Russell J. Donnelly
Saturated Vapor Pressure
Calculated Thermodynamic Properties of Superfluid Helium-4
4.3.c. The Gibbs Free Energy
The Gibbs free energy has a ground state Part o (P) and an excitation part E(P, T) given by the double integral in table I. o(P) can be determined by integrating the expression d = VdP.. The results determine o at T =0 K to within an additive constant Lo as described in section 4.3.b. above.
13. The Helmholtz free energy of the excitations as a function of pressure
and temperature. The Gibbs free energy has substantially the same appearance.
The Gibbs free energy is not directly accessible experimentally. The data of table V come from integration of the equation of state of Abraham et al.  and hence should be reliable. The excitation part of the Gibbs free energy is tabulated in table 7: the appearance of this energy surface is similar to the Helmholtz energy in figure 13. We have checked the integration by comparing to ( / T )p to S. The differences are less than 0.3% over most of the T-P plane and never exceed 1%.
4.3.d. The Enthalpy
enthalpy W has a ground state part Wo (P) and an excitation
part WE (P,
T) as given by the double integral in table I. The ground state
part Wo (P) = o(P)
is tabulated in table V in section 4.3.c. above. The enthalpy of the excitations
is tabulated in table 8 and illustrated in figure 14.
4.3.e. The Specific Heat
specific heat at constant pressure,
obtained by the integral expression in table I. Below 1 K this worked
satisfactorily; above 1 K we experienced difficulty in getting a smooth
table. We therefore used (29) directly, selecting five local entropy values,
fitting a quadratic function by least squares, and finding
from that function. The resulting values are listed in table 10 and plotted
in figure 15.
The ratio of specific heats is compiled from
The term in is subject to an accumulation of errors and is perhaps as much as 50% in error in some regions. Values of ? are tabulated in table 12.
Figure 14. The enthalpy of the excitations as a function and pressure and temperature, as calculated from the expression in table I.
The specific heat at constant volume CV = CP / is listed in table 11. Although is quite uncertain, the correction is generally small. Comparing with the data of Wiebes , we find the deviations are as follows (T 1.6 K):
Table I shows an integral expression for CV . We have used the correction of eq (30) rather than our theoretical expression because the latter involves a derivative at constant volume.
Figure 15. The specific heat at constant pressure as a function of pressure and temperature, calculated from the expression in table I. Data, Wiebes, .
The density of thermal excitations,
calculated by numerical integration, has been separated into a phonon
part NP and a roton part Nr by defining (quite arbitrarily) excitations
with wave number less than 1.1 Å-1 phonons. For phonons, we have
for the number density [from eq (1)]
and for rotons
where qmax=1.1 Å-1 qf =3.0 Å-1. Tabulations of Np and Nr appear in tables 13 and 14, respectively. No comparison of the results can be made since the number densities are not directly accessible experimentally.
4.4.b. The Normal Fluid and Superfluid Densities
The normal fluid density may be computed from Landau's expression
ps = p - pn,
one may also obtain the ratios pn/p and
ps/p. The quantities pn/p, ps,
and ps/p appear in tables 16, 17, and 18
respectively, and pn/p is
plotted in figure 16.
The oscillating disk data of Romer and Duffy , principally above 1.6 K, averages about 4-7% low compared to our calculations.
4.4.c. The Velocity of Second Sound
The velocity of second sound, neglecting thermal expansion, is
uII was used to correct
the velocity of first sound in section 4.2.a above. Allowing for thermal
expansion the velocity of second sound is given by [40, 41].
Figure 16. The normal fluid fraction as a function of temperature for different pressures calculated from eq (33). The data points are from Maynard .
We have used equation (34) to compute u2, with results given in table 19 and figure 17. The data shown in figure 17 is that of Heiserman et al. . Comparison of our results with the data of  is as follows:
Below 0.8 K, the calculated values show marked effects of phonon dispersion, as discussed by Brooks and Donnelly .
Figure 17. The velocity of second sound as a function of temperature, at different pressures. Solid lines, eq (34); data points, from Heiserman et al. . Some of the unevenness of the calculated curves may be the result of numerical problems; note how many derived quantities appear in eq (34).
4.4.d. The Velocity of Fourth Sound
The velocity of fourth sound is, to good approximation [40, 41]
We have used equation (35) to compute u4, with results given in table 20 and figure 18. The deviations, compared with the data of Heiserman et al.  are:
have provided tables of the thermodynamic and superfluid properties of
helium II computed, when possible, by theoretical methods. Due to inadequacies
in the present theory, we were not able to achieve absolute agreement
with the experimental data in all cases. The accuracy of our computed
values is, however, generally very good, and we hope that the tables will
provide a ready reference for theoretical and experimental research in
helium II .
Figure 18. The velocity of fourth sound as a function of pressure and temperature calculated from eq (35). The data points are from Heiserman et al. .
This research was supported by the Air Force Office of Scientific Research under grant AFOSR-76-2880 and by the National Science Foundation under grant DMR-72-3221/A01. This work was begun when the authors were guests at the Niels Bohr Institute, Copenhagen, and we thank Professor Aage Bohr for the hospitality of the Institute. We would also like to thank Mr. James Gibbons for much help in the preparation of the computer results for our initial report, and for the entire task of preparing the present tables using recent data and the formulae of table I. We are indebted to Professors D. L. Goodstein, R. P. Feynman, R. M. Mazo, and P. H. Roberts for discussions and collaboration on the theory. Professor I. Rudnick, Dr. Jay Maynard, and Dr. C. Van Degrift have made data available to us in advance of publication. We have appreciated the support and patience of the editors and staff of this Journal in the preparation of the manuscript.
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