In the case of refinement of X-ray data, the situation is a little different: the relationships between the calculated quantity, F_calc, and the adjustable parameters are known (what might be comparable to knowing m and b in the previous example); the adjustable parameters are the x, y and z-coordinates and the temperature factors for each atom plus an overall scale factor. The calculated structure factor, F_c, is not linear in these “unknowns”:

The relationship can be made linear by approximating the function as a Taylor series:

If the Taylor series is truncated after the first power in Δx, the structure factor expression is linear with respect to the Δx_i, the shifts in the parameters needed to improve the model based on the approximate parameters obtained from the initial solution of the structure.

Similar to the earlier example, the quantity to be minimized is

(with X-ray data, it is common to assign a weighting factor, w_hkl, that reflects the reliability of the individual observed intensities - large intensities generally have smaller errors and, thus, larger weighting factors).