Explicit expressions for structure factor least-squares

3 Explicit Expressions for Structure Factor Least-Squares 

In the least-squares treatment of diffraction intensity data, the experimental quantities ft are usually defined as either the square of the structure factor, F2, or the structure factor, F. The discrepancy between observation and calculation is then 

Unlike the treatment described in the previous section, the structure factors are not linear functions of the unknowns Xj, as was the case for the observations described by expression (4.3). The equations of the preceding section can only be retained by the approximation that all the second and higher derivatives are zero. The price paid for this assumption is that the equations are no longer exact, so that a single calculation no longer leads to the minimum of the error function. If the deviations from linearity are not pronounced, the minimum may still be reached, but a number of iterations may be necessary to achieve convergence. 

To linearize the equations, the slope of the error function S, S j may be developed as a Taylor series in the current values of the unknowns Xj. For the y th unknown, we obtain 

Linearization is accomplished by truncating the series of Eq. (4.14) after the first derivative, or 

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