Improved methods for the determination of the spherical aberration coefficient in high-resolution electron microscopy from micrographs of an amorphous object

New and improved statistical procedures are proposed for determination of the spherical aberration coefficient Cs in high-resolution electron microscopy (HREM) from micrographs of an amorphous object. A least-squares technique applied to the experimental zeros of the microscope's transfer function with a parabola instead of a straight line as fitting curve represents the major key to improved determination of the spherical aberration coefficient. A smaller further improvement is realised by application of a second iteration step for the parabola method. Another improvement is obtained by using a focal series of HREM images instead of a single micrograph. With the multifocus iterated parabola method, a standard deviation (sd) below 1% can be reached for the spherical aberration coefficient. The methods are applied to the determination of Cs for the Philips CM30-SuperTWIN and the Philips CM20-UltraTWIN microscopes. Apart from an error in the calibration of the magnification, which leads to an estimated error of 1% on Cs, we obtained for the CM30-Super- TWIN Cs = 1.210 mm with sd = 0.008 mm, and for the CM20-UltraTWIN Cs = 0.494 mm with sd = 0.004 mm.