Some words of Wisdom / Caution B. Holzer is allowed to spend 5 % of his (precious) time to act as consultant for the lattice design and beam optics of the ESS accumulator ring. C.F. Gauss

Input … until now … 1.) a highly sophisticated design drawing T. E.

Input … until now: 2.) Thesis from Jakob Jonnerby

… defining the geometry field map of a storage ring dipole magnet ρ α ds The angle swept out in one revolution must be 2π, so … for a full circle … using 64 dipoles all in all 2 per cell  dipole fill factor = 35 % The lattice is not highly packed -> space for additional elements

… scaling the focusing properties Aperture Requirements σ = 6*18mm = 108mm Pole Tip Field: B 0 = 0.73 T Larger Aperture requirements will lead to longer quadrupole lenses, -> l q =0.5m … for the 2π disp suppr. Assuming: … scaling the beta functions

… scaling the focusing properties Present work: Jonnerby / Wildner

And the Space Charge Effect Problem: strong defoc. effect limits the single bunch intensity Booster Parameters: E kin =50 MeV N p = 3*10 12 ΔQ ≈ 0.5 (50 MeV) Scaling to ESS Accumulator Ring … ΔQ ≈ 0.2 PSB tune diagramdefoc space charge force We are still in a space charge dominated regime.

Space Charge Effect in the RCS: (court. M. Fitterer) C 0 = 170m, N p = 3*10 12, E INJ =160MeV Difference between (vertical) emittance of the undisturbed lattice and the space charge dominated rms emittance determined from the tracked distribution … after 200 turns. ε increases quadratically with the intensity. Simulation results quadratic fit anlytical estimation

Lattice Optimisation for Highest Space Charge Performance FODO -> very flexible / robust / compact beam size for high energy machines relatively large variation of the beam size Doublet: -> more space in the lattice, optimum for focusing of highly non-spherical beams (mini-β in electron colliders) can lead to large changes in the transverse beam size and so in the non-linear space-charge kick. Triplet: -> very smooth variation of the beam size, and in particular small variations of the ratio between the two transverse sizes, (e.g. mini-β in p-colliders) almost uniformly distributed space-charge field

Dispersion Suppressor Schemes: The 2π Scheme For an arc structure with (modulo) 2π Phase advance the dispersion is oscillating around its Periodic values and vanishes automatically after 360 degrees. + very compact + no additional hardware -Very rigid, fixes the focusing of the structure -Might lead to sourious dispersion in the straights - Dispersion is overshooting 2 empty cells, 4*90 0 arc cells

Example: phase advance in the arc Φ C = 90° number of suppr. cells n = 2 If we require for the dipole field at the end of th arc We can add n suppressor cells so that the phase advance Adds up to a odd multiple of π Dispersion Suppressor Schemes: The missing / half-bend scheme: Adding celles at the end of the arc with half strong (or long) dipoles reduces (for special phase advances) the dispersion to zero. + very elegant + no additional hardware (some dipole modification) + no influence on the size of the dispersion -Needs space

Matching to the straights: Assuming that a FODO is an adequate solution for the arc, a smooth matching to the straights is needed. Even empty FODO cells in the straight have a quite different periodic optics that has to be matched (due to the missing 1/ρ 2 term). For “round” beams (i.e. β x =β y ) in a long straight, a triplet provides the best results in both transverse planes at the same time. Example: LHC high beta insertion (β*=200m), PS-Booster Lattice LHC 200m optics

Tune trim quadrupoles 90 0 cells -> q-pairs do the job Tune optimisation via the arc quadrupoles will spoil the optics & especially the 2π dispersion suppressor Ideal: keep the 90 0 arc structure untouched, Add two pairs of trim quadrupoles at the end or in a straight section for tune modifications In first order: no influence on the β-function

1.) Dependence of emittance growth on the lattice cell … and thus the variation of the beam size. MADX/PTC-Orbit, C 0 =157m, 160 MeV inj energy, N=2*10 12 Space Charge Tracking: Examples from the RCS Design Study, (court. M. Fitterer)

2π Disp suppr. Mismatch durch 1/ρ 2 … corr. via individual quads Lattices with a 2π dispersion suppression show a larger and more irregular dispersion in the arc than the regular lattices  large variation of the horizontal beam size & possible emittance growth for horizontal working points near the integer resonance, where near means values as large as 0.3. The dispersion beating, being the source of the phenomena, can be reduced with a dispersion suppressor scheme or additional individual quadrupoles 2.) Dispersion Suppressor Schemes Space Charge Tracking:

Time evolution of vertical rms emittance * for the 24 cell FODO lattices with 2 × 2 cells of straight section with only two quadrupole families (no ind), * with additional individual quadrupoles for beta-beating correction (ind) * with a half missing bend dispersion suppression scheme (disp) * for the regular24 cell FODO lattice (FODO 24 cells). For all simulations the initial distribution has been created taking the linear and second order dispersion into account. 3.) FOFO structure: Optics Mismatch Space Charge Tracking:

Time evolution of (vertical) rms emittance for symmetry 1, 2, 4 and 16 4.) Symmetry of the Ring Lattice Space Charge Tracking: Remember: PS-Booster = symmetry 16 RCS Symmetry broken by shortening a dipole magnet (at certain locations of the machine) & rematch of the optics Original optics after symmetry break optics re-established

The next steps: Finalise the linear lattice, include some flexibility via additional quadrupoles for beta-beat correction / dispersion suppressor do we need sextupoles ? … yes if ΔQ Δp/p ≈ ΔQ space charge Input from RF collimators H- injection requirements extraction elements Establish space charge tracking -> compare to other designs / compare cells decide how comfortable we have to be in the lattice design Impedance budget does it play a role ?  instabilities ? Tolerances (errors in alignment, field & multipoles) misalignment  orbit correction schemes influences beam loss studies

1.) Given the size of the space charge problem … a FODO should do 2.) provided that the lattice delivers a certain flexibility tune trim / match empty cells / arc cells 3.) spurious dispersion has to be avoided still … a 2π dispersion suppressor should work, if supported by individual quadrupoles 4.) a triplet to match to the straights is recommended 5.) keep the lattice as symmetric as possible 6.) tools are available and ready to use MADX for lattice design PTC-Orbit to do a fair space charge tracking Resume :

Lattice Optimisation for Highest Space Charge Performance

… what we need in the end: Dynamic aperture studies aperture requirements: r 0 = 12 * σ LHC: DA depends on beam size, nonlinear fields, Non-linear effects, storage time, lattice

dynamic aperture for Nb3Sn case: full error table (red) b3 reduced to 50% (green) b3 reduced to 25% (violett) b3 = 0 and to compare with: present LHC injection dyn aperture injection optics, minimum of 60 seeds for the experts: there is not much difference between b3=0 and perfect Nb 3 Sn magnets !! A scan in b 3 values has been performed and shows that values up to b 3 < 20 units are ok. Option 2.) determine tolerance limits for the b3 at injection... and try to improve the technical design