Gaussian integration

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. 606-607

Section 6.3.3.4. Gaussian integration

E. N. Maslen^a ^‡

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

6.3.3.4. Gaussian integration

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The integral in the transmission factor in equation (6.3.3.1) may be approximated by a sum over grid points spaced at intervals through the crystal volume. It is usually convenient to orient the grid parallel to the crystallographic axes. The grid is non-isometric, the points being chosen weighted by Gaussian constants to minimize the difference between the weighted sum at those points and the exact value of the integral.

Thus, an integral such as $[\int^b_a\,f(y)\,{\rm d} y]$ may be approximated (Stroud & Secrest, 1966) by $[\int\limits^b_a f(y)\,{\rm d} y = {b-a\over 2}\sum^n_{i=1}\; w_i\; f(y_i)+R_n, \eqno (6.3.3.33)]$ where $[y_i= \left({b-a\over 2}\right) X_i+\left({b+a \over2}\right),]$ [X_i] is the ith zero of the Legendre polynomial [P_n(X),] $[w_i={2\over (1-X^2_i)}[P'_n(X_i)]^2, \eqno (6.3.3.34)]$ and $[R_n ={(b-a)^{2n+1}(n!)^4 \over(2n+1)[(2n!)]^3}\, 2^{2n+1}f^{(2n)}(\xi), \quad -1 \lt \xi \lt 1. \eqno (6.3.3.35)]$ When applying this to the calculation of a transmission coefficient (Coppens, 1970), we commence with the a-axis grid points [x_i] selected such that $[x_i=x_{\rm min}+(x_{\rm max}-x_{\rm min})X_i, \eqno (6.3.3.36)]$ where the [X_i] are the Gaussian constants.

For each [x_i] , a line is drawn parallel to b and points are then selected such that $[y_{ij}=y_{\rm min}(x_i)+[y_{\rm max}(x_i)-y_{\rm min}(x_i)]\,X_j. \eqno (6.3.3.37)]$ The procedure is repeated for the c direction, yielding $[z_{ijk}= z_{\rm min}(x_i,y_j)+[z_{\rm max}(s_i,y_j)-z_{\rm min}(x_i,y_j)]\,X_k. \eqno (6.3.3.38)]$

To calculate the absorption corrections, the incident and diffracted wavevectors are determined. For each grid point, the sum $[T_{ijk}]$ of the path lengths for the incident and diffracted beams is evaluated. The sum that approximates the transmission coefficient is then $[A=1/V \textstyle\sum\limits_{i,\, j,\, k}\; w_iw_jw_k\exp(-\mu T_{ijk}). \eqno (6.3.3.39)]$ Gaussian constants are tabulated by Abramowitz & Stegun (1964).

Alternative schemes based on Monte Carlo and three-dimensional parabolic integration are described by de Graaff (1973, 1977).

References

Abramowitz, M. & Stegun, I. A. (1964). Handbook of mathematical functions, p. 916. National Bureau of Standards Publication AMS 55.Google Scholar

Coppens, P. (1970). The evaluation of absorption and extinction in single crystal structure analysis. Crystallographic computing, edited by F. R. Ahmed, S. R. Hall & C. P. Huber, pp. 255–270. Copenhagen: Munksgaard.Google Scholar

Graaff, R. A. G. de (1973). A Monte Carlo method for the calculation of transmission factors. Acta Cryst. A29, 298–301.Google Scholar

Graaff, R. A. G. de (1977). On the calculation of transmission factors. Acta Cryst. A33, 859.Google Scholar

Stroud, A. H. & Secrest, D. (1966). Gaussian quadrature formulas. New Jersey: Prentice-Hall.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.3, pp. 606-607