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The Observed Properties of Liquid Helium
at the Saturated Vapor Pressure

Chapter 14. Structure Factor

Adopted Database

Author(s)	Key #	Method	Range(Å-1)
Robkoff and Hallock	1	X-ray diffraction	0.213 Q 5.129
Svensson, et. al.	2	Neutron diffraction.	0.79 Q 6.66

Comments and Key to Authors
1) Ref. 9.
2) Ref. 10.
3) We adopted the theoretical value of the structure factor at zero wave number

S(0)=K_yT /cm²=5.1x10^-2

where is the ratio of specific heats, m is the mass of the 4He atom and c is the velocity of sound. The wave number Q used here is expressed in reciprocal ?ngstroms (=108 cm).
4.) The Authors of Refs. 9 and 10 are unable to explain the discrepancy in the two sets of data in the region of the peak. Our spline passes near the average of the two sets of data in the region of the peak. If a choice needs to be made, the X-ray data could be used for Q < 1, and the neutron data for Q > 1.

Table 14.1. Adopted database used for structure factor of liquid ⁴He.

Q (Å-1)	S(Q)	Key	Q (Å-1)	S(Q)	Key
0.000E+0	5.10000E-2	3	2.315E+0	1.24900E+0	1
2.130E-1	8.10000e-2	1	2.370E+0	1.14200E+0	2
2.670E-1	9.10000E-2	1	2.450E+0	1.09500E+0	2
3.200E-1	9.90000E-2	1	2.468E+0	1.10500E+0	1
3.730E-1	1.10000E-1	1	2.560E+0	1.04000E+0	2
4.270E-1	1.22000E-1	1	2.620E+0	1.04000E+0	1
4.800E-1	1.31000E-1	1	2.700E+0	9.89000E-1	2
5.330E-1	1.42000E-1	1	2.771E+0	9.86000E-1	1
5.860E-1	1.54000E-1	1	2.850E+0	9.59000E-1	2
6.400E-1	1.64000E-1	1	2.921E+0	9.96000E-1	1
6.930E-1	1.75000E-1	1	3.000E+0	9.52000E-1	2
7.460E-1	1.86000E-1	1	3.068E+0	9.64000E-1	1
7.900E-1	1.56000E-1	2	3.140E+0	9.37000E-1	2
7.990E-1	2.00000E-1	1	3.217E+0	9.65000E-1	1
8.520E-1	2.15000E-1	1	3.290E+0	9.55000E-1	2
8.600E-1	1.92000E-1	2	3.364E+0	9.56000E-1	1
9.400E-1	2.28000E-1	2	3.430E+0	9.57000E-1	2
1.010E+0	2.49000E-1	2	3.509E+0	9.80000E-1	1
1.011E+0	2.65000E-1	1	3.570E+0	9.79000E-1	2
1.090E+0	2.83000E-1	2	3.653E+0	9.64000E-1	1
1.169E+0	3.33000E-1	1	3.720E+0	9.95000E-1	2
1.170E+0	3.29000E-1	2	3.795E+0	9.56000E-1	1
1.240E+0	3.59000E-1	2	3.860E+0	1.00100E+0	2
1.320E+0	4.00000E-1	2	3.936E+0	9.74000E-1	1
1.328E+0	4.24000E-1	1	4.000E+0	1.01200E+0	2
1.400E+0	4.64000E-1	2	4.075E+0	1.02800E+0	1
1.470E+0	5.32000E-1	2	4.140E+0	1.02400E+0	2
1.485E+0	5.63000E-1	1	4.213E+0	9.92000E-1	1
1.550E+0	6.24000E-1	2	4.280E+0	1.03200E+0	2
1.620E+0	7.28000E-1	2	4.349E+0	9.73000E-1	1
1.642E+0	7.32000E-1	1	4.410E+0	1.00300E+0	2
1.700E+0	8.60000E-1	2	4.484E+0	1.03200E+0	1
1.770E+0	1.00300E+0	2	4.550E+0	1.01100E+0	2
1.799E+0	1.07200E+0	1	4.617E+0	9.89000E-1	1
1.850E+0	1.17700E+0	2	4.690E+0	1.00700E+0	2
1.851E+0	1.20100E+0	1	4.748E+0	1.00500E+0	1
1.903E+0	1.33700E+0	1	4.820E+0	1.00400E+0	2
1.920E+0	1.32900E+0	2	4.877E+0	1.00200E+0	1
1.955E+0	1.39900E+0	1	4.990E+0	1.00000E+0	2
2.000E+0	1.37100E+0	2	5.004E+0	1.00000E+0	1
2.006E+0	1.43300E+0	1	5.129E+0	1.00700E+0	1
2.058E+0	1.45800E+0	1	5.180E+0	1.00600E+0	2
2.070E+0	1.37600E+0	2	5.380E+0	9.95000E-1	2
2.109E+0	1.44800E+0	1	5.570E+0	9.98000E-1	2
2.150E+0	1.33500E+0	2	5.760E+0	1.00500E+0	2
2.161E+0	1.41200E+0	1	5.940E+0	1.00500E+0	2
2.212E+0	1.35200E+0	1	6.130E+0	1.00400E+0	2
2.220E+0	1.27300E+0	2	6.310E+0	1.00400E+0	2
2.264E+0	1.29000E+0	1	6.480E+0	1.00800E+0	2
2.300E+0	1.20800E+0	2	6.660E+0	1.00500E+0	2

Figure 14.1. The recommended values for the structure factor of liquid ⁴He as a function of wave number.

Table 14.2. Knots and coefficients for the spline fit of structure factor.

Knots	Coefficients
K(1)=0.000000E+00	C(1)=5.098474E-02
K(2)=0.000000E+00	C(2)=8.713372E-02
K(3)=0.000000E+00	C(3)=1.381343E-01
K(4)=0.000000E+00	C(4)=2.211584E-01
K(5)=7.467679E-01	C(5)=3.034749E-01
K(6)=8.843175E-01	C(6)=5.022945E-01
K(7)=1.171682E+00	C(7)=7.050370E-01
K(8)=1.483348E+00	C(8)=1.644316E+00
K(9)=1.615547E+00	C(9)=1.146399E+00
K(10)=1.954501E+00	C(10)=8.365880E-01
K(11)=2.269539E+00	C(11)=1.045785E+00
K(12)=2.738304E+00	C(12)=9.733701E-01
K(13)=4.266671E+00	C(13)=1.020061E+00
K(14)=4.686186E+00	C(14)=1.004961E+00
K(15)=6.660000E+00
K(16)=6.660000E+00
K(17)=6.660000E+00
K(18)=6.660000E+00

Figure 14.2. The fractional deviation of values of the adopted database from the recommended values for the structure factor expressed in percent.

Table 14.3. Recommended values of the structure factor of liquid ⁴He.

Q(Å-1)	S(Q)	Q(Å-1)	S(Q)
0.000	5.098E-2	2.500	1.089
0.050	5.834E-2	2.550	1.065
0.100	6.588E-2	2.600	1.045
0.150	7.362E-2	2.650	1.028
0.200	8.154E-2	2.700	1.013
0.250	8.967E-2	2.750	1.000
0.300	9.799E-2	2.800	9.889E-1
0.350	1.065E-1	2.850	9.793E-1
0.400	1.152E-1	2.900	9.712E-1
0.450	1.242E-1	2.950	9.644E-1
0.500	1.333E-1	3.000	9.590E-1
0.550	1.427E-1	3.050	9.549E-1
0.600	1.522E-1	3.100	9.519E-1
0.650	1.620E-1	3.150	9.499E-1
0.700	1.720E-1	3.200	9.490E-1
0.750	1.822E-1	3.250	9.490E-1
0.800	1.928E-1	3.300	9.498E-1
0.850	2.040E-1	3.350	9.513E-1
0.900	2.164E-1	3.400	9.535E-1
0.950	2.303E-1	3.450	9.563E-1
1.000	2.463E-1	3.500	9.596E-1
1.050	2.648E-1	3.550	9.634E-1
1.100	2.862E-1	3.600	9.675E-1
1.150	3.109E-1	3.650	9.718E-1
1.200	3.395E-1	3.700	9.764E-1
1.250	3.716E-1	3.750	9.810E-1
1.300	4.066E-1	3.800	9.857E-1
1.350	4.438E-1	3.850	9.903E-1
1.400	4.825E-1	3.900	9.948E-1
1.450	5.219E-1	3.950	9.991E-1
1.500	5.614E-1	4.000	1.003
1.550	6.049E-1	4.050	1.007
1.600	6.633E-1	4.100	1.010
1.650	7.466E-1	4.150	1.012
1.700	8.529E-1	4.200	1.014
1.750	9.721E-1	4.250	1.016
1.800	1.094	4.300	1.016
1.850	1.210	4.350	1.016
1.900	1.308	4.400	1.015
1.950	1.380	4.450	1.014
2.000	1.418	4.500	1.012
2.050	1.424	4.550	1.010
2.100	1.405	4.600	1.008
2.150	1.369	4.650	1.006
2.200	1.323	4.700	1.004
2.250	1.273	4.750	1.003
2.300	1.226	4.800	1.001
2.350	1.184	5.700	1.002
2.400	1.148	5.750	1.003
2.450	1.116	5.800	1.004
4.850	1.000	5.850	1.004
4.900	9.993E-1	5.900	1.005
4.950	9.986E-1	5.950	1.006
5.000	9.980E-1	6.000	1.006
5.050	9.976E-1	6.050	1.007
5.100	9.973E-1	6.100	1.008
5.150	9.972E-1	6.150	1.008
5.200	9.972E-1	6.200	1.008
5.250	9.973E-1	6.250	1.008
5.300	9.976E-1	6.300	1.009
5.350	9.979E-1	6.350	1.009
5.400	9.984E-1	6.400	1.008
5.450	9.989E-1	6.450	1.008
5.500	9.995E-1	6.500	1.008
5.550	1.000	6.550	1.007
5.600	1.001	6.600	1.006
5.650	1.001

Chronological Bibliography for Mutual Friction

1	C. F. A. Beaumont and J. Reekie, "Atomic Distribution in Liquid Helium," Proc. R. Soc., London Ser. A 228, 363-376 (1955).
2	D. G. Hurst and D. G. Henshaw, “Atomic Distribution in Liquid Helium by Neutron Diffraction,” Phys. Rev. 100, 994-1002 (1955).
3	W. L. Gordon, C. H. Shaw, and J. G. Duant, “Radial Atomic Distribution in Liquid 4He by X-Ray Scattering,” J. Phys. Chem. Solids 5, 117-128 (1958).
4	D. G. Henshaw, “Effect of the Lambda Transition on the Atomic Distribution in Liquid Helium by Neutron Diffraction,” Phys. Rev. 119, 9-13 (1960).
5	E. K. Achter and L. L. Meyer, “X-Ray Scattering from Liquid Helium,” Phys. Rev. 188, 291-300 (1969).
6	R. B. Hallock, “X-Ray Scattering from Liquid 4He,” Phys. Rev. A 5, 320-330 (1972).
7	H. N. Robkoff, D. A. Ewen, and R. B. Hallock, “Temperature-Dependent Spatial Order in Superfluid and Normal 4He,” Phys. Rev. Lett. 43, 2006-2009 (1979).
8	V. F. Sears and E. C. Svenson, “Pair Correlations and the Condensate Fraction in Superfluid 4He,” Phys. Rev. Lett. 43, 2009-2012 (1979).
9	H. N. Robkoff and R. B. Hallock, “Structure Factor Measurements at Saturated Vapor Pressure for 1.38 < T < 4.24 K,” Phys. Rev. B 24, 159-182 (1981).
10	E. C. Svensson, V. F. Sears, A. D. B. Woods, and P. Martel, “Neutron-Diffraction Study of the Static Structure Factor and Pair Correlations in Liquid 4He,” Phys. Rev. B 21, 3638-3651 (1989).
11	V. F. Sears, E. C. Svensson, A. D. B. Woods, and P. Martel, AECL Ltd. Report No. AECL-6779, (unpublished)