Crystal Imperfection of the Second Kind

For a one-dimensional crystal, Equation (3.35) can be rewritten, by grouping the first neighbor pairs, second neighbor pairs, etc., separately, as 

The placement of atoms in the one-dimensional lattice is characterized by the probability function px(x) specifying that the probability of finding a nearest-neighbor pair at a distance between x and x + dx is equal to p (x) dx. The function p (x) is normalized to unity and the mean distance d between first neighbors is given by 

Equation (3.47) implies that pi(x) is defined only for positive values of x. We also define a function p (x) by 

The probability p2(x) dx of finding a second neighbor pair at a distance between x and x -f- dx is then equal to 

In going from the second to the third member of (3.49), p (x) is regarded equal to zero for negative values of x. By a similar reasoning it can be seen that the probability function pm(x) describing the distance between mth nearest neighbors is given by 

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