Some remarks about the integrated reflection power ratio formulae for single-crystal slabs

Lipson, H.; Langford, J. I.; Hu, H.-C.

doi:10.1107/97809553602060000601

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

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International Tables for Crystallography (2006). Vol. C. ch. 6.2, p. 598

Section 6.2.6. Some remarks about the integrated reflection power ratio formulae for single-crystal slabs

H. Lipson,^a ^‡ J. I. Langford^a and H.-C. Hu^b

^a School of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England, and ^bChina Institute of Atomic Energy, PO Box 275 (18), Beijing 102413, People's Republic of China

6.2.6. Some remarks about the integrated reflection power ratio formulae for single-crystal slabs

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The transfer equations for intensity may be rewritten in the form of one-dimensional power transfer equations (Hu & Fang, 1993). The $[P_{H}/P_{0}]$ in (b) and (c) for a mosaic crystal slab under symmetrical and unsymmetrical Bragg and Laue geometries are the general solutions of power transfer equations employing three dimensionless parameters b, ξ and τ. For a crystal slab with a rectangular mosaic distribution, considering multiple reflection, the integrated reflection power ratio, ρ′, can be obtained by substituting σ₀ for σ in the formulae for $[P_{H}/P_{0}]$ and multiplying the result by the mosaic width. However, for crystals with other kinds of mosaic distribution, the corresponding ρ′ can be obtained only by integrating the expression for $[P_{H}(\Delta \theta _{0})/P_{0}]$ over the whole range of $[\Delta \theta _{0}]$ . Formulae (1)–(3) listed in Table 6.3.3.1 , i.e. the transmission coefficient A multiplied by Q, QA, are identical to those of (b) and (c) for the case of $[\mu /\sigma _{0}>> 1]$ , which is the integrated reflection power ratio for a crystal slab based on the kinematic approximation without consideration of multiple reflection.

The secondary extinction factor for X-ray or neutron diffraction in a mosaic crystal slab can be obtained as ρ′/(QA), in which the integrated reflection power ratio with consideration of multiple reflections can be obtained as described above.

Both the transmission power ratio $[P_{T}/P_{0}]$ and the absorption power ratio $[P_{A}/P_{0}]$ can also be obtained by solving the power transfer equations. For details, see Hu (1997a,b), Werner & Arrott (1965) and Werner, Arrott, King & Kendrick (1966).

References

Hu, H.-C. (1997b). A universal treatment of X-ray and neutron diffraction in crystals. II. Extinction. Acta Cryst. A53, 493–504.Google Scholar

Hu, H.-C. & Fang, Y. (1993). Neutron diffraction in flat and bent mosaic crystals for asymmetric geometry. J. Appl. Cryst. 26, 251–257.Google Scholar

Werner, S. A. & Arrott, A. (1965). Propagation of Bragg-reflected neutrons in large mosaic crystals and the efficiency of monochromators. Phys. Rev. 140, A675–A686.Google Scholar

Werner, S. A., Arrott, A., King, J. S. & Kendrick, H. (1966). Propagation of Bragg-reflected neutrons in bounded mosaic crystals. J. Appl. Phys. 37, 2343–2350.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.2, p. 598