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Radiation transport in numerical astrophysics

Description: In this article, we discuss some of the numerical techniques developed by Jim Wilson and co-workers for the calculation of time-dependent radiation flow. Difference equations for multifrequency transport are given for both a discrete-angle representation of radiation transport and a Fick's law-like representation. These methods have the important property that they correctly describe both the streaming and diffusion limits of transport theory in problems where the mean free path divided by characteristic distances varies from much less than one to much greater than one. They are also stable for timesteps comparable to the changes in physical variables, rather than being limited by stability requirements.
Date: February 1, 1983
Creator: Lund, C.M.
Partner: UNT Libraries Government Documents Department

Proof that stable monotonic equilibrium distributions in a continuous focusing channel are necessarily axisymmetric

Description: The transverse Vlasov equilibrium distribution function of an unbunched ion beam propagating in a continuous focusing channel is specified by a function f{perpendicular} (H{perpendicular}), where H{perpendicular} is the single-particle Hamiltonian. In standard treatments of continuous focusing equilibria in Vlasov-Poisson electrostatic models, it is assumed that a stable beam equilibrium specified by monotonic f{perpendicular}(H{perpendicular}) with {partial_derivative}f{perpendicular}(H{perpendicular})/{partial_derivative}H{perpendicular} {le} 0 is axisymmetric (no variation in azimuthal angle, i.e., with {partial_derivative}/{partial_derivative}{theta} = 0). In this paper a simple, but rigorous, proof is presented that only axisymmetric equilibrium solutions are possible in Vlasov-Poisson models for any physical choice of f{perpendicular}(H{perpendicular}) with {partial_derivative}f{perpendicular}(H{perpendicular})/{partial_derivative}H{perpendicular} {le} 0 if the confining boundary of the system (the beam pipe) is axisymmetric or if the geometry is radially unbounded.
Date: March 28, 2007
Creator: Lund, S M
Partner: UNT Libraries Government Documents Department

NUMERICAL SOLUTION FOR THE POTENTIAL AND DENSITY PROFILE OF A THERMAL EQUILIBRIUM SHEET BEAM

Description: In a recent paper, S. M. Lund, A. Friedman, and G. Bazouin, Sheet beam model for intense space-charge: with application to Debye screening and the distribution of particle oscillation frequencies in a thermal equilibrium beam, in press, Phys. Rev. Special Topics - Accel. and Beams (2011), a 1D sheet beam model was extensively analyzed. In this complementary paper, we present details of a numerical procedure developed to construct the self-consistent electrostatic potential and density profile of a thermal equilibrium sheet beam distribution. This procedure effectively circumvents pathologies which can prevent use of standard numerical integration techniques when space-charge intensity is high. The procedure employs transformations and is straightforward to implement with standard numerical methods and produces accurate solutions which can be applied to thermal equilibria with arbitrarily strong space-charge intensity up to the applied focusing limit.
Date: March 29, 2011
Creator: Lund, S M & Bazouin, G
Partner: UNT Libraries Government Documents Department

Space-charge transport limits of ion beams in periodic quadrupole focusing channels

Description: It has been empirically observed in both experiments and particle-in-cell simulations that space-charge-dominated beams suffer strong growth in statistical phase-space area (degraded quality) and particle losses in alternating gradient quadrupole transport channels when the undepressed phase advance {sigma}{sub 0} increases beyond about 85{sup o} per lattice period. Although this criterion has been used extensively in practical designs of strong focusing intense beam transport lattices, the origin of the limit has not been understood.We propose a mechanism for the transport limit resulting from classes of halo particle resonances near the core of the beam that allow near-edge particles to rapidly increase in oscillation amplitude when the space-charge intensity and the utter of the matched beam envelope are both sufficiently large. When coupled with a diffuse beam edge and/or perturbations internal to the beam core that can drive particles outside the edge, this mechanism can result in large and rapid halo-driven increases in the statistical phase-space area of the beam, lost particles, and degraded transport. A core-particle model is applied to parametrically analyze his process. Extensive self-consistent particle in cell simulations are employed to better quantify space-charge limit and verify core-particle model predictions.
Date: March 3, 2006
Creator: Lund, S M & Chawla, S R
Partner: UNT Libraries Government Documents Department

Theory of longitudinal beam halo in RF linacs: II. envelope-particle resonances

Description: Using the core/test-particle model described in a companion paper in these proceedings (``Theory of Longitudinal Halo in rf Linacs: I. Core/Test Particle Formulation,`` by J. J. Barnard and S. M. Lund), we analyze longitudinal beam halo produced by resonant self-field interactions in intense, ion-beam rf linacs. It is shown that particles moving in the presence of the space-charge forces of an oscillating, mismatched ellipsoidal beam bunch can be resonantly driven to large longitudinal amplitude. This resonantly produced halo is first analyzed in a limit where it is most simply understood, with particles moving purely longitudinally and with linear rf focusing. Then modifications of the resonance induced by nonlinear rf and transverse-longitudinal coupling are explored.
Date: May 1, 1997
Creator: Lund, S.M. & Barnard, J.J.
Partner: UNT Libraries Government Documents Department

Theory of longitudinal beam halo in RF linacs: I. core/test-particle formulation

Description: For intense beams, the analysis of tenuous halo components of the particle distribution that surround the main core of the distribution can be challenging. So-called core/test particle models in which a test particle is evolved in the applied and space-charge forces of the beam core have been instrumental in understanding the structure and extent of transverse beam halo produced by resonant particle interactions with the oscillating space-charge forces of a mismatched beam core. Here we present a core/test particle model developed for the analysis of longitudinal beam halo in intense, ion-beam rf linacs. Equations of motion are derived for a test particle moving interior to, and exterior to, a uniform density ellipsoidal beam bunch. Coupled transverse-longitudinal mismatch modes of the ellipsoidal beam envelope are analyzed. Typical parameters suggest the possibility of a low-order resonant interaction between longitudinal particle oscillations and a low-frequency envelope mode. Properties of this resonance are in an accompanying paper by the authors in these proceedings.
Date: May 1, 1997
Creator: Barnard, J.J. & Lund, S.M.
Partner: UNT Libraries Government Documents Department

Course Notes: United States Particle Accelerator School Beam Physics with Intense Space-Charge

Description: The purpose of this course is to provide a comprehensive introduction to the physics of beams with intense space charge. This course is suitable for graduate students and researchers interested in accelerator systems that require sufficient high intensity where mutual particle interactions in the beam can no longer be neglected. This course is intended to give the student a broad overview of the dynamics of beams with strong space charge. The emphasis is on theoretical and analytical methods of describing the acceleration and transport of beams. Some aspects of numerical and experimental methods will also be covered. Students will become familiar with standard methods employed to understand the transverse and longitudinal evolution of beams with strong space charge. The material covered will provide a foundation to design practical architectures. In this course, we will introduce you to the physics of intense charged particle beams, focusing on the role of space charge. The topics include: particle equations of motion, the paraxial ray equation, and the Vlasov equation; 4-D and 2-D equilibrium distribution functions (such as the Kapchinskij-Vladimirskij, thermal equilibrium, and Neuffer distributions), reduced moment and envelope equation formulations of beam evolution; transport limits and focusing methods; the concept of emittance and the calculation of its growth from mismatches in beam envelope and from space-charge non-uniformities using system conservation constraints; the role of space-charge in producing beam halos; longitudinal space-charge effects including small amplitude and rarefaction waves; stable and unstable oscillation modes of beams (including envelope and kinetic modes); the role of space charge in the injector; and algorithms to calculate space-charge effects in particle codes. Examples of intense beams will be given primarily from the ion and proton accelerator communities with applications from, for example, heavy-ion fusion, spallation neutron sources, nuclear waste transmutation, etc.
Date: May 30, 2008
Creator: Barnard, J.J. & Lund, S.M.
Partner: UNT Libraries Government Documents Department

Sheet beam model for intense space-charge: with application to Debye screening and the distribution of particle oscillation frequencies in a thermal equilibrium beam

Description: A one-dimensional Vlasov-Poisson model for sheet beams is reviewed and extended to provide a simple framework for analysis of space-charge effects. Centroid and rms envelope equations including image charge effects are derived and reasonable parameter equivalences with commonly employed 2D transverse models of unbunched beams are established. This sheet beam model is then applied to analyze several problems of fundamental interest. A sheet beam thermal equilibrium distribution in a continuous focusing channel is constructed and shown to have analogous properties to two- d three-dimensional thermal equilibrium models in terms of the equilibrium structure and Deybe screening properties. The simpler formulation for sheet beams is exploited to explicitly calculate the distribution of particle oscillation frequencies within a thermal equilibrium beam. It is shown that as space-charge intensity increases, the frequency distribution becomes broad, suggesting that beams with strong space-charge can have improved stability.
Date: January 10, 2011
Creator: Lund, Steven M.; Friedman, Alex & Bazouin, Guillaume
Partner: UNT Libraries Government Documents Department

Generation of Initial Kinetic Distributions for Simulation of Long-Pulse Charged Particle Beams with High Space-Charge intensity

Description: Self-consistent Vlasov-Poisson simulations of beams with high space-charge intensity often require specification of initial phase-space distributions that reflect properties of a beam that is well adapted to the transport channel--both in terms of low-order rms (envelope) properties as well as the higher-order phase-space structure. Here, we first review broad classes of kinetic distributions commonly in use as initial Vlasov distributions in simulations of unbunched or weakly bunched beams with intense space-charge fields including: the Kapchinskij-Vladimirskij (KV) equilibrium, continuous-focusing equilibria with specific detailed examples, and various non-equilibrium distributions, such as the semi-Gaussian distribution and distributions formed from specified functions of linear-field Courant-Snyder invariants. Important practical details necessary to specify these distributions in terms of usual accelerator inputs are presented in a unified format. Building on this presentation, a new class of approximate initial kinetic distributions are constructed using transformations that preserve linear-focusing single-particle Courant-Snyder invariants to map initial continuous-focusing equilibrium distributions to a form more appropriate for non-continuous focusing channels. Self-consistent particle-in-cell simulations are employed to show that the approximate initial distributions generated in this manner are better adapted to the focusing channels for beams with high space-charge intensity. This improved capability enables simulation applications that more precisely probe intrinsic stability properties and machine performance.
Date: April 3, 2007
Creator: Lund, Steven M.; Kikuchi, Takashi & Davidson, Ronald C.
Partner: UNT Libraries Government Documents Department

Efficient computation of matched solutions of the KV envelope equations for periodic focusing lattices.

Description: A new iterative method is developed to numerically calculate the periodic, matched beam envelope solution of the coupled Kapchinskij-Vladimirskij (KV) equations describing the transverse evolution of a beam in a periodic, linear focusing lattice of arbitrary complexity. Implementation of the method is straightforward. It is highly convergent and can be applied to all usual parameterizations of the matched envelope solutions. The method is applicable to all classes of linear focusing lattices without skew couplings, and also applies to parameters where the matched beam envelope is strongly unstable. Example applications are presented for periodic solenoidal and quadrupole focusing lattices. Convergence properties are summarized over a wide range of system parameters.
Date: January 17, 2006
Creator: Lund, S M; Chilton, S H & Lee, E P
Partner: UNT Libraries Government Documents Department

A Core-Particle Model for Periodically Focused Ion Beams with Intense Space-Charge

Description: A core-particle model is derived to analyze transverse orbits of test particles evolving in the presence of a core ion beam described by the KV distribution. The core beam has uniform density within an elliptical cross-section and can be applied to model both quadrupole and solenoidal focused beams in periodic or aperiodic lattices. Efficient analytical descriptions of electrostatic space-charge fields external to the beam core are derived to simplify model equations. Image charge effects are analyzed for an elliptical beam centered in a round, conducting pipe to estimate model corrections resulting from image charge nonlinearities. Transformations are employed to remove coherent utter motion associated with oscillations of the ion beam core due to rapidly varying, linear applied focusing forces. Diagnostics for particle trajectories, Poincare phase-space projections, and single-particle emittances based on these transformations better illustrate the effects of nonlinear forces acting on particles evolving outside the core. A numerical code has been written based on this model. Example applications illustrate model characteristics. The core-particle model described has recently been applied to identify physical processes leading to space-charge transport limits for an rms matched beam in a periodic quadrupole focusing channel [Lund and Chawla, Nuc. Instr. and Meth. A 561, 203 (2006)]. Further characteristics of these processes are presented here.
Date: August 2, 2006
Creator: Lund, S M; Barnard, J J; Bukh, B; Chawla, S R & Chilton, S H
Partner: UNT Libraries Government Documents Department

Space-charge transport limits of ion beams in periodic quadrupolefocusing channels

Description: It has been empirically observed in both experiments and particle-in-cell simulations that space-charge-dominated beams suffer strong growth in statistical phase-space area (degraded quality) and particle losses in alternating gradient quadrupole transport channels when the undepressed phase advance {sigma}{sub 0} increases beyond about 85{sup o} per lattice period. Although this criterion has been used extensively in practical designs of strong focusing intense beam transport lattices, the origin of the limit has not been understood. We propose a mechanism for the transport limit resulting from classes of halo particle resonances near the core of the beam that allow near-edge particles to rapidly increase in oscillation amplitude when the space-charge intensity and the utter of the matched beam envelope are both sufficiently large. When coupled with a diffuse beam edge and/or perturbations internal to the beam core that can drive particles outside the edge, this mechanism can result in large and rapid halo-driven increases in the statistical phase-space area of the beam, lost particles, and degraded transport. A core-particle model is applied to parametrically analyze this process. Extensive self-consistent particle in cell simulations are employed to better quantify properties of the space-charge limits and to verify core-particle model predictions.
Date: February 23, 2006
Creator: Lund, S. M. & Chawla, S. R.
Partner: UNT Libraries Government Documents Department

Truncated Thermal Equilibrium Distribution for Intense Beam Propagation

Description: An intense charged-particle beam with directed kinetic energy ({lambda}{sub b}-1)m{sub b}c{sup 2} propagates in the z-direction through an applied focusing field with transverse focusing force modeled by F{sub foc} = -{lambda}{sub b}m{sub b}{omega}{sub beta}{sup 2} {perpendicular} x {perpendicular} in the smooth focusing approximation. This paper examines properties of the axisymmetric, truncated thermal equilibrium distribution F(sub)b(r,p perpendicular) = A exp (-H Perpendicular/T perpendicular (sub)b) = (H perpendicular-E(sub)b), where A, T perpendicular (sub)b, and E (sub)b are positive constants, and H perpendicular is the Hamiltonian for transverse particle motion. The equilibrium profiles for beam number density, n(sub)b(r) = * d{sup 2}pF(sub)b(r,p perpendicular), and transverse temperature, T perpendicular (sub)b(r) = * d{sup 2}p(p{sup 2} perpendicular/2 lambda (sbu)bm (sub)b)F(sub)b(r,p perpendicular), are calculated self-consistently including space-charge effects. Several properties of the equilibrium profiles are noteworthy. For example, the beam has a sharp outer edge radius r(sub)b with n(sub)b(r greater than or equal to rb) = 0, where r(sub)b depends on the value of E(sub)b/T (sub)perpendicular(sub)b. In addition, unlike the choice of a semi-Gaussian distribution, F{sup SG}(sub)b = A exp (-p{sup 2}(sub)perpendicular/2lambda(sub)bm(sub)bTperpendicular(sub)b) = (r-r(sub)b), the truncated thermal equilibrium distribution F(sub)b(r,p) depends on (r,p) only through the single-particle constant of the motion Hperpendiuclar and is therefore a true steady-state solution (*/*t = 0) of the nonlinear Vlasov-Maxwell equations.
Date: February 26, 2003
Creator: Davidson, Ronald C.; Qin, Hong & Lund, Steven M.
Partner: UNT Libraries Government Documents Department

On the relaxation of semi-Gaussian and K-V beams to thermal equilibrium

Description: A beam propagating in a continuous, linear focusing channel tends to relax to a thermal equilibrium state. We employ nonlinear conservation constraints to theoretically analyze changes in quantities that characterize both an initial semi-Gaussian beam with a matched rms beam envelope and a K-V beam under a relaxation to thermal equilibrium. Results from particle-in-cell simulations are compared to the theoretical predictions.
Date: May 2, 1995
Creator: Lund, S.M.; Barnard, J.J. & Miller, J.M.
Partner: UNT Libraries Government Documents Department

Self-consistent 3D simulations of longitudinal halo in rf -linacs

Description: In order to prevent activation of the beam pipe walls and components of a high power ion accelera- tor: beam loss must be minimized. Here we present self-consistent, 3D particle-in-cell simulations of longi- tudinally mismatched beams including the effects of rf non-linearities using parameters based on the Acceler- ator Production of Tritium linac design. In particular, we explore the evolution of the longitudinal halo distri- bution, i.e., the distribution of particles in longitudinal phase space with oscillation amplitudes significantly larger than amplitudes of particles in the main body or ''core'' of the beam. When a particle reaches a suf- ficiently large amplitude longitudinally it can he lost from the rf bucket and consequently loses synchro- nism with thr rf wave. Such particles will lose energy and so be poorly matched to the transverse focusing field and consequently can be lost transversely. We compare the present simulations in which all particles contribute self-consistently to the self-field to predic- tions of a core/test particle model in which the core distribution has uniformly distributed charge and does not evolve self-consistently. Effects of self-consistent, non-linear space-charge forces, non-linear rf focusing on envelope mismatch induced beam halo are explored through comparisons of both models.
Date: August 19, 1998
Creator: Barnard, J J; Lund, S M & Ryne, R D
Partner: UNT Libraries Government Documents Department

3D particle simulations of space-charge-dominated beams in HIF accelerator experiments

Description: The development of a high current, heavy-ion beam for inertial confinement fusion requires a detailed understanding of the behavior of the beam, including effects of the large self-fields. This necessity makes particle-in-cell (PIC) simulation the appropriate tool, and for this reason, the three-dimensional PIC/accelerator code WARP3d is being developed. WARP3d has been used extensively to study the creation and propagation of ion beams both to support experiments and for the understanding of basic beam physics. An overview of the structure of the code is presented along with a discussion of features that make the code an effective tool in the understanding of space-charge dominated beam behavior. A number of applications where WARP3d has played an important role is discussed, emphasizing the need of three-dimensional, first principles simulations. Results and comparisons with experiment are presented.
Date: May 1, 1997
Creator: Grote, D.P.; Friedman, A.; Lund, S.M. & Haber, I.
Partner: UNT Libraries Government Documents Department

Iron free permanent magnet systems for charged particle beam optics

Description: The strength and astounding simplicity of certain permanent magnet materials allow a wide variety of simple, compact configurations of high field strength and quality multipole magnets. Here we analyze the important class of iron-free permanent magnet systems for charged particle beam optics. The theory of conventional segmented multipole magnets formed from uniformly magnetized block magnets placed in regular arrays about a circular magnet aperture is reviewed. Practical multipole configurations resulting are presented that are capable of high and intermediate aperture field strengths. A new class of elliptical aperture magnets is presented within a model with continuously varying magnetization angle. Segmented versions of these magnets promise practical high field dipole and quadrupole magnets with an increased range of applicability.
Date: September 3, 1995
Creator: Lund, S. M. & Halbach, K.
Partner: UNT Libraries Government Documents Department