Electron microscopy
Lenz Model for Elastic Scattering Distribution
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In the literature, various relatively simple models of elastic scattering such as a free-atom Thomas–Fermi model with partial wave expansion [7] and Mott cross-sections 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],
          Lenz model ------------------------- [964a]
          characteristic angle of elastic scatteringcharacteristic angle of elastic scattering
          here, θ0 -- the characteristic screening angle of elastic scattering. For the binary materials, θ0 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,
          a0 -- the Bohr radius.

Noted that θ0 ≈ 100θE. Here, θE stands for the characteristic angular range of inelastically scattered electrons as given by θE = E/γm0v2 with the usual relativistic factor γ.

The elastic differential cross-section in the Lenz model can be given by,
          Lenz model ------------------------- [964b]
          e -- the elementary charge,
          E0 -- the energy of the incident electrons.          

In combination of Equations 964a and 964b, one is able to obtain,
          Lenz model ------------------------- [964c]

The Lenz model can also give the elastic-scattering intensity dIe within an angular range dθ by, [5]
          Lenz model ------------------------- [964d]     
          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 power-law dependence,
         Lenz atomic model of scattering --------------------- [964e]
However, more sophisticated calculations suggested: [4]
         Lenz atomic model of scattering --------------------- [964f]
The exponent x actually depends on collection semi-angle.

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 energy-loss spectrum.

The Lenz atomic model predicted that:
        i) the total cross section per atom σt 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 solid-state 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 core-loss and low-loss electrons. [1]   

Table 964. Single and multiple electron scattering.

Single scattering
Lenz model, Mott cross-sections
Multiple scattering
Poisson distribution
















[1] K. Wong and R. F. Egerton, Correction for the effects of elastic scattering in core-loss quantification, Journal of Microscopy, 178(3), (1995) pp. 198-207.
[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. 47-62.
[5] S. C. Cheng and R. F. Egerton, Elemental Analysis of Thick Amorphous EELS, Micron, Vol. 24, No. 3, pp. 251 256, (1993).
[6] Huai-Ruo Zhang, Ray F. Egerton, Marek Malac, Local thickness measurement through scattering contrast and electron energy-loss spectroscopy, Micron 43 (2012) 8-15.
[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.