Particles, Particles, Particles — Question 8

Question 8

Attenuation distinguishes a mean free path from a geometric thickness

A particle beam traverses a uniform slab with interaction mean free path $\lambda=20.0$ cm. Independent interactions give survival $P_0=e^{-x/\lambda}$. Slab thickness is 5.00 cm.

1. Find the probability of at least one interaction and compare with the thin-slab approximation x/lambda.

2. Find the thickness that gives a 50 percent interaction probability.

Question 8 - Solution

Attenuation distinguishes a mean free path from a geometric thickness

(a) Method and calculation

$$\boxed{P_{\ge1}=1-e^{-5/20}=.221199}.$$$$P_{\mathrm{thin}}=x/\lambda=.250,\quad\boxed{(P_{\mathrm{thin}}-P_{\ge1})/P_{\ge1}=.130203}.$$

(b) Interpretation and further calculation

$$1-e^{-x_{50}/\lambda}=1/2\Rightarrow x_{50}=\lambda\ln2.$$$$\boxed{x_{50}=13.8629\,\mathrm{cm}}.$$

Physical check

The linear approximation overcounts when repeated-interaction probability is no longer negligible. The exponential models a constant interaction coefficient.