Chapter IV Liquid and Solid3He

CHAPTER I V

LIQUID AND SOLID 3He BY

E. R. GRILLY

AND

E. F. HAMMELT

UNIVERSITY OF CALIFORNIA, Los ALAMOS SCIENTIFIC LABORATORY LOSALAMOS,NEW MEXICO CONTENTS:1. Theories of liquid 3He, 113. - 2. Theories of solid SHe, 118.

-

3. Pressure-volume-temperature relations, 119. - 4. Thermal properties. 129. 5. Transport properties of liquid and solid SHe, 134. - 6. Nuclear spin relaxation in condensed SHe, 138. - 7. Velocity of sound in SHe, 143. - 8. Summary, 147.

1. Theories of Liquid 3He

Since the early and unsuccessful efforts to describe the properties of liquid 3He in terms of an ideal Fermi-Dirac gas1, numerous attempts have been made to obtain a more accurate theoretical description of this liquid. Although discussion of these theories in detail is not the purpose of this article, it is important for a full appreciation of the experimental results that they be related to theory whenever possible. To facilitate this comparison and simplify references, a very brief summary of several theories of 3He is presented below.Q1 1.1. LANDAU’S THEORY OF LIQUID3He

The theory of an isotropic Fermi liquid was developed by Landau2p3. and extended by Abrikosov and Khalatnikov5*6, 7. Landau’s theory is based upon two fundamental assumptions: a) There is a one to one correspondence between the energy levels in a Fermi liquid (with interactions) and those in an ideal Fermi gas, i.e. the “switching on” of the interactions is adiabatic; the number of quasi-particles thus equals the number of atoms. b) The energy of a quasi-particle in a selfconsistent field is determined by the state of the surrounding particles ; the energy of the system is therefore not equal to the sum of the t Work performed under the auspices of the U.S. Atomic Energy Commission. References

9 . 150

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E. R. GRILLY A N D E. F. HAMMEL

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1

energies of the quasi-particles but is instead, a functional of their distribution function. From these assumptions, using a simple generalization of the Hartree-Fock method, Landau has been able to derive the properties of a so-called Fermi liquid in terms of integrals over the interaction potential. It is assumed that 3He is such a liquid on the basis oi its apparent failure (to date) to exhibit superfluidity. For Landau's theory to be valid, the excitation energies, of the order of KT, must be considerably greater than the quantum indeterminancy of the hit, where z is the time between collisions and varies energy, i.e. kT as T-2. In order to satisfy this inequality it is required that

>

T

< 0.3 "K

6*7.

For the specific heat C and entropy S Landau finds, in the limit T = 0, C = S = AT where A is the ideal Fermi gas constant with the particle mass m replaced by the effective mass m* of the quasi-particle. The nuclear magnetic susceptibility expression for a Fermi liquid, although including a large exchange interaction term favoring parallel spin orientation, predicts a net paramagnetic susceptibility which becomes constant in the limit T = 0 if the average exchange interaction is less than a certain critical value. The Fermi liquid viscosity and thermal conductivity were obtained by Abrikosov and Khalatnikov6,'. Their results confirm the predictions of Pomeranchuks made poise earlier, and give for the viscosity 7 = m/T2 (am degz) and for the thermal conductivity x = /?/T (b m lo2 - lo3 erg cm-lsec-l). Landau3 has also suggested that a new type of wave motion called zero sound should be observable in a Fermi liquid. This topic will be discussed in Section 7 . 3 . Although comparison of theory with experiment will be made in more detail in subsequent sections of this review, it may be stated here that the properties of liquid 9He are, with the exception of the thermal conductivity, either in agreement with, or at least consistent with Landau's predicted properties of a Fermi liquid 7 . If this theory of a Fermi liquid is applied to 3He at temperatures above a few tenths ot a degree Kelvin (but still much less than the ideal Fermi degeneracy temperature of 5 OK) it is found that the experimental and theoretical results diverge. Khalatnikov and Abrikosov5 have, however, shown that the Fermi liquid formalism can be used to 7 See footnote p. References p . 150

134.

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improve the fit to the experimental data for the specific heat and magnetic susceptibility up to 2 OK approximately, if the ideal gas type ) p2/2m* is replaced by a shell-like distribution in mospectrum ~ ( p= mentum space of the quasi-particles with E ( $ ) = (fi - pJ2/2m* where p , is the radius of the shell. Usiisghas pointed out that such a spectrum as well as the ideal gas type used by Landau leads to a positive thermal expansion coefficient in the limit T = 0, in apparent contradiction with experiment, and suggests an alternative spectrum E(+) = d + p2/2m* which can yield the correct sign of the expansion coefficient.

+

1 . 2 . THE BRUECKNER AND GAMMELTHEORY OF LIQUID3He

Brueckner and Gammels have developed a theory of liquid 3He valid in the limit T = 0 and based on Brueckner’slO general theory for a many-body system. For the interaction between pairs of atoms, the potential of Yntema and Schneiderll was used. A summary of the results of the B-G theory for liquid 3He at 0 OK are as follows: a) The computed binding energy is about -k of the observed value and extremely sensitive to the potential used. Small changes in either the attractive or repulsive part of the potential used could easily improve the fit to experiment. b) The predicted compressibility is about 5.3 %/atm. c) The limiting specific heat is C = C,m*/nz where C, is the specific heat of an ideal Fermi gas with density equal to that of the liquid and m*/m is the ratio of effective mass to actual mass. m*/mwas evaluated as 1.84, giving C = 3.78 T cal mol-l degl. Brueckner and Gammel also predicted the correct variation, qualitatively, of C with pressure (through the increase of m*/m with pressure). d) For magnetic susceptibility, B-G derived X = X,E,(F)/Eswhere X, is the ideal Fermi gas susceptibility and E,(F)/E, is the ratio of magnetization energies for the gas and liquid, respectively. The energy ratio obtained was 12, which is in good agreement with that derived from susceptibility measurements. Such close agreement is probably fortuitous, however, as E,(F)/E, is very sensitive to parameters. The theory also predicts an increase of X with pressure and at sufficiently high compressions antiparallel spin alignment may become energetically unstable with respect to parallel spin orientation. e) Brueckner and Atkins12 and de Boerso extended the B-G calculations to show that the liquid expansion coefficient at zero pressure should be about -0.1 T degl.

References

p.

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[CH. IV,

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1

1 . 3 . GOLDSTEIN’STHEORY OF LIQUID3He

In order to correlate nuclear magnetic susceptibility measurements with other properties of condensed 3He, Goldstein13-ls devised a theory of the nuclear-spin system. He set the fraction per unit volume of atoms or spin moments which escape orientation by the internal field equal to the ratio of X/X,, where X is the actual susceptibility and X, is the limiting Langevin susceptibility which the liquid would have at its actual density if it were an ideal paramagnet. Since X I X , gives the fraction of free spin configurations, the entropy of the spin system is defined as S , = ( X / X o ) R In 2. Through classical statistical thermodynamic formulas other properties, such as specific heat, volume, and volume derivatives, can be derived for the spin system. Such a property, of course, is to be considered only as a portion of the total or observed property. However, at low temperature, where other influences die out, the behavior of the total property should approach that of the partial spin property. From the susceptibility data down to 0.23 OK, Goldstein obtained the following: a) in the limit T = 0, the temperature derivatives of the spin entropy and the spin specific heat approach 2.31 R ; this value sets a lower limit for the T = 0 slopes of the corresponding total properties. In this same limit, the contribution of the spin system to the expansion coefficient, a = l / V ( a V /aT),, tends to -0.126 T . From an estimate of the non-spin contribution to a,-0.103 T was obtained for the total expansion coefficient as T + 0. b) As T increases, ( aC/ W), and ( as/W), [= - aV] increase from zero, pass through maxima, and eventually become negative. At vapor pressures, these coefficients become negative at about 0.15 OK and 0.5 O K respectively. Since these and other derived results by Goldstein either agree with or are consistent with those obtained from experiment, the implicit assumption that exchange and other interaction effects in the spin system are automatically accounted for by using the experimental susceptibilities appears justified within the stated limitations of this model. 1 . 4 . PAIRCORRELATION THEORIESOF LIQUID3He

Although London’slgconjecture, that Bose statistics are essential for the formation of the superfluid phase, received support in subsequent theories20 of liquid 4He, it was not until recently that the apparently contradictory superfluid behavior of a-system of fermions, i.e. the electron gas in many metals, was resolved by the theory of Bardeen, Refevences

p . 150

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IV, 3 11

LIQUID AND SOLID

3He

117

Cooper and Schriefferzl. The success of this approach prompted similar investigations of other strongly interacting fermion systems, the first of which was carried out by Cooper, Mills and Sessler22.Besides introducing a pair correlation function in the wave function and using the usual approximation21 that only pairs of given total spin and total momentum are strongly correlated, it was also assumed that the “normal” fluid corresponded to an uncorrelated “phase”, i.e. the normal fluid was treated as a system of independent fermions in a momentum dependent potential. It is in the nature of such a theory to search for bound states of “Cooper pairs”. I n this first work, Cooper et al. tacitly assumed the bound state to be of S wave character in angular momentum. A number of trial S state solutions were examined, but no transition to a correlated superfluid phase was found nor did a transition appear likely. Such a transition was not definitely excluded however. Emery and Sessler23 and simultaneously Brueckner, Soda, Anderson, and Morel24 extended the Cooper et aLZ2 treatment to include pair interactions for states of higher angular momentum. It was found that for states of relative angular momentum I > 0 (more probably 1 > 1) cooperative effects arise leading to a phase transition in the temperature range 0.05-0.1 OK. Immediately above the transition temperature, the specific heat should be proportional to the temperature with C m 2 C,, where C, is the ideal Fermi gas specific heat23. At the transition temperature T,, a discontinuity in the specific heat is predicted. The actual AC will depend upon the relative contributions from the various 1 and m modes. For example, for 1 = 2, m = 0 , AC = 0.71 C , where C, is the “normal” fluid specific heat at T,. Below the transition temperature, the particle pairs will be correlated for arbitrary directions in the medium. Since the correlation range is only about 400 A, the unperturbed liquid should break up into randomly oriented cells of roughly this dimension, in each of which a correlation axis will exist. In order to observe certain correlation dependent properties, it will be necessary to establish macroscopic polarization axes in some fashion, e.g. by viscous interaction of the flowing fluid with the walls. The properties then measured will be angle dependent. Below the transition temperature, an enhanced fluidity rather than perfect superfluidity should be observed in liquid 3He. It is well known that the latter phenomenon requires the existence of an energy gap, whereas the formation of a correlated phase involves only a reduction in level densityg4. References p . 150

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2. Theories of Solid 3He 2 . 1. QUALITATIVE

Pomeranchuk*, assuming negligible exchange effects from nuclear spin in the solid, concluded that the spin alignment temperature should be about lo-’ OK. Although the correct value proved to be about 0.3 OK (see Sect. 3.3a), Pomeranchuck’s discussion disclosed the important anomaly in 3He at T < w 1 “K that the solid entropy should be greater than the liquid entropy, which is observed as the negative thermal effect of melting and the minimum in the melting P-T curve (see Sect. 3.3a). Primakoff z5 predicted that nuclear spin alignment in the solid should decrease with incresing pressure. The corresponding entropy increase, at sufficiently low temperatures, led to the prediction that solid 3He will have a negative expansion coefficient (see Sect. 3 .3a). Goldstein 15, 16, 1 7 extended .his theory of the partial spin properties from the liquid (Sect. 1.3) to the solid, with the result that parallel behavior would be exhibited among certain thermal and PVT properties. 2 . 2 . THEORY OF BEKNARDES AND PRIMAKOFF

A quantitative analysis of the properties of solid 3He was made by Bernardes and Primakoff26, who began with a gas-phase LennardJones “12-6” potential modified at small interatomic distances. I n contrast with Porneranchuk, they concluded that exchange effects represent the predominant mechanism for spin alignment in the solid. Their calculations for P M 30 atm and T w 1 “K led to the following conclusions: a) The cohesive energy per atom is about R X 2.5 “ K ; b) the root-mean-square deviation of an atom from its lattice site is about 0.36 times the nearest neighbor distance; (c) the nuclear magnetic susceptibility X follows the Curie-Weiss law X = c/(T - 0) with a Weiss constant e of antiferromagnetic sign 0 w - 0.1 OK; d) the decrease of - 19 w T , with increasing pressure corresponds with a possible transition to ferromagnetic behavior at fi w 150 atm, which could be connected with an observed crystallographic transition (see Sect. 3.3b); e) at T,, the specific heat and susceptibility exhibit singularities (cusp-like or otherwise well-defined maxima) associated with the alignment of the nuclear spins; f ) the thermal expansion coefficient becomes negative below about 0.6 OK; and g) the melting Refersnrrs

p.

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LIQUID AND SOLID

3He

119

curve is characterized by a minimum at T w 0.37 OK and a maximum at T M 0.08 OK. The most striking predictions of this theory appear to be: 1) The singularities in specific heat and susceptibility; and (2) the maximum in the melting curve. The apparent absence of 1) in the liquid is ascribed ultimately to the difference in character between the associated quasi-particles (phonons and magnons or spin waves in solid; individual atoms with m* # m in liquid).

3. Pressure-Volume-Temperature Relations A rather large amount of work has gone into PVT studies of condensed 3He, beginning in 1949 with the vapor pressure and the density of saturated liquid27. The reasons for this great effort lie in 1) the inherent importance of determining the behavior of a second quantum liquid, whose properties were expected to be significantly different from thcse of 4He; 2) the technical need of knowing how to handle the substance in the course of many experiments; 3) the rapid development in the entire field of 3He studies, which naturally brought on simultaneous duplicative investigations. The total effort now covers the tempxature range 0.3 to 3.2 OK between vapor pressures and melting pressures and up to 30 OK along the melting curve. The measurements involved from 0.02 to 12 cm3 of liquid, used a variety of techniques, and usually attempted to obtain high accuracy. At present, therefore, the PVT data on liquid 3He are comparable in extent and quality to those on the much more available and “older” 4He, which in turn has received greater attention than most liquids. The studies on solid 3He have been limited for the most part to the region of the melting curve. 3 . 1 . AT VAPORPRESSURES

The vapor pressure of liquid 3He was measured originally by Sydoriak, Grilly and HammelZ7, then more accurately over the range 1.O-3.3 OK by Abraham, Osborne, and Weinstock28. Measurements were extended down to 0.45 OK by Sydoriak and R0berts2~,who cooled the sample in a liquid 3He bath and determined temperatures from the susceptibilities of two different paramagnetic salts. Sydoriak and Roberts also recomputed the data of Abraham et al. to derivea single equation accurate over the entire range of 0.45 to 3.327 O K (the critical point) and fairly reliable down to 0.28 OK. This equation is References

p.

150

120

E. R. GRILLY AND E. F. HAMMEL

+

[CH. I\’,

$3

In P(mmHg) = 2.3214 In T - 2.53853/T 4.8153 - 0.20644 T 0.08640 T2 - 0,00919 T3,

+

where T = T , is based on the “55E” scale of Clement30. The vapor pressure as T -+0 OK can be calculated from another equation of Sydoriak and Roberts provided that the spin entropy integral can be evaluated. The first serious attempt at high accuracy in saturated liquid density was made over 1.3-3.2 OK by Kerr3I, who tried to limit the error to 0.2 yo.P t ~ k h followed a ~ ~ with a technique that, unfortunately, allowed a possible error of 1 %, but her results agreed with Kerr’s within 0.2 yo up to 2.2” K, then jumped to 0.6% greater in molar volumes. Peshkov’s measurement^^^, through refractive index observation, yielded changes in density, particularly with pressure, more accurately than absolute values. Sherman and E d e s k ~ t yundertook ~~ an ambitious program to determine all of the PVT surface between the vaporization and melting curves from 0.96 to 3.32 OK with great accuracy. Their estimated possible error was less than 0.1% in molar volume, but their results are consistently higher than all the others, at both vapor and melting pressures (the latter comparison being made with the Grilly and Mills35 data)?. A t the time of this writing, Taylor and Kerr36 are remeasuring the molar volumes of the saturated liquid, particularly to determine the behavior of the expansion coefficient below 1 OK. It seems highly desirable to present here a consistent and “bestvalue” summary of PVT data. For molar volumes of saturated liquid, there appear to be three attempts to obtain high accuracy. In the region of overlap, 1.2-1.6 OK, the volumes of Taylor and Kerr are higher than those of K e n by 0.2Sy0and lower than those of Sherman and Edeskuty by 0.28%. It seems adequate, therefore, simply to average the data from these three sources. The resulting numbers are shown in Table 1. For the thermal expansion coefficient ct = V-l( aV/ aT),, the values in Table 1 were chosen as follows: for 0.3 to 1.2 OK, those of Taylor and Kerr; for 1.2 to 3.0 OK, those of Sherman and Edeskuty. The dividing temperature of 1.2 OK was selected because here both sets of data yield the same value and above it the conversion of the V-T slopes of Taylor and Kerr to isobaric derivatives becomes too uncertain. The low-temperature anomalous behaviour of ci is illustrated in Fig. 1, i.e., the values become negative at sufficiently low temperatures. t For explanation, References p . 150

see footnote on p. 122.

CH. IV,

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LIQUID AND SOLID

3He

121

TABLE1 P V T relations of liquid 3He at vapor pressures

TE (OK)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.7 2.8 2.9 3.0 3.1 3.2

P (mmHg at 0 ' C)

(cw3mole-I)

0.00001 0.001 50 0.024 05 0.141 8 0.4985 1.291 2.744 5.092 8.564 19.765 38.03 64.91 101.93 150.55 212.28 288.60 381.02 433.73 491.00 553.01 619.92 691.88 769.04

36.785 36.746 36.722 36.713 36.719 36.741 36.777 36.830 36.899 37.089 37.354 37.705 38.18 38.83 39.67 40.75 42.15 43.03 44.05 45.30 46.90 49.00 52.30

V

loa a

102 ,B

(deg-')

(ah-')

- 8.46 -4.29 -0.13 4.12 8.50 13.1 17.9 23.1 34.0 48.7 66.5 89.4 118.3 155.4 207.4 287.1 344.0 439.0 593.0 932.0

3.68 3.69 3.70 3.73 3.77 3.82 3.89 3.96 4.18 4.49 4.94 5.61 6.58 8.03 10.36 14.43 17.9 23.4 33.6 58.8

The values of compressibility = -V-l( aV/ aP), of saturated liquid 3He come from these sources: the conventional PVT measurements over 1.0-3.0 OK of Sherman and Edeskuty; the index of refraction observations over 1.6-3.1 OK of Peshkov; and the velocity of sound measurements of Laquer, Sydoriak, and Roberts3' over 0.343.14 "K and of Atkins and Flicker3* over 1.2-3.2 OK. The last method gives adiabatic compressibility, which must be multiplied by C,/C,; therefore the uncertainty becomes excessive above 2 OK. The measurements were taken within ?n atmosphere of the vapor pressure except those of Sherman and Edeskuty, which were extended to the melting pressure. All the results are in remarkable agreement, i.e., within 1yo, except Peshkov's, which are lower by as much as 10% at T < 2.7 OK and still more at T > 2.7 O K . The results of the others for saturated liquid are combined in Table 1.

References

p . 150

1'1'1

E. R. GRILL\'

AND E. F. HAMMEL

Temperature

[CH. IV.

$3

(OK)

Fig. 1. Thermal expansion coefficient versus temperature for liquid 3He a t various pressures [from Lee and Fairbank*O]. The dotted curve represents additional results of Taylor and Kerra6 a t vapor pressures.

3 . 2 . AT INTERMEDIATE PRESSURES

Although the PVT area bounded by the vaporization curve, the melting curve, T = 1.0 OK, and T = 3.3 OK was measured by Sherman and E d e ~ k u t y ~there ~ , exists the possibility that their molar volumes are too high. Near vapor pressures, this excess might amount to 0.28% below 1.6 O K and somewhat more above 1.6 OK. Near melting pressures, their volumes are greater than those of Grilly and Mil1s3j by 0.3% in the region of 2.0-2.8 O K and by greater amounts above and below this region. Presenting their extensive array of data is only possible in tabIes such as theirs 7 . Below 1 OK, the major interest has been in the behavior of the thermal expansion coefficient a. From A T / d P measurements of adiabatic expansions covering pressures 1.7 to 22 ?tm, Brewer and Daunt39 derived a values over the range 0.15 to 0.6"K, all of whichwere negative. Up to 1.15 OK, they also obtained T where a = 0. Thus they showed that : a was negative below a temperature which monotonically t .4 recalibration of the cell volume shows that the molar volumes of Sherman and Edeskuty should be lowered by 0.30%. The corrected molar volumes are within 0.08% of those a t vapor pressures by Taylor and Kerr and of those a t melting pressures by Griily and Mills up t o 2.8 "K. Therefore, in Table 1 the values of V at T > 1.6 "K are now derived solely from the results of Sherman and Edeskuty. References

p . I50

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LIQUID AND SOLID

3He

123

increased with pressure; and a had, at 0.2 O K , minimal values which increased with pressure. More direct values of u, through dielectric constant measurements, were obtained by Lee and Fairbank40 between 0.2 and 29.5 atm over a 0.15 to 1.2 OK range. The behavior was similar to that seen by Brewer and Daunt. In Fig. 1 the data of Lee and Fairbank are compared with those along the vapor pressure curve derived from data of Taylor and From all the present results, it appears that CI ( P = 0) could approach T = 0 approximately as cc = -0.1 T , which was computed by de Boereo, by Goldsteinl6P1*, and by Brueckner and Atkins12. 3.3. AT MELTINGPRESSURES a) Low Region The melting curve of 3He was measured by Weinstock, Abraham, and Osborne41from 1.5 down to 0.16 OK by using the blocked capillary method. They found the melting pressure leveling off to 29.3 atm below 0.4 “K, which effect is to be expected when their technique is used below the temperature of a pressure minimum. Lee, Fairbank, and 40 used a dielectric constant measurement to distinguish between liquid and solid. They concluded there is a minimuminmelting at 29.1 atm and 0.32 OK. The first to report a detailed pressure, Pmin, study of the minimum were Baum, Brewer, Daunt and Edwards43, who measured pressures with a strain gauge cemented to the cell containing sealed-off 3He. The gauge was sensitive to f 0.02 atm and had been calibrated to f 0.1 atm when the vessel contained liquid 3He. The cooling and thermometry were accomplished through paramagnetic salts. The measurements showed Pmin to occur at 29.3 f 0.1 atm and 0.32 OK. Sydoriak, Mills, and Grilly4*used a system in which pressure on the solid could always be measured as the sum of a spring pressure plus liquid 3He pressure. Before and after 3He measurements, their bourdon gauges were calibrated, in sitw, to i 0.02 atm. Cooling and thermometry involved separate compartments of liquid 3He. The melting pressures were within 0.1 atm of those of Sherman and E d e ~ k u t yin~ the ~ short region of overlap between 1.0 and 1.2 OK. Sydoriak et al. found the pressure minimum at 0.330 & 0.005 OK and 28.91 f 0.02 atm, and their points below 0.5 OK were consistent to & 0.02 atm with the empirical equation (where T = T , of ref.29) P(atm) = 28.91 Referenres

p . 150

+ 32.2 (T - 0.330)2.

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E. R . GRILLY AND E. F. HAMMEL

[CH. IV,

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They also observed at 0.308 "K the heating effect of melting connected with dP/dT < 0. The results of the various investigations are given in smoothed form in Table 2. The question of a minimum in the melting curve of 3He arose in 1950 when Pomeranchuks suggested its possibility (see Sect. 2 . 1 ) . Fairbank and Walters45 were first to observe the reversal in the heat of melting, at T M 0.4 O K , which corresponds to a negative dP/dT. TABLE2 3He melting pressures (atm) below 1.2 "K

T

(OK)

0.12 0.16 0.20 0.25 0.30 0.33 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

Ref." 29.3 29.3 29.3 29.3 30.1 31.5 33.2 35.2 37.4 39.9 42.6 45.7

Ref..4s 30.8 30.2 29.8 29.5 29.3 29.3 29.45 30.15 31.4 33.1

Hef.44

Ref.4o

28.94 28.91 29.07 29.80 31.02 32.02 34.50 36.79 39.30 42.08 45.10

29.1 29.1 29.1 30.0 31.5 33.2 35.6

Their nuclear magnetic susceptibility measurements showed that alignment in the solid occurs at about 0.3 OK in contrast with Pomeranchuk's estimate of lo-' OK. From this behavior, Bernardes and Primakoff 26 concluded that solid 3He is a nuclear antiferromagnetic, in the paramagnetic region its Weiss constant O( m - T,) being about - 0.1 OK. The corresponding entropy value led to a predicted minimum in the melting curve at 0.37 OK where P, = 29.1 atm. Below the minimum, their calculated pressures agreed with the measurements of Baum et a1.43,while those computed from T , = 10V "K were much higher. Furthermore, they predicted a maximum in the melting curve at 0.08 OK and 31.7 atm, which did not arise from Pomeranchuk's theory. Such region has not yet been investigated experimentally. To obtain volume measurements below 1 OK and in the vicinity of the melting curve minimum, Sydoriak, Mills and grill^*^ used the apparatus already described for melting curve measurements. The Refcremes p . 150

$ 31

CH. IV,

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LIQUID AND SOLID

125

TABLE3 Molar volume of melting liquid (V,) and volume change of melting ( A V,) for SHe below 1.2 "K ~

T("K) Vl(cm3/mole) d V , (cm3/mole)

0.5 25.84 1.19

0.33 25.99 1.20

0.6 25.65 1.18

0.8 25.17 1.14

1.0 24.63 1.10

1.2 24.07 1.06

molar volume of liquid Vl and volume change on melting AVm, with estimated possible errors of 0.1 yo and 1yo,respectively, are presented in Table 3. They joined smoothly the results of Grilly and Mills35 above 1.2 OK. The expansion coefficient is shown in Fig. 2. The anomalous negative values of m1 found a t lower pressures are seen to persist up to the melting pressure. From the relation

and the observation that dVl/dPm was approximately constant as P + Pmin, it appeared likely that ml + 0 as T -+Tmin.Furthermore at Tminthe term in parentheses of eq. (1)seemed to be zero, which would make dorl/dT = 0 , This behavior of ccl does not readily fit in with that expected at low pressure, i.e., the vanishing of al at T = 0 with finite negative slope, and therefore deserves more study. In contrast, the

I

5

m

d o

T 2

U

-5 -10 0.2

0.01

0.2

1.0

0.6

1

I

0.6

I

Tmt deg K

I

I .o

1.4

I

I 1.4

Fig. 2. Thermal expansion coefficient (upper figure) and compressibility coefficient (lower figure) of 3He a t melting pressures [from Sydoriak, Mills and grill^^^]. A Grilly Sherman and Edeskuty34; @ calculated from a , using eq. (1). Broken and Mills35; curves represent gS calculated from eq. (2) - - - : assuming pS = p , ; - - - - - - : assuming ps = 0.99 p,.

References

p . 150

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E. R. GRILLY A N D E. F. HAMMEL

normal behavior of the liquid compressibility

[CH. IV,

53

is also shown in Fig. 2 ,

as measured and as calculated from eq. (1).

The thermal expansion of the solid was calculated from g,

1 dV, dV, 1 - __ -- + +-

=

V, d T

V, dT

dPm

( B E

- 81) dT

(2)

with the reasonable assumption that 8, 2 /I,. As shown in Fig. 2, it was concluded that g becomes negative at a T, of 1.0 to 1.1 OK, in qualitative agreement with the theories of Goldstein1a.17 and of Bernardes and Primakoff 25,26. b) High Region The melting curve in the region 1.2-31 OK and 50-3400 atm was measured by Mills and grill^*^, using the blocked capillary method under a procedure that insured obtaining equilibrium values within very narrow limits. Estimated possible errors in meltin: pressure were 0.02 and 0.2 atm below and above 240 atm, respectively; temperatures were significant within O.O0lo up to 5", within 0.1' between 5' and 14", and within 0.01' above 14". The results are reproduced by three equations :

+ 15.5053 T2 - 1.35019 T 3 for 1.2 < T < 3.148 = 3.748 + 29.5713 T + 3.95049 T 2 for 3.148 < T < 4.4

P(atm) = 26.379 = 24.35

-

0.62615 T

+ 19.4362 T1.517'38

tor 1.9 < T

OK; O K ;

< 31 OK.

Sherman and E d e ~ k u t obtained y~~ similar results and give the equation P(atm) = 24.559 16.639 T 2 - 2.0659 T3 0.11212 T4 for 1.07 < T < 3.1 O K . Several volume relations near the melting curve were examined by Grilly and Mills35 over the range of 1.3-31 'K and 50-3500 atm. Measured directly were: the molar volume of fluid Vf to 0.1 yo;the volume change on melting d V m to 0.5%; the thermal expansion and compressibility of fluid, and Bf, respectively, to 5%. The A V , measurements led to the unexpected conclusion that there exist two forms of solid 3He. The d V of transition was found, indirectly, to be about 10% of AV,. The P-T curve of transition was determined from the sudden change in compressibility accompanying the phase change. The phase diagram is shown in Fig. 3, and some properties of the transition are listed in Table 4. Subsequently, X-ray diffraction

+

References

p . 150

+

CH. IV,

3 31

LIQUID A N D SOLID

3He

127

T (deg K )

Fig. 3. Phase diagram for condensed 3He [from Grilly and MillsS5].

studies by Schuch, Grilly, and Mills4' showed that the cr-solid, existing at lower pressures, has the body-centered-cubic structure and the ,&solid has the hexagonal-close-packed structure. Furthermore, the lengths of unit cell axes yielded a density equal to 0.154 f 0.004 g/cm3 a t 1.9 "K and 96.8 atm and density equal to 0.172 f 0.004 g/cm3 at 3.3 OK and 177 atm, which are in good agreement with the values derived by extrapolation from the direct measurement^^^ along the melting curve. TABLE4 Properties of the

T (OK)

1.8 2.2 3.6 3.0 3.148

References

p . 150

+ 01 transition in in solid 8He dPjdT (atmideg)

AV

(aW

(cm3/mole)

AS (cal/deg/mole)

107.9 113.2 120.8 131.3 135.9

10.7 16.0 22.4 39.9 33.0

0.068 0.094 0.116 0.123 0.125

0.017 0.035 0.061 0.087 0.098

P

128

E. R. GRILLY AND E. F. HAMMEL

[CH. IV,

33

The results on Vf and AVm(cm3/mol)are reasonably represented as functions of melting pressure (atm) by:

V , = -3.248 + 50.841 ( P + 1.04)-0.1a1532for 50 < P < 3440; AV, = 1.55910 - 0.39023 log,, (P - 29.033) for 50 < P < 135.92; A V , = 1.506 15 - 0.30825 log,, ( P - 41.212) for 135.92 < P < 3440.

Since AV,(P > 136 atm) was observed to decrease with pressure in a regular fashion, it was interesting to examine the behavior of the

Fig. 4. The thermal expansion coefficient, q ,and the compressibility coefficient, of fluid SHe along the melting curve [from Grilly and Millsss].

pi,

corresponding entropy change AS,, which could be computed from AV, and dP/dT through the Clapeyron equation. A formula for A S , as a iunction of P gave a maximum in A S , at 4080 atm, which is only slightly higher than the experimental range, and indicated that A S , = 0 at 77 x lo3 atm (T = 235 OK). Therefore, while a critical point in a melting curve has never been seen, the requirements of one, A S , = dV, = 0 , could possibly be met in 3He, both in principle and technically. References

p . 150

CH. IV,

3 41

LIQUID AND SOLID

3He

129

The compressibility of fluid seems to behave normally aU along the melting curve, i.e., it decreases regularly with increasing pressure and temperature and never changes sign (see Figs. 2b and 4). Previous discussions brought out the anomalous negative values of thermal expansion below 1.2 OK (see Fig. 2 4 . At higher temperatures, ccI first rises to a maximum at 3.1 OK and 140 atm and thereafter falls in a regular way, as shown in Figs. 4 and 5.

I 0

2000

P,(kg C m " )

3000

Fig. 5. The thermal expansion coefficient of fluid SHeand *He along the melting curve [from Grilly and MillsSS].

4. Thermal Properties 4.1. SPECIFIC HEAT

The history of specific heat measurements on liquid We has reflected the interest in trying to answer the questions: 1) Is there a lambda or other type anomaly in the specific heat-temperature curve ? 2) How does the specific heat extrapolate to 0 OK ? In seeking answers to these questions, investigators successively lowered the temperature limit of measurements. The early measurements of de Vries and Daunt 48 from 0.57 to 2.3 OK were improved and extended by Roberts and Sydoriak'g to 0.37 OK, by Abraham, Osborne and Weinstock50 to 0.23 OK, and by References

p.

150

130

E. R. GRILLY AND E. F. HAMMEL

[CH. IV,

54

Brewer, Sreedhar, Kramers and DauntK1to 0.085 OK. The first three series obtained Csst,a specific heat for a change of state while the liquid remains saturated (and thus involves three variables : P,V,T) while the last measured C,, where P was constant at 6 to 14 ern Hg. Qualitative, as well as quantitative, differences can occur between the various specific heats, as shown by Goldstein16 in Fig. 6.

TVK)

Fig. 6. Various molar heat capacities of liquid SHea t vapor pressures [from G o l d ~ t e i n ~ ~ ] The dotted curve represents additional results of Brewer et al."'.

In the region of 0.5 to 1.7 OK, the data of Roberts and S y d ~ r i a k ~ ~ were assigned probable errors of 1.5 to 2.0% and fit the empirical formula C,,, = 0.577 0.388 T 0.0613 T3 cal mol-1 deg-l

+

+

with a mean deviation of 1.0%. Below 1 OK, the merging of C,,, and

C, permits a direct comparison in Fig. 7. These results, combined with

the early warm-up observations up to 3.21 "K by Sydonak and Hamme16*, permit us to conclude there is no btype transition in liquid 3He down to 0.085 OK. Furthermore, no maximum of any kind, except possibly in C , at 2.5 "K, appears. Below 0.7 OK, the behavior of C is interesting in that the very small variation down to 0.2 OK rapidly changes, so that the extrapolation of C at 0.085 OK to C = 0 at 0 OK by Brewer et aL61appeared reasonable and consistent with the linearity predicted by the theories of G ~ l d s t e i nl6, ~ ~Landau29 . 4, and Brueckner and GammeP. References p . 150

131

LIQUID AND SOLID 3He

CH. IV, 10

I

09 08

2 07 go6 r05 J

204 0 03

a? 0 0

01

a2

04

03

TEMPERATURE (K)

06

05

07

Fig. 7. C , for liquid *He versus temperature [from Brewer, Daunt, and Sreedhara']. rn De Vries and Daunt48. Roberts and Sydoriak". x Abraham, Osborne and Weinstock &O.

+

TABLE5 Specific heat of liquid SHe in cal mole-' deg-l as a function of P (atmospheres) and T (OK)

P T

< 0.1

0.10 0.12 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75

References p . 150

V.P. 4.00 T 0.400 0.472 0.555 0.640 0.684 0.714 0.737 0.757 0.777 0.793 0.807 0.823 0.845 0.867 0.890

-

15

29

0.510 0.565 0.609 0.630 0.640 0.648 0.654 0.662 0.671 0.683 0.693

0.522 0.571 0.602 0.612 0.617 0.622 0.627 0.637 0.647 0.659 0.673

5 ~

0.494 0.560 0.626 0.662 0.685 0.698 0.706 0.717 0.732 0.748 0.764

. -

132

E. R . GRILLY AND E. I;. HAMMEL

[CH. IV,

34

Brewer, Daunt and S r e e d h a ~measured -~~ C , at pressures up to the melting pressure between 0.12 and 0.6 OK. The results, partially reproduced in Table 5, show that ( aC,/ aP),is negative above T w 0.16 "K and positive below this temperature. Near 0 OK, a positive value was predicted by Brueckner and Gammels, Hammel et u Z . ~ ~ , and Goldstein16,whereas a negative value arises from an ideal Fermi-gas model. The lowest temperature, 0.12 "K, was not sufficient to allow reliable extrapolation of C, to 0 "K (but see Sec. 4 . 2 on the entropy). 4 . 2 . ENTROPY

The early m e a s ~ r e m e n t s ~50~ ~of specific heat were not at low enough temperatures to allow extrapolation to 0 "K so as to yield absolute entropies directly. Brewer et uLsl linearly extrapolated their specific heat results (equivalent to Csat) from 0.085 OK to 0 OK thereby obtaining Saatto i 0.03 cal d e g l mole-l. The possible error inherent in this procedure is emphasized by the p r e d i ~ t i o n 2 ~of9 ~a~specific heat anomaly below 0.1 OK. Such an anomaly would influence the limiting slope of C and S, and it might change the values of S above the anomaly temperature. However, the data of Roberts and S y d ~ r i a k ~ ~ , who obtained their absolute entropy values from the thermodynamic vapor pressure equation1, agree with those of Brewer et ~ 1 . Another ~ ~ . way of deriving S,,, is through combination of calculated vapor entropy and measured vaporization heat AH,,, which was done by Abraham, Osborne and W e i n ~ t o c kTheir ~ ~ . measured 499

AH,

= 10.39 & 0.02 cal

mole-1 at TL5S=1.5 "K,

from which Seat(1.5OK) = 2.614 f 0.03 cal d e g l mole-l.

Combining this value with their specific heat measurements 5 0 , which are quite consistent with those of other investigators (see Fig. 7), one finds that their entropy values down to 0.23 OK are higher than those of Brewer et al. by 0.10 f 0.06 cal d e g l mole-l. As Table 6 is based on the Ssatvalues of Brewer et al., one should understand from the above discussion that there is a slight uncertainty in the reference zero of the data presented. At higher pressures, extrapolation of C, to 0 OK was more uncertain. Therefore, C, was used only to derive AS,, which was combined with entropy of compression (S, - Sgat).The latter was computed originally References p . 150

CH. IV,

9: 41

LIQUID A N D SOLID

3He

133

by Brewer and Daunt 39 from their thermal expansion results to yield values of S , as a function of pressure and temperature up to 22 atm and 1 OK, respectively. However, their values were slightly altered, using the more direct expansion coefficients of Lee and Fairbank40, to those given in Table 6. TABLE 6

Entropy of liquid SHe in cal mole-' deg-' as a function of P (atmospheres) and T ("K) P T

V.P.

5

10

15

22

4.00 T 0.476 0.594 0.766 0.914 1.042 1.153 1.253 1.344 1.426 1.503 1.573 1.703 1.822 1.933 2.036

4.44 T 0.516 0.635 0.807 0.951 1.073 1.180 1.273 1.357 1.432 1.603 1.568 1.684 1.792 1.890 1.982

4.77 T 0.547 0.666 0.836 0.977 1.097 1.200 1.290 1.370 1.443 1.511 1.574 1.686 1.790 1.881 1.966

5.12 T 0.581 0.701 0.871 1.010 1.125 1.225 1.312 1.389 1.460 1.524 1.585 1.696 1.792 1.881 1.962

5.55 T 0.619 0.740 0.910 1.046 1.161 1.258 1.342 1.417 1.487 1.549 1.607 1.712 I. 805 1.887 1.964

______

T+O 0.12 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.70 0.80 0.90 1.00

Examination of these -Jta led Brewer and Daunt to t,,e conclusion that as T + 0, SPIT= y = Cp/T,where y is a constant. This observation, then, lends support to the Fermi-liquid model theories of Landau23 4 (see calculations by Khalatnikov and Abrikosov5! ') and of Brueckner and Gammel9 as well as to the nuclear spin theory of Goldstein13r15.From the theory of Brueckner and Gammel, y is expected to be 3.78 cal mole-1 d e g 2 at Psatand to increase with pressure. Using the relation of G01dstein~~. l5 between nuclear magnetic susceptibility and spin entropy, Brewer and Daunt obtained y values of 4, 5, and 6 at 0, 11.2, and 27.6 atm, respectively, which are close to their observed values for total entropy. In Table 6, one can see that the normal variation of entropy with pressure is reversed at low temperatures, starting at T M 1 OK for the higher pressures, and becoming completely reversed at T < 0.6 OK. This behavior is also consistent with the predictions of Goldstein15 and of Brueckner and Gamnielg. References p . 150

13.4

E. R. GRILLY AND E. F. HAMMEL

[CH. IV,

$5

5. Transport Properties of Liquid and Solid 3He

LIQUID Several measurements have been made of both the thermal conductivity and viscosity of liquid 3He with the result that the data now extend from approximately 0.26 OK to the vicinity of the boiling point, as shown in Figs. 8 and 9. Although some slight discrepancies exist between the experimental values from different laboratories, the 5.1. THERMAL CONDUCTIVITY AND

VISCOSITY OF

0.3

0

A,0,

TPK) Fig. 8. The thermal conductivity of liquid SHe. Lee and Fairbanks*, p = 3 a t m ; Challis and Wilks6'.

+

temperature dependence over the indicated temperature range of each of these quantities now appears to be well established. In the thermal conductivity measurements of Lee and F a i ~ - b a n k ~ ~ , anomalous x values were observed at first below the density maximum for high heat fluxes. This was attributed to a contribution from convective heat transfer in the liquid sample ; consequently below this temperature (0.48 OK at 8 atm) and for the higher heat currents, the direction of the heat flux was inverted with respect to the gravitational field. A t high temperatures although the heat transport through the walls of the containing tube was larger than the heat flow through the sample, accurate corrections were made. At low temperatures, where the corrections were smaller, they were slightly less well known due to the perturbing effect of the Kapitza boundary resistance 62 t . The t A t the VIIth International Conference on Low Temperature Physics, University of Toronto, 29 August-3 September, 1960 (see the Programme, p. 22), J. Jeener and References p . 150

LIQUID AND SOLID 3He

CH. IV, 5 51

135

thermal conductivity values of Challis and WilksS7 have not been corrected for the thermal boundary resistance. These authors estimate

T W )

Fig. 9. The viscosity of liquid aHe from Peshkov and Zinov’eva68. Osborne and Abraham68; Taylor and Dashao;0, x Zinov’evas’.

+

A Weinstock,

>

F’

2

11#

. . ..

I

1.

Fig. 10. The ratio of the thermal conductivity t o the product of viscosity and specific heat a t constant volume as a function of temperature for liquid *He and liquid 4He. The kinetic theory value of this ratio, 2.5, is shown b y the horizontal line. The upper curves are calculated using the viscosity data of Zinov’evaal; the lower using the viscosity values of Taylor and DashGo.See footnote p. 134.

that the application of this correction would increase their x values of the order of 10% and hence bring them into closer agreement with G. Seidel showed that the boundary resistance corrections would raise the conductivities

of Lee and Fairbank as T falls below 0.5 “K (by 30% at 0.25 OK). Accordingly, x

increases as T decreases.

Riferefices p . 150

136

E. R. GRILLY AND E. F. HAMMEL

[CH. IV,

5

those of Lee and Fairbank in the range of overlap. Since tabulations and discussion of the viscosity values can be found in the original articles and also in the review article by Peshkov and Z i n ~ v ’ e v a ~ ~ , these data will not be further reviewed here. Attempts to correlate the properties of liquids usually derive from the assumed similarity of this phase either to a disordered solid or a highly compressed gas. The high zero point energy of liquid 3He and its associated expanded structure suggest that the gas model should be the more applicable. A t “high’’ temperatures, this assumption appears to be justified by the fit of x , 7, and C, to the kinetic equation x = 5 / 1 2 C,q as shown in Fig. 10 taken from the work of Lee and F a i r b a ~ ~ k ~ ~ . The apparent failure of this equation below 1 OK suggests that some new process may be contributing in this temperature region to the transport of energy or momentum. Finally it is of interest to note that below 1 OK the marked change in temperature dependence of the viscosity, which has been interpreted as the beginning of a transition to the expected T-2 dependence of 7 for a Fermi liquid’, is not reproduced by the thermal conductivity (see footnote pag. 134). 5.2. HEATTRANSPORT IN SOLID3He

The heat conductivity in solid 3He has been measured by E. J. Walker and H. A. Fairbank63. For a dielectric solid at low temperatures the thermal conductivity should be given by the expression x = ATne-elbT’,

and the results (shown in Fig. 11) demonstrate that this relationship is obeyed by solid 3He. The changes in slope for the lower density curves have been tentatively identified with the change in the sign of the expansion coefficient of the solid reported by Sydoriak, Mills and Grilly44. The discontinuity in the curve BB’ is attributed to the a+ phase change which occurs at this density as the temperature is reduced (see Fig. 3). 5 . 3 . SELF-DIFFUSION COEFFICIENTFOR LIQUID3He

The coefficient of self-diffusion has been measured in liquid 3He, using spin echo techniques, by Garwin and Reich6*and by Hart and Wheatley‘j5t.The former authors determined the pressure as we1 as the t These measurements have recently been extended to 0.03 “K by Anderson, Hart, and Wheatleye2. The magnetic susceptibility was simultaneously determined.

References 9. 150

CH. IV,

4 51

LIQUID AND SOLID

I

I

I

I

3He

I

1

137

I

VT PK-~)

Fig. 11. Thermal conductivity of solid SHe from E. J . Walker and H. A . Fairbankss.

temperature dependence of D.Within the experimental error the coefficient of self-diffusion for pure liquid 3He is given as a function both of T and e, by the empirical equation

D

=

5.9 In

0.16 (T) exp (T/2.8)

applicable between about 1.5 and 4 OK for pressures from 2.4 to 67 atm. A t about 0.55 OK the diffusion coefficient in the saturated liquid passes

through a minimum and increases rapidly below 0.2 OK as shown in Fig. 12, taken from the work of Hart and Wheatley. Above approximately 0.6 O K , it is apparent from eq. (3) that the References p. 150

138

E. R. GRILLY AND E. F. HAMMEL

[CH. IV,

36

diffusion process in liquid 3He is neither thermally activated [requiring an exp (-T,/T) type temperature dependence] nor gas-like (for which D = IAV cc TIJa). Garwin and Reich suggest that the observed dependence is explicable qualitatively by considering the diffusion of 3He to be a quantum mechanical tunneling through potential barriers. The increase in D at low temperatures is, according to Hart and Wheatley, probably caused by a decrease in the probability of atomic scattering processes, (equivalent to an increase in the excitation mean free path predicted by Pomeranchuk* and Landau2).

*-

VERAGED DATA, 5-6 CM-HG ATURATED VAPOR PRESSUR GARWIN AND REICH. 238ATM 2

5

Fig. 12. Logarithm of the self-diffusion coefficient of liquid W e vs the logarithm of the temperature. The data of Garwin and Reich are taken from ref.u4.

5 . 4 . SELF-DIFFUSION COEFFICIENTFOR SOLID3He

The preliminary data available is discussed in Sect. 6 . 2 . 6. Nuclear Spin Relaxation in Condensed 3He 6 . 1 . LIQUID3He

The spin-lattice or longitudinal relaxation time T , of a spin system is defined as the time necessary for all but l/e of the spins, following an instantaneous change of state, to reach thermal equilibrium with the other degrees of freedom of the medium containing the nuclei in question. T , is therefore a measure of the coupling or interaction between the nuclear spin system and the “lattice”. According to the References p , 150

CH.

IV,

9 61

LIQUID AND SOLID

3He

139

theory by Bloembergen, Purcell and Pound66,the spin relaxation of a given nucleus in a pure liquid is caused by the Fourier components, at the Larmor frequency, of the fluctuating magnetic fields generated at a given nucleus by the thermal or Brownian motions of adjacent nuclei. The associated relaxation time is given by

where y is the gyromagnetic ratio, b the average interspin distance, z, is the "correlation time" of the motion (z, is a measure of the time interval during which molecular orientation persists and the local field at the nucleus is approximately constant) and w o is the precession 1, a condition fulfrequency in the field Ho(wo= yH,). For mot, filled in all of the spin relaxation experiments carried out in liquid 3He to date (t,M 10-12 sec, w o rn lo7), eq. (4)can be simplified to yield Ti1 = 0.9 '/41i2b-'tc.

<

For monatomic liquids with diffusion coefficient D , this equation becomes

Ti'

=

0.3ny4@No/aD

(5)

where a is the atomic radius and N o is the number of magnetic moments per unit volume. The Stokes-Einstein expression relating the diffusion and viscosity coefficients may also be used to give? Ti' = 9n2y4?i2qNo/5kT.

(6)

Equations ( 5 ) and (6) in principle permit a comparison of T , values computed from the BPP theory, using either experimental viscosity599 60, or diffusion coefficient 64 data, with experimentally determined values of the same quantity 67-75. Although some uncertainty remains with respect to the appropriate numerical coefficient to be used in the application of these equations to liquid 3He7097679, this problem is at the moment of much less import than that of resolving the widely divergent reported results for T , in liquid 3He. These results initially appeared so confusing that the present authors sought information from the investigators involved, all of whom have respond7 However the gas-kinetic relationship, e D / q = constant, describes the relationship between q and D for liquid 3He over a wider temperature range. References

p . 150

140

E. K. GRILLY AlYD E. F. HAMMEL

[CH. IV,

9

6

ed. Their contribution to this section is gratefully acknowledged. The following is then, a summary of the status of spin-lattice relaxation time results based upon these communications and published reports. a) On the basis of our present understanding of spin relaxation processes, systematic errors in the measurement of T , should produce shorter rather than longer values of T I . Consequently from the available experimental results, those measurements yielding consistently the largest values of T , should most closely approximate the true value. Figure 13 shows one such spin-lattice relaxation time measurement along the saturated vapor line. These results were obtained by R ~ m e r and ~ ~ ,represent the largest T,’values reported to date (see ref. 93 however). b) Wall relaxation processes have been shown, particularly by Careri, Modena and Santini’O. 71, to yield spuriously short relaxation times. These processes were probably absent in R o r n e r ’ ~work ~ ~ since 900

I

I

I

I

I

1

1

‘

1

I

/

u I0 I25

‘075

15

-

175 2 0 225 250 TEMPERATURE ( O K )

2 275 30 325

Fig. 13, Relaxation time, T , V ~ Y S U Stemperature for various fields, [from Rorner7*]

his measured T,’s were found to be independent of container surface to volume ratios. Romer also used Pyrex glass containers, the walls of which are known to be poor spin relaxing surfaces. c) Bulk impurities (of the order of 1 part in l o 7 oxygen or other paramagnetic impurities in the liquid) can also yield spuriously short relaxation times. Hence even longer T , values than those measured by Romer are not excluded, and the results shown in Fig. 13 must therefore still be regarded as tentative. References

p . 150

CH. IV,

5 61

LIQUID AND SOLID

3He

141

d) R ~ m e r ’ sT ~, ~ results are qualitatively consistent with the predictions of the BPP theory, i.e. equations ( 5 ) and (6). e) The BPP theory predicts that T , should be independent of magnetic field. Romer’s results (Fig. 13) demonstrate this independence from about 1-3.2 OK, and for fields from 1560-12 200 gauss. According to recent work by Low and Rorschache3 field dependent T , values probably originate in field dependent wall or bulk liquid impurity relaxation processes. f) The BPP theory also predicts that, provided coot, 1, the spinlattice relaxation time T , should be equal to the spin-spin relaxation time T2. Schwettman, Low and Rorschachso have measured T , and T , in liquid 3He over the temperature interval 1.2-2.5’ K and found that T , w ilaT1. The values of T , so found were about 30 sec, independent of chamber size and the value of T,. This result is obviously not in agreement with the BPP theory. On the other hand, neither has it been shown that T , is unaffected by bulk impurities (see also ref. 93). g) Garwin and Reich72 have reported T , as a function of pressure. Their results at 2.38 atm (the lowest pressure studied), showrrin Fig. 13, are considerably lower than R ~ m e r ’ and s ~ ~show a different temperature dependence. For reasons noted above, it is probable that both wall and bulk impurity processes produced, in the low pressure results, the low T , values. As the pressure was increased the temperature dependence of TI at low temperatures changed sign and T , passed, with increasing temperature, through a maximum which shifted progressively to higher temperatures. At the highest pressure studied, T , increased with temperature over almost the entire liquid range (2.0-4 OK). In the region 1.5-3.2 OK, T , was found to decrease with increasing pressure. h) Schwettman, Low and Rorschachso observed a decrease in T , with time in the course of a single run. They have also, when the same sample chamber is used for several successive runs, observed a monotonic increase in values of T , from one run to the next. Observations such as these and those of Careri et aZ.70v71 emphasize the caution which must be exercised in obtaining reliable data from any given experimental apparatus or procedure. RorschachB1has also suggested that by postulating a wall relaxation time rn 1/D, and a bulk relaxation time M D,most of the published results on T , can be understood (see also ref. 93).

<<

Refercwces p . 150

142

E. li. GRILLY -4SD E. F. HAMMEL

[CH. IV,

96

From the above it is clear that although considerable progress has been made in clarifying this subject, additional work is still required to elucidate the nature of wall relaxation processes and the degree of applicability of the BPP theory to liquid He3. And finally in view of Hart and Wheatley's 65 results on the temperature dependence of the diffusion coefficient below 0.5 "K, it is clear that measurements of T , and T , at progressively lower temperatures will prove extremely interesting. 6 . 2 . SOLID3He

Reichs2 and Goodkind and Fairbanks3 have recently measured T , and T , in solid 3He as a function of temperature and pressure, and Reich, in addition, has measured D. Their results are summarized below. a) In the u-phase T , M T , in the high temperature region (wt 1) as would be expected according to the BPP theory of nuclear spin relaxation modified for relaxation by translational diffusion by Torre^'^. Upon reducing the temperature at constant solid density T , and T , both decrease and then diverge, T , increasing and T , decreasing. After passing through the minimum and entering upon the 1, TIincreases initially, as exp (T,/T) and then region where ~t becomes constant for all densities. A t constant temperature (in the vicinity of 2 OK) a discontinuity in T , and T , (TI increasing, T , decreasing) is observed upon passing from the a to ?j 3He solid. At lower temperatures, Goodkind and Fairbank find this discontinuity replaced by a discontinuity in slope of the In T , and In T , curves vs 1/T. b) In the high temperature region (i.e., COT 1) for which T is greater than about Tm/2and in the a-phase, Reich's results indicate that

<

>

<

D

=

Do exp (--T0/T).

(7)

To was found to increase with increasing density according to To = 5.2Tm.Do was shown to be approximately equal to 3.5 x cm2/sec, independent of density. At the lowest density, D becomes constant as the temperature is further reduced. The results of both investigators are tabulated below.

References p . 150

CH. IV,

9 71

LIQUID AND SOLID

3He

143

TABLE 7 Activation energies for diffusion in solid a

Approx.

e

klcvn3)

____

0.13407 0.13881. 0.148* 0.150* 0.1500t 0.1561t

t

- 8He (see eq. (7))

_. -- -

...

.

7.68 9.79 12.5

14.0 13.70

15.45

ReichBe,* Goodkind and Fairbankas

7. Velocity of Sound in 3He

7.1. EXPERIMENTAL RESULTS

Direct measurements of the velocity of sound in liquid 3He along the saturated vapor pressure curve as well as at higher pressures have been carried out by Laquer, Sydoriak, and Roberts37 and by Atkins 84. The sound velocity has also been measured by the and

TEMPERATURE CK)

Fig. 14. Sound velocities in liquid SHe and 4He [from ref.871. 3He data at 5 MHz from Laquer, Sydoriak and Roberts37;4He data from Van Itterbeek and Forrezs6 and from Atkins and Stasiorsa. References

p. 150

144

[CH. IV, 9

E. R. GRILLY AND E. F. HAMMEL

7

TABLE8 Sound velocity in fluid 3He

T

("K) 0.0

0.5 1.0

1.5 2.0 2.5 3.0 3.2 3.40 3.87 4.22

8

b C

d

U,(P) (in m/sec) (from ref.S4)at indicated Bressures i n atmos.

LT,(saf) (iiz m/sec)

from ref.s7) 183.4 182.4 177.8 170.0 158.5 141.8 115.0 99.0 -

1

1938 191b 178 16Ob

136 1286

-

2

4

-

21Oa 206b 197 184b 167 160 -

2348 232b 228 221b 210 204 200c 185C 172bC

6

2588 255b 251 246b 236 231 -

8

9

-

-

-

2748 272b 269 265b 260 257 -

-

2788 270 266 -

Extrapolated from smoothed published data. Interpolated from smoothed published data. Gas values at 3.99 atm (see original paper for additional data). Inconsistent with value of 99 m/sec a t saturated vapor pressure from refs.s7 andS'J.

1.5. The velocity of 14 MHz sound in liquid SHe as a function of temperature at various constant pressures [from Atkins and Flicker 841.

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latter authors above the critical temperature in the gass4. Along the saturated vapor pressure curve there is effectively no difference, within the experimental error, between the data from the two laboratories (see note d of Table 8, however). The results are displayed in Figures 14 and 15 and in Table 8. 7 . 2 . ATTENUATION

Sound attenuation measurements in 3He have not yet been reported. Although the attenuation in principle was obtainable from measurement of successive echo amplitudes in the sound transmission cell of fixed path length used in the velocity measurements, both groups of investigators found that these amplitudes were too poorly defined to permit their accurate measurement. In part this was due to the small liquid samples available which accentuated wall effects, and to the fixed geometry of the sound cell which prevented in sit% alignment of the crystal transducers. The theoretical implications of the variation of the attenuation with temperature have been discussed by Goldstein 15, Landau3, and by Abrikosov and Khalatnikov7. Pellam and Squire8' first measured sound attenuation in 4He I, and showed that in this liquid the classical expression

was obeyed from about 3.2" to 4.5 O K . Below 3.2 OK an extra attenuation was observed, increasing to a value many times the classical at the A-point. For 3He there is no a ?riori reason for expecting a deviation from the above expression until temperatures of the order of 0.1 OK are reached. Below 1 OK, the attenuation due to the thermal conductivity should be negligible due t o the smallness of the factor (C,/C, - 1). According to Abrikosov and Khalatnikov7 the first viscosity 7 5, the second viscosity for 3He, and the attenuation below about 1 "K should therefore be given almost entirely by the first viscosity term. At sufficiently low temperatures (i.e., less than 0.05-0.1 OK) as has already been pointed out by Pomeranchuck* and by Abrikosov and Khalatnikov' the viscosity and hence also the attenuation of liquid 3He should vary inversely as T 2 .

>

References p . 150

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7 . 3 . ZERO SOUND

[CH. I V ,

fi

7

A t some temperature below 0.3 OK, the quantum statistical properties of liquid 3He should also manifest themselves in its sound transmission behavior. Landau3 has suggested that ordinary compressional waves of sound will continue to propagate in the liquid provided the wave length is long compared with the mean free path of the 1 where t is the liquid relaxation time. In quasi-particles, i.e. ~t this region the classical attenuation (eq. (8)) should continue to be obeyed. For a Fermi liquid z cc T-2 however, so that for any given frequency there will be some temperature below which the above inequality will no longer be fulfilled; the wave length of the sound will approach that of the mean free path of the quasi-particles, and the sound wave will be strongly attenuated. At higher frequencies or lower temperatures, for which at 1, Landau predicts the existence of a new type of sound termed zero sound. Since the wave length of zero sound is very much less than the mean free path of the quasiparticles, collisions between the quasi particles are neither essential for its propagation nor capable of establishing local thermodynamic equilibrium in the path of the sound wave. Zero sound is thus a nonequilibrium type of wave propagation. It is characterized analytically by a periodic deformation of the Fermi surface (ie., a time variation in the distribution function). An example would consist of an extension of the Fermi surface at maximum amplitude in the direction of the wave motion and a lesser flattening of the surface in the opposite direction. Half a cycle later the deformation is reversed. The velocity of zero sound in liquid 3Hein the limit T -+0 is estimated to be slightly larger than that of first sound, namely 192 m/sec. Although in principlc, zero sound modes which differ from one another in their angular dependence of both velocity and amplitude are possible in a Fermi liquid, Landau considers it improbable that such modes can be propagated in liquid 3He. Experimentally, zero sound in liquid 3He should be equivalent to an ordinary compression-rarefaction wave in the medium and should be demonstrable by suitable ultrasonic techniques. to The attenuation of zero sound will be proportional to T2 (h., the number of collisions of the quasi-particles, which in turn result in absorption of the sound quanta), and independent of the frequency provided the energy qf the sound quanta is small in comparison with that of the quasi-particles, i.e., tiw kT. In addition both these latter

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>

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Referemes p . 160

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quantities must be large with respect to the quantum uncertainty in Kw hz-l, the latter condition the energy of the quasi-particles: kT being a general one for applicability of the whole theory of a Fermi liquid. As the temperature is lowered further or the frequency increased, fio will become equal to or greater than kT. In this region a quantum calculation shows the attenuation t o be dependent only on the square of the frequency. Because of the high frequencies necessary for the propagation of zero sound, Kapitza’ suggested that this phenomenon might be effectively investigated using the satellite lines (Brillouin doublets) from the Rayleigh scattering of visible light. Investigation of this idea led to the conclusion that the frequency shift of the satellite lines would be related t o the velocity of zero sound by the expression d o = f qzc, where u is the zero sound velocity and q = (2o/c) sin #Iwhere , 8 is the scattering angle. There is some question, however, whether the intensity of the scattered beam will be sufficient to permit an accurate measurement of the effect.

> >

7.4. SOUND PROPAGATION IN LIQUID3He BELOW THE “PHASE TRANSITION”

According to pair correlation theories, at the transition temperature the attenuation will increase strongly. For temperatures below T , and low enough so that the number of quasi-particles is small, it has been predictedss that ordinary sound will again be propagated in the “superfluid” with a velocity

where p , is the momentum at the Fermi surface and p is the mass of a 3He atom. In the correlated phase the attenuation will be small (similar to *He) and will decrease with decreasing temperature due to the decreasing density of excitations. 8. Summary Since the writing of the article on 3He for this series in 1955,not only has much more experimental work appeared, but also theoretical descriptions of the liquid and solid have become much more sophisticated. In 1955, although the difficulties inherent in the simple ideal FermiDirac description of 3Hewere beginning to be recognized, no alternative References p . 150

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[CH. IV,

08

treatment had yet appeared. Subsequently, several attempts to introduce the effects of interactions between the “particles” were made, with the result that, at the present time, our understanding of liquid and solid 3He has progressed considerably. Formidable mathematical difficulties still stand in the way of a quantitative theoretical description of 3He, however, and consequently assumptions, approximations, and experimental data have been required to derive theoretical predictions of new 3He phenomena. The degree to which the theoretical conclusions depend upon these approximations and assumptions is as yet not well established and for those cases in which experimental data is available to compare with theory, the correspondence, although sometimes impressive, is more often only fair. But it is probably naive at the present time to expect any theory to provide a complete and quantitative description of 3He. Hence if the different theories arc viewed by experimentalists as alternative approaches to an exceedingly difficult problem, and if the comparison of theoretical predictions with experiment is used by the theoreticians to draw conclusions concerning the validity of the various approaches employed, 3Hewill continue to be a rich and rewarding raw material for both experimental investigation and its complementary theoretical interpretation for some time to come. In summary, it appears that : a) The incipient linear temperature dependence of the specific heat as T -+ 0 provides a satisfactory agreement between quasi-particle theory and experiment, at least at low pressures. The prediction of a sharp maximum or discontinuity in specific heat at 0.03 < T < 0.08 by the pair correlation theories still lacks an experimental check. Even if a transition to a correlated phase is subsequently demonstrated, the quasi-particle description may still be valid in the temperature range T , < T T,. Above 0.2 OK, the specific heat lacks a basic explanation in much the same sense as in all other theories of the liquid state. Further specific heat work, both experimental and theoretical, is warranted on the compressed liquid and on the solid, including for the latter an investigation of the predicted singularity in specific heat at about 0.1 OK. b) In general, the experimental PVT relations of the liquid are fairly well established. The locus of the minimum in thermal expansion with respect to temperature and pressure requires further definition, however. For both liquid and solid along the melting curve, it seems that an anomaly in thermal expansion ( a = 0 ) might occur at about

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Referewes p . 150

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0.3 OK. A possible maximum in melting pressure at about 0.08 OK has been predicted but not yet observed. The experimental observation that the cr-type solid is not close-packed was unexpected theoretically and is not understood. c) Experimental determinations of transport properties in general tend to support Landau’s theory of a Fermi liquid. Although the predicted variation of viscosity as T-2 is not inconsistent with experimental data obtained to date, the thermal conductivity was found to have a positive? variation with T in the corresponding temperature range instead of tending toward the predicted T-l. Measurements to still lower temperatures are obviously required. The most recent data on the variation of the diffusion coefficient with temperature tend to support Landau’s predictions. d) Sound absorption in liquid 3He has not yet been the subject of an experimental investigation, and in view of the predictions of “zero sound” this topic appears to be a rewarding if difficult research problem. e) Because the nuclear magnetic susceptibility of liquid and solid 3He had been adequately discussed in previous reviews5*,this subject was omitted from the present article. Recently however the results of two new investigations by Low and Rorschachs3 and by Adams, Meyer, and Fairbankg5have appeared. The former presents additional susceptibility data in the liquid, and the latter work includes new measurements on nuclear resonance in both the liquid and solid phases, including a discussion of ferromagnetism in the compressed liquid and in the solid. Finally, new measurements of the liquid susceptibility to 0.03 OK have also been reported by Anderson, Hart, and Wheatleysz.

Acknowledgement The authors wish to express their appreciation to their many colleagues who contributed to this article by discussion, letter, or the sending of preprints of recently completed research. In particular the authors wish to thank Professor Robert Brout for many helpful comments on the theoretical section.

t

See footnote p. 134.

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77