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Next: 2.2 The Application of Direct Methods Atoms Up: 2. Special Methods Previous: 2. Special Methods

2.1 Linear Structure Factor Equations

In many cases two coordinates (e.g. xj, yj) 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 (zj). The method of linear structure factor equations (SFE) by Kutschabsky2 and by Kutschabsky and Höhne3 allows us to calculate these atomic coordinates (zj) 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

\begin{displaymath}
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})\end{displaymath} (2.1)

where P is the number of equipoints in the unit cell, N is the number of symmetrically independent atoms, fsj 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 Rs and the translation ts we obtain:

\begin{displaymath}
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)]\end{displaymath} (2.2)

The separation into the components leads to

\begin{displaymath}
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)]\end{displaymath} (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

\begin{displaymath}
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)}\end{displaymath} (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 aj, bj, cj and dj.

aj = aj(h, k, fj, xj, yj) etc. where the exact form of dependence on h, k, fj, xj and yj 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 zj 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 + (Sj(1))2 = 1. The accuracy of the values zj may be further improved by using F(hkl) with L larger than one.


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Next: 2.2 The Application of Direct Methods Atoms Up: 2. Special Methods Previous: 2. Special Methods

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