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Submitted on 1 Jan 1978
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DIFFUSION COEFFICIENT OF 3He IN SOLID 4He
A. Allen, M. Richards, J. Schratter
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8, Tome 39, aoiit 1978, page C6-113
DIFFUSION COEFFICIENT OF
3 ~ eIN SOLID
4 ~ eA.R. Allen, M.G. Richards and J. Schratter.
Physics Laboratory, University of Sussex, Brighton, BNI 9@, England.
R6sum6.- Ayant am6lior6 la mesure du coefficient de diffusion D pour des solutions diludes de 'He dan; 'He solide, on prdsente de nouveaux r6sultats pour des concentrations x3 en ,He comprises entre 10- et Le produit Dx3 n'est constant que si l'on a x,< 3 x
Abstract.- Im roved measurements of the diffusion coefficient D are reported for dilute solutions of
I:
3 ~ e in solid He at fractional impurity content (x,) from to
lo-'.
Dx, does not become constant until x, < 3 x lo-'.In 1969, Andreev and Lifshitz/l/ showed that any impurity (e.g. vacancy, interstitial, substitu- tional impurity) that can tunnel to neighbouring equivalent sites in a crystal lattice will become delocalised at low temperatures and move as a quasi- particle in a solid state band of width % M/T where
T is the tunnelling time. The only system where this effect has so far been clearly observed is in the motion of 3 ~ e impurity in solid 'He using NMR methods.
In 1972, Richards et a1 121 reported measure- ments of the spin diffusion coefficient D of ,He in solid 4 ~ e as a function of x,. and found that D
x
;'. This result, later confirmed by Grigoriev et a1 / 3 / , has presented two problems. A kinetic theo- ry for the motion of the impuritons, behaving as a gas of quasiparticles interacting with cross section
0 leads I41 to the expression
D = 1T~~~a'/12 x ,a. (1
where a is the lattice spacing and J3,/2r the tun- nelling frequency. The first problem was that in- dependent measurements of N14R line width suggested 'that 0 % a and this led to values of J3'1 that were improbably low. This problem has been resolved by showing /4,5/ that the model used for the linewidth was inadequate and that
a
is probably betxeen 10a2 and 100a2, leading to an acceptable value for J34' The second problem concerns the effect that the interactions between impurity atoms would have. At sufficiently low concentractions they simply determine the cross section U and the quasi-parti- cles spend most of the time moving freely. At higher concentrations, the impuritons are interacting most of the time and various arguments /6,7/ have
led to the law
for the regime where the impurity-impurity interac- tion at spacing r is given by V(r) = vo(a/r),.
It now seems widely agreed that Eq.(l) is ap- plicable at sufficiently low x3 and Eq.(2) at some higher x,. What is not clear is
(i) What is the concentration below which Eq.(I)is followed.
(ii) Are the.early published data /2/ for which
3 x < x, < 5 x to be interpreted by Eq.(l)
or Eq.(2), bearing in mind that they were not accu-
-1 -413
rate enough to distinguish between x 3 and x3
.
It should be noted that impuritons are well defined excitations in the region covered by Eq.(2). This follows from studying the temperature depen- dence of D at different x, which shows /8,9/ regions where D falls as T rises.due to thermal excitations
(probably phonons) scattering the impuritons. This' shows that the impuriton mean free path for x3 < lC3 and T < 1K is greater than a, hence the lifetime is
-
1greater than J34
.
The measurements reported here represent an attempt to establish the dependence of D on x3 with greater accuracy. The method used is the same as in the earlier work /lo/, i.e. the damping ofNMR spin echoes due to a magnetic field gradient, using both the Carr-Purcell and sriuiulated echo techniques. EX-
tra care was taken over sdmple preparation, gradient calibration and repeated-data was taken so that er-. sors bars represent scatter of data taken many days apart, sometimes on different samples.
Figure 1 shows the data taken at two diffe- rent molar volumes. The temperature was between 0.5
K and 0.6 K where it'has been shown/3/ that D is
independent of temperature. The operating frequency was 2MHz and sample concentrations were taken from the gas mixtures from which the solid solutions were formed. These concentrations agreed within 10% with those given by the equilibrium NMR signal heights which at a given tmeperature should be pro- protional to x,. A calibrator in the spectrometer normalised the signals for variations in receiver gain and in quality factor of the receiver coil.
Fig. 1 : The spin diffusion coefficient D of 3 ~ e in solid 4 ~ e as a function of x,, the fractional 3 ~ e concentration. Circles refer to molar volume 20.95 cm3 and squares to 20.70 cm3.
Throughout the range of x,, the data fit ap-
-1.5
proximately the law D a x3
,
but it is probably more realistic to view the high x3 data as approa- ching the minimum in D that occurs Ill/ between x3 = 10-I and and low x, data as exhibiting the onset of the weakly interacting impuriton re--
1gime for which D a x,
.
The solid line on the fi-gure represents other recent data /9/ taken on the same apparatus but with different samples. There are two limited regions where the slope aRnD/aRn x, has a slope of -413 as predicted by Eq.(2). One is
-
4around 5 x and the other around 5 x 10
.
Between these two regions the slope reaches a value of -2.0.The values of Dx, given in the literature
-11 -1
vary from 0.8 to 2.8 x 10 cm2s
.
Most of this variation probably comes from fitting data points-
1to the law D a X, in regions where this investi-
gation suggests such.a law is not followed. The
effect of this is to obtain larger values of Dx, as the region of investigation is moved to lower x,. The data reported here suggests that assuming the slope aRn~/aRnx, does become equal to -1 at low enough x3 then the value of Dx, will be 3.0
*
0.3 x 10-l1 cm2s-I and the region of validity is x, <-4
3 x 10
.
These two numbers are internally consis- tent and agree with published estimates 151 of- 3
MJ34/Vo, namely about 3 x 10
.
The molar volume dependence of D is more di- fficult to explain. The two sets of values in figu- re 1 are for molar volumes 20.95 and 20.70 cm3 and show a difference of approximately a factor of 2 throughout the range. This implies a value of aLnD/a!?,nV of about 70 which is too large to be ac- counted for by the known /lo/ density dependence of J34'
References
/I/ Adreev, A.F. and Lifshitz, I.M., Sov. Phys. JETP
29
(1969) 1107; see also Guyer, R.A. and Zane, L.I., Phys. Rev. Lett.24
(1970) 660.121 Richards, M.G., Pope, J. and Widom, A., Phys. Rev. Lett.
2
(1972) 708./3/ Grigor'ev, V.N., Esel'son, B.N., Mikheev, V.A. and Shul'man, Yu. E., J.E.T.P. Lett. (1973) 16.
/ 4 / Sacco, J.E., Widom, A., Locke, D. and Richards, M.G., Phys. .Rev. Lett.
37
(1 976) 760./5/ Huang, W., Goldberg, H.A. and Guyer, R.A., Phys. Rev. (1975) 3374.
161 Landesman, A., and Winter, J.M., Proc. of the 13th Conf. on Low Temp. Phys. (LT13) Vo1.2, New York, Plenum Press, 1974, p.73.
/7/ Andreev, A.F., Sov. Phys. Usp.
19
(1 976) 137./ 8 / Mikheev, V.A., Esel'son, B.N., Grigor'ev, V.N.
and Mikhin, N.P., Sov. J. Low Temp. Phys.
2
(1977) 186./9/ Allen, A.R. and Richards, M.G., Phys. Lett. 65A (1978) 36.
-
/lo/ Richards, M.G., Pope, J., Tofts, P.S. and Smith J.H., J. Low Temp. Phys.
