this paper we report an analysis of our linewidth data for xenon [9], first correcting the data for the background specific heat and thermal conductivity.

The remaining singular part of the linewidth will then be compared with the Kawasaki theory, both in the hydrodynamic region and also in the region very near the critical point where the dynamics of the density fluctuations are no longer adequately described by hydrodynamics. We will conclude with a consideration of the modifications to the theory when Kawasaki's general integral equation for the linewidth is evaluated with correlation functions other than the Ornstein- Zernike form.

non-singular background contributions must be sub-

tracted off if data are to be analyzed over extended

''.

summer

r

'

very different from the true asymptotic behavior which

r = - - .

is presumably describable by the simple exponential P~,(P, T, q)

laws (see the discussion in the 1967 review by Fish- We assume that any physical property of a system er [16]). Sengers and Keyes 1171 have shown that near the critical point, which we write as h(x, q) (where when the carbon dioxide data are analyzed with the x could, for example, be the density or temperature background thermal conductivity taken into account, or magnetization), can be written as the sum of two

THE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT _CI-83 terms, h = h,

+

is the value the property would have in the absence of any critical anomaly, and (2) the singular part, h,, which (for q ⁼ 0) diverges to infinity or converges to zero as the critical point is approached [16],

h, (x, q = 0) ⁼ ho xa[l

+

+

4 - 1

+

where 45 is the exponent that characterizes the singula- rity and ho is the amplitude of the singularity ; in general the higher order terms are unknown and are not necessarily simple.

Separating I and c, into regular and singular parts, we can write

Kawasaki's expression for the asymptotic behavior of the singular part of the linewidth (which applies to all values of qt) is 1141 :

where A

=

In terms of the Kawasaki equation the singular part of the linewidth can be written (assuming that

rK

-

rK[l +

p(c,),

In our analysis of the xenon linewidth data we have taken into account the effect of the regular parts of the thermodynamic quantities 1, and (c,),, as we shall show, but we have assumed that the higher order terms in the singular part of the linewidth (5), which are of unknown form, are negligible. Also, since the shear viscosity of pure fluids appears to have a very weak critical anomaly, if any, we have treated A as an adjus- table constant, but we shall return to this point later.

(The variation in the temperature factor T in A is negligible in the temperature range investigated.) Assuming the Ornstein-Zernike form for the q- dependence of c,,

we obtain the final expression used in the linewidth data analysis :

For q5 2 1, the Kawasaki function K becomes K(y) =

...

+

In the limit q5 9 1, K(y) = (3 n/8) y3 ; hence

Equation (7) for the linewidth is a complicated function of ^E and q ; for graphing purposes it is conve- nient to rewrite (7) as follows :

Written in this way the theory can be seen to predict that, since TK/q3 = (8 A/3 n) K(q()/(qt)3 is a function of q t alone, the experimental data for T/q3, after being modified by the background corrections, should all fall on a single universal curve as a function of q(.

The linewidth data were fitted t o (7) [or, equivalently, to (9)] using a nonlinear least-squares computer fitting routine in which A,

to,

111. The background contributions in xenon. - We estimate the regular part of the the~mal conducti- vity following the method used by Sengers and Keyes [17] to analyze the CO, data. This procedure is based on the consideration of the cc excess ^N (or (c residual ^B) thermal conductivity,

which measures the excess of the thermal conductivity A(p, T) at density p over the dilute gas value 1(0, T) at the same temperature T. A well-known empirical result, which has been demonstrated to high precision for many fluids, is that away from the critical point the excess thermal conductivity is a function of density alone [15] ; this fact is frequently utilized in the inter- pretation of data in the engineering literature 1211, 1221.

It is reasonable to assume, and this is the crucial assumption in the Sengers-Keyes analysis, that the regular part of the thermal conductivity in the critical region is given by (10) with the excess thermal conducti- vity determined from data away from the critical point.

Thus on the critical isochore we have

where the r-subscript on

?,

C1-84 H. L. SWINNEY, D. L. HENRY AND H. Z. CUMMINS

thermal conductivity is to be determined from data outside of the critical region.

Since for argon there is an abundance of thermal conductivity data, we first obtain L,(p,, T) for argon and then employ a corresponding states relation to obtain A,(p,, T) for xenon.

The results for X(p) for argon were deduced from the data of Bailey and Kellner [23] and Michels et al. [24], and a few representative points are shown in figure 1, illustrating the temperature-independence of X(p) far from the critical point (T, = 150.8 OK). The curve represents an empirical equation which Bailey and Kellner claim reproduces their data to within 3 %

and which also reproduces the Michels data to better than 3 %. Only those data which were obtained far from the critical point were used for the Bailey and Kellner equation, since an anomalous increase in the thermal conductivity was indeed observed near the critical point.

-

:

- j

B

I

-

-

:

j

B P

/-'

-200 -150 -MO -50 0 M lw

1 ( " C )

FIG. 2. -The dilute gas thermal conductivity [I(O, T)] data for argon. Bailey and Kellner [23] ; Michels et al. 1241.

The corresponding states relation used in determin- ing A, for xenon from the results for argon is [21], [25]

A~ where C, which for argon is C"' = 2.97, is assumed to

x 50 oc be the same for all gases of the same type, such as the

25

" ⁰ inert gases. The xenon dilute gas data of Saxena and

• - 70 Saxena [26] and Kannuluik and Carman [27] were

.

and the result for Axe(O, T,), vvhen combined with the argon result for C, gives

A$(p,, Tc) = 1 630 ~:rgs/cm. s . deg

.

FIG. 1. - The excess thermal conductivity

for argon. The data are those of Bailey and Kellner [23] and Michels et al. [24], and the curve represents an empirical equation which Bailey and Kellner found reproduces their data

to within 3 %.

Similarly, Bailey and Kellner fit an empirical equa- tion to their dilute gas data [23], which are shown in figure 2 along with the dilute gas data of Michels et al. [24]. (Several other investigators have measured the thermal conductivity of argon, but because none of those experiments were as extensive as the two des- cribed here and because their data show considerable scatter, we chose to ignore them.) Since the data from the two experiments agree very well with each other, we used the Bailey and Kellner equations to deduce

* A,(p,) ⁼ 1 911 ergs/cm . s . deg and

4 0 , T,) = 972 ergslcm . s

.

.

This indirect route to a value for A, for xenon has recently become unnecessary because Tufeu, Le Neindre, and Vodar [28] have just completed extensive thermal conductivity measurements on xenon. Sen- gers' [28] analysis of these new data, together with our results computed for LXe(O, T'), yields the final result used in the xenon linewidth data analysis :

L,(p,, T) = [l 614

+

and

Although the inert gas data are often cited as the basis for assuming the validity of the law of correspond- ing states for the thermal conductivity of gases of the same type, there has been relatively little data available to support that assumption. [Cf. Owens and Tho- dos [21] and Hendricks et al. [22] ; note that until very recently no distinction was made between A and A,.]

We have presented the argon calculations because the excellent agreement between the values for C obtained from the very thorough argon and xenon data lends strong support to the widely-used corresponding states law for the thermal conductivity [21], [22], [25].

(Uncertainties in the values for C have not been stated Hence (1 1) yields Ir(pc, T,) ⁼ 2 883 ergs/cm. s .deg. since they are very difficult to estimate reliably ;

THE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT C1-85 however, the excellent agreement for A(0, T) and ;i@)

between two independent, careful studies indicates that CA* is probably known with an accuracy of at least 10 %.)

We now consider the constant pressure specific heat, beginning with the thermodynamic relation (q ⁼ 0) :

where rc, is the isothermal compressibility, assumed to be described (for p = p,) in the temperature region of interest by the singular part alone,

where

r,

The data of Edwards et al. [29] were used for c, in (1 3). Near Tc the c , termin (13) is negligible compared to the compressibility term, and even at the highest temperature of our data, T - Tc ⁼ 5.8 OC, the c,

term contributes only 3 % to the value of c,.

The derivative (aP/aT), was obtained by fitting the P-T data of Habgood and Schneider [30] and Beattie et al. [31] to the lowest-order symmetry form of the extended scaling law equation of Green, Cooper, and Levelt Sengers [32] which in the form applicable to the critical isochore is

The parameters obtained in a nonlinear least-squares analysis of the P-T data are given in TableI. (The value for the exponent p is from [33].)

P-T equation [eq. (15)] parameters on the critical isochore of xenon. Data from 1301, [31], temperature range, 0

<

<

a = 0.06 c1 = 5.925 048 PI = 6.483 4 pc 23, = - 8.666 2 Std. error = 0.001 6 %

The only remaining quantity needed for the line- width equation is (c,),, which is given from (l3), (14), and (15) by

P c2

(c,)~ = lim (c,) = -s

rT

.

T-r T, PC ^Tc

IV. Experimental results. - We now present the results of the analysis of the xenon linewidth data [in the temperature range

and for scattering angles 420 d 0 d 13801, considering first the parameters in Table 11, which are the para- meters for which the Kawasaki theory [with back- ground corrections, eq. (7)] best represents the line- width data. The uncertainties stated in Table 11, which are much larger than the statistical uncertainties, arise primarily (as we shall see) from the uncertainty in the compressibility parameters y and T , and from the weak dependence of the best-fit parameters on the temperature range of the data analyzed.

The values of the parameters which give the best jit of the xenon linewidth data to the Kawasaki theory [corrected for the background contributions, as in eq. (7)]. The constants to and A are highly correlated with the value of v ; the stated uncertainties are for v jixed, v = 0.64.

The xenon linewidth data, modified by the back- ground corrections, is compared with the theory in figure 3 for the parameters given in Table 11. As can be seen, the linewidth data at different temperatures and scattering angles do follow a single curve when plotted as a function of l / q t [see eq. (9)], and furthermore the agreement with the predicted functional form is quite good.

As a function of the exponent v, the error has a shallow minimum near v = 0.64, as illustrated in Table 111 for a particular choice of the compressibility parameters, y and

r,.

Dependence of the parameters determined from the linewidth data on the correlation length exponent v.

(y = 1.237 9,

rT

v

to

- - - -

0.61 1.625 0 4.207 4.320 %

0.62 1.5296 4.250 4.005

0.63 1.444 8 4.307 3.835

0.64 1.363 7 4.358 3.831

0.65 1.281 6 4.391 3.991

0.66 1.207 0 4.434 4.293

C1-86 H. L. SWINNEY, D. L. HENRY AND H. Z. CUMMINS

(C~)KAWASAKI THEORY V=0.64

6 ~ 1 . 3 6 3 7

A = ~ . ~ ~ ~ X I O - ~ C M ~

(b) XENON DATA

.

FIG. 3. - A comparison of the Kawasaki theory with the line- width data for xenon on the critical isochore (a). A plot of the Kawasaki linewidth eq. (3) for the parameters shown (b).

The deviation of the xenon linewidth data from the theoretical curve shown in (a). The linewidth data have been corrected for the background using eq. (9) with y = 1.237 9 and

r~ ⁼ 0.068 68 [34] and with Lr and cp as given in the text. The 153 data points are in the temperature range 0.003 < (T - Tc) d 5.8 OC and for scattering angles

4 2 O < 8 < 138O. The standard error is 3.83 %

.

Table IV illustrates the dependence of 5, and A on the choice of the compressibility parameters y and

rT

<

<

Type of Region - all

hydrodynamic all

hydrodynamic intermediate critical

Dependence of the parameters determined from the linewidth data on the choice of compressibility para- meters y and

r,.

A Std.

Ref. y TT 10-12 crn3,s error

- - - - -

34 1.237 9 0.068 68 1.363 7 4.358 3.831 % 33 1.26 0.058 70 1.388 2 4.397 4.076 35 1.277 5 0.052 1.388 9 4.406 4.335

V. Comparison with other experiments. - The Kawasaki theory predicts a simple relation between the high frequency shear viscosity y

*

According to Kawasaki [36], the value of q * should lie in the interval q, < y* < (y, -k y,). The viscosity of pure fluids and mixtures is expected to have a weak critical anomaly, and indeed a logarithmic divergence in y has been observed for aniline-cyclohexane [37] ; however, for pure fluids the limited amount of data available near the critical point (C0,-Kestin et al. [38] ; xenon-Zollweg et al. [19]) indicate that the amplitude of the singularity in these systems must be small ; that is, it appears that y, 4 y, except possibly very near T,.

Hence we would expect that y

*

The regular part of the viscosity can be analyzed by the same method used for the thermal conductivity.

Away from the critical point the excess viscosity,

like the excess thermal conduct.ivity, depends only on the density, as figure 4 illustrates for xenon. Figure 4 was obtained by combining the high density data of Reynes and Thodos [39] and Kestin and Leiden- frost [40] with the dilute gas clata plotted in figure 5 (Kestin and Leidenfrost [40], Rimkine [41], and Trautz and Heberling [42]). The data in figures 4 and 5 were fit

Dependence of the parameters determinedfi*om the linewidth data on the q( range of the data ; f o r y = 1.26, T , ⁼ 0.058 701 1331, and v == 0.64

No. of Points 153 105 135 87 39 8

50

( 4 0 ^,ax

-

(4

-

4.5 1.382 2

0.32 1.382 2 (fixed)

4.5 1.361 4

0.32 1.361 4 (fixed)

1.2 1.349 4

4.5 1.369 1

Std.

error - 4.076 %

3.935 3.822 3.492 4.389 4.744

THE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT (21-87

FIG. 4. -The excess shear viscosity [q * = q(p, T) - q(0, T)]

for xenon, deduced from the high density data of Reynes and Thodos [39] and Kestin and Leidenfrost 1401, and the dilute gas

data shown in figure 5.

to fourth degree polynomials, and the final result for the regular part of the shear viscosity of xenon is q,@,, T) = [5.2

+

x poise

.

Xe

KESTIN A N D LEIDENFROSI

* RANKINE

.

FIG. 5. - The dilute gas shear viscosity [q(0, T)] data for xenon. Kestin and Leidenfrost 1401, Rankine 1411, Trautz

and Heberling [42].

y* is x 30 % larger than y,, apparently reflecting a larger amplitude viscosity anomaly for those systems.

In Table VII we compare our results for the correla- tion length with the values determined by Zollweg et al. [19] from measurements of the angular depen- dence of the scattering intensity. A comparison of the actual magnitudes of 5 at different temperatures, as shown on the right in Table VII, is perhaps more meaningful than a comparison of the highly correlated values for 5, and v. Near the critical point the results of the two experiments agree, but at T - T, = 1 OC the value for l deduced from fitting the linewidth data to the Kawasaki theory is only two-thirds as large as the value determined from the angular dependence of the scattering intensity.

This result is compared in Table VI with y* and with VI. Modifications to the theory. - The general the value for the viscosity obtained by Zollweg et integral expression derived by Kawasaki for the al. [I91 from measurements of the spectrum of light Rayleigh linewidth (the decay rate of the density scattered from xenon surface waves near the critical fluctuations) is [14]

point. Our result for y* is only slightly (10 %) larger

than y,, as expected ; for the binary mixtures aniline-

r -

cyclohexane [43] and nitroethane-3 methyl pentane [44], - 8 7c3 q* G ( 4 )

Shear viscosity of xenon near the critical point Quantity Measured

-

q* = high freq. shear visc.

y, = regular part of shear visc.

q / p = kinematic visc.

Author Shear visc. poise)

- -

this paper 5.73 5 0.26

Reynes & Thodos [39] 5.2

+

Zollweg [I91 4.97

+

TABLE VII

The correlation length for xenon on the critical isochore : 5 ⁼ lo ^E-'

Correlation length (A)

Measurement Author 5,

(4

- - - - - -

intensity Zollweg [I91 3.07 0.57

+

linewidth this paper 1.36

+

+

C1-88 H. L. SWINNEY, D. L. HENRY AND H. 2. CUMMINS

where q q ) = e i ~ - r , and G(r) is the density- where q in

e3

1

density correlation function. (An alternative derivation

q-space form (ref. [46], eq. 6.19) of Gl(r) cc e-'lr/r '+l,

of an for the linewidth (I6), which is often suggested as a form which may describe involving an integration over r rather than q, has been

given by Ferrell [45].) Kawasaki evaluated the integral the departures from 0. Z. behavior ; z 2 , which is some- in (16) using the Ornstein-Zernike (0. Z.) form for the times (incorrectly) called the Fourier transform of correlation function, G(q - k) cc

[t-2 +

83

theory q = 0. (We use the conventional notation in In the limiting regions q t

<

to detect experimentally ; however, a lower bound for q function, the value of y, and the particular limit taken.

significantly greater than zero is expected on the basis This modification to the linewidth is qualitatively of the following inequality rigorously proved (for similar to a correction recently calculated by Lo and ferromagnets) by Fisher [47] : Kawasaki [49], who investigated the contribution of the simplest vertex corrections to eq. (16) [all vertex 3(b - 1)

y 2 2 - --- corrections were ignored in obtaining (16)] ; Lo and (6

+

<

critical isotherm. A recent scaling law analysis [33] Experimental evidence for the modification of (3) of the data in the critical region of a number of fluids due to the correlation function correction or the vertex and ferromagnets yielded 8 4.4 ; hence ~ i ~ correction would be manifested in two ways for line- h ~ ~ ' ~ inequality implies y 2 0.1. width data that are analyzed (as we have done) in

We have evaluated the integral in (16) for the follow- terms eq. (3) :

ing three forms for the correlation function z(q) : (1) Since

to

8,

<

<

- 2 -(I-q/2) A for qt % 1) ; however, the values obtained for

2 2 CC (q2

+

and values which would be obtained if the data were fit to

the modified linewidth equations. The size of this

A ( K 2

+ v2

G3 cc (17~) correction is shown in Table VIII.

5-2

+

+

The size of the correction to the correlation length coefJicient

to

Modification * *

[(50)corr

-

(174 GI

+

+

(17b) G2

+

+

( 1 7 ~ ) G3 (9 = 0.1) - 1.82

+

11. Vertex correction - 2.83

+

THE RAYLEIGH LINEWIDTH IN XENON NEAR THE CRITICAL POINT C1-89

the intermediate region where q t

-

-

region in their linewidth measurements on aniline- cyclohexane.] The predicted departure of the data from the theory is of the same order as the precision of our linewidth data ; however, an effect of this size is well within the state of the art in linewidth measurements, so it will be interesting to look carefully for such a departure in future experiments.

rements for xenon, when corrected for the nonanoma- lous background contributions to the thermal conducti- vity and specific heat, agree with the Kawasaki equa- tion within the experimental accuracy. Furthermore, the value 0.64 found for the exponent by adjusting the parameters in Kawasaki's equation to produce a best fit to the data is now in agreement with theoretical expectations and with experimental results for many other systems. These results, in conjunction with Sengers and Keyes' analysis of the CO, data with thermal conductivity background corrections, bring virtually all Rayleigh linewidth measurements into agreement with each other and with the theory. Only the case of SF, remains to be subjected to a similar quantitative analysis - an undertaking which is now also possible with the help of recent measurements of the shear viscosity by Wu and Webb [50].

Finally, we see that possible residual errors in Kawasaki's expression (3) resulting from failure to include vertex corrections, or from departure of the correlation function from the Ornstein-Zernike form, should not change the functional shape of K(q5) by more than a few percent, and then only in the crossover region q t

-

FIG. 6. - The correction to the linewidth calculated by evaluat-

ing the Kawasaki integral eq. (16) using different forms for Acknowledgements. - We thank Dr. P. C. Hohen-

the correlation function. ^{r K} is the result for the 0. Z. correlation berg, prof. M. E. ~ j ~ h ~ ~ , and prof. K. ~ ~for ~ ~ ~ ~ k i function and rcorr is the result for the correlation function

forms given in (17), assuming that the parameters 51, and A stimulating discussions, and Dr. B. A. Saleh for his in each case are adjusted to give the same value for the line- thorough compute1 analysis of the linewidth data. We width in the limits q5 < ¹ and q< 9 1. also thank Dr. Levelt Sengers for her suggestions concerning the calculation of the background correc- VII. Conclusion. - In conclusion, we have shown tions, and Prof. J. V. Sengers for furnishing us with the that our previously reported Rayleigh linewidth measu- xenon thermal conductivity results prior to publication.

References

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