International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 7.4, pp. 656-657
Section 7.4.2.3. TDS correction factor for thermal neutrons (single crystals)
B. T. M. Willisd
|
The neutron treatment of the correction factor lies along similar lines to that for X-rays. The principal difference arises from the different topologies of the one-phonon `scattering surfaces' for X-rays and neutrons. These surfaces represent the locus in reciprocal space of the end-points of the phonon wavevectors q (for fixed crystal orientation and fixed incident wavevector k0) when the wavevector k of the scattered radiation is allowed to vary. We shall not discuss the theory for pulsed neutrons, where the incident wavelength varies (see Popa & Willis, 1994).
The scattering surfaces are determined by the conservation laws for momentum transfer, and for energy transfer,
where
is the neutron mass and
is the phonon energy. ɛ is either +1 or −1, where ɛ = +1 corresponds to phonon emission (or phonon creation) in the crystal and a loss in energy of the neutrons after scattering, and ɛ = −1 corresponds to phonon absorption (or phonon annihilation) in the crystal and a gain in neutron energy. In the X-ray case, the phonon energy is negligible compared with the energy of the X-ray photon, so that (7.4.2.15)
reduces to
and the scattering surface is the Ewald sphere. For neutron scattering,
is comparable with the energy of a thermal neutron, and so the topology of the scattering surface is more complicated. For one-phonon scattering by long-wavelength acoustic modes with
, (7.4.2.15)
reduces to
where β
is the ratio of the sound velocity in the crystal and the neutron velocity. If the Ewald sphere in the neighbourhood of a reciprocal-lattice point is replaced by its tangent plane, the scattering surface becomes a conic section with eccentricity 1/β. For
, the conic section is a hyperboloid of two sheets with the reciprocal-lattice point P at one focus. The phonon wavevectors on one sheet correspond to scattering with phonon emission and on the other sheet to phonon absorption. For
, the conic section is an ellipsoid with P at one focus. Scattering now occurs either by emission or by absorption, but not by both together (Fig. 7.4.2.3
).
To evaluate the TDS correction, with q restricted to lie along the scattering surfaces, separate treatments are required for faster-than-sound and for slower-than-sound
neutrons. The final results can be summarized as follows (Willis, 1970
; Cooper, 1971
):
The sharp distinction between cases (a) and (b) has been confirmed experimentally using the neutron Laue technique on single-crystal silicon (Willis, Carlile & Ward, 1986).
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