Chapter 4
WEIGHTING
4.1 Why weighting of observation is necessary
To obtain reasonable distribution of densities, it is reported that observed diffraction data need to be weighted with a proper function. Without weighting, residuals of observed and calculated structure factors, \((F_\mathrm {o} – F_\mathrm {c})\), of low-\(Q\) (large \(d\)-spacing) region tend to be very large.
The necessity of weighting in MEM for crystallography comes from intrinsic nature of crystals. Although the information theory assumes that the probabilities of individual events to occur are independent of each other, electron/nuclear densities or their Patterson functions in crystals are not random but smooth, having strong correlation with vicinal positions. Therefore, the left-hand values of Eq. (2.4) is also smooth in real space. In reciprocal space, such nature of crystals causes systematic decrease of structure factor amplitudes with decreasing \(d\)-spacing, and amplitudes of the left-hand values of Eq. (2.5) also decrease with decreasing \(d\)-spacing. On the other hand, the right-hand values of Eqs. (2.4) and (2.5) are functions of \(\Delta F\), which should not depend on \(d\)-spacing but reflect random errors of observations. MEM formalism forces these two different kinds of terms to be equivalent, and this is why a proper weighting is necessary to suppress systematic dependency of \(\Delta F\) to \(d\)-spacing.
4.2 Weighting based on exponential of \(d\)-spacing
4.3 Weighting based on \(n\)-th power of \(d\)-spacing
Weighting in the \(F\) constraint on the basis of the lattice-plane spacing was first proposed by de Vries et al [21]. Its effectiveness was later confirmed by some other researchers including Hofmann et al. [22].
Let \(\bm {s}_j\) be the reciprocal-lattice vector (\(= h \bm {a}^* + k \bm {b}^* + l \bm {c}^*\)) for reflection \(j\), \(x\) the real number for weighting, and \(d_j\) the lattice-plane spacing (\(= 1/|\bm {s}_j |\)). Then, the weighting factors, \(w_j\), in Eq. (3.9) is given by \begin {equation} \begin {split} w_j &= \frac {1}{|\bm {s}_j|^x} \left [ \sum _{i=1}^{N_F} \frac {1}{|\bm {s}_i|^x}\right ] \\ &= {d_j^x} \left [ \sum _{i=1}^{N_F} {d_i^x}\right ] . \end {split} \label {eq:w_j} \end {equation}
If \(x = 0\), no weight is imposed in the same way as with PRIMA [9].
This weighting scheme is suitable for X-ray diffraction where parts of low-\(Q\) reflections often have very large \(\Delta F_j\) values. Values of around 2 are usually recommended in X-ray powder diffraction; \(x\) is increased as the quality of intensity data is improved. In the case of single-crystal X-ray diffraction, de Vries et al. [21] empirically found \(x=4\) to be optimum. On the other hand, this approach is not suited for the analysis of neutron diffraction data, where \(b_\mathrm {c}\) values of constituent elements remain constant regardless \(Q\), particularly those measured at low temperature. Therefore, \(x\) is usually set at 0 in neutron diffraction.
Ad-hoc weighting described above will be effective, particularly, in MEM analysis from X-ray diffraction data, where \(\sigma _j (\bm {h}_j)\)’s tend to be estimated at unreasonably large values for low-\(Q\) reflections.
