Theory of hydrogen diffusion in metals. Quantum isotope effects

J. Php. Chem. Solids Vol. 52, No. 2, pp. 351-355, 1991 Printed in Great Britain.

W22-3697/91 53.00 + 0.00 Q 1991 Pergamon Press plc

THEORY OF HYDROGEN DIFFUSION IN METALS. QUANTUM ISOTOPE EFFECTS S. FUJITA and A. GARCIA? Department of Physics and Astronomy, State University of New York at Buffalo, Buffalo, NY 14260, U.S.A. (Received

27 November

1989; accepted in revised form 25 July 1990)

Abstract-A statistical mechanical theory of hydrogen diffusion in metals is developed. The mystery of why heavier deuterons can diffuse faster than hydrogens in Pd is resolved by computing the quantum mechanical zero-point energy associated with the confining potential perpendicular to the interstitial channel. An excellent quantitative agreement between theory and experiment is obtained for fee and bee host metals. Keyworh: Hydrogen diffusion in metals, statistical mechanics of interstitial diffusion, quantum isotope effect, theory of atomic diffusion.

1. INTRODUCTION

Numerous experiments [l] show that the diffusion coefficient D for hydrogen in face-centered-cubic (fee) metals like Pd, Ni and Cu, obey the Arrhenius law D = D, exp( - E,/k,

T),

where E, is the activation energy and D,, the preexponential factor. Isotope measurements [l] indicate that both parameters (E,, Do) depend on the species (H, D, T). First, the pre-exponential factor D,, obeys the law: ,

(cc,p = species).

(2)

This law can be derived and interpreted by the classical mechanical theory of interstitial diffusion previously developed by us [2] (summarized in Section 2) as well as by the currently predominant theory based on the hopping model [3]. Second, the activation energy E, also depends on the isotopes, and the inequalities

were observed; this latter effect is stronger at low temperatures, and can yield a remarkable reversed isotope dependence such that heavier deuterons may diffuse faster than lighter protons below a certain temperature

(SOOC for Pd).

t Work in part based on the thesis submitted in partial fulfilment of the requirements for the degree of Master of Arts at the State University of New York at Buffalo. PCS 52,*--A

The main purpose of the present article is to present a quantum-mechanical theory of diffusion, and derive the inequalities (3), which appear to have been unexplained in the past. A preliminary qualitative outline of this theory was reported earlier [4]. Briefly, the great majority of protons are trapped at the equlibrium interstitial sites (which also form a fee lattice for the fee metal), and only a very few protons with energies high enough to overcome the energy barrier 6 along the straight channel lines connecting the equilibrium sites can migrate and thus contribute to diffusion. The fraction of the migrating protons is proportional to exp( -c/k, T), which generates a relation: E. = 6 between the experimentally-determined activation energy E, and the microscopic barrier energy c. Classical-mechanically, the barrier energy E, which is the difference between the maximum and minimum energies along the channel, can be determined completely by the crystal potential V(r) for a proton. In closer detail a typical proton may migrate not strictly on the channel line but with a certain amount of transverse motion. This motion perpendicular to the channel direction is dictated strongly by the crystal potential at a saddle point if such a thing exists, see below. The quantum-mechanical zero-point energy associated with the transverse motion at the saddle point may be significant if the migrating particle is as light as protons (and deuterons). Since the zero-point energy depends on the diffuser’s mass, this generates a quantum isotope effect on the activation energy. In the present work, we evaluate the crystal potential V with the assumption of a screened Coulomb interaction [eqn (14)] between a proton and each lattice ion, and demonstrate explicitly the equilibrium points and the saddle points for a 351

352

S. FUJITAand A. GARCIA

fee metal. Using the harmonic approximation to the confining potential, we compute the zeropoint energy and then compare the results with experiments. Our theory of interstitial diffusion [5], reviewed in Section 2, is distinct from the currently prevalent theories [3,6] based on the hopping picture and the transition-state arguments [7]. Connections and comparison with most of the other theories were discussed in Ref. 5, and will not be repeated here. The embedded-atom theory based on the quasi-atom scheme developed by Daw and Baskes [8] aims to compute the crystal potential after a great deal of numerical work with sophisticated choice and fits of parameters. The calculated potential indicates that the equilibrium sites for protons are the interstitital sites, (f a,, 0,O) and equivalent, which are in agreement with the present calculations with the assumption of the screened Coulomb potential. The channels along which protons are expected to move distances a few times the lattice constant a,,, see Section 2, are, by definition, straight lines which pass through the equilibrium interstitial sites and along which the crystal potential is of a restoring type. For a fee metal, only one type of channel in (110) is allowed because of the restoring-potential type restriction. At the mid-point of the nearest interstitial sites, the potential I’ has a saddle-point structure, that is, the potential has a maximum along the channel direction and a minimum in directions perpendicular to the channel. It is interesting to note that the saddle-points in the real lattice space, say at (’4 a,, a a,, 0), mentioned here are distinct from the saddle points in the transition-state theory for the hopping model, the points where the potential difference is the smallest over the whole set of paths connecting the two equilibrium sites. The saddle points in this sense are located at about two-thirds of the way in the direction [ 1111from the octahedral site (f a,, 0,O) to the tetrahedral site (i a,, $ a,,, a Q,) according to the calculations of Daw and Baskes [8]. The idea that the reversed isotope effect may arise from the transverse vibration frequency at the saddle point appears to have been proposed almost 50 years ago [9]. This idea was re-expressed by Vijlkl and Alefeld [l] and possibly others [lo]. The saddle points here are in the context of the transitionstate theory. The same idea applied to the saddle points in the channel model (which can be recognized without much calculation, see Section 3) will be used, in the present work, to resolve the puzzling isotope effect. In Section 2, we briefly review the classical theory of interstitial diffusion [2]. The isotope effect for the pre-exponential factor (2), is derived here. In Section 3, we present a quantum theory of the isotope effect (3) on the activation energy. We apply our theory, in Section 4, to hydrogen diffusion in fee metals and compare the results with experiments. In Section 5, we summarize our results and briefly comment on

hydrogen diffusion in body-centered-cubic (bee) metals which exhibits different isotope effects. 2. CLASSICAL THEORY OF INTERSTITIAL DIFFUSION

Hydrogens are thought to migrate interstitially in a crystal. Recently, we laid lown a microscopic foundation of the Arrhenius law (I) for interstitial diffusion [2] with the aid of correlated walk model with the trap possibility [ll]. In summary, the diffusion coefficient D can be written as D = $ vlq,

(4)

where v, 1 and q are the average migration speed, the mean straight path and the migration participation ratio, respectively. When the barrier potential height E for the interstitial channel is much greater than the thermal energy k,T L B k,T,

(5)

the ratio q, which is the fraction of those migrating hydrogens with energies greater than 6, can be computed simply, and is approximately given by q = q* exp(-c/ksT),

(6)

where q * is a numerical factor of order 1. Under the same condition (5) the migration speed is calculated to be a = YVT,

(7)

vr = (2k, T/nM)li2

(8)

where

is the thermal speed, and y is a numerical factor a little greater than unity. Combination of eqns (4), (6) and (7) and comparison with eqn (1) yields E, = E Do = f q*yvTi.

(9) (10)

The relations (8) and (10) simply lead to the isotope effect expressed by eqn (1) for the pre-exponential factor D,. Physically, lighter diffusers migrate faster than heavier ones, and this affects the pre-exponential factor since D,, av,aM-“2. In the case of hydrogen diffusion in Ni (fee) the experimental observations [1] indicate that E,,/k, = T, = 4870 K and D,, = 6.9 x lo-’ m2 s-i. At a typical temperature of observation T = 500 K, the migration participation ratio q m exp( -4870/ 500) x low5 is a very small number. If we assume 4 * = 1 and y = 2, we can estimate the mean straight 1 path along the channel to be about 0.57 nm at

Theory of H diffusion in metals T = 500 K (v = 3600 m s-‘), which is a little greater than the lattice constant 0.35 nm for this crystal.

3. QUANTUM ISOTOPE EFFECT ON THE ACTIVATION

and tritons. Then, the main contribution to the crystal potential should come from the screened Coulomb interaction between H+ and host ions of charge Ze. Let us represent it in a standard form:

ENERGY

U = e(Ze)(4ngK)-‘r-‘exp(-q,r)

According to eqn (9) the activation energy E, is simply equal to the barrier energy e, that is, the difference between the maximum and minimum potential energies along the interstitial channel, E,=c

=Emax-Emi,.

(11)

Since the channel potential arising from the interaction between diffuser and host crystal is the same for all isotopes, our classical theory predicts the same activation energy for all in contrast to the observed inequalities (3). By applying quantum theory, we may account for these inequalities in the following manner. For a fee crystal, the interstices also form a fee lattice parallel to the original lattice. The crystal potential at every interstitial site has the same minimum value. The obvious interstitial channels are along the nearest-neighbor directions, e.g. (110). At the mid-point (f a,,, $ u,,, 0) of the interstitial step from (5 a,, 030) to (a o, f a,, 0), the crystal potential has a saddle-point configuration; it has a maximum along the channel direction (110); since only two Ni*+ at (fao,t a 0,O) and (a,, 0,O) are located nearby at the distance (2,/Z))‘a,. These ions generate a steep potential minimum perpendicular to the channel direction through the screened Coulomb interaction between H+ and NiZ+. The quantum-mechanical zero-point energy computed to the harmonic approximation for this confining potential scales as h4-“*, and therefore should generate the following inequalities: Emax. H ’ &ax, D ’ &ax, T .

(12)

In contrast, the crystal potential near the interstices where the energy is lowest, should have a flatter minimum since six nearest ions are located farther apart at equal distance and with cubic symmetry. The zero-point energy therefore should have negligible differences for the three isotopes: Emin,H = Emin,D

353

=

Edn,

T.

(13)

Combination of eqns (1 l-l 3) then yields the desired inequalities (3), which will be numerically computed in the next section.

4. APPLICATIONS AND COMPARISON EXPERIMENTS

WITH

Hydrogen isotopes are thought to move (in the crystal) as ions, that is, as protons, deuterons

c e2(4ns)-‘Cr-’

exp(-q,r),

(14)

where q;’ is the Debye screening length, which arises from the (Coulomb) interaction between H+ and conduction electrons; and K is the dielectric constant arising from the (Coulomb) interaction between H+ and atom-core electrons. This potential U is isotropic and, therefore, it is only an approximation; but it is hard to improve upon in any simple manner. The two parameters Z and ICgenerate the strength of the potential, c

EZK-‘.

(15)

The screening constant qo, according to the Thomas-Fermi theory [12], can be related to the electron density n in the form: q. = 2.95(u,/r,)“*

A-‘,

(4n/3)ri s n-‘,

(16)

where ua is the Bohr radius = 0.529 A. The crystal potential V can be calculated in terms of U, and it is given by V(r) = 14.421;c ]c I

riI-'exP[-qoIr-riIl,

(17)

where the summation i runs over the lattice sites; the numerical constants were adjusted such that if q. is given in A-’ and Iri- r[ in angstroms, the potential energy is given in electron volts. The confining potential W at the saddle point (~uo,~a,,,O) is given by W(x) = e2C(47cQ)-‘[(d + x)-’ + (d -x)-l

e-mcd+*

e-m(d-x)] a constant + fk,,x*

(18)

with k, = 462Cd-‘[l + (1 + qod)*]e-“4

(19)

where d( = 2-‘I* UJ is half the distance between the neighboring ions at (f G, i Q, 0) and (u,,, 0,O). In the harmonic approximation, the zero-point energy E. is given by E0 = f ho,

w s (k,/M)‘“.

(20)

S. FUJITA and

354

A. GARCIA

Table 1. Quantum isotope effects for fee metals Z

Host metal

31

Pd CU Ni

3.89 3.61 3.52

2 1 2

&tS (eV)

i

QZEt

(FG)

(eV)

7:;e

1.953 1.806 2.052

0.243 0.403 0.409

0.35 0.42 0.49

0.019 0.021 0.008

0.0186 0.0249 0.0264

t Experimental values. $ Calculated values.

The quantum isotope deuterons defined by

effect

for

protons

QIE = K,H- 4,~

metals listed, Ni is ferromagnetic and has a Curie temperature, T,= 627 K. It appears that the diffusion data can be fitted by two sets of (D,, E,) below and (21) above the Curie temperature [l]: and

can be calculated through

E, = 0.42eV,

QZE = fh(k0/m,)“2[1 - 2-“2],

(22)

where mp is the proton mass. We are now ready to compare between theory and experiment. We computer-generated a fee crystal with the known lattice constant a, and put 24 neighboring ions in three contiguous planes. We adjusted the strength [ by matching the calculated barrier energy t with the observed activation energy E,.We then computed QZE through eqn (22). The results are given in Table 1. Notice a remarkable agreement between theory and experiment with the exception for Ni in spite of our crude model. In the present analysis, [ = ZK-’ is the only adjustable parameter, which may be viewed as the effective charge parameter with K = 1. The ionic valence Z and the screening constant q. are correlated through the conduction electron density n. If Z is assumed to take a greater value than those indicated in the table, the screening constant q,, becomes greater, which roughly compensates the effect of the former. Therefore, the calculated QZE are found to be quite stable with respect to the variations of (Z, qo). The dielectric constant K must, by definition, be greater than one, and its values calculated from the table for Pd, Cu and Ni are 5.7, 2.4, and 4.1, which appears to be reasonable. These values may be reduced further if the local deformations (expansions) of the lattices, which are neglected in our calculations, are taken into account. As noted earlier, the theoretical QZES for Ni is far from the experimental QZEt. Unlike the other fee

E,=O.409

eV,

D, = 6.9 x 10m3cm* s-l,

T > T,

D,=4.8

T
x 10-3cm2s-‘,

This makes the analysis of the QZE more complicated. Whether and how the ferromagnetic interaction may influence the hydrogen diffusion is not clear. It is, however, reasonable to assume that there are two activated processes for this metal. One is the interstitial migration which we have assumed throughout the present work. The mean straight path I at 500 K for Ni is estimated to be 5.7 w which is only marginally greater than the lattice constant u0 = 3.5 A, see Section 2. Then the usual hopping mode for which the jump distance is simply the distance between the two equilibrium sites, 1 = 2-l’* a,, and the activation energy is the smallest potential energy difference over any curved paths between the two sites, may also contribute to diffusion. The saddle point defined in this sense, briefly discussed in Section 1, should have the energy difference distinct from that associated with the saddle point in the channel model, and should have a less severe transverse confinement. These two activated processes should be in action concurrently. If so, the QZE for the total processes should be less than what we computed for the interstitial mode only. At any rate, further studies of the isotope effects in Ni are necessary.

5. CONCLUSIONS We calculated the QZE on the activation energy for hydrogen diffusion in fee metals with the assumption

Table 2. Quantum isotope effect for bee metals Z

Host metal Ta Nb V

$1, 3.31 3.30 3.01

5 5 5

t Experimental values. $ Calculated values.

2.20 2.20 2.28

E,tS (eV) 0.14 0.106 0.05

l 3.12 2.35 0.92

F;e -0.023 -0.023 -0.030

y;;e - 0.032 - 0.028 -0.021

Theory of H diffusion in metals that protons and deuterons migrate interstitially in the crystal potential computed with the screened Coulomb interaction between proton and host ions. These microscopic calculations give results in good agreement with the experimental data. This QIE arises from the quantum zero-point energy associated with the confining potential at the saddle point in the interstitial channel. The diffusion data [l] for bee metals such as Ta, Nb and V show that the activation energy E, depends on the isotopes but the inequalities (3) are reversed. This can be explained as follows. The expected interstitial channel is in the direction (100) and passes through the O-site (energy ~nimum), T-site (energy maximum), O-site and so on. The crystal potential at the O-sites, e.g. (0, f a,,, 0) and (f%,t a 0, 0), is of a confining type while that at the T-sites, e.g. (auO, ia,, 0), is not. The QZE, therefore, is significant only at the O-sites, that is, at the minimum energy sites, which yields the desired inequalities for the activation energy in view of eqn (1 I). The quantitative calculations and relevant data are given in Table 2. Here again, the agreement between theory and experiment is quite remarkable.

Acknowledgements-One

of the authors (A.G.), wishes to thank CONACYT (Mexico) for financial support.

3.55 REFERENCES

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