Fermilab http://www.fnal.gov Chromaticity Correction for a Muon - - PowerPoint PPT Presentation

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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

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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 (Uturn = q2β3γ4/3ǫ0R)

The small lifetime (τ = γ 2.2 µs) requires

• large number of muons be produced
• and 6D cooled and quickly accelerated.

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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. Nb × Nµ 1 × 20 · 1011 10 × 1 · 1011 ∆p/p 0.1% 1% ǫN 25 µm 2 µm In order to reach a luminosity of ≃ 1034 cm2 s−1, very small β∗ is required. β∗=1 cm in both planes has been chosen as a compromise between luminosity and feasibility.

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Design Issues

β(s) = β∗ + s2/β∗

500 1000 1500 2000 1 2 3 4 5 6 7 8

β [m] s [m] β∗= 0.005 m β∗= 0.010 m β∗= 0.50 m

Large β at strong quadrupoles:

• large sensitivity to misalignments

and field errors

• large chromatic effects limit the

momentum acceptance and require strong correction sextupoles

• large non-linearities limit the

Dynamic Aperture Hourglass effect limits σℓ ≤ 1 cm. For achieving such short bunches with a reasonable voltage |αp| 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..

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Constraints for present design

Design constraints β∗

x, β∗ y (ǫx = ǫy)

10 mm free space around IP ± 6 m |αp| ≤ 1 × 10−4 ˆ g ≤ 260 Tm−1 ˆ B 10 T (8 T in the IR) Moreover: ℓB ≤ 6 m, ℓQ ≤ 3 m. Energy for this design: 750 GeV per beam. Solutions for 1.5 TeV per beam are under study.

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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 Bz ≡ 1 β(0)

z

∂βz ∂δ Az ≡ ∂αz ∂δ − α(0)

z Bz

(z = x/y)

• bey the equations

dBz ds = −2Az dµz ds and dAz ds = 2Bz dµz ds − β(0)

z k

k ≡ ±(KQuad − DxKSext) (+/ − for x/y) Second order chromaticity ξ(2)

z

= 1 8π C ds

• −kBz ± 2KSext

dDx dδ

• β(0)

z

− ξ(1)

z

ξ(1)

z

≡ linear chromaticity

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“Classical” approach

• Chromatic IR beta-wave is corrected with sextupole families in the arcs

(Dx = 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-

• rthogonality 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.

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“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

• n this scheme had large momentum range and DA.

ˆ βy=900 km, very strong sextupoles and their large number of families (22) make this (very instructive) design likely un-feasible as it is.

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Local chromatic correction

dBz ds = −2Az dµz ds and dAz ds = 2Bz dµz ds − β(0)

z k

• Az becomes non zero when the low-β quadrupoles are encountered, but

as long as the phase advance does not change, Bz 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 Dx=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.

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IR optimization for chromatic correction

• Non-symmetric IR design: ˆ

βy ≫ ˆ βx (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.

50 100 150 200 250 300 50 100 150 200 √β[m], Dx[cm] s [m] √βx √βy Dx

• 2000
• 1000

1000 2000 3000 4000 5000 6000 50 100 150 200 Mad-X chromatic functions s [m] S1 S2 S3 S4 Wx Wy dDx/dδ [cm]

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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 Dx=0 regions. MAD-8 STATIC detuning coefficients dQ1/dE1 0.60× 104 m−1 dQ1/dE2 0.20× 102 m−1 dQ2/dE2

• 0.50× 104 m−1

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Arc cell

The collider includes two identical IPs (twofold symmetric optics). The IR having a large positive contribution to αp, arcs must give a negative contribution so to get |αp| ≤ 10−4. Proposed cell

• Almost
• rthogonal

chromaticity correction (one family/plane).

• 300 deg phase advance/cell: can-

cellation over 6 cells.

• αp and its dependence on momen-

tum

a controlled through middle

quadrupole and sextupole.

a 1 L

• ds[ 1

ρ ∂Dx ∂δp + 1 2D′ x]

Collider circumference (including matching sections): 2727 m.

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Lattice performance

Fractional tunes just above half-integer chosen for orbit and β-beat at low-β quads considerations.

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

• 0.01 -0.005

0.005 0.01

dp/p

qx qy

Tunes (fractional part) vs. dp/p

• 9e-05
• 8e-05
• 7e-05
• 6e-05
• 5e-05
• 4e-05
• 3e-05
• 2e-05
• 1e-05
• 0.01 -0.005

0.005 0.01

dp/p

αp

αp vs. dp/p

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Dynamic Aperture determined by tracking particles over 1000 turns (time needed for the beam current to decay by a factor ≃ 2).

900 800 700 600 500 400 300 200 100 900 800 700 600 500 400 300 200 100 γ Ay

2 (µm)

γ Ax

2 (µm)

stable lost

MAD8 DA (on energy, w/o synchrotron oscillations) A ≡ oscillation amplitude

• #σ=
• γA2

ǫN

• DA is 5.7σ for ǫN=25 µm

(3σ needed)

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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

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(A. Netepenko) ǫN=10 µm

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Fringe fields effect DA in terms of amplitudes ...or starting coordinates at IP

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Phase space trajectories (x0=0)

• y0=6 µm
• y0=12 µm
• y0=18 µm
• y0=24 µm

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Phase space trajectories (x0=6 µm)

• y0=6 µm
• y0=12 µm
• y0=18 µm
• y0=24 µm

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Summary

• Owing to the large chromaticity, the IR optics of a high luminosity Muon

Collider must be designed having non-linear corrections in mind.

• In the scheme here presented the larger vertical chromatic wave is cor-

rected “in loco”, while a pseudo −I insertion provide convenient locations for the correction of the horizontal chromatic wave.

• The energy range, within the machine is stable, is about ±1.2%. The Dy-

namic Aperture for the nominal optics and on-energy particles is large with-

• ut having to resort to an error prone non-interleaved sextupole scheme

in the arcs.

• Preliminary studies of multipole errors, fringe fields and beam-beam effects

indicate a tolerable reduction of the DA.