=================================================================================
Electron beam diameter (d)
is a major factor
in all aspects of electron microscopes, including TEM, STEM, and SEM imaging, where we use a fine focused beam. The measurement of d
is not straightforward. We can either calculate or measure it experimentally. The former is easy
but imprecise, the latter is difficult and can be equally
imprecise. Lack of universally accepted definition of the beam
diameter is one of the major problems on determination of d. The microscope manufacturers give the users a list of nominal
beam sizes for each setting of the C1 lens (condenser lens 1). Unfortunately, these
values are calculated and may significantly differ from the actual
beam size. One reason is that the calculation are carried out by assuming
the electronintensity distribution of the electron beam is
a Gaussian form, and the beam diameter is defined as the
fullwidth at halfmaximum (FWHM) of the Gaussian
distribution. However, the misalignments of electron beam and apertures, and astigmatism in the condenser lenses prevent formation of
real Gaussian intensity distribution. Especially, there may be
six different C1 lens excitations, each of which gives a
different calculated beam size, giving more complexity. Furthermore, if a too small aperture is selected the intensity distribution will be truncated
at a fraction of the full Gaussian curve. If a too large aperture is selected the actual beam will extend
out well beyond the calculated size.
Table 4954. Operating parameters and characteristics of electron sources
Type of source 
Tungsten
thermionic 
LaB_{6}
thermionic 
Schottky
emission 
Cold field
emission 

Material 
W 
LaB_{6} 
ZrO/W 
W 

d_{s} (µm) 
≈40 
≈10 
≈0.02 
≈0.01 

ΔE (eV) 
1.5 
1.0 
0.5 
0.3 

φ (eV) 
4.5 
2.7 
2.8 
4.5 

T (K) 
2700 
1800 
1800 
300 

E (V/m) 
Low 
Low 
≈10^{8} 
>10^{9} 

J_{e} (A/m^{2}) 
≈10^{4} 
≈10^{6} 
≈10^{7} 
≈10^{9} 

β (Am^{2}/sr^{1}) 
≈10^{9} 
≈10^{10} 
≈10^{11} 
≈10^{12} 

Vacuum (Pa) 
<10^{2} 
<10^{4} 
<10^{7} 
≈10^{8} 

Lifetime (hours) 
100 
1000 
10^{4} 
10^{4} 

** d_{s}  Effective (or virtual) source
diameter; ΔE  Energy spread of the electron beam; φ  Work function; T  Operating temperature; E  Electric field; J_{e}  current density of electron beam; β  Electronoptical brightness at the cathode.
If no apertures are considered, the theoretical limit of the resolution of an electrooptical system is defined by diffraction of the electron (dominated by wavelength λ), and spherical and chromatic aberration of the electron lenses (mainly the objective lenses). Generally speaking, for a Gaussian beam, the beam diameter d is defined by [1]:
 [4954a]
where i is the beam current, β is the brightness of the electron source, α is the convergence semiangle of the electron beam, C_{c} and C_{s} are the chromatic and spherical aberration coefficients of the system, and E_{0} and ΔE are the average energy and the energy spread of the electrons in the beam.
To simplify Equation [4954a], we can make a firstorder calculation of the beam size by considering ΔE = 0, so that,
d = (d_{g}^{2}+d_{s}^{2}+d_{d}^{2})^{1/2}  [4954b]
where d_{g} is the initial Gaussian
diameter at the gun. d_{s} beam broadening effect of spherical aberration in the beamforming
lens, and d_{d} diffraction effect at the final aperture.
 [4954c]
 [4954d]
 [4954e]
i is the current in the electron probe and α is the aperture angle. Equation [4954c] suggests beam diameter (d) increases with beam intensity (i). Equation [4954d] indicates the diameter of the disc with minimum confusion caused by spherical
aberration. Clearly, this term
is not Gaussian unless the beam is correctly apertured,
which, as we mentioned above, is not always possible. However,
C_{s} correction can reduce this contribution to the beam
broadening. The diameter due to
diffraction (d_{d}) is the Rayleigh criterion and refers to a spacing between two
overlapping images of the beam.
When the energy spread of electron beam (ΔE) is not 0, the chromatic aberration may not be ignored. So Equation 4954b becomes,
d = (d_{g}^{2}+d_{s}^{2}+d_{c}^{2}+d_{d}^{2})^{1/2}  [4954f]
d_{c} = C_{c}α(ΔE/E_{0})  [4954g]
The diagram in Fig. 4954 shows how these sources contribute in a typical column according to Equation [4954b]. In systems with thermionic sources (big source), spherical aberrations tend to be the limiting factor for beam diameter, while chromatic aberrations dominate in field emission systems (small source).
Figure 4954. Diameter of the electronbeam as a function of beamconvergence semiangle.
1: big electron source. 2: small electron source.
In scanning electron microscope (SEM), spherical aberration of the probeforming lens is increased with the increase of working distance, resulting in a larger electronprobe size. Therefore, the need to have better spatial resolution of SEM leads to shorten the working distance and consequently to change the SEM detector position.
Many modern aberrationcorrected STEMs have electron probes less than one atom dimension in diameter so that atom column resolution for composition analysis can be obtained.
In order to improve the energy spread of the electron sources in EMs, monochromators have recently been introduced. In the best cases, the energy spread with a monochromator for any source can be reduced to < ~100 meV. There are two basic setups: the Wien filter with crossed electric and magnetic fields and the electrostatic Omega filter. It is very common that both types improve the energy width to about 0.2 eV at an acceptable beam current of several 100 pA. The single Wien filter limits the probe size to about 2 nm, whereas there is no such limitation in the case of the Omega filter.
[1] J. I. Goldstein, Practical scanning electron microscopy. Electron and ion microprobe analysis, 1975.
