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A NEW METHOD FOR DETERMINING COMPLEX CRYSTAL STRUCTURES
In the paper under review¹ an exceptionally simple method is developed for determining the signs of structure amplitudes, making it possible, after a short processing of the experimental data, to proceed to the construction of Fourier series of the electron density of the crystal. The method is completely free from any assumptions about the structure—it does not even require knowledge of the chemical formula of the compound; only information about the unit-cell composition of the substance under investigation is desirable.
The essence of the method is as follows. From the intensities of X-ray reflections the structure amplitudes \(F_H\) are calculated by well-known methods; the values of \(F\) are put on an absolute scale; and the average atomic factor is found by the usual averaging methods. After this, for each reflection we have knowledge of the absolute magnitudes of the unitary structure amplitudes \(U_H\), i.e., of quantities which are equal to unity if all atoms scatter in the same phase.
It can be shown that the following equality holds:
\[ \left(|U_H|+|U_K|\right)^3 = 1-D_{HK}+S_H S_K\left(U_{H+K}+U_{H-K}\right)+U_{H+K}U_{H-K}. \]
Here \(H\) and \(K\) symbolize triples of diffraction indices \(h_1h_2h_3\) and \(k_1k_2k_3\), while \(S_H\) and \(S_K\) are the signs of \(U_H\) and \(U_K\). The quantity \(D_{HK}\) is a positive quantity less than 2.
If \(D_{HK}\) is unknown, the expression written above can be used as an inequality that, in some cases, makes it possible to determine \(S_H\) and \(S_K\), provided that the signs of \(U_{H+K}\) and \(U_{H-K}\) are known. This circumstance has already been used in the work of Patterson (see, for example, review²), but the calculations proved complicated and for the most part did not lead to success.
In the paper under review attention was drawn to the fact that the quantity \(D_{HK}\) has a very sharp maximum at \(D_{HK}=1\). Even for a small number of pairs of indices \(H_iK_i\), the mean value \(\overline{D_{H_iK_i}}=1\). If so, then
\[ \left(\overline{|U_{K_i}|+|U_{H+K_i}|}\right)^3 = \overline{ S_{K_i}S_{H+K_i}U_H + S_{K_i}S_{K_i+H}U_{H+2K_i} + U_{H+2K_i}U_H }. \]
The right-hand side of the equality must be positive. It is not difficult to show that in a number of cases this is possible only when the first term on the right-hand side is positive, i.e., when
\[ S_H = S\left(\overline{S_{K_i}S_{H+K_i}}\right). \]
Let us denote by \(\sigma\) the root-mean-square value of the unit structural amplitude, and call “large” those values of structural amplitudes that exceed \(1.5\sigma\).
It can be shown that, for \(\sigma > 0.20\), the mean value \(\overline{|S_{K_i}S^{*}_{H+K_i}|}\), taken over pairs of large structural amplitudes, is equal to 1. This means that for each pair of large structural amplitudes, with probability practically equal to unity,
\[ S_H = S_{K_i} S_{H+K_i}. \]
In the case of a Gaussian distribution, structural amplitudes with large values will constitute approximately \(12\text{–}15\%\) of the total number of reflections. This number is undoubtedly sufficient so that, from combinations \(K_i\) and \(H+K_i\), taken among the large amplitudes, the signs of all the remaining ones may be determined.
The equality written above is used at the first stages of the work to determine the signs of large structural amplitudes (\(H\), \(K\), and \(H+K\) are selected among the large amplitudes). We have the right to assign arbitrary signs to three amplitudes of general-type indices (corresponding to the three possibilities of choosing the origin of coordinates at the center of symmetry). Next, we denote by conventional digits the signs of 10–15 amplitudes; after this we begin to establish the signs by means of the equality \(S_{H+K} = S_H S_K\). For example, if the signs \(+\) are assigned to 057 and 311, then the reflection 368 must also have the sign \(+\). In the process of applying the equality, the conventional digits are deciphered, and over a period of about 10–12 working days it is possible to establish the signs of several hundred structural amplitudes.
It should be emphasized that the new method of work frees us from the trial-and-error method, from the method of multi-dimensional vectors, from the calculation of signs of structural amplitudes, and permits us, after a small amount of work, to proceed directly to constructing a series of electron density. The method probably saves at least \(3/4\) of the time hitherto spent on determining the structure and, most importantly, permits us to undertake the solution of a structural problem even in the case when we do not have the slightest idea of how the atoms are arranged in the cell (we do not possess a rough trial structure).
We mentioned above that the equality \(S_{H+K} = S_H S_K\) is realized with probability practically equal to unity if \(\sigma > 0.20\). What structures does this condition correspond to? The value of \(\sigma\) is related to the structure in the following way:
\[ \sigma^2 = \frac{ \displaystyle \sum_{1}^{N} Z_j^2 }{ \left( \displaystyle \sum_{1}^{N} Z_j \right)^2 }, \]
where \(Z_j\) is the number of electrons of the \(j\)-th atom. If all atoms are identical, then
\[ \sigma = \frac{1}{\sqrt{N}}. \]
Consequently, the condition \(\sigma > 0.20\) corresponds to the requirement that there be no more than 25 atoms in the cell. Structures such as, for example, the structure of anthracene (28 carbon atoms in the cell) will be solved exceptionally easily by this method.
However, the method is also applicable to more complicated cases—for a number of atoms in the cell equal to 200. And this (see, for example, ²) is the limit of the possibilities for direct determination of a structure by the methods of X-ray structural analysis.
The point is that, by forming the combinations \(S_H S_K\), we can determine the sign \(S_{H+K}\) from many pairs \(S_H S_K\) (as a rule, in 5 to 20 ways). If \(\sigma < 0.20\), then the equality \(S_{H+K} = S_H S_K\) will hold with a probability less than 1, but still sufficiently high. With a number of atoms in the cell of about 200, this equality will hold on average in two cases out of three. Thus, the new method can be applied to centrosymmetric crystal structures of any degree of complexity.
The method was applied with complete success to establish the structure of the monoclinic form of metaboric acid (12 molecules in the cell!). All methods that existed up to now proved powerless in this case.
Thus, the method reviewed here has greatly expanded the possibilities of X-ray structural analysis.
A. I. Kitaigorodskii
References
- W. H. Zachariasen, Acta Cryst. 5, 68, 1952.
- A. I. Kitaigorodskii, UFN, No. 1, 1952.