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[Legenda]

Classical differential cross section = MACRO-m3c-diff

Scattering events can be described in terms of total cross sections

[Compare with the total cross section described elsewhere (link type: `compared'; target: a macroscopic module)]

, differential cross sections or double differential cross sections

[Compare with the double differential cross section described elsewhere (link type: `compared'; target: a macroscopic module)]

Cross sections contain all information about momentum, energy (electronic, vibrational and rotational) and angular momentum transfer.

The differential cross section $d \sigma_p$ is defined as the number of events of a particular type p per unit of time, divided by the total flux F, where F is the number of particles crossing a unit of transverse area per unit of time.

For the process of elastic scattering sketched in figure MACRO-m3c-diff-F1, the differential cross section per unit solid angle is given by

[click to unfold details] The number of particles passing through the transverse area $2 \pi b db$ per time unit is given by $2 \pi b db F$ . The number of particles is conserved in the scattering process, such that all particles passing through the ring $2 \pi b db$ must be scattered through an angle between $\theta$ and $\theta + d\theta$ :
$\begin{displaymath}2 \pi b db F = d \sigma_{el}(\theta) F. \end{displaymath}$ The solid angle $d\Omega$ is given by

This leads to
$\begin{displaymath}\big( \frac{d \sigma}{d \Omega} \big)_{el} = \frac{b}{\sin \theta} \frac{db}{d \theta}, \end{displaymath}$ Because ${d \sigma}{d \Omega}$ is the number of events in some area (it has the dimension of area), it is always positive. On the other hand $\frac{db}{d \theta}$ is always negative, because $\theta$ decreases for increasing b. Therefore the differential cross section is given by the absolute value in equation 0.41.

$\begin{displaymath} \big( \frac{d \sigma}{d \Omega} \big)_{el} = \Sigma_{b_i} \frac{b}{\sin \theta} \vert\frac{db}{d \theta}\vert, \end{displaymath}$ where $\theta$ is the angle over which the particle incoming with an impact parameter b is scattered.

Thus the differential cross section can be calculated if the deflection function $\theta (b)$ is known.