1 Amy Bug, Melaku Muluneh and Jillian Waldman Dept. of Physics and Astronomy, Swarthmore College, U.S.A. Philip Sterne Lawrence Livermore National Laboratory, U.S.A. Positronium in Solids: Computer simulation of Pick-off and Self-Annihilation
2 Ps forms and thermalizes in void spaces (defects, cages, bubbles, …) in insulating materials 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 + )
3 Simple Tao-Eldrup models are commonly used... Data from various molecular solids (Jean, 1995) -1 = -1 [ R / (R+ R) + (1/2 ) sin(2 R / (R+ R) ) ] Data typically fit with R =1.66 Å, 2ns Simple models cannot account for... irregular pore geometry ionic substitution framework content presence of adsorbates irregular pore geometry ionic substitution framework content presence of adsorbates (Brandt et al, 1960; Eldrup et al, 1981) (ns) ( cages?) 1.8MS-5A 2.1MS-4A 1.6MS-3A (ns) ( cages?) zeolite (Mohamed and El-Sayed, 1997) R MS-3A+Kr( MS-3A+Kr( 7 I 2 (%) 6 I 3 (%) 4.6 (ns) ( cages?) 1.4 MS-3A (ns) ( cages?) zeolite (Ito et al, 1982) Extensions to model: Itoh et al, 1999; Gidley et al, 1999; Gorowek et al, 2002)
4 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”
5 Comparison of PIMC with finite-element results: e+ lifetime in solid Cs Charge density - from LDA DFT code: superimposed atomic charges V+ = V coul + V corr ( - (r)) - and V+ fit with cubic spline (21 3 nodes sufficient for BCC Cs, a = 11.4 au) - (r)] from Arponen-Pajanne uniform e- gas P = 120 T = 0.1 au = 382 ps (all enhanced) cf LLNL finite element code: = 385 ps (all enhanced) = 414 ps (valence enhanced) cf experiment : 418 ps V+
6 PIMC can incorporate thermal effects: e+ in solid Cs with a monovacancy T = 0.1 au ≈ 390 ps T = 0.01 au ≈ 420 ps V+ (1 of 16 atoms deleted) Binding energy into vacancy ≈.02 au
7 PIMC predictions for a spherical pore: o-Ps lifetime and internal contact density, R (a.u.) T-E R c = 10, e+ of Ps R c = 6, e+ of Ps x R c = 6, e+ alone New predictions result from a 2-particle model of Ps. symbols: calculation curves: T-E g.s. theory R c = 5, e+ of Ps (Larrimore et al, 2000) “quasi”Ps exists in bound state self ~
8 Lifetimes depend on temperature; occupation of higher-energy states Lifetime, (ns) Sphere radius, r (nm) Ps in a spherical pore: Explicit sum over ground and excited-state contributions cf. PIMC, in which finite-temperature excited-state contributions are incorporated automatically
9 Excited states affect lifetime in a mesoscopic pore T = 600K R = 46.9 a.u. The lifetime decrease owing to greater mass is more than offset by having a realistic electron/positron system in the pore.
10 Ps in Argon Why argon? Pseudopotentials well-worked out Literature on PIMC of e- and Ps (effective particle) in Ar fluid and clusters Relevance of noble gases in metals and zeolites Ar-e+ DCS Ar-e- 0 Space et al, 1992 Potentials: Ar-e- and Ar-e+ Note: Ar polarizability in presence of Ps is not modeled
11 Ps in Bulk and Monovacant Ar 1.9 (2) ns.53 (3) ns p-o perfect monovacant Ar-e+ bead correlation function
12 Goals for of Ps in microporous solids Goals for simulation of Ps in microporous solids Correlate the annihilation rate with –pore size and shape –ionic composition, acidity Study annihilation in the presence of guests (noble gas, hydrogen, organics, …)
13 o-Ps Lifetimes in Si-Sodalite and Si-Faujasite Experiment: (dehydrated) SOD ns (“) high Si-FAU ns Simple 1/r 12 repulsion for e+ and e- with zeolitic oxygens Calculations down to au (T R ) = 4.3 (insulator model based on silica) n (n = 3, 4) T-E-type model: SOD (R c = 9.4 au)2.5 ns FAU (R c = 15.4 au)9.8 ns (T=0), 9.2 ns (T R ), 5.0 ns (10 T R ) PIMC result: SOD2.7 ns FAU (R c = 15.4 au) 9.5 (+ 3.0) ns (T R ), 4.6 ns (10 T R )
14 Sodalite: Despite numerical agreement … some different physics? e+ density lower near wall than TE model would predict e- density enters calculation differently than T-E calculated
15 Faujasite: Despite numerical agreement … some different physics? CM positions of e+ chain in FAU at T=10T R At this temperature, Ps is readily able to exist between cages. Confinement in a single cage over many lifetimes may be the wrong picture …
16 Positronium distribution in faujasite Bead positions in FAU Future work Statistical characterization of distribution among cages Future work: Statistical characterization of e+ distribution among cages
17 Future direction: Rate of transport of Ps in materials Extended or Localized ? (e+ in metal: Sterne, 2000)
18 Polarizability : = 36 (3) E Shielding/polarization of Ps reduces self-annihilation rate and modifies hyperfine splitting energy. Future direction: Electrostatic shielding and polarization
19 Future direction: Fluid-filled pore spaces Argon-type atom / spherical pore
20 Many thanks to... Colleagues: Roy Pollock (LLNL), Richard Howell (LLNL) P. Asoka-Kumar (LLNL), Thomas Gibson (Texas Tech U), Terrence Reese (Southern U), David Schoepf (Bucknell U) Students at Swarthmore: Lisa Larrimore, Robert McFarland, Peter Hastings, Gabriel Benjamin- Fernandez, Amanda Bonfitto (Earlham Coll.) Funding agencies: Department of Energy ACS Petrolium Research Fund Faculty research fund of Swarthmore College The Organizing Committee and Participants of ICPA-13