Soft-Edge Quadrupole

This is a modification of the test for a matched electron beam propagating through a stable FODO cell, in which the quadrupoles have been replaced with soft-edge quadrupole elements. The on-axis field profile in this example has been chosen to correspond to the hard-edge limit, so the two tests should coincide.

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

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_quadrupole_softedge.py or

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

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

Python: Script

Listing 36 You can copy this file from examples/quadrupole_softedge/run_quadrupole_softedge.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 = 1  # number of slices per ds in the element

quad1 = elements.SoftQuadrupole(
    ds=1.0,
    gscale=1.0,
    cos_coefficients=[2],
    sin_coefficients=[0],
    mapsteps=400,
    nslice=ns,
)

quad2 = elements.SoftQuadrupole(
    ds=1.0,
    gscale=-1.0,
    cos_coefficients=[2],
    sin_coefficients=[0],
    mapsteps=200,
    nslice=ns,
)

drift1 = elements.Drift(ds=0.25, nslice=ns)
drift2 = elements.Drift(ds=0.5, nslice=ns)

# assign a fodo segment
sim.lattice.extend([monitor, drift1, quad1, drift2, quad2, drift1, monitor])

# run simulation
sim.evolve()

# clean shutdown
sim.finalize()

Executable: Input File

Listing 37 You can copy this file from examples/quadrupole_softedge/input_quadrupole_softedge.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 quad1 drift2 quad2 drift1 monitor
lattice.nslice = 1

monitor.type = beam_monitor
monitor.backend = h5

drift1.type = drift
drift1.ds = 0.25

quad1.type = quadrupole_softedge
quad1.ds = 1.0
quad1.gscale = 1.0
quad1.cos_coefficients = 2.0
quad1.sin_coefficients = 0.0
quad1.mapsteps = 400

drift2.type = drift
drift2.ds = 0.5

quad2.type = quadrupole_softedge
quad2.ds = 1.0
quad2.gscale = -1.0
quad2.cos_coefficients = 2.0
quad2.sin_coefficients = 0.0
quad2.mapsteps = 400


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


###############################################################################
# Diagnostics
###############################################################################
diag.slice_step_diagnostics = false

Analyze

We run the following script to analyze correctness:

Script analysis_quadrupole_softedge.py

Listing 38 You can copy this file from examples/quadrupole_softedge/analysis_quadrupole_softedge.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,
)