Collective Effects in Cesr TA

July 6, 2008
 
Use /home/dlr/developmnt7_linux/collective/test.f90 to compute beam size 
from intra beam scattering and touschek lifetime. Input parameters to IBS calculation are
bunch charge, ratio of coupling to total vertical emittance. Input to Touschek calculation
include energy aperture.

I'm not sure what the energy aperture is. There is that associated with physical aperture,
dynamic aperture and energy bucket. The relatively large momentum compaction in Cesr
will limit energy aperture, since a relatively small energy offset will correspond to a 
big longitudinal position change, and the possibility that the particle will not make it
into the bucket. Also the voltage figures.

Here is a plot of lifetime vs current for various intial emittances, and couplings
with energy aperture of 1%  Lifetime (1%) 

And here is the result if the energy aperture is only 0.5%
 Lifetime (0.5%) 

Meanwhile we have some measurements. Details are in the machine studies log for June 21
and June 24, 2008.

Single bunch and 9X1  positron  lifetime
Single bunch  electrons  lifetime, same conditions as
positrons

And another of single bunch  positrons  with 
artificially increased vertical emittance to about 0.43nm

Comparison of theory and measurements shows that the lifetime is very sensitive to 
energy aperture and the measurement is consistent with something between 1% and 0.5%.

Here is the calculation for energy aperture of 0.75%  Lifetime (0.75%) 

And here it is for energy aperture 0.75% with initial vertical emittances of 100pm,300pm,500pm 
Lifetime (0.75% 100-300-500pm)

And here it is for energy aperture 0.75% with initial vertical emittances of 10pm,30pm,50pm
and r=0.1,0.2 which is closer to the measurement since then the vertical emittance was due
presumably mostly to vertical dispersion. 
Lifetime (0.75% 10-30-50pm ratio 0.1 0.2)

And here it is for energy aperture 0.75% with initial vertical emittances of 100pm,300pm,500pm
and r=0.1,0.2 which is closer to the measurement since then the vertical emittance was due
presumably mostly to vertical dispersion. 
Lifetime (0.75% 100-300-500pm ratio 0.1 0.2)

Energy aperture 0.9%. Initial vertical emittances of 10pm,30pm,50pm. r=0.1,0.2.
Lifetime (0.9% 10-30-50pm ratio 0.1 0.2)

Energy aperture 0.85%. Initial vertical emittances of 10pm,30pm,50pm. r=0.1,0.2.
Lifetime (0.85% 10-30-50pm ratio 0.1 0.2)

Energy aperture 0.9%. Initial vertical emittances of 5pm,15pm,25pm. r=0.05,0.1.
Lifetime (0.9% 5-15-25pm ratio 0.05, 0.1)
-------------------------------
According to Sands p.95 equation 3.61, with parameters from bmad_6wig_nosol_2085.lat
  alpha = 1.154e-2
  U_0=0.107MeV
  k=1281
  E_0=2085MeV
  V=6MV  (at f_s = 35kHz, V=6.8MV)

  Delta E/E ~ 0.011

So I guess the 1% estimate is good, unless the limit is dynamic or physics aperture.
Note that the energy aperture scales roughly as (V_RF/U_0)**0.5.

Measurements of lifetime vs RF voltage might be illuminating.
And when we have all 12 wigglers, that will double U_0 and reduce energy aperture by
0.7

But the 6 wiggler data does not really fit unless the energy aperture < 0.85%.
Is that possible?