International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 4.1, p. 405   | 1 | 2 |

Section 4.1.5.1. Measurement with X-rays

B. T. M. Willisa*

aChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England
Correspondence e-mail: bertram.willis@chemcryst.ox.ac.uk

4.1.5.1. Measurement with X-rays

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Frequencies can be derived indirectly with X-rays from the intensity of the thermal diffuse scattering. For a monatomic crystal with one atom per primitive cell, there are no optic modes and the one-phonon TDS intensity, equation (4.1.3.6[link]), reduces to[{{{\rm d}\sigma}\over{{\rm d}\Omega}}=Q^2f^2e^{-2W}\sum_{j=1}^{3}{{E_j({\bf q})}\over{\omega^2({\bf q})}}\cos^2\alpha_j({\bf q})\delta({\bf Q}\pm{\bf q}-{\bf Q}_{\bf h}),\eqno(4.1.5.1)] where [\alpha_j({\bf q})] is the angle between Q and the direction of polarization of the mode (j q). There are three acoustic modes associated with each wavevector q, but along certain directions of Q it is possible to isolate the intensities contributed by the individual modes by choosing [\alpha_j({\bf q})] to be close to 0 or 90°. Equation (4.1.5.1[link]) can then be employed to derive the frequency [\omega_j({\bf q})] for just one mode. The measured intensity must be corrected for multi-phonon and Compton scattering, both of which can exceed the intensity of the one-phonon scattering. The correction for two-phonon scattering involves an integration over the entire Brillouin zone, and this in turn requires an approximate knowledge of the dispersion relations. The correction for Compton scattering can be made by repeating the measurements at low temperature.

The X-ray method is hardly feasible for systems with several atoms in the primitive cell. It comes into its own for those few materials which cannot be examined by neutrons. These include boron, cadmium and samarium with high absorption cross sections for thermal neutrons, and vanadium with a very small coherent (and a large incoherent) cross section for the scattering of neutrons. An important feature of TDS measurements with X-rays is in providing an independent check on interatomic or intermolecular force constants derived from measurements with inelastic neutron scattering. The force model is used to generate phonon frequencies and eigenvectors, which are then employed to compute the one-phonon and multi-phonon contributions to the X-ray TDS. Any discrepancy between calculated and observed X-ray intensities might be ascribed to such features as ionic deformation (Buyers et al., 1968[link]) or anharmonicity (Schuster & Weymouth, 1971[link]).

References

First citation Buyers, W. J. L., Pirie, J. D. & Smith, T. (1968). X-ray scattering from deformable ions. Phys. Rev. 165, 999–1005.Google Scholar
First citation Schuster, S. L. & Weymouth, J. W. (1971). Study of thermal diffuse X-ray scattering from lead single crystals. Phys. Rev. B, 3, 4143–4153.Google Scholar








































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