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The Observed Properties of Liquid Helium
at the Saturated Vapor Pressure


Chapter 13. Dispersion Curve

Adopted Database

Author(s)

Key #

Method

Range(Å-1)
Svensson et al. 1 Neutron scattering 0.3 Q 1.13
Cowley & Woods 2 " 0.2 Q 3.6
Woods et al. 3 " 1.9 Q 1.96
Stirling et al. 4 " 0.089 Q 0.8925
Stirling 5 " 1.88 Q 1.97

Comments and Key to Authors
1) Ref. 10.
2) Ref. 3.
3) Ref. 7.
4) Ref. 9.
5) Ref. 13.
6) The accuracy of neutron scattering measurements depends very much on the wavenumber range. It is probably best appreciated by inspecting the deviation plot in Fig. 13.3. The quantity E is a measure of the energy of an elementary excitation at a wave number Q. The wave number Q used here is expressed in reciprocal Ongstroms (=108).

Table 13.1. The adopted database used for the dispersion curve for liquid 4He.

Q(-1)

E/k (K)

Key

Q(-1)

E/k (K)

Key
0.08940 1.61309 4 1.100 13.8000 2
0.09460 1.71754 4 1.130 13.8200 1
0.1150 2.10050 4 1.200 13.7500 2
0.1210 2.25137 4 1.300 13.5000 2
0.1390 2.61112 4 1.400 12.9500 2
0.1430 2.63433 4 1.500 12.2000 2
0.1594 2.97088 4 1.600 11.2000 2
0.1767 3.29582 4 1.700 10.2500 2
0.1818 3.38866 4 1.800 9.25000 2
0.1938 3.63236 4 1.880 8.69400 5
0.1990 3.73681 4 1.890 8.65700 5
0.2000 3.70000 2 1.900 8.70000 2
0.2110 3.96891 4 1.900 8.65400 3
0.2162 4.08496 4 1.900 8.63400 5
0.2278 4.28224 4 1.910 8.63500 3
0.2329 4.38669 4 1.910 8.61600 5
0.2445 4.60718 4 1.915 8.61100 5
0.2495 4.71163 4 1.920 8.62600 3
0.2611 4.92052 4 1.920 8.61000 5
0.2776 5.23385 4 1.925 8.60600 5
0.2825 5.32669 4 1.930 8.62600 3
0.2938 5.52398 4 1.930 8.60600 5
0.2988 5.62842 4 1.935 8.63000 3
0.3000 5.57000 1 1.935 8.61200 5
0.3000 5.65000 2 1.940 8.63000 3
0.4000 7.40000 2 1.940 8.60900 5
0.4036 7.63609 4 1.950 8.65000 3
0.4082 7.71732 4 1.950 8.63300 5
0.4187 7.91461 4 1.960 8.68300 3
0.4232 7.99584 4 1.960 8.67200 5
0.4355 8.18152 4 1.970 8.69500 5
0.4498 8.37881 4 2.000 8.95000 2
0.4643 8.64572 4 2.100 10.0000 2
0.4785 8.86622 4 2.200 11.6500 2
0.4926 9.10992 4 2.300 13.5500 2
0.5000 9.15000 2 2.400 15.5000 2
0.5605 10.1544 4 2.500 16.4500 2
0.6000 10.7500 2 2.600 17.0000 2
0.6243 11.0015 4 2.700 17.3000 2
0.6965 11.8023 4 2.800 17.5000 2
0.7000 11.7500 2 2.900 17.7000 2
0.7649 12.4173 4 3.000 17.8500 2
0.8000 12.7200 1 3.100 18.0000 2
0.8000 12.6500 2 3.200 18.1500 2
0.8300 12.8815 4 3.300 18.3000 2
0.8925 13.2297 4 3.400 18.3500 2
0.9000 13.1500 2 3.500 18.4000 2
1.000 13.5500 2 3.600 18.4500 2

Figure 13.1. Energy E in degrees Kelvin of elementary excitations in helium II as a function of wave number Q in reciprocal ?ngstroms(Ref.12).

Table 13.2. Knots and coefficients for the spline fit of the dispersion curve of liquid 4He.

Knots

Coefficients
K( 1) = 0.0894 C( 1) =1.53895
K( 2) = 0.0894 C( 2) = 1.932
K( 3) = 0.0894 C( 3) = 4.8
K( 4) = 0.0894 C( 4) = 14.85
K( 5) = 0.15 C( 5) = 14.88
K( 6) = 0.510 C( 6) = 5.9384
K( 7) = 1.60 C( 7) = 16.5014
K( 8) = 2.023 C( 8) = 17.72455
K( 9) = 2.42 C( 9) = 18.43656
K(10) = 2.665 C(10) = 18.43545
K(11) = 3.60  
K(12) = 3.60  
K(13) = 3.60  
K(14) = 3.60  

Figure 13.2. Group velocity of elementary excitations in helium II as a function of wave number Q in reciprocal ?ngstroms (Ref.12).

Table 13.3. Recommended values of the dispersion curve of elementary excitations in liquid 4He as a function of wavenumber Q.


Q (-1)

E/k (K)

Q (-1)

E/k (K)
0.00 0.00* 1.85 8.866
0.05 0.804 1.90 8.667
0.10 1.747 1.95 8.644
0.15 2.757 2.00 8.833
0.20 3.772 2.05 9.271
0.25 4.772 2.10 9.941
0.30 5.749 2.15 10.784
0.35 6.694 2.20 11.742
0.40 7.598 2.25 12.751
0.45 8.452 2.30 13.753
0.50 9.249 2.35 14.687
0.55 9.979 2.40 15.492
0.60 10.641 2.45 16.111
0.65 11.237 2.50 16.544
0.70 11.767 2.55 16.832
0.75 12.232 2.60 17.019
0.80 12.633 2.65 17.147
0.85 12.971 2.70 17.258
0.90 13.248 2.75 17.366
0.95 13.464 2.80 17.471
1.00 13.62 2.85 17.573
1.05 13.717 2.90 17.671
1.10 13.757 2.95 17.765
1.15 13.74 3.00 17.855
1.20 13.667 3.05 17.94
1.25 13.54 3.10 18.019
1.30 13.359 3.15 18.093
1.35 13.125 3.20 18.161
1.40 12.839 3.25 18.222
1.45 12.503 3.30 18.276
1.50 12.118 3.35 18.323
1.55 11.684 3.40 18.363
1.60 11.202 3.45 18.394
1.65 10.68 3.50 18.417
1.70 10.148 3.55 18.431
1.75 9.644 3.60 18.435
1.80 9.204    

* The dispersion curve must be zero at zero wavenumber. Our spline returns
-2.485941E-4 K, which is incorrect, but negligible.

Figure 13.3. The fractional deviation of values of the adopted database from the recommended values for the dispersion curve for liquid 4He, expressed in percent.

 

Chronological Bibliography for Mutual Friction


1

.P. J. Bendt, R. D. Cowan, and J. L. Yarnell, "Excitations in Liquid Helium: Thermodynamic Calculations," Phys. Rev. 113, 1386-1395 (1959).
2 J. L. Yarnell, G. P. Arnold, P. J. Bendt, and E. C. Kerr, “Excitations in Liquid Helium: Neutron Scattering Measurements,” Phys. Rev. 113, 1379-1385 (1959).
3 R. A. Cowley and A. D. B. Woods, “Inelastic Scattering of Thermal Neutrons from Liquid Helium,” Can. J. Phys. 49, 177-200 (1971).
4 O. W. Dietrich, E. H. Graf, C. H. Huang, and L. Passell, “Neutron Scattering by Rotons in Liquid Helium,” Phys. Rev. A 5, 1377-1391 (1972).
5 E. C. Svensson, P. Martel, V. F. Sears, and A. D. B. Woods, “Neutron Scattering Studies of the Dynamic Structure of Liquid 4He,” Can. J. Phys. 54, 2178-2192 (1976).
6 H. Maris, “Phonon-Phonon Interactions in Liquid Helium,” Rev. Mod. Phys. 49, 341-359 (1977).
7 A. D. B. Woods, P. A. Hilton, R. Scherm, and W. G. Stirling, “A New Determination of the Roton Energy in Superfluid Liquid Helium,” J. Phys. C 10, 145-149 (1977).
8 D. S. Greywall, “Specific Heat and Phonon Dispersion Curve of Liquid 4He,” Phys. Rev B 18, 2127-214 (1978).
9 W. G. Stirling, J. R. D. Copley, and P. A. Hilton, “High Resolution Study of the Phonon Dispersion at Small Wave Vectors in Superfluid 4He,” in IAEA Symposium, Neutron Inelastic Scattering (Vienna, 1978), Vol. III, pp. 45-52.
10 E. C. Svensson, R. Scherm, and A. D. B. Woods, “Temperature Dependence of Phonons in Liquid Helium,” J. Phys. C 39, 211-214 (1978).
11 J. A. Tarvin and L. Passell, “Analysis of Line Shapes Observed in the Scattering of Thermal Neutrons from Superfluid 4He,” Phys. Rev. B 19, 1458-1462 (1979).
12 R. J. Donnelly, J. A. Donnelly, and R. N. Hills, “Specific Heat and Dispersion Curve for Helium II,” J. Low Temp. Phys. 44, 471-489 (1981).
13 W. G. Stirling, “New High-Resolution Neutron Scattering Investigations of Excitations in Liquid 4He,” presented at the Second International Conference on Phonon Physics (unpublished)

 

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