Intensity of X-ray Diffraction by a One-dimensionally Disordered Crystal

• Published in: Journal of the Physical Society of Japan Vol. 9; no. 2; p. 177
• Main Authors: Kakinoki, Jiro, Komura, Yukitomo
• Format: Journal Article
• Language: English
• Published: Tokyo The Physical Society of Japan 01-03-1954
• ISSN: 0031-9015; 1347-4073

The general intensity equation for X-rays diffracted by a one-dimensionally disordered crystal with any value of the correlation range s i.e. I=NSpur VF+∑\limitsn=1N-1(N-n)Spur VFQn+conj. was applied to the case of the close-packed structure. The obtained result is I=V0V0*\Bigg{N∑ν=1lcν0\frac(1-xν02)cosρν+2xν0sinρνsin(\varphi-θν)1+xν02-2xν0cos(\varphi-θν) +∑ν=1lcν0(Hν(1)cosρν+Hν(2)sinρν)\Bigg} for 2h+k=3m±1, in which V0 is the layer from factor, N the number of layers, l=2s-1, xν=xν0eiθν, which are the l roots of the Characteristic equation of det[x2\ bi1+xαβ+{αβ+(\ bi1-α)βu}{αβ-(\ bi1-α)βu}]=0, cν=cν0eiρν which are the roots of the l simultaneous equations of ∑\limitsi=1lxiλci=\frac32sp. xλ-\frac12 (λ: 0.1,..., l-1) and two H's the higher terms which, when |xν|=1, give the Laue function whose maximum is at \varphi=θν. The characteristic equation is derived from the general difference equation xn+xn-1αβ+xn-2{αβ+(\ bi1-α)βu}{αβ-(\ bi1-α)βu}=hαβ, from which we can easily obtain the difference equations that Wilson and Jagodzinski derived in the cases of s=2 and 3 by the use of rather complicated genealogical tables. We showed the correspondence of our results in the cases of s=2 and 3 with those obtained by Wilson, Jagodzinski, Hendricks and Teller. The above matrices are those of l/2-th order. The elements in xn are the probabilities of finding the same kind of layer at the (j+n)-th position as that at the j-th position and h=x0. Those in αβ and (\ bi1-α)βu are the probabilities of finding the layer after the other layers in the cubic and hexagonal manners respectively. When 2h+k=3m, I=V0V0*(sin2N\varphi/2)/(sin2\varphi/2) without giving any diffuse line.