

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 ^{4}He 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 temperaturedependent 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
d^{3}=[T
ln( P / P_{0})
where
P_{0} 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/d^{4},
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
d^{3}=[T
ln( P / P_{0})
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, 161163 (1962). 
2 
R.
S. Kagiwada, J. C. Fraser, I. Rudnick, and D. Bergman, “Superflow
in Helium Films: Third Sound Measurements,” Phys. Rev. Lett.
22, 338342 (1968). 
3 
I.
Rudnick, R. S. Kagiwada, J. C. Fraser, and E. Guyon, “Third
Sound in Adsorbed Superfluid Films,” Phys. Rev. Lett. 20, 430434
(1968). 
4 
D.
Bergman, “Hydrodynamics and Third Sound in Thin He II Films,”
Phys. Rev. 188, 370384 (1969). 
5 
K.
A. Pickar and K. R. Atkins, “Critical Velocity of a Superflowing
LiquidHelium Film Using Third Sound,” Phys. Rev. 178, 389399
(1969). 
6 
C.
H. Anderson and E. S. Sabiskey, “Phonon Interference in Thin
Films of Liquid Helium,” Phys. Rev. Lett. 24, 10491052 (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. 3776. 
8 
D.
Bergman, “Third Sound in Superfluid Helium Films of Arbitrary
Thickness,” Phys. Rev. A 3, 20532056 (1971). 
9 
T.
Wang and I. Rudnick, “Anomalous Attenuation of Third Sound,”
J. Low Temp. Phys. 9, 425433 (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, 147151 (1974). 
11 
K.
Telschow, T. Wang, and I. Rudnick, “Observation of the Critical
Velocity Peak in Superfluid Films,” Phys. Rev. Lett. 32, 12921295
(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, 4363 (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, 147163 (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, 240243 (1978). 
15 
D.
T. Ekholm and R. B. Hallock, “Film Thinning in Unstaturated
Superfluid 4He Films During Persistent Flow,” Phys. Rev. B 19,
24852487 (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, 437440
(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, 339361 (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 u_{1},u_{2}
and u_{4}.
2) Ref. 9. Simultaneously measuredu_{1},u_{2}
and u_{4} . Data of both sets of authors agree to
within a few tenths percent for u_{4}.
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.
T_{90} (K) 
u_{1} (m/s) 
Key 
T_{90} 
u_{1} (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)
= 1.786538E02 
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.
T_{90} (K) 
u_{1} (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

9.322E1 
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, 962965 (1959). 
2 
K.
A. Shapiro and I. Rudnick, “Experimental Determination of the
Fourth Sound Velocity in Helium II,” Phys. Rev. A 137, 13831391
(1965). 
3 
I.
Rudnik, H. Kojima, W. Veith, and R. S. Kagiwada, “Observation
of SuperfluidHelium Persistent Current by DopplerShifted Splitting
of Fourth Sound Resonance,” Phys. Rev. Lett 23, 12201223 (1969). 
4 
M.
Kriss and I. Rudnick, “Size Effects in Helium II as Measured
by Fourth Sound,” J. Low Temp. Phys. 3, 339357 (1970). 
5 
H.
Kojima, W. Veith, S. Putterman, E. Guyon, and I. Rudnick, “VortexFree
Landau State in Rotating Superfluid Helium,” Phys. Rev. Lett.
27, 714718 (1971). 
6 
H.
Kojima, W. Veith, E. Guyon, and I. Rudnick, “Persistent Current
States in Rotating Superfluid Helium,” J. Low Temp. Phys. 8,
187193 (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. 279282. 
8 
J.
Heiserman, J. P. Hulin, J. Maynard, and I. Rudnick, “Precision
SoundVelocity Measurements in Helium II,” Phys. Rev. B 14,
38623867 (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, 173190 (1987) 

