# Longitudinal Emittance Growth
from Transition Crossing

### A. G. Ruggiero and S. M. Pruss

### September 11, 1989

A suggestion was made that much higher
intensities (>10^{14}) might be desirable from the Main
Injector. among the many problems such high intensities present is
longitudinal emittance blow-up for the standard main injector design
with no transition jump. Using a value of 5 Ohms for the imaginary part
of z/n, we find tat as the intensity increases there is a minimum value
of longitudinal emittance after transition that can be achieved. If one
starts with a very small emittance, it blows up to very large. As the
emittance at injection increases, the amount of blow-up decreases such
that the resultant emittance after transition decreases. As the
emittance at injection continues to increase eventually the emittance
after transition passes through a minimum and starts to increase again.
The minimum after transition emittance occurs for a blow-up of about
40%. The minimum after transition emittance scales about as 1 eV-sec/10^{11}
per bunch for 500 bunches in the main injector. Thus 3 × 10^{13}
from the main injector probably will have 0.6 eV-sec rather than 0.4
eV-sec quoted in the standard main injector parameter list. The
attached table and graph show this data in more detail and illustrate
how these conclusions were reached. If a transition jump is added to the
standard main injector wit a 100msec jump speed, then the minimum after
transition emittance is reduced by a factor of four. This calculation
took space charge and the realistic wall impedance into account, but
neglected any possible problems from non-uniform beam loading of the RF
system (i.e. the missing 90 bunches for the abort kicker risetime gap.)