All Problems Diffraction of Light
Problem 5.153
A beam of X-rays impinges on a three-dimensional rectangular array whose periods are \(a, b,\) and \(c .\) The direction of the incident beam coincides with the direction along which the array period is equal to \(a\). Find the directions to the diffraction maxima and the wavelengths at which these maxima will be observed.
5.153. Suppose \(\alpha, \beta,\) and \(\gamma\) are the angles between the direction to the diffraction maximum and the directions of the array along the periods \(a, b,\) and \(c\) respectively. Then the values of these angles can be found from the following conditions: \(a(1-\cos \alpha)=k_{1} \lambda\) \(b \cos \beta=k_{2} \lambda,\) and \(c \cos \gamma=k_{3} \lambda .\) Recalling that \(\cos ^{2} \alpha+\cos ^{2} \beta+\) \(+\cos ^{2} \gamma=1,\) we obtain \[ \lambda=\frac{2 k_{1} / a}{\left(k_{1} / a\right)^{2}+\left(k_{2} / b\right)^{2}+\left(k_{3} / c\right)^{2}} \]