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

2. 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 10-20 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. 3 Some Formulas 04/15/2015Y.Alexahin & Y.Ji | E-Cloud Impedance from POSINST simulations

4. 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. 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  10 13 /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. 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. 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 k15711131719 Z  /L, M  /m 2 1.1+1.3i1.1-0.3i1.8+0.0i2.8+1.9i0.4+1.4i4.3-4.6i5.6-1.9i 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. 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: 2 3 45 6 12 3 45 6 x 1 xx “slow wave” x “fast wave” bunch displacement at z=0

9. 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. 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