A Formula of the Electron Cloud Linear Map Coefficient in a Strong Dipole
aa r X i v : . [ phy s i c s . acc - ph ] S e p A FORMULA OF THE ELECTRON CLOUD LINEAR MAPCOEFFICIENT IN A STRONG DIPOLE
S. Petracca, A. Stabile, University of Sannio, Benevento (Italy),T. Demma, INFN-LNF, Frascati (Italy), G. Rumolo, CERN, Geneva (CH)
Abstract
Electron cloud effects have recognized as as one of themost serious bottleneck for reaching design performancesin presently running and proposed future storage rings. Theanalysis of these effects is usually performed with verytime consuming simulation codes. An alternative analyticapproach, based on a cubic map model for the bunch-to-bunch evolution of the electron cloud density, could beuseful to determine regions in parameters space compati-ble with safe machine operations. In this communicationwe derive a simple approximate formula relating the linearcoefficient in the electron cloud density map to the parame-ters relevant for the electron cloud evolution with particularreference to the LHC dipoles.
INTRODUCTION
In [1] it has been shown that, the evolution of the elec-tron cloud density can be followed from bunch to bunchintroducing a cubic map of the form: ρ m +1 = a ρ m + b ρ m + c ρ m (1)where ρ l is the average line electron density between twosuccessive bunches, and the coefficients a , b and c are ex-trapolated from simulations and are function of the beamparameters and of the beam pipe characteristics. An ana-lytic expression for the linear map coefficient that describesthe particle behavior has been derived from first principlesin [2, 3].In this paper we generalize the model presented in [3] inorder to take into account the vertical symmetry in the elec-tron cloud distribution induced by the strong vertical mag-netic field. We consider N m ( x ) quasi-stationary electrons,where x is the distance from the bunch, uniformly dis-tributed in a vertical stripe of the transverse cross-sectionof the beam pipe (Fig. 1). The bunch m accelerates the N m ( x ) electrons initially at rest to an energy E g ( x ) . Af-ter the first wall collision two new jets are created: thebackscattered one with energy E g , and the true secondarieswith energy E sec ≈ eV . The sum over these jets givesthe number of surviving electrons N m +1 , and the linear co-efficient a is obtained by a = N m +1 /N m .In the next section we compute the electron energy gain E g ( x ) due to the passage of a bunch in the presence ofa magnetic dipolar field and compute the number of sec-ondary electrons produced after an electron-wall collisionas a function of the electron energy. Then, following [4], we calculate the linear map coefficient a ( x ) for the case ofan LHC-like dipole. (cid:1) (cid:2) (cid:3)(cid:0)(cid:4) (cid:5)(cid:6) Figure 1: Cross section of the LHC screen (solid line) andthe circular cross section of the pipe used in this communi-cation (dashed line).
ELECTRON MOTION
In Fig. 1 the actual cross section of the LHC beam-screen together with the circular model used in this paperare shown. In the presence of a high vertical magnetic fieldwe can consider only the transverse vertical motion of theelectrons. We approximate the elicoidal trajectories as ver-tical straight lines since the cyclotron radius of the singleparticle motion is very small with respect to the transversebeam-pipe radius R . In this approximation, the time offlight for an electron with energy E is: T f ( E, x ) = r m e ( R − x ) E (2)where m e is the electron mass.Usually it is possible to distinguish two regimes of theelectron cloud. One corresponds to electrons outside thebeam core (kick regime), the other to electrons that aretrapped within the beam core (autonomous regime) [5].The former first one represents the electron motion out-side the bunch during a bunch passage, and in the second (cid:7) (cid:8) (cid:9) (cid:10) (cid:11) N b R c (cid:12) R Figure 2: Critical radius as a function of bunch population.one the electrons in the beam pipe perform harmonic oscil-lations. The critical radius R c separating these two regimesis defined as the radial distance for which the time for thebunch to pass is equal to a quarter of the oscillation period( R c = c T / ). Hence R c is given as follows R c = π s σ r σ z r e N b (3)and it is shown in Fig. 2. The energy gain E g ( x ) is evalu-ated by averaging on the surface of the stripe < E g > = S − Z S dS E ( x, y ) (4)where S is the surface of the vertical stripe. Since R c is ofthe same order as the considered beam pipe radius, the av-erage energy gain, for N m electrons uniformly distributedin a vertical stripe of the beam pipe, can be written as: E g ( x ) = Θ( x − R c ) L Z L dyE kick ( x, y ) +Θ( R c − x ) L (cid:26)Z LL c dyE kick ( x, y ) + Z L − L c dy E aut ( x, y )2 (cid:27) (5)where E kick ( x, y ) = m e c N b r e y ( x + y ) (6) E aut ( x, y ) = m e c π y R c (7) L = √ R − x , L c = p R c − x , N b is the bunchpopulation, r e is the classical electron radius, σ r and σ z are transverse and longitudinal beam dimensions respec-tively. Θ( x ) is the Heaviside function. The function E g ( x ) is shown in Fig. 3 for different values of the bunch popula-tion. (cid:13) R E g (cid:14) m e c (cid:15) Figure 3: Energy gain as a function of the transverse dis-tance for different values of the bunch population N b . LINEAR MAP
The electrons multiplication in the beam pipe wall isparameterized by the so-called Secondary Emission Yield(SEY or δ ). The total yield is the sum of the”’true secon-daries”’ δ t ( E g ) and ”reflected” δ r ( E g ) electrons.The N m ( x ) electrons gain an energy E g ( x ) , during thepassage of bunch-m, hit the chamber wall and produce N m ( x ) δ t true secondaries and N m ( x ) δ r reflected elec-trons. The reflected electrons travel vertically across thestripe with energy E g ( x ) and perform a number of colli-sions, n ( x ) , with the chamber wall, between two consecu-tive bunches, that is n ( x ) = t sb − T f ( E g , x ) T f ( E g , x ) (8)where t sb is the bunch spacing. Hence, the total numberof reflected, high energy electrons at the passage of bunch- ( m + 1) is N r ( x ) = N m ( x ) δ r ( E g ) n ( x ) (9)The true secondaries electrons produced after the first wallcollision give rise to a low energy jet ( E sec ≈ eV ). Forthis jet there is no distinction between the true secondariesand reflected, since all are produced with the same energy.After the i th wall collision the number of electrons is N t ( x ) = N m ( x ) δ t ( E g ) δ r ( E g ) i − δ k i ( x ) sec (10)where δ sec = δ r ( E sec ) + δ t ( E sec ) and k i ( x ) = t sb − iT f ( E sec , x ) T f ( E sec , x ) (11)is the number of collisions after the i th collision. The lowenergy electrons at the passage of bunch- ( m + 1) is N s ( x ) = N m ( x ) δ t ( E g ) n ( x ) X i =1 δ r ( E g ) i +1 δ k i ( x ) sec (12)inally the total number of surviving electrons at bunchpassage ( m + 1) is obtained taking into account both thehigh and low energy contributions N m +1 ( x ) = N m ( x ) δ n ( x ) r + δ t n ( x ) X i =1 δ i +1 r δ k i ( x ) sec (13)and the linear term can be written in the form a ( x ) = N m +1 ( x ) N m ( x ) = δ n ( x ) r + δ t δ ηsec δ n ( x ) ηsec − δ n ( x ) r δ ηsec − δ n ( x ) r (14)where η = p E sec /E g . In Fig. 4 and 5, respectively, thelinear map coefficient is displayed for different values of N b and δ max . Table 1: Parameters for LHCparameter unit valuebeam particle energy GeV 7000bunch spacing m 7.48bunch length m 0.075number of bunches N b – 72number of particles per bunch N B T 8.4length of bending magnet m 1vacuum screen half height m 0.018vacuum screen half width m 0.022circumference m 27000primary photo-emission yield - . · − maximum SEY δ max - 1.3 to 2.5energy for max.
SEY E max eV 237.125energy width for secondary e − eV 1.8 x (cid:16) R a Figure 4: Linear map coefficient a ( x ) for different valuesof N b . CONCLUSIONS