In general, two main features characterize the passage of charged particles through matter: a loss of energy by the particle and a deflection of the particle from its incident direction. These effects are primarily the results of two proc-esses, respectively: inelastic collisions with the atomic electrons of the material and elastic scattering from nuclei. These reactions occur many times per unit path length and it is their cumulative result which accounts for the two principal effects observed.
Of the two electromagnetic processes, the inelastic collisions are almost solely responsible for the energy loss of heavy particles in the matter. In these collisions energy is transferred from the particle to the atom causing a ionization or excitation of the latter. The amount transferred in each collision is generally a very small fraction of the kinetic energy of the particle; however, in normally dense matter, the number of collisions per unit path length is so large, that a sub-stantial cumulative energy loss is observed even in relatively thin layers of mate-rial. These atomic collisions are customarily divided into two groups: soft colli-sions in which only an excitation results, and hard collicolli-sions in which the energy transferred is sufficient to cause ionization. In some of the hard reactions, enough energy is, in fact, transferred such that the electron itself causes substan-tial secondary ionization. These high-energy recoil electrons are sometimes re-ferred to as δ-rays or knock-on electrons [122]. As discussed in the previous
A.1.1 The Bethe-Block formulation
151 chapter, the energy and spatial distribution of the electrons produced are finally responsible of the quality of the radiation in a particular point of the field, be-cause they characterize the structure of the primary particle track.
Elastic scattering from nuclei also occurs frequently although not as often as electron collisions. In general very little energy is transferred in these collisions since the masses of the nuclei of most materials are usually large, compared to the incident particle.
The inelastic collisions are, of course, statistical in nature, occurring with a certain quantum mechanical probability. However, because their number per macroscopic path length is generally large, the fluctuations in totally energy loss are small and it is possible to work with the average energy loss per unit path length. This quantity, called stopping power de/dx, was first calculated by Bohr in 1913 using classical arguments [123], and later by Bethe [27], Block[29] and others[30] using quantum mechanics.
A.1.1 The Bethe-Block formulation
The correct quantum-mechanical calculation was first performed by Bethe, Bloch and other authors. In the calculation the energy transfer is parameterized in terms of momentum transfer rather than the impact parameter. The formula obtained is:
commonly known as Bethe-Block formula, valid for heavy charged particles for energies ranging from MeV up to GeV. In Table A.1 all the variables used in the equation are summarized.
The last two terms in the formula, δ and C, are respectively the density effect correction and the shell correction.
The density effect correction is important at high energy. It arises from the fact that the electric field of the particle also tends to polarize the atoms along its path. Because of this polarization, electrons far from the path of the particle will be shielded from the full electric field intensity.
152
Symbol Definition Units or value
NA Avogadro’s Number 6.0221367 × 1023 mol-1
(
0 2)
2 4 mc
e
re = πε e Classical electron radius 2.817940325 × 10-13cm mec2 Electron energy at rest 0.510998918 MeV ρ Density of absorbing material g × cm-3
Z Atomic number of absorbing ma-terial
A Atomic weight of absorbing
mate-rial g × mol-1 z Charge of incident particle in unit
of e Coulomb-1
c
=v
β Relativistic term
v Velocity of incident particle cm/s 1 2
1 β
γ = − Relativistic term
Wmax Maximum energy transfer in one
collision eV
I Mean excitation potential eV δ Density effect correction
Z
C Shell effect correction
Table A.1: Variables use in the equation (A.1)
Collisions with these outer lying electrons will therefore contribute less to the total energy loss than predicted by Bethe-Bloch formula. This effect be-comes more important as the particle energy increases. Moreover, it is clear that this effect depends on the density of the material (hence the term “density” ef-fect), since the induced polarization will be greater in condensed materials than in lighter substances such as gas.The density effect strongly depends on dielec-tric properties of the traversed medium and, moreover, it decreases when the atomic number of the medium increases, because in this case electrons result closer and less contribute to the polarization effect. Sternheimer derived the value of δ for a series of materials and obtained results that agree very good to the experimental data except in the low energy region where, however, density
A.1.1 The Bethe-Block formulation
153 effects are negligible [124].
The shell effect is important at low energies. It accounts for effects which arise when the velocity of the incident particle is comparable or smaller than the orbital velocity of the bound electrons. At such energies, the assumption that electron is stationary with respect to the incident particle is no longer valid and the Bethe-Block formula breaks down. The correction is generally small and its description is usually done by using empirical formulae [125].
In addition to the shell and density effects, the validity and accuracy of the Bethe-Bloch formula may be extended by including a number of other correc-tions pertaining to radiation effects at ultra-relativistic velocities, kinematical ef-fects due to the assumption of an infinite mass of the projectile, higher-order terms in the scattering cross section, correction for the internal structure of the particle and electron capture at very low energy [126][127]. However, with the exception of the latter, which could be not negligible with very heavy ions, the others are usually of the order of ≈ 1%.
The maximum energy transfer Wmax is that produced by knock-on collisions.
Starting from kinematics considerations and considering incident particles with mass M » me, it is given by:
This quantity, which is responsible of the kinetic energy of the electron pro-duced by primary particles, is also strictly connected to the track diameter. In-deed, it clearly shows, from a quantitative point of view, that electrons produced by particles travelling with high velocity v are in turn emitted with high kinetic energy, giving rise to a more wide diameter track (section 1.2.1).
An important parameter in the Bethe-Block formula is the mean excitation potential I, which is very difficult to theoretically calculate. Indeed values of I for several materials have been deduced from actual measurements of de/dx and semi-empirical formulae [128]. The ionization potential varies with the atomic number of the absorbing medium Z in a complicated manner, because of local irregularities due to the closing of certain atomic shells. This parameter is crucial in hadrontherapy because Bragg peak position is sensibly influenced by
consid-154
ering its slight variations. The standard value for the ionization potential of water widespread used is 75 eV, according to the ICRU (International Commission on Radiation Units) recommendations [129], but values up to 80 eV can be found elsewhere [130]. When Monte Carlo codes are used for the transport of thera-peutic ion beams in the matter, the accurate definition of I actually represents an open issue and usually it has to be slightly modified in order to tune the simu-lated ranges of the primary particles (i.e. the Bragg peak position) with the ex-perimental data [131].
At non relativistic energies, de/dx is dominated by the overall 1/β2 factor and decreases with increasing velocity until about v ≈ 0.96c, where a maximum is reached. Particles at this point are known as minimum ionizing. The minimum value of de/dx is almost the same for all particles of the same range. As the en-ergy increases beyond this point, the term 1/β2 becomes almost constant and de/dx rises again due to the logarithmic dependence. However the relativistic rise is partially compensated by density correction.
For energy below the minimum ionizing value, each particle exhibits a de/dx curve which, in most cases, is distinct from the other particle types. At very low energy, the stopping power reaches a maximum and then drops rapidly, due to the tendency of particles to pick up electrons.