Mathematical formulation of the contrast from a crystal defect

2 Mathematical formulation of the contrast from a crystal defect 

The potential at a point r in a perfect crystal is given by K(r). If the crystal is now deformed by the presence of a defect, then the potential at point r will be different from V(t) because the potential at a point depends on the positions of the atoms in the neighborhood. If we assume that the deformation is not too severe (i.e., the deformation is a slowly varying function of position), then the potential at point r in the deformed crystal will be equivalent to the potential at point (r—R) in the undeformed crystal. R, in general, is a function of position and is called the displacement function. Thus, in a deformed crystal, the potential at a point r is given by 

As before, this can be expressed formally as a Fourier series  

The Fourier coefficients are now functions of r instead of being constants. 

For the two-beam approximation there is only one operating reflection g, and for the deformed crystal the corresponding Fourier coefficient is 

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