1. Computer Simulations of
Positronium
Lisa Larrimore and Robert McFarland
Dept. of Physics and Astronomy
Swarthmore College

2. Phenomenon of Pair Annihilation
Why are we doing this?
Phenomenon of Pair Annihilation
The GIF at right illustrates positronium annihilation. When an e+ and e- come into contact, they may interact and form positronium (Ps) before annihilating into a pair of photons
GIF courtesy of C.A Quarles of TCU

3. Inject positron into a solid
Why are we doing this? (continued)
Inject positron into a solid
“Pick-off” annihilation with solid electrons
GIF courtesy of C.A Quarles of TCU
The annihilation time tells much about the structure of a material.

4. How are we doing this?
We are working in computational physics, creating computer models of particles.
We are using a technique called monte carlo path integration to simulate the behavior of a quantum mechanical system.
This method involves representing a single particle as a chain of many beads, and a pair of interacting particles as two interacting chains.

5. Each particle is represented as a chain of beads.
Simulation Methods:
Each particle is represented as a chain of beads.

6. Two interacting chains = Ps
Positron
Positronium
Electron
All these beads are not really there. In any real measurement, you will find only one particle.
The density of beads in a particular region tell us how likely we are to find the particle in that region.
Knowing where the positron beads are helps us predict what should happen in a positron annihilation experiment.

7. Program testing: Hydrogen
We tested our program using one string of beads to model one particle, the electron in a hydrogen atom.
Hydrogen is tricky: computers are stupid and will crash the electron into the nucleus since the Coulomb potential goes to -∞ at the nucleus.
Instead of the Coulomb potential, we use the Yukawa potential, developed by Muser and Berne for this use in 1997.

8. Hydrogen and Positronium
We construct the radial density, the probability of the electron and positron beads being separated by a certain distance.
(For hydrogen, this is the distance from the electron to the nucleus.)

9. Positronium in a Cavity
6.0 a.u.

10. This is LTA, a zeolite, one of the many solids with cavities one might want to probe using positrons.

12. Energy Positronium -0.25000 (theory)
Positronium ± (calculated)
Ps in 6.0 a.u. cavity ± (calculated)
Ps in 4.0 a.u. cavity ionized!
• As a cavity squeezes positronium, the energy rises.
• A cavity of 4.0 a.u. is so small that positronium can no longer exist.

13. Methods note: How do we calculate the energy?
There is a constant a associated with the Yukawa potential, which describes how the Yukawa potential is to the Coulomb potential.
The Yukawa potential becomes a better approximation as a goes to zero and as the number of beads goes to infinity.
We extrapolate our data to this ideal.

14. Extrapolation

15. The Stark Effect z No Electric Field Weak Electric Field Dipole Moment
= e • z z

16. Visualizing the Stark Effect Using IDL
Probability densities for positronium.
No Electric Field
Weak Electric Field

17. Simulation of Ps in Electric Field
Polarizability:
a = 36 (theory)
a ≈ 39 (simulation)