On the temperature dependence of hydrogen site occupancy in rare earth dihydrides

281

Journal of the Less-Common Metals, 138 (1988) 281 - 287

ON THE TE~P~~T~RE DEPENDENCE OF HYDROGEN OCCUPANCY IN RARE EARTH DIHYDRIDES

SITE

J. D. PATTERSON ~10~~

of Tech~oiogy,

Institute

~e~&ourne,

FL 32901-6995

{U.S.A.)

NM 87185-5800

(U.S.A.)

PETER M. RICHARDS Sandia National

Laboratories,

(Received June 10, 1987;in

Albuquerque,

revised form August l&1987)

summary Previous work claimed that the observed anomalous tern~~t~e dependence of octahedral(o)-site occupation in f.c.c. rare earth dihydrides could be explained by interaction effects not involving optic mode vibrations. We show that this conclusion was based on an erroneous assumption about the role of the chemical potential in dete~~~g the thermal equilibrium solution to lattice-gas-model equations. A correct derivation is given, and we conclude that such interactions cannot be the source of the anomaly. We examine some other possibilities.

1. Introduction How hydrogen occupies interstitial sites in a metal hydride lattice is of fundamental interest as welI as having important implications for questions of hydrogen storage. The rare earth dihydrides (REHZ_6 with 6 < 1) have been studied extensively in this regard [l]. Conventional wisdom suggests that the tetrahedral (t) sites in the f.c.c. metal lattice are preferentially occupied, thus giving a simple explanation to the hydrogen-to-metal ratio [H]/[M ] = 2 at stoichiometry, since there are two t sites per metal ion. Recent work [2,3] has addressed the question of how much octahedral(o)site occupation X, (also referred to as intrinsic disorder) exists by direct measurements involving inelastic neutron scattering. Earlier studies (referred to in refs. 2 and 3) made less direct determinations of the intrinsic disorder. A remarkable feature of the results is that in all the systems examined [3] (RE = Y, La, Ce) for which a non-zero z, could be found, X, either decreased or essentially stayed constant as the temperature T increased between about 15 K and 400 K. Furthermore, these relatively low temperature data all showed w,, values greater than the values inferred from pressure-composition 0022-5088/88:$3,50

@ Elsevier Sequoia/Printed in The Netherlands

282

10-5 1 0

I

AU=

0.2 ev

AS=

-1 kg

1 200

400

800

800

I 1000

T(K) Fig. 1. Octahedral site occupation vs. temperature with different values of energy difference AU and entropy difference AS between o site and t site. Optic vibrational effects are the same for both curves and are as used in Goldstone et af. [ 21. Ground state interaction is neglected (J, = 0). Data are from Gotdstone et al. f2] for YHs, with the full circles from inelastic neutron scattering and the open circle from the thermodynamic data of Dantzer and Kleppa [4 J. Data have been replotted noting that, for [ H]/[M ] ratio = 2 and x0 -Z 1, xg is twice the fraction of hydrogen on the o-sites shown in ref. 2.

isotherms at about 900 K (see Fig. 1 and ref. 3). Thus, contrary to reasonable intuition, intrinsic disorder does not increase with temperature. One of us [ 53 showed that x, could be expected to decrease with Tat high temperature because large vibrational amplitudes of the t-site protons are inhibited if a neighbor o site is occupied. Vibrational entropy increases faster than site-disorder entropy decreases if o sites become depopulated with increasing T, and one’s faith in the second law of thermodynamics is restored. Additional support of this model may come from nuclear magnetic resonance (NMR) data [ 61. These data are interpreted as a decrease in the proton hopping rate with T above about 700 K. It has been shown that this same vibrational effect which limits o-site occupation also causes the hopping rate to decrease [ 71. However, since the mechanism requires excitation of vibrational modes whose energy is about 1400 K in temperature units, in the original theory [ 51 X, increased with T below about 500 K and reached

283

a maximum value X, x $ before decreasing. Both the initial increase and the large maximum value were ruled out in the work of Goldstone et al. [ 21. It was claimed by Goldstone et al. 123 that metal lattice vibrations and proton-proton interactions could explain the behavior below 400 K. They argued that because the metal lattice contracts as o sites are occupied [l] (deduced from the fact that REH,,, has a smaller lattice constant than REHz -6 ), and it expands with t-site occupation, metal lattice entropy favors t-site occupation. Although this showed that the second law could be satisfied at lower temperatures with acoustic metal lattice vibrations, just as it had been at higher temperatures with the optic modes, the effect was not large enough to explain the data based on reasonable estimates of the entropy difference. Goldstone et al. [2] thus required additional interactions. We point out here that although their assessment of the interactions may have been correct, they erred in finding the state of true the~od~~ic equilibrium. The correct equilibrium state presents the original dilemma of an X, value which can only increase with T at low temperature. This paper shows the nature of the error and corrects it. We then reassess the situation with regard to the data of refs. 2 and 3 and alternative explanations, none of which seem entirely satisfactory.

2. Lattice gas theory Goldstone et al. 121 used a lattice gas model [S] which has successfully been employed to describe hydride phases [l, 91 and is generally accepted as providing a reasonable description of hydrogen in metals. The problem is not with the basic model but with their assumption that, when the transcendental equations give more than one solution for x0, thermal equilibrium corresponds to the solution with the lowest chemical potential. The chemical potential enters the problem as a convenience by allowing the number of particles to fluctuate in a grand canonical ensemble; however, the canonical ensemble of a lattice gas in the standard treatment has constant volume (i.e. number of lattice sites), temperature and number of particles. Thus the Helmholtz free energy F must still be the quantity to minimize in spite of this use of the chemical potential or Gibbs free energy per particle. We show below that the solution of Goldstone et al. [Z] gives a maximum F and thus must be ruled out. This eliminates two conceptual problems with the solution. The decrease in x, with T was accomplished with interactions alone, without specifically invoking the effect of metal lattice entropy which would seem to be necessary to satisfy the second law. In addition, the solution which had the lower chemical potential depended on an additional condition on the interactions which could be specified without altering the values of X, in the solutions. The role of the chemical potential can also be seen in an alternative way of treating the problem, as done when considering hydrogen concentration in the metal as a function of pressure. In that case hydrogen in the metal is

284

in equilibrium with molecular hydrogen gas in a container of fixed volume. Requiring F of the combined system to be a minimum gives the well-known result that the chemical potentials of hydrogen in the gas and metal must be equal in equilibrium. The problem is then solved by the known relation between pressure and chemical potential for the gas. To be more explicit in our comment regarding the solution of Goldstone et al. [ 21 we consider the lattice gas in some detail. The hamiltonian is H = ($)xJijPiPj ij

- x.ViPi

(1)

i

where Pi = 0 or 1 is the number of hydrogens on site i, Vi is the site energy and Jij an interaction energy. For the problem at hand there are two types of sites o and t so that Vi has the value U,, or U, and Jij GUI be of the form J 00, Jtt or J,,t depending on the type of sites involved. As is common, Goldstone et al. [2] solved for x, = and xt = (Pi,> by using a grand canonical ensemble with the chemical potential p determined from the condition x, + 2x, = 2 at stoichiometry 6 = 0. A mean-field-like treatment [lo] was employed. Since we wish to investigate minimization of F = E - TS, we write the entropy S in a mean-field (MF) approximation, indicated by the subscript MF, as S MF = 2NkBxt ln((1 -xt)/xt} -NkB

ln(1 -x0)

- 2NkB ln(1 -xt)

+Nkgx,

ln{(l

-x,)/x0} (2)

The above is the standard expression [l] for the disorder entropy when 2N t sites and N o sites are occupied with fractions xt and x, respectively, all configurations with the same xt and x, being assumed to give the same energy in the MF approximation. The mean-field energy EMF may be expressed as E MF = N(AUx,

+ (+)J,xo2}

(3)

where AU = U, - U, + Jot - Jtt and J, = Jo, + (k)J,, -Jot. Here xt = 1 x,/2 has been used assuming stoichiometry, a constant term has been ignored, J,,, = ZjoJio~o where i, and j, are o sites, and the other J parameters are defined in a similar manner. A plot of the mean field FMMF = EMF - TSMF is given in Fig. 2 based on eqns. (2) and (3) with J, < 0, AU > 0, xt = 1 x,/2 as used by Goldstone et al. [2]. It is evident that the intermediate x, value at which aF&ijx, = 0 corresponds to maximum Helmholtz free energy. Since eqn. (2) gives -%&ax, = 2NkB In x, for x, + 0, it follows that aF/i3x, < 0 for x, + 0. Thus the lowest root to aFlax,, = 0 must give a local minimum F value, and therefore the next highest root must give a local maximum. This argument requires only that aFlax, be dominated by the logarithmic term for x, + 0, so it should be generally valid rather than restricted to the MF approximation (thus the MF subscripts have been left off). The condition aFMF/axo = 0 at xt = 1 -x,/2 leads to the transcendental equation

285

x0 Fig. 2. Free energy us, x0 (note the change in scale at FMF = 0 for x0 < 5 X 10W4 and that the right-hand scale is to be used for x0 > 5 X 10e4). The curve is from eqns. (2) and (3) &J/k, = 3800 K, Je = 2.7 x lo4 K, T = 200 K. with xt = 1 -x0/2,

x02 = 2 exp{-&AU

+ J&j

(1 -x0/2)(

1 -x0)

(4)

with p = l/k&Z’. The same equation results from direct calculation of X, and 3tt with a grand canonical ensemble. It is also the same as used by Goldstone et al. [ 21 except that there, because of a modified MF theory [lo], J, could be considered temperature dependent and X, 4 1 was assumed so that the last two terms in parentheses could be taken as unity. They also included a term exp(-@AGL) with AGL the increase in metal lattice free energy when a hydrogen is moved from a t to an o site to account for the effect of acoustic vibrations. They recognized that two solutions with Z, Q 1 were possible when, as expected, J, < 0, and that the larger of the two had the desired feature of giving agreement with the data. They were misled (actually only Richards was misled; experimentalists Goldstone, Eckert and Venturini should be held blameless) by their observation that the resulting chemical potential is given by P = c+>W. + ucl -

Jtt - Jot1+ W2)lJoo - Jttl

(5)

for x, 4 1. This has the feature that if Jo, < Jtt, the root with the larger X, value has the lower chemical potential cz and was thus claimed to be the “correct” solution. (Goldstone et aE. [2] used the temperature dependent J parameters of modified MF, but the conclusion with regard to the relative sizes of the interactions is the same.) Our statement on the basis of the above is that eqn. (5) is irrelevant and that, irrespective of the relative

286

sizes of the interactions, the higher of the two X, Q 1 roots maximum F value and thus is unstable.

always has a

3. Other mechanisms We are thus left without an explanation for the observed lack of increase in x, with T below 400 K. The original theory [5] remains viable at the higher temperatures as long as the optic modes have the assumed strong effect. Some discussion of the low temperature behavior and possible explanations follow. First, it should be noted that since most lattice entropy should effectively be zero for T = 15 K or less, corresponding to the lowest measuring temperatures, any theory must show an increase in the equilibrium x, value with T below about 100 K, if the ground state really has an o-site occupation x, = 0. Earlier work [ll] noted that equilibration times would be far too long at low temperatures to expect to see a true equilibrium value. The lowest temperature To at which the observed x, value would correspond to equilibrium was estimated to be about 150 K based on extrapolation of the proton jump rate inferred from NMR. One should see a nearly temperature independent x, below To. To explain the data of Goldstone et al. [2] would require To = 400 K. Assuming an activated hop time r = r. exp(E&,T) gives EA = 1.2 eV for a prefactor 7. = 10-i’ s and the condition r = lo3 s at To. This is at least a factor of two larger than NMR activation energies and thus seems unreasonable. If diffusion is sufficiently rapid to make To < 100 K, one might have to question the assumption that the ground state has an o-site occupation x, = 0. Equation (3) shows that EMF could be minimized with a small non-zero x, value if AU is less than zero and is small and J, is greater than zero and is large. As noted by Goldstone et al. [2], however, long-range elastic interactions [ 121 are expected to make J,,,, Jtt < 0 and Jot > 0, and the strongest short-range repulsion should be the nearest-neighbor o-t one. Thus J, < 0 is most likely, which would rule out a lowest energy with small non-zero x,. Formation of a ground state o-site superlattice, which is not within the scope of simple MF, might still be possible. A large metal lattice entropy contribution to AGL in eqn. (4) could combine with a small AU value to produce the desired effect. The entropy difference would have to be AS < -512, to make x, < 0.1 at 100 K and to ensure x, does not increase with T. This seems unreasonably large if the hydrides behave like metals where typical defect entropies [13] are about lka, but not if they are like alkali halides where 5 lOk, is common [13]. Figure 1 shows that reasonable agreement can be obtained in this way. Another possibility is the existence of “unfavorable” t sites caused by impurities. Imagine that 2 t sites in the neighborhood of an unspecified impurity atom have their site energies raised (lowered in magnitude) above those of both the normal t and o sites. If the impurities are in concentration c per metal atom, but the [H]/[M] ratio is still 2, the lowest energy

287

corresponds to xt = 1 - c’/Z, xi = 0, x, = c’, where xt and xi are the occupations of “normal” and high energy or unfavorable t sites respectively, and where c’ = Zc, assuming Zc 4 1 and a random distribution of impurities. These values are maintained as temperature is increased until the point where thermal energy allows occupation of the remaining high energy t sites and/or further o sites. With suitable choice of c and energy differences, a curve similar to the AU = 0 one of Fig. 1 results. However, c = 1% is required, which seems unreasonable for good starting materials. It should be noted, however, that the impurities could be foreign gas atoms such as carbon as well as metal atoms. Such an impurity effect would also provide a natural explanation for why [ 3] no o-site occupation is seen at an [H] /[M] ratio of 1.9. At very high temperatures hydrogen can escape from the metal lattice, whereby both x0 and xt would decrease. However, this is not believed to be important in the data of Goldstone et al. [ 21.

4. Conclusions In summary we have shown that a decrease in octahedral site occupation with temperature cannot occur with ground-state interactions alone. This is contrary to the claim of Goldstone et al. [2] which was based on an incorrect the~odyn~ic ~te~retation, The work of ref. 5 in which o-site occupation was shown to decrease above about 500 K as a result of interactions when optic modes are excited does, however, remain a plausible explanation for the high temperature behavior, both of o-site occupation and NMR relaxation [6, 71. We have discussed the likelihoods of the observed decrease below 400 K being caused by the following: (i) a ground state with non-zero o-site occupation, (ii) an exceptionally large energy barrier for motion from o to t site, (iii) a large metal lattice entropy effect, and (iv) impurities. We have reached no firm conclusions.

Helpful conversations with G. L. Jones are gratefully acknowledged. The work of J.D.P. arose from a summer spent at Sandia National Laboratories sponsored by Associated Western Universities, Inc. The work at Sandia National Laboratories was supported by the U.S. Department of Energy under Contract DC-AC-04-76DP00789.

References 1 W. M, Mueller, J. P. Blacklege and G. P. Libowitz, Metal Hydrides, New York, 1968.

Academic Press,

288 2 J. A. Goldstone, J. Eckert, P. M. Richards and E. L. Venturini, Solid State Commun., 49 (1984) 475. 3 J. A. Goldstone, J. Eckert, P. M. Richards and E. L. Venturini, Physica B, 136 (1986) 183. 4 P. G. Dantzer and 0. J. Kleppa, J. Chem. Phys., 73 (1980) 5259. 5 P. M. Richards, J. Solid State Chem., 43 (1982) 5. 6 R. G. Barnes et al., Phys. Rev. B, 35 (1987) 890. 7 P. M. Richards, Phys. Rev. B, 36 (1987) 7417. 8 T. D. Lee and C. N. Yang, Phys. Rev., 87 (1952) 410. 9 C. K. Hall, in P. Jena and C. B. Satterthwaite (eds.), Electronic Structure and Properties of Hydrogen in Metals, Plenum, New York, 1983, p. 11. 10 P. M. Richards, Phys. Rev. B, 28 (1983) 300. 11 E. L. Venturini, J. Less-Common Met., 74 (1980) 45. 12 H. Wagner, in G. Alefeld and J. Voelkl (eds.), Hydrogen in MetaZs I, Topics in Applied Physics, Vol. 28, Springer, New York, 1978, p. 5. 13 C. P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford, 1972.