Discovering the Nucleus — Question 5

Question 5

A thin-foil count rate measures differential cross section

A foil has areal nuclear density $n_t=2.00\times10^{22}\,\mathrm{m^{-2}}$. The beam rate is $F=10^6\,\mathrm{s^{-1}}$. A small detector subtends $\Delta\Omega=10^{-3}\,\mathrm{sr}$ and has efficiency .800. At its angle, $d\sigma/d\Omega=5.00\times10^{-28}\,\mathrm{m^2/sr}$. Assume single scattering.

1. Derive and evaluate detected rate.

2. How long is needed for a 5 percent relative Poisson uncertainty in the signal if background is negligible?

Question 5 - Solution

A thin-foil count rate measures differential cross section

(a) Method and calculation

$$P=n_t\frac{d\sigma}{d\Omega}\Delta\Omega,\qquad R=\varepsilon FP.$$$$\boxed{R=.800(10^6)(2\times10^{22})(5\times10^{-28})(10^{-3})=.00800\,\mathrm{s^{-1}}}.$$

(b) Interpretation and further calculation

$$\sigma_N/N=1/\sqrt N=.0500\Rightarrow\boxed{N=400}.$$$$\boxed{t=N/R=5.00\times10^4\,\mathrm s=13.89\,\mathrm h}.$$

Physical check

The small scattering probability justifies neglecting repeated scattering into the detector. Counting uncertainty depends on detected counts, not on incident beam particles.