Energy Distribution/Spread of Electron Source
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The electron beams used in EMs have an energy spread (ΔE) in the range 0.3 to 3 eV, depending on the type of electron source as shown in a table in page1409.

For a field emission source, the beam of electrons emitted presents an energy profile described by the Fowler-Nordheim (F-N) Distribution [1],

          Energy Distribution / Spread of Electron Source------------- [4772a]

          Δ = µ - E ---------------------- [4772b]

          b = 6.8 x 107α ---------------------- [4772c]

          µ -- The chemical potential (Typical value ~ 11 eV),
          φ -- The work function (for a tungsten tip as 4.5 eV),
          α -- The image correction term (Typical value ~ 0.85),
          F -- The electric field strength (Typical value ~ 3 x 107 V/m),
          E -- The accelerating voltage on the incident electrons.

The F-N energy distribution of emitted electrons includes two distinct regions [2]: a low energy Fermi tail and a high energy tunnelling tail. The low energy Fermi tail is a property of the Fermi surface of the tip material (e.g. tungesten) and is independent of the extraction voltage. The slope of the high energy tunnelling tail is determined by the field strength. Those tails cause beam energy-broadening in field emission guns.

The width of the zero-loss peak in EELS spectrum, typically 0.2 – 2 eV, reflects mainly the energy distribution of the electron source. Note that a monochromator for the electron source or data deconvolution is necessary in the frontiers of TEM-EELS if the energy spread of the available electron source in TEM is larger than the intrinsic fine structures of spectra.

In STEM, the fluctuations in accelerating voltage or the current flow in the probe-forming electromagnetic lenses contributes to the chromatic aberration that leads a cutoff to the highest spatial frequency. In HRTEM, the cut-off frequency corresponds to the information limit. In STEM, the effect of the fluctuations induces additional contribution to the probe size [3],

        effect of the fluctuations induces additional contribution to the probe size ------------- [4772d]

        Cc -- The chromatic coefficient,
        E0 -- The energy of the electron beam (the accelerating voltage),
        I -- The current in the probe-forming lens,
        ΔE -- The spread in energy of the beam,
        ΔE0 -- The fluctuation in the accelerating voltage,
        ΔI -- The fluctuation of the lens current.

In summary, the energy spread of the electron beam is mainly induced by:
         i) The HT (high tension).
         ii) The energy spread in the electron source.
         iii) The interaction between the incident electrons and specimen. Even though there is an energy spread (ΔE, 0.3 ~ 1.5 eV) in the electron sources, the major factor to the chromatic aberration (Cs) is the large energy loss ΔE (normally < 2 keV) induced by inelastic scattering when electrons pass through the specimen.

Due to the significant energy spread in the electron beam, the chromatic aberration associated with the lenses will lead to image degradation in TEM as listed in Table 4772. On the other hand, according to Equation 4772d, i) and ii) above induces also an increase in probe size in the scanning modes (e.g. SEM and STEM). This effect of the energy spread (ΔE) is proportional to ΔE/E0.

Table 4772. Causes of energy spreads and the image degradation induced by the associated lenses.

Causes of energy spread
Lenses inducing image degradation
Variation of HT (high tension) Both pre-field of objective lens and post-fields of objective lens
Energy spread in electron source Both pre-field of objective lens and post-fields of objective lens
Interaction between incident electrons and specimen Post-fields of objective lens

Furthermore, for TEM analysis, the size of electron source must be large to achieve a large area of illumination with sufficient electron density. From this point of view, thermionic sources are more than capable of meeting these criteria. However, this relatively large source size in addition to a large electron energy spread means that thermionic sources are not applicable to scanning probe microscopy and EELS microanalysis.





[1] Fowler, R.H., Nordheim, L. Electron emission in intense electric fields. Proc. R. Soc. A 119, 173-181 (1928).
[2] Kimoto, K. et al. 0.23eV energy resolution obtained using a cold field-emission gun and a streak imaging technique. Micron 36, 465-469 (2005).
[3] Spence JCH. High resolution electron microscopy. 3rd ed. Oxford: Clarendon Press; 2003.