Abstract
The highly ordered matter at a molecular atomic level gives origin to crystals. In the history of modern science, the proof of the crystalline (ordered) form of matter occurred simultaneously with the demonstration that X-radiation was an electromagnetic radiation. One of the most important areas of applied physics, X-ray crystallography, was started by the diffraction of X-rays in a NaCl crystal (Bragg and Bragg 1913; Max von Laue 1913). To it we owe our current knowledge on the atomic structure of materials and proteins, including the structure of the double helix of deoxyribonucleic acid (DNA) (Franklin and Gosling 1953; Watson and Crick 1953; Wilkins et al. 1953).
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- 1.
The smallest of the crystal volume and the volume defined by radiation coherence lengths. For the sake of textual simplification, we will refer to diffracting volume only as crystal volume.
- 2.
Please, do not confuse this k index with the module k of the wavevector, which are distinguished by the used font and by the context in which they appear in the text.
- 3.
The Bragg’s law is many times written as \(2d\,\sin \theta = n\lambda\) where d is the interplanar distance, \(\theta\) the incidence angle with the planes, and n = 1, 2, 3, …etc. Note that different values of n are equivalent to consider different RLPs along the plane’s normal direction.
- 4.
RLPs are also called nodes in order to emphatically differentiate them from any other point of the reciprocal space to which indices hkl are not integer numbers.
- 5.
The number (2π)3 disappears from (4.13) because the convolution operation, “ * ”, stands for an integration in dV q , see (1.32), but becomes one in dV Q = (2π)3dV q , which means, \((2\pi )^{3}\int \delta (2\pi {\boldsymbol q}\,' -{\boldsymbol Q}_{\mathrm{hkl}})\,W(2\pi {\boldsymbol q} - 2\pi {\boldsymbol q}\,')\,\mathrm{d}V _{q'} =\int \delta ({\boldsymbol Q}\,' -{\boldsymbol Q}_{\mathrm{hkl}})\,W({\boldsymbol Q} -{\boldsymbol Q}\,')\,\mathrm{d}V _{Q'} = W({\boldsymbol Q} -{\boldsymbol Q}_{\mathrm{hkl}})\).
- 6.
See, for example, Giacovazzo (2002, Ch. 9).
- 7.
Several semiconductors have cubic unit cell with zinc blend or diamond structures, such as GaAs and Si, space groups F\(\bar{4}3\) m and Fd\(\bar{\mathrm{3}}\) m, respectively. See Hahn (2006, Ch. 7.1, pp. 112–717).
- 8.
See Friedel and Bijvoet pairs, e.g. Giacovazzo (2002, pp. 170 and 475).
- 9.
There are experimental methods based on a dynamical theory of X-ray diffraction, which are susceptible to the value of \(\Psi _{T}\), e.g. Morelhão et al. (2011).
- 10.
Temporal coherence: longitudinal coherence C L , (1.15), divided by the speed of light, e.g. \(\Delta \lambda /\lambda = 10^{-4}\) and \(\lambda = 1.54\,\text{\AA }\, \Rightarrow \, C_{L}/c = 2.6\) fs of temporal coherence.
- 11.
Further details about atomic disorder parameters are available in Authier (2006, pp. 228–242).
- 12.
Although (4.21) has an analytical solution, the routine debye.m allows numerical verification in one dimension.
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Morelhão, S.L. (2016). Crystals. In: Computer Simulation Tools for X-ray Analysis. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-19554-4_4
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