Problem: Finite Size Effects

    Consider a finite size, simple cubic crystal of atoms, with lattice spacing a.

  (a) For , plot the -dependence of the intensity of the scattered X-rays (defined as , where is the amplitude of the scattered radiation) around the (0,0,1) point in reciprocal space. Show that there is a peak at (0,0,1) and that the peak width is finite. How is the width related to the size of the crystal ? How does the height of the peak depend on ? How does the integrated intensity behave as a function of ? Assume .

(b) What happens to the line width and the peak height if we have a collection of small crystallites, which are all oriented parallel to each other, but their positions are random?