Abstract
Diffraction of X-rays by crystals offers the unique possibility to determine at atomic resolution the three-dimensional structure of solids. This opportunity is exploited by solid state chemists or by mineralogists to study the structure of crystals, by chemists to determine the structure of molecules and by molecular biologists to investigate structure and properties of living organisms molecules, like proteins, nucleic acids etc. In this chapter, after a brief introduction on crystals and crystal symmetries, the general principles of diffraction are introduced, then the reader is made familiar with the mathematical formalism of scattering and diffraction. The physical principles of diffraction by crystals are then introduced, and finally the phase problem in crystallography is presented and the main methods to solve it are shortly outlined. Few notes on the refinement process and on powder diffraction are also reported.
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Notes
- 1.
The diffraction of other types of radiations, such as neutrons or electrons, may be formally treated in a similar way.
- 2.
One of us (DV) had the privilege of attending a course in Parma in 1969 and a meeting in Buffalo in 1974 where Paul Ewald gave memorable lectures on the theory of diffraction by crystals. In these notes the same general approach will be used.
- 3.
For partially polarized radiation the polarization factor becomes
$$\begin{aligned} P = P_o - P' \end{aligned}$$(8.42)where, if \(\psi \) is the azimutal angle on the detector,
$$\begin{aligned} P' = \frac{\tau '}{2} \; \cos 2\psi \, \sin ^2 2\theta \end{aligned}$$(8.43)and
$$\begin{aligned} \tau ' = \frac{\alpha (1+\tau )-(1-\tau )}{\alpha (1+\tau )+(1-\tau )} \end{aligned}$$(8.44)where \(\alpha \) depends from the monochromator crystal (for a perfect crystal \(\alpha = \cos 2 \theta _M\), \(\theta _M\) being the Bragg angle of the monochromator) and
$$\begin{aligned} \tau = \frac{I_\parallel - I_\perp }{I_\parallel + I_\perp } \end{aligned}$$(8.45)in which \(I_\parallel \) is the intensity of the source in a direction parallel to the beam (the electric field vector is along \(y\) in Fig. 8.9) and \(I_\perp \) is the intensity in a perpendicular direction (electric field vector along \(z\)).
- 4.
This approximation is assumed valid for most applications; only when very accurate diffraction measurements are available the deviations from the spherical symmetry are taken into account for detailed charge density studies.
- 5.
A more detailed description of the experimental techniques is given in Chap. 10.
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Viterbo, D., Zanotti, G. (2015). X-Ray Diffraction by Crystalline Materials. In: Mobilio, S., Boscherini, F., Meneghini, C. (eds) Synchrotron Radiation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55315-8_8
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