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Amy Bug, Melaku Muluneh and Jillian WaldmanDept. of Physics and Astronomy, Swarthmore College, U.S.A.

Philip SterneLawrence Livermore National Laboratory, U.S.A.

Positronium in Solids: Positronium in Solids: Computer simulation of Computer simulation of

Pick-off and Self-Pick-off and Self-AnnihilationAnnihilation

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PsPs forms and thermalizes in forms and thermalizes in void spacesvoid spaces(defects, cages, bubbles, …) in insulating (defects, cages, bubbles, …) in insulating

materialsmaterials

PALS and ACAR indicate size distribution, contents, and chemical nature of voids

p) ~ ∑n | ∫ dr e -ip .r (r) n(r) √[( r)] | 2

-1 ≈ re2 c ∫ dr dr+ (r+) (r) [( r)] r - r+)

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Simple Tao-Eldrup models are commonly used ...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 ...Simple models cannot account for ...• • irregular pore geometry • ionic substitution • framework content • irregular pore geometry • ionic substitution • framework content • presence of adsorbatespresence of adsorbates

(Brandt et al, 1960; Eldrup et al, 1981)

9.5

4.4

3.5

(ns)( cages?)

1.8MS-5A

2.1MS-4A

1.6MS-3A

(ns)( cages?)

zeolite

(Mohamed and El-Sayed, 1997)

R

113.671.5MS-3A+Kr(

74.791.2MS-3A+Kr(

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I2(%)

6

I3 (%)

4.6

(ns)( cages?)

1.4MS-3A

(ns

)(

cages?)

zeolite

(Ito et al, 1982)

Extensions to model: Itoh et al, 1999;

Gidley et al, 1999; Gorowek et al, 2002)

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We We simulatesimulate Ps in materials with two- Ps in materials with two-chainchain

Path Integral Monte Carlo ( Path Integral Monte Carlo (PIMCPIMC))

• The Quantum density matrix: () = exp( - H) is represented in the position basis:

<r| () |r’> = ∫ <r| () | r1> < r1| () | r2> ... <rP-1| () |r’> d r1 … rP-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”

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Comparison of PIMC with finite-element results:Comparison of PIMC with finite-element results: e+ e+ lifetime in solidlifetime in solid CsCs

• Charge density - from LDA DFT code: superimposed atomic charges

• V+ = Vcoul + Vcorr(- (r))

• - and V+ fit with cubic spline (213 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+

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PIMC can incorporate thermal effects: PIMC can incorporate thermal effects: e+e+ in solid in solid CsCs with awith a monovacancymonovacancy

T = 0.1 au≈ 390 ps

T = 0.01 au≈ 420 ps

V+

(1 of 16 atoms deleted)

Binding energy into vacancy ≈ .02 au

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0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10

P+ , r10square well theoryP+, r6square well theoryPsingle, r6P+, r5square well theory

P(r)

r, radial position (a.u.)

PIMC predictions for a spherical pore:PIMC predictions for a spherical pore: o-o-Ps lifetimePs lifetime and and internal contact internal contact

densitydensity, ,

R (a.u.) T-E 10 1.16 1.05

8 1.30 1.10

6 1.22 1.20

5 1.11 1.45

•Rc = 10, e+ of Ps

•Rc = 6 , e+ of Ps

x Rc = 6 , e+ alone

New predictions result from a 2-particle model of Ps.New predictions result from a 2-particle model of Ps.

symbols: calculationcurves: T-E g.s. theory

Rc = 5, e+ of Ps

(Larrimore et al, 2000)

“quasi”Ps exists inbound state

self ~

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Lifetimes depend on temperature; Lifetimes depend on temperature; occupation of higher-energy statesoccupation of higher-energy states

Lif

eti

me, (n

s)

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

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Excited states affect lifetimeExcited states affect lifetime

in a mesoscopic porein a mesoscopic pore

T = 600KR = 46.9 a.u.

The lifetimedecrease owing togreater mass ismore than offsetby having a realisticelectron/positron system in the pore.

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Ps in ArgonPs in ArgonWhy 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

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 20 40 60 80 100 120 140 160

exp't (approx)DCS_reidwadDCS_gibson1.0

log(DCS)

angle(deg)

Ar-e+ DCS Ar-e- 0

Space et al, 1992

Potentials: Ar-e- and Ar-e+

Note: Ar polarizabilityin presence of Ps is not modeled

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Ps in Bulk and Monovacant ArPs in Bulk and Monovacant Ar

1.9 (2) ns.53 (3) nsp-

o

perfectmonovacant

Ar-e+ bead correlation function

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Goals for Goals for simulation of Ps in microporous of Ps in microporous solidssolids

• Correlate the annihilation rate with – pore size and shape– ionic composition, acidity

• Study annihilation in the presence of guests (noble gas, hydrogen, organics, …)

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o-Ps Lifetimes in Si-Sodalite and Si-Faujasiteo-Ps Lifetimes in Si-Sodalite and Si-Faujasite

Experiment: (dehydrated) SOD 1.5 - 2.5 ns

(“) high Si-FAU 5 - 15 ns

• Simple 1/r12 repulsion for e+ and e- with zeolitic oxygens• Calculations down to 0.001 au (TR)• = 4.3 (insulator model based on silica)

n (n = 3, 4)

T-E-type model: SOD (Rc = 9.4 au) 2.5 ns

FAU (Rc = 15.4 au) 9.8 ns (T=0), 9.2 ns (TR), 5.0 ns (10 TR)

PIMC result: SOD 2.7 ns

FAU (Rc = 15.4 au) 9.5 (+ 3.0) ns (TR), 4.6 ns (10 TR)

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Sodalite: Despite numerical agreement … some Sodalite: Despite numerical agreement … some different physics?different physics?

e+ density lower near wall than TE model would predict

e- density enters calculation differently than

T-E

calculated

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Faujasite: Despite numerical agreement … some Faujasite: Despite numerical agreement … some different physics?different physics?

CM positions of e+ chain in FAU at T=10TR

At this temperature, Ps is readily able to exist between cages. Confinement in a single cage over many lifetimes may be the wrong picture …

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Positronium distribution in Positronium distribution in faujasitefaujasite

Bead positions in FAU

Future workFuture work: Statistical c Statistical characterization ofharacterization of e+ distribution among distribution among cagescages

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Future direction:Future direction: Rate of transport of Rate of transport of Ps in materialsPs in materials

Extended or Localized ?

(e+ in metal: Sterne, 2000)

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Polarizability:= 36 (3)

E

Shielding/polarization of Ps reduces self-annihilation rate and modifies hyperfine splitting energy.

Future direction:Future direction: Electrostatic Electrostatic shielding and polarizationshielding and polarization

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Future direction:Future direction: Fluid-filled pore Fluid-filled pore spacesspaces

Argon-type atom / spherical pore

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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 FundFaculty research fund of Swarthmore College

The Organizing Committee and Participants

of ICPA-13