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

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 http://gamma.is.tcu.edu/~quarles/research.html

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 http://gamma.is.tcu.edu/~quarles/research.html The annihilation time tells much about the structure of a material.

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.

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

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.

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.

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.)

Positronium in a Cavity 6.0 a.u.

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

Energy Positronium -0.25000 (theory) Positronium -0.242 ± 0.004 (calculated) Ps in 6.0 a.u. cavity -0.163 ± 0.008 (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.

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.

Extrapolation

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

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

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