Denton Woods's PhD Defense

Denton Woods

NSF support provided under grant no. PHYS. 968638Computational resources provided by UNT's High Performance Computing Initiative

August 6, 2015

denton.woods@unt.edu

Variational Calculations of PositroniumScattering with

Hydrogen

Major Professor: Dr. Quintanilla

Acknowledgments

Denton Woods (University of North Texas) Positronium-Hydrogen Collisions August 6, 2015 3 / 58

I would like to thank:• My PhD supervisor, Dr. Quintanilla (Ward)• Our collaborator and my minor professor, Dr. Van Reeth• My committee: Dr. Weathers, Dr. Ordonez, and Dr. Shiner

I also acknowledge:• NSF for grant no. PHYS-968638 and a UNT faculty research

grant• Computational resources provided by UNT’s High

Performance Computing Services (http://hpc.unt.edu)• Figures and data from our accepted Physical Review A

article [10]

Publications / Presentations

PublicationsDenton Woods, S. J. Ward, and P. Van Reeth, accepted by Phys. Rev. A

Presentations• Poster at 45th DAMOP Meeting – June 2014• Contributed talk at 23rd CAARI – May 2014• Contributed talk at APS March Meeting 2014• Poster at 44th DAMOP Meeting – June 2013• Contributed talk at APS March Meeting 2013• Invited talk at 22nd CAARI – August 2012• Contributed talk at 43rd DAMOP Meeting – June 2012• Poster at 42nd DAMOP Meeting – June 2011• Poster at 41st DAMOP Meeting – May 2010

Open Science• All codes (multiple languages) on GitHub (https://github.com/DentonW/)• Notes on figshare (http://figshare.com/authors/Denton_Woods/581638)• Interactive versions of plots on plotly (https://plot.ly/~Denton)• All linked on my personal site (www.dentonwoods.com)

Topics

• Introduction• Positronium Hydride• Scattering Theory and Computational Methods• Results• Phase Shifts• Effective Range Theory• Cross Sections

Introduction

Positrons / Positronium

First experimental evidence of a positron - Carl D. Anderson

Positrons

Positronium• Exotic atom: positron and electron bounded• Lifetime of ~10-10 s for para-Ps and ~10-7 for ortho-Ps

• Predicted by Paul Dirac in 1931• Positrons first observed in 1932 by Carl D. Anderson• Same properties as electrons (spin, mass) but with

positive charge

Positrons and positronium study important for astrophysics, condensed matter physics and medical physics

Importance

Positronium Beams• Positron Group at University College London

• Energy-tunable Ps beam• Ps-gas cross sections for He, Ar, H2, CO2 and other targets

• Australian National University looking at creating Ps beamSimilarity to e- Scattering

Unexpected result: Ps-target scattering is similar to e--target scattering• Ps neutral and 2x mass of e- !• e+ seems to play only a small role

Positronium Hydride

• Exotic “molecule”• Singlet bound state• First observed in 1990 (Pareja and Gonzalez)• Lifetime of 0.5 ns Figure from our paper [1]

Positronium Hydride

• Exotic “molecule”• Single bound state (singlet)• First observed in 1990 (Pareja and Gonzalez)• Lifetime of 0.5 ns

Hamiltonian

Hylleraas-Type Short-Range Terms

Terms are included such that

Positronium Hydride

Rayleigh-Ritz Variational Method

Can be written as a generalized eigenvalue problem:

Operation of the Hamiltonian

Hamiltonian acting on the short-range terms is complicated:

S-Wave

•“Three-electron” or “four-body” integrals•Two methods:• Asymptotic expansion [Drake and Yan 1995]• Recursion relations [Pachucki et al. 2004]

Short-Range Integrals

Positronium Hydride Code

• Written from scratch in Fortran

• Project uses C++, Fortran, MATLAB, Mathematica and Python

• Fully quadruple precision

• Matrix element integrals largest bottleneck

• Solving generalized eigenvalue equation is much faster

• Typical run of 2.9 million matrix elements

• “Embarrassingly” parallel

• OpenMP directives to parallelize

Positronium Hydride: Results

1S N(ω) ECurrent work 1505 -0.789 190Hylleraas(Yan / Ho) [6] 5741 -0.789 196

Scattering Theory and Computational Methods

General Scattering Theory

[Adapted from http://commons.wikimedia.org/wiki/File:ScatteringDiagram.svg]

(f is the scattering amplitude)

• Kohn-type variational method• Close coupling• Confined variational method• Diffusion Monte Carlo• Stochastic variational method• Static exchange

Scattering Theory

Scattering Methods

• Expand wavefunction in Legendre polynomials:

• Each term in the summation is a partial wave (denoted by )

• At low energies, only a few partial waves required

• Main goal is to get phase shifts,

• Gives a measure of the interaction with scattering center

Scattering Theory

Partial Wave Expansion

Kohn Variational Method

• Variants include inverse Kohn, complex (generalized) Kohn and generalized Kohn [7,8,9]

• Phase shift code implements many Kohn-type methods

• Can give accurate calculations

• Variants can be generalized to

Hamiltonian

Trial Wavefunction

Long-Range Terms

Hylleraas-Type Short-Range Terms

Terms are included such that

S-Wave Wavefunction

Kohn-type functionals stationary with respect to variations in linear parameters, i.e.

and where

giving:

Scattering parameter solved for by

Kohn-Type Variational Methods

e.g., Kohn

Kohn-Type Matrix

IntegrationsTwo types of computational techniques:

• Gaussian quadratures• Four-body integrations (asymptotic expansion / recursion relations)

Gaussian quadratures Gaussian quadraturesFour-body integrals

General Partial Waves (Four-Body Integrals)

Two methods:• Rotation and integration over external angles to reduce to

S-wave form• Drake and Yan general method for arbitrary angular

momentum with asymptotic expansion

General Short-Range Integrals

Long-Range – Short-Range and Long-Range — Long-Range Integrations

After analytic integration over the 3 external angles, integrals are of the form

• Gauss-Laguerre, Gauss-Legendre and Gauss-Chebyshev quadrature for integrals:

Long-Range Integrals

Gauss-Laguerre

• Cusp in r2 and r3 integrands• Cannot be solved as accurately• ~ 2 billion integration points total for each 6-D integral• Code written in extended precision C++

Linear Dependence

• Biggest problem is linear dependence

• Finding where linear dependence occurs is tricky

• No exact bound for system (empirical bound)

• Use Todd’s method [1,23]

• Runs with multiple Kohn-type methods

• Asymptotic expansion gives accuracy of ~1 part in 1020

• Gaussian quadratures only ~1 part in 106

Phase Divergence in Kohn-Type Methods

Figure from our paper [1]

UNT Talon Cluster

• 250 individual compute nodes (Dell R420 servers)• 4096 processor cores• Intel Xeon E5-2450 and E5-4640 8 core processors• 32 GB, 64 GB and 512 GB nodes• 16 GP-GPU nodes• 1.5 PB total storage• InfiniBand interconnects (56 Gb/s)• http://hpc.unt.edu

UNT Talon Cluster

Results(mainly)

S-Wave Singlet Comparisons

Comparisons with other calculations Figure from our paper [1]

S-Wave Results

[Dashed lines show resonance positions from Zong-Chao Yan and Y. K. Ho, Phys. Rev. A 59, 2697 (1999)]Figure from our paper [1]

P-Wave Results

[Dashed lines show resonance positions from Zong-Chao Yan and Y. K. Ho, Phys. Rev. A 57, R2270 (1998)]Figure from our paper [1]

Two Resonances (S-Wave / P-Wave)

• Smooth polynomial background• Breit-Wigner resonance terms• Parameters fit using MATLAB’s nlinfit with all 8 weightings• Interfaced to Python using mlabwrap in IPython

Resonance Fitting

S-WaveCurrent work 4.0065 0.0955 5.0272 0.0608Complex Rotation (Yan / Ho) 4.0058 0.0952 4.9479 0.0585

CC (Walters et al.) 4.149 0.103 4.877 0.0164

Resonance Fitting

P-WaveCurrent work 4.2856 0.0445 5.0577 0.0459Complex Rotation (Yan / Ho) 4.2850 0.0435 5.0540 0.0585

CC (Walters et al.) 4.475 0.0827 4.905 0.0043

Two Resonances (S-Wave / P-Wave)

D-Wave Results

[Dashed line shows resonance position from Zong-Chao Yan and Y. K. Ho, J. Phys. B 31, L877 (1998)]Figure from our paper [1]

General Code

• Generalized short-range and long-range codes for = 0 through 5 for first 2 symmetries

• Results for ω = 5 (924 terms) for

• Through H-wave, full Kohn calculations much more accurate than Born-Oppenheimer approximation

F-Wave

Effective Range Theory

Definition

Approximation

Scattering Length

4.3306 4.3306 2.1363 2.1363

• Describes scattering at low energy

with a.u.

Short-Range Interaction

Including the van der Waals Potential

Flannery (2000)

Gao (1998)

Blatt & Jackson (1949)Bethe (1949)

Martin & Fraser (1980)

Hickelmann &Spruch (1971)

Table from our paper [1]

Cross Sections

• Gives strength of the interaction [22]

Partial

Integrated

Momentum transfer

Differential

(Spin-weighting)(Spin-weighting)

Cross Sections: Triplet

Cross Sections: Singlet

Elastic Integrated Cross Sections

Elastic Differential Cross Section

Gives information about angular and energy dependence

Differential Cross Section

Momentum Transfer Cross Section

Cross Section Comparisons

0.46 eV

isotropic

0.46 eV

isotropic

Comparisons

Data from [24]

Summary

• Kohn-type variational calculations (past[4,5] and present[1]) have provided results for low-energy elastic Ps-H scattering• Phase shifts for S-wave through H-wave• Highly accurate results for S-wave and P-wave• Effective ranges and scattering lengths• Integrated, differential and momentum transfer cross sections

• This project has given experience in multiple aspects of computational physics• Multiple programming languages• Parallel programming techniques• Database administration• Using computers to solve a physical problem

• Dissertation: http://bit.ly/1CT0VJy and http://www.dentonwoods.com

References

1. Denton Woods, S. J. Ward, and P. Van Reeth, Phys. Rev. A (in press)2. http://www.myvmc.com/investigations/pet-scan-positron-emission-tomography/3. Carl D. Anderson, Phys. Rev. 43, 491 (1933).4. P. Van Reeth and J. W. Humberston, J. Phys. B 36, 1923 (2003).5. P. Van Reeth and J. W. Humberston, Nucl. Instr. and Meth. in Phys. Res. B 221, 140

(2004).6. Y. K. Ho and Zong-Chao Yan, J. Phys. B 31, L877 (1998).7. E. A. G. Armour and J. W. Humberston, Phys. Rep. 204, 165 (1991).8. J. N. Cooper, M. Plummer, and E. A. G. Armour, J. Phys. A 43, 175302 (2010).9. J. N. Cooper, E. A. G. Armour, and M. Plummer, J. Phys. A Math. Theor. 42, 095207

(2009).10. https://en.wikipedia.org/wiki/Scattering_length11. J. Blackwood, M. McAlinden, and H. R. J. Walters, Physical Review A 65, 030502(R)

(2002).12. H. R. J. Walters, A. C. H. Yu, S. Sahoo, and S. Gilmore, Nucl. Instr. and Meth. in Phys.

Res. B 221, 149 (2004).13. I. A. Ivanov, J. Mitroy, and K. Varga, Phys. Rev. A 65, 032703 (2002).14. D. W. Martin and P. A. Fraser, J. Phys. B 13, 3383 (1980).15. J. M. Blatt and J. D. Jackson, Phys. Rev. 76, 18 (1949).16. H. A. Bethe, Phys. Rev. 76, 38 (1949).17. O. Hinckelmann and L. Spruch, Phys. Rev. A 3, 642 (1971).

References

19. M. R. Flannery, Springer Handbook of Atomic, Molecular, and Optical Physics, 2nd ed., edited by G. W. F. Drake (Springer, New York, NY, 2006) p. 668.

20. B. Gao, Phys. Rev. A 58, 1728 (1998).21. B. Gao, Phys. Rev. A 58, 4222 (1998).22. B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Pearson Education

Limited, Harlow, England, 2003).23. A. Todd, Ph.D. thesis, The University of Nottingham, (2007), unpublished.24. P. Van Reeth, private communication.25. P. Van Reeth, Ph.D. thesis, University College London, (1994) unpublished.26. G. W. F. Drake and Zong-Chao Yan, Phys. Rev. A 52, 3681 (1995).27. Zong-Chao Yan and G. W. F. Drake, J. Phys. B 30, 4723 (1997).28. Y. K. Ho and Zong-Chao Yan, J. Phys. B 31, L877 (1998).29. K. Pachucki, M. Puchalski, and E. Remiddi, Phys. Rev. A 70, 032502 (2004).

S-Wave Triplet Comparisons

Comparisons with other calculations

Gauss-Laguerre Quadraturer1 Integrand

• Rough fit to integrand

Rescaling Gauss-Laguerre

• Slow convergence in r1, r2 and r3 coordinates• More structure near origin• Adding more integration points can increase the run time to

unmanageable levels

• Our solution: rescale for more points near origin and less farther out

• Convergence of matrix element integrations is accelerated

Rescaling Gauss-Laguerre(M

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