Denton Woods's PhD Defense

Denton Woods
NSF support provided under grant no. PHYS. 968638
Computational resources provided by UNT's High Performance Computing Initiative
August 6, 2015
denton.woods@unt.edu
Variational Calculations of Positronium
Scattering with Hydrogen
Major Professor: Dr. Quintanilla

Acknowledgments
Denton Woods (University of North Texas) Positronium-Hydrogen Collisions August 6, 2015 2 / 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
Publications
Denton 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
OpenScience
• 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
PositroniumBeams
• 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 beam
Similaritytoe- 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

PositroniumHydride

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
e−
pe+
e−
r1
r2
r3
r23
r12
r13
ρ

Hamiltonian
Hylleraas-TypeShort-RangeTerms
Terms are included such that
Positronium Hydride
Rayleigh-RitzVariational Method
Can be written as a generalized eigenvalue problem:
with

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) E
Current work 1505 -0.789 190
Hylleraas
(Yan / Ho) [6]
5741 -0.789 196

ScatteringTheoryand
ComputationalMethods

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
ScatteringTheory
ScatteringMethods

• Expand wavefunction in Legendre polynomials:
• Each term in the summation is a partial wave (denoted by ℓ)
• At low energies, only a few partial waves required
• Main goal is to get phase shifts, 𝛿ℓ
• Gives a measure of the interaction with scattering center
ScatteringTheory
PartialWaveExpansion

Kohn Variational Method
KohnVariationalMethod
• 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
TrialWavefunction
Long-RangeTerms
Hylleraas-TypeShort-RangeTerms
Terms are included such that
S-Wave Wavefunction

Kohn-type functionals stationary with respect to variations in
linear parameters, i.e.
and where
giving:
or
Scattering parameter solved for by
Kohn-Type Variational Methods
e.g., Kohn

Kohn-Type Matrix
Integrations
Two types of computational techniques:
• Gaussian quadratures
• Four-body integrations (asymptotic expansion / recursion relations)
Gaussian quadratures Gaussian quadraturesFour-body integrals

GeneralPartialWaves(Four-BodyIntegrals)
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-RangeandLong-Range—Long-RangeIntegrations
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)]

P-Wave Results
[Dashed lines show resonance positions from Zong-Chao Yan and Y. K. Ho, Phys. Rev. A 57, R2270 (1998)]

TwoResonances(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-Wave
Current work 4.0065 0.0955 5.0272 0.0608
Complex 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-Wave
Current work 4.2856 0.0445 5.0577 0.0459
Complex Rotation
(Yan / Ho)
4.2850 0.0435 5.0540 0.0585
CC (Walters et al.) 4.475 0.0827 4.905 0.0043
TwoResonances(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)]

General Code
• Generalized short-range and long-range codes for ℓ = 0 through 5
for first 2 symmetries
• Results for ω = 5 (924 terms) for ℓ > 2
• Through H-wave, full Kohn calculations much more accurate than
Born-Oppenheimer approximation

F-Wave

Effective RangeTheory
Definition
Approximation
ScatteringLength
4.3306 4.3306 2.1363 2.1363
• Describes scattering at low energy

with a.u.
Short-RangeInteraction
IncludingthevanderWaalsPotential
Flannery (2000)
Gao (1998)
Blatt & Jackson (1949)
Bethe (1949)
Martin & Fraser (1980)
Hickelmann &
Spruch (1971)

Effective Range Theory
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 IntegratedCross Sections

Elastic Differential Cross Section
Gives information about angular and energy dependence

Differential Cross Section

MomentumTransfer 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_length
11. 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 Quadrature
r1 Integrand
• Rough fit to integrand

RescalingGauss-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

RescalingGauss-Laguerre(Magnitudesunimportant)

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