Error analysis

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

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. 8.7, p. 729

Section 8.7.4.4.7. Error analysis

P. Coppens,^a Z. Su^b and P. J. Becker^c

^a 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,^bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and ^cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.4.4.7. Error analysis

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In the most general case, it is not possible to obtain x, and thus M(h) directly from R. Moreover, it is unlikely that all Bragg spots within the reflection sphere could be measured. Modelling of M(h) is thus of crucial importance. The analysis of data must proceed through a least-squares routine fitting $[R_{\rm calc}]$ to $[R_{\rm obs}]$ , minimizing the error function $[\varepsilon = \displaystyle\sum\limits_{{\bf h} \atop{\rm observed}}\, \displaystyle{1\over \sigma^2(R)}\, [R_{\rm obs}({\bf h})- R_{\rm calc}({\bf h})]^2, \eqno (8.7.4.48)]$ where $[R_{\rm calc}]$ corresponds to a model and σ²(R) is the standard uncertainty for R.

If the same counting time for $[I_\uparrow]$ and for $[I_\downarrow]$ is assumed, only the counting statistical error may be considered important in the estimate of R, as most systematic effects cancel. In the simple case where α = π/2, and the structure is centrosymmetric, a straightforward calculation leads to $[{\sigma^2(x)\over x^2} = {\sigma^2(R) \over R^2}\, {R \over (R-1)^2}; \eqno (8.7.4.49)]$ with $[{\sigma^2(R) \over R^2} \sim {1\over I_{\uparrow}} + {1\over I_\downarrow}, \eqno (8.7.4.50)]$ one obtains the result $[{\sigma^2(x) \over x^2} = \textstyle{1\over8}\, \displaystyle{(F^2_N+M^2) \over (F_NM)^2}. \eqno (8.7.4.51)]$ In the common case where $[x\ll 1]$ , this reduces to $[{\sigma^2(x) \over x^2} \sim \textstyle{1\over 8}\, \displaystyle{1\over M^2} = {1\over 8F^2_N}\, {1\over x^2}. \eqno (8.7.4.52)]$ In addition to this estimate, care should be taken of extinction effects.

The real interest is in M(h), rather than x: $[{\sigma^2(M) \over M^2} = {\sigma^2(x) \over x^2} + {\sigma^2(F_N) \over F^2_N}. \eqno (8.7.4.53)]$ If [F_N] is obtained by a nuclear neutron scattering experiment, $[\sigma^2(F_N)\sim a+bF^2_N,]$ where a accounts for counting statistics and b for systematic effects.

The first term in (8.7.4.53) is the leading one in many situations. Any systematic error in [F_N] can have a dramatic effect on the estimate of M(h).

References

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 729