International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 729
Section 8.7.4.4.7. Error analysisa 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 |
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 to , minimizing the error function where corresponds to a model and σ2(R) is the standard uncertainty for R.
If the same counting time for and for 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 with one obtains the result In the common case where , this reduces to In addition to this estimate, care should be taken of extinction effects.
The real interest is in M(h), rather than x: If is obtained by a nuclear neutron scattering experiment, 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 can have a dramatic effect on the estimate of M(h).