 Solution

For a paraxial orbit $z^{\prime }\simeq p_z/p_s$, where $p_z$ is transverse momentum and $p_s$ longitudinal momentum. Acceleration mainly increases $p_s$, so a given transverse kick produces a smaller angular deviation. In an ideal matched accelerator without scattering or radiation, the conserved transverse canonical phase-space area becomes a smaller area in the coordinates $(z,z^{\prime })$. This is adiabatic damping of geometric emittance, not loss of canonical phase-space volume.

The invariant normalized emittance isHere ${\beta }_{\mathrm{rel}}$ is speed in units of $c$, distinct from the lattice beta function. Neglecting dilution mechanisms, each acceleration stage thus supplies a beam with smaller angular spread and geometric emittance for injection into the next stage.

A chain of increasingly energetic particle accelerators lets each ring operate over a manageable range of magnetic rigidity and radio-frequency settings. It also avoids injecting a large-emittance low-energy beam directly into the small-aperture high-field main ring. Higher injection energy reduces space-charge effects and improves capture and transmission. Emittance dilution, mismatching and collective instabilities can oppose the ideal damping; acceleration alone does not reduce normalized emittance.