Binding Energy of Positronium Hydride and Low-Energy Positronium-Hydrogen Collisions Denton Woods

University of North Texas Dr. S. J. Quintanilla

University of North Texas Dr. Peter Van Reeth

University College London April 19, 2011 Colloquium Famously predicted by Paul Dirac in 1931

Observed in 1932 by Carl D. Anderson

Has same properties as electron (spin ½, mass) but with a positive charge of +1e Positron Positronium Figure 1. First experimental evidence of a positron Exotic atom

Positron and electron bound

Lifetime of ~10 s for para-Ps and ~10 s for ortho-Ps

Positrons and positronium study important for astrophysics, condensed matter physics, medical physics (most notably PET scans) Figure 2. Positronium coordinates Figure 3. PET scan -10 -7 Motivation: Experiment Gas-scattering experiment

Uses Na as positron source (β decay)

Positron can form positronium after losing energy

Any gas can be pumped into chamber

Positronium scatters off gas atoms/molecules until it annihilates

Plan to measure low-energy positronium scattering off alkali atoms + Positron Research Group at St. Olaf College (Northfield, MN) 3 22 Part I

Bound State:

Positronium Hydride Positronium Hydride Positronium Hydride coordinate system Figure 3. Proton is considered infinitely heavy, so origin is at proton. Positronium Hydride (PsH) first predicted by A. Ore in 1951

First observed by Pareja and Gonzalez in 1990

Lifetime of 0.5 ns

Fundamental four-body Coulomb system

Can be described as having both an atomic and a molecular structure [24] Hylleraas Trial Function Wavefunction All terms with non-negative k , l , etc. are included such that i i . Hamiltonian Number of terms with respect to ω Figure 4. Positron Group at University College London 28 Gas-scattering experiment

Have measured Ps-gas total cross-sections for He, Ar, H , CO 2 2 Rayleigh-Ritz Variational Method The Rayleigh-Ritz variational method gives the ground state energy: Can be written in matrix notation as a generalized eigenvalue problem: where , and is the vector of coefficients. Linear Dependence 15 Variational method gives lower bound on energy

More terms (larger ) give higher precision

Problem: Larger matrices in the generalized eigenvalue problem can have near linear dependence. Solution: Algorithm developed by Allan Todd to remove problematic terms 16 19 - LAPACK is used to solve eigenvalue problem

- LAPACK errors start occurring at = 6 Bound State: Energy Results General Scattering Problem Part II

Scattering Calculations:

Positronium-Hydrogen Partial Wave Expansion Expand wavefunction as a series of Legendre polynomials 21 Each term in the summation is a partial wave (denoted by their respective l)

At low energies, only a few partial waves required 21 Once is found, the partial wave elastic cross-section and scattering length can easily be determined. Scattering Coordinates S-Wave Scattering Kohn Variational Method Numerical Integration Methods Figure 7. Ps and H scattering coordinates Scattering Trial Wavefunction Long-Range Terms Short-Range Terms Hamiltonian Need approximation method to obtain phase shift,

Chose to use the Kohn variational method

Good results at low energies

Does not give exact bounds for phase shifts

Gives exact upper bound on scattering length - Similar derivation as Rayleigh-Ritz variational method - In practice, empirically bounded Variational equation for Kohn method where k is the momentum of the incoming particle(s) and The Kohn functional is stationary with respect to variations in the linear parameters, i.e. and . Performing these variations leads to a system of N+1 independent equations, which can be written as Other Kohn Methods Short-Range - Short-Range These integrals are a special Hylleraas-type

Drake and Yan algorithm (asymptotic expansion) evaluates these extremely accurately

Same type of integration as bound state Long-Range - Short-Range and Long-Range - Long-Range Gauss-Legendre quadrature for integrals of type Gauss-Laguerre quadrature for integrals of type A mixture of Gauss-Legendre and Gauss-Laguerre is used for these elements. An example integration (after integrating over external angles) is Kohn method can be modified to generate multiple new methods - Inverse Kohn

- Complex Kohn (S-Matrix and T-Matrix)

- Generalized Kohn Provides a way to test numerical accuracy and linear dependence

Helps avoid Schwartz singularities (spurious resonances) S-Wave Results and Comparisons [13] Van Reeth and Humberston (2003)

[11] Blackwood, McAlinden and Walters (2002)

[25] Ray and Ghosh (1996)

[27] Adhikari and Biswas (2008) UNT Talon Cluster 224 individual compute nodes

1792 processor cores

32 large-memory nodes with 64GB of RAM and 2TB of disk space

64 medium-memory nodes with 32GB of RAM and 2TB of disk space

128 small-memory nodes with 24GB of RAM and 1TB of disk space

200TB of high-performance disk storage

Processor capacity: 20 Teraflops maximum Hardware Parallel Computing OpenMP (Open Multi-Processing) Open MPI (Open Message Passing Interface) Intranode communication

Shared memory

Talon has 8 cores per node (4 cores / processor x 2 processors/node) Internode communication

Distributed memory

Talon limit of 8 nodes for program (limit of Talon, not MPI) P-Wave Wavefunction Long-Range Terms Short-Range Terms Conclusions References [1]http://www.ucl.ac.uk/phys/amopp/research/collisions

[2] http://www.virtualmedicalcentre.com/healthinvestigations.asp?sid=7

[3] http://www.stolaf.edu/academics/positron/index.htm

[4] A. Ore, Phys. Rev. 83, 665-665 (1951).

[5] Sergiy Bubin and Ludwik Adamowicz, Physical Review A 74, 052502 (2006).

[6] J. Mitroy, Physical Review A 73, 7-9 (2006).

[7] Zong-Chao Yan and Y. K. Ho, Physical Review A 59, 2697-2701 (1999).

[8] J. Mitroy, M.W.J. Bromley and G.C. Ryzhikh, in New Directions in Antimatter Chemistry and Physics, pp. 21-43, 2001 Kluwer Academic Publishers.

[9] I.A. Ivanov, J. Mitroy and K. Varga, Phys. Rev. Lett. 87, 063201-63201-4(2001).

[10] I.A. Ivanov, J. Mitroy and K. Varga, Phys. Rev. A 65, 032703-1 - 032703-8 (2002).

[11] Jennifer E. Blackwood, Mary T. McAlinden, and H.R.J. Walters, Phys. Rev.A 65, 032517-1 - 032517-10 (2002).

[12] H.R.J. Walters, A.C.H. Yu, S. Sahoo and Sharon Gilmore, Nucl. Instr. and Meths. in Phys. Res. B 221, 149-159 (2004).

[13] Peter Van Reeth and J. W. Humberston, J. Phys. B 36, 1923-1932 (2003).

[14] Peter Van Reeth and J. W. Humberston, Nucl. Instr. and Meths. in Phys. Res. B 221, 140-143 (2004).

[15] http://en.citizendium.org/wiki/Rayleigh-Ritz_method

[16] A. Todd, Ph.D. thesis, The University of Nottingham, (2007), unpublished.

[17] G. W. F. Drake and Zong-Chao Yan, Phys. Rev. A 52, 3681–3685 (1995).

[18] Sadhan Adhikari, Variational Principles and the Numerical Solution of Scattering Problems, pp. 107-114, 1998,Wiley-Interscience.

[19] LAPACK (www.netlib.org/lapack)

[20] Arne Lüchow and Heinz Kleindienst, Chemical Physics Letters 197, 105-107 (1992).

[21] B.H. Bransden and C.J. Joachain, Physics of Atoms and Molecules, pp. 571-589, 2002, Prentice Hall.

[22] P. Van Reeth, Ph.D. thesis,

[23] A.H. Stroud and Don Secret, Gaussian Quadrature Formulas, pp. 17-36,1966, Prentice-Hall.

[24] Shiro L. Saito, Nucl. Instr. And Meths. In Phys. Res. B 171, 60-66 (2000)

[25] H. Ray and A. S. Ghosh, J. Phys. B 29 5505-5511 (1996).

[26] P. A. Fraser, Proc. Phys. Soc. 78 329 (1961).

[27] Sadhan Adhikari and P. K. Biswas (2008).

[28] S.J. Brawley, A.I. Williams, M. Shipman and G. Laricchia, Physical Review Letters 105, 263401 (2010) Linear dependence a very important problem

Todd method and Kohn variants provide test of accuracy of calculation

Parallel programming provides a way to include more terms

Ps-H system requires highly correlated short-range terms

Good agreement for PsH binding energy

Good agreement with elaborate close-coupling approximation and other Kohn methods

P-Wave and D-Wave calculations will build on the S-Wave work Talon Cluster Communication between nodes (computers) via Open MPI Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 CPU 1 CPU 2 Core 1 Core 2 Core 3 Core 4 Core 3 Core 1 Core 2 Core 4 Communication between CPUs and cores performed by OpenMP Positronium Hydride (PsH) first predicted by A. Ore in 1951

First observed by Pareja and Gonzalez in 1990

Lifetime of 0.5 ns

Normally theoretically described as an atomic structure but also more accurately described as molecular [24]

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# April 19, 2011 Colloquium

15 minute colloquium talk given on April 19, 2011 at the University of North Texas over my research

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