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• W55555140 abstract "With in view the design of the CLIC long transfer lines, we develop a formal approach for the optimisation of a straight FODO line. Optimum phase advance and cell length depending on beam parameters are derived for power consumption, overall cost and sensitivity to quadrupole misalignment. INTRODUCTION In the CLIC project, the drive beam and the main beam will travel from their central production sites towards the head of the main linacs over a straight distance L > 20 km [1]. With such long lines, a systematic approach must be used in order to optimize many parameters. This includes optics, beam dynamics, operational issues like reliability, diagnostics and safety with respect to beam losses. Finally, the cost of the equipment and the power consumption are major issues. In this paper, an attempt to optimize a long line is presented. We consider a line made of FODO cells and search for the minimum of a few figure of merits which depend on optics functions. We identified the sensitivity of the beam emittance to quadrupoles misalignment, the number of cells and the electrical power which supply the magnets. We briefly discuss further steps of optimization which require a study of parameters which are beyond pure optics studies. We finally discuss some choices of parameters which are partly specific to the CLIC transfer lines. FODO BASICS We consider a symmetric FODO cell of length Lc, which starts and ends in the middle of focussing quadrupoles (QF). A de-focussing quadrupole (QD) is located at the center of the cell. The gradient of the quadrupoles are equal in magnitude, i.e k = kQF = −kQD. We further consider a thin length approximation, with the focal length f = 1/kl. QF and QD are separated by a drift (D) of length L c/2. The cell contains no dipoles, because we consider a straight beam line. A derivation of several formulae can be found in [2] and a summary in [3], with in both cases slightly different notations than here. The length of the line is L.. The transfer matrix of the horizontal plane through the cell is obtained with M = Mqf/2MdMqdMdMqf/2 which expands to MFF = ( m11 m12 m21 m22 ) = ( 1− L 2 c 8f2 Lc(1 + Lc 4f ) − Lc 4f2 (1− Lc 4f ) 1− L 2 c 8f2 ) (1) The right side of Eq. (1) is then identified with the parametrization of Courant-Snyder for a periodic and symmetric structure (α = 0) MFF = ( cosμ β sinμ sinμ β cosμ ) (2) With Eqs. (1) and (2), cosμ = 1− L 2 c 8f2 and sin μ 2 = Lc 4f . (3) The β-functions reaches its maximum at the extremities of the cell. We write β = β and identify m12 in Eqs. (1) and (2) using Eq. (3). We get β = Lc 1 + sin μ2 sinμ . (4) Considering the transfer matrix from QD to QD, β = Lc 1− sin μ2 sinμ . (5) At this point, we note that Eq. 2 contains two free parameters, namely β and μ. While other choices might be considered, optimization shall preferably be made with using optic functions proper, here β and μ, instead of say, L c and f . We will see below that this choice is indeed good. It allows to express all the useful quantities and our figures of merit with functions which allow to separate the variables, i.e. A(β, μ) = F (β)G(μ). The focal length is expressed as a function of β and μ with Eqs. (3) and (4) : f = β sinμ 4 sin μ2 (1 + sin μ 2 ) (6) and the number of cells is N = L Lc = L β 1 + sin μ2 sinμ . (7) The chromaticity of a cell is Ccell = − tan μ2 /π [2, 3]. The chromaticity of the line is C = NCcell. With Eq. (7), we obtain C = − L 2πβ 1 + sin μ2 cos2 μ2 (8) OPTIMIZATION In order to preserve the emittance at best, the displacement Δ of the beam at the end of the line must be minimized. The main source of displacement comes from quadrupole displacement (’parasitic kicks’). If the momentum width ±δp of the beam is not negligible, a parasitic dispersion D is associated to Δ and smears the beam. A THPC017 Proceedings of EPAC08, Genoa, Italy 05 Beam Dynamics and Electromagnetic Fields 3014 D01 Beam Optics Lattices, Correction Schemes, Transport large chromaticity aggravates the smearing by filamenting the beam. The effect of parasitic kicks is computed below, while the chromaticity C(β, μ) is given by Eq. 8. The number of cells N is given by Eq. 7. We do not explore explicitely the cost optimization of the vacuum system, because it is quite case specific. If the need for beam aperture is small, the radius of the vacuum chamber may be fixed to a larger value for adequate conductance. In other cases, the radius is fixed by collective effect issues. We therefore limit our evaluation to different hypothesis for the choice of the radius, see below the section on magnet power. Parasitic Kicks The displacement Δ of the beam at the end of the line (β(L) = β), which results from random kicks δx ′ = (kl)δx = δx/f associated to a r.m.s. displacement δx of the quadrupoles is" @default.
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• W55555140 date "2008-08-27" @default.
• W55555140 modified "2023-09-27" @default.
• W55555140 title "Optimisation of a Beam Transfer FODO Line" @default.
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