Lattice Structure and Miller Indices

The coordinate system we will use to describe the molecule surface system has its z-axis perpendicular to the surface, i.e., in the direction of the surface normal (see Fig. 2.1). The crystal shape is determined by the unit cell, which is defined by the values of a, b, and c specifying the length along three axes a, b, and c. For the simple lattices considered here the axes are perpendicular to each other and have therefore been denoted x, y, and z. In order to characterize the directions and atomic planes in the crystal one introduces various notations. 

Here we shall just mention the so-called Miller indices, which will be denoted h, k, 1). These three indices indicate the intercepts of a plane with the crystal axes. Additional indices are needed in noncubic lattices and yet another notation (i jk) is used to characterize the direction in a lattice. For a discussion of this the reader is referred to more specialized literature. 

FIGURE 2.1 The coordinate system used to describe the dynamics is placed with the z-axis perpendicular to the surface. 

FIGURE 2.2 The planes (100) (plane 1) and (111) (plane 2) for a simple cubic crystal. 

Returning to the Miller indices, consider, for example, a simple cubic lattice (Fig. 2.2) we see that plane 1 cuts the x, y and z-axes at a, oo and oo respectively, whereas plane 2 cuts the axes at a, b and c. Thus the Miller indices (h, k, 1) [15] are defined as  

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