Lattice, surface

Diffraction is not limited to periodic structures [1]. Non-periodic imperfections such as defects or vibrations, as well as sample-size or domain effects, are inevitable in practice but do not cause much difSculty or can be taken into account when studying the ordered part of a structure. Some other forms of disorder can also be handled quite well in their own right, such as lattice-gas disorder in which a given site in the unit cell is randomly occupied with less than 100% probability. At surfaces, lattice-gas disorder is very connnon when atoms or molecules are adsorbed on a substrate. The local adsorption structure in the given site can be studied in detail. 

In LEED experunents, the matrix M is detennined by visual inspection of the diffraction pattern, thereby defining the periodicity of the surface structure the relationship between surface lattice and diffraction pattern will be described in more detail in the next section. 

In this section, we concentrate on the relationship between diffraction pattern and surface lattice [5], In direct analogy with the tln-ee-dimensional bulk case, the surface lattice is defined by two vectors a and b parallel to the surface (defined already above), subtended by an angle y a and b together specify one unit cell, as illustrated in figure B1.21.4. Withm that unit cell atoms are arranged according to a basis, which is the list of atomic coordinates within drat unit cell we need not know these positions for the purposes of this discussion. Note that this unit cell can be viewed as being infinitely deep in the third dimension (perpendicular to the surface), so as to include all atoms below the surface to arbitrary depth. 

The diffraction of low-energy electrons (and any other particles, like x-rays and neutrons) is governed by the translational syimnetry of the surface, i.e. the surface lattice. In particular, the directions of emergence of the diffracted beams are detemiined by conservation of the linear momentum parallel to the surface, bk,. Here k... 

A. Patrykiejew, S. Sokolowski, K. Binder, Phase transitions in adsorbed layers formed on crystals of square and rectangular surface lattice, Surf. Sci. Reports 37, 207 (2000). 

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

Figure 6.8 Definition and properties of the two-dimensional reciprocal lattice a, and a2 are base vectors of the surface lattice, and a and a2 are the base vectors of the reciprocal lattice. The latter is equivalent to the LEED pattern.

Figure 6.8 summarizes the most important properties of the reciprocal lattice. It is important that the base vectors of the surface lattice form the smallest parallelogram from which the lattice may be constructed through translations. Figure 6.9 shows the five possible surface lattices and their corresponding reciprocal lattices, which are equivalent to the shape of the respective LEED patterns. The unit cells of both the real and the reciprocal lattices are indicated. Note that the actual dimensions of the reciprocal unit cell are irrelevant only the shape is important. 

Figure 6.9 The five different surface lattices, base vectors of the real and reciprocal lattices, and the corresponding LEED patterns.

Surface lattice structure Density of active surface atoms and reactivity of the surface determined by the crystalline orientation of silicon/electrolyte interface... 

Figure 24. Schematic illustrations of the conditions of surface lattice structure (a) amorphous-like surface with no identity of orientation, (b) surface with kinks, steps and terraces characteristic of certain crystalline orientation and (c) surface with no identity of the lattice structure of the crystal due to the coverage of an amorphous oxide film. 

The dissolution of quartz is accelerated by bi- or multidentate ligands such as oxalate or citrate at neutral pH-values. The effect is due to surface complex formation of these ligands to the Si02-surface (Bennett, 1991). In the higher pH-range the dissolution of quartz is increased by alkali cations (Bennett, 1991). Most likely these cations can form inner-spheric complexes with the =SiO groups. Such a complex formation is accompanied by a deprotonation of the oxygen atoms in the surface lattice (see Examples 2.4 and 5.1). This increase in C H leads to an increase in dissolution rate (see Fig. 5.9c). 

This idea can be extended to Fe(II) centers within or near the surface lattice of Fe(II) silicates. White and Yee (1985) have shown that the equilibrium... 

The specific interactions between a water molecule and the metal atoms. The pair interaction H2O-M is further modified by the metal surface lattice structure and by defects and the electrons in the metal, as discussed earlier. 

In addition, the direct electrostatic interaction between adsorbates has been treated . At intermediates distances of the order of a surface lattice constant, Norskov, Holloway, and Lang report that this interaction can give rise to substantial (> 0.1 eV) interaction energies, when both adsorbates in question induce electron transfer to or from the surface or have a large internal electron transfer. 

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