In the literature, various relatively simple models of elastic scattering such as a freeatom Thomas–Fermi
model with partial wave expansion [7]
and Mott crosssections for a screened Coulomb potential
[8] had been proposed. On the other hand, Lenz model is a very simple atomic model. In this model, the single elastic scattering distribution is
given by [1, 3],
 [964a]
where,
here, θ_{0}  the characteristic screening angle of elastic
scattering. For the
binary materials, θ_{0} is taken as the mean characteristic
angle of each element weighted by the atomic fraction.
R  is the screening radius of
atomic electrons ( ).
Z  the atomic number for a single element, or the effective atomic number for a component.
λ  the electron wavelength,
a_{0}  the Bohr radius.
Noted that θ_{0} ≈ 100θ_{E}. Here, θ_{E} stands for the characteristic angular range of inelastically scattered electrons as given by θ_{E} = E/γm_{0}v^{2} with the usual relativistic factor γ.
The elastic differential crosssection in the Lenz model can be given by,
 [964b]
where,
e  the elementary
charge,
E_{0}  the energy of the incident electrons.
In combination of Equations 964a and 964b, one is able to obtain,
 [964c]
The Lenz model can also give the elasticscattering intensity dI_{e}
within an angular range dθ by, [5]
 [964d]
where,
C  a constant.
The simple Lenz atomic model of scattering suggested that the cross section (per atom) for inelastic scattering can be given by a powerlaw dependence,
 [964e]
However, more sophisticated calculations suggested: [4]
 [964f]
The exponent x actually depends on collection semiangle.
The Lenz
atomic model shows that:
i) the Lenz
values are mostly higher than the experimental ones at low angles,
but lower at high angles (no objective aperture). [6]
ii) the consideration of multiple scattering and aperture effects (represented by θ) is important to the correct interpretation of an energyloss
spectrum.
The Lenz
atomic model predicted that:
i) the total cross section per atom σ_{t} increases with increasing atomic number. [6]
The elastic Lenz model had been used to:
i) correct the effects of elastic scattering in amorphous materials. [1] However, since the Lenz model is atomic in nature and thus it does not
account for solidstate effects such as Bragg scattering and channelling.
ii) calculate the effective contrast thickness in transmission electron microscopy. [2]
iii) discuss the effects of elastic
scattering on the coreloss and lowloss electrons. [1]
Table 964. Single and multiple electron scattering.
[1] K. Wong and R. F. Egerton, Correction for the effects of elastic scattering in coreloss quantification, Journal of Microscopy, 178(3), (1995) pp. 198207.
[2] L. Reimer, Transmission Electron Microscopy, Physics of Image Formation, and Microanalysis (Springer, Berlin, 1989).
[3] F. Lenz, Z. Naturforsch. Teil A 9, 185 (1954).
[4] Crewe, A.V., Langmore, J.P., and Isaacson, M.S. (1975) Resolution and contrast in the scanning electron microscope. In: Physical Aspects of Electron Microscopy and Microbeam Analysis. B.M. Siege1 and D.R. Beaman, eds. Wiley, New York, pp. 4762.
[5] S. C. Cheng and R. F. Egerton, Elemental Analysis of Thick Amorphous EELS, Micron, Vol. 24, No. 3, pp. 251 256, (1993).
[6] HuaiRuo Zhang, Ray F. Egerton, Marek Malac, Local thickness measurement through scattering contrast and electron energyloss spectroscopy, Micron 43 (2012) 815.
[7] Ichimura, S., Aratama, M., Shimizu, R., Monte Carlo calculation
approach to quantitative Auger electron spectroscopy. J. Appl. Phys.
51, (1980) 2853–2860.
[8] Shimizu, R., Kataoka, Y., Matsukawa, T., Energy distribution
measurement of transmitted electrons and Monte Carlo simulation for
kilovolt electron. J. Phys. D 8, (1975) 820–828.
