Chapter SIC CCLRTOR HSICS The guiding and focuing of a charged particle bea in a circular accelerator rel on a erie of agnetic eleent, eparated b field-free drift pace, that for the accelerator lattice. The deign of the lattice i one of the firt tak for the accelerator deigner. One norall tart with a iplified tructure, containing onl ideal agnetic dipole and quadrupole. Thi baic tructure i referred to a the linear lattice and once it ha been deterined the character of the achine i ore or le fixed. In a eparated function nchrotron, dipole agnet bend the particle onto circular trajectorie and quadrupole agnet are needed for the focuing of particle with all deviation copared to the ideal trajector. The equilibriu orbit can be defined a the orbit of a particle with the deign oentu p that cloe upon itelf after one turn and i table. The otion of a particle in an accelerator i then convenientl decribed b the deviation of it trajector with repect to thi equilibriu orbit.. Coordinate te Generall in a nchrotron, there i bending in onl one plane referred to a the horiontal plane which contain the equilibriu orbit. For a decription of particle otion a righthanded curvilinear coordinate te (x,, ) following the equilibriu orbit i ued. The aiuthal coordinate i directed along the tangent of the orbit. The radial tranvere direction x i defined a noral to the orbit in the horiontal plane and the vertical tranvere direction i given b the cro product of the unit vector xˆ and ŝ. The vector ŝ and ẑ then define the vertical plane. The local radiu of curvature U and the bending angle /U are defined a poitive for anticlockwie rotation when viewed fro poitive. The choice of the coordinate te i hown in Figure.. particle trajector x equilibriu orbit ŝ ẑ xˆ U horiontal plane local centre of curvature Figure.. Curvilinear coordinate te. 7
Chapter aic accelerator phic. ccelerator agnet For thi tud, the agnetic eleent are conidered a purel tranvere, twodienional field in the curvilinear coordinate te that follow the equilibriu orbit. It i aued that each eleent can be decopoed into a haronic erie that extend onl a far a the extupole coponent and that each eleent can be repreented b a block of field that i dicontinuou in the axial direction. Thi i the o-called hard-edge approxiation. For a final deign, end-field correction will have to be applied to iprove the accurac of the hard-edged odel. The equilibriu orbit correpond to a particle with the noinal oentu and tarting condition. article of the ae oentu, but with all patial deviation will ocillate about thi orbit. The equilibriu orbit i varioul known a the central orbit, the reference orbit and the cloed orbit. In the hard-edged odel, thi orbit i a erie of traight line connecting circular arc of cclotron otion (ee Figure.). v avv nev /U U Figure.. Cclotron otion for a particle of a av with charge ne. The force experienced b a charged particle, oving in a agnetic field i given b the Loren equation a dp F q v u, (.) dt where v i the particle velocit, q it charge and the agnetic field. Since the Loren force alwa act perpendicular to the particle velocit, onl the direction of oveent change while the agnitude of the velocit i contant. The noinal trajector in a circular accelerator i defined b the ain bending agnet which provide a dipolar field, ˆ. (.) quating the agnetic deflection to the centrifugal force (calar value give, nev av U v Ÿ U avv, (.3) ne where i the atoic a nuber, av i the average relativitic a per nucleon, n i the charge tate of the particle, e i the eleentar charge, and U i the radiu of the cclotron otion. The reluctance of the particle to be deviated i characteried b U which i known a the agnetic rigidit. 8
aic accelerator phic Chapter The application of a ign convention for the agnetic rigidit i uuall avoided and an engineering forula i quoted in which the oentu i alwa taken a poitive. Magnetic rigidit: [ T ] U [ ] pav [GeV/c] (.4).9979458 n where p av i the average oentu per nucleon. Fro (.4) it can be een that the higher the particle oentu, the higher the agnetic field needed to keep the particle on the noinal trajector. Therefore, the agnetic field in an accelerator have to be continuoul adjuted, according to the actual particle oentu during acceleration. In order to avoid the oentu dependenc when characteriing agnetic eleent in an accelerator, the field are noralied w.r.t. the agnetic rigidit (particle oentu). For a dipole agnet the noralied bending trength h, i given b U h. (.5) The abolute field level required to keep a particle with a certain oentu on the noinal trajector i then ipl found b ultipling (.5) with the agnetic rigidit (.4). The equilibriu orbit in a nchrotron i defined b the ain dipole agnet. In general particle trajectorie will deviate lightl fro thi ideal deign orbit. To aintain tabilit about the deign orbit, focuing force are needed. Thee are provided b quadrupole agnet. quadrupole agnet ha four pole with a hperbolic contour and the agnetic field in the current-free region of the agnet gap can be derived fro a calar potential [], ( x, G x, (.6) ) ) where G i known a the quadrupole gradient, being defined a: G d dx. (.7) The equipotential line are the hperbolae x = cont and the field line are perpendicular to the. The horiontal and vertical agnetic field in a quadrupole agnet are linear in the deviation fro the agnet centre: x G and G x. (.8) The tranvere force acting on a particle with a deviation (x,) fro the equilibriu orbit are obtained with (.) a and F F x qv ( x, ) qvgx (.9) qv ( x, qvg. (.) x ) 9
Chapter aic accelerator phic epending on the ign of G, the force will be focuing in the horiontal direction and defocuing in the vertical direction or vice vera. quadrupole that focue horiontall (G < ), i called a focuing quadrupole and a quadrupole that defocue horiontall (G > ), i called a defocuing quadrupole. alternating focuing and defocuing quadrupole an overall focuing effect can be obtained in both the horiontal and the vertical plane (alternating gradient focuing). n iportant propert of a quadrupole agnet i that the horiontal force depend onl on the horiontal and not on the vertical poition of the particle trajector. Siilarl, the vertical coponent of the Lorent force depend onl on the vertical poition. The conequence i that in a linear achine, containing onl dipolar and quadrupolar field, the horiontal and vertical otion are copletel uncoupled. In analog to the noralied bending trength of a dipole agnet (.5), the quadrupole gradient can be noralied w.r.t. the agnetic rigidit, thu defining the noralied quadrupole trength, G k. (.) U.3 Tranvere optic When analing the tranvere otion in an accelerator, it i practical to ue the ditance eaured along the trajector, intead of the tie a independent variable, d dt d d d dt d v. (.) d It i uual to conider onl dipole and quadrupole field (linear lattice), which lead to uncoupled otion in the two tranvere plane. The equation of otion are obtained fro (.) while retaining onl firt order ter in the tranvere coordinate [3]. The horiontal and vertical otion can be repreented b one iple expreion valid for both plane, where, for the horiontal otion, and, for the vertical otion, c ( K( ( (.3) x { x and K( K ( h( k( (.4) { and K( K ( k(. (.5) For the hard-edged odel, the focuing function K ( i piecewie contant along the trajector and for a circular achine, it ha the periodicit of the lattice, K ( L) K (, (.6) where the accelerator i copoed of N identical ection or cell, with C = NL and C being the circuference of the achine.
aic accelerator phic Chapter The otion of ono-energetic particle about their equilibriu orbit i known a the betatron otion and it i uual to paraeterie the otion o that it peudo-haronic behaviour i brought into evidence. The firt tage i to expre the otion in a for developed b Courant and Snder [4], ª dv º etatron otion: ( co, (.7) «³ V where ( repreent either tranvere coordinate a a function of the ditance along the equilibriu orbit, and are contant depending on the tarting condition and ( i the betatron aplitude function (dienion length). The phae ( of the peudo ocillation i given a» ¼ ³ dv. (.8) V To coplete thi decription, the derivative of ( i added in the relation, and d d (.9) J. (.) The expreion (.7)-(.) all depend on a knowledge of (. lthough the analtic olution for ( i ore coplicated than the original otion equation, it i reaonabl ea to evaluate thi function nuericall and to tabulate it for an lattice. The paraeter (, ( and J( are collectivel known a the Courant and Snder paraeter, or ore uuall the Twi paraeter. The above paraeteriation i now o coonplace that it i the tarting point for nearl all lattice deign. The peudo-haronic otion can be further tranfored into a iple haronic otion b returning to equation (.7), introducing and differentiating, to obtain, ( co (.) co in. (.) c The phae ter can be extracted and ued to define new coordinate () and c() that are known a noralied coordinate, co (.3) c in c. (.4)
Chapter aic accelerator phic It i ueful to repreent real-pace coordinate b lower cae bol and noralied coordinate b upper cae bol (x o X, etc.). Noralied coordinate ue the phae advance a independent variable and real-pace coordinate ue the ditance. The tranforation between the two te are convenientl expreed in atrix for a, / Real to noralied: d / d / d / d (.5) Noralied to real: d /d / / d /d. (.6) The eliination of the phae advance fro equation (.3) and (.4) ield an invariant of the otion, c J c c ( contant). (.7) quation (.7) i in fact the equation of an ellipe in (, c) phae pace. The contant equal the (area/s) of the ellipe decribed b the betatron otion in either the noralied (, c) or the real (, c) phae pace (ee Figure.3). When referring to a ingle particle, thi area i oetie called the ingle-particle eittance. When referring to a bea it i known a the eittance. The ituation with a bea i coplicated b the definition ued for deciding the liiting ellipe that define the area. Thi a be related to a nuber of tandard deviation of the bea ditribution, or the overall axiu. It i ueful to note that the eittance in real and noralied phae pace are equal and to reexpre (.7) a, S ittance: ˆ S ˆ œ ˆ S and ˆ S, (.8) where ŷ and ˆ are the axiu excurion in real and noralied phae pace that define the bea-ie at the oberver poition..5 c [rad].5 c [ / ] ( J /S) - ( /S ) ( +c ) [] [ -.5.5 -.5.5 / ] -.5 ( /S) = ŷ -.5 ( /S) =ˆ Figure.3. hae-pace ellipe in real and noralied phae pace.
aic accelerator phic Chapter It hould be noted that during acceleration the eittance decreae and therefore the phae pace denit change. Thi i not againt Liouville theore, but due to the definition of the tranvere phae pace. The c-coordinate, decribing the divergence of a particle with repect to the noinal orbit, c d d d v dt v v, (.9) i not a generalied oentu a defined in claical echanic. long a the oentu i contant, the difference i jut a caling factor but during acceleration thi factor change and o doe the eittance. The phical reaon i the increae of the longitudinal oentu during acceleration, wherea the tranvere oentu i contant. Therefore the divergence v /v decreae and the bea hrink, which i known a adiabatic daping. Noraliing the eittance w.r.t. the particle oentu give a contant of the otion, the o called noralied eittance,, (.3) n, relj rel where rel and J rel are the relativitic paraeter..4 Tranfer atrix forali nother baic expreion that i needed for the preent tud i the general tranfer atrix. Thi can be derived b expanding (.) into two ter, co in (.3) where and are new contant. ifferentiation of (.3) with repect to give, co in co in. (.3) c The contant and can be replaced uing the initial condition at =, chooing = and c, (.33) to give the general tranfer atrix fro poition to poition. The phae advance fro to i written a ', General tranfer atrix : co ' M o in ' in ' > in ' co ' @ co ' in '. (.34) 3
Chapter aic accelerator phic The phae-pace coordinate at poition are then given b M ( o ). (.35) c c When equation (.35) i applied to a full turn in a ring, the input condition equal the output condition ( = =, = =, ' = SQ), o that co SQ in SQ in SQ M turn, (.36) J in SQ co SQ in SQ where Q i known a the betatron tune and i equal to the nuber of tranvere ocillation the particle ake during one turn in the achine. xpreion (.36) decribe the evolution of the phae-pace coordinate of a particle at a certain poition in the achine. plot of the coordinate for a large nuber of turn give a phae-pace trajector. In a linear achine phae-pace trajectorie are alwa of elliptical hape; the orientation of the ellipe at an poition in the achine i deterined b the local Twi paraeter and the bea ie i found with the eittance according to Figure.3. The general tranfer atrix for noralied coordinate i ipl a x rotation atrix decribing a clockwie rotation b the phae advance ' between poition and, co ' in ' M N ( o ). (.37) in ' co ' The ingle-turn atrix for noralied coordinate i cosq in SQ M N, turn. (.38) insq cosq The x tranfer atrix forali i coonl known a Twi-atrix forali..5 Off-oentu particle and diperion function iple extenion of (.7) allow the otion of particle with different oenta to be decribed, ( ª dv º Gp ( co ³» ( (.39) ( p ««etatron otion» ¼ iperion otion where ( i known a the diperion function and Gp/p = (p part. - p )/p i the relative oentu deviation of the particle. The diperion i created b the oentu dependenc of the bending radiu in dipole agnet and appear therefore onl in the plane of bending (generall the horiontal plane). The equilibriu orbit for an off-oentu particle i to firt order diplaced fro the central orbit b the product of the diperion function and the oentu deviation, Gp Gp Q.O and c Q.O c. (.4) p p 4
aic accelerator phic Chapter n analtic derivation of the diperion function i given in [4], but it i again coon practice to rel on lattice progra to uppl nuerical liting of ( and it derivative with ditance, a for the betatron aplitude function. The oentu deviation Gp/p i treated a a quai-variable and particle are tranferred through the lattice with 3u3 atrice of the for, G p 3 c 3 c / p Gp / p (.4) where,, and are the coefficient of the general tranfer atrix (.34) and 3, 3 are additional diperion coefficient (ee Section 3.). The diperion vector in the for (, c, Gp/p=) alo propagate through the lattice according to (.4). The noralied for of the diperion function ( n, cn), iilar to (.5), will alo be frequentl ued, n d n /d / / d / d. (.4) It hould be entioned that (.4) i trictl applicable to onl all value of Gp/p and to linear lattice. For trajectorie with ore than a few per il oentu deviation, or for trajectorie that pa through non-linear agnetic lene, it i adviable to perfor a nuerical tracking if the orbit poition i required to a high preciion. xactl thi ituation arie when calculating the poition and angle of the eparatrice for a reonant extraction [5] and it i ueful to be able to incorporate the ore exact tracking ethod into the general tranfer atrix a, M o > c @ Q, Q, Q, G > c c Q, Q, Q, @/ Gp / p / p / p (.43) where Q and cq are the poition and angle of the tracked off-oentu equilibriu orbit. ore detailed and coplete introduction to accelerator phic can be found in Reference [3,6,7]. 5