Cross-Section of Elastic-Scattering of Electrons
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The cross section of elastic scattering of an electron, when divided by the actual area of an atom, gives the probability that a scattering event will occur.

Rutherford cross-section gives convenient formulas to calculate the elastic scattering angle and total elastic cross section so that it is often used to describe the elastic scattering of electron. However, its accuracy becomes worse in low-energy region or in high atomic-number materials.

Figure 4667a shows schematically electron scattering by an atom. Ω here is solid angle.

Schematic of electron scattering

Figure 4667a. Schematic of electron scattering.

The cross-section of elastic-scattering is given by,

      cross-section of elastic-scattering ------------------------------------------- [4667a]

where is a function, which defines the scattering properties by

      f(K) --[4667b]

The element of solid angle (Ω) can be written as,

      dΩ= 2πsinθdθ --------------------------------------------------------------------[4667c]

Scattering vectoris the scattering vector, given in Figure 4667b. ki is the incident wave vector, ke the scatteredwave vector. For elastic scattering, |ke| = |ki| = 1/λ (λ is wavelength) and scattering vector. The Bragg angle Φ and the scattering angle θ are also shown in the figure.

Scattering vector of electron by atoms for elastic scattering

Figure 4667b. scattering vector for elastic scattering.

Based on Turner–Doyle scattering factors [1], σe can be plotted by the curve in Figure 4667c. On the other hand, by simple estimation, σe can be plotted by the straight line using Lenz model given by [2],

      cross section of elastic scattering of electrons by atoms ------------------- [4667d]

The elastic scattering cross-section σe plotted as a function of atomic number based on different models

Figure 4667c. The elastic scattering cross-section σe plotted as a function of atomic number based on different models. The strength of the scattering increases with atomic number. The two models are almost consistent. Both scales are logarithmic. [3]

Figure 4667d below shows the cross section of elastic scattering process in Al as a function of the incident electron energy at the scattering angles of ~ 0°. For comparison, the figure also shows cross sections for the various inelastic scattering processes.

Cross sections for the various elastic and inelastic scattering processes in Al as a function of the incident electron energy

Figure 4667d. Cross sections for the elastic and various inelastic scattering processes in Al as a function of
the incident electron energy at the scattering angles of ~ 0°.

 

[1] Humphreys, C. J., Hart-Davis, A., and Spencer, J. P. (1974). Optimizing the signal/noise in the dark field imaging of single atoms. Proc. 8th Int. Congr. Electron. Microsc., Canberra, p. 248.
[2] Lenz, F. (1954). Zur Streung mittleschneller elektronen in klienste Winkel. Z. Naturforsch. 9A, 185.

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