To calculate the intensity of the scattered radiation from a crystal after irradiating with X-rays, one can add up all electromagnetic fields of the oscillating electrons (calculated using the Liénard–Wiechert potential). Taking the time-average of the norm of the Poynting vector of the scattered em-field, leads to an expression for the resulting scattered intensity. If the incident radiation is an electromagnetic plane wave with electric field(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\bar{E}(t,\bar{x})=\bar{E}_{0}e^{-i(\bar{k}\cdot\bar{x}-\omega t)}

[/tex]

then the term [itex]\cos^{2}(\bar{k}\cdot\bar{x}-\omega t+c^{te})[/itex] is the only time dependent term of the Poynting vector and its average is [itex]1/2[/itex]. However, the electrons are moving in the atom, the atoms undergo thermal motion and the [itex]\bar{E}_{0}[/itex] is also time dependent (e.g. unpolarized light). Therefore the appear more time-dependent terms in the Poynting vector. This is implicitly dealt with in many text books by considering the time average of each of these processes separately. The time average of the electron motion leads to the atomic scattering factor, the time average of the atomic thermal motion leads to the Debye Waller factor and the time average of [itex]\bar{E}_{0}[/itex] leads to the polarization factor. But why can these be averaged independently? Mathematically, it is like saying that "the integral of a product of functions" is "the product of the integrals of these functions". When is this a valid approximation?

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# Time-average Poynting vector of crystal scattering

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