# 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

## 1 Answer

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.

In particle physics, we can ask similar questions. What is the expected rate at which a particle will scatter off a gas of other particles? The cross-section quantifies this rate: if we know the interaction cross-section of a particle with a target particle, and we know how many target particles there are per unit volume, and we know their relative velocity, we can calculate how many scatterings there will be per unit time. However, this is only an averaged rate: the likelihood that any two particles scatter off one another is a quantum mechanical question, so our only answer will be in the form of expectation values.

• 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 see. To clarify, are you saying that "collisional cross section is an effective area for collision of particles. However, it is often expressed in terms of probability"? Thanks! – user7852656 Aug 12 '20 at 23:18
• 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
• I see. Thanks for sharing the insights! – user7852656 Aug 12 '20 at 23:23