First‐order treatment of combined effect of space‐charge and external fields on plane‐symmetric finite‐emittance beams of charged particles

1976 ◽  
Vol 47 (12) ◽  
pp. 5292-5296 ◽  
Author(s):  
G. Bosi
Keyword(s):  
Space Charge ◽  
Charged Particles ◽  
Combined Effect ◽  
External Fields ◽  
Plane Symmetric ◽  
First Order

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.

scholarly journals Approximate Noether symmetries from geodetic Lagrangian for plane symmetric spacetimes

2015 ◽  
Vol 12 (10) ◽  
pp. 1550124 ◽  
Author(s):  
Farhad Ali ◽  
Tooba Feroze
Keyword(s):  
Conformal Factor ◽  
Noether Symmetries ◽  
Plane Symmetric ◽  
First Order ◽  
Symmetric Spacetimes ◽  
General Plane ◽  
Symmetric Spacetime ◽  
Conformal Plane ◽  
General Time

Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time-conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed,5–6 where they obtained approximate Noether symmetries from Lagrangian for a particular plane symmetric static spacetime. In the present paper, the most general plane symmetric static spacetime is considered and perturbed it by introducing a general time-conformal factor eϵf(t), where ϵ is very small which causes the perturbation in the spacetime. Taking the perturbation up to the first-order, we find all Lagrangian for plane symmetric spacetimes for which approximate Noether symmetries exist.

scholarly journals Gauge dependence in QED amplitudes in expanding de Sitter space

2016 ◽  
Vol 31 (10) ◽  
pp. 1650050 ◽  
Author(s):  
Nistor Nicolaevici
Keyword(s):  
External Field ◽  
Gauge Invariance ◽  
De Sitter Space ◽  
External Fields ◽  
De Sitter ◽  
First Order ◽  
Gauge Dependence ◽  
Invariance Violation ◽  
The Face ◽  
First Order Transition

We consider first-order transition amplitudes in external fields in QED in the expanding de Sitter space and point out that they are gauge dependent quantities. We examine the gauge variations of the amplitudes assuming a decoupling of the interaction at large times, which allows to conclude that the source of the problem lies in the fact that the frequencies of the modes in the infinite future become independent of the comoving momenta. We show that a possibility to assure the gauge invariance of the external field amplitudes is to restrict to potentials which vanish sufficiently fast at infinite times, and briefly discuss a number of options in the face of the possible gauge invariance violation in the full interacting theory.

Correction of the deterministic part of space–charge interaction in momentum microscopy of charged particles

2015 ◽  
Vol 159 ◽  
pp. 488-496 ◽  
Author(s):  
G. Schönhense ◽  
K. Medjanik ◽  
C. Tusche ◽  
M. de Loos ◽  
B. van der Geer ◽  
...  
Keyword(s):  
Space Charge ◽  
Charged Particles ◽  
Charge Interaction
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