X-ray diffraction.

If an X-ray beam is directed at a row of equally spaced atoms, as represented in the top picture at the left, each atom becomes a source of scattered waves spreading spherically and reinforce in certain directions to produce the zero-, first-, second-, and higher order diffracted beams.

A row of atoms has infinite rotational symmetry along the axes passing through it. Therefore in three dimensions those reinforcement directions of different order are represented with the cones of corresponding order as it is shown in the picture at the bottom left.
Two-dimensional array of equally spaced atoms consequently produces scattered waves which reinforce along the lines of the cross section of two sets of corresponding cones oriented along the coordinate axes. In three-dimensional case the set of the cones oriented along the third coordinate axes causes the reinforcement of scattered waves (constructive interference) to occur at certain locations. Those locations are the points of cross section of all three sets of cones, oriented along three coordinate axes of the crystal.

W. L. Bragg made the assumption that the sheets of planes of atoms in the crystal behave like perfect reflectors for X-rays, so that interference was caused by multiple reflections from these planes in the same manner as a stack of glass plates produces interference by multiple reflection. Let us consider a set of partially reflecting planes spaced at an interval d as in the figure below..

We wish to know the path difference between two reflected rays:

D = LM + MN = 2d sinq

In order to get constructive interference, D must be equal to nl (n is the called the order of interference and is equal to 0, ±1, ±2 etc.). Therefore:

nl = 2d sinq

This is Bragg Law.

On diffraction, part of the energy in the incident beam with wave-vector k_i, will be scattered into a beam marked k_r in the figure. How does k_i differ from k_r? It is only the component of k_i, perpendicular to the planes AA' and BB' which is changed. The change of wave-vector upon diffraction is perpendicular to the planes and its magnitude for the diffraction of the first order is :

2(2p/l)sinq = 2k sinq = 2p/d

Where wave-vector k = 2p/l, by definition.

Term 2p/d is equal to the magnitude of the reciprocal lattice vector. Therefore in the X-ray diffraction experiment where we control the magnitude and the direction of incident wave-vector we can directly map the reciprocal lattice of the crystal.

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