Errors and aberrations: general discussion

Parrish, W.; Wilson, A. J. C.; Langford, J. I.

doi:10.1107/97809553602060000596

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

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

Section 5.2.1.2. Errors and aberrations: general discussion

W. Parrish,^a ^‡ A. J. C. Wilson^b ^‡ and J. I. Langford^c

^a IBM 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

5.2.1.2. 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: $[\lambda=2d\sin\theta. \eqno (5.2.1.1)]$ 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 $[d^{-2}=a^{-2}(h^2+k^2+l^2), \eqno (5.2.1.2)]$ where a is the single lattice parameter. The general relation is $[\eqalignno{ d^{-2} &=G^{-1}(abc){}^{-2}[A(hbc){}^2+B(kca){}^2+C(lab){}^2 \cr &\quad+ 2abc(Dkla+Elhb+Fhkc)], &(5.2.1.3)}]$ 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 5.2.1.1.

Table 5.2.1.1| top | pdf |
Functions of the cell angles in equation (5.2.1.3) for the possible unit cells

Function	Cell
Function	Cubic tetragonal orthorhombic	Hexagonal	Monoclinic (c unique)	Rhombohedral	Triclinic
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 (5.2.1.1) shows that the errors in the measurement of d are related to the errors in the measurement of λ and θ by the equation $[(\Delta d)/d=(\Delta\lambda)/\lambda-\cot \theta(\Delta\theta). \eqno (5.2.1.4)]$ Wavelength and related problems are discussed in Section 5.2.2 and geometrical and other aberration problems in Section 5.2.3.

References

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