Space Charge Effects in the TESLA Damping Ring Winfried Decking DESY -MPY- Damping Ring Parameters Direct Space Charge The Cure Summary July 2001
injection ejection Damping Ring - Introduction Long TESLA bunch train (2820 bunches, 337 ns bunch-spacing) would require a 280 km circumference damping ring compress bunch train with smaller bunch spacing in damping ring Circumference is now given by the achievable kicker raise/fall time Assume kicker raise/fall time of 20 ns circumference > 2820 20ns c 17 km To avoid excessive additional tunnel cost build most part of the ring in the linac tunnel : DOG-BONE + e to IP LINAC tunnel RF wiggler straight section wiggler arc arc Note: Because of the TESLA positron source scheme the position of an ejected bunch is filled again after 1.5 turns
εy,inj./εy,ej. = 500000 requires 7 damping times = damping time τd 28 ms εx,inj. = 1 10 2 m εy,inj. = 1 10 2 m σl,inj. = σe,inj. = 1 % = 200 ms (frep. = 5 Hz) εx,ej. = 8 10 6 m εy,ej. = 2 10 8 m σl,ej. = 6 mm σe,ej. = 0.13 % 0.02 0.01 0 0.01 0.02 x [m] 0.02 0.01 0 0.01 0.02 x [m] 0.02 0.02 0.01 0.01 y [m] 0 y [m] 0 0.01 0.01 0.02 0.02 Injected positron beam Ejected positron beam Damping Ring - Introduction
Tunnel Layout TESLA Damping Ring -W. Decking-
Incoherent Space Charge Tune Shift Space charge force is z 2 2reNee 2σ z 2 Fsc;x y(x y, z) x y 2πγ 3 σx y (σx + σy)σz Tune shift due to space charge force Qsc = 1 4π βf sc Large ring length and relative low energy leads to huge incoherent space charge tune shift: Q sc;x y z 2 LreNee 2σ z 2 (2π) 3 2γ 2 εx,nε x y,n σz With z = z0 cos(2π#turnsνz) particle tune oscillates with twice synchrotron frequency
0 0.25 0.2 0.15 0.1 0.05 0 Q y,inc. (Q s,picture = 1/15*Q s,nominal ) 1 Number of Damping Times N τd 2 3 4 0*σ p 1*σ p 3*σ p 5 6 7 Incoherent tune shift versus damping time and initial longitudinal deviation: Incoherent Space Charge Tune Shift
QX 38.9 72 72.1 72.2 72.3 72.4 72.5 39 39.1 QY 39.2 39.3 satellites 3. ord. 2. ord. 39.4 nqx + m(qy ± 2Qs) = int. Avoid resonances up to third order (always a good idea) and their satellites at twice the synchrotron frequency: Tune Diagram
vertical amplitude growth with space charge 0.1 0.05 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 p/p [σ p ] p/p [σ p ] ε y [ε y;0 ] 10 1 QY 0.2 0.15 0.1 0.25 100 0.3 no space charge space charge included no space charge space charge included 1000 0.35 Average CS Invariant versus initial p/p Vertical Tune versus initial p/p Tunes at Qx = 72.32, Qy = 39.30 Include misalignment and orbit distortion (0.2 % coupling) calculate average Courant-Synder invariant as measure of emittance increase tracking with (non-linear) space charge kick at each element ( weak-strong model) Inc. Space Charge Tune Shift - More Results
How to Cure the Space Charge Tune Shift Space charge force is F sc,x/y (x/y) x/y γ 3 σ x/y (σx+σy)σz 1 Increase ring energy γ 3 needs lattice redesign reason to increase DR energy from 3.2 GeV to 5.0 GeV Scaling including constant normalized emittance and lattice change shows only weak dependence on γ 2 Increase bunch volume through local vertical dispersion Vertical dispersion has negative impact on IBS emittance growth 3 Increase bunch volume through local coupling in long straights Reduce Fsc by the ratio ɛ y Larc + Lstraight ɛx L 5 Additional coupling in a low-coupling ring
Local Beam Blow Up Use special beam optics transformation to create beam with vortex distribution (Y. Derbenev) transformation can be realized with skew quadrupole triplet Beam transformed back with inverse transformation Drift between the two insertions has to fulfill µx = µy = no residual coupling
[*10**( -3)] [*10**( -3)] [*10**( -3)] Table name = FRED Table name = MID2 Table name = FRED -1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 x (m) -1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 x (m) -1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 x (m) -1.0-1.0-1.0-0.8-0.8-0.8-0.6-0.6-0.6-0.4-0.4-0.4-0.2-0.2-0.2 0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 y (m) [*10**( -3)] 0.6 0.8 y (m) [*10**( -3)] 0.6 0.8 y (m) [*10**( -3)] 0.6 0.8 1.0 TESLA DOGBONE DR HP/UX version 8.23/0 27/01/:0 17.51.01 1.0 TESLA DOGBONE DR HP/UX version 8.23/0 27/01/:0 17.52.18 1.0 TESLA DOGBONE DR HP/UX version 8.23/0 27/01/:0 17.53.17 before insertion between insertions after insertions Table name = ENVELOPE Table name = ENVELOPE 0.0 50. 100. 150. 200. 250. 300. 350. 400. 450. 500. 550. 600. S 0.0 50. 100. 150. 200. 250. 300. 350. 400. 450. 500. 550. 600. S 0.0 0.0 0.05 0.05 0.10 0.10 0.15 0.15 0.20 0.20 0.25 0.25 0.30 0.30 SIG [*10**( -3)] 0.35 0.40 0.35 SIGX SIGY SIG [*10**( -3)] SIGX SIGY 0.45 TESLA DOGBONE DR HP/UX version 8.23/0 27/01/:0 17.49.04 0.40 TESLA DOGBONE DR HP/UX version 8.23/0 27/01/:0 17.48.24 insertion off insertion on Local Beam Blow Up
0.1 0 0.5 1 1.5 2 2.5 3 3.5 p/p * σ p 0.06 0 0.5 1 1.5 2 2.5 3 3.5 p/p * σ p 0.08 0.1 rel. emittance change 1 0.12 0.14 10 QY 0.16 0.18 0.2 100 0.22 0.24 no rotation with rotation 0.26 no rotation with rotation 1000 0.28 no vertical emittance increase in straight sections vertical emittance increased in straight sections with local coupling bump Average CS Invariant versus initial p/p Vertical Tune versus initial p/p Tunes at Qx = 72.32, Qy = 39.30 Include misalignment and orbit distortion (0.2 % coupling) Effects of Emittance Blow Up
Qsc up to 0.1 looks tolerable p/p [ σ p ] p/p [σ p ] 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 0.1 0 1 0.05 10 0.1 no coupling ε x = 9e-6 m ε x = 2e-6 m ε x = 1e-6 m ε y [ ε y;0 ] 100 υ y 0.15 1000 0.2 10000 no coupling ε x = 9e-6 m ε x = 2e-6 m ε x = 1e-6 m 0.25 100000 0.3 no space charge Maximum CS Invariant versus initial p/p Vertical Tune versus initial p/p Decrease horizontal emittance (increase space charge force) Tunes at Qx = 72.32, Qy = 39.27 Include misalignment and orbit distortion (0.2 % coupling) Effects of Smaller Emittance
Intra Beam Scattering Intra-beam scattering denotes the effect of many small angle Coulomb scatterings between particles in the bunch leading to diffusion. Exact theory difficult, lets try some simplified scalings The diffusion rates are: 1 τx,y;ibs NeHx,y γσ zεx,y;n(εx;nεy;n) 3/4 1 τz;ibs Ne γ 3/2 σzσ 2 ɛ (ε x;nεy;n) 3/4 The horizontal emittance is roughly proportional to Hx which means that with a decrease of the horizontal emittance the IBS scattering rates scale as (εx;n) 3/4 The final equilibrium emittance is: εx,ibs εx,0 = 1 1 τ x;d τ x;ibs
20 10 4 10 2 ε 10 0 10 2 x /(8e 6) relative emittance change [%] 0 20 40 60 hor. hor.*ε x 3/4 1/τ ε IBS x ver. long. 80 Horizontal IBS scattering rate scaled with emittance (dashed curve) and scaling with ε 3/4 x (magenta curve) also given Calculation of the emittance growth due to IBS (assuming the damping times to be constant) with Bjoerken-Mitwinga theory using present TESLA DR Calculation of Intra Beam Scattering emittance Growth
Summary Incoherent space charge tune shift at TESLA damping ring can be up to 0.23 even at 5 GeV Qinc;SC < 0.1 seems to be tolerable Reduce space charge with local beam blow-up Simulations show that local coupling bump is successful