The thermodynamic properties of liquid 3He-4He mixtures between 0 and 20 atm in the limit of absolute zero temperature

Physica 144B (1987) 127-144 North-Holland, Amsterdam

THE THERMODYNAMIC PROPERTIES OF LIQUID 3He-4He MIXTURES BETWEEN 0 AND 20 ATM IN THE LIMIT OF ABSOLUTE ZERO TEMPERATURE R. DE B R U Y N O U B O T E R Kamerlingh Onnes Laboratorium der Rijksuniversiteit Leiden, Nieuwsteeg 18, 2311 SB Leiden, The Netherlands

Chen Ning YANG Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook NY 11794, USA Received 10 October 1986

The presently existing thermodynamic data of liquid 3He-4He mixtures between 0 and 20 atm in the limit of absolute zero temperature are presented in a coherent way, without reference to models, through constructing the thermodynamic surface e(n3, n4), where e is the energy density and n 3, n 4 are the molar densities. The chemical potentials are the partial derivative of e. The construction is made through the information contained in the existing thermodynamical data. The geometrical meaning of the surface whose coordinates are the energy density and the molar densities of both components is explained. In the pressure range considered the energy density and the chemical potentials are determined as a function of the molar densities. Details of the thermodynamic functions are summarized in the tables at the end of the article. Special attention is given to the saturated solution curve and the one-phase region in which the energy density appears to be expandable not in integral but in fractional powers of the 3He density.

1. Introduction In this article available thermodynamic data are collected and the thermodynamic functions of liquid 3He-4He mixtures are calculated in the limit of zero temperature in the pressure range between 0 and 20 atm for both the one- and two-phase regions. After the discovery by Edwards et al. [1] in 1965 that 3He has a finite solubility in 4He near absolute zero, many accurate experimental determinations of several thermodynamic properties, such as the molar volume, the specific heat, the heat of mixing and the osmotic pressure have been made in the subsequent years, mainly by the research group of the Ohio State University. They are excellently reviewed by E b n e r and Edwards [9]. The properties in the dilute onephase region were mainly described in a dilute solution picture where the 3He quasi-particles can move nearly independent of one another in a superfluid 4He background, with the aid of the L a n d a u - P o m e r a n c h u k model. When the 3He content increases, quasi-particle interactions be-

come increasingly important and are, for instance, treated in the B a r d e e n - B a y m - P i n e s theory. It is the purpose of this article to present the presently existing thermodynamic data in a coherent way, without reference to models, and free from theoretical suppositions. The energy density e -- U / V and the chemical potentials /z3 and /z4 will be determined as a function of the molar densities n 3 ~ - X / V and n 4 ----(1 -- X ) / V , where U, V and X are, respectively, the internal molar energy, the molar volume and the molar concentration N 3 / ( N 3 + N4) of the mixture. Use will be made of the following relations valid at T = 0: U = - p V + X i z 3 + (1

-

X)[.£

4 ,

d U = - p dV + (/z 3 - / x 4 ) d X , e = - - p + n3P, 3 + n4/.t 4 ,

de = / z 3 dn 3 + / x 4 dn 4 , V d p = X d / ~ 3 -t- (1 - X )

dp = n 3 d/z 3 + n 4 d/x4 .

0378-4363/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

d/z 4 ,

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

128

Photograph 1. View of a schematic model of the e(n3, n 4 ) surface at T = 0. Shown are isobars at p = 0, 4, 8, 12, 16 and 20 atm with the p = 0 isobar labelled as A B C . The curved lines A A ' and CC' in the co-ordinate planes are the lines e°3(n°) and e°(n °) for the pure components. T h e curved line B B ' is the saturated solution curve S. A A ' B ' B represents the two-phase region II. C C ' B ' B represents the one-phase region I. C o m p a r e figs. 7 and 8. According to general principles of thermodynamics, the surface has continuous first derivatives satisfying de = ~1'/'3dn3 +/x4 dn4 T h e surface is the envelope of all its tangent planes, which must lie below the surface. Furthermore, because of the relationship

Oe --p

:

e

--

n3/d. 3 --

n4/& 4 =

e

--

n 3 - -On 3

3e n 4 - 0n 4

the intercept of each tangent plane on the e axis is at e = - p . T h u s all tangent planes intercept the e-axis on the - e side. In region II each tangent plane is tangent to the e - n 3 - n 4 surface along an entire isobar which is a mathematical straight line. T h u s the plane d A B is tangent to the surface along the whole straight line A B at zero pressure. Similarly for the plane O ' A ' B ' at 20 arm. (The difference between the points O and O ' is 20 atm or about 2.03 J/cm3). Notice that they are also tangent at B and B', respectively, to the region I part of the surface (i.e. the C C ' B ' B part). T h e tangent planes in region I are each tangent to the surface at only one point. The slope of the intersecting lines of the tangent planes with the vertical co-ordinate planes are equal to their respective chemical potentials.

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

The geometrical meaning of these equations is explained in the captions of photograph 1 which presents a model of the e - n 3 - n 4 surface. The model is made with some features exaggerated to bring out special properties of the H e 4He thermodynamical surface. In this approach we follow the ideas of Gibbs (The Scientific Papers of J. Williard Gibbs, Dover Publications, Vol. 1, p. 85) who was especially fond of the geometrical meaning of thermodynamic functions. [See M.J. Klein in the Dictionary of Scientific Biography (Scribner) entry on Gibbs, Vol. V, p. 386.] The following thermodynamic functions will be discussed: The molar volumes and densities and the saturated solution curve at absolute zero in section 2; the energy density and the chemical potential of the pure components in section 3; the energy density and the chemical potentials in the one- and two-phase region in section 4 and the analytic or non-analytic description of e(n3) in the one-phase region in section 5. The variation of the measured molar volumes, molar internal energies and osmotic pressures with temperature are known so that their values in the ground state at absolute zero were determined by extrapolation. Details of the thermodynamic functions are summarized in the tables at the end of the article.

2. The molar volumes and densities and the saturated solution curve at absolute zero

Already in 1957 Kerr [3] found that his molar volume determinations at saturated vapour pressure in the He lI region at temperatures above 1.2 K satisfied the relation V ( T , X) = (1 - X ) V ° ( T ) + XV3(T ) ,

in which the partial molar volume V3 - V + (1 X ) J V / a X was slightly less than the molar volume V ° of pure liquid 3He and independent o f the concentration. Later on very careful measure-

ments of the molar volume of dilute mixtures were performed by Ifft, et al [4] at saturated

129

vapour pressure down to 0.025 K and by Watson et al. [5] for mixtures up to 10% 3He concentration down to 0.05 K at pressures up to 22 atm. Also the measurements in this pressure range of Abraham et al. [6] and of Boghosian and Meyer [7] have to be mentioned. In the review article of Ebner and Edwards [9] these measurements are discussed extensively. In their table 1 on p. 100 values for V ° and (V3 - V°4)/V ° at absolute zero are given in the pressure range between 0 and 20 atm. From these values the partial molar volume V3(p) is calculated and plotted in fig. 1 and is given in table II. The molar densities n 3 and n 4 are then determined. The empirical fact that in the one-phase region (I) V3 is independent of the concentration X greatly simplifies matters. What it means is that the isobars in the one-phase region, when projected into the n a - n 3 plane, are straight lines: n4/n ° = 1 - n3V3 ,

see fig. 8a. In fact, the intercept O A = 1 / V 3 (fig. 8b). The slopes of the isobars are defined by the parameter f l i ( p ) = - - ( a n 4 / O n 3 ) p = V 3 / V °, which is plotted in fig. 4a. In fig. 1, the (partial) molar volumes are plotted at absolute zero as a function of pressure: the ~artial molar volume V3 of the 3He solute in the He surrounding in the dilute one-phase region, the molar volumes V ° [2] and V ° [1] of the pure components. At zero pressure, V3 35.41cm3/mol and V ° ~36.87 cm3/mol. From fig. 1 one observes that the difference between Vs° and V3 gradually decreases at increasing pressure and even changes sign. Therefore V3 and V ° are plotted in fig. 2 as a function of pressure for the pressure range between 10 and 20 atm, using a much more sensitive scale for the molar volumes. That this is a real effect directly follows from the observation of the saturated 3He concentration X s ( p ) at absolute zero (fig. 3). The saturated 3He concentration Xs as a function of pressure has been measured by Landau et al. [10], by Watson et al. [5] and by Abraham et al. [6] and is reviewed by Ebner and Edwards [9] on pp. 106-107, fig. 12, of their article. In fig. 3,

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

130

30 20 E E

uE 29

V3

o.

30

1c V~

V3

28 1

2C

3

J

J 10 plotm

I

i 20

10

15

C 0.06

20

.,,,"T

J

0.07

Q08

I

0.09

X

Dlotm

Fig. 1. (Left) The (partial) molar volumes at absolute zero are plotted as a function of the pressure: the partial molar volume V3 [3-7, 9] of the 3He solute in the one-phase region and the molar volumes V ° and V ° of the pure components [2, 9]. Experiment [3-7, 9] shows that in the dilute one-phase region (I) just above absolute zero the partial molar volume V3 is independent of the concentration X. One observes that the difference between V ° and V3 in the range from 0 to 10 atm gradually decreases at increasing pressure and at higher pressures even changes of sign. Fig. 2. (Center) V3 and V ° are plotted as a function of pressure in the range between 10 and 20 atm, using a much more sensitive scale for the molar volumes as used in. fig. 1. Fig. 3. (Right) The saturated 3He concentration X, at absolute zero is plotted as a function of the pressure p [5, 6, 9, 10]. The saturated 3He concentration at zero pressure is equal to 6.6%. As is shown in the text, figs. 1, 2 and 3 are thermodynamically consistent with each other. V~ and V4° are tabulated for different pressures in table I, V3 and X, in table II.

X , ( p ) is p l o t t e d . A t z e r o p r e s s u r e it is e q u a l t o 6 . 6 % . A t i n c r e a s i n g p r e s s u r e , u p to a b o u t 10 a t m it i n c r e a s e s , a n d t h e n slowly d e c r e a s e s in t h e p r e s s u r e r a n g e b e t w e e n 10 a n d 20 a t m . T h e v a r i a t i o n o f X , w i t h p r e s s u r e c a n also b e calculated by equating the chemical potential (/t3~) o f t h e o n e - p h a s e r e g i o n w i t h t h a t (/t°3) o f t h e p u r e 3He p h a s e . T h e l a t t e r ( / t ° ) is e v a l u a t e d with t h e e q u a t i o n d / t ° = V ° d p , o r P

go(p)

= / t 3o( P = O ) +

f

Vo 3dp.

0

T h e f o r m e r (/t3~) is e v a l u a t e d b y i n t e g r a t i n g t h e equation d/t3 dp

1 n3

n4 d / t 4 n 3 dp

a l o n g a c u r v e at fixed X = n3/(n 3 + n4). N o w , approximately, d/t4/d p = 1/n°(p), Thus d/t 3

dp

1 ~--

n4 --

n 3

6 =V3

or

n3n4 p

/t3~(P) = / t a ( P = O, Xs( p * 0 ) ) + J V3 dp . 0

Fig. 8d s h o w s t h e c o n s t r u c t i o n o f X , ( p ) b y this m e t h o d . A s l o n g as V ° > V3 t h e s a t u r a t e d c o n c e n t r a t i o n will i n c r e a s e with i n c r e a s i n g p r e s s u r e , w h e n e v e n t u a l l y V o3 < V3, t h e s a t u r a t e d c o n c e n t r a t i o n X , d e c r e a s e s . F o r t u n a t e l y , t h e 3He c h e m i c a l p o t e n t i a l at z e r o p r e s s u r e a n d t e m p e r a t u r e is d e t e r m i n e d e x p e r i m e n t a l l y [11] n o t o n l y in t h e c o n c e n t r a t i o n r a n g e 0 < X < A s ( p = 0) = 6 . 6 % , b u t also in t h e s u p e r s a t u r a t e d r a n g e f r o m 6 . 6 % u p to n e a r l y 1 5 % , see t h e r e v i e w b y E b n e r

R. de Bruyn Ouboter and C.N. Yang / Liquid aHe-4He mixtures at absolute zero

and Edwards [9] on pp. 100-101, fig. 9. This allows for the construction method of fig. 8d. The results presented in the figs. 1, 2 and 3, and given in tables I and II, are thermodynamically consistent with each other. Besides making use of the 3 H e chemical potentials in the onephase region and in the supersaturated range, also the saturated solution data X s ( p ) for pressure values between 0 and 10 atm are used to calculate /z3(X> Xs, p = 0) for the corresponding concentration range between 6.6% and 9.49%. Subsequently these results are used to calculate (V3 - V °) at higher pressures in the range between 10 and 20 atm where the saturated concentration decreases. Thus determined partial molar volume data V3(p), with respect to the molar volume data V ° ( p ) of pure 3 H e which are left here unchanged, are plotted in fig. 2 and given in table II. They only give rise to very small corrections. Fig. 1 shows that as the pressure increases, V ° - V 3 changes sign. This is not entirely unreasonable. The interatomic attractive potentials, as a function of distance, are almost the same for 3 H e and 4 H e , b u t the zero-point energy varies roughly inversely proportional to the atomic mass. Due to this difference in zero-point energy, pure 3He has a larger molar volume and a larger compressibility than pure liquid 4 H e . On the other hand, in a dilute mixture in the onephase region, 3He in a 4He surrounding, at zero pressure, the partial molar volume V3 is only slightly3 less than the molar volume V3° of pure liquid He, which can be understood on the basis of the equilibrium between the weak attractive and the repulsive zero-point forces [3]. By the same reason the compressibility of pure 3 H e is slightly larger than that of a 3He cell in a 4 H e surrounding. Consequently the curves V ° ( p ) and V3(p) must cross each other. The molar densities n 3 and n 4 for the 3He and 4He components, respectively, are determined by the equations n 3 =-- X / V

and

n 4 ----(1 - X ) / V

;

the molar densities of the pure components by n3° = 1 / V °

and

n 40 = I l V 0

131

1.oI 1.3 -

~

1.2 1 3 ~ 0

1.1o

i

i

IO

i

i

20

P/arm

,g 58

E -~ 5 6

"~ 5 4

52 b [

I

10

I

I

20

Platm.

Fig. 4. (a) fl(p) ~ -(an4/an3) ~ is plotted as a function of the pressure p. In the two-phase region (II) in a n 4 versus n3 plot the lines at constant pressure are straight by definition, as is shown in fig. 7b, with slope//~i. In the one-phase region (I) experiment shows [3-7, 9] that these lines are straight also with a slope: /3~(p) = Va/V °, since the partial molar volume V3 in the one-phase region is independent of concentration. The difference between /3~ and fll in the range from 0 to 10 atm gradually decreases at increasing pressure and at higher pressures even changes sign. This is due to the fact that V3 - V 03 changes sign when p increases (see the figs. 1, 2 and 8). (b) axi(p ) ~ (de/an3) p for the two-phase region (II) is plotted as a function of the pressure p. In the two-phase region in a e versus n 3 plot the lines at constant pressure are straight by definition, as is shown in fig. 7d, with slope a~. In the one-phase region (I) these lines are not straight, as is shown clearly in fig. 9 where the heat of mixing data are plotted [11], although the contribution of the heat of mixing, u E / v , to the total energy density e is very small. ~i,/3n and a n are tabulated for different pressures in table II.

R. de Bruyn Ouboter and C.N. Yang I Liquid 3He-4He mixtures at absolute zero

132

0

__

0

__

-20.5615 joule/mole [14] a n d U 4 -- - L 4 - 5 9 . 6 1 4 3 j o u l e / m o l e [9], (see for c o m m e n t a r y table 1). T h e internal e n e r g y per unit v o l u m e e ° = U°/V ° of the pure c o m p o n e n t at z e r o pressure is equal to: e 03 = --0.558 J / c , 3 a n d e 40 = - 2 . 1 6 J / c m 3. T h e internal m o l a r e n e r g y at finite pressure is d e t e r m i n e d by integration: U ° ( p ) = U ° ( p = 0 ) - fop p d V and the e n e r g y per unit v o l u m e e°(p) = U ° ( p ) / V ° ( p ) , using the V ° and n o values of table I. T h e chemical p o t e n t i a l / . 0 o f the pure c o m p o nent at absolute zero and zero pressure is identical with the m o l a r internal energy, h e n c e

F o r V ° ( p ) at absolute zero the experimental data o f A b r a h a m and O s b o r n e and o f Greywall [2] were used. F o r V ° ( p ) and V3(p) at absolute zero, the values as tabulated by E b n e r and E d wards [9] were used after small corrections applied to achieve consistency with the saturated solution curve X s ( p ) , see figs. 1 and 2 and tables I and II. In fig. 7a and b for the one- and two-phase region, the 4He m o l a r density n 4 versus the 3He m o l a r density n 3 are plotted, respectively, at 0, 2, 4, 6 , . . . , 18 and 20 atm. In the o n e - p h a s e region the lines o f constant pressure are empirically straight with slope flI--- ( a n 4 / a n 3 ) p = V3/V °, and in the two-phase region these lines are straight by definition with s l o p e / 3 n = - ( a n 4 / a n 3 ) p. The two slopes fli and /3n are plotted versus the pressure in fig. 4a.

O(p = o) = v ° , ( p = o) = - L ° 4 = -59.6143 J/mol,

O(p = 0 ) = U ° ( p = 0 ) =

3. T h e e n e r g y d e n s i t y a n d t h e c h e m i c a l p o t e n t i a l of the pure components

-L

°

= -20.5615 J/mol. T h e chemical potential o f the p u r e c o m p o n e n t at finite pressure is d e t e r m i n e d by integration: /z°(p) = / x ° ( p = 0) + f0p V ° d p , using the V ° values of table I.

T h e m o l a r internal e n e r g y U ° of the pure c o m p o n e n t at absolute zero and zero pressure is equal to minus the heat o f vaporization L ° at absolute zero and equal to: U ° = - L ° =

4c 0

P/arm 10 i

2o I

E~(p)

C O

E

E-1

5 -2C 0

~9 i

pO4(p)

0

-40

E,°(p)

-60 0

I

L 10 P/arm

I

I 2o

• • e30 and e4o and the chemical potentials/L 30 and/~40 of the pure components are plotted as a function of Fig. 5. The energy densmes the pressure p between 0 and 20 a t . . The molar energies U ° and the chemical potentials/o are increasing when the pressure increases in contrast with the energy density e °.

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

-O.5

"~ 4C

133

/

/ "~ "5-0.6: 0

0

-0.

o ~

0

I

I

c

-2C

[

~)

I

0.030 0.035 n O I ( m o l e 3He I cm 3)

I

I

I

0.030 0.035 n O l ( m o l e 3He I c m 3) O

u

~

5 0-4C

-o-

o(n4) o

.o -2.3 oj

o

~t<0 ~ -2.4

0.035

I

I

0.040

-6C I

I

0.035

n ° / (mole 4He Icm 3)

I

I

0.040 n ° I ( m o l e 4He Icm 3)

Fig. 6. The energy d e n s i t i e s a n d t h e c h e m i c a l p o t e n t i a l s of t h e p u r e c o m p o n e n t s are p l o t t e d as a f u n c t i o n of t h e i r m o l a r d e n s i t i e s : e 03 ( n 03 ) , / %0( n 30) , e ,0( n 40 ) , [ A0, 4 ( n 04 ) . F o r p u r e 3He, in c o n t r a s t w i t h p u r e 4He, o n e o b s e r v e s a m i n i m u m in its e n e r g y d e n s i t y e ° ( n ° ) . F o r l o w d e n s i t i e s , I%o = Oea/an3 o o < 0, w h e r e a s for h i g h d e n s i t i e s , / x ° a~3/0n • 3 0 0 > 0. O n t h e o t h e r h a n d for p u r e 4He,/.4, 40= 0 £ 4o/ a n o4 <~ 0 m 0 0 0 0 0 0 t h e c o m p l e t e d e n s i t y r a n g e in q u e s t i o n , e4,/~4, n4, e3,/~3 a n d n 3 are t a b u l a t e d for d i f f e r e n t p r e s s u r e s in t a b l e I. =

The results are given in table I and are presented in figs. 5 and 6. In fig. 5 the energy densities 0 0 • • 0 0 % and e 4 and the chemical potentmls/% and/~4 are plotted as a function of the pressure between 0 and 20 atm. In fig. 6 the energy densities and the chemical potentials are plotted as a function • • • o 0 0 o' o o of their molar densmes: e3(n3),/~3(n3), e4(n4) and /J,4(n4). 0 0 For pure 3He, in contrast with pure 4He, one observes a minimum in its energy density e3(n3). 0 0 For low densities, /%0 = 0 0 o Oea/On3 0 . On the other hand for pure 4He, o 0 0 P,4 = ae4/an4 < 0 in the complete density range in question• The molar energies U ° and the chemical potentials /x° are increasing functions when the pressure increases, in contrast with the energy density e ° as is obvious from the figures.

4. The energy density and the chemical potentials in the one- and two-phase regions The one-phase region at zero pressure has been investigated with great care by the Ohio State University group: (1) As mentioned in the preceding sections the molar volumes and the saturated solution curve were determined. (2) Seligman et al. [11] measured the heat of mlxang together with the specific heat so that they were able to determine the heat of mixing at absolute zero. Their results are shown in fig. 9a together with some recent modifications [12] applied to the data by using values for the/x 4 chemical potential and recent specific heat data. (3) Landau et al. [10] measured the osmotic

134

R. de Bruyn Ouboter and C.N. Yang

pressure down to 0.027 K, The data can be extrapolated to absolute zero. From these measurements the chemical potential /x4 of the solvent can be directly determined in the one-phase region. Their results are shown in fig. 9c. Besides, at finite pressures osmotic pressure measurements were also performed. All these measurements were reviewed by Ebner and Edwards [9]. The heat of mixing, the excess molar internal energy ( u E ) , is defined as the heat required to add ( + ) or to withdraw ( - ) in order to keep the temperature constant when X mole of pure 3He is added to ( 1 - X ) mole pure 4He, hence, U=- X U ° +

(I-

Liquid 3He-4He mixtures at absolute zero

0.042L

a

O.Q£-

k\\Xoo, m

oo

oo, x.S I

8k\\\"q ~ 0.0~=

0

0.OO2

Ootm.

C}

0004

20otto

O.O1 0.02 0.03 n 31 (mole ~,-leIcm 3)

n 31 (mole 3Helcm3)

X)U°4 + u E ( x ) .

In the one-phase region, U E is positive at temperatures above about a quarter of a degree. Below this temperature it becomes negative, in agreement with the observation that the onephase region ( X < Xs) is stable at absolute zero against phase separation into pure components. In fig. 9a the data obtained by extrapolation to absolute zero are plotted. The internal molar energy at absolute zero U(X, p = 0) is calculated using the values for U°a(p = O) = -L°a and U ° ( p = O) = - L ° given in table I, and the heat of mixing d a t a U E ( p = 0, X). The energy density e is obtained by the relation e = U/V. From fig. 9a it is immediately clear that the e versus n 3 curves in the one-phase region are not straight lines, although the contribution of the heat of mixing density ( u E / v ) to the total energy density is very small. The investigation on the analytic or nonanalytic behaviour of e(n3) in the one-phase region will be postponed to the following section. H e r e an over-view is given of e(n3) at different pressures in the one- and two-phase regions. The internal molar energy U(X, p) at finite pressures p is determined again by integration: U(X, p) = U(X, p = O) - fop p d V ( X , p), using the values at zero pressure of the internal energy. The energy per unit volume e(n 3, p), or e(X, p), is determined by U(X, p ) / V ( X , p). In fig. 7c, e is plotted as a function of n 3 for the

-z°l-

)

s

Iodm/>,,

o.ot

0.02

0.03

d d

c _o,

to -2.3

to

-

-iie~rr" H ~ '~

c&2 o.ob4 -zo n31 (m°le 3H~Icm3)

-2.5

Fig. 7. (a, b) For both the one- and two-phase regions the 4He molar density n 4 is plotted versus the 3He molar density n 3 at different pressures: 0, 2, 4 , . . . , 18, 20 atm. For a threedimensional model of the isobars, with features schematically exaggerated, see photograph 1, see also fig. 8. (a) In the one-phase region (I) the lines of constant pressure are straight with slope/3~ ~- - ( a n J O n 3 ) p = V3/V ° due to the experimental fact that the partial molar volume V3 in this region is independent of the concentration. (b) In the two-phase region (II) these lines at constant pressure are straight by definition, with slope/3 n = - - ( a n 4 / a n 3 ) p. The two slopes/31 and ~I1 are plotted versus the pressure in fig. 4a. (c, d) For both the one- and two-phase regions the energy density e = U / V is plotted versus the 3He molar density n 3 at different pressures: 0, 2, 4 . . . . . 18, 20 atm. (c) In the onephase region (I) these lines of constant pressure are not straight, as is clearly shown in fig. 9 where at zero pressure the heat of mixing density data U E / V are plotted [11]. (d) In the two-phase region (II) these lines of constant pressure are straight by definition, with slope a~ =-(at/an3) p. a n is plotted versus the pressure in fig. 4b. s is the saturated solution c u r v e Es(n3,s) , n3,s, n4,s, s~,/3~,/3 n and ct11 are tabulated for different pressures between 0 and 20 atm in table II.

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

one-phase region at different pressures: 0, 2, 4 . . . . 18, 20 atm, the values e s obtained at the saturated density rt3, s included. In the two-phase region, the e(n3) curves are straight lines by definition. They connect the saturated es(n 3 s) values with the pure 3He val0 0 ' ues e3(n3) and are plotted in fig. 7d for 0, 2, 4 . , . 18, 20 atm The slopes of the isobars Otii ( p ) =- (ae/an3) p are determined and plotted in fig. 4b as a function of pressure. In fig. 8 a qualitative geometrical picture of the e(n 3, n4) surface is given, the main features are very much exaggerated: (1) The "pinched effect" of the isobars when projected into the n a - n 4 plane where the saturated solution curve (s) separates the one-phase from the two-phase region (see fig. 8a). (2) The non-linear behaviour of the energy e(n3) in the one-phase region (see fig. 8c). The "pinched effect" is due to the fact that Va - V ° changes sign when p increases. In photograph 1, a three-dimensional schematic model is shown for the surface whose coordinates are the energy density and the molar densities of the 3He and 4He components. The lines of constant pressure in the one- and twophase regions are constructed together with the saturated solution curve. The curves e°(n °) and 0 0 e4(n4) are indicated in the coordinate planes of the pure components. In the two-phase region (II), the straight lines i n t h e e - n a - n 4 space have, as already mentioned before, two slopes a l i ( P ) = ( a e / a n 3 ) p and fill(P) -~ --(dn41cgn3)p. One has the following relations: _

0

n 4 -- fill(n3

E =

-- n3)

,

0 0 E 3 -- Otli(n 3 -- n3).

After differentiation, and realizing that e = - p + n3/z3 + n4/z4 and de = / z 3 dn 3 +/i,4 dn4 and that a , / 3 , / z 3 and/24 are in the two-phase region only functions of p and not of n 3, one obtains d°~ll

/z4 =

dfin

_

/-t 0 - - 0~ii fiII

135

!

E

~dp

. . . . . .

It

. . . . . . .

~(~O)~V-----~T ::

/ II

~yV,,,p, _×

C

p

.

~(p=o,

r Xs(P=O) =6.60/0

/

(3

f13

b s

lI

N3

@l/v~

Fig. 8. A qualitative geometrical picture of the e(n3, n4) surface, the main features are very much exaggerated (compare photograph 1 and fig. 7): (a) the "pinched effect" of the isobars when projected into the n3-n 4 plane where the saturated solution curve (s) separates the one- from the two-phase region, (c) the non-linear behaviour of the energy e(n3) in the one-phase region. The "pinched effect" is due to the fact that Va-V°3 changes sign when p increases (see figs. 1, 2 and 4a). The one-phase region with negative UE/V (see fig. 8c) is stable against separation, the curvature is exaggerated, in reality the largest deviation from linear extrapolation at the saturated solution is only about 0.1%. (d) The construction of the saturated solution curve Xs(p) in a schematic way.

The chemical potential /x4(p, X) of the solvent in the one-phase region (I) can be determined from the osmotic pressure H0(P, X) at absolute zero by means of the relation ~4(P, X) - - / z ° ( p ) = - H o ( p , X ) V ° ( p ) . As an example, in fig. 9c the zero pressure result for (/z 4 -/z4) is plotted versus n 3 which is derived from the osmotic pressure data [10]. It appears that the chemical potential /z4 in the one-phase region at the saturated n3, s density derived from the osmotic pressure data matches perfectly the two-phase result obtained by means

136

R . d e B r u y n O u b o t e r a n d C . N . Y a n g / L i q u i d 3 H e - 4 H e mixtures at absolute z e r o

of the relation /~4 = ( / 0 _ O~ll)/flll. This is indicated (s) in fig. 9c and also in fig. 10a by the little square). A more detailed discussion of the chemical potentials /z3 and /z4 in the one-phase region at different pressures will be given in the following section.

This gives 3"l = v ° E

3"2=3'2 ,

+ 3" E 1 ,

lim (UE/n3 V )

n3---~0

5. Analytic or non-analytic description of e(n 3) in the one-phase region It is the purpose of this section to investigate the analytic or the non-analytic behaviour of the e(n3) dependence at different pressures in the one-phase region. This investigation starts with an expansion of the energy density e ( p , n 3 ) in integral powers of n3: e ( p , n3) = 3'0(P) + TI(P)n3 + 3'2(P) n2 + 3'3(P)n~ + 3'4(P)n 4 .

Obviously, 3'0(P) = e 40( p ) . The coefficients 3"1, 3"2, 3'3, and 3'4 can be determined from a limited amount of available experimental data of sufficient accuracy: (i) at zero pressure the heat of mixing data U E (fi[g. 9a, ref. 11) and the osmotic pressure data H U (fig. 9c, ref. 10), (ii) at finite pressures only the osmotic pressure data (fig. 10a, ref. 10) at 10 and 20 atm (iii) the saturated solution curve n3,s(p) with the equilibrium condition/.t3.,(p) = / z ° ( p ) . The data of (i) and (ii) are available over the complete concentration range of the one-phase region. In order to determine the 3"s and their relation with the chemical potentials one preceeds as follows: The heat of mixing density U E / V is expanded in integral powers of n3: uE/v

= 3E E 2 . E 3 E 4 l n 3 + 3 ' 2 n 3 -t- 3 ' 3 n 3 + 3 ' 4 n 3 .

The energy density e = U / V = n3 U° + n4 U° + ( u E / v ) , with n 4 = n o - flin3 . Hence, 0 0 E E 2 E 3 = e4 + ( U 0 -- f l l U 4 + 3"1 ) n 3 + 3 " 2 n 3 + 3 ' 3 n 3 E 4 + 3'4n3 •

E

3"3=3'3 ,

E

3'4=3'4 ,

= U3(n3----> 0 ) -

and

U 0 = 3'1E .

At the same time, e = - p + n3/z3 + n4/~4. After making the following expansion for the chemical potentials: 0

].L4 - - [,1~4 = x l n 3

2

3

4

+ XEn3 + x3n3 + x4n3 ,

/~3 -- /'t30 = ¢'0 + ~bln3 +

~b2n2 +

,

and using the G i b b s - D u h e m relation n 3 d/z 3 = - n 4 d/z 4 with dp = 0, one obtains, for instance, 0 E the following relations: X1 = 0 , X2 = -V43"2 = -V°~bl/2, etc. A complete list of all the relations obtained in this way are given in the left column of the Review. From the left column of the Review it is clear that there exist simple direct relations between the coefficients of the squared, cubic and higher terms 3'2, 3'3, 3'4 in the expansion of the energy density and the corresponding coefficients )(2, )(3 and g4 from the expansion of the osmotic pressure - IIoV°4 = / z 4 - / z °. Furthermore, we have for the coefficients in the expansion of 0 ~ 3 - / x 3 a phase matching condition when the 0 solution is saturated: n 3 = n3, s then/%.s -/~3 = 0. This facilitates the analysis at zero pressure where we have both the heat of mixing and the osmotic pressure data, together with n3. s. In fig. 9c the best fit of the osmotic pressure data is shown [10] at zero pressure. The values of X2, )(3 and )(4 so determined are used to calculate E E E 3'2,3' 3 and 3' 4 and are used in the best fit of the heat of mixing data [11] shown in fig. 9a and b. The dashed curve in fig. 9b in a U E / n 3 V versus n 3 plot shows the best fit using the osmotic pressure data. One observes that the 3E obtained from this fit is not in agreement with what the experiment shows in the limit n 3--->0, and if one uses the extrapolated value (indicated by 77E) in the limit of n3---->0 and the osmotic pressure data one obtains the dotted curve of

R. de Bruyn Ouboter and C . N . Y a . g I Liquid 3He-4He mixtures at absolute zero

137

Review of the t h e r m o d y n a m i c relations obtained in an analytic and in a non-analytic description of the energy density e If in the one-phase region (I) e(n3) is an analytic surface t h e n e(n3) can be expanded in integral powers of "3 e = To + Tin3 + T2n2 + "Y3"3 + % n4 0 0 0 ")tO= E4 -- - - P + n 4 ~ 4

UE/V =

%

= u3

0

E

E 2

ts, v ,

0

+

")/1E = U 3 ( n 3 " - ' ) 0 ) -

"3

= no + ~ , " 3 + n . " ] '3 + n2"~3 + n : 0 3 '3

no = : = - p E 3

E 4

' Y l n 3 + ~ t 2 " 3 + ' Y 3 " 3 + '}/4n3 -

In the one-phase region (I) e(n3) is expandable in fractional powers in

E.

~,,, t3, =

vjv

°

003 = 2 ~ (uE/v"3)a'

"~2 = "y E, 'y3 = 'y 3E, ~/4 = ~ 4E

+ o;

E 5/3 ..[_ E 2 -- E 8/3 '~IE"3 + ~ a " 3 712"3 ~- T]b"3

UE/V= 0

0

7h = U 3 - fl~U,, + ~ ,yE = U 3 ( , 3 _ . . ~ 0 )

__

E

U 0 = in~m0 (oE/vri3)a)

E -- E E 'Y~a = 'Y~a , "t~2 -- 'l*~2 , 'f~b = "~b

0 0 2 3 4 /£4 -- $/'4= -V4-r/o = X2"3 + X 3 " 3 + X4"3

]'£4 -- ['1"40= - - V ° F ~ o : 0a"53/3 + 02 " 2 ~'- ~b n8/3

with Xl = 0b)

~ = -(3/2)0./V °

T2 = --X2 Iv°

n~ = - o J v

73

=

--X312V°

°

nb= - - ( 3 / 5 ) 0 b / v °

"Y4 = -X413V03 ~'3 - ~ 0 = *0 + * : ' 3 + ~ : : 3 + ~'3"3

m - ~,03= ¢0 + ~a.2/3 + ~,"3 + ¢ 5 / .

t,bo =/.t3(n3---~ 0) -- ~3°

60 = m(.3-, o) - ~,~ ~, = - ( 5 / 2 ) 0 , / v

°

I~, = - 2 0 2 / V ° ~ = t~,O./V ° - ( 8 / 5 ) a J V

o

qfl = - 2 x 2 / V ° ~b2= fl,Xz - ( 3 / 2 ) 0 3 / v ° ~3 = /31X3 --

(4/3)x4/V°4

") Ua(n3---,0) = Ua(X---'0), where Ua -~ U + (1 - X ) aU/OX. b) If we a s s u m e Xl # 0, one is forced to accept a n a In n 3 term in u E / v and e with the consequence of a logarithmic singularity in the limit n : - * 0 . O n the other h a n d , one expects e to be finite when n3--,0,

fig. 9a, completely in disagreement with the experiment. A n analytic description with an expansion of e in integral powers of n 3 has immediately as a consequence that gl = 0, hence the squared term with coefficient )(2 is dominant and must describe the osmotic pressure data correctly in the limit n 3 --->0. Figs. 9c and 9f show that this is not the case. We conclude that the experimental data indicate that e(n3) in the one-phase region is nonanalytic in n 3. For nonanalytic forms we turn to theoretical models. The thermodynamic properties of liquid 3 H e -

4He mixtures

in the dilute one-phase region usually have been discussed on the basis of the L a n d a u - P o m e r a n c h u k model and the BardeenBaym-Pines theory [8, 9]. An energy spectrum associated with the solute 3He atom is supposed 2 , to be E = E03 + p / 2 m 3 , where p 2 / 2 m ~ is the kinetic energy associated with the translational motion through the superfluid 4He of the 3He atom with an effective mass m I*. The distribution function of the 3He quasi-particles with spin t m~ contains an interaction with the superfiuid 4He background.

138

R. de Bruyn Ouboter and C.N. Yang / Liquid aHe-4He mixtures at absolute zero

n3/(molelcrr?) 0.001 0.003

-o

P=O

-0.02 ,

_e~o mE° -1 ~

a

~ -0.04

b

~. -2 '

~ -Q04 .o

=

w~

>, - 0 o 8 vD

-0.0~ •....

0

,_.q~

s

' 0,.44,.., ' X

I

0

O.C~D2 i

n3/(mole/c m 3) 0 @

I0-~

10 d

~4

f

P=O v

0

E

E £J w~ 0

> .

E

-1"11 0."

,

1 0 -4

,

,

.....

I

10"3

,

|

i

0 0.002 n3/(mole/cm3)

'

t0]

. . . . . . . .

I

10-3 n3/(mole/cm3)

Fig. 9. Different plots of the heat of mixing (UE)- and osmotic pressure (H0)-data at zero pressure in the one-phase region. (a) Full curve: T h e heat of mixing data (U ~) of Seligman et al. [11] at absolute zero are plotted as a function of the molar concentration X. Dashed curve: A correction applied by Kuerten et al. to the data of ref. 11 by taking into account the osmotic pressure data [10] and the more recent specific heat and entropy data leading to better phase matching agreement. If one tries to fit this result of the full- and the dashed curve where it deviates from the full curve by an analytic description with an expansion in integral powers of n 3 one obtains: UE/V = - 2 . 2 3 5 n 3 + 726n 2 - 66900n~, where the coefficients of the squared and cubic terms are directly obtained from the best fit of the osmotic pressure data as is shown by the dashed curve in fig. 9c. However, the coefficient of the linear term, y~ = -2.235, does not fit the experimental data as is shown in the UE/n3V = U E / X versus n 3 plot of fig. 9b by the dashed curve. Dotted curve: This result is obtained again in an analytic description with an expansion in integral powers of n3: U E / V = - 2 . 5 9 n 3 + 726n32 - 66900n~, where the coefficient of the linear term is now obtained from the extrapolation in the limit n3 ~ 0 of the direct heat of mixing data in the UE/n3V versus n 3 plot of fig. 9b and the coefficients of the squared and cubic terms are obtained again from the osmotic pressure data of fig. 9c. From the figs. 9a, b and c the failure of the analytic description is clear. (b) UE/Vna versus n3. Circles: results obtained directly from the heat of mixing data [11]. Dashed curve: corresponds to the analytic description with an expansion in integral powers of n3: U~/n3V = -2.235 + 726n3 - 66900n~ + 198000n33. As is presented with the full- and dashed-curve in fig. 9a, the last term is added in order to fulfill the phase-matching condition. Full curve: This result is obtained with the non-analytic description with fractional powers in n 3 of table III. The coefficients */1,r/.,r/2 , ~E E and r/bE are given in table III; 7/,, ~/2EE and r/bEare obtained directly from the corresponding fit of the osmotic pressure data (0,, 02, ao) as given in fig. 10a by the dotted points (at p = 0) and in fig. 9I by the full curve.

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

½ is governed by the Pauli exclusion principle So that at absolute zero all quasi-particle states up to a maximum momentum PF and energy k T F are filled. The Fermi temperature of the quasi-particle gas is given by k T F = ( h Z / 2 m ~ ) ( 3 H 2 n 3 ) 2Is and hence U 3 ~ - N E o 3 + R T F. In the limit n3---~0 the Fermi energy of this quasi-particle gas goes to zero. Hence U 3 ( n 3 - - - ~ O ) = N E o 3 . In particular, the quantity U3(n3---~0) has been determined in an analysis based on these models [9] and will be determined in this investigation again (see later, results are given in table III). The LandauPomeranchuk model has been very successful in describing4 the thermodynamic properties of dilute 3He- He mixtures [9]. As the concentration in the dilute mixture increases, 3He quasi-particle interactions become important also and the solution is not dilute enough to be treated by the Landau Pomeranchuk model. It is found that INE03 [ increases with concentration [9, 13] indicating an attractive quasi-particle interaction (Bardeen-Baym-Pines theory [8]). A consequence of these considerations is that the heat of mixing density u E / v = n 3 ( U 3 - U °) + n 4 ( U 4 - U ° ) contains a term proportional to 5/3 2/3 n 3 , o r U E / n 3 V a t e r m proportional t o n 3 o f a degenerate quasi-particle Fermi gas. The same is true for the osmotic pressure. For instance one uses for the osmotic pressure at absolute zero and zero pressure the expression [15, 12] / / 0 ( n 3 ) = 3 . 0 9 2 × 1 0 5 ( n 3 V ° ) 5/3 5

0 2

X 10 ( n 3 V 4 )

-6.91x

-

-

1.32 5

0 8/3

10 ( n 3 V 4 )

Pa.

139

It is obvious immediately from fig. 9c that n 53/ 3 probably will be a better approximation than n23 for the first term in the expansion for [£4 - - /.gO ~-0 - V 4 I I o in the limit n3----~0. Suggested by these considerations regarding Fermi liquid theories, we try an expansion of the following form: ,"

\

5/3

e ( p , n3) = ~/0(P) + ~71(P)n3 + ~ia[p)n3

+ ~T2(p)n2 + *lb(P)n~/3 . Obviously, ~/o(P) = e°4(P) • The coefficients 1~1,~a, ~2 and ~/b can be determined again from the heat of mixing and osmotic pressure data. The heat of mixing density u E / v is expanded in fractional powers of n3: uE/v=lTEn3 +

E 5/3 E 2 E 8/3 "r/a n 3 + 1"/2 n'l3 + l~b n3 ,

and the following expansion is made for the osmotic pressure II0: iz 4 - i~ 4o = _ V O n o = O Van 35,3 + 02n 32 + o n ,3 ,

and for the 3He chemical potential: 0

J-

2/3

.,-

5/3

Again by using the Gibbs-Duhem relation one obtains relations between ~/a and 0a, between 172 and 02, etc. A complete list of all the relations obtained in this way are given in the right column of the Review.

Fig. 9 (continued). (c) o _ -V4FIo - ~4 - ~4o versus n 3. Full curve: The osmotic pressure data (Ho) at zero pressure of Landau et al. [10]. Dashed curve: The best fit of the osmotic pressure data by an analytic description: ;z4 _ / o = - 2 . 0 0 2 x 104n32+ 3.69 x 106n~- 1.64 x 107n~. s: This point gives the phase matching result obtained by the relation/z4, ~ = (/~0 - a)//3 (saturated solution). (d) the non-analytic description with fractional powers in n 3 leads to a logarithmic plot of {(UE/Vn3) - ~/~} versus ha, the full curve indicates an exponential behaviour of 2/3 in the limit n 3~ 0. Circles: Results obtained from the measured heat of mixing data [11] and from "O~ as determined in fig. 9b, indicated by art arrow. When making a comparison with the corresponding plot of fig. 9f for the osmotic pressure data one observes that in the case of the heat of mixing the data are available at lower n 3 densities. (e) UE/Vn3 versus n 3. Circles: results obtained directly from the heat of mixing data [11]. Full curves: results obtained with the non-analytic description with fractional powers in n 3 of table III; r/~, ~/~, and r/~ are obtained directly from the corresponding fit of the osmotic pressure data (0,, 02, 0b) as given in fig. 9f by the full curve (see also fig. 9b). As mentioned in table III two values for r;t~ are used: -2.5941 and -2.5349. The latter is in agreement with phase-matching and corresponds with the calculations of ref. [12]. (f) The logarithmic plot of //o = (#0 _ 1~4)/Vo versus n 3. Circles: the osmotic pressure data of ref. [10]. In the n 3 density range an exponent ~ 1.53 is found. From the corresponding fit of the heat of mixing data at lower values of n 3 as shown in fig. 9d one may expect for the osmotic pressure an exponent equal to 5/3-~ 1.67 when going to lower n 3 densities.

140

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

n3/(mole/cm 3) 0.001 0,003

n~/(mole/c rn 3) 0.001 0.003

5 -0.02 E

-0.5

- 0.04

~-1.0 E -1.5

0

0

°,~ - 0 . 0 6

a

0 rn

-2£

o-0.08

i

:£ -2.~

Y -

-~o

~o\

0.10

-3.( ~-go

S

0.001

n3I(mole/c m 3) 0.002

0.003

0.004

-0.001 C

0

"E -0.002 u

_e

~ -0.oo3 ~ - -0.004 D

2ootm

$

-0.005

lO-~

10-2 p=lOatm

p=2Ootm

d

e

4~ E u "e 10-3

u£ 10 .3

0

D

.o_.

¢, 10-4 10-4

. . . . . . . .

I

10-3 %/(mole/cm 3)

'

'

10"

10-4

,

,

i

,

, ,,,[

,

t

10 -3

n3/(mole/crn3)

Fig. I0. The chemical potentials and the heat of mixing at 0, 10 and 20 atm using the non-analytic description as given in table III. (a) - V ° l l o = P-4 - / o versus n 3. Full curves: The osmotic pressure data (//0) at 0, 10 and 20 atm. of Landau et al. [10]. s: The saturated solutions. Square: The phase matching result at zero pressure. The points indicate the results obtained with the non-analytic fit with fractional powers in n 3. The coefficients 0,, 02, and 0b are given in table III and are used in the description of IL3 - 1~° and U E / V in the figs. 10b and 10c. (b)/~3 - / o versus n 3 calculated at 0, I0 and 20 atm using the osmotic pressure data of fig. 10a and the heat of mixing data at zero pressure of fig. 9e, lower curve. The results arc obtained in such a way that phase-matching results (s the saturated solution). The cocfficents ~0, ~,, ~:1 and ~b are given in table III.

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

N o w it is possible to obtain perfect fits of the osmotic pressure data as is shown in fig. 10a in a (1~4 _ p o) = _ VO Ho versus n 3 plot of the existing osmotic pressure data at 0, 10 and 2 0 a t m of L a n d a u et al. [10]. T h e coefficients 0~, 02 and /to are determined and given in table III. F r o m these coefficients the coefficients of the higher terms */E, 7/2E and */E from the expansion of the heat of mixing density U E / V are determined and used to fit the heat of mixing data at zero pressure of Seligman et al. [11]. As is shown in fig. 9b and e, this fit gives the coefficient of the first t e r m 7/1E. = U .3 ( n 3 .- - * 0 ). - U3° = lim n3-,0 ( UE/ Vn3). The fit is satisfactory. N o w It is possible to plot v e r s u s n 3 on a double logarithmic scale as is done in fig. 9d. T h e data indicate a dominant t e r m in the limit n3--*0 proportional to n 2/3 3 . Finally the coefficients 0 ~0, ~ , ~1 and ~b of the expansion o f / % - / ~ 3 are determined with the relations given in the right column of the Review, and with the data already obtained for the O's and */'s as given in table III. The results for zero pressure are presented in 0 fig. 10b as a /~3-/x3 versus n 3 plot and the ~'s are given in table III. T w o values for */z were used in the calculation: -2.5491 and -2.5349. The latter is in agreement with phase matching: 0 /x3 - / ~ 3 = 0 at n 3 = n3. ~. In fig. 9e the two curves calculated with */~, E */2e and */bE and with b o t h values of */E are presented together with the results obtained directly f r o m the heat of mixing data. The value for */E _-- --2.5349 is used in the rest of the calculation because of its therm o d y n a m i c consistency. In fig. 9f the osmotic pressure data at zero pressure are given in a double logarithmic plot of H 0 versus n 3. In the n 3 density range an exponent ~ 1.53 is found. One would expect an exponent equal to 5 / 3 ~ 1.67. H o w e v e r , one has to realize that the osmotic pressure data are only available at higher n 3

141

densities than the heat of mixing data; compare figs. 9d and 9f with each other. At 10 and 20 atm, as mentioned before, only osmotic pressure data (Ho) and the saturated solution density (n3,s) a r e available. The coefficients 0a, 02 and 0b are given in table III, and from E E these coefficients */E, */2, */b, ~a, ~1, and ~b are calculated and given in table III. Finally the determination of 7/1E and ~0. The following relations are obtained for /~3(X, p) - / z 30( p ) and f o r / z 4 ( X , p) - / x ° ( p ) : /~a(X, p) _ p O(p) = [/z3(X, p = O) - / z 3o( p = 0)]

(uE/vn3--*/E1)

P

+ f (V3 - V~) d p , 0

/~4(X, p) - / z ° ( p ) = [/~4(X, p = 0) - / z 4 0( p = 0)1 P

+ f (v, - v °) dp. 0

H e n c e the (~4 - /z°) versus X curves at different pressures nearly coincide (and the better the smaller the concentration) and the ( ~ - / x °) versus X curves are parallel for different pressures. F u r t h e r m o r e , they have to satisfy the phase matching condition: /z3(p, X = X s ( p ) ) = O ( p ) . In the limit X---~ 0 or n 3 ~ 0 the following relation for ~0 is obtained: ~0(P) =/~3(P, n3--~ 0) - / i f ( P ) P

= ~o(P = 0 ) + f (V3 - V °) d p , 0

and for */1: E

Fig. 10 (continued). (c) UE/V versus n3 calculated at 10 and 20 atm. The measured heat of mixing curve at zero pressure [11] is also given (s the saturated solutions). The coefficients "qIE,*1dE,'12Eand *1b~are given in table III. r/a,*12,E ~ and *1bEare determined directly by the osmotic pressure data of fig. 10a, (d and e) The logarithmic plot of Ho = (/.t4° - #4)/V ° versus n 3 at 10 and 20 atm. Circles: the osmotic pressure data (H0) of ref. 10. Fig. 9f is the corresponding figure at zero pressure. In the same n 3 density range again an exponent ~ 1.53 is found.

142

R. de Bruyn Ouboter and C.N. Yang / Liquid aHe-4He mixtures at absolute zero

E

ves immediately that between 0 and 10 atm Co decreases because V3 < V3° in this pressure range. At higher pressures between 10 and 20 atm, Co increases a small amount because in this pressure range V3 > V °. Furthermore, at zero pressure ~ and Co are equal to each other because p = 0. At 10 atm Co and ~ are nearly equal to each other because V3 ~ V °, and at 20 atm 7/1r < Co because

"q, ( p ) = U3(p, n3---~0) - U ° ( p ) = tx3(p, n3---~0) - / x ° ( p ) - p(V3(p) - V°(p)) =

Co - P ( V 3 ( P ) - V ° ( P ) ) .

7/1 is determined by the relation: 71 = U ° -

v3 > v °

t3iv ° +

Finally the osmotic pressure data are plotted at 10 and 20 atm in figs. 10d and 10e in a double logarithmic plot of H 0 versus n 3. The same results, for the same n 3 density range, are obtained as at zero pressure; compare fig. 9f with figs. 10d and 10e. It would be desirable to extend these measurements to a lower n3 density range.

The values obtained for C0, ~ and 171 at 0, 10 and 20 atm are given in table III. Fi~,s. 10a, 10b and 10c show the results for (/x 4 - / x 4 ) , (/z3,/z3°) and U E / V versus n 3 at the pressures 0, 10 and 20 atm. Making a comparison between the values of C0 and ~71 z obtained at 0, 10 and 20 atm, one obserTable I Tabulated are the

components 3He smoothness

molar volume (V°), the molar density (n°), the energy density (e °) and the chemical potential (/o) of the pure (T=0) for different pressures (p) (the numbers between brackets are given for

and 4He at absolute zero

only)

p

(atm)

2

4

6

8

10

V~

(cm3/mol) 27.5(80) (mol/cm3) 0.0362(58) (J/mol) -59.61(43)-) (J/cm 3) -2.16(15)-) (J/mol) -59.61(43)-)

26.9(59) 0.0370(93) -59.55(14) -2.20(90) -54.12(68)

26.4(27) 0.0378(40) -59.38(7) -2.24(73) -48.72(24)

25.9(61) 0.0385(19) -59.15(36) -2.27(86) -43.40(12)

25.5(49) 0.0391(40) -58.86(14) -2.30(39) -38.16(31)

25.1(80) 0.0397(14) -58.52(49) -2.32(43) -33.00(82)

U~ e° /o

(cm3/mol) 36.8(73) (mol/cm3) 0.0271(20) (J/mol) -20.56(15)-) (J/cm3) 0.557(63)-) (J/mol) -20.56(15)-)

34.6(38) 0.0288(70) -20.33(50) -0.587(07) -13.31(57)

33.1(32) 0.0301(82) -19.87(73) -0.599(94) -6.44(89)

31.9(94) 0.0312(56) -19.00(7) -0.603(26) +0.15(00)

31.0(78) 0.0321(77) -18.65(10) -0.600(14) +6.54(08)

30.3(13) 0.0329(89) -17.95(34) -0.592(27) +12.76(13)

p

(atm)

14

16

18

20

Ref.

V~

(cm3/mol) 24.8(43) (mol/cm3) 0.0402(53) (J/mol) -58.14(93) (J/cm 3) -2.34(07) (J/mol) -27.93(64)

24.5(35) 0.0407(58) -57.74(35) -2.35(35) -22.94(78)

24.2(52) 0.0412(34) -57.31(34) -2.36(32) -18.04(23)

23.9(89) 0.0416(86) -56.86(04) -2.37(03) -13.13(68)

23.7(44) [1, 9] 0.0421(16) -56.38(87) -2.37(49) [9] -8.31(44) [9]

n~=l/V~

U~

e~ ~ V3° n~=l/V °

o o n4=l/V 4

U~ e~ ~ V~

0

12

(cm3/mol) 29.6(53) 29.0(86) 28.5(68) 28.1(07) 27.6(89) [2] (mol/cm3) 0.0337(21) 0.0343(88) 0.0350(04) 0.0355(78) 0.0361(15) [2] U~ (J/mol) -17.22(00) -16.46(26) -15.68(44) -14.89(03) -14.08(56) e~ (J/cm3) -0.580(68) -0.566(11) -0.549(02) -0.529(77) -0.508(71) [14] ~ (J/mol) +18.83(75) +24.78(88) +30.63(00) +36.37(26) +42.02(61) [14] a)These values correspond with a molar heat of vaporizationL ° at absolute zero equal to L°/R =2.473 K and L°JR=7.17 K. Here are used those values which are consistent with the data used for the mixtures. In recent temperature scales (EP'r76) slightly different values are used: L°/R=2.492 K and L°/R=7.23 K. n~ = 1/V~

R. de Bruyn Ouboter and C.N. Yang / Liquid 3He-4He mixtures at absolute zero

143

Table II Tabulated are for the one-phase region (I): the partial molar volume (V3) and the ratio fll=--(dn4/On3)p=V3/V°; for the saturated solution (s): the molar concentration of the 3ae (.As), the molar densities (n3.~) and (n4,~) of both components, the energy density (e~) and the 4He chemical potential (/z4,~); and for the two-phase region (II): a n = (0e/On3)p and/3 n = -(On4/cgn3)p for different pressures ( p ) at absolute zero ( T = 0 ) p V3 fll =

(atm)

0

2

4

6

8

10

(cm3/mol)

35.4(13) 1.2840

33.9(68) 1.2600

32.7(69) 1.2398

31.8(28) 1.2261

31.0(16) 1.2137

30.3(13) -) 1.2038 -)

V3/VO

X~ n3,s n4,~ es P,4,~

0.0660 (mol3He/cm 3) 0.00234900 (mol4He/cm 3) 0.033242 (J/cm 3) -2.0320 (J/mol) -59.6(746)

0.0780 0.0028357 0.033520 -2.0564 -54.1(765)

0.0860 0.0031885 0.033887 -2.0791 -48.7(867)

0.0910 0.0034346 0.034308 -2.0997 -43.4(962)

0.0939 0.0036029 0.034767 -2.1179 -38.2(795)

0.0949 0.0036974 0.035263 -2.1336 -33.1(092)

~II fill

(J/mol)

59.5(200) 1.34(197)

56.4(382) 1.28(753)

54.7(969) 1.25(538)

53.7(874) 1.23(315)

53.1(166) 1.21(673)

52.6(202) 1.20(386)

p

(atm)

12

14

16

18

20

ref.

V3 [3x = V3/V °

(cm3/mol)

29.6(90) -) 1.1952 ~)

29.1(58) a) 1.1885 a)

28.6(83) -) 1.1826 -)

28.2(24) a) 1.1768 a)

27.8(58) ") 1.1735 ")

[4-7, 9] [4-7, 9]

0.0943 a) 0.0037156 0.035812 -2.1503 -28.0(463)

0.0927 -) 0.0037055 0.036354 -2.1643 -23.0(404)

0.0905 a) 0.0036630 0.036902 -2.1769 -18.0(994)

0.0880 0.0036122 0.037435 -2.1868 -13.2(055)

0.0855 0.0035483 0.037952 -2.1948 -8.3(642)

[10, 5, 6, 91

52.3(113) 1.19(352)

52.0(880) 1.18(484)

51.9(409) 1.17(744)

51.8(376) 1.17(110)

51.7(734) 1.16(536)

Xs nn,~ n4,~ e~ /J,4,~

(mol 3He/cm3) (mol 4He/cm3) (J/cm 3) (J/mol)

an /3n

(J/mol)

") Included is a small correction in order that the data are in agreement with the saturated solution curve X~(p). Table III In the one-phase region (I) is e(n3) a non-analytic surface, e(n3) is expanded with fractional powers in n3: e =,7o+,1~n3 +,7.n~ '3 +,72n3~ +,l~n~+,Tbn~ '3 E

E

E 5/3

U /V=Tiln3+~l~n3 ]1,4 -- [.gO = --VOllo

E 2 E 8/3 -['-T/2 n3 --~T]b n3

= 0an5/3 .~. 02/,/2 _[. 0bn83/3

P~3 -- ;U'0 = 60 + ~/~an23/3+ ~/~ln3 dr 6bn53/3

p

(atm)

0

10

20

Ref.

Oa 02 ~,

-2.147 x 103 2.77 × 103 1.321 105

-2.010 x 103 4.84 x 103 3.781 104

- 1.657 x 103 2.63 x 103 7.661 104

[10, 9]

rt[ = U3(n3-"O) - U~ = lim (Ur~/Vn3) .tie =71. = _(3/2)09/V o -3-0 ~72 =*/2 = -02/V°4 E -0 ~b --rib-- --(3/5)0b/V4 U3° U3(n3--~0) = U° +*/[

-2.5349 -) 116.8 -100 --2.87 X 103 -20.56(15) -23.09(64)

-2.9383 119.7 -192 --901 - 17.95(34) -20.89(17)

-3.1938 104.7 -111 --1.941 103 - 14.08(56) -17.27(94)

[11, 9] [11, 9, 10] [11, 9, 10] [11, 9, 10] [14]

144

R. de Bruyn Ouboter and C.N. Yang

Liquid 3He-4He mixtures at absolute zero

Table III (continued) p

(atm)

0

10

20

Ref.

60 = P-3(n3---'0)-/z3° 6, = -(5/2)0,/V~ 61 = --202/vO

-2-5349") 194.6 --201

-2.9384 199.6 -384

-2.8513 174.5 -222

[11]

- 1.04 x 104

-4.82 x 103 (-4.72 X 103) b)

-7.11 X 103 (-5.57 x 103) b)

[11, 10]

~b = ~xOa- (8/5)g,V ° po P-3(na"-}O)= /z°+ ~o

-20.56(15) -23.09(64)

+12.76(13) +9.82(29)

+42.02(61) +39.17(48)

•% = eo4 rh = U ° - ~i U° +r/1E fl~ = g3 / g°4 U~ /o

-2.16(15) 53.449 1.2840 -59.61(43) -59.61(43)

-2.32(43) 49.564 1.2038 -58.52(49) -33.00(82)

-2.37(49) 48.884 1.1735 -56.38(87) -8.31(44)

[11] [9]

[9]

a) Two values for ~/1Eare used: -2.5941 and -2.5349, the latter is in agreement with phase-matching. b) Values adopted in order to obtain phase-matching.

Acknowledgements It is a great pleasure to thank Mr. J.J.F. Scheffer for his help in performing the calculations. One of us (C.N.Y.) acknowledges the partial support of the U.S. NSF under grant no. PHY85-07627. References [1] D.O. Edwards, D.F. Brewer, P. Seligman, M. Skertic and M. Yagub, Phys. Rev. Lett. 15 (1965) 773. See also, J. Wilks, The properties of Liquid and Solid Helium (Clarendon Press, Oxford, 1967) and K.R. Atkins, Liqnid Helium (Cambridge Univ. Press, London, 1959). [2] B.M. Abraham and D.W. Osborne, J. Low Temp. Phys. 5 (1971) 335. B.M. Abraham etal., J. Low Temp. Phys. 6 (1972) 521. D.S. Greywal, Phys. Rev. 27 (1983) 2747, table VII p. 2765. [3] E.C. Kerr, Proc. 5th Int. Conf. LOw Temp. Phys., Madison (1957) p. 158. E.C. Kerr and R.D. Taylor, Ann. Phys. (NY) 26 (1964) 292. See also discussion in: K.W. Taconis and R. de Bruyn Ouboter, Progr. In Low Temp. Phys., Vol. IV, C.J. Gorter, ed. (North-Holland, Amsterdam, 1964) ch. 2. [4] E.M. lift, D.O. Edwards, R.E. Sarwinski and M.M. Skertic, Phys. Rev. Lett. 19 (1967) 831; Phys. Rev. 177 (1969) 283. D.O. Edwards, E.M. Ifft and R.E. Sarwinsky, Phys. Rev. 177 (1969) 380. [5] G.E. Watson, J.D. Reppy and R.C. Richardson, Phys. Rev. 188 (1969) 384.

[6] B.M. Abraham, O.G. Brandt and Y. Eckstein, Proc. 12th Int. Conf. Low Temp. Phys., Kyoto, Japan (1970) p. 161. [7] C. Boghosian and H. Meyer, Phys. Lett. 25A (1967) 352. [8] J. Bardeen, G. Baym and D. Pines, Phys. Rev. Lett. 17 (1966) 372; Phys. Rev. 156(1) (1967) 207, L.D. Landau and I.J. Pomeranchuk, Doklady Adad. Nauk. SSSR 59 (1948) 669. [9] C. Ebner and D.O. Edwards, Phys. Rep. 2C (1971) 77154. [10] J. Landau, J.T. Tough, N.R. Brubaker and D.O. Edwards, Phys. Rev. Lett. 23 (1969) 283; Phys. Rev. 2A (1970) 2472. [11] P. Seligman, D.O. Edwards, R.E. Sarwinski and J.T. Tough, Phys. Rev. 181 (1969) 415. D.O. Edwards, D.F. Brewer, P. Seligman, M. Skertic and M. Yagub, Phys. Rev. Lett. 15 (1965) 773. [12] J.G.M. Kuerten, C.A.M. Castelijns, A.T.A.M. de Waele and H.M. Gijsman, Physica 128B (1985) 197; Cryogenics 25 (1985) 419. [13] R. de Bruyn Ouboter, K.W. Taeonis, C. le Pair and J.J.M. Beenakker, Physica 26 (1960) 853. D.O. Edwards, D.F. Brewer, P. Seligman and M. Yagub, Phys. Rev. Lett. 15 (1965) 773. D.S. Greywall, Phys. Rev. Lett. 41 (1978) 177. A.C. Anderson, D.O. Edwards, W.R. Roach, R.E. Sarwinski and J.C. Wheatley, Phys. Rev. Lett. 17 (1966) 367. [14] T.R. Roberts, R.H. Sherman and S.G. Sydoriak, J. Res. Natl. Bur. Std. (US) 68A (1964) 567. [15] A. Ghozlan and E. Varoquanx, Ann. de Phys. 3 (1979) 239.