# Difference between cross section and probability of interaction

When I read some textbooks I find that the definition of cross section is the probability. Of incident neutron interact with a matter .but I find many other (cross section of fission, scattering, absorption..) And I find another definition of probability of interaction in distance $$l$$ which is given by

$$P(l)=1-e^{\frac{l}{\lambda}}$$

Where $$\lambda=\frac{1}{\Sigma}$$ And $$\Sigma$$ is the macroscopic cross section $$\Sigma=N \sigma$$, $$N$$ is the density of nuclear and $$\sigma$$ is the croos section

My question is what is the difference between the two definition?and if we want to know if an incident particle is interact with nuclear who is the better and give experiment result?

For example if we send neutron into $$Po^{209}$$ what is all the cross section that we should take it into consideration

• Before answering your question, can you please comment on the limit where $l\rightarrow\infty$ because $P(l)\rightarrow -\infty$ in that case! Shouldn't $0\leq P(l)\leq 1$? Should it be $P(l)=1-e^{-\frac{l}{\lambda}}$? Commented Feb 1, 2022 at 2:25
• @Newbie is correct. Please see my answer. Commented Feb 1, 2022 at 3:52
• @JohnDarby Just upvoted it. Good answer. Commented Feb 1, 2022 at 3:54
• @Newbie is correct. Equation is missing an $-l$ in the exponent Commented May 1 at 11:46

For nuclear reactions the following definitions are typically used.

The cross section $$\sigma$$ for a particular reaction can be defined as "the fraction of the incident particles which undergo the specific reaction, divided by the number of target nuclei per unit area of a thin target". [Evans, The Atomic Nucleus] The cross section is the effective area per single target nucleus for the reaction. [Glasstone and Edlund, The Elements of Nuclear Reactor Theory] The cross section has units of area, typically expressed in barns where one barn = $$10^{-24} cm^2$$. The cross section is sometimes called the "microscopic cross section" to distinguish it from the "macroscopic cross section" defined below.

The product of the (microscopic) cross section $$\sigma$$ and the density of target nuclei $$N$$ is called the "macroscopic cross section" $$\Sigma = \sigma N$$. $$\Sigma$$ is the probability of the reaction per unit path length and has units of inverse distance. The macroscopic cross section is typically expressed in $$cm^{-1}$$.

For incident particles with density $$n$$ and velocity $$v$$, the flux is defined as $$\phi = nv$$. The reaction rate for a particular reaction with target nuclei is $$\Sigma \phi$$ reactions per $$cm^3 \over sec$$.

The probability that an incident particle will have a first interaction over a distance $$l$$ is $$\int_{0}^{l}e^{-\Sigma x} \Sigma dx = 1 - e^{-\Sigma l}$$. $$\Sigma$$ is sometimes denoted as $$1/\lambda$$ where $$\lambda$$ is the mean free path for the reaction in $$cm$$.

If you bombard $$Po^{209}$$ neutrons, $$\sigma_a$$ is the basic dimension needed for the reaction type $$a$$, but you also need to know $$N_{Po^{209}}$$ and $$\phi$$ for your particular condition to evaluate the reaction rate $$\Sigma_{a} \phi$$.