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

This is a reprint with minor revisions of an article which appeared in Volume 6 of the Journal of Physical and Chemical Reference Data in 1977, pages 51-104. The pages in the contents below refer to the actual publication. It is being made available as a Report of the Department of Physics, University of Oregon as part of a project to make these tables available electronically. Reprints are still available from the Journal. James Brooks is now at the Department of Physics, Florida State University, Tallahassee. New tables giving the observed properties of liquid helium at the saturated vapor pressure are available from the Department based on the T90 scale of temperatures



Abstract

Comprehensive tables of the primary thermodynamic properties of superfluid helium-4, such as the specific heat and entropy, are presented as computed from the Landau quasiparticle model, with the aid of inelastic neutron scattering data. The neutron data are represented by continuous functions of temperature, pressure, and wave number and certain excitation properties such as number density, normal and superfluid densities are calculated directly from it. A discussion of the methods used in our computations is included, and comparisons of computed and experimental results are made where applicable. Certain inadequacies of present theoretical methods to describe the thermodynamic properties are reported, and the use of an effective spectrum is introduced to offset some of these difficulties. Considerable experimental effort is also needed to improve the present situation. Key words: Computed thermodynamic properties; entropy; equation of state; excitation spectrum; hehum-4; normal fluid helium-4; phonons; rotons; specific heat; superfluid helium-4.

Contents

List of Figures...................................................................................................
List of Auxiliary Tables in Text............................................................................
List of Tables in Appendix A...............................................................................
Nomenclature...................................................................................................
1. Introduction..................................................................................................
2. Theoretical Background.................................................................................
3. Experimental Data Used in the Analysis.........................................................
51
52
52
52
53
55
56

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

56
 
a. The Phonon Branch.....................................................................
b. The Maxon Branch......................................................................
c. The Roton Minimum....................................................................
d. The Shoulder Beyond the Roton Minimum.....................................
e. A Polynomial Representation for the Excitation Spectrum...............
f. The Effective Sharp Spectrum for Thermodynamics.........................		 56
57
57
59
59
60

3.2. The Thermodynamic Data..............................................................		 61
    a. The Equation of State..................................................................
b. The Calorimetric Data .................................................................		 61
61
4. Computational Methods and Comparison of Computed Values With Experiment 62
  4.1. Generation of the Effective Spectrum 62...........................................
4.2. The Equation of State....................................................................
 62
63
    a. The Velocity of Sound .................................................................
b. The Isothermal Compressibility.....................................................
c. The Grüneisen Constant...............................................................
d. The Coefficient of Thermal Expansion............................................
 63
66
66
66
  4.3. The Fundamental Thermodynamic Functions................................... 67
    a. The Entropy................................................................................
b. The Helmholtz Free Energy...........................................................
c. The Gibbs Free Energy ...............................................................
d. The Enthalpy...............................................................................
e. The Specific Heat.........................................................................
 67
67
68
69
69
  4.4. Superfluid Properties...................................................................... 71
    a. The Excitation Number Densities..................................................
b. The Normal Fluid and Superfluid Densities.....................................
c. The Velocity of Second Sound .....................................................
d. The Velocity of Fourth found.........................................................
 71
71
72
73
73
5. Conclusions..................................................................................................
6. Acknowledgments.........................................................................................
7. References ...................................................................................................
Appendix A. Tables of the Calculated Properties of Helium II.................................
Appendix B. Selected Helium Data: A Reference Guide.......................................
 73
73
74
76
103

List of Figures


1. A schematic diagram of the excitation spectrum for helium II ...........................
2. A schematic phase diagram for helium-4 ........................................................
3. The experimentally determined excitation spectrum at 1.1 K, SVP....................
4. The density dependence of the maxon peak ...................................................
5. Least squares fit of eq (11) to the experimental roton energy gap ? ....................
6. Least squares fit of eq (12) to the experimental roton effective mass ?................
7. The excitation spectrum at two pressures at 1.1 K...........................................
8. The relative change in molar volume given by the expression in table I ...............
9. The PVT surface of helium II ..........................................................................
10. The velocity of first sound as a function of temperature for different pressures....
11. The thermal expansion coefficient as a function of pressure and temperature....
12. The entropy of helium IIas a function of pressure and temperature calculated
from eq (2.)......................................................................................................
13. The Helmholtz free energy of the excitations as a function of pressure and
temperature.....................................................................................................
14. The enthalpy of the excitations as a function of pressure and temperature, as
calculated from the expression in table l..............................................................
15. The specific heat at constant pressure as a function of pressure and temperature, as calculated from the expression in table I......................................
16. The normal fluid ratio as a function of temperature for different pressures, calculated from eq (33) .....................................................................................
17. The velocity of second sound as a function of temperature at different pressures
18. The velocity of fourth sound as a function of pressure and temperature..............
54
54
55
57
58
59
60
64
65
66
67

68

69

70

71

72
73
74



List of Auxiliary Tables in Text


I. Expressions for thermodynamic quantities........................................................
II. Density differences and expansion coefficients corrected to P=0 as determined by Van Degrift [20]. ro= 0.145119 g cm-3..............................................................
III. The Grüneisen constantUG............................................................................
IV. The. ground state Helmholtz free energy J g-1...........
V. The ground state Gibbs free energy J g-1................
56

61
66

68
64
 

List of Tables in Appendix


1. Density (g cm-3).........................................................................................
2. Molar volume V (cm-3 mol-1)............................................................................
3. First sound velocity u1, (m s-1)........................................................................
4. Isothermal compressibility (cm2 dyne-1)........................................................
5. Thermal expansion coefficient  (K-1).............................................................
6. Helmholtz free energy of excitations E (J g-1)...................................................
7. Gibbs free energy of excitations E (J g-1).......................................................
8. Enthalpy of excitations WE (J g-1)...................................................................
9. entropy S (J g-1 K-1)........................................................................................
10. Specific heat at constant pressure Cp(J g-1 K-1).............................................
11. Specific heat at constant volume Cp(J g-1 K-1)...............................................
12. Ratio of specific heats CP/CV........................................................................
13. Phonon number density Np (cm-3)..................................................................
14. Roton number density Nr (g cm-3)..................................................................
15. Normal fluid density (g cm-3)....................................................................
16. Normal fluid ratio ...............................................................................
17. Superfluid density (g cm-3)..........................................................................
18. Superfluid ratio ...................................................................................
19. Second sound velocity (m s-1)....................................................................
20. Fourth sound velocity  (m s-1).....................................................................
21. Energy of maxon peak ...............................................................
22. Thermal roton energy gap ...............................................................
23. Thermal roton effective mass ................................................................
24. Wave number at roton minimum.......................................................
25. Coefficients of model dispersion curve series and ............................
77
78

79
80
81
82

83
84
85
86

87
88
89

90
91
92
93
94
95

96
97
98
99

100
101

 


Symbol or expression
Physical quantity
Unit symbol or value
m Helium-4 mass 6.646 ´10-24 g
k Boltzmann’s constant 1.380658X10-23 J/K
h, ( ) Planck’s constant, 6.260755X10-34J+g
1.05457266X10-34J-g
P Pressure 1.01325X105Pascal
V Molar volume cm3/mole
T Temperature K
PVT Pressure-Volume-Temperature, the equation of state  
SVP Saturated vapor pressure  
Tl(P) The temperature at which liquid helium-4 becomes a superfluid for a given pressure K
helium I
(He I)
 The non-superfluid state of liquid helium-4, T > T gm.cm-3
helium II
(He II)
 The superfluid state of liquid helium-4, T < T
 gm.cm-3
p Density gm.cm-3
Pn Density of the normal component K
Ps Density of the superfluid component
/ K Energy of an excitation ?k
Frequency of an excitation rad
P Excitation momentum g.cm.s-1
q=p/ Wave number of excitation Å-1(1Å-1=1010m-1)
p0 Momentum at the roton minimum g.cm.s-1
pc Momentum at which g.cm.s-1
n(p) Distribution of excitations as a function of momentum cm-3

 


Symbol or expression
Physical quantity
Unit symbol or value
/ k Roton energy/k k
t / k “Thermal” roton energy/k k
u Roton effective mass m
ut “Thermal” roton effective mass m
/K Half width of scattered neutron distribution ? k k
S(q , w) Dynamic structure factor  
S Entropy J/g-K
u1, u2, u4 Velocity of first, second, fourth sound
uI, uII Velocity of first, second sound, uncorrected
Energy of maxon peak ? k k
Nr Roton number density cm-3
Np Phonon number density cm-3
an Coefficients of excitation energy series Ån
aP Coefficient of thermal expansion K-1
kT Isothermal compressibility cm2 dyne-1
UG Grüneisen constant  
L Latent heat
F Helmholtz free energy
F Gibbs free energy
W Enthalpy
Specific heat at constant pressure
Cv Specific heat at constant volume
Ratio of specific heats  
V0, L0, F0, 0 W0, etc. Ground state (T=0) values of quantities  
Finite temperature values of quantities due to excitations only  
     


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