FODO Cell with RF

Stable FODO cell with short RF (buncher) cavities added for longitudinal focusing. The phase advance in all three phase planes is between 86-89 degrees.

The matched Twiss parameters at entry are:

  • \(\beta_\mathrm{x} = 9.80910407\) m

  • \(\alpha_\mathrm{x} = 0.0\)

  • \(\beta_\mathrm{y} = 1.31893788\) m

  • \(\alpha_\mathrm{y} = 0.0\)

  • \(\beta_\mathrm{t} = 4.6652668782\) m

  • \(\alpha_\mathrm{t} = 0.0\)

We use a 250 MeV proton beam with initial unnormalized rms emittance of 1 mm-mrad in all three phase planes.

The second moments of the particle distribution after the FODO cell should coincide with the second moments of the particle distribution before the FODO cell, to within the level expected due to noise due to statistical sampling.

In this test, the initial and final values of \(\lambda_x\), \(\lambda_y\), \(\lambda_t\), \(\epsilon_x\), \(\epsilon_y\), and \(\epsilon_t\) must agree with nominal values.

Run

This example can be run either as:

  • Python script: python3 run_fodo_rf.py or

  • ImpactX executable using an input file: impactx input_fodo_rf.in

For MPI-parallel runs, prefix these lines with mpiexec -n 4 ... or srun -n 4 ..., depending on the system.

Listing 21 You can copy this file from examples/fodo_rf/run_fodo_rf.py.
#!/usr/bin/env python3
#
# Copyright 2022-2023 ImpactX contributors
# Authors: Marco Garten, Axel Huebl, Chad Mitchell
# License: BSD-3-Clause-LBNL
#
# -*- coding: utf-8 -*-

from impactx import ImpactX, distribution, elements

sim = ImpactX()

# set numerical parameters and IO control
sim.particle_shape = 2  # B-spline order
sim.space_charge = False
# sim.diagnostics = False  # benchmarking
sim.slice_step_diagnostics = True

# domain decomposition & space charge mesh
sim.init_grids()

# load a 250 MeV proton beam with an initial
# unnormalized rms emittance of 1 mm-mrad in all
# three phase planes
kin_energy_MeV = 250.0  # reference energy
bunch_charge_C = 1.0e-9  # used with space charge
npart = 10000  # number of macro particles

#   reference particle
ref = sim.particle_container().ref_particle()
ref.set_charge_qe(1.0).set_mass_MeV(938.27208816).set_kin_energy_MeV(kin_energy_MeV)

#   particle bunch
distr = distribution.Waterbag(
    lambdaX=3.131948925200e-3,
    lambdaY=1.148450209423e-3,
    lambdaT=2.159922887089e-3,
    lambdaPx=3.192900088357e-4,
    lambdaPy=8.707386631090e-4,
    lambdaPt=4.62979491526e-4,
)
sim.add_particles(bunch_charge_C, distr, npart)

# add beam diagnostics
monitor = elements.BeamMonitor("monitor", backend="h5")

# design the accelerator lattice
sim.lattice.append(monitor)
#   Quad elements
quad1 = elements.Quad(ds=0.15, k=2.5)
quad2 = elements.Quad(ds=0.3, k=-2.5)
#   Drift element
drift1 = elements.Drift(ds=1.0)
#   Short RF cavity element
shortrf1 = elements.Buncher(V=0.01, k=15.0)

lattice_no_drifts = [quad1, shortrf1, quad2, shortrf1, quad1]
#   set first lattice element
sim.lattice.append(lattice_no_drifts[0])
#   intersperse all remaining elements of the lattice with a drift element
for element in lattice_no_drifts[1:]:
    sim.lattice.extend([drift1, element])

sim.lattice.append(monitor)

# run simulation
sim.evolve()

# clean shutdown
sim.finalize()
Listing 22 You can copy this file from examples/fodo_rf/input_fodo_rf.in.
###############################################################################
# Particle Beam(s)
###############################################################################
beam.npart = 10000
beam.units = static
beam.kin_energy = 250.0
beam.charge = 1.0e-9
beam.particle = proton
beam.distribution = waterbag
beam.lambdaX = 3.131948925200e-3
beam.lambdaY = 1.148450209423e-3
beam.lambdaT = 2.159922887089e-3
beam.lambdaPx = 3.192900088357e-4
beam.lambdaPy = 8.707386631090e-4
beam.lambdaPt = 4.62979491526e-4
beam.muxpx = 0.0
beam.muypy = 0.0
beam.mutpt = 0.0


###############################################################################
# Beamline: lattice elements and segments
###############################################################################
lattice.elements = monitor quad1 drift1 shortrf1 drift1 quad2 drift1          \
                   shortrf1 drift1 quad1 monitor

monitor.type = beam_monitor
monitor.backend = h5

quad1.type = quad
quad1.ds = 0.15
quad1.k = 2.5

drift1.type = drift
drift1.ds = 1.0

shortrf1.type = buncher
shortrf1.V = 0.01
shortrf1.k = 15.0

quad2.type = quad
quad2.ds = 0.3
quad2.k = -2.5


###############################################################################
# Algorithms
###############################################################################
algo.particle_shape = 2
algo.space_charge = false

Analyze

We run the following script to analyze correctness:

Script analysis_fodo_rf.py
Listing 23 You can copy this file from examples/fodo_rf/analysis_fodo_rf.py.
#!/usr/bin/env python3
#
# Copyright 2022-2023 ImpactX contributors
# Authors: Axel Huebl, Chad Mitchell
# License: BSD-3-Clause-LBNL
#

import numpy as np
import openpmd_api as io
from scipy.stats import moment


def get_moments(beam):
    """Calculate standard deviations of beam position & momenta
    and emittance values

    Returns
    -------
    sigx, sigy, sigt, emittance_x, emittance_y, emittance_t
    """
    sigx = moment(beam["position_x"], moment=2) ** 0.5  # variance -> std dev.
    sigpx = moment(beam["momentum_x"], moment=2) ** 0.5
    sigy = moment(beam["position_y"], moment=2) ** 0.5
    sigpy = moment(beam["momentum_y"], moment=2) ** 0.5
    sigt = moment(beam["position_t"], moment=2) ** 0.5
    sigpt = moment(beam["momentum_t"], moment=2) ** 0.5

    epstrms = beam.cov(ddof=0)
    emittance_x = (sigx**2 * sigpx**2 - epstrms["position_x"]["momentum_x"] ** 2) ** 0.5
    emittance_y = (sigy**2 * sigpy**2 - epstrms["position_y"]["momentum_y"] ** 2) ** 0.5
    emittance_t = (sigt**2 * sigpt**2 - epstrms["position_t"]["momentum_t"] ** 2) ** 0.5

    return (sigx, sigy, sigt, emittance_x, emittance_y, emittance_t)


# initial/final beam
series = io.Series("diags/openPMD/monitor.h5", io.Access.read_only)
last_step = list(series.iterations)[-1]
initial = series.iterations[1].particles["beam"].to_df()
final = series.iterations[last_step].particles["beam"].to_df()

# compare number of particles
num_particles = 10000
assert num_particles == len(initial)
assert num_particles == len(final)

print("Initial Beam:")
sigx, sigy, sigt, emittance_x, emittance_y, emittance_t = get_moments(initial)
print(f"  sigx={sigx:e} sigy={sigy:e} sigt={sigt:e}")
print(
    f"  emittance_x={emittance_x:e} emittance_y={emittance_y:e} emittance_t={emittance_t:e}"
)

atol = 0.0  # ignored
rtol = 2.2 * num_particles**-0.5  # from random sampling of a smooth distribution
print(f"  rtol={rtol} (ignored: atol~={atol})")

assert np.allclose(
    [sigx, sigy, sigt, emittance_x, emittance_y, emittance_t],
    [
        3.145694e-03,
        1.153344e-03,
        2.155082e-03,
        9.979770e-07,
        1.008751e-06,
        1.000691e-06,
    ],
    rtol=rtol,
    atol=atol,
)


print("")
print("Final Beam:")
sigx, sigy, sigt, emittance_x, emittance_y, emittance_t = get_moments(final)
print(f"  sigx={sigx:e} sigy={sigy:e} sigt={sigt:e}")
print(
    f"  emittance_x={emittance_x:e} emittance_y={emittance_y:e} emittance_t={emittance_t:e}"
)

atol = 0.0  # ignored
rtol = 2.2 * num_particles**-0.5  # from random sampling of a smooth distribution
print(f"  rtol={rtol} (ignored: atol~={atol})")

assert np.allclose(
    [sigx, sigy, sigt, emittance_x, emittance_y, emittance_t],
    [
        3.112318e-03,
        1.153322e-03,
        2.166501e-03,
        9.979770e-07,
        1.008751e-06,
        1.000691e-06,
    ],
    rtol=rtol,
    atol=atol,
)