Theory of physisorption interactions

194

Surface Science 125 (1983) 1944217 North-Holland Publishing Company

THEORY OF PHYSISORF’TION INTERACTIONS

*

L.W. BRUCH Department

Received

of Physics,

of

University

15 June 1982; accepted

Wisconsin,

for publication

Madison,

Wisconsrn

23 August

53706,

USA

1982

Information on the single-atom holding potential of physical adsorption and the lateral interactions of the adatoms is discussed and summarized. Results on the substrate-mediated dispersion energy and on the interaction of adsorption dipoles are consolidated.

1. Introduction 1.1. Scope Effects of physisorption interactions are apparent in atom-surface scattering experiments [l-3], in the structures of physisorbed arrays of atoms observed with a variety of diffraction techniques [4-61, and in thermal excitation processes measured with macroscopic thermodynamic methods [7]. Much of the current interest in the physisorption systems stems from the variety of ordered phases and transitions between the phases which are evident in the calorimetric and diffraction experiments. However, the interactions themselves reflect the charge rearrangements which take place in adsorption and the role of the substrate in the properties of the adsorbed layers. As such, physical adsorption, which is sometimes termed “weak adsorption”, reflects many of the processes which are included in the more complex case of chemical adsorption. For physical adsorption, the lateral interaction energies of the adatoms among themselves are frequently a larger fraction of the total adsorption energy than in the case of chemisorption, so that their effects are a more prominent part of the total observation. Similarly, the structures of the physisorbed arrays show marked variations with temperature and also yield information on the lateral interactions. The purpose of this paper is to survey the present state of information on the physisorption interactions and to consolidate the material on the

* Research

supported

in part by the National

0039~6028/83/0000-0000/$03.00

Science Foundation

0 1983 North-Holland

through

grant DMR-7920400.

L. W. Bruch / Theory ojphysisorption interactions

195

substrate-mediated dispersion forces. This paper is not a definitive review of the field as a whole, but the bibliography which is assembled here may make the field more accessible. process in which the single atom Physisorption is a “weak” adsorption holding energy is a few tens of meV’s and in which there is no apparent chemical bond formation. The major parts of the substrate-mediated forces appear to be non-resonant with respect to the chemical structure of the adsorbate and only weakly dependent on the crystal structure of the substrate. The latter is the origin of the periodic components of the adatom-substrate interaction, which determine the registry and commensurate structures observed in physisorbed layers. While there may be site-wise adsorption, the lateral interaction energies of near neighbors are generally larger than differences between site energies, although the reverse is probably true for the adsorption of krypton on the (111) face of silicon [Kr/Si(ll l)] [S]. The principal experimental systems considered here under the heading of physical adsorption are the inert gases on graphite [4,5], alkali halides [2], and such metals as silver [9- 111, copper [5], and aluminum [ 121. For these systems, there are lateral attractions among the adatoms which lead to island formation at low temperatures and submonolayer coverages. There are apparently systematic differences for the adsorption of inert gases on transition metals such as tungsten [ 13,141 and palladium [ 15- 171, and mechanisms involving some chemical bonding character have been proposed [ 181 in those cases. For physisorption interactions it is convenient to separate terms into the single-atom holding potential and lateral interactions of the adatoms. Processes arising from the effect of one adatom on the interaction of another adatom with the substrate are then included in the effective lateral interactions of adatoms. Another reason for proceeding this way is that many of the considerations about the lateral interactions depend on the structure of the complex formed in the adsorption of a single atom. The organization of this paper is: Section 1 continues with a summary of concepts of intermolecular forces and of results for the interaction of closed cell atoms in three dimensions. Section 2 contains a review of the single-atom holding potential and section 3 contains a review of the lateral interactions of adatoms. An appendix contains the evaluation of a dipole tensor product used in the theory of multibody interactions. 1.2. The theory of intermolecular forces The interactions discussed in the theory of physical adsorption are defined in the context of the Born-Oppenheimer approximation [ 191, as is the case for the interactions of closed shell atoms in three dimensions. In principle, the potential energies of interaction are to be obtained from the ground state solution of the appropriate Schrodinger equation for the electronic motions in

196

L. W. Bruch / Theory

of physisorptioninteractions

the field of fixed nuclei. A full accurate solution of such a problem is so nearly intractable that most applications consist of combining limiting cases of the theory with empirical information or of making further drastic approximations to the treatment of the correlation energy and of self-consistency requirements in the theory. The conventional language for the intermolecular forces [ 19,201 between closed shell atoms and for physical adsorption refers to dispersion energies and to exchange-overlap forces. These are the dominant processes in the full theory for well-separated subsystems and for strongly overlapping subsystems, respectively. More detailed results are available for these two limiting cases than for the intermediate separations where the minimum of the pair potential occurs. For well-separated subsystems, in the absence of permanent multipole moments, the Van der Waals attraction is the dominant long-range interaction [9,20]. It arises from the correlated motion of fluctuating electric multipole moments on the subsystems, with the electric dipoles giving the leading term. It can be evaluated with a perturbation theory for the correlated response of the subsystems and the result is expressed in terms of the dynamic response functions (polarizability and dielectric function) of the separated species [21, 221, using the Casimir-Polder identities [23]. Experimental data for the response functions are frequently available [24] and enable fairly accurate evaluations of the strength coefficients in the dispersion energy. The resulting interaction energy for two closed-shell ground-state atoms in three dimensions varies as the inverse sixth power of the interatomic separation (London); for one atom interacting with a planar semi-infinite solid, the interaction varies as the inverse third power of the perpendicular distance (Lennard-Jones [25], Bardeen [26], Lifshitz [27]). The case of two atoms interacting with a semi-infinite solid was treated in these terms by McLachlan [28], who simplified a calculation of Sinanoglu and Pitzer [29]. Such analyses typically retain only the leading multipole terms and use the linear response of the subsystems. The exchange-overlap force between two atoms is a result of combining an electrostatic Hamiltonian with the requirement of totally antisymmetric electronic wave functions. For two closed shell atoms, in the Hartree-Fock approximation, the result is a repulsion [19,20] which varies approximately exponentially with separation (Slater). To achieve a potential well in the ground state interaction of two closed shell atoms, the theory must be carried beyond the Hartree-Fock approximation [30-321; with the multi-configuration-self-consistent field (MCSCF) method, Wahl and co-workers [30,3 l] reproduced the empirical depth of the potential minima for He, and Ne, to about 5%. Even the Hartree-Fock calculation for a physisorbed atom interacting with a semi-infinite solid is lacking; some results are available [33] for He interacting with a cluster of ions at surfaces of LiH. In the absence of detailed calculations, one method of estimating the net interaction is to add together [34] the functions obtained in the long-range and

L. W. Bruch / Theory o/physisorption interactions

short-range limits; that is, each functional form is extrapolated beyond regime where it is dominant. For two helium atoms in three dimensions, result is of the form

V(R)=,4

emuR-(C,/R").

197

the the

(1.1)

where R is the interatomic separation (Slater and Kirkwood). Modern constructions of this type [35,36] dampen the growth of the dispersion energy ( C,/R6) at intermediate separations and incorporate functional forms based on the interatomic potential in the electron spin triplet state of molecular hydrogen. For one atom interacting with a semi-infinite metal an approximation similar to eq. (1.1) was used by Kleiman and Landman [37]. The additivity of the dispersion energy and the overlap energy was tested in other work, for a simple model of the physisorption of atomic hydrogen by a jellium metal, where-the detailed numerical solution was also obtained: in the region of the potential minimum, the result obtained with the additivity assumption differed by only 10% from the detailed solution (Bruch and Ruijgrok) [38]. A second method of estimating the interactions is based on the formal theory of Hohenberg, Kohn and Sham for the ground state energy of an inhomogeneous electron gas [39], the “density functional theories”. The form (actually, an interpolation) for the exchange and correlation energy as a function of density of a uniform jellium is retained, the local approximation for exchange and correlation. The kinetic energy is treated either with a gradient approximation (frequently, the leading order, Thomas-Fermi kinetic energy proportional to the 5/3 power of the electron density, is used) or with the orbitals of Kohn and Sham. Clugston [40] has reviewed applications to atomic pairs in three dimensions in which the density of the interacting systems is taken to be equal to the sum of the densities of the separate systems (no charge rearrangement, the Gordon-Kim approximation): corresponding approximations for physisorption were made by Freeman [41,42] and by Van Himbergen and Silbey [43]. There is also one calculation using the Kohn-Sham orbitals and allowing for charge rearrangement, for one atom interacting with semi-infinite jellium (Lang) [44]. 1.3. Results from three dimensions The potential energy of like pairs of inert gas atoms is known empirically [45], in the region of the potential minimum, to 5% in the well depth and 1% in the spacing. There is some theoretical input to this determination in the form of multipole coefficients for the long-range dispersion forces, but it is primarily the result of constructions using thermodynamic data, transport properties, spectroscopic data, and molecular beam scattering cross sections. The constructions of Barker and co-workers [45] were followed by constructions with somewhat simpler functional forms for the potential by Az.iz and co-workers WI.

198

L. W. Bruch / Theory of physisorption

interactrons

Even with the multiparameter functional forms which are adjusted to give fits to the variety of experimental data, the empirical result [45] is that the interactions of like pairs of inert gases have a quite similar shape in the region of the potential minimum. The potential well is then characterized by two parameters, one for the energy scale and one for the length scale. Thus the assumption made for the potential in the classical theory of corresponding states (191 is fulfilled. The total potential energy of an assembly of inert gas atoms in a condensed phase is not precisely equal to the sum of the potential energies of atomic paires; this is termed the non-pair-additivity of the intermolecular potentials [ 19,45,47]. It is found empirically [45] that the non-pair-additive dispersion energy terms, notably the triple dipole energy [48] of Axilrod, Teller and Muto, account for most of the missing energy in condensed phases of the inert gases. The triple dipole energy for solid xenon is 10% of the total cohesive energy of xenon, with smafler fractional contributions for the lighter inert gases [45]. Overlap-dependent, short-range three-body energy terms are known to exist for the inert gases [47], but their magnitude at typical interatomic distances in the liquid and solid is known only from appreciable extrapolations of shortseparation calculations. The usual form of the triple dipole energy is [49] fzC3) = V,&( 1 + 3 cos 8, cos e, cos 8,)/(

‘SQ3,

(1.2)

where r, s and t are the three sides and 6,, 19,and 8, are the three interior angles of the triangle formed by particles a, b and c. For the discussion of adsorption, a more convenient form of eC3)is C13) = _r 3~abcTrfT,2-

63.

where the dipole tensors

T,=

(1.3)

T3,)>

Tj are given by

v,y,;.

(1.4)

‘J

2. Single-atom holding potential 2. I. General aspects The potential energy of interaction of one adatom with the planar surface of a crystalline solid, u(r), may be written [SO] as the sum of a laterally averaged term +(z) and terms with the two-dimensional periodicity of the substrate surface:

u(~)=u~(z)+

(2.1)

C eiC.puG(t). G*O

In eq. (2.1), the G’s are the reciprocal

lattice

vectors

of the surface,

z is the

L.. W. Bruch / Theory of physisor~tion interactions

199

adatom coordinate perpendicular to the surface and p are the adatom coordinates parallel to the surface. For incommensurate adsorbate phases and for systems with high adatom lateral mobility, when Q(Z) has a sharply defined minimum, the adatom motions are neariy confined to a plane. At large perpendicular distances z, the effects of the discrete structure of the surface become extremely small; model calculations [50,51] show the dependence of z+(z) on z is then approximately exponential: U&Z) a ePGs.

(2.2)

What little information there is on the properties of U&Z) comes mainly from model calculations [ 1,50,51] and from measurements of intensities in atom-surface scattering [52] and of adatom band structure effects in selective adsorption scattering [53]. As-is the case for pairs of inert gas atoms [21,22], the most quantitative theoretical results on uO(z) are for large separations; this work is summarized in section 2.2. Information for the ~nimum of u*(z) is summarized in section 2.3. 2.2. The Van der Waals a~~ra~rion For large distances z, but not yet so large that retardation effects are significant, the laterally averaged holding potential becomes 125-273 Ug( z) = - Cj/Z3,

(2.3) also termed the polarization potential [54]. Lennard-Jones 125) was the first to derive this form, for a model in which the substrate perfectly imaged the fluctuating dipole of the adatom. Bardeen 1261 then included the finite response of a metal substrate, that is, allowing for the incomplete screening of the high frequency components of the fluctuating dipole field, The strength coefficient C, can be written [27,54] in terms of the dynamic polarizability of the adatom cu(io) and the dynamic linear dielectric function of the substrate c(iw), both at pure imaginary frequencies: (2.4) in atomic units. The dielectric function is given in terms of the measured absorptive part of the dielectric constant at real frequencies by a Kramers-Kronig relation, c(io) = 1 + $l”$$$

dx.

(2.5)

For the basal plane surface of graphite, which is electronically anisotropic, the dielectric function becomes a geometric average of the dielectric functions for

200

L. W. Bruch / Theory

electric fields polarized (2.4) is replaced by 3

parallel

of physisorpfion interactions

and perpendicular

to the surface [54]; E in eq.

g(iw) = {e,, (iw) eI (iw)}“2.

(2.6)

The representation of C, provided by eqs. (2.4) to (2.6) enables evaluations using experimental data for the responses and without specific models for the electronic structure of either subsystem. Values for several adatom-substrate combinations obtained in this way are assembled in table 1; further cases are

Table 1 Dispersion

energy coefficients

Ag

Cll

C,: H H, ei He Ne Ar Kr Xe Cs, H Hag’ He Ne Ar Kr Xe G2

a.b)

0.124 0.169 0.0583 0.121 0.402 0.569 0.841

0.126 0.174 0.0618 0.130 0.420 0.602 0.87 1

AU

0.135 0.188 0.0678 0.143 0.458 0.642 0.941

Al

Pd

0.123 0.166 0.0544 0.110 0.384 0.546 0.812

Gr ‘)

0.112 0.152 0.0523 0.108 0.362 0.512 0.758

0.090 0.121 0.043 0.086 0.30 0.43 0.6 1

d).

. 4.72 7.88 0.754 3.02 39.4 82.1 191

4.66 7.85 0.773 3.13 39.7 82.2 191

4.90 8.33 0.837 3.41 42.5 87.6 202

4.87 8.07 0.744 2.93 39.7 83.3 195

4.3 I 7.16 0.680 2.72 35.7 74.5 174

3.55 5.80 0.57 2.23 29.7 63 142

3.76 5.99 0.503

3.62 5.81 0.501 1.94 27.8 59.3 141

3.93 6.39 0.568 2.22 31.0 65.5 155

4.06 6.43 0.527 2.01 30.2 65.0 156

3.20 5.04 0.417 1.60 23.7 51.1 123

2.16 3.30 0.31 1.18 17.6 38 89

dt.

H H, e, He Ne Ar Kr Xe

.

1.93 28.3 60.8 146

a) Entries in atomic units. For Cs, C’s, and Cs,: 1 au = 0.9573 X lo-l2 erg A6 = 597.5 meV Ae’. For C,: 1 au = 6.460 x lo- ‘a erg A’ = 4032 meV A3. b, Atomic polarizability calculated with data from refs. [21,22]. Substrate response calculated with data from refs. 1241 and 1541. ‘) Graphite with basal plane surface. dt For comparison the C, values from ref. [22) are: H, 6.50; Ha, 11.49; He, 1.46; Ne, 6.43; Ar, 64.2; Kr, 127.9; Xe, 290.5. ‘) Spherical average coefficient; see ref. [22].

L. W. Bruch / Theory ofphysisorptioninteractions

201

presented in the original papers [54,55]. The entries of table 1 are estimated to be accurate to 5 to 10%. The theory has also been generalized to include relativistic retardation of u0 is an effects at large distances 123,273, where the distance dependence inverse fourth power, and to include spatially non-local dielectric {27,56] functions, c(k, w). The latter effect appears to become significant only at such short distances z that the adatom is in the region of the electronic spillout from a jellium surface [57]. Zaremba and Kahn 1571 included effects of the spatial width of the electron density at the substrate surface by adjusting the zero from which the distance z in eq. (2.3) is measured. The polarization potential at large distances is probed with atom-surface scattering experiments [2,3,58]. In early experiments [58] on the deflection of an atom& beam of cesium atoms by a gold cylinder the atomic turning points were approximately 500 A from the gold surface; at such separations retardation effects make significant contributions and reduce the effective value of C,. However, there is an unresolved discrepancy [54,58,59] between the calculated and the derived values, with the magnitude of the interaction estimated from the experiments being only half that of the calculated value. Agreement is much better for values of C, derived from resonance energies in selective adsorption experiments [Z], where the calculated values [53] for He and I-I interacting with LiF and NaF and the derived values agree to within the combined uncertainties of the two determinations. A less direct comparison of calculation with experiment occurs for multilayer adsorption data. Singleton and Halsey [60] give as an approximation for the pressure p, in an adsorption isotherm at which the n th layer condenses ln( A/P,)

= - [ C,(A,

X) - C,(&

A)] /d3n3kJ’.

(2.7)

In eq. (2.7) the notation C,(A, X) denotes the coefficient for an adatom A interacting with a substrate of X atoms; d is the interlayer spacing of the adsorbate, T is the absolute temperature, and k, is Boltzmann’s constant. To obtain this form requires assumptions that there is little relaxation of the lattice constant in a multilayer and that entropy effects in the adsorbed film are small. The latter assumption is supported in an analysis [lo] of a lattice gas model for multilayer adsorption at low enough temperatures that there are sharp condensation transitions of the successive layers. For basal plane graphite, the n = 2 to 5 condensations of krypton [61] at 77 K, the n = 2 to 4 condensations of xenon 1611 at 109 K and the n = 2 to 5 condensations of argon [62] at 64 K all follow this functional behavior for the pressures. However, the calculated values [54] for the differences in the C, coefficient in eq. (2.7) are only about half the values derived from the data. There is no definite resolution of this discrepancy yet. Steele [63] analyzed computer simulations of the multilayer absorption of argon on graphite at much higher

202

L. W. Bruch / Theory

ofphysisorprioninteractions

temperatures and found that the adsorption did not filling or by a continuous thickening of a slab sequential layer filling was expected to be a more temperatures. Cole, Frank1 and Goodstein did note (2.7) in observations of helium films on graphite.

occur by sequential layer of liquid; however, the realistic model at lower [l] some support for eq.

2.3. The potential minimum in u,,(z) The values of the laterally averaged holding potential U,,(Z) near the potential minimum largely determine the low coverage heats of adsorption, the low coverage adsorption isotherms, and the mean square adatom vibration amplitude perpendicular to the surface. Values for the depth of the holding potential z0 and the angular frequency w of perpendicular vibrations which appear in a parabolic fit to the potential minimum,

(2.8) (m is the adatom mass) are presented in table 2 for argon, krypton and xenon on graphite and on the (111) face of silver. The values for graphite are derived from data of Halsey and co-workers [64,65]. The values for silver are mainly from electron diffraction experiments [9,10]. The width of the potential well as a function of energy can also be constructed from the vibrational energy levels, using the semi-classical limit of quantum mechanics. This is the RKR construction of diatomic potential wells in three dimensions [66]. The analogous construction has been performed [67] for atomic hydrogen and helium interacting with LiF and NaF, using energy levels observed in selective adsorption experiments. More frequently, the potential well is determined from such data, as for helium on graphite [68], by

Table 2 Parameters

of the holding

potential

minimum

‘) w

Xe Kr Ar

172” 106-111 ‘) 65% 68”

(10 12

s-I

)

Grb'

A&111)

Grb'

158 120 90

4.6 d) 7.0 d 6.6 e’

5.3 6.3 8.0

‘) Parameters in parabolic fit uo(z) = - co + fmw*(r - ro)*, where m is adatom mass. b, Values for graphite from an analysis of data of Halsey [64] by Ross and Olivier [65]. ‘) Values derived from electron diffraction experiments; see refs. [IO] and [ 1I]. d’ Values derived from Debye-Waller factor data [IO]. e, From curvature of potential minimum calculated by Lang [44].

L. W. Bruch /

Theory of physisorption interaciions

203

adjusting parameters in some assumed functional form for u,, until a fit to the observed levels is achieved. Theoretical deter~nations of z+,(z) have been in the context of density functional theories. Lang [44] determined the charge rearrangement in the adsorption of argon on a jellium approximation to silver; his potential depth was within 20% of the value derived [lo] from electron diffraction experiments for Ar/Ag( 111). Lang’s self-consistent solution for the Kohn-Sham orbitals is the most detailed application of density functional approximations to physisorption to date. Van Himbergen and Silbey (431 in their application of a density functional approximation, with no rearrangement of charge in the adsorption, found values of e0 for argon on jellium which were about one-half the values derived from adsorption on metals. Freeman [41] in a similar application to inert gas adsorption on graphite found values of e0 which were much smaller than experimental values. Most calculations of the holding potential u(r) are based on summing model potentials for the interaction of adatom-substrate atom pairs [50], in close analogy to the calculation of the potential energy of inert gas solids by summing pair potentials. Although probably most applicable to adsorption on inert gas solid [50,69], the most important current applications of the technique are to the adsorption of inert gases on graphite. There are quite efficient transformations [50,5I] of the pair sums to generate the functions u0 and uG. TWO sorts of questions arise in the application of this technique: (1) the role of multibody interactions in the holding potential [70], which in part reflects the ambiguities in resolving the holding potential into a sum of pair interactions; (2) an apparent need to use anisotropic pair potentials [68,71] to reproduce the values of uG derived from selective adsorption band-structure splittings observed for helium on graphite. The second topic has been discussed extensively by Cole and co-workers [ 1,681. A few remarks are made here on the first topic. At one time it was suggested [70], based on an analysis with a Drude oscillator model [72] for the substrate atoms and the atoms and using inert gas polarizabilities, that the decomposition of the adatom-substrate interaction into the result of sum~ng pairs, triples, quadruples, etc, is only slowly convergent. However, the work of Nijboer and Renne [73,74] does not support this result. The holding potential can be formed for this model using eq. (1.4) for atomic triples and generalizations of that for higher multibody interactions. The multibody sums can be approximated with a result of Renne [73,74]: (2.9) solid In eq. (2.9), the sum on the index “j” runs over a planar semi-infinite lattice, of density ps; the index “i ” denotes a site in the solid and the index “0” denotes a position outside the solid. The approximation results from replacing the sum over atomic lattice sites by a continuous integration over the half-space

204

L. W. Bruch / Theory of physisorption

interactions

occupied by the solid. With the use of eq. (2.9), the following is obtained [73,74]

expression

for C,

(2.10) In eq. (2.10), (Y~and wi denote the static polarizability and resonant angular frequency of the Drude oscillators for the adatom (i = A) and the solid (i = S). The series in multibody interactions is recovered from eq. (2.10) by expanding it in powers of the “small parameter” (2?rpsas/3). For the case of argon adsorbed on xenon, successive terms are down by approximately a factor of 10 and the series is rapidly convergent. The derivation of eq. (2.9) is discussed in the appendix. Vidali et al. [75] have attempted to summarize the information on the physisorption holding potential in the form of corresponding states scalings. They explore the hypothesis that the basic shape of the physisorption potential remains the same, at least within some families of adatom-substrate combinations, They consider a variety of data and conclude that there is a common shape for light gases interacting with insulators, but that some systematic differences in the shape occur when other substrates are included in the comparison.

3. The lateral interactions of physisorbed atoms 3.1. General aspects In three dimensions, an inert gas atom retains its identity in condensed phases and the pair potential which determines properties of the dilute gas phase is supplemented to a limited degree by many-body interactions in the total potential energy of dense phases [45]. There are a great many physisorption systems where the primary lateral interaction of adatoms is again the pair potential From the dilute 3D gas and where the main supplementary terms in the substrate-mediated interactions can be identified. The inert gases adsorbed on graphite and such metals as silver, copper and aluminum appear to be such systems [5,10,76,77]. There remains the possibility that the subtle correlations which give rise to attractive forces between inert gas atoms can be substantially disrupted by the adsorption process; the adsorption of xenon on transition metals may have some chemical bonding aspects [ 14,181 and an associated larger lateral repulsion between adatoms. There is little firm information on the latter possibility and this section summarizes the information obtained from the analysis of the adsorption on graphite and on silver. The components of the total lateral energy of the physisorbed layer include:

L W. Bruch / Theory of physisorption interactions

205

the isolated pair potential, known [45,46] from the analysis of three-dimensional gas and condensed phase data; the substrate-mediated dispersion energy [28,29], proposed by Sinanoglu, Pitzer and McLachlan; the electrostatic energies of the adsorption-induced dipoles [78], which contribute to adsorptiondependent work function changes; the overlap of the adsorption complexes [42], which also includes screening oscillations of the substrate electrons [79,80]; the energy of adsorbate-driven elastic distortion of the substrate [81]; the energy of elastic distortion of the adsorbed layer [82], driven by the periodic components of the adatom-substrate potential and determining orientational epitaxy; and many-body interactions within the layer, such as the triple-dipole energy. The total energy includes the zero-point energy of adatom vibrations, which is easily calculated for a nearly classical lattice such as formed from argon, krypton, or xenon atoms. If the adsorbed layer is treated as a purely two-dimensional array, modifications to the effective potential by averaging over adatom vibrations perpendicular to the substrate must be considered [83-851. There is very little first principles theory in this part of the subject of physisorption interactions. There are no complete model solutions for the lateral interactions including both overlap and correlation processes. The substrate-mediated dispersion energy is formulated with a perturbation theory for the adatoms [28] and dielectric imaging for the substrate response. The only application of the density functional approximations to the lateral interactions is work of Freeman [42] for inert gases on graphite. The lateral interactions have prominent effects [ 10,11,76,77] in the structural properties of adsorbed layers and in their thermodynamic functions. The effects are also observed in the electronic properties of the adsorbed layers [16]: angle-resolved photoemission experiments for xenon and krypton adsorbed on the (100) face of palladium show a dispersion of electron energies agreeing with tight-binding band structure calculations for the adatom lattice. The electron overlap terms in the tight-binding band theory are closely related to matrix elements in an approximate Hartree-Fock theory of the overlap repulsion. The dispersion of the electron energies depends on the total overlap matrix elements of adatom electrons and is not specific to additional interactions in adsorption. It is not known how much difference there would be in the observed dispersion if the adatoms formed a two-dimensional liquid rather than a two-dimensional solid. The organization of the remainder of this section is: information on substrate-mediated dispersion energies is consolidated in section 3.2; the properties of the adsorption dipoles are reviewed in section 3.3; other mechanisms for lateral interactions are reviewed in section 3.4; a final summary is given in section 3.5.

206

L. W. Bruch /

3.2. Substrate-mediated

Theory

of physisorption

interactions

dispersion energies

The lateral interaction energy is defined to be the difference between the energy for the two adatoms in the presence of the substrate and the sum of the separate interaction energies of the adatoms with the substrate. In terms of the multibody interactions discussed for three-dimensional inert gas solids, the substrate may be described as an extended third body interacting with the two adatoms. Such a description has been used in the analysis of inert gas adsorption on inert gas solids [86]. For the adsorption on substrates composed of quite polarizable constituents another description in terms of the response of the substrate to external fields [28] is much more convenient and provides a formalism where experimental response data [24] can be incorporated directly into the calculation. Sinanoglu and Pitzer [29] began the perturbation theory of the effect of the substrate on the adatom-adatom dispersion forces. They explicitly treated the response of the adatoms and of the substrate atoms. To make quantitative estimates of the effects, they made several simplifying assumptions about the adsorption geometry and about the atomic responses. They included effects which contribute to the adsorption-induced dipole moments. It is now advantageous, for the analysis of realistic physisorption systems, to take the net adatom dipole moment from work function measurements [78] and to treat separately the dispersion energies of adatoms. McLachlan [28] expressed the response of the substrate to the fields of the fluctuating adatom dipoles using an electromagnetic boundary condition. Then the interaction energy of the adatoms is calculated with second order perturbation theory for the interactions of the adatom dipoles and their electrostatic images in the substrate (no relativistic retardation effects), allowing for a

Fig. 1. Coordinates for the McLachlan substrate-mediated dispersion energy, section 3.2. The angles and lengths are used in eq. (3.1). In eq. (3.4), for both adatoms at the same perpendicular distance from the substrate, the distance I, is denoted L and the distance R,, is denoted r, with R,e, = (r* +4L*)“*.

L. W. Bruch / Theory ojphysisorption rnreracrions

207

frequency-dependent response of the substrate. At intermediate separations this energy is larger than the direct interaction of the adsorption dipoles because the root-mean-square dipole moment on the adatom is much larger than the mean dipole moment. The result of the perturbation theory is expressed in terms of the geometry of the adatoms and the substrate, the dynamic polarizability of the adatoms, and the frequency-dependent dielectric constant of the substrate. The McLachlan expressions for the substrate-mediated dispersion energy for two identical adatoms are. for the geometry shown in fig. 1,

9=

cs,

(3.1)

with (atomic

units) (3.2)

(3.3) For the frequently treated geometry lar distance L from the substrate energy is

in which both adatoms and lateral separation

are at perpendicur the McLachlan

(3.4) The London-Van separation r is h.vdW

=

-

der Waals

C6/rht

energy

of an isolated

pair

of the atoms

at

(3.5)

with C, =

ilrn dw

[a(iw)12.

(3.6)

Values for the strength coefficients for several adatom-substrate combinations are presented in table 1. These are calculated with bounding approximations for the adatom polarizability functions [21] and with experimental data for the dielectric response (241, using eq. (2.5). Because the resonant frequencies of inert gas atoms are large compared to plasmon frequencies, the response of the metal in the range of interband transitions is required. This is evident in some semi-empirical expressions [55] for adsorption strength coefficients where the substrate frequency parameter which is used is much higher than the known plasmon frequencies.

208

L. W. Bruch / Theory ojphysisorption interactions

To apply eq. (3.1) to an adsorbed layer requires information on the overlayer distance L. For a few cases [6,10,87,88], the distance of the adatoms above the ion cores in the surface layer of the substrate has been measured. Some adjustments are necessary to convert this length to the distance L from the image plane. For adsorption on metals the image plane calculated for jellium at the substrate density and for fluctuating fields 1571 has been used [83]. For an insulator another prescription [57] is to take the image plane as one-half an interplanar spacing outside the surface layer of the substrate. In the absence of experimental information, the overlayer distance may be estimated from the pair-wise summation calculation of the holding potential (section 2.3) [50,89]. ‘l’wo other methods have been used to derive eq. (3.1). Duniec and Ninham [90] derived it by evaluating the dispersion energy in terms of the shifts of the zero-point energies of the modes of coupled dipole oscillators on the subsystems. MacRury and Linder [86] showed that approximations to the McLachlan strength coefficients are obtained by summing multibody interactions involving two adatoms and one or two substrate atoms; their analysis for the triple-dipole is discussed in the appendix. In the McLachlan derivation [28] the screening response of the substrate is approximated by image charge fields. For application to physisorbed layers, the response for a disturbance at short distances, near atomic spacings, is required. There is now some evidence that this approximation does not introduce large errors. In addition to some model calculations (91-931, core hole shifts for xenon adsorbed on Pd(100) measured [94] with core-level spectroscopy and Auger spectroscopy agree with the use of image fields for a charge in the first few adsorbed layers. Some of the estimates of the dispersion energy for adsorbed layers have involved quite drastic simplifications [77,9.5] of the Sinanoglu-Pitzer-McLachlan energy. One such appro~mation [77] is to replace eq. (3.1.) by S(r)

= - & t&/G,

(3.7)

where [ is the adatom polarizability, PO is the (negative) one-atom holding potential and r is the lateral separation of the adatoms. Trends for adsorption calculated with eq. (3.7) frequently differ from those calculated with eq. (3.4). For krypton at lateral separations of 4 to 4.5 A, the energy calculated from eq. (3.7) is larger than from (3.4) on graphite and smaller on silver. For xenon at separations of 4.3 to 5 A the energy calculated with eq. (3.7) is larger than with eq. (3.4) for both graphite and silver, but the percent difference is larger for graphite than for silver. A scaling factor reducing the Sinanoglu-Pitzer energy, eq. (3.7), invoked in some applications [77], in part reflects this overestimate. Freeman [42] calculated the short and intermediate range lateral interaction of inert gases on graphite using a density functional approximation. There is a repulsion for two argon atoms at separations of 3 to 4 A which is in fair

L. W. Bruch / Theory

of physisorption inferactions

209

agreement with the result of eq. (3.4) [84]. Since these two approaches treat opposite limiting cases of the same physical process, this is a quite encouraging comparison. 3.3. Interaction

of adsorption-induced dipoles

Adsorption-induced dipoles for krypton and xenon on metals are inferred from changes in the work function of the metal with the adsorbate coverage [78]. Presumably such dipoles occur for lighter gases and for adsorption on graphite, but they have not been observed yet. The value of the adsorption dipole is obtained from the work function change S@ for a layer of n dipoles per unit area (each of value p): S6p= 4mp.

(3.8)

Kohn and Lau [96] derived such a result for “arbitrary adatoms on arbitrary metal surfaces, even when they produce large perturbations and/or are situated in the surface region rather than outside of it.” The adsorption dipole reflects the charge rearrangement in the processes which give the single-atom holding potential. A perturbation theory for the rearrangement of charge associated with the polarization potential, eq. (2.3), was developed by Antoniewicz [97] and then by Zaremba [98] and by Linder and Kromhout [99]. Kromhout and Linder [ 1001 concluded this mechanism did not give adsorption dipoles of the magnitude observed on metals. The relation of the dispersion force dipole moment to collision-induced dipole moments on pairs and triples of atoms was treated by Bruch and Osawa [lot 1. Bruch and Ruijgrok f38] treated an idealized model for the physisorption of atomic hydrogen which included effects similar to a rearrangement of charge driven by overlap repulsions. However, the only self-consistent treatment of the adatom and metal charge distribution is the calculation of Lang [44] within the electron density functional formalism; he has evaluated the dipole moment for an argon atom interacting with jellium metal at densities corresponding to silver and to aluminum. Flynn and Chen [ 181 discuss the contribution of polar excited configurations to the physisorption dipole. The potential energy of a two-dimensional lattice of polarizable point dipoles and the corresponding variation of the dipole moment per adatom with density (the Topping formula) was derived by Miller [ IOZ]. Palmberg [15] found his data for the work function of Xe/Pd(l~) following the Topping formula with an atomic polarizability twice that of isolated xenon. However, Miller’s derivation did not explicitly include the effect of surface screening charges (image charge fields) in the lateral interactions. A first indication of the importance of electrostatic image effects is that the lateral interaction energy of a pair of adsorption dipoles is double the value for

L. W. Bruch / Theory of physisorption

210

an isolated

pair of parallel

+ = 2pL,Mr3.

interactions

dipoles of the same values (3.9)

This was shown in general terms by Kohn and Lau [96] and is understood in terms of the field of the image dipole in the metal. That there should be image terms in the depolarizing field determining the dipole moment per adatom is shown by calculations of Antoniewicz [103] for dipoles in the presence of a perfectly screening metal. For adatoms at separations large compared to the overlayer distances, the image field terms effectively double the adatom polarizability, consistent with Palmberg’s fitting [ 151. The Topping formula for the dependence of the depolarizing field on coverage applies to the case of a homogeneous adsorbate phase, rather than to a mixture of dense adsorbate islands and dilute two-dimensional gas. Depolarizing field effects and image charge effects are readily computed [76] when the adsorbed layer forms a two-dimensional lattice. The results for a disordered two-dimensional fluid of polarizable dipoles are much more limited [ 1041. A good knowledge of the charge redistribution in the adsorption process is also necessary for reliable calculation of the depolarizing fields. Wang and Gomer [14] interpret their data for xenon adsorbed on tungsten as showing that a second layer of adsorbed atoms makes negligible contribution to the work function change. This is consistent with the idea that the charge rearrangement giving the dipole occurs only for the more strongly bound atoms in the first layer. 3.4. Other mechanisms There are several other mechanisms for the lateral interactions which have been proposed and have been investigated for inert gases adsorbed on graphite and silver. They are generally smaller than the electrostatic and dispersion energies discussed in sections 3.2 and 3.3, but there are circumstances where they become significant. The polarization of the electrons in a jellium substrate by overlap with the adatom electrons leads to an oscillatory (Friedel) lateral interaction energy of two adatoms [79]. Although estimates for xenon adsorbed on silver show [76] it to be smaller than the terms of sections 3.2 and 3.3, this process is believed to be significant in the lateral interactions of chemisorbed atoms [80]. The forces which cause the atomic adsorption also cause stresses on the substrate lattice. The substrate distortion has been treated with continuum elasticity theory, including effects of elastic anisotropy [81]. For two like adatoms, an effective repulsion is obtained which varies as the inverse cube of the lateral separation. The components uG of the holding potential, eq. (2.1), cause elastic distortions in a monolayer solid which is not commensurate with the substrate.

L. W. Bruch / Theory

ofphysisorpiioninteractions

211

Novaco and McTague [82] treated this effect with perturbation theory and predicted the occurrence of orientational registry for incommensurate adsorbed solids. Estimates of the orientational registry energy with values of uG obtained by the pairwise-summation method show that it contributes significantly in the lateral compressibility of argon on graphite at substantial thermal expansion [lo]. There are also multibody interactions within the adsorbed layer. The triple dipole energy, the largest of these, is a much smaller fraction of the total lateral of the adsorbed layer [83] than of the dense three-dimensional phases because of the smaller number of triangles per atom on a 2D lattice than on a 3D lattice. 3.5. Summary The facts that the 3D pair potentials follow the corresponding states form and that the pair sum is the largest part of the calculated 2D energies for adsorption on graphite and on silver led to an attempt [lo] to make a common scaling of the observed properties of the 2D solids. This was done [lo] for the lateral compressibility and for a heat of adsorption difference related to the lateral stress in the layer, using data for Ar, Kr and Xe on Ag( 111). The length and energy scales were the values for the 3D pair potentials. The scaled results, particularly for the lateral compressibility, became a single experimental curve, with scatter consistent with known uncertainties in the data. Lateral compressibility data for argon on graphite, when the effect of the orientational registry Table 3 Hierarchy

of terms in the ground

state energy of 2D solid Xe Xe/Ag(

q,(expt) a’ L, b’ E la, ‘) x2 d, McL @ Dipole n Triple-dipole s) Zero point energy a) b, ‘) d, ‘) 9 @

meV/atom A meV/atom

111)

225&5 4.425 - 54.2 - 77.0 12.4 6.3 1.5 2.6

Monolayer latent heat of adsorption; see ref. [lo] Calculated zero temperature nearest neighbor distance. Net energy of lateral interactions, calculated as in ref. [76]. Net potential energy from sum of gas phase pair potential X2 [45]. Sinanoght-Piker-McLachlan dispersion energy. Lateral interaction of adsorption-induced dipoles. Axilrod-Teller-Muto triple-dipole energy of 3 xenons.

Xe/Gr 239+4 4.405 -63.3 -77.5 10.0 -(7) 1.5 2.7

212

L.W. Bruch / Theory

ofphysisorption interactions

energy at the larger lattice constants is included, also follow the same trend [lo]. The lateral compressibility derived from data on compressed xenon on graphite deviates from the trend, but in a sense which is consistent with being caused by registry energy terms [lo]. There is much theoretical work aimed at identifying the dominant term in the interaction of adatoms at large lateral separations. The interaction of the adsorption-induced dipoles [96] and the interaction through elastic deformation of the substrate [81], both of which vary as the inverse cube of the distance, dominate over the Sinano~u-Pitzer-~cLachlan dispersion energy there. Even the Friedel screening term [79] is formally dominant over the dispersion energy there. However, in calculating the energy of condensed monolayers of inert gases on silver and on graphite, it is the Sinanoglu-Pitzer-McLachlan dispersion energy which is the quantitatively dominant [ IO.761 substrate-mediated energy. AR enumeration of the calculated contributions to the lateral interaction energy in the ground state of xenon on Ag(ll1) and on graphite is given in table 3. A similar enumeration for Ar, Kr and Xe on Ag(ll1) has been reported previously [lo]. For such systems, the major components of the interactions in the monolayer, apart from terms dependent on details of the structure of the adsorption complex, are now recognized in nature and relative magnitude. Especially for xenon, the use of these interactions in calculations of the properties of the 2D solid, of its sublimation, and of the formation of the bilayer solid gives extensive agreement with experimental observations [lo,1 1,76,105].

Added note Selective adsorption resonances have been observed for He on Ag by Tommasini [ 1071 and for He on Cu by Lapujoulade [ 1081 and their co-workers. The holding potentials U&Z) inferred for these metals are quite similar to the results of calculations of Zaremba and Kohn (ref. [109], see also ref. [43]). The parameters for the potential wells are very different from the values for He on graphite [ 11, in marked contrast to the comparisons for the heavier inert gases shown in table 2.

It is a pleasure to acknowledge again the contributions of my collaborators, particularly Professors M.B. Webb and J.M. Phillips, in the work on which this paper is based.

L. W. Bruch / Theory of physisorpiion interactions

213

Appendix Integration

of dipole tensor product [I 061

In the theory of multibody dispersion energies, sums of products of dipole tensors occur. If the discrete sums are replaced by integrations, explicit expressions for the total can be obtained. Such identities have been reported by Nijboer and Renne [73,74], but most of the derivations are complex. A simple derivation can be given by exploiting an electrostatic analogy [106]; this is presented here for the case of the sum required in the derivation of the leading term in the McLachlan interaction [28] from the triple-dipole energy [48]. Consider the total triple-dipole energy for two atoms, a and b, external to a semi-infinite planar solid lattice of atoms c (fig. 2) E(3) = c ( rJabc/3) Tr( 5, * G, * Ta 1. Approximate density p)

the sum by an integral

c Tb,. ?a = p/ i

P(vavb)/dr3 v3

Then,

over the volume

of the solid (number

dr, Tb3 . T& = 7.

For this geometry, neither manipulated to give

=

(A4

with Gauss’

(A.3

rbs nor r,, vanishes

and the integral

in (A.2) can be

64.3)

*

theorem,

the integral

becomes

a surface

integral

with the

A

C

C

ccc

s +

c

c3

Fig. 2. Schematic illustration showing atomic positions for the triple-dipole energy sum, eq. (A. 1). The adatoms are A and B; the planar surface of the solid is denoted S,; the discrete substrate lattice positions are denoted C. The point B’ is the mirror image of adatom B with respect to the surfce S,.

214

vector element

L. W. Bruch / Theory

of surface area directed

ofphysisorptioninteractions “outward”

from the volume of the solid (A.4)

The integral in eq. (A.4) is difficult under direct reduction; Renne [74] evaluated such integrals using Bessel function representations. However, it is the same integral which gives the potential arising from the screening charge density induced on a planar metal surface by an external point charge. Hence the result can be written immediately in terms of image charge potentials: (A.5) where the prime in rab. means that this distance is computed of a to the mirror image of b in the surface. The total triple-dipole energy, eq. (A.l), is then Ec3) = ( GJ~)

Tr( Tab .T ).

The dependence on the positions of particles a and leading (C,,) term of eq. (3.1). Eq. (A.6) corresponds the strength coefficient obtained by retaining only expansion of the Clausius-Mossotti equation for the stant e(iw) - 1 = 47rpas(iw).

from the position

(A.61 b is the same as for the to an approximation to the leading term in an substrate dielectric con-

(A.7)

This relation between the McLachlan energy and the triple-dipole energy sums was recognized in numerical work of MacRury and Linder [86]. The reduction from eq. (A.2) to (A.5) is modified when one of the atoms a or b is part of the solid lattice. Then a small spherical volume about that atom is excluded [73,74] from the integral in eq. (A.2) and the result is eq. (2.9).

References [1] [2] [3] [4]

M.W. Cole, D.R. Frank1 and D.L. Goodstein, Rev. Mod. Phys. 53 (1981) 199. H. Hoinkes, Rev. Mod. Phys. 52 (1980) 933. M. Cardillo, Ann. Rev. Phys. Chem. 32 (1981) 331. P.A. Heiney, R.J. Birgeneau, G.S. Brown, P.M. Horn, D.E. Moncton and P.W. Stephens, Phys. Rev. Letters 48 (1982) 104. [5] P.S. Schabes-Retchkiman and J.A. Venables, Surface Sci. 105 (1981) 536; S. Calisti, J. Suzanne and J.A. Venables, Surface Sci. 115 (1982) 455; S.C. Fain, Jr., M.D. Chinn and R.D. Diehl, Phys. Rev. B21 (1980) 4170; A. Glachant, M. Jaubert, M. Bienfait and G. Boato, Surface. Sci. 115 (1982) 219. [6] K. Carneiro, L. Passell, W. Thomlinson and H. Taub, Phys. Rev. B24 (1981) 1170. [7] A. Thorny, X. Duval and J. Regnier, Surface Sci. Rept. 1 (1981) 1;

l.. W. Bruch / Theory of physisorption interactions

18) [9] [IO] [1 I] [ 12) [ 13) [14]

[IS] [ 161 [ 171 [ 181

[ 191 1201 [Zl] [22]

123)

[24]

[25] 126) [27] (281 1291 [30]

215

O.E. Vilches, Ann. Rev. Phys. Chem. 31 (1980) 463; J.G. Dash. Films on Solid Surfaces (Academic Press. New York, 1975); F. Millot, Y. Larher and C. Tessier, J. Chem. Phys. 76 (1982) 3327. E. Conrad and MB. Webb, private communication. J. Unguris, L.W. Btuch, E.R. Moog and M.B. Webb, Surface Sci. 87 (1979) 415. J. Unguris, L.W. Bruch. E.R. Moog and M.B. Webb, Surface Sci. 109 (1981) 522. J. Unguris. L.W. Bruch, M.B. Webb and J.M. Phillips, Surface Sci. I15 (1982) 219. T.-C. Chiang, G. Kaindl and D.E. Eastman, Solid State Commun. 41 (1982) 661. T. Engel, F. Bornemann and E. Bauer, Surface Sci. 81 (1979) 252. C. Wang and R. Gomer. Surface Sci. 91 (1980) 533. P.W. Palmberg, Surface Sci. 25 (1971) 598. K. Horn, M. Scheffler and A.M. Bradshaw. Phys. Rev. Letters 41 (1978) 822; K. Hermann, J. Nofke and K. Horn, Phys. Rev. B22 (1980) 1022. K. Wandelt and J.E. Hulse, to be published. C.P. Flynn and Y.C. Chen. Phys. Rev. Letters 46 (1981) 447, and references contained therein: N.D. Lang, A.R. Williams, F.J. Himpsel, B. Reihl and D.E. Eastman, Phys. Rev. B26 (1982) 1478. J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, The Molecular Theory of Gases and Liquids (Wiley, New York, 1965). P.R. Certain and L.W. Bruch, in: MTP International Review of Science. Physical Chemistry Series One, Vol. I, Theoretical Chemistry, Ed. W.B. Brown (Butterworths, London. 1972). K.T. Tang, J.M. Norbeck and P.R. Certain, J. Chem. Phys. 64 (1976) 3063. P.J. Leonard and J.A. Barker, in: Theoretical Chemistry - Advances and Perspectives, Vol. I. Eds. H. Eyring and D. Henderson (Academic Press, New York, 1975). pp. 117-156: P.W. Langhoff, R.G. Gordon and M. Karplus, J. Chem. Phys. 55 (1971) 2126. H.B.G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360; I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii. Advan. Phys. 10 (1961) 165: A.D. McLachlan, Proc. Roy. Sot. (London) A271 (1963) 387. H.-J. Hagemann, W. Gudat and C. Kunz. J. Opt. Sot. Am. 65 (1975) 742: H.-J. Hagemann, W. Gudat and C. Kunz, DESY fntemaf Report SR 74/7 (Hamburg, 1974): J.H. Weaver and R.L. Benbow, Phys. Rev. B12 (1975) 3509; see also J.H. Weaver, C. Krafka D.W. Lynch and E.E. Koch, Optical Properties of Metals, Physics Data, No. I8 (Fachinformationszentrum. Karlsruhe, 1981). J.E. Lennard-Jones, Trans. Faraday Sot. 28 (1932) 334. J. Bardeen, Phys. Rev. 58 (1940) 727. E.M. Liftshitz, Soviet Phys.-JETP 2 (1956) 73; YuS. Barash and V.L. Ginzburg, Soviet Phys.-Usp. 18 f 1975) 305. A.D. MacLachlan, Mol. Phys. 7 (1964) 381. 0. Sinanoghr and KS. Pitzer, J. Chem. Phys. 32 (1960) 1279. P. Bertoncini and A.C. Wahl, Phys. Rev. Letters 25 (1970) 991; J. Chem. Phys. 58 (1973) 1259.

(311 W.J. Stevens, A.C. Wahl, M.A. Gardner and A.M. Karo, J. Chem. Phys. 60 (1974) 2195. [32] R.E. Lowther and R.L. Coldwell, Phys. Rev. A22 (1980) 14, and references contained therein. [33) J.C. Wood, Surface Sci. 71 (1978) 548. 1341 J.N. Murrell and G. Shaw, J. Chem. Phys. 49 (1969) 4731. [35] R. Ahhichs, R. Pence and G. Stoles, Chem. Phys. 19 (1977) 119. [36] K.T. Tang and J.P. Toennies. J. Chem. Phys. 66 (1977) 1496. [37) G.G. KJeiman and U. Landman, Phys. Rev. B8 (1973) 5485; Solid State Commun. I8 (1976) 819. (381 L.W. Bruch and Th. W. RuiJgrok, Surface Sci. 79 (1979) 509.

216

L. W. Bruch /

[39] W. Kohn

[40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [SS]

[59] [60] [61] [62] [63] [63] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75]

Theory

of physisorption

interactions

and P. Vashishta, to be published. M.J. Clugston, Advan. Phys. 27 (1978) 893. D.L. Freeman, J. Chem. Phys. 62 (1975) 941. D.L. Freeman, J. Chem. Phys. 62 (1975) 4300. J.E. van Himbergen and R. Silbey, Solid State Commun. 23 (1977) 623. N.D. Lang, Phys. Rev. Letters 46 (1981) 842. J.A. Barker, in: Rare Gas Solids, Vol. 1, Eds. M.L. Klein and J.A. Venables (Academic Press, New York, 1976) ch. 4. R.A. Aziz and H.H. Chen, J. Chem. Phys. 67 (1977) 5719; R.A. Aziz, Mol. Phys. 38 (1979) 177. 0. Novaro, Kinam 2 (1980) 175. B.M. Axilrod and E. Teller, J. Chem. Phys. 11 (1943) 299; Y. Muto, Proc. Phys. Math. Sot. Japan 17 (1943) 629. The coefficients tabulated in refs. [21] and [22] are related by vabc = 3Z,,,(l, 1, 1). W.A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974). W.A. Steele, Surface Sci. 36 (1973) 317. T.H. Ellis, S. Iannotta, G. Stoles and U. Valbusa, Phys. Rev. B24 (1981) 2307; B.F. Mason and B.R. Williams, Phys. Rev. Letters 46 (1981) 1138. G. Boato, P. Cantini, C. Guidi, R. Tatarek and G.P. Felcher, Phys. Rev. B20 (1979) 3957; G.D. Derry, D. Wesner, G. Vidali, T. Thwaites and D.R. Frankl, Surface Sci. 94 (1980) 221. L.W. Bruch and H. Watanabe, Surface Sci. 65 (1977) 619. G. Vidali and M.W. Cole, Surface Sci. 107 (1981) L374; 110 (1981) 10; S. Rauber, J.R. Klein, M.W. Cole and L.W. Bruch, Surface Sci. 123 (1982) 173. E.J.R. Prosen, R.G. Sachs and E. Teller, Phys. Rev. 57 (1940) 1066; E.J.R. Prosen and R.G. Sachs, Phys. Rev. 61 (1942) 65. N.D. Lang and W. Kohn, Phys. Rev. B7 (1973) 3541; E. Zaremba and W. Kohn, Phys. Rev. B13 (1976) 2270. D. Raskin and P. Kusch, Phys. Rev. 179 (1969) 712; A. Shih, D. Raskin and P. Kusch, Phys. Rev. A9 (1974) 652; A. Shih and V.A. Parsegian, Phys. Rev. Al2 (1975) 835. M.J. Mehl and W.L. Schaich, Phys. Rev. A21 (1980) 1177; see also A.M. Marvin and F. Toigo, Phys. Rev. A25 (1982) 803. J.H. Singleton and G.D. Halsey, Jr., Can. J. Chem. 33 (1955) 184. A. Thorny and X. Duval, J. Chim. Physique 67 (1970) 286. B. Gilquin, Thesis, University of Nancy (1979). W.A. Steele, J. Colloid Interface Sci. 75 (1980) 13. J.R. Sams, Jr., G. Constabaris and G.D. Halsey, Jr., J. Phys. Chem. 64 (1960) 1689; 66 (1962) 2154. S. Ross and J.P. Olivier, On Physical Adsorption (Interscience, New York, 1964) especially section 7.1 .F. E.g., E.A. Mason and L. Monchick, Advan. Chem. Phys. 12 (1967) 329, especially section IILA. C. Schwartz, M.W. Cole and J. Pliva, Surface Sci. 75 (1978) 1. W.E. Carlos and M.W. Cole, Surface Sci. 91 (1980) 339. A.J. Bennett, Phys. Rev. B9 (1974) 741. W.M. Gersbacher, Jr. and F.J. Milford, J. Low Temp. Phys. 9 (1972) 189. A.D. Crowell, Surface Sci. 111 (1981) L667. A.A. Lucas, Physica 39 (1968) 5. B.R.A. Nijboer and M.J. Renne, Chem. Phys. Letters 2 (1969) 35. M.J. Renne, Thesis, University of Utrecht (1971). G. Vidali, S. Rauber, M.W. Cole and J.R. Klein, to be published.

L. W. Bruch / Theory of physisorprion interactions

217

[76] L.W. Bruch and J.M. Phillips, Surface Sci. 91 (1980) 1. 1771 G.L. Price and J.A. Venables, Surface Sci. 59 (1976) 506; J.A. Venables and P.S. Schabes-Retch~man, Surface Sci. 71 (1978) 27. [?S] M.A. Chesters, M. Hussain and J. Pritchard, Surface Sci. 35 (1973) 161. and references contained therein. [79] K.H. Lau and W. Kohn, Surface Sci. 75 (1978) 69, and references contained therein. [80] T.L. Einstein, CRC Critical Rev. Solid State Mater. Sci. 7 (1978) 261. [Sl] K.H. Lau, Solid State Commun. 28 (1978) 757; W. Kappus, 2. Physik B29 (1978) 239. [82] A.D. Novaco and J.P. McTague, Phys. Rev. Letters 38 (1977) 1286; J.P. McTague and A.D. Novaco, Phys. Rev. B19 (1979) 5299. [83] L.W. Bruch, PI. Cohen and M.B. Webb, Surface Sci. 59 (1976) 1. [84] G. Vidali and M.W. Cole, Phys. Rev. B22 (1980) 4661; B23 (1981) 5649 (E). [SS] P.A. Monson, M.W. Cole, F. Toigo and W.A. Steele, Surface Sci. 122 (1982) 401. [86] T.B. MacRury and B. Linder, J. Chem. Phys. 54 (1971) 2056; 56 (1972) 4368. [87] PI. Cohen, J. Unguris and M.B. Webb, Surface Sci. 58 (1978) 429; N. Stoner, M.A. Van Hove, S.Y. Tong and M.B. Webb, Phys. Rev. Letters 40 (1978) 243. [SS] C.G. Shaw, SC. Fain, Jr., M.D. Chinn and M.F. Toney, Surface Sci. 97 (1980) 128. [89] M.W. Cole and J.R. Klein, to be published. [90] J.T. Duniec and B.W. Ninham, J. Chem. Sot. Faraday Trans. II, 72 (1976) 1513. [Sl] S.C. Ying, J.R. Smith and W. Kahn, Phys. Rev. Bll (1975) 1483; N.D. Lang and A.R. Williams, Phys. Rev. Letters 34 (1975) 531. 1921 D.M. Newns, J. Chem. Phys. 50 (1969) 4572. 1931 A.P. Lehnen and L.W. Bruch, Phys. Rev. B21 (1980) 3193. 1941 G. Kaindl, T.-C, Cbiang, D.E. Eastman and F.J. Himpsel, Phys. Rev. Letters 45 (1980) 1808. [95] D.H. Everett, Disc. Faraday Sot. 40 (1965) 177. [96] W. Kohn and K.H. Lau, Solid State Commun. 18 (1976) 553. [97] P.R. Antoniewicz, Phys. Rev. Letters 32 (1974) 1424. [98] E. Zaremba, Phys. Letters A57 (1976) 156. f99] B. Linder and R.A. Kromhout, Phys. Rev. B13 (1976) 1532. [100] R.A. Kromhout and B. Linder, Chem. Phys. Letters 61 (1979) 283. [loll L.W. Bruch and T. Osawa, Mol. Phys. 40 (1980) 491. [102] A.R. Miller, Proc. Cambridge Phil. Sot. 42 (1946) 292. [ 1031 P.R. Antoniewicz, Phys. Status Solidi (b) 86 (1978) 645; see also the appendix of ref. [38]. [104] G.D. Mahan and A.A. Lucas, J. Chem. Phys. 68 (1978) 1344. [105] M.S. Wei and L.W. Bruch, J. Chem. Phys. 75 (1981) 4130. [ 1061 Based on unpublished work of C.J. Goebel. [ 1071 F. Tommasini, private ~mmunication; A. Luntz, L. Mattera, M. Rocca, F. Tommasini and U. Valbusa, Surface Sci. 120 (1982) L447. [ 1081 J. Perreau and J. Lapujoulade, Surface Sci. 119 (1982) L292. [lOS] E. Zaremba and W. Kohn, Phys. Rev. B15 (1977) 1769.