##
1Theory of forward glory scattering for chemical reactions
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Theory of forward glory scattering for chemical reactions

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The theory of forward glory scattering is developed for a state-to-state chemical reaction whose scattering amplitude can be expanded in a Legendre partial wave series.

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Two transitional approximations are derived that are valid for angles on, and close to, the axial caustic associated with the glory.

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These are the integral transitional approximation (ITA) and the semiclassical transitional approximation (STA), which is obtained when the stationary phase method is applied to the ITA.

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Both the ITA and STA predict that the scattering amplitude for glory scattering is proportional to a Legendre function of real degree or, to a very good approximation, a Bessel function of order zero.

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A primitive semiclassical approximation (PSA) is also derived that is valid at larger angles, away from the caustic direction, but which is singular on the caustic.

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The PSA demonstrates that glory structure arises from nearside–farside (NF) interference, in an analogous way to the two-slit experiment.

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The main result of the paper is a uniform semiclassical approximation (USA) that correctly interpolates between small angles, where the ITA and STA are valid, and larger angles where the PSA is valid.

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The USA expresses the scattering amplitude in terms of Bessel functions of order zero and unity, together with N and F cross sections and phases.

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In addition, various subsidiary approximations are derived.

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The input to the theory consists of accurate quantum scattering matrix elements.

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The theory also has the important attribute that it provides physical insight by bringing out semiclassical and NF aspects of the scattering.

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The theory is used to show that the enhanced small angle scattering in the F + H

_{2}(*v*_{i}= 0,*j*_{i}= 0,*m*_{i}= 0) → FH(*v*_{f}= 3,*j*_{f}= 3,*m*_{f}= 0) + H reaction is a forward glory, 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.
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The forward angle scattering for the H + D

_{2}(*v*_{i}= 0,*j*_{i}= 0,*m*_{i}= 0) → HD(*v*_{f}= 3,*j*_{f}= 0,*m*_{f}= 0) + D reaction is also analysed and shown to be a forward glory, in agreement with a simpler treatment by D. Sokolovski (*Chem. Phys. Lett.*, 2003,**370**, 805), which is a special case of the STA.
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## Introduction

15

The differential cross section for a state-to-state reactive molecular collision often exhibits a complicated interference pattern.

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An important problem, which is discussed in recent reviews,

^{1–13}is to analyse this scattering pattern in order to obtain information on the reaction dynamics.
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One theoretical approach for understanding structured angular distributions, and which is used in this paper, applies a nearside–farside (NF) decomposition to the partial wave series (PWS) representation of the scattering amplitude.

^{2,5,12,14–31}
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The aim of a NF analysis is to decompose the PWS exactly into the sum of two subamplitudes that have simpler properties, namely a N PWS subamplitude and a F PWS subamplitude.

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The complicated interference structure in the angular distribution can then often be interpreted as arising from interference between the N and F subamplitudes, or from the N subamplitude, or from the F subamplitude.

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NF theory has been developed for PWS expanded in a basis set of Legendre polynomials, associated Legendre functions or reduced rotation matrix elements.

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Applications have reported for the reactions: Cl + HCl → ClH + Cl,

^{14,16,29,30}F + H_{2}→ FH + H,^{22}H + D_{2}→ HD + H,^{22}F + HD → FH + D (or FD + H),^{22}and I + HI → IH + I.^{25,26,31}
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22

The F + H

_{2}reaction provides a good example of the physical insight supplied by a NF analysis: the NF decomposition of the scattering amplitude for a state-to-state angular distribution suggested^{22}that the oscillatory forward angle scattering arises from a short-lived decaying Regge state with a lifeangle of ≈30°–40°.
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This suggestion that a short-lived resonance state contributes to the forward angle scattering was contrary to conventional wisdom at the time it was made (see,

*e.g.*,^{ref. 32}) but was later verified by Sokolovski*et al.*^{33,34}
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A second, and more fundamental, theoretical procedure for understanding structure in an angular distribution is to take the

*ħ*→ 0 limit of the scattering amplitude.^{35}
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The scattering pattern is again interpreted as the interference between a small number of simpler (semiclassical) subamplitudes.

^{2,29–31,33–38}
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Unfortunately, carrying out a rigorous (uniform) asymptotic analysis of the scattering amplitude for a reactive collision is very difficult: it remains an important unsolved problem at the present time.

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However, progress has been made for a Legendre PWS whose scattering matrix element, when analytically continued, possesses poles in the complex angular momentum (CAM) plane.

^{2,29–31,33–38}
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A common observation in the angular distributions of state-to-state reactions is an enhancement of the forward angle scattering, which is often accompanied by oscillations.

^{1–13}
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29

The semiclassical theory developed in

^{refs. 29 and 30}for zero helicity state-to-state transitions shows that the forward angle scattering is associated with the factor 1/sin*θ*_{R}, where*θ*_{R}is the reactive scattering angle.
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30

The factor 1/sin

*θ*_{R}is characteristic of an axial caustic,^{39–43}which suggests that the enhanced intensity at forward angles is an example of glory scattering.
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31

Unfortunately, the semiclassical CAM theory presented in

^{refs. 29 and 30}is not uniformly valid at*θ*_{R}= 0.
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32

Sokolovski has partially overcome this defect in a letter whose main emphasis is the relation between CAM pole structure and forward angle scattering.

^{38}
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33

In particular, Sokolovski has developed a transitional semiclassical approximation, based on one given by Landau and Lifshitz (

^{ref. 44}, p. 523), which he has applied to the H + D_{2}reaction,^{38}obtaining good agreement with the exact angular distribution at small values of*θ*_{R}.
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The present paper extends this research of Sokolovski,

^{38}and makes the following contributions to understanding glories in the forward angle scattering of reactive collisions:
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(1) Two new transitional approximations are derived: the integral transitional approximation (ITA) and the semiclassical transitional approximation (STA), which is a stationary-phase approximation to the ITA.

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A further approximation to the STA yields the glory formula used by Sokolovski.

^{38}
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Transitional approximations are valid for angles on, and close to, the axial caustic,

*i.e.*, for*θ*_{R}≈ 0, but they become increasingly inaccurate at larger angles.
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(2) A primitive semiclassical approximation (PSA) is obtained, which is valid away from the caustic direction at larger angles, but which is singular (infinite) at

*θ*_{R}= 0.
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39

The PSA angular distribution is analogous to the interference pattern in the famous two-slit experiment and clearly demonstrates that the forward-angle glory scattering arises from NF interference.

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If the interference term in the PSA is neglected, we obtain the classical semiclassical approximation (CSA), which lets us make contact with quasiclassical trajectory calculations of the angular scattering.

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(3) A uniform semiclassical approximation (USA) is derived which correctly interpolates between

*θ*_{R}≈ 0, where the transitional approximations are valid, and larger angles where the PSA is valid.
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The derivation of the USA makes use of mathematical techniques for the (uniform) asymptotic evaluation of oscillating integrals.

^{45–51}
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The ITA, STA, PSA, CSA and USA, together with some subsidiary approximations, are derived in section II.

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An important property of all these approximations is that the semiclassical picture is clearly evident, even though their input consists of accurate scattering matrix elements,

*i.e.*it is not necessary to assume the Wentzel–Kramers–Brillouin (WKB) approximation for the scattering matrix elements (although this approximation, and other ones, could be used if convenient).
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The theory of section II is applied in sections III and IV to two benchmark state-to-state reactions that exhibit forward angle scattering.

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These are the F + H

_{2}→ FH + H and H + D_{2}→ HD + D reactions respectively.
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Section V contains our conclusions.

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The appendix examines five different ways to approximate a Legendre function by a Bessel function of order zero that have been used in the molecular collision theory literature.

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## Theory of forward glory scattering for reactive molecular collisions

### Introduction

49

We consider a state-to-state chemical reaction of the type A + BC(

*v*_{i},*j*_{i},*m*_{i}= 0) → AB(*v*_{f},*j*_{f},*m*_{f}= 0) + Cwhere*v*_{i},*j*_{i},*m*_{i}and*v*_{f},*j*_{f},*m*_{f}are the vibrational, rotational and helicity quantum numbers for the initial and final states, respectively.
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Because the helicity quantum numbers are equal to zero, the scattering amplitude,

*f*(*θ*_{R}), can be written as a Legendre partial wave series (PWS) where*k*is the initial translational wavenumber,*J*is the total angular momentum quantum number,*S*_{J}is a scattering matrix element,*P*_{J}(•) is a Legendre polynomial of degree*J*and*θ*_{R}∈ [0,π] is the reactive scattering angle,*i.e.*, the angle between the outgoing AB molecule and the incoming A atom.
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The subscript,

*v*_{i},*j*_{i},*m*_{i}→*v*_{f},*j*_{f},*m*_{f}, has been omitted from*f*(*θ*_{R}) and*S*_{J}for notational simplicity, as has the subscript,*v*_{i},*j*_{i}, from*k*.
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The series (1) also occurs in some approximate theories of chemical reactions that expand the PWS in a basis set of Legendre polynomials (see, for example,

^{refs. 14, 16, 25, 26, 29, 30, 52 and 53}).
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53

The differential cross section is given by

*σ*(*θ*_{R}) = |*f*(*θ*_{R})|^{2}.For the theoretical development that follows, it is convenient to first write the PWS (1) in the standard form with*θ*_{R}∈ [0,π], where the argument of the Legendre polynomial is cos*θ*_{R}rather than cos(π –*θ*_{R}).
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54

The modified scattering matrix element

*S̃*_{J}in eqn. (3) is defined by*S̃*_{J}= exp(iπ*J*)*S*_{J},*J*= 0,1,2,…
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We begin the derivation by applying the Poisson summation formula (

*e.g.*,^{ref. 39}) to the PWS (3), which yields the exact result for*θ*_{R}∈ [0,π) where*λ*=*J*+ 1/2 and the notation*S̃*_{J}=*S̃*_{λ–1/2}≡*S̃*(*λ*)has been used.
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56

In eqn. (4),

*S̃*(*λ*) and*P*_{λ–1/2}(cos*θ*_{R}) have been interpolated (and extrapolated where necessary) from half-integer values of*λ*= 1/2,3/2,5/2,… to continuous values,*λ*∈ [0,∞).
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Sections III and IV describe the interpolation used for

*S̃*(*λ*).
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In practice, separate interpolations are used for |

*S̃*(*λ*)| and arg*S̃*(*λ*), where arg does not in general denote the principal value, rather arg*S̃*(*λ*) is a continuous function with arg*S̃*(*λ*) ∈ (–∞,∞).
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In eqn. (4),

*P*_{λ–1/2}(cos*θ*_{R}) is a Legendre function of the first kind of real degree,*λ*– 1/2, with*θ*_{R}∈ [0,π).
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60

[

*n*.
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61

*b*.,

*P*

_{λ–1/2}(cos

*θ*

_{R}) is singular at

*θ*

_{R}= π, except for

*λ*= 1/2,3/2,5/2,…]

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62

Under semiclassical conditions,

*kR*≫ 1, where*R*is the reaction radius, the integrands in the series (4) are highly oscillatory functions.
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We assume, as is commonly the case,

^{39–41}that any points of stationary phase occur only for the*m*= 0 integrand.
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We then have for

*θ*_{R}∈ [0,π)The upper limit,*λ*_{max}+ 1/2 =*J*_{max}+ 1, on the integral in eqn. (6) recognises the fact that the PWS (3) has, in practice, a finite number of non-zero terms,^{54}with*S̃*_{J}≡ 0 for*J*>*J*_{max}.
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65

Note: if we numerically evaluate the integral (6) and compare it with the result from summing the PWS (3), we can then test the accuracy of neglecting terms with

*m*≠ 0 in the Poisson series (4).
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Eqn. (6) is the starting point for the approximations derived below.

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67

In the derivations, we shall often expand arg

*S̃*(*λ*) about a real point*λ*=*λ*_{s}in a Taylor's series to second order arg*S̃*(*λ*) = arg*S̃*(*λ*_{s}) +**(*λ*_{s}) (*λ*–*λ*_{s}) + ½**′(*λ*_{s}) (*λ*–*λ*_{s})^{2}wherewill be called the*quantum deflection function*.
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The phrase “deflection function” indicates that

**(*λ*) is analogous to the classical deflection function often used in the semiclassical theory of elastic scattering.^{39–41}
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69

The adjective “quantum” indicates that {

*S̃*_{J}}, and its continuation*S̃*(*λ*), are in principle exact quantum quantities (although, if convenient, approximate scattering matrix elements could be used).
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70

In addition, the notations andhave been employed in the Taylor expansion (7).

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71

Figs. 1(a) and 1(b) show, for the F + H

_{2}reaction of section III, the behaviour of arg*S̃*(*λ*) and**(*λ*) around the point*λ*=*λ*_{g}where arg*S̃*(*λ*) is stationary (a maximum) so that**(*λ*_{g}) = 0.
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In addition, Fig. 1(b) illustrates the quantities

*λ*_{±}(*θ*_{R}) and*λ*(*θ*_{R};*φ*) with*φ*∈ [0,π] which play an important rôle in the derivations below.
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### Transitional approximations (TA)

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In this section, two transitional approximations are derived for the forward glory scattering.

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A TA is valid on, and close to, the axial caustic,

*i.e.*, for values of*θ*_{R}close to*θ*_{R}= 0.
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#### Integral transitional approximation (ITA)

75

When

*θ*_{R}= 0, we have*P*_{λ–1/2}(1) = 1 for all*λ*∈ [0,∞).
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For

*θ*_{R}close to*θ*_{R}= 0, we assume that*P*_{λ–1/2}(cos*θ*_{R}) is slowly varying relative to the oscillations of*S̃*(*λ*) and can be removed from the integrand of eqn. (6).
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The (real) value used for

*λ*in the removal of*P*_{λ–1/2}(cos*θ*_{R}) is that given by the stationary phase condition**(*λ*_{g}) = 0which defines the glory angular momentum variable,*λ*_{g}, see Fig. 1.
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78

In practice, there may be more than one root to eqn. (9)-see sections III and IV for examples.

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79

In this situation, we use the

*λ*_{g}that makes the largest contribution to the integral in eqn. (6), when the stationary phase approximation is employed (see section IIB.2).
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80

We can now approximate eqn. (6) by

*f*_{ITA}(*θ*_{R}) =*GP*_{λg–1/2}(cos*θ*_{R})where*G*denotes the complex-valued integralThe corresponding differential cross section is*σ*_{ITA}(*θ*_{R}) = |*G*|^{2}*P*_{λg–1/2}(cos*θ*_{R})^{2}Eqns. (10)–(12) define the ITA for forward glory scattering.
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81

Note that the value of the cross section at

*θ*_{R}= 0 is finite, being*σ*_{ITA}(*θ*_{R}= 0) = |*G*|^{2}.
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#### Semiclassical transitional approximation (STA)

82

The ITA can be further approximated if we evaluate the integral (11) by the stationary phase method.

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83

We again assume that the stationary phase condition is given by eqn. (9) and that the situation in Fig. 1 applies.

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84

Then with the help of the stationary phase result [

^{ref. 55}, eqns. (4.1) and (4.12)] where*F*′(*λ*_{s}) = 0,*a*<*λ*_{s}<*b*,*G*(*λ*_{s}) ≠ 0,∞ and*F*″(*λ*_{s}) ≠ 0,∞, we obtain for the STA The corresponding differential cross section is.
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85

Eqns. (14) and (15) define the STA for forward glory scattering.

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86

At

*θ*_{R}= 0, the value of the cross section is Note that eqn. (7) becomes for this case arg*S̃*(*λ*) = arg*S̃*(*λ*_{g}) + ½**′(*λ*_{g}) (*λ*–*λ*_{g})^{2}where**′(*λ*_{g}) is negative,*i.e.*, arg*S̃*(*λ*) has a maximum at*λ*=*λ*_{g}as illustrated in Fig. 1.
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#### Remarks

87

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88

(

*i*) The ITA is a*global*approximation, in that the input consists of {*S̃*_{J}}, and its continuation*S̃*(*λ*), which contain all the information on the dynamics of the reaction.
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89

In contrast, the STA is a

*local*approximation, since it only requires dynamical information at*λ*=*λ*_{g}.
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90

(

*ii*) Both the ITA and STA predict that the angular distribution for forward glory scattering is proportional to*P*_{λg–1/2}(cos*θ*_{R})^{2}.
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91

However conversely, if we find that the angular scattering at

*θ*_{R}≈ 0 can be fitted to the form constant ×*P*_{λt–1/2}(cos*θ*_{R})^{2}say, then it does not follow that we have a forward glory without further investigation,*e.g.*, we must check that**(*λ*_{t}) = 0 is satisfied.
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92

For there may be other dynamical mechanisms that result in the integral (11) being dominated by a small range of

*λ*values around*λ*=*λ*_{t}, but with the property,**(*λ*_{t}) ≠ 0.
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93

(

*iii*) Following Ford and Wheeler,^{56,57}it has been common in the semiclassical theory of glory scattering for elastic collisions to approximate Legendre polynomials by Bessel functions of order zero.
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94

Eqns. (12) and (15) demonstrate that this approximation is unnecessary [this last result has also been obtained by Child, but not commented on: see eqn. (5.52) of

^{ref. 40}].
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95

The appendix examines five ways in which a Legendre function can be approximated by a Bessel function of order zero.

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96

The Hilb approximation is the one recommended for use: which has an error,

*O*(*λ*^{–3/2}), that is uniform with respect to*θ*_{R}∈ [0,π –*ε*] with*ε*> 0.
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Advantage: None |
Novelty: None |
ConceptID: Mod9

97

At

*θ*_{R}= 0, where eqn. (17) takes the indeterminate form 0/0, we use the well-known limit,*θ*_{R}/sin*θ*_{R}→ 1 as*θ*_{R}→ 0.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod9

98

When the Hilb relation (17) is used to replace the Legendre functions in the ITA and STA, the resulting approximations will be denoted ITA/

*J*_{0}and STA/*J*_{0}respectively.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod9

99

If we make the additional approximation of replacing the factor (

*θ*_{R}/sin*θ*_{R})^{1/2}by unity for*θ*_{R}≈ 0 in the STA/*J*_{0}, we obtain the forward glory approximation used by Sokolovski-see eqn. (17) of .^{ref. 38}
Type: Result |
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ConceptID: Res7

100

(

*iv*) The ITA and STA both predict that the cross section is zero at the roots of the equation,*P*_{λg–1/2}(cos*θ*_{R}) = 0.
Type: Result |
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ConceptID: Res11

101

Similarly the ITA/

*J*_{0}and STA/*J*_{0}predict zero cross sections at the roots of*J*_{0}(*λ*_{g}*θ*_{R}) = 0 [note that the first five roots of*J*_{0}(*x*) = 0 are x = 2.405, 5.520, 8.654, 11.792 and 14.931, from^{ref. 58}. p. x, table II).
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res11

### Primitive semiclassical approximation (PSA)

102

As

*θ*_{R}moves away from*θ*_{R}= 0, the assumption of the ITA and STA that*P*_{λ–1/2}(cos*θ*_{R}) is slowly varying becomes less valid.
Type: Result |
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ConceptID: Res12

103

Instead, we must use the asymptotic approximation which is valid for

*λ*sin*θ*_{R}≫ 1.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

104

The two exponentials in eqn. (18) have a physical interpretation: The term exp(+i

*λθ*_{R}) describes an angular wave travelling anti-clockwise in*θ*_{R}and corresponds to farside (F) scattering in a semiclassical treatment, whilst the term exp(–i*λθ*_{R}) represents an angular wave travelling clockwise in*θ*_{R}and corresponds to nearside (N) scattering.^{14–30}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

105

Substituting approximation (18) into the integral (5) lets us write

*f*(*θ*_{R}) =*f*_{+}(*θ*_{R}) +*f*_{–}(*θ*_{R}) where the F and N subamplitudes are with
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

106

Next we apply the stationary phase approximation to the

*I*_{±}(*θ*_{R}) integrals, assuming that |*S̃*(*λ*)| is slowly varying.
Type: Model |
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ConceptID: Mod10

107

The stationary phase condition is or

**(*λ*) = ∓*θ*_{R}for*λ*=*λ*_{±}(*θ*_{R}) where we have assumed that there exist a unique pair of real valued roots to eqn. (23).
Type: Model |
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ConceptID: Mod10

108

The

*λ*_{±}(*θ*_{R}) are illustrated in Fig. 1(b).
Type: Observation |
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ConceptID: Obs1

109

Note that

*λ*_{–}(*θ*_{R}) and*λ*_{+}(*θ*_{R}) are coincident at*θ*_{R}= 0, since*λ*_{g}=*λ*_{±}(*θ*_{R}= 0).
Type: Observation |
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ConceptID: Obs1

110

Also Fig. 1(b) shows that

**′(*λ*_{±}(*θ*_{R})) < 0.
Type: Observation |
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ConceptID: Obs1

111

Explicit formulae for the

*λ*_{±}(*θ*_{R}) can be obtained when the quadratic approximation (16) is made for arg*S̃*(*λ*).
Type: Model |
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ConceptID: Mod7

112

For

**(*λ*), we have**(*λ*) = –|**′(*λ*_{g})|(*λ*–*λ*_{g}) and the stationary phase eqn. (23) then gives
Type: Model |
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ConceptID: Mod11

113

Applying the stationary phase approximation (13) to the integrals (21) yields The term exp[+i

*λ*_{±}(*θ*_{R})*θ*_{R}] in eqn. (25) describes an angular wave travelling anti-clockwise in*θ*_{R}, so that*f*_{+}(*θ*_{R}) corresponds to the F scattering, whilst the term exp[–i*λ*_{±}(*θ*_{R})*θ*_{R}] represents an angular wave travelling clockwise in*θ*_{R}, so that*f*_{–}(*θ*_{R}) corresponds to the N scattering.
Type: Model |
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ConceptID: Mod11

114

The formulae for the

*f*_{±}(*θ*_{R}) subamplitudes take a simpler form if we first define the F and N cross sections The*σ*_{±}(*θ*_{R}) are analogous to classical cross sections.
Type: Model |
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ConceptID: Mod11

115

This analogy can be made more explicit by defining the impact parameter variable,

*b*=*λ*/*k*together with*b*_{±}(*θ*_{R}) =*λ*_{±}(*θ*_{R})/*k*.
Type: Model |
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ConceptID: Mod12

116

We can then write eqn. (26) in the form Substituting eqns. (25) and (26) into eqn. (20), we obtain for the F and N subamplitudes

*f*_{+}(*θ*_{R}) = –[*σ*_{+}(*θ*_{R})]^{1/2}exp[i*β*_{+}(*θ*_{R})]and*f*_{–}(*θ*_{R}) = –i[*σ*_{–}(*θ*_{R})]^{1/2}exp[i*β*_{–}(*θ*_{R})]where the F and N phases*β*_{±}(*θ*_{R}) are defined by*β*_{±}(*θ*_{R}) = arg*S̃*(*λ*_{±}(*θ*_{R})) ±*λ*_{±}(*θ*_{R})*θ*_{R}.
Type: Model |
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ConceptID: Mod12

117

The full PSA scattering amplitude is then given by

*f*_{PSA}(*θ*_{R}) = –[*σ*_{+}(*θ*_{R})]^{1/2}exp[i*β*_{+}(*θ*_{R})] – i[*σ*_{–}(*θ*_{R})]^{1/2}exp[i*β*_{–}(*θ*_{R})].
Type: Model |
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ConceptID: Mod12

118

The extra factor i in eqn. (28) compared to eqn. (27), arises from the phase difference exp(iπ/2) between the F and N angular waves in eqn. (18).

Type: Model |
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ConceptID: Mod12

119

Eqns. (26)–(30) define the PSA for the scattering amplitude.

Type: Model |
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ConceptID: Mod12

120

The corresponding PSA differential cross section for glory scattering is given by

*σ*_{PSA}(*θ*_{R}) =*σ*_{+}(*θ*_{R}) +*σ*_{–}(*θ*_{R}) + 2[*σ*_{+}(*θ*_{R})*σ*_{–}(*θ*_{R})]^{1/2}sin[*β*_{+}(*θ*_{R}) –*β*_{–}(*θ*_{R})].
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ConceptID: Mod12

121

Eqn. (31) has the typical form of a two slit interference pattern for the F and N scattering.

Type: Result |
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ConceptID: Res2

### Classical semiclassical approximation (CSA)

122

If the sinusoidal term in eqn. (31) is neglected, we obtain a result that will be called the CSA

*σ*_{CSA}(*θ*_{R}) =*σ*_{+}(*θ*_{R}) +*σ*_{–}(*θ*_{R})*σ*_{CSA}(*θ*_{R}) can be estimated by quasiclassical trajectory (QCT) calculations.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod13

123

If the |

*S̃*(*λ*_{±}(*θ*_{R}))|^{2}in eqn. (26) contain a sizable contribution from tunnelling, then we might expect the QCT method to underestimate*σ*_{CSA}(*θ*_{R}).
Type: Hypothesis |
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ConceptID: Hyp4

124

On the other hand, the QCT binning procedure used to “quantize” the final states, has a weak theoretical justification, and could result in an overestimate of

*σ*_{CSA}(*θ*_{R}).
Type: Hypothesis |
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ConceptID: Hyp5

125

It is interesting to take the

*θ*_{R}→ 0 limit of eqn. (32).
Type: Model |
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ConceptID: Mod14

126

We find The singularity at

*θ*_{R}= 0 can be clearly seen in eqn. (33).
Type: Model |
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ConceptID: Mod14

127

Its origin is the (non-uniform) asymptotic approximation (18), which is not valid for

*θ*_{R}→ 0.
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ConceptID: Mod14

128

Eqn. (33) is analogous to the classical result for elastic scattering from a central potential.

Type: Result |
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ConceptID: Res13

129

[

*n*.
Type: Background |
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ConceptID: Bac11

130

*b*., there are errors in the formulae given by Pauly in eqn. (2.15) of

^{ref. 59}and in eqn. (42) of

^{ref. 60}].

Type: Background |
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ConceptID: Bac11

131

We can also derive eqn. (33) from the STA intensity (15), by averaging over the oscillations in

*P*_{λg–1/2}(cos*θ*_{R})^{2}.
Type: Model |
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ConceptID: Mod15

132

With the help of eqn. (18), we find 〈

*P*_{λg–1/2}(cos*θ*_{R})^{2}〉 ≈ 1/(π*λ*_{g}sin*θ*_{R}) so that eqn. (15) reduces to eqn. (33).
Type: Model |
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Novelty: None |
ConceptID: Mod15

133

An alternative derivation of eqn. (33) uses the Hilb approximation (17) together with the average 〈

*J*_{0}(*λ*_{g}*θ*_{R})^{2}〉 ≈ 1/(π*λ*_{g}*θ*_{R}).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod16

### Uniform semiclassical approximation (USA)

#### Introduction

134

The ITA and STA do not merge smoothly with the PSA when

*θ*_{R}increases from*θ*_{R}= 0 to larger angles.
Type: Result |
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ConceptID: Res14

135

We can rectify this defect by deriving a USA which does change smoothly as

*θ*_{R}moves away from the axial caustic and which contains the ITA/*J*_{0}and PSA as limiting cases.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod17

136

In particular, we generalise to reactive collisions the uniform semiclassical treatment of forward glory elastic scattering given by Berry

^{45}(see also^{refs. 46–51}for more general uniform approximations).
Type: Goal |
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ConceptID: Goa1

137

The derivation of the USA proceeds by applying uniform asymptotic techniques to a two-dimensional integral representation for

*f*(*θ*_{R}).
Type: Method |
Advantage: None |
Novelty: Old |
ConceptID: Met5

138

The first step in the derivation inserts the Hilb approximation (17) for

*P*_{λ–1/2}(cos*θ*_{R}) into the integrand of eqn. (5) giving Now the Bessel function,*J*_{0}(*x*), has the following integral representation, [^{ref. 58}, p. 57, eqn. (4.3), together with*J*_{0}(*x*) =*J*_{0}(–*x*)].
Type: Model |
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ConceptID: Mod17

139

In the second step, we insert eqn. (35) into eqn. (34) obtaining the following two-dimensional integral for

*f*(*θ*_{R}) where the phase is defined by*B*(*θ*_{R};*λ*,*φ*) = arg*S̃*(*λ*) –*λθ*_{R}cos*φ*.
Type: Model |
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ConceptID: Mod17

140

Before proceeding, it is first helpful to examine the stationary points of

*B*(*θ*_{R};*λ*,*φ*).
Type: Model |
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ConceptID: Mod18

#### Stationary points of *B*(*θ*_{R};*λ*,*φ*)

141

The stationary phase equations for the two dimensional integral (36) are where eqn. (8) has been used.

Type: Model |
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ConceptID: Mod18

142

For

*θ*_{R}≠ 0 the two roots of eqn. (39) for*φ*∈ [0,π] (*i.e.*, for*φ*in the range of integration) are*φ*= 0 and*φ*= π.
Type: Model |
Advantage: None |
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ConceptID: Mod18

143

Substituting the root

*φ*= 0 into eqn. (38) gives**(*λ*_{–}(*θ*_{R})) = +*θ*_{R}whilst substituting the root*φ*= π into eqn. (38) leads to**(*λ*_{+}(*θ*_{R})) = –*θ*_{R}, which agree with eqn. (23).
Type: Model |
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ConceptID: Mod18

144

The stationary points of

*B*(*θ*_{R};*λ*,*φ*) are therefore (*λ*_{–}(*θ*_{R}),*φ*= 0) and (*λ*_{+}(*θ*_{R}),*φ*= π).
Type: Model |
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ConceptID: Mod18

145

The values of

*B*(*θ*_{R};*λ*,*φ*) at the stationary points are*B*(*θ*_{R};*λ*_{±}(*θ*_{R}),*φ*= π,0) = arg*S̃*(*λ*_{±}(*θ*_{R})) ±*λ*_{±}(*θ*_{R})*θ*_{R}≡*β*_{±}(*θ*_{R})where the definition of the*β*_{±}(*θ*_{R}) in eqn. (29) has been used.
Type: Model |
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ConceptID: Mod18

146

It is convenient to continue to assume

*θ*_{R}≠ 0 in the following calculations of stationary phase points.
Type: Model |
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Novelty: None |
ConceptID: Mod18

#### Stationary phase integration over λ

147

The phase

*B*(*θ*_{R};*λ*,*φ*), defined by eqn. (37), is a rapidly varying function of*λ*.
Type: Result |
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ConceptID: Res15

148

It is therefore legitimate to apply the stationary phase approximation to the integral over

*λ*in eqn. (36).
Type: Model |
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Novelty: None |
ConceptID: Mod19

149

There is one stationary point,

*λ*=*λ*(*θ*_{R};*φ*), given by Note that*λ*(*θ*_{R};*φ*) varies from*λ*(*θ*_{R};*φ*= 0) ≡*λ*_{–}(*θ*_{R}) to*λ*(*θ*_{R};*φ*= π) ≡*λ*_{+}(*θ*_{R}) as*φ*changes from 0 to π, as illustrated in Fig. 1(b).
Type: Model |
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ConceptID: Mod19

150

Applying the stationary phase formula (13) to eqn. (36), and noting that together with

**′(*λ*(*θ*_{R};*φ*)) < 0 [see Fig. 1(b)], we obtain*Remark*:
Type: Model |
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ConceptID: Mod19

151

When

*θ*_{R}≈ 0, the phase*B*(*θ*_{R};*λ*(*θ*_{R};*φ*),*φ*) in eqn. (42) is a slowing varying function of*φ*, so that it is not legitimate to apply the stationary phase approximation again.
Type: Hypothesis |
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ConceptID: Hyp6

152

However, as

*θ*_{R}moves to larger angles,*B*(*θ*_{R};*λ*(*θ*_{R};*φ*),*φ*) varies more rapidly and we can once more apply the stationary phase technique, which provides us with a check on the PSA result (30).
Type: Hypothesis |
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ConceptID: Hyp6

153

The stationary phase condition is which simplifies, with the help of eqn. (41), to The two stationary points of eqn. (44) are

*φ*= 0 and*φ*= π for*φ*∈ [0,π].
Type: Model |
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ConceptID: Mod20

154

This means that the stationary points coincide with the end points of integration in eqn. (42), so the stationary phase result, eqn. (13), is not immediately applicable, since it assumed

*a*<*λ*_{s}<*b*.
Type: Result |
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ConceptID: Res16

155

However, the modification when a stationary point coincides with an end point is simple: the result in eqn. (13) is multiplied by 1/2.

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ConceptID: Res16

156

[

^{ref. 55}, p. 77].
Type: Result |
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ConceptID: Res16

157

We then find that the stationary phase evaluation of the integral (42) results in the PSA, eqn. (30) for

*f*(*θ*_{R}).
Type: Result |
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ConceptID: Res17

#### Exact change of variable

158

When

*θ*_{R}≈ 0, the phase of the integrand in eqn. (42), namely*B*(*θ*_{R};*λ*(*θ*_{R};*φ*),*φ*) = arg*S̃*(*λ*(*θ*_{R};*φ*)) –*λ*(*θ*_{R};*φ*)*θ*_{R}cos*φ*would be the same as the phase of the Bessel function of order zero, eqn. (35), were it not for the slow dependence of*λ*(*θ*_{R};*φ*) on*φ*.
Type: Hypothesis |
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ConceptID: Hyp7

159

This suggests that we make the following

*exact one-to-one change of variable**B*(*θ*_{R};*λ*(*θ*_{R};*φ*),*φ*) =*A*(*θ*_{R}) –*ζ*(*θ*_{R})cos*ψ*where the new parameters*A*(*θ*_{R}) and*ζ*(*θ*_{R}) are functions of the old parameter*θ*_{R}, whilst the new variable*ψ*=*ψ*(*θ*_{R};*φ*) is a function of both*θ*_{R}and the old variable*φ*[and inversely,*φ*=*φ*(*θ*_{R};*ψ*)].
Type: Model |
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ConceptID: Mod21

160

The rhs of eqn. (45) can also be regarded as the unfolding of a germ of infinite codimension.

Type: Model |
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ConceptID: Mod21

161

The stationary phase condition for the lhs of eqn. (45) has already been written down in eqns. (43) and (44).

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ConceptID: Mod21

162

The two stationary points are

*φ*= 0 and*φ*= π for*φ*∈ [0,π].
Type: Model |
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ConceptID: Mod21

163

For the rhs of eqn. (45), the stationary phase condition is

*ζ*(*θ*_{R})sin*ψ*= 0 and the two stationary points are*ψ*= 0 and*ψ*= π for*ψ*∈ [0,π].
Type: Model |
Advantage: None |
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ConceptID: Mod21

164

The requirement that the transformation (45) be one-to-one, and hence that d

*φ*/d*ψ*≠ 0,∞, is used to determine*A*(*θ*_{R}) and*ζ*(*θ*_{R}).
Type: Model |
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ConceptID: Mod21

165

Differentiating both sides of eqn. (45) with respect to

*ψ*and using the lhs of eqn. (44) gives In order that d*φ*/d*ψ*≠ 0,∞ in eqn. (46), we evidently must have the correspondences.
Type: Result |
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ConceptID: Res18

166

Note that d

*φ*/d*ψ*in eqn. (46) becomes 0/0,*i.e.*, indeterminate, when eqns. (47) are substituted into it and a more detailed analysis is necessary to determine the explicit form of d*φ*/d*ψ*at*ψ*= 0 and*ψ*= π.
Type: Result |
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ConceptID: Res18

167

This is done in section IID.6 below.

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ConceptID: Res18

168

Inserting the correspondence

*φ*= 0 ↔*ψ*= 0 into eqn. (45) gives*β*_{–}(*θ*_{R}) =*A*(*θ*_{R}) –*ζ*(*θ*_{R}) where eqn. (40) has been used.
Type: Model |
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ConceptID: Mod22

169

Similarly, inserting the correspondence

*φ*= π ↔*ψ*= π into eqn. (45) yields*β*_{+}(*θ*_{R}) =*A*(*θ*_{R}) +*ζ*(*θ*_{R}).
Type: Model |
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ConceptID: Mod22

170

Solving the linear eqns. (48) and (49) results in

*A*(*θ*_{R}) = ½[*β*_{+}(*θ*_{R}) +*β*_{–}(*θ*_{R})] and*ζ*(*θ*_{R}) = ½[*β*_{+}(*θ*_{R}) –*β*_{–}(*θ*_{R})].
Type: Model |
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ConceptID: Mod22

171

Eqns. (50) and (51) are the sought-for relations which express

*A*(*θ*_{R}) and*ζ*(*θ*_{R}) in terms of the F and N phases,*β*_{±}(*θ*_{R}).
Type: Result |
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ConceptID: Res19

172

Next we use eqn. (45) and the correspondences (47) to change variable from

*φ*to*ψ*in the integral (42) for*f*(*θ*_{R}).
Type: Model |
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ConceptID: Mod23

173

We obtain where

*φ*is to be regarded as a function of*ψ*,*i.e.*,*φ*=*φ*(*θ*_{R};*ψ*).
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ConceptID: Mod23

#### Approximation of the pre-exponential factor

174

Next we express the pre-exponential factor in eqn. (52) in the form{•} =

*p*(*θ*_{R}) +*q*(*θ*_{R})cos*ψ*+*r*(*θ*_{R};*ψ*)sin*ψ*involving the functions*p*(*θ*_{R}),*q*(*θ*_{R}) and*r*(*θ*_{R};*ψ*).
Type: Model |
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ConceptID: Mod24

175

When

*θ*_{R}≈ 0, the factor {•} is approximately constant since*λ*(*θ*_{R}≈ 0;*φ*) ≈*λ*_{g}[see Fig. 1(b)], whereas when*θ*_{R}moves away from zero, the integrand becomes oscillatory, receiving its contributions from the two stationary points,*ψ*= 0 and*ψ*= π, where the third term on the rhs of eqn. (53) is zero.
Type: Model |
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ConceptID: Mod24

176

These considerations suggest that we neglect the term

*r*(*θ*_{R};*ψ*)sin*ψ*in eqn. (53) and approximate the pre-exponential factor by with*φ*=*φ*(*θ*_{R};*ψ*).
Type: Model |
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ConceptID: Mod24

177

The integral representation (35) for

*J*_{0}(*x*) and the identity [^{ref. 58}, p. 2, eqn. (1.5)] lets us write eqns. (52) and (54) in terms of Bessel functions of order 0 and 1, namely
Type: Model |
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ConceptID: Mod24

#### Determination of *p*(*θ*_{R}) and *q*(*θ*_{R})

178

To determine

*p*(*θ*_{R}) and*q*(*θ*_{R}), we substitute the correspondences*φ*= 0 ↔*ψ*= 0 and then*φ*= π ↔*ψ*= π into eqn. (54), obtaining andwhere the relations*λ*(*θ*_{R};*φ*= 0) ≡*λ*_{–}(*θ*_{R}) and*λ*(*θ*_{R};*φ*= π) ≡*λ*_{+}(*θ*_{R}) have also been used.
Type: Model |
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ConceptID: Mod25

179

The two linear eqns. (56) and (57) have solutions andFinally we must obtain explicit expressions for d

*φ*/d*ψ*at*ψ*= 0 and*ψ*= π.
Type: Model |
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ConceptID: Mod25

180

To do this we rewrite eqn. (46) in the form and differentiate with respect to

*ψ*, obtaining.
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ConceptID: Mod25

181

Substituting the correspondences

*φ*= 0 ↔*ψ*= 0 and*φ*= π ↔*ψ*= π into eqn. (60), then gives and where the results*λ*(*θ*_{R};*φ*= 0) ≡*λ*_{–}(*θ*_{R}) and*λ*(*θ*_{R};*φ*= π) ≡*λ*_{+}(*θ*_{R}) have again been used.
Type: Model |
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182

Substituting eqns. (61) and (62) into eqns. (58) and (59) and using eqns. (26), lets us write

*p*(*θ*_{R}) and*q*(*θ*_{R}) in the form and
Type: Model |
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ConceptID: Mod25

#### Uniform approximation

183

The USA scattering amplitude,

*f*_{USA}(*θ*_{R}), is obtained upon combining eqns. (55), (63), (64) where*A*(*θ*_{R}) and*ζ*(*θ*_{R}) are defined by eqns. (50) and (51) respectively.
Type: Model |
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ConceptID: Mod26

184

The corresponding USA differential cross section is given by.

Type: Model |
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ConceptID: Mod26

185

Eqns. (65) and (66) express the USA scattering amplitude and cross section in terms of the F and N cross sections,

*σ*_{±}(*θ*_{R}), and phases,*β*_{±}(*θ*_{R}).
Type: Model |
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ConceptID: Mod26

186

Now the equations

*J*_{0}(*x*) = 0 and*J*_{1}(*x*) = 0 have an infinite number of simple roots (zeros), with no roots in common (^{ref. 58}, p. 105).
Type: Result |
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ConceptID: Res20

187

This implies that

*σ*_{USA}(*θ*_{R}), in contrast to the ITA and STA, is never equal to zero unless at some*θ*_{R}, we have*σ*_{+}(*θ*_{R}) =*σ*_{–}(*θ*_{R}) and*J*_{0}(*ζ*(*θ*_{R})) = 0.
Type: Result |
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ConceptID: Res20

#### Limiting cases

188

We must also check that eqn. (65) contains as limiting cases the non-uniform PSA and STA/

*J*_{0}scattering amplitudes.
Type: Hypothesis |
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ConceptID: Hyp8

189

The PSA is valid when

*θ*_{R}moves to larger angles, where we have*ζ*(*θ*_{R}) ≫ 1.
Type: Hypothesis |
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ConceptID: Hyp8

190

In this limit, we can replace the two Bessel functions in eqn. (65) by their asymptotic forms [

^{ref. 58}, p. 83, eqn. (5.34)] and [^{ref. 58}, p. 83, eqn. (5.36), together with*J*_{1}(*x*) = –d*J*_{0}(*x*)/d*x*] We then find, with the help of eqns. (50) and (51), that the USA scattering amplitude, eqn. (65), reduces to the PSA scattering amplitude, eqn. (30).
Type: Result |
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191

Next we consider the STA/

*J*_{0}, which is valid for*θ*_{R}→ 0.
Type: Hypothesis |
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192

We make the following three approximations to deduce the limiting form of eqn. (65) when

*θ*_{R}→ 0:
Type: Model |
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193

(

*a*) Since*λ*_{±}(*θ*_{R}) →*λ*_{g}[see, Fig. 1(b)] and therefore,*σ*_{+}(*θ*_{R}) →*σ*_{–}(*θ*_{R}) when*θ*_{R}→ 0, we can neglect the term containing*J*_{1}(*ζ*(*θ*_{R})) in eqn. (65).
Type: Model |
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ConceptID: Mod27

194

[

*N*.*b*., we also have*ζ*(*θ*_{R}) → 0 for the argument of the Bessel function].
Type: Model |
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ConceptID: Mod27

195

(

*b*) For*A*(*θ*_{R}), we similarly find that*A*(*θ*_{R}) → arg*S̃*(*λ*_{g}) as*θ*_{R}→ 0.
Type: Model |
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ConceptID: Mod27

196

(

*c*) We also require the leading term of*ζ*(*θ*_{R}) as*θ*_{R}→ 0.
Type: Model |
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ConceptID: Mod27

197

We can find it using the quadratic approximation (16) for arg

*S̃*(*λ*) together with the definitions (29) and (51) for*β*_{±}(*θ*_{R}) and*ζ*(*θ*_{R}) respectively.
Type: Model |
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ConceptID: Mod7

198

The F and N angular momentum variables are given by eqn. (24), namely We then find that the F and N phases in the quadratic approximation are

*β*_{±}(*θ*_{R}) = arg*S̃*(*λ*_{g}) + ½*θ*2R/|**′(*λ*_{g})| ±*λ*_{g}*θ*_{R}which shows that*ζ*(*θ*_{R}) →*λ*_{g}*θ*_{R}as*θ*_{R}→ 0.
Type: Model |
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ConceptID: Mod11

199

Substituting these three limits into eqn. (65), we obtain which is the scattering amplitude for the STA/

*J*_{0}, see eqns. (14) and (17).
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ConceptID: Mod27

### Forward glory scattering in the F + H_{2} → FH + H reaction

200

The F + H

_{2}reaction is an example of a benchmark exoergic reaction that has been extensively studied by theory and experiment.
Type: Background |
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ConceptID: Bac2

201

In particular, the forward angle scattering associated with the FH(

*v*_{f}= 3) final vibrational state has been of much interest, as discussed in the reviews,^{refs. 1, 10, 11, 61 and 62}(for other recent research on this reaction, see .^{refs. 63–79})
Type: Background |
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ConceptID: Bac2

202

In this section we apply the glory theory of section II to the angular scattering of the state-to-state transition F + H

_{2}(*v*_{i}= 0,*j*_{i}= 0,*m*_{i}= 0) → FH(*v*_{f}= 3,*j*_{f}= 3,*m*_{f}= 0) + H
Type: Goal |
Advantage: None |
Novelty: None |
ConceptID: Goa2

### Input scattering matrix data

203

The input to the glory analysis consists of a set of accurate quantum scattering matrix elements, {

*S̃*_{J}} for*J*= 0(1)23 (and {*S̃*_{J}≡ 0} for*J*> 23), computed for the Stark-Werner potential energy surface^{80}at a total energy of*E*= 0.3872 eV, measured with respect to the classical minimum of the H_{2}potential energy curve.^{22,33}
Type: Method |
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Novelty: Old |
ConceptID: Met1

204

The translational energy is 0.119 eV, which corresponds to a translational wavenumber of

*k*= 10.2 Å^{–1}.
Type: Model |
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ConceptID: Mod28

205

Fig. 2(a) shows the modulus of the scattering matrix input data, |

*S̃*_{λ–1/2}|, for*λ*= ½,,,…,23½ (solid circles), plotted*versus**λ*, together with its real valued continuation, |*S̃*(*λ*)|, drawn as a solid line.
Type: Result |
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Novelty: None |
ConceptID: Res22

206

The continuation is achieved by interpolating the discrete data (and extrapolating to

*λ*= 0) with polynomials of degree 6.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod29

207

It can be seen that |

*S̃*(*λ*)| possesses two peaks at*λ*≈ 10 and*λ*≈ 16, and a minimum at*λ*≈ 135.
Type: Observation |
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ConceptID: Obs2

208

Fig. 2(b) displays the analogous plot for the input phase data, arg

*S̃*_{λ–1/2}*versus**λ*(solid circles) for*λ*= ½,,,…,23½, as well as its continuation, arg*S̃*(*λ*)*versus**λ*(solid curve).
Type: Result |
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ConceptID: Res23

209

We recall from section IIA that arg does not denote the principal value, which would produce phases/rad ∈ (–π,π]; in fact it can be seen in Fig. 2(b) that arg

*S̃*(*λ*)/rad ∈ (0,40).
Type: Result |
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ConceptID: Res23

210

The phase arg

*S̃*_{λ–1/2}is calculated by adding multiples of 2π to the phases computed as principal values, until two successive values, arg*S̃*_{λ–1/2}and arg*S̃*_{λ+1/2}differ by less than π.
Type: Method |
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Novelty: Old |
ConceptID: Met1

211

The continuation again uses polynomials of degree 6.

Type: Model |
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ConceptID: Mod29

212

It can be seen that arg

*S̃*(*λ*) has a maximum at*λ*≈ 16.
Type: Observation |
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Novelty: None |
ConceptID: Obs3

213

The region around the maximum in Fig. 2(b), from

*λ*= 13½ to*λ*= 17½, has already been shown in Fig. 1(a) as an aid to the derivations in section II.
Type: Observation |
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ConceptID: Obs3

### Nearside–farside analysis

214

Before applying the glory theory of section II, it is first helpful to perform a NF analysis of the angular scattering.

^{14,15}
Type: Model |
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ConceptID: Mod30

215

The NF decomposition for the Legendre PWS (3) is (for

*θ*_{R}≠ 0,π)*f*(*θ*_{R}) =*f*_{N}(*θ*_{R}) +*f*_{F}(*θ*_{R}) where the NF subamplitudes are defined by in which*Q*_{J}(cos*θ*_{R}) is a Legendre function of the second kind.
Type: Model |
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ConceptID: Mod30

216

The corresponding NF angular distributions are given by

*σ*_{N,F}(*θ*_{R}) = |*f*_{N,F}(*θ*_{R})|^{2}.
Type: Model |
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ConceptID: Mod30

217

*Note*: it is common practice in the literature to use a superscript “–” to represent the N scattering, and a superscript “+” for the F scattering; however this notation has not been employed in eqns. (67)–(69) to avoid confusion with the “±” used in the semiclassical NF theory of sect. IIB-IID.

Type: Background |
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ConceptID: Bac12

218

Fig. 3 shows the results of the NF analysis for the PWS scattering amplitude, eqn. (3).

Type: Result |
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ConceptID: Res24

219

It displays log plots of

*σ*(*θ*_{R}),*σ*_{N}(*θ*_{R}) and*σ*_{F}(*θ*_{R})*versus**θ*_{R}.
Type: Result |
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ConceptID: Res24

220

At backward angles the scattering is N dominated, with the oscillations arising from NF interference.

Type: Result |
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ConceptID: Res24

221

In this angular range, the main features of the scattering can be understood using simple classical-like models, such as the semiclassical optical model (which is an approximate N theory).

^{22,25}
Type: Model |
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Novelty: None |
ConceptID: Mod31

222

In contrast, the scattering at forward angles contains substantial contributions from both the N and F subamplitudes (with F dominant), resulting in pronounced NF oscillations.

Type: Result |
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ConceptID: Res24

223

The change in mechanism for the forward and backward angle scattering occurs around

*θ*_{R}= 40°, which implies that the glory theory of section II should only be used for*θ*_{R}≲ 40° (and in particular, the accuracy of the USA glory equations is expected to degrade as*θ*_{R}increases from*θ*_{R}= 0° to*θ*_{R}≈ 40°).
Type: Result |
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Novelty: None |
ConceptID: Res24

224

The

*σ*(*θ*_{R}),*σ*_{N}(*θ*_{R}) and*σ*_{F}(*θ*_{R}) cross sections plotted in Fig. 3 are similar in their structure (generic) to the corresponding angular distributions for the Cl + HCl → ClH + Cl reaction, which have been analysed in detail in^{refs. 29 and 30}using complex angular momentum techniques.
Type: Result |
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ConceptID: Res24

225

In particular, the mechanism for the forward angle scattering in the Cl + HCl reaction is the decay of a short-lived Regge state.

^{29,30}
Type: Background |
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ConceptID: Bac13

226

The genericity of the cross sections led Dobbyn

*et al*^{22}. to suggest that the same mechanism also occurs for the F + H_{2}reaction.
Type: Background |
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Novelty: None |
ConceptID: Bac14

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac3

Type: Background |
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Novelty: None |
ConceptID: Bac15

229

In a time-dependent experiment (or calculation), the forward angle scattering will be time-delayed relative to the backward angle scattering, as discussed in more detail for the H + D

_{2}reaction in sect. IV.
Type: Background |
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Novelty: None |
ConceptID: Bac16

### Glory analysis of forward scattering

230

The radius of the interaction region for the F + H

_{2}reaction is approximately*R*= 2 Å, so we have*kR*≈ 20.
Type: Model |
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Novelty: None |
ConceptID: Mod32

231

This confirms that

*kR*≫ 1, which is the condition for the semiclassical theory of section II to be valid.
Type: Model |
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ConceptID: Mod32

232

We begin by examining the validity of retaining just the

*m*= 0 integral in the Poisson series, eqn. (4), since this approximation is fundamental to the analysis of sect. II.
Type: Model |
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ConceptID: Mod32

233

For

*θ*_{R}= 0, at which angle*P*_{λ–1/2}(cos 0) = 1 for all*λ*∈ [0,∞), the PWS, eqn. (3) has the value*f*(0) = 0.213 + 0.293i Å, which agrees closely with the result, 0.215 + 0.291i Å, obtained by numerical quadrature of the integral in eqn. (6).
Type: Result |
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Novelty: None |
ConceptID: Res25

234

More generally, the angular distribution computed from eqn. (6) by quadrature for

*θ*_{R}∈ [0°,90°] agrees with the PWS cross section, eqn. (2), to within graphical accuracy (for*θ*_{R}> 90°, integrals with*m*= ±1 in the Poisson series, eqn. (4), start to become important numerically).
Type: Result |
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ConceptID: Res26

235

These tests therefore justify the neglect of integrals with

*m*≠ 0 in the Poisson series, eqn. (4), at forward angles.
Type: Result |
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Novelty: None |
ConceptID: Res26

236

The quantum deflection function,

**(*λ*), defined by eqn. (8), plays a key role in the glory theory of section II.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod7

237

Fig. 4 shows a plot of

**(*λ*)*versus**λ*for the F + H_{2}reaction.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs4

238

It can be seen that eqn. (9) has a (glory) root at

*λ*_{g}≈ 16.0 together with additional roots at*λ*_{g}≈ 18.4,20.3 and 23.2.
Type: Observation |
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Novelty: None |
ConceptID: Obs4

239

However |

*S̃*(*λ*)|^{2}is very small at these additional roots, being ≈10^{–4}, ≈10^{–6}and ≈10^{–8}respectively; their contribution has therefore been neglected.
Type: Result |
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ConceptID: Res27

240

The ITA, defined by eqns. (10)–(12), and the STA, defined by eqns. (14) and (15), require an accurate value for

*λ*_{g}.
Type: Model |
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Novelty: None |
ConceptID: Mod33

241

This was obtained by fitting a polynomial of degree 4 to the input phase data, arg

*S̃*_{λ–1/2}, for*λ*= 13½(1)17½, as illustrated in Fig. 1(a).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod33

242

We then find on differentiation and solving eqn. (9) that

*λ*_{g}= 16.02.
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ConceptID: Res28

243

We also find that

**′(*λ*_{g}) = –0.522.
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ConceptID: Res28

244

Fig. 5 compares the PWS cross section of eqn. (2) with the ITA angular distribution for the angular range, 0° ≤

*θ*_{R}≤ 40°.
Type: Observation |
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ConceptID: Obs5

245

It can be seen there is close agreement between the PWS and ITA angular distributions for the forward peak up to

*θ*_{R}≈ 10°; then the ITA becomes less accurate as*θ*_{R}increases.
Type: Result |
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Novelty: None |
ConceptID: Res29

246

This loss of accuracy is expected since (a) the ITA is, by construction, a transitional approximation designed to be most reliable at

*θ*_{R}= 0°, and (b) the change in mechanism discussed in section IIIB becomes increasingly important as*θ*_{R}moves towards*θ*_{R}= 40°.
Type: Result |
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ConceptID: Res29

247

The ITA/

*J*_{0}, STA and STA/*J*_{0}approximations are not shown separately in Fig. 5 since they all agree closely with the ITA, as does the transitional approximation used by Sokolovski for the H + D_{2}reaction*i.e.*, STA/*J*_{0}with replacement of the factor (*θ*_{R}/sin*θ*_{R})^{1/2}by unity (see also the discussion in the Appendix).
Type: Result |
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Novelty: None |
ConceptID: Res30

248

Fig. 5 also displays the glory scattering predicted by the USA.

Type: Result |
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ConceptID: Res31

249

It can be seen that the USA is more accurate than the ITA, in particular for the amplitudes of the subsidiary glory oscillations.

Type: Result |
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ConceptID: Res31

250

The change in mechanism at

*θ*_{R}≈ 40° is likely to account, in part, for the discrepancies between the USA and PWS angular distributions as*θ*_{R}increases.
Type: Result |
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Novelty: None |
ConceptID: Res31

251

The PSA agrees closely with the USA for

*θ*_{R}≳ 4°; it is not shown separately in Fig. 5.
Type: Result |
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ConceptID: Res32

252

Finally, the CSA is also plotted in Fig. 5.

Type: Observation |
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ConceptID: Obs6

253

As expected, it is monotonic and passes through the glory oscillations; the divergence of the CSA as

*θ*_{R}→ 0° can be clearly seen.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res33

### Forward glory scattering in the H + D_{2} → HD + D reaction

254

The H + D

_{2}collision system is another example of a benchmark chemical reaction that has received extensive attention from theory and experiment.
Type: Background |
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Novelty: None |
ConceptID: Bac2

255

In particular, a time-delayed mechanism for the forward angle scattering of the HD(

*v*_{f}= 3,*j*_{f}= 0) final rovibrational state has been of much interest^{1,37,38,82,83}(for other recent research on this reaction, see .^{ref. 84–96})
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

256

The next section applies the glory theory of section II to the angular scattering of the state-to-state transition H + D

_{2}(*v*_{i}= 0,*j*_{i}= 0,*m*_{i}= 0) → HD(*v*_{f}= 3,*j*_{f}= 0,*m*_{f}= 0) + D
Type: Goal |
Advantage: None |
Novelty: None |
ConceptID: Goa2

### Input scattering matrix data

257

The input to the glory analysis consists of a set of accurate quantum scattering matrix elements, {

*S̃*_{J}} for*J*= 0(1)30, with {*S̃*_{J}≡ 0} for*J*> 30, computed for the Boothroyd–Keogh–Martin–Peterson potential energy surface^{97}number 2 at a total energy of*E*= 2.00 eV, measured with respect to the classical minimum of the D_{2}potential energy curve.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod34

258

The corresponding translational energy is 1.81 eV, with the translational wavenumber being

*k*= 26.4 Å^{–1}.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod34

259

The scattering matrix elements were computed by a new time dependent plane wavepacket method

^{82,83,98–100}and were kindly supplied to the author by Dr S. C. Althorpe.
Type: Method |
Advantage: None |
Novelty: New |
ConceptID: Met7

260

Fig. 6(a) shows the modulus input data, |

*S̃*_{λ–1/2}|, for*λ*= ½,,,…,30½, plotted*versus**λ*(solid circles), together with its real valued continuation, |*S̃*(*λ*)| (solid curve).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod34

261

Polynomials of degree three have been used to interpolate the discrete data (and to extrapolate to

*λ*= 0).
Type: Model |
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Novelty: None |
ConceptID: Mod34

262

It can be seen that |

*S̃*(*λ*)| possesses two pronounced maxima at*λ*≈ 15.7 and*λ*≈ 22.7, and two pronounced minima at*λ*≈ 13.1 and*λ*≈ 19.0.
Type: Observation |
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Novelty: None |
ConceptID: Obs7

263

Thus the |

*S̃*(*λ*)| plot for the H + D_{2}reaction is more complicated than that for the F + H_{2}system in Fig. 2(a).
Type: Result |
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Novelty: None |
ConceptID: Res34

264

The plot for the input phase data, arg

*S̃*_{λ–1/2}*versus**λ*(solid circles) for*λ*= ½,,,…,30½, as well as its continuation, arg*S̃*(*λ*)*versus**λ*(solid curve), is shown in Fig. 6(b).
Type: Result |
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Novelty: None |
ConceptID: Res34

265

The continuation also uses polynomials of degree three.

Type: Model |
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Novelty: None |
ConceptID: Mod34

266

It can be seen that arg

*S̃*(*λ*) has a maximum at*λ*≈ 24.
Type: Observation |
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Novelty: None |
ConceptID: Obs8

267

The noticeable drop in arg

*S̃*_{λ–1/2}on going from*λ*= 18½ to*λ*= 19½ is associated with the corresponding near zero in |*S̃*_{λ–1/2}| evident in Fig. 6(a).
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res35

### Nearside–farside analysis

268

Fig. 7 shows log plots of

*σ*(*θ*_{R}),*σ*_{N}(*θ*_{R}) and*σ*_{F}(*θ*_{R})*versus**θ*_{R}.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res36

269

At backward angles the scattering is N dominated, with the weak oscillations arising from NF interference.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res36

270

Simple classical-like models, such as the semiclassical optical model—an approximate N theory—can be used to understand the main features of the scattering in this angular range.

^{22,25}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod31

271

The semiclassical optical model assumes direct dynamics and works best for rebound reactions in which the backward angle scattering arises from repulsive interactions between the reactants.

Type: Model |
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ConceptID: Mod31

272

Fig. 7 shows that the scattering at forward angles contains significant contributions from both the N and F subamplitudes (with N dominant), resulting in strong NF oscillations.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs9

273

At around

*θ*_{R}= 40°, there is a change in mechanism for the forward and backward angle scattering, which implies, similar to the F + H_{2}case, that the glory theory of sect. II will become less accurate as*θ*_{R}increases towards 40°.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res37

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac17

275

At the International Complutense Seminar on Quantum Reactive Scattering, held at San Lorenzo de El Escorial, Spain, 20–23 June 2003, Althorpe showed a movie of the H + D

_{2}reaction.^{82,83}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac18

276

This movie, which I viewed, showed a wavepacket (containing initial translational energies in the range 1.3–2.2 eV, equally weighted) as it evolved for a duration of 100 fs from the initial state, H + D

_{2}(*v*_{i}= 0,*j*_{i}= 0), to the final state HD(*v*_{f}= 3,*j*_{f}= 0) + D.
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277

The movie showed that the HD(

*v*_{f}= 3,*j*_{f}= 0) molecule is formed by two distinct reaction mechanisms: the backward angle scattering arises from a direct recoil mechanism, whilst the forward angle scattering is formed by an indirect mechanism, being delayed by about 25 fs.^{82,83}
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278

Subsequently, I had private viewings of the movie, courtesy of Dr Althorpe.

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279

I have also examined clips and stills from the movie.

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280

The NF analysis in Fig. 7 is consistent with the two mechanism picture of the reaction dynamics depicted in the movie.

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281

Thus N scattering of the reactants, which dominates backward angles, is clearly visible in the movie and corresponds to the direct recoil mechanism.

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282

In contrast, the N and F scattering in the angular range,

*θ*_{R}≲ 40°, which arise from decaying Regge states (surface waves),^{37,38}correspond to the indirect time-delayed mechanism.
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283

Indeed, the movie shows surface-like waves propagating around a circle of radius 1.9 Å (approximately the size of the transition state region), which then decay into the forward direction.

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284

If we compare the NF analysis for the H + D

_{2}reaction in Fig. 7 with that for the F + H_{2}reaction in Fig. 3, we see that they are qualitatively similar.
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285

Thus we expect that the discussion given above for the H + D

_{2}reaction will also largely apply to the F + H_{2}system (and*vice versa*).
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286

Note that one difference is that H + D

_{2}is N dominated for*θ*_{R}≲ 40°, whereas the F + H_{2}reaction is F dominated in this angular range.
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### Glory analysis of forward scattering

287

The H + D

_{2}reaction has a radius of interaction of approximately*R*= 2 Å.
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288

Thus

*kR*≈ 50, which means that the semiclassical theory of section II is applicable.
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289

We first examine the validity of retaining just the

*m*= 0 integral in the Poisson series (4).
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290

For

*θ*_{R}= 0, where*P*_{λ–1/2}(cos 0) = 1 for all*λ*∈ [0,∞), the PWS (3) has the value*f*(0) = –0.0640 – 0.0765i Å, in close agreement with the value –0.0647 – 0.0763i Å, obtained from the integral (6) by numerical quadrature.
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291

In fact, the angular distribution computed by quadrature from eqn. (6) agrees with the PWS cross section (2) to within graphical accuracy for

*θ*_{R}∈ [0°,90°].
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292

As for the F + H

_{2}reaction, integrals with*m*= ±1 in the Poisson series (4) only become important numerically when*θ*_{R}> 90°.
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293

Thus, at forward angles, we can safely neglect integrals with

*m*≠ 0 in the Poisson series (4).
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294

Fig. 8 shows a plot of

**(*λ*)*versus**λ*for the H + D_{2}reaction.
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295

The plot is more irregular than the one for the F + H

_{2}example of Fig. 4.
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296

This may be because (a) the H + D

_{2}reaction is more complicated dynamically than is the F + H_{2}system: compare Fig. 2 with Fig. 6, and/or (b) there are different numerical errors in the scattering matrix computations for the two reactions.
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297

Fig. 8 shows that eqn. (9) has a (glory) root at

*λ*_{g}≈ 24 together with additional roots at*λ*_{g}≈ 18.5,19.6 and 29.7.
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298

The first two roots are associated with the near-zero of

*S̃*(*λ*) at*λ*≈ 19.0.
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299

The absolute values of the slope, |d

**(*λ*)/d*λ*|, are also very large at these two roots and their contributions in the stationary phase approximation of eqn. (25) will be small—they have been neglected in the following.
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300

The root near

*λ*= 29.7 has |*S̃*(*λ*_{g})|^{2}≈ 10^{–8}; its contribution has also been neglected.
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301

Next we apply the glory approximations derived in Section II.

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302

We recall that the ITA requires a value for

*λ*_{g}, the STA requires values for*λ*_{g}and the slope of**(*λ*) at*λ*=*λ*_{g}, whilst the USA, PSA and CSA require values for*λ*_{±}(*θ*_{R}) and the slopes of**(*λ*) at*λ*=*λ*_{±}(*θ*_{R}) for*θ*_{R}≲ 40°.
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303

The irregularities in the

**(*λ*) curve of Fig. 8 near*λ*≈ 24 make it difficult to extract accurate values for these quantities.
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304

To avoid this difficulty, we interpolated the input phase data, arg

*S̃*_{λ–1/2}for*λ*= 21½(1)25½ to determine*λ*_{g}, and then approximated**(*λ*) by a straight line—in effect the input data has been smoothed.
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305

This procedure resulted in

*λ*_{g}= 24.19 and**′(*λ*_{g}) = –0.233.
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306

These values are very close to those obtained by Sokolovski, namely

*λ*_{g}= 24.2 and**′(*λ*_{g}) = –0.24, using a smoothed Padé reconstruction.
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307

The straight-line approximation for

**(*λ*) is drawn as a dashed line in Fig. 8 for**(*λ*)/deg ∈ [–40,40].
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308

Values for the

*λ*_{±}(*θ*_{R}) needed in the USA, PSA and CSA are then given by eqn. (24).
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309

Fig. 9 compares the PWS cross section of eqn. (2) with the ITA, USA and CSA glory angular distributions for the angular ranges, 0° ≤

*θ*_{R}≤ 10° and 10° ≤*θ*_{R}≤ 40° (two angular ranges are used in order to show more clearly the errors of the different approximations).
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310

There is close agreement between the PWS and ITA angular distributions for the forward peak up to

*θ*_{R}≈ 12°, with the ITA then becoming less accurate as*θ*_{R}increases.
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311

As for the F + H

_{2}example, this loss of accuracy is expected since (a) the ITA is a transitional approximation which is most reliable at*θ*_{R}= 0°, and (b) the change in mechanism discussed in sect. IVB becomes increasingly significant as*θ*_{R}→ 40°.
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312

The ITA/

*J*_{0}, STA and STA/*J*_{0}curves have not been plotted in Fig. 9 since they all agree closely with the ITA curve, as does the transitional approximation used by Sokolovski,^{38}*i.e.*, STA/*J*_{0}with the factor (*θ*_{R}/sin*θ*_{R})^{1/2}replaced by unity (see the Appendix for some additional discussion).
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313

Fig. 9 also displays the glory scattering predicted by the USA.

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314

It can be seen that the USA agrees closely with the PWS and ITA cross sections at smaller angles, but is more accurate than the ITA at larger angles, in particular, for the amplitudes of the subsidiary glory oscillations.

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315

The change in mechanism at

*θ*_{R}≈ 40° is likely to account, in part, for the discrepancies between the USA and PWS angular distributions as*θ*_{R}increases.
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316

The PSA is not shown separately in Fig. 9 since it agrees closely with the USA for

*θ*_{R}≳ 3°.
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317

The cross section for the CSA in Fig. 9 is seen to be monotonic.

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318

It passes through the glory oscillations and is divergent as

*θ*_{R}→ 0°.
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319

Finally, we note that Althorpe has introduced

^{82,83,99,100}the concepts of a time-dependent scattering amplitude and a time dependent cross section, and has demonstrated their value for understanding the dynamics of reactions in his wavepacket calculations.
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320

The NF and glory theory developed in this paper can also be used to understand structure in these time dependent quantities, when the time-dependent scattering amplitude can be expanded in a basis set of Legendre polynomials.

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## Conclusions

321

The paper has developed a theory of forward glory scattering for a state-to-state chemical reaction whose scattering amplitude can be written as a Legendre PWS.

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322

The most important result derived in the paper is the USA, which correctly interpolates between small angles where the ITA and STA are valid, and larger angles where the PSA is valid.

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323

The USA expresses the scattering amplitude in terms of N and F cross sections and phases, together with Bessel functions of order zero and unity.

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324

The PSA clearly demonstrates that glory structure arises from nearside–farside (NF) interference, in an analogous way to the two-slit experiment.

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325

The theory has the important attribute that it provides physical insight by bringing out semiclassical and NF aspects of the scattering, even though no WKB phases are used.

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326

Rather, the input to the theory consists of accurate quantum scattering matrix elements.

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327

The theory has been used to show that the enhanced small angle scattering in the F + H

_{2}(*v*_{i}= 0,*j*_{i}= 0,*m*_{i}= 0) → FH(*v*_{f}= 3,*j*_{f}= 3,*m*_{f}= 0) + H reaction is a forward glory.
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328

And the same is also true for the H + D

_{2}(*v*_{i}= 0,*j*_{i}= 0) → HD(*v*_{f}= 3,*j*_{f}= 0) + D reaction, in agreement with a simpler treatment by Sokolovski^{38}which is a special case of the present theory.
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