513.001 Molecular and Solid State Physics

Ewald's sphere - 2

A simple cublic crystal with a lattice constant of a=3Å is analyzed by x-ray diffraction. The crystal is oriented such that the (100) plane is on the surface. An x-ray beam penetrates with an angle of 30° relative to the surface of the crystal. The detector scans in the (001) plane.

(a) Make a sketch of the experimental setup.
(b) Draw the Ewald-construction (here: a sphere) under the assumption that the primary x-ray beam has a wavelength of 1.57Å
(c) Which wavelength is requires to catch the (220) reflection peak? How big is the scattering angle 2$\theta$? Which angle does the primary x-ray beam enclose to the (220) plane? Which angle does the reflected x-ray beam enclose to the (220) plane?
(d) Show that Bragg's law is fulfilled for the diffraction on the (220) plane by calculating the interplanar distance $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$

Hint: Draw the real space lattice with the lattice points u,v,0 ( 0 means zero) together with the reciprocal space lattice points h,k,0 (0 means zero) by using the same origin of both lattices.