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

International Tables for Crystallography (2006). Vol. C, ch. 6.4, p. 612

Section 6.4.13. Asymptotic behaviour of the integrated intensity

T. M. Sabinea

aANSTO, Private Mail Bag 1, Menai, NSW 2234, Australia

6.4.13. Asymptotic behaviour of the integrated intensity

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From the definition of the extinction factor, the integrated intensity from a non-absorbing crystal in which the block size is sufficiently small, and the mosaic spread is sufficiently large, will approach the kinematic limit. It is instructive to examine the behaviour in the limit of large block size and low mosaic spread. The volume of the mosaic block is v and the volume of the crystal is V. The number of blocks in the crystal is [V/v] [(=L^3/\ell ^3)]. The surface area of the block is [v^{2/3}] and of the crystal is [V^{2/3}]. The subscripts L and B will again be used for the Laue and the Bragg case, respectively. The kinematic integrated intensity is given by [ I^{\rm kin}=Q_\theta V=\lambda^3N_{{c}}^2F^2V/\sin 2\theta. \eqno (6.4.13.1)]

6.4.13.1. Non-absorbing crystal, strong primary extinction

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  • (a) Laue case

    The limiting value of [E_{{L}}] is [(2/\pi){}^{1/2}x{}^{-1/2}]. Hence, [I_{{L}}=(4/5)N_{{c}}\lambda {^2}FVv^{-1/3}/\sin 2\theta. \eqno (6.4.13.2)]

    The dynamical theory has a numerical constant of 1/2 instead of 4/5.

    (b) Bragg case

  • The limiting value of [E_{{B}}] is [x^{-1/2}]. Hence, [I_{{B}}=N_{{c}}\lambda {^2}FVv^{-1/3}/\sin 2\theta. \eqno (6.4.13.3)]This is in exact agreement with the dynamical theory (Ewald solution).

6.4.13.2. Non-absorbing crystal, strong secondary extinction

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For this condition, the limiting values of the integrated intensity are [I_{{L}}=(4/5)g^{-1}V^{2/3}], and [I_{{B}}=g^{-1}V^{2/3}.] In this limit, which was also noted by Bacon & Lowde (1948[link]) and by Hamilton (1957[link]), the intensity is proportional only to the mosaic spread and to the surface area of the crystal. No structural information is obtained from the experiment.

6.4.13.3. The absorbing crystal

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Only the Bragg case for thick crystals will be considered here. The asymptotic values of A, B, and C are [1/(2\mu L^{*})], [1/(\mu L^{*})], and [2/(\mu L^{*})], respectively, so that [ BCx=2N_{{c}}^2\lambda ^2F^2/\mu ^2. \eqno (6.4.13.4)]For BCx small, the integrated intensity, [I_{{B}}], is given by [ I_{{B}}=(Q_\theta /2\mu)[1-(N_{{c}}F/)2]V^{2/3}. \eqno (6.4.13.5)]For BCx large, [ I_{{B}}=(1/2\sqrt {2})[1-(\mu /2\lambda N_{{c}}F)^2]\lambda ^2N_{{\rm c}}FV^{2/3}/\sin 2\theta. \eqno (6.4.13.6)]It can be shown that the parameter g (which has no relation to the parameter g used to describe the mosaic-block distribution) used by Zachariasen (1945[link]) in discussing this case is equal to −μ/2NCF. Hence, on his y scale, [ I_{{B}}=(\pi /2\sqrt {2})[1-g^2]. \eqno (6.4.13.7)]The value he obtained is IB = 8/3[1 − 2|g|], while Sabine & Blair (1992[link]) found IB = 8/3[1 − 2.36|g|].

References

Bacon, G. E. & Lowde, R. D. (1948). Secondary extinction and neutron crystallography. Acta Cryst. 1, 303–314.
Hamilton, W. C. (1957). The effect of crystal shape and setting on secondary extinction. Acta Cryst. 10, 629–634.
Sabine, T. M. & Blair, D. G. (1992). The Ewald and Darwin limits obtained from the Hamilton–Darwin energy transfer equations. Acta Cryst. A48, 98–103.
Zachariasen, W. H. (1945). Theory of X-ray diffraction in crystals. New York: John Wiley, Dover.








































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