Kapchinskij-Vladimirskij Distribution Research Articles

Generalized Courant–Snyder theory and Kapchinskij–Vladimirskij distribution for high-intensity beams in a coupled transverse focusing lattice

The Courant–Snyder (CS) theory and the Kapchinskij–Vladimirskij (KV) distribution for high-intensity beams in an uncoupled focusing lattice are generalized to the case of coupled transverse dynamics. The envelope function is generalized to an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are noncommutative. In an uncoupled lattice, the KV distribution function, first analyzed in 1959, is the only known exact solution of the nonlinear Vlasov-Maxwell equations for high-intensity beams including self-fields in a self-consistent manner. The KV solution is generalized to high-intensity beams in a coupled transverse lattice using the generalized CS invariant. This solution projects to a rotating, pulsating elliptical beam in transverse configuration space. The fully self-consistent solution reduces the nonlinear Vlasov-Maxwell equations to a nonlinear matrix ordinary differential equation for the envelope matrix, which determines the geometry of the pulsating and rotating beam ellipse. These results provide us with a new theoretical tool to investigate the dynamics of high-intensity beams in a coupled transverse lattice. A strongly coupled lattice, a so-called N-rolling lattice, is studied as an example. It is found that strong coupling does not deteriorate the beam quality. Instead, the coupling induces beam rotation and reduces beam pulsation.

Solutions of the matched KV envelope equations for a “smooth” asymmetric focusing channel

In many particle accelerators the applied focusing forces may differ in two or three directions either by design in order to avoid resonances or for other reasons, such as design constraints, bunch compression/expansion, dispersion, etc. At high intensities, space charge effects and related collective forces may cause unwanted emittance growth via instabilities and equipartitioning (relaxation of temperature anisotropy). For the transverse two-dimensional case, such asymmetric (anisotropic) systems are described by the coupled, matched envelope equations of the Kapchinskij–Vladimirskij distribution with different focusing strengths and emittances in the x and y directions, which must be solved numerically for a periodic lattice. In this article, we present results for a “smooth” asymmetric focusing channel, in which case one obtains a set of two coupled algebraic equations for the envelopes X and Y. Though the algebraic equations can easily be solved numerically, the scaling with the physics parameters is usually obscured by the numerical procedures. We derived an approximate solution as well as a general, more accurate solution, both of which represent results that exhibit the scaling with the applied focusing, space-charge, and emittance terms. The accuracy of the approximate solution is in the range of a few percent for a channel with a small degree of asymmetry. The general solution is obtained by solving for the aspect ratio A=Y/X by an iteration method that yields results to any desired degree of accuracy. More importantly, to facilitate the comparison between systems with different particle species and/or operating parameters, the envelope equations in this general treatment are written in dimensionless form. This is accomplished by expressing the envelopes X and Y in terms of the “average radius” as, and by introducing dimensionless parameters, v and w, which measure the degrees of focusing and emittance asymmetries, and the ratios of the space charge to the external focusing forces, defined by the intensity parameter χ. The results are then used to obtain formulas for the frequencies, or wave numbers, of the betatron oscillations and the tune depressions due to space-charge forces in the x and y directions, which are of fundamental importance for understanding the beam physics. These dimensionless relations exhibit the desired beam physics scaling and represent mathematically convenient forms for calculating, tabulating or plotting both exact as well as approximate solutions for the various quantities of interest. Two examples, including figures, are presented to illustrate the practical use of the theoretical relations. Our results should be useful for machine designers, theorists and experimentalists.

Three-dimensional stability theory of a continuous beam with axisymmetric Kapchinskij-Vladimirskij distribution

This work is a three-dimensional stability study based on the modal analysis for a continuous beam with an axisymmetric Kapchinskij-Vladimirskij (KV) distribution. The analysis is carried out self-consistently within the context of linearized Vlasov-Maxwell equations and electrostatic approximation. The emphasis is on investigating the coupling between longitudinal and transverse perturbations in the high-intensity region. The interaction between the transverse modes supported by the KV distribution and the ``usual transverse modes'' is examined. We found two classes of ``coupling modes'' that would not exist if longitudinal and transverse perturbations are treated separately. We also found that some transverse modes can interact among themselves through longitudinal perturbation to cause instability. The effects of wall impedance on beam stability is also studied and numerical examples are presented.

Chaotic particle motion and beam halo formation induced by image-charge effects in a small-aperture alternating-gradient focusing system

It is shown with a test-particle model that image-charge effects induce a new mechanism for chaotic particle motion and halo formation in an intense charged-particle beam propagating through an alternating-gradient focusing channel with a small aperture, circular, perfectly conducting pipe. This mechanism occurs for a well-matched beam with the Kapchinskij–Vladimirskij distribution. The percentage of beam loss to the conductor wall is calculated as a function of propagating distance and aperture size.

Space charge dynamics of bright electron beams

The longitudinal dynamics and its coupling with the transverse dynamics of bunched beams with strong space charge are analyzed. We introduce a self-consistent Vlasov description for the longitudinal phase space similar to the familiar description for the transverse phase space using a Kapchinskij-Vladimirskij distribution. A longitudinal beam envelope equation is derived. An exact solution is then obtained when coupling to the transverse dynamics is ignored. This longitudinal envelope equation is coupled to the transverse envelope equation to form a set of coupled dynamical equations, which is then solved numerically. This analysis is prompted by the surprising results of recent experiments which showed that by driving an intense laser pulse into matter, which in turn creates a plasma, short bright relativistic electron bunches are produced, surprisingly narrowly focused. We find that because the space charge forces weaken with increasing transverse and longitudinal phase space, both the transverse and longitudinal emittance blowouts anticipated of bright compact bunches are mitigated by this coupling. It should be possible to capture these bunches into an rf cavity to accelerate to higher energies.

Simulation of a dc electron beam strongly influenced by self-fields

A method is presented to simulate the transport of a dc charged particle beam strongly influenced by its self-fields. If the initial phase space of the beam is chosen to be a Kapchinskij-Vladimirskij distribution, the simulation is in agreement with the Kapchinskij-Vladimirskij envelope equations. The simulation was initialized with a Gaussian transverse momentum distribution to investigate the origin of current profiles measured in a 130 keV, 635 mA plasma-focused electron beam.

Nonlinear properties of the Kapchinskij-Vladimirskij equilibrium and envelope equation for an intense charged-particle beam in a periodic focusing field.

The nonlinear properties of the Kapchinskij-Vladimirskij (KV) equilibrium and envelope equation are examined for an intense charged-particle beam propagating through an applied periodic solenoidal focusing magnetic field, including the effects of the self-electric and self-magnetic fields associated with the beam space charge and current. It is found that the beam emittance is proportional to the maximum canonical angular momentum achieved by the particles within the KV distribution. The Poincare mapping technique is used to determine systematically the axial dependence of the radius of the matched (equilibrium) beam and to study nonlinear behavior in the nonequilibrium beam envelope oscillations. It is shown that the nonequilibrium beam envelope oscillations exhibit nonlinear resonances and chaotic behavior for periodic focusing magnetic fields and sufficiently high beam densities. Certain correlations are found between the nonlinear resonances and well-known instabilities for the KV equilibrium. It is also shown, in agreement with previous studies, that there exists a uniquely matched beam in the parameter regime of practical interest, i.e., [sigma][sub 0][lt]90[degree], where [sigma][sub 0] is the vacuum phase advance over one axial period of the focusing field. The nonlinear resonances and chaotic behavior in the nonequilibrium beam envelope oscillations may play an important role in mismatched ormore » multiple beam transport, including emittance growth and beam halo formation and evolution.« less

Second-order treatment of the influence of space charge and external fields on rotationally symmetric finite-emittance relativistic beams of charged particles

The present paper reports a relativistic treatment of space-charge effects on rotationally symmetric beams of charged particles under the influence of an external electromagnetic field. The general form of a stationary distribution describing a homogeneous, monokinetic beam is suggested, and the details of a suitable generalization of the Kapchinskij–Vladimirskij distribution are fully developed. It is shown that if second-order terms, which mainly affect the longitudinal motion, are retained, the given distribution actually provides a parabolic charge density throughout a beam cross section while the longitudinal current density remains uniform, as is usually assumed. The relativistic version of the envelope equation, including the magnetic field, is also calculated and its classical limit is compared with an available equation applicable to a beam under the influence of merely an electric field. Emittance-dependent discrepancies, related to the longitudinal effects of the space charge, are detected. The standard geometry of a rotationally symmetric Pierce gun is finally revised to allow for the addition of an external magnetic field.

Steady-State Transport of High-Current Beams in a Focused Channel

By following the orbits of several thousand particles in their self-consistent spacecharge fields, the evolution of an unstable Kapchinskij-Vladimirskij distribution is examined until saturation is reached. A distribution is formed which is stable on a scale of hundreds of quadrupole magnets.