Surfaces with tetragonal symmetry

Fig. 5.3. Surface with tetragonal symmetry.. The example of Cu(OOl) is shown. The lattice in real space. The large dots represent the nuclei in the top layer. The small dots represent the nuclei in the second layer. The reciprocal space is also shown.

An example of surfaces with tetragonal symmetry is the Cu(OOl) surface, as shown in Fig. 5.3. The top-layer nuclei form a two-dimensional square lattice on the x,y plane with lattice constant a. The origin of the coordinate system is chosen to be at one of the top-layer nuclei. The +z direction is defined as pointing into the vacuum. The reciprocal lattice is also shown in Fig. 5.3, with a lattice constant of... 

Fig. 6.5. Close-packed surface with tetragonal symmetry, (a) The square lattice in real space. There is an atom on each lattice point, (b) The reciprocal space.

This treatment can be extended immediately to the case of surfaces with tetragonal symmetry. The Fourier coefficients for the tunneling conductance of a tetragonal lattice is... 

Table 6.2. Independent-state model corrugation amplitudes for surfaces with tetragonal symmetry ...

By comparing this with the results for surfaces with tetragonal symmetry, it is clear that the only difference is the factor of 8 in Eq. (6.29) is replaced by 9/2. With the same lattice constant, the corrugation amplitude of a surface with hexagonal symmetry is smaller than that for a surface with tetragonal symmetry by a factor of 9/16=0.5625. The decay constant of the corrugation is... 

Since the surfaces of crystals have specific symmetries (usually triangular, square, or tetragonal) and indenters have cylindrical, triangular, square, or tetragonal symmetries, the symmetries rarely match, or are rotationally misaligned. Therefore, the indentations are often anisotropic. Also, the surface symmetries of crystals vary with their orientations relative to the crystallographic axes. A result is that crystals cannot be fully characterized by single hardness numbers. 

The symmetry of a material is also only partially revealed by the shape of etch pits on a crystal surface. These are created when a crystal begins to dissolve in a solvent. Initial attack is at a point of enhanced chemical reactivity, often where a dislocation reaches the surface. A pit forms as the crystal is corroded. The shapes of the pits, called etch figures, have a symmetry corresponding to one of the of 10 two-dimensional plane point groups. This will be the point group that corresponds with the symmetry of the face. An etch pit on a (100) face of a cubic crystal will be square, and on a (101) face of a tetragonal crystal will be rectangular. 

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