Introduction The electromagnetic fields generated by the beam will be deformed by the interaction with the surroundings. The deformed field, which is known as the wake field, will disturb the beam dynamics, and under certain conditions, lead to collective instabilities. The impedance and collective effects should be well estimated during the design stage. A thorough evaluation of the impedance is necessary in controlling the total impedance of the ring, which can accordingly prevent the occurrence of the beam instability.
Main parameters of BAPS ParameterSymbol, unitValue Beam energyE, GeV5 CircumferenceC, m1364.8 Beam currentI 0, mA100 Bunch numbernbnb 1836 Number of particles per bunchNbNb 1.55 10 9 Natural bunch length l0, mm 2.9 RF frequencyf rf, MHz500 Harmonic numberh2276 Natural energy spread e0e0 1.5 10 3 Momentum compaction factor pp 4 10 5 Betatron tune x / y 113.4/39.3 Synchrotron tune s 0.004 Damping time (H/V/s) x / y / z, ms 28, 43, 29 Emittance (horz./vert.) x / y, pmrad 10/10 Trans. beam size (horz./vert.) x / y, m 4.4/7.4
What might be important in BAPS Short bunch of l = 2.9 mm –Large loss factor and high beam power loss –Coherent Synchrotron radiation (CSR) Small beam pipe aperture b = 11 mm High resistive wall impedance Transverse resistive wall instability High filling factor n b /h = 1836/2276 with small transverse emittance –Coupled bunch instability due to HOMs –Fast Beam Ion Instability –Intra-beam Scattering (IBS)
Impedance Main impedance components in BAPS –Resistive wall, vacuum transitions, RF cavities, BPMs, vacuum flanges, injection kickers, bellows, vacuum pump, etc. ObjectsDescription Insertion devices inside vacuum Stainless steel + TiN coating, b = 2.5mm Resistive wall, 72 taper transitions Insertion devices outside vacuum Stainless steel, b = 5mm Resistive wall, 72 taper transitions Other part of the ringStainless steel, b = 11mm, Resistive wall RF cavities7
1. Resistive wall For cylindrical beam pipe, the longitudinal and transverse resistive wall impedance are estimated by 1) Longitudinal impedance –With natural bunch length of 2.9 mm, k l = 101.7 V/pC (P loss = 2.5kW). 2) Transverse impedance –Transverse resistive wall impedance can induce coupled bunch instability.
2. Taper transitions Main contribution is from the transition of the insertion device. The impedance and loss factor are calculated by the code ABCI. A slope angle of 5 is suggested as a compromise between the impedance and the longitudinal space occupied. Longitudinal wake potential (b = 11mm, = 5 , = 2.9mm) Longitudinal impedance and power loss with different slope angle.
3. RF cavities The BEPCII superconducting RF cavity will be used in BAPS. With bunch length of l = 2.9 mm, the calculated loss factor for single cavity is k l = 6.4 V/pC (P loss = 0.2 kW) Inductive impedance Z // /n= i0.05
Summary of the impedance study Resistive wall impedance is dominated in both longitudinal and transverse impedances, and RF cavities contribute high inductive impedance. Further studies are needed with more vacuum components are designed. Components Z // /n, Z , M /m Loss factor k l (V/pC) ( l0 =2.9 mm) power P (kW) (54.5 A, 1836) Resistive wall (b=11mm) 18.4(1 i)/ n278(1 i)/ n 101.72.5 Taper transitions (b=11mm, =5 ) 0.03i 0.14i 1.00.03 RF cavities 0.36i 1.55i 451.1
1. Single bunch effects Longitudinal microwave instability –According to the Keil-Schnell criterion, the threshold of longitudinal impedance is |Z // /n| < 0.28 . –This will induce bunch lengthening and energy spread increase. Transverse mode coupling instability (TMCI) –The threshold of transverse impedance is |Z | < 12 M /m. –The equivalent longitudinal impedance is 2.7 , which is much larger than that of the longitudinal instability.
Coherent synchrotron radiation –The threshold of bunch population for coherent synchrotron radiation is –The CSR threshold in BAPS is N 0,Th = 4.3 10 10 >> N b = 1.55 10 9. –CSR is supposed not to be a problem in BAPS.
2. Multi bunch effects Transverse resistive wall instability with pn = 2 f rev (pn b + n + x,y ) The growth rate for the most dangerous instability mode is 6 kHz in the vertical plane with mode number of = 1797. The growth rates are much higher than the transverse radiation damping rate. An efficient transverse feedback system is necessary! With Cu beam pipe and Cu coating on the ID, the growth time will increase to 0.6 ms. Growth rate vs. mode number in the vertical plane
Larger decimal tune are preferred to alleviate the transverse resistive wall instability. Nonzero chromaticity will also help to damp the instability. Instability growth rate vs. vertical tune Instability growth rate vs. chromaticity
Longitudinal/transverse HOMs of RF cavities –The main longitudinal and transverse HOMs’ data of the BEPCII superconducting cavity is used in the calculations. –The instability rising time for the fastest growing longitudinal mode is z = 1.7 s and for the transverse mode is y = 0.5 s. –The instabilities will not be driven by the HOMs of the RF cavities.
3. Ion effects Ion trapping The condition of ion trapping for the beam with a gap T g is –The oscillation frequency of the ion is i = 540 MHz (CO + ). –Including a gap of T g = 0.9 s (~ 20% of the ring circumference), the left side of the equation gives 211. –The ion trapping is not supposed to happen in BAPS.
Fast beam-ion instability (FBII) The growth rate for the trailing bunches can be expressed as –Taking into account the bunch gap, the instability growth time is about 0.02ms. –With multi-trains, the growth time is 0.04 ms for two bunch trains, 0.07 ms for four bunch trains, and 0.1 ms for eight bunch trains. –Feedback system is needed. –The nonlinear effects will introduce spread in the effective ion frequency, which will help to damp the instability. –Detailed simulations should be done to estimate the instability.
Summary of collective effects The growth time of the FBII is too fast to be damped effectively by feedback systems even with a long beam gap. Multi-train scheme should be used along with the feedback system to damp the instability. The coupled bunch instability induced by the transverse resistive wall impedance need to be damped with feedback systems. The longitudinal microwave instability is expected to happen, which will induce bunch lengthening and energy spread increase. TMCI and CSR are not supposed to be problems in BAPS.