Design of Damping Ring
KEKB Review, Kikuchi, M. 2004.2.17•Introduction•

Beam Parameters•

Lattice design•

DR parameters•

From DR to SuperKEKB•

SummaryKEKB Review 2004.2.17

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Introduction
Damping Ring is necessary :• IR: In SuperKEKB βy* is squeezed to 3 mm
- IR design needs lower emittance for the beam (e+)• C-band linac: lower emittance is needed (e+)• Beam background: Injected beam-charge is doubled.
- needed damped beam for smaller energy-tail andemittance-tail. (e+ and e-)
Positron DR is placed before C-band linac: from the layoutof linac the energy should be ~ 1 GeV.
■ Electron DR is preferable from the detector background,though its priority being lower.
■

 I will talk only on the positron DR.
KEKB Review 2004.2.17

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Layout of Beam Lines
1-GeV Damping Ring

0RTL-line(BCS)

Sector-2

50 m

LTR-line(ECS)

Sector-3

Positron target

KEKB Review 2004.2.17

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Beam parameters -1DR emittance:
IR Design assumed the emittance of injected beam to beγε = 3.13 x 10-4 (= emittance of KEKB electron-beam i.e. noelectron DR)==> DR emittance < 160 nm (for 1 GeV)
However, considering the possibility of electron DR in far
future, the emittance of positron DR should be the same asthat of possible electron DR:
Design emittance of positron DR should be smallest value,achievable in the lattice design.

KEKB Review 2004.2.17

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Beam parameters -2ECS before injection to DR
Employing ECS, almost 100% of particles are included
within the energy band-width of ±1.5%(Note momentum acceptance of transport line: ±5%)Before ECS

DensityTransmission

Before ECS

Simulation
σε=0.17%σz=2.3 mmEnergy acceptance

After ECS

After ECS

DensityTransmission

Δp/p

Δp/p

Energy acceptance
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Beam parameters -3Parameters of injected beam
• Assume two-bunch mode for injection
before ECS

after ECS

Energy1.0
Repetion frequency
50Emittance
1.23E-06
Energy spread *)
1.30
0.406
Bunch length *)
2.30
6.05
Number of bunches/pulse
2
Bunch spacing
98
Bunch charge
1.2
ECS cavity voltage
30.5
ECS cavity frequency
2.856
R56 component-0.486*) defined as extension that contains 95.5% divided by 4.

UnitGeVHzm%
mmnsnCMVGHzmKEKB Review 2004.2.17

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Lattice design -1Requirements:• Large acceptance (transverse and longitudinal)• Fast damping
FODO has advantage that
• Fast damping: bends embedded in every drift space• Large dynamic aperture (transverse and longitudinal)
but has drawback• Tendency to higher momentum-compaction

We propose a variant of the FODO:
FODO with alternating bend
that preserves good nature of FODO and have a very low
positive/negative momentum-compaction factor.
KEKB Review 2004.2.17

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Lattice design -2Thin lens model:
2π: bend angle of main bendn(1− r)n : number of cells
ρ : bending radiusr : ratio of reverse bend to main bend
µ : phase advance per cell
E 0 : beam energy in GeV
l1 = 2lq + 4lqb = 2l − 2lb

θ=FODO w/ alternating bend. It reduces
to ordinal FODO when r = -1.
€

Assumptions:• equal phase advance µx= µy=µ• same bending radius for B1 and B2

KEKB Review 2004.2.17

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Lattice design -3■ Momentum compaction factor
α p = G(r, µ)θ 2For fixed r, by adjusting the phaseadvance, low positive or negative αp
can be achieved.

3 − 8r + 3r 2 + (1+ r 2 )cos µG(r, µ) =16sin 2 µ /2
When r > 2 − 3 = 0.268
there exist a combination of r and µthat satisfies α p = 0€€3αapp€2.5

r = 0.5
θ = 0.3927

0.06
0.04
0.02

20µm 1.5
10.5

11.5m

22.5

3

phase advance µ

0.5
00.3

0.4

0.5

rr

0.6

0.7

0.8
KEKB Review 2004.2.179Lattice design -4■ Emittancelθ 2 2ε0 = Cqγ F(r, µ)8ρ
1− r
1µ3µ 
22F(r, µ) = 2(3sin − sin )
 3 − 4r + 3r + (1− 4r + r )cos µ − rsin µ /2sin µ 1+ r
22 

€h.
32 3 mc

with Cq = 55

Since we select µ and r such that α p ≈ 0 , ε0 can be rewritten as€€lθ 2 2ε0 = Cqγ f (µ)8ρ9€8f (µ)• f (µ) has its minimum at µ = 2.1 (r = 0.35)€7fHmL 654
30.5

11.5m

22.53µKEKB Review 2004.2.17

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Lattice design -5■ Optimization of parameters• Damping time
τ=2E 0 C2 ρ 
1− r=2πρ+(nl+2l)
12 
J xU 0 c cCτ J x γ 3 1+ r

Cτ =

€4π
re
3

n : number of cells

2l 2 : length of straight sections

Jx = 1€l 1 : (cell length) - (length of bends)€
as a function of τ, r, and n:
For fixed l 1 and l 2 , ρ is written€ρ = ρ(τ, r, n) = ρ(τ, µ, n)€€θ=2π(1− r)nif αp = 0• Emittance

2l = l 1 + (1+ r ) ρθ

ε0 = ε0 (τ , µ,n)€€€

KEKB Review 2004.2.17

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Lattice design -6The emittance at extraction

εext = ε0 + (εi − ε0 )exp(−2T / τ ) ≡ εext (τ , µ,n)where εi is the emittance at injection and T is store time.

€We optimize the emittance at extraction in (τ, µ, n) space• Assumption: T = 40 ms (i.e., two bunch-trains)• For any n, minimum emittance is obtained around (τ = 12 ms, µ=2.3)

µµµ

τ(ms)

τ(ms)n = 60n = 50n = 40

n = 30

µ

τ(ms)

τ(ms)
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Lattice design -7• Minimum emittance itself depends on number of cells ∝ n-2• To determine the number of cell, the bend field and the circumference
must be taken into account.
Kicker rise/fall time = 100ns
C > 120 m and n > 40◆εext(nm)◆µ=2.3
τ=12msFor n = 40, B < 1.3 T

τ = 12 ms

µC(m)

µ=2.3

B(T)

τ = 12 msµ=2.3nµ
KEKB Review 2004.2.17

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DR parameters
UnitGeVEnergy
Number of bunch trains
Number of bunches/train

1.022Bunch spacing
Bunch charge

98
1.2

nsnC

50
131.3
73
120.0019
40
12.2
1.23
13.7
4.06E-03
6.05
5.29E-04
5.03
0.35
1.932

HzmkeVms

Repetition frequency
Circumference
Energy loss per turnHorizontal damping time
Momentum compaction factor
Number of normal cells
Emittance at equilibrium
Emittance at injection
Emittance at extraction
Energy spread of injected beam
Bunch length of injected beam
Energy spread
Bunch length
Bend-angle ratio of reverse-bendPhase advance/cellnmumnmmmmmrad• Cavity voltage of 0.26MV is
within the spec. of KEKB ARES
cavity (0.5 MV/cavity)

Bend field
Quad field
Sextupole field
Length of straight sections
Length of main bend
Length of reverse bend
Length of quad
Length of sext
Minimum space between magnets

1.267
16.3
426
2x6
0.7286
0.255
0.25
0.1
0.1

Unit
T
T/m
T/m^2mmm
mmm

Cavity voltage for 1.5% bucket heightRF frequency

0.261
509

MVMHz

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DR optics
Optics functions of normal cell

Optics functions of entire ringSDB2

SFQF

B1QD

• Dispersion suppressor: 1 half-length bend + 7 adjustable quads• Chromaticity correction: 2-family sextupoles

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Dynamic aperture -1Tracking simulation results: very large DA• (νx, νy) = (12.24, 4.26)• 4000 turns• RF bucket height = 4%
• Machine errors (just 1 sample):- strength error: 3 x 10-4 for quad, 5 x 10-4 for sext
- misalignments: 0.5 mm, orbit corrected by correctors

2J (µm)w/o error
w/ error
Injected beam(2σ)Δp/p(%)
Initial momentum and action thatsurvived 4000 turns
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Dynamic aperture -2Tune survey results: wide operational tune space
Without machine errors• Third-order resonance driven bysextupoles. The phase advance between
sexts was accidentally near to 2π/3.
• Phase advance/cell, accordingly the bend
ratio r, should be changed to avoid thisresonanceνyνxTune space where the DA islarger than the extent of the beam

KEKB Review 2004.2.17

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From DR to SuperKEKB -1Beam transport in longitudinal phase-space• Bunch length compression to match the subsequent C-band linac.
• Energy spread to meet the aperture of SuperKEKB.
■ 3.5-GeV caseSuperKEKB
e+ DRECSBCS

C-band linac

R56~-5m

LER
BT-line■ 8-GeV caseSuperKEKB
e+ DRHER
R56~-5m

BCS

C-band linac

BT-lineKEKB Review 2004.2.17

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From DR to SuperKEKB -2Longitudinal beam parameters
3.5-GeV case
8.0-GeV case*

Energy spread
Bunch length
Energy spread
Bunch length

DR0.053
5.03
0.053
3.54*

after BCS
0.499
0.59
0.510
0.37

end of linac
0.2860.59
0.145
0.37

after ECS
0.065
2.65

after BT
0.065
4.25
0.145
2.96

Unit%mm%mm

For 8-GeV case, the bunch length in the DR was reduced to 3.54 mm by changing RF voltage to 0.5 MV.

Injection to HER

Injection to LERδδz (mm)The 99.5% of beams are within theassumed acceptance of the ringz (mm)The 98.7% of beams are within theassumed acceptance of the ring
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Summary
1. We found realistic parameters of ECS in the LTR line.
• Employing ECS, almost 100% of particles areincluded within the energy band-width of ±1.5%
2. We have proposed new design for the positron Damping
Ring, based on the ‘Alternating-bend FODO’ lattice.
• Low emittance• Low positive/negative momentum compaction• Very large dynamic aperture• Robust to machine-errors• …
3. We have found consistent longitudinal beam-parametersfrom DR to SuperKEKB rings that satisfy therequirement.
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KB-LERSKB-HER
SKB-LER

Amplitude 2J (µm)

KB-HER

Emittance of injected-beam (nm)• Effective septum thickness: 6 mm
• Ring emittance: 24 nm• βx at septum: 100 m

KEKB Review 2004.2.1721