A charged particle traversing matter exerts electromagnetic forces on atomic electrons and imparts so-called collision energy to them (ICRU 46 1992; Knoll 1989). The energy transferred may be sufficient to knock an electron out of an atom and thus ionise it. Alternatively, it may leave the atom in an excited non-ionised state. When an atom is ionised, the secondary electron produced may have enough energy to cause several more ionisations or excitations along a branched track before being thermalised. The energy releases by particles are discrete events, the spacing of which will depend on the energy and type of the particles.
The average rate of collision energy loss of particles in a medium -dE/dx or collision stopping power can be derived, using relativistic quantum mechanics, from the Bethe formula (1):
In this relation:
Na = Avogadro's number z = charge of the particle e = magnitude ofthe electron charge me = electron rest mass v = speed of the particle p = density of the irradiated medium Z = atomic number of the absorbing atom A = atomic mass of the absorbing atom
B = correction factor, dependent on the medium, particle type and energy
As can be seen from this formula, the stopping power increases with the density of the irradiated medium.
If they have the same energy, heavy particles such as a-particles and protons, are much slower than electrons. Therefore the average rate of energy loss of heavy particles is much greater. Because they are heavier than the atomic electrons with which they collide their energy loss per collision is small and their deflection is almost negligible. Therefore they are gradually slowed down as a result of a large number of small energy losses. They travel along almost straight paths through matter leaving a dense track of ionised and excited atoms in their wake (ICRU 49, 1993).
In contrast, electrons and positrons can lose a large fraction of their energy in a single collision with an atomic electron, thereby suffering relatively large deflections. The energy releases are widely spaced along the particle tracks. Because of their small mass, electrons are frequently scattered through large angles by nuclei. Electrons and positrons generally do not travel through matter in straight lines (ICRU 39, 1984).
The v-2-dependence of the stopping power, (see the Bethe formula above), indicates that for low velocities at the end of the particle trajectories, the deposited energy increases sharply (the v-2-dependence is slightly compensated by a smaller decrease of the factor B with energy). For electrons this absorbed dose increase is substantial only over the last nm of the trajectory, where clusters of ionisations occur. For high-energy protons this region extends over a few mm (the so-called Bragg peak). The Bragg peak of protons forms the basis of the application of these particles in proton radiotherapy, allowing more precise dose distributions in tumours (Courdi 1993; Rosenwald J. 1993).
A high-speed particle can also be sharply decelerated and deflected by an atomic nucleus, causing it to emit the energy lost as electromagnetic radiation in a process called "bremsstrahlung" (braking radiation). The rate of energy loss by this process is proportional with z2Z2/m2, where z and Z are the charge of the particle and the atomic number of the nucleus, and m the mass of the particle. For particles with a heavier mass than electrons bremsstrahlung production is negligible at the energies used for the irradiation of biological samples. For electrons bremsstrahlung production is a second-order effect in the radiolysis of biological materials, because they are low-Z materials. The bremsstrahlung process is in this context only relevant for the production of X-rays with electron accelerators (see next chapter).
The range of a charged particle, i.e. the distance that a particle can penetrate into matter, depends on the initial energy of the particle and the density of the absorber. The reciprocal of the stopping power (including collision and radiative processes) gives the distance travelled per unit energy loss. Therefore, the range R(T) of a particle of kinetic energy T is the integral of this quantity down to zero energy (2):
Because the common unit for stopping powers is MeV cm2 g-1, the range is expressed in g cm-2. To obtain the range in cm R has to be divided by the density p of the absorbing material.
The range of 1 MeV a-particles, protons and electrons in water are respectively 4.6 pm, 39 ^ m and 4.37 mm. For 10 MeV these values are 0.1 mm, 1.2 mm and 49.8 mm. For other materials the range is roughly inversely proportional with the density.
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