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Transverse Beam Transfer Functions via the Vlasov Equation

M. Blaskiewicz, V.H. Ranjbar C-AD BNL • Introduction • Calculating BTFs using particle simulations • Calculations with the Vlasov equation • Comparison

Introduction • Stability and Response properties are encapsulated in Beam

Transfer Functions (BTFs). • Measurements are generally straightforward but can be

challenging. • The data can be quite rich. • Reliable predictions and improvements to the model can

result from an adequate theory. • The problem is important enough to warrant more than one

approach.

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Calculating BTFs using simulations Starter problem: Consider a kicker giving a kick f(τ) as

the bunch passes on turn 0. Let Df(k, τ) be the beam response (dipole moment) to this kick on turns k=0,1,…

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BTF with f(τ) for n=2. Space charge and a Q=5 resonator wake with frτb=6. Solid lines are tracking results, crosses from Vlasov.

Zero chrom, sc and hi freq imped, comp3.plt

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Low frequency resolution BTF taken at frequency equal to inverse bunch length (n=2).

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low frequency BTF (n=0). Has a step function wake for various chromaticities.

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BTF with same wake and chrom but with f=1/4τb.(n=1) upper and lower sidebands differ.

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Like previous but now including space charge.

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Conclusions • Transverse beam transfer functions can be calculated reliably

using either simulations or the Vlasov equation. • Only 2 tracking runs are needed to get the full TBTF for

bunches short compared to the machine circumference. • The Vlasov approach requires more CPU power than tracking

for comparable results. This is especially true with large space charge.

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Discretize with linear interpolation The matrix element is left as a 2D integral. The driving terms for upper and lower sidebands are

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BTF with f(τ) for n=1. Solid lines are tracking results. Crosses are from Vlasov. Space charge is the collective effect.