Revisiting the Longitudinal 90° Limit in High Intensity Linear Accelerators

Parametric envelope and sum envelope resonances are analyzed to revisit the validity of an assumed stop band and design limit of high intensity linear accelerators at a longitudinal phase advance of 90° per focusing lattice period. While the 90° limit is unquestioned in the transverse plane, we show here that it can be dropped as longitudinal limit for lattices with two or more rf gaps per focusing period. A new limit arises, however, from a novel transverse-longitudinal parametric sum envelope instability. The resulting sum instability rule allows the phase advance to exceed 90° longitudinally provided that transversely it remains correspondingly under 90°. We suggest that the additional design freedom opens the possibility for larger accelerating gradients and stronger longitudinal focusing with potential length and cost saving in the design of advanced superconducting linear accelerator concepts—as long as technological cavity limits are not reached.

Figure 1

Schematic view of a cryomodule with three focusing lattice periods (dashed lines), each containing two rf periods (dotted lines).

Figure 2

First two transverse lattice periods with matched envelopes in $x$ and $z$ for FODO lattice (top graph) and solenoid lattice (center graph), both with fixed $k_{0,x,y}=60°$, $k_{0,z}=120°$ per transverse focusing period. Bottom graph: Relative growth of saturated envelopes in $z$ as function of $k_z$ for FODO (no growth) and solenoid channels; also, FODO with slightly unsymmetric rf gap locations.

Figure 3

Relative growth of saturated KV envelopes in FODO lattice and for fixed $k_{0,z}=120°$. Top graph: $xy–z$ sum parametric instability as function of $k_z$, with fixed $k_{0,x,y}=89°$ (showing $z$ envelope growth). Bottom graph: Decreasing $k_{0,x,y}$ from 120°, but fixed current, shows, first, the $xy$ envelope instability, then the $xy–z$ sum parametric instability (showing is only $z$ envelope growth).

Figure 4

KV envelopes vs cell number of FODO lattice for sum resonance at $k_{0,x,y}=85°$ and $k_{0,z}=120°$ ($k_z=100°$).

Figure 5

Particle-in-cell results for sum resonance case of Fig. 4. Top graph: rms emittances. Center graph: rms tunes $k_x$, $k_y$, $k_z$ vs cell number. Bottom graph: phase space plots $x-x^{\prime }$ and $z-z^{\prime }$ at cell 100.

Figure 6

Schematic stability chart in the longitudinal-transverse plane for a lattice with two rf gaps per transverse focusing period, showing the extended design region (dashed green border); also shown by an arrow (red) is the scan of Fig 3.