Betatron amplitude function
= Betatron amplitude function
{title2=$\beta(s)$}
For rms geometric <beam emittance> $\epsilon$, the transverse beam variance is $\sigma^2(s)=\epsilon\beta(s)$. At a beam waist, the angular variance is $\epsilon/\beta$. Reducing both transverse amplitude functions by a factor $a$ at fixed emittances increases the ideal head-on <collider luminosity> by $a$; practical finite-bunch-length and crossing-angle effects can limit this gain.