# How could the collision cross section be a probability?

I am new to learning about the concept about the collision cross section. I am having a hard time understanding the collision cross section (defined as $$\sigma_{AB}=\pi(r_A+r_B)^2$$ in this link ).

But then the verbal definition of collision cross section is "an "effective area" that quantifies the likelihood of a scattering event when an incident species strikes a target species" indicating that it is a probability. I am missing some connections between the two.

Then the practice problem proceeds to ask about the "cross section", which really asks for $$\sigma_{AB}=\pi(r_A+r_B)^2$$ where $$r_A$$ and $$r_B$$ are given. This makes me think that collision cross section is an area rather than a probability.

I'm confused as to whether collision cross section is an area or probability. If it is a probability, what are some variables that contribute to the probability?

• Just by looking at your formula, you can see that the cross section has the dimensions of an area. Probabilities have no dimensions. – G. Smith Aug 13 '20 at 0:25

Consider the case of a solid sphere: it has cross-section $$\sigma = \pi R^2$$. We now throw it into a box filled with point particles, and we ask "what is the rate at which this sphere intersects points?" The answer is stated as an average rate $$\Gamma$$. We do not need to know about the exact positions of the particles in the box, all we need to know is the number of particles per unit volume, the number-density $$n$$. In a tube of radius $$R$$, the number of point particles is $$\pi R^2 n = \sigma n$$. Then the number of particles passing through the sphere per unit time is $$n\sigma v$$, where $$v$$ is the relative velocity of the particles and the sphere.
• Are the two mentions of $R$ the same variable, or different? Why assign the average rate a symbol ($\Gamma$) if it is not being referenced again later? – electronpusher Aug 12 '20 at 23:14
• Yes, both instances of $R$ are the same variable. I introduce $\Gamma$ because it is, at the very least, common terminology that is useful for a new physicist to know. – David Aug 12 '20 at 23:16
• I can ask, for instance, what is the probability that two particles will scatter with one another within a given area $A$. Then the interaction probability is $\sigma/A$. In this sense, the cross-section has a straight forward interpretation as a probability. However, the choice of $A$ is not clear, so the use of rates is often more physical. – David Aug 12 '20 at 23:22