X-RAY SCATTERING IN TWO DIMENSIONS FROM SHAPES WITH AN INCLUSIONother half, and then expand the composite function as a Fourier series. The principal difficulty caused by the inclusion is that the expansion coefficients do not decrease as rapidly as in the case without an inclusion. The method will be applied to a physical three-dimensional case in a future publication.We wish to thank Mr M. Woodcock for advice and assistance with the programming. Phase expansion starting from a few initial phases is investigated with reference to the size of the starting set, phase errors in the starting set, the lower limit of the E value in expansion and the different phase-determining formulae. The results stress the need for a sufficient size of the initial phase set with small phase errors for subsequent application of a phase-expanding procedure. The common basis of phase-expansion procedures is shown to consist of a cyclic modification of the preliminary structure and explains the impossibility of correcting substantial errors in already known phases associated with larger E values by subsequent phase determination for smaller E values. The phenomenon of losing structure information by careless application of the tangent formula and consequently the appearance of partial structures is pointed out. This information-destroying phase expansion is shown not to exist in the 'phase-correction' procedure.
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