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) vertical orbit distortions 1 / 12

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 A. Bogomyagkov (BINP) vertical orbit distortions 2 / 12

Introduction For flat orbits only E [ MeV ] = 440 . 64843 ( 3 ) × ν . Approximation (R. Assmann, J.P . Koutchouk) ∆ ν = ν 2 cot πν � α 2 i , 8 π α i are the orbit rotation angles. � z 2 � (assuming that � z � = 0), Using observed vertical orbit RMS number of quadrupole lenses N with average focal length F ∆ ν = ν 2 cot( πν ) z 2 � � N . F 2 8 π A. Bogomyagkov (BINP) vertical orbit distortions 3 / 12

Validity of approximation 50 E, keV ∆ 40 30 20 10 0 τ Ψ ’ 4 4.1 4.2 4.3 4.4 4.5 4.6 ν 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. A. Bogomyagkov (BINP) vertical orbit distortions 4 / 12

General approach Spin tune shift (Kodratenko) | ω k | 2 ∆ ν = 1 � 2 ν − k k Spin harmonics 2 π ω k = 1 � ν z ′′ exp [ − i (Φ( θ ) − νθ ) − ik θ ] d θ , 2 π 0 d 2 z z ′′ = 1 d θ 2 , Π = 2 π R , ds = Rd θ , R � θ ν RK 0 ( θ ′ ) d θ ′ Φ( θ ) = 0 A. Bogomyagkov (BINP) vertical orbit distortions 5 / 12

Individual harmonics Alignment harmonic k 2 π s � � z = A k sin = A k sin( k θ ) , Π where A k is harmonic amplitude. Spin harmonics assuming no straight sections ( Φ( θ ) = νθ ) ω k = − 1 2 i ν A k R k 2 , � ν k 4 A 2 ν − k + ν k 4 ∆ ν = 1 � k . 2 i ν A k 1 R 2 ν 8 ν + k R k 2 . ω − k = A. Bogomyagkov (BINP) vertical orbit distortions 6 / 12

Numerical results for individual harmonics Beam energy E = 45 . 6 GeV, ν = 103 . 484, Π = 100 km A k = 15 · 10 − 3 m A k = 3 · 10 − 4 m k ∆ ν/ν k ∆ ν/ν 1 . 2 · 10 − 13 5 · 10 − 17 1 1 1 . 9 · 10 − 12 8 · 10 − 16 2 2 9 . 7 · 10 − 12 4 · 10 − 15 3 3 3 . 1 · 10 − 11 1 · 10 − 14 4 4 1 . 3 · 10 − 9 5 · 10 − 13 10 10 1 . 4 · 10 − 6 6 · 10 − 10 50 50 3 . 5 · 10 − 4 1 . 4 · 10 − 7 100 100 2 . 8 · 10 − 3 1 . 1 · 10 − 6 103 103 A. Bogomyagkov (BINP) vertical orbit distortions 7 / 12

Vertical orbit Assumptions and definitions No straight sections: Φ( θ ) = νθ Constant vertical beta function: β z = const = � β z � Average over circumference �� , average over orbits ¯ j = f 2 δ ij Random and uniform kicks F i F ∗ Results � z 2 � ∞ ∆ ν = ν 2 k 4 � ( ν 2 2 Q z − k 2 ) 2 ( ν − k ) k = −∞ cot πν z + π 2 π csc 2 πν z Q = 2 ν 3 2 ν 2 z z σ ∆ ν = ν 2 √ � ∞ � z 2 � � k 8 3 � � � 2 ν ( ν 2 z − k 2 ) 4 ( ν − k ) 2 ( ν + k ) 2 Q k = −∞ A. Bogomyagkov (BINP) vertical orbit distortions 8 / 12

Comparison with simulation 5 E, keV 4 ∆ 3 2 1 0 -1 τ Ψ ’ -2 4 4.1 4.2 4.3 4.4 4.5 4.6 ν 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. A. Bogomyagkov (BINP) vertical orbit distortions 9 / 12

FCCee at E = 45 . 6 GeV, σ z = 1 mm E, GeV E, GeV 45.43 45.47 45.52 45.56 45.61 45.65 45.7 45.74 45.78 45.43 45.47 45.52 45.56 45.61 45.65 45.7 45.74 45.78 - 83.5 127.0 Δ E ( σ z = 1 mm ) σΔ E ( σ z = 1 mm ) 126.5 - 84.0 126.0 σΔ E, keV Δ E, keV - 84.5 125.5 125.0 - 85.0 124.5 - 85.5 124.0 123.5 103.1 103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9 103.1 103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9 ν ν A. Bogomyagkov (BINP) vertical orbit distortions 10 / 12

Tables for Z and W E , GeV 45.6 78.65 81.3 �� z 2 � , mm 0.6 0.28 0.27 ν z 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 − 7 · 10 − 8 − 7 · 10 − 7 − 7 · 10 − 7 E σ ∆ E 1 · 10 − 6 1 · 10 − 6 1 · 10 − 6 E Beam energy shift needs to be added to the actual value of the beam energy, uncertainty is unavoidable and sets the minimum error. A. Bogomyagkov (BINP) vertical orbit distortions 11 / 12

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 orbit 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. A. Bogomyagkov (BINP) vertical orbit distortions 12 / 12