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## Homework Statement

A beam of ##^{50}##Ti nuclei impinged on a ##0.5 mg/cm^2## thick ##^{208}##Pb target in an experiment lasting 176 hours. During that time a total of ##3260 \; ^{257}##Rf nuclei were detected. The beam intensity was throughout the experiment constant at ##6.6\mu A##. Each ##^{50}##Ti beam ion carried a charge of ##q=+15##. The efficiency for detecting ##^{257}##Rf was ##3\%##. Calculate the reaction cross section.

## Homework Equations

Reaction rate

##R=N_0\sigma I##

where ##N_0## is the number of target particles ##\sigma## the cross section and ##I## the current of incoming particles (I assume).

## The Attempt at a Solution

As I understand it ##3\%## of the true number of ##^{257}##Rf were detected so we don't have to worry about any decay.

Secondly I assume the cross section I should calculate is actually the cross section of the entire target i.e. ##\sigma_{tot} = N_0 \sigma## so the thickness doesn't matter. I don't see how I could calculate anything otherwise since I don't know how big the total sample is.

In that case the number of ##^{50}##Ti hitting the target in one second is

##N_{Ti} = \frac{6.6 \mu A}{15e} = \frac{6.6 \cdot 10^{-6}}{15 \cdot 1.6 \cdot 10^{-19}} = 2.75\cdot 10^{12}.##

And the number of ##^{257}##Rf created in one second is

##N_{Rf} = \frac{3260}{0.03\cdot 176\cdot 3600} = 0.17##

this is then also the reaction rate.

Then ##\sigma_{tot} = \frac{R}{I} = \frac{N_{Rf}}{N_{Ti}} = 6.24\cdot 10^{-14}##.

Now I have no idea if what I did was correct or what I should be doing. There's also a question about units, my cross sections seems to be unit less but shouldn't the unit of a cross section be barn i.e. an area unit? Or in the case of the macroscopic cross section ##m^{-1}##.

Perhaps what I should calculate is the macroscopic cross section but again I don't see how since I don't know the size of the sample.