Solenoid channel

Proton beam undergoing 90 deg X-Y rotation in an ideal solenoid channel.

The matched Twiss parameters at entry are:

  • \(\beta_\mathrm{x} = 2.4321374875\) m

  • \(\alpha_\mathrm{x} = 0.0\)

  • \(\beta_\mathrm{y} = 2.4321374875\) m

  • \(\alpha_\mathrm{y} = 0.0\)

We use a 250 MeV proton beam with initial unnormalized rms emittance of 1 micron in the horizontal plane, and 2 micron in the vertical plane.

The solenoid magnetic field corresponds to B = 2 T.

The second moments of the particle distribution after the solenoid channel are rotated by 90 degrees: the final horizontal moments should coincide with the initial vertical moments, and vice-versa, 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_solenoid.py or

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

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

Listing 30 You can copy this file from examples/solenoid/run_solenoid.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
# horizontal rms emittance of 1 um and an
# initial vertical rms emittance of 2 um
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=1.559531175539e-3,
    lambdaY=2.205510139392e-3,
    lambdaT=1.0e-3,
    lambdaPx=6.41218345413e-4,
    lambdaPy=9.06819680526e-4,
    lambdaPt=1.0e-3,
)
sim.add_particles(bunch_charge_C, distr, npart)

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

# design the accelerator lattice
sim.lattice.extend(
    [
        monitor,
        elements.Sol(ds=3.820395, ks=0.8223219329893234),
        monitor,
    ]
)

# run simulation
sim.evolve()

# clean shutdown
sim.finalize()
Listing 31 You can copy this file from examples/solenoid/input_solenoid.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 = 1.559531175539e-3
beam.lambdaY = 2.205510139392e-3
beam.lambdaT = 1.0e-3
beam.lambdaPx = 6.41218345413e-4
beam.lambdaPy = 9.06819680526e-4
beam.lambdaPt = 1.0e-3
beam.muxpx = 0.0
beam.muypy = 0.0
beam.mutpt = 0.0


###############################################################################
# Beamline: lattice elements and segments
###############################################################################
lattice.elements = monitor sol1 monitor
lattice.nslice = 1

monitor.type = beam_monitor
monitor.backend = h5

sol1.type = solenoid
sol1.ds = 3.820395
sol1.ks = 0.8223219329893234


###############################################################################
# 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_solenoid.py
Listing 32 You can copy this file from examples/solenoid/analysis_solenoid.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 = 1.3 * 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],
    [
        1.559531175539e-3,
        2.205510139392e-3,
        1.0e-3,
        1.0e-6,
        2.0e-6,
        1.0e-6,
    ],
    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 = 1.3 * 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],
    [
        2.205510139392e-3,
        1.559531175539e-3,
        6.404930308742e-3,
        2.0e-6,
        1.0e-6,
        1.0e-6,
    ],
    rtol=rtol,
    atol=atol,
)