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The specific heat of dilute 3He-4He mixtures measured with a second-sound Helmholtz resonator
Physica 142B (1986) 187-199 North-Holland, Amsterdam
T H E SPECIFIC HEAT OF DILUTE 3He-4He M I X T U R E S MEASURED W I T H A SECOND-SOUND H E L M H O L T Z R E S O N A T O R H . C . M . VAN D E R Z E E U W * , R.F. M U D D E and H. VAN B E E L E N Kamerlingh Onnes Laboratorium der Rijksuniversiteit Leiden, The Netherlands Received 10 June 1986 An alternative way of measuring the specific heat of superttuid 4He and dilute 3He-4He mixtures is described. Specific heat data are derived from the measured resonance curves for the temperature amplitude in a second-sound Helmholtz resonator. It is shown that the nature of the losses that govern the shape of these curves, need not to be known for the analysis. The applicability of the method is first demonstrated by a series of measurements on pure 4He in the temperature range between 1.4-2.0 K; the results are in good agreement with existing specific heat data. Next, some measurements performed in the range from 0.3 K to 0.7 K on two mixtures of molar concentrations X= 6.0 x 10 -3 and X = 8.5 x 10-3 are presented. Although the thermal anchoring of the resonator to the mixing chamber of the Leyden dilution refrigerator did not appear to be optimal, so that the precision of the results was not high, the feasibility of the method also for the case of mixtures could be shown. It is further shown that the analysis of the data is somewhat more difficult for mixtures than for pure 4He; this is due to less direct coupling between the amplitudes of the energy and the temperature, caused by the accompanying oscillations in the concentration X.
1. The connection between second-sound resonances and the specific heat 1.1. Introduction
Most of the m e a s u r e m e n t s on the specific heat of pure 4He or dilute 3 H e - a H e mixtures have been carried out by means of the well-known calorimeter technique. With this technique the heat capacity of a thermally isolated cell, filled with the liquid, is deduced from the increase of the t e m p e r a t u r e due to the supply of an accurately known a m o u n t of heat to the cell. After subtracting the calibrated heat capacity of the e m p t y cell, the specific heat of the liquid is calculated. A completely different way of measuring the specific heat of these quantum liquids is described in ref 1. For this method the calorimeter cell is connected by means of a thermal link to the bath. After the t e m p e r a t u r e of the cell is raised uniformly by a heater, the heater is *Present address: Philips International B.V., Elcoma-BAE 1, Eindhoven, The Netherlands.
switched off and the subsequent relaxation of the t e m p e r a t u r e back to the bath value is registered. F r o m the relaxation time r of the exponential decay and the calibrated thermal conductance K of the heat link, the heat capacity C of the cell follows as C = KT. By again subtracting the heat capacity of the e m p t y cell, the specific heat of the liquid can be obtained. In ref. 1 the utility of this method has been demonstrated for various dilute 3 H e - 4 H e mixtures. A third, unusual, method of determining the specific heat of the helium liquids, in which explicit use is m a d e of their superfluid character, will be described in this article. The m e t h o d was originally proposed to us by K r a m e r s [2] and involves the m e a s u r e m e n t of the second-sound resonance curve of the t e m p e r a t u r e amplitude in a Helmholtz oscillator. The specific heat appears in the proportionality factor between the mechanical energy of the second-sound oscillations and the square of their t e m p e r a t u r e amplitudes in the end-volume. The mechanical energy at resonance can in turn be deduced directly from the height and the width of the resonance curve for the the t e m p e r a t u r e response at a given
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
188
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
driving power, irrespective of the damping mechanisms that are responsible for the broadening of the curve, at least as long as the quality of the oscillator does not become too small. In section 1 of this article it will be shown in detail how the specific heat can be derived from the measurements, first for the case of pure 4He and subsequently for a mixture. In section 2 the experimental set-up will be discussed and the experimental results, as obtained for pure 4He and subsequently for two dilute 3He-nile mixtures with molar 3He concentrations X = 6.0 x 10 -3 and X = 8.5 × 10 -3 will be presented. The article will be dosed with some concluding remarks regarding the usefulness of the method.
1.2. Second sound in a Helmholtz oscillator Second-sound resonances in a Helmholtz oscillator have been used before to study the properties of pure 4He [3, 4] and of dilute 3He-4He mixtures [5] Helmholtz oscillators in their simplest form, consist of a reservoir connected by a tube to a bath. The volume V of the reservoir is chosen large as compared to the volume of the tube, A L , where L is its length and A its crosssectional area (see fig. 1). When second sound has to be excited in the oscillator, which involves
I
bath
A
i
TM
a temperature oscillation in the reservoir, a direct thermal contact between the reservoir and the bath has to be avoided. In principle, the reservoir and tube must therefore be surrounded by a vacuum can. One can, for instance, study the frequency and decay of free second-sound oscillations. Such an oscillation corresponds to a standing wave of second sound in the tube, which transports entropy, carried by the normal component, periodically into and out of the reservoir. It will thus show up as a temperature oscillation in the reservoir. From the measured frequency the second-sound velocity is obtained, while the decay rate provides information on the dissipative processes that are driven by the oscillation. To the damping contribute not only the various dissipative transport processes taking place in the bulk of the liquid itself, but also the irreversible thermal conduction via the walls and the dissipative effects within a viscous penetration depth near the wall of the tube. Usually the latter appear to dominate the damping [3-5]. Instead of studying free oscillations, the same information is usually obtained by using the oscillator as a resonator. A second-sound oscillation is then maintained by a periodic heat supply via a heater in the reservoir. Its temperature amplitude for a fixed heating power is registered as function of the driving frequency. As long as the damping rates are not too high, the location of the top of this resonance curve will of course correspond to the frequency of the free oscillations, the width of the curve yields again the damping. If these data are combined with the third quantity that can directly be obtained from the curve, namely the temperature amplitude at resonance for the given heating power, the specific heat can be deduced as was stated in the previous introduction.
L
1.3. The Helmholtz oscillator and the two-fluid description
Fig. 1 Schematic drawing of a Helmholtz oscillator. T h e volume V is connected with a bath by m e a n s of a tube of length L and cross section A.
The hydrodynamic properties of the superfluid helium mixtures are described successfully by a two-fluid model [6]. In this model the liquid is composed of a superfluid and a normal fluid
189
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
fraction, with mass densities p~ and p. and independent velocity fields v~ and v.. The 3He quasiparticles, with effective hydrodynamic mass my, are assumed to participate in the motion of the normal fluid so that
as
U102= C 21(0 + p_.&Spnax2x2)
first sound
(7)
u2o = Co/
second sound,
(8)
-- axX P.
V3 : Vn ,
P = Ps "~ P. ---~Ps "~ (Pn4 + Pn3)"
(1)
Acoustic phenomena, which correspond to propagating small disturbances of the global equilibrium state, are covered by the set of linearized two-fluid equations. On the non-dissipative level this set reads as 0_£p+ p~V- v~ + p,V. v, = 0
Ot
mass balance,
(2)
@__y_x+ oxV. v = O at mass balance 3He,
(3)
aps + psV. v. = 0 at
where the following abbreviations are used: ¢10 ~- k ' ~ - e )
2
0V s
*4-s-x
ps--~ + p n - ~ + V P=O momentum balance, aV s
at
(5)
+Vt~4 = 0
equation of motion for as.
(6)
The potential /~4 that drives the superfluid, equals the chemical potential per unit mass of 4He in the solution, as it is determined through the relations for local equilibrium by the local values of the three independent thermodynamic variables. For these the pressure P, the temperature T and the 3He-mass-concentration x will be chosen. Two different longitudinal plane wave modes satisfy the above set of equations, i.e. first and second sound. When the extremely small effects due to thermal expansion are neglected, their speeds of propagation u~0 and U2o can be written
l(ap)
7x
in which c34 --~ T(ds/OT)e,x is the specific heat per unit mass of mixture. In order to obtain ul0 and uz0 in the compact forms (7) and (8), use is made of the fact that it appears quantitatively 2 2 that Cl0 >> c20. First sound corresponds to ordinary sound in classical fluids; the two components move in phase with their velocity amplitudes v~ and v', related by the equation
(4)
OVn
Ps
(9) (as)
pslJs + P n V . ~
entropy balance,
I°s ~
pVs/(1 - axX).
(10)
Relation (10) implies that in mixtures the normal fluid moves considerably faster than the superfluid does (v'.> v~), especially at temperatures well below 1 K where p, ~ Pn3 = (my/ms)PX. In second sound the two components move in opposite directions. For the case of pure 4He the net mass flow is zero, the waves manifest themselves as temperature waves. In mixtures, however, a small mass flow will result in addition, almost proportional to the 3He concentration: PsV~ + PnV" "~
PaxX v'~.
1 -- OtxX
(11)
This is a clear consequence of the difference in mass and molar volume of the two isotopes, resulting in a value of ax considerably different from zero. Acoustic oscillators of the Helmholtz type have now been in use for classical fluids for more than a century [7]. Their first application to the case of superfluid helium was reported in 1960 [8]. In view of the variety of sounds that can be
190
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
excited in superfluid helium, not only first-sound and second-sound oscillators have since been investigated, but also fourth-sound and even third-sound oscillators, In the case of fourth sound the tube between the reservoir and the bath is very narrow or is replaced by a superleak, so that the normal component is practically immobile [9]; in the case of third sound, the inner walls of the tube and reservoir are covered by an unsaturated helium film [10]. The theory of Helmholtz oscillators filled with a superflmd H e - He mixture has been discussed extensively in ref. 5. Here it suffices to show how their behaviour is described in principle by the simplified set of two-fluid equations represented by eqs. (2)-(6). Integration of the linearized balance equations (2), (3) and (4) over the volume of the reservoir yields •
3
4
*
V~t = -A(psv' s + p.v'),
(12)
V c~px c~t -Apxv'n
(13)
V aps c~t
(14)
-Apsv',
where p, px and ps represent the mean densities in the reservoir and v's and v', the mean transport velocities away from the reservoir. These equations can now be combined with eq. (5) and (6) applied to the tube and with eq. (10) or (11), describing the supply through the tube for the case of first sound and second sound, respectively. Considering first the limiting case, in which the variations in the densities over the volume of the reservoir and in the velocities o's and o'. as well as in the gradients V/z4, VP, VT and Vx along the tube are fully neglected, one obtains immediately for the time variation of all relevant quantities, for instance for o's, the harmonic oscillator equation: +
A
v's = o .
(15)
In eq. (15) u 0 represents the speed of first or of second sound, as are given by eqs. (7) and (8), respectively. When the variations along the tube
and reservoir (assumed to be cylindrical with length LR) due to the finite acoustic wave-length are taken into account, the eigenfrequency aJr~0 is corrected up to first order in the small ratio A L / V into
2
tOH0 =
2 A[
AL( L~]] 1 - - ~ - 1 + L2 ] j .
U0 ~
(16)
In the above derivation all dissipative effects have been ignored. They will of course lead to damping of the free oscillations, and can be taken into account by allowing a~H0to become a complex quantity ~oH. For those contributions to the damping arising from the attenuation of the sound wave in the tube, this is done by replacing u 0 in eq. (16) by a complex quantity u. As has been demonstrated before [3-5], the damping in a second-sound oscillator is in practice often dominated by the viscous dissipation occurring within the viscous penetration depth ~n(to) near the wall of the tube of radius R:
-=
,
(17)
where ~1 is the shear viscosity• This dissipation leads to u2 = u : o 1 - ~ ( i + 1 )
(18)
and to the damping rate 3,, of the free oscillations 1 ps~n Y" = 2 t°n° p R "
(19)
Both quantities u 2 and 7, are given here up to first order in the ratio ~,/R, which is made small by the design of the oscillator• Contributions to the damping arising from the other dissipative processes that are driven by the oscillations will simply add to 3'~ to yield the total damping rate 7. Helmholtz oscillators can also be used as resonators. Second-sound resonances can be obtained by means of a periodic supply of heat
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
191
is found from the resonance curve, since as long as Ato ~ too toO
"Tre5~ I
Q--A---dw"
_rw "I/2V-2T~es< AtO>
1,4. Determination of the mechanical energy at resonance from the resonance curve The periodic heat supply W(t) of eq. (20) will eventually lead to a stationary situation in which the temperature of the reservoir T R will vary periodically and can be written as
f
%
co
Fig. 2. Resonance curve with width Aw at The resonance angular frequency is wo.
t 2 T R~ _--Tres/X/~.
W(t) to the reservoir. In practice one has W(t) = Wo(1 + coso~t).
(20)
Variation of to at constant amplitude W0, yields a resonance curve for the amplitude of the periodic temperature response in the reservoir, T~, like is shown in fig. 2. The peak in this curve, T~es, will be located at an angular frequency too equal to the real part of o~. From too, with the use of eq. (16), the speed of second sound in the tube follows as the real part of uz. The width Ato of the peak at a height Try(to) = T~es/X/2 equals twice the decay rate y of the free oscillations, while y in turn will also be equal to the attenuation rate of the second-sound wave in the tube, as long as damping contributions coming from outside the tube can be neglected. Possible differences between y and y~ can be found by confrontation of speed and attenuation in eq. (18), using the bulk speed for u20. Finally, the quality Q of the oscillator, defined as
total mechanical energy of the oscillation at resonance a = 2,tr dissipation of energy per period Eres
= too ( P ) r ~
(22)
TR(t ) = TO+ A TO+ Tl~ cos(tot - ~b).
In eq. (23) TO is the temperature of the bath, A TOthe constant temperature difference that will arise due to the average rate W0 at which thermal energy is transported through the various parallel thermal resistances between the reservoir and the bath, and T'r(to) and ~b(to) the amplitude and phase of the periodic temperature response. The mean rate at which entropy is supplied to the reservoir by the heater is (W/TR) while the mean flow of entropy that reaches the bath equals Wo/T o. The mean rate of entropy production ( S ) is therefore given by
( 3 ) - w0 To W o ( l + costot) + A TO
I
TI~ cos(tot- ~b)
-Z-0+Z
(24) irrespective of the dissipative mechanisms that are responsible. Up to first order in the small ratio's ATo/T o and T~/T o expression (24) simplifies to
(3)
(21)
(23)
w°ar°
- --T~
Wor
+ ~
cos~b ,
(25)
where the first term stems from the dc heat current through the tube. At resonance cos~b = 1
192
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
so that ( P ) .... the mean rate of dissipation of mechanical energy at resonance, can be obtained from the measured values of T'~s at given W0, according to
to be written as the volume integral of the potential energy density Em = /-l_p
_~_(- 1
x(~lb4/~X)P, ~ T c34) ~ T t2d V .
Tt
res
(P)res = T0(S)r,~ = 2T0 W°"
By using relations (21) and (22), the mechanical energy at resonance can subsequently be deduced from the measured width of the resonance curve, zato, yielding Ere~_ (P)res a _ too
(30)
(26)
T~esWo 2 T OAto "
(27)
Since the distribution of T' over the volume is known, the integration can be carried out straightforwardly. When the effects of the finite wavelength in the tube and reservoir are neglected by assuming a linear temperature profile along the tube and constant amplitude T~ in the reservoir, one obtains
1 c34 ( A L ) Em= ~ p -~- V 1+ -~ 1.5. The relation between the mechanical energy and the temperature amplitude at resonance The driven oscillations can be considered as standing waves of second sound in the tube and reservoir, having a wavelength which is large compared to the lengths L and L R. The two parts of these waves are matched to each other at the boundary between the tube and the reservoir by the condition for continuity of the temperature and the mass transports. The total mechanical energy of the oscillations equals the volume integral of the maximum kinetic-energy density 1 12 E m = f ( T1p ~ v st2 + 2PnVn )dV.
(28)
The local velocity amplitudes, related to each other by eq. (11), can of course also be expressed in the temperature amplitude T' of the second sound. From the two-fluid equations (2)2 2 (6) one obtains (again neglecting (C2o/Clo)X with respect to one)
p~v's = -pnv'. =-p,
1+
(~s/Pn)O/xX
1 - a~x
(1+ Ps axX)u2 --
C34
Pn
S4
(29) T
Substitution of eq. (29) into eq. (28) allows E m
× (1 -
X( O~t4/OX) p, T $24T
c34) T~ 2 .
(31)
The error, introduced by this simplification, will not exceed a few percent for the dimensions of the devices used in the experiments and for frequencies close to the resonance frequency*. Combining eqs. (27) and (31) with the experimentally determined values, one obtains the thermodynamic entity C34( 1
=
X(OId'4/tgg)e'T C34)
s24T
1 Wo . pV(l + ( A L / a V ) ) T'resAto
(32)
In the case of pure 4He, for which the concentration x is obviously zero, one thus determines the specific heat c 4 directly. For a mixture, however, the additional factor can be considerably different from one. A fair estimate can be obtained by applying the Landau-Pomeranchuk model [11]. Using this model the dilute solution can be described as an ideal mixture of a quasi*The main error stems from the cosine distribution of T~ in the reservoir. This introduces an additional factor (1 + ~ ( A L / 3 V ) ( L 2 / L 2 ) ) into relation (31), with fl between the values - 1 and +2, depending on where in the reservoir T~ is registered.
193
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
ideal Fermi gas, formed by the 3He quasiparticles, and pure superfluid 4He. The Gibbs function per unit mass of the solution is represented by
in the experiment, the value of (37) can be expected to be not much different from 2.5.
2. The experiment g = x ~3o + ~ 3 lnX
2.1. The experimental set-up
RT + (1 - x){ ~4o + ~--~-4In(1 - X)} .
(33)
Both /z30 and /x40 have to be evaluated at the pressure and temperature of the mixture. The molar concentration X is related to the mass concentration x as xM 4
4
(34)
S = x M 4 q- (1 - x ) M 3 ~ -3 x .
It now follows immediately that x ( Olx4~
:
\ aX /P,T
RT x M 4 1-x
dX d---x ~
RT x M3 ,
(35)
while s 4 attains the value R R s 4 = s40 - -;7-, In(1 - X) ~- x .
(36)
M3
1vl 4
The specific heat measurements in pure 4He have been carried out in the threefold Helmholtz resonator, used before by van der Boog et al. [5]. For the measurements in the 3 H e - a H e mixtures a similar resonator of identical dimensions was built, differing only from the first one by the use of Stycast 1266, instead of Epocast 202, as working material. A drawing of the resonator is presented in fig. 3. The volumes V1,2, 3 a r e cylindrical and have an inner diameter of 1.40cm and a length of 0.60cm. The tubes L1,2, are both 0.90cm long and have an innerdiameter of 0.30 cm. The figure also shows how the resonator is connected to the mixing chamber of a Leyden dilution refrigerator [13]. The connection, formed by a thick copper strip, coated on both sides with a silver layer, provides for good thermal contact. The part of the strip inside volume V2 of the resonator, is sintered with silver powder, in
The approximations made in the second part of both equations are justified in view of the low temperature range (0.3-0.7 K) and low concentrations ( X < 0.01) prevailing in the present experiment. One thus obtains 1
x(am/ax), s24T
(R)
c34~ 1 + c34/ x ~
.
(37)
It is pointed out that the results (35), (36) and (37) do not depend on the particular form of the expression assumed for/%0; deviations from the parabolic Landau-Pomeranchuk spectrum, for instance, that would account for the additional contribution to c34 reported by Greywall [12] and by van der Zeeuw et al. [1] will not affect them. If, for the purpose of obtaining a fair numerical estimate, we substitute for c~, in expression (37) the Landau-Pomeranchuk limit c34-----> LP ~xR/M3 it follows that for the conditions prevailing
i! /" O ~ 2cm Fig. 3. The resonator and its connection to the mixing chamber. HE: heat exchanger: $1: superleak: Mu: upper flange of the mixing chamber.
194
H . C . M , van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
order to increase the heat-exchanging surface area. In a symmetric, threefold Helmholtz oscillator, two types of Helmholtz resonances can be generated, as is discussed by van der Boog et al. [5]. One, the symmetrical mode, is characterized by temperature oscillations in V1 and V3 that are in phase and of equal amplitude and a temperature oscillation in V2 of opposite phase. For the case that the three volumes are of equal size, the amplitude in V 2 will be twice that in 1:1 and V3. The other mode, the anti-symmetrical one, is characterized by temperature oscillations in V1 and V3 that are of opposite phase and equal amplitude, while no oscillation of the temperature occurs in V2. In the present experiments only the antisymmetrical mode could be used, in view of the thermal anchoring of the device via V2. Its temperature distribution is sketched in fig. 4. The second-sound transmitters (SST) in both resonators are made out of resistance strip. The second-sound detectors (SSD) in the Epocast resonator are formed by a layer of aquadag painted on small strips of paper. In the Stycast resonator polished Matshushita resistors are used.
2.2. The measuring procedure The second-sound oscillations are generated by an ac.-voltage applied to one of the transmitters. Different from the situation in the experiments of van der Boog et al. [5], the accurate values of the supplied driving powers W have to
o!
.
O I
iI
I
;
I i I
i I
V1
-TR
ALl 2 ~
i
2V1.3(1 + ~ /
_i
I V2
be known. Both the voltage across and the current through the transmitter must therefore be measured. The current is determined from the voltage measured across a calibrated resistor connected in series with the transmitter. The induced second-sound oscillations will have twice the frequency of the driving voltages. They are registered by means of a detector in V1 or in V3, its resistance being modulated by the temperature oscillations. For this purpose a small dc measuring current is applied to the detector and the resulting ac-voltage is amplified and detected with a lock-in amplifier. From the so determined amplitude R' of the resistance variations, the temperature amplitude T' is obtained by means of a polynomial fit through the calibration data of the detector resistance as function of temperature. The values of the driving power were chosen as small as was permitted by the detection accuracy, in order to keep the disturbance caused by the unavoidable dc component to the heat flow as small as possible. The resonance curves for the temperature response at fixed driving amplitudes, of which the heights T~eS and widths Ato need to be determined, are obtained from twelve to sixteen measuring points taken around the resonance frequency of the anti-symmetric mode, within the range where (1/2)T',e s < T ' < T',e,. The values of T'~ and Ato are substituted into the right-hand side of eq. (32), in which an extra factor of two must be added to the denominator in view of the fact that eq. (32) was derived for a single Helmholtz oscillator. Taking into account the 1% thermal contraction in the linear dimensions of the device and applying a small correction for the volumes of the transmitters and detector, the effective volume V' that enters into the denominator of eq. (32) becomes for the threefold oscillator
V3
Fig. 4. The temperature distribution of the antisymmetric mode in the resonator.
--=V' = (1.79 -+ 0.05) cm 3.
2.3. Experimental determination of the specific heat of pure 4He The measurements on superfluid 4He have
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator I
I
I
'
I 1.4-
i
I
5C
I
195
i
I
'
I 1.8
,
600
T = 1.533K V O = 8 6 . 4 4 Hz 1.Ol HZ 4C
575
x."
::t
3Q ~/
~_ 4 0 #~-
30
20
I 86.00
I 86.50
I 87.00
I
I
I
~
& "13
o 550T = 1.915K Vo= 80.185 Hz AV= 0.41 Hz
3
525
4/
[ 80.00
J 80.25
I 80.50
<
V(Hz) Fig. 5. Examples of resonance curves in pure 4He. T h e circles represent the experimental points, renormalized for Wo = 0 . 1 mW.
been carried out in the teinperature range between 1.4K and 2.0K at saturated vapour pressure. In this temperature range the dilution refrigerator is of course not in operation. The resonator is cooled by the surrounding 4He bath by means of 4He-gas admitted into the vacuum can. Thermometers and second-sound detectors are calibrated against the vapour pressure of the 4He bath. Some examples of the resonance curves obtained for different bath temperatures are shown in fig. 5. Mean driving powers W0 between 10-4W and 2 x 10-4W were employed, resulting in temperature amplitudes T'res varying from
500
I i 1.6 T(K)
2.0
Fig. 6. The angular resonance frequency as function of temperature. The lower curve is drawn through the measured resonance frequencies. The upper curve represents the calculated values for tOHo=C2o(2A/V'L1) ~/2 as function of temperature. 1.5
'
[
'
I
i
I
'
I
,
1.0 O
3 <~ O 0.5
5 0 / x K at T = 1.45 K to 8 5 / z K at T = 1.90K.
The observed resonance frequencies are plotted as a function of temperature in fig. 6. The variation is mainly determined by the temperature dependence of the second-sound velocity. This is demonstrated by the curve for toll0 = C2o(2A/V'L1) 1/2, calculated according to eq. (16) with the thermodynamic quantities in C2o of eq. (9) taken from the literature [14]. In fig. 7 the observed damping, represented by
O
I 1.4
,
1.6 T(K)
1.8
2.0
Fig. 7. The damping factor, (Atoltoo) as function of temperature. The lower curve represents the calculated values for (p,/p)(8~/R); ( © ) and (E]) represent data obtained from the temperature in volume V 1 and I,'3, respectively.
196
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
20
1
'
I
i
I
form
,
V4Wo 15-
//
'.~+"TO
E I0
3 ,¢ U
/
c 4 -
I 1.4
,
I i 1.6 T(K)
I 1.8
(38)
with V4 the molar volume, are shown in fig. 8. Error bars, corresponding to an uncertainty of about 5% seem to be appropriate in view of the uncertainties in V' and T',e~ and of the assumption that T're~ represents the average temperature amplitude in V1.3.. Furthermore, the uncertainty in the absolute temperature is significant because of the steep variation of C a with temperature. In the figure the results are compared with the specific heat data, tabulated by Maynard [14]. It demonstrates the applicability of the resonance method as well as its limitations set by the measuring accuracy.
/ 0
V,T;esAo.)
I
2.0
Fig. 8. The results for the specific heat of pure 4He; (O) and (F]) represent data obtained from the temperature response in volume VI and V3, respectively. The drawn line is taken from tabulated values by Maynard [14].
the quantity ('ttO/OOHO)=(2T/OOHO), is plotted against temperature. The lower curve in the figure corresponds to calculated values* of (Ps/ p)(8~/R), i.e. the expected damping in case the viscous dissipation in the penetration depth near the wall of the tube dominates the damping (see eqs. (18) and (19)). Comparison with the data shows that, particularly towards lower temperatures, other contributions to the damping become relatively less and less important. Although the observed damping thus confirms the expected behaviour it is repeated here that such understanding is not really necessary to obtain the specific heat c a from the observed resonance curves. The results for the molar specific heat, C4, obtained from eq. (32) in the
* For the calculations values of -t/as reported in refs. 15 and 16 were used; values for ps/p were taken from ret. 14.
2.4. The experimental results for the dilute 3He-4He mixtures Due to a number of circumstances, only a few resonance curves for mixtures can be presented here. Nevertheless, the few results obtained, demonstrate the feasibility of the method. The analysis offers a number of suggestions by which the method could be further improved. For the mixture of molar concentration X = 6.0 × 10 -3 the measurements are restricted to only one curve, taken at T = 683 mK. Seven curves, spread evenly over the temperature interval 0.3-0.7 K were taken with a mixture of which the concentration was at first unknown, but of which the results could be proved to be compatible with the specific heat data reported in literature for a mixture with X = 8.5 x 10 -3, as will be shown in this section. All measurements were done at P = 0. In fig. 9 some examples of the measured resonance curves for the mixture with X = 8.5 × 10 - 3 are shown. Driving powers W0, ranging from 5 / x W at the lowest temperature T = 344 mK to 12 ~ W at T = 683 mK, were required. They resuited in temperature amplitudes T',os between 7 / z K and 15/.~K. These values of W0 also resulted in a considerable dc temperature difference
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
12
~ / ~{
10
=
~ "1~
X=8.5x 10-3 T = 0.560K Vo =133.48 HZZ~v 6.25Hz
I 13.5
I 140
-
/
~
-
8
,~ -~
-r" 125
I 130 I
I
I
×=8.5×10 -3 T = 0.374- K Vo =109.45 Hz AV= 5.80 HZ
1
the second series of measurements extending over the entire temperature region. When comparing the latter results with the earlier one a large discrepancy appeared, unfortunately, as if, due to some unexplained effect, the concentration of the mixture had changed from X = 6.0 × 10 -3 to X = 8.5 × 10 -3 during the recycling process. Secondly, the rather large dc component W0 to the heat flow introduced by the second-sound transmitter, not only leads to the steady temperature difference A T, but will also give rise to a concentration difference A X by the well-known heat-flush effect. According to the two-fluid description 3He will move with the normal fluid away from the heater until due to diffusion in the growing concentration gradient a steady state is reached. If one defines for this state an effective thermal conductivity Kelf by
Lwo
(39)
reef --- A A T 6 -(
I 105
I 110 V (Hz)
f 115
120
Fig. 9. Examples of resonance curves in the dilute 3He-4He mixture with molar concentration X = 8.5 x 10 -3. The circles represent the experimental points.
between reservoirs V1 and V2,3, with a typical value A T - - 30 mK. The resonator temperatures, quoted in this section, correspond to averages over the three volumes. All thermometers were calibrated against the vapour pressure of 3He after the resonator was filled with mixture. Before deducing the specific heat values c34 from the measured curves a few remarks should be made to explain the scarcity of results. Firstly, the original mixture was made at room temperature with a molar concentration X = 6.0 x 10 -3. After the first resonance curve at T = 683 m K was measured, the refilling tube of the 3He cryostat became clogged up and the whole device had to be warmed up to room temperature. The mixture was regained and recondensed again into the resonator to carry out
197
the values one obtains for K~effall in the range of a few tenths of a W / K m . Such values are very well compatible with the results for X = 1.3 x 10 -2 and X = 5 × 10 -2 obtained by Abel and Wheatley [17] in a thermal conduction experiment with a tube of 2.5 mm diameter. In this description one can obtain the value of A X that corresponds to A T from the condition V/x4 = 0 (imposed by the equation of motion for the superfluid, eq (6)) together with the assumption that in the steady state VP is negligible. From the expression for /~4 introduced in eq. (33) it then follows that VX _ 1- X
M4s4° RT
VT
(40) "
If we insert numerical values, for instance for the measurements at T = 462 m K and X = 8.5 × 10 -3 for which A T = 30 mK, one obtains A X = 6.0 × 10 -4. The heat-flush effect is thus considerable and it therefore puts severe restrictions on the accuracy by which the specific-heat values can be derived from the measuring runs. We therefore decided not to extend the measurements to more concentrations nor to lower temperatures, but to
198
H . C . M . van der Z e e u w et al. / Dilute 3He-aHe mixtures in a Helmholtz resonator
Table I Experimental results Molar concentration X
T (mK)
Wo (~W)
T~es (gK)
~o (rad/s)
A~ (rad/s)
AT (mK)
6.0 x 10 3
683
11.6
15.0
872
41.8
-70
X 10 -3
344 374 418 462 512 560 622
5.3 8.0 7.7 8.2 8.2 9.9 11.7
7.3 11.7 11.3 11.4 10.8 12.6 13.2
664 688 729 766 806 839 873
39.0 36.4 36.4 37.7 38.6 39.3 39.9
21 32 28 30 30 38 45
8.5
content ourselves with the analysis of the present runs and with some suggestions how the heatflush effect can be reduced by improving the construction and the thermal anchoring of the resonator. In table I all the experimental results obtained for the two mixtures are compiled. In order to obtain the specific heat from these data we first rewrite eq. (32) in molar form. In the approximation of an ideal mixture, i.e. using eq. (33), eq. (32) becomes RC34
C34(1 ~- ~XXxI- [M4s40 --.-RR-In(I -
1.5
X)]2)/V34(X)
Wo -
V' T',es Ato
.
(41)
//
qi 0
E 1.0
Values for the molar volume V34(X) and s40 can be obtained from the literature [18], so that from the datapoint for X = 6 . 0 x 10-, the corresponding value of the molar specific heat can be solved directly. The result is plotted in fig. 10. In this figure the curve calculated for this mixture from the relation (see refs. 1 and 12) 6 3 4 = C 4 -[- C LP ht-
the small additional term ( 3 / 2 ) X R S ( T - To) , as LP is demonstrated by the curve for C34, also drawn in the figure. For the second mixture the unknown concentration X and C34(X, T) cannot both be deduced
3XRS(T- To)
(42)
_
/
Z
X = 6.0,~ 10-a (D 0.5
04/
I
0.2
I
0.4
0.6
0.8
T (K)
(in which S = 0 for T < ~ T o = 0 . 2 5 K and S = 0.2 K - ' for T > To) has been drawn for comparison. The agreement is quite good. The uncertainty in the measured value, however, is too large to confirm unequivocally the existence of
Fig. 10. The molar specific heat C34, derived from the measurements through eq. (41) for X = 6.0 x 10 -3 (O) and X = 8.5 x 10 -3 (O). The drawn curves through the data obey eq. LP (42); the dashed curves represent C34(-=C4+ cLP). The values for C4 are obtained from ref. 14.
H.C.M. van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
from the data. One can, however, check the compatibility of the results with the description given by eq. (42). This is done by solving X numerically for each individual data point from eq. (41), by substituting eq. (42) for (?34 and by checking subsequently whether the spread in the values of X is acceptable. The m e a n value of X one finds in this way is X = 8.5 x 10 -3. Fig. 10 shows the experimental results for C34 as obtained by using this value for X, as well as the corresponding curve calculated from eq. (42). In view of the measuring accuracy the agreement is surprisingly good. That this agreement seems better than with the related curve for C3La dotted in the figure has little meaning, since a comparison of the data with the LP-description would require a redetermination of the value of X.
2.5.
Concluding remarks
The results obtained for pure superfluid 4He as well as for the dilute mixtures have shown the feasibility of K r a m e r s ' original suggestion that in a second-sound Helmholtz-resonator experiment the specific heat of these q u a n t u m liquids can be determined in situ from the resonant behaviour. The applicability of the method could perhaps be extended beyond this goal if a n u m b e r of technical i m p r o v e m e n t s were made, in particular those concerning the reduction of the heat-flush effect. Since the value of T'r~s is determined by W0 through the quality factor of the resonances, W0 can only be reduced by improving this quality, for instance by suppressing the viscous damping through the use of wider tubes. In addition, this would increase the effective heat conductance at the same time. A further reduction of the heat flush can perhaps be realized by thermally an-
199
choring both outer reservoirs 1t"1, and V3 instead of V2, and then supplying a ' d u m m y ' dc heat input W0 also to V3.
References [1] H.C.M. van der Zeeuw, L.P.J. Husson, M. Durieux and R. de Bruyn Ouboter, Proc. LT-17 Conf., U. Eckern, A. Schmid, W. Weber and H. Wfihl, eds. (Elsevier Science Publ., Amsterdam, 1984) p. 1245. H.C.M. van der Zeeuw, Thesis, Leiden (1985). [2] H.C. Kramers, private communication. [3] R.R. IJsselstein, M.P. de Goeje and H.C. Kramers, Physica 96B (1979) 312. [4] M.P. de Ooeje and H. van Beelen, Physica 133B (1985) 109. [5] A.G.M. van der Boog, E E L Husson, Y. Disatnik and H.C. Kramers, Physica 104B (1981) 285. [6] I.M. Khalatnikov, An introduction to the Theory of Superfluidity (W.A. Benjamin, New York, 1965). [7] H.L.F. yon Helmholtz, Crelle Bd. 57 (1860); On the Sensations of Tone (2nd ed. reprinted (1954) by Dover, Publ., New York.) [8] H.C. Kramers, T.M. Wiarda and A. Broese van Groenou, Proc. Int. Conf. on Low Temp. Phys. LT-7, G.M. Graham and A.C. Hallett, eds. (North-Holland, Amsterdam, 1961) p. 562. [9] I. Rudnick, Proc. Int. School of Physics "Enrico Fermi" LXIII (1976) p. 112. [10] G. Bannink, M.G.M. Brocken, I. van Andel and H. van Beelen, Physica 124B (1984) 1. [11] L.D. Landau and I.J. Pomeranchuk, Dokl. Akad. Nauk. USSR 59 (1948) 669. [12] D.S. Greywall, Phys. Rev. Lett. 41 (1978) 177. [13] K.W. Taconis, N.H. Pennings, E Das and R. de Bruyn Ouboter, Physica 56 (1971) 168. [14] J. Maynard, Phys. Rev. B14 (1976) 3868. [15] A.D.B. Woods and A.C. Hollis-Hallett, Can. J. Phys. 41 (1963) 576. [16] W.J. Heikkila and A.C. Hollis-Hallett, J. Phys. 33 (1955) 420. [17] W.R. Abel and J.C. Wheatley, Phys. Rev. Lett. 21 (1968) 1231. [18] R. Radebaugh, Nat. Bur. Stand. (US) Techn. Note no. 362 (1967).
T H E SPECIFIC HEAT OF DILUTE 3He-4He M I X T U R E S MEASURED W I T H A SECOND-SOUND H E L M H O L T Z R E S O N A T O R H . C . M . VAN D E R Z E E U W * , R.F. M U D D E and H. VAN B E E L E N Kamerlingh Onnes Laboratorium der Rijksuniversiteit Leiden, The Netherlands Received 10 June 1986 An alternative way of measuring the specific heat of superttuid 4He and dilute 3He-4He mixtures is described. Specific heat data are derived from the measured resonance curves for the temperature amplitude in a second-sound Helmholtz resonator. It is shown that the nature of the losses that govern the shape of these curves, need not to be known for the analysis. The applicability of the method is first demonstrated by a series of measurements on pure 4He in the temperature range between 1.4-2.0 K; the results are in good agreement with existing specific heat data. Next, some measurements performed in the range from 0.3 K to 0.7 K on two mixtures of molar concentrations X= 6.0 x 10 -3 and X = 8.5 x 10-3 are presented. Although the thermal anchoring of the resonator to the mixing chamber of the Leyden dilution refrigerator did not appear to be optimal, so that the precision of the results was not high, the feasibility of the method also for the case of mixtures could be shown. It is further shown that the analysis of the data is somewhat more difficult for mixtures than for pure 4He; this is due to less direct coupling between the amplitudes of the energy and the temperature, caused by the accompanying oscillations in the concentration X.
1. The connection between second-sound resonances and the specific heat 1.1. Introduction
Most of the m e a s u r e m e n t s on the specific heat of pure 4He or dilute 3 H e - a H e mixtures have been carried out by means of the well-known calorimeter technique. With this technique the heat capacity of a thermally isolated cell, filled with the liquid, is deduced from the increase of the t e m p e r a t u r e due to the supply of an accurately known a m o u n t of heat to the cell. After subtracting the calibrated heat capacity of the e m p t y cell, the specific heat of the liquid is calculated. A completely different way of measuring the specific heat of these quantum liquids is described in ref 1. For this method the calorimeter cell is connected by means of a thermal link to the bath. After the t e m p e r a t u r e of the cell is raised uniformly by a heater, the heater is *Present address: Philips International B.V., Elcoma-BAE 1, Eindhoven, The Netherlands.
switched off and the subsequent relaxation of the t e m p e r a t u r e back to the bath value is registered. F r o m the relaxation time r of the exponential decay and the calibrated thermal conductance K of the heat link, the heat capacity C of the cell follows as C = KT. By again subtracting the heat capacity of the e m p t y cell, the specific heat of the liquid can be obtained. In ref. 1 the utility of this method has been demonstrated for various dilute 3 H e - 4 H e mixtures. A third, unusual, method of determining the specific heat of the helium liquids, in which explicit use is m a d e of their superfluid character, will be described in this article. The m e t h o d was originally proposed to us by K r a m e r s [2] and involves the m e a s u r e m e n t of the second-sound resonance curve of the t e m p e r a t u r e amplitude in a Helmholtz oscillator. The specific heat appears in the proportionality factor between the mechanical energy of the second-sound oscillations and the square of their t e m p e r a t u r e amplitudes in the end-volume. The mechanical energy at resonance can in turn be deduced directly from the height and the width of the resonance curve for the the t e m p e r a t u r e response at a given
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
188
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
driving power, irrespective of the damping mechanisms that are responsible for the broadening of the curve, at least as long as the quality of the oscillator does not become too small. In section 1 of this article it will be shown in detail how the specific heat can be derived from the measurements, first for the case of pure 4He and subsequently for a mixture. In section 2 the experimental set-up will be discussed and the experimental results, as obtained for pure 4He and subsequently for two dilute 3He-nile mixtures with molar 3He concentrations X = 6.0 x 10 -3 and X = 8.5 × 10 -3 will be presented. The article will be dosed with some concluding remarks regarding the usefulness of the method.
1.2. Second sound in a Helmholtz oscillator Second-sound resonances in a Helmholtz oscillator have been used before to study the properties of pure 4He [3, 4] and of dilute 3He-4He mixtures [5] Helmholtz oscillators in their simplest form, consist of a reservoir connected by a tube to a bath. The volume V of the reservoir is chosen large as compared to the volume of the tube, A L , where L is its length and A its crosssectional area (see fig. 1). When second sound has to be excited in the oscillator, which involves
I
bath
A
i
TM
a temperature oscillation in the reservoir, a direct thermal contact between the reservoir and the bath has to be avoided. In principle, the reservoir and tube must therefore be surrounded by a vacuum can. One can, for instance, study the frequency and decay of free second-sound oscillations. Such an oscillation corresponds to a standing wave of second sound in the tube, which transports entropy, carried by the normal component, periodically into and out of the reservoir. It will thus show up as a temperature oscillation in the reservoir. From the measured frequency the second-sound velocity is obtained, while the decay rate provides information on the dissipative processes that are driven by the oscillation. To the damping contribute not only the various dissipative transport processes taking place in the bulk of the liquid itself, but also the irreversible thermal conduction via the walls and the dissipative effects within a viscous penetration depth near the wall of the tube. Usually the latter appear to dominate the damping [3-5]. Instead of studying free oscillations, the same information is usually obtained by using the oscillator as a resonator. A second-sound oscillation is then maintained by a periodic heat supply via a heater in the reservoir. Its temperature amplitude for a fixed heating power is registered as function of the driving frequency. As long as the damping rates are not too high, the location of the top of this resonance curve will of course correspond to the frequency of the free oscillations, the width of the curve yields again the damping. If these data are combined with the third quantity that can directly be obtained from the curve, namely the temperature amplitude at resonance for the given heating power, the specific heat can be deduced as was stated in the previous introduction.
L
1.3. The Helmholtz oscillator and the two-fluid description
Fig. 1 Schematic drawing of a Helmholtz oscillator. T h e volume V is connected with a bath by m e a n s of a tube of length L and cross section A.
The hydrodynamic properties of the superfluid helium mixtures are described successfully by a two-fluid model [6]. In this model the liquid is composed of a superfluid and a normal fluid
189
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
fraction, with mass densities p~ and p. and independent velocity fields v~ and v.. The 3He quasiparticles, with effective hydrodynamic mass my, are assumed to participate in the motion of the normal fluid so that
as
U102= C 21(0 + p_.&Spnax2x2)
first sound
(7)
u2o = Co/
second sound,
(8)
-- axX P.
V3 : Vn ,
P = Ps "~ P. ---~Ps "~ (Pn4 + Pn3)"
(1)
Acoustic phenomena, which correspond to propagating small disturbances of the global equilibrium state, are covered by the set of linearized two-fluid equations. On the non-dissipative level this set reads as 0_£p+ p~V- v~ + p,V. v, = 0
Ot
mass balance,
(2)
@__y_x+ oxV. v = O at mass balance 3He,
(3)
aps + psV. v. = 0 at
where the following abbreviations are used: ¢10 ~- k ' ~ - e )
2
0V s
*4-s-x
ps--~ + p n - ~ + V P=O momentum balance, aV s
at
(5)
+Vt~4 = 0
equation of motion for as.
(6)
The potential /~4 that drives the superfluid, equals the chemical potential per unit mass of 4He in the solution, as it is determined through the relations for local equilibrium by the local values of the three independent thermodynamic variables. For these the pressure P, the temperature T and the 3He-mass-concentration x will be chosen. Two different longitudinal plane wave modes satisfy the above set of equations, i.e. first and second sound. When the extremely small effects due to thermal expansion are neglected, their speeds of propagation u~0 and U2o can be written
l(ap)
7x
in which c34 --~ T(ds/OT)e,x is the specific heat per unit mass of mixture. In order to obtain ul0 and uz0 in the compact forms (7) and (8), use is made of the fact that it appears quantitatively 2 2 that Cl0 >> c20. First sound corresponds to ordinary sound in classical fluids; the two components move in phase with their velocity amplitudes v~ and v', related by the equation
(4)
OVn
Ps
(9) (as)
pslJs + P n V . ~
entropy balance,
I°s ~
pVs/(1 - axX).
(10)
Relation (10) implies that in mixtures the normal fluid moves considerably faster than the superfluid does (v'.> v~), especially at temperatures well below 1 K where p, ~ Pn3 = (my/ms)PX. In second sound the two components move in opposite directions. For the case of pure 4He the net mass flow is zero, the waves manifest themselves as temperature waves. In mixtures, however, a small mass flow will result in addition, almost proportional to the 3He concentration: PsV~ + PnV" "~
PaxX v'~.
1 -- OtxX
(11)
This is a clear consequence of the difference in mass and molar volume of the two isotopes, resulting in a value of ax considerably different from zero. Acoustic oscillators of the Helmholtz type have now been in use for classical fluids for more than a century [7]. Their first application to the case of superfluid helium was reported in 1960 [8]. In view of the variety of sounds that can be
190
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
excited in superfluid helium, not only first-sound and second-sound oscillators have since been investigated, but also fourth-sound and even third-sound oscillators, In the case of fourth sound the tube between the reservoir and the bath is very narrow or is replaced by a superleak, so that the normal component is practically immobile [9]; in the case of third sound, the inner walls of the tube and reservoir are covered by an unsaturated helium film [10]. The theory of Helmholtz oscillators filled with a superflmd H e - He mixture has been discussed extensively in ref. 5. Here it suffices to show how their behaviour is described in principle by the simplified set of two-fluid equations represented by eqs. (2)-(6). Integration of the linearized balance equations (2), (3) and (4) over the volume of the reservoir yields •
3
4
*
V~t = -A(psv' s + p.v'),
(12)
V c~px c~t -Apxv'n
(13)
V aps c~t
(14)
-Apsv',
where p, px and ps represent the mean densities in the reservoir and v's and v', the mean transport velocities away from the reservoir. These equations can now be combined with eq. (5) and (6) applied to the tube and with eq. (10) or (11), describing the supply through the tube for the case of first sound and second sound, respectively. Considering first the limiting case, in which the variations in the densities over the volume of the reservoir and in the velocities o's and o'. as well as in the gradients V/z4, VP, VT and Vx along the tube are fully neglected, one obtains immediately for the time variation of all relevant quantities, for instance for o's, the harmonic oscillator equation: +
A
v's = o .
(15)
In eq. (15) u 0 represents the speed of first or of second sound, as are given by eqs. (7) and (8), respectively. When the variations along the tube
and reservoir (assumed to be cylindrical with length LR) due to the finite acoustic wave-length are taken into account, the eigenfrequency aJr~0 is corrected up to first order in the small ratio A L / V into
2
tOH0 =
2 A[
AL( L~]] 1 - - ~ - 1 + L2 ] j .
U0 ~
(16)
In the above derivation all dissipative effects have been ignored. They will of course lead to damping of the free oscillations, and can be taken into account by allowing a~H0to become a complex quantity ~oH. For those contributions to the damping arising from the attenuation of the sound wave in the tube, this is done by replacing u 0 in eq. (16) by a complex quantity u. As has been demonstrated before [3-5], the damping in a second-sound oscillator is in practice often dominated by the viscous dissipation occurring within the viscous penetration depth ~n(to) near the wall of the tube of radius R:
-=
,
(17)
where ~1 is the shear viscosity• This dissipation leads to u2 = u : o 1 - ~ ( i + 1 )
(18)
and to the damping rate 3,, of the free oscillations 1 ps~n Y" = 2 t°n° p R "
(19)
Both quantities u 2 and 7, are given here up to first order in the ratio ~,/R, which is made small by the design of the oscillator• Contributions to the damping arising from the other dissipative processes that are driven by the oscillations will simply add to 3'~ to yield the total damping rate 7. Helmholtz oscillators can also be used as resonators. Second-sound resonances can be obtained by means of a periodic supply of heat
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
191
is found from the resonance curve, since as long as Ato ~ too toO
"Tre5~ I
Q--A---dw"
_rw "I/2V-2T~es< AtO>
1,4. Determination of the mechanical energy at resonance from the resonance curve The periodic heat supply W(t) of eq. (20) will eventually lead to a stationary situation in which the temperature of the reservoir T R will vary periodically and can be written as
f
%
co
Fig. 2. Resonance curve with width Aw at The resonance angular frequency is wo.
t 2 T R~ _--Tres/X/~.
W(t) to the reservoir. In practice one has W(t) = Wo(1 + coso~t).
(20)
Variation of to at constant amplitude W0, yields a resonance curve for the amplitude of the periodic temperature response in the reservoir, T~, like is shown in fig. 2. The peak in this curve, T~es, will be located at an angular frequency too equal to the real part of o~. From too, with the use of eq. (16), the speed of second sound in the tube follows as the real part of uz. The width Ato of the peak at a height Try(to) = T~es/X/2 equals twice the decay rate y of the free oscillations, while y in turn will also be equal to the attenuation rate of the second-sound wave in the tube, as long as damping contributions coming from outside the tube can be neglected. Possible differences between y and y~ can be found by confrontation of speed and attenuation in eq. (18), using the bulk speed for u20. Finally, the quality Q of the oscillator, defined as
total mechanical energy of the oscillation at resonance a = 2,tr dissipation of energy per period Eres
= too ( P ) r ~
(22)
TR(t ) = TO+ A TO+ Tl~ cos(tot - ~b).
In eq. (23) TO is the temperature of the bath, A TOthe constant temperature difference that will arise due to the average rate W0 at which thermal energy is transported through the various parallel thermal resistances between the reservoir and the bath, and T'r(to) and ~b(to) the amplitude and phase of the periodic temperature response. The mean rate at which entropy is supplied to the reservoir by the heater is (W/TR) while the mean flow of entropy that reaches the bath equals Wo/T o. The mean rate of entropy production ( S ) is therefore given by
( 3 ) - w0 To W o ( l + costot) + A TO
I
TI~ cos(tot- ~b)
-Z-0+Z
(24) irrespective of the dissipative mechanisms that are responsible. Up to first order in the small ratio's ATo/T o and T~/T o expression (24) simplifies to
(3)
(21)
(23)
w°ar°
- --T~
Wor
+ ~
cos~b ,
(25)
where the first term stems from the dc heat current through the tube. At resonance cos~b = 1
192
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
so that ( P ) .... the mean rate of dissipation of mechanical energy at resonance, can be obtained from the measured values of T'~s at given W0, according to
to be written as the volume integral of the potential energy density Em = /-l_p
_~_(- 1
x(~lb4/~X)P, ~ T c34) ~ T t2d V .
Tt
res
(P)res = T0(S)r,~ = 2T0 W°"
By using relations (21) and (22), the mechanical energy at resonance can subsequently be deduced from the measured width of the resonance curve, zato, yielding Ere~_ (P)res a _ too
(30)
(26)
T~esWo 2 T OAto "
(27)
Since the distribution of T' over the volume is known, the integration can be carried out straightforwardly. When the effects of the finite wavelength in the tube and reservoir are neglected by assuming a linear temperature profile along the tube and constant amplitude T~ in the reservoir, one obtains
1 c34 ( A L ) Em= ~ p -~- V 1+ -~ 1.5. The relation between the mechanical energy and the temperature amplitude at resonance The driven oscillations can be considered as standing waves of second sound in the tube and reservoir, having a wavelength which is large compared to the lengths L and L R. The two parts of these waves are matched to each other at the boundary between the tube and the reservoir by the condition for continuity of the temperature and the mass transports. The total mechanical energy of the oscillations equals the volume integral of the maximum kinetic-energy density 1 12 E m = f ( T1p ~ v st2 + 2PnVn )dV.
(28)
The local velocity amplitudes, related to each other by eq. (11), can of course also be expressed in the temperature amplitude T' of the second sound. From the two-fluid equations (2)2 2 (6) one obtains (again neglecting (C2o/Clo)X with respect to one)
p~v's = -pnv'. =-p,
1+
(~s/Pn)O/xX
1 - a~x
(1+ Ps axX)u2 --
C34
Pn
S4
(29) T
Substitution of eq. (29) into eq. (28) allows E m
× (1 -
X( O~t4/OX) p, T $24T
c34) T~ 2 .
(31)
The error, introduced by this simplification, will not exceed a few percent for the dimensions of the devices used in the experiments and for frequencies close to the resonance frequency*. Combining eqs. (27) and (31) with the experimentally determined values, one obtains the thermodynamic entity C34( 1
=
X(OId'4/tgg)e'T C34)
s24T
1 Wo . pV(l + ( A L / a V ) ) T'resAto
(32)
In the case of pure 4He, for which the concentration x is obviously zero, one thus determines the specific heat c 4 directly. For a mixture, however, the additional factor can be considerably different from one. A fair estimate can be obtained by applying the Landau-Pomeranchuk model [11]. Using this model the dilute solution can be described as an ideal mixture of a quasi*The main error stems from the cosine distribution of T~ in the reservoir. This introduces an additional factor (1 + ~ ( A L / 3 V ) ( L 2 / L 2 ) ) into relation (31), with fl between the values - 1 and +2, depending on where in the reservoir T~ is registered.
193
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
ideal Fermi gas, formed by the 3He quasiparticles, and pure superfluid 4He. The Gibbs function per unit mass of the solution is represented by
in the experiment, the value of (37) can be expected to be not much different from 2.5.
2. The experiment g = x ~3o + ~ 3 lnX
2.1. The experimental set-up
RT + (1 - x){ ~4o + ~--~-4In(1 - X)} .
(33)
Both /z30 and /x40 have to be evaluated at the pressure and temperature of the mixture. The molar concentration X is related to the mass concentration x as xM 4
4
(34)
S = x M 4 q- (1 - x ) M 3 ~ -3 x .
It now follows immediately that x ( Olx4~
:
\ aX /P,T
RT x M 4 1-x
dX d---x ~
RT x M3 ,
(35)
while s 4 attains the value R R s 4 = s40 - -;7-, In(1 - X) ~- x .
(36)
M3
1vl 4
The specific heat measurements in pure 4He have been carried out in the threefold Helmholtz resonator, used before by van der Boog et al. [5]. For the measurements in the 3 H e - a H e mixtures a similar resonator of identical dimensions was built, differing only from the first one by the use of Stycast 1266, instead of Epocast 202, as working material. A drawing of the resonator is presented in fig. 3. The volumes V1,2, 3 a r e cylindrical and have an inner diameter of 1.40cm and a length of 0.60cm. The tubes L1,2, are both 0.90cm long and have an innerdiameter of 0.30 cm. The figure also shows how the resonator is connected to the mixing chamber of a Leyden dilution refrigerator [13]. The connection, formed by a thick copper strip, coated on both sides with a silver layer, provides for good thermal contact. The part of the strip inside volume V2 of the resonator, is sintered with silver powder, in
The approximations made in the second part of both equations are justified in view of the low temperature range (0.3-0.7 K) and low concentrations ( X < 0.01) prevailing in the present experiment. One thus obtains 1
x(am/ax), s24T
(R)
c34~ 1 + c34/ x ~
.
(37)
It is pointed out that the results (35), (36) and (37) do not depend on the particular form of the expression assumed for/%0; deviations from the parabolic Landau-Pomeranchuk spectrum, for instance, that would account for the additional contribution to c34 reported by Greywall [12] and by van der Zeeuw et al. [1] will not affect them. If, for the purpose of obtaining a fair numerical estimate, we substitute for c~, in expression (37) the Landau-Pomeranchuk limit c34-----> LP ~xR/M3 it follows that for the conditions prevailing
i! /" O ~ 2cm Fig. 3. The resonator and its connection to the mixing chamber. HE: heat exchanger: $1: superleak: Mu: upper flange of the mixing chamber.
194
H . C . M , van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
order to increase the heat-exchanging surface area. In a symmetric, threefold Helmholtz oscillator, two types of Helmholtz resonances can be generated, as is discussed by van der Boog et al. [5]. One, the symmetrical mode, is characterized by temperature oscillations in V1 and V3 that are in phase and of equal amplitude and a temperature oscillation in V2 of opposite phase. For the case that the three volumes are of equal size, the amplitude in V 2 will be twice that in 1:1 and V3. The other mode, the anti-symmetrical one, is characterized by temperature oscillations in V1 and V3 that are of opposite phase and equal amplitude, while no oscillation of the temperature occurs in V2. In the present experiments only the antisymmetrical mode could be used, in view of the thermal anchoring of the device via V2. Its temperature distribution is sketched in fig. 4. The second-sound transmitters (SST) in both resonators are made out of resistance strip. The second-sound detectors (SSD) in the Epocast resonator are formed by a layer of aquadag painted on small strips of paper. In the Stycast resonator polished Matshushita resistors are used.
2.2. The measuring procedure The second-sound oscillations are generated by an ac.-voltage applied to one of the transmitters. Different from the situation in the experiments of van der Boog et al. [5], the accurate values of the supplied driving powers W have to
o!
.
O I
iI
I
;
I i I
i I
V1
-TR
ALl 2 ~
i
2V1.3(1 + ~ /
_i
I V2
be known. Both the voltage across and the current through the transmitter must therefore be measured. The current is determined from the voltage measured across a calibrated resistor connected in series with the transmitter. The induced second-sound oscillations will have twice the frequency of the driving voltages. They are registered by means of a detector in V1 or in V3, its resistance being modulated by the temperature oscillations. For this purpose a small dc measuring current is applied to the detector and the resulting ac-voltage is amplified and detected with a lock-in amplifier. From the so determined amplitude R' of the resistance variations, the temperature amplitude T' is obtained by means of a polynomial fit through the calibration data of the detector resistance as function of temperature. The values of the driving power were chosen as small as was permitted by the detection accuracy, in order to keep the disturbance caused by the unavoidable dc component to the heat flow as small as possible. The resonance curves for the temperature response at fixed driving amplitudes, of which the heights T~eS and widths Ato need to be determined, are obtained from twelve to sixteen measuring points taken around the resonance frequency of the anti-symmetric mode, within the range where (1/2)T',e s < T ' < T',e,. The values of T'~ and Ato are substituted into the right-hand side of eq. (32), in which an extra factor of two must be added to the denominator in view of the fact that eq. (32) was derived for a single Helmholtz oscillator. Taking into account the 1% thermal contraction in the linear dimensions of the device and applying a small correction for the volumes of the transmitters and detector, the effective volume V' that enters into the denominator of eq. (32) becomes for the threefold oscillator
V3
Fig. 4. The temperature distribution of the antisymmetric mode in the resonator.
--=V' = (1.79 -+ 0.05) cm 3.
2.3. Experimental determination of the specific heat of pure 4He The measurements on superfluid 4He have
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator I
I
I
'
I 1.4-
i
I
5C
I
195
i
I
'
I 1.8
,
600
T = 1.533K V O = 8 6 . 4 4 Hz 1.Ol HZ 4C
575
x."
::t
3Q ~/
~_ 4 0 #~-
30
20
I 86.00
I 86.50
I 87.00
I
I
I
~
& "13
o 550T = 1.915K Vo= 80.185 Hz AV= 0.41 Hz
3
525
4/
[ 80.00
J 80.25
I 80.50
<
V(Hz) Fig. 5. Examples of resonance curves in pure 4He. T h e circles represent the experimental points, renormalized for Wo = 0 . 1 mW.
been carried out in the teinperature range between 1.4K and 2.0K at saturated vapour pressure. In this temperature range the dilution refrigerator is of course not in operation. The resonator is cooled by the surrounding 4He bath by means of 4He-gas admitted into the vacuum can. Thermometers and second-sound detectors are calibrated against the vapour pressure of the 4He bath. Some examples of the resonance curves obtained for different bath temperatures are shown in fig. 5. Mean driving powers W0 between 10-4W and 2 x 10-4W were employed, resulting in temperature amplitudes T'res varying from
500
I i 1.6 T(K)
2.0
Fig. 6. The angular resonance frequency as function of temperature. The lower curve is drawn through the measured resonance frequencies. The upper curve represents the calculated values for tOHo=C2o(2A/V'L1) ~/2 as function of temperature. 1.5
'
[
'
I
i
I
'
I
,
1.0 O
3 <~ O 0.5
5 0 / x K at T = 1.45 K to 8 5 / z K at T = 1.90K.
The observed resonance frequencies are plotted as a function of temperature in fig. 6. The variation is mainly determined by the temperature dependence of the second-sound velocity. This is demonstrated by the curve for toll0 = C2o(2A/V'L1) 1/2, calculated according to eq. (16) with the thermodynamic quantities in C2o of eq. (9) taken from the literature [14]. In fig. 7 the observed damping, represented by
O
I 1.4
,
1.6 T(K)
1.8
2.0
Fig. 7. The damping factor, (Atoltoo) as function of temperature. The lower curve represents the calculated values for (p,/p)(8~/R); ( © ) and (E]) represent data obtained from the temperature in volume V 1 and I,'3, respectively.
196
H . C . M . van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
20
1
'
I
i
I
form
,
V4Wo 15-
//
'.~+"TO
E I0
3 ,¢ U
/
c 4 -
I 1.4
,
I i 1.6 T(K)
I 1.8
(38)
with V4 the molar volume, are shown in fig. 8. Error bars, corresponding to an uncertainty of about 5% seem to be appropriate in view of the uncertainties in V' and T',e~ and of the assumption that T're~ represents the average temperature amplitude in V1.3.. Furthermore, the uncertainty in the absolute temperature is significant because of the steep variation of C a with temperature. In the figure the results are compared with the specific heat data, tabulated by Maynard [14]. It demonstrates the applicability of the resonance method as well as its limitations set by the measuring accuracy.
/ 0
V,T;esAo.)
I
2.0
Fig. 8. The results for the specific heat of pure 4He; (O) and (F]) represent data obtained from the temperature response in volume VI and V3, respectively. The drawn line is taken from tabulated values by Maynard [14].
the quantity ('ttO/OOHO)=(2T/OOHO), is plotted against temperature. The lower curve in the figure corresponds to calculated values* of (Ps/ p)(8~/R), i.e. the expected damping in case the viscous dissipation in the penetration depth near the wall of the tube dominates the damping (see eqs. (18) and (19)). Comparison with the data shows that, particularly towards lower temperatures, other contributions to the damping become relatively less and less important. Although the observed damping thus confirms the expected behaviour it is repeated here that such understanding is not really necessary to obtain the specific heat c a from the observed resonance curves. The results for the molar specific heat, C4, obtained from eq. (32) in the
* For the calculations values of -t/as reported in refs. 15 and 16 were used; values for ps/p were taken from ret. 14.
2.4. The experimental results for the dilute 3He-4He mixtures Due to a number of circumstances, only a few resonance curves for mixtures can be presented here. Nevertheless, the few results obtained, demonstrate the feasibility of the method. The analysis offers a number of suggestions by which the method could be further improved. For the mixture of molar concentration X = 6.0 × 10 -3 the measurements are restricted to only one curve, taken at T = 683 mK. Seven curves, spread evenly over the temperature interval 0.3-0.7 K were taken with a mixture of which the concentration was at first unknown, but of which the results could be proved to be compatible with the specific heat data reported in literature for a mixture with X = 8.5 x 10 -3, as will be shown in this section. All measurements were done at P = 0. In fig. 9 some examples of the measured resonance curves for the mixture with X = 8.5 × 10 - 3 are shown. Driving powers W0, ranging from 5 / x W at the lowest temperature T = 344 mK to 12 ~ W at T = 683 mK, were required. They resuited in temperature amplitudes T',os between 7 / z K and 15/.~K. These values of W0 also resulted in a considerable dc temperature difference
H . C . M . van der Z e e u w et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
12
~ / ~{
10
=
~ "1~
X=8.5x 10-3 T = 0.560K Vo =133.48 HZZ~v 6.25Hz
I 13.5
I 140
-
/
~
-
8
,~ -~
-r" 125
I 130 I
I
I
×=8.5×10 -3 T = 0.374- K Vo =109.45 Hz AV= 5.80 HZ
1
the second series of measurements extending over the entire temperature region. When comparing the latter results with the earlier one a large discrepancy appeared, unfortunately, as if, due to some unexplained effect, the concentration of the mixture had changed from X = 6.0 × 10 -3 to X = 8.5 × 10 -3 during the recycling process. Secondly, the rather large dc component W0 to the heat flow introduced by the second-sound transmitter, not only leads to the steady temperature difference A T, but will also give rise to a concentration difference A X by the well-known heat-flush effect. According to the two-fluid description 3He will move with the normal fluid away from the heater until due to diffusion in the growing concentration gradient a steady state is reached. If one defines for this state an effective thermal conductivity Kelf by
Lwo
(39)
reef --- A A T 6 -(
I 105
I 110 V (Hz)
f 115
120
Fig. 9. Examples of resonance curves in the dilute 3He-4He mixture with molar concentration X = 8.5 x 10 -3. The circles represent the experimental points.
between reservoirs V1 and V2,3, with a typical value A T - - 30 mK. The resonator temperatures, quoted in this section, correspond to averages over the three volumes. All thermometers were calibrated against the vapour pressure of 3He after the resonator was filled with mixture. Before deducing the specific heat values c34 from the measured curves a few remarks should be made to explain the scarcity of results. Firstly, the original mixture was made at room temperature with a molar concentration X = 6.0 x 10 -3. After the first resonance curve at T = 683 m K was measured, the refilling tube of the 3He cryostat became clogged up and the whole device had to be warmed up to room temperature. The mixture was regained and recondensed again into the resonator to carry out
197
the values one obtains for K~effall in the range of a few tenths of a W / K m . Such values are very well compatible with the results for X = 1.3 x 10 -2 and X = 5 × 10 -2 obtained by Abel and Wheatley [17] in a thermal conduction experiment with a tube of 2.5 mm diameter. In this description one can obtain the value of A X that corresponds to A T from the condition V/x4 = 0 (imposed by the equation of motion for the superfluid, eq (6)) together with the assumption that in the steady state VP is negligible. From the expression for /~4 introduced in eq. (33) it then follows that VX _ 1- X
M4s4° RT
VT
(40) "
If we insert numerical values, for instance for the measurements at T = 462 m K and X = 8.5 × 10 -3 for which A T = 30 mK, one obtains A X = 6.0 × 10 -4. The heat-flush effect is thus considerable and it therefore puts severe restrictions on the accuracy by which the specific-heat values can be derived from the measuring runs. We therefore decided not to extend the measurements to more concentrations nor to lower temperatures, but to
198
H . C . M . van der Z e e u w et al. / Dilute 3He-aHe mixtures in a Helmholtz resonator
Table I Experimental results Molar concentration X
T (mK)
Wo (~W)
T~es (gK)
~o (rad/s)
A~ (rad/s)
AT (mK)
6.0 x 10 3
683
11.6
15.0
872
41.8
-70
X 10 -3
344 374 418 462 512 560 622
5.3 8.0 7.7 8.2 8.2 9.9 11.7
7.3 11.7 11.3 11.4 10.8 12.6 13.2
664 688 729 766 806 839 873
39.0 36.4 36.4 37.7 38.6 39.3 39.9
21 32 28 30 30 38 45
8.5
content ourselves with the analysis of the present runs and with some suggestions how the heatflush effect can be reduced by improving the construction and the thermal anchoring of the resonator. In table I all the experimental results obtained for the two mixtures are compiled. In order to obtain the specific heat from these data we first rewrite eq. (32) in molar form. In the approximation of an ideal mixture, i.e. using eq. (33), eq. (32) becomes RC34
C34(1 ~- ~XXxI- [M4s40 --.-RR-In(I -
1.5
X)]2)/V34(X)
Wo -
V' T',es Ato
.
(41)
//
qi 0
E 1.0
Values for the molar volume V34(X) and s40 can be obtained from the literature [18], so that from the datapoint for X = 6 . 0 x 10-, the corresponding value of the molar specific heat can be solved directly. The result is plotted in fig. 10. In this figure the curve calculated for this mixture from the relation (see refs. 1 and 12) 6 3 4 = C 4 -[- C LP ht-
the small additional term ( 3 / 2 ) X R S ( T - To) , as LP is demonstrated by the curve for C34, also drawn in the figure. For the second mixture the unknown concentration X and C34(X, T) cannot both be deduced
3XRS(T- To)
(42)
_
/
Z
X = 6.0,~ 10-a (D 0.5
04/
I
0.2
I
0.4
0.6
0.8
T (K)
(in which S = 0 for T < ~ T o = 0 . 2 5 K and S = 0.2 K - ' for T > To) has been drawn for comparison. The agreement is quite good. The uncertainty in the measured value, however, is too large to confirm unequivocally the existence of
Fig. 10. The molar specific heat C34, derived from the measurements through eq. (41) for X = 6.0 x 10 -3 (O) and X = 8.5 x 10 -3 (O). The drawn curves through the data obey eq. LP (42); the dashed curves represent C34(-=C4+ cLP). The values for C4 are obtained from ref. 14.
H.C.M. van der Zeeuw et al. / Dilute 3He-4He mixtures in a Helmholtz resonator
from the data. One can, however, check the compatibility of the results with the description given by eq. (42). This is done by solving X numerically for each individual data point from eq. (41), by substituting eq. (42) for (?34 and by checking subsequently whether the spread in the values of X is acceptable. The m e a n value of X one finds in this way is X = 8.5 x 10 -3. Fig. 10 shows the experimental results for C34 as obtained by using this value for X, as well as the corresponding curve calculated from eq. (42). In view of the measuring accuracy the agreement is surprisingly good. That this agreement seems better than with the related curve for C3La dotted in the figure has little meaning, since a comparison of the data with the LP-description would require a redetermination of the value of X.
2.5.
Concluding remarks
The results obtained for pure superfluid 4He as well as for the dilute mixtures have shown the feasibility of K r a m e r s ' original suggestion that in a second-sound Helmholtz-resonator experiment the specific heat of these q u a n t u m liquids can be determined in situ from the resonant behaviour. The applicability of the method could perhaps be extended beyond this goal if a n u m b e r of technical i m p r o v e m e n t s were made, in particular those concerning the reduction of the heat-flush effect. Since the value of T'r~s is determined by W0 through the quality factor of the resonances, W0 can only be reduced by improving this quality, for instance by suppressing the viscous damping through the use of wider tubes. In addition, this would increase the effective heat conductance at the same time. A further reduction of the heat flush can perhaps be realized by thermally an-
199
choring both outer reservoirs 1t"1, and V3 instead of V2, and then supplying a ' d u m m y ' dc heat input W0 also to V3.
References [1] H.C.M. van der Zeeuw, L.P.J. Husson, M. Durieux and R. de Bruyn Ouboter, Proc. LT-17 Conf., U. Eckern, A. Schmid, W. Weber and H. Wfihl, eds. (Elsevier Science Publ., Amsterdam, 1984) p. 1245. H.C.M. van der Zeeuw, Thesis, Leiden (1985). [2] H.C. Kramers, private communication. [3] R.R. IJsselstein, M.P. de Goeje and H.C. Kramers, Physica 96B (1979) 312. [4] M.P. de Ooeje and H. van Beelen, Physica 133B (1985) 109. [5] A.G.M. van der Boog, E E L Husson, Y. Disatnik and H.C. Kramers, Physica 104B (1981) 285. [6] I.M. Khalatnikov, An introduction to the Theory of Superfluidity (W.A. Benjamin, New York, 1965). [7] H.L.F. yon Helmholtz, Crelle Bd. 57 (1860); On the Sensations of Tone (2nd ed. reprinted (1954) by Dover, Publ., New York.) [8] H.C. Kramers, T.M. Wiarda and A. Broese van Groenou, Proc. Int. Conf. on Low Temp. Phys. LT-7, G.M. Graham and A.C. Hallett, eds. (North-Holland, Amsterdam, 1961) p. 562. [9] I. Rudnick, Proc. Int. School of Physics "Enrico Fermi" LXIII (1976) p. 112. [10] G. Bannink, M.G.M. Brocken, I. van Andel and H. van Beelen, Physica 124B (1984) 1. [11] L.D. Landau and I.J. Pomeranchuk, Dokl. Akad. Nauk. USSR 59 (1948) 669. [12] D.S. Greywall, Phys. Rev. Lett. 41 (1978) 177. [13] K.W. Taconis, N.H. Pennings, E Das and R. de Bruyn Ouboter, Physica 56 (1971) 168. [14] J. Maynard, Phys. Rev. B14 (1976) 3868. [15] A.D.B. Woods and A.C. Hollis-Hallett, Can. J. Phys. 41 (1963) 576. [16] W.J. Heikkila and A.C. Hollis-Hallett, J. Phys. 33 (1955) 420. [17] W.R. Abel and J.C. Wheatley, Phys. Rev. Lett. 21 (1968) 1231. [18] R. Radebaugh, Nat. Bur. Stand. (US) Techn. Note no. 362 (1967).