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,a, b, and c.c . The direction of the incident beam coincides with the direction along which the array period is equal to aa. Find the directions to the diffraction maxima and the wavelengths at which these maxima will be observed.

Reveal Answer
5.153. Suppose α,β,\alpha, \beta, and γ\gamma are the angles between the direction to the diffraction maximum and the directions of the array along the periods a,b,a, b, and cc respectively. Then the values of these angles can be found from the following conditions: a(1−cos⁡α)=k1λa(1-\cos \alpha)=k_{1} \lambda bcos⁡β=k2λ,b \cos \beta=k_{2} \lambda, and ccos⁡γ=k3λ.c \cos \gamma=k_{3} \lambda . Recalling that cos⁡2α+cos⁡2β+\cos ^{2} \alpha+\cos ^{2} \beta+ +cos⁡2γ=1,+\cos ^{2} \gamma=1, we obtain λ=2k1/a(k1/a)2+(k2/b)2+(k3/c)2 \lambda=\frac{2 k_{1} / a}{\left(k_{1} / a\right)^{2}+\left(k_{2} / b\right)^{2}+\left(k_{3} / c\right)^{2}}
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