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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(Å-1)	E/k (K)	Key	Q(Å-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 (Å-1)	E/k (K)	Q (Å-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)