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The Observed Properties of Liquid Helium
at the Saturated Vapor Pressure

Chapter 2. Superfluid and Normal Fluid Densities

Author(s)	Key #	Method	Range (K)	Uncertainty %
Landau Theory	1	Integration	0.1 T 1.25
Maynard	2	u2 & u4	1.2 T 2.15	< 5
Tam & Ahlers	3	u2 & u4	1.553 T 2.15917	< 5
Singsass & Ahlers	4	Asymptotic formula	T > 2.207	see paper

1) We have generated 24 points of p_n, the normal fluid density from 0.1 to 1.25 K by integrating over the dispersion curve of Chapter 13. This method is very good at low temperatures, but by 1 K starts to degrade because the spectrum becomes temperature dependent. We used the total density from Chapter 1 to generate the superfluid density .
2) Data from Reference 22.
3) Data from Reference 25.
4) Singsass and Ahlers, Ref. 24, give the asymptotic formula

with t=1-(T/T), k₀=2.403, k1=-1.46, Dp=0.33, =0.6717 and =0.5. The formula is based on new precision entropy data. We generated 18 points from 2.1 to 2.1768 K.
5) The total density from chapter 1 is used to compute quantities such as .
6) The densities are tabulated in g/cm3. Multiply the entries by 1000 to convert to kg/m³.

Table 2.1. The adopted database for the superfluid density of helium II as a function of temperature.

T90(K)	ps(g/cm)3	key	T90(K)	ps(g/cm)3	key
0 0.	14514	1	1.85411900	0.09193	2
0.10036490	0.1451	1	1.90415100	0.08359	2
0.15053830	0.1451	1	1.90415100	0.08419	3
0.20070610	0.1451	1	1.90415100	0.08421	3
0.25086810	0.1451	1	1.95418900	0.0742	2
0.30102560	0.1451	1	1.95418900	0.07493	3
0.35117790	0.1451	1	1.95418900	0.07499	3
0.40132560	0.1451	1	2.00423400	0.0636	2
0.45146850	0.1451	1	2.00423400	0.06421	3
0.50160770	0.1451	1	2.05429300	0.05145	2
0.55174260	0.1451	1	2.05429300	0.05164	3
0.60187360	0.1451	1	2.05429300	0.05168	3
0.65199130	0.1451	1	2.07680000	0.04419	4
0.70212460	0.1451	1	2.08434800	0.04297	3
0.75224480	0.145	1	2.10439900	0.03669	2
0.80236180	0.145	1	2.10439900	0.03651	3
0.85247570	0.1449	1	2.10439900	0.03652	3
0.90258630	0.1447	1	2.11442900	0.03302	3
0.95269420	0.1444	1	2.12446200	0.02935	3
1.00277700	0.1441	1	2.12680000	0.02821	4
1.05286100	0.1436	1	2.13449700	0.02539	3
1.10295500	0.143	1	2.14453100	0.02108	3
1.15302900	0.1422	1	2.14954700	0.01873	3
1.20326700	0.141	2	2.15456300	0.01633	3
1.20326700	0.1411	1	2.15456300	0.01628	3
1.25320300	0.1399	1	2.15456300	0.0167	2
1.25320300	0.1398	2	2.15857500	0.01409	3
1.30332400	0.1383	2	2.16058000	0.01308	3
1.35344300	0.1364	2	2.16258600	0.01218	3
1.40351500	0.1343	2	2.16358900	0.0112	3
1.45355700	0.1317	2	2.16680000	0.00959	4
1.50359800	0.1287	2	2.17080000	0.00679	4
1.55363700	0.1253	2	2.17380000	0.00425	4
1.55664000	0.1244	3	2.17580000	0.00202	4
1.60369000	0.1208	3	2.17620000	0.00143	4
1.60369000	0.1213	2	2.17650000	8.99048E-4	4
1.65377900	0.1168	2	2.17670000	4.29195E-4	4
1.65377900	0.1166	3	2.17674000	3.04392E-4	4
1.70388200	0.1117	2	2.17677000	1.90995E-4	4
1.70388200	0.1116	3	2.17679000	9.12685E-5	4
1.75398100	0.1059	2	2.17679500	5.72837E-5	4
1.75398100	0.106	3	2.17679900	1.94274E-5	4
1.80406500	0.09933	2	2.17679950	1.21951E-5	4
1.80406500	0.09958	3	2.17679990	4.13664E-6	4
1.85411900	0.09241	3	2.17680000	0

Table 2.2. Knots and coefficients of the spline fit of the superfluid density of helium II.

Knots	Coefficients
K(1)=0.0	C(1)=1.451275432822459E-1
K(2)=0.0	C(2)=1.451334563362309E-1
K(3)=0.0	C(3)=1.449759191497576E-1
K(4)=0.0	C(4)=1.455008000684433E-1
K(5)=0.443	C(5)=1.4075E-1
K(6)=0.9012	C(6)=1.095E-1
K(7)=1.5419	C(7)=8.15E-2
K(8)=1.7540	C(8)=5.30E-2
K(9)=1.918	C(9)=2.1E-2
K(10)=2.111	C(10)=8.904576E-3
K(11)=2.156991	C(11)=3.053214E-3
K(12)=2.173218	C(12)=1.494043E-3
K(13)=2.175647	C(13)=8.342826E-4
K(14)=2.176358	C(14)=5.10686E-4
K(15)=2.176568	C(15)=2.8379E-4
K(16)=2.176692	C(16)=1.287426E-4
K(17)=2.176766	C(17)=5.202569E-5
K(18)=2.176791	C(18)=2.153580E-5
K(19)=2.176798	C(19)=8.564206E-6
K(20)=2.176799	C(20)=3.567958E-6
K(21)=2.17679999	C(21)=0
K(22)=2.1768
K(23)=2.1768
K(24)=2.1768
K(25)=2.1768

Figure 2.1. Recommended values of the superfluid density of helium II as a function of temperature.

Figure 2.2. Recommended values of the superfluid density of helium II near the lambda transition.

Figure 2.3. The fractional deviation of the adopted database from the recommended values for the superfluid density of helium II expressed in percent.

Table 2.3. Recommended values of the normal and superfluid densities of helium II.

T90(K)	ps(g/cm3)	pn(g/cm3)	T90(K)	ps(g/cm3)	pn(g/cm3)
0	0.14513	0	1.45	0.13199	0.01316
0.05	0.14513	2.95E-9	1.5	0.12900	0.01617
0.1	0.14512	5.9E-9	1.55	0.12556	0.01963
0.15	0.14511	8.85E-9	1.6	0.12163	0.02358
0.2	0.14510	2.771E-8	1.65	0.11715	0.02809
0.25	0.14509	6.687E-8	1.7	0.11206	0.03321
0.3	0.14509	1.368E-7	1.75	0.10630	0.03900
0.35	0.14508	2.502E-7	1.8	0.09982	0.04554
0.4	0.14508	4.242E-7	1.85	0.09254	0.05286
0.45	0.14509	7.03E-7	1.9	0.08444	0.06103
0.5	0.14511	1.249E-6	1.95	0.07542	0.07012
0.55	0.14512	2.588E-6	2	0.06507	0.08055
0.6	0.14513	6.069E-6	2.05	0.05275	0.09297
0.65	0.14513	1.452E-5	2.1	0.03779	0.10804
0.7	0.14511	3.295E-5	2.11	0.03443	0.11143
0.75	0.14506	6.923E-5	2.12	0.03091	0.11497
0.8	0.14498	1.345E-4	2.13	0.02718	0.11874
0.85	0.14486	2.431E-4	2.14	0.02311	0.12284
0.9	0.14469	4.127E-4	2.15	0.01862	0.12737
0.95	0.14446	6.636E-4	2.16	0.01359	0.13243
1	0.14414	0.00102	2.17	0.00729	0.13878
1.05	0.14371	0.00141	2.171	0.00652	0.13955
1.1	0.14313	0.00199	2.172	0.00572	0.14036
1.15	0.14235	0.00276	2.173	0.00489	0.14120
1.2	0.14137	0.00375	2.174	0.00401	0.14208
1.25	0.14012	0.00499	2.175	0.00300	0.14310
1.3	0.13860	0.00652	2.176	0.00175	0.14436
1.35	0.13676	0.00837	2.1768	0	0.14611
1.4	0.13457	0.01057

Table 2.4. Recommended values of superfluid and normal fluid density ratios
for helium II as a function of temperature.

T90(K)	ps / p	pn / p	T90(K)	ps / p	pn / p
0	1.000	0	1.45	0.909	0.091
0.05	1.000	2.03E-8	1.50	0.889	0.111
0.1	1.000	4.06E-8	1.55	0.865	0.135
0.15	1.000	6.10E-8	1.60	0.838	0.162
0.2	1.000	1.91E-7	1.65	0.807	0.193
0.25	1.000	4.60E-7	1.70	0.771	0.229
0.3	1.000	9.42E-7	1.75	0.732	0.268
0.35	1.000	1.72E-6	1.80	0.687	0.313
0.4	1.000	2.92E-6	1.85	0.636	0.364
0.45	1.000	4.84E-6	1.90	0.580	0.420
0.5	1.000	8.61E-6	1.95	0.518	0.482
0.55	1.000	1.78E-5	2.00	0.447	0.553
0.6	1.000	4.18E-5	2.05	0.362	0.638
0.65	1.000	1.00E-4	2.10	0.259	0.741
0.7	1.000	2.27E-4	2.11	0.236	0.764
0.75	1.000	4.77E-4	2.12	0.212	0.788
0.8	0.999	9.27E-4	2.13	0.186	0.814
0.85	0.998	2.00E-3	2.14	0.158	0.842
0.9	0.997	3.00E-3	2.15	0.128	0.872
0.95	0.995	5.00E-3	2.16	0.093	0.907
1	0.993	7.00E-3	2.17	0.050	0.950
1.05	0.990	1.00E-2	2.171	0.045	0.955
1.1	0.986	1.40-2	2.172	0.039	0.961
1.15	0.981	1.9E-2	2.173	0.033	0.967
1.2	0.974	2.6E-2	2.174	0.027	0.973
1.25	0.966	3.4E-2	2.175	0.021	0.979
1.3	0.955	4.5E-2	2.176	0.012	0.988
1.35	0.942	5.8E-2	2.1768	0	1.000

Chronological Bibliography for Superfluid Density

1	E. L. Andronikashvili, "Direct Observation of Two Types of Motion in Helium II," Zhur. Ekxp. Theor. Fiz 16, 780-785 (1946).
2	V. P. Peshkov, “Determination of the Velocity of Propagation of the Second Sound in Helium II,” Sov. Phys. USSR 10, 389-398 (1946).
3	E. L. Andronikashvili, “Temperature Dependence of the Normal Density of Helium II,” Zhur. Eksp. Theor. Fiz. 18, 424-428 (1948).
4	J. G. Dash and R. D. Taylor, “Hydrodynamics of Oscillating Disks in Viscous Fluids: Density and Viscosity of Normal Fluid in Pure 4He from 1.2K to the Lambda Point,” Phys. Rev. 105, 7-24 (1957).
5	P. J. Bendt, R. D. Cowan, and J. L. Yarnell, “Excitations in Liquid Helium: Thermodynamic Calculations,” Phys. Rev. 113, 1386-1395 (1959).
6	J. T. Tough, W. D. McCormick, and J. G. Dash, “Viscosity of Liquid Helium II,” Phys. Rev. 132, 2373-2378 (1963).
7	J. D. Reppy, “Application of Superfluid Gyroscope to the Study of Critical Velocities in Liquid Helium Near the Lambda Transition,” Phys. Rev. 14, 733-735 (1965).
8	K. A. Shapiro and I. Rudnick, “Experimental Determination of the Fourth Sound Velocity in Helium II,” Phys. Rev. A 137, 1383-1391 (1965).
9	J. R. Clow and J. D. Reppy, “Temperature Dependence of the Superfluid Density in Helium II Near T Lambda,” Phys. Rev. Lett. 16, 887-889 (1966).
10	J. A. Tyson and D. H. Douglass Jr., “Superfluid Density and Scaling Laws for Liquid Helium Near T Lambda,” Phys. Rev. Lett. 17, 472-474 (1966).
11	C. J. Pearce, J. A. Lipa, and M. J. Buckingham, “Velocity of Sound Near the Lambda Point of Helium,” Phys. Rev. Lett. 20, 1471-1473 (1968).
12	J. A. Tyson, “Superfluid Density of Helium II in the Critical Region,” Phys. Rev. 166, 166-176 (1968).
13	J. A. Tyson and D. H. Douglass Jr., “Critical-Region Second-Sound Velocity in Helium II,” Phys. Rev. Lett. 21, 1308-1310 (1968).
14	R. H. Romer and R. J. Duffy, “Normal-Fluid Densities in Liquid Helium II Under Pressure,” Phys. Rev. 186, 255-262 (1969).
15	M. Kriss and I. Rudnick, “Size Effects in Helium II as Measured by Fourth Sound,” J. Low Temp. Phys. 3, 339-357 (1970).
16	J. R. Clow and J. D. Reppy, “Persistent-Current Measurements of the Superfluid Density and Critical Velocity,” Phys. Rev. A 5, 424-438 (1972).
17	D. S. Greywall and G. Ahlers, “Second Sound Velocity, Scaling, and Universality in Helium II Near the Superfluid Transition,” Phys. Rev. Lett. 28, 1251-1254 (1972).
18	S. A. Scott, E. Guyon, and I. Rudnick, “Specific Heat and Superfluid Density of Liquid Helium in Porous Media,” J. Low. Temp. Phys. 9, 389-407 (1972).
19	D. S. Greywall and G. Ahlers, “Second Sound Velocity and Superfluid Density in 4He Under Pressure Near T Lambda,” Phys. Rev. A 7, 2145-2162 (1973).
20	A. Ikushima and G. Terui, “Superfluid Density Near the Lambda Point of 4He Under Pressure,” J. Low Temp. Phys. 10, 397-405 (1973).
21	G. Ahlers, “Experiments Near the Superfluid Transition in 4He,” in The Physics of Liquid and Solid Helium, edited by K. H. Bennemann and J. B. Ketterson (Wiley, New York, 1976).
22	J. Maynard, “Determination of the Thermodynamics of Helium II from Sound Velocity Data,” Phys. Rev. B 14, 3868-3890 (1976).
23	A. L. Singsaas, “Entropy of Helium II and Universality Near the Superfluid Transition in 4He,” Ph.D. Thesis, University of California, 1984 (unpublished).
24	A. L. Singsaas and G. Ahlers, “Universality of Static Properties Near the Superfluid Transition in 4He,” Phys. Rev. B 30, 5103-5115 (1984).
25	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)