Practical Electron Microscopy and Database

The electrons in the beam in EMs undergo different phase shifts when propagating through the objective lens because every electron source has a finite energy width. The maximum phase shift due to the energy width of the electron source is given by,

           -------------------- [4158a]

where,
          C_c -- Chromatic aberration coefficient
          ΔE -- Energy width of the electron source
          E₀ -- Accelerating voltage of the electron beam
          λ -- Wavelength of the electron beam
          k -- Spatial frequency

Conventionally, the phase shift (phase distortion function) due to the objective lens can be combined into a single phase factor χ, given by,

          -------------- [4158b.a]

                          -------------- [4158b.b]

where,
          C_s -- The spherical aberration coefficient, defining the quality of objective lens,
          Δf -- The defocus value,
          |g| -- Also the spatial frequency, the same as k above,
          α -- The convergence semi-angle.

Considering the non-constant wavelength of an incident electron beam and the astigmatism, the phase shift function can be described by,

          --- [4158c]

The last and the second terms describe the effects of the non-constant wavelength and astigmatism, respectively.

Note that for an ideal lens, the phase shift induced by objective lens, χ(α), is equal to 0.