Charged particles traversing a medium lose their energy by interaction predominately with the electrons of the material. The ability of the target medium to slow down the moving particles is expressed by the stopping powerS, which is defined as the energy loss per length:
S =−dT
dx . (2.1)
At energies above 1.5keV/u, the contributing interaction processes are mainly inelastic collisions of the projectile with the target medium. The projectile loses its kinetic energy due to momentum transfer to the electrons of the medium, leading to an excitation or ionisation of the target atoms or molecules. This kind of energy loss is described by the electronic stopping power Se. Below energies of about 1.5keV/u, the energy loss of the projectile predominately occurs through elastic collisions with the atoms or molecules of the traversed medium which is described by the nuclear stopping power Sn. The sum of
3
these two contributions gives the collisional stopping power
Sc=Se+Sn. (2.2)
Charged particles can additionally lose their kinetic energy by emission of bremsstrahlung in the electric field of the target nuclei. This energy loss is described by the radiative stopping power Sr. The total stopping powerS can then be written as the sum ofSc and Sr:
S =Sc+Sr. (2.3)
It is related to the stopping cross section Sσ via Sσ = V
NS (2.4)
with the number of atoms or molecules N in a volume V. As the radiative stopping only occurs at very high projectile energies, it will be not further considered in this work.
The stopping power does not only depend on the physical properties of the target, such as its electronic structure and density, but also on the mass and charge of the projectile.
Furthermore, the stopping power is a function of the kinetic energy of the projectile. An example of the dependence of the components of the collisional stopping power on the kinetic energy T is given in figure 2.1. When coming from the high–energy side, the stopping power increases with decreasing projectile energy T until it reaches its maximal value at energies around 300keV/u. At the left hand side of the maximum, the stopping power shows a steep decrease with decreasing projectile kinetic energy. The energy of a projectile with an incident energy of T0 in a certain depth dof the traversed medium can be calculated by
T(d) = T0− ˆx=d
x=0
S(T(x0))dx0. (2.5)
Figure 2.1(b) shows the stopping power of water for carbon ions with an initial energy of T0 = 1.3 GeV as function of the penetration depth d calculated using equation 2.5.
The stopping power data used for the calculation, shown in figure 2.1(a) were taken from SRIM2013 [99]. At the end of the projectile range a sharp maximum is built up which is called Bragg peak. Its location in depth depends on the initial energy of the projectiles as well as on the stopping power of the target medium.
In principle, the stopping power can be determined by transmitting projectiles with known energy T through a thin slice of material with a definite thickness dx and then measuring their final energy (see also figure 2.2). This transmission method is, however, limited to those materials which are thin enough to enable a differential measurement and at the same time are stable in an evacuated environment. The maximal thickness of the target that can be accepted for a transmission measurement can be estimated according
2.1 STOPPING POWER 5
Figure 2.1: (a) Stopping power of carbon ions in water calculated with SRIM2013[99]. (b) Stop-ping power as function of the depth in water for carbon ions with a starting energy of1.3 GeV.
dx
T - dT T
Figure 2.2: Illustration of the transmission method for the measurement of the stopping power.
to the average residual rangeR of the projectiles:
R(T0) = −
Figure 2.3 shows the average residual range of carbon projectiles in water in dependence on their initial kinetic energy T calculated using equation 2.6. As it can be seen from figure 2.3, the residual range of projectiles is rather small in the vicinity of the Bragg peak area.
In the case of water, this measurement has to be performed at atmospheric pressure.
This is necessary to prevent a variation of the target thickness during the measurement and the slow down of the projectiles in the evaporated water. Consequently, the target must be sealed against the vacuum of the beam line. If the primary particles are not available from a radioactive source as in the case for protons or carbon ions, the projectiles have to be produced by an accelerator with an evacuated beam line. The entrance window of the target sealing against the beam line must have a mechanical strength to withstand the pressure difference between the atmosphere and the vacuum. Usually, this mechanical strength requires a window thickness in the same order as the range of the incident
/ MeV
0 500 1000 1500T 2000 2500 3000
/ cmR
Figure 2.3: Residual range of carbon ions stopped in water in dependence on the kinetic energy.
The marked region covers the energy interval in which the stopping power reaches the maximal value.
projectiles. Therefore, the application of the transmission method for the measurement of the stopping power of water for projectile energies of a few hundredkeV/uis not feasible.
As mentioned above, the stopping power only describes the energy loss of a projectile per length. In the field of dosimetry of ionising radiation, the term Linear Energy Transfer (LET) is commonly used. It describes the energy transfer to the medium per path length.
The LET is defined as
L∆ = dT∆
dx and lim
∆→∞L∆(T) = Sc(T). (2.7) It gives the energy transfer up to energies of ∆. In the limit of ∆ = ∞ the LET and stopping power are equal.
Radiobiological experiments indicate that the LET is correlated to the relative bio-logical effectiveness (RBE) of ionising radiation [23]. It has to be mentioned that RBE not only depends on the projectile and the LET but also on the fractionation of the irra-diation, the cell type and cell cycle, the environment and oxygen saturation, and so on.
However, it can be assumed that the radiation damage increases with the LET due to the larger amount of energy deposited along the same length. For the quantification of the radiation damage in case of treatment planning, the stopping power of the projectiles at all depths in the healthy tissue and malignant neoplasm (cancer) has to be known.