Example Form Factor of a Parallelepiped

Consider diffraction by a single transparent parallelepiped with edge lengths A, B, C, Fig. 5.11a. 

Assume density p=const within the parallelepiped and p=0 outside of its volume. According to Eq. 5.17, the scattering amplitude is 

The plot of scattered field amplitude is shown in the upper part of Fig. 5.1 lb. It is the so-called sine-integral function. The scattering intensity is shown in the lower part of the figure. Integrating over the y and z co-ordinates we obtain the three-dimensional scattering amplitude F(q)= pFA AyA and intensity 7(q)= p F2(A,AyA,)2. 

Note that, for infinitely thick parallelepiped (A oo), there is no diffraction, only directly transmitted beam is left and the integral becomes 8-function. Generally, the larger parallelepiped dimensions the narrower is the central peak. We shall come back to this point when discussing the diffraction on thin layers of a smectic A liquid crystal. 

In the top left sketch, the parallelepiped is degenerated into the infinitely thin plane with dimensions A oo, B- oo, C 8(z). All its density is concentrated in 

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