Effect of He4 on exchange in h.c.p. solid He3

Solid State Communications,

Vol. 7, pp. 1521—1525, 1969.

Pergamon Press.

Printed in Great Britain

4 ON EXCHANGE IN h.c.p. SOLID He3

EFFECT OF He

W.N. Yu* and H.A. Reich IBM Watson Laboratory New York, New York 10025

(Received 28 August 1969 by J.A. Krumhansl)

NMR measurements of the exchange-lattice relaxatiQn time of h.c.p. solid He3 containing He4 impurity atoms show two time-constants. The frequency and temperature dependence of the first time-constant is characteristic of pure He3. The second time-constant is associated with impurity He4 atoms and may be analysed to give an average increased exchange parameter, as well as the number of He3 atoms influenced by each He4. A model previously proposed for b.c.c. He3 can be used to explain these results, but certain inadequacies appear.

IN A RECENT paper, Bernier and Landesman’ introduce a specific model for a He4 atom in a 3 lattice in order to explain the large observed He increase2 - ~ of exchange specific heat caused by He4 atoms in solid He3. BL extend the threebath model for pure He3 by dividing the lattice into two parts: (a) those He3 atoms near a He4, which are assumed to have a different value of the exchange parameter, J 4, and (b) the remainder of the lattice, assumed to have the pure bulk value, J3. See Fig. 1. In their analysis of the 3, BLfurther assume measurements on b.c.c. He that the two exchange baths are tightly coupled (via spin or atomic diffusion), so that they reach

4 SYSTEM

Z 4 i I

~---.~

I

i

~

L

I

I

I

I

3 SYSTEM

,,-‘

Z 3

a common temperature in a time short compared to TZE or T EL , the Zeeman-exchange and exchange-lattice relaxation times,

I

E3

FIG. 1. Three-bath model extended to two separated exchange baths. The dotted lines show relaxation processes which do not occur under bath, conditions this paper. is the Zeeman E the inexchange bath,Z and

In this letter we report measurements 3 containing He4 impurity atoms, on which h.c.p. Hethat the two exchange baths can remain indicate uncoupled for times much longer than TEL. The

1. the lattice. average value of J 4 number in the neighborhood each 4 and to give the of He3 atomsofhaving He this value. However, the model is inadequate to

spin—echo method was used. The observed longitudinal relaxation time T

1

explain all of the results.

can be analyzed in a simple way to yield an *

solid He3 is prepared (i) at a We perform the 4) following experiments: Pure (<5p.p.m. molar volume of He 19.65cm/mole. At a frequency

Physics Department, Columbia University, New York, N.Y. 1521

4 ON EXCHANGE IN h.c.p. SOLID He3

1522

Vol.7, No. 21

EFFECT OF He sec 40

-

o5pr,mHe4 0-Il 35 • -1520

~

~

.53K

~

t:I~Il~lIl2I

sec

f2’(MHz)2

3-

.532 K

~76K

-10

-8

-6

liii -4 -2

0

111111111111111 2 4 6 8 10

12

14

6

f2 .(MHz)2

FIG. 2. Long (T, 1’s) and short (7~)exchange-latticerelaxation times vs. ~~, showing relaxation characteristic o? isolated baths, at a molar volume of 19.65 cm3/mol. Temperatures and impurity content as shown. Points without error bars indicate small error, typically 1—2 per cent.

near 2.85MHz, and a temperature near 0.5 K, TEL 40 sec and TZE 0.1 sec. We measure J by observing the recovery of magnetization after a single3TzE, 90°pulse, the observations so thatstarting the Zeeman and exchange after -~ systems have a common temperature. The recovery of magnetization is represented by

M(t)

=

M(~’o)(1

—

Aet /T)

(1)

For TZE << TEL /

=

C

/

212

TEL (~i+~L)= TEL (~i+)

(2)

9~

E

2/kT2 is the for the h.c.p. lattice. N(}I’f) wi)2/kT2 Zeeman specific heat, C~ CE ~(—3/32)Nz is the exchange specific heat, I and J are the =

=

Zeeman and exchange frequencies, and Z is the

number of nearest neighbors. The plot of T 1 vs. f2 yields a straight line, from which we obtain the ratio C~/CE and TEL , the exchangelattice relaxation time. We also observe that TEL depends on temperature as TEL ~ T7 below ~0.6K. A = CE/(CE + Cz) may also be used to deduce CE /Cz, with consistent results. (ii) The same experiment with, say, 1135 p.p.m. He4 yields two time constants for the exchangelattice relaxation in the same frequency and temperature range. The observed form is

M(t)

=

M(~o)(B

-

C 3 et /T3

-

C4et /T4). (3)

This form of relaxation has been previously observed,3 and attributed to an ‘extra’ bath in series with the three bath model.

4 ON EXCHANGE IN h.c.p. SOLID He3

Vol.7, No.21

1523

EFFECT OF He

IOC

i

1

sec

I

I

5ppm He4 +~-

60 625

T~

o.r

FIG. 3. Long

I 1.6

I 1.7

I 1.8 1.9 I/I (log scale)

I 2.0

0K1

(T 1, T3) and short (1’4) exchange-lattice relaxation times vs. temperature. The slope of the straight lines is 7.

3) and In T Fig. 2 we plot T1 (for nearly pure4He vs. j2 3 different for various concentrations of He for two temperatures. Using Eq. 2 we observe that T 2, as 3 yields 2the same value are of Jthe T thus deduced 1. All of Jand do not depend on the same as the for values pure He3 He4 concentration, x.

recovery leads to the conclusion that cross relaxation, either between the two different Zeeman baths, or the two different exchange baths, is slow compared to the longest relaxation time, l’s. In Fig. 3 we plot the 1/T dependence of T

1, T3, and 1’4, taken at a fixed frequency of We plot 1’4 vs. ~2 for several concentrations and temperatures in the lower part of Fig. 2. Evidently is obeyed of temperature and we(2)thus deduce independent another J value, j 4, on 2 = (2/9)12 where ~2 is the intercept using the j2 Jaxis. Observation of two time-constant

2.852MHz, on a log—log scale. All values of T3, and the one available point of T1,7fall within line. 1’4 also experimental error on the same T follows a T ~ law, ality depends on x. but the constant of proportion-

4 ON EXCHANGE IN h.c.p. SOLID He3

1524

Vol.7, No.21

EFFECT OF He

Table 1. Values of exchange parameter, J, and number of He3 atoms, N, associated with one He4 atom as deduced from the amplitudes B, C 3, and C4 of the two-time-constant exchange-lattice recovery x

C3

B*

C4

p.p.m.

J/27T

N

MHz

1135

0.978

0.287

0.377

1.25

364

±44

6

8

8

0.10

0.094 8

0.386 8

1.46

35% 439 30%

1520 ±34

0.910 28

0.06

at Sp.p.m., J/27T 35p.p.m., J = 0.69 ±0.065 at 1l 3/27T = 0.70 ±0.15 *

B is an instrumental parameter which would be unity if both 90—180°pairs had the same separation. They are intentionally unequal, but this introduces no error in the analysis.

The model predicts at low temperatures two relaxation times TZE3 and TZE4. From the known dependence on J, TZE4 should be of the order of 2Omsec. We have looked for this component of TZE, but have observed that only the component TZE3 is apparently always3. present and has the same value as for pure He The parameters C 3 and C4 in (3) depend on the initial temperature of the 3 and 4 baths, which in turn depend on the pulse sequence used to observe the spin echo. Two pairs of 90—180° pulses separated by a time t are used, starting from equilibrium. After is the~, first pulse,thermal the Zeeman temperature and90° the exchange temperature is that of the lattice. After a time t 3TZE 3 << T3, T4, each exchange bath comes to equilibrium with its corresponding Zeeman bath, Neglecting the portion of the energy leaking into the lattice, from energy conservation we have /

=

~i +

~_

F

L

C

\~‘

(1

=

\

EJ

+

p)~’

(4)

L

and

M(t)

where N3 and N4 are the number of spins in the 3 and 4 systems, TF and TL are the final and lattice temperatures, and p is the ratio of specific heats. We thus have -1

-i

B—C3 —C4 =(1—xN)(1+p3) +xN(1+p4) (6) where N is number of He3 atoms influenced by one He4 atom. Table 1 lists the results obtained for two different impurity concentrations. We may regard the two N-values as identical for the present purposes. The number of atoms in a sphere out to the fourth nearest-neighbor is 7 has calculated the shift in position 322. Glyde of a second-nearest-neighbor He3 atom due to a He4 atom in a b.c.c. lattice and obtained 0.6% with negligible shift further out. If we assume similar results for the h.c.p. lattice, this is too small an effect to explain N values. 4 atomsour arelarge separated by 8 At xatoms = 1135 p.p.m., He on the average, so that for N = 300 half of the He3 atoms are influenced by He4. For higher concentrations, a greater fraction is influenced, and for the concentration used previously by Garwin and Reich,2 there is indeed no He3 atom farther than twice the unit cell dimension a from a He4 atom, thus explaining the absence of the

B

M(~’c)

—

C 3

=

N +N ~

—

C4 (1

+

two time constants in their TEL data.

P3)

+

N +N (1 ~

+

P4 )~ (5)

One major flaw appears in this 5-bath model. 4-concentrations We have that atbylow He each He4assumed is surrounded a sphere of fixed size,

Vol.7, No.21

4 ON EXCHANGE IN h.c.p. SOLID He3

EFFECT OF He

containing -‘~300atoms, having a common temperature and an average J 4 value different 3. If we measure J from that of bulk He 4 by the ratio of specific heat, CZ/CE, then we should obtain value of x for small x, We should aalso findindependent the temperature dependence of

T4 to be the same for all x. Instead, we find J to be increasing almost linearly with x, and thewould magnitude of 1’4 tofrom be 2,as be expected proportional to J~ Griffiths’ two-phonon theory8 for a Debye solid, The increased relaxation rate, however, cannot be attributed to a disturbance of the phonon spectrum, since J 3 and 1’3 are unchanged, A conjecture that spin diffusion may be quenched9 as a mechanism for coupling the two exchange baths is apparently satisfied in the h.c.p. phase. In the b.c.c. phase, the BL assumption of rapid equilibrium of the 3 and 4 exchange baths is not required to explain the absence of two component recovery. It is well 6

established

that atomic diffusion provides a

1525

mechanism for relaxing exchange energy, and indeed if the mean jump-time were the same for both baths, both would be observed to relax with the same, single time constant TEL. On the of 7 dependence other hand, from the observed T both T 3 and T4, we may conclude that atomic diffusion is not important in the h.c.p. phase, and hence the two baths can relax at different rates, governed by their respective J values. In summary, we have deduced that two separated exchange baths exist in solid He3 containing impurity He4. From the frequency variation of the two components of exchangelattice relaxation two J values may be obtained, 3 one the observed be identical to that ofdependence bulk He and other to larger. The temperature of both components is T7. A model which attempts to explain the increased J near a He4 atom fails to account for the fact that the observed increase depends on the concentration of He4 atoms present. Acknowledgement — We wish to acknowledge stimulating conversations with R.L. Garwin.

REFERENCES 1.

BERNIER M. and LANDESMAN A., Solid State Commun. 7, 529 (1969). Hereafter referred to as BL. GARWIN R.L. and REICH H.A., Phys. Rev. Lett. 12, 354 (1964).

2. 3. BEAL B.T., GIFFARD R.P., HATTON J. and RICHARDS M.G., Phys. Rev. Lett. 12, 393 (1964). 4. GIFFARD R.P. and HATTON J., Phys. Rev. Lett. 18, 1106 (1967). 5. RICHARDS M.G., HATTON J. and GIFFARD R.P., Phys. Rev. 139, A91 (1965). 6.

GARWIN R.L. and LANDESMAN A., Phys. Rev. 133, A1503 (1964).

7.

GLYDE H.R., Phys. Rev. 177, 272 (1969).

8.

GRIFFITHS R.B., Phys. Rev. 124, 1023 (1961).

9.

REDFIELD A.G. and YU W.N., Phys. Rev. 169, 443 (1968).

Des mesures, par NMR, du temps de relaxation d’~changedu réseau, pour He3 solide sour la forme h.c.p. et contenant des impuret~s de He4, indique l’existence de deux constantes de temps. La variation de la premidre constante de temps avec la fréquence et La temperature est caractéristique de l’He3 ~ l’état pur. La seconde constante de temps est associée avec les impuretés d’He4 et son analyse conduit a un paramétre d’échange effectif plus élevé et permet d’obtenir egalement le nombre d’atomes d’He3 influences par chaque atome d’He4. Un moddle propose précédemment pour l’He3 b.c.c. peut être utilise pour expliquer ces résultats, mais certaines insuffisances apparaissent.