The major limitations on TEM performance are spherical aberration (also known as aberration), chromatic aberration, and astigmatism. The spherical aberration and chromatic aberration limit the resolution of traditional TEM. Both of these defects are inevitable when using static rotational symmetric electromagnetic fields. Ball aberration is the most important factor determining the performance of an objective lens. For thicker samples, the color difference will be more severe. To reduce this issue, it is best to make thinner samples. Scattering can affect the focusing ability of the image, but it can be completely corrected. Spherical aberration is caused by the uneven effect of the lens field on off-axis rays. In other words, light rays that are "parallel" to the optical axis but at different distances from the axis cannot converge at the same point. The more electrons deviate from the axis, the stronger they bend towards the axis. Therefore, point like objects are imaged as a finite sized disk, which limits the ability to zoom in on details. The following figure shows the impact of ball deviation. Point P is imaged as the disk with the smallest radius in the "least confusion" plane, and as PI in the imaging plane. The central bright area is strong, and there is a halo around it.

The expression for calculating the spherical aberration disk radius (rsph) on the image plane is as follows: 

Rsph=Csβ3

where Cs is a constant for a specific lens, called the spherical aberration coefficient, and β is the maximum convergence half angle of the objective aperture. According to this derivation, Cs has a length dimension, usually approximately equal to the focal length. In TEM, the focal length is usually about 3 millimeters, but in HRTEM it is much smaller than 1 millimeter. One of the methods to minimize aberrations is to use short focal length lenses (i.e. lenses with small spherical aberration coefficients). The following image is an example of a point light source imaged by a system with negative spherical aberration (top), zero spherical aberration (center), and positive spherical aberration (bottom). Only the center point is a point, and the image above and below it is displayed as a disk.

The impact of spherical aberration on point light sources. The image on the left side of the center is defocused inward; The image on the right side of the center is defocused towards the outside.

Color difference: The term color difference is related to the energy of electrons, which are not monochromatic (introduced from optics, electrons can be understood as having no energy fluctuations). Electrons are emitted from the electron gun with various energies, and the objective lens will bend them to varying degrees; Electrons with lower energy (greater loss) bend more severely. Therefore, electrons from a certain point on the sample once again form a disk image, just like spherical aberration. The radius of the disk (rchr) is obtained by the following formula: where Cc is the chromatic aberration coefficient (length) of the lens, Δ E is the energy loss of electrons, Eo is the initial electron beam energy, and β is the collection half angle of the lens.
The Δ E in the incident electron beam is less than 1 eV. For most electrons passing through samples with a thickness of 50-100 nm, the Δ E is typically 15-25 eV. The thicker the sample, the greater the color difference, as the proportion of inelastic scattered electrons is higher and may be affected by color difference.

Astigmatism: Astigmatism occurs when the cross-section of the electron beam is not completely circular.

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