Keywords:
single particle, circular motion, finite difference effects
This example shows how to simulate the uniform circular motion of a single electron in a constant, uniform magnetic field in VSim. The electron is loaded inside a cylindrical capacitor with grounded walls to eliminate any stray electric fields. The electron is loaded far from the walls to reduce any effects from image charges. The magnetic field points down the positive z-axis.
Due to the finite difference algorithm utilized by VSim, two corrections must be made in order to get the electron to take a true circular trajectory. The first correction is to the cyclotron frequency. In the finite difference world of VSim, the electron does not move along a circular arc from time step to time step, instead it moves along a straight line. To correct for this we need to set our \(\omega_{cyclo_{FD}} = \frac{2}{\Delta T} \arctan \left( \frac{\omega_{cyclo} \Delta T}{2} \right)\) [1] (see chapter 4 section 3).
The next correction is to account for the implementation of the Boris Method [1], the algorithm used in VSim to push particles. In the Boris Method, the position of the particle, \(\vec{x}(t)\), is defined at full time steps, while the velocity, \(\vec{v}(t)\), is defined at half time steps. This scheme of ‘well-centered’ derivatives means that VSim is automatically accurate to second order, but it means we have to be careful about our initial conditions for the electron’s velocity. The initial velocity is set under Particle Dynamics → Kinetic Particle → electrons0 → particleLoader0 then velocity distribution. VSim will assume that this is the particle’s velocity a ONE HALF time step before the start of the simulation, so we must load the particle with the velocity it would have a half time step before the start of the simulation.
This simulation can be performed with a VSimBase license.
The Single Particle Circular Motion example is accessed from within VSimComposer by the following actions:
All of the properties and values that create the simulation are
now available in the Setup Window as shown in
Fig. 491.
You can expand the tree elements and navigate through the various
properties, making any changes you desire. The right pane shows
a 3D view of the geometry, if any, as well as the grid, if
actively shown. To show or hide the grid, expand the Grid
element and select or deselect the box next to Grid
.
The Single Particle Circular Motion example includes some constants for easy adjustment of simulation properties:
After performing the above actions, continue as follows:
In serial, this simulation only takes seconds to run.
After performing the above actions, continue as follows:
Proceed to the Visualize Window by pressing the Visualize button in the left column of buttons.
Simulations are correct only to some accuracy. The corrections we made to the cyclotron frequency and the initial velocity make this simulation correct to second order. By looking at the phase space plot, we can explore the second order accuracy of this simulation. Navigate to the Visualize Window, select ‘Phase Space’ from the ‘Data View’ drop down menu, and plot ‘electrons0_r’ vs ‘electrons0_ur.’
As you scroll through the dumps for the first time (with the ‘Auto Reset’ box UN-checked), the axes will adjust. The particle is taking an elliptical path in phase space. In a perfect simulation, the electron would remain at the same position in phase space with constant radius and zero radial velocity. Instead, the electron oscillates between the positions \(r = 0.50500\) m and \(r = 0.50542\) m for a \(\Delta r = 0.00042\) m. Cut the time step in half and double the number of timesteps taken (so that the simulation runs through the same amount of time). Now look at the phase space plot again. By approximately what factor did \(\Delta r\) drop? Since the simulation is correct to second order, dropping the time step by a factor of 2 should drop the error by a factor of 4.
Other things you can play around with:
[1] Birdsall, C. K., & Langdon, A. B. (1985). Plasma Physics via Computer Simulation. New York: McGraw-Hill.