Range - Aetherwisp

In particle physics, the range of a particle in the distance it can travel in a medium before losing all its kinetic energy due to the medium's stopping power. It is defined as

R(K)=\int_{0}^{K_{0}} \frac{dK}{S(K)}

where $R$ is the mean range, $K_{0}$ is the initial energy of the particle, and $S(K)$ is the stopping power. It is typically measured in $\text{g}/\text{cm}^{2}$ .

Provided different particles are given the same starting kinetic energy $K_{0}$ , the range measured from their collisions onto a material is strongly representative of their mass. As such, range is a powerful tool for determining the mass of particle when we know its kinetic energy.

The definition follows from expressing the range as an integral and playing with the differentials to turn it into an energy integral:

R(K)=\int_{0}^{R}dx=\int_{K_{0}}^{0} \frac{dK}{dK} dx=-\int_{K_{0}}^{0} \frac{dK}{S(K)}=\int_{0}^{K_{0}} \frac{dK}{S(K)}

Since the amount of energy lost in the material is random (energy straggling), the range is also random and described by a Probability distribution parameterized by kinetic energy. This phenomenon is called range straggling.

Nonrelativistic behavior#

For a particle of charge $z$ , mass $M$ , and nonrelativistic kinetic energy $K_{0}$ , we can approximate the stopping power using the Bethe-Bloch formula. In nonrelativistic range, the logarithm is essentially constant. Then, the only functional dependence in $dK/dx$ is $z^{2}/\beta ^{2}$ and everything else is a constant. Then, we can state

\frac{dK}{dx} \simeq C \frac{z^{2}}{\beta^{2}}=C\frac{z^{2}c^{2}(M/2)}{v^{2}(M/2)} = C'\frac{Mz^{2}}{K}

where $M$ is the mass of the particle and $C,C'$ are some common constant to group up all the other constants. Then range becomes

R(K)=\int_{K_{0}}^{0} \frac{dK}{dK}dx\simeq \frac{C'}{Mz^{2}}\int_{K_{0}}^{0}KdK=\frac{C''}{Mz^{2}}K_{0}^{2}

(where $C''=-C'$ ). Given an initial kinetic energy $K_{0}$ , the range varies with mass according to $1/M$ . The heavier the particle, the smaller the range and greater the energy loss. Just like the Bethe-Bloch curve, this fact is very useful for measuring particle mass, as the path covered by a particle before decaying or stopping is often relatively easy to measure by, say, analyzing tracks in a cloud chamber or having a layered set of detectors that count the passage of particles.

Diagram Layered range measurement.svg|80%

Experimental setup to measure range. A layer of detectors that absorb energy is setup, each with some length $l$ . If a particle is absorbed is some detector, it will be recorded. Each detector will therefore count the number of particles that stop in it. This provides a histogram of distances covered and thus an estimate of the range distribution. If the range is bigger then the total length, $R(K_\text{in})>L$ , then we'll see few measurements and no peak. If the range is less, $R(K_\text{in})<L$ , then we'll see the full range distribution with a peak in one of the detectors and very few $N_\text{out}$ .