Bravais lattice, surface

Figure A.2 The five surface Bravais lattices square, primitive rectangular, centered rectangular, hexagonal, and oblique.

Problem 2.1. Draw the arrangement of atoms in the (111) surface of (a) a bulk centered cubic (bcc) crystal (b) a face centered cubic (fee) crystal. Show the surface unit cell and indicate the [110] and [112] axes in both cases. What symmetry do the surface Bravais lattices have ... 

Superlattice A surface Bravais lattice characterizing a surface reconstruction or an ordered arrangement of adsorbed atoms. 

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

The common periodic structures displayed by surfaces are described by a two-dimensional lattice. Any point in this lattice is reached by a suitable combination of two basis vectors. Two unit vectors describe the smallest cell in which an identical arrangement of the atoms is found. The lattice is then constructed by moving this unit cell over any linear combination of the unit vectors. These vectors form the Bravais lattices, which is the set of vectors by which all points in the lattice can be reached. 

In two dimensions, five different lattices exist, see Fig. 5.6. One recognizes the hexagonal Bravais lattice as the unit cell of the cubic (111) and hep (001) surfaces, the centered rectangular cell as the unit cell of the bcc and fee (110) surfaces, and... 

Compared with the problem in three-dimensional space, which has 14 Bravais lattices and 230 space groups, the problem of surface symmetry is tmly a dwarf It only has 5 Bravais lattices and 17 different groups. The five Bravais lattices are listed in Table E.l. 

We now remove the inconvenience of the translation subgroup, and consequently the Bravais lattice, being infinite by supposing that the crystal is a parallelepiped of sides Aja,-where ay, j 1,2,3, are the fundamental translations. The number of lattice points, N1N2N3, is equal to the number of unit cells in the crystal, N. To eliminate surface effects we imagine the crystal to be one of an infinite number of replicas, which together constitute an infinite system. Then... 

Crystalline surfaces can be divided into five Bravais lattices (Fig. 8.3) according to their symmetry. They are characterized by the lattice angle a and the lengths of the lattice vectors ai and a2. The position vectors of all individual surface atoms can be indicated by... 

Crystalline surfaces can be classified using the five two-dimensional Bravais lattices and a basis. Depending on the surfaces structure, the basis may include more than just the first surface layer. The substrate structure of a surface is given by the bulk structure of the material and the cutting plane. The surface structure may differ from the substrate structure due to surface relaxation or surface reconstruction. Adsorbates often form superlattices on top of the surface lattice. 

If the 10 point groups allowed are arranged in nonredundant patterns allowed by the five 2D Bravais lattices, 17 unique two-dimensional space groups, called plane groups, are obtained (Fedorov, 1891a). Surface structures are usually referred to the underlying bulk crystal structure. For example, translation between lattice points on the crystal lattice plane beneath and parallel to the surface (termed the substrate) can be described by an equation identical to Eq 1.10 ... 

A solid surface is intrinsically an imperfection of a crystalline solid by destroying the three-dimensional (3D) periodicity of the structure. That is, the unit cell of a crystal is usually chosen such that two vectors are parallel to the surface and the third vector is normal or oblique to the surface. Since there is no periodicity in the direction normal or oblique on the surface, a surface has a 2D periodicity that is parallel to the surface. By considering the symmetry properties of 2D lattices, one obtains the possible five 2D Bravais lattices shown in Figure 2. The combination of these five Bravais lattices with the 10 possible point groups leads to the possible 17 2D space groups. The symmetry of the surface is described by one of these 17 2D symmetry groups. 

Figure 21.31 shows a two-dimensional representation of this kind of line defect. Plane defects are usually seen at the surfaces of crystals or at interfaces between two smaller crystals in a larger piece of solid material, as seen in Figure 21.32. Plane defects can also exist between two different Bravais lattices of the same compound.

Because of the regularity of atoms and molecules in a crystal, many of the possible planes of atoms can diffract X rays. We use a system called Miller indices to label which plane of atoms is diffracting X rays, and different Bravais lattices have different planes, with characteristic Miller indices, that diffract. In this way, we can differentiate unit cells by their characteristic X-ray diffraction patterns. In the next chapter, we will find that Miller indices are also useful in describing the orientation of the surface of the crystal. 

Besides the 2D Bravais lattice (2.3) in radius vector space, it is worthwhile to introduce a lattice in the space of wave vectors parallel to the surface. A basis in such a space can be determined as... 

In the above example of the Ni(100)-0 surface, the two primitive translation vectors ai 2 of the corresponding point lattice (Figure 4.8f) were chosen intuitively. Yet, their definition is by no means unique, as shown by the three examples displayed in Figure 4.9a for a hexagonal point lattice. The request for primitive vectors reflects the fact that by = nia-i + nai (m, n = —oo,..., -Foo) aU lattice points (and only them) must be reached (Bravais lattice). The arrangement of the lattice points has... 

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