Abstract
With the exception of point targets, such as a free electron, the target in the elastic Coulomb scatter of a charged particle is typically distributed and screened, such as the atomic nucleus. The degree of spatial variation and Coulomb screening perceived by the projectile will be dependent upon its de Broglie wavelength of both the atom and the nucleus. In this chapter, we extend the previous discussion of elastic Coulomb scatter from a point charge to the more realistic case of a distributed and screened target. This will begin with the definition of the form factor for a spatially distributed charge that the projectile is incident to. Examples such as a hard-edge homogeneous sphere and a sphere with a charge distribution with a Gaussian radial charge distribution are considered in the calculation of these form factors.
Screened target are next considered and in much detail. We begin with the simple model of continuous screening using a Yukawa-type potential to represent that of the screened nuclear potential. A key parameter used in this model is the atomic electron screening parameter, which is the coefficient of the spatial variable in the exponent of the Yukawa form. The Thomas–Fermi statistical model of the atom is developed and its results used to derive a functional form of the atomic electron screening parameter.
We then proceed to the model of discrete screening in which the potential perceived by the projectile is the summation of those due to the atomic electrons and that due to the atomic nucleus. From this calculation is extracted the atomic elastic scattering form factor.
The outcomes of the continuous and discrete atomic electron screening models are differential cross sections which display modifications from the Rutherford (unscreened) Coulomb differential cross section which avoid the divergence at zero scattering angle.
We conclude with a numerical analysis of these results.
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Notes
- 1.
This feature makes the measurement of elastic scattering cross sections as a useful means of discerning the internal structure of a target.
- 2.
The discussions of the screening provided in this chapter are of the target only. The projectile is considered to be ‘bare’, i.e. it is not an ion with its own atomic electrons. This particular case is discussed later within the context of energy loss.
- 3.
Carbon, at Z = 6, is selected as it represents the smallest atomic number atom that can be justifiably modelled using the Thomas–Fermi method from which we derive a screening parameter.
- 4.
For practical calculational purposes, this assumption is valid over a surprisingly long range. For example, sin θ ≈ θ to within 10 % for 0o < θ <45°.
- 5.
Ions will be considered inter alia in the derivation, but are not a focus as they are not relevant targets in medical radiation physics considerations.
- 6.
See, for example, Hoffman (1993)
- 7.
In addition, carbon would reflect the lowest atomic number that could be expected to be treated as valid in the Thomas–Fermi statistical atomic model which requires Z ≫ 1.
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McParland, B.J. (2014). Elastic Coulomb Scatter from Distributed and Screened Charges. In: Medical Radiation Dosimetry. Springer, London. https://doi.org/10.1007/978-1-4471-5403-7_5
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