Subject: Dependence of B parameter on positron current
From: "James A. Crittenden"
Date: Wed, 03 Aug 2005 17:22:04 -0400
To: Dave Rice , David Rubin , Stu Peck , Sasha Temnykh , mjf7 , Mike Billing , John Sikora , Mark Palmer sag , David Sagan , Jessica Darland


I have included the B parameter in the BBI calculations.
The definition of the B parameter can be found in ST's
minimac talk. It is the sum over the parasitic crossings
(excluding the north IP) of

Beta_Y / (Separation/Beam size)*2

multiplied by

(5.3 / Beam energy) * Bunch current / 10

If its value exceeds 1.2 +- 0.2, it has been found
empirically that the beam lifetime will suffer. This
value of 1.2 was determined using the zero-current values
for Beta_Y, separation and beam size.

Beta_Y, the horizontal separation and the beam size
all depend on the positron current in the weak/strong
approximation, so B depends on current in a complicated
way. The value of 1.2 was calculated using the zero-current
values. My calculations show that it can change by several
factors if the current-dependent values of Beta_Y, separation
and beam size are used, and that it does so very differently
in different lattices and for different electron bunches.

As an example, consider the 5.3 GeV lattice in use
in June, 2001. At zero positron current, I find B to be 1.2.
The p.c. with the largest contribution for e- t1.b1 is the one
nearest the  IP (2.1 m away), contributing 0.54. At zero
current, Beta_Y is 80 m, the separation is 1.7 cm, and the
beam size is 1.7 mm.  At a bunch current of 6 mA, Beta_Y is 100 m,
the separation is 1.5 cm and the beam size is 2.1 mm.
The B parameter is 2.0 and the contribution from this p.c.
is 1.2.

For now I am using the electron beam size in the calculation
of B, so it varies with positron current.

The best way to present the dependence of B on positron current
took some thought, since it is defined to be proportional to
current. The departure from this linear behavior is then a measure
of the BBI effects. I chose to remove the factor of the current,
plotting instead "B parameter per mA":

http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_866/

The zero-current value for this quantity is 0.2. So by multiplying
it with the maximum operational bunch current of 6 mA/bunch, one obtains
the design value of 1.2. This plot shows that e- bunch 1 is in fact anomalous,
and the other bunches show higher values for the B parameter.
The plotted value at 6 mA for e- t1.b1 is 0.33, corresponding to a B value of 2.0.
The other bunches show B values exceeding 3 at 6 mA/bunch.

The present 1.9 GeV lattice and the proposed anti-solenoid 1.9 GeV lattice
show very different and interesting behavior. I have made plots for the
following cases:

1.9 GeV / 8x5 / IP BBI on:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_870/
1.9 GeV / 9x5 / IP BBI on:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_871/

1.9 GeV anti-solenoid / 8x5 / IP BBI on:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_872/
1.9 GeV anti-solenoid / 9x5 / IP BBI on:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_873/

----------------------------
To isolate the effect of the BBI at the IP, I have repeated these
calculations with it turned off:

5.3 GeV / 9x5 / IP BBI off:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_883

1.9 GeV / 8x5 / IP BBI off:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_879/
1.9 GeV / 9x5 / IP BBI off:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_880/

1.9 GeV anti-solenoid / 8x5 / IP BBI off:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_874/
1.9 GeV anti-solenoid / 9x5 / IP BBI off:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/2aug05/bparam_881/

--------------------------------

The full complement of plots showing the dependence of lattice quantities
on positron bunch current can be found at the web addresses with
"bunch" substituting "bparam."

Four new quantities have been added to these plots:
1) the horizontal and vertical eta*
2) the horizontal and vertical beam size at the IP
   including the contributions from both the beta
   function and the energy spread.

-- Jim

========================================================
James Crittenden                   Tel. (607) 255-9424
Wilson Synchrotron Laboratory      Fax  (607) 255-8062
Cornell University
Ithaca, New York 14853-8001
========================================================