- Email: [email protected]
Measurement of diffusion in 3He-4He solutions and determination of the 3He-roton cross section
Volume 40A, number 1
PHYSICS LETTERS
19 June 1972
MEASUREMENT OF DIFFUSION IN 3He-4He SOLUTIONS AND DETERMINATION OF THE 3He-ROTON CROSS SECTION* G.A. HERZLINGER
and J.G. KING
Department of Physics and Research Laboratory of Electronics, Massachusetts Institite of Technology, Cambridge, Mass. 02139, USA
Received 14 April 1972
The diffusion constant of very dilute 3He-4He solutions has been measured directly using a new technique. The results show that the 3He-roton cross section is energy dependent, increasing with temperature over the temperature range of the experiment.
We have investigated the 3He-roton interaction by measuring the diffusion constant in dilute 3He-4He mixtures at temperatures from 1.27” to 1.69” at concentrations of 1.45 X 10e4. (Higher concentrations were also used.) Our method differs from that used in all previous determinations of diffusion in 3He-4He mixtures in that we determine the diffusion constant from the time decay of an applied 3He concentration gradient (rather than from the decay of nuclear polarization in a non-uniform magnetic field [l ,2] or from measurement of thermal conduction [3]). 3He-3He collisions are unimportant in the diffusion since mass transport rather than spin transport is involved. Phonons do not contribute to transport processes in 3He-4He mixtures above 0.6’K [4] and the measurements yield the effective 3He-roton cross section. (In addition, at the low 3He concentrations used, the 3He number density is much less than that of the rotons throughout the temperature range covered, so that 3He atoms are probably unimportant in limiting the free paths of either other 3He atoms or rotons.) In the experiment, a 3He gradient is produced by exploiting the “heat flush” effect in a controlled way. A thin plane heater at the bottom of the diffusion chamber produces a current of thermal excitations. The chamber is thermally isolated from an outer 4He bath except for a thin perforated copper disc located just below the liquid surface; the only thermal path from the heater to the bath is through the mixture to the copper disc. The resulting upward convective flow * Research sponsored by U.S. Armed Forces Joint Services Electronics Program under Contract no. DAAB07-71-C-0300.
of excitations interacts with the solute 3He atoms forcing them to the region near the top of the chamber. When the flow of 3He due to the thermal current just balances the back diffusion current due to the build up of the 3He concentration, a steady state exponential concencentration distribution results [ 51. After the steady state has been established, the heater is turned off, and the exponential distribution decays into a uniform distribution as the 3He atoms diffuse through the He II in the chamber. The diffusion is kept nearly l-dimensional by a network of vertical channels consisting of corrugated and plane stainless foil which have been coiled together. During the diffusion process, the 3He concentration is monitored by continuosly sampling the vapor just above the liquid through a small capillary which is connected to a high uacuum system and an omegatron mass spectrometer. The vapor concentration monitored in this way follows the decay of the liquid concentration, since: 1) the diffusion constant of the saturated vapor in dilute 3He-4He mixtures is known to be of the order of 50 times the values observed for the liquid [6], i.e., the diffusion time in the vapor is negligible for the temperatures and vapor pressures of interest. 2) The vapor concentration above sufficiently dilute 3He-4He solutions is proportional to the liquid concentration [e.g. 71. The effect of the sampling process on the diffusion is negligible both in terms of depleting the 3He and in terms of producing an unwanted “heat flush” effect. The time dependence of the 3He decay was observed to be in excellent accord with that obtained by solving the diffusion equation for the system, and values for D, the diffusion constant, are obtained from 65
53-
I0.6 -
Ii
11
4_
2-
19 June 1972
PHYSICS LETTERS
Volume 40A, number 1
x I’
4 “E Y
11’
“b ;;
I I
r
I=
0.6 0.4 -
1
I
pj,,
0.3 0.58
I 0.60
81
6 0.62
lb 0.64
I 0.66
I
I II 0.66 0.70 I/T WI
I
I 0.72
I
I 0.74
If
3 0.76
I 0.76
the data. These are shown as a function of temperature in fig. 1. The error brackets shown are estimates of the relative uncertainty when comparing one data point with another. In addition, there is an overall uncertainty of about 10%1 S% in absolute value for the curve as a whole. The observed temperature variation of D reflects the temperature dependence of the roton number density, the average velocity of a 3He quasi-particle, and any energy dependence of the 3He-roton cross section. The curve is in fact steeper than can be accounted for by the roton density and velocity factors, and an energy dependent diffusion cross section, u;, is implied. The relation between D and u& is given by the Khalatnikov-Zharkov expression [4] (valid for T > 0.6”K): (l/o&+)
(1)
where the brackets refer to a weighted average over momentum space. Forthe concentrations in the experiment, n3 4 rzr, and the normal fluid density, p,, is nearly the same as that of pure He II at the same temperature, pno Eq. (1) may be approximated to give: 4, = KT/(r+m3b.+D)
- exp(A/T)ID
.
(2)
A plot of a;, versus temperature is shown in fig. 2. The empirical relation of Yarnell et al. [8] derived from neutron scattering data, A= 8.68-0.0084T7, has been assumed. Sample error brackets reflecting the uncertainty in D are given. The error associated with the empirical expression for A(T) is not known, and for comparison the effective cross section assuming a constant A, equal to 8.65”, is also shown. The approximation (2) may be used to estimate the absolute value of the cross section. Using m; = 2.7m3, 66
1.2
1.3
1.4 TEMPERATURE
Fig. 1. Diffusion constant plotted on a semiiog scale versus l/T.
D = (PnJP$(KTlnrm3)
, , , , , , ( I,,
, , /, 1.5
, ,,
I.6
1.7
(‘K)
Fig. 2. Effective jHe-roton cross section. The x’s were derived using the empirical relation for the roton energy gap, A = 8.68 - 0.0084T’ 181. Tine 3’s show the same curve assuming a constant energy gap A = 8.65”. p = 0. 16m4 for the 3He and roton effective masses,
and the above empirical relation for A(T), the range of values for the cross section is: u;, = 1.6 X 10-14cm2 at T = 1.27”K, a;, = 2.35 X 10-14cm2 at T = 1.69”K. Although there is some uncertainty in the exact form of the u& versus T curve, the greatest increase in usr occurs below 1 .S”, rather than at the higher temperatures of the experiment, and thus the increase in cross section is not due simply to an increase in density of the thermal excitation gas. We have calculated [9] the diffusion cross section, assuming that the scattering arises from the interaction between the “backflow” velocity fields of the 3He atom and the roton. The result is that u& is about 1 X 10-14cm2, and increases with temperature, but at a slower rate than observed. We are thus able to account qualitatively for some of the features of the data. References [ 11 R.L. Garwin and H.A. Reich, Phys. Rev. 115 (1959) 1478. [ 21 J.E. Qpfer, K. Luszczynski and R.E. Nordberg, Phys. Rev. 172 (1968) 192. [3] T.P. Ptukha, Zh. Eksp. i Teor. Fiz. 40 (1969) 1583; Sov. Phys. JETP 13 (1961) 1112. [4] LM. Khalatnikov and V.N. Zharkov, Zh. Eksp. i Teor. Fiz. 32 (1957) 1108; Sov. Phys. JETP 5 (1957) 905. (51 J.J.M. Beenakker, K.W. Taconis, E.L. Lynton, Z. Dokoupil and G. Van Soest, Physica 18 (1952) 433. [6] P.J. Bendt, Phys. Rev. 110 (1958) 85. (71 H.S. Sommers, Phys. Rev. 88 (1959) 113. [S] J.L. YarneB, G.P. Arnold, P.J. Bendt and E.C. Kerr, Phys. Rev. 113 (1959) 1379. [9] G.A. Herzlinger, R.L.E. Quarterly Progress Report, April 1972 (unpublished).
PHYSICS LETTERS
19 June 1972
MEASUREMENT OF DIFFUSION IN 3He-4He SOLUTIONS AND DETERMINATION OF THE 3He-ROTON CROSS SECTION* G.A. HERZLINGER
and J.G. KING
Department of Physics and Research Laboratory of Electronics, Massachusetts Institite of Technology, Cambridge, Mass. 02139, USA
Received 14 April 1972
The diffusion constant of very dilute 3He-4He solutions has been measured directly using a new technique. The results show that the 3He-roton cross section is energy dependent, increasing with temperature over the temperature range of the experiment.
We have investigated the 3He-roton interaction by measuring the diffusion constant in dilute 3He-4He mixtures at temperatures from 1.27” to 1.69” at concentrations of 1.45 X 10e4. (Higher concentrations were also used.) Our method differs from that used in all previous determinations of diffusion in 3He-4He mixtures in that we determine the diffusion constant from the time decay of an applied 3He concentration gradient (rather than from the decay of nuclear polarization in a non-uniform magnetic field [l ,2] or from measurement of thermal conduction [3]). 3He-3He collisions are unimportant in the diffusion since mass transport rather than spin transport is involved. Phonons do not contribute to transport processes in 3He-4He mixtures above 0.6’K [4] and the measurements yield the effective 3He-roton cross section. (In addition, at the low 3He concentrations used, the 3He number density is much less than that of the rotons throughout the temperature range covered, so that 3He atoms are probably unimportant in limiting the free paths of either other 3He atoms or rotons.) In the experiment, a 3He gradient is produced by exploiting the “heat flush” effect in a controlled way. A thin plane heater at the bottom of the diffusion chamber produces a current of thermal excitations. The chamber is thermally isolated from an outer 4He bath except for a thin perforated copper disc located just below the liquid surface; the only thermal path from the heater to the bath is through the mixture to the copper disc. The resulting upward convective flow * Research sponsored by U.S. Armed Forces Joint Services Electronics Program under Contract no. DAAB07-71-C-0300.
of excitations interacts with the solute 3He atoms forcing them to the region near the top of the chamber. When the flow of 3He due to the thermal current just balances the back diffusion current due to the build up of the 3He concentration, a steady state exponential concencentration distribution results [ 51. After the steady state has been established, the heater is turned off, and the exponential distribution decays into a uniform distribution as the 3He atoms diffuse through the He II in the chamber. The diffusion is kept nearly l-dimensional by a network of vertical channels consisting of corrugated and plane stainless foil which have been coiled together. During the diffusion process, the 3He concentration is monitored by continuosly sampling the vapor just above the liquid through a small capillary which is connected to a high uacuum system and an omegatron mass spectrometer. The vapor concentration monitored in this way follows the decay of the liquid concentration, since: 1) the diffusion constant of the saturated vapor in dilute 3He-4He mixtures is known to be of the order of 50 times the values observed for the liquid [6], i.e., the diffusion time in the vapor is negligible for the temperatures and vapor pressures of interest. 2) The vapor concentration above sufficiently dilute 3He-4He solutions is proportional to the liquid concentration [e.g. 71. The effect of the sampling process on the diffusion is negligible both in terms of depleting the 3He and in terms of producing an unwanted “heat flush” effect. The time dependence of the 3He decay was observed to be in excellent accord with that obtained by solving the diffusion equation for the system, and values for D, the diffusion constant, are obtained from 65
53-
I0.6 -
Ii
11
4_
2-
19 June 1972
PHYSICS LETTERS
Volume 40A, number 1
x I’
4 “E Y
11’
“b ;;
I I
r
I=
0.6 0.4 -
1
I
pj,,
0.3 0.58
I 0.60
81
6 0.62
lb 0.64
I 0.66
I
I II 0.66 0.70 I/T WI
I
I 0.72
I
I 0.74
If
3 0.76
I 0.76
the data. These are shown as a function of temperature in fig. 1. The error brackets shown are estimates of the relative uncertainty when comparing one data point with another. In addition, there is an overall uncertainty of about 10%1 S% in absolute value for the curve as a whole. The observed temperature variation of D reflects the temperature dependence of the roton number density, the average velocity of a 3He quasi-particle, and any energy dependence of the 3He-roton cross section. The curve is in fact steeper than can be accounted for by the roton density and velocity factors, and an energy dependent diffusion cross section, u;, is implied. The relation between D and u& is given by the Khalatnikov-Zharkov expression [4] (valid for T > 0.6”K): (l/o&+)
(1)
where the brackets refer to a weighted average over momentum space. Forthe concentrations in the experiment, n3 4 rzr, and the normal fluid density, p,, is nearly the same as that of pure He II at the same temperature, pno Eq. (1) may be approximated to give: 4, = KT/(r+m3b.+D)
- exp(A/T)ID
.
(2)
A plot of a;, versus temperature is shown in fig. 2. The empirical relation of Yarnell et al. [8] derived from neutron scattering data, A= 8.68-0.0084T7, has been assumed. Sample error brackets reflecting the uncertainty in D are given. The error associated with the empirical expression for A(T) is not known, and for comparison the effective cross section assuming a constant A, equal to 8.65”, is also shown. The approximation (2) may be used to estimate the absolute value of the cross section. Using m; = 2.7m3, 66
1.2
1.3
1.4 TEMPERATURE
Fig. 1. Diffusion constant plotted on a semiiog scale versus l/T.
D = (PnJP$(KTlnrm3)
, , , , , , ( I,,
, , /, 1.5
, ,,
I.6
1.7
(‘K)
Fig. 2. Effective jHe-roton cross section. The x’s were derived using the empirical relation for the roton energy gap, A = 8.68 - 0.0084T’ 181. Tine 3’s show the same curve assuming a constant energy gap A = 8.65”. p = 0. 16m4 for the 3He and roton effective masses,
and the above empirical relation for A(T), the range of values for the cross section is: u;, = 1.6 X 10-14cm2 at T = 1.27”K, a;, = 2.35 X 10-14cm2 at T = 1.69”K. Although there is some uncertainty in the exact form of the u& versus T curve, the greatest increase in usr occurs below 1 .S”, rather than at the higher temperatures of the experiment, and thus the increase in cross section is not due simply to an increase in density of the thermal excitation gas. We have calculated [9] the diffusion cross section, assuming that the scattering arises from the interaction between the “backflow” velocity fields of the 3He atom and the roton. The result is that u& is about 1 X 10-14cm2, and increases with temperature, but at a slower rate than observed. We are thus able to account qualitatively for some of the features of the data. References [ 11 R.L. Garwin and H.A. Reich, Phys. Rev. 115 (1959) 1478. [ 21 J.E. Qpfer, K. Luszczynski and R.E. Nordberg, Phys. Rev. 172 (1968) 192. [3] T.P. Ptukha, Zh. Eksp. i Teor. Fiz. 40 (1969) 1583; Sov. Phys. JETP 13 (1961) 1112. [4] LM. Khalatnikov and V.N. Zharkov, Zh. Eksp. i Teor. Fiz. 32 (1957) 1108; Sov. Phys. JETP 5 (1957) 905. (51 J.J.M. Beenakker, K.W. Taconis, E.L. Lynton, Z. Dokoupil and G. Van Soest, Physica 18 (1952) 433. [6] P.J. Bendt, Phys. Rev. 110 (1958) 85. (71 H.S. Sommers, Phys. Rev. 88 (1959) 113. [S] J.L. YarneB, G.P. Arnold, P.J. Bendt and E.C. Kerr, Phys. Rev. 113 (1959) 1379. [9] G.A. Herzlinger, R.L.E. Quarterly Progress Report, April 1972 (unpublished).