Equally Spaced Points Along a Line

DIFFRACTION FROM EQUALLY SPACED POINTS ALONG A LINE 

To accomplish our task, we must formulate an expression for the scattering of waves by a point lattice. Let us begin by considering the scattering of waves by a one-dimensional periodic lattice, a line of points separated by constant distances a, that is, points related in a periodic manner by a vector a, where a has a magnitude comparable to that of X. 

In general, (a s) will not be integral, so 2mt(a s) will be multiples of some nonzero phase angle. F-s will therefore be the summation of a long series of N waves, each out of phase with the one before and after by a constant amount. 

as we saw in Chapter 3, is a Dirac delta function, which always sums to zero except when (a s) becomes integral, that is, when a s is a multiple of X. Because a is invariant, only when o and k articulate certain relationships that result in specific directions for s will that be true. When s takes on those unique directions, then waves scattered by all N points have relative phases of zero and thus constructively interfere. The amplitudes of all of the N scattered waves then add arithmetically. 

This is a one-dimensional illustration of Bragg s law. The most important consequence of this example is to show that the diffraction pattern of a periodic distribution of points is 

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