A New Rainbow: Angular Scattering of the F + H₂(v_i= 0,j_i= 0) → FH(v_f= 3,j_f= 3) + H Reaction

Abstract

The angular scattering of a state-to-state chemical reaction contains fundamental information on its dynamics. Often the angular distributions are highly structured and the physical interpretation of this structure is an important and difficult problem. Here, we report a surprising finding for the benchmark F + H(2) --> FH + H reaction, when the product molecule FH is in a vibrational state with quantum number = 3 and a rotational state with quantum number = 3. We demonstrate that the differential cross section (DCS) is an example of (attractive) rainbow scattering, being characterized by an Airy function and its derivative. The rainbow reveals its presence in the DCS by interference with the repulsive (or nearside) scattering producing characteristic diffraction oscillations. The rainbow is broad, which explains why it has not been recognized in the many earlier theoretical and experimental investigations of this reaction. There is an angular region in the DCS where the rainbow dominates, but with the unusual property that the DCS is less intense than in adjoining angular regions. The reaction investigated is F + H(2)(v(i) = 0, j(i) = 0, m(i) = 0) --> FH(v(f) = 3, j(f) = 3, m(f) = 0) + H, where v(i), j(i), m(i) and v(f), j(f), m(f) are initial and final vibrational, rotational and helicity quantum numbers, respectively. The relative translational energy is 0.119 eV. We use rigorous semiclassical (asymptotic) techniques that provide physical insight as well as a mathematically sound and numerically accurate description of the angular scattering. The semiclassical DCS agrees very closely with the exact quantum DCS. The semiclassical scattering amplitude is used to assess the physical effectiveness of the Fuller nearside-farside decomposition for the partial wave series of the F + H(2) reaction, including the effect of one resummation. We also compare the semiclassical and exact quantum nearside, farside, and full local angular momenta and find good agreement. Although our new rainbow has unusual and unexpected properties, similar rainbows are predicted to occur in the DCSs of many state-to-state chemical reactions, since the semiclassical analysis is generic and not specific to the present F + H(2) example.

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

“…In contrast, σ (Pmod) (θ R ) has oscillations over the whole angular range, which are particularly pronounced for θ R 70 • . As a result of Qmod,Pmod interference in the fundamental identity (22), the oscillations in σ (θ R ) arise mainly from the Pmod subamplitude.…”

“…32,33 It is also known that the scattering at larger angles is an example of a broad attractive F rainbow with unusual properties. 22 The rainbow angle occurs at θ r R = 35.2 • , with the bright side being θ R < θ r R and the dark side, θ R > θ r R . Figure 3 shows plots versus J of the unmodified |S (Q) J |, |, or |S J |, as expected from the discussion in Sec.…”

“…Section III describes the theory of the QP decomposition for the S matrix and the scattering amplitude, and makes an application to the F + H 2 reaction using quantum numerical S matrix elements. 6 The techniques of nearside-farside (NF) decomposition 8,18,19,22,23, and Local Angular Momentum (LAM) analysis 22,23,31,34,[36][37][38][39][40][41][42][43][45][46][47][48] for resummed Legendre PWS 22,23,30,31,36,38,[42][43][44][45][46][47] are also employed in Sec. III.…”

“…If there is Airy-type rainbow in the DCS, then we also expect Re J 0 ≈ J r , where J r is the rainbow angular momentum variable, which separates the bright and dark sides of the rainbow. 5,22,23,27,55,56 The imaginary part of J 0 is related to the life-angle (written 0 θ R ) by 0 θ R = 1/ (2ImJ 0 ); it controls the decay of a surface (or creeping) wave which propagates around the interaction zone. Note that the life-angle can be very small (short-lived or direct scattering), or very large (for long-lived collisions), as well as taking any value in between.…”

“…(31)- (34), the args are, in general, not principal values in order that the derivatives be well defined. The Qmod and Pmod PWS subamplitudes (19) and (20) can also be resummed if desired 22,23,30,31,36,38,[42][43][44][45][46][47] before making the NF decompositions, since it is known this helps "clean" the N and F un-resummed DCSs and LAMs of unphysical structure. The resummation equations have been described in detail by Totenhofer et al; 42 they are easily adapted to Eqs.…”