Rutherford (Elastic) Scattering
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The model of Rutherford scattering of α particles on gold nuclei was based on four assumptions: i)  The gold nucleus mass (M) is much larger than the α particle mass mα (M >> mα); ii) Scattering of α particles on atomic electrons is negligible because mα >> me (me is the electron mass); iii) The α particle does not penetrate the nucleus (meaning no nuclear reactions); iv) The classical relationship for the kinetic energy EK of the α particle (EK = mαυ2/2) is valid (υ is the velocity of the α particle).

When the α particle (positive charge ze) approaches the nucleus (positive charge Ze) the interaction between them is a repulsive Coulomb interaction, and thus, the α particle follows a hyperbolic trajectory, as shown in Figure 4417. The nucleus is in the outer focus of the hyperbola because of the repulsive interaction. The relationship between the impact parameter b and the scattering angle θ may be derived by determining two independent expressions for the change in momentum Δp of the scattered α particle. υ is the initial velocity of the α particle and p is the initial momentum of the α particle. The momentum transfer is along a line that bisects the angle π − θ. The magnitude of the Coulomb force Fcoul acting on the α particle is given by,

       Rutherford (Elastic) Scattering ------------------------------------ [4417a]

where, r -- The distance between the α particle and the nucleus M
           z -- The atomic number of the α particle (for helium z = 2)
           Z -- The atomic number of the absorber (for gold Z = 79)

Schematic diagram for scattering of an α particle on a nucleus

Figure 4417. Schematic diagram for scattering of an α particle on a nucleus.

The momentum transfer Δp is given by,

       momentum transfer Δp ------------------------------------ [4417b]

The impact parameter b may be written as,

        impact parameter b ----------------------------------------- [4417c]

In HAADF STEM, the "high-angle" means beyond the angle at which diffraction maxima (spots) can be found. High-angle scattered electrons are few in number of elections and are mostly induced by Rutherford scattering.

Table 4417 shows that electrons interact with 1 electron, many electrons, 1 nucleus, and many nuclei in solids.

Table 4417. Effects of interactions of electrons in solids.
  Interaction with electron(s) Interaction with nucleus/nuclei
  1 electron Many electrons 1 nucleus Many nuclei
Scattering type Inelastic Inelastic Quasi-elastic Elastic Inelastic
Scattering effect Electron Compton effect; electron excitation (from 50 eV to a few keV: EDS and EELS) Plasmon excitation (< 50 eV, ~100 nm TEM specimen); Cerenkov effect Rutherford scattering; phonon scattering (< 1 eV, heat) Bragg scattering Bremsstrahlung



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