The scattering of electrons by three-dimensional potential fields, and, in particular, the potential fields associated with a crystal lattice, is considered in terms of the new approach to physical optics recently proposed by Cowley & Moodie. The three-dimensional potential field is approximated by a large number of closely spaced two-dimensional potential distributions. An expression is obtained for the wave function on an arbitrary plane of observation for a point source of electrons at a finite distance or at infinity (parallel irradiation). Particular cases considered are the wave function at the exit surface of a crystal, corresponding to the image produced by an ideal electron microscope, and the diffraction pattern, or angular scattering function, of a crystal.Two methods of approximation to the general expressions are discussed. In the first the wave function on the plane of observation is expressed as the sum of the contributions of electron waves scattered 1, 2 ..... n .... times. The contribution from singly scattered waves is shown to be equivalent to the amplitude distribution given by the usual kinematic theory of scattering.The second method of approximation corresponds to the successive increase in the number of two-dimensional distributions by which the three-dimensional potential field is approximated.It is shown that for the special case, in which only the incident beam and one diffracted beam have appreciable amplitude in the crystal, the present theory gives essentially the same result as the dynamical theory of Bethe.The present theory is particularly suited to the study of the diffraction of electrons by very thin crystals and crystals containing imperfections. Its applications to matters of practical importance in this field will be considered in a future publication.

(sodium)= 0.8685; B(fluorine)=0.8671 A2], using force constants derived by fitting a shell model to measured dispersion curves. This shows that the treatment of the extinction-affected reflexions in the least-squares refinement was entirely satisfactory. The author's thanks are due te the X-ray diffraction group of the University of St. Andrews, Scotland, for the use of a Siemens four-circle diffractometer. His thanks are also due to the referee for his constructive criticism. A method for the numerical evaluation of N-beam diffraction amplitudes and intensities which has been successfully employed over the last few years is described. This derives from the multi-slice formulation of Cowley and Moodie. The physical basis of the method and practical approaches to calculation are described. References

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General conditions are given for the continued complete absence of kinematically space-group forbidden refiexions when the scattering is dynamic. It is shown that reflexions which are forbidden because of glide planes or twofold screw axes will remain forbidden under dynamic conditions if suitable restrictions are placed on the direction of the incident beam. These restrictions can be represented by a cross, one arm of which corresponds to the exact Bragg condition, the other arm being normal to the first and to the screw axis or glide direction. When only zero-layer interactions are taken into account the locus of zero intensity includes both arms. When higher-layer interactions are included the locus of zero intensity will be one or other arm of the cross depending on whether the incident beam is perpendicular to a screw axis or parallel to a glide plane. The case in which the incident beam is perpendicular to a glide plane, though trivial for zero-layer interactions, leads to a condition for zero intensity satisfied only at the intersection of the arms of the cross when higher layer interactions are included.

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