Beam alignment and related problems of spherical aberration corrected high-resolution TEM images

For spherical aberration corrected transmission electron microscopes recently developed, the Scherzer resolution in proportion to C(s) 1/4 lambda 3/4 is still inevitably limited by the influence of some other electron optical factors, such as chromatic aberration coefficient of objective lens (C(c)), beam divergence and alignment, incident electron energy spread (deltaE), and the instability of accelerating voltage (deltaV) and of lens current (deltaI). Depending on the image resolution guaranteed by C(s) correction, the defocus spread caused by C(c), deltaV, deltaI and deltaE would need to be reduced. The effect of beam alignment as a phase shift in contrast transfer function is also studied for the C(s) corrected microscopes with different values of C(s), defocus and spatial frequency. There is a relationship between the phase shift and defocus that allows us to find a series of 'alignment-free' focal conditions. A triangular relation among C(s), defocus and lattice spacing is established for proper image contrast and 'alignment-free' imaging with the C(s) corrected microscope.