Crystal surface simple model

In the next section we describe a very simple model, which we shall term the crystalline model , which is taken to represent the real, complicated crystal. Some additional, more physical, properties are included in the later calculations of the well-established theories (see Sect. 3.6 and 3.7.2), however, they are treated as perturbations about this basic model, and depend upon its being a good first approximation. Then, Sect. 2.1 deals with the information which one would hope to obtain from equilibrium crystals — this includes bulk and surface properties and their relationship to a crystal s melting temperature. Even here, using only thermodynamic arguments, there is no common line of approach to the interpretation of the data, yet this fundamental problem does not appear to have received the attention it warrants. The concluding section of this chapter summarizes and contrasts some further assumptions made about the model, which then lead to the various growth theories. The details of the way in which these assumptions are applied will be dealt with in Sects. 3 and 4. 

We now describe a relatively simple MD model of a low-index crystal surface, which was conceived for the purpose of studying the rate of mass transport (8). The effect of temperature on surface transport involves several competing processes. A rough surface structure complicates the trajectories somewhat, and the diffusion of clusters of atoms must be considered. In order to simplify the model as much as possible, but retain the essential dynamics of the mobile atoms, we will consider a model in which the atoms move on a "substrate" represented by an analytic potential energy function that is adjusted to match that of a surface of a (100) face-centered cubic crystal composed of atoms interacting with a Lennard-Jones... 

Figure 2.4 A simple model depicting the various types of defects formed on a single-crystal oxide surface.

The geochemical fate of most reactive substances (trace metals, pollutants) is controlled by the reaction of solutes with solid surfaces. Simple chemical models for the residence time of reactive elements in oceans, lakes, sediment, and soil systems are based on the partitioning of chemical species between the aqueous solution and the particle surface. The rates of processes involved in precipitation (heterogeneous nucleation, crystal growth) and dissolution of mineral phases, of importance in the weathering of rocks, in the formation of soils, and sediment diagenesis, are critically dependent on surface species and their structural identity. 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

Networks of steps, seen in STM observations of vicinal surfaces on Au and Pt (110), are analyzed. A simple model is introduced for the calculation of the free energy of the networks as function of the slope parameters, valid at low step densities. It predicts that the networks are unstable, or at least metastable, against faceting and gives an equilibrium crystal shape with sharp edges either between the (110) facet and rounded regions or between two rounded regions. Experimental observations of the equilibrium shapes of Au or Pt crystals at sufficiently low temperatures, i.e. below the deconstruction temperature of the (110) facet, could check the validity of these predictions. 

For two and three dimensions, it provides a crude but useful picture for electronic states on surfaces or in crystals, respectively. Free motion within a spherical volume gives rise to eigenfunctions that are used in nuclear physics to describe the motions of neutrons and protons in nuclei. In the so-called shell model of nuclei, the neutrons and protons fill separate s, p, d, etc orbitals with each type of nucleon forced to obey the Pauli principle. These orbitals are not the same in their radial shapes as the s, p, d, etc orbitals of atoms because, in atoms, there is an additional radial potential V(r) = -Ze2/r present. However, their angular shapes are the same as in atomic structure because, in both cases, the potential is independent of 0 and (f>. This same spherical box model has been used to describe the orbitals of valence electrons in clusters of mono-valent metal atoms such as Csn, Cu , Na and their positive and negative ions. Because of the metallic nature of these species, their valence electrons are sufficiently delocalized to render this simple model rather effective (see T. P. Martin, T. Bergmann, H. Gohlich, and T. Lange, J. Phys. Chem. 95, 6421 (1991)). 

Tlhe importance of zeolites in research on heterogeneous catalysis is A based mainly on the fact that the structure of the active surface is a defined part of the crystal structure and does not represent a more or less severe lattice defect as most catalyst surfaces do. The crystal structure, and therefore the structure of the zeolite surface, can be determined by x-ray diffraction. Knowledge of this structure allows the construction of simple models of the distribution of electric fields in the holes of the zeolite by which wide ranges of experimental results can be explained, as is shown by the pioneering work of Barrer 1-5) and Kiselev 6-9) on calculation of the heats of adsorption of various substances. 

Section III deals with the surface excitations of the anthracene crystal, confined in the first (SJ, second (S2) and third (S3) (001) lattice planes. The experimental observations are briefly summarized. A simple model shows how the fast radiative decay arises and how the underlying bulk reflection modulates this superradiant emission, as well as why gas condensation on the crystal surface strongly narrows this emission, thus accounting for the observed structures. An intrinsic process is proposed to explain the surface-to-bulk relaxation at low temperatures, observed in spite of the very weak surface-to-bulk coupling for k 0 states. 

The following describes results of three, relatively simple chemical reactions involving hydrocarbons on model single crystal metal catalysts that illustrate this general approach, namely, acetylene cyclotrimerization and the hydrogenation of acetylene and ethylene, all catalyzed by palladium. The selected reactions fulfdl the above conditions since they occur in ultrahigh vacuum, while the measured catalytic reaction kinetics on single crystal surfaces mimic those on reahstic supported catalysts. While these are all chemically relatively simple reactions, their apparent simplicity belies rather complex surface chemistry. 

Abstract The surfaces of model metal oxides offer many fundamental examples where the outcome of a specific chemical reaction might be linked to the surface structure and local electronic properties. In this work the reaction of simple molecules such as ammonia, alcohols, carboxylic and amino acids is studied on two metal oxide single crystals rutile TiO CllO) and (001) and fluorite UOj(l 11). Studies are conducted with XPS, TPD, and Plane Wave Density Functional Theory (DFT). The effect of surface structure is outlined by comparing the TiOj(llO) rutile surface to those of TiOjCOOl), while the effect of surface point defects is mainly discussed in the case of stoichiometric and substoichiometric UOjClll). 

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