Russell J. Donnelly
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Table
1 : : 2 : : 3
: : 4 : : 5
Saturated Vapor Pressure
The
Calculated Thermodynamic Properties of Superfluid Helium-4
James S. Brooks and Russell J. Donnelly
3.1.f. The Effective Sharp Spectrum for Thermodynamics
If
we use the neutron model dispersion curves described in the previous section
to compute, say, the entropy from table I, we discover that the agreement
with the calorimetrically determined entropy is quite good at low temperatures
for all pressures. At higher temperatures, however, the calculated entropy
lies markedly higher. For example, at the vapor pressure, the calculated
entropy exceeds the experimental by 17% at ~ 2.1 K; at 15 atmospheres
the calculated entropy is 11% high at ~ 1.9 K, and at 20 atmospheres it
is 23% high at ~ 1.8 K. This same trend may be seen in figure 17 of Dietrich
et al. [3] who, however, made a correction to the formula for S by an
additional integration over the linewidth. The departures between calculated
and measured entropy occur at about the same value of (T
-T) at all pressures,
and correspond to the temperatures at which the linewidths of the scattered
neutron distributions start to grow rapidly. The resulting problem in
interpretation has been referred to in section 2 above.
Following Brooks [18] we have investigated the idea of constructing an
effective dispersion spectrum, which represents the energies which yield
thermodynamic results in accord with experiment. This is done by noting
that the only constraint among eqs (16)-(19) which can be varied at all
readily within experimental uncertainty is the exact value of in (18) [and also of ? since it is computed from
eq (9)]. We therefore allow to float, retaining all other constraints as before,
and produce a dispersion curve identical with the neutron data except
for the precise value of , computing from eq (9). This spectrum assumes there is some
effective sharp frequency (that is, 
)
for each value of q; since there
is but one adjustment, the resulting effective spectrum is unique. An
extensive numerical investigation has demonstrated the great utility of
such a spectrum.
We find:
(1) The effective spectrum is capable of yielding accurate thermodynamic
results over all pressures and temperatures up to ~ (T-0.1
K). All quantities calculated from table I and this spectrum appear
to be in reasonable agreement with experiment.
(2) The effective spectrum is consistent with the observed functional
form of the spectrum determined neutron scattering. It coincides with
the neutron spectrum within experimental error at low temperatures, and
at high temperatures the difference between the observed and effective
values of 
is such that (effective
-neutron)<
.
(3) Because the effective spectrum is unique, the exact value of
can be chosen by reference to more than one: thermodynamic quantity. We
have used entropy, expansion coefficient, and normal fluid density.
(4) The points listed above allow one to guess that the effective spectrum
is probably a reasonable representation of the energies of elementary
excitations of helium II.3.2. The Thermodynamic Data
3.2.
The Thermodynamic Data
In order to use the formulas
provided by the Landau theory, we need an equation of state to relate
the pressure, volume (or density),.and temperature, and from which we
may obtain the velocity of first sound. Likewise, to test the results
of our computations, we need calorimetric data, primarily entropy and
specific heat. We now discuss these experimental data.
3.2.a. The Equation of State
Many
of the parameters of the excitation spectrum depend on the density, and
terms in the density, or molar volume, appear in expressions of the Landau
theory (table I). It is therefore imperative to have a suitable equation
of state or "PVT" relation. Our relationship is based upon the
work of Abraham, Eckstein, Ketterson, Kuchnir, and Roach [31], who showed
that as T 
0, the equation of state can be written
P=A0(p-p0)
+ A1(p-p0)2 + A2(p-p0)3,
where
p0
is the density at P =
0, T = 0; Ao
= 560 atm g-1cm3, Al
= 1.097 X 104 atm g-2cm6, and A2= 7.33 x 104atm
g-3cm9. The "ground state"
molar volume, Vo(P) 
V(0,
P) is calculated from eq. (21) by a root searching technique.
Before the methods which led to the expressions in table I were developed,
an empirical equation of state generalizing (21) was developed by Brooks
[18]. This empirical equation of state was based on (21) and the data
of Boghosian and Meyer [32], Elwell and Meyer [33], and Kerr and Taylor
[34]. It was sufficiently accurate to give a good account of the density
and isothermal compressibility, but was not sufficiently accurate to calculate
the expansion coefficient.
Since this work was begun, a thesis has appeared by Craig Van Degrift
[20] which employs the dielectric method for determining the density and
expansion coefficient at the vapor pressure. Dr. Van Degrift has kindly
supplied to us a table of data corrected to zero pressure. We have abstracted
some of his data in table II. His complete results are in the process
of preparation for publication. We should like to encourage the acquisition
of data of comparable accuracy at a series of higher pressures.
Table
II. Density differences and expansion coefficients corrects to P=0 as
determined by Van Degrift [20].po =0.145119 g/cm3
T |
(p-p0)/p0X106 |
 p(K-1)X106 |
| 0.30 | -2.255 | 29.49 |
| 0.35 | -4.122 | 45.96 |
| 0.40 | -6.931 | 67.28 |
| 0.45 | -10.94 | 93.93 |
| 0.50 | -16.42 | 126.3 |
| 0.55 | -23.67 | 164.5 |
| 0.60 | -32.95 | 207.7 |
| 0.65 | -44.49 | 254.2 |
| 0.70 | -58.37 | 300.5 |
| 0.75 | -74.46 | 342.1 |
| 0.80 | -92.40 | 373.3 |
| 0.85 | -111.5 | 387.9 |
| 0.90 | -130.8 | 380.2 |
| 0.95 | -149.1 | 345.4 |
| 1.00 | -164.8 | 279.8 |
| 1.05 | -176.5 | 181.0 |
| 1.10 | -182.4 | 47.48 |
| 1.15 | -180.6 | -122.3 |
| 1.20 | -169.5 | -330.4 |
| 1.25 | -146.9 | -580.2 |
| 1.30 | -110.7 | -876.4 |
| 1.35 | -58.43 | -1223.0 |
| 1.40 | 12.50 | -1622.0 |
| 1.45 | 104.5 | -2062.0 |
| 1.50 | 218.9 | -2511.0 |
| 1.55 | 356.7 | -3045.0 |
| 1.60 | 524.0 | -3652.0 |
| 1.65 | 722.8 | -4304.0 |
| 1.70 | 955.7 | -5015.0 |
| 1.75 | 1226.0 | -5806.0 |
| 1.80 | 1539.0 | -6699.0 |
| 1.85 | 1899.0 | -7724.0 |
| 1.90 | 2315.0 | -8922.0 |
| 1.95 | 2797.0 | -10360.0 |
| 2.00 | 3360.0 | -12140.0 |
| 2.05 | 4025.0 | -14480.0 |
| 2.10 | 4833.0 | -17990.0 |
| 2.15 | 5896.0 | -25880.0 |
3.2.b. The Calorimetric Data
The most fundamental calorimetric property directly accessible experimentally is the entropy S, which can be measured in an unusual way by employing the thermomechanical effect. The entropy is calculated from the relation.
where P and T are the differences in pressure and temperature between two chambers of helium II connected by a superleak. This relation, which is a direct result of the two fluid nature of superfluid helium, is discussed in the standard references [11, 12, 25, 26].
The specific heat C may be obtained by conventional calorimetric methods,
and the entropy may be computed from the results by the relation
The
entropy is available from the thermomechanical effect data of Van den
Meijdenberg, Taconis, and De Bruyn Ouboter [35] in the temperature range
and pressure
range 
. The
specific heat capacity measurements of Wiebes [36] cover the same range
of pressure, but the temperature range 
.
We have used these two sets of data in our analysis, since they cover
nearly the entire superfluid phase, and are in reasonable mutual agreement.
The specific heat has also been measured by Phillips, Waterfield, and
Hoffer [28]. Through the kindness of Professor Phillips and Dr. Hoffer
we have had access to some of their original calorimetric measurements,
which are in satisfactory agreement with those of Wiebes, (see, for example,
Brooks and Donnelly [17], figure 2). The publication of the final data
from these experiments is still awaited.
4. Computational Methods and Comparison of Computed Values With Experiment
In
this section we discuss the way in which each of the tabulated properties
given in Appendix A is obtained, and describe to what degree of accuracy
the computed values agree with the corresponding experimental data. Direct
references are made to the tables.
4.1. Generation of the Effective Spectrum
We
discussed in section 3.l.f above, the concept of an effective sharp spectrum
for thermodynamics. It differs from the spectrum of section 3.1.e. only
in allowing the value of 
(po)=to
float, its exact value being chosen by comparison of the calculated quantities
with experiment. In the original tables computed by Brooks [18], the procedure
was to adjust
and thus the effective mass through eq (9), so that the calculated entropy
agrees with the experimental entropy at all temperatures and pressures
at which the roton part of the spectrum contributes significantly. This
method had the disadvantage that even with quite accurate values of S,
the expansion coefficient calculated from (
S/ P)Tcould
be quite poor. Dr. Jay Maynard of UCLA drew our attention to the fact
that the normal fluid density (cf. section 4.4.b.) is weighted heavily
toward higher momenta, and hence is also a useful measure of the effective
roton gap. We decided, therefore, that for the generation of the effective
spectrum for this work, we would endeavor to employ a method which would
incorporate all available thermodynamic evidence: from entropy, expansion
coefficient, and normal fluid density.
James Gibbons undertook the job of computing the effective spectrum. This
proved to be an arduous task because of the great sensitivity of the thermodynamics
to minor changes of the spectrum in the vicinity of the roton minimum.
The first step was a weighted fit to experimental values of S and , making use of the Maxwell relations (V
/ T)p=-(S
/ T)T.
This produced a set of polynomials in the pressure at 0.1 K temperature
intervals. Since the tables are tabulated in 0.05 K intervals, this data
was interpolated by a second degree polynomial in temperature fitted to
three local points to find the best fit to data at 0.05 K intervals. A
new set of polynomials in pressure were then generated from T= 1 K to
T= 2.2 K in 0.05 K intervals. Special care had to be taken near the lambda
line because of the existence of large high-order derivatives. An iterative
root-searching method was then used to find what one could call (S,)from
eq (2), in all cases calculating 
from
eq (9).
The second step was to interpolate the experimental data on pn/p
, which also exists chiefly on 0.1 K increments, by a procedure analogous
to that used for S and .
The end result of a similar root search was a set of values named
, pn/p at 0.05 K intervals.
The third step was to adjust the values of
at zero pressure to the compromise [(pn/p)
+ (S,)]/2
retaining the curvature of the polynomials determined in the first step
at higher pressures. The result of this step might be called (S,,pn/p
).
The fourth and final step was a smoothing operation on the energy data
of step three by means of a power series in Nr
the roton number density. This step was essential to encourage the monotonic
behavior of the values of as T is reduced below 1 K. Once the smoothed table
of was available, 
was calculated
by eq (9). We have named these final values of the "thermal" roton gap t, and the corresponding thermal effective masses t. They
are listed in tables 22 and 23. Having obtained t
and t, the effective
spectrum (20) is computed by the methods described in section 3.1.e. In
eqs (16) and (19), the isothermal velocity of sound was used, eq (24)
below. The errors resulting from this approximation are negligible. Attempts
to use eq (25) for ul led to severe problems in numerical instability.
It is interesting to note that the differences between the neutron roton
energy gap and the thermal roton gap t
are not random but systematic. We find that, in general, t
lies above
at higher temperatures and pressures. The two agree within experimental
uncertainties low temperatures. The differences (t-)<<
except at temperatures
near the lambda line.
may be calculated from the expression of Roberts and Donnelly[37]:
/k
= 0.65 X10-33Nr/(1/2p2/9T1/6)
K.
At
the vapor pressure, the neutron determinations of Yarnell et al. [9] give
/k
= 8.65 
0.04 K at 1.1 K, while Cowley and Woods [2] obtained /k
= 8.67 0.04
K at the same temperature, and Dietrich et al. [3] find /k = 8.54 K at T = 1.26 K, P = 1 atm, which extrapolated
zero pressure gives /k
= 8.64 K. There is a completely independent way of finding .
It is known that Raman scattering from liquid helium at low temperatures
gives a peak at
16.97 0.03
K. This peak has been identified by Greytak and his collaborators as a
loosely bound state of two rotons [38]. An exact quantum mechanical solution
to one model for this state by Roberts and Pardee [39] gives the binding
energy the rotons as 0.290 K. Thus the roton energy should be (16.97
0.290)/2 = 8.63
0.015 K, in good agreement with the neutron measurements. We find t
= 8.622 K at 1.1 K, P=0, in satisfactory
agreement with all methods.
4.2. The Equation of State
The
integral expression for V in table I is, in contrast to our earlier empirical
equation of state [18, 19], completely consistent with the expressions
for P
and kT. By making use of the empirical equation of
state [19], the equation of state for our tables 1 and 2 was obtained
in a single iteration. One can also start from absolute zero data for
V and kT, and iterate to find substantially the same results. As
we mentioned in section 3.2.a above, Vo (P) comes from eq
(21).
The low temperature behavior of the molar volume is shown in figure 8,
plotted in the form VE (T, P) = V(T, P) - V(0, P)
vs. temperature for 2.5 atmosphere increments in pressure. A nonlinear
scale has been deliberately chosen to emphasize the characteristic maximum
in the molar volume near 1 K. We show in figure 9 the entire temperature
and pressure range of V, plotted with the data of Boghosian and Meyer
[32], and Elwell and Meyer [33]. The deviations of our equation of state
from the experimental data used in our analysis varies across the P-T
plane. If we define a fractional deviation for the molar volume V=
(Vcalculated - Vmeasured)/Vmeasured, we can get
a rough idea of the variations by averaging V over
temperature at each pressure and denoting the result by 
expressed
in percent. The results are given in tabular form below, separated to
display positive and negative deviations at six pressures, and we see
that the temperature-averaged deviations are at most + 0.42% - 0% from
the data.
The density and expansion coefficient have been investigated recently,
but only at the SVP. The reference is J. J. Niemela and R. J. Donnelly,
J. Low Temperature Physics 98 1-16, (1995).
P(atm) |
0 |
5 |
10 |
15 |
20 |
25 |
 |
0.01 | 0.17 | 0.19 | 0.29 | 0.39 | 0.42 |
 |
0 | 0 | 0 | 0 | 0 | 0 |
4.2.a. The Velocity of Sound
There
are several other quantities which may be calculated from the equation
of state, such as the, isothermal velocity of sound:
where
kT
is defined in (26) below. A first order correction for thermal expansion
using computed values of 
(discussed
in section 4.3.e.) gives
The actual expression used for the calculations reported here carries a higher order correction for thermal expansion [40, 41]
where
uII
is defined in section 4.4.c. below. The corrected velocity of first sound
ul is given in table 3 and illustrated in figure 10. At T= 0 K, the deviations
from the data of Abraham et al. [31] are less than 0.09%. At higher temperatures,
(> 1 K), the deviations
ul
=(ul ca1c - ul meas)/ul meas meas from some preliminary data of Maynard
and Rudnick [21] are as follows:
P(atm) |
0 |
5 |
10 |
15 |
20 |
25 |
|
0.8 | 0.3 | 0.5 | 0.8 | 1.0 | 0.5 |
|
0 | 0 | 0 | 0 | 0 | 0 |
Below 1.6 K, the deviations are much less than 1%. Only the highest few temperatures depart significantly from the data.
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