Amy Bug, Tim Cronin and Zach Wolfson Dept. of Physics and Astronomy, Swarthmore College, U.S.A. Philip Sterne Lawrence Livermore National Laboratory, U.S.A. Simulation of Ps: Toward more realistic models of void spaces in materials

In insulating materials: Ps forms, thermalizes in defects, cages, bubbles …, and annihilates either by pickoff or self-annihilation PALS and ACAR indicate size distribution, contents, and chemical nature of voids  p) ~ ∑ n | ∫ dr e -ip.r   (r)  n (r) √  [   ( r)] | 2  -1 ≈  r e 2 c ∫ dr  dr +   (r + )   (r  )  [   ( r  )]    r  - r + )

This talk: Application of a two-particle model of Ps to three situations of interest … Pore within a polarizable, 1 Pore within a polarizable, dielectric medium Fluid-filled, 3 Fluid-filled, cylindrical pores cylindrical pores Cylindrical 2 Cylindrical geometries geometries Suzuki et al, 2001

Ground state + Single particle + hard sphere = Tao-Eldrup (TE) model Data from molecular solids (Jean, 1995)  -1 =   -1 [  / R c + (1/2  ) sin(2  (R c -  / R c )] Mixture of states + Single particle + Single particle + hard parallelpiped = Extended Tao-Eldrup (RTE) model Brandt et al, 1960; Tao, 1972; Eldrup et al, Itoh et al, 1999; Gidley et al, 1999; Gorowek et al, 2002 These are single particle-in-a-box (SPIB) models … Data from silicas and zeolites (Dull, 2001) RcRc Data typically fit with  =1.66 Å,    0.5ns

We simulate Ps in materials with two-chain Path Integral Monte Carlo (PIMC) The Quantum density matrix:  (  ) = exp( -  H) is represented in the position basis: = ∫... d r 1 … r P-1 (  P) The solution of the Bloch equation for Ps is instantiated by two chains of “beads” which have become analogous to two interacting, harmonic, ring polymers. The location of each e+ bead is determined by the likelihood of measuring e+ at this location in the solid. Ps wave packet (cf. single-chain model: Miller, Reese et al, 1996, 2002) e- e+ Ps “chains” Beyond SPIB models …

Positions of beads in Ps

Positions of beads in Ps

Positions of beads in Ps

Positions of beads in Ps

Positions of beads in Ps

Positions of beads in Ps

Positions of beads in Ps

Positions of beads in Ps

Application of a two-particle model of Ps to … Pore within a polarizable, dielectric medium 1 Pore within a polarizable, dielectric medium Wave function and lifetime are a function of dielectric constant, k o Even a non-polar medium interacts electrically with e+ Ps is polarized and attracted to surface of dielectric Polarizability:  = 36  E k o > 1

Annihilation / Polarization model k o > 1 RcRc

Dielectric material polarizes Ps and alters distribution of e + within spherical void ground state, R c = 10 au k o = 1 k o = 15

In absence of polarization: Lifetimes predicted by two-particle model of Ps are dramatically increased over SPIB predictions … T = 1053K  = 300) T = 2106K  = 150) 1 particle PIMC SPIB x 2 particle PIMC

Radial distribution function for e + ground state, R c = 20 au k o = 3 k o = 1 SPIB

Pickoff lifetime varies with R c in a way which is sensitive to k o Single positron in pore Ps k o = 3 k o = 1 + (ns)

Pickoff lifetime varies with k o in a way which is sensitive to R c (ns)

Application of a two-particle model of Ps to … Cylindrical geometries 2 Cylindrical geometries SPIB and two-chain Ps lifetimes are functions of pore radius and temperature Many interesting materials contain void spaces which are well-modelled as cylindrical pores Bamford et al, 2001

SPIB: For ground state behavior, a scaling relationship maps the Tao-Eldrup lifetime in a sphere directly to lifetime in a cylinder cylinder radius R Sphere radius R Sphere radius  R See poster by T. Cronin et al. for details …

Ps: Orbital in cylindrical pore: Probablility density for relative coordinate in terms of r, z, and  R c = 25 au

 = 71 ns  = 52 ns  = 46 ns Ps: Probablility density for e+ coordinate shows that lifetime decreases with temperature …

61  1/2 free R Q Orbital becomes anisotropic … but is compressed in axial and radial directions

Application of a two-particle model of Ps to … Fluid-filled, cylindrical pores 3 Fluid-filled, cylindrical pores Cylindrical pores are good models for connected void spaces which can adsorb fluid Fluid atoms and pore walls compete to annihilate Ps Iannacchione et al, 1996

MMC simulation is performed at constant fluid density and temperature Argon-lepton potential U (r) (au)

Snapshots from MMC simulation at constant fluid density and temperature before equilibration in equilibrium

Radial distribution function for e+ : R c = 16 au, T =632K At intermediate densities, Ar excludes volume and Ps expands outward in pore … compressed at higher densities into “bubble”

Snapshots from MMC simulation at two densities  * = 0.35  * = 0.25

Pickoff rate with wall and with fluid atoms as a function of fluid density R c =16 au, T = 632 K

Ps orients preferentially in pore which is cylindrical or spherical Ps orients preferentially in pore which is cylindrical or spherical

Ps orients preferentially in a spherical geometry … k o does not strongly effect orientation

In conclusion … Two particle, PIMC calculations of Ps in pores taking into account temperature dielectric polarizability of the bulk solid elongated (cylindrical) geometry of pores the presence of fluid atoms yield lifetimes and information on the structure of Ps. It is hoped that encorporating these realistic features of materials will allow these (fairly simple) calculations to further refine our understanding of results of PALS experiments.

Many thanks to... Colleagues: Roy Pollock (LLNL), Terrence Reese (Southern U) Students at Swarthmore: Lisa Larrimore, Robert McFarland, Peter Hastings, Jillian Waldman Funding agencies: Provost’s office of Swarthmore College Petrolium Research fund of the ACS U.S. DOE The Organizing Committee (Adriano de Lima,Chair) and Participants of PPC-8