Influence of vertical orbit distortions on energy calibration - - PowerPoint PPT Presentation

Influence of vertical orbit distortions on energy calibration accuracy

• A. Bogomyagkov

Budker Institute of Nuclear Physics Novosibirsk

June 20th, 2018

• A. Bogomyagkov (BINP)

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References

A.M. Kondratenko. Doctoral Thesis. Novosibirsk, 1982.

• R. Assmann, J.P

. Koutchouk, CERN SL/94-13 (AP). A.V. Bogomyagkov, S.A. Nikitin, A.G. Shamov, MOAP02,RuPAC 2006, Novosibirsk, Russia, http://accelconf.web.cern.ch/AccelConf/r06/PAPERS/MOAP02.PDF https://arxiv.org/abs/1801.01227

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Introduction

For flat orbits only

E[MeV] = 440.64843(3) × ν .

Approximation (R. Assmann, J.P . Koutchouk)

∆ν = ν2 cot πν 8π

• α2

i ,

αi are the orbit rotation angles. Using observed vertical orbit RMS

• z2

(assuming that z = 0), number of quadrupole lenses N with average focal length F ∆ν = ν2 cot(πν) 8π N

• z2

F 2 .

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Validity of approximation

ν 4 4.1 4.2 4.3 4.4 4.5 4.6 E, keV ∆ 10 20 30 40 50

τ ’ Ψ

Energy shift versus spin tune at 1 mm vertical orbit RMS for VEPP-4M. Triangles are calculations by approximate expression (Assmann), circles with error bars are results of the simulation.

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General approach

Spin tune shift (Kodratenko)

∆ν = 1 2

• k

|ωk|2 ν − k

Spin harmonics

ωk = 1 2π

2π

• νz′′ exp [−i(Φ(θ) − νθ) − ikθ]dθ ,

z′′ = 1 R d2z dθ2 , Π = 2πR , ds = Rdθ , Φ(θ) = θ νRK0(θ′)dθ′

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Individual harmonics

Alignment harmonic

z = Ak sin

• k2π s

Π

• = Ak sin(kθ) ,

where Ak is harmonic amplitude.

Spin harmonics

assuming no straight sections (Φ(θ) = νθ) ωk = − 1 2i ν Ak R k2 , ω−k = 1 2i ν Ak R k2 . ∆ν ν = 1 8 A2

k

R2 νk4 ν − k + νk4 ν + k

• .
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Numerical results for individual harmonics

Beam energy E = 45.6 GeV, ν = 103.484, Π = 100 km Ak = 15 · 10−3 m k ∆ν/ν 1 1.2 · 10−13 2 1.9 · 10−12 3 9.7 · 10−12 4 3.1 · 10−11 10 1.3 · 10−9 50 1.4 · 10−6 100 3.5 · 10−4 103 2.8 · 10−3 Ak = 3 · 10−4 m k ∆ν/ν 1 5 · 10−17 2 8 · 10−16 3 4 · 10−15 4 1 · 10−14 10 5 · 10−13 50 6 · 10−10 100 1.4 · 10−7 103 1.1 · 10−6

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Vertical orbit

Assumptions and definitions

No straight sections: Φ(θ) = νθ Constant vertical beta function: βz = const = βz Average over circumference , average over orbits¯ Random and uniform kicks FiF ∗

j = f 2δij

Results

∆ν = ν2 2

• z2

Q

∞

• k=−∞

k4 (ν2

z − k2)2(ν − k)

Q = π 2ν3

z

cot πνz + π2 2ν2

z

csc2 πνz σ∆ν = ν2√ 3 2

• z2

Q

• 2ν

∞

• k=−∞

k8 (ν2

z − k2)4(ν − k)2(ν + k)

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Comparison with simulation

ν 4 4.1 4.2 4.3 4.4 4.5 4.6 E, keV ∆

• 2
• 1

1 2 3 4 5

τ ’ Ψ

Energy shift versus spin tune at 1 mm vertical orbit RMS for VEPP-4M. Solid and dashed lines are the spin tune shift and its uncertainty, circles with error bars are results of the simulation.

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FCCee at E = 45.6 GeV, σz = 1 mm

ΔE(σz=1 mm) 103.1 103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9

• 85.5
• 85.0
• 84.5
• 84.0
• 83.5

45.43 45.47 45.52 45.56 45.61 45.65 45.7 45.74 45.78 ν ΔE, keV E, GeV σΔE(σz=1 mm) 103.1 103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9 123.5 124.0 124.5 125.0 125.5 126.0 126.5 127.0 45.43 45.47 45.52 45.56 45.61 45.65 45.7 45.74 45.78 ν σΔE, keV E, GeV

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Tables for Z and W

E , GeV 45.6 78.65 81.3

• z2

, mm 0.6 0.28 0.27 νz 269.22 269.2 269.2 ν 103.484 178.487 184.5 ∆E, keV

• 31
• 54
• 56

σ∆E, keV 46 82 85

∆E E

−7 · 10−8 −7 · 10−7 −7 · 10−7

σ∆E E

1 · 10−6 1 · 10−6 1 · 10−6 Beam energy shift needs to be added to the actual value of the beam energy, uncertainty is unavoidable and sets the minimum error.

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Conclusion

Vertical orbit distortions produce beam energy shift. Vertical orbit distortions produce uncertainty of the beam energy. We may divide the problem into alignment of the ring and vertical

• rbit correction.

To estimate alignment requirements we need spectrum. The best approach is to calculate harmonics for existing machine (LHC, LEP) and apply results for FCC.

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