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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

$${\rho}(r) = \frac {1}{V} \sum_H F_H \exp (-2{\pi}iH{\cdot}r)$$ (17)

and upon squaring this function becomes

$$\rho^2(r)=\left[\frac{1}{V} \sum_L F_L \exp (-2{\pi}iL{\cdot... ...ime}}F_LF_{L^{\prime}} \exp (-2{\pi}i(L + L^{\prime}){\cdot}r).$$ (18)

(18) is rewritten by setting $H = L + L^{\prime}$ and $K = L^{\prime}$ to

$$\rho^2(r) = \frac{1}{V^2} \sum_H \sum_K F_K F_{H-K} \exp (-2{\pi}iH{\cdot}r).$$ (19)

Because $\rho^2(r)$ is also a periodic function it can be written, by analogy with (17), as

$$\rho^2(r) = \frac{1}{V} \sum_H G_H \exp (-2{\pi}iH{\cdot}r)$$ (20)

in which G_H is the structure factor of the squared structure. Comparing (19) and (20) it follows that

$$G_H = \frac{1}{V} \sum_K F_K F_{H-K}.$$ (21)

The structure factor G_H is:

$$G_H = \sum^N_{j=1} g_j \exp 2 {\pi}i(H{\cdot}r_j)$$ (22)

in which g_j is the form factor of the squared atoms. For equal atoms (22) reduces to

$$G_H = g \sum^N_{j=1} \exp 2 {\pi}i(H{\cdot}r_j).$$ (23)

The normal structure factor for equal atoms is

$$F_H = f \sum^N_{j=1} \exp 2{\pi}i(H{\cdot}r_j).$$ (24)

Thus from (23) and (24) we obtain

$$G_H = \frac{g}{f} F_H.$$ (25)

Finally from (21) and (25) it follows that

$$F_H = \frac{f}{g} \frac{1}{V} \sum_K F_KF_{H-K}$$ (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 |F_KF_H-K|. Rewriting (26) to

$$\vert F_H\vert\exp i{\phi}_H = \frac{f}{gV} \sum_K \vert F_KF_{H-K}\vert \exp i({\phi}_K + {\phi}_{H-K})$$ (27)

and considering a reflection with large |F_H| it can therefore be assumed that the terms with large |F_KF_H-K| have their angular part approximately equal to the angular part of |F_H| itself, illustrated in Fig. 8. For one strong |F_KF_H-K| this leads to:

$$\exp i\phi_H \approx \exp i(\phi_K + \phi_{H-K})$$ (28)

or

$$\phi_H \approx \phi_K + \phi_{H-K}$$

or

$$\phi_{-H} + \phi_K + \phi_{H-K} \approx 0.$$ (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

$$\exp i\phi_H = \frac{\sum_k \vert F_KF_{H-K}\vert\exp i(\phi... ...ert\sum_K\vert F_KF_{H-K}\vert\exp i(\phi_K + \phi_{H-K})\vert}$$ (30)

in which K ranges over a limited number of terms may be very helpful.

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

$$\tan \phi_H = \frac{\sum_K E_3 \sin (\phi_K + \phi_{H-K})}{\sum_K E_3 \cos (\phi_K + \phi_{H-K})}$$ (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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