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The Observed Properties of Liquid Helium
at the Saturated Vapor Pressure

Chapter 6. Third Sound Velocity

The velocity of third sound in helium II for wavelength long compared to the thickness of the film, d, is given to good approximation by

where <p_s>/p is the effective superfluid density in the film, f is the restoring force per unit mass, S is the entropy, T the thermodynamic temperature and L the latent heat. The restoring force is

where is the van der Waals's attraction between a ⁴He atom and the substrate and is a retardation parameter.
The effective superfluid density is less than the bulk density because of healing effects near the walls, such that

where is the bulk value and D is a temperature-dependent parameter which has been determined to have the form

D=a+bTp/p_s

with a = 0.5 layers/K and b=1.13 for glass. Here D and d are in units of atomic layers. Both constants are determined from experiment. The thickness scale is determined by the partial pressure P in the sample chamber from the relationship

d³=[T ln( P / P₀)

where P₀ is the saturated vapor pressure at temperature T.
For very thin films (d < 10 atomic layers, where 1 atomic layer = 3.6 angstroms) retardation effects are negligible and the restoring force can be approximated by f=3/d⁴, so that to first order the velocity becomes

The parameter D can then be determined by making a plot of

as a function of d with slope unity and intercept D.
For a substrate which is reasonably smooth on the microscopic scale, the third sound velocity is relatively independent of the substrate on which it is measured because

and

d³=[T ln( P / P₀)

so that the first order is independent of .

Chronological Bibliography for Third Sound Velocity

1	C. W. F. Everitt, K. R. Atkins, and A. Denenstein, "Detection of Third Sound in Liquid Helium," Phys. Rev. Lett. 8, 161-163 (1962).
2	R. S. Kagiwada, J. C. Fraser, I. Rudnick, and D. Bergman, “Superflow in Helium Films: Third Sound Measurements,” Phys. Rev. Lett. 22, 338-342 (1968).
3	I. Rudnick, R. S. Kagiwada, J. C. Fraser, and E. Guyon, “Third Sound in Adsorbed Superfluid Films,” Phys. Rev. Lett. 20, 430-434 (1968).
4	D. Bergman, “Hydrodynamics and Third Sound in Thin He II Films,” Phys. Rev. 188, 370-384 (1969).
5	K. A. Pickar and K. R. Atkins, “Critical Velocity of a Superflowing Liquid-Helium Film Using Third Sound,” Phys. Rev. 178, 389-399 (1969).
6	C. H. Anderson and E. S. Sabiskey, “Phonon Interference in Thin Films of Liquid Helium,” Phys. Rev. Lett. 24, 1049-1052 (1970).
7	K. R. Atkins and I. Rudnick, “Third Sound,” in Progress in Low Temperature Physics, edited by C. G. Gorter (North Holland Publishing, 1970), Vol. 6, pp. 37-76.
8	D. Bergman, “Third Sound in Superfluid Helium Films of Arbitrary Thickness,” Phys. Rev. A 3, 2053-2056 (1971).
9	T. Wang and I. Rudnick, “Anomalous Attenuation of Third Sound,” J. Low Temp. Phys. 9, 425-433 (1972).
10	J. Scholtz, E. MacLean, and I. Rudnick, “Third Sound and the Healing Length of Helium II Films as Thin as 2.1 Atomic Layers,” Phys. Rev. Lett. 32, 147-151 (1974).
11	K. Telschow, T. Wang, and I. Rudnick, “Observation of the Critical Velocity Peak in Superfluid Films,” Phys. Rev. Lett. 32, 1292-1295 (1974).
12	K. Telschow, I. Rudnick, and T. Wang, “An Experiment on the Bernoulli Thinning Effect in Unsaturated Superfluid Films,” J. Low Temp. Phys. 18, 43-63 (1975).
13	R. K. Galiewicz, K. L. Telschow, and R. B. Hallock, “Persistent Currents in Saturated Superfluid 4He Films in the Presence of Third Sound Resonances,” J. Low Temp. Phys. 26, 147-163 (1977).
14	J. S. Brooks, F. M. Ellis, and R. B. Hallock, “Direct Observation of Pulsed Third Sound Mass Displacement Waves in Unsaturated 4He Films,” Phys. Rev. Lett. 40, 240-243 (1978).
15	D. T. Ekholm and R. B. Hallock, “Film Thinning in Unstaturated Superfluid 4He Films During Persistent Flow,” Phys. Rev. B 19, 2485-2487 (1979).
16	P. H. Roberts, R. N. Hills, and R. J. Donnelly, “Calculation of the Static Healing Length in Helium II,” Phys. Lett. A 70, 437-440 (1979).
17	D. T. Ekholm and R. B. Hallock, “Thickness Measurements of Unsaturated Superfluid 4He Films Under Driven and Persistent Flow,” J. Low Temp. Phys. 42, 339-361 (1981).

Chapter 6. Fourth Sound Velocity

Adopted Database

Author(s)	Key #	Range (K)
Heiserman et al.	1	1.187 T 2.15
Tam & Ahlers	2	1.553 T2.15917

Comments and Key to Authors

1) Ref. 8. Simultaneously measured u₁,u₂ and u₄.
2) Ref. 9. Simultaneously measuredu₁,u₂ and u₄ . Data of both sets of authors agree to
within a few tenths percent for u₄.

Figure 6.1. The recommended values for the velocity of fourth sound as a function of temperature at the saturated vapor pressure.

Figure 6.2. The fractional deviation of values of the adopted database from the recommended values for the velocity of fourth sound in helium II expressed in percent.

Table 6.1. Adopted database for fourth sound velocity in helium II.

T90 (K)	u1 (m/s)	Key	T90	u1 (m/s)	Key
1.1901	2.347E+2	1	1.9542	1.636E+2	1
1.2462	2.333E+2	1	1.9542	1.643E+2	2
1.3013	2.317E+2	1	1.9542	1.644E+2	2
1.3544	2.298E+2	1	2.0042	1.512E+2	1
1.4055	2.274E+2	1	2.0042	1.513E+2	2
1.4546	2.250E+2	1	2.0543	1.340E+2	1
1.5046	2.226E+2	1	2.0543	1.348E+2	2
1.5536	2.183E+2	1	2.0843	1.224E+2	2
1.5566	2.176E+2	2	2.1044	1.118E+2	1
1.6037	2.141E+2	2	2.1044	1.125E+2	2
1.6047	2.147E+2	1	2.1144	1.068E+2	2
1.6538	2.098E+2	2	2.1245	1.005E+2	2
1.6588	2.094E+2	1	2.1345	9.330E+1	2
1.7039	2.047E+2	2	2.1445	8.490E+1	2
1.7069	2.046E+2	1	2.1495	7.990E+1	2
1.7540	1.989E+2	2	2.1546	7.440E+1	2
1.7570	1.981E+2	1	2.1546	7.450E+1	2
1.8041	1.921E+2	2	2.1546	7.690E+1	1
1.8081	1.915E+2	1	2.1586	6.920E+1	2
1.8541	1.835E+2	1	2.1606	6.660E+1	2
1.8541	1.843E+2	2	2.1616	6.420E+1	2
1.9042	1.747E+2	1	2.1636	6.160E+1	2
1.9042	1.751E+2	2	2.1768	0.000
1.9042	1.752E+2	2

Table 6.2. Knots and coefficients for the spline fit of the velocity of fourth sound.

Knots	Coefficients
K(1) = 1.190093	C(1) = 2.346992E+2
K(2) = 1.190093	C(2) = 2.317649E+2
K(3) = 1.190093	C(3) = 2.235997E+2
K(4) = 1.190093	C(4) = 1.972721E+2
K(5) = 1.622630	C(5) = 1.746328E+2
K(6) = 1.770580	C(6) = 1.437044E+2
K(7) = 1.936770	C(7) = 1.120775E+2
K(8) = 2.042310	C(8) = 7.758563E+1
K(9) = 2.129610	C(9) = 4.689201E+1
K(10) = 2.161789	C(10) = -1.786538E-02
K(11) = 2.176800
K(12) = 2.176800
K(13) = 2.176800
K(14) = 2.176800

Table 6.3. Recommended values of fourth sound velocity in helium II.

T90 (K)	u1 (m/s)
1.2000	2.345E+2
1.2500	2.333E+2
1.3000	2.318E+2
1.3500	2.300E+2
1.4000	2.278E+2
1.4500	2.252E+2
1.5000	2.222E+2
1.5500	2.187E+2
1.6000	2.147E+2
1.6500	2.102E+2
1.7000	2.051E+2
1.7500	1.993E+2
1.8000	1.926E+2
1.8500	1.848E+2
1.9000	1.757E+2
1.9500	1.651E+2
2.0000	1.523E+2
2.0500	1.363E+2
2.1000	1.146E+2
2.1500	8.000E+1
2.1761	6.321E+0
2.1762	5.447E+0
2.1763	4.563E+0
2.1764	3.670E+0
2.1765	2.767E+0
2.1766	1.855E+0
2.1767	9.322E-1
2.1768	0.000E+0

Chronological Bibliography for Fourth Sound Velocity

1	K. R. Atkins, "Third and Fourth Sound in Liquid Helium II," Phys. Rev. 113, 962-965 (1959).
2	K. A. Shapiro and I. Rudnick, “Experimental Determination of the Fourth Sound Velocity in Helium II,” Phys. Rev. A 137, 1383-1391 (1965).
3	I. Rudnik, H. Kojima, W. Veith, and R. S. Kagiwada, “Observation of Superfluid-Helium Persistent Current by Doppler-Shifted Splitting of Fourth Sound Resonance,” Phys. Rev. Lett 23, 1220-1223 (1969).
4	M. Kriss and I. Rudnick, “Size Effects in Helium II as Measured by Fourth Sound,” J. Low Temp. Phys. 3, 339-357 (1970).
5	H. Kojima, W. Veith, S. Putterman, E. Guyon, and I. Rudnick, “Vortex-Free Landau State in Rotating Superfluid Helium,” Phys. Rev. Lett. 27, 714-718 (1971).
6	H. Kojima, W. Veith, E. Guyon, and I. Rudnick, “Persistent Current States in Rotating Superfluid Helium,” J. Low Temp. Phys. 8, 187-193 (1972).
7	H. Kojima, W. Veith, E. Guyon, and I. Rudnick, “Decay of Saturated and Unsaturated Persistent Currents in Superfluid Helium,” in 13th International Conference on Low Temperature Physics, edited by K. D. Timmerhaus, W. J. O'Sullivan, and E. F. Hammel (Plenum Press, 1974), Vol. I, pp. 279-282.
8	J. Heiserman, J. P. Hulin, J. Maynard, and I. Rudnick, “Precision Sound-Velocity Measurements in Helium II,” Phys. Rev. B 14, 3862-3867 (1976).
9	W. Y. Tam and G. Ahlers, “Superfluid Fraction of 4He from 1.5K to T Lambda (P) and from Vapor Pressure to the Melting Curve,” J. Low Temp. Phys. 66, 173-190 (1987)