1Y. SatoIndiana University; SNS, ORNL Electron Cloud Feedback Workshop Indiana University, Bloomington 03/15-19/2004 Simulation of e-Cloud using ORBIT Yoichi Sato Indiana University, Bloomington; SNS/ORNL J. Holmes, A. Shishlo, S. Danilov, S. Cousineau, S. Henderson SNS/ORNL Y. SatoIndiana University; SNS, ORNL

2 What we are doing Pipe Electron Cloud Region Proton Bunch L=248 m and about 1000 turns We have to simulate a building up an electron cloud, its dynamics, its effect on a proton bunch during the whole accumulation period or at least for several turns to detect the development of instability. Proton beam 3D SC potential grid Electron Cloud Grids with few (may be only one) longitudinal slices 2Y. SatoIndiana University; SNS, ORNL

3 Surface Model The secondary emission surface under a phenomenological model --- simplified one from Furman and Pivi’s: PRST-AB (2002) Removes a macroparticle hitting the surface Adds a macroparticle whose macrosize is multiplied by the secondary emission yield d and energy is determined by model spectrum with Monte Carlo method (x,y) (n_x,n_y) q0q0 MacroSize, E 0 MacroSize *d, E q d(E 0 ) = ( secondary current ) ( incident electron beam current ) 3 We can keep the same number of electron-macroparticles through the secondary emission process

4Y. SatoIndiana University; SNS, ORNL Surface Model, cont.  (E 0 ) = d el + d rd + d ts d el = ( elastic backscattered current )/( incident current ) d rd = ( rediffused current )/( incident current ) d ts = ( true secondary current )/( incident current ) Elastic backscattering emission: (d el /d) Rediffuing emission: (d rd /d) True secondary emission: 1b n b M emiss (d ts /d)*( P n,ts / S P i,ts ) ; P n,ts = ( M emiss C n )*(d ts / M emiss ) *(1- d ts / M emiss ) i=1 M emiss n M emiss -n Each component has particular model spectrum. With following probabilities we choose the type of emission and get emitted energy with its spectrum. For getting energy of true secondary, we use E0 p E to simplify the model 4

5Y. SatoIndiana University; SNS, ORNL Secondary Emission Surface Spectrum True secondaries rediffused backscattered Gaussian distribution around E0 in the data corresponds to energy resolution of the detector The ORBIT spectrum matches Furman and Pivi’s simulation PRST-AB (2002) Including the response around E0

6Y. SatoIndiana University; SNS, ORNL Secondary Emission Surface Spectrum, cont. The low E0 response of the stainless steel also matches Furman and Pivi’s results (Courtesy of M. Pivi)

7Y. SatoIndiana University; SNS, ORNL Pipe Electron Cloud Region Proton Bunch No kick on the proton bunch to compare the results with Pivi and Furman’s PRST-AB (2003) EC peak height is sensitive to the low energy SEY d(E 0 =0). E-Cloud Development (ORBIT Simulation)

8Y. SatoIndiana University; SNS, ORNL Analytic Electron Cloud Model in KV proton beam p-bunch a p,b p e-cloud a e ~a p, b e ~b p Ref: D. Neuffer et. al. NIM A321 p1 (1992) Centroid oscillation model of uniform line densities of proton and electron y p_c = A p Exp[ i*( n q - w t )], y e_c = A e Exp[ i*( n q - w t )] Dispersion relation (no frequency spread): ( w e - w ) { w b + w p - ( n w 0 - w ) } = w e w p ep freq. rev. freq. betatron freq. n = longitudinal harmonic of ep mode ep freq. w p,V = 4l e r p c / g b e (a e + b e ) w e,V = 4l p r e c / b p (a p + b p ) The eqs. of motion are valid for the inside of streams 22 22

9Y. SatoIndiana University; SNS, ORNL Analytic Electron Cloud Model in KV proton beam, cont. ( w e - w ) { w b + w p - ( n w 0 - w ) } = w e w p ep freq. rev. freq. betatron freq. ep freq. The dispersion relation has complex solutions (instability) near w ~ w e and w ~ (n w 0 - w b ), slow wave, under satisfying the threshold condition w p t (w b /w e ) |nw 0 - w e - w | For a e =b e =a p =b p =30 mm, 1GeV proton beam, betatron tune Q x =Q y =6.2 and revolution frequency w 0 =2p/T=6.646[1/ms], Q e = w e /w 0 = Q p = w p /w 0 = *f e ; f e = neutralization factor and the most unstable at the longitudinal harmonic number n = 178. n = 178 has 4 roots of w: w 1 /w 0 = |A e /A p | = 1.56E6 w 2 /w 0 = – 0.716i|A e /A p | = where A e /A p = Q e /(Q e - (w/w 0 ) ) w 3 /w 0 = i w 4 /w 0 = |A e /A p | = So, if we set the initial electron cloud and proton beam are the slow waves having n=178 modulation in benchmark, we can expect EC centroid oscillation as the superposition of the last 3 eigen modes (w = w 2, w 3, w 4 ). 1/2 222

10Y. SatoIndiana University; SNS, ORNL Two stream benchmark (ORBIT Simulation) Growth rate is given by Im(w)>0 Instability threshold is found by solving Im(w)=0 y p_c = A Exp[ i*( n q - w t )], y e_c = B Exp[ i*( n q - w t )] Up to t = 35ns we can say the centroid oscillation is the superposition of the two eigen modes of w. If there is no kick on proton beam, the EC centroid does not move.

11Y. SatoIndiana University; SNS, ORNL Two stream benchmark (ORBIT Simulation), cont. The solvable model is valid when the EC is overlapping the proton beam. We can apply the model upto t ~ 37ns If there is no kick on proton beam, the EC keep the same radius.

12Y. SatoIndiana University; SNS, ORNL Conclusion 1.The whole new e-cloud module has been integrated in the ORBIT simulation code. 2.Secondary emission surface model based on M. Pivi and M.Furman’s shows matching spectrum results with theirs. PRST-AB (2002), PRST-AB (2003) 3.The benchmark of the code with the two stream instabilities example is in progress. Initial benchmarks with simplest models that can be solved analytically have been done.