Russell J. Donnelly
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Table 1 : : 2 : : 3 : : 4 : : 5

The Observed Properties of Liquid Helium at the
Saturated Vapor Pressure

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

3.1.c. The Roton Minimum

The region of the spectrum denoted (c) in figure 1 is called the "roton minimum" and can be very well represented by Landau's parabolic expression

ronton=+ ( p-po)2/2u

where is the minimum roton energy, or energy gap, p0 is the roton momentum at minimum energy, and ??is the effective mass of excitation near p0. These three quantities which describe the roton minimum are called the "Landau parameters", and have the typical values at 1.I K, SVP of /k = 8.68 K, po/= 1.91-1, and . The temperature, pressure, and density dependence of these parameters are best known from the experimental work of Dietrich et al. [3], who find that and decrease with increasing temperature and/or pressure, but that p0 is a function of density only:

po/ =3.64p1/3A-1

Values of eq (6) are given in table 24.
Donnelly [15] has provided simple relations for the density dependence of and at
T = 0, and also for the relation between and at finite temperatures:

(p,0)/k=(16.99-57.31p)(K)

u(p,0)=(0.32-1.103p)(m)

An expression which describes the temperature and density dependence of(,T)/k has been given by Brooks and Donnelly [16]:

where is the normal fluid density and Nr is the roton number density (both quantities will be discussed in section 4). Here, a =8.75 X 10-23 cm3 K. Equation (10) gives a good qualitative description of the data of Dietrich et al. [3] to within 20% for [and through eq (9)] below the lambda point. These expressions are discussed in detail by Brooks [18].

Recently, motivated by (10), we have performed a least squares fit on the data for, and , using expressions of the form

(p,T)/k = 1+ 2+ 3et T/p+ 4e2t P and

u(p,T)=U1+ U2et T + U3p + U4et T + (U5 + U6p) e3t (m)

Here,t = -(p,0)/kT, and the coefficients of eqs (11) and (12) are:

=17.4167 (K)

2= -60.48823 (K g-1 cm3)

3 = -0.5307478 (g cm-3)

4 = 1.81726 X 104 (K g-1 cm3)

5 = -1.351398 X 107 (K)

6 = 1.621499 X 108 (K g-1 cm3)

7= -5.062661 X 108 (K g-2 cm6)

and U1 = 0.342061 (m)

U2 = 1.239037 (m K-1)

U3 = -1.238153 (m g-1 cm3)

U4 = -0.2234561 (m K-1 g cm-3)

U5 = -13429.95 (m)

U6 =632197.06 (m g-1 cm3)

The experimental roton energy gap and effective mass are shown in figures 5 and 6 with the results of eqs (11) and (12) respectively. Although eqs (11) and (12) suggest an expansion in terms of the roton number density, no particular theoretical significance can be attached to the terms in these equations. We will see in section 3.1.f below that the experimental values of and described in this section will have to be adjusted slightly to obtain the accurate thermodynamic information from the Landau theory.


Figure 5. Least squares fit of eq (11) to the experimental roton energy gap . Data at SVP, Henshaw and Woods [5] (solid circles), Cowley and Woods [2] (open circle); data at higher pressures, Dietrich et al. [3].


Figure 6. Least squares fit of eq (12) to the experimental roton effective mass . Data point at SVP, Cowley and Woods [2]: data at higher pressures, Dietrich et al. [3].

3.1d. The Shoulder beyond the Roton Minimum

Recent neutron scattering experimental results (Graf, Minkiewicz, Møller, and Passell [6] indicate that for momenta larger than po/the slope of the spectrum approaches the velocity of sound at a momentum , as shown at point (d) in figure 1 and by he arrow in figure 3. The spectrum then bends over and approaches twice the roton energy (curve B in figure 1), finally terminating at a momentum p'. The experimental spectrum in figure 3 also has this qualitative feature. This behavior for large momenta was predicted by Pitaevskii [29] to be

p(p)=2- exp[-/(p'-p)]

where and a are constants to be determined. Neutron measurements have not yet been made over a sufficient range to provide information for the dependence of eq (13), and in particular and a, on temperature and pressure. Fortunately, like the maxon peak, this region of the spectrum has little thermodynamic content, and it is sufficient for our purposes to locate the momentum pc to the right of p0 at which from eq (5) reaches the slope of the velocity of first sound u1, and to represent the dispersion curve as a straight line of slope u1 above pc, terminating at . This is curve C in figure 1, which is given by

(p)=u1(p-pe) + (pe)p pe


where Pe=
u1 +p0.

This approximation is also indicated in figure 3 as the straight line above pe/

3.1.e. A Polynomial Representation for the Excitation Spectrum

We may summarize the behavior of the excitation spectrum described in the previous sections in the following way:
The excitation spectrum starts out at zero momentum and energy with the slope . Hence

(0)=0;'(0)=u1

at point (a) of figure 1. Here' (p)=d(p)/dp . For momentum increasing from zero, the spectrum attains its first maximum at 1.1-1, and

(1.1-1)=max; (1.1-1)=0,

at point (b) in figure 1. Continuing down to the roton minimum, we find that the parabolic representation near po provides us with the relations

(p0)=;' (p0)=0;'' (p0)=1/;

at point (c) of figure 1. Beyond the roton minimum, for momentum pc where the slope approaches the velocity at sound, we finally have

' (pe)=u1

at point (d) in figure 1.
Clearly, eqs (16) through (19) represent the most important features of the excitation spectrum. We have discovered, partially guided by theoretical considerations (see Feenberg [27]), that a polynomial in momentum without a quadratic term

(p)/k= u1P + 3P3 + 4P4+ 5P5 + 6P6 + 7P7 + 8P8,

is an excellent representation for the excitation spectrum in the momentum interval and that by using eq (14) for the interval pe/ p/ 3.0-1 , one has a continuous expression for the spectrum which may be used to great advantage in the computation of thermodynamic properties based upon the Landau theory. Note that if eq (20) is continued above pc it diverges negatively, as indicated by the dotted line in figure 1. Hence care must be used to apply eq (14) above pc

The coefficients an of eq (20) may be obtained for any temperature and pressure by applying the conditions imposed by eqs (16) through (19), which in turn will depend on T and P. Since there is no constant term, and the first coefficient must be u1, the problem of determining the coefficients an is reduced to solving six equations with six unknowns, at any temperature and pressure.

The degree to which eqs (20) and (14) fit the neutron scattering data is. shown as the solid line in figure 3 at 1.1 K, SVP, and again by the solid lines in figure 7 at 1.1 K, at SVP, and at 25.3 atm. The experimental values were used in eqs (16) through (19) to obtain these plots. One can appreciate. from figure 7 how the spectrum changes with pressure.
.


Figure 7. The excitation spectrum at two pressures at 1.1 K. Dots, SVP, open circles, 25.3 atm, Henshaw and Woods [4]; triangles, 1.25 K, 24.26 atm, Dietrich et al.[3]; solid lines, eqs (20) and (14).

As mentioned in section 3.1.a, a controversy surrounds the exact form that eq (20) should have for small momenta. We make here several comments concerning our choice of eq (20). First, we found that a series with no quadratic term was by far the best fit to the neutron data. A detailed comparison of several series is given by Brooks [18]. Secondly, one can see by inspection of the results of such calculations that the coefficient of the first nonlinear term a3 changes sign upon increasing pressure. We find this to be consistent with the results of Phillips et al. [28] (see Brooks and Donnelly [17]). Finally, the form of eq (20) has prompted us to make some calculations concerning the low temperature behavior of the second sound velocity (Brooks and Donnelly [30]). The second sound velocity is quite sensitive to the leading terms in (20) and a careful measurement would serve as a check on the form we have chosen [30].



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