InterviewSolution
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InterviewSolution
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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. |
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Answer» 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`. |
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