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Charged Particle Interactions with Matter

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Nuclear Medicine Radiation Dosimetry

Abstract

In nuclear medicine, moving charged particles are released in tissue through either the radioactive decay processes of Chap. 4 or as a result of the photon–matter interactions of Chap. 6. Being charged, these particles interact significantly with the medium, transferring their kinetic energy resulting in an absorbed dose to the medium as they slow down to thermal energies. Hence, the study of charged particle interactions with matter is the fundamental core of ionizing radiation dosimetry. In this chapter, the two mechanisms of energy loss are presented. Collision energy losses between the particle and atomic electrons are derived through the Bohr classical and the Bethe quantum-mechanical means; hard collisions losses are derived independently from various quantum-mechanical results. Radiative energy losses resulting from bremsstrahlung are initially derived from classical theory which then progresses to the Bethe–Heitler quantum-mechanical theory. The polarization effects of a charged particle upon the medium will limit the collision energy losses and are derived. As energy loss is inherently stochastic, energy straggling models are also presented. In particular, the Vavilov theory of energy straggling is derived as are the Gaussian and Landau results which are treated limiting conditions to that theory. Multiple scatter strongly affects electrons and positrons and the Fermi–Eyges theory is derived as a means of justifying the Gaussian model. The Goudsmit–Saunderson and Moliére theories of multiple scattering are derived. Finally, the mechanisms through which a positron can annihilate on an electron are derived.

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Notes

  1. 1.

    The photon mean free path is the reciprocal of the linear attenuation coefficient. The exact range of an electron in a medium is more difficult to define as, due to multiple scatter, its path will be tortuous.

  2. 2.

    Assuming a constancy of photon speed ignores the frequency dependence of the medium’s index of refraction. Such an effect, however, is negligible for photons with sufficient energy to ionize.

  3. 3.

    Although it is possible for an α particle to penetrate the Coulomb barrier through quantum tunneling, but the likelihood of this occurring at kinetic energies of a few MeV typical of therapeutic nuclear medicine is extremely small.

  4. 4.

    It has been argued that this nomenclature incorrect as the ratio has the units of force (i.e., 1 N = 1 J/m) rather than power (i.e., 1 W = 1 J/s). While this proposal is dimensionally correct, it does not seem realistic to accept it given the decades-long use of the term stopping power in the context of a charged particle slowing down.

  5. 5.

    Also called the electronic stopping power.

  6. 6.

    Protons will also figure in these derivations due to their historical significance.

  7. 7.

    While, strictly speaking, such an interaction would be considered inelastic, a hard collision can be modeled as being elastic if the projectile’s incident kinetic energy sufficiently exceeds the electron binding energy such that the latter can be ignored.

  8. 8.

    This implicitly neglects any electrostatic correlations between the atomic electrons.

  9. 9.

    The presence of electrons in an ion projectile will have two effects upon the rate of energy loss. The first is that the effective charge will be reduced to the screening by these electrons. The second is that the excitation or ionization of the projectile itself will provide an additional energy loss channel.

  10. 10.

    These transformed values of the electric field result from the Lorentz transformation corresponding to the boost along an axis with a speed βc for the particle in one reference frame to that containing the electron at rest. These are provided here without derivation, but one may refer to, for example, that provided by Jackson (1999).

  11. 11.

    Although the stopping power is also written as S and the mass collision stopping power as \( {\hbox{$S$} \mathord{\left/{\vphantom {S \rho }}\right.}\hbox{$\rho $}} \), it will be written here as a differential.

  12. 12.

    Reviews of the derivation of the Bethe theory can be found in Fano (1964), Inokuti (1971) and Ahlen (1980).

  13. 13.

    The plasma frequency describes the oscillatory motion of free electrons in a plasma displaced from a uniform background of ions. The equation of motion for an electron in the simplest case is given by \( {m_e}{\hbox{${{d^2}x}$} \mathord{\left/{\vphantom {{{d^2}x} {d{t^2}}}}\right.}\hbox{${d{t^2}}$}} = - eE - \left( {{\hbox{${{\rho_e}{e^2}}$} \mathord{\left/{\vphantom {{{\rho_e}{e^2}} {{\varepsilon_0}}}}\right.}\hbox{${{\varepsilon_0}}$}}} \right)x \) where we have taken the restoring electric field to be equal to \( {\hbox{${ - {\mathbf{P}}}$} \mathord{\left/{\vphantom {{ - {\mathbf{P}}} {{\varepsilon_0}}}}\right.}\hbox{${{\varepsilon_0}}$}} \) where P is the polarization. Solving this equation yields the plasma frequency, \( {\omega_p} = \sqrt {{{\hbox{${{\rho_e}{e^2}}$} \mathord{\left/{\vphantom {{{\rho_e}{e^2}} {{\varepsilon_0}{m_e}}}}\right.}\hbox{${{\varepsilon_0}{m_e}}$}}}} \).

  14. 14.

    This is also referred to as the digamma function.

  15. 15.

    Because the interatomic spacings in a gas are greater than those in a solid or liquid, the wider dispersion of atoms in the gaseous phase will limit the dielectric response such that the mass collision stopping power for a given medium will be greater for the condensed phases than for the vapor phase and, hence, this interest in this phenomenon in tissue.

  16. 16.

    Should this not be the case, the electric susceptibility and dielectric permittivity scalars would be replaced by tensors.

  17. 17.

    ICRU Report 49 (1993) provides a comprehensive historical summary of the various experimental techniques used to measure the stopping power.

  18. 18.

    A closely related quantity is the linear energy transfer, or LET, which is simply the restricted linear collision stopping power.

  19. 19.

    Both classes represent conceptual difficulties: a Class I code does not link the sampling of the energy loss pdf with secondary recoil electrons and the Class II code neglects any energy straggling associated with soft collisions.

  20. 20.

    Further discussion of the practical matters of the Monte Carlo simulation of the transport of charged particles can be found in the articles by Salvat et al. (1999) and Chibani (2002).

  21. 21.

    See, for example, Sauli’s description of multiwire proportional chambers (1977).

  22. 22.

    In an inhomogeneous medium (such as the body), the scattering power would also be a function of x and y. That level of complexity is not required for this discussion.

  23. 23.

    Fernández-Varea et al. (1993) have provided an additional and shorter derivation of the theory beginning with the Goudsmit–Saunderson result.

  24. 24.

    Unlike Molière, Bethe included the contributions of the atomic electrons and assumed these to be coherent so that Z is replaced by Z(Z + 1). This is repeated here.

  25. 25.

    Molière achieved this result using an expansion of Hankel functions.

  26. 26.

    This limitation is reasonable for nuclear medicine purposes as the maximum electron energy resulting from the Compton scatter of a 511 keV photon is 340 keV for a backscattered photon and the maximum β− kinetic energies of isotopes typical of clinical nuclear medicine interest are below a couple of MeV.

  27. 27.

    There are two classical arguments that will allow electron-electron bremsstrahlung to be neglected. On the simplest level, in the dipole approximation, the energy radiated away by an accelerated charged particle is proportional to the dipole moment. As the dipole moment is also proportional to the center-of-mass (which is stationary of particles of identical mass), our first approximation is that electron-electron bremsstrahlung will be zero. One can also think of the accelerations of an electron projectile and electron target resulting in electromagnetic radiations of equal magnitude but opposite phase resulting in cancellation.

  28. 28.

    As only the instantaneous power flow is being considered, the 1/2 multiplicative factor is excluded.

  29. 29.

    There is also a radiation yield calculation associated with positrons, although this is not considered here. Customarily, the in-flight e−e+ → 2γ is excluded from the calculation of the positron radiation yield.

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    Brian J. McParland BASc MSc PhD

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McParland, B.J. (2010). Charged Particle Interactions with Matter. In: Nuclear Medicine Radiation Dosimetry. Springer, London. https://doi.org/10.1007/978-1-84882-126-2_7

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