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. Language: English