Section 7.4.2.1. Glossary of symbols

Direction cosines of ej(q) e j (q) Polarization vector of normal mode (jq) E j (q) Energy of mode (jq) E meas Total integrated intensity measured under Bragg peak E 0 Integrated intensity from Bragg scattering E 1 Integrated intensity from one-phonon scattering F (h) Structure factor Planck's constant h divided by 2π 2πh Reciprocal-lattice vector H Scattering vector j Label for branch of dispersion relation k 0 Wavevector of incident radiation k Wavevector of scattered radiation Boltzmann's constant Neutron mass m Mass of unit cell N Number of unit cells in crystal q Wavevector of normal mode of vibration Radius of scanning sphere in reciprocal space V Volume of unit cell Elastic wave velocity for branch j Mean velocity of elastic waves α TDS correction factor Scattering angle Bragg angle Differential cross section for Bragg scattering Differential cross section for one-phonon scattering ρ Density of crystal Frequency of normal mode (jq)

Thermal diffuse scattering (TDS) is a process in which the radiation is scattered inelastically, so that the incident X-ray photon (or neutron) exchanges one or more quanta of vibrational energy with the crystal. The vibrational quantum is known as a phonon, and the TDS can be distinguished as one-phonon (first-order), two-phonon (second-order), scattering according to the number of phonons exchanged.

The normal modes of vibration of a crystal are characterized as either acoustic modes, for which the frequency ω(q) goes to zero as the wavevector q approaches zero, or optic modes, for which the frequency remains finite for all values of q [see Section 4.1.1 of IT B (2001)]. The one-phonon scattering by the acoustic modes rises to a maximum at the reciprocal-lattice points and so is not entirely subtracted with the background measured on either side of the reflection. This gives rise to the `TDS error' in estimating Bragg intensities. The remaining contributions to the TDS – the two-phonon and multiphonon acoustic mode scattering and all kinds of scattering by the optic modes – are largely removed with the background.

It is not easy in an X-ray experiment to separate the elastic (Bragg) and the inelastic thermal scattering by energy analysis, as the energy difference is only a few parts per million. However, this has been achieved by Dorner, Burkel, Illini & Peisl (1987) using extremely high energy resolution. The separation is also possible using Mössbauer spectroscopy. Fig. 7.4.2.1 shows the elastic and inelastic components from the 060 reflection of LiNbO₃ (Krec, Steiner, Pongratz & Skalicky, 1984), measured with γ-radiation from a ⁵⁷Co Mössbauer source. The TDS makes a substantial contribution to the measured integrated intensity; in Fig. 7.4.2.1, it is 10% of the total intensity, but it can be much larger for higher-order reflections. On the other hand, for the extremely sharp Bragg peaks obtained with synchrotron radiation, the TDS error may be reduced to negligible proportions (Bachmann, Kohler, Schulz & Weber, 1985).

Figure 7.4.2.1| top | pdf | 060 reflection of LiNbO3 (Mössbauer diffraction). Inelastic (triangles), elastic (crosses), total (squares) and background (pluses) intensity (after Krec, Steiner, Pongratz & Skalicky, 1984).

Let E_meas represent the total integrated intensity measured in a diffraction experiment, with the contribution from Bragg scattering and that from (one-phonon) TDS. Then, where α is the ratio and is known as the `TDS correction factor'. α can be evaluated in terms of the properties of the crystal (elastic constants, temperature) and the experimental conditions of measurement. In the following, it is implied that the intensities are measured using a single-crystal diffractometer with incident radiation of a fixed wavelength. We shall treat separately the calculation of α for X-rays and for thermal neutrons.