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2.1 Linear Structure Factor Equations

In many cases two coordinates (e.g. x_j, y_j) of any atom in the real structure are the same as those of the superposition structure. There arises the task of determining the third atomic coordinate (z_j). The method of linear structure factor equations (SFE) by Kutschabsky² and by Kutschabsky and Höhne³ allows us to calculate these atomic coordinates (z_j) directly, using the reflection of the first level of the reciprocal lattice (F(hkl)).

The basic relation follows directly from the formula for the structure factor

$$F(\textbf{H}) = \sum_{s = 1}^p \sum_{j = 1}^N (f_{sj}(\textb... ...bf{Hr}_{sj} + if_{sj}(\textbf{H}) \sin 2 {\pi}\textbf{Hr}_{sj})$$ (2.1)

where P is the number of equipoints in the unit cell, N is the number of symmetrically independent atoms, f_sj is the atomic scattering factor and $\textbf{r}_{sj}$ is the radius vector of the centre of atom s, j.

Vector $\textbf{H}$ is defined by $h\textbf{a}$* + $k\textbf{b}$* + $l\textbf{c}$*, where ($\textbf{a}$*, $\textbf{b}$*, $\textbf{c}$*) are the basic reciprocal vectors.

Using the symmetry matrices R_s and the translation t_s we obtain:

$$F(H) = \sum^p_{s = 1} \sum^N_{j = 1} f_{sj}(\textbf{H})[\cos... ...\textbf{t}_s) + i \sin 2{\pi}H(R_s\textbf{r}_j + \textbf{t}_s)]$$ (2.2)

The separation into the components leads to

$$F(H) = \sum^p_{s = 1} \sum^N_{j = 1} {\gamma}_{sj}(\textbf{H... ...pi}({\alpha}_sx_j + {\beta}_sy_j + {\gamma}_sz_j + {\delta}_s)]$$ (2.3)

(2.4)

In the most important cases the factor ${\gamma}_s$ depends only on l (not h or k). In those cases we obtain for structure factors P(hkl) with constant L

$$F(\textbf{H}) = \sum^N_{j = 1} (a_j + ib_j) C^{(L)}_j + \sum^N_{j = 1} (c_j + id_j)s_j^{(L)}$$ (2.5)

where the unknown variables cos 2${\pi}Lz_j$ and sin 2${\pi}Lz_j$ have been denoted by C^(L)_j and S^(L)_j, respectively, and their known coefficients by a_j, b_j, c_j and d_j.

a_j = a_j(h, k, f_j, x_j, y_j) etc. where the exact form of dependence on h, k, f_j, x_j and y_j may be obtained from Table 4 of International Tables for X-ray Crystallography, Vol. 1.

The 2N unknown variables C^(L)_j and S^(L)_j may be determined by a system of linear equations using $F_{\mbox{obs}}(\textbf{H}$)for $F(\textbf{H}$).

If the phases of the $F_{\mbox{obs}}(\textbf{H})$ are unknown the unobserved reflections may be used to obtain a system of homogeneous linear equations. Often it is of advantage to use in addition to the homogeneous equations one equation belonging to a strong structure factor whose phase may be fixed arbitrarily in centrosymmetrical space groups, and in the non-centrosymmetrical space groups in which the origin may have any position in the z-direction.

Because the coefficients of these equations are inaccurate and, moreover, the structure factors are zero only approximately a more accurate solution for the values C^(L)_j and S^(L)_j may be obtained by using more equations than there are variables and by minimizing the sum of the squares of the deviations $\sum\vert F_{\mbox{obs}}(\textbf{H})-F_{\mbox{calc}}(\textbf{H})\vert^2$,where $F_{\mbox{calc}}(\textbf{H})$ stands for the right side of equation (2.5) and F(H) is to be replaced by $F_{\mbox{obs}}(\textbf{H})$ in this relation. The C^(L)_j and S^(L)_j from the first calculation may be used to determine the phases of further structure factors. Taking these equations in addition to those used already, the number of equations increases and thus the accuracy of C^(L)_j and S^(L)_j is improved.

If the F(hkl) with L = 1 are used, the atomic parameters z_j of all atoms resolved in the (x, y)-projection follow from C⁽¹⁾_j= cos 2${\pi}z_j$ and S⁽¹⁾_j = sin 2${\pi}z_j$ and (C⁽¹⁾_j)² + (S_j⁽¹⁾)² = 1. The accuracy of the values z_j may be further improved by using F(hkl) with L larger than one.

Copyright © 1981, 1998 International Union of Crystallography