The Theory of Crystal Structure Analysis de A. Kitaigorodskii

The Theory of Crystal Structure Analysis

A. Kitaigorodskii

Structure analysis is based on the phenomena of the diffraction of radia tion by materials. In the first ten to twenty years after Laue's discovery, a very complete theory was developed for the diffraction of x-rays and, later, of electrons. This theory led to equations by means of which it was possible to compute the intensity pattern for a given structure. The theory of structure analysis came to mean that of the diffraction of radiation. In 1935, Patterson pointed out a way leading to the solution of the inverse problem: the finding of the structure from a given intensity distribution pattern. At first the conservatism of researchers, and then the war, hampered the de velopment and broad application of the ideas set forth in this work. It was only during the last ten years that all the rich possibilities of the Patterson method - the method of the analysis of the convolution of the electron density - were brought to light and applied in practice.

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Cuprins

I Mathematical Introduction.- 1 The Fourier integral and reciprocal space.- 2 Fourier transforms.- 3 Special cases of transforms.- 4 Projections and sections of transforms.- 5 Convolution of functions.- 6 Self-convolutions of functions.- II Principles of the Theory.- 1 Scattering of x-rays.- 2 Reciprocal space as the space of scattering functions.- 3 The time average of the electron density.- 4 The object considered as a system of atoms.- 5 The object as a system of particles.- 6 The form factor.- 7 Infinite o-lattices.- 8 Finite o-lattices.- 9 The ideal finite crystalline lattice.- 10 The real crystal.- 11 Structure amplitudes and products.- 12 A comparison of x-ray, electron, and neutron structure analyses.- III Structure Amplitudes and Products as Random Quantities.- 1 Statement of the problem.- 2 The distribution function of a sum of independent quantities.- 3 Gaussian representation of the distribution function of the structure amplitude of a centrosymmetric crystal.- 4 The dependence of the amplitude distribution function on the structure.- 5 Deviation of a crystal from centrosymmetry “Complete” loss of the inversion center.- 6 Statistical differences between centrosymmetric crystals and crystals without inversion centers.- 7 The distribution functions of structure amplitudes and hypersymmetry.- 8 The mean value of the structure factor and the problem of finding the unitary amplitudes.- 9 Concerning the possibility of determining the space group from intensity statistics.- 10 The distribution function of the third-order structure product.- 11 The Gaussian distribution of the structure products.- 12 The probability of a positive sign for the structure product within the limits of applicability of the Gaussian distribution.- 13 Peculiarities of incompleteaveraging of structure products.- IV The Theory of the Relations between Structure Amplitudes.- 1 Statement of the problem.- 2 Reliably positive structure products.- 3 The centrosymmetric crystal with one atom in a general position.- 4 Relationships between amplitudes and their components.- 5 Concerning the averaging of the relationships between unitary structure amplitudes and their components.- 6 The simplest inequality relating unitary structure amplitudes.- 7 Derivation of the relationships between structure amplitudes using Cauchy’s inequality. The first method.- 8 The second method of applying Cauchy’s inequality.- 9 The fundamental equation connecting structure amplitudes.- 10 Form of the connecting determinants of low orders.- 11 The determination of signs.- 12 Capabilities of the basic connecting equations for determining signs.- 13 The predominant positivity of the structure product.- 14 Finding positive structure products.- 15 Relations between the absolute magnitudes of structure amplitudes.- 16 Signs of structure products and amplitudes.- 17 The limits of possible values of structure amplitudes.- 18 Graphical representation of the connecting equations for the simplest Dm.- 19 The complete theory of the relationships between structure amplitudes.- 20 The probability of a positive sign of the structure product.- 21 Procedures for direct structure analyses.- V Analysis of the Convolution of the Electron Density.- 1 The convolution as a sum of interatomic functions.- 2 The form of the interatomic functions.- 3 Picking out the interatomic vector system from the convolution.- 4 The convolution of the electron density and crystal symmetry.- 5 Picking the structure out of the convolution.- 6 The difference inversions.- 7 Sharpening of convolutions.- VI Methodsof Obtaining Agreement between the Measured and Calculated Structure Amplitude.- 1 Antagonistic reflections.- 2 The R-factor and the correlation coefficient.- 3 General features of approximation methods.- 4 Booth’s method of “steepest descent”.- 5 Method of least squares.- 6 The differential method.- 7 The method of difference series.- 8 Accuracy factors.- 9 Computation of the errors in the determination of atomic coordinates.- 10 An approximate evaluation of the sums that enter into the error formulas.- Conclusion.