Primary extinction

Sabine, T. M.

doi:10.1107/97809553602060000603

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. 6.4, p. 610

Section 6.4.5. Primary extinction

T. M. Sabine^a

^a ANSTO, Private Mail Bag 1, Menai, NSW 2234, Australia

6.4.5. Primary extinction

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Zachariasen (1967) introduced the concept of using the kinematic result in the small-crystal limit for σ, while Sabine (1985, 1988) showed that only the Lorentzian or Fresnellian forms of the small crystal intensity distribution are appropriate for calculations of the energy flow in the case of primary extinction. Thus, $[ \sigma (\Delta k)= {Q_kT \over 1+(\pi T\Delta k)^2}, \eqno (6.4.5.1)]$ where [Q_kV] is the kinematic integrated intensity on the k scale $[(k=2\sin \theta /\lambda)]$ , $[Q_k=(N_{{c}}\lambda F)^2/\sin \theta]$ , and T is the volume average of the thickness of the crystal normal to the diffracting plane (Wilson, 1949). To include absorption effects, which modify the diffraction profile of the small crystal, it is necessary to replace T by TC, where $[ C= {\tanh (\mu D/2)\over(\mu D/2)}. \eqno (6.4.5.2)]$ To determine the extinction factor, E, the explicit expression for $[\sigma (\Delta k)]$ [equation (6.4.5.1)] is inserted into equations (6.4.4.3) and (6.4.4.4), and integration is carried out over Δk. The limits of integration are $[+\infty]$ and $[-\infty]$ . The notation $[E_{{L}}]$ and $[E_{{B}}]$ is used for the extinction factors at 2θ = 0 and 2θ = π rad, respectively.

After integration and division by $[I^{\rm kin},]$ it is found that $[\eqalignno{ E_{{L}} &=\exp (-y)\{ [1-(x/2)+(x^2/4) \cr &\quad -(5x^3/48)+(7x^4/192)]\}, \quad x\leq 1, &(6.4.5.3) \cr E_{{L}} &=\exp (-y)[2/(\pi x)]^{1/2}\{ 1-[1/(8x)] \cr &\quad -[3/(128x^2)]-[15/(1024x^3)]\},\quad x\gt 1,&(6.4.5.4) \cr E_{{B}} &=A/(1+Bx)^{1/2},&(6.4.5.5)\cr A &=\exp (-y)\sinh y/y, & (6.4.5.6)}%fd6.4.5.4fd6.4.5.5fd6.4.5.6]$ and $[B=(1/y)-\exp (-y)/\sinh y=A {{\rm d}(A^{-1})\over{\rm d}y}. \eqno (6.4.5.7)]$ In these equations, [x=Q_kTCD] and y = μD.

References

Sabine, T. M. (1985). Extinction in polycrystalline materials. Aust. J. Phys. 38, 507–518.Google Scholar

Sabine, T. M. (1988). A reconciliation of extinction theories. Acta Cryst. A44, 368–373.Google Scholar

Wilson, A. J. C. (1949). X-ray optics. London: Methuen.Google Scholar

Zachariasen, W. H. (1967). A general theory of X-ray diffraction in crystals. Acta Cryst. 23, 558–564.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.4, p. 610