International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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

7.4.2.3. TDS correction factor for thermal neutrons (single crystals)

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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[link]).

The scattering surfaces are determined by the conservation laws for momentum transfer, [{\bf H}={\bf k}-{\bf k}_0=2\pi {\bf h}+{\bf q},]and for energy transfer, [\hbar ^2(k^2-k_0^2) /2m_n=-\varepsilon \hbar \omega _j({\bf q}), \eqno (7.4.2.15)]where [m_n] is the neutron mass and [\hbar \omega _j({\bf q})] 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)[link] reduces to [k=k_0,]and the scattering surface is the Ewald sphere. For neutron scattering, [\hbar \omega _j\left ({\bf q}\right) ] 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 [q\ll k_0], (7.4.2.15)[link] reduces to [ k=k_0-\varepsilon \beta q,]where β [\left (={\bf v}_L/{\bf v}_n\right) ] 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 [\beta \lt1], 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 [\beta \gt1], 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[link]).

[Figure 7.4.2.3]

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Scattering surfaces for one-phonon scattering of neutrons: (a) for neutrons faster than sound (β < 1); (b) for neutrons slower than sound (β > 1). The scattering surface for X-rays is the Ewald sphere. P0, P1, etc. are different positions of the reciprocal-lattice point with respect to the Ewald sphere, and the scattering surfaces are numbered to correspond with the appropriate position of P.

To evaluate the TDS correction, with q restricted to lie along the scattering surfaces, separate treatments are required for faster-than-sound [(\beta \lt1)] and for slower-than-sound [(\beta \gt1)] neutrons. The final results can be summarized as follows (Willis, 1970[link]; Cooper, 1971[link]):

  • (a) For faster-than-sound neutrons, the TDS rises to a maximum, just as for X-rays, and the correction factor is given by (7.4.2.13)[link], which applies to the X-ray case. (This is a remarkable result in view of the marked difference in the one-phonon scattering surfaces for X-rays and neutrons.)

  • (b) For slower-than-sound neutrons, the correction factor depends on the velocity (wavelength) of the neutrons and is more difficult to evaluate than in (a). However, α will always be less than that calculated for X-rays of the same wavelength, and under certain conditions the TDS does not rise to a maximum at all so that α is then zero.

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[link]).

References

First citation Cooper, M. J. (1971). The evaluation of thermal diffuse scattering of neutrons for a one velocity model. Acta Cryst. A27, 148–157.Google Scholar
First citation Popa, N. C. & Willis, B. T. M. (1994). Thermal diffuse scattering in time-of-flight diffractometry. Acta Cryst. A50, 57–63.Google Scholar
First citation Willis, B. T. M. (1970). The correction of measured neutron structure factors for thermal diffuse scattering. Acta Cryst. A26, 396–401.Google Scholar
First citation Willis, B. T. M., Carlile, C. J. & Ward, R. C. (1986). Neutron diffraction from single-crystal silicon: the dependence of the thermal diffuse scattering on the velocity of sound. Acta Cryst. A42, 188–191.Google Scholar








































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