Showing papers in "Physical Review Special Topics-accelerators and Beams in 2002"

Probabilistic model for the simulation of secondary electron emission

Abstract: We provide a detailed description of a model and its computational algorithm for the secondary electron emission process. The model is based on a broad phenomenological fit to data for the secondary-emission yield and the emitted-energy spectrum. We provide two sets of values for the parameters by fitting our model to two particular data sets, one for copper and the other one for stainless steel.

Zettawatt-exawatt lasers and their applications in ultrastrong-field physics

Abstract: Since its birth, the laser has been extraordinarily effective in the study and applications of laser-matter interaction at the atomic and molecular level and in the nonlinear optics of the bound electron. In its early life, the laser was associated with the physics of electron volts and of the chemical bond. Over the past fifteen years, however, we have seen a surge in our ability to produce high intensities, 5 to 6 orders of magnitude higher than was possible before. At these intensities, particles, electrons, and protons acquire kinetic energy in the megaelectron-volt range through interaction with intense laser fields. This opens a new age for the laser, the age of nonlinear relativistic optics coupling even with nuclear physics. We suggest a path to reach an extremely high-intensity level ${10}^{26--28}\mathrm{W}/{\mathrm{cm}}^{2}$ in the coming decade, much beyond the current and near future intensity regime ${10}^{23}\mathrm{W}/{\mathrm{cm}}^{2}$, taking advantage of the megajoule laser facilities. Such a laser at extreme high intensity could accelerate particles to frontiers of high energy, teraelectron volt, and petaelectron volt, and would become a tool of fundamental physics encompassing particle physics, gravitational physics, nonlinear field theory, ultrahigh-pressure physics, astrophysics, and cosmology. We focus our attention on high-energy applications, in particular, and the possibility of merged reinforcement of high-energy physics and ultraintense laser.

Energetic ions generated by laser pulses: A detailed study on target properties

Abstract: We present the results of a detailed study on the acceleration of intense ion beams by relativistic laser plasmas. The experiments were performed at the 100 TW laser at the Laboratoire pour L'Utilisation des Lasers Intenses. We investigated the dependence of the ion beams on the target conditions based on theoretical predictions by the target normal sheath acceleration mechanism. A strong dependence of the ion beam parameters on the conditions on the target rear surface was found. We succeeded in shaping the ion beam by the appropriate tailoring of the target geometry and we performed a characterization of the ion beam quality. The production of a heavy ion beam could be achieved by suppressing the amount of protons at the target surfaces. Finally, we demonstrated the use of short pulse laser driven ion beams for radiography of thick samples with high resolution.

Processing of interlaced images in 4–10 MeV dual energy customs system for material recognition

Abstract: The aim of this article is to demonstrate the practical value of radioscopic differentiation of materials in the 1--10 MeV energy range to the work of customs services. The proposed method for achieving singling out and identifying four basic groups of materials according to an atomic number is complex. Atomic numbers are identified using high- and low-energy profiles obtained through the irradiation of materials on an alternate pulse-by-pulse basis. This is done using a bremsstrahlung beam with $8\text{ }\mathrm{MeV}/4\text{ }\mathrm{MeV}$ dual boundary energies and by using scintillating crystals coupled with silicon photodiodes as detecting elements. An image segmentation technique is then used to discern the distribution of an atomic number on any given image. The color visualization of integral absorption and a material's atomic composition is carried out according to the intensity hue saturation (IHS) colorization scheme. The experiments were carried out on a full-scale prototype of an 8 MeV customs inspection system developed by the Efremov Research Institute.

Coherent synchrotron radiation instability in a bunch compressor

Abstract: The coherent synchrotron radiation of a bunch in a bunch compressor may lead to the microwave instability producing longitudinal modulation of the bunch with wavelengths small compared to the bunch length. It can also be a source of an undesirable emittance growth in the compressor. We derive and analyze the equation that describes linear evolution of the microwave modulation taking into account incoherent energy spread and finite emittance of the beam. Numerical solution of this equation for the Linac Coherent Light Source bunch compressor gives the amplification factor for different wavelengths of the beam microbunching.

Formulas for coherent synchrotron radiation microbunching in a bunch compressor chicane

Abstract: A microbunching instability driven by coherent synchrotron radiation (CSR) in a bunch compressor chicane is studied using an iterative solution of the integral equation that governs this process. By including both one-stage and two-stage amplifications, we obtain analytical expressions for CSR microbunching that are valid in both low-gain and high-gain regimes. These formulas can be used to explore the dependence of CSR microbunching on compressed beam current, energy spread, and emittance, and to design stable bunch compressors required for an x-ray free-electron laser.

Space-charge effects in high brightness electron beam emittance measurements

Abstract: The measurement of emittance in space-charge dominated, high brightness beam systems is investigated from conceptual, computational, and experimental viewpoints. As the self-field-induced collective motion in the low energy, high brightness beams emitted from photoinjector rf guns are more important in determining the macroscopic beam evolution than thermal spreads in transverse velocity; traditional methods for phase space diagnosis fail in these systems. We discuss the role of space charge forces in a traditional measurement of transverse emittance, the quadrupole scan. The mitigation of these effects by use of multislit- or pepper-pot-based techniques is explained. The results of a direct experimental comparison between quadrupole scanning and slit-based determination of the emittance of a 5 MeV high brightness electron beam are presented. These data are interpreted with the aid of both envelope and multiparticle simulation codes. It is shown that the ratio of the beam's $\ensuremath{\beta}$ function to its transverse plasma wavelength plays a central role in the quadrupole scan results. Methods of determining the presence of systematic errors in quadrupole scan data are discussed.

Beam instability and microbunching due to coherent synchrotron radiation

Abstract: A relativistic electron beam moving in a circular orbit in free space can radiate coherently if the wavelength of the synchrotron radiation exceeds the length of the bunch. In accelerators coherent synchrotron radiation of the bunch is usually suppressed by the shielding effect of the conducting walls of the vacuum chamber. However an initial density fluctuation with a characteristic length much shorter than the bunch length can radiate coherently. If the radiation reaction force results in the growth of the initial fluctuation, one can expect an instability which leads to microbunching of the beam and an increased coherent radiation at short wavelengths. Such an instability is studied theoretically in this paper.

Electron scattering and acceleration by a tightly focused laser beam

Abstract: By numerically solving the relativistic equations of motion of a single electron in laser fields modeled by those of a Gaussian beam, we demonstrate electron capture by, reflection from, and transmission through the beam. In modeling the fields, terms of order up to ${ϵ}^{5}$, where $ϵ$ is the diffraction angle, are retained. All cases of capture are accompanied by energy gain that may reach a few GeV, from fields of present-day intensities. Reflection and transmission, on the other hand, result sometimes in no gain or even in a loss of energy. It is shown that a laboratory static magnetic field may be used to eject a captured electron, a process that sometimes results in even more energy gain. For example, a 2.5 T uniform magnetic field suffices to eject a 3.633 MeV electron injected at $6\ifmmode^\circ\else\textdegree\fi{}$ to the axis of a linearly polarized beam of a 10 PW power output and aimed at a point near the focus. Such an electron gains 1128 MeV from the laser field alone. However, it emerges with a 1230 MeV net energy gain under the additional action of the small magnetic field.

Beam extraction studies at 900 GeV using a channeling crystal

Abstract: Luminosity-driven channeling extraction has been observed for the first time in a 900 GeV study at the Fermilab Tevatron. This experiment, Fermilab E853, demonstrated that useful TeV level beams can be extracted from a superconducting accelerator during high luminosity collider operations without unduly affecting the background at the collider detectors. Multipass extraction was found to increase the efficiency of the process significantly. The beam extraction efficiency was about 25%. Studies of time dependent effects found that the turn-to-turn structure was governed mainly by accelerator beam dynamics. Based on the results of this experiment, it is feasible to construct a parasitic 5--10 MHz proton beam from the Tevatron collider.

Transverse to longitudinal emittance exchange

Abstract: A scheme is proposed to exchange the transverse and longitudinal emittances of an electron bunch. A general analysis is presented and a specific beamline is used as an example where the emittance exchange is achieved by placing a transverse deflecting mode radio-frequency cavity in a magnetic chicane. In addition to reducing the transverse emittance, the bunch length is also simultaneously compressed. The scheme has the potential to introduce an added flexibility to the control of electron beams and to provide some contingency for the achievement of emittance and peak-current goals in free-electron lasers.

Experimental study of rf pulsed heating on oxygen free electronic copper

Abstract: When the thermal stresses induced by RF pulsed heating are larger than the elastic limit, microcracks and surface roughening will occur due to cyclic fatigue. Therefore, pulsed heating limits the maximum surface magnetic field and through it the maximum achievable accelerating gradient. An experiment using circularly cylindrical cavities operating in the TE{sub 011} mode at a resonant frequency of 11.424 GHz was designed to study pulsed heating on Oxygen Free Electronic (OFE) copper. An X-band klystron delivered up to 10 MW to the cavities in 1.5 {micro}s pulses at 60 Hz repetition rate. One run was executed at a temperature rise of 120 K for 56 x 10{sup 6} pulses. Cracks at grain boundaries, slip bands and cracks associated with these slip bands were observed. The second run consisted of 86 x 10{sup 6} pulses with a temperature rise of 82 K, and cracks at grain boundaries and slip bands were seen. Additional information can be derived from the power-coupling iris, and we conclude that a pulsed temperature rise of 250 K for several million pulses leads to destruction of copper. These results can be applied to any mode of any OFE copper cavity.

Electron cloud simulations: beam instabilities and wakefields

Abstract: HEADTAIL is a simulation program developed at CERN which is aimed at studying the single-bunch instability arising from the interaction on successive turns of a single bunch with the cloud generated by the previous bunches. The code includes chromaticity, space charge tune spread, broad-band impedance, and detuning with amplitude for more realistic simulation. Examples of application are shown. Transverse and longitudinal wake functions are also outputs of the HEADTAIL code.

Energy doubler for a linear collider

Abstract: At 2 miles long and 50 GeV, the Stanford Linear Collider (SLC) is the highest energy linear accelerator in the world, and the only linear collider. Along with the Large Electron Positron Collider (LEP) at CERN, it has succeeded in unveiling much of the detailed physics of the standard model of elementary particles and fields. However, the Higgs boson and the ultimate test of the standard model appear now to lie above 100 GeV and therefore out of the reach of the SLC. The results of the last runs of LEP were suggestive of the fact that the discovery of the Higgs may have been just beyond the reach of that machine [1,2]. In this report, we describe a scenario for doubling the energy of a collider by using a plasma wake field accelerator section several meters long placed at the end of each beam line just before the collision point. Such a doubling scheme could be used to extend the high-energy physics reach of the SLC or a future linear collider. The concept of a plasma wake field accelerator has received considerable attention recently [3‐ 7]. In a plasma wake field accelerator, the space charge of a particle bunch displaces the electrons of a preformed quiescent plasma to produce a large plasma wake field that can accelerate a subsequent bunch at a very high rate. In this report, 3D simulation models are used to show that the amplitude of the accelerating wake scales with the inverse square of the bunch length and that this scaling continues to hold for parameters far exceeding the linear theory from which it is derived. This leads us to propose the concept of a plasma afterburner — a specifically designed plasma that accelerates as well as focuses each beam from a linear collider in a single, short, final stage. Finally, we outline the critical issues that remain to be addressed in order to realize this concept. The afterburner concept is illustrated schematically in Fig. 1. Electrons and positrons are accelerated to the collider’s nominal operating energy (e.g., 50 GeV for the SLC example), overcompressed to form two microbunches each, then the trailing half-bunches are doubled in energy over a few meters in the plasma afterburner. To sustain the luminosity at the interaction point (IP) at the nominal level of the original collider without the plasma, the reduction in number of particles must be offset by a smaller spot size at the IP. Reduction of the spot size is possible in the strong focusing fields of the plasma; thus, higher density plasma lenses are added to the design just before the interaction point. To guide the discussion of the simulations to follow, we begin by reviewing key features of plasma wake field excitation in linear theory [8]. The linear response of a plasma to a Gaussian bunch is optimized when the plasma density no is chosen such that the bunch length and plasma wavelength are matched; more precisely, for kpsz p 2,

Transient beam loading effects in harmonic rf systems for light sources

Abstract: Harmonic cavities have been used in storage rings to lengthen bunches and increase beam lifetimes dominated by Touschek scattering. Transient beam loading in the harmonic cavities generated by asymmetries in the fill pattern causes significant variation of the bunch synchronous phase and bunch length along the bunch train when the longitudinal restoring force has been reduced. This results in a significant reduction in the mean bunch lengthening and potential lifetime increase. We describe how beam current modulations give rise to transient effects much larger than expected from the linear model of the beam cavity interaction. We also develop a tracking simulation to predict results and apply this simulation to an analysis of the beam loading transients for the case of passive and active normal and superconducting third harmonic rf systems using Advanced Light Source parameters.

Luminosity optimization near the beam-beam limit by increasing bunch length or crossing angle

Abstract: We discuss the choice of bunch length and crossing angle near the beam-beam limit in a storage-ring collider. First, we derive expressions for the tune shifts of either bunched or continuous round beams which are induced by a single collision with arbitrary crossing angle and bunch length and for the associated luminosities. Then, considering two collision points with alternating planes of crossing, we demonstrate that, if the total beam-beam tune shift is held constant, the collider luminosity increases as a function of bunch length and crossing angle. This implies a corresponding increase in the bunch intensity. As an illustration, we present numerical examples for a Large Hadron Collider upgrade and for the Very Large Hadron Collider.

Determination of longitudinal bunch shape by means of coherent Smith-Purcell radiation

Abstract: Coherent enhancement of the Smith-Purcell radiation produced from the interaction of a 1.8 MeV electron beam with a grating has been observed. The emitted radiation has been measured at angles in the 40± to 120± range, which correspond to wavelengths from 0.65 to 4 mm, approximately. The radiated power was 320 mW at 90±. Its angular distribution agrees well with the description of the process in terms of induced surface currents and has been used to infer the longitudinal profile of the electron bunch. It is concluded that the bunch has an approximately triangular profile, with 85% of the bunch particles contained within 14 ps. The possibilities of the technique as a bunch-shape diagnostic tool are also discussed.

Nonlinear dynamics in a SPEAR wiggler

Abstract: BL11, the most recently installed wiggler in the SPEAR storage ring at the Stanford Synchrotron Radiation Laboratory, produces a large nonlinear perturbation of the electron beam dynamics, which was not directly evident in the integrated magnetic field measurements. Measurements of tune shifts with betatron oscillation amplitude and closed orbit shifts were used to characterize the nonlinear fields. Because of the narrow pole width in BL11, the nonlinear fields seen along the wiggling electron trajectory are dramatically different from the magnetic measurements made along a straight line with a stretched wire. This difference explains the tune shift measurements and the observed degradation in dynamic aperture. Because of the relatively large dispersion (1.2 m) at BL11, the nonlinearities particularly reduced the off-energy dynamic aperture. Because of the nature of these nonlinear fields, it is impossible, even theoretically, to cancel them completely with short multipole correctors. Magic finger corrector magnets were built, however, that partially correct the nonlinear perturbation, greatly improving the storage ring performance.

Beam halo definitions based upon moments of the particle distribution

Abstract: Two different parameters for the quantitative description of beam halo are discussed. Both are based on moments of the particle distribution and represent a convenient and model-independent method for quantifying the magnitude of beam halo observed in either spatial or phase-space projections. One parameter is a measure of spatial profile of the beam and has been defined by Wangler and Crandall previously. The current authors defined a new parameter using kinematic invariants to quantify halo formation in 2D phase space. Here we expand the development and present detailed numerical results. Although the spatial-profile parameter and the phase-space halo parameter both reduce to the same value when the distribution has the elliptical symmetry, in general these parameters are not equal. Halo in the 1D spatial profiles is relatively easily measured, but is variable as the beam distribution evolves and can hide as it rotates in phase space. The 2D phase-space halo is more difficult to measure, but it varies more smoothly as the halo evolves. It provides a more reliable characterization of the halo as an intrinsic property of the beam.

Algorithm for loading shot noise microbunching in multidimensional, free-electron laser simulation codes

Abstract: We discuss the underlying reasoning behind and the details of the numerical algorithm used in the GINGER free-electron laser(FEL) simulation code to load the initial shot noise microbunching on the electron beam. In particular, we point out that there are some additional subtleties which must be followed for multi-dimensional codes which are not necessary for one-dimensional formulations. Moreover, requiring that the higher harmonics of the microbunching also be properly initialized with the correct statistics leads to additional complexities. We present some numerical results including the predicted incoherent, spontaneous emission as tests of the shot noise algorithm's correctness.

Strong-strong beam-beam simulation using a green function approach

Abstract: In this paper we present a news approach, based on a shifted Green function, to evaluate the electromagnetic field in a simulation of colliding beams. Unlike a conventional particle-mesh code, we use a method in which the computational mesh covers only the largest of the two colliding beams. This allows us to study long-range parasitic collisions accurately and efficiently. We have implemented this algorithm in a new parallel strong-strong beam-beam simulation code. As an application, we present a study of a beam sweeping scheme for the LBNL luminosity monitor of the Large Hadron Collider.

Multimoded rf delay line distribution system for the Next Linear Collider

Abstract: The delay line distribution system is an alternative to conventional pulse compression, which enhances the peak power of rf sources while matching the long pulse of those sources to the shorter filling time of accelerator structures. We present an implementation of this scheme that combines pairs of parallel delay lines of the system into single lines. The power of several sources is combined into a single waveguide delay line using a multimode launcher. The output mode of the launcher is determined by the phase coding of the input signals. The combined power is extracted from the delay line using mode-selective extractors, each of which extracts a single mode. Hence, the phase coding of the sources controls the output port of the combined power. The power is then fed to the local accelerator structures. We present a detailed design of such a system, including several implementation methods for the launchers, extractors, and ancillary high power rf components. The system is designed so that it can handle the 600 MW peak power required by the Next Linear Collider design while maintaining high efficiency.

Normal form of particle motion under the influence of an AC dipole

Abstract: AC dipoles in accelerators are used to excite coherent betatron oscillations at a drive frequency close to the tune. These beam oscillations may last arbitrarily long and, in principle, there is no significant emittance growth if the AC dipole is adiabatically turned on and off. Therefore the AC dipole seems to be an adequate tool for non–linear diagnostics provided the particle motion is well described in presence of the AC dipole and non–linearities. Normal Forms and Lie algebra are powerful tools to study the non–linear content of an accelerator lattice. In this article a way to obtain the Normal Form of the Hamiltonian of an accelerator with an AC dipole is described. The particle motion to first order in the non– linearities is derived using Lie algebra techniques. The dependence of the Hamiltonian terms on the longitudinal coordinate is studied showing that they vary differently depending on the AC dipole parameters. The relation is given between the lines of the Fourier spectrum of the turn–by–turn motion and the Hamiltonian terms.

Macroparticle simulation studies of a proton beam halo experiment

Abstract: We report macroparticle simulations for comparison with measured results from a proton beam halo experiment in a 52-quadrupole periodic-focusing channel. An important issue is that the input phase-space distribution is not experimentally known. Three different initial distributions with different shapes predict different beam profiles in the transport system. Simulations have been fairly successful in reproducing the core of the measured matched-beam profiles and the trend of emittance growth as a function of the mismatch factor, but underestimate the growth rate of halo and emittance for mismatched beams. In this study, we find that knowledge of the Courant-Snyder parameters and emittances of the input beam is not sufficient for reliable prediction of the halo. Input distributions with greater population in the tails produce larger rates of emittance growth, a result that is qualitatively consistent with the particle-core model of halo formation in mismatched beams.

Induction core alloys for heavy-ion inertial fusion-energy accelerators

Abstract: Induction core alloys are evaluated that are appropriate for heavy-ion induction accelerators to drive heavy-ion inertial fusion (HIF) power plants. Parameters evaluated include the usable flux swing and the energy loss over a range of magnetization rates of $\ensuremath{\sim}{10}^{5}--{10}^{7}\mathrm{T}/\mathrm{s}$, corresponding to pulse durations of $\ensuremath{\sim}20$ to $0.2\ensuremath{\mu}\mathrm{s}$, respectively. The usable flux swing, for minimum core losses, extends from near the reversed remanent field to about 80% of the saturation field. The usable flux swing is enhanced, with little increase in losses, by annealing the core after winding. Maintaining low energy loss at high magnetization rates requires insulation to block interlaminar eddy currents. To obtain annealed cores with a high ratio of remanent to saturation magnetic field, the insulation must withstand annealing temperatures and apply minimum mechanical stress to the core during cooldown. We find that commercially available insulating coatings for amorphous metals either break down near ${10}^{6}\mathrm{T}/\mathrm{s}$ (a factor of 10 below the requirement), or do not achieve the maximum remanent field and hence the usable flux swing after annealing. More satisfactory coatings are available for silicon steel and nanocrystalline alloys, which could have applications in HIF. Amorphous alloys are capable of meeting most HIF needs, especially with improved coatings.

Tune shifts of bunch trains due to resistive vacuum chambers without circular symmetry

Abstract: We present an explanation for the opposite signs of the horizontal and vertical tune shifts of bunch trains which have been observed recently in several high-energy storage rings. This result can be understood in terms of the long-range quadrupolar wakes of noncircular vacuum chambers with finite resistivity. In vacuum chambers with circular cross section, the dominant transverse wake driven by a leading particle and seen by a trailing test particle is dipolar and is proportional only to the transverse offset of the driving particle. The contributions of preceding bunches or previous turns tend to cancel, as they add with oscillatory factors. On the other hand, quadrupolar wakes are independent of the offset of the driving particle, and thus the contributions of preceding bunches and turns are strictly additive. Since quadrupole fields are focusing in one plane and defocusing in the plane orthogonal to it, their effects on tune shifts in these planes have opposite signs. Their cumulative effect also explains the large values of the tune shifts measured in PEP-II, which exceeded estimates from other impedance sources by factors of 3 to 4. Our analysis also offers a connection to the familiar Laslett tune shift.

Development of high power X-band semiconductor microwave switch for pulse compression systems of future linear colliders

Abstract: We describe development of semiconductor X-band high-power RF switches. The target applications are high-power RF pulse compression systems for future linear colliders. We describe the design methodology of the architecture of the whole switch systems. We present the scaling law that governs the relation between power handling capability and number of elements. We designed and built several active waveguide windows for the active element. The waveguide window is a silicon wafer with an array of four hundred PIN/NIP diodes covering the surface of the window. This waveguide window is located in an over-moded TE01 circular waveguide. The results of high power RF measurements of the active waveguide window are presented. The experiment is performed at power levels of a few megawatts at X-band.

Efficiency of a Boris-like integration scheme with spatial stepping

Abstract: A modified Boris-like integration, in which the spatial coordinate is the independent variable, is derived. This spatial-Boris integration method is useful for beam simulations, in which the independent variable is often the distance along the beam. The new integration method is second order accurate, requires only one force calculation per particle per step, and preserves conserved quantities more accurately over long distances than a Runge-Kutta integration scheme. Results from the spatial-Boris integration method and a Runge-Kutta scheme are compared for two simulations: (i) a particle in a uniform solenoid field and (ii) a particle in a sinusoidally varying solenoid field. In the uniform solenoid case, the spatial-Boris scheme is shown to perfectly conserve for any step size quantities such as the gyroradius and the perpendicular momentum. The Runge-Kutta integrator produces damping in these conserved quantities. In the sinusoidally varying case, the conserved quantity of canonical angular momentum is used to measure the accuracy of the two schemes. For the sinusoidally varying field simulations, error analysis is used to determine the integration distance beyond which the spatial-Boris integration method is more efficient than a fourth-order Runge-Kutta scheme. For beam physics applications where statistical quantities such as beam emittance are important, these results imply the spatial-Boris scheme is 3 times more efficient.

Computer simulations of a single-laser double-gas-jet wakefield accelerator concept

Abstract: We report in this paper on full scale 2D particle-in-cell simulations investigating laser wakefield acceleration First we describe our findings of electron beam generation by a laser propagating through a single gas jet Using realistic parameters which are relevant for the experimental setup in our laboratory we find that the electron beam resulting after the propagation of a $08\ensuremath{\mu}\mathrm{m}$, 50 fs laser through a 15 mm gas jet has properties that would make it useful for further acceleration Our simulations show that the electron beam is generated when the laser exits the gas jet, and the properties of the generated beam, especially its energy, depend only weakly on most properties of the gas jet We therefore propose to use the first gas jet as a plasma cathode and then use a second gas jet placed immediately behind the first to provide additional acceleration Our simulations of this proposed setup indicate the feasibility of this idea and also suggest ways to optimize the quality of the resulting beam

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

Abstract: 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.