KAHIBARO
Table of Contents
Meaning and Definition
When a charged particle moves through matter, it loses energy by interacting with the atoms of the material. The rate at which the particle loses energy as it travels is called stopping power.
If a particle has energy $E$ and travels a small distance $dx$, then the stopping power is written as
$$
S = -\frac{dE}{dx}
$$
The minus sign appears because the particle’s energy decreases as distance increases.
Stopping power tells us how strongly a material slows down a charged particle. A large stopping power means the particle loses energy quickly. A small stopping power means it can travel farther before stopping.
Stopping power is the energy lost per unit path length:
$$
S = -\frac{dE}{dx}
$$
Its sign is usually taken as positive in words, even though the mathematical expression contains a minus sign.
Physical Picture
A charged particle, such as an alpha particle, proton, or electron, passes near atomic electrons and nuclei in a material. Because of electric forces, it transfers some of its energy to the material. This energy can go into ionization, excitation, and other processes.
For heavy charged particles, most of the energy loss comes from interactions with atomic electrons. The particle gradually slows down as it penetrates deeper into the material. The stopping power describes how intense this slowing down is at each point along the path.
Units
Stopping power is often measured in units such as:
| Quantity | Common unit |
|---|---|
| Stopping power | $\text{MeV/cm}$ |
| Stopping power | $\text{keV}/\mu\text{m}$ |
| SI form | $\text{J/m}$ |
Sometimes physicists divide by the material density $\rho$ to compare different substances more fairly. This gives the mass stopping power:
$$
\frac{1}{\rho}\left(-\frac{dE}{dx}\right)
$$
Typical units for mass stopping power are
$$
\text{MeV cm}^2/\text{g}
$$
Dependence on Particle Speed
Stopping power is not constant. It depends strongly on the speed of the incoming particle.
For many heavy charged particles, the stopping power is relatively large when the particle is slow, becomes smaller at intermediate speed, and then changes again at higher energies. A very important trend for beginners is that as a heavy charged particle slows down near the end of its path, it often loses energy more rapidly per unit distance.
This is why the energy loss becomes especially intense near the end of the track.
Stopping power depends on both the particle and the material. It is not a fixed property of the particle alone.
Dependence on Charge and Material
A more highly charged particle generally causes stronger electric interactions with the electrons in the material, so it usually has a larger stopping power.
The material also matters. Denser materials, or materials with more electrons per unit volume, usually cause greater energy loss. That means the same particle may travel far in air but only a short distance in metal.
In a simple qualitative sense:
| Factor increased | General effect on stopping power |
|---|---|
| Particle charge | Increases stopping power |
| Electron density of material | Increases stopping power |
| Particle speed | Changes stopping power in a non-simple way |
Collision Stopping and Radiative Stopping
For charged particles, energy loss can come from different mechanisms. In this chapter, the main idea is the total energy loss per unit distance, but it is useful to separate two important contributions.
For heavy charged particles, the dominant part is usually collision stopping power, caused by ionization and excitation of atoms.
For light particles such as electrons, there can also be significant radiative stopping power, where energy is lost by emitting electromagnetic radiation when the particle is accelerated in the electric field of nuclei. This is called bremsstrahlung.
So the total stopping power can be thought of as
$$
\left(-\frac{dE}{dx}\right)_{\text{total}}
=
\left(-\frac{dE}{dx}\right)_{\text{coll}}
+
\left(-\frac{dE}{dx}\right)_{\text{rad}}
$$
For heavy particles, the radiative term is usually much smaller than the collisional term.
Relation to Penetration in Matter
Stopping power is closely connected to how far a particle can go in a material. If the stopping power is large, the particle loses energy quickly and stops after a shorter path. If it is small, the particle penetrates more deeply.
Because stopping power changes as the particle slows down, the path to rest is found by combining the energy loss over the full motion. This idea leads naturally to the concept of range, which is treated separately.
Typical Behavior for Heavy Charged Particles
Heavy charged particles, such as alpha particles and protons, usually travel in fairly straight paths and lose energy continuously. As they slow down, the stopping power tends to rise strongly near the end of the track. This behavior explains why the deposited energy becomes concentrated near the end of the path.
A qualitative graph looks like this:
This sharp rise near the end is closely related to the Bragg peak, which is discussed in its own chapter.
Simple Interpretation of the Formula
Suppose a particle loses $2 \, \text{MeV}$ of energy while traveling $1 \, \text{cm}$ in a material. Then its average stopping power over that interval is
$$
S \approx \frac{2 \, \text{MeV}}{1 \, \text{cm}} = 2 \, \text{MeV/cm}
$$
If in another material it loses the same $2 \, \text{MeV}$ in only $0.2 \, \text{cm}$, then
$$
S \approx \frac{2 \, \text{MeV}}{0.2 \, \text{cm}} = 10 \, \text{MeV/cm}
$$
So the second material stops the particle much more strongly.
Why Stopping Power Matters
Stopping power is important because it helps us predict how radiation deposits energy in matter. This matters in detector design, shielding, and medical physics. It tells us where the particle slows down most strongly and how quickly its energy is transferred to the medium.
In practical work, tables and models of stopping power are used to estimate particle behavior in different materials.
A large stopping power means rapid energy loss and usually shorter penetration depth.
A small stopping power means slower energy loss and usually greater penetration depth.
Summary
Stopping power is the energy lost by a charged particle per unit distance traveled in a material. It is written as
$$
-\frac{dE}{dx}
$$
and depends on the particle’s charge, speed, and the properties of the material. For heavy charged particles, it mainly comes from ionization and excitation of atoms. It is a central quantity for understanding how radiation slows down and deposits energy in matter.