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Next: The Positive Quartet Relation Up: An Introduction to Direct Methods. The Previous: Large |EH|, |EK| and |E-H-K|: The

The Triplet Relation from Sayre's Equation

The earliest formulation of the triplet-relation (10) for the centrosymmetric case was via Sayre's equation (Sayre, 1952). This equation can be derived from Fourier theory as follows.

The electron density can be written as

\begin{displaymath}
{\rho}(r) = \frac {1}{V} \sum_H F_H \exp (-2{\pi}iH{\cdot}r)\end{displaymath} (17)
and upon squaring this function becomes

\begin{displaymath}
\rho^2(r)=\left[\frac{1}{V} \sum_L F_L \exp (-2{\pi}iL{\cdot...
 ...ime}}F_LF_{L^{\prime}} \exp (-2{\pi}i(L +
L^{\prime}){\cdot}r).\end{displaymath} (18)
(18) is rewritten by setting $H = L + L^{\prime}$ and $K = L^{\prime}$ to

\begin{displaymath}
\rho^2(r) = \frac{1}{V^2} \sum_H \sum_K F_K F_{H-K} \exp (-2{\pi}iH{\cdot}r).\end{displaymath} (19)
Because $\rho^2(r)$ is also a periodic function it can be written, by analogy with (17), as

\begin{displaymath}
\rho^2(r) = \frac{1}{V} \sum_H G_H \exp (-2{\pi}iH{\cdot}r)\end{displaymath} (20)
in which GH is the structure factor of the squared structure. Comparing (19) and (20) it follows that

\begin{displaymath}
G_H = \frac{1}{V} \sum_K F_K F_{H-K}.\end{displaymath} (21)
The structure factor GH is:

\begin{displaymath}
G_H = \sum^N_{j=1} g_j \exp 2 {\pi}i(H{\cdot}r_j)\end{displaymath} (22)
in which gj is the form factor of the squared atoms. For equal atoms (22) reduces to

\begin{displaymath}
G_H = g \sum^N_{j=1} \exp 2 {\pi}i(H{\cdot}r_j).\end{displaymath} (23)
The normal structure factor for equal atoms is

\begin{displaymath}
F_H = f \sum^N_{j=1} \exp 2{\pi}i(H{\cdot}r_j).\end{displaymath} (24)
Thus from (23) and (24) we obtain

\begin{displaymath}
G_H = \frac{g}{f} F_H.\end{displaymath} (25)
Finally from (21) and (25) it follows that

\begin{displaymath}
F_H = \frac{f}{g} \frac{1}{V} \sum_K F_KF_{H-K}\end{displaymath} (26)
which is known as Sayre's Equation. It is emphasised that, given an equal-atom structure, Sayre's equation is exact. The summation (26) contains a large number of terms; however, in general it will be dominated by a smaller number of large |FKFH-K|. Rewriting (26) to

\begin{displaymath}
\vert F_H\vert\exp i{\phi}_H = \frac{f}{gV} \sum_K \vert F_KF_{H-K}\vert \exp i({\phi}_K +
{\phi}_{H-K})\end{displaymath} (27)
and considering a reflection with large |FH| it can therefore be assumed that the terms with large |FKFH-K| have their angular part approximately equal to the angular part of |FH| itself, illustrated in Fig. 8. For one strong |FKFH-K| this leads to:

\begin{displaymath}
\exp i\phi_H \approx \exp i(\phi_K + \phi_{H-K})\end{displaymath} (28)
or

\begin{displaymath}
\phi_H \approx \phi_K + \phi_{H-K}\end{displaymath}

or

\begin{displaymath}
\phi_{-H} + \phi_K + \phi_{H-K} \approx 0.\end{displaymath} (29)
Relation (29) is identical to (16), the triplet relation. Thus by introducing the obvious argument that the most important terms in Sayre's equation (27) must reflect the phase $\phi_H$ the triplet relation is found.

In the event that only a number of larger terms in (27) are available the scaling constant f/gV has no meaning. Nevertheless most likely the phase information included in these terms is correct and thus an expression such as

\begin{displaymath}
\exp i\phi_H = \frac{\sum_k \vert F_KF_{H-K}\vert\exp i(\phi...
 ...ert\sum_K\vert F_KF_{H-K}\vert\exp i(\phi_K + \phi_{H-K})\vert}\end{displaymath} (30)
in which K ranges over a limited number of terms may be very helpful.

The so called tangent formula (Karle and Hauptman, 1956)

\begin{displaymath}
\tan \phi_H = \frac{\sum_K E_3 \sin (\phi_K + \phi_{H-K})}{\sum_K E_3 \cos (\phi_K
+ \phi_{H-K})}\end{displaymath} (31)
in which the signs of numerator and denominator are used to determine the quadrant of the phase $\phi_H$, is closely related to (30). This formula is used in almost all direct method procedures.


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Next: The Positive Quartet Relation Up: An Introduction to Direct Methods. The Previous: Large |EH|, |EK| and |E-H-K|: The

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