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Nuclear magnetic resonance study of hydrogen diffusion in palladium and palladium-cerium alloys
43
Journal of the Less-Common Metab, 39 (1975) 43 - 54 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
NUCLEAR MAGNETIC RESONANCE STUDY OF HYDROGEN DIFFUSION IN PALLADIUM AND PALLADIUM-CERIUM ALLOYS
DAVID A. CORNELL* Physics Department, (Received
June
and E. F. W. SEYMOUR University of Warwick, Coventry CV4 7AL (Ct. Britain)
20, 1974)
Summary Proton spin-lattice relaxation measurements have been carried out on /3-PdH, and (Pd-Ce)H, in the temperature range 100 - 300 K and at resonance frequencies of 7, 11 and 47 MHz. Contributions to the relaxation rate due to conduction electrons and due to hydrogen diffusion are identified. Measurements on fl-PdHo.T, undertaken both as a reference for the work on the alloys and also in an attempt to resolve an apparent discrepancy between earlier NMR and neutron scattering (Beg and Ross 1970) determinations of hydrogen diffusion rates, yield an activation energy for diffusion of 0.228 f 0.010 eV for temperatures above 195 K; below 195 K interpretation of the data is uncertain. The relation with neutron scattering data is discussed. The difference between diffusion coefficients for the pure metal, and alloys containing 3% and 6% cerium, is less than 30% and it is concluded that the considerably higher hydrogen permeation rates in the alloys (Wise et al., 1972) are due to differences in surface processes.
1. Introduction The motivation for the present study was the observation by Wise et al. that alloying palladium with a few percent. of cerium not only increases the mechanical stability during hydrogen cycling below the critical temperature at which the Pd-H miscibility gap disappears, but it also significantly increases the rate at which hydrogen is able to permeate through a foil of the metal. Such foils are thus of interest as commercial diffusion membranes [2] for the purification of hydrogen and possibly for fuel cells. The increase in permeability suggests that the hydrogen bulk diffusion coefficient may be considerably larger in the alloys than in pure palladium; indeed, the fact that the lattice parameter increases by several percent. on alloying might lead one to expect that this would be so. Proton spin-lattice relaxation measurements have been made in many transition metal-hydrogen systems, including palladium-hydrogen [3 - 61, [l]
* Permanent
address:
Principia
College,
Elsah, 111.62028,
U.S.A.
44
as a means of deducing hydrogen diffusion jump times and, hence, diffusion coefficients. It was decided therefore to use this technique to investigate alloys in which the o-p phase immiscibility for the hydrides is partly removed (Pda.s, CeO.ca) and totally removed (Pd,-,s4 Cec.es). To serve as a reference for the observations on the alloys, it was necessary to make measurements also on the palladium-hydrogen system itself because of some mutual inconsistency among the results of the earlier workers. Results for PdH, are also of considerable interest since earlier results obtained by nuclear magnetic resonance led to values of hydrogen jump times in apparent disagreement with those deduced from quasielastic neutron scattering measurements [ 71. The method relies on the fluctuating dipole-dipole’interaction between diffusing nuclei, which produces spin-lattice relaxation characterized by a time, TID, given by 1 -=+1(1+ T 1D
1) {J&.+)
+Jz(2w.)}
(1)
where y is the nuclear gyromagnetic ratio and I the nuclear spin, h = h (Planck’s constant)/2n, the J’s are spectral densities of the power frequency spectrum of the fluctuations and o,/2n is the proton resonance frequency. In our case, only interactions between protons need be considered since both the moment and abundance of lo5Pd are small and cerium nuclei do not carry a magnetic moment. Assuming that protons jump between neighbouring octah+dral interstitial sites of the f.c.c. lattice with a mean time between jumps, 7, we may use Torrey ‘s [ 81 evaluation of eqn. (1) 1 -=
8n y4h2W + 1) ,__s;s;sw
TED
cp(k,y) 0
where c is the fraction of octahedral sites occupied, n, = 4 is the number of sites in a unit cell, a is the lattice constant, 1 = a/42, the jump length, k a constant equal to 0.7428 for a f.c.c. lattice, y = w,,r/2 and cp(k,y) is a function tabulated by Torrey. In this case, c = X, the hydrogen-metal atom ratio, since there is one octahedral site per metal atom. The jump time, T, is expected to follow an Arrhenius relation T = T- exp(E,/k,T)
(3)
and is related to the diffusion coefficient by D = 12/6r. The temperature dependence of TlD arises almost entirely from that of 7. Relaxation is most efficient when w,~ = 1.06 where q(k,y) is a maximum; in the low-temperature limit (a.7 B l), T,D is proportional to w 27 and in the high-temperature limit (0.7 < l), T1D is proportional to 7-l. Other competing relaxation processes must be allowed for before analysing the results in this fashion. Most importantly, the protons interact with the conduction electrons to produce a relaxation rate Tie’, where it is expected that T1,T = constant. The observed rate becomes _LL+L. 7’1 T,D
(4) Tie
45
It is also possible that paramagnetic relaxation rate [9] .
impurities
may contribute
to the total
2. Experimental The palladium samples consisted of Johnson Matthey spectrographicallystandardized sponge. The alloy samples were filed from ingots made from the pure metals in an argon-arc furnace by Dr. I.R. Harris of The Department of Physical Metallurgy and Science of Materials, Birmingham University. The powders, of particle size less than 25 pm, were mixed with powdered quartz to ensure penetration of the r.f. field, loaded with hydrogen by exposure to the gas and sealed in glass tubes. The proton magnetization was measured following a 180 - 90 ’ radiofrequency pulse sequence, as a function of the time interval between pulses. In all cases, the magnetization recovered in a simple exponential fashion indicating that all the nuclei whose resonance was observed can be characterized by the same relaxation time, T1 . Figure 1 shows T1 in PdHe.,a, as a function of temperature, for three values of resonance frequency, w,. The curves show the characteristic frequency-dependent minimum due to diffusion, and converge at high- and low temperatures where relaxation is governed mainly by interaction with conduction electrons. The failure of the values measured at 7 MHz below 140 K to converge towards those measured at higher frequencies appears to be real and is not understood. Similar behaviour to that displayed in Fig. 1 was shown by specimens with lower hydrogen contents, except that the
100
a00
200
T
(K)
Fig. 1. Proton spin-lattice relaxation times in PdH 0.70 as a function of temperature at resonance frequencies of 7 MHz (+), 11 MHz (0) and 47 MHz (0).
46 loo0
300-
0
l+*O000 100+*Oo
T, tmg)
9 O” . 00
10
I
I
I
100
200
300 T (K)
Fig. 2. Proton spin-lattice relaxation times in (Pdo_g*Ceo& Ho.53 as a function of temperature at resonance frequencies of 7 MHz (+), 11 MHz (*) and 47 MHz (0).
e 1@ l
100
0
l
0 0
50
0
%r
00 0
s?@ OQO
loi T
(K)
Fig. 3. Proton spin-lattice relaxation times in PdHO.70 (0) and (PdOg, Ceo.03) Ho.72 (e) as a function of temperature at a resonance frequency of 11 MHz. minima were less deep, as expected from eqn. (2), and were displaced to lower temperat~es, confirming earlier observations that diffusion is more rapid in palladium with lower hydrogen contents. (An example is shown in Fig. 4.) Figure 2 shows corresponding data for (Pd0.s4 Ceo,06) HoS6s,an alloy in which the hydrogen solubility is less than for pure palladium. Rgures 3
47
o
I
I
I
loo
200
300 T(K)
Fig. 4. Proton spin-lattice relaxation times in PdH 0.56 to) and Wo.64 as a function of temperature at a resonance frequency of 11 MHz.
Ce0.06)
Ho.53
to)
and 4 show the effects of alloying with cerium for (almost) fixed hydrogen concentration; comparisons are made between PdHO.,O and (Pdo.s7 Ceo.os)H o.72, and between PdHo.56 and (Pdo.a4Ceo.06)Ho.5s. The hydrogen concentrations are rather uncertain and can probably not be relied on to better than 0.03. All the alloys are single phase except PdHa.56 which is likely to be a mixture of (11and p phases but in which the great majority of the hydrogen is in the /3 phase of composition about PdHa.64; it is this phase which provides, essentially, all the observed signal from this sample. 3. Discussion 3.1. Palladium-hydrogen We consider first the results for PdHO.‘IO. The simplest method to evaluate the jump time, 7, and its temperature dependence is to use the theoretical result that w,r = 1.06 at the temperatures corresponding to minima of T1. For this purpose it is not necessary to extract the relatively-weakly temperature-dependent T,, from the observed T, values. It is found immediately from Fig. 1 that r is 24 ns at 230 K, 15 ns at 234 K and 3.6 ns at 270 K; these results can be combined to provide a rough value of the activation energy, E,, for diffusion of (0.24 + 0.02) eV. The minima observed by Gil’manov and Bikchantayev [6] at 16 and 32 MHz, and by Burger et al. [5] at 30 MHz, are consistent with this result. It is noteworthy that the value of the activation energy obtained in this way is not dependent on the details of Torrey’s model; it relies solely on the result that the minimum of T, occurs for some fixed value of w ,r independent of frequency.
48
With these values of 7 and E, one can at once derive a value for 7i1 from eqn. (3). The result is 7Z.l - 7.5 X 1012 Hz, or, if one uses a more precise value of E, = 0.228 eV, derived from the whole of our data as will be -’ = 4.8 X 1012 Hz. It is of interest to compare this described shortly, rm jump attempt frequency with the frequency of “optical” vibration of hydrogen in the palladium lattice (v,,~~= 13.5 X 1012 Hz) as determined by Bergsma and Goedkoop [lo] from neutron cross-section measurements and Chowdhury [ 111 from inelastic neutron scattering. As discussed by Korn [ 121 (who analysed the case of TiH,) one can expect 7Z1 s (1 -c)LJ,~~ where (1 - c) is the probability of finding any given neighbouring site unoccupied. In our case (1 -c) = 0.3 and (1 -c)vopt = 4 X 1012 Hz. The agreement with 7Z1 is perhaps fortuitously good; nevertheless, it provides confidence in the interpretation of the minimum of Tr. To make use of the data over the whole temperature range it is necessary to make an allowance for Ti,. Now Ti, is unknown, but one can attempt to deduce it from the measured [9] proton Knight shift, K, and the Korringa relation. Assuming that the whole of K and T,, arise from a direct Fermi contact interaction with conduction electrons in an s-like band (in PdHa.,e it is thought that the 4d band is filled), and inserting a Korringa enhancement factor of 0.77, reflecting the effect of electron-electron interactions appropriate to the conduction electron density, one finds Ti, = 1.74 s at 180 K. This value is far too large to fit our results and we must conclude that either K or C,, or both, are not entirely produced by the direct contact interaction, or, alternatively, the zero from which K was measured was inappropriate. As an alternative procedure we have carried out a computer analysis of the Ti data at temperatures well away from the minimum to produce the best fit to eqn. (4) with Tr,T = constant and Tin exponentially dependent on reciprocal temperature. The best fit is obtained with Tr,T = 68 f 2 s K, independent of frequency as it should be. It is then possible to compare the minimum values of Tin observed with the predictions of eqn. (2) in which (since we are dealing with the minima) there are no adjustable parameters. Using a = 0.404 nm and cp(h,y),,, = 0.4387, we find the results given in Table I. Within the model, the chief uncertainty in the theoretical result comes from that in the hydrogen concentration (about 5%). Agreement is good, though the theory slightly overestimates the relaxation rate (as has also been found [14] for TiHi.ss and &Hi.,). As a final test of the model we may use eqn. (2) to deduce values of y and hence T over the whole range of temperature covered by the measurements. Since (T1n)min is slightly different from the theoretical value, y(T) was deduced by equating experimental values of TID/(Trn)m~ to 0.4387/q(h,y), equivalent to treating the concentration, x, as a (slightly) adjustable parameter. Figure 5 shows an Arrhenius plot of log 7 against reciprocal temperature from the data of Fig. 1 for PdHa.,e at all three frequencies. From 195 K to 330 K (a range including the minima of 2’1) the data are well fitted by an activation energy, E, = 0.228 + 0.010 eV. Below 195 K the scatter is much worse but it seems clear that at about this temperature there is a definite break in the
49
TABLE
I
Experimental minimum values of 2’1~ for PdHO.,o compared with theoretical values from Torrey’s random walk model Frequency
Temperature
(Tl)n,in observed
(Tl~)ti
expt. corrected for T1,
(MHz)
W)
(ms)
(ms)
(ms)
47 11 7
270 234 230
59 _+5 18.2 ?r 0.8 11.4 f 0.4
77 +6 19.4 + 0.8 11.9 f 0.4
71 16.6 10.5
slope of the curve; if one leaves aside the 7 MHz data below 140 K as “anomalous” one could describe the results in the range 110 - 195 K by an activation energy, E, = 0.10 + 0.02 eV, but this is clearly much less reliable than the derived value for E, above 195 K. A similar discontinuity in NMR data at about this temperature has been observed before by Torrey [4] and by Burger et al. [5]. The physical reason for the change at this point is unknown. It is possible that the break indicates some change in the diffusion mechanism. It is well-known that it is in the “slow motion” regime below the Tr minimum that NMR is particularly sensitive to the details of the diffusion process. No other properties appear to show discontinuities at this
I 3
1
5
6
7
s
8
WOO/l
I W
(K-‘)
Fig. 5. Proton jump time, T, as a function of reciprocal temperature in PdHo.70, derived from 2’1 values (Fig. 1) measured at ‘7 MHz (+), 11 MHz (0) and 47 MHz (0). The hatched region indicates the range in which T1 minima occur.
50
temperature however. A transition, possibly an order-disorder transition, is known to occur in PdH,, but at too low a temperature (50 K) to be relevant here. Moreover, a change in the sign of the isotope dependence of diffusion in PdH, has been found in the neighbourhood of 100 K [15] but this also appears to occur at too low a temperature to be related to the discontinuity in the NMR data. Very recently, Arons et al. [ 161 have found no evidence of any discontinuity in the related quantity T,, (rotating frame spin-lattice relaxation time) for protons in p-PdH down to 170 K. All this may point to the alternative possibility that there is no change of E, at this temperature but that either the temperature dependence of Tl, changes or, more likely, some additional relaxation rate which increases as the temperature decreases is present. The latter contribution could arise from a small amount of paramagnetic impurity, e.g., dissolved iron, in the sample (though it is diffic~t to envisage at what stage the impurity entered the originally pure material). A relaxation rate of order 0.2 s-l to 1 s-’ at 130 K and varying as Tl (impurity) a T would account, roughly, for the observed behaviour. The scatter between breakaway temperatures (195 - 230 K) found by different authors [ 4, 51 might then be accounted for by differing impurity contents. We now consider briefly the effect of changing hydrogen concen~tion. The minimum of T,, changes in depth and temperature with changing concentration, X. According to eqn. (Z), (TzD)min is inversely proportional to 3~.From Figs. 1 and 4 the ratio of the minimum values for PdHovTOand PdH0.s4 is 18.2/20.2 2 0.90 (the correction for T1, hardly affects the ratio) in agreement with the inverse ratio of concen~ations 0.64/0.70 r 0.91. Within experimental error, the change in the temperature of the minimum can be accounted for solely by the change in the factor (1- c) occurring in T:‘, without any change in Vopt or E,. Moreover, one can again fit the whole T1 curve for PdH0.s4 down to 195 K assuming TI,T and E, retain the same values as for x = 0.70. Of course, one might expect both of these quantities to change with x due to changing conduction electron density and lattice parameter; experimental accuracy enables one to say that the former changes by less than 10% and the latter by less than 5% over the rather small accessible range of x in the p phase. Although in the temperature range 195 - 330 K an internally consistent interpretation has been achieved in terms of nearest-neighbour octahedraloctahedral jumps, a serious difficulty remains in that there is a large discrepancy between all the NMR data and the quasielastic neutron scattering data [ 7) obtained in overlapping temperature ranges. A summary of some of the available data for E, is given in Table II. Apart from the discrepancy between the activation energies derived by the two methods, and between r values (our value of r at 293 K is 40 times larger than that of Beg and Ross [7] at the same temperature), the neutron scattering data are not fitted by a model involving only nearest neighbour octahedral-octahedral jumps. The suggestion of Beg and Ross that NMR may be observing a minority of protons for which 7 is much greater than the average is not now tenable in
51
TABLE
II
Experimentally
determined
activation
energies
for hydrogen
diffusion
Method
Temperature (R)
Activation energy (eV)
Source
NMR: 7-r
195 - 330 100 - 195
0.228 f 0.010 0.10 f 0.02
present
work
NMR: T1
> 220 < 220
0.24 0.08
Torrey
[4 ]
NMR: T1
230 - 320 180 - 230
0.21 0.06
Burger et al. [ 5 ]
NMR: T1
180 - 350
0.198
Gil’manov
in &PdH
et al. [ 61
NMR: T1,
170 - 300
0.225
Arons et al. [ 161
Neutron scattering
290 - 470
0.146
Beg and Ross [ 71
Macroscopic diffusion
290 - 400
0.25 i: 0.01
Wicke and Bohmholdt
[ 171
view of the agreement displayed in Table I, where the theoretical values of Vi ~)mti assume that all the protons are involved; further, around the temperature of the minimum any protons not efficiently relaxed by diffusion would give rise to a tail on the observed signals with a decay constant of T1,. Such tails were looked for but not observed. We have not analysed in detail a much more complicated situation envisaged by Beg and Ross, where the fact that most octahedral sites are occupied is assumed to encourage both rapid octahedral-tetrahedral jumps and jumps between nonnearest neighbour sites as well as nearest octahedral-octahedral jumps, but we stress that here again the observed NMR signals cannot be produced by a minority jump process with long 7 in the presence of a majority jump process with much shorter 7. To reconcile the NMR and neutron data is thus still a problem. In the meantime there are several points worthy of comment. Firstly, 7 values derived by the two methods may differ genuinely since, in neutron scattering, one is concerned with a simple position vector correlation function and in NMR with a more complicated tensor correlation function, but the difference on this account will not exceed a factor of three. Secondly, the Torrey theory contains some approximations, notably that of isotropic diffusion, and this can, in principle, cause errors in the 7 values deduced. However Sholl [ 181 has shown that the exact formulation of the theory for an f.c.c. lattice gives r values, for given TID, which are only a few per cent. different from the Torrey values. It may be noted that the Sholl formulation, besides being exact within the model, is also more readily adaptable to the discussion of more complicated jump processes. Thirdly, consideration of temporal correlations between successive jumps of the type discussed by Eisenstadt and Redfield [ 191 and by Wolf [ 201 can change the r values
52
deduced, and since their effect can be temperature dependent, can change the value of E, also, but neither effect can be expected to be substantial, particularly in this case where the concentration of vacancies in the hydrogen lattice is relatively large (2 30%), so that the distinction between a “jump” of a proton and an “encounter” with a vacancy becomes unimportant. Fourthly, because NMR observations involve generally longer r values than neutron scattering observations, the approximation of neglect of the time actually spent by the proton in jumping between sites (- lo-l3 s) is a particularly good one in interpreting NMR data. Fifthly, comparison between NMR and neutron scattering data has been made [21] for one other system, /3-NbH, and agreement in that case is good, so that the problem may be peculiar to palladium. 3.2. Palladium-cerium-hydrogen The remaining uncertainties in the interpretation of the data on fl-PdH need not prevent us from making a comparison of Tin values for P-PdH and (Pd-Ce)H, at any rate near ( T1,),,. A somewhat surprising result, immediately evident from Figs. 3 and 4, is that the temperatures of the minima are the same, within experimental error, for the alloys as for pure palladium. The 11 MHz data, for instance, yield the following temperatures of the minima: PdHo.70,
239 * 3 K
(Pdo.e7Ceo.es)Ho.72,236
+4 K
PdHo.56, (Pdo.e&eo.os)Ho.53,222
225 + 2 K * 2 K
and the minima shift with frequency as before. This means, at these temperatures at any rate, that the diffusion coefficient is not much increased, if at all, by addition of cerium; taking full advantage of the errors quoted above, an upper limit for any change can be set at 30%. This is in strong contrast with the increased permeation rates observed (such experiments are necessarily made at higher temperatures) by Wise et al. [l] in these alloys, and suggests that the more rapid permeation through membranes is not an effect of increase in lattice constant on bulk diffusion rates, but of more rapid surface reactions. A complete analysis of the T1 - T curves for the alloys is a matter of some difficulty. From the evidence of the undisplaced T,, minima, it is natural to expect that the activation energy, E,, is also unchanged between the pairs of alloys compared in each of Figs. 3 and 4. If we allocate the relaxation rate which remains after subtracting the corresponding values of TIDel to a conduction electron term as before, the apparent T1,T values judged from the higher temperature data are about 2/3 and l/4 of that for PdH0.70 for the 3% and 6% Ce samples, respectively. The constancy of T1,T above 195 K is not by any means so convincing as for PdH,, and the deviations below about 195 K are considerably greater. The faster conduction-electron-induced relaxation rates are unexpected, since susceptibility data [ 221 on palladium-cerium solid solutions indicate that cerium is non-
53
magnetic, merely donating four electrons per atom to the conduction band. If this is true also for (Pd-Ce)H, we might expect Ce to have very little effect on T,,. However, since we do not understand T1, in PdH, quantitatively, we cannot be sure. If the extensive deviations below about 195 K are not associated with TID, then there is a third relaxation contribution to explain. The most likely candidate is a paramagnetic impurity which might be traces of PdsCe, with which a 4f virtual bound state is associated [ 231. However, the additional relaxation rate does not appear to be inversely proportional to temperature as was roughly the case for PdH, below 195 K and for strongly paramagnetic rare earth-hydrogen alloys [9]. In view of these uncertainties, extraction of T1 ,,-l values from our data is only justified near the minimum where TIDM1 dominates. 4. Conclusion NMR measurements of T1 in P-PdH, are quantitatively explained at temperatures above 195 K by a model involving diffusion by octahedraloctahedral jumps. The activation energy deduced is in reasonable agreement with the results of macroscopic diffusion measurements, but in serious disagreement with neutron scattering results. At room temperature the macroscopic results of Wicke and Bohmholdt [ 171 yield a value of D of low6 cm2 s-l; the neutron scattering value is about 3 times larger and our NMR value about 10 times smaller than this. These discrepancies remain to be explained. Below 195 K, interpretation of our data is uncertain probably due to the effects of paramagnetic impurities. Although a full analysis of the T1 data for (Pd-Ce)H, has not been possible at all temperatures, the minima of T1 are unshifted within experimental error from the corresponding minima in PdH,. It is concluded that D is not increased by more than 30% by alloying with cerium so that the known permeation increase is probably associated with surface phenomena. Acknowledgements This work was supported by a grant from the Science Research Council. We are grateful to Dr. I. R. Harris of The University of Birmingham who supplied the alloy specimens and we wish to acknowledge helpful discussions with Mr. D. K. Ross, Professor D. Zamir, Dr. C. A. Sholl and Dr. P. P. Davis. References 1 M. L. H. Wise, J. P. G. Farr, I. R. Harris and J. R. Hirst, L’Hydrogene dans les Metaux, l(l972) 1. 2 J. P. G. Farr and I. R. Harris, Brit. Pat. Appl. 25512/70. 3 R. E. Norberg, Pbys. Rev., 86 (1952) 745. 4 H. C. Torrey, Nuovo Cimento, Suppl. IX, (1958) 95. 5 J. P. Burger, N. P. Poulis and W. P. A. Hass, Pbysica, 27 (1961) 514. 6 A. N. Gil’manov and I. G. Bikchantayev, Phys. Metals Metallgr., 31 (3) (1970) 64.
54 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
M. M. Beg and D. K. Ross, J. Phys. C, Solid State Phys., 3 (1970) 2487. H. A. Resingand H. C. Torrey, Phys. Rev., 131 (1963) 1102. L. Shen, J. P. Kopp and D. S. Schreiber, Phys. Letters, 29A (1969) 438. J. Bergsma and J. A. Goedkoop, Physica, 26 (1960) 744. M. R. Chowdhury, Ph. D. Thesis, Univ. Birmingham, 1971. C. Kom, Ph. D. Thesis, Weizmann Institute of Science, Rehovot, 1971. A. Fert and P. Averbuch, J. Phys. Radium, 25 (1964) 20. H. T. Weaver and J. P. VanDyke, Phys. Rev., B6 (1972) 694. Y. de Ribaupierre and F. D. Manchester, J. Phys. C, Solid State Phys., 6 (1973) L391. R. R. Arons, H. G. Bohn and H. Lutgemeier, Solid State Commun., 14 (1974) 1203. E. Wicke and G. Bohmholdt, 2. Physik. Chem. Frankfurt am Main, 42 (1964) 115. C. A. Sholl, J. Phys. C, Solid State Phys., 7 (1974) 3378. M. Eisenstadt and A. G. Redfield, Phys. Rev., 132 (1963) 635. D. Wolf, Z. Naturforsch., 26a (1971) 1816. B. Alefeld, H. G. Bohn and N. Stump, Ber. Bunsenges. Phys. Chem., 76 (1972) 781. I. R. Harris and M. Norman, J. Less-Common Metals, 15 (1968) 285. W. E. Gardner, J. Penfold, T. F. Smith and I. R. Harris, J. Phys. F., Metal Phys., 2 (1972) 133.
Journal of the Less-Common Metab, 39 (1975) 43 - 54 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
NUCLEAR MAGNETIC RESONANCE STUDY OF HYDROGEN DIFFUSION IN PALLADIUM AND PALLADIUM-CERIUM ALLOYS
DAVID A. CORNELL* Physics Department, (Received
June
and E. F. W. SEYMOUR University of Warwick, Coventry CV4 7AL (Ct. Britain)
20, 1974)
Summary Proton spin-lattice relaxation measurements have been carried out on /3-PdH, and (Pd-Ce)H, in the temperature range 100 - 300 K and at resonance frequencies of 7, 11 and 47 MHz. Contributions to the relaxation rate due to conduction electrons and due to hydrogen diffusion are identified. Measurements on fl-PdHo.T, undertaken both as a reference for the work on the alloys and also in an attempt to resolve an apparent discrepancy between earlier NMR and neutron scattering (Beg and Ross 1970) determinations of hydrogen diffusion rates, yield an activation energy for diffusion of 0.228 f 0.010 eV for temperatures above 195 K; below 195 K interpretation of the data is uncertain. The relation with neutron scattering data is discussed. The difference between diffusion coefficients for the pure metal, and alloys containing 3% and 6% cerium, is less than 30% and it is concluded that the considerably higher hydrogen permeation rates in the alloys (Wise et al., 1972) are due to differences in surface processes.
1. Introduction The motivation for the present study was the observation by Wise et al. that alloying palladium with a few percent. of cerium not only increases the mechanical stability during hydrogen cycling below the critical temperature at which the Pd-H miscibility gap disappears, but it also significantly increases the rate at which hydrogen is able to permeate through a foil of the metal. Such foils are thus of interest as commercial diffusion membranes [2] for the purification of hydrogen and possibly for fuel cells. The increase in permeability suggests that the hydrogen bulk diffusion coefficient may be considerably larger in the alloys than in pure palladium; indeed, the fact that the lattice parameter increases by several percent. on alloying might lead one to expect that this would be so. Proton spin-lattice relaxation measurements have been made in many transition metal-hydrogen systems, including palladium-hydrogen [3 - 61, [l]
* Permanent
address:
Principia
College,
Elsah, 111.62028,
U.S.A.
44
as a means of deducing hydrogen diffusion jump times and, hence, diffusion coefficients. It was decided therefore to use this technique to investigate alloys in which the o-p phase immiscibility for the hydrides is partly removed (Pda.s, CeO.ca) and totally removed (Pd,-,s4 Cec.es). To serve as a reference for the observations on the alloys, it was necessary to make measurements also on the palladium-hydrogen system itself because of some mutual inconsistency among the results of the earlier workers. Results for PdH, are also of considerable interest since earlier results obtained by nuclear magnetic resonance led to values of hydrogen jump times in apparent disagreement with those deduced from quasielastic neutron scattering measurements [ 71. The method relies on the fluctuating dipole-dipole’interaction between diffusing nuclei, which produces spin-lattice relaxation characterized by a time, TID, given by 1 -=+1(1+ T 1D
1) {J&.+)
+Jz(2w.)}
(1)
where y is the nuclear gyromagnetic ratio and I the nuclear spin, h = h (Planck’s constant)/2n, the J’s are spectral densities of the power frequency spectrum of the fluctuations and o,/2n is the proton resonance frequency. In our case, only interactions between protons need be considered since both the moment and abundance of lo5Pd are small and cerium nuclei do not carry a magnetic moment. Assuming that protons jump between neighbouring octah+dral interstitial sites of the f.c.c. lattice with a mean time between jumps, 7, we may use Torrey ‘s [ 81 evaluation of eqn. (1) 1 -=
8n y4h2W + 1) ,__s;s;sw
TED
cp(k,y) 0
where c is the fraction of octahedral sites occupied, n, = 4 is the number of sites in a unit cell, a is the lattice constant, 1 = a/42, the jump length, k a constant equal to 0.7428 for a f.c.c. lattice, y = w,,r/2 and cp(k,y) is a function tabulated by Torrey. In this case, c = X, the hydrogen-metal atom ratio, since there is one octahedral site per metal atom. The jump time, T, is expected to follow an Arrhenius relation T = T- exp(E,/k,T)
(3)
and is related to the diffusion coefficient by D = 12/6r. The temperature dependence of TlD arises almost entirely from that of 7. Relaxation is most efficient when w,~ = 1.06 where q(k,y) is a maximum; in the low-temperature limit (a.7 B l), T,D is proportional to w 27 and in the high-temperature limit (0.7 < l), T1D is proportional to 7-l. Other competing relaxation processes must be allowed for before analysing the results in this fashion. Most importantly, the protons interact with the conduction electrons to produce a relaxation rate Tie’, where it is expected that T1,T = constant. The observed rate becomes _LL+L. 7’1 T,D
(4) Tie
45
It is also possible that paramagnetic relaxation rate [9] .
impurities
may contribute
to the total
2. Experimental The palladium samples consisted of Johnson Matthey spectrographicallystandardized sponge. The alloy samples were filed from ingots made from the pure metals in an argon-arc furnace by Dr. I.R. Harris of The Department of Physical Metallurgy and Science of Materials, Birmingham University. The powders, of particle size less than 25 pm, were mixed with powdered quartz to ensure penetration of the r.f. field, loaded with hydrogen by exposure to the gas and sealed in glass tubes. The proton magnetization was measured following a 180 - 90 ’ radiofrequency pulse sequence, as a function of the time interval between pulses. In all cases, the magnetization recovered in a simple exponential fashion indicating that all the nuclei whose resonance was observed can be characterized by the same relaxation time, T1 . Figure 1 shows T1 in PdHe.,a, as a function of temperature, for three values of resonance frequency, w,. The curves show the characteristic frequency-dependent minimum due to diffusion, and converge at high- and low temperatures where relaxation is governed mainly by interaction with conduction electrons. The failure of the values measured at 7 MHz below 140 K to converge towards those measured at higher frequencies appears to be real and is not understood. Similar behaviour to that displayed in Fig. 1 was shown by specimens with lower hydrogen contents, except that the
100
a00
200
T
(K)
Fig. 1. Proton spin-lattice relaxation times in PdH 0.70 as a function of temperature at resonance frequencies of 7 MHz (+), 11 MHz (0) and 47 MHz (0).
46 loo0
300-
0
l+*O000 100+*Oo
T, tmg)
9 O” . 00
10
I
I
I
100
200
300 T (K)
Fig. 2. Proton spin-lattice relaxation times in (Pdo_g*Ceo& Ho.53 as a function of temperature at resonance frequencies of 7 MHz (+), 11 MHz (*) and 47 MHz (0).
e 1@ l
100
0
l
0 0
50
0
%r
00 0
s?@ OQO
loi T
(K)
Fig. 3. Proton spin-lattice relaxation times in PdHO.70 (0) and (PdOg, Ceo.03) Ho.72 (e) as a function of temperature at a resonance frequency of 11 MHz. minima were less deep, as expected from eqn. (2), and were displaced to lower temperat~es, confirming earlier observations that diffusion is more rapid in palladium with lower hydrogen contents. (An example is shown in Fig. 4.) Figure 2 shows corresponding data for (Pd0.s4 Ceo,06) HoS6s,an alloy in which the hydrogen solubility is less than for pure palladium. Rgures 3
47
o
I
I
I
loo
200
300 T(K)
Fig. 4. Proton spin-lattice relaxation times in PdH 0.56 to) and Wo.64 as a function of temperature at a resonance frequency of 11 MHz.
Ce0.06)
Ho.53
to)
and 4 show the effects of alloying with cerium for (almost) fixed hydrogen concentration; comparisons are made between PdHO.,O and (Pdo.s7 Ceo.os)H o.72, and between PdHo.56 and (Pdo.a4Ceo.06)Ho.5s. The hydrogen concentrations are rather uncertain and can probably not be relied on to better than 0.03. All the alloys are single phase except PdHa.56 which is likely to be a mixture of (11and p phases but in which the great majority of the hydrogen is in the /3 phase of composition about PdHa.64; it is this phase which provides, essentially, all the observed signal from this sample. 3. Discussion 3.1. Palladium-hydrogen We consider first the results for PdHO.‘IO. The simplest method to evaluate the jump time, 7, and its temperature dependence is to use the theoretical result that w,r = 1.06 at the temperatures corresponding to minima of T1. For this purpose it is not necessary to extract the relatively-weakly temperature-dependent T,, from the observed T, values. It is found immediately from Fig. 1 that r is 24 ns at 230 K, 15 ns at 234 K and 3.6 ns at 270 K; these results can be combined to provide a rough value of the activation energy, E,, for diffusion of (0.24 + 0.02) eV. The minima observed by Gil’manov and Bikchantayev [6] at 16 and 32 MHz, and by Burger et al. [5] at 30 MHz, are consistent with this result. It is noteworthy that the value of the activation energy obtained in this way is not dependent on the details of Torrey’s model; it relies solely on the result that the minimum of T, occurs for some fixed value of w ,r independent of frequency.
48
With these values of 7 and E, one can at once derive a value for 7i1 from eqn. (3). The result is 7Z.l - 7.5 X 1012 Hz, or, if one uses a more precise value of E, = 0.228 eV, derived from the whole of our data as will be -’ = 4.8 X 1012 Hz. It is of interest to compare this described shortly, rm jump attempt frequency with the frequency of “optical” vibration of hydrogen in the palladium lattice (v,,~~= 13.5 X 1012 Hz) as determined by Bergsma and Goedkoop [lo] from neutron cross-section measurements and Chowdhury [ 111 from inelastic neutron scattering. As discussed by Korn [ 121 (who analysed the case of TiH,) one can expect 7Z1 s (1 -c)LJ,~~ where (1 - c) is the probability of finding any given neighbouring site unoccupied. In our case (1 -c) = 0.3 and (1 -c)vopt = 4 X 1012 Hz. The agreement with 7Z1 is perhaps fortuitously good; nevertheless, it provides confidence in the interpretation of the minimum of Tr. To make use of the data over the whole temperature range it is necessary to make an allowance for Ti,. Now Ti, is unknown, but one can attempt to deduce it from the measured [9] proton Knight shift, K, and the Korringa relation. Assuming that the whole of K and T,, arise from a direct Fermi contact interaction with conduction electrons in an s-like band (in PdHa.,e it is thought that the 4d band is filled), and inserting a Korringa enhancement factor of 0.77, reflecting the effect of electron-electron interactions appropriate to the conduction electron density, one finds Ti, = 1.74 s at 180 K. This value is far too large to fit our results and we must conclude that either K or C,, or both, are not entirely produced by the direct contact interaction, or, alternatively, the zero from which K was measured was inappropriate. As an alternative procedure we have carried out a computer analysis of the Ti data at temperatures well away from the minimum to produce the best fit to eqn. (4) with Tr,T = constant and Tin exponentially dependent on reciprocal temperature. The best fit is obtained with Tr,T = 68 f 2 s K, independent of frequency as it should be. It is then possible to compare the minimum values of Tin observed with the predictions of eqn. (2) in which (since we are dealing with the minima) there are no adjustable parameters. Using a = 0.404 nm and cp(h,y),,, = 0.4387, we find the results given in Table I. Within the model, the chief uncertainty in the theoretical result comes from that in the hydrogen concentration (about 5%). Agreement is good, though the theory slightly overestimates the relaxation rate (as has also been found [14] for TiHi.ss and &Hi.,). As a final test of the model we may use eqn. (2) to deduce values of y and hence T over the whole range of temperature covered by the measurements. Since (T1n)min is slightly different from the theoretical value, y(T) was deduced by equating experimental values of TID/(Trn)m~ to 0.4387/q(h,y), equivalent to treating the concentration, x, as a (slightly) adjustable parameter. Figure 5 shows an Arrhenius plot of log 7 against reciprocal temperature from the data of Fig. 1 for PdHa.,e at all three frequencies. From 195 K to 330 K (a range including the minima of 2’1) the data are well fitted by an activation energy, E, = 0.228 + 0.010 eV. Below 195 K the scatter is much worse but it seems clear that at about this temperature there is a definite break in the
49
TABLE
I
Experimental minimum values of 2’1~ for PdHO.,o compared with theoretical values from Torrey’s random walk model Frequency
Temperature
(Tl)n,in observed
(Tl~)ti
expt. corrected for T1,
(MHz)
W)
(ms)
(ms)
(ms)
47 11 7
270 234 230
59 _+5 18.2 ?r 0.8 11.4 f 0.4
77 +6 19.4 + 0.8 11.9 f 0.4
71 16.6 10.5
slope of the curve; if one leaves aside the 7 MHz data below 140 K as “anomalous” one could describe the results in the range 110 - 195 K by an activation energy, E, = 0.10 + 0.02 eV, but this is clearly much less reliable than the derived value for E, above 195 K. A similar discontinuity in NMR data at about this temperature has been observed before by Torrey [4] and by Burger et al. [5]. The physical reason for the change at this point is unknown. It is possible that the break indicates some change in the diffusion mechanism. It is well-known that it is in the “slow motion” regime below the Tr minimum that NMR is particularly sensitive to the details of the diffusion process. No other properties appear to show discontinuities at this
I 3
1
5
6
7
s
8
WOO/l
I W
(K-‘)
Fig. 5. Proton jump time, T, as a function of reciprocal temperature in PdHo.70, derived from 2’1 values (Fig. 1) measured at ‘7 MHz (+), 11 MHz (0) and 47 MHz (0). The hatched region indicates the range in which T1 minima occur.
50
temperature however. A transition, possibly an order-disorder transition, is known to occur in PdH,, but at too low a temperature (50 K) to be relevant here. Moreover, a change in the sign of the isotope dependence of diffusion in PdH, has been found in the neighbourhood of 100 K [15] but this also appears to occur at too low a temperature to be related to the discontinuity in the NMR data. Very recently, Arons et al. [ 161 have found no evidence of any discontinuity in the related quantity T,, (rotating frame spin-lattice relaxation time) for protons in p-PdH down to 170 K. All this may point to the alternative possibility that there is no change of E, at this temperature but that either the temperature dependence of Tl, changes or, more likely, some additional relaxation rate which increases as the temperature decreases is present. The latter contribution could arise from a small amount of paramagnetic impurity, e.g., dissolved iron, in the sample (though it is diffic~t to envisage at what stage the impurity entered the originally pure material). A relaxation rate of order 0.2 s-l to 1 s-’ at 130 K and varying as Tl (impurity) a T would account, roughly, for the observed behaviour. The scatter between breakaway temperatures (195 - 230 K) found by different authors [ 4, 51 might then be accounted for by differing impurity contents. We now consider briefly the effect of changing hydrogen concen~tion. The minimum of T,, changes in depth and temperature with changing concentration, X. According to eqn. (Z), (TzD)min is inversely proportional to 3~.From Figs. 1 and 4 the ratio of the minimum values for PdHovTOand PdH0.s4 is 18.2/20.2 2 0.90 (the correction for T1, hardly affects the ratio) in agreement with the inverse ratio of concen~ations 0.64/0.70 r 0.91. Within experimental error, the change in the temperature of the minimum can be accounted for solely by the change in the factor (1- c) occurring in T:‘, without any change in Vopt or E,. Moreover, one can again fit the whole T1 curve for PdH0.s4 down to 195 K assuming TI,T and E, retain the same values as for x = 0.70. Of course, one might expect both of these quantities to change with x due to changing conduction electron density and lattice parameter; experimental accuracy enables one to say that the former changes by less than 10% and the latter by less than 5% over the rather small accessible range of x in the p phase. Although in the temperature range 195 - 330 K an internally consistent interpretation has been achieved in terms of nearest-neighbour octahedraloctahedral jumps, a serious difficulty remains in that there is a large discrepancy between all the NMR data and the quasielastic neutron scattering data [ 7) obtained in overlapping temperature ranges. A summary of some of the available data for E, is given in Table II. Apart from the discrepancy between the activation energies derived by the two methods, and between r values (our value of r at 293 K is 40 times larger than that of Beg and Ross [7] at the same temperature), the neutron scattering data are not fitted by a model involving only nearest neighbour octahedral-octahedral jumps. The suggestion of Beg and Ross that NMR may be observing a minority of protons for which 7 is much greater than the average is not now tenable in
51
TABLE
II
Experimentally
determined
activation
energies
for hydrogen
diffusion
Method
Temperature (R)
Activation energy (eV)
Source
NMR: 7-r
195 - 330 100 - 195
0.228 f 0.010 0.10 f 0.02
present
work
NMR: T1
> 220 < 220
0.24 0.08
Torrey
[4 ]
NMR: T1
230 - 320 180 - 230
0.21 0.06
Burger et al. [ 5 ]
NMR: T1
180 - 350
0.198
Gil’manov
in &PdH
et al. [ 61
NMR: T1,
170 - 300
0.225
Arons et al. [ 161
Neutron scattering
290 - 470
0.146
Beg and Ross [ 71
Macroscopic diffusion
290 - 400
0.25 i: 0.01
Wicke and Bohmholdt
[ 171
view of the agreement displayed in Table I, where the theoretical values of Vi ~)mti assume that all the protons are involved; further, around the temperature of the minimum any protons not efficiently relaxed by diffusion would give rise to a tail on the observed signals with a decay constant of T1,. Such tails were looked for but not observed. We have not analysed in detail a much more complicated situation envisaged by Beg and Ross, where the fact that most octahedral sites are occupied is assumed to encourage both rapid octahedral-tetrahedral jumps and jumps between nonnearest neighbour sites as well as nearest octahedral-octahedral jumps, but we stress that here again the observed NMR signals cannot be produced by a minority jump process with long 7 in the presence of a majority jump process with much shorter 7. To reconcile the NMR and neutron data is thus still a problem. In the meantime there are several points worthy of comment. Firstly, 7 values derived by the two methods may differ genuinely since, in neutron scattering, one is concerned with a simple position vector correlation function and in NMR with a more complicated tensor correlation function, but the difference on this account will not exceed a factor of three. Secondly, the Torrey theory contains some approximations, notably that of isotropic diffusion, and this can, in principle, cause errors in the 7 values deduced. However Sholl [ 181 has shown that the exact formulation of the theory for an f.c.c. lattice gives r values, for given TID, which are only a few per cent. different from the Torrey values. It may be noted that the Sholl formulation, besides being exact within the model, is also more readily adaptable to the discussion of more complicated jump processes. Thirdly, consideration of temporal correlations between successive jumps of the type discussed by Eisenstadt and Redfield [ 191 and by Wolf [ 201 can change the r values
52
deduced, and since their effect can be temperature dependent, can change the value of E, also, but neither effect can be expected to be substantial, particularly in this case where the concentration of vacancies in the hydrogen lattice is relatively large (2 30%), so that the distinction between a “jump” of a proton and an “encounter” with a vacancy becomes unimportant. Fourthly, because NMR observations involve generally longer r values than neutron scattering observations, the approximation of neglect of the time actually spent by the proton in jumping between sites (- lo-l3 s) is a particularly good one in interpreting NMR data. Fifthly, comparison between NMR and neutron scattering data has been made [21] for one other system, /3-NbH, and agreement in that case is good, so that the problem may be peculiar to palladium. 3.2. Palladium-cerium-hydrogen The remaining uncertainties in the interpretation of the data on fl-PdH need not prevent us from making a comparison of Tin values for P-PdH and (Pd-Ce)H, at any rate near ( T1,),,. A somewhat surprising result, immediately evident from Figs. 3 and 4, is that the temperatures of the minima are the same, within experimental error, for the alloys as for pure palladium. The 11 MHz data, for instance, yield the following temperatures of the minima: PdHo.70,
239 * 3 K
(Pdo.e7Ceo.es)Ho.72,236
+4 K
PdHo.56, (Pdo.e&eo.os)Ho.53,222
225 + 2 K * 2 K
and the minima shift with frequency as before. This means, at these temperatures at any rate, that the diffusion coefficient is not much increased, if at all, by addition of cerium; taking full advantage of the errors quoted above, an upper limit for any change can be set at 30%. This is in strong contrast with the increased permeation rates observed (such experiments are necessarily made at higher temperatures) by Wise et al. [l] in these alloys, and suggests that the more rapid permeation through membranes is not an effect of increase in lattice constant on bulk diffusion rates, but of more rapid surface reactions. A complete analysis of the T1 - T curves for the alloys is a matter of some difficulty. From the evidence of the undisplaced T,, minima, it is natural to expect that the activation energy, E,, is also unchanged between the pairs of alloys compared in each of Figs. 3 and 4. If we allocate the relaxation rate which remains after subtracting the corresponding values of TIDel to a conduction electron term as before, the apparent T1,T values judged from the higher temperature data are about 2/3 and l/4 of that for PdH0.70 for the 3% and 6% Ce samples, respectively. The constancy of T1,T above 195 K is not by any means so convincing as for PdH,, and the deviations below about 195 K are considerably greater. The faster conduction-electron-induced relaxation rates are unexpected, since susceptibility data [ 221 on palladium-cerium solid solutions indicate that cerium is non-
53
magnetic, merely donating four electrons per atom to the conduction band. If this is true also for (Pd-Ce)H, we might expect Ce to have very little effect on T,,. However, since we do not understand T1, in PdH, quantitatively, we cannot be sure. If the extensive deviations below about 195 K are not associated with TID, then there is a third relaxation contribution to explain. The most likely candidate is a paramagnetic impurity which might be traces of PdsCe, with which a 4f virtual bound state is associated [ 231. However, the additional relaxation rate does not appear to be inversely proportional to temperature as was roughly the case for PdH, below 195 K and for strongly paramagnetic rare earth-hydrogen alloys [9]. In view of these uncertainties, extraction of T1 ,,-l values from our data is only justified near the minimum where TIDM1 dominates. 4. Conclusion NMR measurements of T1 in P-PdH, are quantitatively explained at temperatures above 195 K by a model involving diffusion by octahedraloctahedral jumps. The activation energy deduced is in reasonable agreement with the results of macroscopic diffusion measurements, but in serious disagreement with neutron scattering results. At room temperature the macroscopic results of Wicke and Bohmholdt [ 171 yield a value of D of low6 cm2 s-l; the neutron scattering value is about 3 times larger and our NMR value about 10 times smaller than this. These discrepancies remain to be explained. Below 195 K, interpretation of our data is uncertain probably due to the effects of paramagnetic impurities. Although a full analysis of the T1 data for (Pd-Ce)H, has not been possible at all temperatures, the minima of T1 are unshifted within experimental error from the corresponding minima in PdH,. It is concluded that D is not increased by more than 30% by alloying with cerium so that the known permeation increase is probably associated with surface phenomena. Acknowledgements This work was supported by a grant from the Science Research Council. We are grateful to Dr. I. R. Harris of The University of Birmingham who supplied the alloy specimens and we wish to acknowledge helpful discussions with Mr. D. K. Ross, Professor D. Zamir, Dr. C. A. Sholl and Dr. P. P. Davis. References 1 M. L. H. Wise, J. P. G. Farr, I. R. Harris and J. R. Hirst, L’Hydrogene dans les Metaux, l(l972) 1. 2 J. P. G. Farr and I. R. Harris, Brit. Pat. Appl. 25512/70. 3 R. E. Norberg, Pbys. Rev., 86 (1952) 745. 4 H. C. Torrey, Nuovo Cimento, Suppl. IX, (1958) 95. 5 J. P. Burger, N. P. Poulis and W. P. A. Hass, Pbysica, 27 (1961) 514. 6 A. N. Gil’manov and I. G. Bikchantayev, Phys. Metals Metallgr., 31 (3) (1970) 64.
54 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
M. M. Beg and D. K. Ross, J. Phys. C, Solid State Phys., 3 (1970) 2487. H. A. Resingand H. C. Torrey, Phys. Rev., 131 (1963) 1102. L. Shen, J. P. Kopp and D. S. Schreiber, Phys. Letters, 29A (1969) 438. J. Bergsma and J. A. Goedkoop, Physica, 26 (1960) 744. M. R. Chowdhury, Ph. D. Thesis, Univ. Birmingham, 1971. C. Kom, Ph. D. Thesis, Weizmann Institute of Science, Rehovot, 1971. A. Fert and P. Averbuch, J. Phys. Radium, 25 (1964) 20. H. T. Weaver and J. P. VanDyke, Phys. Rev., B6 (1972) 694. Y. de Ribaupierre and F. D. Manchester, J. Phys. C, Solid State Phys., 6 (1973) L391. R. R. Arons, H. G. Bohn and H. Lutgemeier, Solid State Commun., 14 (1974) 1203. E. Wicke and G. Bohmholdt, 2. Physik. Chem. Frankfurt am Main, 42 (1964) 115. C. A. Sholl, J. Phys. C, Solid State Phys., 7 (1974) 3378. M. Eisenstadt and A. G. Redfield, Phys. Rev., 132 (1963) 635. D. Wolf, Z. Naturforsch., 26a (1971) 1816. B. Alefeld, H. G. Bohn and N. Stump, Ber. Bunsenges. Phys. Chem., 76 (1972) 781. I. R. Harris and M. Norman, J. Less-Common Metals, 15 (1968) 285. W. E. Gardner, J. Penfold, T. F. Smith and I. R. Harris, J. Phys. F., Metal Phys., 2 (1972) 133.