Calculation of angular distributions in complex angular momentum theories of elastic scattering

Connor, J. N. L.; Mackay, D.C.

doi:10.1080/00268977900101261

Cited by 48 publications

(10 citation statements)

References 25 publications

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“…The pole DCS becomes smaller at large angles, where it possesses the characteristic oscillations of a backward glory. [143][144][145][146][147] If f pole (θ R ) is omitted from the CAM representation of Eq. (44), then it is found that all the oscillations in the full CAM DCS in Fig.…”

Section: H Results For the F + H 2 Reactionmentioning

confidence: 99%

Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

Connor

2013

The Journal of Chemical Physics

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Three new contributions to the complex angular momentum (CAM) theory of differential cross sections (DCSs) for chemical reactions are reported. They exploit recent advances in the Padé reconstruction of a scattering (S) matrix in a region surrounding the ReJ axis, where J is the total angular momentum quantum variable, starting from the discrete values, J = 0, 1, 2, .... In particular, use is made of Padé continuations obtained by Sokolovski, Castillo, and Tully [Chem. Phys. Lett. 313, 225 (1999)] for the S matrix of the benchmark F + H2(v(i) = 0, j(i) = 0, m(i) = 0) → FH(v(f) = 3, j(f) = 3, m(f) = 0) + H reaction. Here v(i), j(i), m(i) and v(f), j(f), m(f) are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. The three contributions are: (1) A new exact decomposition of the partial wave (PW) S matrix is introduced, which is called the QP decomposition. The P part contains information on the Regge poles. The Q part is then constructed exactly by subtracting a rapidly oscillating phase and the PW P matrix from the input PW S matrix. After a simple modification, it is found that the corresponding scattering subamplitudes provide insight into the angular-scattering dynamics using simple partial wave series (PWS) computations. It is shown that the leading n = 0 Regge pole contributes to the small-angle scattering in the centre-of-mass frame. (2) The Q matrix part of the QP decomposition has simpler properties than the input S matrix. This fact is exploited to deduce a parametrized (analytic) formula for the PW S matrix in which all terms have a direct physical interpretation. This is a long sort-after goal in reaction dynamics, and in particular for the state-to-state F + H2 reaction. (3) The first definitive test is reported for the accuracy of a uniform semiclassical (asymptotic) CAM theory for a DCS based on the Watson transformation. The parametrized S matrix obtained in contribution (2) is used in both the PW and semiclassical parts of the calculation. Powerful uniform asymptotic approximations are employed for the background integral; they allow for the proximity of a Regge pole and a saddle point. The CAM DCS agrees well with the PWS DCS, across the whole angular range, except close to the forward and backward directions, where, as expected, the CAM theory becomes non-uniform. At small angles, θ(R) ≲ 40°, the PWS DCS can be reproduced using a nearside semiclassical subamplitude, which allows for a pole being close to a saddle point, plus the farside surface wave of the n = 0 pole sub-subamplitude, with the oscillations in the DCS arising from nearside-farside interference. This proves that the n = 0 Regge resonance pole contributes to the small-angle scattering.

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Section: H Results For the F + H 2 Reactionmentioning

confidence: 99%

Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

Connor

2013

The Journal of Chemical Physics

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“…Next we examine Figure , which shows plots of the DCSs for the pole, direct, and refc subamplitudes together with the full uniform Watson/CAM DCSs for the three transitions. We note the following: The DCS for the pole subamplitude is small at large angles, where it possesses the characteristic oscillations of a backward glory (not visible in Figure c). − At smaller angles, the oscillations damp out and the pole DCS becomes monotonic and comparable in magnitude to the full CAM DCS. The direct DCS is small at angles close to the forward direction but becomes comparable to the full CAM DCS at sideward and large angles. For the 300 and 310 cases in Figure a,b, the DCSs for the refc subamplitude are significant at small and sideward angles but then become very small at large angles. This is because the residue and erfc subamplitudes cancel in this angular range; the cancellation occurs when | u n (θ R )| ≫ 1 (see the appendix of ref ).…”

Section: Application To the F + H2 Reactionmentioning

confidence: 88%

State-to-State F + H₂ Reaction at E_trans = 0.04088 eV: QP Decomposition, Parametrized S Matrix Incorporating Regge Poles, and Uniform Asymptotic Complex Angular Momentum Analysis of the Angular Scattering

Xiao

Connor

2016

J. Phys. Chem. A

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We report two new contributions for understanding the quantum dynamics of the benchmark state-to-state reaction, F + H2(vi, ji, mi) → FH(vf, jf, mf) + H, where (vi, ji, mi) and (vf, jf, mf) are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. We analyze product differential cross sections (DCSs) for the transitions, 000 → 300, 000 → 310, and 000 → 320, at a translational energy of 0.04088 eV using the potential energy surface of Fu-Xu-Zhang. The two new contributions are as follows: (1) We exploit the recently introduced QP decomposition of J. N. L. Connor [ J. Chem. Phys . 2013 , 138 , 124310 ] to transform numerical partial-wave scattering (S) matrix elements for the three transitions into parametrized (analytic) formulas, in which all terms in the three parametrized S matrices have a direct physical interpretation. In particular, they contain the positions and residues of Regge poles in the first quadrant of the complex angular momentum (CAM) plane. We obtain very close agreement between the values of the parametrized and numerical S matrix elements. (2) We then apply a uniform asymptotic Watson/CAM theory, which allows a Regge pole to be close to a saddle point. It uses the parametrized S matrices and is applied to the partial wave series (PWS) representation for the scattering amplitude to understand structure in a DCS in terms of three contributing subamplitudes. We prove using this powerful CAM theory that resonance Regge poles contribute to the small-angle scattering in the DCSs for all three transitions, with the oscillations at larger angles arising from nearside-farside interference. We obtain very good agreement between the uniform asymptotic Watson/CAM DCSs and the corresponding PWS DCSs, except for angles close to the forward and backward directions, where (as expected) the Watson/CAM formulas become nonuniform.

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“…Formulas (2.15) and (2.18) are used in the first-order as well as higher-order phase-integral calculations of l and r in the pole sum (2.11). As for the Legendre function, which is given in [27], where the accuracy is also discussed in detail.…”

Section: 'mentioning

confidence: 99%

Regge-pole description of rainbow scattering by means of the phase-integral method

Amaha¹,

Thylwe

1994

Phys. Rev. A

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In the present paper we study the effects in the rainbow differential cross sections of using higherorder phase-integral residues in the pole sum of the scattering amplitude. We have recently found that already in the first-order approximation the uniform residue formula, derived by Froman and Froman, provides an important means for an accurate calculation of the rainbow differential cross section where reliable numerical quantum methods are not available. From the present investigation it is clear that the use of higher-order approximations is not the crucial step to further reveal the rainbow structures in the cross section. For that purpose, it is instead essential to increase the purely numerical precision (in any order) of the uniform phase-integral residue calculations.PACS number(s): 34.40. +n, 03.65.Nk, 03.65.Sq

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Calculation of angular distributions in complex angular momentum theories of elastic scattering

Cited by 48 publications

References 25 publications

Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

State-to-State F + H₂ Reaction at E_trans = 0.04088 eV: QP Decomposition, Parametrized S Matrix Incorporating Regge Poles, and Uniform Asymptotic Complex Angular Momentum Analysis of the Angular Scattering

Regge-pole description of rainbow scattering by means of the phase-integral method

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Calculation of angular distributions in complex angular momentum theories of elastic scattering

Cited by 48 publications

References 25 publications

Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

State-to-State F + H2 Reaction at Etrans = 0.04088 eV: QP Decomposition, Parametrized S Matrix Incorporating Regge Poles, and Uniform Asymptotic Complex Angular Momentum Analysis of the Angular Scattering

Regge-pole description of rainbow scattering by means of the phase-integral method

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State-to-State F + H₂ Reaction at E_trans = 0.04088 eV: QP Decomposition, Parametrized S Matrix Incorporating Regge Poles, and Uniform Asymptotic Complex Angular Momentum Analysis of the Angular Scattering