D.Rubin
last updated June 9, 2005 Beambeam Simulations

Beambeam simulations

Beambeam simulations

Physics

Results

On going

Physics of the simulation

Simulations of CESR luminosity are based on a machine model built from BMAD subroutines that incorporates the following physics:

Beambeam interaction
The beam beam interaction is modeled as weak-strong and self consistent. The weak beam consists of typically 500 particles. The dimensions of the strong beam, assumed gaussian, are updated to be equal to the gaussian fitted dimensions of the weak beam every n turns, where n~1000. The strong beam is sliced longitudinally so that bunch length and crossing angle dependence are included.

The kick imparted by strong beam to the particles in the weak beam is based on the assumption that the strong beam can beam can be represented by a gaussian distribution. In order to test this assumption we compute the chi2 of the fitted gaussian to the weak beam distribution and consider its dependence on bunch current for the 1.89 GeV CESR-c configuration. We find that the chi2 per degree of freedom of the vertical distribution does indeed increase with current, indicating that the vertical distribution is increasingly non gaussian. ( Presumably the vertical distribution is developing non gaussian tails.) Horizontal and longitudinal distributions are gaussian independent of bunch current. Note that the low current distribution is gaussian by construction.

Effects of finite bunch length and crossing angle depend on the number of longitudinal slices. In optics with artificially reduced momentum compaction we compare luminosity vs bunch current with 5 and 10 slices. There is no significant difference.

We compare results with 1, 5 and 10 slices in the 12 wiggler cesr-c optics, hibetainj_20040628_v01.lat and find that computed luminosity increases with number of slices and that there is no significant difference going from 5 to 10.

Any coherent motion of the particles in the bunch is also precluded in the weak strong regime. Such motion might coincide with a non gaussian distortion of the distribution. In any event, we anticipate that if there is coherent motion that our calculation of luminosity will be an overestimate.

Tracking
The weak beam particles are tracked through the individual elements of the arc, including, interaction region solenoid and coupling elements, arc sextupoles, wigglers, RF cavities, and electrostatic separators. Tracking through the superconducting wigglers is via a third order, symplectified, taylor map.

Radiation damping and excitation
Radiation damping and excititation takes place in all relevant guide field elements including bends, and wigglers, and quadrupoles if the beam is off axis.

Parasitic interactions
Parasitic interactions are represented as beam beam kicks with zero length.

Number of turns
We typically track for about 5 damping times, finding that the luminosity has equilibrated in that time. The plot shows the luminosity computed after every 1000 turns for several currents, has equilibrated after 100000 turns.

Number of particles
Calculation of luminosity vs number of weak beam particles indicates convergence with 500 particles. The plot shows luminosity computed tracking 200,500,1000, and 2000 particles for various currents. Except at the very highest current, it is clear that as long as the number of particles is greater than 500, the luminosity is independent of the number.

Free parameters
There are no free parameters. The zero current beam size is based strictly on particle tracking and radiation and damping as described above.

Tuning
Input parameters include tunes, bunch current, and bunch pattern. Note that electron and positron trajectories are determined by optics, including electrostatic separators and in general, the trajectories do not coincide at the IP when all parasitic interactions are included. Prior to the start of tracking, beams are brought into collision. In CESR, the electron-positron vertical orbit difference at the IP is tuned to zero by manipulating the closure (separator kick and phase advance), of the half-wave electrostatic vertical bump that separates bunches at the point diametrically opposite the interaction point. Horizontal orbit differences are zeroed by adjustment of the pair of horizontal separators nearest the IP. In terms of control groups, VNOSEING 1,are VNOSEING 2 varied to set differential vertical angle and offset to zero at the interaction point. PRETZING 13 is adjusted to zero differential horizontal offset. As much as possible, initialization of the machine model mimics initialization in the real machine.

Lifetime/lost particles
The lifetime is Tau = N/(dN/dt). The loss of one of the N_part particles in the weak beam in N_turn turns indicates a lifetime of order, Tau = (N_turn)(N_part)/(f_rev), where f_rev is the revolution frequency. We typically track 500 particles for 100k turns. The loss of a single particle corresponds to a lifetime ~ 128 seconds and indicates an upper limit on the bunch current imposed by the beambeam interaction. We improve the sensitivity to finite lifetime by increasing the number of macroparticles to 5000, compute the current dependence of limiting lifetime in the CESR-c configuration. We find that lifetime limit depends on the number of bunches per train. With 9 trains of 4 bunches in the strong beam we compute a bunch current limit of less than 3.2mA. (Inverse lifetime with 9X4 ). The lifetime at 3.2mA/bunch is about 7 minutes and greater than 21 minutes (our resolution given 5000 particles and 100k turns) at less than 3mA. Whereas, with 5 bunches per train the bunch current is limited to something less than 2.6mA (Inverse lifetime with 9X5 ). In practice bunch current is limited to about 2.4mA/bunch and 2.0mA/bunch in the 9X4 and 9X5 configurations respectively at a lifetime of ~37 minutes (CLEO's threshold to turn on the detector). In the simulation, particles are lost on collision with the horizontal aperture, consistent with the sensitivity to pretzel amplitude. Note that there are no optical or alignment errors in the simulation.

Simulation results


Comparison of simulation and measurements. We compare the simulation of luminosity versus bunch current in the two configurations in which we have spent enough time tuning to be confident that the conditions in the machine are optimized including Phase II CESR at 5.3 GeV, and CESR-c at 1.9GeV/beam. The 5.3GeV data includes was collected near the end of the run on the Upsilon(4s), in the spring of 2001. The CESR-c measurements were taken at the end of October 2004. Both data sets correspond to the highest specific luminosity recorded in those respective conditions. Agreement between measurements and simulation is good. details

Energy dependence of solenoid compensation and tune shift. We find that the depressed tuneshift limit that we compute with the beambeam simulation for the low energy (CESR-c) optics is a result of the strong dependence of solenoid compensation on energy. The energy dependence of the coupling compensation is characteristic of the Phase III (superconduting/ permanent magnet hybrid) design and is not specific to the cesr-c incarnation. The simulation indicates that the tune shift parameter would be increased by more than 50% in a solenoid off, compensation off configuration. details

Correcting IR coupling with Compensating Solenoid and Tune Shift Limit. The energy dependence of the solenoid compensation with the Phase III IR optics dilutes the vertical equilibrium emittance, compromises off energy dynamic aperture and limits the beambeam tune shift parameter. We find that the energy dependence can be very nearly eliminated by including a compensating solenoid in a configuration very similar to that developed for the DAPHNE collider. In our model, the CESR compensating solenoid is located in the straight adjacent to the IR quad cryostat. The integrated field of the solenoid is set equal in magnitude but of opposite sign to the integrated field of the CLEO solenoid over half if its length. We suppose that the compensating solenoid has a length of 0.95m, extending between 3.75 and 4.7m from the IP, and with Bz~1.85T. (We assume that the field of the CLEO solenoid is 1T with effective length 3.51m) details

Minimizing energy dependence of Solenoid Compensation. As noted in an earlier document , there is not a unique implementation of an anti-solenoid. Using 4-pair compensation, with integrated anti-solenoid opposite integrated CLEO, we find that if the four pairs are Q00 tilt, sc_sk_q01, sc_sk_q02, and sk_q02, that we get significantly better luminosity than if the four pairs are sc_sk_q01, sc_sk_q02, sk_q02, sk_q04 with Q00 tilt fixed at 4.5. We define a new energy dependent coupling constraint so that an IR can be systematically designed to minimize energy dependence. details

Dependence of luminosity on alpha*. We compute the dependence of luminosity on SCMATING 1 (alpha *). We find that a 1% change in alpha* corresponds to a 10% change in specific luminosity, and that the optimum luminosity is achieved for Delta(alpha*)=0.01, (and not 0). details

Distributed radiation. In order to understand the effect of the localized radiation source of the wigglers, we consider alternative optics that have no wigglers at all. In the model, all of the arc bending magnets are replaced with five bends, each with 1/5 of the bending radius. There is negligible effect on the machine geometry. Having increased the field of the dipoles by a factor of 5, the radiation damping time is reduced to 20ms, essentially the same as in the 12 wiggler cesr-c optics. The simulated specific luminosity is nearly twice that in 12 wiggler cesr-c conditions. details

Wiggler nonlinearity The superconducting damping wigglers have strong nonlinearities. The vertical cubic nonlinearity that is characteristic of all wigglers, introduces a significant amplitude dependent tune shift. Dependence of vertical field on horizontal displacement results from the finite pole width. The wiggler map in our standard machine model includes all of the field nonlinearity. We explore the effect of wiggler nonlinearity on luminosityy by replacing the precised wiggler map with a linearized verison. We find no significant different in specific luminosity. details

Dependence on Pretzel and parasitic beambeam interactions In the standard cesr-c optics, hibetainj_20040628_v01.lat, we find no significant effect of crossing angle, pretzel, and parasitic crossings on specific luminosity. details

Dependence on differential vertical displacement at IP

Reduce momentum compaction, short bunch. In order to explore the dependence on bunch length and synchrotron tune we introduce an artificial element to adjust momentum compaction without effecting optics. A first order taylor element with zero length is introduced at L3. The nonzero matrix elements are R(i,i)=1, and R(5,6) = 6. The value of R(5,6) was chosen so that when Qz=0.049, sigma_l=13.3mm. (When R(5,6)=0, sigma_l=13.2mm with Qz=0.089.) Alternatively, if Qz=-0.089, sigma_l=7.3mm. We first consider the effect of bunch length. We set Qz=0.089, (the standard for CESR-c) so that sigma_l=7.3mm. details

Wiggler off optics. The lattice bmad_nowig_250505.lat has energy 1.89GeV and no wigglers. Damping time is 1/2 second, energy spread 0.22% and the emittance is 20nm. With a synchrotron tune of 0.049 the bunch length is 6.2mm. Tunes, solenoid compensation, pretzel, etc. are similar to standard cesr-c 12 wiggler optics (hibetainj_20040628_v01.lat). Details

Dependence on differential vertical displacement at IP

On going

July 29, 2005
  • The lattice bmad_q0_c_040305.lat gives best performance of any optics with anti-solenoid. It requires rotation of the permanent magnet to 1.9 deg. Specific luminosity is not quite at the level of no solenoid at all. Perhaps that is because the tune has not been optimized. Example initialization
    • Scan tunes 0.515 < Qx < 0.535, 0.58 < Qy < 0.6 in increments of 0.002. Synchrotron tune at -0.089, 2mA/bunch,9X5.
    • At the optimum tunes, scan current from 0.5 to 2.5 in 0.25 ma steps.

  • 1.4T wigglers. With 1.4T vs 2.1T energy spread is smaller. The lattice is bmad_14kg_021505.lat . Simulation with Qz=-0.089 shows very slight improvement compared to nominal 2.1T optics. Perhaps it will do better at lower synch tune. Do a current scan with Qz=-0.07. (That gives the same bunch length as 0.089 with 2.1T wigglers). Of course to find the optimum we may need a tune scan as well. Let's first see if there is any advantage to changing synch tune alone. Example initialization with Qs=-0.089

  • The optics hibetainj_20040628_v01_dl6.lat artificially reduces momentum compaction and allows for lower synchrotron tune or shorter bunch. We have used to to calculate luminosity once with Qs=0.049 and sigma_l=13.2mm and also with Qs=0.089 and sigma_l=7.3mm. ( details ) The results were nearly the same, suggesting that it is something like the geometric mean of bunch length and Qs that matters. To test the hypothesis, do current scans with Qs=0.04,0.1,0.01, keeping Qx and Qy the same. Example initialization with Qs=-.089