denton woods's phd defense
TRANSCRIPT
Denton Woods
NSF support provided under grant no. PHYS. 968638Computational resources provided by UNT's High Performance Computing Initiative
August 6, 2015
Variational Calculations of PositroniumScattering with
Hydrogen
Major Professor: Dr. Quintanilla
Acknowledgments
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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
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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
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• Introduction• Positronium Hydride• Scattering Theory and Computational Methods• Results• Phase Shifts• Effective Range Theory• Cross Sections
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Introduction
Positrons / Positronium
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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
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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
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Positronium Hydride
Positronium Hydride
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• Exotic “molecule”• Singlet bound state• First observed in 1990 (Pareja and Gonzalez)• Lifetime of 0.5 ns Figure from our paper [1]
Positronium Hydride
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• 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
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Hylleraas-Type Short-Range Terms
Terms are included such that
Positronium Hydride
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Rayleigh-Ritz Variational Method
Can be written as a generalized eigenvalue problem:
with
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Operation of the Hamiltonian
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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
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Positronium Hydride Code
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• 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
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Positronium Hydride: Results
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1S N(ω) ECurrent work 1505 -0.789 190Hylleraas(Yan / Ho) [6] 5741 -0.789 196
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Scattering Theory and Computational Methods
General Scattering Theory
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[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
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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
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Partial Wave Expansion
Kohn Variational Method
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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
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Trial Wavefunction
Long-Range Terms
Hylleraas-Type Short-Range Terms
Terms are included such that
S-Wave Wavefunction
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Kohn-type functionals stationary with respect to variations in linear parameters, i.e.
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and where
giving:
or
Scattering parameter solved for by
Kohn-Type Variational Methods
e.g., Kohn
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Kohn-Type Matrix
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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
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Long-Range – Short-Range and Long-Range — Long-Range Integrations
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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
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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
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• 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
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Figure from our paper [1]
UNT Talon Cluster
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• 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
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UNT Talon Cluster
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Results(mainly)
S-Wave Singlet Comparisons
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Comparisons with other calculations Figure from our paper [1]
S-Wave Results
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[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
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[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
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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
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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)
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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
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• 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
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Figure from our paper [1]
Effective Range Theory
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Definition
Approximation
Scattering Length
4.3306 4.3306 2.1363 2.1363
• Describes scattering at low energy
Effective Range Theory
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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)
Effective Range Theory
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Figure from our paper [1]
Effective Range Theory
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Table from our paper [1]
Effective Range Theory
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Table from our paper [1]
Cross Sections
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• Gives strength of the interaction [22]
Partial
Integrated
Momentum transfer
Differential
(Spin-weighting)(Spin-weighting)
Cross Sections: Triplet
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Cross Sections: Singlet
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Figure from our paper [1]
Elastic Integrated Cross Sections
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Figure from our paper [1]
Elastic Differential Cross Section
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Gives information about angular and energy dependence
Figure from our paper [1]
Differential Cross Section
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Figure from our paper [1]
Differential Cross Section
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Figure from our paper [1]
Differential Cross Section
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Figure from our paper [1]
Momentum Transfer Cross Section
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Cross Section Comparisons
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0.46 eV
isotropic
Figure from our paper [1]
Differential Cross Section
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0.46 eV
isotropic
Figure from our paper [1]
Comparisons
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Data from [24]
Summary
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• 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
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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
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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
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Comparisons with other calculations
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Gauss-Laguerre Quadraturer1 Integrand
• Rough fit to integrand
Rescaling Gauss-Laguerre
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• 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
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Rescaling Gauss-Laguerre(M
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