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Chapter XI Solid Helium
CHAPTER X I SOLID HELIUM BY
C. DOMB KING'SCOLLEGE, UNIVERSITY OF LONDOM AND
J. S. DUGDALE DIVISIONOF PUREPHYSICS, NATIONAL RESEARCH COUNCIL, OTTAWA CONTENTS:Part I , 338. - 1. Introduction, 338. - 2. General Thermodynamic Properties of Solid He4, 340. - 3. Thermal Conductivity of Solid Helium, 346. - 4. Melting Curve and Melting ParameCers of He4, 349. - 5. Melting Parameters and General Properties of Solid He8, 355. - Part 11, 356. Theoretical Aspects, 356.
I. EXPERIMENTAL ASPECTS
1. Introduction Before the last war, the thermodynamic properties of solid helium had been investigated only in the temperature region 1.5 to 4 ' K and under pressures up to about 140 atmospheres The melting characteristics of solid helium had been more extensively studied: the latent heat of melting had been measured at several temperatures 1, and the melting curve had been determined by Simon and his coworkers up to a temperature of about 43' K, corresponding to more than 8 times the liquid-gas critical temperature. The melting curve for helium is shown in Fig. 1. The unique importance of solid helium in the study of melting has been stressed by Simon 5, S1, whose argument may be summarized in the following way. Experiments on many substances show that their melting curves may all be represented by an equation of the form6
For a given class of substance (e.g. the solidsed gases, the alkali metals, etc.) c has approximately the same value, so that there exists
SOLID HELIUM
339
for each class a reduced melting curve in which a and To, the characteristic temperature and pressure, are the reducing parameters. Of all substances helium has the lowest value of a and of To and so, for given laboratory resources, is capable of yieIding the most extensive information about the behaviour of the melting curve of a typical substance at ‘high’ temperatures and pressures.
TEMPERATURE f ’ K f
Fig. 1.
The melting curve of He4; T,is the critical temperature. The inset shows the low temperature phase diagram of He4.
Of great practical importance is the fact that the melting properties of helium can be studied in the region of low temperatures where the heat capacities of the necessary pressure vessels are conveniently small. This of course applies equally to other thermal measurements on solid helium under pressure. The effect on these properties of quite modest pressures is drastic. For example, by applying to solid helium pressures of only a few thousand atmospheres, its volume can be halved and its Debye temperature changed by a factor of 5. From a more detailed theoretical point of view, the properties of the solid have a double interest: the helium atom, having central
340
C. DOMB AND J. 9. DUGDALE
additive short-range forces, makes the solid a suitable substance for theoretical treatment, while the small mass of the helium atom and its relatively weak binding force make manifest, in a remarkable way, the effects of the zero-point motions of the atoms in the solid. All these considerations which are true of solid He4 apply even more strongly to solid He3, and they have formed the basis of much of the recent work on solid helium. This work we shall now discuss in more detail.
2. The General Thermodynamic Properties of Solid He4 2.1.
SPECIFICHEATS
Keesom and Keesom in 1936 measured the heat capacity of solid ' K. helium at low densities in the temperature range from 1.5" to 4 The more recent work has, therefore, been concentrated on the temperature regions above and below this. Webb, Wilkinson and Wilks? were primarily interested in the thermal conductivity of solid helium and wished to correlate their conductivity measurements with the characteristic temperature of the lattice. I n order to do so, they measured the specific heat of the solid at temperatures between 0.6 and 1.5" K at three different densities. Their method was to enclose both the paramagnetic salt (which was used as the cooling agent) and the solid helium in a calorimeter and to measure the total heat capacity. The salt was used as a thermometer and heat was supplied electrically. The heat capacity of the helium was obtained by subtracting from the total the estimated heat capacity of the calorimeter and salt. Their results are shown in Fig. 2 . Dugdale and Simon* measured the heat capacity of solid helium in the higher range of temperatures and densities, covering the region from 5' K to the melting temperature at molar volumes between 16 and 10.6 CCS. They used a vacuum calorimeter designed to stand pressures up to 3000 atmospheres, and measured temperatures with a constantan thermometer. Their results at four densities are shown in Fig. 3. The measurements of Keesom and Keesom may be compared directly with those of Webb, Wilkinson and Wilks. The two sets of results are in reasonable agreement and indicate that the Debye temperature a t these densities rises with falling temperature. I n Fig. 4 the Debye temperatures for the two lowest densities are derived
341
SOLID HELIUM
from the measurements of Keesom and Keesom. At the still lower temperatures reached in the experiments of Webb et al. the 6, reaches a maximum and begins to fall. The Debye temperatures, calculated from Dugdale and Simon's measurements, are also shown in Fig. 4: they are almost constant although falling somewhat as the melting point is approached.
0.5
I
1.0 temperature (" K )
I
1.5
Fig. 2. The specific heat at constant volume of solid He4 at three Merent densities. I. 0.194 g/cms; 11. 0.205 gfcms; 111. 0.214 g/cm3.
Unfortunately the low temperature measurements of Keesom and Keesom and of Webb et al. on the one hand and the higher temperature measurements of Dugdale and Simon on the other do not sufficiently overlap in density and temperature range t o allow a very close comparison and more measurements would be valuable, particularly at high densities in the liquid helium temperature range. 2.2. THE EQUATION OF STATE OF THE SOLID Dugdale and Simon found that their results could be adequately represented by a Griineisen equation of state. That is, the specific
342
C. DOMB AND J . S. DUODALE
Fig. 3. The heat capacity of solid He4 at four molar volumes. 0 , 10.6 ema; 0 , 11.7 emS; f, 13.0 cm3; A, 14.4 em3.
heat at constant volume was found to be a function of T/p,where p, is a characteristic temperature depending only on the volume. Furthermore, for the purpose of calculating certain thermodynamic properties of the solid from the specific heats, the results of Keesom and Keesom could also, though less closely, be represented in this way. The values of p, calculated from their results are shown in table I as a function of volume. rn*DTrn
T
___ 10.5
11 12 13 14
113 101.5 83.2 68.7 57.4
15 16 17 18 19 20
48.5 41.4 35.7 31.1 27.2 24.2
343
SOLID HELmM 140
I
0
5
I
10
I
IS
I
20
TEMPERATURE (‘KI
Fig. 4.
50
Fig. 5.
The Debye 8’s of solid He4 at six molar volumes.
I
5
10 15 temperature (OK)
I
20
The isochores of solid He4 at eight molar volumes. - - - - - transition line in the solid.
I
25
344
C. DOMB AND J. S. DUGDALE
From the specific heats it is then possible to calculate the entropy,
8, and the thermal contribution to the internal energy, U - U,. Here U is the internal energy at temperature T and volume V , and U , the internal energy at the same volume and at absolute zero. Table I1 gives G,, S and (U-U,)/T as functions of T / q derived in this way. T A B L E I1
C,, S and ( U - Uo)/T as a function of T/tp. (All in cal/" C per mole)
0.04 0.06
0.08 0.10 0.12 0.14 0.16 0.18 0.20
0.02 0.08 0.19 0.38 0.64 0.96 1.33 1.73 2.21
0.010 0.017 0.043 0.091 0.158 0.248 0.359 0.491 0.646
0.008 0.023 0.060 0.121 0.213 0.333 0.483 0.662 0.870
The specific heats as a function of temperature and volume do not by themselves form a thermodynamically complete set of data. For completeness one p , V , T relationship which traverses the same range of temperature and volume is needed and the melting curve together with the volume of the solid on melting provides this additional information (see 5 4.1 below). On this basis it is now possible to calculate the isochores of the solid by integrating the relationship dlnq 'Yc o where y = dlnV
which is valid if Go is a function of T/p,and q depends only on volume. The isochores at eight densities are shown in Fig. 4 together with the equilibrium line between the two solid modifications ( 5 2.3). This figure shows that the isochores are only very slightly temperature dependent: indeed, as a first approximation, it might be said that the pressure in the solid is independent of temperature and dependent only on the volume *.
* It is partly for this reason that solid helium can be a useful pressure trensmitting medium at low temperatures. The presrsure can be applied at such a temperature that the helium k fluid. Thereafter the system may be cooled at constant volume to the working temperature without serious loss of pressure.
345
SOLID HELIUM
From the data in Fig. 5, the compressibility, B, at O°K can be derived and by using Eq. (2) the thermal expansion coefficient, a, can be calculated. These quantities are listed in tables 111and IV.On the same basis the internal energy at 0' K can be found and hence, by subtracting from this the zero point energy, the lattice energy can be deduced*. T A B L E 111
The compressibility of solid helium at 0°K volume bmS)
preseure (atm)
1vp
volume
(atm-1)
10.6 11.0 12.0 13.0 14.0
2170 1660 1070 695 460
10 12 17 26 38
15.0 16.0 17.0 18.0 19.0
1vp
pressure (atm)
(atm-l)
295 200 136 88 50
54 76 103 140 190
TABLE IV
The thermal expansion of solid helium; the volume expansion coefficient &B a function of temperature 10.6 cms 104 a
T (OK) 0 2
0
-
4 8 12
l6 20
.
0.09 0.93 3.4 7.3 12.6
12 cm8 104 a 0 -
0.48 4.9 14.6
-
15 cm8 104 a 0 1.05 10.2
-
-
2.3. LATTICE STRUCTURE AND TRANSITION Dugdale and Simon in the course of their investigation of the heat capacity of solid helium found an anomaly in the specific heate between 15 and 17'K, the actual temperature depending on the volume. They concluded from their experiments that this was a first order, reversible transition involving an entropy change of about 5 x 10-s cals/"K per mole *. Furthermore, since substances with shortrange forces of the van der Waals type are stable only in close-packed
* Since all such quantities associated with the transit,ion are small (e.g. the volume change is estimated at 4 x ccs/Mole) their contribution to the general thermodynamic properties of the solid has been neglected.
346
C. DOME AND J. S. DUQDALE
lattices, they also concluded that the transition was due to a change from the close-packed hexagonal structure stable at the lower temperatures (i.e. the so-called oc-phase) to the close-packed cubic structure stable at the higher temperatures (the p-phase).
3. The Thermal Conductivity of Solid Helium 3.1. EXPERIMENTAL RESULTS 3.1.1,
Introduction
I n discussing the thermal conductivity, x , of a dielectric it is convenient to define a mean free path, 1, by the relationship x = QC.v.rZ
where C is the lattice heat capacity per unit volume and v the velocity of the lattice waves. At sufficiently low temperatures, the thermal conductivity of a chemically pure and physically perfect dielectric is limited only by the dimensions of the specimen. If, for example, the specimen has the form of a long, narrow cylinder then the mean free path is under these conditions approximately equal to the diameter of the cylinder. Consequently the thermal conductivity varies directly with the heat capacity, C , which means in general that x is proportional to the cube of the absolute temperature at very low temperatures. The growth of thermal conductivity in an ideal crystal with rising temperature is ultimately arrested by so-called ‘Umklapp’ processes, a concept introduced by Peierls l’Jtll. I n an Umklapp process, two normal modes interact to produce a resultant whose wave number lies outside the basic cell of the reciprocal lattice. This means that its wave number is opposite in direction to that of its components. According to Peierls it is just these Umklapp processes which give rise to thermal resistivity in ideal insulators, and he estimates that the rate at which they occur at low temperatures is proportional to exp ( - B/bT); here 0 is the characteristic Debye temperature of the lattice and b is a numerical factor greater than 1 l2. It thus seems reasonable to assume that the mean free path at such temperatures should be given by A = A exp (B/bT) (3) where A is a parameter which depends on the coupling of the lattice waves and may indeed include a temperature factor.
347
SOLID HELIUM
3.1.2. The Measurements
Measurements of the thermal conductivity of solid helium have been made by Webb, Wilkinson and Wilks7.13 and by Webb and Wilks 14. The conductivity was measured at densities between 0.345 and 0.194 gmslcc in the helium temperature range; at the lowest density, the measurements were extended down to 0.3"K. This change of density altered the Debye temperature of the lattice from about 25" K to about 90" K and increased the thermal conductivity at 2 ° K by a factor of over 300. Webb, Wilkinson and Wilks made the measurements at the lower densities (0.194 to 0.218 gmslcc). For most of the experiments the specimen holder was a German silver tube of internal diameter 6 mm: for the measurements below 1" K the holder had a diameter of 0.5 mm. Their results at three densities are shown in Table V. TABLE V
The thermal conductivity of solid helium Density
(gmsicm7
0.194 0.208 0.218
*
Temperature in 0.25 20"
-
0.50 85" -
1.0 107
-
*
1.2 24" -
-
O
1.4 11 58 160
K
1.7 5 19 52
8
2.0
2.3 -
22
11
These size-dependent values were obtained with specimens of 0.5 mm diameter.
At the higher densities, the measurements were made by Webb and Wilks using two experimental arrangements which were essentially modifications of the previous apparatus. Up to a density of 0.28 gms/cc the results of this work were self-consistent and consistent with the earlier measurements. These results and some from the earlier work are shown in Pig. 6. For densities above 0.28 gms/cc however the two apparatuses gave different results. These differences were not simply attributable to specimen size effects and moreover the results from each apparatus singly were somewhat irregular. It is therefore difficult to make any deductions from these measurements. 3.1.3. Compurison with Theory
The experimental values of thermal conductivity and specific heat enable the mean free path of lattice waves, 1, to be calculated (the
348
C. DOME AND J. S. DUGDALE
velocity of sound may be deduced from the appropriate value of 6). The mean free paths so deduced are shown plotted as a function of 8/T at various densities up to 0.218 gms/cc in Fig. 7 '. Two conclusions can be drawn. I n the first place, provided that size effects are absent, log 1 is proportional to 8lT as the theory
Temperature O K Fig. 6.
.,
T h e hhermd conductivity of solid He* st four densities.
A, 0.218 g/cm3; 0 , 0.262 g/cms; x, 0.276 g/oms;
0.282 g/cms.
demands. The constant b of Eq. (3) has the value 2.3 and the parameter, A, of the same equation appears t o be independent of density, having the value 6 x cms. I n the second place, size effects become evident at very low temperatures (large values of e/T)and 1 becomes almost constant, although even at the lowest temperature it is only about half of the theoretically
349
SOLID HELIUM
expected value, 5 x cms., which is the diameter of the specimen. It should also be noted that the thermal conductivity, in the temperature range where il is limited by size, varies as T2e3and not T3 as theory predicts. I n a recent paper Leibfreid and Schlomann have attempted for the first time to calculate the absolute value of the thermal conductivity of dielectric crystals. The anharmonic terms in the expansion of the lattice potential energy are taken account of by means of Gruneisen's y. The authors find that the thermal conductivity is given by x,f(B/T) where f is a universal function, and x, is proportional to 8Ba/y2, 6 being the lattice spacing of nearest neighbours in the crystal. When detailed values are substituted for solid helium reasonable agreement in absolute values is obtained over a considerable range of pressures.
lfJ-
/4
*? */ ?
-i
c
cc
';1
-
A 9
4, 12-
I
I
I
I
I
el T
Fig. 7. The mean free path of lattice waves in solid He'.
4.
The Melting Curve and Melting Parameters of He4
4.1. EXPERIMENTAL RESULTS 4.1.1. The Melting Curve
Because helium can remain liquid down to the lowest temperatures, the melting curve of solid helium does not join the vapour pressure
w cn
T A B L E VI
0
Measurements of the melting curve of He4 Author
Date
Temperature Range
Pressure Range
Estimated Accuracy
Keesom l5 Simon, Ruhemann and Edwards3 Simon, Ruhemann and Edwards4 Keesom and Keesom l6 Holland, Huggill, Jones and Simon 17. Simon and Swenson l9. 20 Swenson 21
1926 1929
1.2" to 4.2" K 12.2" to 20.2' K
25-140 ats. 800 to 1800 ats.
1929
20" to 42" K
1800 to 5500 ats.
1933 1950
1.15" to 1.8" K 35' K to 50" K
2&30 ats. 4000 to 7500 ats.
0.1" K, 0.1 ats. O.l'K, 1 yo of pressure 0.loK, 1 yo of pressure 0.01" K, 0.01 ats. 0.1" K f 100 ats.
Density measurements. Blocked capillary
1950
1.0" to 1.8" K
25.0 to 27.0 ats.
.Ole K, 0.01 ats.
Blocked capillary.
1952
1.6' to 4" K
27 to 130 ats.
Piston displacement *.
Swenson 22
1953
1.5' to 4" K
26 to 130 ats.
Dugdale and Simons Robinson 23
1953 1953
4.2" to 12" K 20" to 55" K
140 to 800 ats. 9,000 ats.
Mills and Grilly z4
1955
2" to 30" K
35-3,500 ats.
.Ol" K, 0.1 yo of pressure .01" K, 0.1 yo of pressure 0.1" K, & 3 ats. f 0.5" K, f 5 % of pressure .01" K, .05 yo of pressure
Method used Blocked capillary. Heating curve method. Blocked capillary.
Blocked capillary *. Blocked capillary. Moving pellet. Blocked capillary,
* Swenson 22 has discussed in detail t,he relative merits of the blocked capillary and piston displacement methods of measuring melting curves.
351
SOLID HELIUM
curve at a triple point. Instead the melting pressure at very low temperatures tends to become constant, its dope vanishing in accordance with the third law of thermodynamics. There have been many measurements made on the melting curve of He4 and these are summarized in Table VI, which shows the range of measurements in temperature and pressure, the method used and the estimated accuracy achieved. According to Swenson 20, the melting pressure between 1.0 and 1.4"K can be represented by P=25.00+0.053T8 ats; and between 2.1 and 4" K by P + 5 . 6 13.458 ~ T1."3074ats 21. Values of P up to 6" K are given in Table VII. All workers agree that their results above 4°K can be represented by a Simon-type expression as given in Eq. (1). The Oxford workers have found that the melting curve can be expressed in the form T 1.6644 P -= -1 (4) 16.46 (.992) where P is measured in atmospheres, and T in degrees Kelvin. Mills and Grilly gave their results as
P + 17.80 = 17.315 T1.5554
(5)
with P in Kg/cm*, which may be re-written in the same form and units as Eq. (4) T 1.6664 -P(6) 17.23 - (1.0179)
-'
4.1.2. The Melting Parameters
Swenson 22 has made very accurate measurements of the melting parameters of He4 between 1 and 4" K. A free piston gauge was used to measure the melting pressure, and the volume change on melting was measured by the displacement of the piston. Dugdale and Simon, using the apparatus already described for measuring specific heats under pressure, measured aIso the melting parameters of solid helium at higher temperatures. The combined results of these experiments and those of Swenson are given in Table VII, and shown in Fig. 8 and 9. The melting parameters, AS, A U and A V a t very low temperatures are of particular interest 19. The application of the Clausius-Clapeyron Equation to the melting curve enables us t o determine A S / A V , and according to the measurements of Simon and Swenson l9 previously
352
C. DOMB AND J. 5. DUODALE
TEMPERATURE P K I
Fig. 8. The entropy of He4 on melting and solidification. The fine lines m e lines of constant volume; - - transition line in the solid.
--
FLUID
I
I 1 5
SOL ID 10
20
TEMPERATURE ('K)
Fig. 9. The molar volume of He4 on melting and solidification. _ - _ _ _ transition line in the solid.
353
SOLID HELIUM TABLE VII
The entropy and molar volume of He4 dong the melting line. Entropies ( S ) in cals/"K per mole, volumes ( V ) in cms/mole. (The volumes above 4" K are specified more accurately than is warranted by the experiments in order to give V correctly)
T
O
P (ats.)
K
25.00 25.05 25.22 25.81 26.99 37.08 56.3 78.1 128.6 255
0 1.0 1.2 1.4 1.6 2.0 2.5 3.0 4.0 6.0 10.0 14.0 18.0 22.0 26.0
*
Below 4°K S,
'&Aid*
'Fluid
V@aUd
0 0.03 0.04 0.05 0.06 0.07 0.10 0.13 0.18 0.29 0.48 0.63 0.78 0.94 1.08
0 0.05, 0.ll6 0.26 0.60 1.24 1.37 1.47 1.59 1.79 2.00 2.21 2.42 2.66 2.88
21.18 21.17 21.15 21.05 20.95 20.26 19.32 18.52 17.37 15.53 13.45 12.30 11.42 10.75 10.10
and V,
'FlUid
*
23.25 23.24 23.22 23.11 22.84 21.66 20.60 19.72 18.40 16.42 14.13 12.88 11.95 11.24 10.57
are from the measurements of Keesom and
Keesom 1.16.
referred to, this is of order T7 for small T . From the third law of thermodynamics, AS, the entropy difference between the solid and liquid phase, must vanish as the temperature approaches the absolute zero; in fact these experimental results on solid helium provide a striking confirmation of the third law. (The latent heat of melting (e=T A S ) must vanish even more rapidly.) Since TAS =A U +pA V , two possibilities exist, remembering that p does not vanish. Either AU and AV must both tend to zero with AS so that the two phases become identical, or else A U = - pA V in the limit as TAB vanishes. It is the second alternative which is found experimentally (see Fig. 10) implying that the phase with the larger volume has the smaller energy. Since liquid helium has always a greater volume than the solid, the liquid ultimately has the lower energy and so becomes the only known example of a liquid having an internal energy lower than that of the solid with which it is in equilibrium. The high-temperature trends of the melting parameters are of interest also, because they may give an answer to the general problem
354
C. DOMB AND J. 8. DUGDALE
of whether or not a solid-fluid critioal point exists. It is seen that although the volume difference between the phases decreases with rising temperature, the entropy differenoe continues slowly to increase and one must conclude that there is certainly no evidence here for the existence of such a critical point.
6-
5-
w
J
4-
0
I
> 3A
a
2-
.------ ------
I-
0
-I
------------1
0
I
I
I
2
I
I
3
1
I
4
Fig. 10. The melting parameters of solid He4 below 4OK. e is the latent heat of melting. P is the pressure. AV is the change of volume on melting. AU is the internal energy change.
Another feature t o which Simon has drawn attention is the increase in entropy of solid helium along the melting line, a behaviour which characterizes dl other substances for which this entropy trend can be estimated. If it is true of all solids at sufficiently high pressures that the entropy at the melting point increases with temperature then it is impossible to melt a substance by adiabatic reversible compression, a fact that may be of significance in geophysics and atrophysics.
355
SOLID HELmM
The findemann Melting Formuh Applied to He4 According to the Lindemann melting formula, 4.1.3.
where 8 is the characteristic temperature of the lattice at volume V and T , is the melting temperature; M is the molecular weight of the substance and c is a constant. The measurements on solid He4 provide, for the first time, the data necessary to test this formula over a substantial region of the melting curve of one single substance. Table VIII gives the values of c calculated for He4 at several different temperatures and it is seen that they are indeed very nearly constant. The value of c for He4 is however considerably smaller than for the heavier inert gases for which c is approximately 160. Some further comments on the Lindemann melting formula are made in 8 6.4 below. TABLE VIII
Application of the Lindemann Melting
110 92 72 55 32
23.3 17.3 11.3 7.9 3.1
FOI'Ind8
to He4
10.6 11.6 13.1 14.4 18.3
101 100 102 96 96
5. The Melting mrameters and General Properties of Solid HeS As yet there is no detailed knowledge of the thermodynamic properties of solid He3. Its melting curve, however, has been measured from 0.2 to 35' K. Abraham; Osborne and Weinstock have measured the curve in the low temperature region 25, 26 and Mills and Grilly at high temperatures 27* 28. Both used the blocked-capillary technique. The data can be represented by the following equations
P = 26.8+ 13.1 T2 atm. between 0.5 and 1.5"K and
P- 25.16 = 20.082 T1~51'~ kg/cm2 between 2 and 35" K.
The second of these expressions may be compared with Eq. (5) above which represents the melting curve of He4 over similar temperature and pressure range (see also $5 6.2 and 6.5 below).
356
C. DOMB AND J. 5. DUGDALE
The low temperature end of the melting curve is again, as with
He4, of particular interest but for a different reason. The He3 nucleus, unlike that of He4, has a nuclear spin and magnetic susceptibility experiments have shown that in the liquid state the nuclear spins are 31. Pomeranchuk 32 pointed already partially aligned at 0.4" K out that if the nuclear spins were aligned in the liquid but remained randomly oriented in the solid until dipole-dipole interaction produced alignment then below a certain temperature the liquid would have a lower entropy than the solid. Consequently assuming no dramatic change in the volume relationship, the slope of the melting curve would vanish at this temperature and become negative below it *. Before finally vanishing it might even become positive again when the nuclear spins became aligned in the solid. Pomeranchuk suggested that this alignment should occur a t about 10-7 OK, but Primakoff 52 has pointed out that the special properties of solid helium would make this temperature in the neighbourhood of 10-30K. Experimental evidence on whether the melting curve does in fact change sign is still lacking. Indeed it has been pointed out that the blocked capillary technique could not record such a change since it will always block at the minimum pressure 33. If, however, the melting curve does indeed have a minimum we shall then have the surprising instance of a liquid which freezes as it is heated at constant pressure! 293
11. THEORETICAL ASPECTS
INTRODUCTION The behaviour of solid helium is dominated by the effect of zero
6.1.
point energys8. In detailed character zero point energy consists of random fluctuations of the atoms about their equilibrium positions in the lattice, and these fluctuations are responsible for the fact that helium is a liquid at the absolute zero. As another consequence of the zero point energy the volume of the solid which forms when liquid helium is subjected to a pressure of 25 atmospheres is well over twice the volume corresponding to the minimum of the potential energy
* If this assumption is correct, one may estimate from the entropy diagramS1of HeS that the pressure rise on the low temperature side of the minimum should be about 4 atmospheres, assuming that the volume difference between the phases is comparable with that found in He4.
SOLID HELIUM
357
of the lattice. Thus even in the solid at absolute zero the atoms execute vibrations of relatively large amplitude about their equilibrium positions, and the normal approximation of lattice dynamics which expands the potential up to terms of the second order about static equilibrium ceases to be valid. The first theoretical problem posed by solid helium is therefore to find a modified treatment which can be applied to a system with large zero point vibrations. As mentioned in the experimental section the high compressibility of solid helium has enabled measurement of melting properties and thermodynamic functions to be made over a range corresponding to a factor of two in volume. This is a far greater range than can be achieved with any other substance, and the data can be used to cast some light on the general properties of substances under pressure. Because of the zero point effects helium cannot be regarded as a perfect model substance. Nevertheless, for some purposes it may be possible to take these effects into account and draw general conclusions; and in other cases an analysis of the differences between the behaviour of classical substances and helium can be illuminating. I n particular the data on the melting of solid helium over such a wide range of volumes provide considerable insight into the general properties of the melting curve at high pressures. The polymorphic transition discovered by Dugdale and Simon has also stimulated theoretical work on the cubic and hexagonal packed lattices and their relative stability at differentvolumes and temperatures. Many of the theoretical developments to be described in the following sections cannot be regarded as complete, but do seem to indicate the possibility of accounting adequately for the experimentally observed properties of solid helium. THE STATIC LATTICE AT ABSOLUTE ZERO If a negative pressure could be applied to a solid at absolute zero so that its volume expanded, the solid would cease to be elastically stable at a volume for which dp/dv becomes zero. For solids consisting of the heavier inert gases, where the effect of zero point energy is small, this volume corresponds to about 1.4 times the normal equilibrium volume. Now zero point energy has the same effect as a negative pressure, and expands the solid at absolute zero to a larger equilibrium volume; for helium this volume is more than twice the volume corresponding to the minimum of the potential energy of the lattice, 6.2.
358
C. DOMB AND J. S . DUODALE
and yet the solid is perfectly stable elastically. This is because zero point energy does not correspond to a static negative pressure but to a dynamic negative pressure arising from the motions of the atoms, and its value is volume dependent, decreasing with increasing volume of the solid. This point can be more clearly illustrated if a theoretical attempt is made to calculate Debye 8 values for solid helium from the data on intermolecular forces. The detailed calculation of the frequency spectrum of normal vibrations of a crystal lattice is a problem of some complexity which has engaged the attention of many authors. However from these, calculations we may draw the general conclusion that deviations from the Debye theory do not usually amount to more than a few per cent, and if we confine our attention to the high temperature expansion of the specific heat, we can readily and simply obtain a formula for 8 34. Thus if ~ ( ris) the static lattice energy per particle, the limiting value of 8 at high temperatures is given by
where m is the mass of an atom of the crystal. When the solid is subjected to pressure we assume that vibrations take place about the new equilibrium position with a changed force constant, and this is accounted for by the term ~ " ( r )Eq. . (7)thus represents the variation of 8 with volume. When we substitute the values of the lattice energy corresponding to helium in Eq. (7) the lowest curve in Fig. 11 results, and this is clearly in serious disagreement with the experimental values represented by the upper curve. The calculated &values become zero at a molar volume of 14.9 cm3 whereas finite experimental values are observed right up to the melting volume of 21.18 om3. This discrepancy provides a confirmation of Born's conteiitioii 35, 36 that classical lattice dynamics, based on vibrations about a position of static equilibrium, has only a restricted region of validity. Born has indicated that one should rather consider vibrations about dynamic (or more generally thermodynamic) equilibrium, and has suggested this as a basis for a modified lattice dynamics which has subsequently been developed by Hooton We can take account of Born's suggestion in an approximate manner by replacing the lattice energy 373
SOLID HELIUM
369
in Eq. (7) by the free energy, which at absolute zero reduces to v(r)+$$a. We thus obtain for 8, the equation
This is a non-linear differential equation whose detailed properties are quite complex. But to test whether the correction is in the right direction we have substituted the experimental value of a
y ( = -Vmln8,)
into the right hand side of (8). The result is again shown in Fig. 11 (upper dotted curve) and it will be seen that the calculated values
120
-
5
Fig. 11. The Debye 0 aa a function of volume in solid He4. experimental values.
We now discuss briefly some of the properties of the differential Eq.(8). As it is a second order equation we might a$ first expect two arbitrary constants to appear; in fact however we are looking for a singular solution, and this is determined by the physical property that for sufficiently small volumes the effect of zero point energy is
360
C. DOMB AND J. 9. DUQDALE
negligible. This property of the physical solution makes a numerical approach at integration rather difficult. The equation may alternatively be regarded as determining the internal energy, Eo, at absolute zero, as a function of volume, and following Dugdale and MacDonald40 we can write it in the form
where a is a constant. Dugdale and MacDonald consider the particular case when y ( r ) is a Mie-Lennard Jones function A B y ( r ) =--Tl0 fl
a
They are then able to derive a solution in the form of a series expansion in powers of r. However, the series is only of use for small a, corresponding to small zero point effects, and cannot be applied to helium. They alternatively suggest an approximate solution in closed form which agrees with the initial terms of the series expansion, and o m be applied over a much wider range. An important feature of this solution ia the disappearance of the minimum in Eo at a particular value of a (Fig. 12). This seems to correspond to an interesting dif-
Fig. 12. The internal energy aa a function of volume for various values of the parameter a.
SOLID HELIUM
361
ference between the solid phases of He4 and He3: the internal pressure of the former as determined from the Simon melting equation (see $$ 4 and 5 ) is positive as for all normal substances whereas the internal pressure of the latter is negative. As mentioned above a more fundamental approach to a modified lattice dynamics when anharmonic effects are large was initiated by Born and developed by Hooton. The nature of Born’s approach can be illustrated by the following simple example. In the problem of the vibrations of a simple diatomic molecule one has to solve the Schrodinger Equation for a potential well of shape Fig. 13(a). For small vibrations (energy level as shown) only the bottom of the well is relevant, and a harmonic approximation is reasonable. If however the well is of a different shape Fig. 13(b), such as is experienced by an atom in the interior of a crystal with a large atomic spacing, the harmonic approximation breaks down completely. Nevertheless a pseudo-harmonic approximation is still possible, using a harmonic oscillator wave function and fitting the frequency parameter by the variation principle.
Fig. 13
Hooton develops a lattice dynamics in this manner, the equilibrium positions of atoms, normal frequencies, and normal co-ordinates being parameters which are fitted so as to give a minimum value to the thermodynamic free energy. These parameters are then dependent on temperature as well as volume, although the temperature dependence is small and can be neglected to a first approximation, I n principle the method is capable of determining the complete vibration spectrum (including its volume and temperature dependence) of a lattice with large anharmonic vibrations ; in practice, however, the equations become extremely complicated. By using an approx-
362
0. DOMB AND J. S. DUGDALE
imation of Debye ,type Hooton is led to an equation similar to (8), although he suggests that the use of Debye functions instandard manner is no longer appropriate. A simple and direct approach, which is nevertheless capable of providing useful information, is to consider how the shape of the potential well in which each atom finds itself changes aa the volume of the solid is varied. This method wa8 used by Henkel 41 to account for the temperature dependence of the elastic constants of solids, and is really a development of the Einstein approximation. Although it would not be expected to account correctly for the thermal behaviour at low temperatures, it should provide a reasonable approximation to the volume dependence of the lowest energy state. Thus as the volume increases, the shape of the potential curve becomes rather similar to a three-dimensional square well, and this can be used as a first approximation. This approach is being further investigated by I. J. Zucker at King’s College, London. 6.3. THE TRANSITION IN SOLID HELIUM As mentioned in 9 2.3 Dugdale and Simon discovered a polymorphic transition in solid helium which they conjectured to be a transition from hexagonal close packed to cubic close packed structure. It is of interest to consider the relative stability of these two structures when the interatomic forces are central forces of van der Waals type. For the static lattices at absolute zero calculations have been made independently by several authors 43,431 44. The general conclusion is that for all reasonable forms of potential the hexagonal structure is more stable, the difference in energy, however, being only about 0.01 %. Under very high pressures, corresponding to a reduction of a factor of 2 in volume, the cubic structure becomes the more stable. Barron and Domb 44 then considered the properties of the two lattices at elevated temperatures. The equivalent Debye 0 at the absolute zero is smaller for the cubic lattice, the difference being about 1 %. Hence, ignoring zero point energy, we should expect a transition to occur from hexagonal to cubic at an elevated temperature. The estimated temperature and energy of the transition are of the same order of magnitude as those observed experimentally in solid helium. However the calculations can only be regarded as indicating the possibility of an explanation, since lattice dynamics based on a
363
SOLID HELIUM
harmonic approximation is not adequate to deal with the large anharmonic vibrations occurring in solid helium. 6.4. ZERO POINT ENERGY AND MELTING The experimental values of 8 determined by Dugdale and Simon 8 enable us to estimate vibration amplitudes and various related properties for a considerable length of the melting curve. Assuming a Debye model the mean square vibration amplitude is given by
The results are Shawn in Table IX. The 4th column, T,/e, represents the degree of degeneracy at the melting point, and it will be observed that melting becomes steadily less degenerate as the melting temTABLE IX
Molar Volume in oms *21.18 20.0 18.0 16.0 14.0 12.5 11.5 10.5
Melting Temp. "K
"K
0 2.12 3.40 5.35 8.65 13.10 17.65 23.55
21 24 31.5 42.5 57.0 76.3 92.5 114
e
T,/O .O .088 .lo8 .I26 .152 .172 .191 .207
Zero point Thermal energy energy Ratio Cals/Mole Cala/Mole a = l?/a 46.9 53.6 70.4 95.0 127.0 170.0 207.0 255.0
0 0.17 0.49 1.19 3.18 6.43 11.2 17.8
.310 .303 ,277 .251 .228 .212 .201 .190
Ratio a' 0 .067 .074 .077 .083
.086 .088 .089
* This value corresponds t o the liquid phase He 11,the remaining values correspond to H e I . perature rises. I n the 5th and 6th columns we have tabulated the zero point energy and thermal energy, and a comparison of their magnitudes illustrates strikingly a fact first pointed out by Simon 64. The zero point energy is much larger than the thermal energy, and if we consider, for example, the heating of helium at a constant molar volume of 14.0 cm3 from the absolute zero, we find that 127 calories of zero point energy are incapable of melting the solid, whereas an additional 3.18 calories of thermal energy are sufficient to induce melting.
364
C. DOMB AND J. 5. DUGDALE
In the 7th column we have tabulated the ratio a of the r.m.s. vibration amplitude of the atoms, 1/2, to the distance between nearest neighbours in the lattice. According to the Lindemann melting formula, which is well satisfied experimentally by classical substances, this ratio should nearly always be in the neighbourhood of 1/10. However, it will be seen that for solid helium this ratio varies continuously along the melting curve from three times to twice the classical value. If we consider the vibrational energy, the difference from classical behaviour is still more striking. The molar volume in the last line of Table IX corresponds to a nearest neighbour spacing about equal to the distance of the minimum in the potential energy curve. By analogy with the heavy inert gases we can estimate that if the melting of helium were purely classical, about 25 calories of thermal energy would then be necessary to produce melting at this volume; in fact the actual vibrational energy at the melting point is more than ten times this quantity. In the final column we have tabulated the corresponding ratio o‘, taking account only of thermal vibrational energy. It will be seen that the values rapidly increase at first and steadily approach the classical value of 1/10. The experimental data thus demonstrate clearly that zero point energy is much less effective than thermal energy at producing melting. The two types of energy differ from one another only in spectral distribution, the former being proportional to v3 (on the Debye model) and the latter to ~ ~ / ( e ” ” ‘1). ~ -However small T is, the denominator in the thermal energy term becomes equal to hv/kT for sufficiently small v, and hence the thermal energy is proportional to v2 for small v. On general physical grounds we might expect that it is long waves which are responsible for melting; for example it is known that sufficiently short transverse waves can be propagated in liquids, and it is long waves which destroy long range order in two dimensional lattices 11. The ineffectiveness of zero point energy seems thus to be due to the fact that it is concentrated in the short wave region of the spectrum, whereas thermal energy has a larger proportion concentrated in the long wave region. These results also demonstrate that it is not vibrational amplitude which should be correlated with melting, and hence that the derivation of the Lindemann melting formula by means of vibration amplitudes cannot be regarded as satisfactory even empirically. An alternative
SOLID HELIUM
365
approach is needed t,a account for the wide range of agreement of this formula with experiment. 6.5. THE SIMON MELTING EQUATION Perhaps the most important aspect of the experimental work on solid helium is the light which it throws on the general properties of the melting curve, and its confirmation of the Simon melting formula (1) over an extensive range. Although we are still far from an adequate theory of melting, even crude approximations have begun t o provide evidence of theoretical justification for the formula. Thus Domb 45 and de Boer 46, using the theory of Lennard Jones and Devonshire, were able to derive a, formula of type ( l ) ,the constant c beingrelated to the repulsive power in the intermolecular potential energy. By making the assumptions of the Griineisen theory of solids, Salter4' was able to show that the Simon formula was equivalent to the Lindemann melting formula, the constant c being related to Griineisen's constant. Finally I. 2. Fisher @,49,50 in considering the limits of stability of the solid phase has obtained a theoretical derivation of the formula (1). There is still a good deal of controversy regarding the possibility of a solid-fluid critical point, but there seem to be strong theoretical reasons for rejecting the idea". It is therefore interesting to note that the experimental results also show no evidence of an approach to a critical point. The recent extensive experimental work on solid He3 by Mills and Grillym is also of considerable theoretical interest. We have seen that for He4 the degeneracy decreases steadfly along the melting curve. At a sufficiently high temperature along the melting curve we might therefore expect the behaviour to become classical, so that the isotopic masses would have no relevance and the melting curves of He4 and He3 would become identical. It is perhaps significant in-this connection that although the melting curves of He4 and He3 run almost parallel for a large range, the formulae of Mills and Grilly indicate that they approach one another a t higher temperatures. The Simon melting formulae to which Mills and Grilly fitted their experimental results would intersect at a much higher temperature. We have seen that as a result of quantum effects helium cannot be regarded as a model substance in the completely rigorous sense. Nevertheless it may fairly be claimed that these quantum effects
366
C. DOMB AND J. S. DUODALE
do not mask the general conclusions which can be drawn, and that the experiments on solid helium have provided great insight into the properties of matter at high pressures and densities. The authors are very grateful to Sir Francis Simon for his interest and for helpful correspondence. They also wish to express their thanks to Dr. T. H. K. Barron and Dr. D. K. C. MacDonald for reading the manuscript. REFERENCES W. H. Keesom and A. P. Keesom, Physica, 3, 105 (1936); Comm. Leiden 240b. F. Simon and F. Steckel, Z. Phys. Chem., Bodenstein-Festband, 737 (1931). F. Simon, M. Ruhemann and W. A. M. Edwards, Z. Phys. Chem. B, 2, 340 (1929).
F. Simon, M. Ruhemann and W. A. M. Edwards, Z. Phys. Chem. B,6, 62 (1929).
'I
lo
l1 l2 l8
l4 IS
l7
l0 *O
a2
23 24
*6
F. Simon, Trans. Farad. Soc., 33, 65 (1937). F. Simon and G. Glatzel, Z. anorg. Chem., 178, 309 (1929). F. J. Webb, K. R. Wilkinson and J. Wilks, Proc. Roy. SOC.A, 214, 546 (1952). J. S. Dugdale and F. E. Simon, Proc. Roy. SOC.A,218, 291 (1953). W. H. Keesom end K. W. Taconis, Physica, 5, 161 (1938); Comm. Leiden 2500. R. E. Peierls, Ann. Phys. Lpz., 3, 1055 (1929). R. E. Peierls, Ann. Inst. Poincare, 5, 177 (1935). R. E. Peierls, Quantum Theory of Solids, p. 43 (1955). K. R. Wilkinson and J. Wilks, Proc. Phys. SOC.64A, 89 (1951). F. J. Webb and J. Wilks, Phil. Mag., 44, 663 (1953). W. H. Keesom, Cornm. Leiden 184b (1926). W. H. Keesom and A. P. Keesom, Proc. Roy. Acad. Amsterdam, 36. 612 (1933); Comm. Leiden 224. F. A. Holland, J. A. W. Huggill, G. 0. Jones and F. E. Simon, Nature, 165, 147 (1950). F. A. Holland, J. A. W. Huggill and G. 0. Jones, Proc. Roy. SOC. A, 207, 268 (1951). F. E. Simon and C. A. Swenson, Nature, 165, 829 (1950). C. A. Swenson, Phys. Rev., 79, 626 (1950). C. A. Swenson, Phys. Rev., 86, 870 (1952). C. A. Swenson, Phys. Rev., 89, 538 (1953). D. W. Robinson, Thesis, University of Oxford (1952). R. L. Mills and E. R. Grilly, Phys. Rev., 99, 480 (1955). D. W. Osborne, B. M. Abraham and B. Weinstock, Phys. Rev., 82, 263 (1951). B. M. Abraham, B. Weinstock and D. W. Osborne, Proc. Internat. Cod. on Low Temp. Phys., p. 29, Oxford (1951).
SOLID HELIUM 37
@
29
30 81
R. L. Mills and E. R. Grilly, Proc. Internat. Cod. on Low Temp. Phys., p. 360, Paris (1955). R. L. Mills and E. R. Grilly, Phys. Rev., 99, 480 (1955). W. M. Fairbank, W. Gordy and S. R. Williams, Phys. Rev., 92, 208 (1963). W. M. Fairbank, W. B. Ard and G. K. Walters, Phys. Rev., 95, 660 (1964). B. M. Abraham, D. W. Osborne and B. Weinstock, Phys. Rev., 98, 661 (1955).
8% 33 84
0
so 8' 88
3P 40 41 42 43 44
a a 47
a 49
10
61 61
6s 54
66
367
I. Pomeranchuk, J. Exp. and Theor. U.S.S.R., 19, 42 (1960). T. R. Roberts and S. G. Sydoriak, Phys. Rev., 93, 1418 (1964). C. Domb and L. S. Salter, Phil. Mag., 43, 1083 (1962). M. Born, Changements de Phases, p. 334, Paris (1952). C. Domb, Changements de Phases, p. 338. D. J. Hooton, Phil. Mag., 46, 422 (1955). D. J. Hooton, Phil. Mag., 46, 433 (1955). D. J. Hooton, Phil. Mag., 46, 486 (1955). J. 8. Dugdale and D. K. C. MacDonald, Phil. Mag., 45, 811 (1964). J. H. Henkel, J. Chem. Phys., 23, 681 (1965). T. Kihara and S. Koba, J. Phys. SOC.Japan, 7, 348 (1962). J. A. Prins, J. M. Dumare and L. T. Tjoan, Physica, 18, 307 (1962). T. H. K. Barron and C. Domb, Proc. Roy, SOC.,227, 447 (1965). C. Domb, Phil. Mag., 42, 1310 (1951). J. de Boer, Proc. Roy. SOC.A, 215, 4 (1952). L. Salter, Phil. Mag., 45, 369 (1954). I. Z. Fisher, Soviet Physics, 1, 154 (1954). I. Z. Fisher, Soviet Physics, 1, 273 (1965). I. Z. Fisher, Soviet Physics, 1, 280 (1966). F. E. Simon, Roc. American Acad. of Arts and Sciences, 82, 319 (1963). H. Primakoff, Private Communication. F. E. Simon, Nature, 133, 629 (1934). F. E. Simon, Changements de Phase, p. 329. G. Leibfried and E. Schlomann, Gottingen Nachrichten, p. 4 1 (1964).
C. DOMB KING'SCOLLEGE, UNIVERSITY OF LONDOM AND
J. S. DUGDALE DIVISIONOF PUREPHYSICS, NATIONAL RESEARCH COUNCIL, OTTAWA CONTENTS:Part I , 338. - 1. Introduction, 338. - 2. General Thermodynamic Properties of Solid He4, 340. - 3. Thermal Conductivity of Solid Helium, 346. - 4. Melting Curve and Melting ParameCers of He4, 349. - 5. Melting Parameters and General Properties of Solid He8, 355. - Part 11, 356. Theoretical Aspects, 356.
I. EXPERIMENTAL ASPECTS
1. Introduction Before the last war, the thermodynamic properties of solid helium had been investigated only in the temperature region 1.5 to 4 ' K and under pressures up to about 140 atmospheres The melting characteristics of solid helium had been more extensively studied: the latent heat of melting had been measured at several temperatures 1, and the melting curve had been determined by Simon and his coworkers up to a temperature of about 43' K, corresponding to more than 8 times the liquid-gas critical temperature. The melting curve for helium is shown in Fig. 1. The unique importance of solid helium in the study of melting has been stressed by Simon 5, S1, whose argument may be summarized in the following way. Experiments on many substances show that their melting curves may all be represented by an equation of the form6
For a given class of substance (e.g. the solidsed gases, the alkali metals, etc.) c has approximately the same value, so that there exists
SOLID HELIUM
339
for each class a reduced melting curve in which a and To, the characteristic temperature and pressure, are the reducing parameters. Of all substances helium has the lowest value of a and of To and so, for given laboratory resources, is capable of yieIding the most extensive information about the behaviour of the melting curve of a typical substance at ‘high’ temperatures and pressures.
TEMPERATURE f ’ K f
Fig. 1.
The melting curve of He4; T,is the critical temperature. The inset shows the low temperature phase diagram of He4.
Of great practical importance is the fact that the melting properties of helium can be studied in the region of low temperatures where the heat capacities of the necessary pressure vessels are conveniently small. This of course applies equally to other thermal measurements on solid helium under pressure. The effect on these properties of quite modest pressures is drastic. For example, by applying to solid helium pressures of only a few thousand atmospheres, its volume can be halved and its Debye temperature changed by a factor of 5. From a more detailed theoretical point of view, the properties of the solid have a double interest: the helium atom, having central
340
C. DOMB AND J. 9. DUGDALE
additive short-range forces, makes the solid a suitable substance for theoretical treatment, while the small mass of the helium atom and its relatively weak binding force make manifest, in a remarkable way, the effects of the zero-point motions of the atoms in the solid. All these considerations which are true of solid He4 apply even more strongly to solid He3, and they have formed the basis of much of the recent work on solid helium. This work we shall now discuss in more detail.
2. The General Thermodynamic Properties of Solid He4 2.1.
SPECIFICHEATS
Keesom and Keesom in 1936 measured the heat capacity of solid ' K. helium at low densities in the temperature range from 1.5" to 4 The more recent work has, therefore, been concentrated on the temperature regions above and below this. Webb, Wilkinson and Wilks? were primarily interested in the thermal conductivity of solid helium and wished to correlate their conductivity measurements with the characteristic temperature of the lattice. I n order to do so, they measured the specific heat of the solid at temperatures between 0.6 and 1.5" K at three different densities. Their method was to enclose both the paramagnetic salt (which was used as the cooling agent) and the solid helium in a calorimeter and to measure the total heat capacity. The salt was used as a thermometer and heat was supplied electrically. The heat capacity of the helium was obtained by subtracting from the total the estimated heat capacity of the calorimeter and salt. Their results are shown in Fig. 2 . Dugdale and Simon* measured the heat capacity of solid helium in the higher range of temperatures and densities, covering the region from 5' K to the melting temperature at molar volumes between 16 and 10.6 CCS. They used a vacuum calorimeter designed to stand pressures up to 3000 atmospheres, and measured temperatures with a constantan thermometer. Their results at four densities are shown in Fig. 3. The measurements of Keesom and Keesom may be compared directly with those of Webb, Wilkinson and Wilks. The two sets of results are in reasonable agreement and indicate that the Debye temperature a t these densities rises with falling temperature. I n Fig. 4 the Debye temperatures for the two lowest densities are derived
341
SOLID HELIUM
from the measurements of Keesom and Keesom. At the still lower temperatures reached in the experiments of Webb et al. the 6, reaches a maximum and begins to fall. The Debye temperatures, calculated from Dugdale and Simon's measurements, are also shown in Fig. 4: they are almost constant although falling somewhat as the melting point is approached.
0.5
I
1.0 temperature (" K )
I
1.5
Fig. 2. The specific heat at constant volume of solid He4 at three Merent densities. I. 0.194 g/cms; 11. 0.205 gfcms; 111. 0.214 g/cm3.
Unfortunately the low temperature measurements of Keesom and Keesom and of Webb et al. on the one hand and the higher temperature measurements of Dugdale and Simon on the other do not sufficiently overlap in density and temperature range t o allow a very close comparison and more measurements would be valuable, particularly at high densities in the liquid helium temperature range. 2.2. THE EQUATION OF STATE OF THE SOLID Dugdale and Simon found that their results could be adequately represented by a Griineisen equation of state. That is, the specific
342
C. DOMB AND J . S. DUODALE
Fig. 3. The heat capacity of solid He4 at four molar volumes. 0 , 10.6 ema; 0 , 11.7 emS; f, 13.0 cm3; A, 14.4 em3.
heat at constant volume was found to be a function of T/p,where p, is a characteristic temperature depending only on the volume. Furthermore, for the purpose of calculating certain thermodynamic properties of the solid from the specific heats, the results of Keesom and Keesom could also, though less closely, be represented in this way. The values of p, calculated from their results are shown in table I as a function of volume. rn*DTrn
T
___ 10.5
11 12 13 14
113 101.5 83.2 68.7 57.4
15 16 17 18 19 20
48.5 41.4 35.7 31.1 27.2 24.2
343
SOLID HELmM 140
I
0
5
I
10
I
IS
I
20
TEMPERATURE (‘KI
Fig. 4.
50
Fig. 5.
The Debye 8’s of solid He4 at six molar volumes.
I
5
10 15 temperature (OK)
I
20
The isochores of solid He4 at eight molar volumes. - - - - - transition line in the solid.
I
25
344
C. DOMB AND J. S. DUGDALE
From the specific heats it is then possible to calculate the entropy,
8, and the thermal contribution to the internal energy, U - U,. Here U is the internal energy at temperature T and volume V , and U , the internal energy at the same volume and at absolute zero. Table I1 gives G,, S and (U-U,)/T as functions of T / q derived in this way. T A B L E I1
C,, S and ( U - Uo)/T as a function of T/tp. (All in cal/" C per mole)
0.04 0.06
0.08 0.10 0.12 0.14 0.16 0.18 0.20
0.02 0.08 0.19 0.38 0.64 0.96 1.33 1.73 2.21
0.010 0.017 0.043 0.091 0.158 0.248 0.359 0.491 0.646
0.008 0.023 0.060 0.121 0.213 0.333 0.483 0.662 0.870
The specific heats as a function of temperature and volume do not by themselves form a thermodynamically complete set of data. For completeness one p , V , T relationship which traverses the same range of temperature and volume is needed and the melting curve together with the volume of the solid on melting provides this additional information (see 5 4.1 below). On this basis it is now possible to calculate the isochores of the solid by integrating the relationship dlnq 'Yc o where y = dlnV
which is valid if Go is a function of T/p,and q depends only on volume. The isochores at eight densities are shown in Fig. 4 together with the equilibrium line between the two solid modifications ( 5 2.3). This figure shows that the isochores are only very slightly temperature dependent: indeed, as a first approximation, it might be said that the pressure in the solid is independent of temperature and dependent only on the volume *.
* It is partly for this reason that solid helium can be a useful pressure trensmitting medium at low temperatures. The presrsure can be applied at such a temperature that the helium k fluid. Thereafter the system may be cooled at constant volume to the working temperature without serious loss of pressure.
345
SOLID HELIUM
From the data in Fig. 5, the compressibility, B, at O°K can be derived and by using Eq. (2) the thermal expansion coefficient, a, can be calculated. These quantities are listed in tables 111and IV.On the same basis the internal energy at 0' K can be found and hence, by subtracting from this the zero point energy, the lattice energy can be deduced*. T A B L E 111
The compressibility of solid helium at 0°K volume bmS)
preseure (atm)
1vp
volume
(atm-1)
10.6 11.0 12.0 13.0 14.0
2170 1660 1070 695 460
10 12 17 26 38
15.0 16.0 17.0 18.0 19.0
1vp
pressure (atm)
(atm-l)
295 200 136 88 50
54 76 103 140 190
TABLE IV
The thermal expansion of solid helium; the volume expansion coefficient &B a function of temperature 10.6 cms 104 a
T (OK) 0 2
0
-
4 8 12
l6 20
.
0.09 0.93 3.4 7.3 12.6
12 cm8 104 a 0 -
0.48 4.9 14.6
-
15 cm8 104 a 0 1.05 10.2
-
-
2.3. LATTICE STRUCTURE AND TRANSITION Dugdale and Simon in the course of their investigation of the heat capacity of solid helium found an anomaly in the specific heate between 15 and 17'K, the actual temperature depending on the volume. They concluded from their experiments that this was a first order, reversible transition involving an entropy change of about 5 x 10-s cals/"K per mole *. Furthermore, since substances with shortrange forces of the van der Waals type are stable only in close-packed
* Since all such quantities associated with the transit,ion are small (e.g. the volume change is estimated at 4 x ccs/Mole) their contribution to the general thermodynamic properties of the solid has been neglected.
346
C. DOME AND J. S. DUQDALE
lattices, they also concluded that the transition was due to a change from the close-packed hexagonal structure stable at the lower temperatures (i.e. the so-called oc-phase) to the close-packed cubic structure stable at the higher temperatures (the p-phase).
3. The Thermal Conductivity of Solid Helium 3.1. EXPERIMENTAL RESULTS 3.1.1,
Introduction
I n discussing the thermal conductivity, x , of a dielectric it is convenient to define a mean free path, 1, by the relationship x = QC.v.rZ
where C is the lattice heat capacity per unit volume and v the velocity of the lattice waves. At sufficiently low temperatures, the thermal conductivity of a chemically pure and physically perfect dielectric is limited only by the dimensions of the specimen. If, for example, the specimen has the form of a long, narrow cylinder then the mean free path is under these conditions approximately equal to the diameter of the cylinder. Consequently the thermal conductivity varies directly with the heat capacity, C , which means in general that x is proportional to the cube of the absolute temperature at very low temperatures. The growth of thermal conductivity in an ideal crystal with rising temperature is ultimately arrested by so-called ‘Umklapp’ processes, a concept introduced by Peierls l’Jtll. I n an Umklapp process, two normal modes interact to produce a resultant whose wave number lies outside the basic cell of the reciprocal lattice. This means that its wave number is opposite in direction to that of its components. According to Peierls it is just these Umklapp processes which give rise to thermal resistivity in ideal insulators, and he estimates that the rate at which they occur at low temperatures is proportional to exp ( - B/bT); here 0 is the characteristic Debye temperature of the lattice and b is a numerical factor greater than 1 l2. It thus seems reasonable to assume that the mean free path at such temperatures should be given by A = A exp (B/bT) (3) where A is a parameter which depends on the coupling of the lattice waves and may indeed include a temperature factor.
347
SOLID HELIUM
3.1.2. The Measurements
Measurements of the thermal conductivity of solid helium have been made by Webb, Wilkinson and Wilks7.13 and by Webb and Wilks 14. The conductivity was measured at densities between 0.345 and 0.194 gmslcc in the helium temperature range; at the lowest density, the measurements were extended down to 0.3"K. This change of density altered the Debye temperature of the lattice from about 25" K to about 90" K and increased the thermal conductivity at 2 ° K by a factor of over 300. Webb, Wilkinson and Wilks made the measurements at the lower densities (0.194 to 0.218 gmslcc). For most of the experiments the specimen holder was a German silver tube of internal diameter 6 mm: for the measurements below 1" K the holder had a diameter of 0.5 mm. Their results at three densities are shown in Table V. TABLE V
The thermal conductivity of solid helium Density
(gmsicm7
0.194 0.208 0.218
*
Temperature in 0.25 20"
-
0.50 85" -
1.0 107
-
*
1.2 24" -
-
O
1.4 11 58 160
K
1.7 5 19 52
8
2.0
2.3 -
22
11
These size-dependent values were obtained with specimens of 0.5 mm diameter.
At the higher densities, the measurements were made by Webb and Wilks using two experimental arrangements which were essentially modifications of the previous apparatus. Up to a density of 0.28 gms/cc the results of this work were self-consistent and consistent with the earlier measurements. These results and some from the earlier work are shown in Pig. 6. For densities above 0.28 gms/cc however the two apparatuses gave different results. These differences were not simply attributable to specimen size effects and moreover the results from each apparatus singly were somewhat irregular. It is therefore difficult to make any deductions from these measurements. 3.1.3. Compurison with Theory
The experimental values of thermal conductivity and specific heat enable the mean free path of lattice waves, 1, to be calculated (the
348
C. DOME AND J. S. DUGDALE
velocity of sound may be deduced from the appropriate value of 6). The mean free paths so deduced are shown plotted as a function of 8/T at various densities up to 0.218 gms/cc in Fig. 7 '. Two conclusions can be drawn. I n the first place, provided that size effects are absent, log 1 is proportional to 8lT as the theory
Temperature O K Fig. 6.
.,
T h e hhermd conductivity of solid He* st four densities.
A, 0.218 g/cm3; 0 , 0.262 g/cms; x, 0.276 g/oms;
0.282 g/cms.
demands. The constant b of Eq. (3) has the value 2.3 and the parameter, A, of the same equation appears t o be independent of density, having the value 6 x cms. I n the second place, size effects become evident at very low temperatures (large values of e/T)and 1 becomes almost constant, although even at the lowest temperature it is only about half of the theoretically
349
SOLID HELIUM
expected value, 5 x cms., which is the diameter of the specimen. It should also be noted that the thermal conductivity, in the temperature range where il is limited by size, varies as T2e3and not T3 as theory predicts. I n a recent paper Leibfreid and Schlomann have attempted for the first time to calculate the absolute value of the thermal conductivity of dielectric crystals. The anharmonic terms in the expansion of the lattice potential energy are taken account of by means of Gruneisen's y. The authors find that the thermal conductivity is given by x,f(B/T) where f is a universal function, and x, is proportional to 8Ba/y2, 6 being the lattice spacing of nearest neighbours in the crystal. When detailed values are substituted for solid helium reasonable agreement in absolute values is obtained over a considerable range of pressures.
lfJ-
/4
*? */ ?
-i
c
cc
';1
-
A 9
4, 12-
I
I
I
I
I
el T
Fig. 7. The mean free path of lattice waves in solid He'.
4.
The Melting Curve and Melting Parameters of He4
4.1. EXPERIMENTAL RESULTS 4.1.1. The Melting Curve
Because helium can remain liquid down to the lowest temperatures, the melting curve of solid helium does not join the vapour pressure
w cn
T A B L E VI
0
Measurements of the melting curve of He4 Author
Date
Temperature Range
Pressure Range
Estimated Accuracy
Keesom l5 Simon, Ruhemann and Edwards3 Simon, Ruhemann and Edwards4 Keesom and Keesom l6 Holland, Huggill, Jones and Simon 17. Simon and Swenson l9. 20 Swenson 21
1926 1929
1.2" to 4.2" K 12.2" to 20.2' K
25-140 ats. 800 to 1800 ats.
1929
20" to 42" K
1800 to 5500 ats.
1933 1950
1.15" to 1.8" K 35' K to 50" K
2&30 ats. 4000 to 7500 ats.
0.1" K, 0.1 ats. O.l'K, 1 yo of pressure 0.loK, 1 yo of pressure 0.01" K, 0.01 ats. 0.1" K f 100 ats.
Density measurements. Blocked capillary
1950
1.0" to 1.8" K
25.0 to 27.0 ats.
.Ole K, 0.01 ats.
Blocked capillary.
1952
1.6' to 4" K
27 to 130 ats.
Piston displacement *.
Swenson 22
1953
1.5' to 4" K
26 to 130 ats.
Dugdale and Simons Robinson 23
1953 1953
4.2" to 12" K 20" to 55" K
140 to 800 ats. 9,000 ats.
Mills and Grilly z4
1955
2" to 30" K
35-3,500 ats.
.Ol" K, 0.1 yo of pressure .01" K, 0.1 yo of pressure 0.1" K, & 3 ats. f 0.5" K, f 5 % of pressure .01" K, .05 yo of pressure
Method used Blocked capillary. Heating curve method. Blocked capillary.
Blocked capillary *. Blocked capillary. Moving pellet. Blocked capillary,
* Swenson 22 has discussed in detail t,he relative merits of the blocked capillary and piston displacement methods of measuring melting curves.
351
SOLID HELIUM
curve at a triple point. Instead the melting pressure at very low temperatures tends to become constant, its dope vanishing in accordance with the third law of thermodynamics. There have been many measurements made on the melting curve of He4 and these are summarized in Table VI, which shows the range of measurements in temperature and pressure, the method used and the estimated accuracy achieved. According to Swenson 20, the melting pressure between 1.0 and 1.4"K can be represented by P=25.00+0.053T8 ats; and between 2.1 and 4" K by P + 5 . 6 13.458 ~ T1."3074ats 21. Values of P up to 6" K are given in Table VII. All workers agree that their results above 4°K can be represented by a Simon-type expression as given in Eq. (1). The Oxford workers have found that the melting curve can be expressed in the form T 1.6644 P -= -1 (4) 16.46 (.992) where P is measured in atmospheres, and T in degrees Kelvin. Mills and Grilly gave their results as
P + 17.80 = 17.315 T1.5554
(5)
with P in Kg/cm*, which may be re-written in the same form and units as Eq. (4) T 1.6664 -P(6) 17.23 - (1.0179)
-'
4.1.2. The Melting Parameters
Swenson 22 has made very accurate measurements of the melting parameters of He4 between 1 and 4" K. A free piston gauge was used to measure the melting pressure, and the volume change on melting was measured by the displacement of the piston. Dugdale and Simon, using the apparatus already described for measuring specific heats under pressure, measured aIso the melting parameters of solid helium at higher temperatures. The combined results of these experiments and those of Swenson are given in Table VII, and shown in Fig. 8 and 9. The melting parameters, AS, A U and A V a t very low temperatures are of particular interest 19. The application of the Clausius-Clapeyron Equation to the melting curve enables us t o determine A S / A V , and according to the measurements of Simon and Swenson l9 previously
352
C. DOMB AND J. 5. DUODALE
TEMPERATURE P K I
Fig. 8. The entropy of He4 on melting and solidification. The fine lines m e lines of constant volume; - - transition line in the solid.
--
FLUID
I
I 1 5
SOL ID 10
20
TEMPERATURE ('K)
Fig. 9. The molar volume of He4 on melting and solidification. _ - _ _ _ transition line in the solid.
353
SOLID HELIUM TABLE VII
The entropy and molar volume of He4 dong the melting line. Entropies ( S ) in cals/"K per mole, volumes ( V ) in cms/mole. (The volumes above 4" K are specified more accurately than is warranted by the experiments in order to give V correctly)
T
O
P (ats.)
K
25.00 25.05 25.22 25.81 26.99 37.08 56.3 78.1 128.6 255
0 1.0 1.2 1.4 1.6 2.0 2.5 3.0 4.0 6.0 10.0 14.0 18.0 22.0 26.0
*
Below 4°K S,
'&Aid*
'Fluid
V@aUd
0 0.03 0.04 0.05 0.06 0.07 0.10 0.13 0.18 0.29 0.48 0.63 0.78 0.94 1.08
0 0.05, 0.ll6 0.26 0.60 1.24 1.37 1.47 1.59 1.79 2.00 2.21 2.42 2.66 2.88
21.18 21.17 21.15 21.05 20.95 20.26 19.32 18.52 17.37 15.53 13.45 12.30 11.42 10.75 10.10
and V,
'FlUid
*
23.25 23.24 23.22 23.11 22.84 21.66 20.60 19.72 18.40 16.42 14.13 12.88 11.95 11.24 10.57
are from the measurements of Keesom and
Keesom 1.16.
referred to, this is of order T7 for small T . From the third law of thermodynamics, AS, the entropy difference between the solid and liquid phase, must vanish as the temperature approaches the absolute zero; in fact these experimental results on solid helium provide a striking confirmation of the third law. (The latent heat of melting (e=T A S ) must vanish even more rapidly.) Since TAS =A U +pA V , two possibilities exist, remembering that p does not vanish. Either AU and AV must both tend to zero with AS so that the two phases become identical, or else A U = - pA V in the limit as TAB vanishes. It is the second alternative which is found experimentally (see Fig. 10) implying that the phase with the larger volume has the smaller energy. Since liquid helium has always a greater volume than the solid, the liquid ultimately has the lower energy and so becomes the only known example of a liquid having an internal energy lower than that of the solid with which it is in equilibrium. The high-temperature trends of the melting parameters are of interest also, because they may give an answer to the general problem
354
C. DOMB AND J. 8. DUGDALE
of whether or not a solid-fluid critioal point exists. It is seen that although the volume difference between the phases decreases with rising temperature, the entropy differenoe continues slowly to increase and one must conclude that there is certainly no evidence here for the existence of such a critical point.
6-
5-
w
J
4-
0
I
> 3A
a
2-
.------ ------
I-
0
-I
------------1
0
I
I
I
2
I
I
3
1
I
4
Fig. 10. The melting parameters of solid He4 below 4OK. e is the latent heat of melting. P is the pressure. AV is the change of volume on melting. AU is the internal energy change.
Another feature t o which Simon has drawn attention is the increase in entropy of solid helium along the melting line, a behaviour which characterizes dl other substances for which this entropy trend can be estimated. If it is true of all solids at sufficiently high pressures that the entropy at the melting point increases with temperature then it is impossible to melt a substance by adiabatic reversible compression, a fact that may be of significance in geophysics and atrophysics.
355
SOLID HELmM
The findemann Melting Formuh Applied to He4 According to the Lindemann melting formula, 4.1.3.
where 8 is the characteristic temperature of the lattice at volume V and T , is the melting temperature; M is the molecular weight of the substance and c is a constant. The measurements on solid He4 provide, for the first time, the data necessary to test this formula over a substantial region of the melting curve of one single substance. Table VIII gives the values of c calculated for He4 at several different temperatures and it is seen that they are indeed very nearly constant. The value of c for He4 is however considerably smaller than for the heavier inert gases for which c is approximately 160. Some further comments on the Lindemann melting formula are made in 8 6.4 below. TABLE VIII
Application of the Lindemann Melting
110 92 72 55 32
23.3 17.3 11.3 7.9 3.1
FOI'Ind8
to He4
10.6 11.6 13.1 14.4 18.3
101 100 102 96 96
5. The Melting mrameters and General Properties of Solid HeS As yet there is no detailed knowledge of the thermodynamic properties of solid He3. Its melting curve, however, has been measured from 0.2 to 35' K. Abraham; Osborne and Weinstock have measured the curve in the low temperature region 25, 26 and Mills and Grilly at high temperatures 27* 28. Both used the blocked-capillary technique. The data can be represented by the following equations
P = 26.8+ 13.1 T2 atm. between 0.5 and 1.5"K and
P- 25.16 = 20.082 T1~51'~ kg/cm2 between 2 and 35" K.
The second of these expressions may be compared with Eq. (5) above which represents the melting curve of He4 over similar temperature and pressure range (see also $5 6.2 and 6.5 below).
356
C. DOMB AND J. 5. DUGDALE
The low temperature end of the melting curve is again, as with
He4, of particular interest but for a different reason. The He3 nucleus, unlike that of He4, has a nuclear spin and magnetic susceptibility experiments have shown that in the liquid state the nuclear spins are 31. Pomeranchuk 32 pointed already partially aligned at 0.4" K out that if the nuclear spins were aligned in the liquid but remained randomly oriented in the solid until dipole-dipole interaction produced alignment then below a certain temperature the liquid would have a lower entropy than the solid. Consequently assuming no dramatic change in the volume relationship, the slope of the melting curve would vanish at this temperature and become negative below it *. Before finally vanishing it might even become positive again when the nuclear spins became aligned in the solid. Pomeranchuk suggested that this alignment should occur a t about 10-7 OK, but Primakoff 52 has pointed out that the special properties of solid helium would make this temperature in the neighbourhood of 10-30K. Experimental evidence on whether the melting curve does in fact change sign is still lacking. Indeed it has been pointed out that the blocked capillary technique could not record such a change since it will always block at the minimum pressure 33. If, however, the melting curve does indeed have a minimum we shall then have the surprising instance of a liquid which freezes as it is heated at constant pressure! 293
11. THEORETICAL ASPECTS
INTRODUCTION The behaviour of solid helium is dominated by the effect of zero
6.1.
point energys8. In detailed character zero point energy consists of random fluctuations of the atoms about their equilibrium positions in the lattice, and these fluctuations are responsible for the fact that helium is a liquid at the absolute zero. As another consequence of the zero point energy the volume of the solid which forms when liquid helium is subjected to a pressure of 25 atmospheres is well over twice the volume corresponding to the minimum of the potential energy
* If this assumption is correct, one may estimate from the entropy diagramS1of HeS that the pressure rise on the low temperature side of the minimum should be about 4 atmospheres, assuming that the volume difference between the phases is comparable with that found in He4.
SOLID HELIUM
357
of the lattice. Thus even in the solid at absolute zero the atoms execute vibrations of relatively large amplitude about their equilibrium positions, and the normal approximation of lattice dynamics which expands the potential up to terms of the second order about static equilibrium ceases to be valid. The first theoretical problem posed by solid helium is therefore to find a modified treatment which can be applied to a system with large zero point vibrations. As mentioned in the experimental section the high compressibility of solid helium has enabled measurement of melting properties and thermodynamic functions to be made over a range corresponding to a factor of two in volume. This is a far greater range than can be achieved with any other substance, and the data can be used to cast some light on the general properties of substances under pressure. Because of the zero point effects helium cannot be regarded as a perfect model substance. Nevertheless, for some purposes it may be possible to take these effects into account and draw general conclusions; and in other cases an analysis of the differences between the behaviour of classical substances and helium can be illuminating. I n particular the data on the melting of solid helium over such a wide range of volumes provide considerable insight into the general properties of the melting curve at high pressures. The polymorphic transition discovered by Dugdale and Simon has also stimulated theoretical work on the cubic and hexagonal packed lattices and their relative stability at differentvolumes and temperatures. Many of the theoretical developments to be described in the following sections cannot be regarded as complete, but do seem to indicate the possibility of accounting adequately for the experimentally observed properties of solid helium. THE STATIC LATTICE AT ABSOLUTE ZERO If a negative pressure could be applied to a solid at absolute zero so that its volume expanded, the solid would cease to be elastically stable at a volume for which dp/dv becomes zero. For solids consisting of the heavier inert gases, where the effect of zero point energy is small, this volume corresponds to about 1.4 times the normal equilibrium volume. Now zero point energy has the same effect as a negative pressure, and expands the solid at absolute zero to a larger equilibrium volume; for helium this volume is more than twice the volume corresponding to the minimum of the potential energy of the lattice, 6.2.
358
C. DOMB AND J. S . DUODALE
and yet the solid is perfectly stable elastically. This is because zero point energy does not correspond to a static negative pressure but to a dynamic negative pressure arising from the motions of the atoms, and its value is volume dependent, decreasing with increasing volume of the solid. This point can be more clearly illustrated if a theoretical attempt is made to calculate Debye 8 values for solid helium from the data on intermolecular forces. The detailed calculation of the frequency spectrum of normal vibrations of a crystal lattice is a problem of some complexity which has engaged the attention of many authors. However from these, calculations we may draw the general conclusion that deviations from the Debye theory do not usually amount to more than a few per cent, and if we confine our attention to the high temperature expansion of the specific heat, we can readily and simply obtain a formula for 8 34. Thus if ~ ( ris) the static lattice energy per particle, the limiting value of 8 at high temperatures is given by
where m is the mass of an atom of the crystal. When the solid is subjected to pressure we assume that vibrations take place about the new equilibrium position with a changed force constant, and this is accounted for by the term ~ " ( r )Eq. . (7)thus represents the variation of 8 with volume. When we substitute the values of the lattice energy corresponding to helium in Eq. (7) the lowest curve in Fig. 11 results, and this is clearly in serious disagreement with the experimental values represented by the upper curve. The calculated &values become zero at a molar volume of 14.9 cm3 whereas finite experimental values are observed right up to the melting volume of 21.18 om3. This discrepancy provides a confirmation of Born's conteiitioii 35, 36 that classical lattice dynamics, based on vibrations about a position of static equilibrium, has only a restricted region of validity. Born has indicated that one should rather consider vibrations about dynamic (or more generally thermodynamic) equilibrium, and has suggested this as a basis for a modified lattice dynamics which has subsequently been developed by Hooton We can take account of Born's suggestion in an approximate manner by replacing the lattice energy 373
SOLID HELIUM
369
in Eq. (7) by the free energy, which at absolute zero reduces to v(r)+$$a. We thus obtain for 8, the equation
This is a non-linear differential equation whose detailed properties are quite complex. But to test whether the correction is in the right direction we have substituted the experimental value of a
y ( = -Vmln8,)
into the right hand side of (8). The result is again shown in Fig. 11 (upper dotted curve) and it will be seen that the calculated values
120
-
5
Fig. 11. The Debye 0 aa a function of volume in solid He4. experimental values.
We now discuss briefly some of the properties of the differential Eq.(8). As it is a second order equation we might a$ first expect two arbitrary constants to appear; in fact however we are looking for a singular solution, and this is determined by the physical property that for sufficiently small volumes the effect of zero point energy is
360
C. DOMB AND J. 9. DUQDALE
negligible. This property of the physical solution makes a numerical approach at integration rather difficult. The equation may alternatively be regarded as determining the internal energy, Eo, at absolute zero, as a function of volume, and following Dugdale and MacDonald40 we can write it in the form
where a is a constant. Dugdale and MacDonald consider the particular case when y ( r ) is a Mie-Lennard Jones function A B y ( r ) =--Tl0 fl
a
They are then able to derive a solution in the form of a series expansion in powers of r. However, the series is only of use for small a, corresponding to small zero point effects, and cannot be applied to helium. They alternatively suggest an approximate solution in closed form which agrees with the initial terms of the series expansion, and o m be applied over a much wider range. An important feature of this solution ia the disappearance of the minimum in Eo at a particular value of a (Fig. 12). This seems to correspond to an interesting dif-
Fig. 12. The internal energy aa a function of volume for various values of the parameter a.
SOLID HELIUM
361
ference between the solid phases of He4 and He3: the internal pressure of the former as determined from the Simon melting equation (see $$ 4 and 5 ) is positive as for all normal substances whereas the internal pressure of the latter is negative. As mentioned above a more fundamental approach to a modified lattice dynamics when anharmonic effects are large was initiated by Born and developed by Hooton. The nature of Born’s approach can be illustrated by the following simple example. In the problem of the vibrations of a simple diatomic molecule one has to solve the Schrodinger Equation for a potential well of shape Fig. 13(a). For small vibrations (energy level as shown) only the bottom of the well is relevant, and a harmonic approximation is reasonable. If however the well is of a different shape Fig. 13(b), such as is experienced by an atom in the interior of a crystal with a large atomic spacing, the harmonic approximation breaks down completely. Nevertheless a pseudo-harmonic approximation is still possible, using a harmonic oscillator wave function and fitting the frequency parameter by the variation principle.
Fig. 13
Hooton develops a lattice dynamics in this manner, the equilibrium positions of atoms, normal frequencies, and normal co-ordinates being parameters which are fitted so as to give a minimum value to the thermodynamic free energy. These parameters are then dependent on temperature as well as volume, although the temperature dependence is small and can be neglected to a first approximation, I n principle the method is capable of determining the complete vibration spectrum (including its volume and temperature dependence) of a lattice with large anharmonic vibrations ; in practice, however, the equations become extremely complicated. By using an approx-
362
0. DOMB AND J. S. DUGDALE
imation of Debye ,type Hooton is led to an equation similar to (8), although he suggests that the use of Debye functions instandard manner is no longer appropriate. A simple and direct approach, which is nevertheless capable of providing useful information, is to consider how the shape of the potential well in which each atom finds itself changes aa the volume of the solid is varied. This method wa8 used by Henkel 41 to account for the temperature dependence of the elastic constants of solids, and is really a development of the Einstein approximation. Although it would not be expected to account correctly for the thermal behaviour at low temperatures, it should provide a reasonable approximation to the volume dependence of the lowest energy state. Thus as the volume increases, the shape of the potential curve becomes rather similar to a three-dimensional square well, and this can be used as a first approximation. This approach is being further investigated by I. J. Zucker at King’s College, London. 6.3. THE TRANSITION IN SOLID HELIUM As mentioned in 9 2.3 Dugdale and Simon discovered a polymorphic transition in solid helium which they conjectured to be a transition from hexagonal close packed to cubic close packed structure. It is of interest to consider the relative stability of these two structures when the interatomic forces are central forces of van der Waals type. For the static lattices at absolute zero calculations have been made independently by several authors 43,431 44. The general conclusion is that for all reasonable forms of potential the hexagonal structure is more stable, the difference in energy, however, being only about 0.01 %. Under very high pressures, corresponding to a reduction of a factor of 2 in volume, the cubic structure becomes the more stable. Barron and Domb 44 then considered the properties of the two lattices at elevated temperatures. The equivalent Debye 0 at the absolute zero is smaller for the cubic lattice, the difference being about 1 %. Hence, ignoring zero point energy, we should expect a transition to occur from hexagonal to cubic at an elevated temperature. The estimated temperature and energy of the transition are of the same order of magnitude as those observed experimentally in solid helium. However the calculations can only be regarded as indicating the possibility of an explanation, since lattice dynamics based on a
363
SOLID HELIUM
harmonic approximation is not adequate to deal with the large anharmonic vibrations occurring in solid helium. 6.4. ZERO POINT ENERGY AND MELTING The experimental values of 8 determined by Dugdale and Simon 8 enable us to estimate vibration amplitudes and various related properties for a considerable length of the melting curve. Assuming a Debye model the mean square vibration amplitude is given by
The results are Shawn in Table IX. The 4th column, T,/e, represents the degree of degeneracy at the melting point, and it will be observed that melting becomes steadily less degenerate as the melting temTABLE IX
Molar Volume in oms *21.18 20.0 18.0 16.0 14.0 12.5 11.5 10.5
Melting Temp. "K
"K
0 2.12 3.40 5.35 8.65 13.10 17.65 23.55
21 24 31.5 42.5 57.0 76.3 92.5 114
e
T,/O .O .088 .lo8 .I26 .152 .172 .191 .207
Zero point Thermal energy energy Ratio Cals/Mole Cala/Mole a = l?/a 46.9 53.6 70.4 95.0 127.0 170.0 207.0 255.0
0 0.17 0.49 1.19 3.18 6.43 11.2 17.8
.310 .303 ,277 .251 .228 .212 .201 .190
Ratio a' 0 .067 .074 .077 .083
.086 .088 .089
* This value corresponds t o the liquid phase He 11,the remaining values correspond to H e I . perature rises. I n the 5th and 6th columns we have tabulated the zero point energy and thermal energy, and a comparison of their magnitudes illustrates strikingly a fact first pointed out by Simon 64. The zero point energy is much larger than the thermal energy, and if we consider, for example, the heating of helium at a constant molar volume of 14.0 cm3 from the absolute zero, we find that 127 calories of zero point energy are incapable of melting the solid, whereas an additional 3.18 calories of thermal energy are sufficient to induce melting.
364
C. DOMB AND J. 5. DUGDALE
In the 7th column we have tabulated the ratio a of the r.m.s. vibration amplitude of the atoms, 1/2, to the distance between nearest neighbours in the lattice. According to the Lindemann melting formula, which is well satisfied experimentally by classical substances, this ratio should nearly always be in the neighbourhood of 1/10. However, it will be seen that for solid helium this ratio varies continuously along the melting curve from three times to twice the classical value. If we consider the vibrational energy, the difference from classical behaviour is still more striking. The molar volume in the last line of Table IX corresponds to a nearest neighbour spacing about equal to the distance of the minimum in the potential energy curve. By analogy with the heavy inert gases we can estimate that if the melting of helium were purely classical, about 25 calories of thermal energy would then be necessary to produce melting at this volume; in fact the actual vibrational energy at the melting point is more than ten times this quantity. In the final column we have tabulated the corresponding ratio o‘, taking account only of thermal vibrational energy. It will be seen that the values rapidly increase at first and steadily approach the classical value of 1/10. The experimental data thus demonstrate clearly that zero point energy is much less effective than thermal energy at producing melting. The two types of energy differ from one another only in spectral distribution, the former being proportional to v3 (on the Debye model) and the latter to ~ ~ / ( e ” ” ‘1). ~ -However small T is, the denominator in the thermal energy term becomes equal to hv/kT for sufficiently small v, and hence the thermal energy is proportional to v2 for small v. On general physical grounds we might expect that it is long waves which are responsible for melting; for example it is known that sufficiently short transverse waves can be propagated in liquids, and it is long waves which destroy long range order in two dimensional lattices 11. The ineffectiveness of zero point energy seems thus to be due to the fact that it is concentrated in the short wave region of the spectrum, whereas thermal energy has a larger proportion concentrated in the long wave region. These results also demonstrate that it is not vibrational amplitude which should be correlated with melting, and hence that the derivation of the Lindemann melting formula by means of vibration amplitudes cannot be regarded as satisfactory even empirically. An alternative
SOLID HELIUM
365
approach is needed t,a account for the wide range of agreement of this formula with experiment. 6.5. THE SIMON MELTING EQUATION Perhaps the most important aspect of the experimental work on solid helium is the light which it throws on the general properties of the melting curve, and its confirmation of the Simon melting formula (1) over an extensive range. Although we are still far from an adequate theory of melting, even crude approximations have begun t o provide evidence of theoretical justification for the formula. Thus Domb 45 and de Boer 46, using the theory of Lennard Jones and Devonshire, were able to derive a, formula of type ( l ) ,the constant c beingrelated to the repulsive power in the intermolecular potential energy. By making the assumptions of the Griineisen theory of solids, Salter4' was able to show that the Simon formula was equivalent to the Lindemann melting formula, the constant c being related to Griineisen's constant. Finally I. 2. Fisher @,49,50 in considering the limits of stability of the solid phase has obtained a theoretical derivation of the formula (1). There is still a good deal of controversy regarding the possibility of a solid-fluid critical point, but there seem to be strong theoretical reasons for rejecting the idea". It is therefore interesting to note that the experimental results also show no evidence of an approach to a critical point. The recent extensive experimental work on solid He3 by Mills and Grillym is also of considerable theoretical interest. We have seen that for He4 the degeneracy decreases steadfly along the melting curve. At a sufficiently high temperature along the melting curve we might therefore expect the behaviour to become classical, so that the isotopic masses would have no relevance and the melting curves of He4 and He3 would become identical. It is perhaps significant in-this connection that although the melting curves of He4 and He3 run almost parallel for a large range, the formulae of Mills and Grilly indicate that they approach one another a t higher temperatures. The Simon melting formulae to which Mills and Grilly fitted their experimental results would intersect at a much higher temperature. We have seen that as a result of quantum effects helium cannot be regarded as a model substance in the completely rigorous sense. Nevertheless it may fairly be claimed that these quantum effects
366
C. DOMB AND J. S. DUODALE
do not mask the general conclusions which can be drawn, and that the experiments on solid helium have provided great insight into the properties of matter at high pressures and densities. The authors are very grateful to Sir Francis Simon for his interest and for helpful correspondence. They also wish to express their thanks to Dr. T. H. K. Barron and Dr. D. K. C. MacDonald for reading the manuscript. REFERENCES W. H. Keesom and A. P. Keesom, Physica, 3, 105 (1936); Comm. Leiden 240b. F. Simon and F. Steckel, Z. Phys. Chem., Bodenstein-Festband, 737 (1931). F. Simon, M. Ruhemann and W. A. M. Edwards, Z. Phys. Chem. B, 2, 340 (1929).
F. Simon, M. Ruhemann and W. A. M. Edwards, Z. Phys. Chem. B,6, 62 (1929).
'I
lo
l1 l2 l8
l4 IS
l7
l0 *O
a2
23 24
*6
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SOLID HELIUM 37
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