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

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. These ones 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. 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) (3) Here B f – bunching factor, N – number of particle, C – circumference, L i – distance between successive BB elements,  – relativistic factor.

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 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 = Δ y = N bb = 197 Δ x = Δ y = These results are in rather good agreement.

Number of particle Kinetic energy, MeV400 Initial momentum spread,  p /p rms emittance,  mm  mrad RF voltage amplitude, MV 0.05  0.4 Transition energy, GeV5.114 Tunes ( x, y ) (6.703, 6.812) Harmonic number84 Circumference, m Direct space charge in Booster Beam and Booster Parameters The simulation has been fulfilled for beam and booster parameters corresponding to ones during injection into the booster.

Direct space charge in Booster Bunching Factor and Momentum Spread 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. The dependence of the bunching factor on number of turn has been found by integrating the equations of longitudinal motion for 5000 particles during 250 turns after injection. The amplitude of the RF voltage increased linearly from 0.05 MV to 0.4 MV during 200 turns and after that kept constant.

Direct space charge in Booster Bunching Factor and Momentum Spread The final value of the bunching factor is in a good agreement with result of measurements /1/. 1. Xiaobiao Huang. Beam diagnosis and lattice modeling of the Fermilab booster. Ph.D. Thesis. Fermilab-thesis

Direct space charge in Booster Space Charge Tune Shifts Lines 2 x,y =13 – tune shifts corresponding to parametric resonances Betatron tunes The working point crosses the lines of the parametric resonance of the both horizontal and vertical betatron oscillations. Thereby the linear lattice functions are unstable while bunching factor B f is greater than 1.3.

Direct space charge in Booster Beta-functions Beta functions at one period of booster without (dashed line) and with (solid line) space charge

Direct space charge in Booster Magnet Nonlinearities The measured nonlinearities of the bending magnets have been included into MAD file of the booster. Each bending magnet has been sliced up to five parts and thin multipole lens has been installed at its exit. The values of the nonlinearities b n and corresponding MAD parameters of thin multipole lens KnL are shown below. *) B 0 – bending magnetic field

Direct space charge in Booster Random Quadrupole Gradient Errors As the parametric resonance driving term is defined by amplitude of 13-th harmonic of focusing field gradients the random errors have been introduced into the bending magnets quadrupole components K1. The errors have been distributed in accordance with Gauss law with  K1 /K1 = The changes of tunes have been avoided with thin multipole lenses described in the previous section. The quadrupole errors lead to additional phase advance between adjacent beam position monitor (BPM) as measured at booster /1/. The magnitudes of these ones are in rather good agreement with results of measurements.

Direct space charge in Booster Dynamic Aperture To examine influence of space charge on the parametric resonance the horizontal tune has been moved to the value x = 6.52 closely to the resonance line. The dynamic aperture defined by particles dynamics during 400 turns have been computed for three cases. quadrupole errors space charge quadrupole errors + space charge The space charge itself does not excite the resonance. The space charge prevents the resonance growth of almost all considered oscillation amplitudes. The dynamic aperture for non-tuned working point in the presence also of the magnets and the chromaticity sextupoles nonlinearities coincides with one shown in central Figure.

Direct space charge in Booster Tracking Initially 1000 particles have Gaussian distribution in the transverse phase space in accordance with the lattice Twiss functions at the beginning point of simulation and uniform distribution in longitudinal momentum During tracking over 250 turns the parameters of the BB elements were recalculated at every turn in accordance with changing of the bunching factor B f and momentum spread. Besides the strength of the orbit bump magnet which shift the orbit during injection went down to zero during 30 turns. All parameters variations have been done by using the Mathematica program. The final particle distributions are shown below. They don’t differ significantly from initial ones.

The particle losses take place mainly during initial 30 turns. This may be explained by influence of the strong magnetic field nonlinearities of the orbit bump magnets and multiple crossing the parametric resonance 2 x = 13. Nearby turn 100 the losses may be explained by crossing the resonance 2 y = 13. Total particle loss is less than 2%. Direct space charge in Booster Tracking Number of particles

Direct space charge in Booster Tracking Both horizontal and vertical emittances of the beam slightly increased. The relative emittance growth is not greater than 15%.

Direct space charge in Booster Tracking The emittance have been evaluated by fitting the integral of distribution function with: where I – is action variable. Simulation with changing from turn to turn emittance of the beam. Fit at the first turn

Direct space charge in Booster Tracking Simulation with changing from turn to turn emittance of the beam. Nominal parameter of the beam as in previous case Number of particle Beam emittance Red – horizontal, blue – vertical

Direct space charge in Booster Tracking Simulation with changing from turn to turn emittance of the beam. Intensity increased twofold in 5 turns, then RF capture as in the previous cases Number of particleBeam emittance Red – horizontal, blue – vertical Particle losses increased in two times in comparison with two previous case. Horizontal emittance is in two time greater than initial one.

Direct space charge in Booster Tracking Simulation with changing from turn to turn emittance of the beam. Fit at the last turn Beam has non-Gaussian form

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 capture into the separatrix of longitudinal motion has given reasonable values of the bunching factor and the momentum spread. The magnetic field nonlinearities and quadrupole errors have been included into MAD file for the booster optics. The simulation of the beam dynamic in the presence of the beam space charge, magnetic field nonlinearities and quadrupole errors has been performed.