E-Cloud Impedance from POSINST simulations Yuri Alexahin, Yichen Ji APC meeting on e-cloud theory & simulations 15 April 2015

04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations2 Recycler Instability Last year (before shutdown) Recycler suffered from fast horizontal instability –had turns growth rate –affected bunches in the second half of the batch –depended strongly on bunch length –had no obvious tune / chromaticity dependence –could be averted by first weak batch To get 15 turns rise-time the impedance must be Z  ~ 0.5G  /m (A. Burov) turnx ½ synchrotron period Onset of instability after injection of a single batch of nominal intensity 4  10 12

3 Some Formulas 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations

4 POSINST Simulations 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations “Flip-flop” due to e- cloud space-charge repulsive effect Ey at the origin (V/m) e-cloud density within 1  (1/m^3) Ex at the origin (V/m) Nm=6 bunches/period Np=5.26e10/bunch x0=1mm  x=3.6mm,  y=1.6mm  ||=73.5cm

5 Closer look at 1 period (6 bunches) 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations I peak = 1.36 A n e (ref) = 3  /m 3 x 0 = 1mm t (ns) n e (max) is delayed from I peak by ~2ns t (ns) I 1 / (I peak x 0 ) E x (V/m) I / I peak n e / n e (ref)

6 Fourier in moving 1-period window 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations k=2*Nm+1 k=3*Nm+1 k=Nm-1 k=2*Nm-1 Despite apparent stochasticity in Ex there is some consistency in Fourier coefficients Im( F  Ex) k=Nm+1 Re( F  Ex) k=3*Nm+1

7 Impedance With total length of focusing quads L= 684m |Z  |~ 1  2 G  /m (for low harmonics) 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations harmonic k Z  /L, M  /m i i i i i i i Ex Fourier harmonics for Nm=6 bunches/period: red dots – average over 1-period sliding window, blue dots – FFT for the entire interval t  (1.3  s, 4.0  s). For exactly periodic Ex would survive only k=j*Nm  1, j=0,1,2,…, these harmonics are indicated by arrows. k Ex(k)Ex(k)

8 Effect on the Beam – Multibunch Modes 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations phase  can be considered as function of z and t: what about x=dx/dz? propagation along z: x 1 xx “slow wave” x “fast wave” bunch displacement at z=0

9 Effect on the Beam – Hydrodynamical Approximation 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations multiply by I 0 and use dI 0 /dz  0 average over period T using commutativity of d/dz and  dt for periodic functions multiply by x From simulations ReZ eff /L  1.3 M  /m 2 =0 use z as the independent variable

10 Growth Rate 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations for parameters used here the increment of an “absolute” instability would be Parameter scan is underway (number of bunches/period, “snake” amplitude, beam current, SEY) Electric field gradient also will be analysed to estimate Landau damping