Past exam of the condensed matter physics course of the University of Oxford / 2022 / 1 / b

The primitive cubic lattice has $G_{hkl}=(2\pi /a)\sqrt{h^2+k^2+l^2}$. Combining this with Bragg's law gives the Bragg angleUsing $a=0.375\mathrm{nm}$ and $\lambda =0.154\mathrm{nm}$ yieldsThese are the full scattering angles; the Bragg angles themselves are half as large.

At each reflection, the argument of the atomic scattering factor isSubstitution into the given atomic scattering factor approximation givesThus the structure factors are $S_{110}=36.2620$ and $S_{111}=113.7340$. The powder diffraction multiplicity is $m_{110}=12$: choose which Miller index is zero in three ways and the signs of the other two in four ways. For $\{111\}$, the eight sign choices give $m_{111}=8$. With the specified instrumental and angular factors omitted, the integrated powder diffraction wave intensity is proportional to $m_{hkl}|S_{hkl}{|}^2$, soIt would be incorrect to use the same atomic scattering factors at the two different scattering vectors, or to omit the powder diffraction multiplicity.