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The interaction of Xe with the Pt (111) surface
Volume 188, number 5.6
CHEMICAL PHYSICS LETTERS
17 January. 1992
The interaction of Xe with the Pt ( 111 ) surface J.A. B a r k e r , C.T. R e t t n e r a n d D . S . B e t h u n e IBM Research Division, Almaden Research Center, K31/803, 650 Harry Road, San Jose. CA 95120-6099. USA
Received 12 September 1991; in final form 24 October 1991
We describe the determination of an empirical potential energy function for the interaction of xenon with the Pt ( 111 ) surface using a novel analytic form. The potential is consistent with a wide range of dynamical and equilibrium experimental data. The equilibrium position for a single xenon atom lies directly above a surface platinum atom at a height of 3.3/k.
The potential energy function describing the interaction o f an a t o m or molecule with a surface holds the key for u n d e r s t a n d i n g such processes as energy exchange and a c c o m m o d a t i o n , trapping a n d desorption, as well as the structures and phase diagrams o f c o m m e n s u r a t e and i n c o m m e n s u r a t e surface phases. These p h e n o m e n a are i m p o r t a n t in contexts as diverse as catalysis, crystal and thin-film growth, and hypersonic flight [1]. There is no gas-surface system for which an accurate potential energy function valid over a wide range of energies is known. We have therefore d e t e r m i n e d an empirical potential function for the system X e / P t ( 1 11 ), chosen because of the wealth o f experimental data o f several different kinds which is available, for Xe energies ranging from a few meV to m o r e than 14 eV. Previous empirical studies have used experimental d a t a o f only one or two kinds [ 1 ], and the resulting potential functions were therefore far from uniquely d e t e r m i n e d . By contrast, the present work makes c o m p a r i s o n with a wide range o f different kinds of data, including m e a s u r e d d e s o r p t i o n rates [ 2 ], the frequency o f v i b r a t i o n of Xe n o r m a l to the surface [3], t r a p p i n g probabilities [4], scattering m e a s u r e m e n t s (including angular distributions and angle-dependent energy m e a s u r e m e n t s ) for incident energies between 0.5 and 14.3 eV [5], a n d a heuristic estimate [6] o f the corrugation o f the potential based on the observed "energy j u m p " at the commensurate to partially i n c o m m e n s u r a t e phase transition. Because o f this, our potential energy function is relatively well d e t e r m i n e d . In particular, we are Elsevier Science Publishers B.V.
able to conclude that the equilibrium site for a single Xe a t o m is very probably above the " o n t o p " site (at a height o f 3.3 A ) , and that no potential with the equilibrium site above the threefold hollow site can fit all the experimental facts. We represent the interaction between a xenon atom and the solid by a sum o f nonspherical pairwise-additive potentials (the s u m m a t i o n in eq. ( 1 ), see below), together with an additional term V which depends only on the n o r m a l distance Zgv of the xenon a t o m from the local average surface. This term is int e n d e d to describe the interaction with the delocalized conduction electrons:
U= ~ [u(Rg~)+v(Rg,)]+V(zg").
(1)
t
The nonspherical pair potential is m o d e l e d by a spherical part centered on the Pt atom, a function o f Rg,, and a part centered a distance h above the Pt ! atom, a function o f Rgi, where R,g2 = (Xg_X,)a + (yg _y,)2_}_ ( 2 g - - Z , - h )
2.
(2)
Also z~" is given by zga v = ~. g - - g7 asv ,
(3)
where the position o f the "local average" surface Zg[ is a weighted average over surface a t o m s z~:=
Z',z'~O(Rgi)
E', ¢(Rg,)
(4)
The primes indicate that the s u m m a t i o n are taken 471
Volume 188,number 5,6
CHEMICALPHYSICSLETTERS
over surface atoms. The weighting function 0(R) is given by
O ( R ) = e x p ( - 6 R z) .
(5)
where the parameter 6 is to be chosen empirically. The spherical function u ( R ) has the form
u ( R ) = A 2 exp(-o~2R) .
(6)
For convenience we define a function F by F(C, 7. E: z ) =
EC exp ( - 7z) E+ Cexp(-Tz) "
(7)
In terms of this, the functions v and V are given by /.~(R') = F ( A , a, B~ R ' ) - C 6 G ( R ' ) / R '6 ,
(8)
V ( z ) = F ( A I , ol,, W.; z) ,
(9)
where the "damping function" G(R) is defined by G ( R ) = 1,
R>RI,
G(R)=exp[-(DR~/R-1)2],
R<~R~,
(10)
with D equal to 1.45 and R~ equal to 3.65 /i,. The form of G(R) is taken from Aziz and Slaman [7]. The Pt atoms were assumed to interact with a nearest-neighbor central harmonic interaction. The force constant was assigned the value 4.56 × 104 erg cm -2, which reproduces the compressibility and the surface phonon frequency. The rationale for the use of the "local average surface" and eqs. ( 3 ) - ( 5 ) is as follows: That part of the force on the xenon atom which is due to the delocalized conduction electrons must ultimately be transmitted to the nuclei of the solid. If this is not correctly taken into account, a part of the energy exchange between the scattering atom and the surface is ignored, with the practical consequence that it is not possible to make the surface "flat" enough to reproduce the narrow angular distributions in low-energy scattering without making the energy transfer too small. In reality, it is clear that the nearest surface nuclei will receive most of the force and the effect will decay with distance in a way that a sufficiently detailed electronic calculation would describe. In the absence of such detailed information, we must make a model, essentially a model of the dependence of the potential on the coordinates of the surface atoms. This must certainly be consistent with the fact that a wholesale displacement of the solid normal to 472
17 January 1992
the surface would simply displace the whole potential. The model embodied in eqs. ( 3 ) - ( 5 ) is perhaps the simplest that satisfies these conditions. Tully [ 8 ] has used a potential with a term analogous to V but without provision for energy exchange via that term, apparently unnecessary in his case because most of the repulsion came from the pair-additive term. We chose our form for the non-spherical potential rather than, for example, that used earlier for He/ graphite by Cole et al. [ 9 ] because the spherical part u ( R ) provides a short-ranged repulsion which is almost decoupled from the term v(R'gi ). Thus, one can retain the good agreement with desorption data found by Bethune et al. [10] by using for v(R'gi) a form close to their spherical potential; the shift by the distance h normal to the surface does not affect the calculated desorption rates. We emphasize that v(R'g, ) is chosen to describe the three-dimensional dependence of the potential on the position of the gas atom as indicated by the experiments. It is not expected by itself to have a transparent physical interpretation except at large distances where it has the right form to model the dispersion interaction. Because this function is finite and has (effectively) zero derivative at R~, = 0, the total potential function is smoothly varying everywhere. The form of the potential given in eqs. ( 1 ) - ( 1 0 ) can describe either a potential with minima over the ontop sites or one with minima over the threefold hollow sites. We shall use "hollow potential" and "ontop potential" to refer to potentials with minima over hollow and ontop sites, respectively. For the hollow potentials, the repulsive interaction at low energies is provided primarily by the pair-additive terms in eq. ( 1 ), while for ontop potentials the "flat" repulsive term V(z~") is dominant at low energies. In either case, the repulsive interaction at high energies comes from the central interaction u(R). To make connection with the experimental data, desorption rates were calculated by the classical transition-state theory as described by Bethune et al. [ 10] and Grimmelmann et al. [ 11 ]. Angular and energy distributions in scattering and trapping probabilities were calculated by standard molecular dynamics methods. The energies of commensurate and incommensurate surface phases at 75 K were calculated by the Metropolis Monte Carlo method with the Pt displacements neglected. For the Xe-Xe in-
Volume 188, number 5,6
CHEMICAL PHYSICS LETTERS
t e r a c t i o n , we used the p o t e n t i a l o f A z i z a n d S l a m a n [ 7 ]. T h e " e n e r g y j u m p " at the c o m m e n s u r a t e - t o - i n c o m m e n s u r a t e t r a n s i t i o n was t h e n d e t e r m i n e d by a m e t h o d s i m i l a r to that d e s c r i b e d by G o t t l i e b [ 1 2 ] a n d G o t t l i e b a n d B r u c h [ 1 3 ] , e x c e p t that we used f i n i t e - t e m p e r a t u r e M o n t e C a r l o r a t h e r t h a n energy m i n i m i z a t i o n a n d a d i f f e r e n t X e - X e potential. W i t h an o p t i m i z e d o n t o p p o t e n t i a l , w h o s e p a r a m eters are listed in table 1, it was possible to fit all o f the d a t a r a t h e r well, as is s h o w n in fig. 1 for the des o r p t i o n data, figs. 2 a n d 3 for the scattering data, a n d fig. 4 for the t r a p p i n g data. A g r e e m e n t w i t h exp e r i m e n t s i m i l a r to that in figs. 2 a n d 3 was also f o u n d for scattering d a t a with i n c i d e n t e n e r g y / a n g l e 1.17 e V / 3 0 ° a n d 14.3 e V / 4 5 °, with surface t e m p e r a t u r e 800 K in each case. F o r the " e n e r g y j u m p " at the c o m m e n s u r a t e - i n c o m m e n s u r a t e transition, the calculated v a l u e was 31 m e V c o m p a r e d w i t h the exTable 1 Parameters for ontop potential "~ A1=2.50× 108 eV ~, =4.25 ~ - ' 14"t= 1.0 eV
A2=3.85X 10t3 eV a2=9.00 A-~ 5=0.17 ~-2
17 January 1992
I
I
P
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I
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I
Xe / Pt(111) x
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Xe
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X
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X
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Xe
LM X cU_
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X
° o°
--).
o
O
co
10
20
40
30
Scattered
50
60
70
90
80
Angle (deg)
Fig. 2. Calculated scattering intensity ((3) and average energy ( × ) compared with experimental values (the lower curve for intensity and dots for energy) as function of scattering angle. Incident energy 6.8 eV, incident angle 30 °, surface temperature 800 K.
C6=25.66 eVA 6 h= 3.02 ,~ 0.80
I
I
I
[
I
I
Xe / Pt(111)
a) The parameter A is zero, and c~and Ware therefore irrelevant.
¢-. -m
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> 6
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72
ol
90
S c a ~ e r e d Angle (deg) Fig. 3. Calculated scattering intensity (O) and average energy ( × ) compared with experimental values (the lower curve for intensity and dots for energy) as function of scattering angle. Incident energy 0.5 eV, incident angle 45 °, surface temperature 800 K.
-2
-4
I 0
IN
14
1000/Ts (K-1) Fig. 1. Arrhenius plot of thermal desorption rate constant. The line is calculated for the ontop potential of table 1 and the circles are experimental values [2].
p e r i m e n t a l j u m p in isosteric heat o f 3 0 + 15 m e V m e a s u r e d by K e r n et al. [ 6 ]. T h e c a l c u l a t e d isosteric h e a t for the c o m m e n s u r a t e phase was 310 m e V , in 473
Volume 188, number 5,6
1.0 ~
I
CHEMICAL PHYSICS LETTERS
]
I
Xenon/Pt(111 ) o.8!
v 0.6 Q-
0
~_ 0.4
•
m
0.2 1
,~ A o
0.0 0.0
L
0.2
I
0.4
0.6 0.8 Ei cosl6e~ (eV)
I 1.0
1.2
1.4
Fig. 4. Trapping probability at normal incidence; C)=0 ° are the calculated values and • =45 °, 0 = 6 0 °, 11=30 c are measured values extrapolated to normal incidence using the empirical observation that the trapping probability is a function of E~cos~6( 0~). The surface temperature is 85 K. excellent agreement with the experimental value 312 +_7 meV. The calculated equilibrium height of the Xe was 3.3 A., reasonably close to the value 3.0 found by Mfiller [14] from an "ab initio" electronic calculation and within the range 3.1-3.5 suggested by Black and Janzen [ 15,16 ]. The calculated frequency of the normal vibration was 3.8 meV compared with the experimental value [3] 3.7 meV. We have also made calculations including the nonadditive substrate-mediated McLachlan interaction, for which the form is given by Bruch [ 17 ] and the parameters by Black and Bopp [ 18 ]. The resulting value of the isosteric heat for the commensurate phase was 301 meV, a little outside the experimental error bars. The corresponding value of the "energy j u m p " was 56 meV, also outside the experimental error bars. This effect is surprisingly large, since the non-additive interaction reduces the effective strength of the X e - X e interaction by only about 15%. Preliminary results indicate that a reduction of the corrugation of the X e - P t potential can bring the "energy j u m p " calculated this way into agreement with the experimental result without spoiling the agreement with other experimental data. 474
17 January. 1992
The dependence of the ontop potential on the position o f the gas atom is indicated in fig. 5, which shows the potential as a function of Zg for the ontop, bridge and hollow sites. It is noteworthy that the curves cross near U = 0 and % = 2 . 7 ~, so that the lowest energy site is the ontop site near the minim u m and the hollow site at smaller values of Zg. It is this feature which permits satisfactory description of the experimental data, which require relatively large corrugation near the m i n i m u m and at high energies, with small corrugation at intermediate energies. This change-over is unique to ontop potentials, assuming only that at high energies the potential approaches a sum of repulsive pair potentials centered on the Pt atoms. The "glitches" near z = 5 are due to the limited (though rather large) flexibility of our potential form and to the absence of experiments providing control in that region. For the hollow potential, it was impossible to fit simultaneously the intermediate-energy scattering data and the trapping data. Furthermore, the "energy-jump" calculated for the hollow potential has the wrong sign ( - 5 7 meV without and - 3 2 meV with the non-additive McLachlan interaction). This is not surprising in view of the results of Gottlieb [12], who showed that a much larger overall cor-
0.8 0.6
"'I
- ~
hoIIov~\
°nt°p 00 ~
0.4 (13
-0.2
0.2
2.0
>
4.0
6.0
8.0
0.0 N~,. hollow -0.2 -0.4
2.0
ontop~,,~ "~ .......,.~.. I
I
I
2.5
3.0 z/~ngstrom
3.5
4.0
Fig. 5. Variation of Xe/Pt interaction with height of Xe. The solid curve is for the ontop site, the short-dashed line for the hollow site, and the long-dashed for the bridge site.
Volume 188, number 5,6
CHEMICALPHYSICSLETTERS
rugation is required to stabilize the commensurate phase for hollow potentials than for ontop potentials. The overall corrugation (absolute magnitude of the energy difference between the minima at hollow and ontop sites) is about 25% of the maximum well-depth both for our ontop potential and for our hollow potentials. A value much larger than that seems highly unlikely. Alternatively, a non-additive interaction about three times as large as the McLachlan interaction would produce agreement with the observed energy jump, but this too is unlikely; apart from the lack of theoretical basis for such a large non-additive interaction, the calculated isosteric heat for the commensurate phase would be about 283 meV. well outside the experimental error bars. Thus, we conclude that the equilibrium position for Xe is very probably above the ontop site, in agreement with the conclusion based on diffraction evidence of Gottlieb [ 12]. Two notes of caution are needed here: The first is that Zeppenfeld et al. [ 19 ] have pointed out that the diffraction evidence used by Gottlieb [ 12 ] to favor adsorption at ontop sites can equally be consistent with adsorption at hollow sites if one of the two sublattices of hollow sites is preferentially occupied, either totally or partially. In the fcc crystal, one sublattice lies directly over an atom in the second layer while the other does not, so that some preference would not be surprising. The second is that Andersen and Rejto [20] were able to construct a hollow potential reproducing the observed surfacephase diagram as well as the normal vibration frequency using the gas-phase Xe-Xe interaction if the McLachlan interaction was included, but not otherwise. That potential, however, took no account of the scattering data which we have used as input. Mfiller [14] has made a cluster-model local-density-functional electronic calculation of the potential of interaction of Xe with a static Pt( 111 ) surface in the region of the attractive well. Our ontop potential is in qualitative agreement with his results as to the location of the equilibrium adsorption site. Quantitatively, the well-depth of Mfiller's potential is about 20% larger than ours (which is certainly correct to better than 5%), the corrugation at the minimum is smaller by a factor of about 2, and the frequency of the normal vibration is larger than the experimental monolayer value which we have used by a factor of
17 January 1992
2.3 (this means a factor of 5 in the second derivative of the potential). The only experimental test of the local-density-functional method as applied to a physisorption system is that of Lang [ 21 ], who used a jellium model for the metal and found agreement to within about 15% with the experimental adsorption energy of argon on silver. This gives no guide to the accuracy to be expected in quantities like vibration frequencies and corrugation derived from cluster calculations. MiJller ascribes the difference between the calculated and experimental normal vibration frequencies to a calculated non-additive Xe-Xe surface interaction which would lower the monolayer frequency. Against this must be set the fact that our isolated Xesurface interaction, unequivocally determined at its minimum by the desorption measurements, together with the known gas-phase Xe-Xe interaction, reproduces within 3 meV the isosteric heat of the commensurate phase. At least part of the difference between the calculated frequency and the experimental monolayer value must arise from the fact that the use of local exchange-correlation makes the potential decay too rapidly to zero at large z (exponentially rather than by inverse cube law). Our potential reproduces the inverse cube behavior because of the term proportional to I / R 6 in the pair potential. The values of Mfiller's potential at the minimum with respect to z are described rather accurately by the single-shell Fourier function ofGottlieb (eq. ( 1 ) ofref. [ 12] ) with Vo= - 2 8 9 meV and l,g= - 3 meV. Calculation of the "energy jump" using this potential together with the Xe-Xe potential of Aziz and Slaman [7] gave the value - 9 meV, the sign being opposite to that of the experimental value. The corresponding value of the isosteric heat of the commensurate phase at 75 K was 367 meV, about 18% higher than the experimental value [6]. With the non-additive McLachlan interaction included, the isosteric heat was 359 meV and the "'energy jump" was 16 meV, the latter being just within the experimental error bars on the low side. In summary, we have determined an empirical gassurface potential energy function for a prototypical physisorption system, namely Xe interacting with the Pt( 111 ) surface. The potential is consistent with a range of experimental data of different kinds, involving both attractive and strongly repulsive inter475
Volume 188, number 5,6
CHEMICAL PHYSICS LETTERS
a c t i o n s w i t h t h e surface. F o r o u r b e s t p o t e n t i a l , t h e e q u i l i b r i u m p o s i t i o n o f t h e X e is a b o v e t h e o n t o p site at a h e i g h t o f 3.3 ~ .
References [1 ] J.A. Barker and D.J. Auerbach, Surface Sci. Reports 4 (1984) 1. [2] C.T, Rettner, D.S. Bethune and E.K. Schweizer, J. Chem. Phys. 92 (1990) 1442. [3] B. Hall, D.L. Mills, P. Zeppenfeld, K. Kern and G. Comsa, Phys. Rev. B 40 (1989) 6326. [4] C.T. Rettner, D.S. Bethune and D.J. Auerbach, J. Chem. Phys. 91 (1989) 1942. [ 5 ] C.T, Renner, J.A. Barker and D.S. Bethune, to be published. [6] K. Kern, R. David, P. Zeppenfeld and G. Comsa, Surface Sci. 195 (1988) 353.
476
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[7] R.A. Aziz and M.J. Slaman, Mol. Phys. 57 (1986) 825. [8] J.C. Tully, Surface Sci. 111 (1981) 461. [9] M.W. Cole, D.R. Frankl and D.L. Goodstein, Rev. Mod. Phys. 53 (1981) 199. [ 10 ] D.S. Bethune, J.A. Barker and C.R. Rettner, J. Chem. Phys. 92 (1990) 6847. [ 11 ] E.K. Grimmelmann, J.C. Tully and E. Helfand, J. Chem. Phys. 74 (1981) 5300. 12] J.M. Gottlieb, Phys. Rev. B 42 (1990) 5377. 13 ] J.M. Gottlieb and L.W. Bruch, Phys. Rev. B 40 ( 1989 ) 148. 14] J.E. Miiller, Phys. Rev. Letters 65 (1990) 3021. 15 ] J.E. Black and R.A. Janzen, Langmuir 5 ( 1989 ) 558. 16] J.E. Black and R.A. Janzen, Surface Sci. 217 (1989) 199. 17] L.W. Bruch, Surface Sci. 125 (1983) 194. 18] J.E. Black and P. Bopp, Surface Sci. 182 (1987) 98. 19] P. Zeppenfeld, G. Comsa and J.A. Barker, to be published. 20 ] H.C. Andersen and P. Rejto, to be published, [21 ] N.D. Lang, Phys. Rev. Letters 46 ( 1981 ) 842.
CHEMICAL PHYSICS LETTERS
17 January. 1992
The interaction of Xe with the Pt ( 111 ) surface J.A. B a r k e r , C.T. R e t t n e r a n d D . S . B e t h u n e IBM Research Division, Almaden Research Center, K31/803, 650 Harry Road, San Jose. CA 95120-6099. USA
Received 12 September 1991; in final form 24 October 1991
We describe the determination of an empirical potential energy function for the interaction of xenon with the Pt ( 111 ) surface using a novel analytic form. The potential is consistent with a wide range of dynamical and equilibrium experimental data. The equilibrium position for a single xenon atom lies directly above a surface platinum atom at a height of 3.3/k.
The potential energy function describing the interaction o f an a t o m or molecule with a surface holds the key for u n d e r s t a n d i n g such processes as energy exchange and a c c o m m o d a t i o n , trapping a n d desorption, as well as the structures and phase diagrams o f c o m m e n s u r a t e and i n c o m m e n s u r a t e surface phases. These p h e n o m e n a are i m p o r t a n t in contexts as diverse as catalysis, crystal and thin-film growth, and hypersonic flight [1]. There is no gas-surface system for which an accurate potential energy function valid over a wide range of energies is known. We have therefore d e t e r m i n e d an empirical potential function for the system X e / P t ( 1 11 ), chosen because of the wealth o f experimental data o f several different kinds which is available, for Xe energies ranging from a few meV to m o r e than 14 eV. Previous empirical studies have used experimental d a t a o f only one or two kinds [ 1 ], and the resulting potential functions were therefore far from uniquely d e t e r m i n e d . By contrast, the present work makes c o m p a r i s o n with a wide range o f different kinds of data, including m e a s u r e d d e s o r p t i o n rates [ 2 ], the frequency o f v i b r a t i o n of Xe n o r m a l to the surface [3], t r a p p i n g probabilities [4], scattering m e a s u r e m e n t s (including angular distributions and angle-dependent energy m e a s u r e m e n t s ) for incident energies between 0.5 and 14.3 eV [5], a n d a heuristic estimate [6] o f the corrugation o f the potential based on the observed "energy j u m p " at the commensurate to partially i n c o m m e n s u r a t e phase transition. Because o f this, our potential energy function is relatively well d e t e r m i n e d . In particular, we are Elsevier Science Publishers B.V.
able to conclude that the equilibrium site for a single Xe a t o m is very probably above the " o n t o p " site (at a height o f 3.3 A ) , and that no potential with the equilibrium site above the threefold hollow site can fit all the experimental facts. We represent the interaction between a xenon atom and the solid by a sum o f nonspherical pairwise-additive potentials (the s u m m a t i o n in eq. ( 1 ), see below), together with an additional term V which depends only on the n o r m a l distance Zgv of the xenon a t o m from the local average surface. This term is int e n d e d to describe the interaction with the delocalized conduction electrons:
U= ~ [u(Rg~)+v(Rg,)]+V(zg").
(1)
t
The nonspherical pair potential is m o d e l e d by a spherical part centered on the Pt atom, a function o f Rg,, and a part centered a distance h above the Pt ! atom, a function o f Rgi, where R,g2 = (Xg_X,)a + (yg _y,)2_}_ ( 2 g - - Z , - h )
2.
(2)
Also z~" is given by zga v = ~. g - - g7 asv ,
(3)
where the position o f the "local average" surface Zg[ is a weighted average over surface a t o m s z~:=
Z',z'~O(Rgi)
E', ¢(Rg,)
(4)
The primes indicate that the s u m m a t i o n are taken 471
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CHEMICALPHYSICSLETTERS
over surface atoms. The weighting function 0(R) is given by
O ( R ) = e x p ( - 6 R z) .
(5)
where the parameter 6 is to be chosen empirically. The spherical function u ( R ) has the form
u ( R ) = A 2 exp(-o~2R) .
(6)
For convenience we define a function F by F(C, 7. E: z ) =
EC exp ( - 7z) E+ Cexp(-Tz) "
(7)
In terms of this, the functions v and V are given by /.~(R') = F ( A , a, B~ R ' ) - C 6 G ( R ' ) / R '6 ,
(8)
V ( z ) = F ( A I , ol,, W.; z) ,
(9)
where the "damping function" G(R) is defined by G ( R ) = 1,
R>RI,
G(R)=exp[-(DR~/R-1)2],
R<~R~,
(10)
with D equal to 1.45 and R~ equal to 3.65 /i,. The form of G(R) is taken from Aziz and Slaman [7]. The Pt atoms were assumed to interact with a nearest-neighbor central harmonic interaction. The force constant was assigned the value 4.56 × 104 erg cm -2, which reproduces the compressibility and the surface phonon frequency. The rationale for the use of the "local average surface" and eqs. ( 3 ) - ( 5 ) is as follows: That part of the force on the xenon atom which is due to the delocalized conduction electrons must ultimately be transmitted to the nuclei of the solid. If this is not correctly taken into account, a part of the energy exchange between the scattering atom and the surface is ignored, with the practical consequence that it is not possible to make the surface "flat" enough to reproduce the narrow angular distributions in low-energy scattering without making the energy transfer too small. In reality, it is clear that the nearest surface nuclei will receive most of the force and the effect will decay with distance in a way that a sufficiently detailed electronic calculation would describe. In the absence of such detailed information, we must make a model, essentially a model of the dependence of the potential on the coordinates of the surface atoms. This must certainly be consistent with the fact that a wholesale displacement of the solid normal to 472
17 January 1992
the surface would simply displace the whole potential. The model embodied in eqs. ( 3 ) - ( 5 ) is perhaps the simplest that satisfies these conditions. Tully [ 8 ] has used a potential with a term analogous to V but without provision for energy exchange via that term, apparently unnecessary in his case because most of the repulsion came from the pair-additive term. We chose our form for the non-spherical potential rather than, for example, that used earlier for He/ graphite by Cole et al. [ 9 ] because the spherical part u ( R ) provides a short-ranged repulsion which is almost decoupled from the term v(R'gi ). Thus, one can retain the good agreement with desorption data found by Bethune et al. [10] by using for v(R'gi) a form close to their spherical potential; the shift by the distance h normal to the surface does not affect the calculated desorption rates. We emphasize that v(R'g, ) is chosen to describe the three-dimensional dependence of the potential on the position of the gas atom as indicated by the experiments. It is not expected by itself to have a transparent physical interpretation except at large distances where it has the right form to model the dispersion interaction. Because this function is finite and has (effectively) zero derivative at R~, = 0, the total potential function is smoothly varying everywhere. The form of the potential given in eqs. ( 1 ) - ( 1 0 ) can describe either a potential with minima over the ontop sites or one with minima over the threefold hollow sites. We shall use "hollow potential" and "ontop potential" to refer to potentials with minima over hollow and ontop sites, respectively. For the hollow potentials, the repulsive interaction at low energies is provided primarily by the pair-additive terms in eq. ( 1 ), while for ontop potentials the "flat" repulsive term V(z~") is dominant at low energies. In either case, the repulsive interaction at high energies comes from the central interaction u(R). To make connection with the experimental data, desorption rates were calculated by the classical transition-state theory as described by Bethune et al. [ 10] and Grimmelmann et al. [ 11 ]. Angular and energy distributions in scattering and trapping probabilities were calculated by standard molecular dynamics methods. The energies of commensurate and incommensurate surface phases at 75 K were calculated by the Metropolis Monte Carlo method with the Pt displacements neglected. For the Xe-Xe in-
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CHEMICAL PHYSICS LETTERS
t e r a c t i o n , we used the p o t e n t i a l o f A z i z a n d S l a m a n [ 7 ]. T h e " e n e r g y j u m p " at the c o m m e n s u r a t e - t o - i n c o m m e n s u r a t e t r a n s i t i o n was t h e n d e t e r m i n e d by a m e t h o d s i m i l a r to that d e s c r i b e d by G o t t l i e b [ 1 2 ] a n d G o t t l i e b a n d B r u c h [ 1 3 ] , e x c e p t that we used f i n i t e - t e m p e r a t u r e M o n t e C a r l o r a t h e r t h a n energy m i n i m i z a t i o n a n d a d i f f e r e n t X e - X e potential. W i t h an o p t i m i z e d o n t o p p o t e n t i a l , w h o s e p a r a m eters are listed in table 1, it was possible to fit all o f the d a t a r a t h e r well, as is s h o w n in fig. 1 for the des o r p t i o n data, figs. 2 a n d 3 for the scattering data, a n d fig. 4 for the t r a p p i n g data. A g r e e m e n t w i t h exp e r i m e n t s i m i l a r to that in figs. 2 a n d 3 was also f o u n d for scattering d a t a with i n c i d e n t e n e r g y / a n g l e 1.17 e V / 3 0 ° a n d 14.3 e V / 4 5 °, with surface t e m p e r a t u r e 800 K in each case. F o r the " e n e r g y j u m p " at the c o m m e n s u r a t e - i n c o m m e n s u r a t e transition, the calculated v a l u e was 31 m e V c o m p a r e d w i t h the exTable 1 Parameters for ontop potential "~ A1=2.50× 108 eV ~, =4.25 ~ - ' 14"t= 1.0 eV
A2=3.85X 10t3 eV a2=9.00 A-~ 5=0.17 ~-2
17 January 1992
I
I
P
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Xe / Pt(111) x
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X
° o°
--).
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10
20
40
30
Scattered
50
60
70
90
80
Angle (deg)
Fig. 2. Calculated scattering intensity ((3) and average energy ( × ) compared with experimental values (the lower curve for intensity and dots for energy) as function of scattering angle. Incident energy 6.8 eV, incident angle 30 °, surface temperature 800 K.
C6=25.66 eVA 6 h= 3.02 ,~ 0.80
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a) The parameter A is zero, and c~and Ware therefore irrelevant.
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90
S c a ~ e r e d Angle (deg) Fig. 3. Calculated scattering intensity (O) and average energy ( × ) compared with experimental values (the lower curve for intensity and dots for energy) as function of scattering angle. Incident energy 0.5 eV, incident angle 45 °, surface temperature 800 K.
-2
-4
I 0
IN
14
1000/Ts (K-1) Fig. 1. Arrhenius plot of thermal desorption rate constant. The line is calculated for the ontop potential of table 1 and the circles are experimental values [2].
p e r i m e n t a l j u m p in isosteric heat o f 3 0 + 15 m e V m e a s u r e d by K e r n et al. [ 6 ]. T h e c a l c u l a t e d isosteric h e a t for the c o m m e n s u r a t e phase was 310 m e V , in 473
Volume 188, number 5,6
1.0 ~
I
CHEMICAL PHYSICS LETTERS
]
I
Xenon/Pt(111 ) o.8!
v 0.6 Q-
0
~_ 0.4
•
m
0.2 1
,~ A o
0.0 0.0
L
0.2
I
0.4
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I 1.0
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1.4
Fig. 4. Trapping probability at normal incidence; C)=0 ° are the calculated values and • =45 °, 0 = 6 0 °, 11=30 c are measured values extrapolated to normal incidence using the empirical observation that the trapping probability is a function of E~cos~6( 0~). The surface temperature is 85 K. excellent agreement with the experimental value 312 +_7 meV. The calculated equilibrium height of the Xe was 3.3 A., reasonably close to the value 3.0 found by Mfiller [14] from an "ab initio" electronic calculation and within the range 3.1-3.5 suggested by Black and Janzen [ 15,16 ]. The calculated frequency of the normal vibration was 3.8 meV compared with the experimental value [3] 3.7 meV. We have also made calculations including the nonadditive substrate-mediated McLachlan interaction, for which the form is given by Bruch [ 17 ] and the parameters by Black and Bopp [ 18 ]. The resulting value of the isosteric heat for the commensurate phase was 301 meV, a little outside the experimental error bars. The corresponding value of the "energy j u m p " was 56 meV, also outside the experimental error bars. This effect is surprisingly large, since the non-additive interaction reduces the effective strength of the X e - X e interaction by only about 15%. Preliminary results indicate that a reduction of the corrugation of the X e - P t potential can bring the "energy j u m p " calculated this way into agreement with the experimental result without spoiling the agreement with other experimental data. 474
17 January. 1992
The dependence of the ontop potential on the position o f the gas atom is indicated in fig. 5, which shows the potential as a function of Zg for the ontop, bridge and hollow sites. It is noteworthy that the curves cross near U = 0 and % = 2 . 7 ~, so that the lowest energy site is the ontop site near the minim u m and the hollow site at smaller values of Zg. It is this feature which permits satisfactory description of the experimental data, which require relatively large corrugation near the m i n i m u m and at high energies, with small corrugation at intermediate energies. This change-over is unique to ontop potentials, assuming only that at high energies the potential approaches a sum of repulsive pair potentials centered on the Pt atoms. The "glitches" near z = 5 are due to the limited (though rather large) flexibility of our potential form and to the absence of experiments providing control in that region. For the hollow potential, it was impossible to fit simultaneously the intermediate-energy scattering data and the trapping data. Furthermore, the "energy-jump" calculated for the hollow potential has the wrong sign ( - 5 7 meV without and - 3 2 meV with the non-additive McLachlan interaction). This is not surprising in view of the results of Gottlieb [12], who showed that a much larger overall cor-
0.8 0.6
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- ~
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°nt°p 00 ~
0.4 (13
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0.2
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6.0
8.0
0.0 N~,. hollow -0.2 -0.4
2.0
ontop~,,~ "~ .......,.~.. I
I
I
2.5
3.0 z/~ngstrom
3.5
4.0
Fig. 5. Variation of Xe/Pt interaction with height of Xe. The solid curve is for the ontop site, the short-dashed line for the hollow site, and the long-dashed for the bridge site.
Volume 188, number 5,6
CHEMICALPHYSICSLETTERS
rugation is required to stabilize the commensurate phase for hollow potentials than for ontop potentials. The overall corrugation (absolute magnitude of the energy difference between the minima at hollow and ontop sites) is about 25% of the maximum well-depth both for our ontop potential and for our hollow potentials. A value much larger than that seems highly unlikely. Alternatively, a non-additive interaction about three times as large as the McLachlan interaction would produce agreement with the observed energy jump, but this too is unlikely; apart from the lack of theoretical basis for such a large non-additive interaction, the calculated isosteric heat for the commensurate phase would be about 283 meV. well outside the experimental error bars. Thus, we conclude that the equilibrium position for Xe is very probably above the ontop site, in agreement with the conclusion based on diffraction evidence of Gottlieb [ 12]. Two notes of caution are needed here: The first is that Zeppenfeld et al. [ 19 ] have pointed out that the diffraction evidence used by Gottlieb [ 12 ] to favor adsorption at ontop sites can equally be consistent with adsorption at hollow sites if one of the two sublattices of hollow sites is preferentially occupied, either totally or partially. In the fcc crystal, one sublattice lies directly over an atom in the second layer while the other does not, so that some preference would not be surprising. The second is that Andersen and Rejto [20] were able to construct a hollow potential reproducing the observed surfacephase diagram as well as the normal vibration frequency using the gas-phase Xe-Xe interaction if the McLachlan interaction was included, but not otherwise. That potential, however, took no account of the scattering data which we have used as input. Mfiller [14] has made a cluster-model local-density-functional electronic calculation of the potential of interaction of Xe with a static Pt( 111 ) surface in the region of the attractive well. Our ontop potential is in qualitative agreement with his results as to the location of the equilibrium adsorption site. Quantitatively, the well-depth of Mfiller's potential is about 20% larger than ours (which is certainly correct to better than 5%), the corrugation at the minimum is smaller by a factor of about 2, and the frequency of the normal vibration is larger than the experimental monolayer value which we have used by a factor of
17 January 1992
2.3 (this means a factor of 5 in the second derivative of the potential). The only experimental test of the local-density-functional method as applied to a physisorption system is that of Lang [ 21 ], who used a jellium model for the metal and found agreement to within about 15% with the experimental adsorption energy of argon on silver. This gives no guide to the accuracy to be expected in quantities like vibration frequencies and corrugation derived from cluster calculations. MiJller ascribes the difference between the calculated and experimental normal vibration frequencies to a calculated non-additive Xe-Xe surface interaction which would lower the monolayer frequency. Against this must be set the fact that our isolated Xesurface interaction, unequivocally determined at its minimum by the desorption measurements, together with the known gas-phase Xe-Xe interaction, reproduces within 3 meV the isosteric heat of the commensurate phase. At least part of the difference between the calculated frequency and the experimental monolayer value must arise from the fact that the use of local exchange-correlation makes the potential decay too rapidly to zero at large z (exponentially rather than by inverse cube law). Our potential reproduces the inverse cube behavior because of the term proportional to I / R 6 in the pair potential. The values of Mfiller's potential at the minimum with respect to z are described rather accurately by the single-shell Fourier function ofGottlieb (eq. ( 1 ) ofref. [ 12] ) with Vo= - 2 8 9 meV and l,g= - 3 meV. Calculation of the "energy jump" using this potential together with the Xe-Xe potential of Aziz and Slaman [7] gave the value - 9 meV, the sign being opposite to that of the experimental value. The corresponding value of the isosteric heat of the commensurate phase at 75 K was 367 meV, about 18% higher than the experimental value [6]. With the non-additive McLachlan interaction included, the isosteric heat was 359 meV and the "'energy jump" was 16 meV, the latter being just within the experimental error bars on the low side. In summary, we have determined an empirical gassurface potential energy function for a prototypical physisorption system, namely Xe interacting with the Pt( 111 ) surface. The potential is consistent with a range of experimental data of different kinds, involving both attractive and strongly repulsive inter475
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a c t i o n s w i t h t h e surface. F o r o u r b e s t p o t e n t i a l , t h e e q u i l i b r i u m p o s i t i o n o f t h e X e is a b o v e t h e o n t o p site at a h e i g h t o f 3.3 ~ .
References [1 ] J.A. Barker and D.J. Auerbach, Surface Sci. Reports 4 (1984) 1. [2] C.T, Rettner, D.S. Bethune and E.K. Schweizer, J. Chem. Phys. 92 (1990) 1442. [3] B. Hall, D.L. Mills, P. Zeppenfeld, K. Kern and G. Comsa, Phys. Rev. B 40 (1989) 6326. [4] C.T. Rettner, D.S. Bethune and D.J. Auerbach, J. Chem. Phys. 91 (1989) 1942. [ 5 ] C.T, Renner, J.A. Barker and D.S. Bethune, to be published. [6] K. Kern, R. David, P. Zeppenfeld and G. Comsa, Surface Sci. 195 (1988) 353.
476
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[7] R.A. Aziz and M.J. Slaman, Mol. Phys. 57 (1986) 825. [8] J.C. Tully, Surface Sci. 111 (1981) 461. [9] M.W. Cole, D.R. Frankl and D.L. Goodstein, Rev. Mod. Phys. 53 (1981) 199. [ 10 ] D.S. Bethune, J.A. Barker and C.R. Rettner, J. Chem. Phys. 92 (1990) 6847. [ 11 ] E.K. Grimmelmann, J.C. Tully and E. Helfand, J. Chem. Phys. 74 (1981) 5300. 12] J.M. Gottlieb, Phys. Rev. B 42 (1990) 5377. 13 ] J.M. Gottlieb and L.W. Bruch, Phys. Rev. B 40 ( 1989 ) 148. 14] J.E. Miiller, Phys. Rev. Letters 65 (1990) 3021. 15 ] J.E. Black and R.A. Janzen, Langmuir 5 ( 1989 ) 558. 16] J.E. Black and R.A. Janzen, Surface Sci. 217 (1989) 199. 17] L.W. Bruch, Surface Sci. 125 (1983) 194. 18] J.E. Black and P. Bopp, Surface Sci. 182 (1987) 98. 19] P. Zeppenfeld, G. Comsa and J.A. Barker, to be published. 20 ] H.C. Andersen and P. Rejto, to be published, [21 ] N.D. Lang, Phys. Rev. Letters 46 ( 1981 ) 842.