Table 2. Scattering parameter comparisons
1
This work (ω = 7) Complex Rotation [1] Kohn [2] Kohn [3] 14H14P s CS [4] 14H14Ps + H - CS [5] ε (eV) a + a − 1 E R Γ 1 2 E R Γ 2 1.0664 02 4.329 2.137 4.0052 0.0948 4.9754 0.0501 1.0665 97 − − 4.0058 0.0952 4.9479 0.0585 1.06 55 4.31 1 2.12 6 − − − − 1.06 55 − − 4.00 72 0.09 56 5.02 67 0.05 97 0.99 4 4.41 2.06 4.37 0.10 − − 1.03 4.327 − 4.149 0.103 4.877 0.0164 Table 2. Scattering parameter comparisons Low-energy S-wave Positronium-Hydrogen collisions Denton Woods , S. J. Ward and P. Van Reeth † University of North Texas, † University College London Introduction Positronium-Hydrogen (Ps-H) elastic scattering is of interest since it is a fundamental four-body Coulomb process. We have computed 1 S and 3 S phase shifts for low-energy elastic Ps-H scattering. We used the Kohn variational method and a number of variants on the method, namely, the inverse Kohn, generalized Kohn and the complex Kohn. We use atomic units throughout, unless stated otherwise. Acknowledgements The authors acknowledge the UNT Computing and Information Technology Center's High Performance Computing Initiative for providing resources on the Talon HPC cluster. S.J.W. acknowledges support from NSF under grant no. PHYS- 968638. References [1] Zong-Chao Yan and Y. K. Ho, Phys. Rev. A 59, 2697 (1999). [2] P. Van Reeth and J. W. Humberston, J. Phys. B 36 (2003) 1923-1932. [3] P. Van Reeth and J. W. Humberston, Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 140-143. [4] Blackwood, McAlinden and Walters, Phys. Rev. A 65 032517 (2002). [5] H.R.J. Walters et al, Nucl. Instr. and Meth. in Phys. Res. B 221 (2003) 149-159. [6] A. Todd, Ph.D. thesis, The University of Nottingham, Results Conclusion Our phase shifts and resonance parameters compare well with previous Kohn calculations [2,3] and coupled-state (CS) approximations [4,5]. We have started to consider P-wave Ps-H scattering. Figure 2. Phase shifts for 1 S and 3 S Ps-H scattering (ω = 7) Trial Wavefunction The short-range terms are of a Hylleraas type, where , and k i , l i , m i , n i , p i , and q i are non-negative integers. Table 1. Phase shift comparisons as a function of κ κ δ + (ω = 7) δ + [5] δ + [7] δ − (ω = 7) δ − [5] δ − [10] 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 -0.427 -0.820 -1.161 -1.446 -1.678 -1.858 -1.964 - 0.425 - 0.817 - 1.158 - 1.443 - 1.674 - 1.852 - 1.959 - 0.428 - 0.825 - 1.167 - 1.453 - 1.685 - 1.867 - 1.992 -0.215 -0.432 -0.646 -0.850 -1.041 -1.217 -1.376 -0.214 -0.431 -0.645 -0.849 -1.038 -1.211 -1.366 - 0.206 - 0.414 - 0.624 - 0.838 - 1.037 - 1.213 - 1.367 e − p e + e − r 1 r 2 r 3 r 23 r 1 2 r 13 ρ Figure 1. Coordinate System Kohn Variational Method The generalized Kohn functional is [9] , where τ is an adjustable parameter (0−). Special cases of τ are τ = 0 for the Kohn and τ = π/2 for the inverse Kohn. The stationary property of this functional yields the phase shifts. The complex Kohn functional for the T- matrix is . The agreement between the phase shifts for multiple values of τ is taken as a measure of the accuracy of the results and of the absence of linear dependence. Furthermore, we have implemented the procedure of Todd [7], which allows us to remove in a systematic manner specific short-range terms in the wavefunction which are responsible for linear dependence.
description
Low-energy S-wave Positronium-Hydrogen collisions Denton Woods , S. J. Ward and P. Van Reeth † University of North Texas, † University College London. ρ. Introduction - PowerPoint PPT Presentation
Transcript of Table 2. Scattering parameter comparisons
This work(ω = 7)
Complex Rotation [1] Kohn [2] Kohn [3]
14H14Ps CS [4]
14H14Ps + H- CS [5]
ε (eV)a+
a−
1ER
Γ12ER
Γ2
1.0664024.3292.1374.00520.09484.97540.0501
1.066597−−
4.00580.09524.94790.0585
1.06554.3112.126
−−−−
1.0655−−
4.00720.09565.02670.0597
0.9944.412.064.370.10
−−
1.034.327
−4.1490.1034.8770.0164
Table 2. Scattering parameter comparisons
Low-energy S-wave Positronium-Hydrogen collisionsDenton Woods, S. J. Ward and P. Van Reeth†
University of North Texas, † University College London
Introduction Positronium-Hydrogen (Ps-H) elastic scattering is of interest since it is a fundamental four-body Coulomb process. We have computed 1S and 3S phase shifts for low-energy elastic Ps-H scattering. We used the Kohn variational method and a number of variants on the method, namely, the inverse Kohn, generalized Kohn and the complex Kohn. We use atomic units throughout, unless stated otherwise.
Acknowledgements The authors acknowledge the UNT Computing and Information Technology Center's High Performance Computing Initiative for providing resources on the Talon HPC cluster. S.J.W. acknowledges support from NSF under grant no. PHYS-968638.
References[1] Zong-Chao Yan and Y. K. Ho, Phys. Rev. A 59, 2697 (1999).[2] P. Van Reeth and J. W. Humberston, J. Phys. B 36 (2003) 1923-1932.[3] P. Van Reeth and J. W. Humberston, Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 140-143.[4] Blackwood, McAlinden and Walters, Phys. Rev. A 65 032517 (2002).[5] H.R.J. Walters et al, Nucl. Instr. and Meth. in Phys. Res. B 221 (2003) 149-159.[6] A. Todd, Ph.D. thesis, The University of Nottingham, (2007), unpublished.[7] J. N. Cooper et al, J. Phys. A. 43 (2010) 175302.
Results
Conclusion Our phase shifts and resonance parameters compare well with previous Kohn calculations [2,3] and coupled-state (CS) approximations [4,5]. We have started to consider P-wavePs-H scattering.
Figure 2. Phase shifts for 1S and 3S Ps-H scattering (ω = 7)
Trial Wavefunction
The short-range terms are of a Hylleraas type, where , and ki, li, mi, ni, pi, and qi are non-negative integers.
Table 1. Phase shift comparisons as a function of κ
κ δ+ (ω = 7) δ+ [5] δ+ [7] δ− (ω = 7) δ− [5] δ− [10]0.10.20.30.40.50.60.7
-0.427-0.820-1.161-1.446-1.678-1.858-1.964
-0.425-0.817-1.158-1.443-1.674-1.852-1.959
-0.428-0.825-1.167-1.453-1.685-1.867-1.992
-0.215-0.432-0.646-0.850-1.041-1.217-1.376
-0.214-0.431-0.645-0.849-1.038-1.211-1.366
-0.206-0.414-0.624-0.838-1.037-1.213-1.367
e−p
e+e−
r1
r2
r3
r23
r12
r13
ρ
Figure 1. Coordinate System
Kohn Variational Method The generalized Kohn functional is [9]
,where τ is an adjustable parameter (0−). Special cases of τ are τ = 0 for the Kohn and τ = π/2 for the inverse Kohn. The stationary property of this functional yields the phase shifts. The complex Kohn functional for the T-matrix is
. The agreement between the phase shifts for multiple values of τ is taken as a measure of the accuracy of the results and of the absence of linear dependence. Furthermore, we have implemented the procedure of Todd [7], which allows us to remove in a systematic manner specific short-range terms in the wavefunction which are responsible for linear dependence.
