Journalof Low TemperaturePhysics, Vol. 16, Nos. 5[6, 1974

Intensity of Critical Opalescence of Helium-4 Akira Tominaga Department o f Appl!ed Physics, Tokyo University o f Education, Tokyo, Japan

(Received March 19, 1974) We present measurements of the critical opalescence of helium-4. The results are analyzed by the Einstein and Ornstein-Zernike theory and the power laws. We obtain 7 = 7 ' = 1.17+_0.02, v = v' = 0.62 + 0.1, F / F ' = 4 . 5 + _ 0 . 3 , Pc = 1706.008 mm Hg, and T c = 5,189.863 m K (Ts8). The critical behavior of helium-4 is almost the same as that of classical fluids and the influence of the quantum nature of helium-4 is not as evident as has been claimed.

1. INTRODUCTION Much theoretical and experimental work has been devoted to the study of the critical behavior near the critical point of continuous phase transitions. Near the gas-liquid critical point of a fluid the fluctuation of the order parameter which becomes enormously large as the critical point is approached is believed to be the fluctuation of the density. Recent progress in experimental techniques has made it possible to study the density fluctuation of helium-4. For classical fluids, Xe, CO2, etc., light scattering experiments near the critical point have given almost the same results. 1-8 For helium-4, however, some groups 9'1° have reported apparently different results (also based on light scattering experiments) and have pointed out 1° that the quantum nature of helium-4 has a remarkable influence on its critical behavior. Since their results are apparently in conflict with P V T measurements, it we must consider the following three possibilities : (1) the theory of light scattering is inadequate for helium-4, (2) large systematic errors exist in the P V T measurements, (3) large systematic errors exist in the light scattering experiments. We have carried out light scattering experiments and analyzed our data by the ordinary theory of light scattering and obtained results consistent with the P V T m e a s u r e m e n t s . We may say that helium-4 is not so different from 571 © 1974 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission o f the publisher.

Akira Tominaga

572

so-called classical fluids as far as gas-liquid critical phenomena are concerned. Our results were partly reported earlier. 12 Recent research on this system includes that of Roach, 1~ Moldover, ~3 Garfunkel et al., 9 Kagoshima et al., 1° and Kierstead. 14 We first give a brief review of critical opalescence. The description is restricted to the static aspects of the critical phenomena. Recent progress on the dynamical aspects are of course very important; however, our experiments are related only to the static phenomena. Section 3 describes the experimental apparatus, procedure, data, and analysis. Section 4 is devoted to a discussion and comparison of results with those of other groups. A simple phenomenological theory of the density fluctuation related to the spinodal line is also developed and is compared with the scaling law equation of state. 2. BRIEF REVIEW OF CRITICAL OPALESCENCE AND CRITICAL P H E N O M E N A

Critical opalescence, which was discovered by Andrews 15 in the course of his measurements of the critical behavior of carbon dioxide, provides one of the striking manifestations of the critical point. It was showed qualitatively by Smoluchovsky 16 and quantitatively by Einstein 17 that critical opalescence is due to a density fluctuation and the scattered intensities are linearly proportional to the Fourier transform of the density~:lensity correlation function Zpp(q), as long as multiple scattering is ignored. Using the statistical explanation of entropy due to Boltzmann, Einstein derived the well-known fluctuation theorem lim Xpp(q) = kB T/(~3#/c3p)T

q~0

(1)

As the critical point is approached, (~p/c3p) T decreases to zero and the intensity of the scattered light becomes large. The first qualitative explanation of the huge increase of forward scattering, the q dependence of Xop(q), was due to Ornstein and Zernike: is They introduced a correlation function e_Kl r

G(r) ,.~ 4~zpR2

r

(2)

where R is called the Debye persistence length or the direct correlation length, and they obtained the following expression :

xo,,(q) = R-~('~ + q~)-~

(3)

Intensity of Critical Opalescence of Helium-4

573

By using the above-mentioned fluctuation theorem, (R~q) 2 = (~/~/~p)r(k~T) -~

(4)

Zpp(q) = kBT(O#/gP)T i( 1 + q2~2)-i

(5)

and

are obtained, where the long-range correlation length ~ is defined by ~c1~ = 1 :

~2 = kBTR2(a#/~p)T1

(6)

As the critical temperature is approached, ~ becomes large and the q dependence is enhanced. Fisher ~9 studied the Ornstein-Zernike theory (OZ theory) for a two-dimensional system and found that the OZ theory predicts that at the critical point the correlation increases with distance r. This is clearly an important failure. He modified the OZ theory by introducing an index q:

G(r) .-~ ( 4 n p R Z ) - i r Z - a - " e x p ( - x i r ) ,

0 < r/ < 1

(7)

where d is a dimension of the system. From this modified correlation function, the following result was obtained for small qZ : Zpp(q) = R:-20c21 + q2)2-a+,/2

(8)

Since the fluctuation theorem demands lim Zpp(q) = R-2/c2(2-d+n/2)

q~0

=

kBT(Ol.t/~p)~. 1

(9)

he obtained, finally, Zpo(q) = kBT(O,u/~p)T i( 1 + q2~2)z-a+,7/2

(10)

~2 = [knTR2/(Op/~p)T]i/(a-2-,1/2)

(i1)

and

The value of the index t/is believed to be small. Meanwhile, the critical region of fluids has long been a subject of great interest, both theoretically andexperimentally: Although the van der Waals equation gives a satisfactory qualitative description of the critical phenomena, it has been realized for many years that it is quantitatively incorrect. For example, Guggenheim 1° realized that the coexistence curve of a fluid system is not parabolic. Moreover, all the mean field theories give the same results as the van der Waals theory. In the early 1960's it came to be recognized that the critical point exponents were significant. But we have no theory which predicts the values of the critical exponents exactly for real systems. On the other hand, many critical exponents were introduced

574

Akira Tominaga

and the relations among certain exponents were studied. The definitions of the exponents vary slightly from author to author. We use the following definitions : (D1)

IP - Pcl/Pc = Bit[ t~

coexistence curve

(D2)

(~3p/~3p) r = F - l t r

P = Pc,

(03)

( ~ # / S p ) r = AI(p - p c ) / p J - 1

T = T~

(D4)

(d#/~P)r = r ' - l l t l ~ '

coexistence curve

(D5)

~ = ~ o t-~

P=Pc,

(D6)

~ = ~ltl-V'

coexistence curve

(D7)

pcv=

P =Pc,

(D8)

pcv = A'~Ltl -~ + A'2

A l t -~ + A2

T > T~

T>

T>

T~

T~

coexistence curve

where t = T/T~ - 1. Many other indices are of no interest to us. Numerous relations among the critical exponents have been derived. Among them the most rigorous are the thermodynamic inequalities, ~' + 2/3 + ¢ > 2 ~' +/3(1 + 6) _> 2 7'(6 + 1) _> (2 - ~')(6 - 1)

(Refs. 19 and 21)

(12)

(Ref. 22)

(13)

(Ref. 22)

(14)

The third inequality is less general than the other two, but is believed to be correct. As a matter of experimental observation, it appears likely that the inequalities are actually equalities. If this is true, the equality /3(6 - 1) = ~' (15) follows. Widom's 23 homogeneous function hypothesis predicts that, as far as the thermodynamic valiables are concerned, the primed and unprimed critical exponents are equal and the inequalities hold as equalities. Kadanoff's scaling law 24 supports the homogeneous function hypothesis. His work predicts not only the relations for the thermodynamic functions, but also certain relations among the correlation functions, for example, 3v=2-~ v(2 - 7) = 7

(16) (17)

and 6 - 1 37' 36 + 1 - 213+ 7' - 2 - ~/

(18)

Intensity of Critical Opalescence of Helium-4

575

The corresponding inequalities are 3v>2-~,

3v' > 2 - ~ '

(2 - q)v > 7

(Ref. 25)

(19)

(Ref. 26)

(20)

(Ref. 27)

(21)

and 36 - 1 > 2 - ~/, 6+1 -

3 ~ ' ~, > 2 - q 2/~+

respectively. Kadanoff divided the Ising spin system spatially so that each cell contains many spins but the characteristic length of the cell is much smaller than the long-range correlation length. Each cell~has some effective spin moments. The interaction between cells is described by the interaction between the effective spin moments. He assumed that the functional form of the free energy corresponding to the above-mentioned effective Hamiltonian is independent of the volume of the cell and obtained the result that the free energy is described by a homogeneous function. Recently Wilson 28 succeeded in explaining the origin of the critical singularities by the renormalization group method and developed the original Kadanoff cell model by dividing the phase space of the magnetic system into blocks. This method gives a microscopic approach to the sealing law and universality of the critical indices and gives some way of performing precise calculations of critical exponents. This method explains how the dimension of the system affects the critical behavior. If the dimension of the system is larger than four, the system behaves classically. For two-dimensional Ising system, we have the famous exact solution given by Onsager. It is interesting that for the real three-dimensional system we have no exact solution. 3, EXPERIMENTAL 3.1. Description of the Apparatus The incident beam from a single-mode helium-neon laser was attenuated to above 3 mW, its polarization was defined by an attenuator made of two polarizers, and it was focused to the center of a sample cell in the cryostat (Fig. 1). Part of the incident beam reflected by a half-mirror between the attenuator and the focussing lens was introduced to a photoconducting cell made of CdSe, which monitored the intensity of the incident beam. The scattered light at a right angle from helium in the sample cell was introduced to a cooled photomultiplier through an aperture. Photoelectron pulses from the photomultiplier were amplified by a homemade solid-state video amplifier and a discriminator and were counted by a digital counter.

576

Akira Tominaga

RADIATIONS H I E L ~ M LASE

SHROUD

FOCUSSING/ "'"-,t::::~:z-'~ /\SAMPLE MONITOR

~

CELL --~ DIAPHRAGM

SCALER DISCRIMINATOR Fig. 1. Schematic diagram of the optical system; the Brewster window of the laser is twisted in the actual system.

The outer radiation shield at liquid nitrogen temperature has four glass windows in Order to avoid heating of the sample cell by room-temperature radiation. The inner radiation shield at liquid helium temperature, however, has no window glass. The sample cell was made of OFHC copper with four indium-sealed windows (Fig. 2). All the windows at the vacuum shroud,

SAMPLE LINE

] IIIllllllll 0 5

10 mm

Fig. 2. Horizontal cross section of the sample cell made of oxide-free high-conductivity copper.

Intensity of Critical Opalescence of Helium-4

577

outer radiation shield, and sample cell are flat to better than 2/4 and were antireflection-coated with dielectric multilayers. Their reflection coefficients were less than 0.1 ~ at r o o m temperature. The sample cell has two diaphragms of 2 m m i.d. at each light path. Such a configuration is useful in eliminating stray light by scattering at windows and in minimizing the background. A 500-f~ heater and a germanium thermometer resistor were buried in the sample cell and were Used to regulate the temperature. 3° The electric leads from the thermometer resistor are thermally anchored at the sample cell to avoid variation of heat leak to the resistor. The temperature of the cell was regulated 3° to better than 20 # K for the whole range of temperature we measured. An identical system has been used in our laboratory in the course of a Second sound experiment near the lambda transition, where the temperature stability was better than a few microkelvin. The difference might be due to the difference of the thermal conductivity of the sample. The fill line of the sample gas is a stainless steel capillary, which is connected to a variable volume (about 0.2 liter) and a precision pressure gauge with a Bourdon tube made of fused quartz. All the electric leads and the sample capillary in the cryostat are placed in a vacuum space to avoid the influence from the variation of liquid level in the main helium bath. Between the sample cell and the main helium bath there is a 0.5-liter helium can, which serves as a temperature buffer when liquid helium is supplied into the main helium bath. In order to maintain a small cooling power some fine copper lines are connected between the 0.5-liter helium can and the sample cell. (See Figs. 3 and 4.)

Fig. 3. Location of the sample cell in the cryostat. HB1, helium bath; HB2, 0.5 liter helium bath; NB, liquid nitrogen bath; RS1, radiation.shield at 77K; RS2, radiation shield at 4.2 K; SC, sample cell; FL, fill line of sample gas ; SC is supported by three stainless steel screws from the bottom of HB2.

Akira Tominaga

57,8

TOSAMPLECELL

~ ~

ME

G ~UUARRDTO ZNTUBE BO ] PURI ~He CONTFAIEIND ER

(~)~- ~[CHA ]TRAP RCOAL ~[~He BOMB. ] Fig. 4. Gas handling system.

3.2. Procedure

In two-phase region below the critical temperature the gas-liquid interface was adjusted by the variable volume so that the laser light passes through the cell just below the interface. After setting at some temperature by our electronic regulator, the sample pressure varied very slowly to the final equilibrium value. This took from a few hours to one day. In the temperature region It[ < 10 -3 the intensity of the scattered light is sufficiently sensitive so as to indicate whether the system reaches equilibrium or not, and we could observe a systematic Variation of the scattered intensity when the system was not in equilibrium. In other words our experimental definition of equilibrium was such that we could not observe any systematic variation of the temperature, the pressure, and the scattered intensity. In the singlephase region beyond T~the time to reach equilibrium is not longer than 5 h. The fact that the relaxation time is smaller in the single-phase region than in the two-phase regio n is consistent with observations in heat capacity experiments, 3°31 but our relaxation time is longer. Since our system has dead volume at room temperature, a small gas flow in the capillary tube must exist for equilibrium to be reached. This may be one of the reasons for an abnormally long relaxation time. In the single-phase region, since we wanted to carry out the experiment along the critical isochore, the density of the sample should be measured. Instead of doing this, we used another method. Along the isotherm we measured the intensity of the scattered light as a function of pressure, which

Intensity of Critical Opalescence of Helium-4

579

was finely varied by the variable volume at r o o m temperature; then the intensity ~m a x i m u m occurred at some pressure. We assumed this ocCurred at the critical isochore because of following reasons. 1. F o r helium-4 we can calculate (~p/Op)T from P V T d a t a 11 and the m a x i m u m (c~p/c~#) T along the isotherms is found to occur at p = Pc within the experimental uncertainty. 2. F o r helium-3 Wallace and Meyer 32 make the same statement. 3. The scaling law equation of state 33'34 predicts that the (Op/c~#) T are symmetric around p = Pc4. For a van der Waals fluid we calculated (cqp/c~#) T and obtained the following: (c~p/c~#) T = [(3V - 1)2/24V 3] I T - T~(V)]- 1

T~(V) = 1 - ~ V

- 1)2 + ~ V - 1)3 + - - -

(22) (23)

Here the temperature T and the molar volume V are in units of T~ and V~ respectively. T~(V) is a projection of the spinodal line to the V - T plane. In the case that the molar volume Vis near unity, it follows that (OP/OP)r = ~{1 - (39/4)(V - 1)2}/{T - 1 + ~ V - 1)z}

(24)

Apparently, near the critical temperature (c~p/c~#)T is symmetric around the critical volume V = 1.

3.3. Data and Analysis I m p o r t a n t for this type of experiment are the thermometry and the estimation of the background. The latter was estimated by observing the empty cell at about T~and the critical pressure cell at liquid nitrogen temperature. Both gave 10.6 Hz. Since dark counts were below 1 Hz, almost all the background was due to the stray light from scattering at window surfaces. Since we measured the sample pressure, we can convert it to temperature. Kierstead 14 proposed an empirical equation (c3P/6~T)o c = D + B t + A t ln [tl

(25)

After the integration we obtain t = { P - Pc + (B/2 ~ A / 4 + A(ln Itl)/Z)t 2}

(26)

Using this formula and the values of constants reported by Kierstead, we estimated t from our measured valpe of pressure. Note that the temperature scale used by him was essentially the NBS provisional scale. Below T~ we also used the 1958 temperature scale. As is mentioned later, both results agree within the experimental uncertainty. But the NBS scale seems to be

580

Akira Tominaga

favorable for the scaling law which predicts that the primed and unprimed exponents are equal. The critical point was determined by observing maximum scattered intensities" For the first run, where the experiment was mainly concerned with the two-phase region, Pc = 1706.008 mm Hg

and

T~ = 5189.863 m K (T58)

and for the second run, the single-pha~se region, Pc = 1705.857 mm Hg

and

T~ = 5189.743 m K (T58)

We checked the zero point of the pressure gauge before and after each run. For the first run we could not find any drift of the zero point ; :however, for the second run we found a zero-point shift of about 0.25 m m H g . The above values of Pc are not corrected. We believe the value of Pc in the first run is more accurate than that of the second run (for a comparison of the critical parameter, refer to Ref. 14). Note that our final results are insensitive to the absolute values of critical parameters, because what is important is P - Pc. Figure 5 gives our results after Subtraction of the background. Departures from straight lines are due tO the q2 term in the Ornstein Zernike theory. From the tangents of the two straight parts we obtain 7(TNBs) = 1.171 ___0.02 7'(TNBs) = 1.170 __ 0.01

(27)

7'(T58) = 1.180 + 0.01 From the separation of two straight lines we obtain F/F' = 4.5 __ 0.3

(if 7 = 7')

(28)

Generally speaking, incident light passing through the fluid is attenuated. In the critical region, as the scattering cross section becomes large, the attenuation of the incident beam is not negligible when the path length in the fluid is too long. Puglieli and Ford 35 obtained the following equation for the turbidity z: z = An

( ~ ) r [ 2~z- +2~+~3 l ln( 1 + 2 ~ )

2(1~2+~!1 k B T

(29)

(30) = 2ko~

(31)

where e is the dielectric constant of the fluid. The turbidity is a decreasing

Intensity of Critical Opalescence of Helium-4

b

J

i

,

--.-

,

~

~

,

i

.~.

i

i

581

~

i

i

i

4H

10 4

~0

c~ o

10 5

>-

T
10,~

11

1 010-5

I

I

I

I

I

1°-4

Ill

I

I t I 10-3

10-2

10-1

Fig. 5. The intensity of the scattered light at a right angle versus reduced temperature. Data below T~ are along the liquid side of the coexistence curve. Data above T~ are along the critical isochore. The solid lines correspond to the zero angle limit and 7 = 7' = 1.171.

function of ~ and in the limit of small ~, z becomes (32)

z o = (8/3)~A(c3p/c~#)TkBT

In our case A = 9.2 x i0 ~6 cm 3 g - 2 and ( @ / ~ g ) r k ~ T = F t - ~kB T

"

= 1 0 - 2 1 g2 ClTI-3

at

t = 10 -5

where we use Edward's data for c~e/t?p and Roach's data for the value of (0p/0#) r. Thus the uppe r limit of the turbidity is z O = 10 = 3 c m

at

t=

10 -5

Since the light path length of our cell is 1 = 1.2 cm, the attenuation of an incident b e a m is smaller than exp ( - z o l ) = 3 x 10 -3. Thus, we canneglect the temperature dependence of the incident beam and can obtain the correlation length. Figure 6 shows the temperature dependence o f ' t h e correlation length. Fitting t h e data to the defining relations (D5) and (D6),

582

AkiraTominaga

104

°.~

oT>T¢ o T < Tc

3

o

z L~ -J 10 2 z

2 t.0 xlt I- O63"~'h

cc o

%-5

i

i

I

i

i

I ii

I

I

i

i

i

i i

163

10- ~

REDUCED TEMPERATURE I t l

Fig. 6. Temperature dependence of the longrange correlation length. The data points were obtained assuming the Ornstein-Zernike theory and 7 = 7' = 1.171.

we o b t a i n 4o = 2.2 _ 0.6/~,

v = 0.63 ± 0.1

~ = 1.0 ___ 0.3 A,

v' = 0.62 ± 0.1

If we a s s u m e v = v', then we o b t a i n v = v' = 0.62 ± 0.1 4o = 2.24 ___ 0.6 A,

¢~ = 1.0 ± 0.3 A,

¢o/¢~ = 2.2 ± 0.4

4. D I S C U S S I O N

4.1. Comparison with Other Results O u r result y = y' = 1.171 + 0.02 is consistent with t h o s e o f R o a c h from P V T m e a s u r e m e n t s a n d of M o l d o v e r ' s specific heat e x p e r i m e n t (see T a b l e I). G a r f u n k e l et al.'s results f r o m light s c a t t e r i n g e x p e r i m e n t s are i n c o n s i s t e n t with ours. But their e s t i m a t i o n of the b a c k g r o u n d was wrong, as we m e n t i o n e d at the 13th I n t e r n a t i o n a l Conference o n L o w T e m p e r a t u r e Physics. 41 K a g o s h i m a et al. o b t a i n e d a value larger t h a n t h a t for the socalled classical fluid a n d p o i n t e d o u t that the q u a n t u m n a t u r e of 4He has a r e m a r k a b l e influence on the critical b e h a v i o r of 4He. W e c a n n o t explain why they o b t a i n e d such large values of y a n d y'. W e only guess that there exist some large systematic errors in their experiment.

4.2. Comparison with Scaling Law Since o u r value of the critical e x p o n e n t ~,' a n d the m o s t precise values 13,1 ¢ of # a n d ~' satisfy the R u s h b r o o k e - F i s h e r i n e q u a l i t y ~'+2#+y'>2

Intensity of Critical Opalescence of Helium-4

>~ " ~ o ~

I I

r ~÷r

÷~

("4

E

~4

I

i i

F~-~'~ .~' C'4

e~

~1 -

~'~I~ ~ , ' ~--. ÷~ I-

I

~1

JI

I

583

584

Akira Tominaga

almost as an equality, we can estimate other exponents from the scaling law" ¢t=a'=

2-(2fl+2)=0.12+0.2

a = y//~ + 1 = 4.30 + 0.07 v = v' = (2 - a)/3 = 0.627 + 0.007 rl = 2 -

7/v

= 0.13 + 0.06

The value of 6 is very close to Kiang's calculation 36 on the liquid-droplet model. T h e values of v and r/are consistent with our experiments.

4.3. Critical Coefficient As mentioned in Section 2, the modified O r n s t e i n - Z e r n i k e theory with scaling law predicts r/r'

= (~o/¢b) r/v = (~o/¢;) 2 - .

(33)

O u r results satisfy this relationship. Schofield e t al. 34'37 p r o p o s e d a parametric representation of the t h e r m o d y n a m i c functions called a linear model. In this model the critical coefficients are determined by the critical exponents. A m o n g them there is a relation F 2 1 F-- = (b 2 - 1)~- 1 1 - b2(1 - 2/~) The right-hand side, as a function of is given by

(34)

b 2, has a m i n i m u m value when b 2

b 2 = (7 - 2/~)/~(1 - 2/~) They analyzed m u c h experimental data and found that the value of the lefth a n d side is close to the m i n i m u m value of the right-hand side, O u r experimental value of y and our choice of the value *4 of fl lead to F / F ' = 4.1, which is smaller than our observation of 4.5. Recently H u a n g and H o 38 p r o p o s e d a n o t h e r equation of state and obtained r/r'=

(3/2fl - 1)~

(35)

which also gives a smaller value than our observation.

4.4. Relation to Spinodal Line As is well known, the van der Waals equation predicts the existence of the spinodal line, where the compressibility becomes infinite, and the projection of the s p i n o d a l line to the p - T plane, T = T~(p), is a p a r a b o l a t h r o u g h critical point like that of the coexistence curve T = T~(p).

Intensity of Critical Opalescence of Helium-4.

585

If we assume (A1)

~

r = F

(A2)

T~ -

~

]

and T~(p) = E(T~ -

Ts(p))

where F and E are some constants independent of p and T, then ~P

=F

"

P =Pc,

T > T~

(36)

T

and along the coexistence curve

Comparing with (D2) and (D4), we obtain 7' = 7

(38)

and F/F' = (E - l y

(39)

Moreover, using the empirical law (D1)

o -

pc

pc

_ Bird.

~

-

rBI

T~ /

we obtain along the critical isotherm (Sp/a#)z = F B r / P E - e J ( p

-

pc)/pc[ -~/~

Comparing with (D3), we obtain A = F-1B-~/~Er

(40)

and 3-

(41)

1 = y/fl

For the heat capacity per unit volume in two-phase region we obtain IT~

TB~2~+7 - 2

after rather tedious calculations. Comparing with (D8), we obtain cz' + 2,8 + ~ = 2

(42)

586

Akira Tominaga

If we assume

(A3) sx= ~°( T --T~(P)/[-~ then along the critical isochore

(43) and along the coexistence curve (44) Comparing with (D5) and (D6),

~0/~; =

(E -

(45)

ly

and v'= v

(46)

The results (38), (41), (42), and (46) are exactly the same as that of the scaling law, and the relations (39), (40), and (45) are correct for a van der Waals fluid, for which the values of E, fl, and 7 are 3, 0.5, and 1.0, respectively. From Eqs. (39) and (45) we obtain F/F' = (~o/~y/v

(47)

which is exactly the same as the prediction of the Ornstein-Zernike theory and the modified theory with the scaling law. [see Eq. (33)]. If we assume (A4)

E = 3/(2fi)

then Eq. (39) is the same as the prediction (35) by Huang and Ho. 38 From assumptions (A2) and (A3) and definition (D1) we obtain along" the critical isotherm = ~oE-VBWI~J(P

-

Pc)/Pc[-~/~

=

--

lac)lla~[-~/~

~oE-'B~/e[(#

(48)

where #c denotes the chemical potential at the critical point. Since Fisher's inequality suggests v

6-1

-->

~

-

(2 -

6+1 .)~

3~

the result (48) is exactly the same as the new inequality proposed by Liu and Stanley. 39

Intensity of Critical Opalescence of Helium-4

o

<

+I

-l-i

~ll

-

+I

-l-i

II

o

~ o +io

+i

+i +I ~

+i

0

+O~I ~', ~°

+i?I* ? + +

587

588

Akira Tominaga

These considerations suggest that the critical anomalies are related not only to the critical point but also to the spinodal line. 5. C O N C L U S I O N S Measurements of the right-angle light scattering by helium-4 near the critical point T~ have been made as a function of temperature. The results obtained here are 7 = 1.171 _ 0.02, ~' = 1.170 _+ 0.01, ~ = (2.24 _ 0.6) t - ° ' 6 3 + ° ' a A , ~ ' = (1.0 _+ 0 . 3 ) t - ° ' 6 2 + ° ' l A , F / F ' = 4.5 _ 0.3, and ~0/¢~ = 2.2 _ 0.4. These results are consistent with P V T d a t a . The critical behavior of helium-4 is almost the same as that of the classical fluids (Table II). We cannot accept the experiments and the results obtained by Kagoshima et al., who claimed that the influence of the quantum nature of helium-4 was evident. ACKNOWLEDGMENTS The author wishes to thank Prof. Y. N a r a h a r a and the members of his laboratory, especially Mr. K. M a t s u m o t o and Mr. A. Nakazawa, for encouragement and assistance during the course of this work. The author is also indebted to Prof. T. Soda and Dr. T. Osawa for helpful discussions and to Prof. K. K u r i y a m a for planning a part of the photon-counting system.

ADDENDUM Dr. K. Ohbayashi told us recently that Dr. S. Kagoshima has repeated the measurements of the intensities of critical opalescence of helium-4 along the critical isochore above the critical temperature and obtained the results 7 = 1.18 ___ 0.02 and v = 0.59 _+ 0.02, which agree well with our experiments.

REFERENCES 1. I. W. Smith, M. Giglio, and G. B. Benedek, Phys. Rev. Letters27, 1556 (1971). 2. H. D. Bale, B. C. Dobbs, J. S. Lin, and Paul W. Schmidt, Phys. Rev. Letters25, 1556 (1970). 3. G. T. Feke, G. A. Hawkins , J. B. Lastovka, and G. B. Benedek, Phys. Rev. Letters 27, 1780 (1971). 4. J. H. Lunacek and D. S. Cannell, Phys. Rev. Letters 27, 841 (197i). 5. J. A. White and B. S. Maccabee, Phys. Rev. Letters 26, 1468 (1971). 6. V. G. Puglielli and N. C. Ford, Jr., Phys. Rev. Letters 25, 143 (1970.). 7. M. Giglio and G. B. Benedek, Phys. Rev. Letters 23, 1145, (1969). 8. G. T. Feke, J. B. Lastovka, G. B. Benedek, K. H. Langley, and P. B. Elterman, Optics Communications 7, 13 (1973).

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9. M. P. Garfunkel; W. I. Goldburg, and C. M. Huang, in Proc. 12th Int. Conf. Low Temp. Phys., Kyoto, 1970 (Academic Press of Japan, Kyoto, 1971), p. 59. 10. S. Kagoshima, K. Ohbayashi, and A. Ikushima, Phys. Letters 41A, 463 (1972). 11. P. R. Roach, Phys. Rev. 170, 213 (1968). 12. A. Tominaga and Y. Narahara, Phys. Letters 41A, 353 (1972). 13. M. R. Moldover, Phys. Rev. 182, 342 (1969). 14. H. A. Kierstead, Phys. Rev. A 7, 242 (1973); 3, 329 (1971). 15. T. Andrews, Phil. Trans. Soc. 159, 575 (1869). 16. M. v. Smoluchovsky, Ann. der Phys. 25, 205 (1908). 17. A. Einstein, Ann. der Phys. 33, 1275 (1910). 18. L. S. Ornstein and F. Zernike, Proc. See. Sci. K. med. Akad. Wet. 17, 793 (1914). 19. M. E. Fisher, J. Math. Phys. 5, 944 (1964). 20. E. A. Guggenheim, J. Chem. Phys. 13, 253 (I945). 21. G. S. Rushbrooke, J. Chem. Phys. 43, 3439 (1965). 22. R. B. Griffiths, J. Chem. Phys. 43, 1958 (1965). 23. B. Widom, J. Chem. Phys. 43, 3892 (1965). 24. L. P. Kadanoff, Physics 2, 263 (1966). 25. B. D. Josephson, Proc. Phys. Soc. 92, 269, 276 (1967). 26. M. E. Fisher, Phys. Rev. 180, 594 (1969). 27. M. J. Buckingham and J. D. Gunton, Phys. Rev. 178, 848 (1969). 28. K. G. Wilson, Phys. Rev. B 4, 3174, 3184 (1971). 29, A. Tominaga, Cryogenics 12, 389 (1972). 30. G. R. Brown and H. Meyer, Phys. Rev+ A 6, 364 (1972). 31. D. Dahl and M. R. Moldover, Phys. Rev. A 6, 1915 (1972). 32. 13. Wallace, Jr. and H. Meyer, Phys. Rev. A 2, 1563 (1970). 33. M. Vicentini, J. M. H. Levelt Sengers, and M. S. Green, Phys. Rev. Letters 22, 389 (1969). 34. P. Schofield, or. D. Litster, and J. T. Ho, Phys. Rev. Letters 23, 1098 (1969). 35. V. G. Puglieli and N. C. Ford, Jr., Phys. Rev. Letters 25, 143 (1970). 36. C. S. Kiang, Phys. Rev. Letters 24, 47 (1970). 37. P. Schofield, Phys. Rev. Letters 22, 606 (1969). 38. C. C. Huang and J. T. Ho, Phys. Rev. A 7, 1304 (1973). 39. L. L. Liu and H. E. Stanley, Phys. Rev. B7, 3241 (1973). 40. T. K. Lira, H. L. Swinney, K. H. Langley, and T. A. Kachnowski, Phys. Rev. Letters 27, 1776 (1971). 41. A. Tominaga and Y. Narahara, in Low Temperature Physics LT-13 (Proc. 13th Int. Conf. Low Temp. Phys., Boulder, 1972), p. 377 (Plenum Press, New York, 1974).