Kurth Distribution in a Constant Focusing Channel
Stationary Kurth distribution in a constant focusing channel (without space charge).
The distribution is radially symmetric in (x,y,t) space, and matched to a radially symmetric constant linear focusing.
We use a 2 GeV proton beam with initial unnormalized rms emittance of 1 um in all three phase planes.
The particle distribution should remain unchanged, to within the level expected due to numerical particle noise. This fact is independent of the length of the channel. This is tested using the second moments of the distribution.
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 as a Python script (python3 run_kurth.py) or with an app with an input file (impactx input_kurth.in).
Each can also be prefixed with an MPI executor, such as mpiexec -n 4 ... or srun -n 4 ..., depending on the system.
#!/usr/bin/env python3
#
# Copyright 2022 ImpactX contributors
# Authors: Ryan Sandberg, Axel Huebl, Chad Mitchell
# License: BSD-3-Clause-LBNL
#
# -*- coding: utf-8 -*-
import amrex
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 proton beam with an initial
# unnormalized rms emittance of 1 um in each
# coordinate plane
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(938.27208816).set_energy_MeV(energy_MeV)
# particle bunch
distr = distribution.Kurth6D(
sigmaX=1.0e-3,
sigmaY=1.0e-3,
sigmaT=3.369701494258956e-4,
sigmaPx=1.0e-3,
sigmaPy=1.0e-3,
sigmaPt=2.9676219145931020e-3,
)
sim.add_particles(bunch_charge_C, distr, npart)
# design the accelerator lattice: here we just assign a single element
sim.lattice.append(elements.ConstF(ds=2.0, kx=1.0, ky=1.0, kt=1.0))
# run simulation
sim.evolve()
# clean shutdown
del sim
amrex.finalize()
###############################################################################
# Particle Beam(s)
###############################################################################
beam.npart = 10000
beam.units = static
beam.energy = 2.0e3
beam.charge = 1.0e-9
beam.particle = proton
beam.distribution = kurth6d
beam.sigmaX = 1.0e-3
beam.sigmaY = 1.0e-3
beam.sigmaT = 3.369701494258956e-4
beam.sigmaPx = 1.0e-3
beam.sigmaPy = 1.0e-3
beam.sigmaPt = 2.9676219145931020e-3
beam.muxpx = 0.0
beam.muypy = 0.0
beam.mutpt = 0.0
###############################################################################
# Beamline: lattice elements and segments
###############################################################################
lattice.elements = constf1
constf1.type = constf
constf1.ds = 2.0
constf1.kx = 1.0
constf1.ky = 1.0
constf1.kt = 1.0
###############################################################################
# Algorithms
###############################################################################
algo.particle_shape = 2
algo.space_charge = false
Analyze
We run the following script to analyze correctness:
Script analysis_kurth.py
#!/usr/bin/env python3
#
# Copyright 2022 ImpactX contributors
# Authors: Axel Huebl, Chad Mitchell
# License: BSD-3-Clause-LBNL
#
import glob
import numpy as np
import pandas as pd
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["x"], moment=2) ** 0.5 # variance -> std dev.
sigpx = moment(beam["px"], moment=2) ** 0.5
sigy = moment(beam["y"], moment=2) ** 0.5
sigpy = moment(beam["py"], moment=2) ** 0.5
sigt = moment(beam["t"], moment=2) ** 0.5
sigpt = moment(beam["pt"], moment=2) ** 0.5
epstrms = beam.cov(ddof=0)
emittance_x = (sigx**2 * sigpx**2 - epstrms["x"]["px"] ** 2) ** 0.5
emittance_y = (sigy**2 * sigpy**2 - epstrms["y"]["py"] ** 2) ** 0.5
emittance_t = (sigt**2 * sigpt**2 - epstrms["t"]["pt"] ** 2) ** 0.5
return (sigx, sigy, sigt, emittance_x, emittance_y, emittance_t)
def read_all_files(file_pattern):
"""Read in all CSV files from each MPI rank (and potentially OpenMP
thread). Concatenate into one Pandas dataframe.
Returns
-------
pandas.DataFrame
"""
return pd.concat(
(
pd.read_csv(filename, delimiter=r"\s+")
for filename in glob.glob(file_pattern)
),
axis=0,
ignore_index=True,
).set_index("id")
# initial/final beam on rank zero
initial = read_all_files("diags/beam_000000.*")
final = read_all_files("diags/beam_final.*")
# 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 # a big number
rtol = 1.5 * 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],
[
1.0e-03,
1.0e-03,
3.369701494258956e-4,
1.0e-6,
1.0e-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 # a big number
rtol = 1.5 * 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],
[
1.0e-03,
1.0e-03,
3.369701494258956e-4,
1.0e-6,
1.0e-6,
1.0e-6,
],
rtol=rtol,
atol=atol,
)