MatSE 405: Microstructure Determination mselogo.jpg

Problem set #6; due Thursday February 28, in class.
  1. In lecture, I used geometry in the complex plane to show that the diffraction intensity from two point sources is I = 4A12cos2(e/2), where A1 is the amplitude of scattering from one point source and e is the phase difference between the two point sources. Derive the same result using algebra. Start with the sum A = A1+A1exp(ie), expand the exponential in terms of cos(e) and sin(e), calculate I = |A|2, and use the half-angle formula, cos(e)+1 = 2cos2(e/2)
  2. Modify diffract_1d_lattice.m to create a plot of the diffraction intensity as a function of sinq for a 1D arrangement of point sources with N=5, l = 0.5 microns, a=0.1 mm. What is the peak intensity? What is the full-width-half-maximum of the peaks?
  3. Use a geometrical construction in the complex plane to illustrate the what happens at a maximum and at a minimum of the diffraction intensity. (Do this by paper-and-pencil but you can look at complex_plane_animate.m if you find it helpful.)
  4. Show how to position this object, a lens of focal length f=100 mm, and a CCD camera to observe a sharp diffraction pattern. What will be the distance between the diffraction intensities on the CCD camera?
  5. (a) Derive an expression for the positions and relative intensities of diffraction peaks from a 1D crystal (lattice constant a) with a basis of 2 identical point sources with separation b. Make sure that you understand that the diffraction pattern of this 1D crystal is the diffraction pattern from the 1D lattice multiplied by the diffraction from the basis of two point sources.

File translated from TEX by TTH, version 3.74.
On 21 Feb 2008, 09:21.