Electron Interactions with Matter

Introduction

When an energetic charged particle, such as an electron or positron, passes through a medium, it undergoes a large number of interactions with the nuclei and orbital electrons of atoms in the absorber. These interactions lead to the loss of kinetic energy of the particle, which is transferred to the medium in the form of ionization, excitation, or radiation. The primary interactions are:

Ionization: Ejection of an orbital electron from the atom.
Excitation: A transfer of energy to an orbital electron, raising it to a higher energy state.
Bremsstrahlung (Radiation loss): Production of X-rays due to inelastic scattering with the nuclei of the absorber.

Types of Stopping Power

The energy loss of a charged particle in a medium is characterized by two main types of stopping power:

Collision Stopping Power (\(s_{\text{col}}\)): Energy loss due to interactions with orbital electrons of the absorber, resulting in ionization or excitation.
Radiation Stopping Power (\(s_{\text{rad}}\)): Energy loss due to interactions with the nuclei of the absorber, primarily through the production of bremsstrahlung radiation.

The total stopping power (\(s_{\text{tot}}\)) is the sum of these two components:

\( s_{\text{tot}} = s_{\text{col}} + s_{\text{rad}} \)

Electron-Orbital Interactions

Coulomb interactions between an incident electron and the orbital electrons of the absorber result in ionization and excitation. These interactions are described as follows:

Ionization: When an electron transfers enough energy to eject an orbital electron from an atom, creating an ion.
Excitation: When the electron transfers energy to excite an orbital electron to a higher energy level without ejection from the atom.

The energy loss from ionization and excitation contributes to the collision stopping power (\(s_{\text{col}}\)).

Electron-Nucleus Interactions

When an energetic charged particle interacts with the nuclei of an absorber, the interaction is governed by Coulomb forces. Most of these interactions are elastic, meaning the particle does not lose energy, but in some cases, the interaction is inelastic and the particle loses energy in the form of bremsstrahlung radiation (X-ray photons).

The energy loss due to bremsstrahlung radiation is characterized by the radiation stopping power (\(s_{\text{rad}}\)) and is described by the Larmor relationship, which states that the rate of energy loss is proportional to the square of the particle’s acceleration and the square of the particle’s charge.

\( \frac{dE}{dx} = \frac{z^2 e^2}{4 \pi \epsilon_0} \frac{v^2}{c^2} \)

Where:

\(z\) is the charge of the incident particle,
\(e\) is the electron charge,
\(\epsilon_0\) is the permittivity of free space,
\(v\) is the velocity of the particle,
\(c\) is the speed of light.

Example Calculation: Total Stopping Power

Let's consider an example where we calculate the total stopping power for a 1 MeV electron in a medium.

Collision stopping power (\(s_{\text{col}}\)) = \( 0.4 \, \text{MeV/cm} \)
Radiation stopping power (\(s_{\text{rad}}\)) = \( 0.1 \, \text{MeV/cm} \)

We can calculate the total stopping power using the formula:

\( s_{\text{tot}} = s_{\text{col}} + s_{\text{rad}} = 0.4 \, \text{MeV/cm} + 0.1 \, \text{MeV/cm} \)

So, the total stopping power is:

\( s_{\text{tot}} = 0.5 \, \text{MeV/cm} \)

This means that the electron will lose \( 0.5 \, \text{MeV} \) of energy for every centimeter it travels through the medium.

Important Notes

Note: The total energy loss of a charged particle in a medium depends not only on the particle’s properties (mass, charge, energy) but also on the properties of the absorbing material (density, atomic number). In materials with higher atomic numbers, the radiation stopping power tends to increase due to more frequent interactions with the nuclei.