- Email: [email protected]
Inverse isotope effects in the surface diffusion of atoms
ii
r
surface science ELSEVIER
Surface Science363 (1996) 403-408
Inverse isotope effects in the surface diffusion of atoms T a k a s h i M i y a k e *, K o i c h i K u s a k a b e , S h i n j i T s u n e y u k i Institute for Solid State Physics, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo 106, Japan
Received 7 August 1995;accepted for publication 23 November 1995
Abstract
The inverse isotope effectin hydrogen diffusion at metal surfaces has been investigated with the Quantum Transition State Theory using a two-dimensional model potential with confinementorthogonal to the diffusion path. It is clarified by quantum Monte Carlo simulation that a decrease in the hopping rate by confinement, competingwith an increase by tunneling, induces the inverse isotope effect within a certain temperature range above a classical~luantum crossovertemperature. We also show that, at high temperature, the confinementeffect can be included in the potential barrier for diffusion and that the inverse isotope effect can be described with a reduced one-dimensional model within Feynman'svariational method. Keywords: Computer simulations; Diffusion and migration; Hydrogen; Models of surface kinetics; Quantum effects; Semi-empirical models and model calculations; Surface diffusion;Tungsten; Tunneling
Diffusion of atomic hydrogen on the surface and in the bulk of metals has been a stimulating problem not only from a practical point of view but also as a fundamental problem of quantum mechanics, since a significant quantum effect [ 1-7] is expected for the lightest element. At high temperature, diffusion occurs classically by thermal activation so that the diffusion rate decreases with temperature, following the Arrhenius' equation. When the temperature decreases down to around 100 K on metal surfaces, we observe a classicalquantum crossover into a tunneling region where the hopping rate shows less temperature dependence [ 8 - 1 2 ] . It is also known that the hopping rate in the bulk diffusion increases again at much lower temperature due to coherent tunneling I-13], * Corresponding author. Fax: +81 3 340 281 74; e-mail: [email protected].
although this temperature range is beyond the scope of this paper. The classical-quantum crossover has been precisely investigated theoretically with a path integral formulation, which is one of the most powerful techniques to give a unifying description of a hopping in both classical and quantum regions. Path integral formulation leads to the well known fact that the crossover occurs distinctly in a narrow temperature region [ 14-16]. In this formulation, the crossover temperature is definitely characterized by d e f o r m a t i o n of a Feynmann path mainly contributing to the hopping, i.e. emergence of instanton path describing tunneling [17]. When we consider hydrogen diffusion, we can utilize three isotopes of hydrogen to observe an interesting effect in the vicinity of t h e crossover temperature arising from the difference in mass of the hopping particles. Usually, an isotope effect is
0039-6028/96/$15.00 Copyright© 1996Elsevier ScienceB.V. All rights reserved PI1 S0039-6028 (96)00168-9
404
T. Miyake et al./Surfaee Science 363 (1996) 403-408
seen so that lighter particles diffuse more easily. It has been reported, however, that hydrogen diffusion on W ( l l 0 ) shows an anomalous isotope effect in that the heavier particle diffuses faster just above the classical-quantum crossover [8]. This inversion of the mass dependence of the diffusion constant is called the inverse isotope effect. The same effect is also observed in the bulk diffusion below a certain temperature (~800 K for Pd) [12]. The inverse isotope effect is peculiar for the following reason. When we consider the diffusion from a microscopic view, an elementary process is thought to be the hopping of a particle (hydrogen) from a stable site to a neighboring site, except for at very high temperatures larger than the barrier height. At high temperature the particle hops classically over the potential barrier. Since the potential is the same for any isotope, mass dependence comes only from the attempt frequency, which leads to the hopping rate, F, being proportional to m -1/2. At low temperature where the particle tunnels quantum mechanically, F is expected to increase exponentially with decreasing m; this is easily derived from WKB approximation. Naively, we expect that at around the crossover temperature, the mass dependence of F will be intermediate between the high temperature m -a/2 dependence and the low temperature exponential behavior. Thus the inverse isotope effect is unlikely to exist at any temperature. However, as a result of numerical simulations using the Quantum Transition State Theory (QTST) [16], which is based on path integral formulation, Rick et al. found the inverse isotope effect for the hydrogen diffusion from a P d ( l l l ) surface into the subsurface layer in the intermediate temperature region [18]. They interpreted it as follows, When the particle reaches the top of the potential barrier, it is highly confined spatially in the directions orthogonal to the diffusion path due to the surrounding metal atoms, a n d will have much larger zero-point energy than it will :in the well bottoms, which suppresses the hopping. This confinement effect has larger influence on lighter particles, so that the inverse isotope effect could be induced. The confinement effect in the diffusion has also been pointed out by other authors [ 19,20]: Although their explanation is intuitively plausi-
ble, there remains several points to be made clearer: (1) Does confinement actually induce the inverse isotope effect? (2) Is it possible to explain it as a change in the potential barrier? (3) What determines the temperature range of the inverse isotope effect? To clarify the role of confinement in the isotope effect, it is sufficient to use a model potential. Hence, in the present work, we introduce a potential with the essence of confinement in order to investigate the inverse isotope effect. As a result of a quantum Monte Carlo simulation (QMC), it will be clarified that the inverse isotope effect does occur. Moreover, its mechanism is discussed with a variational approach of Feynman, which gives a clear description of the isotope effect at high temperature. In order to discuss both the classical and the quantum region consistently, we calculated a hopping rate following QTST, which is concerned with a particle confined in a double-well potential [ 16]. Here the phonons are treated as a sluggish bath, which means that their frequencies are negligible compared to that of the particle. In QTST, the center of mass of the cyclic Feynman path called the centroid plays an important role. The hopping rate is calculated from the probability distribution of the centroid, P(x), F-
2q-~ax
~)P(x*),
(1)
where 13 is the inverse temperature, x* is the coordinate of the transition state, and Ax is a width of P(x) at the transition state. The factor is a slowly varying function of the temperature from fi/flo (the high temperature limit) to 1 (at the low temperature limit) [ 16]. Here flo is the inverse of the classical-quantum crossover temperature, which is related to the frequency at the barrier top cob as fl~=2~/hcob [14]. Thus the temperature dependence of F comes mainly from P(x*). As the first step towards an exact description, we investigated the confinement effect with a model potential which includes the essence of the confinement, thereby introducing a parameter which controls the magnitude of the confinement. It is sufficient to consider the model in two-dimension:
T. Miyakeet aL/SurfaceScience363 (1996) 403-408 one coordinate is the reaction coordinate x, and the other is the orthogonal one causing the confinement, y. When y = 0, we assume a symmetric quartic potential in the x-direction with minima located at y = __do. In the y-direction, the potential is expanded to the second order around the minim u m (y = 0). In summation, the potential V(x,y) in a two-dimensional space is given by
V(x,y) = v(x) + f ( x ) y 2,
(2)
v(x)=Vo I(~O) 2 - - 1 12,
(3)
where f ( x ) determines the degree of the confinement. The confinement should be the strongest at the barrier top (x = 0). Thus we set f ( x ) as 2 + (8oo2 + 2coo~& o ) e-Xa/a:), f ( x ) = -mn T (oo~
(4)
where mn is the mass of hydrogen. It takes a m a x i m u m ml~(Co~+&o)2/2 at x = 0 , and approaches mnco2/2 with increasing Ix] with the decay length of 2. When f ( x ) is a constant (~o9 = 0), the p a t h integral of y(r) separates from that of x(z) and adds only a constant extra potential to v(x). Under these conditions no confinement effect exist. The confinement effect arises from the variation of f ( x ) with x. In the present model, its strength is characterized by 8co. In the following discussion, we use a special system of units defined by h = 1, e 2 = 2000, and m~ = 1/2000, where e is the charge unit and m, the electron mass. In this system of units, unit of energy, length and mass are 13.61 meV, 0.5292 and 1.822 x 10 -27 Kg, respectively. In the model potential, we fix parameters except the confinement strength. We use Vo = 200.0 meV = 14.70 (in our units) and do = 1.500 * = 2.835 hereafter, which are typical values for hydrogen diffusion on metal surfaces [-1 lJ. As for the coupling to the bath, the relaxation energy is 48 meV in H/Ni(100) according to the embedded a t o m method [20]. The relaxation energy is roughly equal to ~odo2 where ~o = ZxC2/(2rnzc°za) is defined by the coupling constant Ca, mass mz and frequency ooz of the 2-th harmonic oscillator composing the sluggish bath [,16]. Thus, we set c~odo 2 to be 5 0 . 0 0 m e V = 3.675.
405
As for other parameters, coo~ is determined by 2 2 __ mneo~odo/2 - %. The decay length 2 is set equal to do. In this system, the inverse of the crossover temperature is flCH= 2.225 for hydrogen and flop = 3.147 for deuterium. The hopping rate is obtained from Eq. (1). Since a path integral is difficult to calculate in general, P(x*) is evaluated using path integral Monte Carlo simulation where each Feynman path is represented by a set of Np discrete points. Here we need to increase N v with increasing fl from N v = 8 to Np = 32 for numerical convergence. In Fig. 1 we show the ratio of hopping rate of hydrogen to that of deuterium, F~,io. When the F~tio is less than unity, deuterium diffuses faster, i.e. the inverse isotope effect is observed. We find a temperature region where F~t~o increases rapidly, while XFrati o changes slowly both a b o v e and below this region (inset). This rapid increase is due to the difference in the crossover temperature of hydrogen and deuterium. Since the crossover temperature is proportional to m -~/2, there exists a temperature region where the hopping process is different for two particles: tunneling for hydrogen and thermal activation process for deuterium. Because the former process is less temperaturedependent, F~tio increases with increasing ft. N o w we turn to the confinement effect. Since classical behavior is realized at fl = 0 , F~atio= ~ D / m n = V~ at that point. In the case of &o = 0, G,tio increases monotonously with fi, reflecting the tunneling effect. As &o increases, however, the 3.0
2.0
20
. . . .
15
~Sm=O
"
. . . . ~
"
. . . .
"
"
"
" /
-?
,o o~=~o 5
~ 1.5 ¢m 1.0 --
o 0
.~ ,
2
,
a
0.5
0.0
1:7:.-77.>~ ~ ~ _ 7 ~?- 7 7 ............. ~i ..... ® . . . . . ~ . ~ ....
-0.5 ~ . . . .
,
- "00.0
F i g . 1. T h e deuterium, inverse
ratio
. . . .
0.5 of the
G~tlo, with
temperature.
4
1.0 hopping
rate
8¢o = 0 , 1 0 a n d
. . . .
i
1.5''' of hydrogen
2.0 to
20 as a function
that
of
of the
406
T. M i y a k e et al./Surface Science 363 (1996) 4 0 3 - 4 0 8
slope of F~atio decreases a t / / ~ 0, and finally there appears a temperature region where F~atio is less than unity. This is the inverse isotope effect. Results for various 6co are summarized in Fig. 2. It is found that the inverse isotope effect appears when the confinement is as strong as 6o9 > 15. In this case, the difference in the zero point energy of hydrogen in the y-direction, between the barrier top and at the well bottoms, is more than 90 meV. This value is not too large to be realized, because it is comparable with or less than the typical energy scale in hydrogen diffusion on metal surfaces. We now proceed to the second question: is it possible to describe the inverse isotope effect by treating quantum effects as the correction to the classical description of the hopping rate? In particular, we are interested in the high temperature region, since the region of the inverse isotope effect is located above the crossover temperature. The confinement effect cannot simply be written as the change in the potential barrier in the hopping direction. This is definitely seen in the path integral formulation as follows. Consider a calculation of a partition function in n-dimensional space. If we first perform the path integral for (n - 1)-dimensional motion except for the reaction coordinate, then we obtain the Lagrangian containing a functional of a one-dimensional path for
the hopping motion. This means that the problem cannot be reduced to a one-dimensional one. Nevertheless we show that, at high temperature, it is possible to reduce the confinement effect approximately to the change in the potential barrier for hopping using Feynman's variational approach [21]. At high temperature, the total path integral can be evaluated by integration of the path centroid and the small fluctuating paths around it. Hence we are allowed to replace the potential part by an effective potential for the centroid which includes quantum effects. In other words, this substitution corresponds to assuming that the estimated variation of the potential term by the fluctuation around the centroid neglects the correlation. Then, path integrals of each degree of freedom are separated from each other and can be performed independently. Now that the effect arising from motion in the y-direction can be included in the effective potential in the x-direction simply by integrating the y component of the centroid. In our case, due to the quadratic form of the potential in the y-direction, we can obtain analytically the effective potential U(x), in which the particle hops classically, as U(x)
= ax 4 -
6ax2- b (fih2) bx 2 + g(x) + - \T£m/
3a ( ~ h 2 7 20
~, A
©
0
0
0
~
X
0
0
0
0
&
X
X
z~
z~
A
z~
x
(5)
+7
15
g(x) = fi N/24m[(6m/[lh 2) + (1/22)-I
10
xexplxZ(_
5 .
0.0
.
.
.
.
.
.
.
0.5
.
.
.
.
.
.
1.0
,
1~
.
.
.
.
,
.
2.0
.
.
.
.
.
.
.
2.5
.
3.0
Fig. 2. Phase diagram of the normal and inverse isotope effect. The abscissa is the inverse temperature and the ordinate the confinement strength. The results were obtained by QMC: (O) inverse isotope effect, (x) normal isotope effect and (A) indistinguishable within statistical error. A phase boundary obtained by Feynman's variational method is also shown by a solid line: the inverse isotope effect appears above this line.
+
m~co~Bh2 , 24m
1 1 ~-~-I-[22 q_ ( 6~nfc4/flh2)]) ] (6)
where a - V o i d 4 and b - 2 V o / d 2 satisfy v(x)= ax 4 - - bx 2, and fi - rnn(Scoz + 2co~o8co)/2. Here U(x) consists of five terms. The first and the second are the original double-well potential v(x). The third term g(x) increases as 6co increases. This term represents the confinement effect and decreases the
T Miyake et al./Surface Science 363 (1996) 403-408
hopping rate through the increase in the potential barrier. The fourth and the fifth reflect the tunneling effect. They increase the hopping rate through a decrease of the potential barrier. In this way, at high temperature, Feynman's variational method enables us to treat quantum effects, i.e. the confinement effect and the tunneling effect, as corrections to the potential in the hopping direction. We now reexamine the confinement effect with the effective potential obtained above. At the high temperature limit, no quantum effects exist. There, log F is linear in /? with the slope of the barrier height following Arrhenius' equation. As the temperature decreases, quantum effects appear. If there is no confinement, the hopping rate deviates upward from the classical straight line owing to the increase due to tunneling. Conversely, the confinement effect which is another quantum effect works against the tunneling effect, and it decreases the hopping rate. In the case of strong confinement, it surpasses tunneling and the hopping rate deviates downward from the classical straight line. This competition between the confinement effect and the tunneling effect is crucial for the inverse isotope effect. At the high temperature limit (/3 = 0), the hopping rate of hydrogen/'H is V~ times as large as that of deuterium, FD. As/3 increases, the hopping rate deviates either upward or downward depending on 509, with larger deviation for hydrogen than deuterium, since both the confinement effect and the tunneling effect change U(x) in proportion to m -1 and/3 at high temperature. If the confinement is strong enough, finally FH becomes smaller than FD. Thus the inverse isotope effect appears. In a computer simulation of hydrogen diffusion at Ni(100), Mattsson et al. found that the quantum hopping rate is less than the classical value at 200 K [20]. This indicates that the confinement effect is stronger than the tunneling effect in the classical over barrier region in that system, and implies that 6co is ,,~ 15 on Ni(100). We expect that the confinement is stronger on W(110) and some other systems, so that the inverse isotope effect is induced. In Fig. 2 we show the phase boundary between the normal and the inverse isotope effect obtained
407
by the present variational approach. At high temperature, the boundary is in good agreement with the QMC result. In other words, the appearance of the inverse isotope effect can be predicted rather accurately by treating quantum effects as corrections. However, the variational approach predicts that the region of the inverse isotope effect extends at low temperature, in contrast to the QMC result. This is due to the limitation of t h e variational method. The lower-temperature-boundary of the inverse isotope effect is determined by the growth of quantum effects with decreasing temperature, of which the Feynman's variational method cannot give an accurate description. In fact, the tunneling effect grows much more rapidly than the variational method predicts and this strongly suppresses the inverse isotope effect. This is an answer to the question regarding the temperature range of the inverse isotope effect. In conclusion, we have clarified that spatial confinement at the barrier top competing with tunneling does induce the inverse isotope effect in atomic diffusion above the classical-quantum crossover temperature. The role of the confinement has been discussed with Feynman's variational method, which gives a reliable description of the quantum effects at high temperature and enables us to include, approximately, the confinement effect in the potential barrier. Although at low temperature it underestimates the tunneling effect and fails to predict the lower-temperature-boundary of the inverse isotope effect, the variational method might be useful for predicting the appearance of the inverse isotope effect quite easily without quantum Monte Carlo simulation. As one can see from the above discussion, the inverse isotope effect is not a phenomenon specific to surfaces. In fact, it is observed in bulk systems of several fcc metals such as Pd, Ni and Cu. Although confinement might be even smaller at metal surfaces, however, it is advantageous that we can control potential parameters by changing surface indices and the direction of diffusion as well as substrate materials. Since confinement is highly dependent on the detail of the potential energy surfaces, it might be fruitful to perform experiments systematically with a variety of sur-
408
T Miyake et al./Surface Science 363 (1996) 403-408
faces. Theoretical investigation with more realistic potentials is also a future problem.
[9]
Acknowledgements
[10] [11]
This work is supported partly by a N E D O International Joint Research Grant. The computations reported in this paper were carried out on a DEC3000/600.
[12] [13]
References [1] R. Nieminen, Nature 356 (1992) 289. [2] M.J. Puska, R.M. Nieminen, M. Manninen, B. Chakraborty, S. Holloway and J.K. Norskov, Phys. Rev. Lett. 51 (1983) 1081; M.J. Puska and R.M. Nieminen, Surf. Sci. 157 (1985) 413. [3] C.M. Mate and G.A. Somorjai, Phys. Rev. B 34 (1986) 7417. [4] C. Astaldi, A. Bianco, S. Modesti and E. Tosatti, Phys. Rev. Lett. 68 (1992) 90. [5] C-H. Hsu, B.E. Larson, M. E1-Batanouny, C.R. Willis and K.M. Martini, Phys. Rev. Lett. 66 (1991) 3164. [6] O. Grizzi, M. Shi, H. Bu, J.W. Rabalais, R.R. Rye and P. Nordlander, Phys. Rev. Lett. 63 (1989) 1408. [7] S.W. Rick and J.D. Doll, Surf. Sci. 302 (1994) L305. [8] R. DiFoggio and R. Gomer, Phys. Rev. B 25 (1982) 3490; S.C. Wang and R. Gomer, J. Chem. Phys. 83 (1985) 4193;
[14] [15]
[16] [17] [18] [19] [20]
[21]
A. Auerbach, K.F. Freed and R. Gomer, J. Chem. Phys. 86 (1987) 2356. E.A. Daniels, J.C. Lin and R. Gomer, Surf. Sci. 204 (1988) 129. T.-S. Lin and R. Gomer, Surf. Sci. 255 (1991) 41. X.D. Zhu, A. Lee, A. Wong and U. Linke, Phys. Rev. Lett. 68 (1992) 1862; A. Lee, X.D. Zhu, L. Deng and U. Linke, Phys. Rev. B 46 (1992) 15472; A. Lee, X.D. Zhu, A. Wong, L, Deng and U. Linke, Phys. Rev. B 48 (1993) 11256. J. VSlkl and G. Alefeld, in: Hydrogen in metals I, Eds. G. Alefeld and J. VBlkl (Springer-Verlag, Berlin, 1978). D. Steinbinder, H. Wipf, A. Magerl, D. Richter, A.-J. Dianoux and K. Neumaier, Europhys. Lett. 6 (1988) 535. I. Affleck, Phys. Rev. Lett. 46 (1981) 388. H. Grabert and U. Weiss, Phys. Rev. Lett. 53 (1984) 1787; H. Grabert, P. Olschowski and U. Weiss, Phys. Rev. B 36 (1987) 1931. M.J. Gillan, J. Phys. C 20 (1987) 3621. S. Coleman, in: The Whys of Subnuclear Physics, Ed. A. Ziehichi (Plenum, New York, 1979). S.W. Rick, D.L. Lynch and J.D. Doll, J. Chem. Phys. 99 (1993) 8183. K. Muttalib and J. Sethna, Phys. Rev. B 32 (1985) 3462. T.R. Mattsson, U. Engberg and G. WahnstrOm, Phys. Rev. Lett. 71 (1993) 2615; T.R. Mattsson and G. Wahnstr6m, Phys. Rev. B 51 (1995) 1885. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
r
surface science ELSEVIER
Surface Science363 (1996) 403-408
Inverse isotope effects in the surface diffusion of atoms T a k a s h i M i y a k e *, K o i c h i K u s a k a b e , S h i n j i T s u n e y u k i Institute for Solid State Physics, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo 106, Japan
Received 7 August 1995;accepted for publication 23 November 1995
Abstract
The inverse isotope effectin hydrogen diffusion at metal surfaces has been investigated with the Quantum Transition State Theory using a two-dimensional model potential with confinementorthogonal to the diffusion path. It is clarified by quantum Monte Carlo simulation that a decrease in the hopping rate by confinement, competingwith an increase by tunneling, induces the inverse isotope effect within a certain temperature range above a classical~luantum crossovertemperature. We also show that, at high temperature, the confinementeffect can be included in the potential barrier for diffusion and that the inverse isotope effect can be described with a reduced one-dimensional model within Feynman'svariational method. Keywords: Computer simulations; Diffusion and migration; Hydrogen; Models of surface kinetics; Quantum effects; Semi-empirical models and model calculations; Surface diffusion;Tungsten; Tunneling
Diffusion of atomic hydrogen on the surface and in the bulk of metals has been a stimulating problem not only from a practical point of view but also as a fundamental problem of quantum mechanics, since a significant quantum effect [ 1-7] is expected for the lightest element. At high temperature, diffusion occurs classically by thermal activation so that the diffusion rate decreases with temperature, following the Arrhenius' equation. When the temperature decreases down to around 100 K on metal surfaces, we observe a classicalquantum crossover into a tunneling region where the hopping rate shows less temperature dependence [ 8 - 1 2 ] . It is also known that the hopping rate in the bulk diffusion increases again at much lower temperature due to coherent tunneling I-13], * Corresponding author. Fax: +81 3 340 281 74; e-mail: [email protected].
although this temperature range is beyond the scope of this paper. The classical-quantum crossover has been precisely investigated theoretically with a path integral formulation, which is one of the most powerful techniques to give a unifying description of a hopping in both classical and quantum regions. Path integral formulation leads to the well known fact that the crossover occurs distinctly in a narrow temperature region [ 14-16]. In this formulation, the crossover temperature is definitely characterized by d e f o r m a t i o n of a Feynmann path mainly contributing to the hopping, i.e. emergence of instanton path describing tunneling [17]. When we consider hydrogen diffusion, we can utilize three isotopes of hydrogen to observe an interesting effect in the vicinity of t h e crossover temperature arising from the difference in mass of the hopping particles. Usually, an isotope effect is
0039-6028/96/$15.00 Copyright© 1996Elsevier ScienceB.V. All rights reserved PI1 S0039-6028 (96)00168-9
404
T. Miyake et al./Surfaee Science 363 (1996) 403-408
seen so that lighter particles diffuse more easily. It has been reported, however, that hydrogen diffusion on W ( l l 0 ) shows an anomalous isotope effect in that the heavier particle diffuses faster just above the classical-quantum crossover [8]. This inversion of the mass dependence of the diffusion constant is called the inverse isotope effect. The same effect is also observed in the bulk diffusion below a certain temperature (~800 K for Pd) [12]. The inverse isotope effect is peculiar for the following reason. When we consider the diffusion from a microscopic view, an elementary process is thought to be the hopping of a particle (hydrogen) from a stable site to a neighboring site, except for at very high temperatures larger than the barrier height. At high temperature the particle hops classically over the potential barrier. Since the potential is the same for any isotope, mass dependence comes only from the attempt frequency, which leads to the hopping rate, F, being proportional to m -1/2. At low temperature where the particle tunnels quantum mechanically, F is expected to increase exponentially with decreasing m; this is easily derived from WKB approximation. Naively, we expect that at around the crossover temperature, the mass dependence of F will be intermediate between the high temperature m -a/2 dependence and the low temperature exponential behavior. Thus the inverse isotope effect is unlikely to exist at any temperature. However, as a result of numerical simulations using the Quantum Transition State Theory (QTST) [16], which is based on path integral formulation, Rick et al. found the inverse isotope effect for the hydrogen diffusion from a P d ( l l l ) surface into the subsurface layer in the intermediate temperature region [18]. They interpreted it as follows, When the particle reaches the top of the potential barrier, it is highly confined spatially in the directions orthogonal to the diffusion path due to the surrounding metal atoms, a n d will have much larger zero-point energy than it will :in the well bottoms, which suppresses the hopping. This confinement effect has larger influence on lighter particles, so that the inverse isotope effect could be induced. The confinement effect in the diffusion has also been pointed out by other authors [ 19,20]: Although their explanation is intuitively plausi-
ble, there remains several points to be made clearer: (1) Does confinement actually induce the inverse isotope effect? (2) Is it possible to explain it as a change in the potential barrier? (3) What determines the temperature range of the inverse isotope effect? To clarify the role of confinement in the isotope effect, it is sufficient to use a model potential. Hence, in the present work, we introduce a potential with the essence of confinement in order to investigate the inverse isotope effect. As a result of a quantum Monte Carlo simulation (QMC), it will be clarified that the inverse isotope effect does occur. Moreover, its mechanism is discussed with a variational approach of Feynman, which gives a clear description of the isotope effect at high temperature. In order to discuss both the classical and the quantum region consistently, we calculated a hopping rate following QTST, which is concerned with a particle confined in a double-well potential [ 16]. Here the phonons are treated as a sluggish bath, which means that their frequencies are negligible compared to that of the particle. In QTST, the center of mass of the cyclic Feynman path called the centroid plays an important role. The hopping rate is calculated from the probability distribution of the centroid, P(x), F-
2q-~ax
~)P(x*),
(1)
where 13 is the inverse temperature, x* is the coordinate of the transition state, and Ax is a width of P(x) at the transition state. The factor is a slowly varying function of the temperature from fi/flo (the high temperature limit) to 1 (at the low temperature limit) [ 16]. Here flo is the inverse of the classical-quantum crossover temperature, which is related to the frequency at the barrier top cob as fl~=2~/hcob [14]. Thus the temperature dependence of F comes mainly from P(x*). As the first step towards an exact description, we investigated the confinement effect with a model potential which includes the essence of the confinement, thereby introducing a parameter which controls the magnitude of the confinement. It is sufficient to consider the model in two-dimension:
T. Miyakeet aL/SurfaceScience363 (1996) 403-408 one coordinate is the reaction coordinate x, and the other is the orthogonal one causing the confinement, y. When y = 0, we assume a symmetric quartic potential in the x-direction with minima located at y = __do. In the y-direction, the potential is expanded to the second order around the minim u m (y = 0). In summation, the potential V(x,y) in a two-dimensional space is given by
V(x,y) = v(x) + f ( x ) y 2,
(2)
v(x)=Vo I(~O) 2 - - 1 12,
(3)
where f ( x ) determines the degree of the confinement. The confinement should be the strongest at the barrier top (x = 0). Thus we set f ( x ) as 2 + (8oo2 + 2coo~& o ) e-Xa/a:), f ( x ) = -mn T (oo~
(4)
where mn is the mass of hydrogen. It takes a m a x i m u m ml~(Co~+&o)2/2 at x = 0 , and approaches mnco2/2 with increasing Ix] with the decay length of 2. When f ( x ) is a constant (~o9 = 0), the p a t h integral of y(r) separates from that of x(z) and adds only a constant extra potential to v(x). Under these conditions no confinement effect exist. The confinement effect arises from the variation of f ( x ) with x. In the present model, its strength is characterized by 8co. In the following discussion, we use a special system of units defined by h = 1, e 2 = 2000, and m~ = 1/2000, where e is the charge unit and m, the electron mass. In this system of units, unit of energy, length and mass are 13.61 meV, 0.5292 and 1.822 x 10 -27 Kg, respectively. In the model potential, we fix parameters except the confinement strength. We use Vo = 200.0 meV = 14.70 (in our units) and do = 1.500 * = 2.835 hereafter, which are typical values for hydrogen diffusion on metal surfaces [-1 lJ. As for the coupling to the bath, the relaxation energy is 48 meV in H/Ni(100) according to the embedded a t o m method [20]. The relaxation energy is roughly equal to ~odo2 where ~o = ZxC2/(2rnzc°za) is defined by the coupling constant Ca, mass mz and frequency ooz of the 2-th harmonic oscillator composing the sluggish bath [,16]. Thus, we set c~odo 2 to be 5 0 . 0 0 m e V = 3.675.
405
As for other parameters, coo~ is determined by 2 2 __ mneo~odo/2 - %. The decay length 2 is set equal to do. In this system, the inverse of the crossover temperature is flCH= 2.225 for hydrogen and flop = 3.147 for deuterium. The hopping rate is obtained from Eq. (1). Since a path integral is difficult to calculate in general, P(x*) is evaluated using path integral Monte Carlo simulation where each Feynman path is represented by a set of Np discrete points. Here we need to increase N v with increasing fl from N v = 8 to Np = 32 for numerical convergence. In Fig. 1 we show the ratio of hopping rate of hydrogen to that of deuterium, F~,io. When the F~tio is less than unity, deuterium diffuses faster, i.e. the inverse isotope effect is observed. We find a temperature region where F~t~o increases rapidly, while XFrati o changes slowly both a b o v e and below this region (inset). This rapid increase is due to the difference in the crossover temperature of hydrogen and deuterium. Since the crossover temperature is proportional to m -~/2, there exists a temperature region where the hopping process is different for two particles: tunneling for hydrogen and thermal activation process for deuterium. Because the former process is less temperaturedependent, F~tio increases with increasing ft. N o w we turn to the confinement effect. Since classical behavior is realized at fl = 0 , F~atio= ~ D / m n = V~ at that point. In the case of &o = 0, G,tio increases monotonously with fi, reflecting the tunneling effect. As &o increases, however, the 3.0
2.0
20
. . . .
15
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"
. . . . ~
"
. . . .
"
"
"
" /
-?
,o o~=~o 5
~ 1.5 ¢m 1.0 --
o 0
.~ ,
2
,
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,
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that
of
of the
406
T. M i y a k e et al./Surface Science 363 (1996) 4 0 3 - 4 0 8
slope of F~atio decreases a t / / ~ 0, and finally there appears a temperature region where F~atio is less than unity. This is the inverse isotope effect. Results for various 6co are summarized in Fig. 2. It is found that the inverse isotope effect appears when the confinement is as strong as 6o9 > 15. In this case, the difference in the zero point energy of hydrogen in the y-direction, between the barrier top and at the well bottoms, is more than 90 meV. This value is not too large to be realized, because it is comparable with or less than the typical energy scale in hydrogen diffusion on metal surfaces. We now proceed to the second question: is it possible to describe the inverse isotope effect by treating quantum effects as the correction to the classical description of the hopping rate? In particular, we are interested in the high temperature region, since the region of the inverse isotope effect is located above the crossover temperature. The confinement effect cannot simply be written as the change in the potential barrier in the hopping direction. This is definitely seen in the path integral formulation as follows. Consider a calculation of a partition function in n-dimensional space. If we first perform the path integral for (n - 1)-dimensional motion except for the reaction coordinate, then we obtain the Lagrangian containing a functional of a one-dimensional path for
the hopping motion. This means that the problem cannot be reduced to a one-dimensional one. Nevertheless we show that, at high temperature, it is possible to reduce the confinement effect approximately to the change in the potential barrier for hopping using Feynman's variational approach [21]. At high temperature, the total path integral can be evaluated by integration of the path centroid and the small fluctuating paths around it. Hence we are allowed to replace the potential part by an effective potential for the centroid which includes quantum effects. In other words, this substitution corresponds to assuming that the estimated variation of the potential term by the fluctuation around the centroid neglects the correlation. Then, path integrals of each degree of freedom are separated from each other and can be performed independently. Now that the effect arising from motion in the y-direction can be included in the effective potential in the x-direction simply by integrating the y component of the centroid. In our case, due to the quadratic form of the potential in the y-direction, we can obtain analytically the effective potential U(x), in which the particle hops classically, as U(x)
= ax 4 -
6ax2- b (fih2) bx 2 + g(x) + - \T£m/
3a ( ~ h 2 7 20
~, A
©
0
0
0
~
X
0
0
0
0
&
X
X
z~
z~
A
z~
x
(5)
+7
15
g(x) = fi N/24m[(6m/[lh 2) + (1/22)-I
10
xexplxZ(_
5 .
0.0
.
.
.
.
.
.
.
0.5
.
.
.
.
.
.
1.0
,
1~
.
.
.
.
,
.
2.0
.
.
.
.
.
.
.
2.5
.
3.0
Fig. 2. Phase diagram of the normal and inverse isotope effect. The abscissa is the inverse temperature and the ordinate the confinement strength. The results were obtained by QMC: (O) inverse isotope effect, (x) normal isotope effect and (A) indistinguishable within statistical error. A phase boundary obtained by Feynman's variational method is also shown by a solid line: the inverse isotope effect appears above this line.
+
m~co~Bh2 , 24m
1 1 ~-~-I-[22 q_ ( 6~nfc4/flh2)]) ] (6)
where a - V o i d 4 and b - 2 V o / d 2 satisfy v(x)= ax 4 - - bx 2, and fi - rnn(Scoz + 2co~o8co)/2. Here U(x) consists of five terms. The first and the second are the original double-well potential v(x). The third term g(x) increases as 6co increases. This term represents the confinement effect and decreases the
T Miyake et al./Surface Science 363 (1996) 403-408
hopping rate through the increase in the potential barrier. The fourth and the fifth reflect the tunneling effect. They increase the hopping rate through a decrease of the potential barrier. In this way, at high temperature, Feynman's variational method enables us to treat quantum effects, i.e. the confinement effect and the tunneling effect, as corrections to the potential in the hopping direction. We now reexamine the confinement effect with the effective potential obtained above. At the high temperature limit, no quantum effects exist. There, log F is linear in /? with the slope of the barrier height following Arrhenius' equation. As the temperature decreases, quantum effects appear. If there is no confinement, the hopping rate deviates upward from the classical straight line owing to the increase due to tunneling. Conversely, the confinement effect which is another quantum effect works against the tunneling effect, and it decreases the hopping rate. In the case of strong confinement, it surpasses tunneling and the hopping rate deviates downward from the classical straight line. This competition between the confinement effect and the tunneling effect is crucial for the inverse isotope effect. At the high temperature limit (/3 = 0), the hopping rate of hydrogen/'H is V~ times as large as that of deuterium, FD. As/3 increases, the hopping rate deviates either upward or downward depending on 509, with larger deviation for hydrogen than deuterium, since both the confinement effect and the tunneling effect change U(x) in proportion to m -1 and/3 at high temperature. If the confinement is strong enough, finally FH becomes smaller than FD. Thus the inverse isotope effect appears. In a computer simulation of hydrogen diffusion at Ni(100), Mattsson et al. found that the quantum hopping rate is less than the classical value at 200 K [20]. This indicates that the confinement effect is stronger than the tunneling effect in the classical over barrier region in that system, and implies that 6co is ,,~ 15 on Ni(100). We expect that the confinement is stronger on W(110) and some other systems, so that the inverse isotope effect is induced. In Fig. 2 we show the phase boundary between the normal and the inverse isotope effect obtained
407
by the present variational approach. At high temperature, the boundary is in good agreement with the QMC result. In other words, the appearance of the inverse isotope effect can be predicted rather accurately by treating quantum effects as corrections. However, the variational approach predicts that the region of the inverse isotope effect extends at low temperature, in contrast to the QMC result. This is due to the limitation of t h e variational method. The lower-temperature-boundary of the inverse isotope effect is determined by the growth of quantum effects with decreasing temperature, of which the Feynman's variational method cannot give an accurate description. In fact, the tunneling effect grows much more rapidly than the variational method predicts and this strongly suppresses the inverse isotope effect. This is an answer to the question regarding the temperature range of the inverse isotope effect. In conclusion, we have clarified that spatial confinement at the barrier top competing with tunneling does induce the inverse isotope effect in atomic diffusion above the classical-quantum crossover temperature. The role of the confinement has been discussed with Feynman's variational method, which gives a reliable description of the quantum effects at high temperature and enables us to include, approximately, the confinement effect in the potential barrier. Although at low temperature it underestimates the tunneling effect and fails to predict the lower-temperature-boundary of the inverse isotope effect, the variational method might be useful for predicting the appearance of the inverse isotope effect quite easily without quantum Monte Carlo simulation. As one can see from the above discussion, the inverse isotope effect is not a phenomenon specific to surfaces. In fact, it is observed in bulk systems of several fcc metals such as Pd, Ni and Cu. Although confinement might be even smaller at metal surfaces, however, it is advantageous that we can control potential parameters by changing surface indices and the direction of diffusion as well as substrate materials. Since confinement is highly dependent on the detail of the potential energy surfaces, it might be fruitful to perform experiments systematically with a variety of sur-
408
T Miyake et al./Surface Science 363 (1996) 403-408
faces. Theoretical investigation with more realistic potentials is also a future problem.
[9]
Acknowledgements
[10] [11]
This work is supported partly by a N E D O International Joint Research Grant. The computations reported in this paper were carried out on a DEC3000/600.
[12] [13]
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