HOM Calculations for Different Cavities and Beam Induced HOM Power Analysis of ESS

For different design of ESS superconducting cavities, the higher order modes (HOMs) of monopoles, dipoles, quadrupoles and sextupoles are found. Their R/Q values are also calculated at their geometric beta. Main HOM related issues are the beam instabilities and the HOM induced power especially from TM monopoles. The analysis for the beam induced HOM voltage and power in this paper shows that, if the HOM frequency is a few kHz away from the beam spectrum, it is not a problem. In order to understand the effects of the beam structure, analytic expressions are developed. With these expressions, the induced HOM voltage and power were calculated by assuming external Q for each HOM. Our analysis confirms that, with the beam structure of ESS and a good cavity design, no special tight tolerances are required for cavity fabrication and no HOM couplers in the cavity beam pipes are planned. INTRODUCTION A time–averaged HOM power spectrum normalized to the cavity’s R/Q has been calculated by Sangho Kim for the Spallation Neutron Source (SNS) beam time structure (single pass) [1], and Haipeng Wang’s paper ‘Beam Induced HOM Power Spectrum in JLab 1MW ERL-FEL’ [2]. The analysis of ESS beam time structures is explained in this paper. Since the most dangerous HOM monopole is close to 5 harmonic of micro-bunch frequency for ESS median-beta cavity, the HOM induced voltage and power analysis based on the 5 harmonic of micro-bunch frequency is also showed in this paper. HOMs are found with the CST MW Studio [3], and HOM properties such as frequencies and R/Q are calculated. We can find the dangerous mode which frequency is close to the beam spectrum from the simulation results. The total HOM induced power can also be calculated by the R/Q values. These works are showed in this paper. ESS BEAM& HOM IDUCED VOLTAGE ANALYSIS Time Structure of ESS Time structure of ESS (50 mA pulse) is showed in Fig. 1. HOM induced voltage analysis of ESS based on Sangho Kim's SNS/AP Technical Note No. 10 and Haipeng Wang’s paper are showed below. Figure 1: Time structure of ESS beam (bunches at 352.21 MHz). The parameters used in the calculation are listed as following:  HOM decay time constant: 2 e e d Q Q T f      SRF cavity’s fundamental mode frequency: f1= 704.42*10 Hz  Total numbers of micro-pulse within one macropulse: N= 1.01*10  Pulse period between each micro-bunch: Tb= 1/ (352.21 MHz) = 2.839*10 s  Bunch period between each macro-pulse: Tm= 71*10 s  Macro-pulse length: Tmb= 2.83*10 s  Macro-pulse spacing: TG= Tm Tmb =68.17*10 s  Single bunch charge (approximate to a point charge): q= 1.42*10 C  R/Q normalized induced voltage by a point charge: q ( ) 2 V q      At the time just after kth macro-pulse: q NT 1-exp( i NT ) (k,Q , ) ( ) T 1 exp( i T )