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Next: 2. Special Methods Up: Crystal Structure Analysis Using the `Superposition' Previous: Crystal Structure Analysis Using the `Superposition'

1. General

Quite a number of crystal structures contain parts (e.g. heavy atoms or building units) with a higher symmetry (e.g. with additional translation or pseudotranslation) compared with the whole structure. In those cases, standard methods may not lead to correct results, this is why special methods may have to be applied.

As an example let the heavy atoms (strongly reflecting atoms) have parameters xs, ys, zs and xs, ys, $\frac{1}{2}$ + zs (Fig. 1). Then they contribute only to the structure factors F(hkl) with l = 2n, since their contribution Fs(hkl) is equal to


(1.1)

Therefore the |F(hkl)|2 with l = 2n are systematically strong, compared with those with l = 2n + 1, i.e. for the mean values $\langle{I}(hkl)\rangle$of the intensities within any region of $\sigma$ values

\begin{displaymath}
\langle{I}(hkl)_{l=2n}\rangle \mbox{ strong}, \qquad \langle{I}(hkl)_{l=2n+1}\rangle \mbox{ weak}.\end{displaymath}


Figure 1: Crystal structure in P1. $\clubsuit$ - symbol for the arrangement of heavy atoms, $\diamondsuit$ and $\Box$ remaining atoms of the structure.
\begin{figure}
\includegraphics {fig1.ps}
\end{figure}

If such a systematic distribution of the intensities or a similar one occurs, it is useful, for methodical reasons, to regard the electron density $\rho(x, y,
z)$ as the sum of two - or if necessary - of several parts.

Thus the electron density function

\begin{displaymath}
{\rho}(x, y, z) = \sum_h \sum_k \sum_l F(hkl) \exp[-2{\pi}i(hx + ky + lz)]\end{displaymath}

may for the case indicated in Fig. 1 be written


(1.2)
(1.2')

where

${\rho}_{\mbox{sup}}(x, y, z)$ denotes a hypothetical structure, called the superposition structure, which is related to the real structure in the following way

\begin{displaymath}
\textstyle{\rho}_{\mbox{sup}}(x, y, z) = \frac{1}{2}[{\rho}(x, y, z) + {\rho}(x, y,
\frac{1}{2} + z)]\end{displaymath} (1.3)

In ${\rho}_{\textrm{sup}}(x, y)$ (Fig. 2) the two heavy atoms ($\clubsuit$) appear with correct weights because they are connected by a translation of c/2, whereas the remaining atoms (squares and triangles) appear with half their weights (marked by shaded symbols) at the original position and at a second point shifted relative to it by c/2.


Figure 2: Superposition structure. $\clubsuit$ - heavy atoms $\heartsuit$,$\spadesuit$ - remaining atoms with half weight.
\begin{figure}
\includegraphics {fig2.ps}
\end{figure}

Generally speaking, the symmetry of ${\rho}_{\mbox{sup}}(x, y, z)$ is identical with the symmetry of the arrangement of the heavy atoms (or other building units with higher symmetry) taken by themselves. This higher symmetry may either be strictly true for the heavy atoms taken by themselves, or only in approximation. In the latter case it may be useful to disregard deviations from the higher symmetry, to start with.

The symmetry of the complementary structure ${\rho}_{\mbox{com}}(x, y, z)$ follows from (1.2) and (1.3):

\begin{displaymath}
\textstyle{\rho}_{\mbox{com}}(x, z, y) =
 \frac{1}{2}[{\rho}(x, y, z) - {\rho}(x, y, \frac{1}{2} + z)]\end{displaymath} (1.4)

Its properties are shown in Fig. 3. Accordingly, the heavy atoms (i.e. those which occur in pairs related to a shift of c/2) are absent in ${\rho}_{\mbox{com}}(x, y, z)$, whereas any other atom appears with half its weight at its real position and with half negative weight at a position shifted by c/2 (Fig. 3). The space group symmetry of the arrangement with positive weights in ${\rho}_{\mbox{com}}(x, y, z)$ is identical with the space group of the real structure $\rho(x, y,
z)$. In a similar way, the introduction of a superposition and a complementary structure may be indicated by systematically strong reflections occurring, e.g. for h = 2n or h + k = 2n or h + k + l = 2n etc.


Figure 3: Complementary structure, $\heartsuit$, $\spadesuit$--atoms with positive half weight. $\flat$. $\natural$--atoms with negative half weight.
\begin{figure}
\includegraphics {fig3.ps}
\end{figure}

If, on the other hand, the heavy atoms (or another part of the structure, taken by itself) possesses a higher symmetry (other than translation) than the structure as a whole, the introduction of other kinds of hypothetical structures, also to be called ${\rho}_{\mbox{sup}}(x, y, z)$ and ${\rho}_{\mbox{com}}(x, y, z)$ may be of use. In this case, no systematically strong and weak reflections result.

As an example, we consider a structure in $P\overline 1$ containing two heavy atoms of the same element per unit cell. The heavy atoms considered by themselves are connected by a centre of symmetry. The structure computed with phases (signs) taken from the heavy atom contribution is necessarily centrosymmetric and is related to the real structure by its superposition with its centrosymmetric image (Fig. 4). If referred to such a partial centre of symmetry as origin, the electron density distribution of the superposition structure ${\rho}_{sup}(x, y,
z)$ may be expressed as

\begin{displaymath}
\textstyle{\rho}_{\mbox{sup}}(x, y, z) = \frac{1}{2}[{\rho}(x, y, z)
 + {\rho}(\overline{x}, \overline{y}, \overline{z})]\end{displaymath} (1.5)

The real structure or at least part of it may in many cases of this kind be obtained using well established chemical knowledge, such as atomic distances, known stereochemistry of molecules or parts of them as of coordination polyhedra etc.


Figure 4: Superposition structure (centrosymmetric). $\clubsuit$ - centrosymmetric arrangement of heavy atoms, $\heartsuit$, $\spadesuit$ - remaining atoms with half weights.
\begin{figure}
\includegraphics {fig4.ps}
\end{figure}


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Next: 2. Special Methods Up: Crystal Structure Analysis Using the `Superposition' Previous: Crystal Structure Analysis Using the `Superposition'

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