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Density-dependence of diffusion constant and nuclear spin lattice relaxation rate of 3He gas
PHYSICS
Volume 33A. number 2
DENSITY-DEPENDENCE NUCLEAR SPIN LATTICE
LETTERS
OF DIFFUSION RELAXATION
5 October 1970
CONSTANT AND RATE OF 3He GAS
J. S. KARRA and G. E. KEMMERER Jr University. Philadelphia, Pa. 19122, USA
Temple
Received 8 August 1970
Diffusion constant and nuclear spin-lattice relaxation rate of 3He gas with 4 % oxygen impurity were measured as a function of density at room temperature using spin-echo. Zero and first density coefficients of the diffusion constant were dekced. In recent years the transport coefficients of moderately dense gases have been expanded in a power series with the density p as the expansion parameter. However, the expressions for corrections to the collision operator are divergent and particularly the inclusion of quadruple events have lead to logarithmic divergences [l, 21. Therefore, instead of a power series form, the diffusion coefficient D is expressed more appropriately in the following way: ~D(P, 2’) = Do@)
(1)
+ pDl b”) +
+ (p21np)Di(T)+p2%(7j
+ . . .
where Do is the zero density coefficient and D1 is the density coefficient, etc. So far, experimental results have been reported on the density dependence of the coefficient of viscosity 77and the coefficient of thermal conductivity X and none whatsoever on the diffusion coefficient.
The purpose of the present experiment is to study the density dependence of the diffusion coefficient D of 3He gas by standard spin echo techniques [3] at room temperature (23’C). The 3He nucleus has a large magnetic moment, but there pure gas has a long spin-lattice relaxation time Tl of - 106 set [4]. To shorten Tl and make the spin echo method feasible, 4% 02 gas (paramagnetic) was added. The nuclear relaxation is due to the dipole-dipole interaction between the 3He and the 02 and the rate is given by [5] l/T1 = (r~‘A)~/~~d~~where y,y’ are the gyromagnetic ratios of 3He and 02 respectively, u is their relative speed, d the distance of closest approach and T is the mean collision time. Fig. 1 is a plot of l,‘Tl versus pressure. The relative concentration of O2 was maintained constant throughout, while the pressure of 3He was varied from 21 to 33 atmospheres (at 23OC) in steps of 3 atm. Under these conditions eq. (1) is still valid since the same type of quadruple
T -
8 u) _t-
-0.5
to Fig. 1. 3He relaxation rate l/T1 versus pressure 4 % O2 impurity.
for
P,
20
Amagat
30
Fig. 2. Plot ofpD versusp for 3He+4 %02, showing the best straight line. Error in intercept indicated by shaded error bar on axis. 105
events occur in a mixture as in pure gas and consequently logarithmic divergences should exist. But the expansion coefficients DO, “1 . . . etc. depend in general on the relative concentration of the gases in the mixture. So long as the same relative concentration is maintained, the expansion coefficients should be independent of concentration. The spin echo height S decays in time as S = = So exp [-Z/3 DyzGzt 3] where 0 is the diffusion constant, G is the magnetic field gradient, established by a pair of anti-Helmholtz coils, and i is the time between the 90- and 180- degree pul-ses. For fixed p and G, echo heights S were measured for various values of I. Approximately 100 echo scans were accumulated in a Varian model 1024 signal averaging computer but even so the signal to noise ratio was only about 5 or 10 to 1. The rms deviation in S, on arbitrary scale, was about 6, compared to values of S ranging from - 10 to 80. The main source of error seemed to be magnetic field drift resulting in detuning of the phase sensitive detection during the long time required for each measurement (about 100 X 10 T,). The results were plotted as log S versus /3 and a least squares fit to a straight line made, from whose slope D and its associated error were calculated. Fig. 2 is a plot of pD versus p, each point being a weighted average of 3 or 4 measurements of II taken at different G-values and the error bars show the rms deviations. To estimate Do,
106
a straight line, obtained by least squares, was extrapolated to p = 0 giving, from the intercept, Do = 0.71 * 0.61 amagat cm2/sec and from the slope, “1 = 0.054 + 0.026 cm2/sec. The experimental points in fig. 2 systematically exclude data taken at the highest field gradients. Our results include maximum field gradients of 1.‘i gauss/cm. Higher gradients cause the echo to be very narrow and to decay rapidly in time, and resulting inaccuracies in measuring the echo introduce much larger errors in the height values the coefficients: IjO = 0.56 i 1.4 and “1 = 0.056 jz 0.05. Further studies are in progress, with better stability in the dc magnetic field with higher 02 c’oncentration, to improve the accuracy of the results and possibly estimate the values of the succeeding terms in the expansion. We gratefully acknowledge the help of J. Anderson and the interest of Drs. K. Kawasaki and H. E. EM Van Leeuwen.
Volume 33A. number 2
DENSITY-DEPENDENCE NUCLEAR SPIN LATTICE
LETTERS
OF DIFFUSION RELAXATION
5 October 1970
CONSTANT AND RATE OF 3He GAS
J. S. KARRA and G. E. KEMMERER Jr University. Philadelphia, Pa. 19122, USA
Temple
Received 8 August 1970
Diffusion constant and nuclear spin-lattice relaxation rate of 3He gas with 4 % oxygen impurity were measured as a function of density at room temperature using spin-echo. Zero and first density coefficients of the diffusion constant were dekced. In recent years the transport coefficients of moderately dense gases have been expanded in a power series with the density p as the expansion parameter. However, the expressions for corrections to the collision operator are divergent and particularly the inclusion of quadruple events have lead to logarithmic divergences [l, 21. Therefore, instead of a power series form, the diffusion coefficient D is expressed more appropriately in the following way: ~D(P, 2’) = Do@)
(1)
+ pDl b”) +
+ (p21np)Di(T)+p2%(7j
+ . . .
where Do is the zero density coefficient and D1 is the density coefficient, etc. So far, experimental results have been reported on the density dependence of the coefficient of viscosity 77and the coefficient of thermal conductivity X and none whatsoever on the diffusion coefficient.
The purpose of the present experiment is to study the density dependence of the diffusion coefficient D of 3He gas by standard spin echo techniques [3] at room temperature (23’C). The 3He nucleus has a large magnetic moment, but there pure gas has a long spin-lattice relaxation time Tl of - 106 set [4]. To shorten Tl and make the spin echo method feasible, 4% 02 gas (paramagnetic) was added. The nuclear relaxation is due to the dipole-dipole interaction between the 3He and the 02 and the rate is given by [5] l/T1 = (r~‘A)~/~~d~~where y,y’ are the gyromagnetic ratios of 3He and 02 respectively, u is their relative speed, d the distance of closest approach and T is the mean collision time. Fig. 1 is a plot of l,‘Tl versus pressure. The relative concentration of O2 was maintained constant throughout, while the pressure of 3He was varied from 21 to 33 atmospheres (at 23OC) in steps of 3 atm. Under these conditions eq. (1) is still valid since the same type of quadruple
T -
8 u) _t-
-0.5
to Fig. 1. 3He relaxation rate l/T1 versus pressure 4 % O2 impurity.
for
P,
20
Amagat
30
Fig. 2. Plot ofpD versusp for 3He+4 %02, showing the best straight line. Error in intercept indicated by shaded error bar on axis. 105
events occur in a mixture as in pure gas and consequently logarithmic divergences should exist. But the expansion coefficients DO, “1 . . . etc. depend in general on the relative concentration of the gases in the mixture. So long as the same relative concentration is maintained, the expansion coefficients should be independent of concentration. The spin echo height S decays in time as S = = So exp [-Z/3 DyzGzt 3] where 0 is the diffusion constant, G is the magnetic field gradient, established by a pair of anti-Helmholtz coils, and i is the time between the 90- and 180- degree pul-ses. For fixed p and G, echo heights S were measured for various values of I. Approximately 100 echo scans were accumulated in a Varian model 1024 signal averaging computer but even so the signal to noise ratio was only about 5 or 10 to 1. The rms deviation in S, on arbitrary scale, was about 6, compared to values of S ranging from - 10 to 80. The main source of error seemed to be magnetic field drift resulting in detuning of the phase sensitive detection during the long time required for each measurement (about 100 X 10 T,). The results were plotted as log S versus /3 and a least squares fit to a straight line made, from whose slope D and its associated error were calculated. Fig. 2 is a plot of pD versus p, each point being a weighted average of 3 or 4 measurements of II taken at different G-values and the error bars show the rms deviations. To estimate Do,
106
a straight line, obtained by least squares, was extrapolated to p = 0 giving, from the intercept, Do = 0.71 * 0.61 amagat cm2/sec and from the slope, “1 = 0.054 + 0.026 cm2/sec. The experimental points in fig. 2 systematically exclude data taken at the highest field gradients. Our results include maximum field gradients of 1.‘i gauss/cm. Higher gradients cause the echo to be very narrow and to decay rapidly in time, and resulting inaccuracies in measuring the echo introduce much larger errors in the height values the coefficients: IjO = 0.56 i 1.4 and “1 = 0.056 jz 0.05. Further studies are in progress, with better stability in the dc magnetic field with higher 02 c’oncentration, to improve the accuracy of the results and possibly estimate the values of the succeeding terms in the expansion. We gratefully acknowledge the help of J. Anderson and the interest of Drs. K. Kawasaki and H. E. EM Van Leeuwen.