Numerical evaluation of electron image phase contrast

This paper is concerned with phase contrast in electron images with emphasis on a periodic scattering object. The Kirchoff diffraction or imaging integral over the back focal plane is formulated in terms of the amplitude and phases of the scattered wave, using coordinates adapted to numerical methods. The integral was programmed for evaluation on the IBM 7090 and the image plane amplitude displayed on a microfilm plotter. The effects of spherical aberration, aperture size, defocus and thermal motion of the scattering atoms on image contrast of atom positions for chains of nickel and of gold atoms were investigated for separations of 2 Å and 8 Å. In the range of atomic separations, spherical aberration is the most destructive single factor in the loss of phase contrast. It appears that an objective lens with a spherical aberration coefficient less than 0.2 mm will be necessary if phase contrast images of atom locations are to be attained. In addition, a practical quarter-wave phase plate is essential for the objective lens system and will be much more effective than defocus contrast. Even so the contrast is marginal for photographic recording except for the case of a thin perfect crystal. Amplitude contrast is very small for atom positions and should be minimized by the use of a large objective aperture. Phase contrast should improve with increasing accelerating potentials due to reduced inelastic cross sections compensating the loss in elastic contrast near 2 × 10
⁵
volts.