Lattice Design in Particle Accelerators

Lattice Design in Particle Accelerators Bernhard Holzer, DESY Historical note:... Particle acceleration where lattice design is not needed 4 N ntz e i N( θ ) = * 4 ( 8πε ) r K sin 0 ( θ / ) uo P Rutherford Scattering, 906 Using radioactive particle sources: α-particles of some ev energy

Lattice design: design and optimisation of the principle elements of an accelerator... the lattice cells 95: Courant, Livingston, Snyder: Theory of strong focusing in particle beams

Lattice Design: how to build a storage ring High energy accelerators circular machines somewhere in the lattice we need a number of dipole magnets, that are bending the design orbit to a closed ring Geometry of the ring: centrifugal force = Lorentz force e * v * B = m v ρ mv e* B = = p / ρ ρ B * ρ = p / e p = momentum of the particle, ρ = curvature radius Bρ= beam rigidity Example: heavy ion storage ring TSR 8 dipole magnets of equal bending strength

Circular Orbit: defining the geometry α ds α = ρ dl ρ ds α = B* dl B* ρ field map of a storage ring dipole magnet The angle swept out in one revolution must be π, so for a full circle Bdl α = = π Bdl = π* B* ρ p q Nota bene: Δ 4 is usually required!! B B 0

Example HERA: 90 GeV Proton storage ring dipole magnets N = 46 l = 8.8m q = + e Bdl N * l * B = π p / q 9 π * 90 * 0 ev B 8 m 46 * 3* 0 * 8. 8m * s e 5. 5 Tesla

Focusing forces and particle trajectories: Equation of motion x'' + K* x = 0 K = k + / ρ K = k hor. plane vert. plane dipole magnet: ρ = quadrupole lens: k B p/ e focal length: f = k*l = g p / e Example: HERA Ring: Circumference: C 0 =6335 m Bending radius: ρ = 580 m Quadrupol Gradient: G= 0 T/m k = 33.64*0-3 /m /ρ =.97 *0-6 /m! For estimates in large accelerators the weak focusing term /ρ can in general be neglected

single particle trajectories y'' + K* y = 0 ρ y Solution for a focusing magnet y ' ys y K s K s K 0 () = 0 *cos( *) + *sin( *) y '( s) = y * K *sin( K * s) + y' *cos( K * s) 0 0 y y Or written more convenient in matrix form: = * y' y' s 0

atrices of lattice elements Hor. focusing Quadrupole agnet QF cos( K * l) sin( K * l) K = K sin( K * l) cos( K * l) Hor. defocusing Quadrupole agnet QD cosh( K * l) sinh( K * l) K = K sinh( K * l) cosh( K * l) Drift space Drift = s 0 = * * * *... lattice QF D QD D QF

Periodic Lattices In the case of periodic lattices the transfer matrix can be expressed as a function of a set of periodic parameters α,, γ cos μ+ αs sin μ βs sin μ () s = γs sin μ cos( μ) αs sin μ For stability of the motion in periodic lattice structures it is required that s+ L μ = s dt β () t µ = phase advance per period: trace( ) < In terms of these new periodic parameters the solution of the equation of motion is ys () = ε * β()*cos( s Φ() s δ) ε y'( s) = * sin( Φ( s) ) + cos( Φ( s) ) β { δ α δ }

The new parameters α,, can be transformed through the lattice via the matrix elements defined above. β C SC S β α = CC ' SC ' + S ' C SS ' * α γ C' S' C' S' γ S 0 Question: What does that mean????... and here starts the lattice design!!!

ost simple example: drift space C = C' S S' l = 0 particle coordinates x l x = * x' 0 x' l 0 x( l) = x + l* x ' x'( l) = x 0 0 0 ' transformation of twiss parameters: β l l β α = 0 l * α γ 0 0 γ l 0 β( s) = β l* α + l * γ 0 0 0 Stability...? trace ( ) = + = A periodic solution doesn t exist in a lattice built exclusively out of drift spaces.

Lattice Design: Arc: regular (periodic) magnet structure: bending magnets define the energy of the ring main focusing & tune control, chromaticity correction, multipoles for higher order corrections Straight sections: drift spaces for injection, dispersion suppressors, low beta insertions, RF cavities, etc...... and the high energy experiments if they cannot be avoided

The FoDo-Lattice A magnet structure consisting of focusing and defocusing quadrupole lenses in alternating order with nothing in between. (Nothing = elements that can be neglected on first sight: drift, bending magnets, RF structures... and especially experiments...) QF QD QF L Starting point for the calculation: in the middle of a focusing quadrupole Phase advance per cell µ = 45, calculate the twiss parameters for a periodic solution

βÿ wœ Š š œ GˆGm k jœ β Output of the optics program: Nr Type Length Strength β x α x φ x β z α z φ z m /m m /π m /π 0 IP 0,000 0,000,6 0,000 0,000 5,95 0,000 0,000 QFH 0,50-0,54,8,54 0,004 5,488-0,78 0,007 QD 3,5 0,54 5,488-0,78 0,070,8,54 0,066 3 QFH 6,00-0,54,6 0,000 0,5 5,95 0,000 0,5 4 IP 6,00 0,000,6 0,000 0,5 5,95 0,000 0,5 QX= 0,5 QZ= 0,5 0. 5* π = 45

Can we understand, what the optics code is doing? matrices QF cos( K* lq) sin( K* lq) K = Ksin( K* lq) cos( K* lq), Drift l = 0 d strength and length of the FoDo elements K = +/- 0.540 m - lq = 0.5 m ld =.5 m The matrix for the complete cell is obtained by multiplication of the element matrices = * * * * FoDo qfh ld qd ld qfh Putting the numbers in and multiplying out... FoDo = 0.707 8.06 0.06 0.707

The transfer matrix for period gives us all the information that we need!.) is the motion stable? trace( FoDo) =.45 cggy.) Phase advance per cell cosμ+ αsin μ βsin μ s () = γsin μ cos( μ) αsin μ cos( μ ) = * ( ) 0.707 trace = μ = arc cos( * trace( )) = 45 3.) hor β-function (, ) β = =.6 sin( μ) m 4.) hor α-function (,) cos( μ) α = = 0 sin( μ)

Can we do it a little bit easier? We can: the thin lens approximation atrix of a focusing quadrupole magnet: QF cos( K * l) sin( K * l) K = K sin( K * l) cos( K * l) If the focal length f is much larger than the length of the quadrupole magnet, f = >> l kl Q Q the transfer matrix can be aproximated using kl = const, l 0 q q 0 = f

m k G Œ š ˆ Ÿ ˆ k ld = L/, L f % = f Calculate the matrix for a half cell, starting in the middle of a foc. quadrupole: = * * halfcell QD / ld QF / halfcell 0 0 ld = * * 0 f f % % halfcell ld l D f = ld l D + f f % % % for the second half cell set f -f

FoDo in thin lens approximation atrix for the complete FoDo cella l D l + l D D l D % % f f = * ld l D ld l D + f f f f % % % % ld l D l ( ) D + % % f f = ld ld ld ( ) 3 f f f % % % Now we know, that the phase advance is related to the transfer matrix by 4l l cos μ = trace ( ) = *( ) = f f D D % % After some beer and with a little bit of trigonometric gymnastics x = x x = x cos( ) cos ( ) sin ( / ) sin ( )

we can calculate the phase advance as a function of the FoDo parameter cos( μ) sin ( μ/ ) = = L sin( μ / ) = ld / f = f % Cell % l f% D sin( μ / ) = L Cell 4 f Example: 45-degree Cell L Cell = l QF + l D + l QD +l D = 0.5m+.5m+0.5m+.5m = 6m /f = k*l Q = 0.5m*0.54 m - = 0.7 m - L Cell sin( μ / ) = 0.405 4 f μ 47.8 β.4m Remember: Exact calculation yields: μ = 45 β =.6m

z ˆ GˆGm k š œš œ Œ FoDo ld l D l ( ) D + f f = ld ld l D ( ) 3 f f f % % % % % Stability requires: trace( ) < SPS Lattice 4 trace( ) = l D < f % f > L cell 4 m Gš ˆ Œ Šˆ Œ Ž ˆšG G Œ ˆ ŽŒ G ˆ ˆG œˆ Œ G Œ ŠŒ Œ Ž HH

Transformation matrix in terms of the Twiss parameters Transformation of the coordinate vector (x,x ) in a lattice ρ xs () x0 = s, s ' x ( s) x 0 x General solution of the equation of motion s s xs () = ε* β()*cos(() s φ s + ϕ) { } x'( s) = ε * α( s)cos( φ( s) + ϕ) + sin( φ( s) + ϕ) β() s Transformation of the coordinate vector (x,x ) expressed as a function of the twiss parameters β (cosφ + α sin φ) ββ sinφ β = ( α α)cos φ ( + αα )sinφ β(cos φ α sin φ) ββ β

% % % { ˆ š Œ G ˆ Ÿ ˆ GˆGm k ŠŒ a halfcell ld l D f = ld l D + f f k L Compare to the twiss parameter form of β (cos φ + α sin φ) ββ sinφ β = ( α α)cos φ ( + αα )sinφ β (cos φ α sin φ) ββ β In the middle of a foc (defoc) quadrupole of the FoDo we allways have α = 0, and the half cell will lead us from β max to β min β μ cos ˆ μ ββsin C S ˆ β = = C' S' μ ˆ β μ sin cos ˆ ββ β

% % Solving for β max and β min and remembering that. μ ld L sin = = % f 4 f μ ˆ + sin S ' β + ld / f = = = C ld / f μ β sin S C ' = ˆ ββ = % f = sin l D μ ˆ β = β = μ (+ sin ) L sin μ μ ( sin ) L sin μ!! The maximum and minimum values of the β-function are solely determined by the phase advance and the length of the cell. Longer cells lead to larger β typical shape of a proton bunch in the HERA FoDo Cell ( Z, X, Y)

Beam dimension: Optimisation of the FoDo Phase advance: In both planes a gaussian particle distribution is assumed, given by the beam emittance ε and the β-function σ = εβ HERA beam size In general proton beams are round in the sense that ε x ε y So for highest aperture we have to minimise the β-function in both planes: r = ε xβ x + ε yβ y typical beam envelope, vacuum chamber and pole shape in a foc. Quadrupole lens in HERA

v š Ž Œ m k ˆšŒ ˆ ˆ ŠŒ r = ε xβ x + ε yβ y search for the phase advance µ that results in a minimum of the sum of the beta s μ μ (+ sin ) * L ( sin ) * L ˆ β + β = + sin μ sin μ ˆ β + β = L sin μ d ( L ) = 0 dμ sin μ 0 βges( μ) 0 0 8 6 4 0 0 30 60 90 0 50 80 0 μ 80 L *cos μ = 0 μ = 90 sin μ

Electrons are different electron beams are usually flat, ε y - 0 % ε x optimise only β hor μ L( + sin ) d ( ˆ d β) = = 0 μ 76 dμ dμ sin μ 30 30 4 β max( μ ) β min ( μ ) 8 red curve: β max blue curve: as a function of the phase advance µ 0 6 0 36.8 7.6 08.4 44. 80 μ 80

Orbit distortions in a periodic lattice field error of a dipole/distorted quadrupole ds Bds δ ( mrad) = = ρ p/ e the particle will follow a new closed trajectory, the distorted orbit: () s x() s = β * β() s cos( φ() s φ() s πq) ds sin πq ρ( s) % % % o % * the orbit amplitude will be large if the β function at the location of the kick is large β(š~ ) indicates the sensitivity of the beam here orbit correctors should be placed in the lattice * the orbit amplitude will be large at places where in the lattice β(s) is large here beam position monitors should be installed

Orbit Correction and Beam Instrumentation in a storage ring Elsa ring, Bonn *

Resumé:.) Dipole strength: p Bds = N * B0 * leff = π q l eff effective magnet length, N number of magnets.) Stability condition: Trace( ) < for periodic structures within the lattice / at least for the transfer matrix of the complete circular machine 3.) Transfer matrix for periodic cell () s = cos μ+ α( s)sin μ β( s)sin μ γ()sin s μ cos( μ) α()sin s μ α,β,γ depend on the position s in the ring, µ (phase advance) is independent of s 4.) Thin lens approximation: QF 0 = f Q f Q = k l Q Q focal length of the quadrupole magnet f Q = /(k Q l Q) >> l Q

5.) Tune (rough estimate): Tune = phase advance in units of π s+ L μ = s dt β () t μ ds πr Q: = N* = * o * R π π = β( s) π β β R, β average radius and β-function Q R β 6.) Phase advance per FoDo cell (thin lens approx) μ sin = L Cell 4 fq L Cell length of the complete FoDo cell, f Q focal length of the quadrupole, µ phase advance per cell 7.) Stability in a FoDo cell f > (thin lens approx) 4 Q L Cell 8.) Beta functions in a FoDo cell (thin lens approx) ˆ β = μ (+ sin ) sin μ L Cell β = μ ( sin ) sin μ L Cell L Cell length of the complete FoDo cell, µ phase advance per cell

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

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

INSERTIONS INSERTIONS