## Abstract

A simplified method of calculating the diffraction patterns of crystals showing mistake broadening is outlined, and used to correct three results published by Wilson in 1943. Wilson's model 3 (mistakes on {110} planes) is extended in the light of probable differences of the energy of interfaces between domains of different types. The apparent particle sizes obtained, when differences in interfacial energy are ignored, are: $$\text{for mistakes on {100} planes} 3N^\frac{1}{2}/2(h+k+l)\delta;$$ $$\text{for mistakes on {110} planes 3(2N)^\frac{1}{2}/4(2h+k)\delta;$$ $$\text{for mistakes on {111} planes 3(3N)^\frac{1}{2}/8h\delta \text{if} h \geqslant k+l,$$ $$3(3N)^\frac{1}{2}/4(h+k+l)\delta \text{if} h < k+l;$$ where $\delta$ is the probability of a mistake per unit length, h, k, l are the positive values of the indices of reflexion arranged in the order h $\geqslant$ k $\geqslant$ 1, and N = h$^2$+k$^2$+l$^2$. When differences in interfacial energy are taken into account the apparent particle size for mistakes on {100} planes becomes $$2N^\frac{1}{2}/{2(h+k+l-u)\delta'+(h+k+l+u)\delta''},$$ where u is the unpaired index, and that for mistakes on {110} planes becomes $$(2N)^\frac{1}{2}/{4h\delta'+2(h+k)\delta''}$$ when h is the unpaired index, and $$(2N)^\frac{1}{2}/{2(h+k)\delta'+(3h+k)\delta''}$$ when k or l is the unpaired index. All {111} interfaces have the same energy in AuCu$_3$, so the expressions for mistakes on {111} planes are not altered.

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