Introduction to Transverse Beam Dynamics I Linear Beam Optics Bernhard Holzer, DESY-HERA
Largest storage ring: The Solar System astronomical unit: average distance earth-sun AE 5 * 6 km Distance Pluto-Sun 4 AE AE
Luminosity Run of a typical storage ring: HERA Storage Ring: Protons accelerated and stored for hours distance of particles travelling at about v c L - km... several times Sun - Pluto and back guide the particles on a well defined orbit ( design orbit ) focus the particles to keep each single particle trajectory within the vacuum chamber of the storage ring, i.e. close to the design orbit.
Transverse Beam Dynamics:.) Introduction and Basic Ideas... in the end and after all it should be a kind of circular machine need transverse deflecting force r r r r Lorentz force F q*( E + v B) typical velocity in high energy machines: v 8 c 3* m s old greek dictum of wisdom: if you are clever, you use magnetic fields in an accelerator wherever it is possible. But remember: magn. fields act allways perpendicular to the velocity of the particle only bending forces, no beam acceleration
The ideal circular orbit ẑ θ s circular coordinate system condition for circular orbit: Lorentz force F e* v* B L centrifugal force F Zentr γ m v p B* e γ mv e* v* B
I.) The Magnetic Guide Field B α μ n I h ds field map of a storage ring dipole magnet p B* e radius of curvature, normalised bending strength B [T ].998 * p [GeV / c] Eample HERA: B 5. Tesla p 9 GeV/c /.778* -3 /m ~ 6m π ~ 3.7 km
Quadrupole Magnets: required: focusing forces to keep trajectories in vicinity of the ideal orbit linear increasing Lorentz force linear increasing magnetic field Bz g * B g * z normalised quadrupole field: gradient of a quadrupole magnet: g μ ni r k g p / e what about the vertical plane: Mawell: r r r r E B j + t B B z z
II.) The equation of motion: Linear approimation: * ideal particle design orbit * any other particle coordinates, z small quantities,z << magnetic guide field: only linear terms in & z of B have to be taken into account Taylor Epansion of the B field: B z db z d B z eg ( ) B z + + + 3 d! d 3! d +... normalise to momentum p/e B B ( ) p / e B B g * p / e! eg p / e 3! eg p / e + + + +...
The Equation of Motion: B ) p / e + k * + m + n! 3! ( 3 +... only terms linear in, z taken into account dipole fields quadrupole fields Separate Function Machines: Split the magnets and optimise them according to their job: bending, focusing etc Eample: heavy ion storage ring TSR * man sieht nur dipole und quads linear
Equation of Motion: ẑ Consider local segment of a particle trajectory... and remember the old days: (Goldstein page 7) θ s z radial acceleration: a r d dθ d Ideal orbit: const, dt dt dt Force: dθ F m mω dt general trajectory: + / F mv F m d dt ( + ) mv + e B z v
v B e mv dt d m F z + + ) ( dt d dt d ) ( + z s ẑ ) ( + remember: mm, m develop for small eb v mv dt d m z ) ( as const Taylor Epansion... ) ( *! ) ( ) ( *! ) ( ) ( ) ( + + + f f f f
guide field in linear appro. B Bz B + z d mv m ( ) ev B + dt B z : m d dt v e v B ( ) m + e v m g independent variable: t s d dt d ds * ds dt d dt d d ds d d * * dt ds dt ds ds ds dt ds dt d dt * v + d ds * dv ds * v v v e v B v ( ) + m e v m g : v
e B ( ) + mv e g mv m v p B + p / e + + + g p / e k + ( k) normalize to momentum of particle B p / e g p / e k * Equation for the vertical motion: no dipoles in general k k quadrupole field changes sign z + k z
Remarks: * + ( k) there seems to be a focusing even without a quadrupole gradient weak focusing of dipole magnets k * even without quadrupoles there is a retriving force (i.e. focusing) in the bending plane of the dipole magnets in large machines it is weak. (!) Mass spectrometer: particles are separated according to their energy and focused due to the / effect of the dipole
III.) Solution of Trajectory Equations Define hor. plane: vert. Plane: K K k k y + K * y Differential Equation of harmonic oscillator with spring constant K Ansatz: ( s) a cos( ω s) + a sin( ω ) s general solution: linear combination of two independent solutions ( s s) a ω sin( ω s) + a ω cos( ω ) ( s) a ω cos( ω s) a ω sin( ω s) ω ( ) ω K s general solution: ( s) a cos( K s) + a sin( K s)
determine a, a by boundary conditions: s () (),, a a K Hor. Focusing Quadrupole K > : ( s) cos( K s) + sin( K s) K ( s) K sin( K s) + cos( K s) For convenience epressed in matri formalism: s s s s s M foc * s M foc cos( K s) sin( K s K sin( ) cos( ) K K s K s
hor. defocusing quadrupole: K * s s s Remember from school: f ( s) cosh( s), f ( s) sinh( s) Ansatz: s) a cosh( ω s) + a sinh( ω ) ( s M defoc cosh Kl sinh Kl K sinh cosh K K l K l drift space: K M drift l! with the assumptions made, the motion in the horizontal and vertical planes are independent... the particle motion in & z is uncoupled
Transformation through a system of lattice elements combine the single element solutions by multiplication of the matrices M M M M M M total QF * D * QD * Bend * D*... ( s,s)* M s s focusing lens dipole magnet defocusing lens court. K. Wille (s) typical values in a strong foc. machine: mm, mrad s
IV.) Orbit & Tune: Tune: number of oscillations per turn 3.9 3.97 Relevant for beam stability: non integer part HERA revolution frequency: 47.3 khz.9*47.3 khz 3.8 khz
Question: what will happen, if the particle performs a second turn?... or a third one or... turns s
The Particle Ensemble: HERA Bunch Pattern: ma τ rev N b * 8 e 3 6 * Cb * s N b * * * 9 8 s. 6 * Cb N 73. * b müssen wir wirklich einen Sack voll Flöhen hüten? do we really have to calculate single particle trajectories????
9th century: Ludwig van Beethoven: Mondschein Sonate Sonate Nr. 4 in cis-moll (op. 7/II, 8)
Astronomer Hill: differential equation for motions with periodic focusing properties Hill s equation Eample: particle motion with periodic coefficient equation of motion: () s k()() s s restoring force const, we epect a kind of quasi harmonic k(s) depending on the position s oscillation: amplitude & phase will depend k(s+l) k(s), periodic function on the position s in the ring.
V.) The Beta Function General solution of Hill s equation: (i) s () ε β()cos( s ψ() s + φ) ε, Φ integration constants determined by initial conditions β(s) periodic function given by focusing properties of the lattice quadrupoles β( s+ L) β( s) Inserting (i) into the equation of motion ψ () s s ds β() s Ψ(s) phase advance of the oscillation between point and s in the lattice. For one complete revolution: number of oscillations per turn Tune Q y ds π β( s) o
Beam Emittance and Phase Space Ellipse general solution of Hill equation () s ( ) ε * β( s) *cos( ψ( s) + φ) ε () ( s) * ( s)*cos( ( s) + ) + sin( ( s) + ) β() s { α ψ φ ψ φ } from () we get s () cos( ψ( s) + φ) ε * β( s) Insert into () and solve for ε α() s β () s + α( s) γ () s β() s ε γ()* s () s+ α()() ss () s+ β() s () s * ε is a constant of the motion it is independent of s * parametric representation of an ellipse in the space * shape and orientation of ellipse are given by α, β, γ
Beam Emittance and Phase Space Ellipse ε γ()* s () s+ α()() ss () s+ β() s () s α ε γ Liouville: in reasonable storage rings area in phase space is constant. εγ α ε β A π*εconst εβ (s) s ε beam emittance woozilycity of the particle ensemble, intrinsic beam parameter, cannot be changed by the foc. properties. Scientifiquely spoken: area covered in transverse, phase space and it is constant!!!
Particle Tracking in a Storage Ring Calculate, for each linear accelerator element according to matri formalism plot, as a function of s t 4 6 8 t y t 4 6 8 t
and now the ellipse: note for each turn, at a given position s and plot in the phase space diagram 5 t 5 5 5 y t