Sector magnets or transverse electromagnetic fields in cylindrical coordinates

Zolkin, Timofey

doi:10.1103/physrevaccelbeams.20.043501

Cited by 5 publications

(5 citation statements)

References 6 publications

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“…But here in equation ( 25), we have constructed a multipole of degree = 3 from six charges of equal magnitude. Actually in the study of cylindrical beams, the related functions r m sin m θ cos mφ are referred to as 2m-poles [9], but these functions grow rather than decrease with r; the poles are considered to be at r = ∞.…”

Section: Planar Multipoles M =mentioning

confidence: 99%

Point charge representations of multipoles

Majic

2022

Eur. J. Phys.

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We generalize the notion of a dipole made of two point charges placed infinitesimally close together to multipoles of all degrees. The charge distributions are derived first by differentiating the delta function for the monopole in the same way that differentiating the potential $1/r$ generates the spherical multipoles $r^{-\ell-1}P_\ell^m(\cos\theta)e^{im\phi}$. This method surprisingly generates messy distributions for spherical multipoles with $m\geq3$. So instead we use a ``bracelet" of $2m$ charges for the planar spherical multipoles of order $m=\ell$, and stack these to construct multipoles for all $m,\ell$. We show that the spherical multipoles of degree $\ell$ can be made of $\ell+1$ charges for $m=0$ and $2m(\ell-m+1)$ charges for $1\leq m\leq \ell$. A similar construction is found for the Cartesian multipoles.

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Section: Planar Multipoles M =mentioning

confidence: 99%

Point charge representations of multipoles

Majic

2022

Eur. J. Phys.

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“…A further option is to develop the field using cylindrical coordinates centred on the "vertical revolution axis," thus using more or less the centre of the ring as reference point (e.g. [65]- [67]). It is expected that these multipoles are correlated, therefore regularisation could be required for a more robust set of coefficients (e.g.…”

Section: On Curvaturementioning

confidence: 99%

Magnets for Low-Emittance Storage Rings an Overview

Schnizer

Bengtsson

2022

IEEE Trans. Appl. Supercond.

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Low emittance machines require lattices with many magnets of short length. Furthermore, successful lattices require strong gradients with strong dipole fields. These lattices are popular today, as MAX IV has demonstrated that a diffraction limited synchrotron light source can be based on such a lattice. This paper gives an overview of the various magnets used by the various diffraction limited light sources -available, under construction or designed. It compares the different magnet design and technology used and presents the magnet parameters in a consistent fashion.

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“…( 83) deals with the actual momentum p of a non-reference particle, we have to find a simple approximation of the fraction p 0 p . We therefore assume a small momentum deviation ∆p = p − p 0 and approximate the reciprocal value 1 p by its first order Taylor expansion…”

Section: Equations Of Motionmentioning

confidence: 99%

Transverse Linear Beam Dynamics

Hillert¹

2021

Preprint

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The subject of this introductory course is transverse dynamics of charged particle beams in linear approximation. Starting with a discussion of the most important types of magnets and defining their multipole strengths, the linearized equations of motion of charged particles in static magnetic fields are derived using an orthogonal reference frame following the design orbit. Analytical solutions are determined for linear elements of a typical beam transfer line (drift, dipole and quadrupole magnets), and stepwise combined by introducing the matrix formalism in which each element's contribution is represented by a single transfer matrix. Application of this formalism allows to calculate single particle's trajectories in linear approximation. After introducing the beam emittance as the area occupied by a particle beam in phase space, a linear treatment of transverse beam dynamics based on appropriately defined optical functions is introduced. The formalism is applied to the concepts of both weak and strong focusing, in particular discussing the properties of the widely-used FODO cell. Specific characteristics of transverse beam dynamics in periodic systems like circular accelerators are studied in detail, emphazising the effects of linear field errors on orbit stability and introducing the phenomena of optical resonances. Finally, the dynamics of off-momentum particles is presented, introducing dispersion functions and explaining effects like chromaticity.

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Sector magnets or transverse electromagnetic fields in cylindrical coordinates

Cited by 5 publications

References 6 publications

Point charge representations of multipoles

Point charge representations of multipoles

Magnets for Low-Emittance Storage Rings an Overview

Transverse Linear Beam Dynamics

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