Fermilab http://www.fnal.gov Chromaticity Correction for a Muon Collider Optics Y. Alexahin, E. Gianfelice, V. Kapin (PAC11, March 28, 2011) Contents: • Introduction • Cromaticity correction • Proposed MC Optics • Lattice performance presented by Eliana GIANFELICE eliana@fnal.gov 1/20 < > � � ⊖ i ≪ ≫ ? P �
Introduction Muon Collider is a promising candidate for the next energy frontier machine. First proposed by Budker (1967), the idea of a MC has been re-launched by recent progress on new ideas for small emittance muon beams. The re-newed interest is testified by the large number of papers presented at this conference. Muons are • point-like as e ± → the whole beam energy is carried by the interacting particles. • but 207 times heavier → no radiation, in practice ( U turn = q 2 β 3 γ 4 / 3 ǫ 0 R ) The small lifetime ( τ = γ 2.2 µ s) requires • large number of muons be produced • and 6D cooled and quickly accelerated. 2/20 < > � � ⊖ i ≪ ≫ ? P �
Machine parameters vary depending on the available number of muons and their emittance. Expectations for two possible cooling scenarios: high transv. emitt. low transv. emitt. 1 × 20 · 10 11 10 × 1 · 10 11 Nb × N µ ∆ p/p 0.1% 1% ǫ N 25 µ m 2 µ m In order to reach a luminosity of ≃ 10 34 cm 2 s − 1 , very small β ∗ is required. β ∗ =1 cm in both planes has been chosen as a compromise between luminosity and feasibility. 3/20 < > � � ⊖ i ≪ ≫ ? P �
Design Issues β ( s ) = β ∗ + s 2 /β ∗ Large β at strong quadrupoles: • large sensitivity to misalignments 2000 and field errors β ∗ = 0.010 m 1500 • large chromatic effects limit the β [m] β ∗ = 0.005 m 1000 momentum acceptance and require 500 β ∗ = 0.50 m strong correction sextupoles 0 0 1 2 3 4 5 6 7 8 • large non-linearities limit the s [m] Dynamic Aperture Hourglass effect limits σ ℓ ≤ 1 cm. For achieving such short bunches with a reasonable voltage | α p | must be as small as possible ( ≤ 1 × 10 − 4 ). Ring circumference should be as small as possible, luminosity being inversely proportional to the collider length. Assuming we are able to accelerate enough muons, the design of the collider ring itself is not trivial either.. 4/20 < > � � ⊖ i ≪ ≫ ? P �
Constraints for present design Design constraints β ∗ x , β ∗ y ( ǫ x = ǫ y ) 10 mm free space around IP ± 6 m ≤ 1 × 10 − 4 | α p | ≤ 260 Tm − 1 g ˆ ˆ B 10 T (8 T in the IR) Moreover: ℓ B ≤ 6 m, ℓ Q ≤ 3 m. Energy for this design: 750 GeV per beam. Solutions for 1.5 TeV per beam are under study. 5/20 < > � � ⊖ i ≪ ≫ ? P �
Chromatic correction To obtain large momentum acceptance it is necessary to correct the depen- dance on momentum of β -functions and tunes. Montague chromatic functions 1 ∂β z A z ≡ ∂α z ∂δ − α (0) B z ≡ z B z ( z = x/y ) β (0) ∂δ z obey the equations dB z dµ z dA z dµ z ds − β (0) = − 2 A z and = 2 B z z k ds ds ds k ≡ ± ( K Quad − D x K Sext ) (+ / − for x/y ) Second order chromaticity � C = 1 dD x ξ (2) β (0) − ξ (1) � � ds − kB z ± 2 K Sext z z z 8 π dδ 0 ξ (1) ≡ linear chromaticity z 6/20 < > � � ⊖ i ≪ ≫ ? P �
“Classical” approach • Chromatic IR beta-wave is corrected with sextupole families in the arcs ( D x � = 0) • Different families are used to correct linear and second order chromatic- ity and possibly the first order dependence of α ( s ) on momentum; non- orthogonality of such corrections result in an increase of the needed sex- tupole strenght. • Constraints on the phase advance between sextupoles of the same family allow to make the lowest order driving terms of the 3th order resonances vanish. This scheme works well in successfully operating colliders as Tevatron and LHC, has been tried for some earlier MC versions but led to extremely small values of the momentum stability range. 7/20 < > � � ⊖ i ≪ ≫ ? P �
“Special sections” approach Owing to the large chromaticity, the IR optics of a high luminosity Muon Collider must be designed having non-linear corrections in mind. The use of “special sections”, with large beta functions and dispersion, next to the low- β region has been suggested: • after the first sextupole located at a knot of the IR chromatic wave, a pseudo − I section is inserted between it and a “twin”sextupole compen- sating the non-linear kick. Two such sections are needed for correcting in both planes. In practice such kind of schemes may be prone to focusing errors. The optics proposed by K. Oide (1996) for a β ∗ =3 mm 2 × 2 TeV MC based on this scheme had large momentum range and DA. ˆ β y =900 km, very strong sextupoles and their large number of families (22) make this (very instructive) design likely un-feasible as it is. 8/20 < > � � ⊖ i ≪ ≫ ? P �
Local chromatic correction dB z dµ z dA z dµ z ds − β (0) = − 2 A z and = 2 B z z k ds ds ds • A z becomes non zero when the low- β quadrupoles are encountered, but as long as the phase advance does not change, B z is unchanged. • At the low- β quadrupoles, the phase advance changes slowly and there is a possibility of correcting the chromatic perturbation before β ( δ ) and µ ( δ ) start differing from the unperturbed values. “Local” correction with sextupoles is possible if the IR dispersion is non- vanishing. If D x = D ′ x ( IP ) =0 the insertion of relatively strong bending mag- nets in the IR region is necessary. They can help avoiding neutrinos hot spots. 9/20 < > � � ⊖ i ≪ ≫ ? P �
IR optimization for chromatic correction • Non-symmetric IR design: ˆ β y ≫ ˆ β x (as in Oide design). • Local chromatic correction for the larger vertical chromatic wave with a single sextupole (S1). • A simultaneous local correction in the horizontal plane being not possible, − I section inserted for accommodating a pair of horizontal sextupoles (S2 and S4). • A 4th sextupole (S3) corrects 2th order dispersion. 6000 W x √β x 300 W y √β y Mad-X chromatic functions 5000 D x dD x /d δ [cm] 250 4000 √β [m], D x [cm] 200 3000 150 2000 1000 100 0 50 S1 S2 S3 S4 -1000 0 -2000 0 50 100 150 200 0 50 100 150 200 s [m] s [m] 10/20 < > � � ⊖ i ≪ ≫ ? P �
Additional octupoles for higher order vertical chromaticity correction are in- serted in the − I section. Detuning with amplitude is compensated by oc- tupoles in the D x =0 regions. MAD-8 STATIC detuning coefficients 0.60 × 10 4 m − 1 dQ 1 /dE 1 0.20 × 10 2 m − 1 dQ 1 /dE 2 -0.50 × 10 4 m − 1 dQ 2 /dE 2 11/20 < > � � ⊖ i ≪ ≫ ? P �
Arc cell The collider includes two identical IPs (twofold symmetric optics). The IR having a large positive contribution to α p , arcs must give a negative contribution so to get | α p | ≤ 10 − 4 . Proposed cell • Almost orthogonal chromaticity correction (one family/plane). • 300 deg phase advance/cell: can- cellation over 6 cells. • α p and its dependence on momen- a controlled through middle tum quadrupole and sextupole. ∂D x 1 ds [ 1 ∂δ p + 1 2 D ′ � x ] a L ρ Collider circumference (including matching sections): 2727 m. 12/20 < > � � ⊖ i ≪ ≫ ? P �
Lattice performance Fractional tunes just above half-integer chosen for orbit and β -beat at low- β quads considerations. 0.9 -1e-05 q x α p 0.85 -2e-05 q y 0.8 -3e-05 -4e-05 0.75 0.7 -5e-05 -6e-05 0.65 0.6 -7e-05 -8e-05 0.55 0.5 -9e-05 -0.01 -0.005 0 0.005 0.01 -0.01 -0.005 0 0.005 0.01 dp/p dp/p Tunes (fractional part) vs. dp/p α p vs. dp/p 13/20 < > � � ⊖ i ≪ ≫ ? P �
Dynamic Aperture determined by tracking particles over 1000 turns (time needed for the beam current to decay by a factor ≃ 2). 900 stable 800 lost 700 600 2 ( µ m) 500 A ≡ oscillation amplitude 400 γ A y � γA 2 � � 300 # σ = ǫ N 200 100 DA is 5.7 σ for ǫ N =25 µ m 0 (3 σ needed) 0 100 200 300 400 500 600 700 800 900 2 ( µ m) γ A x MAD8 DA (on energy, w/o synchrotron oscillations) 14/20 < > � � ⊖ i ≪ ≫ ? P �
We have started looking to • DA with energy offset and synchrotron oscillations • multipole errors in IR magnes • beam-beam effects on chromatic correction and luminosity • fringe field effects • phase space trajectories 15/20 < > � � ⊖ i ≪ ≫ ? P �
(A. Netepenko) ǫ N =10 µ m 16/20 < > � � ⊖ i ≪ ≫ ? P �
Fringe fields effect DA in terms of amplitudes ...or starting coordinates at IP 17/20 < > � � ⊖ i ≪ ≫ ? P �
Phase space trajectories ( x 0 =0) • y 0 =6 µ m • y 0 =12 µ m • y 0 =18 µ m • y 0 =24 µ m 18/20 < > � � ⊖ i ≪ ≫ ? P �
Phase space trajectories ( x 0 =6 µ m) • y 0 =6 µ m • y 0 =12 µ m • y 0 =18 µ m • y 0 =24 µ m 19/20 < > � � ⊖ i ≪ ≫ ? P �
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