Abstract
In a recent publication1 the author has given a brief account of a new method for the direct determination of the components of interatomic distances in crystals. The method is based on the application of the results of the theory of Fourier series. It is the object of the present paper to give a more detailed account of the underlying theory and to describe the application of the method to specific examples. In addition, two new results are presented which promise to increase the usefulness of the method for the investigation of complex structures.
\({G_h} = \sum {m_s}{\text{exp}}.2\pi i(h.{r_s})\)(16)
\({\left| {Gh} \right|^2} = {\text{ }}{\sum _{ss}}r{n_s}{m_S}{\text{cos 2}}\pi (h.({{\text{r}}_{\text{s}}}---{r_S}))\)(31)
Wir sehen daß die Beträge |Gn| nur die Differenzen der Basiskoördinaten, r s — r S ’, enthalten.
P. P. Ewald, Z. Krist.56 (1921) p. 129
Now there has been developed a procedure for the straightforward interpretation of experimental data which provides much definite information about the structure of the crystal, leading in some cases to the immediate determination of the structure and in others to the limitation to a restricted number of possibilities. This method, developed by Patterson after preliminary work by Warren, involves the calculation of Fourier series showing the magnitudes and orientations of vectors drawn between atoms in the crystal or their projections on crystal-lographic planes or lines.
L. Pauling, Current Sci. (1937) p. 21
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References
Patterson, A. L., Physic. Rev. 46 (1934) 372.
See for example Bragg, W. L., and West, J., Z. Kristallogr. 69 (1928) 118.
See Bragg, W. L., Proc. Roy. Soc. London A 123 (1929) 537, for details and earlier references.
See Bragg, W. L., loc. cit. also West, J., Z. Kristallogr. 74 (1930) 306
Patterson, A. L., Z. Kristallogr. 76 (1930) 177, 187.
Such a series might represent the average density of electrons in planes parallel to (100) for an orthorhombic crystal, in which case of course we would have F(h) = F(h00).
The equivalent expressions (4) and (6) are known to the theory of Fourier series as the ‘Faltung’ of p(x). In discussing crystal problems, it has seemed convenient to use the term ‘F 2-series’ for series of the type (6) and ‘F-series’ for the type represented by (1) and (2).
[The following note is taken from Patterson’s paper cited sub1 this Vol. p. 457: The result obtained here is an extension of the application to crystals of the theory of scattering of X-rays in liquids reported by Gingrich and Warren at the Washington meeting of the American Physical Society (B. E. Warren and N. S. Gingrich, Phys. Rev. 46 (1934) 368) and arose in a discussion of that work.’ Ed.]
These f-factors correspond directly to the atomic f-factors of X-ray diffraction theory. The atoms of the present series are assumed to be at rest. The f-factors are of course functions of h, being the coefficients of a distribution of period d having an atom at the origin.
e.g. James, R. W., and Brindley, G. W., Z. Kristallogr. 78 (1931) 470.
The functions f r and t r are complicated functions of the indices (hkl) and of the lattice constants. In practice they are tabulated as functions of sin ϑ/λ, where ϑ is the Bragg angle for the plane in question when a wavelength λ is used. By the Bragg equation, sin ϑ/λ = 1/2d. The spacing of can be calculated from the indices and the lattice constants and thus the appropriate f for the plane can be determined. This property of the f’s which is familiar to those who work in the field will not be expressed in the formulae in the text. Whenever an f -value or a f -value or any quantity derived from these is mentioned in connection with a Fourier coefficient of indices (hkl), the appropriate value derived as sketched above is the value to be used. This process is of course best carried out graphically by reciprocal lattice methods. See for example Bernai, J. D., Proc. Roy. Soc. London B 113 (1926 117.
Polya, G., Z. Kristallogr. 60 (1924) 278
Niggli, P., Z. Kristallogr. 60 (1924) p. 283.
Patterson, A. L., Z. Kristallogr. 90 (1935) 543.
Preliminary results of the application of the new method to two first mentioned substances have already been published (I.c.). Here the method which has been used to derive these results will be emphasized. References to the original structure determinations will be given below. It might be noted here that no new data have been obtained for any of these crystals, and that the results which we obtain are in close agreement with the published structures. This is not surprising as the substances were chosen as well established structures which would illustrate the various aspects of the new method.
Beevers, C. A., and Lipson, H., Philos. Mag. 17 (1934) 855. I am indebted to Prof. W. L. Bragg for the details of this method before its publication.
Beevers, C. A., and Lipson, H., Proc. Roy. Soc. London A 146 (1934) 570. I am very much indebted to these authors for a manuscript of their paper before publication. I am also indebted to Prof. W. L. Bragg, who told me of their work, and suggested this substance as an example for trial by the new method.
I wish to thank Miss. E. L. Knight for her assistance in this computation.
Expressed as fractions of the cell translation.
The letter A indicates the CuS distance, and B the S-S distance. The small letters correspond to Cu-O distances.
The weight of a peak is of course strictly proportional to its area. The heights are, however, much easier to obtain, and we are justified in using this classification as long as too much importance is not attached to it.
That such effects are present is shown by the large negative areas in the plot. These cannot be real, and must be due to the premature cutting off of the series.
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Bijvoet, J.M., Burgers, W.G., Hägg, G. (1972). The Patterson-synthesis. In: Bijvoet, J.M., Burgers, W.G., Hägg, G. (eds) Early Papers on Diffraction of X-rays by Crystals. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6878-0_7
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