research papers Extinction-corrected mean thickness and integral width used in the program UMWEG98

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1 Journal of Applied Crystallography ISSN Received 6 July 1999 Accepted 13 January 2 # 2 International Union of Crystallography Printed in Great Britain ± all rights reserved Extinction-corrected mean thickness and integral width used in the program UMWEG98 Elisabeth Rossmanith Mineralogisch-Petrographisches Institut der UniversitaÈt Hamburg, Grindelallee 48, D-2146 Hamburg, Germany. Correspondence rossmanith@mineralogie.uni-hamburg.de The kinematical upper limit of the extinction-corrected mean thickness as well as the corresponding integral width of the intensity pro les are estimated for spherical crystals. The new expression for the primary-extinction correction is compared with expressions given recently in the literature as well as with those widely used in crystal structure analysis. 1. Introduction UMWEG98, a program for the calculation and graphical representation of multiple diffraction patterns (Rossmanith, 1992, 1998, 1999) can be applied to mosaic as well as perfect crystal spheres. Especially in the second case, the knowledge of the primary-extinction correction for the whole range of values for the scattering angle is indispensable. Furthermore, the extinction correction used in crystallographic software must span a large range of the mean-crystal-thickness to extinction-length ratio, tå/, which, for example in the case of a perfect Si sphere with a radius of about 2 mm evaluates to tå/ ' 4 for the 111 re ection measured with Cu K radiation. It is well known that the width of the intensity pro le depends on the extinction-corrected mean thickness of the crystal. Therefore, the fundamental quantity in pro le analysis, the integral width of the pro le, de ned as the width of a rectangle with the same maximum and the same area, is also analysed. The knowledge of both quantities, the extinction-corrected mean thickness as well as the corresponding integral width, is necessary for the calculation of multiple diffraction patterns. These quantities are therefore extensively used in the program UMWEG98 (see Rossmanith, 1992, 1998). 2. Theory 2.1. The integrated reflectivity of a perfect crystal in the kinematical approach Using the abbreviation 1= ˆ r jf h jk=v cell ; 1 where r is the classical electron radius, is the wavelength of the radiation used for diffraction, F h is the structure factor, V cell is the volume of the unit cell and K is the polarization coef cient for the parallel or perpendicular component of the X-ray electric eld, the dimensionless integrated re ectivity R h can be expressed as [von Laue, 196, equation (18.38) therein] R h ˆ I integral=i! q ˆ Imax=I kin qš! integral ˆ L= 2 t: 2 In (2), I is the incident intensity, q is the cross section of the irradiated crystal volume normal to the incident beam, I! integral is the `integrated intensity', i.e. the energy received by the counter during rotation of the crystal about the! axis with uniform angular velocity, Imax kin and! integral are the maximum value and the integral width of the Bragg re ection pro le obtained with the! scanning technique, respectively, and L is the Lorentz factor. The mean thickness of the crystal, tå, is de ned by t ˆ V cry =q 3 where V cry is the irradiated crystal volume. In the case of perfect crystals considered in this paper, the deviations of the integrated re ectivity from the kinematical expressions (2) are generally handled by primary-extinction theories, i.e. using a theoretical primary-extinction correction factor y p, R ext h ˆ L= 2 ty p ; 4 and consequently, the extinction-corrected re ection-dependent mean thickness of the crystal volume involved in the diffraction process can be de ned as t ext ˆ ty p : Primary-extinction correction for the plane parallel plate: results of the dynamical theory For simplicity, only two special cases, namely the symmetrical Bragg case and the symmetrical Laue case, will be considered here. Using the symbols, L, tå and t ext, introduced above, the formulae for the integrated re ectivity derived in the framework of the dynamical theory [von Laue, 196, expressions (31.2) and (31.31) therein] can be expressed as R h ˆ L= 2 t ext ˆ L= 2 tanh t= Š 6 for the Bragg case, and R h ˆ L= 2 t ext ˆ L= 2 2t= =2 R J z dzš 7 33 Elisabeth Rossmanith Extinction correction J. Appl. Cryst. (2). 33, 33±333

2 for the Laue case. J is the zero-order Bessel function. For tå, the tanh in (6) as well as the integral in (7) converge to unity. The extinction-corrected mean thickness t ext of the crystal volume involved in the diffraction process is limited to the extinction length in the Bragg case and converges to /2 in the Laue case (Fig. 1, dotted curves 3 and 4) Primary-extinction correction for the spherical crystal Results of the kinematical approach: Bragg case (back-reflection). It was pointed out by Rossmanith (2) that in the framework of the kinematical approach the step intensities of the pro le, de ned by equations (7), (8) and (3) of Rossmanith (2), can be given as analytical expressions, i.e. by means of Mathematica (Wolfram, 1999), the power received by the counter for an individual step with the! scanning technique can be calculated according to I kin =I q ˆ t 2 = 2 9=8 f r 2 2 cos 4r 4r sin 4r Š=32 4 r 4 4 g f a ; Figure 1 The extinction-corrected mean thickness of the crystal versus the mean thickness of the crystal: comparison of the results for the plane parallel plate (dynamical theory: dotted curve 3 represents the Bragg case; dotted curve 4 represents the Laue case) with the results for the spherical crystal [solid line 1 represents the kinematical approach; solid line 2, extinction correction according to Chukhovskii et al. (1998) applied for = ]. The horizontal faint solid line corresponds to equation (1). The upper dashed line corresponds to equation (2); the lower dashed line corresponds to equation (21). 8 where r is the radius of the spherical crystal and is the distance between the reciprocal-lattice point and the sphere of re ection in the direction of the re ected beam. The term f(a) depends on the upper limit of integration, a, which is determined by the detector surface (see Fig. 1 of Rossmanith, 2). The lower limit of integration is zero. f(a) is negligibly small in the case of an aperture, large enough to let in the total diffracted intensity. It can easily be shown that in this case expression (8) is equivalent to expression (29) given by Becker & Coppens (1974). For =, expression (8) has to be replaced by the maximum value Imax kin of the Bragg re ection pro le de ned by (Rossmanith, 2) Imax=I kin q ' t 2 = 2 9=8 : 9 Because of energy conservation, the diffracted intensity de ned in (8) and (9) cannot be larger than the intensity incident on the crystal sample, i.e. the relation I kin /I q 1 must be ful lled for any step in the measured! scan. For large Bragg angles, it therefore follows that in the framework of the kinematical approach the total incident intensity is completely diffracted for individual steps in a spherical crystal with a mean thickness t 2 2 1=2 =3Š 1 and can consequently be interpreted as the corresponding kinematical extinction length. For small crystals, i.e. in the case tå < [2(2 1/2 )/3], integration over the pro le de ned in (8) and (9) results in the integrated re ectivity (2) (see Rossmanith, 2). For large crystals, on the other hand, i.e. in the case tå [2(2 1/2 )/3], the integrated re ectivity is limited to R ext h ˆ R I kin ext =I q d! ˆ L= 2 t ext < R h ; 11 where Iext kin /I q is equivalent to (9) in the case I kin /I q 1; otherwise Iext kin /I q = 1. Consequently, in the case that the entrance surface of the incident beam is very nearly equal to the exit surface of the re ected beam, i.e. in the back-re ection case with a Bragg angle ' 9, the extinction correction y p introduced in (4) can be estimated by y p ' 9 ˆR I kin ext =I q d! R I kin =I q d! 12 and the extinction-corrected mean thickness of the crystal can be calculated according to t ext = ˆ =L R Iext kin =I q d!: 13 Curve 1 in Fig. 1 represents the extinction-corrected mean thickness t ext obtained in this way. For Bragg angles <9, the entrance surface of the incident beam does not agree with the exit surface of the re ected beam. Part of the diffracted beam therefore cannot escape the crystal and will be rescattered in the direction of the incident beam. The corresponding process of multiple diffraction will additionally reduce the intensity in the direction of the diffracted beam, i.e. Rh ext de ned in equation (11) will further be diminished. The extinction-corrected mean thickness J. Appl. Cryst. (2). 33, 33±333 Elisabeth Rossmanith Extinction correction 331

3 de ned in equation (13) therefore represents an estimation of the kinematical upper limit for t ext / for all Bragg angles A theoretical approach based on the Takagi±Taupin equations: Laue case (forward diffraction). The expression for the primary extinction y p ˆ 1=V cry Rr R 2 dr R 1 d cos R2 exp t 1 t 2 Š 1 jj 2 t 1 t 2 1=2 =Šj 2 d' 14 derived in the framework of the dynamic theory in the Laue approximation by Chukhovskii et al. (1998), simultaneously takes into account the extinction and absorption effects. The path lengths t 1 and t 2 in expression (14) are de ned by t 1 ˆ r 2 R 2 cos 2 R 2 sin 2 sin 2 ' Š 1=2 R sin cos ' ; t 2 ˆ r 2 R 2 cos 2 R 2 sin 2 sin 2 ' Š 1=2 R sin cos ' ; 15 where is the linear absorption coef cient, J is the zeroorder Bessel function, R, and ' are the spherical coordinates and is the Bragg angle. Neglecting the imaginary part of the anomalous-dispersion correction for the atomic form factor, i.e. in the case = and F h real, equation (14) is identical to the expression (11) given in the paper by Al Haddad & Becker (199). The solid line 2 in Figs. 1 and 3 calculated for =2, and the faint curves in Fig. 2 represent the extinctioncorrected mean thickness t ext calculated with the primaryextinction correction de ned by equation (14) Analytical expressions for the primary-extinction correction. The widely used analytical expressions for the primary-extinction correction factors for a spherical crystal, y p, obtained by Zachariasen (1967) (dashed-dotted line 3 in Fig. 3), y p ˆ 1= 1 2x 1=2 ˆ 1 1 3=2 I kin max=i q Š 1=2 ˆ 1= 1 27=16 t= 2 Š 1=2 ; and Becker & Coppens (1974) (dotted curves in Fig. 3), y p ˆf1 2x A x 2 = 1 B xšg 1=2 ; are given for comparison, where x in (16) and (17) is de ned by x ˆ 3=4 R h = integral 18 and A() = cos 2 and B() =.22.12(.5 cos 2) 2 are Bragg-angle-dependent terms. [Equation (16) corresponds to the revised formulation of Becker & Coppens (1974).] Thorkildsen & Larsen (1998) recently deduced an analytical expression for the primary-extinction correction, in the Figure 2 The extinction-corrected mean thickness of the spherical crystal versus the mean thickness of the crystal: comparison of the result of the kinematical approach (curve 1) with the extinction-corrected mean thickness obtained with y p of Chukhovskii et al. (1998) (faint curves) for = and =, 5, 1, 15,...,9. Figure 3 The extinction-corrected mean thickness of the spherical crystal versus the mean thickness of the crystal. Curve 1: result of the kinematical approach ( ' 9 ). Curve 2: result obtained with y p according to Chukhovskii et al. (1998) ( ' ). Curve 3: result obtained with y p according to Zachariasen (1967). Dotted curves: obtained with y p of Becker & Coppens (1974) for various Bragg angles. 332 Elisabeth Rossmanith Extinction correction J. Appl. Cryst. (2). 33, 33±333

4 framework of the dynamical theory and valid in the whole range of values of the scattering angle, y p ˆ P 1 n f n s r= sin 2 2n ; 19 n where the coef cients f n s [available from Thorkildsen & Larsen (1998) up to the fth order] explicitly depend on the Bragg angle. Owing to the slow convergence of the series representing y p, with the available coef cients f n s the primary-extinction correction can be calculated for the whole range of Bragg angles only for a very small tå/ range (tå/ <.5). For the limiting cases! /2 (Bragg case) and! (Laue case), Larsen & Thorkildsen (1998) presented the asymptotic expressions (dashed lines in Fig. 1) y p ' 3=4r =48r 2 ; 2 for the Bragg case, and y p ' 3=8r f1 = 2r= 3=2 Š cos 4r= 5=4 2 =16r 2 g; 21 for the Laue case. As stated by the authors, these expressions are valid for tå/ > The integral width based on the kinematical approach: Bragg case. For small crystals, i.e. for Imax/I kin q <1, the integral width of the Bragg re ection pro le is de ned by equations (2) and (9):! integral ˆ L= 9t=8 : 22 For tå [2(2 1/2 )/3], on the other hand, because Imax/I kin q =1, the integral width is equal to the extinction-corrected integrated re ectivity de ned in (11) with a limiting value:! integral ˆ Rh ext ˆ L= t ext =!:9L=; 23 according to t ext!.9 (solid line 1 in Figs. 1, 2 and 3) for large tå/ ratios. In this case the kinematical integral width of the intensity pro les is no longer proportional to the reciprocal of the mean thickness, but there is a minimum integral width proportional to the reciprocal of the extinction length. 3. Discussion As a result of the complete re ection of the incident beam in large crystals, for all tå/ ratios, the extinction length is obviously an upper limit for the extinction-corrected mean thickness of the crystal t ext (Fig. 1). This is true for the plane parallel plate considered in the framework of the dynamical theory (dotted curve 3) as well as for the spherical crystal considered in the framework of the kinematical approach (solid line 1). As a result of the Laue approximation, the Chukhovskii et al. y p values, on the other hand, are overestimated at high Bragg angles and large tå/ ratios (Fig. 2). The same holds true for the Becker & Coppens (1974) analytical expressions (Fig. 3). It is clear from Fig. 3 that Zachariasen's primary-extinction correction (dashed-dotted curve 3), which is independent of the scattering angle, can be used only as a very rough rst approximation. The new approach of Thorkildsen & Larsen (1998), although valid for all Bragg angles, in the present state of its development it suffers from the small tå/ range (tå/ <.6 for >6 ) for which values for the primary-extinction factor can be calculated. The asymptotic expressions given for the limiting Bragg and Laue cases by Larsen & Thorkildsen [1998; expressions (7) and (8) in their paper], on the other hand, seem to be questionable. In the Laue case, their asymptotic result [lower dashed line in Fig. 1; equation (8) in their paper] does not agree with the Al Haddad & Becker (199) primary-extinction correction (solid line 2 in Fig. 1). 1 In the Bragg case, their results (upper dashed curve in Fig. 1) exceed the kinematical upper limit (solid line 1 in Fig. 1). Because of the various shortcomings of the extinction corrections discussed above, in the present state of the art the Chukhovskii et al. values presented in Fig. 2 for <5 are implemented in table form in UMWEG98. For! scans with <5, the extinction-corrected mean thickness of the crystal is obtained by interpolation in the program. In all other cases, the primary-extinction correction is evaluated according to the solid curve 1 in each of Figs. 1, 2 and 3. The extinctioncorrected mean thickness obtained in this way, although limited to.9, is overestimated for intermediate Bragg angles and tå/ ratios. Nevertheless, the agreement of calculated spectra obtained with the approximations for the extinction-corrected mean thickness and the integral width introduced in this paper with measured ones is excellent, as can be seen in the gures presented by Rossmanith et al. (1994), Rossmanith & Armbruster (1995), and Rossmanith (1999). This may be regarded as an indication for the justi cation of the approximations made. The author wishes to thank the referees for helpful comments. The development of the program UMWEG98 was funded by the Deutsche Forschungsgemeinschaft. References Al Haddad, M. & Becker, P. (199). Acta Cryst. A46, 112±123. Becker, P. J. & Coppens, P. (1974). Acta Cryst. A3, 129±147. Chukhovskii, F., Hupe, A., Rossmanith, E. & Schmidt, H. (1998). Acta Cryst. A54, 191±198. Larsen, H. B. & Thorkildsen, G. (1998). Acta Cryst. A54, 511±512. Laue, M. von (196). RoÈntgenstrahlinterferenzen. Frankfurt Am Main: Akademische Verlagsgesellschaft. Rossmanith, E. (1992). Acta Cryst. A48, 596±61. Rossmanith, E. (1998). Z. Kristallogr. 213, 563±568. Rossmanith, E. (1999). J. Appl. Cryst. 32, 355±361. Rossmanith, E. (2). J. Appl. Cryst. 33, 323±329. Rossmanith, E., Adiwidjaja, G., Eck, J., Kumpat, G. & Ulrich, G. (1994). J. Appl. Cryst. 27, 51±516. Rossmanith, E. & Armbruster, T. (1995). Z. Kristallogr. 21, 645±649. Thorkildsen, G. & Larsen, H. B. (1998). Acta Cryst. A54, 172±185. Wolfram, S. (1999). The Mathematica Book. Fourth Edition. Wolfram Media/Cambridge University Press. Zachariasen, W. H. (1967). Acta Cryst. 23, 558± Following the comment of one of the referees, one has to include at least one more term in the asymptotic formula for y p in the Laue case. J. Appl. Cryst. (2). 33, 33±333 Elisabeth Rossmanith Extinction correction 333

5 letters to the editor Journal of Applied Crystallography ISSN Comments on Extinction-corrected mean thickness and integral width used in the program UMWEG98 by Rossmanith (2) Received 1 April 2 Accepted 7 August 2 Helge B. Larsen a and Gunnar Thorkildsen b * a Department of Materials Science, Stavanger University College, Ullandhaug, N-491 Stavanger, Norway, and b Department of Mathematics and Natural Science, Stavanger University College, Ullandhaug, N-491 Stavanger, Norway. Correspondence gunnar.thorkildsen@tn.his.no # 2 International Union of Crystallography Printed in Great Britain ± all rights reserved Comments are made on a paper by E. Rossmanith [J. Appl. Cryst. (2), 33, 33±333] concerning the use of asymptotic expressions for the extinctioncorrected mean thickness. Received In a recently 1 April 2 published paper, Rossmanith (2) accounts for Accepted expressions 7 August for the 2extinction-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 extinctioncorrected 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 oh! (pure Laue case) and oh! /2 (pure Bragg case). Here oh denotes the Bragg angle. Rossmanith questions the result for the Laue case because it `does not agree with the Al Haddad & Becker (199) 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 y p x; oh! ' 3=8x f1 = 2x 3=2 Š cos 4x 5=4 1=16x 2 g; 1 where x = R/ 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, as is evident from Fig. 1 of Rossmanith (2). 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 # 2 International the kinematical Union of Crystallography upper limit'. This statement is somewhat confusing Printed Great owing Britain to ± the all fact rightsthat reserved 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 oh! {, /2}, the diffraction geometry is quasi one-dimensional. For these two cases, the expression for the primary-extinction factor for a nite convex crystal of general shape, bathed in the incident beam, becomes y i p ˆ 1=V cry R du dvt jj u; v y plate;i p x ˆ t jj = oh A? 2 i ˆ Laue; Bragg 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. References Al Haddad, M. & Becker, P. (199). Acta Cryst. A46, 112±123. Larsen, H. B. & Thorkildsen, G. (1998). Acta Cryst. A54, 511±512. Rossmanith, E. (2). J. Appl. Cryst. 33, 33±333. Takagi, S. (1962). Acta Cryst. 15, 1311±1312. Takagi, S. (1969). J. Phys. Soc. Jpn, 26, 1239±1253. Thorkildsen, G. & Larsen, H. B. (1999). Acta Cryst. A55, 84±854. J. Appl. Cryst. (2). 33, 1447 Larsen and Thorkildsen Extinction correction 1447