LIGHT SCATTERING FROM SF6 IN THE VICINITY OF THE CRITICAL POINT

HAL Id: jpa-00214908

https://hal.archives-ouvertes.fr/jpa-00214908

Submitted on 1 Jan 1972

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

LIGHT SCATTERING FROM SF6 IN THE VICINITY OF THE CRITICAL POINT

R. Mohr, K. Langley

To cite this version:

R. Mohr, K. Langley. LIGHT SCATTERING FROM SF6 IN THE VICINITY OF THE CRITICAL POINT. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-97-C1-103. �10.1051/jphyscol:1972118�.

�jpa-00214908�

JOURNAL DE

PHYSIQUE Colloque CI, supplement au n°2-3, Tome 33, Fevrier-Mars 1972, page Cl-97

LIGHT SCATTERING FROM SF

⁶

IN THE VICINITY OF THE CRITICAL POINT

R. MOHR (*) and K. H. LANGLEY

Department of Physics and Astronomy, University of Masaschusetts, Amherst, Massachusetts, 01002

Resume. — On a determine 1'attenuation et la dispersion de la Vitesse des hypersons dans SF6 le long de 1'isochore critique, en utilisant des mesures de diffusion Brillouin pour des angles de diffusion de 45°, 70° et 135°. On a compare les spectres Brillouin obtenus dans le domaine de temperature 0,04 =S T— T

^e

< 45 °C, au modele hydrodynamique d'un fluide qui relaxe. Ce modele predit une forme de raie asymetrique qu'on a observee. On a trouve les parties critiques de 1'atte- nuation et de la dispersion en soustrayant aux valeurs observees les valeurs caleulees en tenant compte des contributions de la dispersion classique et de la relaxation moleculaire vibrationnelle.

On compare 1'attenuation et la dispersion critique aux mesures ultrasonores et a la theorie de Kawasaki en utilisant un seul parametre de frequence reduite. On presente des mesures prelimi- naires de la largeur de la raie Rayleigh faites au moyen de la spectroscopie d'autocorrelation de photons. On utilise les resultats de ces mesures pour analyser les donnees sur 1'attenuation et la dispersion.

Abstract. — Using Brillouin scattering measurements at scattering angles of 45°, 70° and 135°, hypersonic sound attenuation and velocity dispersion in SF« have been determined on the critical isochore. Brillouin spectra obtained in the temperature range 0.04 < T— Tc =£ 45 °C were fit to a hydrodynamic model of a relaxing fluid which predicts the observed asymmetric lineshape. The critical parts of the attenuation and velocity dispersion were found by subtracting from the observed values the calculated classical and molecular vibrational relaxation contributions. The critical attenuation and dispersion are compared with ultrasonic measurements and with the theory of Kawasaki using a single reduced frequency parameter. Preliminary measurements of the Rayleigh linewidth made using photon autocorrelation spectroscopy are reported and are used in analyzing the attenuation and dispersion data.

In this paper we wish to present first some prelimi- nary measurements of the Rayleigh linewidth in SF

⁶

along the critical isochore and then to interpret measurements of the sound attenuation and velocity derived from our Brillouin scattering spectra and from ultrasonic data [1], [2] utilizing these results. Rayleigh linewidth measurements in SF

6

have been reported recently by two different groups [3], [4] but the reasons for the different temperature dependence found along the critical isochore are not clear. We felt additional measurements using photon correlation function techniques [5] would be useful. The critical contribu- tions to the sound attenuation and velocity dispersion are discussed in terms of the analysis proposed by Garland, Eden and Mistura [6] and by Cummins and Swinney [7] for the attenuation and dispersion in xenon. In these analyses, based on the work of Fix- man [8], Botch and Fixman [9], Kadanoff and Swift [10], and Kawasaki [11], the critical attenuation per wave- length a

x

(crit) and velocity dispersion depend on both temperature and frequency through a single reduced

(*) Present address : Department of Physics, Catholic Uni- versity of America, Washington, D. C. 20012.

variable co* = w/co

^D

where the characteristic frequency co

^D

= (2A/pC

^p

)£~

²

. A is the thermal conductivity, p is the density, C

^p

is the specific heat per gram and £ is the correlation length. In the hydrodynamic limit q £, -4 1 (q is the scattering wave vector), the Rayleigh linewidth is given by r

R

= (A/pC

p

) q

2

so that if the measured correlation length [12]

i = (72 ± ii) ( r - r

o

)-°-

6 7 ± 0

-

0 2

A is taken together with our Rayleigh linewidth measure- ments, co

D

needed in analysis of the attenuation and dispersion is completely determined.

Experimental methods. — A schematic diagram of the apparatus used to obtain both the Rayleigh line- width and the Brillouin spectra is shown in figure 1.

Temperature stability of the sample cell is achieved by two-stage temperature control. An outer heavy-walled copper container is stabilized to within + 0.05 °C, while the temperature of the copper sample cell is controlled to within one millidegree by a thermister sensor bridge and servo-controlled heater. Filling of the cell to an average density within about 1 % of the critical density was accomplished by overfilling and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972118

CI-98 R. MOHR AND K. H. LANGLEY

LASER SPECTRA PHYSICS 132

DIGITAL CORRELATOR

SPATIAL FILTER

V V [PHOTOMULTIPLIER A \ j *\ RCA 7265

AMPLIFIER- DISCRIMINATOR

MICHELSON INTERFEROMETER

FIG. 1. — Schematic diagram of the apparatus. Two conical reflectors of unequal included angle are immersed in the fluid ; four scattering angles may be studied in a single cell. The remo- vable aperture and prism are inserted during measurements

of the Rayleigh linewidth.

then reducing the density until only a small change in the meniscus height was observed as the temperature was raised from 0.040 °C to 0.020 °C below T

c

. The sample, Air Products and Chemicals instrument-grade SF

6

, showed less than 100 p. p. m. impurities in a gas- chromatograph analysis. The same sample and cell filling was used in all measurements reported here.

The interferometer system used to obtain the Bril- louin spectrum has been described in detail else- where [13], [14] Rayleigh linewidth measurements are made using the same 1 MW helium-neon laser and scattering cell by restricting scattered light passing through the spatial filter to roughly a single coherence area by a movable aperture and inserting a prism to divert the light directly to a photomultiplier. Photo- electron pulses are applied to the input of a correlation computer [15] which uses digital integrated circuit techniques to calculate the autocorrelation function g(j) of the photomultiplier current;

g(%) = < i(t) i(t + T) > .

If the power spectrum of light reaching the photo- multiplier is Lorentzian

p(d) = < P > (r\7t)\[(a) - co

0

)

2

+ r

2

]

where < P > is the average intensity, the correlation function is essentially the sum of a single exponential plus a constant g(x) = A + Bexp(— 2 JV). In our experiments, the correlation function is allowed to build up in the digital memory of the correlator until adequate signal to noise ratio is attained, typically 30 s to a few minutes, then the correlator output is computer fit to extract the Rayleigh linewidth T

R

. Independent runs at the same temperature made two weeks apart

all give an excellent fit to an exponential with the value of r reproducible to within 2-3 %.

Hypersonic sound attenuation and velocity can be found from the Brillouin spectra only by taking proper account of the combined interferometer instrumental response function. The Michelson interferometer acts as a filter with transmittance given by

M(co) = sin

²

(Op

where the free spectral range co

F

is carefully matched to the free spectral range of the Fabry-Perot and to the laser longitudinal mode spacing. The spectrum observ- ed experimentally, S

B

(a)), is the convolution of the Fabry-Perot instrumental response function, F(co), with the true spectrum, S(co), filtered by the Michelson interferometer :

sE(fl>:

»-J:

F(a> - co') M(co') S(co') dco'.

In each spectrum generated, the response function of the Fabry-Perot interferomter is measured directly by the observed profile of the extremely sharp true Rayleigh line which is not completely blocked by the Michelson interferometer filtrer. A spectrum is computer fit in the following way : An analytic expression for the true spectrum containing initial esti- mates of the adjustable parameters is generated and multiplied by the Michelson transmittance. The convo- lution of this expression with the measured Fabry-Perot instrumental response is calculated using the discrete fast Fourier transform [16] and the result least-squares fit to the experimentally observed spectrum using the method of Newton and Raphson to adjust the parameters. In all cases the observed spectrum was fit to within the range of the noise and the adjustable parameters were found to converge to the same value within the limits expected from experimental error, independent of the initial values chosen. This uni queness of the fit results because the adjustable para- meters are directly related to the features of the spectrum e. g. intensity, position of the line center, or linewidth, and are therefore relatively independent.

The form of the true spectrum S(co) we have used is that of a fluid whose bulk viscosity relaxes with a single relaxation time. A hydrodynamic model of such a fluid has been considered by Mountain [17] and by Mon- trose, Solovyev and Litovitz [18]. The spectrum is given by the ratio of two polynomials in the general case, but if the attenuation is small the spectrum may be decomposed into a sum of partial fractions exhibiting the Lorentzian-shaped Rayleigh line and the diffusive mode centered at the incident frequency, and the asymmetric Mandelshtam-Brillouin doublet lines shift- ed up and down by the Brillouin frequency. The largest total attenuation we observed corresponds to oc

x

« 0.2 so the decomposition of the spectrum is well justified.

The additional diffusive mode introduced by the

relaxation in the bulk viscosity would be extremely

LIGHT SCATTERING FROM SFs IN TH [E VICINITY OF THE CRITICAL POINT Cl-99

difficult to distinguish from the tail of the normal

Rayleigh line in our experiment and is not expected to significantly affect the Brillouin linewidth, therefore it is neglected in fitting the spectrum. Presence of the diffusive mode and greater attenuation necessitated use of the polynomial expression for the spectrum in xenon near the critical point [19]. The exact expression which we used for S(o) is

:

rR is held constant at some small value while the four parameters IB/IR, I;/I,, uq, and r, are allowed to vary freely to accomplish a fit to the spectrum. We identify the quantities of interest as the amplitude attenua- tion per wavelength a,

=

2 rcT,/vq and the wave vector dependent phase velocity v. As has been pointed out [18], [20] the form of the spectrum in eq. (1) is exactly the same as that of a normal (non-relaxing) fluid but the equations relating r B , v, IB and If, to thermo- dynamic quantities and transport coefficients are different. The point is that eq. (1) is the simplest form that gives a good fit to the spectrum and allows easy identification of the quantities of interest.

Rayleigh Linewidth. - Recently Kawasaki [ l l ] has used a mode-mode coupling calculation to develop an equation describing the critical part of the linewidth of the light scattered quasi-elastically from the density fluctuations in a fluid near the critical point. His closed expression for the Rayleigh linewidth applies over the entire range of values of q t

k T

rR

⁼

^{A [I} + ^x2 + ^(x3

^-

x-I) tan-'

x ] ,

(2) 8

7Y7t3

where x

=

qt,

v]

is the high-frequency shear viscosity, and kB is the Boltzmann constant. The same result has been derived by Ferrell[21] using the Kubo formula. In the extreme hydrodynamic limit q t

4

1, the linewidth reduces to the result from the linearized equations of hydrodynamics [22] rR

⁼

xq2 where the thermal diffu- sivity x

=

A/pC,

=

k, T/6 nyg. The shear viscosity does not diverge in the Kawasaki theory

;

A,

C,

and 5

diverge as

^{8-$, 8-}

and

e-"

respectively where

so that the thermal diffusivity is proportional to

Y - y .

The above relation for x implies

y

- ^$

⁼

v, a result found by Kadanoff and Swift [lo] from a mode-mode coupling analysis of transport coefficients. For q& 5 I, eq. 2 yields rR

⁼

^xq2[1

^-k

(315) q 2 g2]. In the limit q t

%

1, eq. 2 becomes rR

⁼

(k, T3/16 y) q 3 predicting an asymptotic approach to a temperature independent linewidth as the temperature is lowered toward T,,

This result is also found from dynamical scaling argu- ments [23]. Rayleigh linewidths in agreement with the Kawasaki form in eq. (2) have been observed in xenon [24] as well as in binary mixtures [25].

Our preliminary measurements of the Rayleigh linewidth along the critical isochore at a scattering angle of 450 19' using digital photon autocorrelation spectroscopy are shown in figure 2. Each data point represents 5 to 10 independent measurements at the

SF, RAYLEIGH LINEWIDTH

e = 45O 19'

1031 I

-

^{m l l I t} 1 1 1 1 1

.01 0.1 1.0 10

T-T, ('C)

FIG. 2.

-

Rayleigh linewidth at 4 5 O 19' scattering angle. The solid curve is the Rayleigh linewidth calculated from eq. 2 using experimental values of the viscosity [26] and the correlation lengthlz. The dashed line indicates an apparent power law

dependence

r~

⁼ 9.3

x

^{lO4(T- Tc)0.77} rad/s.

same temperature with all measured values bracketed by the error bars or the size of the data point symbol whichever is larger. The solid curve is rR from eq. (2) calculated using no adjustable parameters

:

Scattering wave vector q is known from the geometry of the experi- ment and is from Puglielli and Ford [12]. The macroscopic cr background

⁾⁾

viscosity measured by Wu and Webb [26] from surface wave scattering, (3.27 + ^{0.5) x} poise, may be used for y. Since both

v]

and

{

are known only to within about 15 %, it is not surprising that the magnitude of our measured linewidth departs from the calculated value near Tc where the linewidth becomes temperature independent, In the hydrodynamic region the linewidth calculated from eq. (2) goes as T

-

T, to the 0.67 power as v

=

0.67 is the exponent in the measured correlation length [12]. For comparison, the dashed line in figure 2 is a straight line fit to the data above T - T,

=

0.05 OC and yields and apparent power law dependence

r,

⁼

9.3 x 104(T - Tc)0.77(rad/s). Certainly the expo-

nent

y

- $

=

0.77 would be decreased if corrections

were made for the non-divergent

⁽⁽

background

^>)

contribution to A. In CO, this correction decreases the

C1-100 R. MOHR AND K. H. LANGLEY

observed exponent

y

- $ from 0.73 to a corrected

value 0.62 [27]

;

it seems reasonable to expect that the correction in SF, is at least as large and could decrease y - $ enough to obey the scaling result y - $

=

v.

Unfortunately, a meaningful estimate of the non- divergent thermal conductivity jn SF, cannot be made from the existing thermal conductivity data [28] as the measurements corresponding to the critical density do not extend far enough away from the critical tempe- rature.

Brillouin scattering.

- The behavior of SF, in the critical region is complicated by the presence of mole- cular degrees of freedom. Translational energy of the molecule is coupled to vibrational degrees of freedom in the collision process, providing a relaxation mecha- nism affecting the frequency dependence of the sound attenuation and velocity. This vibrational relaxation must be accounted for in order to make any compa- rison of light scattering and ultrasonic data, and in order to find the contributions to attenuation and dis- persion due to the critical nature of the fluid. Several authors [29], 1301, [31] have studied the thermal relaxation of vibrational states in SF, using the ultra- sonic dispersion and attenuation and found that the entire dispersion region is well represented by a single characteristic relaxation time. Vibrational modes are weakly coupled to the translational degrees of freedom but energy is spread rapidly among all the modes by secondary collisions. From the relaxation time 5.89 x s measured at one atmosphere [29], 1311, we calculate the relaxation time at the critical density and temperature [32] to be z

=

2.85 x 10-'s.

Below we develop equations for the attenuation and velocity dispersion due to the exchange of energy with vibrational degrees of freedom. In those equations the relaxation is characterized by a time z'

=

z(C,

-

C')/C, where C'

=

15.7 cal mole-' OK-' is the specific heat due to vibration degrees of freedom 1291. Using values of C, found below, we find that the value of z' is temperature dependent, but has a typical value

z

1.5 x lo-' s. In the ultrasonic frequency range < 1 MHz we find that wz' < 0.1 while for the light scatte- ring data (at frequencies above 100 MHz), wz' > 10.

The existing ultrasonic data is at essentially zero fre- quency as far as vibrational relaxation is concerned

;

velocity dispersion is very small and near T, is domina- ted by dispersion due to critical point effects. The vibra- tional contribution to the attenuation is given by the low frequency limit of eq. (4). At the Brillouin fre- quencies vibrational velocity dispersion has saturated and the attenuation is given by the high frequency limit of eq. (4).

We now proceed to calculate the attenuation and dispelsion contributions due to vibrational relaxation.

The procedure is identical to that described, for example, in Herzfeld and Litovitz [32] with the impor- tant distinction from the treatment for driven sound waves that for the light scattering experiments the

scattering wave vector q is real and the frequency w is complex. The resulting equations for the velocity dispersion and attenuation are

a2 u4(o)

v 2 0 - . = ¹ + (C, - C,) C' w2 zt2 u; u; w2 C,(C, - C') 1 + o2 zr2 (3) where a is the attenuation per unit length and

v2 ( C p - C J C ' wz'

a, (vib)

⁼

n

- -

v2(o) CP(C" - C') 1 +

⁰²

zf2

⁽⁴⁾

Neglecting the second term on the left in eq.

(3)

results in negligible error in u2(o)/ug.

All the quantities needed to evaluate the right hand sides of eq. (3) and (4) are known from experimental data on SF,. C, is determined by using the thermo- dynamic identity

together with PVT data [33] and the adiabatic compressibility Ks, from the low frequency (1 kHz) sound velocity measurements of Fritsch [2]. Another identity

together with the isothermal compressibility KT measured by Puglielli and Ford [12] yields values of C,.

Attenuation. -

Following Garland, Eden and Mis- tura [6] we define the critical attenuation as

a, (crit)

⁼

a, (total) - a, (class) - a, (vib) ,

where al (total) is the observecl total attenuation, the classical contribution is given by

where q, is the shear viscosity and qvO is the non- relaxing part of the bulk visco!;ity, and a, (vib) is the contribution from vibrational relaxation. The shear viscosity is expected to have a constant value throu- ghout the critical region

;

recent measurements of Wu and Webb [26] yield a value (3.27

^_+

.5) x poise.

An estimate [34] of qvo using measured values of shear viscosity in the dilute gas region yields 1.9 x poise.

The term in A calculated using our measured values of A/pC, and C,/C, found above are roughly an order of magnitude smaller than the sun1 of the viscosity terms.

At hypersonic frequencies a, (class) makes a contri-

bution of the order of 0.01 vlrhich is a small effect

compared to the dominant critical and vibrational

attenuation. At ultrasonic frequencies a, (class) js small

compared to all values of a, (crit) we consider.

LIGHT SCATTERING

FROM

SF6 IN THE VICINITY OF THE CRlTICAL POINT C1-101

where k, is the Boltzmann constant,

yl

is the exponent in the Fisher correction to the Ornstein-Zernike corre- lation function 1351, vo is the zero frequency sound velocity and x is the product of the correlation length and the wave vector of the fluctuation considered in the integral. This result is essentially identical to an equa- tion obtained by Kawasaki [11,] with an important simplification

:

Using the reduced frequency

An expression for the critical attenuation has been derived by Garland et al. [6] from a modified form

of Fixman's theory [8] 0.25

together with the divergence of

(, C,,

and uo predicted from scaling laws, it can be seen that the fact01 in front of the integral varies slowly with temperature. In particular <- l(a<- ' / a ~ ) ~ -

^N

^{since v -- 213.}

The exponent a in the divergence of C, is expected to be small. Equation 5 is expected to be valid as long as the primary contributions to the integral are from values of x 5 2, a condition satisfied for o* 5 2. We have chosen the Kawasaki form of

+

^45'19' e - 0 70' 53'

A 1 3 4 O 4 1 '

for comparison to our data although our data are certainly not capable of distinguishing between the Kawasaki form and that originally proposed by Fixman [9] K(x)

=

x2(1 + x2).

The factor multiplying the integral in eq. (5) may be calculated directly without adjustable parameters. The exponent

q

is expected to be less than 0.1 and is neglected, thermodynamic quantities are calculated from PVT data and the low frequency sound velocity as indicated previously, (is from Puglielli and Ford [12].

For

vo

we have used the sound velocity calculated from the molecular vibration model at infinite fre- quency from eq. 3 above. At the Brillouin frequency, velocity dispersion due to molecular relaxation is completely saturated and

v ,

(vib) would be the sound velocity in the absence of any critical point effects.

The observed total attenuation obtained from Brillouin scattering is plotted as a function of reduced frequency in figure 3. The data were obtained over the temperature range 0.04 6 T - T, < 35 OC and in the frequency ranges 128 to 218 MHz for 0

⁼

450 19', 198 to 323 MHz for 6

=

700 53', and 331 to 461 MHz for 0

=

1340 41'. Reduced frequencies were obtained using our measured values of A/(pC,) and the correla- tion length measured by Pyglielli and Ford. The solid lines are simply smooth curves drawn through the data.

For each data point a value of a, (vib) was calculated from eq. (4) using thermodynamic quantities at the

FIG. 3. - Solid lines : Total attenuation per wavelength of hypersonic sound obtained from Brillouin spectra. Error bars are about the size of the data point symbols. Dashed lines : Molecular vibrational contribution to attenuation per wave-

length, calculated using eq. 3.

corresponding temperature and o equal to the Bril- louin frequency. It should be noted that the calculated a, (vib) (shown as dashed curves in Fig.

3)

are quite sensitive to the value of

C,,

particularly at small w*

which may explain why a, (vib) appears to cross the 0

=

45O 19' curve for a, (total).

In figure 4 we show a, (crit), the solid curve, as a function of o m calculated from eq. (5). The data points correspond to the data points of figure 3 after subtract- ing the calculated molecular vibrational contribution to a,. The dashed curve in figure 4 is simply the result of subtracting the dashed curve from the solid curve for the data at 450 19' in figure 3.

,

0.05 ^-

K A W A S A K I

0 b

10-3 10-2 10-1

lo0 101 lo2 lo3 104 W / W D

FIG. 4. - Critical attenuation per wavelength. The solid curve is an (crit) calculated from eq. 4 using the Kawasaki form

for K(x), The dashed curve is U A (crit) for 0 = 700 53'.

At high temperatures (small values of o*) we are

clearly in the hydrodynamic regime, and the Kawasaki

theory should work well, but on the other hand this is

the region where

cc,

(vib) is quite sensitive to the value

of C,. Uhfortunately, a, (vib) is not known well enough

to determine if the data for all three scattering angles

do lie on a single curve. A more careful treatment of

a,(class) as a function of frequency brings data corres-

C1-102 R. MOHR AND K. H. LANGLEY

ponding to different angles somewhat closer together.

Closer to the critical point (increasing o * ) at o * 2 2, eq. (5) is no longer in agreement with experi- ment as is expected from the breakdown of the Ornstein-Zernike form of the correlation function for q >

< - I .

A similar disagreement is also observed in xenon [6]. Qualitatively at least, a,(crit) continues to be described for all three scattering angles by a single function of a * . Kawasaki has recently estimated that at very large values of o * the attenuation should become nearly independent of frequency and temperature.

Whether this occurs is not clear from our data.

Existing ultrasonic data has clearly shown the ano- malous behavior of the attenuation in the critical region, but unfortunately it is very difficult to make a quantitative comparison with our Brillouin scattering results. In principle one should be able to calculate the classical and vibrational contributions to the total attenuation at 600 kHz and substract these from the total attenuation observed by Schneider [I] to obtain

a,

(crit) at 600 kHz. The vibrational contribution calculated from eq. (4) which is roughly independent of o * turns out to be a factor of 10 larger than the total attenuation observed by Schneider at small values of o * , preventing any meaningful estimate of a, (crit). The nature of the apparent difficulty is the following

:

at 600 kHz the frequency is well below the vibrational relaxation frequency estimated to be - 10 MHz, and a, (vib) is given by the low frequency limit of the form of eq. (4) appropriate to ultrasonic measurements, i. e.

complex wave vector and real w v2(w) C' a, (vib)

=

n -

-

o r .

4 C"

with the various quantities defined the same as in eq. (5). Swinney and Cumrnir~s [7] have used this equation to analyze the velocity dispersion in xenon.

This result is expected to be valid in the hydrodynamic region

q< 4

1 and in the non-hydrodynamic region as long as the major portion of the integral is contributed by values of x 5 2. This condition is satisfied over a considerably wider range of

^ol*

than is the similar condition on the integral in eq. (5). Even at o *

=

100, 64 % of the integral arises from

x

< 2.

In the same spirit as for the attenuation we take the value of v o to be the infinite frequency sound velocity calculated from eq. (3) for t:he dispersion due to molecular vibration. At the Brillouin frequency there is little dispersion attributable to vibrational relaxa- tion

;

this is evidenced by the fac;t that at a given tempe- rature well above

Tc

the hypersonic sound velocity is independent of the Brillouin frequency. Therefore we take the difference between the velocity at the Brillouin frequency v(w) and the velocity from the infinite frequency limit of eq. (3), v, (vib), as the critical dis- persion to be compared with the dispersion calculated from eq. (6).

Measured values of the critical dispersion are plotted in figure 5 as a function of w*, again using our Rayleigh linewidth results to find o*. The solid curve is eq. (6) calculated with no adjustable parameters. As mentioned Reducing a, (vib) calculated from this equation by a

factor of 10 requires an unreasonable choice of the quantities on the right hand side

;

v, is known to within

^N

2 %, 4 ~ ) cannot be larger than the observed hypersonic velocities, and C' is known to within a few percent from the studies of vibrational relaxation at low density [29]-[31]. Only small velocity dispersion is observed at 1 MHz so the value 2.85 x s taken for

z

must be correct to within a factor of about 2. Our estimates of C, and C, are consistent with the Landau- PIaczek ratio we observe when corrections are made for dispersion 1361

;

any significant change in C, would drastically alter a,(vib) calcluated at the Brillouin frequency. Increasing C, by a factor as large as two would also lower v, (vib) at T

- Tc =

3

^OC

by nearly 20 %, increasing the apparent critical part of the velo- city dispersion to 5 times that predicted by the Kawa- saki theory. At present there does not appear to be any reasonable way to estimate a, (crit) from the ultrasonic data.

Velocity dispersion. - Kawasaki [I 11 has investi- gated the critical sound velocity dispersion near the liquid-gas critical point using the mode-mode coupling theory. He obtains the expression :

FIG. 5.

-

Critical velocity disper:;ion. The solid curve is calculated from eq. 6 using the Kawasaki form for K(x). The plotted points are determined by calculating

[u(4

- U, (vib)

l/u,(vib)

where v(w) is the measured hypersonic sound velocity and u, (vib) is calculated from eq. 3.

previously, all the quantities on the right hand side are known experimentally. Far from the critical point (at small o/o,), the dispersion is small so velocity errors of about 1 % would account for the scatter in the data.

Nearer Tc at o * 2 50 where critical velocity dispersion

is significant, the theory underestimates its magnitude,

but in this region the theory may be expected to break

down. Our data at large o* does strongly support the

LIGHT SCATTERING FROM SF6 IN THE VICINITY OF THE CRITICAL POINT Cl-103

theoretical result that the dispersion and the reduced W e are grateful t o K. Fritsch and E. F. Carome for frequency are both functions of frequency and tempe- their low frequency sound velocity data in SF6 and t o rature in just such a way that data a t different fre- J. V. Sengers, C. W. Garland and N. C . Ford, Jr. for quencies a n d temperatures all fall o n one single curve. helpful discussions.

References

[l]

SCHNEIDER (W. G.),

J. Chem. Phys., 1950, 18, 1300 ; Can. J. Chem., 1951, 29, 243 ; J. Chem. Phys., 1952,20,759.

121

FRITSCH (K.) and CAROME (E. F.), NASA Contractor Report NASA

CR-1670,

National Aeronautics and Space Administration, Wash. D. C.,

1970,

CAROME (E. F.) and FRITSCH (K.), Proceedings of the Fifth Symposium on Thermophysical Properties, American Society of Mechanical Engineers,

1970.

131

BENEDEK (G. B.), in

Polarisation Matikre et Rayonne- ment, Livre de Jubile' en l'honneur du Professeur A. Kastler,

edited by the French Physical Society, Presses Universitaires de France, Paris, France,

1969.

[4]

BRAUN (P.), HAMMER (D.), TSCHARNUTER (W.) and WEINZIERL (P.),

Phys. Letters, 1970, 32A, 390.

[5]

FOORD (R.)

et al. Nature, 1970,227,242.

For a review see for example CUMMINS (H. Z.) and SWIN-

NEY

(H. L.),

Progress in Optics,

Vol. VIII, Ed.

E. Wolf, North-Holland Publishing Co., Amster- dam,

1970.

[6]

GARLAND (C. W.), EDEN (D.) and MISTURA (L.),

Phys. Rev. Letters, 1970, 25,1161.

[7]

CUMMINS (H. Z.) and SWINNEY (H. L.),

Phys. Rev.

Letters, 1970,25, 1 165.

[8]

FIXMAN (M.),

J. Chem. Phys., 1962, 36, 1961.

[9]

BOTCH

(W.) and FIXMAN

(M.),

J. Chem. Phys., 1965, 42, 199.

[lo]

KADANOFF (L. P.) and SWIFT (J.),

Phys. Rev., 1968, 166, 89.

[ l l ]

KAWASAKI (K.),

Phys. Letters, 1969, 30A, 325 ; Phys. Rev., 1970, A 1,1750

and

Ann. Phys.

(N. Y.),

1970,61,1.

1121

PUGLIELLI (V. G . ) and FORD (N. C.),

Jr., Phys. Rev.

Letters, 1970, 25, 143.

1131

LANGLEY (K. H.) and

FORD

(N. C.),

Jr., J. Opt. SOC.

Amer., 1969,59,281.

[14]

MOHR (R.), LANGLEY (K. H.) and FORD

(N.

C.),

Jr.

J. Acoust. Soc. Am., 1971, 49, 1030.

[15]

This correlator was designed and built by FORD (N. C.)

Jr.

and ASCH (R.).

(161

COCHRAN (W. T.), COOLEY (J. W.), FAVIN (D. L.), HELMS (H. D.), KAENEL

(R.

A.), LANG (W. W.), MALING (G. C.), NELSON (D. E.), RADER (C. M.) and WELCH (P. D.),

Proc. IEEE, 1967, 55, 1664.

MOUNTAIN (R. D.),

J. Res. Natl. Bur. Std. ( U . S . ) 1966,70A, 207.

MONTROSE (C. J.), SOLOVYEV (V. A.) and LITO-

VITZ

(T. A.),

J. Acoust. SOC. Am., 1967, 43, 117.

CANNELL (D. S.) and BENEDEK

(G.

B.),

Phys. Rev.

Letters, 1970,25, 1157.

CANNELL (D. S.), thesis, Massachusetts Institute of Technology,

1970.

FERRELL (R.),

Phys. Rev. Letters, 1970, 24, 1169.

MOUNTAIN (R. D.),

Rev. Mod. Phys., 1966, 38, 205.

HALPERIN

(B. I.) and HOHENBERG

(P. C.),

Phys. Rev., 1969, 177, 952.

HENRY (D.

L.),

SWINNEY (H. L.) and CUMMINS (H. Z.),

Phys. Rev. Letters, 1970, 25, 1170.

BERG^ (P.), CALMETTES (P.), LAJ (C.) and VOLO-

CHINE

(B.),

Phys. Rev. Letters, 1969, 23, 693.

BERGB (P.), CALMETTES (P.), LAJ

(C.),

TOURNA-

RIE

(M.) and VOLOCHINE (B.),

Phys. Rev. Letters, 1970, 24, 1223.

WU (E. S.) and WEBB (W. W.), Abstract, International Conference on the Scattering of Light from Fluids, Paris, France,

1971.

SENGERS (J. V.) and KEYES (P. H.),

Phys. Rev. Letters, 1971, 26, 70.

Lrs (J.) and KELLARD (P.

O.), Brit. J. AppZ. Phys., 1965, 16, 1099.

O'CONNOR (C. L.),

J. A C O U S ~ . SOC. Am., 1954,26,361.

HOLMES (R.) and S T O ~ (M. A.),

Brit. J. Appl. Phys.

1968, 1

Ser

2, 1331.

BESTE (K. W.),

Acustica, 1970,23,121 ;

Haebel (E. U.),

Acustica, 1968, 20, 65.

HERZFELD (K. F.) and LITOVITZ (T. A.),

Absorbtion and Dispersion of Ultrasonic Waves

(Academic Press, New York,

1959).

MEARS (W. H.), ROSENTHAL (E.) and SINKA (J.

V.), J. Phys. Chem., 1969,73,2254.

Earlier work on the properties of SF6 is surveyed in this paper.

HERSHFELDER (J. O.), CURTISS (C. E.) and BIRD (R. B.),

Molecular Theory of Gases and Liquids

(John Wiley and Sons, Inc., New York,

1954), 634.

FISHER (M.),

J. Math. Phys., 1964,5, 944.

PINNOW

@.

A,), CANDAU (S. J.), LA MACCHIA (J. T.)

and LITOWTZ (T. A.),

J. Acoust. Soc. Am., 1968, 43, 131.