then make up the missing fraction). Such models contain several parameters, but the argument is made that the structure of how the reactions proceed is not unduly in¯u- enced by the detailed choice of such parameters (Eiswirth et al. 1990, Krischer et al. 1991).

The use of rate and diVusion equations is a powerful tool for modeling chemical kinetic experiments, and other related topics such as population biology. There are many features in common to all these ®elds that would be wonderful to study, if only one weren't limited by time! As a result, many of the developments have proceeded in parallel in the diVerent groups, without interaction. Here we use these techniques to discuss the (arguably simpler) case of epitaxial crystal growth in chapter 5.

Further reading for chapter 4

Bruch, L.W., M. W. Cole & E. Zaremba (1997) Physical Adsorption: Forces and Phenomena (Oxford University Press).

Desjonquères, M.C. & D. Spanjaard (1996) Concepts in Surface Physics (Springer) chapter 6.

Henrich, V.E. & P.A. Cox (1994, 1996) The Surface Science of Metal Oxides (Cambridge University Press) chapter 6.

Hill, T.L. (1960) An Introduction to Statistical Thermodynamics (Addison-Wesley, reprinted by Dover 1986) especially chapters 7±9, 15 and 16.

Hudson, J.B. (1992) Surface Science: an Introduction (Butterworth-Heinemann) chapters 12,13.

Masel, R.I. (1996) Principles of Adsorption and Reaction on Solid Surfaces (John Wiley) chapter 3.

Stanley, H.E. (1971) Introduction to Phase Transitions and Critical Phenomena (Oxford University Press).

Zangwill, A. (1988) Physics at Surfaces (Cambridge University Press) chapters 8, 9, 11 and 14.

Problems and projects for chapter 4

These problems and projects explore ideas about surface crystallography, adsorption potentials and the question of localization in adsorbed layers.

Problem 4.1. Monolayer structures on a honeycomb lattice

Consider rare gas adatoms on graphite, consulting ®gure 1.16 for the incommensurate aligned (IA) phase of Xe/graphite, and ®gure 4.8 for the incommensurate rotated (IR) phase of Ne/graphite, as needed. DiVraction from a square or rectangular lattice does not give any conceptual problems, but the graphite, or honeycomb lattice is more diYcult because the lattice vectors g are not perpendicular to each other, so we have to keep in mind what g´r means.

1424 Surface processes in adsorption

(a)Convince yourself that the 2D unit cells of the real lattice of graphite (lattice

parameter ac ) and of the commensurate (C) phase of adsorbed rare gases (lattice parameter a) are as shaded in the bottom right hand corner of ®gure 1.16 in

chapter 1.

(b)Construct the lowest order region of the reciprocal lattice g5ha*1kb*10c* of both the graphite and the Xe ML C-phase, remembering that reciprocal lattice vectors are perpendicular to real lattice vectors and inversely proportional to the corresponding (hk0) plane spacing. Show that the lowest order (10.0) and (01.0) C- phase diVraction spots are in the center of the triangles formed from the lowest order graphite spots. (The dot in (hk.0), representing2 (h1k), is the fouraxis notation for hexagonal crystals, see e.g. Kelly and Groves 1970.)

(c)Explain which features of the Ne/graphite diVraction pattern (®gure 4.8) indicate that the IR phase has a smaller unit cell than the C-phase, rotated ,618° from the

IA orientation.

Problem 4.2. Adatom energies on graphite

Equation (4.14) gives the formal expression for an expansion of the substrate potential V(r) seen by a single adatom in terms of the average potential V0 and the Fourier coeYcients Vg.

(a)Construct the potential V(r) for the graphite surface if only the six lowest order gs

are important, which have a common value of Vg. Show that the adsorption energy for an adatom in the middle of a graphite hexagon is (V016Vg), whereas at the

bridge position between two carbon atoms is (V0 24Vg), and at the ontop site, over one carbon atom it has the value (V0 23Î3Vg). Given that the lowest energy position is calculated to be in the center of the hexagon, show that Vg is negative, and that the diVusion path is via the bridge site with a diVusion energy Ed5210Vg.

(b)If a calculation gives the adsorption energies in these positions (hexagon center Eh, bridge Eb, on top of carbon Ec), show that the description of V(r) in terms of the Fourier series (4.14) requires three parameters V0, Vg 1 and Vg 2. Make a sensible choice for g2, and set up a matrix to solve for the Fourier coeYcients. If a particular calculation gives Eh521427, Eb521392 and Ec521388 K per atom, calculate V0, Vg 1 and Vg 2 in the same units. Compare this calculation with literature values for Kr/graphite (e.g. Price 1974, Bruch 1991, Bruch et al. 1997, chapter 2),

and note that the calculation does not depend on the height of the adatom above the surface being the same at the three positions.

Problem 4.3. Localized or 2D gas adsorption?

(a)DiVerentiate the same Fourier series (4.14) (with two or three sets of terms) twice,

to derive the diVusion frequency nd when the Kr atom is placed at the hexagon center position on graphite. Using the energy parameters given above, ®nd the value of nd in THz units given that Kr has atomic mass 83.8.

(b)Use the arguments of section 4.2 to discuss whether a low density gas of Kr adatoms should be considered to be localized, vibrating about a particular lattice site, or in a 2D gas. Estimate the transition temperature between these two states.

Project 4.4. Chemisorption on d-band metals

This project should not be attempted until after reading chapter 6 in addition to this chapter.

The Anderson±Grimley±Newns model described in section 4.5.2 is useful for correlating a large range of data, where trends can be analyzed. One such correlation is the eVect of strain in the substrate on reactivity. As shown by several authors (Hammer and Nùrskov 1997, Ruggerone et al. 1997, Mavrikakis et al. 1998) moderate strains are calculated to produce substantial changes in reactivity, with expanded surfaces having higher reactivity. Given the importance of d-band metals as catalysts, explore how this comes about, and how adsorption and dissociation energies can be correlated with movement of the center of the d-band relative to the Fermi level.