Sanford-Wang parameterisation

The Sanford-Wang parameterisation is an empirical formula used to model the production of pions in nuclear interaction of the form p+A → ${\displaystyle \pi ^{+}}$+X where a beam of high-energy protons hit a material.

Its formula for the double-differential cross section with respect to momentum (p) and solid angle (${\displaystyle \Omega }$) is as follows.

${\displaystyle {\frac {d^{2}\sigma (p+A\rightarrow \pi ^{+}+X)}{dpd\Omega }}(p,\theta )=}$ ${\displaystyle c_{1}p^{c_{2}}\left(1-{\frac {p}{p_{beam}}}\right)\exp \left[-c_{3}{\frac {p^{c_{4}}}{p_{beam}^{c_{5}}}}-c_{6}\theta (p-c_{7}p_{beam}(\cos \theta )^{c_{8}})\right]}$

Where p and ${\displaystyle \theta }$ are the momentum of the outgoing pion and its angle from the direction of the incident proton. The numbers ${\displaystyle c_{1}\ldots c_{8}}$ are the Sanford-Wang parameters and are typically varied to give a good fit with experimental data.