General expression for the Fourier coefficients

A tetragonal lattice is the combination of two perpendicular one-dimensional lattices with the same lattice constant and having a reflection symmetry, as shown in Fig. 6.5. Therefore, we start with a one-dimensional case. The tunneling conductance from the nth atom is g(x — na, z). The total tunneling conductance from all the atoms is 

Apparently, it is a periodic function with periodicity a. Therefore, it can be expressed as a Fourier scries. 

As shown, due to the infinite sum in Eq. (6.21), the integration extends over the entire x axis. 

To construct an image including the lowest nontrivial Fourier components, only three terms in the Fourier series are significant. Those are Go(z), G i(z), and Gdz)- Because of the reflection symmetry of the conductance function g(x,z), the last two Fourier coefficients are equal, and are denoted as Gi(z). Up to this term. 

Following the general relation between current images and topographic images, Eq. (5.7), the topographic image is 

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The general expression for the Fourier coefficient for the atom j is explicitly given by ... 

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