Introduction to Transverse Beam Optics. II.) Twiss Parameters & Lattice Design

Transcription

1 Introduction to Transverse Beam Optics Bernhard Holzer, CERN II.) Twiss Parameters & Lattice esign ( Z X Y) Bunch in a storage ring

2 Introduction to Transverse Beam Optics Bernhard Holzer, CERN... don't worry: it's still the "ideal world" Historical note: ε & β... Particle acceleration whithout emittance or beta unction 4 N ntz e i N( * 4 ( 8 ) r K sin ( / ) N(θ) Rutherord Scattering, 9 Using radioactive particle sources: α-particles o some ev energy θ

3 Reminder o Part I Equation o otion: Solution o Trajectory Equations K K K k k hor. plane: vert. Plane: s * s drit l oc cos( K l) K sin( K l) sin( K l) K cos( K l) deoc cosh( K l K sinh( K l sinh( K l K cosh( K l

4 Transormation through a system o lattice elements combine the single element solutions by multiplication o the matrices * * * * total QF Q Bend *... ( s,s)* s s ocusing lens dipole magnet deocusing lens court. K. Wille (s) typical values in a strong oc. machine: mm, mrad s

5 Question: what will happen, i the particle perorms a second turn?... or a third one or... turns s

6 Astronomer Hill: dierential equation or motions with periodic ocusing properties Hill s equation Eample: particle motion with periodic coeicient equation o motion: ( s) k( s) ( s) restoring orce const, k(s) = depending on the position s k(s+l) = k(s), periodic unction we epect a kind o quasi harmonic oscillation: amplitude & phase will depend on the position s in the ring.

7 The Beta Function General solution o Hill s equation: (i) ( s) ( s) cos( ( s) ) ε, Φ = integration constants determined by initial conditions β(s) periodic unction given by ocusing properties o the lattice quadrupoles ( s L) ( s) Inserting (i) into the equation o motion () s s ds () s Ψ(s) = phase advance o the oscillation between point and s in the lattice. For one complete revolution: number o oscillations per turn Tune Q y ds o ( s)

8 8.) The Beam Emittance General solution o Hill s equation: ( s) ( s) cos ( s) ( s) ( s)cos ( s) sin ( s) () s β(s) = periodic unction given by ocusing properties o the lattice ( s L) ( s) ε = constant, determined by initial conditions o the particle ensemble. ( s)* ( s) ( s) ( s) ( s) ( s) ( s) Liouville: in reasonable storage rings area in phase space is constant. A = π*ε=const ε beam emittance = woozilycity o the particle ensemble, intrinsic beam parameter, cannot be changed by the oc. properties. Scientiiquely spoken: area covered in transverse, phase space and it is constant!!!

9 Phase Space Ellipse particel trajectory: ( s) ( s) cos ( s) ma. Amplitude: ˆ ( s) at that position? put ˆ ( s) into ( s)* ( s) ( s) ( s) ( s) ( s) ( s) and solve or / * * A high β-unction means a large beam size and a small beam divergence. et vice versa!!! In the middle o a quadrupole β is maimum, α = zero! and the ellipse is lat

10 Phase Space Ellipse ( s) ( s) ( s) ( s) ( s) ( s) ( s) ( s) ( s) () s ( s) () s solve or, and determine d ˆ via: d ˆ ˆ shape and orientation o the phase space ellipse depend on the Twiss parameters β α γ

11 Emittance o the Particle Ensemble: ( s) ( s) cos( ( s) ) ˆ ( s) ( s) Gauß Particle istribution: N ( ) e e particle at distance σ rom centre 68.3 % o all beam particles single particle trajectories, N per bunch LHC: * 5* m*8m. 3 mm aperture requirements: r = * σ

12 Emittance o the Particle Ensemble: particle bunch ( Z X Y) Eample: HERA beam parameters in the arc ( ) 8 m 7* 9 rad m ( ).75 mm

13 9.) Transer atri yes we had the topic already general solution o Hill s equation ( s) ( s) cos ( s) ( s) ( s)cos ( s) sin ( s) () s remember the trigonometrical gymnastics: sin(a + b) = etc s ( ) s cos s cos sin s sin ( s) cos cos sin sin sin cos cos sin s s s s s s s starting at point s() = s, where we put Ψ() = cos, sin ( ) inserting above

14 ( s) s cos sin sin s s s s s ( s) cos ( )sin cos sin s s s s s s s s which can be epressed... or convenience... in matri orm s s cos sin sin ( )cos ( )sin s s s s s s s s s s cos sin s s s * we can calculate the single particle trajectories between two locations in the ring, i we know the α β γ at these positions. * and nothing but the α β γ at these positions. *! * Äquivalenz der atrizen

15 .) Periodic Lattices transer matri or particle trajectories as a unction o the lattice parameters s cos sin sin ( )cos ( )sin s s s s s s s s s s cos sin s s s This rather ormidable looking matri simpliies considerably i we consider one complete turn elta Electron Storage Ring ( s) cos turn s sin s sin turn turn cos s turn sin s turn sin turn turn s s L ds ( s) ψ turn = phase advance per period ds Q * Tune: Phase advance per turn in units o π ( s)

16 Stability Criterion: Question: what will happen, i we do not make too many mistakes and your particle perorms one complete turn? atri or turn: cos sin sin turn s turn s turn sin cos sin s turn turn s turn cos sin atri or N turns: J N cos J sin cos N J sin N N The motion or N turns remains bounded, i the elements o N remain bounded real cos Trace()

17 stability criterion. proo or the disbelieving collegues!! atri or turn: cos sin sin turn s turn s turn sin cos sin s turn turn s turn cos sin atri or turns: I J I * cos J *sin * I *cos J *sin I * cos cos IJ *cos sin JI *sin cos J sin sin now I I I *J J * I * * I * J J * I J * I I * cos( ) J *sin( ) I *cos( ) J *sin( )

18 .) Transormation o α, β, γ consider two positions in the storage ring: s, s s s where QF Q B rit QF... C C S S since ε = const (Liouville):, y Beta unction in a storage ring

19 epress, as a unction o,. ẑ ρ s s θ s... remember W = CS -SC = s s S S C C S S C C inserting into ε sort via, and compare the coeicients to get... ( C C ) ( S S )( C C ) ( S S )

20 ( s) C SC S ( s) CC ( SC S C) SS ( s) C S C S in matri notation: s C SC S CC SC CS SS C S C S!.) this epression is important.) given the twiss parameters α, β, γ at any point in the lattice we can transorm them and calculate their values at any other point in the ring. 3.) the transer matri is given by the ocusing properties o the lattice elements, the elements o are just those that we used to calculate single particle trajectories. 4.) go back to point.)

21 .) Lattice esign: how to build a storage ring High energy accelerators circular machines somewhere in the lattice we need a number o dipole magnets, that are bending the design orbit to a closed ring Geometry o the ring: centriugal orce = Lorentz orce e* v* B mv mv e* B p / B* p/ e p = momentum o the particle, ρ = curvature radius Bρ= beam rigidity Eample: heavy ion storage ring TSR 8 dipole magnets o equal bending strength

22 Circular Orbit: deining the geometry α ρ ds dl ds B * dl B * ield map o a storage ring dipole magnet The angle swept out in one revolution must be π, so Bdl B* Bdl * p q or a ull circle Nota bene: B B 4 is usually required!!

23 Eample LHC: 7 GeV Proton storage ring dipole magnets N = 3 l = 5 m q = + e B dl N l B p / e 9 7 ev B 8 m 3 5 m 3 s e 8.3 Tesla

24 Focusing orces single particle trajectories ρ y'' K * y K k / hor. plane y K k vert. plane dipole magnet quadrupole magnet k B p / q g p / q Eample: HERA Ring: Bending radius: ρ = 58 m Quadrupol Gradient: g = T/m k = 33.64* -3 /m /ρ =.97 * -6 /m For estimates in large accelerators the weak ocusing term /ρ can in general be neglected Solution or a ocusing magnet y ' y s y K s K s K ( ) *cos( * ) *sin( * ) y '( s) y * K *sin( K * s) y ' *cos( K * s)

25 The Twiss parameters α, β, γ can be transormed through the lattice via the matri elements deined above. S C SC S CC ' SC ' S ' C SS ' * C ' S ' C ' S ' Question: What does that mean????

26 ost simple eample: drit space C C' S S' l particle coordinates l * ' ' l ( l) l * ' '( l) ' transormation o twiss parameters: l l l * l ( s) l * l * Stability...? trace( ) A periodic solution doesn t eist in a lattice built eclusively out o drit spaces.

27 3.) The Foo-Lattice A magnet structure consisting o ocusing and deocusing quadrupole lenses in alternating order with nothing in between. (Nothing = elements that can be neglected on irst sight: drit, bending magnets, RF structures... and especially eperiments...) Starting point or the calculation: in the middle o a ocusing quadrupole Phase advance per cell μ = 45, calculate the twiss parameters or a periodic solution

28 Periodic solution o a Foo Cell QF Q QF y L Output o the optics program: Nr Type Length Strength β α ψ β y α y ψ y m /m m /π m /π IP,,,6,, 5,95,, QFH,5 -,54,8,54,4 5,488 -,78,7 Q 3,5,54 5,488 -,78,7,8,54,66 3 QFH 6, -,54,6,,5 5,95,,5 4 IP 6,,,6,,5 5,95,,5 Q X =,5 Q Y =,5. 5* 45

29 Can we understand, what the optics code is doing? matrices QF cos( K * lq) sin( K * lq) K K sin( K * l ) cos( K * l ) q q, rit l d strength and length o the Foo elements K = +/-.54 m - lq =.5 m ld =.5 m The matri or the complete cell is obtained by multiplication o the element matrices Foo qh * ld * qd * ld * qh Putting the numbers in and multiplying out... Foo

30 The transer matri or period gives us all the inormation that we need!.) is the motion stable? trace( Foo ).45 <.) Phase advance per cell () s cos sin sin sin cos( ) sin cos( ) * ( ).77 trace arc cos( * trace( )) 45 3.) hor β-unction 4.) hor α-unction (,) sin( ).6 m (,) cos( ) sin( )

31 Can we do it a little bit easier? We can: the thin lens approimation atri o a ocusing quadrupole magnet: QF cos( K * l) sin( K * l) K K sin( K * l) cos( K * l) I the ocal length is much larger than the length o the quadrupole magnet, kl Q l Q the transer matri can be aproimated using kl q const, l q

32 Foo in thin lens approimation l l L/, L Calculate the matri or a hal cell, starting in the middle o a oc. quadrupole: * * halcell Q / l QF / halcell l * * halcell l l l l or the second hal cell set -

33 Foo in thin lens approimation atri or the complete Foo cell: l l l l * l l l l l l l ( ) l l l ( ) 3 Now we know, that the phase advance is related to the transer matri by 4l l cos trace ( ) *( ) Ater some beer and with a little bit o trigonometric gymnastics cos( ) cos ( ) sin ( / ) sin ( )

34 we can calculate the phase advance as a unction o the Foo parameter cos( ) sin ( / ) sin( / ) l / L Cell l sin( / ) L Cell 4 Eample: 45-degree Cell L Cell = l QF + l + l Q +l =.5m+.5m+.5m+.5m = 6m / = k*l Q =.5m*.54 m - =.7 m - L Cell sin( / ) m Remember: Eact calculation yields: 45.6m

35 Stability in a Foo structure Foo l l l ( ) l l l ( ) 3 Stability requires: Trace() SPS Lattice 4ld Trace( ) ~ L cell 4 For stability the ocal length has to be larger than a quarter o the cell length!!

36 Transormation atri in Terms o the Twiss parameters Transormation o the coordinate vector (, ) in a lattice ρ ( s) '( s) s, s ' General solution o the equation o motion s s ( s) * ( s) *cos( ( s) ) ' ( s) * ( s) ( s)cos( ( s) ) sin( ( s) ) Transormation o the coordinate vector (, ) epressed as a unction o the twiss parameters ( (cos )cos ( sin ) )sin (cos sin sin )

37 Transer matri or hal a Foo cell: halcell l l ~ ~ l l ~ l L Compare to the twiss parameter orm o ( (cos )cos ( sin ) )sin (cos sin sin ) In the middle o a oc (deoc) quadrupole o the Foo we allways have α =, and the hal cell will lead us rom β ma to β min C S C' S' cos sin sin cos

38 4.) Scaling o the Twiss Paramerters Solving or β ma and β min and remembering that. l L sin 4 ˆ sin S ' l / C l / sin ˆ ( sin ) L sin! S C ' ˆ l sin ( sin ) L sin! The maimum and minimum values o the β-unction are solely determined by the phase advance and the length o the cell. Longer cells lead to larger β typical shape o a proton bunch in the HERA Foo Cell

39 Conclusion: * the arc o a storage ring is usually built out o a periodic sequence o single magnet elements eg. Foo sections * a irst guess o the main parameters o the beam in the arc is obtained by the settings o the quadrupole lenses in this section * we can get an estimate o the beam parameters using a selection o rules o thumb Usually the real beam properties will not dier too much rom these estimates and we will have a nice storage ring and a beautiull beam and everybody is happy around.

40 And then someone comes and spoils it all by saying something stupid like installing a tiny little piece o detector in our machine

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