Muon Collider Lattice Design FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY f Yuri...

Muon Collider Lattice Design

FERMI NATIONAL ACCELERATOR LABORATORY

US DEPARTMENT OF ENERGY

f

Yuri Alexahin, Eliana Gianfelice-Wendt

(Accelerator Physics Center)

Workshop on Dynalic Aperture for Ultimate Storage Rings, IU, Bloomington November 11-12, 2010

MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

The Road to High Luminosity 2

hC

Pfh

Nnf rep

b

~2

1

4

2

0L

The recipe:

Pack as many muons per bunch as the beam-beam effect allows (in practice this means 1 bunch / beam)

Make beams round to maximize the beam-beam limit

Invent new chromatic correction scheme to reduce *

Do not leave free spaces to reduce C (also necessary to spread neutrino radiation)

4

Nr

C – collider circumference (limited from below by available B-field)

– muon lifetime

* – beta-function at IP (limited from below by chromaticity of final focusing and aperture restrictions in IR magnets),

small * requires small z large p / p = || / z

– beam-beam parameter (limited by particle stability, < 0.1/IP ?)

0.5 1 1.5 2

0.6

0.7

0.8

0.9

h

z /

“Hour-glass factor”

P – average muon beam power (limited by the P-driver power )

MC Lattice Design Challenges 3

What we would like to achieve compared to other machines:

MC Tevatron LHC

Beam energy (TeV) 0.75 0.98 7

* (cm) 1 28 55

Momentum spread (%) >0.1 <0.01 0.0113

Bunch length (cm) 1 50 15

Momentum compaction factor (10^-3) 0.05 2.3 0.322

Geometric r.m.s. emittance (nm) 3.5 3 0.5

Particles / bunch (10^11) 20 2.7 1.15

Beam-beam parameter , 0.1 0.0250.01

Muon collider is by far more challenging:

much larger momentum acceptance with much smaller *

~ as large Dynamic Aperture (DA) with much stronger beam-beam effect

- New ideas for IR magnets chromaticity correction needed!


Various MC lattice designs studied

1996 by Carol J., A. Garren

1996 by K.Oide

“Dipole first” (2007) ~ satisfy the requirements

Eliana’s “synthetic” (2009)

Asymmetric dispersion

“Flat top”

“Three sextupoles” scheme – the final solution based on Eliana’s preliminary design.

________________

1996 designs used special -I chromatic correction sections, had very high sensitivity to field errors.

Oide’s lattice: sextupoles arranged in non- interleaved pairs.

“Dipole first” - our first exercise defying conventional wisdom: all sextupoles were interleaved. Decent momentum acceptance and DA, but high sensitivity to beam-beam effect.

4


Chromatic Correction Basics 5

,

,1

,

22xxx

xpx

xxpx

xx

px

BAW

BA

Montague chromatic functions :

Ax,y are created first, and then converted into Bx,y as phase advances x,y grow

K1 , K2 are normalized quadrupole and sextupole gradients, Dx is dispersion function: Dx = dxc.o./dp

The mantra:

Kill A’s before they transform into B’s !

- difficult to achieve in both planes

- requires dispersion generated close to IP or large Dx0 at IP

Bx,y are most important since they determine modulation of phase advance x,y

xxx

xxxxx

AB

KDKBA

2

),(2 21

s

pxppx s

sds

0 ),(1

1),(

x,y = -x,y /2 , x,y are Twiss lattice functions, p is relative momentum deviation.

Equations for chromatic functions


IR design by K.Oide

*=3mm, Ebeam=2TeV

Is this design practical?

- huge max ~ 1000 km

- too small quad aperture?

First quad G=214T/m (NbTi), with Nb3Sn B0=10T a~5cm

but is this enough to accommodate the shielding?

QC1-QC4 have small octupole component


6

“Local” chromatic correction with Oide’s design

y

x

ax

ay

dx /2 - goes to - 4824.5 !

bx

by

dx /2

* = 3mm, max = 901,835 m

hor. CCver. CC

with chromatic correction sections separated from IR there inevitably are places with large chromatic modulation of betatron phase advances – potential for a trouble

chromatic phase (arg ax,y/bx,y) advances by 2 at locations where respective beta-functions are low


7

KEK ATF arc cell with negative c

The defocusing sextupoles in the arc cells have gradients up to 67000 T/m^2 at the beam energy 2TeV. In the 750GeV case, if we reduce the geometrical sizes ~E, the sextupole gradient will actually rise as 1/E^2. - Again, not very practical solution.


8

Momentum acceptance with Oide design

Qx

Qy

c

p

p

Static momentum acceptance is ~0.55%, but change in c sign limits

dynamic acceptance to ~0.4%


9

Dynamic Aperture with Oide’s design

Tracking performed with program SAD that automatically includes fringe fields for all elements


10

MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007

Sensitivity to errors

“Dipole First” lattice had 2 IPs. With 1 IP the probability of losing stability would be lower.

Still it would be difficult to find the stability window

% unstable

relative sextupole field error

Oide

% unstable

relative quadrupole field error

not a single case with “dipole first” lattice

“dipole first”

Oide

Some unstable cases are MAD artefacts - sometimes it cannot find optics for linearly stable lattice

MAD8 simulations of the random field errors (Gaussian distribution truncated at 2.5) effect on linear optics stability for:

1) Oide’s lattice (1 IP, * = 3mm, max = 901.8 km),

2) “Dipole first” lattice (2 IPs, * = 1cm, max = 32.8 km),

With quadrupole errors stability was required for |p| 0.3%.

With sextupole errors stability was required for |p| 0.5%.

11

New paradigm

Chromaticity of the larger -function should be corrected first (before is allowed to change) – and in one kick to reduce sensitivity to errors!

To avoid spherical aberrations it must be y then small x will kill all detuning coefficients and RDTs (this will not happen if y x)

Chromaticity of x should be corrected with a pair of sextupoles separated by -I sectionto control DDx (smallness of y is welcome but not sufficient)

Placing sextupoles in the focal points of the other -function separated from IP by = integer reduces sensitivity to the beam-beam interaction.

These considerations almost uniquely determine the IR layout.

Requirements adopted for the latest version:

full aperture A = 10sigma_max + 2cm (A.Zlobin adds 1cm on top of that)

maximum quad gradient 12% below quench limit at 4.5°K as calculated by A.Zlobin

bending field 8T in large-aperture open-midplane magnets, 10T in the arcs

IR quad length < 2m (split in parts if necessary!) – no shielding from inside

Sufficient space for magnet interconnects (typically 30-40cm)

322

222 ~,~ x

x

xyx

y

y bdE

dQb

dE

dQ


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3 Sextupoles Chromatic Correction Scheme

y

x

Chrom. Correction Block

Wx

Wy

correctors

sextupoles bends

Dx (m)

quads

RF

multipoles for higher order chrom. correction


13

One More Innovation: the Arc Cell

SY

DDx/5

Dx (m)

SX SX

SASY

x

y

Central quad and sextupole SA control the momentum compaction factor and its derivative (via Dx and DDx) w/o significant effect on tunes and chromaticity

Large -functions ratios at SX and SY sextupole locations simplify chromaticity correction

Phase advance 300/ cell spherical aberrations cancelled in groups of 6 cells

Large dipole packing factor small circumference (C=2.5 km with 10T dipole field)

dsDDD

Cd

d

dsD

CC

xx

p

c

Cx

c

0

2

0

)(2

11

,1


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Momentum Acceptance

Fractional parts of the tunes

Static momentum acceptance = 1.2%, while the baseline scheme calls for only 0.3%

Central value of the momentum compaction factor = -1.4510-5, can be made even smaller but we don’t know yet what is practical

With 2 IPs the central tunes are 18.56, 16.58

- good (!) for beam-beam effect - good for the orbit stability and DA

p

c

x*

y*p

Qx Qy

p


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Dynamic Aperture

DA= ( CSI / N )1/2= 4.7 for N=25 m

1024 turns DA (Lie3 tracking method): blue – Courant-Snyder Invariants of stable muons, red – lost muons

Dynamic Aperture is marginally sufficient for N=50 m

DA can be further increased with vertical nonlinear correctors

N=0

Delta(p)/p: 0.00000000 M o d e 1 M o d e 2Fractional tunes: Q1 = 0.55999422 Q2 = 0.57999563First order chromaticity: Q1' = 0.32480253 Q2' = -1.98911112Second order chromaticity: Q1'' = -862.88043160 Q2'' = 2077.02609602Normalized anharmonicities: dQ1/dE1 = -0.24797555E+06 dQ1/dE2 = 0.14326980E+06 dQ2/dE2 = 0.73894849E+05

MAD8 STATIC command output - very imprecise!

CSIx [m]

CSIy [m]


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Dynamic Aperture with Beam-Beam Interaction

“Diagonal” Dynamic Aperture (Ax=Ay) vs. (constant) momentum deviation in the presence of beam-beam effect ( = 0.09/IP) for normalised emittance N=25 m.

Opposing bunch is represented by 12 slices

Only muons at bunch center tracked !

p

DA ()

beam extent


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Muon Collider Parameters

s (TeV) 1.5 3

Av. Luminosity / IP (1034/cm2/s) 1.25* 5

Max. bending field (T) 10 14

Av. bending field in arcs (T) 8.3 12

Circumference (km) 2.5 4

No. of IPs 2 2

Repetition Rate (Hz) 15 12

Beam-beam parameter / IP 0.087 0.087

* (cm) 1 0.5

Bunch length (cm) 1 0.5

No. bunches / beam 1 1

No. muons/bunch (1012) 2 2

Norm. Trans. Emit. (m) 25 25

Energy spread (%) 0.1 0.1

Norm. long. Emit. (m) 0.07 0.07

Total RF voltage (MV) at 800MHz 20 230

----------------------------------------------------------------------- *) With increase by the beam-beam effect

hC

Pfh

Nnf rep

b

~2

1

4

2

0L

P – average muon beam power (~ )

4

Nr

C – collider circumference (~ if B=const)

– muon lifetime (~ )

* – beta-function at IP

– beam-beam parameter

0.5 1 1.5 2

0.6

0.7

0.8

0.9

h

z /

“Hour-glass factor”


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Lesson - Symplecticity matters!


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1000 turns DA for one of the earlier MC versions with 1 IP calculated with MAD8 using different tracking methods. With METHOD=LIE4 not a single particle has survived! Almost all were lost in ~30 turns.

Probable cause for discrepancy: kinematic nonlinearities in long IR dipoles (or roundoff errors?)

Conclusions

Arrangement of sextupoles in non-interleaved -I pairs helps to reduce detuning and improve DA, but is not a “must”.

Vertical chromaticity correction sextupole may be leaved w/o a pair if at its location x <<< y

New arc cell design allows for efficient (and independent!) control of c

and dc /dp and attainment of very low | c |. It turned out that a similar

design was considered by Yuri Senichev for CERN PS2 project (but not accepted).

Fringe-fields are of utmost importance, especially in the case of large ’s at strong large-aperture magnets.

Symplecticity of tracking algorithm may be necessary for as little as 100 turns!

It seems that MAD (at least MAD8) can not properly handle non-linear effects in lattices with very large ’s.


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Muon Collider Lattice Design FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY f Yuri...

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