Classical and Quantum Theories of Dynamic X-Ray Diffraction

Classical and Quantum Theories of Dynamic X-Ray Diffraction

Copyright: © 2014 |Pages: 22
DOI: 10.4018/978-1-4666-4687-2.ch012


The main results of the dynamical theories of X-ray diffraction in their standard form of coupling of Maxwell equations with the Bloch waves on the one hand and the interaction of radiation-crystal in terms of perspective of quantum field on the other are unitarily presented.
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12.2. Semi-Classical Theory

As for the dynamic theory Laue, it will be expressed the electronic density in any point of the crystal as Fourier sum over the reciprocal space associated (Batterman & Cole 1964):

(12.1) where 978-1-4666-4687-2.ch012.m02 is the volume unit, 978-1-4666-4687-2.ch012.m03 is the vector of the reciprocal space expressed based on the Miller indices and the vectors of the its main directions. The reciprocal relation (12.1) will unfold of the structure factor as the Fourier continuous transform of the punctual electronic density over the unit cell:


Need to further consider how is affected the dielectric constant of the medium of the electronic density correlated with the structure factor (12.1), because it links the induction and the electric intensity in the Maxwell equations, allowing their uniform solution, as we shall see.

For an environment with intrinsic polarization 978-1-4666-4687-2.ch012.m05 there is the bond:

(12.3) where 978-1-4666-4687-2.ch012.m07 is the dielectric constant, and 978-1-4666-4687-2.ch012.m08 is the electric permittivity of the vacuum.

If we consider an external sinusoidal electric field of pulsation much higher than its own pulsation of oscillation of the electrons on their intrinsic energetic levels, 978-1-4666-4687-2.ch012.m09, the second equation of Newton, the equation of the restoration of the movement of oscillation in the external field, will give the amplitude this movement:

(12.4) where, 978-1-4666-4687-2.ch012.m11 is the amplitude of the external field applied, and e and 978-1-4666-4687-2.ch012.m12 are the charge and respectively the mass of the electron.

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