With the present Main Injector design transition is one of the major restrictions for high intensity beams (of the order of 1014protons per cycle). At transition the bunch length becomes very small while the momentum spread extends to infinity. In the Main Injector the transition gamma (rt) is 20.56.
Deiter Mohle from CERN and Tom Collins from FNAL suggested that another Main Injector lattice might be examined with an imaginary transition gamma.
The transition occurs during acceleration due to relativistic effects. The transition gamma is defined by:
where "L" is the total length of the accelerator r while AL is a difference in a path of an off momentum parties (p+Ap). Toavoid transition the l/(rt)2 should be less t n zero. The source of dispersion is the dipole where the higher/ wer momentum particles are bent less/more.
Dxi = Di * Dp/p.
The 1/ (rd 2 will be
negative if the sum o the dispersion function through the dipoles is a negative nu er. If
the dispersion function is presented by Floquet's transformation (1) where the coordinates
c(s) (the y-axis) and x(s) (the
x-axis) are defined as:
ci = Di Ö bi
xi= D'i Ö bi + Di * ai / Ö bi
where the a and b are the betatron functions while D is dispersion with the slope of the dispersion D'=dD/ds, t becomes clear that the dipoles should be placed in a lattic in the third and fourth quadrant of the c and x coordinate system. This idea was implemented and an example of a lattice with n imaginary gt is presented below where the gt = i*37.29.
The maximum of the dispersion function in he lattice with an imaginary gt is 1.25 m, the tunes are around 3.688, while the chromaticity is xx =-63.2 and xy =-56.05. The iaxima of the betatron functions bx and by are 74 m, while he minima of the bx and by in the regular FODO cell are 13.23 m. Figure 1 presents the lattice functions within a repetitive ccl . Fifteen of these cells make the whole ring. The straight sections are not designed yet. The lenghth of the ring without straight sections is 3366.6 meters. A rough estimate of the length with the straight sections can be done by adding the drift space in the Main Injector straight to the length of the special low beta cells of this example. This calculation raises the circumference up to 3700 meters. This lattice example has 270 dipoles with respect to the 300 dipoles in the Main Injector design.
1. E.D. Courant and H.S. Snyder, "Theory of the Alternating Gradient Synchrotron," Ann. Phys 3, l(1958).