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Scattering Cross Section


[math]\frac{d\sigma}{d\Omega} = \frac{\left(\frac{number\ of\ particles\ scattered/second}{d\Omega}\right)}{\left(\frac{number\ of\ incoming\ particles/second}{cm^2}\right)}=\frac{dN}{\mathcal L\, d\Omega} =differential\ scattering\ cross\ section[/math]

[math]where\ d\Omega=\sin{\theta}\,d\theta\,d\phi[/math]

[math]\Rightarrow \sigma=\int\limits_{\theta=0}^{\pi} \int\limits_{\phi=0}^{2\pi} \left(\frac{d\sigma}{d\Omega}\right)\ \sin{\theta}\,d\theta\,d\phi =\frac{N}{\mathcal L}\equiv total\ scattering\ cross\ section[/math]

Since this is just a ratio of detected particles to total particles, this gives the cross section as a relative probablity of a scattering, or reaction, to occur.


Luminosity is the quantity that measures the ability of a particle accelerator to produce the required number of interactions and is the proportional to the number of events per second dR/dt and the total cross section σp:

[math]dN=\frac{dR}{dt}=\mathcal L \times \sigma_p[/math]

In order to compute a luminosity for fixed target experiment, it is necessary to take into account the properties of the incoming beam and the stationary target.

[math]\mathcal L = \Phi \rho \ell[/math]


[math]\Phi \equiv [/math]flux, or incoming particles per second

[math]\rho \equiv [/math]target density

[math]\ell \equiv [/math]length of target

For the differential cross-section:

[math]\frac{d\sigma}{d\Omega} =\frac{dN}{\mathcal L d\Omega}=\frac{dN}{\Phi \rho \ell d\Omega}[/math]