Longitudinal Schottky Tomography

1
Longitudinal Schottky Tomography
Alexey Burov
RR Talk June 2008
2
General integral equation

• Longitudinal Schottky signal gives distribution
  function over the energy offset .
• The energy and time offsets are
  canonically conjugated values, with the
  Hamiltonian
• where is the effective mass,
  is the RF Voltage, and is the
  revolution time.
• The measured momentum distribution
  relates to an unknown 2D phase space distribution
  as
• Solution of this equation gives the phase space
  density f .

3
Rectangular Barrier Bucket (RBB)

For this potential well, the original integral
equation transforms into differential form for
the integral distribution

4
Solution of the Integral Equation for RBB

• This equation can be solved starting from
  high Hamiltonians, where the shifted functions
  , leading
  to

Then, a recursive algorithm is applied to find
the function at any smaller Hamiltonian value
5
Resulting Phase Space Density

• With this solution, the integral phase space
  density is found
• Here is a time period for the
  longitudinal motion as a function of the
  Hamiltonian E
• with
  are inside the bucket and
• outside velocities, is
  acceleration inside the barriers.

6
Realization in Mathcad

• Data from Dan (PA1964, 5/29/2008 338 PM)

Regular decline of the tails indicates no
background has to be worried about.
7
Resulting Integral Phase Space Distribution F(H)

8
Calculated Schottky

• Found distribution can be used to calculate
  expected Schottky signal, and then compare it
  with the original data, checking the solution

Red line calculated Schottky Blue dots
original data Brown line DC contribution
9
Conclusions

• Integral equation for the longitudinal Schottky
  tomography is solved for an arbitrary rectangular
  barrier bucket. An exact solution is found in a
  form of recursive analytical procedure with a
  small number of steps.
• This algorithm is realized in MathCad. The code
  is tested for a Gaussian distribution, the
  correct result was found.
• The code was applied for real Schottky data, the
  phase space distribution was obtained. In
  particular, a fraction of the DC beam was found
  2.4 times more compared to the naïve estimation.
• A calculated Schottky signal agrees with the
  initial data.
• The MathCad code looks ready for translation into
  a routinely working console application.

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3.3 hr of cooling

6/12/08, 1336
6/12/08, 1657
Same Pbar beam, same bunch length. Cooling
results in the tails.
11
Beam squeezing, R37 data of June 19/2008 (Lionel)

DC Prediction for Tsep16.4 µs
DC Prediction for Tsep15.26 µs
Initially the beam was 6.4 us, then it was
squeezed to 5.26 us. Time between the two data
sets is 5 min. Back DC prediction is very
accurate, while the forward one overestimates DC
1.7 times. It looks as the distribution of DC
changes for that 5 min.
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Dependence on the frequency span

Comparison of 40 and 50 KHz spans For the same
distribution (f_0/245 KHz). While the far
tails start to differ, the DC portions are close
3.8E-3 vs 4.3E-3.
Schottky data for these two measurements
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Stretching-Squeezing (data June 24, Lionel)

DC calculations for Tsep15.83 µs 3.0E-3 vs
2.9E-3
843 -gt
DC Prediction for Tsep15.83 µs
DC Prediction for Tsep17.35 µs
8.52 -gt
Problem while narrow-to-wide prediction works
with good accuracy, Wide-to-narrow prediction
overestimates DC 2 times.
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Stretching-Squeezing (cont)

Red 5.83µs direct analysis (as above) Blue
7.35µs-based calculation for 5.83µs
Red 7.35µs direct analysis (as above) Blue
5.38µs-based calculation for 7.35µs
15
Dependence on RF Value

The tree plots show how the direct
calculation and prediction depend on the
assumed RF value.
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Ultimate cases of T1sep

Red 0.72µs direct analysis Blue 9.24µs-based
calculation for 0.72µs
Red 9.24µs direct analysis Blue
0.72µs-based calculation for 9.24µs
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Conclusions

• The tomography results were cross-checked by
  comparing direct calculations with the
  predictions based on other width of the
  potential well (other T1sep). The two
  distributions are normally coincided at the core
  and far tails, but disagree by a factor of 2 at
  the AC-DC boundary.
• The calculated distribution is shown to be very
  sensitive to RF 5 of the RF amplitude leads to
  factor of 2 change of the DC portion.
• I guess that the cross-check disagreement relates
  to 5 difference of the real and model
  (rectangular barrier bucket) RF.