FODO Cell, Chromatic

Stable FODO cell with a zero-current phase advance of 67.8 degrees, with chromatic focusing effects included.

The matched Twiss parameters at entry are:

  • \(\beta_\mathrm{x} = 2.82161941\) m

  • \(\alpha_\mathrm{x} = -1.59050035\)

  • \(\beta_\mathrm{y} = 2.82161941\) m

  • \(\alpha_\mathrm{y} = 1.59050035\)

We use a 2 GeV electron beam with initial unnormalized rms emittance of 2 nm.

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 \(\sigma_x\), \(\sigma_y\), \(\sigma_t\), \(\epsilon_x\), \(\epsilon_y\), and \(\epsilon_t\) must agree with nominal values.

Run

This example can be run either as:

  • Python script: python3 run_fodo_chr.py or

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

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

Listing 24 You can copy this file from examples/fodo/run_fodo_chr.py.
#!/usr/bin/env python3
#
# Copyright 2022-2023 ImpactX contributors
# Authors: 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 2 GeV electron beam with an initial
# unnormalized rms emittance of 2 nm
kin_energy_MeV = 2.0e3  # 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(0.510998950).set_kin_energy_MeV(kin_energy_MeV)

#   particle bunch
distr = distribution.Waterbag(
    lambdaX=3.9984884770e-5,
    lambdaY=3.9984884770e-5,
    lambdaT=1.0e-3,
    lambdaPx=2.6623538760e-5,
    lambdaPy=2.6623538760e-5,
    lambdaPt=2.0e-3,
    muxpx=-0.846574929020762,
    muypy=0.846574929020762,
    mutpt=0.0,
)
sim.add_particles(bunch_charge_C, distr, npart)

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

# design the accelerator lattice)
ns = 25  # number of slices per ds in the element
fodo = [
    monitor,
    elements.ChrDrift(ds=0.25, nslice=ns),
    monitor,
    elements.ChrQuad(ds=1.0, k=1.0, nslice=ns),
    monitor,
    elements.ChrDrift(ds=0.5, nslice=ns),
    monitor,
    elements.ChrQuad(ds=1.0, k=-1.0, nslice=ns),
    monitor,
    elements.ChrDrift(ds=0.25, nslice=ns),
    monitor,
]
# assign a fodo segment
sim.lattice.extend(fodo)

# run simulation
sim.evolve()

# clean shutdown
sim.finalize()
Listing 25 You can copy this file from examples/fodo/input_fodo_chr.in.
###############################################################################
# Particle Beam(s)
###############################################################################
beam.npart = 10000
beam.units = static
beam.kin_energy = 2.0e3
beam.charge = 1.0e-9
beam.particle = electron
beam.distribution = waterbag
beam.lambdaX = 3.9984884770e-5
beam.lambdaY = 3.9984884770e-5
beam.lambdaT = 1.0e-3
beam.lambdaPx = 2.6623538760e-5
beam.lambdaPy = 2.6623538760e-5
beam.lambdaPt = 2.0e-3
beam.muxpx = -0.846574929020762
beam.muypy = 0.846574929020762
beam.mutpt = 0.0


###############################################################################
# Beamline: lattice elements and segments
###############################################################################
lattice.elements = monitor drift1 monitor quad1 monitor drift2 monitor quad2 monitor drift3 monitor
lattice.nslice = 25

monitor.type = beam_monitor
monitor.backend = h5

drift1.type = drift_chromatic
drift1.ds = 0.25

quad1.type = quad_chromatic
quad1.ds = 1.0
quad1.k = 1.0

drift2.type = drift_chromatic
drift2.ds = 0.5

quad2.type = quad_chromatic
quad2.ds = 1.0
quad2.k = -1.0

drift3.type = drift_chromatic
drift3.ds = 0.25


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


###############################################################################
# Diagnostics
###############################################################################
diag.slice_step_diagnostics = true

Analyze

We run the following script to analyze correctness:

Script analysis_fodo_chr.py
Listing 26 You can copy this file from examples/fodo/analysis_fodo_chr.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],
    [
        7.5451170454175073e-005,
        7.5441588239210947e-005,
        9.9775878164077539e-004,
        1.9959540393751392e-009,
        2.0175015289132990e-009,
        2.0013820193294972e-006,
    ],
    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],
    [
        7.4790118496224206e-005,
        7.5357525169680140e-005,
        9.9775879288128088e-004,
        1.9959539836392703e-009,
        2.0175014668882125e-009,
        2.0013820380883801e-006,
    ],
    rtol=rtol,
    atol=atol,
)