# Particle stepper¶

An example of PlasmaPy’s particle stepper class, currently in need of a rewrite for speed.

import numpy as np
from astropy import units as u
from plasmapy.classes import Plasma
from plasmapy.simulation import ParticleTracker
from plasmapy.formulary import gyrofrequency


Initialize a plasma. This will be a source of electric and magnetic fields for our particles to move in.

plasma = Plasma(domain_x=np.linspace(-1, 1, 10) * u.m,
domain_y=np.linspace(-1, 1, 10) * u.m,
domain_z=np.linspace(-1, 1, 10) * u.m)


Initialize the fields. We’ll take B in the x direction and E in the y direction, which gets us an E cross B drift in the z direction.

B0 = 4 * u.T
plasma.magnetic_field[0, :, :, :] = np.ones((10, 10, 10)) * B0

E0 = 2 * u.V / u.m
plasma.electric_field[1, :, :, :] = np.ones((10, 10, 10)) * E0


Calculate the timestep. We’ll take one proton p, take its gyrofrequency, invert that to get to the gyroperiod, and resolve that into 10 steps for higher accuracy.

freq = gyrofrequency(B0, 'p').to(u.Hz, equivalencies=u.dimensionless_angles())
gyroperiod = (1/freq).to(u.s)
steps_to_gyroperiod = 10
timestep = gyroperiod / steps_to_gyroperiod


Initialize the trajectory calculation.

number_steps = steps_to_gyroperiod * int(2 * np.pi)
trajectory = ParticleTracker(plasma, 'p', 1, 1, timestep, number_steps)


We still have to initialize the particle’s velocity. We’ll limit ourselves to one in the x direction, parallel to the magnetic field B - that way, it won’t turn in the z direction.

trajectory.v[0][0] = 1 * (u.m / u.s)


Run the pusher and plot the trajectory versus time.

trajectory.run()
trajectory.plot_time_trajectories()


Plot the shape of the trajectory in 3D.

trajectory.plot_trajectories()


As a test, we calculate the mean velocity in the z direction from the velocity and position

vmean = trajectory.velocity_history[:, :, 2].mean()
print(f"The calculated drift velocity is {vmean:.4f} to compare with the"
f"theoretical E0/B0 = {E0/B0:.4f}")


Out:

The calculated drift velocity is 0.5233 m / s to compare with thetheoretical E0/B0 = 0.5000 V / (m T)


and from position:

Vdrift = trajectory.position_history[-1, 0, 2] / (trajectory.NT * trajectory.dt)
print(f"The calculated drift velocity from position is {Vdrift:.4f}")


Out:

The calculated drift velocity from position is 0.5232 m / s


Total running time of the script: ( 0 minutes 0.815 seconds)

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