Rutherford formula

The probability for a ion to scatter in a particular direction is determined by the ion-target interaction, and can be expressed in terms of a cross section For a Coulomb potential, the diflFerential cross section is the well-known Rutherford formula ... 

This is the Rutherford formula Z2e4/4pV sin4 9 multiplied by (1 — v2 sin2 6), which is the correction to the Rutherford formula arising from the spin of the electron. 

Elastic scattering cross sections were caleulated using the Rutherford formula, taking into account the screening parameter given by Moliere [187], The differential cross section (dc/ df2)ei and total cross sections cTei for each molecule are represented by... 

In order for the integral (4.15) not to be close to zero at large q, the wavefunction ipn of the final state must contain a factor —exp (iqx). Such ij/n correspond to an ionized state of a molecule with the ejected electron having momentum hq, that is, of the same type we would have in the case of collisions with free electrons. The cross section in this case is given by the Rutherford formula (4.20). 

The second subregion corresponds to large to and q and is related to close collisions (the knock-on). At very large transferred momenta (qa0> 1) the inelastic scattering by a molecule (an atom) is actually the elastic scattering by a free electron with the cross section given by Rutherford formula. In this case the function /(to, q) can be presented analytically as a delta function ... 

The above reduces to the famous Rutherford formula when the Coulomb amplitude is all that contributes to the scattering amplitude... 

If the projectile energy is chosen properly (below Coulomb barrier and above electronic screening - see Sect. 2.5.2 for shielded Rutherford cross-sections), the scattering yield follows the Rutherford cross-section which is basically proportional to the square of the atomic number of the element and inversely proportional to the square of the projectile energy (Fig. 2.11). If the energy is higher than the upper value, the backscattering cross-section shows deviation from Rutherford formula as discussed in Sect. 2.5. 

When the energy is higher than the upper value, the backscattering cross-section shows deviation from Rutherford formula. 

Figure 4 Typical geometry for Rutherford backscattering. The incident particle loses energy AEi going through the sample to the scattering site, loses energy AE (determined by the Rutherford formula) in the collision and loses energy AE2 on the recoil path to the detector.

Resonant elastic scattering At certain energies, the yield of RBS analysis may be strongly enhanced (or reduced) by interactions with the target nucleus. One example of this is resonant scattering of protons from i.e. C(p,p) C. At an energy of 1.76 0.1 MeV, the cross-section for backscattered protons from increases by a factor of 60 relative to the classical Rutherford formula and this can be exploited to improve the sensitivity for carbon. This is shown in the simulations of Figure 7, for a 10 nm thick layer of carbon on a copper substrate. 

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