Two-body Coulomb scattering and complex scaling

This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 193.6.178.36 This content was downloaded on 05/01/2015 at 13:57 Please note that terms and conditions apply. Two-body Coulomb scattering and complex scaling View the table of contents for this issue, or go to the journal homepage for more 2013 J. Phys.: Conf. Ser. 436 012032 (http://iopscience.iop.org/1742-6596/436/1/012032) Home Search Collections Journals About Contact us My IOPscience Two-body Coulomb scattering and complex scaling Istv´an Horny´ak University of Debrecen, Faculty of Informatics, PO Box 12, 4010 Debrecen, Hungary E-mail: [email protected] Abstract. The two-body Coulomb scattering problem is studied with the complex scaling method. Our splitting method [1], on partial wave level, simplifies the scattering boundary condition. The scattered part of the wave function is square integrable. This property makes the numerical solution easier. 1. Introduction The method of complex scaling (CS) has been successfully applied in many areas of quantum physics, but for scattering problems the CS procedure can only be applied to short-range potentials, which limits the applicability of CS in atomic and nuclear physics. It has been shown in the Ref. [2] that scattering calculations with the exterior CS can be successfully performed for long-range interactions as well. Nevertheless, the exterior CS method has been under scrutiny since an artificial cutoff of some of the interaction may cause troubles. Recently, however, it has been shown [3] that the standard CS can be applied to scattering problems even in the presence of the pure Coulomb interaction. The method is based on the two-potential formalism. An alternative technique is presented below. The full scattering solution is written in the form ψ+(k,r) = φ0(k,r) + ψsc+(k,r),(1) where ψsc+(k,r) is the scattered wave, and φ0(k,r) is a known function. From the Schr¨odinger equation the so called driven Schr¨odinger equation can be derived for the scattered wave: (E−ˆ H)ψsc+(k,r) = S(k,r),(2) where the source term is given by S(k,r) = ( ˆ H−E)φ0(k,r). 2. Theory We have found that the choice of the so called Coulomb-modified plane wave (CMPW), φ0(k,r) = eikr(kr −kr)iγ,(3) does not simplify the boundary condition from the point of view of the complex scaling. We rather split the wave function into incoming and scattered waves at the partial-wave level. The exact incoming wave is ψi,l(k, r) [4] and we write ψ+ l(k, r) = ψi,l(k, r) + ψsc+ l(k, r),(4) 10th International Conference on Clustering Aspects of Nuclear Structure and Dynamics IOP Publishing Journal of Physics: Conference Series 436 (2013) 012032 doi:10.1088/1742-6596/436/1/012032 Content from this work may be used under the terms of theCreative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 where ψi,l(k, r) = ωi,l(k, r) + χl(k, r) (5) and ωi,l(k, r) = e−ikreγπ/2(−1)l+1(2ikr)lU(l+ 1 −iγ, 2l+ 2,2ikr),(6) χl(k, r) = eikr+γπ/2 2ikr (−1)l (2ikr)l Γ(2l+ 1) Γ(l+ 1 −iγ) l X n=0 (−1)n(iγ −l)n (−2l)nn!(2ikr)n,(7) with Ubeing the confluent hypergeometric function. For the driven radial Schr¨odinger equation direct calculation gives the following source term Sl(k, r) = eikr+γπ/2 2r2Γ(−iγ).(8) After CS by scaling angle θ, which satisfies the condition 0 < θ < π, and the transformation hsc+ l,θ (k, r) = rl+1ψsc+ l,θ (k, r), which results in a regular function hsc+ l,θ (k, r) at r= 0) we get the driven radial Schr¨odinger equation "k2 2+e−2iθ 1 2 d2 dr2−e−2iθ l r d dr −e−iθ γk r−Vs(reiθ)#hsc+ l,θ (k, r) = rl+1Stot l,θ (k, r),(9) where the new source term reads Stot l,θ (k, r) = Sl,θ(k, r) + ei3θ/2ψi,l(k, reiθ)Vs(reiθ).(10) The boundary conditions are hsc+ l,θ (k, 0) = 0 (11) and lim r→∞ hsc+ l,θ (k, r)=0.(12) One can show that the phase shift δlcan be expressed as the r→ ∞ limit of a function σl(r), which we call the local representation of the phase shift: e2iσl(r)= (2kreiθ)iγ "e−ikreiθ 2ikre−iθ/2˜ ψsc+ l,θ (k, r) + eγπ/2(−1)l(iγ −l)l Γ(l+ 1 −iγ)#→e2δl(r→ ∞).(13) 3. Numerical Results In numerical calculations the boundary condition (12) is approximated by a similar expression for a large but finite value Rof r, which in in the asymptotic region: hsc+ l,θ (k, R)=0.(14) The finite element method is used as a numerical technique for the solution of Eq. (9). In the calculations equally spaced finite elements of length 1 atomic unit are taken. The degree of the Lobatto shape functions is denoted by N, and the same Nvalue is chosen for each element. A short-range potential Vs= 7.5r2exp(−r) is added to the Coulomb potential. In Figure 1 a case of θ= 0.1 radian is shown, with the angular momentum l= 0 and k= 3. The Coulomb potential is given by 1/r. The local representation of the phase shift is practically constant in a huge region, which can be identified with an approximate phase-shift value. In contrast the local approximation of the phase shift in [5] tends to the exact value by a persistent oscillation with decreasing order of amplitude. 10th International Conference on Clustering Aspects of Nuclear Structure and Dynamics IOP Publishing Journal of Physics: Conference Series 436 (2013) 012032 doi:10.1088/1742-6596/436/1/012032 2 200 400 600 800 1000 -0.18 -0.178 -0.176 phase shift exact R=250 R=1000 200 400 600 800 1000 r 0.9 0.92 0.94 0.96 phase shift Coulomb Coulomb + short range Figure 1. The local representation of the phase shift for the pure Coulomb potential (upper part) and for the Coulomb plus short-range potential (lower part). For the pure Coulomb case the dashed line displays the exact solution. 4. Conclusion The two-body scattering problem has been solved for a pure Coulomb potential as well as for a Coulomb plus finite-range potential using the standard complex scaling method. No cutoff is applied to the tail of the long-range interaction. The CS makes it possible to observe the scattering boundary condition with great accuracy in a fairly simple manner. Reference [1] Hornyak I and Kruppa A T 2012 Phys. Rev. A85 022702 [2] Rescigno T N, Baertschy M, Byrum D and McCurdy C W 1997 Phys. Rev. A55 4253 [3] Kruppa A T, Suzuki R and Kato K 2007 Phys. Rev. C75 044602 [4] van Haeringen H 1976 Il Nuovo Cimento 34B 53 [5] Volkov M V, Elander N, Yarevsky E and Yakovlev S L 2009 Europhys. Lett. 85 30001 10th International Conference on Clustering Aspects of Nuclear Structure and Dynamics IOP Publishing Journal of Physics: Conference Series 436 (2013) 012032 doi:10.1088/1742-6596/436/1/012032 3