Empirical methods

Maslen, E. N.

doi:10.1107/97809553602060000602

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. 6.3, pp. 607-608

Section 6.3.3.5. Empirical methods

E. N. Maslen^a ^‡

^a Crystallography Centre, The University of Western Australia, Nedlands, Western Australia 6009, Australia

6.3.3.5. Empirical methods

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Some crystals do not have regular faces, or cannot be measured because these are obscured by the crystal mounting. If corrections based on measurements of the crystal shape are not feasible, absorption measurements may be estimated, either from the intensities of the same reflection at different azimuthal angles ψ (see Subsection 6.3.3.6), or from measurements of equivalent reflections, by empirical methods.

There are variants of the method related to differences in experimental technique. The principles may be illustrated by reference to the procedure for a four-circle diffractometer (Flack, 1977).

Intensities [H_m] are measurements for a reflection S at the angular positions $[\Omega_m]$ , $[2\theta]$ , $[\chi_m]$ , $[\varphi_m]$ . Corrected intensities [I_m] are to be derived from the measurements by means of a correction factor [A^*_m] such that $[I_m=A^*_mH_m. \eqno (6.3.3.40)]$ It is assumed that the correction can be written in the form of a rapidly converging Fourier series $[\eqalignno {A^*_m &=\textstyle\sum\limits^\infty_{i,\,j,\,k,\, l=-\infty}\; a_{ijkl}\cos(i\Omega+j2\theta+k\chi+l\varphi) \cr & \quad + b_{ijkl}\sin(i\Omega+j2\theta +k\chi+l\varphi). &(6.3.3.41)}]$ The form of the geometrical terms may be simplified by taking advantage of the symmetry of the four-circle diffractometer. If it is assumed that diffraction is invariant to reversal of the incident and diffracted beams, the settings $[\Omega,2\theta,\chi,\varphi]$ ; $[\Omega,-2\theta,-\chi,\pi+\varphi]$ ; $[-\Omega,-2\theta,\pi+\chi,\varphi]$ ; $[-\Omega,2\theta,\pi-\chi,\pi+\varphi]$ ; $[\pi+\Omega,-2\theta,\chi,\varphi]$ ; $[\pi+\Omega,2\theta,-\chi,\pi+\varphi]$ ; $[\pi-\Omega, 2\theta,\pi+\chi,\varphi]$ ; $[\pi -\Omega, -2\theta,\pi-\chi,\pi+\varphi]$ are equivalent. In shorthand notation, the series (6.3.3.41) reduces to $[\eqalignno{ A^*_m &=\sum a_{\rm cccc}+a_{\rm ccsc}+a_{\rm sccc}+a_{\rm scsc} + a_{\rm sssc}+a_{\rm sscc} \cr &\quad +a_{\rm cssc}+a_{\rm cscc} +b_{\rm cccs}+b_{\rm ccss} +b_{\rm sccs} + b_{\rm scss} \cr &\quad+b_{\rm ssss}+b_{\rm sscs}+b_{\rm csss}+b_{\rm cscs}. & (6.3.3.42)}]$ The range of indices for some terms may be restricted by noting other symmetries in the diffraction experiment. Thus, equation (6.3.3.40) will define the absorption correction for measurements of the incident-beam intensity, with $[\Omega=2\theta=0]$ . Since with this geometry the correction will be invariant to rotation about the χ axis, the coefficients for the function involving $[\cos(i\Omega)\cos(\,j2\theta)]$ must vanish if the χ index, k, is non-zero. By similar reasoning with the $[\varphi]$ axis along the incident beam, one may deduce that coefficients for $[\sin(i\Omega)\cos(\,j2\theta)\sin(k\chi)]$ will vanish unless l = 0.

Because for a given reflection all measurements are made at the same Bragg angle, the $[2\theta]$ dependence of the correction cannot be determined by empirical methods. This factor in A is obtained from the absorption correction for a spherical crystal of equivalent radius.

Since an empirical absorption correction is defined only to within a scale factor, the scale must be specified by applying a constraint such that $[{1\over N_{\bf S}}\;\sum_{\bf S} A^*_{\bf S}=1, \eqno (6.3.3.43)]$ where $[N_{\bf S}]$ is the number of independent reflections. Equation (6.3.3.42) may be expressed in the shorthand notation $[A^*_{\bf S}=\textstyle\sum\limits_{p=0}C_p\;f_{p{\bf S}}, \eqno (6.3.3.44)]$ where [C_p] is the coefficient in a term such as $[a_{\rm ccsc}]$ or $[b_{\rm ccss}]$ and $[f_{p{\bf S}}]$ is the corresponding geometrical function. Labelling the constant geometrical term with a value of unity as [f_0] and rearranging leads to $[A^*_{\bf S} = 1+\sum_{p=1}C_p\left\{\, f_{p{\bf S}}- \displaystyle{1\over N_{\bf S}}\;\sum_{\bf S}\, f_{p{\bf S}}\right\}=1+\sum_{p=1}C_pg_{p{\bf S}}, \eqno (6.3.3.45)]$ which defines $[g_{p{\bf S}}]$ .

Equation (6.3.3.40) is now expressed as $[I_{m{\bf S}}=H_{m{\bf S}}+H_{m{\bf S}}\textstyle\sum\limits_{p=1}\;C_p g_{p{\bf S}}, \eqno (6.3.3.46)]$ in which the coefficients [C_p] are to be chosen so that the values of $[I_{m{\bf S}}]$ for each S are as near equal as possible. Since the values within each set will not be exactly equal, we rewrite (6.3.3.46) as $[\Delta_{m{\bf S}}-H_{m{\bf S}}=-I_{\bf S}+H_{m{\bf S}}\textstyle\sum\limits_{p=1}\; C_p g_{p{\bf S}}, \eqno (6.3.3.47)]$ in which the mean intensity $[I_{\bf S}]$ and the [C_p] are chosen to minimize $[\textstyle\sum_{{\bf S},m}w^2_{\bf S}\Delta{}^2_{m{\bf S}}]$ , where $[\Delta_{m{\bf S}}=I_{m{\bf S}}-I_{\bf S}, \eqno (6.3.3.48)]$ and $[w_{\bf S}]$ is the weight for that reflection.

If the equation to be solved $[-w_{\bf S} H_{m{\bf S}}\simeq -w_{\bf S} I_m+\textstyle\sum\limits_{p=1}\,C_p g_{p{\bf S}} w_{\bf S}I_{m{\bf S}} \eqno (6.3.3.49)]$ is written in the shorthand form $[{\bi D=FC}, \eqno (6.3.3.50)]$ in which D corresponds to $[-w_{\bf S} H_{m{\bf S}}]$ , the [I_m] and [C_p] correspond to C, with $[(-w_{\bf S})]$ and $[w_{\bf S} g_{p{\bf S}}H_{m{\bf S}}]$ corresponding to F, the solution to (6.3.3.50) can be determined from the normal equations $[{\bi C}=({\bi F}^T{\bi F})^{-1}{\bi F}^T{\bi D}, \eqno (6.3.3.51)]$ where $[{\bi F}^T]$ is the transpose of F. This procedure suffers from the disadvantages of requiring a matrix inversion whenever the set of trial functions (i.e. those multiplied by the coefficients [C_p] ) is modified. The tedious inversion of the normal equations, described by (6.3.3.51), may be replaced by a simple inversion via the Gram–Schmidt orthogonalizing process, i.e. by calculating a matrix W with mutually orthogonal columns $[{\bi W}_j]$ such that $[{\bi W}_1={\bi F}_1]$ $[{\bi W}_j={\bi F}_j-\textstyle\sum\limits^{j-1}_{k=1}\;({\bi F}_j\cdot{\bi W}_k){\bi W}_k/{\bi W}^2_k. \eqno (6.3.3.52)]$ The minimizing of (D − FC)² is replaced by minimizing (D − WA)². Differentiating with respect to [a_j] yields $[a_j={\bi D}\cdot {\bi W}_j \over {{\bi W}^2_j}. \eqno (6.3.3.53)]$ If equation (6.3.3.52) is written as $[{\bi F}={\bi WB},]$ where the upper triangular matrix B is $[\eqalignno{ b_{ij}={\bi F}_j\cdot{\bi W}_i/{\bi W}^2_i,\quad i \lt j; \quad&b_{ij}=1, i=j; \cr &b_{ij}=0, i \gt j, &(6.3.3.54)}]$ the vector determining the coefficients is $[{\bi C}={\bi B}^{\rm -1}{\bi A}, \eqno (6.3.3.55)]$ in which the inversion of B is straightforward.

In difficult cases, with data affected by errors in addition to absorption, the method described may give physically unreasonable absorption corrections for some reflections. In such cases, it may help to impose the approximate constraints $[\textstyle\sum\limits_{\bf S}\, w^2_{\bf S}\, H_{m{\bf S}}\Big/\textstyle\sum\limits_{\bf S}\;w^2_{\bf S}=\textstyle\sum\limits_{\bf S}\;w^2_{\bf S}\, I_{m{\bf S}}\Big/\textstyle\sum\limits_{\bf S}\;w^2_{\bf S}. \eqno (6.3.3.56)]$ If $[m=1,2,\ldots,M]$ , this reduces to the M constraint equations $[\sum_{p=1}\;C_p\left\{ \displaystyle{\sum_{\bf S}\,w^2_{\bf S}\,H_{m{\bf S}}\,g_{p{\bf S}}\over \sum_{\bf S}\,w^2_{\bf S}}\right\} = \,\sum_{p=1}\;C_p\left\{ \displaystyle{\varepsilon w_m \sum_{\bf S}\,w^2_{\bf S}\, H_{m{\bf S}}\, g_{p{\bf S}}\over \sum_{\bf S}\,w^2_{\bf S}}\right\} =0, \eqno (6.3.3.57)]$ where [w_m] is the square root of the weight for the weighted mean of the equivalent reflections [H_m] , defined as $[H_m=\textstyle\sum\limits_{\bf S} w^2_{\bf S} H_{m{\bf S}}\Big/\textstyle\sum\limits_{\bf S} w^2_{\bf S}\quad \, \hbox{for { } each { }{\bf S}}, \eqno (6.3.3.58)]$ and the multiplier $[\varepsilon]$ controls the strength with which the additional constraints are enforced. With the additional constraint equations, the sum of squares to be minimized, corresponding to (6.3.3.48), becomes $[\textstyle\sum\limits_{{\bf S},m}\, w^2_{\bf S}(I_{m{\bf S}}-I_m)^2+\textstyle\sum\limits_m\, \varepsilon^2w^2_m(H_m-I_m)^2. \eqno (6.3.3.59)]$

A closely related procedure expressing the absorption corrections as Fourier series in polar angles for the incident and diffracted beams is described by Katayama, Sakabe & Sakabe (1972). A similar method minimizing the difference between observed and calculated structure factors is described by Walker & Stuart (1983). Other experimental techniques for measuring data for empirical absorption corrections that could be analysed by the Fourier-series method are described by Kopfmann & Huber (1968), North, Phillips & Mathews (1968), Flack (1974), Stuart & Walker (1979), Lee & Ruble (1977a,b), Schwager, Bartels & Huber (1973), and Santoro & Wlodawer (1980).

References

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Flack, H. D. (1977). An empirical absorption–extinction correction technique. Acta Cryst. A33, 890–898Google Scholar

Katayama, C., Sakabe, N. & Sakabe, K. (1972). A statistical evaluation of absorption. Acta Cryst. A28, 293–295.Google Scholar

Kopfmann, G. & Huber, R. (1968). A method of absorption correction for X-ray intensity measurements. Acta Cryst. A24, 348–351.Google Scholar

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Stuart, D. & Walker, N. (1979). An empirical method for correcting rotation-camera data for absorption and decay effects. Acta Cryst. A35, 925–933.Google Scholar

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International Tables for Crystallography (2006). Vol. C. ch. 6.3, pp. 607-608