Denton Woods NSF support provided under grant no. PHYS. 968638 ational resources provided by UNT's High Performance Computing Initi August 6, 2015 [email protected]Calculations of Positronium Scattering with Hydrogen Major Professor: Dr. Quintanilla
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Denton Woods
NSF support provided under grant no. PHYS. 968638Computational resources provided by UNT's High Performance Computing Initiative
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
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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)
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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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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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
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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).
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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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