Transmission-electron Fourier imaging of crystal lattices using low-voltage field-emission sources: Theory

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Abstract
The low-voltage electron field-emission ‘‘point’’ projection images of Fink, Stocker, and Schmid are analyzed for the case of thin crystalline samples, based on the theory of scanning-transmission-electron-microscopy lattice imaging and Fourier imaging. The formation, by this method, of atomic-resolution images of crystal lattices without lenses or scanning are discussed, as originally proposed by Cowley and Moodie. The existence of high-order Fourier images is established under general multiple-scattering conditions, when a transmission function cannot be used. Computed images are analyzed, and it is found that, because of the rapid change of scattering phase with thickness and angle due to multiple scattering, no simple relationship between image and crystal potential can be established. Resolution is limited by the angular range α over which the wave field striking the crystal is coherent—a large angle being desirable for high resolution. The number of lattice fringes within the Bragg angle (subtended by the detector at the sample) is found to equal the order of the Fourier image, and axial high-order images (corresponding to a large tip-to-sample distance and a large region of periodic averaging) are found to show least image distortion due to multiple scattering. The optimum experimental conditions for atomic-resolution Fourier lattice imaging at low voltages are analyzed, and their uses discussed.

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