Brent A. Chipman and Dr. Ross Spencer, Physics and Astronomy
The motivation for this work comes from the paper “Parametrically Pumped Electron Oscillators” by J. Tan and G. Gabrielse (1). Tan and Gabrielse describe a number of unexplained behavior exhibited by electrons confined in a parametrically pumped penning trap. In order to gain a better understanding of the mechanisms behind the behavior, numerical simulations of the trap are necessary. Though the equations of motion for the electrons are relatively easy to write and solve, the amount of computing time required makes this approach practically impossible. The purpose of my research was to develop a faster method for solving the problem.
The proposed method is this. Electrons in a penning trap oscillate back and forth at a frequency called the penning frequency. Each pair of electrons exchanges energy and phase as they interact through the coulomb force. Each pair of particles interacts once every half penning period, the time for half an oscillation through the trap. Standard solution methods typically require about 40,000 steps to simulate the motion through this period of time. Standard solution methods, such as Runge-Kutta 4, start at an initial time and step the solution of the equations up through small amounts of time until a desired final time is reached. Therefore, if a theory could be developed that simply quantified the exchange process in a single formula, a solution to the equations of motion could proceed with time steps on the order of a half penning period bypassing all the short time steps necessary for other methods to simulate the motion.
Since the proposed simulation will not be based on the exact equations of motion, we require some prior knowledge of what a solution should look like in order to test the validity of a simulation. It is easy to show for a simplified model of the penning trap that the average electron energy should remain constant. Second, as the electrons exchange energy they eventually come to a state of thermal equilibrium. This implies a statistical distribution of energies that can be characterized by the root mean square of the energy or Erms. It can be easily shown that this value should equal the value of the average energy when thermal equilibrium is achieved. These two quantities become the test for a good simulation.
To examine the exchange process, we look first at two particles confined in a penning trap. The motion can be dealt with by separating it into center of mass motion and relative motion in the standard way. We find that the center of mass motion of two particles is simple harmonic motion, simple oscillation at the penning frequency. The relative motion can be described as simple harmonic motion with a small perturbation due to the coulomb interaction. This perturbed equation can be solved using the averaging method (2). The solution of the relative motion is also simple harmonic motion, in the averaging method approximation, with a small frequency shift that can be calculated in terms of the relative motion amplitude, phase and radial separation of the interacting electrons.
The expression for the frequency shift given by the averaging method is all that is needed to calculate exchange processes. For two particles, it is now possible to write down the motion of each particle 55 as a simple function of time. This gives very good results. For many particles, this forms the basis for a faster simulation algorithm. It is as follows:
1. Calculate approximate passing times for each pair of electrons. This allows pairwise interaction to be influenced by what has happened before.
2. Determine parameters necessary for the frequency shift calculation and then calculate the frequency shift.
3. Update the position and velocity estimates of each electron through a time equal to a half penning period.
4. Repeat the process for as many half penning periods as desired.
The results of this simulation show that the model requires more work. The average energy in the simulation tends to creep upward. More significantly, the Erms stays much lower than expected. It can be shown that low values of Erms indicate too few low energy particles compared with the ideal distribution predicted by thermodynamics. The averaging method is an approximation that assumes the coulomb interaction is small. It works poorly when the interaction is large. Particles with low energy do not experience much of the external forces of the penning trap and so the coulomb force becomes dominate. In addition, the statistical distribution of thermal equilibrium shows a greater probability for an electron to be at low energy than high energy. Thus the real situation has a high density of low energy electrons strongly interacting through the coulomb force. This is a situation not handled well by the averaging method frequency shift.
Further research then would include an examination of the dynamics of high density, low energy electron clouds. This is the most likely problem area in the current fast simulation. Though the model described in this report did not yield the desired solution, it simulated the electron motion in just a few minutes while the standard solution methods required several hours to simulate the same motion. The promise of greater speed and ultimate explanation of the phenomena described by Tan still make this an intriguing area of research.
References
- J. Tan and G. Gabrielse, “Parametrically Pumped Electron Oscillators”, Physical Review A, vol. 48, no. 4, October 1993.
- R. Baierlein, Newtonian Dynamics, McGraw-Hill, New York, 1983.