Figure 2.71: Pdot and RF voltage during the MI ramp used in ESME simulations. Figure 2.72: Phase space distribution after transition in MI with zero charge (left) and full charge (right). Beam Stability in the Recycler and MI The main beam parameters used for in the Recycler stability estimate are listed in Table 2.22. At injection ( = 9.5) the space charge is strong transversely, the tune shift is much i.e. Qsc≈0.1 larger than the synchrotron tune the space charge is also important longitudinally, Qs≈0.0034; leading to the synchrotron tune depression of about 20% [51], and to a possibly of longitudinal instability similar to the The longitudinal instability can be significantly “dancing bunches” [54]. exacerbated by coupled-bunch interaction through high order modes (HOM) in the cavities, leading to the growth rate [55]: Here is the shunt impedance, r is the HOM frequency, and  is the rms bunch length in time Rs units. For k, MHz, and =1.9 ns this yields a rather high frequency r/(2)=150  Rs=35 suppression factor , leading to . A narrow-band damper could suppress such slow coupled-bunch motion if it will be required. Figure 2.73: Phase space distribution after transition with full charge and two different sizes of the gamma-t jump. Half jump (left) and whole jump (right). Table 2.22: Beam parameters in the Recycler used in beam stability estimates Bunch population, N Number of bunches 2.5 mm mrad Transverse emittance, norm. rms, n Longitudinal emittance, rms, = 3.6 meV s || E Maximal RF Voltage , 0.125 MV V Transition gamma, 21.6 t The transverse single-bunch instability is described by a growth rate [49] where is the chromaticity factor determined by the head-tail phase , is a bunch-averaged wake function, is the average beta-function. For , the chromaticity factor saturates at its maximum, . For the resistive wall case, with the half-gap and the conductivity the average wake is , b estimated as
Figure 2.71: Pdot and RF voltage during the MI ramp used in ESME simulations. Figure 2.72: Phase space distribution after transition in MI with zero charge (left) and full charge (right). Beam Stability in the Recycler and MI The main beam parameters used for in the Recycler stability estimate are listed in Table 2.22. At injection ( = 9.5) the space charge is strong transversely, i.e. the tune shift Qsc≈0.1 is much larger than the synchrotron tune Qs≈0.0034; the space charge is also important longitudinally, leading to the synchrotron tune depression of about 20% [51], and to a possibly of longitudinal instability similar to the “dancing bunches” [54]. The longitudinal instability can be significantly exacerbated by coupled-bunch interaction through high order modes (HOM) in the cavities, leading to the growth rate [55]: Here Rs is the shunt impedance, r is the HOM frequency, and  is the rms bunch length in time units. For Rs=35 k, r/(2)=150 MHz, and  =1.9 ns this yields a rather high frequency suppression factor , leading to . A narrow-band damper could suppress such slow coupled-bunch motion if it will be required. Figure 2.73: Phase space distribution after transition with full charge and two different sizes of the gamma-t jump. Half jump (left) and whole jump (right). Table 2.22: Beam parameters in the Recycler used in beam stability estimates Bunch population, N Number of bunches Transverse emittance, norm. rms, n 2.5 mm mrad Longitudinal emittance, rms, || = E 3.6 meV s Maximal RF Voltage , V 0.125 MV Transition gamma, t 21.6 The transverse single-bunch instability is described by a growth rate [49] where is the chromaticity factor determined by the head-tail phase , is a bunch-averaged wake function, is the average beta-function. For , the chromaticity factor saturates at its maximum, . For the resistive wall case, with the half-gap b and the conductivity , the average wake is estimated as
PIP-II_RDR_v0.0_MI_RR_IK.docx