What is an inverse femto barn?

Question

I came across the use of the unit barn and inverse barn while reading about the operation of LHC. What is an inverse femtobarn? What does it tell about the experiment being described?

Answer 1

A barn is a unit of area, equal to $10^{-28}\text{ m}^2$. The prefix femto- signifies $10^{-15}$, so a femtobarn is equal to $10^{-43}\text{ m}^2$. When high-energy physicists talk about inverse femtobarns, they mean collisions per femtobarn of beam cross-sectional area. The inverse femtobarn is not a measure of information, but rather of the effectiveness of the particle accelerator.

Here's the reasoning behind that: imagine two beams of particles coming at each other, and for simplicity let's designate one beam as the "target" and the other as the "probe" (it doesn't matter which is which). The scattering cross section $\sigma$ is the effective size of a single target particle, as "seen" by a probe. So the probability of a collision will be just the fraction of the target beam area that is actually occupied by the target particles; in other words, the scattering cross section divided by the average cross-sectional area per particle,

$$P_\text{collision} \approx \frac{\sigma}{A_t/n_t}$$

($t$ for "target"). Here I'm assuming that the beam is highly diffuse, and that we're only talking about one "bunch" (a finite-length section of a particle beam).

The total number of collisions is just this probability times the number of chances for a collision to happen, namely the number of probe particles in a bunch:

$$n_\text{collision} = \sigma \frac{n_t n_p}{A_t}$$

($p$ for "probe"). Finally, since this happens every time two bunches cross paths, to get the overall rate at which collisions occur you need to multiply by the rate of bunch crossings, which is denoted $f$.

$$R_\text{collision} = \sigma \frac{f n_t n_p}{A_t}$$

The quantity $\frac{f n_t n_p}{A_t}$ is defined as the two-beam luminosity $L$. Roughly speaking, it represents the number of potential collisions per unit area per unit time.

To find the actual number of collisions for any particular process, you multiply luminosity by the cross-section for that process and integrate it over time,

$$n_{X\to Y} = \bigl(\sigma_{X\to Y}\bigr)\times\biggl(\int L \,\mathrm{d}t\biggr)$$

This split is convenient because the first factor depends only on the physical process being considered, and the second factor depends only on the design of the accelerator. So that second factor is a good way to characterize the production capacity of a given accelerator. The units of this value are 1/area, and the inverse femtobarn turns out to be roughly the right magnitude for measuring $\int L \,\mathrm{d}t$ at current particle accelerators. This is what particle physicists mean when they talk about inverse femtobarns.