Hi Dave

Mike Billing told me before he left that you had asked him
for suggestions on how we might improve the CESR-c luminosity
during the coming run. He asked me to calculate parasitic BBI
effects in 8x3 versus 8x5 conditions. He explained that
since we operate in a regime where the beam-beam parameter
is not saturated, we may have an advantage in running fewer
bunches with a higher bunch current.

I have obtained a quantitative estimate of the luminosity
gain which might be achieved in this manner. Together with
that calculation, you will find two suggestions of my own
described below. The first pertains to optimizing the pretzel
for the orbit of electron bunch 3 rather than for bunch 1
as we currently do. The second pertains to optimizing the
pretzel at the operating positron current rather than
at zero current as we do now.

8x3 vs 8x5
----------
We believe the parasitic crossings are a prime contributor
to the present current limit of about 2 mA/bunch, since
we reach 4 mA/bunch in 1x1. A measure of the optical
distortion is given by the maximum value of the horizontal
beta function in the ring vs positron bunch current. While
this calculation is performed in a weak-strong approximation,
it is plausible that the total actual distortion limiting
the beam current occurs at the same current level where
the weak-strong limit occurs. DLR's beam shape studies
indicate that coherent effects are not limiting the current
level at present. So we can ask the question: At what
positron bunch current in 8x3 do we reach the same beta
distortion as in 8x5? If the answer is a current higher
than sqrt(5/3) * 2 mA, we may expect a luminosity gain by
lowering the number of bunches.

The two calculations are shown here:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/21nov05/8x3/

The 8x5 case (4260 lattice) is shown on the left. One remark
is that the kink for t1.b1 which occurs at 2 mA in 8x5 occurs
above 3 mA in 8x3. As an estimate of the current limit, let's consider
t1.b5, which is the worst case of rising beta function.
In 8x5 conditions, the beta function value has increase from 49 m
to 58 m at 2 mA. This level of optical distortion is reached
at 3.4 mA in the 8x3 configuration. Such a calculation
suggests that a luminosity increase of a factor
3/5 (3.4/2.0)^2 =  1.7 might be achieved.

Pretzel Optimization for Bunch 3
--------------------------------
The two plots here:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/21nov05/b1b5/
compare the pretzel orbit separations at the parasitic crossing
for electron t1.b1 and t1.b5. At present lattice optimization constraints
are based on t1.b1. The plots show that the separations are very different
for bunch 5, and that its minimum separation reaches 5 sigma, as opposed
to 6.5 for bunch 1. I haven't looked at the other bunches, but it is clear
that the pretzel separations are bunch-dependent at an important level.
Improvement might be obtained by optimizing on the middle bunch rather than
a bunch at an extreme end of the train. The disparity in pretzel bunch
dependence would ALSO be reduced by reducing the number of bunches in
a train, by designing, say, for bunch 2 in an 8x3 configuration.

Pretzel Optimization Accounting for Dynamic Distortions of the Optics
---------------------------------------------------------------------
At present the lattice design criteria do not account for dynamic
distortions of the optics. In particular, the B parameter constraint,
which depends on vertical beta function, electron beam size and
separation of the beams at the parasitic crossing, all of which vary
strongly with bunch current, does not take such distortions into account.
The dependence of the B parameter on positron bunch current is shown here:
http://www.lns.cornell.edu/~critten/cesr/injection/notes/21nov05/b/

The B parameter is proportional to the positron bunch current, so the
bunch current is divided out in the quantity plotted here.
Thus the current dependence in the plotted quantity arises from
the variation with current of orbit separation at the crossing points,
electron beam size, and vertical beta function.

The lattice design criterion is that B is not to exceed 1.2.
Since B/I at zero current is 0.4, one can conclude that the
maximum positron bunch current as estimated in the lattice design
procedure is about 3 mA. Taking the current dependence into account,
the value of 1.2 is reached at a positron bunch current somewhat less
than 2.5 mA/bunch for t1.b1, and at lower values for t3.b1 and t6.b1.

It is certainly feasible to calculate B for optics closer to those
at operating current during the optimization procedure. The result
would be that B would be larger for smaller currents, but the
strength of the BBI at low current is not of concern, whereas
an optimal pretzel at operating currents IS of primary concern.

-- Jim

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