Geometric emittance ($ϵ=\sqrt{⟨z^2⟩⟨z^{\prime 2}⟩-⟨z{z}^{\prime }{⟩}^2}$)

For a centered beam, the displayed rms area in $(z,z^{\prime })$ coordinates. A matched beam has $⟨z^2⟩=ϵ\beta$, where $\beta$ is one of the Courant-Snyder parameters. Under ideal acceleration, adiabatic damping of geometric emittance decreases this area even when the canonical phase-space area is preserved.

For slowly accelerated paraxial motion, $z^{\prime }\simeq p_z/p_s$ decreases as longitudinal momentum grows while transverse canonical phase-space area is conserved. The normalized emittance stays constant in the ideal limit, so the geometric emittance scales inversely with longitudinal momentum. This is kinematic damping rather than dissipation of canonical area.