1. Simulation of direct space charge in Booster by using MAD program Y.Alexahin, N.Kazarinov

2. Direct space charge in Booster The calculation of the direct space charge in MAD program may be fulfilled by using set of BEAMBEAM (BB) elements, which give tune shift: Where β – the Twiss beta-function at BB location, K – kick acting on the particle, N bb – number of BB elements. For Gaussian beam kick can be expressed in terms of complex error function W: As the space charge force is continuous function of distance s the sum in formula (1) has to represent the integral over distance - - and give correct expression for linear tune shift. Thereby the number of particle N’ in fictitious colliding beam must be set as: (1) (2) (3) Here B f – bunching factor, N – number of particle, C – circumference, L i – distance between successive BB elements,  – relativistic factor.

3. Direct space charge in Booster Number of BB elements For correct representation of space charge effect the distance between two successive BB elements should have minimum value as possible. For linear lattice function calculation the BB elements have been placed just after every optical elements of the ring. Long elements such as dipole magnets have been sliced up so that the length of each element was less than 60 cm. Resulting number of BB elements is equal to 1015. Unfortunately TRACK module of MAD does not permit to have more than 200 BB elements. Thereby the particle tracking has been fulfilled with 197 BB elements with average distance between these ones equal to 2.4 m. Comparison of computational results for two sets of BB elements for the same beam parameters gives the following result: N bb = 1015 Δ x = -0.11354 Δ y = -0.299186 N bb = 197 Δ x = -0.11403 Δ y = -0.300516 These results are in rather good agreement.

4. Direct space charge in Booster Bunching factor Bunching factor – the ratio of the beam peak current to the average one - is not constant and changes due to longitudinal oscillation of the particles. Its maximum value may be evaluated with help of simple model. ΔE m – bucket energy height ΔE i – injected beam energy spread

5. Direct space charge in Booster Injected beam parameters Number of particles 2.5 10 12 Kinetic energy 400 MeV Energy spread 10 -3, 3.3 10 -4 Emittance 1.278  mm  mrad

6. Direct space charge in Booster Bunching factor, tunes Bunching factor for two values of injected energy spread Curve 1 – ΔE i / E = 10 -3 Curve 2 – ΔE i / E = 3.310 -4 Betatron tunes for various values of bunching factor(curve 1) V < 0.25 MV Stable solutions for lattice functions do not exist for amplitude of RF voltage greater than 0.25 MV: Parametric resonance for vertical oscillations.

7. Direct space charge in Booster Tune shifts, beta-functions Tune shifts (curve 1, V ≤ 0.25 MV) Twiss β-functions without (black) and with (red) space charge for V = 0.2 MV. Horizontal tune shift is almost constant due to contribution of the momentum spread to the horizontal beam size.

8. Direct space charge in Booster Estimating of the beam emittance growth The vertical tune shifts are inverse proportional to beam emittance. Thereby a small increase in the vertical emittance could keep the tune at the boundary of the parametric resonance stopband and stabilize the oscillations. Vertical tune shift is kept constant by emittance growth for V  0.25 MV Corresponding emittance growth Not very big (< 25%) emittance growth may stabilize vertical oscillations.

9. Direct space charge in Booster Tracking Tracking have been fulfilled for the following parameters: Number of particles 2.5 10 12 Kinetic energy 400 MeV Energy spread 10 -3 Emittance 1.278  mm  mrad RF voltage 0.05 MV B f 1.15 Q x 6.58179 Q y 6.65291 Number of particle in simulation 1000

10. Direct space charge in Booster Tracking 1000 particles have Gaussian distribution in the transverse phase space in accordance with the lattice Twiss functions at the initial point of simulation and uniform distribution in longitudinal momentum. The initial particle distributions in various phase planes is shown below. (x,y) plane(x,x’) plane (y,y’) plane

11. The simulation have been fulfilled for three cases: – the influence on the beam dynamics space charge only – the influence on the beam dynamics nonlinearities caused by damping magnets – the influence on the beam dynamics both space charge and nonlinearities Direct space charge in Booster Tracking Particle losses In the third case the number of lost particle is smaller than in second case.

12. Direct space charge in Booster Dynamic aperture The computation of the dynamic aperture have been performed in all above cases Space chargeNonlinearities Space charge + nonlinearities Space charge increases the dynamic aperture reduced by nonlinearities

13. Direct space charge in Booster Conclusion Within the framework of the MAD program the algorithm of computation of the direct space charge has been created. The simulation of the beam dynamic in the presence of the beam space charge has been performed The space charge may provide a beneficial influence on beam dynamics Near future plans To simulate the beam dynamics with variable from turn to turn space charge taking into account the changing of the bunching factor due to longitudinal oscillation and possible emittances growth. The software for this simulation have been created. To include into simulation the measured nonlinearities of the magnetic elements.