UDC 548.11

CRYSTALLOGRAPHY

V. I. SIMONOV, A. M. VAISBERG

CALCULATION OF THE SIGNS OF STRUCTURE AMPLITUDES FROM A SECTION OF THE DOUBLED FUNCTION OF INTERATOMIC VECTORS

(Presented by Academician N. V. Belov on 22 IX 1969)

The Fourier transform of the minimization function ((^{1,2})) makes it possible to solve the problem of initial phases in the determination of the atomic structure of crystals of a certain complexity ((^{3-6})). The success of this method prompted us to analyze other functions, suitable for this purpose, for extracting the structure from the distribution of interatomic vectors. Very encouraging results have been obtained in calculating the signs of structure amplitudes from special sections of the doubled function of interatomic vectors. The generalized Patterson function, which has been called the doubled function of interatomic vectors, was proposed by D. Sayre ((^{7})):

[
DP(\mathbf{u}_1,\mathbf{u}_2)=\int \rho(\mathbf{r})\rho(\mathbf{r}+\mathbf{u}_1)\rho(\mathbf{r}+\mathbf{u}_2)\,dv =
]

[
= \frac{1}{V^2}\sum_{\mathbf{H}1}\sum2}
F1}F2}F}_1-\mathbf{H}_2
\exp[-2\pi i(\mathbf{H}_1\mathbf{u}_1+\mathbf{H}_2\mathbf{u}_2)]
]

Fig. 1. Parametrization of the vectors (\mathbf{u}_1) and (\mathbf{u}_2) for constructing a section of the doubled function of interatomic vectors

The function is defined in six-dimensional space; for its calculation one needs the experimental moduli of the structure amplitudes and the phases of the structural products (F_{\mathbf{H}1}F2}F_2)) give the mutual arrangement of triples of atoms in the structure, and the weights of the peaks are equal to the product of the atomic numbers forming the corresponding triple of atoms.}_1-\mathbf{H}_2}). The coordinates of the maxima of (DP(\mathbf{u}_1,\mathbf{u

The principal difficulty associated with determining the phases of structural products for centrosymmetric structures can be overcome if one uses the statistical relation of Zachariasen ((^{8})). With an accuracy up to equality, Zachariasen’s relation (F_{\mathbf{H}1}F2}F) can be replaced by the product of the moduli of the corresponding structure amplitudes.}_1-\mathbf{H}_2

Calculation and analysis of (DP(\mathbf{u}_1,\mathbf{u}_2)) in the full volume of the six-dimensional elementary cell are difficult even for the most powerful modern computers.

V. N. Biyushkin and N. V. Belov proposed and demonstrated the promise of using three-dimensional symmetric sections (DP(\mathbf{u})) of the six-dimensional function (DP(\mathbf{u}_1,\mathbf{u}_2)) for the determination of crystal structures ((^{9})).

The main difficulty in interpreting symmetric sections is due to the ambiguity of these syntheses, which is equivalent to the ambiguity of M. Buerger’s implication diagrams ((^{10})). Another class of three-dimensional sections of the function (DP(\mathbf{u}_1,\mathbf{u}_2)) was proposed by V. Hoppe ((^{11})). Hoppe’s section for a centrosymmetric structure singles out a unique image of the structure, but to construct such a section, in addition to the information used in calculating (DP(\mathbf{u}_1,\mathbf{u}_2)), it is necessary to know a non-overlapping vector that connects, in the structure, two atoms related by a center of symmetry. Thus, for the synthesis of Hoppe’s section, information is required that is necessary in constructing a superpositional—

of the synthesis ((^{1})), the Zachariasen statistical relation is also used essentially.

Suppose that we know the vector (2\mathbf r_0), joining atoms related by a center of symmetry. To construct a Hoppé section, let us define the relation between the vectors (\mathbf u_1) and (\mathbf u_2) entering into (DP(\mathbf u_1,\mathbf u_2)) as follows: (\mathbf u_1=-\mathbf r_0-\mathbf u,\ \mathbf u_2=\mathbf r_0-\mathbf u) (Fig. 1). We choose the origin of coordinates at the center of symmetry of the structure, in the middle of the vector (2\mathbf r_0). With this parametrization, the three-dimensional section (DP(\mathbf u)) of the six-dimensional function (DP(\mathbf u_1,\mathbf u_2)) can be written as:

[
DP(\mathbf u)=\frac{1}{V^2}\sum_{\mathbf H_1}\sum_{\mathbf H_2}
F_{\mathbf H_1}F_{\mathbf H_2}F_{-\mathbf H_1-\mathbf H_2}
\exp{-2\pi i[\mathbf H_1(-\mathbf r_0-\mathbf u)+\mathbf H_2(\mathbf r_0-\mathbf u)]}=
]

[
=\frac{1}{V^2}\sum_{\mathbf H_1}\sum_{\mathbf H_2}
F_{\mathbf H_1}F_{\mathbf H_2}F_{-\mathbf H_1-\mathbf H_2}
\exp{2\pi i[(\mathbf H_1-\mathbf H_2)\mathbf r_0+(\mathbf H_1+\mathbf H_2)\mathbf u]}.
]

Making the change of variables (\mathbf H_2=-\mathbf H-\mathbf H_1), we obtain

[
DP(\mathbf u)=\frac{1}{V^2}\sum_{\mathbf H}\sum_{\mathbf H_1}
F_{\mathbf H}F_{\mathbf H_1}F_{-\mathbf H-\mathbf H_1}
\exp[2\pi i(2\mathbf H_1+\mathbf H)\mathbf r_0]\exp[-2\pi i\mathbf H\mathbf u]=
]

[
=\frac{1}{V^2}\sum_{\mathbf H}\widetilde F_{\mathbf H}\exp[-2\pi i\mathbf H\mathbf u],
]

where the role of the Fourier coefficients of the function (DP(\mathbf u)) is played by the quantities

[
\widetilde F_{\mathbf H}=F_{\mathbf H}\sum_{\mathbf H_1}
F_{\mathbf H_1}F_{-\mathbf H-\mathbf H_1}
\exp[2\pi i(2\mathbf H_1+\mathbf H)\mathbf r_0].
]

Taking into account the centrosymmetry of (DP(\mathbf u)) and, consequently, the fact that (\widetilde F_{\mathbf H}) are real numbers, by simple transformations one can obtain:

[
\widetilde F_{\mathbf H}=2F_{\mathbf H}\left{
\left[\sum_{\mathbf H_1}F_{\mathbf H_1}(F_{\mathbf H+\mathbf H_1}+F_{\mathbf H-\mathbf H_1})\cos 2\pi 2\mathbf H_1\mathbf r_0\right]\cos 2\pi\mathbf H\mathbf r_0-
\right.
]

[
\left.
-\left[\sum_{\mathbf H_1}F_{\mathbf H_1}(F_{\mathbf H+\mathbf H_1}-F_{\mathbf H-\mathbf H_1})\sin 2\pi 2\mathbf H_1\mathbf r_0\right]\sin 2\pi\mathbf H\mathbf r_0
\right}=
]

[
=2F_{\mathbf H}\sum_{\mathbf H_1}F_{\mathbf H_1}F_{\mathbf H+\mathbf H_1}
\cos 2\pi(2\mathbf H_1+\mathbf H)\mathbf r_0.
]

To the accuracy with which the Zachariasen relation is satisfied, (\widetilde F) is written, in a form convenient for computations, in terms of the moduli of the structure amplitudes:

[
F_{\mathbf H}^{DP}\approx 2\sum_{\mathbf H_1}|F_{\mathbf H}F_{\mathbf H_1}F_{\mathbf H+\mathbf H_1}|
\cos 2\pi(2\mathbf H_1+\mathbf H)\mathbf r_0.
]

By analogy with superposition synthesis, in the present case one can construct the first approximation to the electron-density distribution in the crystal from the experimental moduli of the structure amplitudes (|F_{\mathbf H}|{\mathrm{exp}}) and the signs of (\widetilde F).

The calculation of (F_{\mathbf H}^{DP}) from a section of the doubled function of interatomic vectors is simpler than obtaining (\widetilde F_{\mathbf H}^{M}) from the minimization function ((^{1})), since it does not require numerical calculation of the Fourier integral from a three-dimensional function.

The fundamental difference in the initial information required for obtaining the signs of structure amplitudes from the minimization function and from the Hoppé section reduces to the use, for constructing the latter, of the principal sign relation of direct methods ((^{8})). Consequently, the proposed method for calculating the signs of structure amplitudes is based on combining the direct and Patterson approaches to solving the phase problem for centrosymmetric structures.

Of interest is a comparison of the Fourier coefficients of the Hoppe section and of Buerger’s product function (^{10}). The possibility of using the product function (\Pi(\mathbf r)) to compute the signs of structural amplitudes was shown in (^{12,13}). The Fourier coefficients of (\Pi(\mathbf r)) can be transformed to the form

[
\tilde F_{\mathbf H}^{\Pi}
=
\sum_{\mathbf H_1}
F_{\mathbf H_1}^{2} F_{\mathbf H+\mathbf H_1}^{2}
\cos 2\pi(2\mathbf H_1+\mathbf H)\mathbf r_0 .
]

The trigonometric parts of the terms composing (\tilde F_{\mathbf H}^{\Pi}) and (\tilde F_{\mathbf H}^{DP}) are the same, but the difference between the coefficients
(\left|F_{\mathbf H}F_{\mathbf H_1}F_{\mathbf H+\mathbf H_1}\right|) and
(F_{\mathbf H_1}^{2}F_{\mathbf H+\mathbf H_1}^{2})
may lead to a discrepancy in the signs of structural amplitudes computed from the Hoppe section and from the product function.

The program for computing (F_{\mathbf H}) from the Hoppe section has so far been written for the M-20 computer only for two-dimensional problems. The effectiveness of the method was tested on the ((x,y)) projection of the structure of synthetic silicate (\mathrm{Na_2Mn_2[Si_2O_7]}) ((a=8.757,\ b=13.294,\ c=5.744\ \text{\AA};\ \beta=90^\circ10';\ P2_1/n;\ Z=4)) (^{4}).

The ((x,y)) projection of this structure is characterized by the symmetry (Pgg). The Mn—Mn distance was used as the known centrosymmetric vector. An independent calculation of the signs of the reflections (hk0) and (\bar h k0), which are in fact related by symmetry, gave 75% correct signs, which coincides with the probability of satisfying the Zachariasen statistical relation (^{14}). The symmetry (Pgg) requires identical signs for (\tilde F_{hk0}) and (\tilde F_{\bar h k0}) when (h+k=2n), and different signs when (h+k=2n+1). The symmetry was taken into account very simply. If the signs of the amplitudes (hk0) and (\bar h k0) proved contradictory, the sign of the amplitude with the larger modulus was chosen. Such allowance for symmetry increased the number of correct signs to 82%. For comparison, the minimization function (M_2(x,y)) was constructed from the Mn—Mn vector; allowance for (Pgg) symmetry made it possible to raise the rank of this function to 4. Fourier inversion of (M_4(x,y)) gave 80% correct signs of the structural amplitudes. Thus, the proposed method for computing the signs of structural amplitudes may prove no less effective than Fourier inversion of the minimization function.

The computation of the signs of the structural amplitudes from the minimization function was carried out by M. I. Sirota, to whom the authors are also indebted for help in compiling the program. At various stages the work was discussed with N. V. Belov, N. P. Zhidkov, V. N. Biyushkin, A. B. Tobis, and B. M. Shchedrin; to all of them the authors express their most sincere gratitude.

Institute of Crystallography
Academy of Sciences of the USSR
Moscow

Received
3 IX 1969

REFERENCES

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8. W. H. Zachariasen, ibid., 5, p. 1, 68 (1952).
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