The atoms in a crystal are vibrating with amplitudes determined by the force constants of the crystal’s normal modes. This motion can never be frozen out because of the persistence of zero-point motion, and it has important consequences for the scattering intensities. Since X-ray scattering (and, to a lesser extent, neutron scattering) is a very fast process, taking place on a time scale of 10−18 s, the photon-matter interaction time is much shorter than the period of a lattice vibration, which is of the order Thus, the recorded X-ray scattering pattern is the sum over the scattering of a large number of 1/v, or ≈10−13s. instantaneous states of the crystal. To an extremely good approximation, the scattering averaged over the instantaneous distributions is equivalent to the scattering of the time-averaged distribution of the scattering matter (Stewart and Feil 1980). The structure factor expression for coherent elastic Bragg scattering of X-rays may therefore be written in terms 〈ρ(r)〉, of the thermally averaged electron density: . . . F(H)=∫unit cell〈ρ(r)〉 exp (2πi H ·r) dr (2.1) . . . The smearing of the electron density due to thermal vibrations reduces the intensity of the diffracted beams, except in the forward |S| = 0 direction, for which all electrons scatter in phase, independent of their distribution. The reduction of the intensity of the Bragg peaks can be understood in terms of the diffraction pattern of a more diffuse electron distribution being more compact, due to the inverse relation between crystal and scattering space, discussed in chapter 1. The reduction in intensity due to thermal motion is accompanied by an increase in the incoherent elastic scattering, ensuring conservation of energy. In this respect, thermal motion is much like disorder, with the Bragg intensities representing the average distribution, and the deviations from the average appearing as a continuous, though not uniform, background, generally referred to as thermal diffuse scattering or TDS. A crystal with n atoms per unit cell has 3nN degrees of freedom, N being the number of unit cells in the crystal.
X-ray phase determination by the principle of minimum charge