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Table 1 : : 2 : : 3 : : 4 : : 5

The Calculated Thermodynamic Properties of Superfluid Helium-4
James S. Brooks and Russell J. Donnelly

1. Introduction

Liquid helium is a rewarding subject for the study of thermodynamic properties, especially because helium II, the lower temperature phase, exhibits the property of superfluidity. The hydrodynamics of this phase are extraordinary: both normal viscous behavior and superflow may be exhibited in closely related experiments. Phenomenologically, one speaks of a "two-fluid" behavior in which the fluid (of density p)¹ acts dynamically as if a fraction of effective density p_n flows as a normal fluid, and a fraction of effective densityp_s = p - p_n flows as an inviscid fluid. This two-fluid motion of helium II leads to some very unusual results, such as the wave-like rather than diffusive propagation of temperature fluctuations (called second sound). Furthermore, the thermodynamic properties are deeply related to the hydrodynamic: for example, the Gibbs free energy is related to the square of the relative velocity between the two fluids. We shall be concerned principally with the static properties of helium IIin which all net flow velocities are zero.

On a deeper level, one has the Landau picture [1] 2 in which the entire fluid is superfluid at absolute zero. As the temperature 3 is raised, the heat content is described entirely in terms of "elementary excitations" of the superfluid. At low temperatures these are "phonons", at higher temperatures more complex excitations called "rotons" are excited.

The energy and momentum p( = q) of the elementary excitations in superfluid helium may be represented by the excitation spectrum (dispersion curve) first proposed by Landau [1] which is shown schematically in figure 1. We will refer again to this figure in subsequent sections. In recent years many of the features, and in some cases the entire dispersion curve, have been measured by inelastic thermal neutron scattering techniques for different pressures and temperatures. (See for example references [2-10].) Landau has put forth a simple but elegant theory which allows one to calculate the thermodynamic properties of He II at low temperatures from the excitation spectrum.

Figure 1 A Schematic diagram of the excitation spectrum for helium II. Details and labels are discussed in the text.

Temperature (K)

Figure 2. A schematic phase diagram for helium-4. The pressure of the liquid-gas transition line has been exaggerated. The hexagonal close-packed and body centered cubic (bcc) phases of the solid are shown, as are the loci of zero thermal expansion (dashed lines).

The purpose of this paper is to give an account of the equilibrium thermodynamic properties of He II and related quantities such as the velocities of first, second, and fourth sounds and properties of the excitations themselves. These quantities are used in a wide variety of contexts, both experimental and theoretical; and it is often important that the data for different properties be thermodynamically consistent. The ideal solution to this problem would be a critical compilation of experimental properties over the entire T-P plane for He II, and some day this will undoubtedly be possible. At the time this project was begun (1972), the tables in the appendices of the books by Wilks [11] and Donnelly [12] were the most complete available, and far from adequate for the task undertaken here.

Another promising avenue is the use of Landau's theory mentioned above. The thermodynamic properties along the vapor pressure curve have been extracted from neutron scattering measurements with reasonable success by Bendt, Cowan, and Yarnell [10]. We have, then, the possibility that direct thermodynamic measurements can be supplemented by the increasing body of neutron data indicated by the references cited. In particular, we were greatly encouraged by the appearance in 1972 of the landmark study of neutron scattering from rotons by Dietrich, Graf, Huang, and Passell at Brookhaven National Laboratory [3].

The superfluid region of the helium P-T phase diagram is shown schematically as the shaded portion of figure 2. It is to this area that our analysis and subsequent tabulations apply. For all pressures in this shaded region, there is a temperature above which the superfluid properties cease to exist, and one has a classical liquid phase (called helium I). The line which separates these two regions is called the lambda line, in recognition of the characteristic anomaly in the specific heat observed along this line. In this paper, we do not treat the thermodynamic properties in the immediate vicinity of this line. Such a treatment has, however, been undertaken in an earlier volume of this Journal, by McCarty [13], who has reported on the thermodynamic properties of helium I; and IUPAC Helium-4 tables by Angus, de Reuck, and McCarty [14] are in the process of completion at the time of writing of the present article.

The results described here are the result of a lengthy series of investigations by our research group at the University of Oregon. The first product of our analysis of the Brookhaven data was a report on the Landau parameters [15]. An early attempt at approximating the dispersion curve in a piecewise fashion was reported at LT13 [16]. This proved to be imprecise and was succeeded by the series representation used in this paper [17]. The results of these investigations were summarized in the Ph. D. thesis of J. S. Brooks, which appeared in 1973 [18]. We issued the Tables from that thesis in the form of a University of Oregon Technical Report in December 1973 [19], requesting contributions from other laboratories. We were gratified to receive a number of suggestions, corrections, and reports of new experimental work. In particular, the thesis of Van Degrift on the expansion coefficient [20], and a systematic study of first, second, and fourth sound by Maynard and Rudnick, began at UCLA, have been extremely useful, as has some independent work on dispersion curves by Dr. Maynard [21]. Since this work was submitted, the UCLA measurements have been completed and sent for publication as shown in [21].

The problem of the proper treatment of the statistical mechanics of interacting Bose excitations has taken considerable effort. Much confusion about the effects of temperature dependent levels and thermal expansion has been removed with the recent appearance of papers by Goodstein, Brooks, Donnelly, and Roberts [22], Roberts and Donnelly [23, 24]. In particular, the latter authors have developed the set of expressions displayed in table I which allow the calculation of the equilibrium properties of He II when the energy levels of excitations are known from experiment. This is the basis of the present calculated tables, which represent a considerable advance in accuracy over the calculations in our early work [18, 19]. For example, the need for an empirical equation of state is eliminated altogether.

The computations reported here have been carried out by James Gibbons on a Hewlett-Packard 2100 A computer.

2. Theoretical Background

Although several authors have discussed the Landau spectrum and theory in detail (see, for example, Wilks [11], Donnelly [12], Keller [25], Khalatnikov [26]) we present here a brief description for completeness. Landau's theory for superfluid helium is based on the assumption that the thermal excitations in the liquid can be described as constituting a weakly interacting gas with the energy spectrum given as the solid line in figure 1. It is also assumed that these excitations obey Bose statistics, and therefore the number density of excitations of a particular momentum,n (p) is given by

From this expression, we see that the low-lying regions of the energy spectrum will contribute predominantly to the thermodynamic properties, and referring to figure 1 we find that there are two such regions on interest. The first is for small momenta, where the spectrum is approximately linear, and is called the phonon branch. The other is the energy minimum about momentum p_o, which is nearly parabolic and is called the roton branch, or roton minimum. Here the energy is much higher, but the density of states is also very large.

The experimentally observed values of the minimum roton energy are large enough that the Boltzmann distribution is an adequate approximation to eq (1). As a rough guide to thinking of the thermodynamics of helium II, it is sufficient to recognize that below 1 K, the phonon excitations are the more numerous and dominate the thermal properties, whereas at higher temperatures the number of rotons increases exponentially with temperature, and rotons become predominant thermodynamically above 1 K.

If one knows the dispersion curve of the excitations, it is straightforward to construct a partition function and proceed by standard statistical mechanical methods to compute the entire set of equilibrium properties of the liquid. As we shall see, this ideal is far from realizable today.

At very low temperatures, when the liquid is relatively free of excitations, inelastically scattered neutrons are observed to form a very sharp spectrum, limited only by the instrumental resolution of the analyzing spectrometer. The location of this spectral line is taken as a measure of the transfer of energy from the incoming neutron beam to single phonons in the liquid, for a given momentum transfer p set by the spectrometer. The result of many such measurements is an excitation spectrum such as is shown in figure 3. It appears theoretically reasonable, and experimentally fairly certain, that the observed excitation spectrum can be identified with the dispersion curve of the elementary excitations in the fluid at low temperatures.

Figure 3. The experimentally determined excitation spectrum at 1.1 K, SVP. The dots are the neutron scattering data of Cowley and Woods [2]; the solid line is eqs (20) and (14). The error bar is the smallest experimental error, and is not to be associated with any one data point.

At higher temperatures, however, Yarnell et al. [9], and Henshaw and Woods [5] observed that the widths of the scattered spectra increase rapidly with temperatures. For rotons (at (c) in fig. 1) the linewidths, as measured by the half-width at half maximum ?, approach the magnitude of the energy itself as approaches (i.e., as ). This means, among other things, that the lifetimes of the excitations are decreasing rapidly with increasing temperature. Furthermore, the roton energy ? is observed to decrease with increasing T and increasing P.

The scattered neutron intensity is proportional to the dynamic structure factor S (q. w) which is in turn related to density-density correlations in the fluid. The unfolding of the experimental scattered neutron spectra in terms of assumed forms for S (q. w) has been discussed in detail by Dietrich et al. [3]. To the best of our knowledge at the time of writing, no one has been able to make a rigorous connection between S (q. w) at finite temperatures and the energies which enter the thermodynamic calculations. Dietrich et al. [3] have included the linewidth ? in their thermodynamic calculations in an ad hoc way; perhaps a complete theory will require considerable information on the observed line shape.

If one were to brush aside the considerations of the last paragraph and try to work with the unfolded dispersion curve , one would still have the problem of treating the statistical mechanics of interacting Bose excitations whose interactions are manifested by an observed dependence of on T and P. Bendt, Cowan, and Yarnell [10] argue that when is a function of T alone, the entropy should be given by

where n is the Bose distribution function (1). Roberts and Donnelly [23] have presented arguments to show that even at arbitrary T and P, (1) and (2) are still valid, provided is available from experiment. They have presented a way in which all the thermodynamic properties may be consistently deduced from experimental dispersion curves [24]. We show in table I the expressions which they have derived for the quantities of interest in the present study. In table I we have omitted the subscripts on and k_T for clarity.

Table I. Expressions for thermodynamic quantities

3. Experimental Data Used in the Analysis

In this section the experimental data underlying the Landau theory is discussed

3.1. Data Obtained by the inelastic Scattering of Thermal Neutrons from Helium II

The energies and line widths of the elementary excitations of helium II are obtained from examination of inelastically scattered thermal neutrons. The techniques are discussed in detail, for example, in a recent review article by Woods and Cowley [8]. We shall use the measured energy spectrum, following Bendt, Cowan, and Yarnell [10], as the basis for our computations of the thermodynamic properties of helium II from the Landau theory. A completely determined experimental excitation spectrum is shown in figure 3 for T = 1.1 K, at the SVP. The shape of this spectrum is a complicated function of pressure and temperature, and to use the Landau theory one must first know the detailed spectrum for every value of P and T. A complete set of data for a single pressure and temperature such as is shown in figure 3 is a major experimental undertaking. Hence it is necessary to find a method of estimating the energy spectrum for general P and T accurate enough that derivatives such as appear in table I may be accurately determined. In the sections below, we discuss the salient features of the excitation spectrum obtained from various kinds of experiments, and in section 3.1.e we describe a method of representing the spectrum as a power series in momentum, with pressure- and temperature-dependent coefficients.

3.1.a. The Phonon Branch

For momentum p decreasing to zero, the phonon branch of the energy spectrum approaches linearity, in accordance with theoretical predictions (cf. Feenberg [27]). Here we take the energy spectrum, as indicated at (a) in figure 1, to be

where u1 is the velocity of ultrasonic (first) sound, a temperature- and pressure-dependent quantity which may be determined experimentally either from the slope of the excitation spectrum in the zero momentum limit, or more conveniently from experimental first sound data (see section 4.2.a).

The first non-linear terms correcting eq. (3) for small, but non-zero, momenta have been the subject of much controversy in the literature in recent years (see Svensson, Woods, and Martel [7], and references there in). These terms may affect the behavior of, for example, the specific heat at temperatures below 0.45 K, as suggested by Phillips, Waterfield, and Hoffer [28]. We will return to this question in section 3.1.e.

3. 1.b. The Maxon Branch

The elementary excitations which have energies at or near the first energy maximum, (b) in figure 1, have come to he called "maxons". This part of the spectrum has been measured at

1.1 K, at the vapor pressure as indicated in figure 3, and the peak position has been determined for various pressures at 1.18 K by Graf, Minkiewicz, Møller, and Passell [6] Measurements have also been made by Henshaw and Woods at the vapor pressure and 25.3 atm [4]. From the existing data on the maxon part of the spectrum, we conclude that ,the peak energy is density dependent only, since no shift in the peak position in momentum space has been observed. Very little data exists on the temperature dependence of the maxon peak. Figure 1 of Bendt, Cowan, and Yarnell [10] and figure 20 of Cowley and Woods [2] suggest it decreases. slowly with temperature, Because the maxon energies are relatively high, their contribution to the thermal properties is very small, and the details of this part of the spectrum are not critical to the computations in this paper.

We represent the density dependence of the maxon energy by the expression

where is the density at T= 0 and the coefficients En are

                                                               E₀ = -216.5672     (K)

                                                                E₁ = 3998.6005    (K g^-1 cm³)

                                                                E₂ = -23028.6027                (K g^-2 cm⁶)

                                                              E₃ = 44199.7232  (K g^-3 cm⁹)

Equation 4a is a fit to the neutron data of Cowley and Woods [2] and Graf et al. [6] and is plotted with the data in figure 4. The rather slow temperature dependence of the peak is represented by

   Values of   are given in table 21, the conversion to pressure being obtained from the equation of state.

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