First-order covariant treatment of the influence of space charge and external fields on finite-emittance relativistic beams of charged particles in generalized curvilinear coordinates

1989 ◽  
Vol 67 (7) ◽  
pp. 678-685
Author(s):  
Giulio Bosi
Keyword(s):  
Distribution Function ◽  
Space Charge ◽  
Charged Particles ◽  
Curvilinear Coordinates ◽  
Coordinate Space ◽  
Transverse Motion ◽  
Metric Tensor ◽  
First Order ◽  
Dimensional Phase Space ◽  
Principal Axes

A first-order covariant treatment of space-charge effects on relativistic beams of charged particles under the influence of applied fields of arbitrary forms is presented. The beams and fields are assumed to fulfill no particular symmetry conditions, and curvilinear orthogonal coordinates are used. Relative to a given distribution function, a suitable choice of both metric tensor and optical axis allows for the selection of appropriate coordinates, reducing the overall transverse motion of a charged particle to a pair of independent motions along the principal axes of a tensor that is related to both geometry and field. Detailed formulas applying to a microcanonical distribution in an eight-dimensional phase space are worked out, and the envelope equations for a monokinetic beam are carried out explicitly. The distribution function is shown to separate into the product of a stationary distribution in transverse-coordinate space and momentum space by using factors carrying the relativistic corrections. As a result, the envelope equations are formally identical to the customary equations describing the envelope of a nonrelativistic beam under the influence of merely transverse forces. A few numerical applications are finally presented; the behavior of beams moving through a magnetic prism and through a cycloidal analyzer is displayed graphically.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

Electromagnetic fluctuations from energetic particles in plasmas

1988 ◽  
Vol 40 (3) ◽  
pp. 407-417 ◽  
Author(s):  
Cheng Chu ◽  
J. L. Sperling
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Charged Particles ◽  
Velocity Distribution Function ◽  
Black Body ◽  
Black Body Radiation ◽  
Specific Model ◽  
Magnetized Plasma ◽  
Plasma Power ◽  
Correlation Techniques

Electromagnetic fluctuations, induced by energetic charged particles, are calculated using correlation techniques for a uniform magnetized plasma. Power emission in the ion-cyclotron range of frequencies (ICRF) is calculated for a specific model of velocity distribution function. The emissive spectra are distinct from that of the black-body radiation and have features that are consistent with experimental observation.

Guidage d'un faisceau de particules chargées à travers une chaîne de solénoïdes cylindriques et toroïdaux alternés

1985 ◽  
Vol 63 (8) ◽  
pp. 1098-1104
Author(s):  
Giulio Bosi ◽  
Alain Durand
Keyword(s):  
Distribution Function ◽  
Space Charge ◽  
Axial Symmetry ◽  
Fringe Field ◽  
Suitable Choice ◽  
Space Charge Effects ◽  
Electron Cooler ◽  
Charge Effects ◽  
Angular Extent ◽  
Transverse Temperature

The present paper is devoted to analyzing the magnetic fringe-field and space-charge effects on a beam of nonrelativistic electrons crossing a sequence of cylindrical and toroidal solenoids, as may be found in an electron cooler. The investigation is mainly aimed at searching for suitable conditions that ensure conservation of the axial symmetry of a given beam throughout the whole system. The need for a vertical steering field, in addition to the longitudinal one provided by each toroidal coil, is emphasized and its form determined. A suitable choice of the angular extent of a torus is shown to suppress axis vibrations at the entrance of the following sector. Finally, the transverse temperature of a cylindrical beam is calculated after specifying the appropriate distribution function.

Interpreting Proper Orthogonal Modes in Vibrations

1997 ◽  
Author(s):  
B. F. Feeny
Keyword(s):  
Distributed System ◽  
Normal Modes ◽  
Single Mode ◽  
Dominant Mode ◽  
Coordinate Space ◽  
Nonlinear Simulation ◽  
Principal Axes ◽  
Linear And Nonlinear ◽  
Proper Orthogonal Modes ◽  
Proper Orthogonal

Abstract We investigate the interpretation of proper orthogonal modes (POMs) of displacements in both linear and nonlinear vibrations. The POMs in undamped linear symmetric systems can represent linear natural modes if the mass distribution is known. This is appoximately true in a distributed system if it is discretized uniformly. If a single mode dominates, the dominant POM approximates the dominant mode. This is also true if a distributed system is discretized arbitrarily. Generally, the POMs represent the principal axes of inertia of the data in the coordinate space. For synchronous nonlinear normal modes, the dominant POM represents a best fit of the nonlinear modal curve. Linear and nonlinear simulation examples are presented.

Transfer Equation in General Curvilinear Coordinates

2011 ◽  
Vol 20 (2) ◽  
Author(s):  
J. Freimanis
Keyword(s):  
Differential Operator ◽  
Radiative Transfer ◽  
Coordinate System ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Dimensional Space ◽  
Polarization Plane ◽  
Curvilinear Coordinates ◽  
Reference Plane ◽  
Metric Tensor

AbstractThe differential operator of the monochromatic polarized radiative transfer equation is studied in case of statistically homogeneous turbid medium in Euclidean three-dimensional space, with arbitrary curvilinear coordinate system defined in it. An apparent rotation of the polarization plane along the light ray with respect to the chosen polarization reference plane generally takes place, due to purely geometric reasons. Using methods of tensor analysis, analytic expressions for the differential operator of the transfer equation depending on the components of the metric tensor and their derivatives are found. Considerable simplifications take place if the coordinate system is orthogonal. As an example, the differential operator of the vector radiative transfer equation in both elliptical conical coordinate system and oblate spheroidal coordinate system is written down. Nonstandard parameterization of standard elliptical conical coordinate system is proposed.

Improved Approach to the Streamline Curvature Method in Turbomachinery

1987 ◽  
Vol 109 (3) ◽  
pp. 213-217 ◽  
Author(s):  
S. Abdallah ◽  
R. E. Henderson
Keyword(s):  
Flow Field ◽  
Velocity Gradient ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
First Order ◽  
Streamline Curvature ◽  
Stators And Rotors ◽  
Streamline Curvature Method

Quasi three dimensional blade-to-blade solutions for stators and rotors of turbomachines are obtained using the Streamline Curvature Method (SLCM). The first-order velocity gradient equation of the SLCM, traditionally solved for the velocity field, is reformulated as a second-order elliptic differential equation and employed in tracing the streamtubes throughout the flow field. The equation of continuity is then used to calculate the velocity. The present method has the following advantages. First, it preserves the ellipticity of the flow field in the solution of the second-order velocity gradient equation. Second, it eliminates the need for curve fitting and strong smoothing under-relaxation in the classical SLCM. Third, the prediction of the stagnation streamlines is a straightforward matter which does not complicate the present procedure. Finally, body-fitted curvilinear coordinates (streamlines and orthogonals or quasi-orthogonals) are naturally generated in the method. Numerical solutions are obtained for inviscid incompressible flow in rotating and non-rotating passages and the results are compared with experimental data.

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