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Comments on Extinction-corrected mean thickness and integral width used in the program UMWEG98 by Rossmanith (2000)

Comments are made on a paper by E. Rossmanith [J. Appl. Cryst. (2000), 33, 330–333] concerning the use of asymptotic expressions for the extinction-corrected mean thickness.

In a recently published paper, Rossmanith (2000) accounts for expressions for the extinction-corrected mean thickness used in the program UMWEG98. A comparison with already existing models for the primary-extinction factor in perfect crystal spheres is also presented.

In particular, Rossmanith's kinematical formula for the extinction-corrected mean thickness as a function of the mean crystal thickness is compared with results based on asymptotic expressions for the primary-extinction factor, y_p, found by the present authors (Larsen & Thorkildsen, 1998) for the limiting cases θ_oh → 0 (pure Laue case) and θ_oh → π/2 (pure Bragg case). Here θ_oh denotes the Bragg angle. Rossmanith questions the result for the Laue case because it `does not agree with the Al Haddad & Becker (1990) primary-extinction correction'. This is owing to a printing error in the expression for the asymptotic primary-extinction factor, equation (8), of Larsen & Thorkildsen (1998). The correct expression is

where x = R/Λ_oh, the ratio between the radius of the sphere and the extinction distance. The sign error in the oscillating term of the erroneous version of equation (1) is equivalent to a phase shift of π, as is evident from Fig. 1 of Rossmanith (2000). We acknowledge Rossmanith for drawing this to attention.

When it comes to the Bragg case, Rossmanith seems to question the result [equation (7) of Larsen & Thorkildsen (1998)] because it `exceeds the kinematical upper limit'. This statement is somewhat confusing owing to the fact that our results are based on dynamical theory as formulated by Takagi (1962, 1969). The equivalence between the Takagi theory and the fundamental theory of dynamical diffraction has been established and demonstrated (Thorkildsen & Larsen, 1999). In the limits θ_oh → {0, π/2}, the diffraction geometry is quasi one-dimensional. For these two cases, the expression for the primary-extinction factor for a finite convex crystal of general shape, bathed in the incident beam, becomes

where A_⊥ denotes the cross section of the crystal projected onto a plane (u, v) normal to the direction of the incident/diffracted beam. The function t_||(u, v) represents the crystal dimension along the incident beam. V_cry is the volume of the crystal. Applying equation (2) to a spherical crystal in the Bragg case gives equation (4) from (3) in the paper by Larsen & Thorkildsen (1998).

In our opinion, corrections for primary extinction, which is a dynamical feature, should be formally handled by a dynamical diffraction theory, rather than a kinematical approach.