Betatron oscillation

Small transverse oscillation about an accelerator's reference orbit. In a linear periodic lattice the displacement obeys Hill's equation $z^{\prime \prime }+k(s)z=0$.

For rms geometric beam emittance $ϵ$, the transverse beam variance is ${\sigma }^2(s)=ϵ\beta (s)$. At a beam waist, the angular variance is $ϵ/\beta$. 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.