Instabilities in single RF systems
Imports
import numpy as np
import matplotlib.pyplot as plt
from mbtrack2 import CavityResonator
from albums import scan_MC_psi_I0
from albums import SystemState
from albums import EquilibriumOptions, TheoryOptions
from albums import (Flow, AlvesEquilibrium, ZeroFrequency, RobinsonCoupling,
HOMCoupledBunch, PTBLHe)
from albums import DEFAULT_FLOWS
Ring and cavity setup
from mbtrack2 import Synchrotron, Electron
from mbtrack2 import Optics
def SOLEIL_II(V_RF=1.7e6):
"""
SOLEIL II lattice v2366 w/ RF parameters from November 2023 RF budget.
Parameters
----------
V_RF : float, optional
Total RF voltage in [V].
The default is 1.7e6.
Returns
-------
ring : Synchrotron object
TDR lattice v2366_v3.
"""
L = 353.97
E0 = 2.75e9
particle = Electron()
ac = 1.05751e-04
tune = np.array([54.2, 18.3])
chro = np.array([1.6, 1.6])
beta = np.array([3.288, 4.003])
alpha = np.array([0, 0])
dispersion = np.array([0, 0, 0, 0])
h = 416
particle = Electron()
mcf_order = np.array([-0.00192226, 0.00061511, 0.00010695])
U0 = 469e3
tau = np.array([7.64e-3, 13.83e-3, 11.64e-3])
sigma_delta = 9.063e-4
emit = np.array([83.7e-12, 83.7e-12*0.01])
optics = Optics(local_beta=beta, local_alpha=alpha,
local_dispersion=dispersion)
ring = Synchrotron(h, optics, particle, L=L, E0=E0, ac=ac, U0=U0, tau=tau,
emit=emit, tune=tune, sigma_delta=sigma_delta, chro=chro,
mcf_order=mcf_order)
tuneS = ring.synchrotron_tune(V_RF)
ring.sigma_0 = (ring.sigma_delta * np.abs(ring.eta())
/ (tuneS * 2 * np.pi * ring. f0))
ring.get_longitudinal_twiss(V_RF)
return ring
ring = SOLEIL_II(V_RF=1.7e6)
I0 = 1.1
MC = CavityResonator(ring, m=1, Rs=5e6, Q=35.7e3, QL=6e3, detune=1e6, Ncav=4)
MC.Vc = 1.7e6
MC.theta = np.arccos(ring.U0/MC.Vc)
MC.set_optimal_detune(I0)
MC.set_generator(I0)
# HC is there just to be able to use scan functions but with RS=1ohm does nothing
HC = CavityResonator(ring, m=4, Rs=1, Q=31e3, QL=31e3, detune=1e6, Ncav=2)
HC.Vg = 0
HC.theta_g = 0
Main cavity tuning vs beam current scan
In the case of a single RF system, the positive tuning side is Robinson unstable (AC Robinson) while for high beam current the negative tuning side is succeptible to zero-frequency instability (DC Robinson or equilibirum phase instability).
Case with synchrotron longitudinal damping
psi_MC_vals = np.linspace(-80, 80, 50)
I0_vals = np.linspace(1e-3, 1.5, 50)
eq_opts = EquilibriumOptions(auto_set_MC_theta=False,
optimal_tunning=False)
result = scan_MC_psi_I0(
SystemState(ring, [MC, HC]),
psi_MC_vals,
I0_vals,
flow=DEFAULT_FLOWS["Alves"],
eq_opts=eq_opts)
result.plot("xi")
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Case without synchrotron longitudinal damping
Without synchrotron damping, the “textbook” picture is obtained.
See Fig.7 of [1] for example.
[1]: Pedersen, F. (1994). Multibunch instabilities. In Frontiers of Particle Beams: Factories with e+ e− Rings (pp. 269-292). Springer, Berlin, Heidelberg.
ring.tau[2] = 100 # Long damping time
psi_MC_vals = np.linspace(-80, 80, 50)
I0_vals = np.linspace(1e-3, 1.5, 50)
eq_opts = EquilibriumOptions(auto_set_MC_theta=False,
optimal_tunning=False)
# Without synchrotron damping, make a custom flow with only ZeroFrequency and
# use RobinsonCoupling to avoid a 'false positive' PTBL because of no synchrotron
# longitudinal damping.
flow_he = Flow(AlvesEquilibrium(),
[ZeroFrequency(), RobinsonCoupling()],
name="Alves")
result = scan_MC_psi_I0(
SystemState(ring, [MC, HC]),
psi_MC_vals,
I0_vals,
flow=flow_he,
eq_opts=eq_opts)
result.plot("xi")
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scan: 98%|█████████▊| 2455/2500 [00:16<00:00, 141.30it/s]
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scan: 100%|█████████▉| 2489/2500 [00:16<00:00, 152.14it/s]
scan: 100%|██████████| 2500/2500 [00:16<00:00, 153.42it/s]
