Beam-beam Force Research Articles

Effects of space charge force on the beam-beam mode coupling instability

Recent e+e− colliders operate with very low emittance and high current. The space-charge tune shift is not negligible in tau-charm and B factories. The vertical space-charge tune shift is on the order of 0.02–0.1 and is comparable to the synchrotron tune and the beam-beam tune shift. The beam-beam mode coupling instability has been studied with and without vacuum environment impedance. The mode coupling is influenced by the space-charge force, which exhibits similar characteristics to the beam-beam force. The combined effect of beam-beam, space charge, and vacuum environment impedance is discussed within the framework of the mode coupling theory. Strong-strong beam-beam simulation, considering the space-charge force and the impedance, are conducted to validate the theory and to study general cases such as asymmetric colliders. Published by the American Physical Society 2024

Instability in crab crossing caused by interaction between beam loading on crab cavities and beam-beam force on colliding beams

Instability in crab crossing caused by interaction between beam loading on crab cavities and beam-beam force on colliding beams

Fundamental beam-beam limit from head-on interaction in the Large Hadron Collider

The beam-beam limit at hadron colliders manifests itself in the form of degraded luminosity lifetime and/or reduced beam lifetime. In particular, for increasing beam intensity, the nonlinear beam-beam force causes incoherent emittance growth, while the (linear) coupling force between the two colliding beams can result in coherent beam-beam instabilities. These phenomena may be enhanced (or suppressed) by lattice errors, external noise, and other perturbations. We investigate the luminosity degradation caused both by incoherent emittance growth and by coherent beam-beam instability. The resulting beam-beam limit for an ideal machine and the of question how it is affected by some of the aforementioned errors are discussed in theory and simulation.

VEPP-2000 Operation with Round Beams in the Energy Range from 1 to 2 GeV

The idea of round-beam collision was proposed more than twenty years ago for the Novosibirsk Phi-factory design [V.V. Danilov et al., in Proc of the EPAC 1996, Sitges, vol. 2, p. 1149]. It requires equal emittances, equal small fractional tunes, equal beta functions at the IP, no betatron coupling in the collider arcs. Such an approach results in conservation of the longitudinal component of angular momentum. As a consequence, it yields an enhancement of dynamical stability, even with nonlinear effects from the beam-beam force taken into account. The Round Beam Concept (RBC) was realized at the electron-positron collider VEPP-2000 and successfully tested at the energy of 510 MeV [D.E. Berkaev et al., in Proc. of the EPAC 2008, Genoa, p. 956]. Despite the low energy, a high single-bunch luminosity of 10 31 cm − 2 s − 1 was achieved together with a maximum tune shift as high as 0.1. At present the work is in progress to increase the energy of the collider to explore the range between 500 MeV and 1 GeV in collision.

A study on dynamic beam-beam effects and their application in BEPCII

Beam-beam interactions cause dynamic behaviour of the charged particle beams in circular colliders, which is directly related to their luminosity. The linear beam-beam force introduces an additional quadrupole that perturbs betatron motion of particles, including dynamic β, dynamic emittance and beam-beam tune-shifts; while the nonlinear force may excite resonances and cause tune-spread. With the weak-strong beam-beam mode, the linear and nonlinear effects are studied, dynamic β and dynamic emittance as functions of tunes are analyzed, and formulas of beam-beam octupole caused tune-spread are derived, starting from beam-beam potential and equations of particle motions, and the measures to increase luminosity of BEPCII are discussed.

Simulation study of beam-beam effects

The Beijing Electron-Positron Collider (BEPC) will be upgraded to BEPCII, and the luminosity will be about 100 times higher. We developed a three-dimensional strong-strong PIC code to study the beam-beam effects in BEPCII. The transportation through the arc is the same as that in Hirata's weak-strong code. The beam-beam force is computed directly by solving the Poisson equation using the Fourier analysis and cyclic reduction method, and the boundary potential is computed by circular convolution. The finite bunch length effect is included by longitudinal slices. An interpolation scheme is used to reduce the required slice number in simulations. The standard message passing interface is used to parallelize the code. The computing time increases linearly with (n+1), where n is the slice number. The calculated luminosity of BEPCII at the design working point is similar to 50% of the design value. The best area in the tune space is near (0.505,0.57) according to the survey, where the luminosity is similar to 80% of the design.

Halo formation due to beam-beam interactions of beams optically mismatched at injection

In this paper, we report on the halo formation and emittance growth driven by a parametric resonance during beam-beam collisions with optically mismatched injection. In the regime of the weak-strong beam-beam interaction, if two beams have the same machine tunes, on-axis head-on collisions between a mismatched strong beam and a weak beam will not cause the formation of halo. However, if the two beams collide with an initial offset, the beam-beam force from the mismatched strong beam can cause halo formation and emittance growth in the weak beam. Meanwhile, if two beams have different machine tunes, for opposite charged colliding beams, when the machine tune of the weak beam is smaller than that of the strong beam, there is emittance growth in the weak beam. When the machine tune of the weak beam is larger than that of the strong beam, there is little emittance growth. In the regime of the strong-strong beam-beam interaction, halo is formed in both beams even when the two beams collide head-on on the axis with equal machine tunes. This puts a strong requirement for a good beam match during the injection to colliders in order to avoid emittance growth if there are no other mechanisms after the injection to decohere the mismatched envelope oscillation.

Equilibrium state of colliding electron beams

We study a nonlinear integral equation that is a necessary condition on the equilibrium phase space distribution function of stored, colliding electron beams. It is analogous to the Haissinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is analyzed for the case of the Chao-Ruth model of the beam-beam interaction in one degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove existence of a unique solution, for sufficiently small beam current, by an application of the implicit function theorem. We have not yet proved that this solution is positive, as would be required to establish existence of an equilibrium. There is, however, numerical evidence of a positive solution. We expect that our analysis can be extended to more realistic models.

Equilibrium beam distribution and quantum lifetime in the presence of a single nonlinear resonance

In the proximity of a nonlinear resonance $\ensuremath{\nu}\ensuremath{\approx}\frac{m}{n}$, the beam distribution in a storage ring is distorted depending on how close is the resonance and how strong is the resonance strength. In the 1-dimensional case, it is well known that the particle motion near the resonance can be described in a smooth approximation by a Hamiltonian of the form $(\ensuremath{\nu}\ensuremath{-}\frac{m}{n})J+{D}_{\ensuremath{\nu}}(J)+{f}_{1}(\ensuremath{\varphi},J)$, where $(\ensuremath{\varphi},J)$ are the phase space angle and action variables, ${D}_{\ensuremath{\nu}}$ is the detuning function, and ${f}_{1}$ is an oscillating resonance term. In a proton storage ring, the equilibrium beam distribution is readily solved to be any function exclusively of the Hamiltonian. For an electron beam, this is not true and the equilibrium distribution is more complicated. This paper solves the Fokker-Planck equation near a single resonance for an electron beam in a storage ring. The result is then applied to obtain the quantum lifetime of an electron beam in the presence of this resonance. Resonances due to multipole fields and due to the beam-beam force are considered as examples.

Estimates of diffusion due to long-range beam-beam collisions

Weak-strong tracking simulations for the Large Hadron Collider have shown that long-range beam-beam collisions give rise to a well-defined diffusive aperture beyond which particles are lost quickly. In order to derive analytical estimates of this stability boundary, we use leading order perturbation theory and the Chirikov resonance overlap criterion applied to a simplified model with a 2-dimensional transverse phase space. In addition, a Fokker-Plank\char21{}type diffusion coefficient is calculated through the nonlinear action kicks imparted by the long-range beam-beam force. The analytical results are compared with the tracking data.

Synchrotron-betatron coupling due to monochromatization.

The effects of large dispersion at the interaction point are discussed within the linear approximation of the beam-beam force. The synchrotron and betatron motions affect each other. Synchrotron and betatron tune shifts as well as bunch-length and energy-spread modifications appear, among many effects. Depending on the parameters, degradations of the luminosity can occur due to the hourglass effect. More serious is an enlargement of the collision energy spread, which may reduce the event rate and invalidate the monochromatization scheme.

Simulations of coherent beam-beam modes at the Large Hadron Collider

The transverse coherent motion of two colliding proton beams at the Large Hadron Collider is studied by multiparticle tracking. We use the beam-beam force for a Gaussian beam distribution with variable barycenters and rms beam sizes, and optionally include the effect of long-range collisions and external impedance. The simulation yields the coherent and incoherent oscillation frequencies, the emittance growth of either beam, and evidence for the existence or lack of Landau damping. For head-on collisions of beams with equal sizes, we find that the $\ensuremath{\pi}$-mode frequency lies outside of the continuum frequency spread, if the ratio of the beam-beam parameters exceeds 0.6, in accordance with predictions. For smaller ratios of the beam-beam parameters, or if, for equal beam-beam parameters, the beam sizes are widely different, the $\ensuremath{\pi}$ mode is Landau damped. When long-range collisions are also included, undamped coherent modes do still exist outside the continuum, both with and without alternating crossing planes at two interaction points. However, separating the tunes of the two beams restores the Landau damping, provided the external impedance is sufficiently small.

Self-consistentβfunctions and emittances of round colliding beams

The flip-flop effect with the linearized beam-beam force is formulated through self-consistent $\ensuremath{\beta}$ functions and equilibrium emittances which are both affected by collision. We give the results of two models of emittance dependence. The effect of finite bunch length is also discussed.

Coherent beam-beam interactions in electron-positron colliders.

We present the results of a new calculational technique that evaluates the beam-beam force due to an arbitrary charge distribution. We find coherent instabilities that dominate at certain operating points and depend strongly on the degree of damping in the system. We conclude that while these resonances may play a significant role for colliders with low damping, with a careful choice of operating points they are unlikely to present a danger to colliders with greater damping, such as the new generation of high-luminosity heavy-quark factories currently under design.

Coherent betatron oscillation modes due to beam-beam interaction

A simple consideration is given of the eigenmodes of the coherent betatron oscillation excited by a series of sinusoidal kicks and influenced by the beam-beam interaction in e +e − colliding storage rings. The differences between tunes corresponding to the eigenmodes are just half of what is expected from rigid point like bunches. The discussion is mainly limited to the case where the beam-beam force is not so strong so that the bunches remain Gaussian and the linear approximation of the force is valid. The nonlinear effect is also discussed briefly.

Simulation of the beam-beam effect for e + e − storage rings

The effect of the beam-beam interaction (“strong-strong”) in e + e − storage rings is simulated by tracking many superparticles per beam, for many turns, through the accelerating system, the accelerator lattice and the beam-beam interaction. The motion of each superparticle is treated in six-dimensional phase space and the effects of quantum excitation and radiation damping are included. Momentum dispersion at the RF cavities and the interaction regions allow computation of single beam and beam-beam synchro-betatron resonances. The longitudinal motion of each superparticle produces a modulation of the betatron phase advance between beam-beam collisions. This modulation also drives beam-beam synchrobetatron resonances. The effects of perturbations on the superperiodicity (errors) and chromaticity are included in the simulation. The beam-beam force is calculated for an elliptical beam with Gaussian charge density distribution in three dimensions. This report is comprised of two main sections; in the first section the simulation technique is described and in the second section a compendium of results obtained from the simulation is presented. The simulation results presented here pertain to the LEP design.

Non-linear coupling from the beam-beam force in electron-positron storage rings

The non-linear beam-beam forces in storage rings can excite the coupling resonance 2 Q v – 2 Q h = m between vertical and radial betatron oscillations, giving rise to an increase in the smaller emittance. It is shown that the magnitude and the sign of the tune difference between vertical and horizontal motion are of particular importance for electron-positron storage rings. Criteria for avoiding this effect are established and the predictions of the theory are compared with some recent experimental results.

Approximate Physical Treatment of the Beam-Beam Effects

When the beam-beam force is approximated as a periodic δ-function non-linear potential kicks, it is expected to excite a continuum(density of rational numbers) of resonances. The totality of resonance within a narrow "width" of the betatron oscillation tune contribute to a diffusion-like amplitude growth as if the kicks are random. A semi-quantitative formulation is derived and applied to electron colliding beams.

Investigation of the Coherent Beam-Beam Effects in the ISR

Although the beam-beam tune shift is relatively small in the ISR (0.001 per intersection) and below the value where stochastic effects might be expected, coherent beam-beam effects can be important. Each high-intensity ISR beam is close to the transverse single beam stability limit. The addition of the beam-beam force has led to the loss of one of the beams accompanied by coherent oscillations. The change of the betatron frequency distribution due to the other beam has been investigated. The resonance excitation by the non-linear beam-beam force has been measured and seems to influence the transverse stability. Quantitative information about the beam-beam force is obtained by exciting a coherent betatron oscillation in one beam and observing the response of the other beam. This beam-beam transfer function depends on the beam separation in the interaction point and can be used to centre the two beams. At the same time the effective beam height is obtained, which together with the beam currents determines the luminosity.

Beam-Beam Simulation for LEP

A multi-particle two-beam (strong-strong) simulation program has been written for investigation of the beam-beam effect in 'LEP'. The motion of the superparticles is treated in six-dimensional phase space and the effects of quantum excitation and radiation damping are included. The effects of perturbations to the superperiodicity (errors) are also included. Non-zero dispersion at the RF cavities allows computation of single beam synchro-betatron resonances. These resonances may also be driven by the beam-beam interaction either by non-zero dispersion at the interaction point or by the longitudinal motion of the interaction 'point' between a particle and the centre of the 'other' beam due to the synchrotron motion of the particle. After each revolution the parameters influencing the beam-beam force (e.g. the beam dimensions and the beam current) are reevaluated in order to simulate a real situation. For the beam-beam force an elliptical beam with Gaussian charge distribution has been assumed. The computation of this force is speeded up by using tabulated values of the complex error function and a fast interpolation procedure. The 'beam-beam limit' is shown to be a function of many machine parameters but is always significantly below the assumed maximum value of 0.06.