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Longitudinal Schottky Tomography

Longitudinal Schottky Tomography. Alexey Burov. RR Talk June 2008. 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

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Longitudinal Schottky Tomography

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  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 3:38 PM): Regular decline of the tails indicates no background has to be worried about.

  7. Resulting Integral Phase Space Distribution F(H)

  8. 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. • The MathCad code looks ready for translation into a routinely working console application.

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