Beam Equilibrium Research Articles

A beam-position monitor system at the VEPP-4M electron−positron collider

The VEPP-4M storage ring has been designed for experiments with colliding electron−positron beams, a synchrotron radiation beam, and extracted γ-ray beams at energies of 1−5 GeV. A correct equilibrium beam orbit must be kept in the storage ring to maintain the required experimental conditions. A beam position monitor system is used for this purpose. It includes 54 beam position monitors, monitor signal-processing modules, and synch signals distributors. In 2013–2016, new signal processing modules with a high time resolution were developed and installed at the VEPP-4M storage ring. These have two main advantages over the old modules. First, the new modules allow separate position measurements of electron and positron bunches at an interval between the bunch signals as long as 18 ns. Second, they are able to perform fullfledged turn-by-turn beam position measurements, which is very important for adjustment of the storagering parameters. Turn-by-turn beam position measurements can be performed during injection of a new beam, beam kicking, beam loss, or at an arbitrary instant of time. The structures and main features of the new modules are described, the measurement accuracy is analyzed, and some results of their operation at the storage ring are presented.

Equilibrium-Based Nonhomogeneous Anisotropic Beam Element

The stiffness matrix and the nodal forces associated with distributed loads are obtained for a nonhomogeneous anisotropic elastic beam element by the use of complementary energy. The element flexibility matrix is obtained by integrating the complementary-energy density corresponding to six beam equilibrium states, and then inverted and expanded to provide the element-stiffness matrix. Distributed element loads are represented via corresponding internal-force distributions in local equilibrium with the loads. The element formulation does not depend on assumed shape functions and can, in principle, include any variation of cross-sectional properties and load variation, provided that these are integrated with sufficient accuracy in the process. The ability to represent variable cross-sectional properties, coupling from anisotropic materials, and distributed element loads is illustrated by numerical examples.

Radial equilibrium of relativistic particle bunches in plasma wakefield accelerators

Drive particle beams in linear or weakly nonlinear regimes of the plasma wakefield accelerator quickly reach a radial equilibrium with the wakefield, which is described in detail for the first time. The equilibrium beam state and self-consistent wakefields are obtained by combining analytical relationships, numerical integration, and first-principles simulations. In the equilibrium state, the beam density is strongly peaked near the axis, the beam radius is constant along most of the beam, and longitudinal variation of the focusing strength is balanced by varying beam emittance. The transverse momentum distribution of beam particles depends on the observation radius and is neither separable nor Gaussian.

Generalized Kapchinskij-Vladimirskij Distribution and Beam Matrix for Phase-Space Manipulations of High-Intensity Beams.

In an uncoupled linear lattice system, the Kapchinskij-Vladimirskij (KV) distribution formulated on the basis of the single-particle Courant-Snyder invariants has served as a fundamental theoretical basis for the analyses of the equilibrium, stability, and transport properties of high-intensity beams for the past several decades. Recent applications of high-intensity beams, however, require beam phase-space manipulations by intentionally introducing strong coupling. In this Letter, we report the full generalization of the KV model by including all of the linear (both external and space-charge) coupling forces, beam energy variations, and arbitrary emittance partition, which all form essential elements for phase-space manipulations. The new generalized KV model yields spatially uniform density profiles and corresponding linear self-field forces as desired. The corresponding matrix envelope equations and beam matrix for the generalized KV model provide important new theoretical tools for the detailed design and analysis of high-intensity beam manipulations, for which previous theoretical models are not easily applicable.

Analytical solution for composite layered beam subjected to uniformly distributed load

ABSTRACTThe article presents an analytical theory for multilayered composite beams subjected to transverse uniformly distributed loads. The formulation is based on a layerwise model characterized by third-order approximation of the axial displacements and fourth-order approximation of the transverse displacements. The layerwise kinematical model is rewritten in terms of generalized variables. The beam equilibrium equations, expressed in terms of stress resultant, allow writing the boundary value governing problem. The layerwise fields are obtained by postprocessing steps. The main advantage is to ensure the accuracy level associated to the layerwise formulations preserving the computational efficiency of the equivalent-single-layer theories.

A model reduction technique for beam analysis with the asymptotic expansion method

In this paper, we apply the asymptotic expansion method to the mechanical problem of beam equilibrium, aiming to derive a new beam model. The asymptotic procedure will lead to a series of mechanical problems at different order, solved successively. For each order, new transverse (in-plane) deformation and warping (out of plane) deformation modes are determined, in function of the applied loads and the limits conditions of the problem. The presented method can be seen as a more simple and efficient alternative to beam model reduction techniques such as POD or PGD methods. At the end of the asymptotic expansion procedure, an enriched kinematic describing the displacement of the beam is obtained, and will be used for the formulation of an exact beam element by solving analytically the arising new equilibrium equations. A surprising result of this work, is that even for concentrated forces (Dirac delta function), we obtain a very good representation of the beam’s deformation with only few additional degrees of freedom.

Electron-Beam Dynamics for an Advanced Flash-Radiography Accelerator

Beam dynamics issues were assessed for a new linear induction electron accelerator being designed for multipulse flash radiography of large explosively driven hydrodynamic experiments. Special attention was paid to equilibrium beam transport, possible emittance growth, and beam stability. Especially problematic would be high-frequency beam instabilities that could blur individual radiographic source spots, low-frequency beam motion that could cause pulse-to-pulse spot displacement, and emittance growth that could enlarge the source spots. Beam physics issues were examined through theoretical analysis and computer simulations, including particle-in-cell codes. Beam instabilities investigated included beam breakup, image displacement, diocotron, parametric envelope, ion hose, and the resistive wall instability. Beam corkscrew motion and emittance growth from beam mismatch were also studied. It was concluded that a beam with radiographic quality equivalent to the present accelerators at Los Alamos National Laboratory will result if the same engineering standards and construction details are upheld.

Refined zigzag theory for homogeneous, laminated composite, and sandwich beams derived from Reissner’s mixed variational principle

A highly accurate and computationally attractive shear-deformation theory for homogeneous, laminated composite, and sandwich laminates is developed for the linearly elastic analysis of planar beams. The theory is derived using the kinematic assumptions of Refined Zigzag Theory (RZT) and a two-step procedure that implements Reissner’s Mixed Variational Theorem (RMVT). The basic expression for the transverse-shear stress that satisfies a priori the equlibrium conditions along the layer interfaces is obtained from Cauchy’s equilibrium equations. The resulting transverse-shear stress consists of second-order derivatives of the two rotation variables of the theory, which subsequently are restated as the unknown stress functions. As the first step in fulfilling RMVT, the Lagrange-multiplier functional is minimized with respect to the unknown stress functions, resulting in the stress functions consisting of first-order derivatives of the kinematic variables. Subsequently, the second term of RMVT is minimized, producing four beam equilibrium equations and consistent boundary conditions. For any number of material layers the new theory maintains only four kinematic variables. The theory is labeled RZT(m), where the superscript (m) stands for mixed formulation. The RZT(m) can accurately model the axial stretch, bending, and transverse-shear deformations, without shear-correction factors. Analytic solutions are derived for simply supported beams subjected to transverse-normal and transverse-shear tractions on the top and bottom surfaces. It is demonstrated that RZT(m) has a wide range of applicability which includes sandwich construction and the laminates with embedded thin compliant layers that can potentially model progression of delaminations. The main advantage of RZT(m) over RZT is in the superior predictions of transverse-shear stresses that are obtained directly from the low-order transverse-shear strain measures of the theory without resorting to a post-processing integration procedure. Importantly, the methodology can be readily extended to plate theory, and it can be applied effectively for developing simple and efficient C0-continuous finite elements.

On torsion of random composite beams

Elastic equilibrium of two-phase random composite beams under torsion, with simply and multiply connected cross-sections, is investigated. The analysis is carried out in order to assess the boundary layer effect exhibited by the shear stress field, expressed in terms of the torsional warping function, near the cross-section boundary. The boundary value problem governing the warping function is formulated in variational terms and solved by a FEM analysis. Standard methods of micromechanics are adopted in order to evaluate the eigenstress fields describing cross-sectional elastic inhomogeneities. The shear stress distribution in a beam with equilateral triangular cross-section is numerically evaluated for different microstructures characterizing the elastic phases of the composite material.

A Monte-Carlo simulation of the equilibrium beam polarization in ultra-high energy electron (positron) storage rings

With the recently emerging global interest in building a next generation of circular electron–positron colliders to study the properties of the Higgs boson, and other important topics in particle physics at ultra-high beam energies, it is also important to pursue the possibility of implementing polarized beams at this energy scale. It is therefore necessary to set up simulation tools to evaluate the beam polarization at these ultra-high beam energies. In this paper, a Monte-Carlo simulation of the equilibrium beam polarization based on the Polymorphic Tracking Code (PTC) (Schmidt et al., 2002 [1]) is described. The simulations are for a model storage ring with parameters similar to those of proposed circular colliders in this energy range, and they are compared with the suggestion (Derbenev et al., 1979 [2]) that there are different regimes for the spin dynamics underlying the polarization of a beam in the presence of synchrotron radiation at ultra-high beam energies. In particular, it has been suggested that the so-called “correlated” crossing of spin resonances during synchrotron oscillations at current energies evolves into “uncorrelated” crossing of spin resonances at ultra-high energies.

Force-based higher-order beam element with flexural–shear–torsional interaction in 3D frames. Part I: Theory

An innovative higher-order beam theory, capable of accurately taking into account flexural–shear–torsional interaction, is originally combined with a force-based formulation to derive the corresponding finite element. The selected set of higher-order deformation modes leads to an explicit and direct interaction between three-dimensional shear and normal stresses. Namely, cross-sectional displacement and strain fields are composed of independent and orthogonal modes, which results in unambiguously defined generalised cross-sectional stress-resultants and in a minimisation of the coupling of equilibrium equations. On the basis of work-equivalency to three-dimensional continuum theory, dual one-dimensional higher-order equilibrium and compatibility equations are derived. The former, which govern an advanced form of beam equilibrium, are strictly satisfied via stress fields arising from the solution of the corresponding systems of coupled differential equations. The formulation, which is numerically validated in a companion paper for both linear and nonlinear material response, inherently avoids shear-locking and accurately accounts for span loads. Finally, the superiority of force-based approaches over displacement-based ones, well established for inelastic behaviour, is also demonstrated for the linear elastic case.

Warping and Ljapounov stability of non-trivial equilibria of non-symmetric open thin-walled beams

We investigate the effects of warping on the dynamic stability of non-trivial equilibrium configurations for non-symmetric open thin-walled beams. We use a direct one-dimensional model coarsely describing warping; the rest of the kinematics is exact. Dynamic derives from the balance of power; constitutive relations are non-linear, hyper-elastic, and distinguish the roles of the centroids and shear centres; inertial actions account for warping, too. By centred finite differences, the warping inertial action is found ineffective on the natural angular frequencies. Then, we follow non-trivial equilibrium paths and investigate their Ljapounov stability, by examining the small superposed oscillations. Results for generic, non-symmetric cross-sections are presented and discussed, showing the effects of warping and of coupling constitutive coefficients.

Design of a 4.8-m ring for inverse Compton scattering x-ray source

In this paper we present the design of a 50 MeV compact electron storage ring with 4.8-meter circumference for the Tsinghua Thomson scattering x-ray source. The ring consists of four dipole magnets with properly adjusted bending radii and edge angles for both horizontal and vertical focusing, and a pair of quadrupole magnets used to adjust the horizontal damping partition number. We find that the dynamic aperture of compact storage rings depends essentially on the intrinsic nonlinearity of the dipole magnets with small bending radius. Hamiltonian dynamics is found to agree well with results from numerical particle tracking. We develop a self-consistent method to estimate the equilibrium beam parameters in the presence of the intrabeam scattering, synchrotron radiation damping, quantum excitation, and residual gas scattering. We also optimize the rf parameters for achieving a maximum x-ray flux.

Analytic Theory and PIC Simulation of Electron Beam Generation under High Voltage

The theoretical analysis and numerical simulation of electron beam generation in foil-less diode are presented. The theories of OAL, CL, and equilibrium beam model are briefly compared. All the theoretical models can give a relatively accurate solution of electron gun, and the equilibrium beam model is more appreciated. A PIC simulation tool UNIPIC is introduced and is used to analyze the behaviors of a foil-less diode. Under the conditions of 180 kV incident voltage wave and 2.5 T axial magnetic fields, we obtained the 250 kV, 1.5 kA output beam current which can be used in high power microwave generators.

Semianalytic Stress-Function Variational Approach for the Interfacial Stresses in Bonded Joints

AbstractHigh interfacial stresses near the edges of bonded joints are responsible for their debonding failure. This paper reports a new semianalytic stress-function variational approach to the interfacial stresses of a bonded joint, which is made of a straight tension bar covered with a reinforcing patch and subjected to mechanical loads and/or uniform change of temperature. The process introduces two interfacial shear and normal stress functions, which are correlated via the approximately same radius of curvature of the slender adherends. All the stress components in the joint are expressed in terms of the interfacial stress functions based on the classic Euler-Bernoulli beam theory and equilibrium equations of elasticity. Deformation compatibility of the joint is satisfied by minimizing the complementary strain energy, which leads to a fourth-order ordinary differential equation (ODE) of the interfacial shear stress function. The interfacial shear and normal stresses are determined explicitly and compar...

Mass-spectrometric study of tellurium-film condensation and evaporation on sapphire

The processes of the condensation and evaporation of a tellurium film produced by a molecular beam of Te2 on a sapphire surface are studied. It is experimentally shown that the desorption of tellurium from a surface upon growth under nonequilibrium conditions exceeds the equilibrium thermal desorption. If the beam intensity is less than the equilibrium value, then desorption is also lower than the equilibrium value. A model of this process with allowance for the transient states of tellurium in the adsorption layer is proposed. Within this model equilibrium and Langmuir beams are expressed in terms of the process parameters. The reflection coefficient of the beam is also a function of the process characteristics. The experimental data make it possible to estimate some of the basic parameters of the process, including the coverage and desorption rate in the adsorption layer. Taking these results into account is important for manufacturing perfect films not only of Te, but also of CdTe.

Influence of finite radial geometry on the growth rate of ion-channel free electron laser

The influence of finite radial geometry on the instability of a tenuous relativistic electron beam propagating in an ion-channel in a waveguide is investigated. The instability analysis is based on the linearized Vlasov-Maxwell equations for the perturbation about a self-consistent beam equilibrium. With the help of characteristic method the dispersion relation for the TE-mode is derived and analyzed through the numerical solutions. It is found that the positioning of the beam radius Rb relative to the waveguide radius Rc, and the ion-channel frequency can have a large influence on the maximum growth rate and corresponding wave number.

Kinetic equilibrium of space charge dominated beams in a misaligned quadrupole focusing channel

The dynamics of intense beam propagation through the misaligned quadrupole focusing channel has been studied in a self-consistent manner using nonlinear Vlasov-Maxwell equations. The equations of motion of the beam centroid have been developed and found to be independent of any specific beam distribution. A Vlasov equilibrium distribution and beam envelope equations have been obtained, which provide us a theoretical tool to investigate the dynamics of intense beam propagating in a misaligned quadrupole focusing channel. It is shown that the displaced quadrupoles only cause the centroid of the beam to wander off axis. The beam envelope around the centroid obeys the familiar Kapchinskij-Vladimirskij envelope equation that is independent of the centroid motion. However, the rotation of the quadrupole about its optical axis affects the beam envelope and causes an increase in the projected emittances in the two transverse planes due to the inter-plane coupling.

Three-dimensional equilibria of nonlinear pre-curved beams using an intrinsic formulation and shooting

This article describes a shooting method for computing three-dimensional equilibria of pre-curved nonlinear beams with axial and shear flexibility using the intrinsic beam formulation. For distributed and concentrated follower loads acting on a cantilevered beam, the method amounts to a direct solution approach requiring only a single shot (zero iterations) to compute the equilibria. This is possible since the system equations are defined in a local coordinate system that rotates and translates with the beam, akin to the follower loads themselves. A general procedure employing nonconservative follower loads, which invokes the Picard–Lindelöf theorem on uniqueness and existence of solutions, is also introduced for finding all solutions for three-dimensional pre-curved beam problems with conservative loading. This is particularly useful in beam buckling problems where multiple stable and unstable solutions exist. Three-dimensional equilibrium solutions are generated for many loading cases and boundary conditions, including three-dimensional helical beams, and are compared to similar solutions where available in the literature. Excellent agreement is documented in all comparison cases. For buckling examples, the stability of the computed solutions is assessed using a dynamic finite element code based on the same intrinsic beam equations. Due to the ability to avoid iteration, the presented approach may find application in model-based control for practical three-dimensional problems such as the control of manipulators utilized in endoscopic surgeries and the control of spacecraft with robotic arms and long cables.

Nonequilibrium electron rings for synchrotron radiation production.

Electron storage rings used for the production of synchrotron radiation (SR) have an output photon brightness that is limited by the equilibrium beam emittance. By using interleaved injection and ejection of bunches from a source with repetition rate greater than 1kHz, we show that it is practicable to overcome this limit in rings of energy ∼1 GeV. Sufficiently short kicker pulse lengths enable effective currents of many milliamperes, which can deliver a significant flux of diffraction-limited soft x-ray photons. Thus, either existing SR facilities may be adapted for nonequilibrium operation, or the technique applied to construct SR rings smaller than their storage ring equivalent.