Diffraction under nonideal conditions

Before going any further, it is important to stop and consider with some care the derivation of the Bragg law given in Sec. 3-2 in order to understand precisely under what conditions it is strictly valid. In our derivation we assumed certain ideal conditions, namely a perfect crystal and an incident beam composed of perfectly parallel and strictly monochromatic radiation. These conditions never actually exist, so we must determine the effect on diffraction of various kinds of departure from the ideal. 

In particular, the way in which destructive interference is produced in all directions except those of the diffracted beams is worth examining in some detail, both because it is fundamental to the theory of diffraction and because it will lead us to a method for estimating the size of very small crystals. We will find that only the infinite crystal is really perfect and that small size alone, of an otherwise perfect crystal, can be considered a crystal imperfection. 

The condition for reinforcement used in Sec. 3-2 is that the waves involved must differ in path length, that is, in phase, by exactly an integral number of wavelengths. But suppose that the angle Q in Fig. 3-2 is such that the path difference for rays scattered by the first and second planes is only a quarter wavelength. These rays do not annul one another but, as we saw in Fig. 3-1, simply unite to form a beam of smaller amplitude than that formed by two rays which are completely in phase. How then does destructive interference take place The answer lies in the contributions from planes deeper in the crystal. Under the assumed 

Suppose, for example, that the crystal has a thickness t measured in a direction perpendicular to a particular set of reflecting planes (Fig. 3-14). Let there be m + 1) planes in this set. We will regard the Bragg angle 0 as a variable and call 6 the angle which exactly satisfies the Bragg law for the particular values of A and d involved, or 

In Fig. 3-14, rays A, D. M make exactly this angle Og with the reflecting planes. Ray D, scattered by the first plane below the surface, is therefore one 

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