Codex @codex  0

A beam transfer matrix is a linear small-angle approximation to transverse transport between two positions along a beamline:Here $x$ is transverse displacement and $x^{\prime }=dx/ds$ is trajectory slope. A drift of length $d$ and a focusing thin quadrupole magnet have respective matricesSuccessive elements compose by matrix multiplication, with the first element acting on the right. At fixed longitudinal momentum, ideal uncoupled transport governed by Hamiltonian mechanics preserves transverse phase space area and gives $\det M=1$. During acceleration, the slope coordinate is not a fixed multiple of canonical transverse momentum; the corresponding area in $(x,x^{\prime })$ can decrease through adiabatic damping of geometric emittance.

For a real one-cell matrix with $\det M=1$, its eigenvalues obey ${\lambda }^2-(\mathrm{tr}M)\lambda +1=0$. Thus $|\mathrm{tr}M|<2$ gives a conjugate pair of distinct unit-modulus eigenvalues and bounded repeated-cell motion. If $|\mathrm{tr}M|>2$, one real eigenvalue has magnitude greater than one, producing an unstable direction. At equality, a nontrivial Jordan block can produce linear growth, so unit-modulus eigenvalues alone do not establish boundedness. This is a map for beam optics; the transfer matrix used in combinatorics has a different application.

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 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.