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Simulation of Positronium in Silica Sodalite Peter Hastings and Amy L.R. Bug Dept. of Physics and Astron., Swarthmore College Philip Sterne Lawrence Livermore National Laboratory 1

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Positron Annihilation Spectroscopy (PAS) provides information on size and chemical content of zeolitic cavities e+ enters zeolite e+ selects electron e+ in Ps is “picked two gamma rays and thermalizes of solid and forms off” prematurely by with characteristic within tens of ps; positronium (Ps) with another electron energy of 511 keV e+ preferentially natural lifetime of of solid are emitted locates in 125 ps (para) or channels and cavities 140 ns (ortho) Lifetime, , of Ps is determined by electronic density - : -1 = r e 2 c ∫ dr - dr + (r + ) (r - ) [ ( r - )] r - - r + ) 2

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Simple “free volume” model is commonly used... Data from various molecular solids including zeolites (Y.C. Jean,1995) -1 = -1 [ R / (R+ R) + (1/2 ) sin(2 R / (R+ R) ) ] Uniform electronic density in layer R. Data typically fit with R = 1.66 Å, 2ns Yet it cannot quantitatively describe diverse data... irregular pore geometry ionic substitution aluminum content presence of adsorbates irregular pore geometry ionic substitution aluminum content presence of adsorbates (Brandt et al, 1960; Eldrup et al, 1981) zeolite (ns) ( cages?) (ns) ( cages?) MS-3A1.63.5 MS-4A2.14.4 MS-5A1.89.5 (Mohamed and El-Sayed, 1997) R zeolite (ns) ( cages?) I 2 (%) (ns) ( cages?) I 3 (%) MS-3A 1.474.66 MS-3A+Kr( 1.294.77 MS-3A+Kr( 1.573.611 (Ito et al., 1982) 3

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Our approach is to simulate Ps in zeolites via Path Integral Monte Carlo (PIMC) e- e+ 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 Ps “chains” 4

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Some messy details : For Coulomb interaction, use either Yukawa potential (Muser and Berne, 1997) or ( Pollock, 1988) exact Coulomb density matrix: ( ) ≈ e - T + + T - ) e - Vext e - P ( ) An atom-based density functional code (Sterne et al., 1999) generates V ext (r + ) and ( r). Try various models (Sterne-Kaiser: metal, Puska et al. : insulator, Independent particle model ) for [ . V ext (r - ) is a sum of exponentials (Space et al., 1992). More work is needed to develop an electronic pseudopotential for zeolite hosts. Codes written in Fortran 9X and run both on Linux workstation and Appleseed Macintosh cluster using Mac MPI libraries. 5

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RESULTS: PIMC yields new prediction even in a spherical cage R (a.u.) free-volume 101.16 81.30 61.22 51.11 R = 10,e+ in Ps R = 6, e+ in Ps x R = 6, e+ alone R = 5,e+ in Ps (Larrimore et al, 2000) The new predictions arise from using a 2-particle model of Ps. 6 symbols: data curves: free volume theory

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RESULTS: Si-sodalite lifetimes are consistent with experiment and simple theory, but with a very different physical interpretation x xx Radial probability distributions in sodalite cage: + (cav pdf), + - chg pdf ). is 4.3 for insulator model. Experiments on sodalite give ≈ 1.5 - 2.5 ns. 7

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RESULTS: Ps at this (high) simulation temperature is encapsulated in sodalite cage after 100K steps: total E: -0.44 a.u., potential E: -0.74 a.u. MC step Ps energy = 50 au, P = 250 + = 50 au,P = 1000 Density averaged over MC run 8

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Single particle and two-particle (preferable) Ps models result in different positron annihilation characteristics. PIMC enhances contributions from the surface of the cage, over the simple “free-volume/uniform electronic density” model. The PIMC approach naturally accounts for non-idealized cage and channel shapes and allows one to faithfully calculate energy, structure, and annihilation rate due to realistic electronic density in the host solid. Ps is stable to ionization, and prefers confinement in a single cage of Si-sodalite. Calculated annihilation rates agree favorably with experiment. More sophisticated potential models, chemical effects, and other zeolite frameworks will be considered in the near future. Conclusions and future directions 9

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Thanks to... The Donors of the Petrolium Research Fund, administered by the American Chemical Society The Faculty Research Support Fund of Swarthmore College This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboritory under Contract No. W-7405-Eng.48 10

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