KAHIBARO
Table of Contents
Meaning of reaction probability
In nuclear physics, reaction probability tells us how likely it is that an incoming particle will interact with a target nucleus in a particular way. For example, a neutron striking a nucleus might be scattered, absorbed, or cause some other nuclear reaction. Reaction probability is the practical idea behind the more formal quantity called the cross section.
If a beam of particles hits a thin target, not every projectile reacts. Some pass through without interaction, while some undergo a specific process. The reaction probability connects what we observe experimentally, the fraction of particles that react, to the microscopic behavior of nuclei.
Probability for a single target nucleus
Imagine one projectile aimed at one nucleus. The chance of reaction depends on how effective that nucleus is at presenting itself to the projectile. This effectiveness is described by the cross section $\sigma$, and the probability also depends on the area over which the projectile could pass.
A larger cross section means a higher chance of interaction. A smaller cross section means the projectile is less likely to react.
A useful mental picture is to think of the nucleus as having an effective target area. This is not always the true geometric area of the nucleus. It is an interaction area that summarizes quantum and nuclear effects.
The reaction probability increases with cross section. In simple terms,
$$\text{larger } \sigma \Rightarrow \text{larger probability of reaction}$$
The cross section is not merely a physical size, it is an effective measure of interaction likelihood.
Probability in a thin target
Suppose a beam passes through a thin foil containing many target nuclei. Let $n$ be the number of target nuclei per unit volume, and let $dx$ be a very small thickness. Then the probability that a projectile reacts while crossing that small thickness is
$$dP = n \sigma \, dx$$
This expression is very important. It shows that the reaction probability for a short path increases when the material contains more nuclei per unit volume, when the cross section is larger, and when the path length is longer.
For a thin layer of material, the differential reaction probability is
$$dP = n \sigma \, dx$$
This formula is valid when the layer is small enough that the probability over that layer is much less than 1.
Survival probability and total reaction probability
Instead of asking for the probability that a reaction happens, we can ask for the probability that no reaction happens after traveling a distance $x$ inside the material. This is called the survival probability.
If reactions occur independently, the survival probability decreases exponentially:
$$P_{\text{survive}}(x) = e^{-n \sigma x}$$
So the probability that at least one reaction has occurred after distance $x$ is
$$P_{\text{react}}(x) = 1 - e^{-n \sigma x}$$
For very small $n \sigma x$, this becomes approximately
$$P_{\text{react}} \approx n \sigma x$$
which matches the thin target expression.
For a uniform target,
$$P_{\text{survive}}(x) = e^{-n \sigma x}$$
and
$$P_{\text{react}}(x) = 1 - e^{-n \sigma x}$$
For thin targets, use the approximation
$$P_{\text{react}} \approx n \sigma x$$
Interpretation of the exponential form
The exponential form means that each small piece of material removes the same fraction of the remaining unreacted particles. This is why the process is random but still follows a precise law when many particles are involved.
This behavior is similar to radioactive decay, except here the change happens with distance traveled through matter rather than with time.
Mean free path
A closely related idea is the mean free path, written as $\lambda$. It is the average distance a projectile travels before reacting.
It is defined by
$$\lambda = \frac{1}{n \sigma}$$
Using this, the survival probability can be written as
$$P_{\text{survive}}(x) = e^{-x/\lambda}$$
and the reaction probability becomes
$$P_{\text{react}}(x) = 1 - e^{-x/\lambda}$$
A small mean free path means reactions are likely to happen quickly. A large mean free path means the projectile can travel farther without interacting.
The mean free path is
$$\lambda = \frac{1}{n \sigma}$$
A shorter mean free path means a higher reaction probability per unit distance.
Reaction probability for different processes
A projectile can undergo more than one type of reaction. For example, it may be elastically scattered, inelastically scattered, or captured. Each process has its own cross section and therefore its own probability.
If the partial cross sections are $\sigma_1$, $\sigma_2$, $\sigma_3$, and so on, then the total cross section is
$$\sigma_{\text{tot}} = \sigma_1 + \sigma_2 + \sigma_3 + \cdots$$
The probability that a reaction occurs through channel $i$ is proportional to $\sigma_i$.
The fraction of reactions going into channel $i$ is
$$\frac{\sigma_i}{\sigma_{\text{tot}}}$$
This is often called the branching fraction for the available reaction channels.
Comparing probabilities
The table below summarizes how physical changes affect reaction probability.
| Change | Effect on reaction probability |
|---|---|
| Larger cross section $\sigma$ | Increases |
| Higher target number density $n$ | Increases |
| Larger thickness $x$ | Increases |
| Smaller mean free path $\lambda$ | Increases |
| Thinner target | Decreases |
Simple geometric picture
A beam entering a target can be pictured as many particles moving through a region filled with nuclei. Only some trajectories lead to interaction.
Some particles pass through, and some interact. The denser the nuclei are, and the larger the effective cross section is, the greater the fraction that react.
Experimental meaning
In experiments, reaction probability is often found by comparing the number of incident particles with the number that react. If $N_0$ particles enter a target and $N$ emerge without the chosen reaction, then for a uniform target of thickness $x$,
$$N = N_0 e^{-n \sigma x}$$
So the fraction that react is
$$\frac{N_0 - N}{N_0} = 1 - e^{-n \sigma x}$$
This connects measurable counting rates to nuclear reaction properties.
Key idea to remember
Reaction probability is the chance that an incident particle undergoes a specified nuclear interaction while passing through a target. It depends on the target density, the distance traveled in the material, and especially the cross section that characterizes the reaction.
The central formulas for reaction probability are
$$dP = n \sigma \, dx$$
$$P_{\text{react}}(x) = 1 - e^{-n \sigma x}$$
$$\lambda = \frac{1}{n \sigma}$$
These equations link microscopic nuclear behavior to measurable reaction rates.