KAHIBARO
Table of Contents
Meaning of Range
When a charged particle travels through matter, it loses energy little by little through many interactions with the atoms of the material. Eventually, the particle slows down enough to stop. The total distance it travels before stopping is called its range.
Range is one of the most useful ideas in radiation physics because it tells us how deeply a particle can penetrate into a material. A particle with higher initial energy usually has a larger range. A denser material usually gives a shorter range, because the particle loses energy more quickly.
Range as a Stopping Distance
If a charged particle enters a material with initial kinetic energy $E_0$, its energy decreases as it moves through the material. If the energy loss per unit distance is written as $\frac{dE}{dx}$, then the range can be found by adding up all the tiny distances traveled while the particle slows from $E_0$ to zero.
Mathematically, the range $R$ is
$$
R = \int_0^{E_0} \frac{dE}{\left|dE/dx\right|}
$$
This equation says that the total path length depends on how strongly the material removes energy at each energy value.
The range is the total distance a charged particle travels in a material before coming to rest.
$$
R = \int_0^{E_0} \frac{dE}{\left|dE/dx\right|}
$$
A larger stopping power generally means a shorter range.
True Range and Projected Range
A real charged particle does not always move in a perfectly straight line. It can be deflected by repeated interactions inside the material. Because of this, there are two useful ideas of range.
The true range is the actual length of the particle's path through the material. The projected range is the depth reached along the original direction of motion. If the path bends a lot, the true range is larger than the projected range.
This distinction is especially important for light charged particles such as electrons, which scatter strongly. Heavier charged particles, such as alpha particles, tend to follow straighter paths, so their true range and projected range are often more similar.
Dependence on Particle Type
Different charged particles with the same kinetic energy do not have the same range. Mass and electric charge both matter. Heavy particles usually move more directly through matter and can have well-defined ranges. Light particles are more easily scattered.
Alpha particles are a classic example of particles with a relatively short and sharp range in matter. Electrons, in contrast, undergo frequent changes in direction, so their stopping distance is less sharply defined.
A simple comparison is shown below.
| Particle type | Typical path behavior | Range character |
|---|---|---|
| Alpha particle | Nearly straight | Short, well-defined |
| Proton | Fairly straight | More well-defined than electrons |
| Electron | Strongly scattered | Less sharp, more spread out |
Dependence on Material
The range also depends strongly on the material the particle enters. A dense material contains more atoms per unit volume, so the particle usually loses energy more rapidly and stops in a shorter distance.
Materials are often compared using mass thickness instead of ordinary length. Mass thickness is the product of density and path length:
$$
\text{mass thickness} = \rho x
$$
where $\rho$ is the density of the material and $x$ is the geometric thickness. This is useful because particles with the same energy often have similar ranges when expressed in units like $\text{g/cm}^2$, even if the physical thickness in centimeters differs.
Range may be expressed as an ordinary length, such as $\text{cm}$, or as a mass thickness, such as $\text{g/cm}^2$.
$$
R_m = \rho R
$$
where $R_m$ is the mass range, $\rho$ is density, and $R$ is geometric range.
Continuous Slowing Down Approximation
A common idealization is to imagine that the particle loses energy smoothly and continuously as it moves. This is called the continuous slowing down approximation, often shortened to CSDA. In this model, the range is called the CSDA range.
In reality, energy loss happens in individual collisions, not as a perfectly smooth process. Still, the CSDA range is very useful because it gives a practical average stopping distance.
The actual distance traveled by real particles is spread around this average because not all particles lose energy in exactly the same way.
Range Straggling
Even if many identical particles enter the same material with the same initial energy, they do not all stop at exactly the same depth. Some stop a little earlier, others a little later. This spread in stopping distance is called range straggling.
Range straggling happens because energy loss is statistical. The particle undergoes many separate collisions, and the number and strength of these collisions vary from one particle to another.
So the range is not a single perfectly exact depth, but a distribution.
Real particle ranges are not identical. There is always a statistical spread called range straggling.
Practical Interpretation
If a beam of charged particles enters an absorber, the number of particles that emerge from the far side depends on whether the absorber thickness is smaller or larger than the particle range. If the absorber is thinner than the range, many particles pass through. If it is thicker than the range, most or all particles are stopped.
This idea is widely used in shielding, detector design, and estimating penetration depth in matter.
Visualizing Range
The figure below shows the idea of a particle entering matter, slowing down, and finally stopping after traveling a finite distance.
Key Idea
Range connects the microscopic process of energy loss to a macroscopic distance in matter. It tells us how far a charged particle can penetrate before stopping, but that stopping distance is usually an average or a distribution rather than a perfectly exact number.
Range is a measure of penetration depth for charged particles in matter.
It depends on the particle's initial energy, the particle type, and the material through which it passes.