Charged Particles - Stopping Power

When a charged particle passes through matter, it loses energy mainly by its electromagnetic interaction with the atomic electrons in the material through which it is moving. The specific loss of energy dE for a charged particle that crosses a medium for a distance dx is given by the Equation 3.1

− dE

dx = Kz²Z A

1

β²L(β) (3.1)

where K/A = 4πN_Ar_e²m_ec² = 0.307075 M eV g⁻¹cm², z is projectile charge, A and Z are the target atomic mass and number, respectively; m_e and r_e are the electron mass and its classical radius; N_A is the Avogadro’s number; c is the speed of light in vacuum. The parameter L(β), is presented in the following form:

L(β) = L₀(β) +

i

∆L_i (3.2)

L₀(β) = ln

2m_ec²β²γ²T_max I²



− β²− δ(βγ)

2 (3.3)

with γ = E_kin+ m₀c²

m₀c² and β =

1− 1/γ² (3.4)

where T_max is the maximum kinetic energy which can be transferred to a free electron in a single collision, I and δ are the mean excitation energy and the density correction, respectively; E_kinis the kinetic energy of the projectile and m₀ is its rest mass. Taking into account only L₀(β), while neglecting the rest of the corrections 

∆L_i, the expression 3.1 is referred to as Bethe-Bloch equation [Blo97]. At non-relativistic energies the ionization energy loss has a 1/β² energy dependence, whereas at relativistic energies the energy loss is slightly growing. Thus, the ionization energy loss has a minimum at a certain energy.

A particle of this energy is called a ’minimum ionizing particle’ (MIP). The Bethe-Bloch formula describes quite precisely the energy losses of projectiles of z = 1, however it fails

21

Lindhard-Sørensen (LS) Correction Lindhard and Sørensen [Lin96] derived a rela-tivistic expression for electronic stopping power of heavy ions taking into account a finite nuclear size. They used the exact solution of the Dirac equation with spherically symmetric potential, which describes the scattering of a free electron by an ion. At moderately rela-tivistic energies (E_kin > 500 M eV /u) the following expression derived for point-like ions is valid:

where η = αz/β with α the fine structure constant, δ_k is a relativistic Coulomb phase shift expressed with the argument of the complex Gamma function, and k is a parameter used in the summation over partial waves. At ultra-relativistic energies, where the de Broglie wave length is much smaller than the projectile nucleus, an asymptotic expression for L(β) is valid:

4πne²/m_e is the plasma frequency, and n is the average density of target electrons. The value of L_ultrareveals a weak dependence on target and projectile parameters.

Partially ionized ions The LS correction gives perfect agreement with the experimental data for bare projectiles. However at lower energies (< 500 M eV /u) the heavy ions are no longer completely stripped. Therefore, in the energy loss description, z is replaced by the effective charge (q_{ef f}) using [Pie68]:

q_{ef f} = z where υ/υ₀ is the projectile velocity in units of the Bohr velocity.

Barkas correction The Barkas effect [Bar63], associated with a z³ correction to the stopping power, is well pronounced at low energies. The correction is due to target polar-ization effects for low-energy distant collisions and can be accounted for by the following expression:

Z. The function F (V ) is the ratio of two integrals within a Thomas-Fermi model of the atom.

3.1. Particle Interaction with Matter 23

The figure inset shows the z² dependence of the stop-ping power for ions of same velocity β.

Shell corrections Shell corrections [Bar64] should be taken into account at lower pro-jectile velocities, when these approaches the velocity of the shell electrons of the target.

The total shell correction can be presented as:

∆L = −C

Z (3.9)

where C is equal to C_K + C_L+ ... and thus, it takes into account the contributions from different atomic shells.

Figure 3.1 presents the electronic stopping power dE/dx of several ions in diamond (ρ_diam = 3.52 g/cm³) in the range of 1 keV/amu-3 GeV/amu. The curves are calculated using ATIMA software [ATIMA], which was developed at GSI for the calculation of various physical quantities characterizing the slowing-down of heavy ions in matter including all presented corrections.

Energy loss of electrons When electrons or positrons pass through matter, they lose energy by ionization like all other charged particles. In addition, they lose energy by bremsstrahlung as a consequence of their small mass.

dE The bremsstrahlung process is caused by the electromagnetic interaction of the electrons with an atomic nucleus, where a photon is radiated from the decelerated electron or positron.

The critical energy (E_c), the energy above which bramsstrahlung dominates above the ionization loss can be estimated from the simplified relation:

E_c = 600 M eV

Z (3.11)

For diamond (Z = 6), the critical energy is about 100 MeV. Figure 3.2 shows the mean energy losses of an electron in diamond and silicon calculated using the ESTAR database

Figure 3.2: The mean energy losses of an electron in diamond (red curves) and silicon (blue curves) versus its kinetic energy.

The solid curves show the to-tal energy loss, the dashed ones correspond to the ionization loss and the dotted correspond to the radiative energy loss. The plot is obtained using the ES-TAR database, after [Kuz06]

[ESTAR]. The minimum energy loss of an electron in diamond is about 1.6 MeV and in silicon about 1.3 MeV .

Restricted energy loss A part of the energy loss by a fast charged particle in matter is converted in fast secondary electrons or high energy photons, which leave the volume of the track. For thin films, the created secondary particles carry out a part of the deposited energy, thus the measured energy deposition within the detector material is smaller than the real energy lost by a primary particle. The restricted energy loss can be expressed in:

− with T_upper = min(T_cut, T_max), and T_cut the cut-off energy above which the secondary parti-cles escape the material. When T_cut > T_maxthe equation 3.12 meets the normal Bethe-Bloch function 3.1.

For standard-diamond thin films of 300-500 µm thick, often a value T_cut = 7.5 keV is assumed [Zha94], which is the energy at which the photons absorption length amounts to λ≈ 500 µm. Using this value the ionization yield for MIPs Q^TM IP^{cut=7.5 keV}/l can be calculated for diamond: energy needed for e-h pair creation in diamond.

Fluctuations of the energy loss - straggling functions In the measurement of dE/dx, there are fluctuations of the average value caused by a small number of collisions that cause large energy transfers. In a thin layer of material the distribution of dE/dx will be asymmetric, with some measurements giving large energy losses. A first descrip-tion of the energy straggling for thin absorbers was given by Landau resulting in so called Landau energy straggling function which depends only on one parameter [Lan44]. One of

3.1. Particle Interaction with Matter 25

Figure 3.3: The Vavilov distribution function Φ as a function of the scaled en-ergy loss λ for various parameter κ and β = 0.98, calculated numerically using the the algorithm given in [Sch74]

the assumptions of the Landau distribution is that incoming particle can transfer all its energy to a single shell electron, which can be described by infinite energy transfer. A generalization of the Landau distribution was given by Vavilov [Vav57]. This distribution takes into account the maximum allowed energy transfer in a single collision of a particle with an atomic electron. The Vavilov distribution is a function of two parameters κ > 0 and 0 < β = ^v_c2² < 1. κ is proportional to the ratio of the mean energy loss over the path length of the particle to the largest energy transfer possible in a single collision with an atomic electron. The Vavilov distribution converges to the Landau distribution as κ → 0 for thin absorbers and large energy transfer. For thick absorbers and/or limited energy transfer, κ > 1 and the Vavilov distribution can be replaced by a Gaussian. Figure 3.3 presents numerical calculation of the Vavilov density function for various κ and β = 0.98 using algorithm of [Sch74].

A comprehensive overview of straggling functions as well as the methods of its numerical calculations can be found in several publications [Bic88, Ait69, Erm77]