Amy Bug, Melaku Muluneh and Jillian Waldman

1
Positronium in Solids Computer simulation of
Pick-off and Self-Annihilation

• 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.

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
t -1 p re2 c ? dr- dr r(r) r-(r-) gr-(
r-) d3(r- - r)
N(p) ?n ? dr e -ip .r f(r) yn(r) vgr-(
r) 2
3
Simple Tao-Eldrup models are commonly used ...
t-1 t?-1 DR / (RDR) (1/2p) sin(2p R /
(RDR) )
(Brandt et al, 1960 Eldrup et al, 1981)
Extensions to model Itoh et al, 1999 Gidley et
al, 1999 Gorowek et al, 2002)
Data typically fit with DR 1.66 Å, t????2ns
Data from various molecular solids (Jean, 1995)
Simple models cannot account for ...
irregular pore geometry ionic substitution
framework content presence of adsorbates
(Mohamed and El-Sayed, 1997)
(Ito et al, 1982)
4
We simulate Ps in materials with two-chain Path
Integral Monte Carlo (PIMC)
(cf. single-chain model Miller, Reese et al,
1996, 2002)
Ps wave packet
Ps chains
The Quantum density matrix r(b) exp( - b H)
is represented in the position basis ltr r(b)
rgt ? ltr r(e) r1gt lt r1 r(e) r2gt ...
ltrP-1 r(e) rgt d r1 rP-1 (e b/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.
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Comparison of PIMC with finite-element results
e lifetime in solid Cs

• Charge density r- from LDA DFT code
  superimposed atomic charges
• V Vcoul Vcorr(r- (r))
• r- and V fit with cubic spline (213 nodes
  sufficient for BCC Cs, a 11.4 au)
• gr- (r) from Arponen-Pajanne uniform e- gas
  

V

• P 120
• T 0.1 au
• t 382 ps (all enhanced)
• cf LLNL finite element code
  t 385 ps (all enhanced) t 414 ps (valence
  enhanced)
• cf experiment 418 ps

r
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PIMC can incorporate thermal effects e in
solid Cs with a monovacancy
r
r
V
T 0.1 au t 390 ps
T 0.01 au t 420 ps
(1 of 16 atoms deleted)
Binding energy into vacancy .02 au
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PIMC predictions for a spherical pore o-Ps
lifetime and internal contact density, k
symbols calculation curves T-E g.s. theory
quasiPs exists in bound state tself k-1
Rc 10, e of Ps
(Larrimore et al, 2000)
Rc 6 , e of Ps
x Rc 6 , e alone
New predictions result from a 2-particle model of
Ps.
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Lifetimes depend on temperature occupation of
higher-energy states
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 lifetimein a mesoscopic
pore
The lifetime decrease owing to greater mass
is more than offset by having a
realistic electron/positron system in the pore.
T 600K R 46.9 a.u.
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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- d0
Ar-e DCS
Space et al, 1992
Potentials Ar-e- and Ar-e
Note Ar polarizability in presence of Ps is not
modeled
r
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Ps in Bulk and Monovacant Ar
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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, )

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

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

tn (n 3, 4)
Experiment (dehydrated) SOD 1.5 - 2.5
ns () high Si-FAU 5 - 15 ns
T-E-type model SOD (Rc 9.4 au) 2.5
ns FAU (Rc 15.4 au) 9.8 ns (T0), 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
different physics?
calculated
T-E
r
e density lower near wall than TE model
would predict e- density enters calculation
differently than
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Faujasite Despite numerical agreement some
different physics?
CM positions of e chain in FAU at T10TR
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 faujasite
Bead positions in FAU
Future work Statistical characterization of e
distribution among cages
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Future direction Rate of transport of Ps in
materials
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Future direction Electrostatic shielding and
polarization
Polarizability a 36 (3)
E
Shielding/polarization of Ps reduces
self-annihilation rate and modifies hyperfine
splitting energy.
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Future direction Fluid-filled pore spaces
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 Fund Faculty research fund of Swarthmore
College The Organizing Committee and
Participants of ICPA-13