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 wavelength at which these maxima will be observed.

Explanation

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 (call them `x, y, & z` axes). Then the value of these angles can be found from the following familiar condiations
`a(1 - cos alpha) = k_(1) lambda`
`b cos beta = k_(2) lambda` and `c cos gamma = k_(3) lambda`
where `k_(1), k_(2), k_(3)` are whole numbers `(+, -,` or `0`)
(These formules are, in effect, Laue equations, see any text book on modern physics). Squaring and adding we get on using `cos^(2) alpha + cos^(2) beta + cos^(2) gamma = 1`
`2-2 cos alpha [((k_(1))/(a))^(2) + ((k_(2))/(b))^(2) + ((k_(3))/(c))^(2)]lambda^(2) = (2k_(1)lambda)/(a)`
Thus `lambda = (2k_(1)a)/([[k_(1)//a)^(2) +(k+_(2)//a)^(2) +(k_(3)//a)^(2)]]`.
Knowing `a, b, c` and the interger `k_(1), k_(2), k_(3)` we can find `alpha, beta, gamma` as well as `lambda`.