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1
Supersolidity and the Thermodynamics of Solid Helium
2
Humphrey J. Maris1, Sebastien Balibar2 3
1Department of Physics, Brown University, Providence, Rhode Island 02912, USA
4
E-mail: Humphrey Maris@Brown.edu
5
2Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 Rue Lhomond,
6
75231 Paris 05, France
7
(Received February 28, 2007; accepted March 27, 2007)
8
The specific heat of solid helium in the temperature range below about 1.2 K 9
has been found to contain a term varying as T7, in addition to the usual T3 10
contribution always found in a crystalline dielectric solid. It has been pro- 11
posed by Anderson, Brinkman and Huse, (Science 310, 1164 (2005)) that the 12
existence of this T7 term supports their theory of supersolidity. However, in 13
this paper we show that corrections to the phonon specific heat arising from 14
phonon dispersion are much larger than expected based on simple order of 15
magnitude estimates and, as a consequence, it is very unlikely that the exis- 16
tence of this T7 term can be considered as evidence for supersolidity.
17
KEY WORDS: solid helium; specific heat; supersolid.
18
PACS Number: # 67.80.
19
1. INTRODUCTION 20
Recently, a number of experiments have reported evidence that suggests 21
that solid helium becomes a superfluid when cooled below a critical tem- 22
perature Tc. Kim and Chan1–3 have performed torsional oscillator exper- 23
iments with solid helium over a wide range of pressure and temperature.
24
They found that below Tc the period of the oscillator decreases indicat- 25
ing that a fraction of the mass of the solid helium has become decoupled 26
from the torsional oscillator. This is referred to as non-classical rotational 27
inertia (NCRI). At the time of writing this article, however, the experimen- 28
tal situation is not entirely clear. Rittner and Reppy4 have also detected 29
a decoupling of mass in a torsional oscillator experiment but report that 30
the effect can be greatly reduced by annealing the solid helium sample 31
at high temperature. This suggests that the effect is associated with some 32
1
© 2007 Springer Science+Business Media, LLC
Journal: JOLT 10909 CMS: 9347 TYPESET DISK LE CP Disp.: 16/4/2005 Pages: 9
2 H. J. Maris and S. Balibar
sort of defect. Initial attempts to detect a steady state superflow, i.e., a dc 33
flow of mass through a solid helium crystal were unsuccessful.5, 6However, 34
a recent experiment by Sasaki et al.7 was able to detect a flow of mass 35
occurs when a crystal contains grain boundaries. This flow had the char- 36
acteristics of a superflow, i.e., the rate of transport of mass was essentially 37
independent of the driving force and appeared to be limited by a critical 38
velocity.
39
There have been a number of theoretical papers that argue that a su- 40
persolid state does not occur in solid helium that is free of defects.8, 9 In 41
contrast, Anderson, Brinkman and Huse10 have presented a theory based 42
on the idea that solid helium is “incommensurate”. In this context, incom- 43
mensurate means that in the ground state the number of lattice sites is 44
greater than the number of atoms. One of the interesting predictions of 45
their theory is that there should be a correction to the low temperature 46
specific heat that varies asT7.This would be in addition to the normalT3 47
term arising from phonons in a dielectric solid at low temperatures. They 48
point out that measurements of the specific heat by Gardner, Hoffer and 49
Phillips11 provide evidence for such a term. Furthermore, they claim that a 50
term with this temperature dependence and magnitude is unlikely to arise 51
from corrections to the low temperature limiting form of the phonon spe- 52
cific heat. Such corrections can arise from two independent mechanisms.
53
Anharmonicity of the phonons does lead to a T7contribution but should 54
be very small. The deviation of the phonon dispersion relation from lin- 55
earity gives a contribution which in lowest order goes as T5. Anderson 56
et al.10 point out that such terms are smaller than the T3 contribution by 57
powers of the parameter (T /D)2, where D is the Debye temperature.
58
Since for solid helium D is 25 K, they expected that the corrections to 59
the specific heat arising from phonon dispersion should be negligible in 60
the temperature range around 1 K. In this paper we examine this question 61
more closely and show that, in fact, the corrections due to dispersion are 62
very unlikely to be small.
63
2. SPECIFIC HEAT OF PHONONS 64
The specific heat of solid helium has been investigated by a number 65
of authors.11–14 Here, we concentrate attention on the data of Gardner 66
et al.11 which are for pressures close to the melting curve and cover the 67
temperature range 0.35 to around 1.2 K. The data for a molar volume of 68
20.96 cm3 are shown in Fig. 1. It can be seen from the figure that these 69
data can be fit very well by the sum of the usual T3 term expected in 70
a dielectric solid at low temperature together with an extra contribution 71
going as T7. Can the existence of this T7 term be understood in terms of 72
0.13
0.12
0.11
1 T4 (K) C / T3 (J mole-1K-4)
0 2
Fig. 1. Specific heat at constant volume of solid helium-4 per mole at a molar volume of 20.96 cm3 mole−1 plotted asC/T3 as a function ofT4.This is taken from Fig. 7 of Ref. 11.
Some of the data points in the range belowT4=0.1 have not been included in this plot.
a phonon contribution or does it mean that there must be an extra effect 73
associated with superfluidity?
74
For a dielectric solid at very low temperatures the specific heat per 75
unit volume is given by the formula 76
CD(T )≈2π2k4T3
5vD3¯h3 =12π4 5 nk
T
D 3
, (1)
77
where n is the number of atoms per unit volume and vD is the Debye 78
velocity, defined as 79
1 v3
D
=
1
vj(θ, φ)3
, (2)
80
withvj(θ, φ)being the velocity of thej-th phonon branch in the direction 81
θ, φ. The Debye temperature is
4 H. J. Maris and S. Balibar
D=
6π2
Vatom 1/3
¯ hvD
k , (3)
82
where Vatom is the volume per atom. The Debye temperature of solid 83
helium at the melting pressure and as T→0 is 26 K.11 This corresponds 84
to a Debye velocity of 2.85×104cm s−1. 85
Here, we investigate whether the effect of phonon dispersion results 86
in an appreciable deviation of the specific heat from the T3 law in the 87
temperature range of interest, i.e., the range in which Gardner et al. have 88
noticed that there is an excess heat capacity that can be fit by a T7 term.
89
This range extends from roughly 0.6 K to 1.2 K. At 1.2 K, for example, the 90
specific heat was found to be approximately 20 % above the T3 value.
91
The general expression for the phonon specific heat neglecting anhar- 92
monicity is 93
C(T )= ¯h2 kT2
ω2g(ω)exp(β¯hω)
[exp(β¯hω)−1]2 dω, (4) 94
where g(ω)is the density of phonon states per unit volume and frequency.
95
The density of states can be calculated from the phonon dispersion curve 96
ωj(q) if this is known for all wave vectors and polarizations. Unfortu- 97
nately, detailed measurements of the phonon dispersion in solid helium 98
have not been made and so it is not possible to determine g(ω) reliably 99
in this way.15 100
To get some idea of the corrections to the specific heat from dis- 101
persion consider first a simplified model in which the dispersion relation 102
is isotropic, and the transverse and longitudinal sound velocities are the 103
same and therefore equal tovD. Suppose that the first correction to linear 104
dispersion is of the form16 105
ω=v
Dq(1−γ3q2+ · · ·). (5) 106
Then it is straightforward to show that the lowest order correction to 107
the specific heat is 108
C3=40π4 7 γ3k6T5
¯ h5v5
D
. (6)
109
Thus, the correction relative to the Debye specific heat CD is 110
C3
CD =100π2 7 γ3
kT
¯ hvD
2
. (7)
111
Is this correction appreciable at 1 K? To answer this, an estimate of the 112
parameter γ3 is needed. We can get an approximate idea of the magnitude 113
ofγ3 from the known values of this parameter in liquid helium. Very accu- 114
rate measurements have been made by Rugar and Foster.16 They find val- 115
ues that are of the order of 1Å2. The value is negative at low pressures 116
but it is known from other experiments and analysis to become positive at 117
higher pressures and to have a value of around 2.2Å2 close to the freezing 118
pressure.17 If we take 1Å2 as a representative value we obtain the result 119
C3
CD≈0.3T2≈200
T
D 2
. (8)
120
Thus, based on this estimate at least, two things are clear. The magnitude 121
of the correction to the specific heat due to non-linear phonon dispersion 122
is significant at 1 K, i.e., of the order of 30%, and in the same gen- 123
eral range as found by Gardner et al. Second, the correction to the spe- 124
cific heat is much larger than the simple order of magnitude estimate of 125
(T/D)2. 126
As a next step, we consider higher order terms in the dispersion.
127
At first sight one might expect that the frequency ω would be a power 128
series containing only odd powers of q. However, in a remarkable paper 129
Kemoklidze and Pitaevski (KP)18 have shown that because of the long 130
range part of the van der Waals force the series contains a q4 term, i.e., 131
ω=vDq(1−γ3q2−γ4q3+ · · ·). (9) 132
They derive an explicit result for the coefficient γ4 133
γ4=π2 24
ρφ v2
Lm2, (10)
134
where ρ is the mass density, vL is the longitudinal sound velocity and m 135
is the atomic mass. φ is the coefficient of the van der Waals force at long 136
distance, i.e., the potential energy associated with the van der Waals force 137
between two atoms is V (r)= −φ/r6. As far as we can see, the KP effect 138
should only apply to longitudinal phonons. Putting in values appropriate 139
for solid helium at the melting pressure19 (ρ=0.1908 g cm−3, vL=4.88× 140
104cm s−1, andφ=1.57×10−60erg cm6) gives γ4=1.2Å3. If this were the 141
only correction term in the dispersion relation, i.e., ifγ3=γ5= · · · =0, then 142
the lowest order correction to can be calculated to be 143
C4=15,120ζ (7) π2 γ4k7T6
¯ h6v6
L
. (11)
144
6 H. J. Maris and S. Balibar
To compare this with the T3 specific heat, note that in an isotropic 145
approximation the Debye velocity is related to the longitudinal vL and 146
transverse vT sound velocities by the formula 147
v−3
D =(v−3
L +2vT−3)/3. (12)
148
Then we find that 149
C4
C1=113,400ζ (7)
π4 γ4 1 1+2vL3/vT3
k3T3
¯
h3vL3 =1.6×10−3T3. (13) 150
Thus in the temperature range up to 1.2 K, this amounts to a small cor- 151
rection. Consequently we will not further consider effects arising from the 152
KP term γ4. 153
Now let us consider how a phonon contribution to the specific heat 154
that goes asT7 might arise. There appear to be two possibilities. The first 155
is to propose that the corrections to the phonon dispersion relation are of 156
the form 157
ω=vDq(1−γ3q2−γ5q4), (14) 158
with negligible higher order terms, and with the coefficient γ3 much 159
smaller than expected. If γ3 is zero, it is straightforward to show that the 160
correction to the specific heat arising from γ5 isC5 where 161
C5
C1=112π4γ5k4T4
¯
h4v4D. (15)
162
To get a T7 term of the magnitude found by Gardner et al would require 163
that γ5=2.0Å4. This is not unreasonably large since it is of the order of 164
the square of the values of γ3 that have been measured in liquid helium.
165
A second possibility that we consider as more likely is that the cubic 166
dispersion term γ3 is positive for some propagation directions and polar- 167
izations and negative for others. We note that calculations of the phonon 168
dispersion in solid helium give anomalous dispersion (i.e., negative γ3)for 169
some directions and polarizations.20 Then when considering all polariza- 170
tions and directions, the T5 term could happen to be small. A T7 term 171
could arise either from the γ5 term already discussed or from the γ3 term 172
in second order. For a single branch with velocity vD, the correction to the 173
specific heat arising from the γ3 term in second order is 174
C3,2=896π6
15 γ32k8T7
¯ h7v7
D
. (16)
175
The total correction coming from all polarizations and directions would 176
then be given by the same expression but with an average value γ32 177
replacing γ2
3. To get agreement with the data of Gardner et al requires 178
γ2
31/2=0.7Å2, which is a reasonable value.
179
We can also investigate the corrections to the T3 law in a simple 180
model crystal. Consider as an example a face-centered cubic crystal with 181
central forces between nearest neighbors described by a spring constantK.
182
Let the atomic mass be M. For this model the Debye temperature is 21 183
D=2.965¯h k
K
M. (17)
184
A straightforward numerical calculation of the phonon density of states 185
followed by an evaluation of the specific heat gives the result 186
C
CD=1+60
T
D 2
+ · · · (18)
187
Thus, again the correction to the specific heat is much larger than the 188
rough estimate of (T/D)2. In Fig. 2 we show the specific heat of the fcc 189
crystal as a function of T/D. The measurements of Gardner et al. cover 190
1 1.2 1.4 1.6
0 0.04 0.08 0.12 T/ D
DC/C
θ
Fig. 2. Specific heatC(T )of a fcc crystal plotted asC(T )/CD versus T/D. CD is the low temperature Debye specific heat proportional toT3.
8 H. J. Maris and S. Balibar
the temperature range up to T/D: 0.05. It can be seen that for the fcc 191
model C/CD varies by 20 % over this temperature range.
192
3. DISCUSSION 193
The above estimates show that one cannot conclude that the excess 194
specific heat seen by Gardner et al.11 is evidence that there is a T7 con- 195
tribution to the specific heat that results from the zero-point vacancies in 196
an incommensurate solid. Phonon dispersion should lead to a contribu- 197
tion to the specific heat that has a magnitude roughly the same as the T7 198
term that has been found experimentally, but at the present time there is 199
no way to calculate the precise magnitude and temperature dependence of 200
this contribution. For example, while the simple model of the fcc crystal 201
that we have considered gives a correction to the phonon specific heat that 202
is positive, if the phonon dispersion is anomalous as it is in liquid helium 203
at low pressure,16, 17 the correction will be negative. An accurate measure- 204
ment of the phonon dispersion curves by neutron scattering, for example, 205
would make it possible to estimate the phonon contribution.
206
Finally, we emphasize that in this paper we have addressed only the 207
phonon contribution to the specific heat. There may also be a significant 208
contribution from thermally- excited vacancies. Through X-ray studies,22 209
the lattice parameter of solid helium has been measured as a function of 210
temperature for samples maintained under conditions of nearly constant 211
volume. The lattice parameter is found to decrease as the temperature is 212
raised and it is assumed that this decrease occurs because vacancies are 213
thermally excited. From the experimental data, an activation energy Ev for 214
vacancy formation can be found. It is then possible to use this value of 215
Ev to calculate the vacancy contributionCvac to the specific heat. Surpris- 216
ingly, the predicted value of Cvac appears to be too large to be consis- 217
tent with the measured specific heat.22 One possible explanation for this 218
discrepancy is that the vacancies are not localized and that as a result 219
they form a band of energies.23–25 It is also possible that in fact the pho- 220
non specific heat is less than has been believed and that as a consequence 221
more of the specific heat that is experimentally measured arises from 222
vacancies.
223
ACKNOWLEDGEMENTS 224
This work was supported in part by the National Science foundation 225
by Grant DMR-0605355 and by the Agence Nationale de la Recherche 226
through grant 05-BLAN-0084–01.
227
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