Simulation of Plasma Particle Trajectories Using LabVIEW

Author(s):
Hans Pfister - Dickinson College Department of Physics and Astronomy
Christopher Graham - Dickinson College Department of Physics and Astronomy

Industry:
University/Education

Products:
LabVIEW

The Challenge:
Demonstrating an alternative venue for LabVIEW as an excellent pedagogical tool.

The Solution:
Using LabVIEW to simulate the motion of plasma particles in several electric and magnetic field configurations

Introducing Students and Researchers to Plasma Physics
In recent years, plasma applications have become increasingly widespread in industrial and commercial manu­facturing processes. This growth fosters a need to educate additional students and researchers about the basic plasma physics concepts that underlie such applications.

Scientists typically describe plasmas, or highly ionized gases, with either magneto­hydro­dyna­mic (MHD) equations or by considering the quasineutral super­heated gases as ensembles of indepen­dently moving, single particles. Plasmas characterized by frequent particle collisions are best described by MHD; at the other extreme, plasmas in which individual particles have few collisions and large mean free paths are best understood by looking at the trajectories of single particles.

Using LabVIEW as a Pedagogical Tool
There are many advantages to using National Instruments LabVIEW as a tool for simulating such single-particle trajectories. Using NI LabVIEW is simple and straightforward, given the array of knobs, dials, and displays that engineers use to easily manipulate parameters. Most importantly, because engineers can perform these adjustments while the program is executing, they can see instantaneously the effect of a parameter change on the behavior of the entire system.

We have illustrated the pedagogical effectiveness of LabVIEW by presenting two basic plasma particle trajectories: a simple Larmor orbit and an EB particle drift. We calculated the trajectories presented with an Euler recursion, a modified Euler recursion, or a fourth-order Runge-Kutta algorithm.

For the sake of simplicity, we started with an Euler recursion. The beauty of this algorithm is that no prior knowledge of the particle’s trajectory is required. That is, we did not theoretically calculate and plot the equation of the orbit. Instead, we followed the particle on its trajectory in a step-by-step fashion. At each step, we used the forces acting on the particle to calculate its acceleration. With this acceleration, we calculated an updated velocity and, consequently, a new position. At this new position, we again calculated a new acceleration and repeated the process until the particle’s path was fully described.

Extending the procedure to 2-D motion was a relatively simple task. We cycled through the steps for the x components (ax, vx, and x) and simultaneously through the corresponding steps for the y components (ay, vy, and y). However, while the parabolic motion of a cannonball, for example, was characterized by an independent motion in the x- and y-directions (i.e., ay does not depend on the motion in the x-direction), circular motions, such as an orbiting planet or the Larmor orbit of a charged particle in a magnetic field, were characterized by the dependence of ax on vy and ay on vx, which necessitated simultaneous calculations for the x and y components.

As powerful and straightforward as the Euler recursion method was, its accuracy was limited by the method’s finite step size, leading to a circular trajectory that does not quite close upon itself. While a reduction of the step size reduced the error in the calcu­lated trajectory, the limited capacity of any computer prevented us from achieving the infinitesimal step size necessary to exactly reproduce the actual particle trajectory. We turned instead to two alternative algorithms.

Two Alternative Algorithms
The first alternate algorithm, which we called the modified Euler recursion, consisted of a slight modification in the order of calculations. In the unmodified Euler algorithm, the program simul­taneously calculated both the x- and y-components of a given time step, using the values of the previous time step.

Conversely, the modified Euler algorithm first calculated all of the x-component quantities at a given step, based on the values of the previous step. Then it updated the y-component values based on the most recently calculated x-component quantities rather than on the values of the previous iteration. This slight change in sequence in the algorithm and the corresponding simple rewiring of the LabVIEW program made the Larmor orbit close upon itself.

The second alternative, which was more accurate but slightly more complex, was a fourth-order Runge-Kutta (RK-4) algorithm. The result was a simulated motion that quite accurately represented the actual motion of the particle, but required considerably more wiring in the LabVIEW program.

Independent of a particular algorithm, we ended with a front panel that showed the particle trajectory and a series of dials, knobs, and sliders, which we used for a quick change of the physical parameters that influenced the particle’s trajectory. For example, increasing the magnetic field by a simple twist of the virtual B-field knob instantaneously resulted in a particle trajectory that had a smaller Larmor radius. Turning the electric field dial changed the drift velocity of the particle. Because the response of the output graph to any change of system parameters was immediate, the students were able to quickly verify their understanding of the underlying physics of the system at hand.

Even for those people not familiar with plasma physics, the particle simulations presented exemplify the usefulness of NI LabVIEW as a valuable pedagogical tool.

For more information, contact:
Christopher Graham or Hans Pfister
Department of Physics and Astronomy
Dickinson College
P.O. Box 17773
Carlisle, PA 17013-1642
Tel: (717) 245-1307
Fax: (717) 245-1642
E-mail: pfister@dickinson.edu

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