Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 5.2, p. 491

Section Errors and aberrations: general discussion

W. Parrish,a A. J. C. Wilsonb and J. I. Langfordc

aIBM Almaden Research Center, San Jose, CA, USA,bSt John's College, Cambridge CB2 1TP, England, and cSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England Errors and aberrations: general discussion

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The relation between the lattice spacing d, the angle of incidence (Bragg angle) θ, and the wavelength λ is Bragg's law: [\lambda=2d\sin\theta. \eqno (]The lattice spacing d is related to the lattice parameters a, b, c, α, β, γ and the indices of reflection h, k, l. In the simple case of cubic crystals, the relation is [d^{-2}=a^{-2}(h^2+k^2+l^2), \eqno (]where a is the single lattice parameter. The general relation is [\eqalignno{ d^{-2} &=G^{-1}(abc){}^{-2}[A(hbc){}^2+B(kca){}^2+C(lab){}^2 \cr &\quad+ 2abc(Dkla+Elhb+Fhkc)], &(}]where a, b, c are the edges of the unit cell, and [A,\ldots,G] are the functions of the angles of the unit cell given in Table[link].

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Functions of the cell angles in equation ([link] for the possible unit cells

HexagonalMonoclinic (c unique)RhombohedralTriclinic
A 1 1 1 [\sin^2\alpha] [\sin^2\alpha]
B 1 1 1 [\sin^2\alpha] [\sin^2\beta]
C 1 [3\over4] [\sin^2\gamma] [\sin^2\alpha] [\sin^2\gamma]
D 0 0 0 [\cos^2\alpha-\cos\alpha] [\cos\beta\cos\gamma-\cos\alpha]
E 0 0 0 [\cos^2\alpha-\cos\alpha] [\cos\gamma\cos\alpha - \cos\beta]
F 0 [1\over2] [-\cos\gamma] [\cos^2\alpha-\cos\alpha] [\cos\alpha\cos\beta - \cos\gamma]
G 1 [3\over4] [\sin^2 \gamma] [1+2\cos^3\alpha-3\cos^2\alpha] [1+2\cos\alpha\cos\beta\cos\gamma] [-\cos^2\alpha-\cos^2\beta-\cos^2\gamma]

Differentiation of ([link] shows that the errors in the measurement of d are related to the errors in the measurement of λ and θ by the equation [(\Delta d)/d=(\Delta\lambda)/\lambda-\cot \theta(\Delta\theta). \eqno (]Wavelength and related problems are discussed in Section 5.2.2[link] and geometrical and other aberration problems in Section 5.2.3[link].

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