The 6Hankel asymptotic approximation for the uniform description of rainbows and glories in the angular scattering of state-to-state chemical reactions: derivation, properties and applications

Abstract

This paper considers the asymptotic (semiclassical) analysis of a forward glory and a rainbow in the differential cross section (DCS) of a state-to-state chemical reaction, whose scattering amplitude is given by a Legendre partial wave series (PWS). A recent paper by C. Xiahou, J. N. L. Connor and D. H. Zhang [Phys. Chem. Chem. Phys., 2011, 13, 12981] stated without proof a new asymptotic formula for the scattering amplitude, which is uniform for a glory and a rainbow in the DCS. The new formula was designated “6Hankel” because it involves six Hankel functions. This paper makes three contributions: (1) we provide a detailed derivation of the 6Hankel approximation. This is done by first generalizing a method described by G. F. Carrier [J. Fluid Mech., 1966, 24, 641] for the uniform asymptotic evaluation of an oscillating integral with two real coalescing stationary phase points, which results in the “2Hankel” approximation (it contains two Hankel functions). Application of the 2Hankel approximation to the PWS results in the 6Hankel approximation for the scattering amplitude. We also test the accuracy of the 2Hankel approximation when it is used to evaluate three oscillating integrals of the cuspoid type. (2) We investigate the properties of the 6Hankel approximation. In particular, it is shown that for angles close to the forward direction, the 6Hankel approximation reduces to the “semiclassical transitional approximation” for glory scattering derived earlier. For scattering close to the rainbow angle, the 6Hankel approximation reduces to the “transitional Airy approximation”, also derived earlier. (3) Using a J-shifted Eckart parameterization for the scattering matrix, we investigate the accuracy of the 6Hankel approximation for a DCS. We also compare with angular scattering results from the “uniform Bessel”, “uniform Airy” and other semiclassical approximations.

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I. Introduction

The angular scattering of a state-to-state chemical reaction contains fundamental information on the dynamics and mechanism of the reaction.¹ However, it has often proven difficult to quantitatively understand the physical content contained within a plot of the differential cross section (DCS) versus reactive scattering angle. Consider, for example, the F + H₂ → FH + H reaction, whose angular scattering was measured in the famous crossed molecular-beam experiments of Neumark et al.² in 1985. It took 19 years before the enhanced forward peak in the small-angle scattering of the product FH(v_f = 3) vibrational state was identified³ as a glory. And it took 24 years before the scattering at larger angles in the DCS was recognized⁴ as a rainbow.

More recently (in 2008), state-of-the-art DCS measurements for the F + H₂ reaction were reported by Wang et al.⁵ using quantum-state-selected crossed molecular beams. An analysis of the angular scattering by ourselves and Zhang⁶ again revealed the presence of glories and rainbows in the FH(v_f = 3) DCSs; they are also accompanied by diffraction oscillations arising from nearside–farside interference.

The analyses in ref. 3, 4 and 6 used powerful asymptotic (semiclassical) techniques to extract physical information from the large number of interfering partial waves which contribute to the scattering amplitude. Two different asymptotic theories were employed: One theory^3,7 led to the uniform glory approximation (and subsidiary approximations), whilst the second theory⁸ resulted in the uniform (and transitional) rainbow approximations. A disadvantage of these theoretical treatments is that the uniform glory approximation becomes non-uniform for rainbow scattering, and vice versa. By a uniform approximation, we mean one in which the error remains approximately constant as a parameter, such as the reactive scattering angle, passes through certain critical values, such as zero (for a forward glory) or the rainbow angle.^3,4,6–8 A transitional approximation is one in which the error remains small in the neighbourhood of these critical values, but the error usually increases as the parameter moves away from the critical values.^3,4,6–8

It is desirable to develop new asymptotic scattering theories that are uniform for both forward glory and rainbow scattering. This was partially done in our paper with Zhang,⁶ where we stated without proof a new asymptotic approximation for the scattering amplitude that is valid for both a glory and a rainbow. We called our new result, the 6Hankel asymptotic approximation, since it contains six Hankel functions.⁶ For the DCS, our new result is a generalization to reactive scattering of a formula given by Miller⁹ for elastic scattering.

The purpose of this paper is: (1) to present a derivation of the 6Hankel approximation, (2) to discuss its properties, and (3) to assess its accuracy for the DCS of a chemical reaction. In our DCS computations, we use a J-shifted Eckart parameterization of the scattering (S) matrix,^10,11 as it allows flexibility in the location of the rainbow angle, thereby allowing us to test the 6Hankel approximation over a wide angular range.

For information on the mathematical description of glories and rainbows, we refer to the extensive review by Adam.¹² Earlier work on the role of forward glories in the DCSs of chemical reactions can be found in ref. 3, 6, 7, 10, 11 and 13–17. Likewise the role of rainbows is discussed in ref. 4 and 6.

In order to describe a rainbow, we first must consider the uniform asymptotic evaluation of a one-dimensional oscillating integral with two coalescing stationary phase points. This is done in Section II. In particular, we generalize a method described by Carrier.¹⁸ He assumes that the phase of the integrand is an odd function of x and that the pre-exponential factor is an even function of x, where x is the integration variable. We extend his result to the case when neither of these symmetry properties holds. We call our generalization, the 2Hankel asymptotic approximation, since it involves two Hankel functions. We also investigate the limit when the stationary phase points coalesce, which leads to the transitional Airy approximation^4,6,8 for rainbow scattering.

Section III applies the 2Hankel approximation to the scattering amplitude for a chemical reaction, when it is given by a Legendre partial wave series, thereby providing a derivation of the 6Hankel approximation. The limit when the scattering angle tends to zero is investigated, resulting in the semiclassical transitional approximation³ previously derived for glory scattering.

The J-shifted Eckart parameterization for the S matrix^10,11 is defined in Section IV. The values of the parameters are chosen so that the rainbow angle occurs at a large value, namely 109.2° in the centre-of-mass reference frame. This allows us to conduct a better test of the accuracy of the 6Hankel approximation than previously, which used numerical S matrix data.⁶ An important point is that the 6Hankel formula is generic, i.e., it also applies to numerous chemical reactions at numerous different energies which have S matrix properties analogous to the J-shifted Eckart parameterization.

Section V describes our results for the DCS using the 6Hankel and other semiclassical approximations. In order to provide additional physical insight into interference structure in the DCS, we have also applied nearside–farside (NF) theory^19,20 and local angular momentum (LAM) theory,^21–24 in both cases including up to three resummations of the partial wave series.^21–24 We also make contact with complex angular momentum (or Regge pole) theory, which has been used to calculate DCSs.^25–27

Our Conclusions are in Section VI. The Appendix describes a test of the 2Hankel approximation when it is applied to three oscillating integrals of the cuspoid type.^28,29

Finally, we emphasize that there is a long tradition in the chemical physics literature of papers concerned with the asymptotic evaluation of oscillating integrals, see for example ref. 3, 4, 6–17, 20, 23, 25–28 and 30–36.

II. Uniform asymptotic integration: the 2Hankel approximation

IIA. Introduction

This section is concerned with the uniform asymptotic evaluation of the oscillating integral:where α is a real parameter and f(α;x) is a real-valued function. In addition, g(x) is slowly varying; it can be a real- or complex-valued function and may also depend on α, although this is not indicated. We assume there exist two real points of stationary phase, denoted x₁(α) and x₂(α), with x₁(α) ≤ x₂(α), which are the roots of f′(α;x) = 0, where the prime indicates differentiation with respect to x. As α varies, it is assumed that the two roots coalesce on the caustic at x = x(α₀) for α = α₀.

The convenient notations x_i = x_i(α), g_i = g(x_i), f_i = f(α;x_i), f_i′ = f′(α;x)|_x=xi, f_i′′ = f′′(α;x)|_x=xi and f_i′′′ = f′′′(α;x)|_x=xi for i = 0, 1, 2 will often be employed in the following. In our application in Section III, f(α;x) has a linear dependence on α of the type, ±αx, which means that the second and third derivatives of f(α;x) are independent of α. But note that f_i′′ and f_i′′′ do depend on α via the x_i = x_i(α).

Given the above assumptions, two cases arise, depending on whether f(α;x) → +∞ or → −∞ as x → +∞ [or equivalently, f(α;x) → −∞ or → +∞ as x → −∞]. These two cases are illustrated in Fig. 1, where f(α;x) is given by simple polynomial functions. The black solid curves for f(α;x) in Fig. 1 possess a local maximum and a local minimum. Also illustrated are the curves (blue solid) when the two stationary points have coalesced for α = α₀. The two cases have the following properties, which we use later:

Fig. 1 Properties of the phase f(α;x) and three of its 3-jets. (a) Case A. Black solid line: f(α;x) = −αx + x³/3 − x⁴/4 + x⁵/5 for α = 1. Its stationary phase points are given by, x₁ ≈ −0.682328 (local maximum) and x₂ = 1 (local minimum). Lower red dashed curve: The 3-jet at x₂, j_x2³f(α = 1;x). Its stationary phase points are denoted, u₂ = 1/4 (local maximum) and x₂ = 1 (local minimum). Upper red dashed curve: The 3-jet at x₁, j_x1³f(α = 1;x). Its stationary phase points are denoted, x₁ ≈ −0.682328 (local maximum) and u₁ ≈ 0.00804454 (local minimum). Blue solid curve: f(α = 0;x) = x³/3 − x⁴/4 + x⁵/5. Its two stationary phase points have coalesced at x₀ = 0. Green dashed curve: The 3-jet at x₀, j_x0³f(α = 0;x) = x³/3. Its two stationary phase points have coalesced at x₀ = 0. (b) Case B. Black solid line: f(α;x) = −(−αx + x³/3 − x⁴/4 + x⁵/5) for α = 1. Its stationary phase points are given by, x₁ ≈ −0.682328 (local minimum) and x₂ = 1 (local maximum). Upper red dashed curve: The 3-jet at x₂, j_x2³f(α = 1;x). Its stationary phase points are denoted, u₂ = 1/4 (local minimum) and x₂ = 1 (local maximum). Lower red dashed curve: The 3-jet at x₁, j_x1³f(α = 1;x). Its stationary phase points are denoted, x₁ ≈ −0.682328 (local minimum) and u₁ ≈ 0.00804454 (local maximum). Blue solid curve: f(α = 0;x) = −(x³/3 − x⁴/4 + x⁵/5). Its two stationary phase points have coalesced at x₀ = 0. Green dashed curve: The 3-jet at x₀, j_x0³f(α = 0;x) = −x³/3. Its two stationary phase points have coalesced at x₀ = 0.

Case A – see Fig. 1(a)

For x₁ < x₂ so that α ≠ α₀

f₁ > f₂, f₁′ = f₂′ = 0,

f₁′′ < 0, f₂′′ > 0, f₁′′′ > 0, f₂′′′ > 0

For x₁ = x₂ = x₀ so that α = α₀

f₀′ = 0, f₀′′ = 0, f₀′′′ > 0

Case B – see Fig. 1(b)

For x₁ < x₂ so that α ≠ α₀

f₁ < f₂, f₁′ = f₂′ = 0,

f₁′′ > 0, f₂′′ < 0, f₁′′′ < 0, f₂′′′ < 0

For x₁ = x₂ = x₀ so that α = α₀

f₀′ = 0, f₀′′ = 0, f₀′′′ < 0

Remarks:

• It is also assumed that g(x) does not possess any singularities or zeros near to the stationary phase points. A modified treatment can be given if this is not the case (e.g., see ref. 37 and 38, for example).

• If f(α;x) = −f(α; −x), i.e., f is an odd function of x, then x₁ = −x₂.

• If f(α;x) possesses more than two real stationary points, then the following derivation is valid locally in the region of the two coalescing stationary points,^8,28 provided that the additional stationary points are well separated from the coalescing pair.

• The mathematical level of our derivations is similar to that of ref. 8 by one of us (JNLC) and Marcus (hereafter referred to as CM). The CM paper applied a different technique, that of Chester, Friedman and Ursell,³⁹ for the uniform asymptotic evaluation of an oscillating integral with two coalescing saddle points. In particular, CM presented the Chester et al. technique in an accessible, yet general, way, making it straightforward to apply to problems in molecular scattering theory.

A comparison of Fig. 1(a) and (b), shows that Cases A and B are related by a minus sign. In the following, we focus only on Case A, since the uniform asymptotic theory for Case B is similar. We have chosen Case A because it is used in our analysis of a reactive DCS in Section III.

IIB. Two limiting cases

Before proceeding, it is helpful to write down results for I(α) in two limiting cases. The results we require can be found in CM. The first limiting case is when x₁ and x₂ are well separated. Then the simple stationary phase approximation (SPA) can be applied at both stationary points, which results in (from ref. 8) Notice that the simple stationary phase approximation (2) does not depend on f_i′′′. Now in practice the sign of f_i′′′can change as x₁ and x₂ move far apart from x₀, i.e., the sign of f_i′′′ is then no longer fixed by Cases A and B. We include this effect in our 2Hankel result in Section IIC.4. The simple stationary phase result (2) is equivalent to making a second-order Taylor series expansion of f(α;x) at each stationary point, together with the approximations, g(x) ≈ g(x₁) and g(x) ≈ g(x₂).

The second limiting case is when α ≈ α₀ so that x₁ ≈ x₀ ≈ x₂. We make a third-order Taylor series expansion of f(α;x) at the point x₀

f(α;x) ≈ f(α;x₀) + f′(α;x₀)(x − x₀) + ⅙f′′′(α;x₀)(x − x₀)³ (3)

where the result f′′(α;x₀) = 0 has been used. Then making the approximation, g(x) ≈ g(x₀), we obtain the transitional Airy approximation (tAiry), namely (see ref. 8)where Ai(•) is the regular Airy function. The result (4) is valid for f₀′ < 0 (two real roots) as well as for f₀′ > 0 (two complex roots). When α = α₀, i.e., on the caustic, we have f₀′ = f′(α₀;x₀) = 0, and eqn (4) simplifies towhere Γ(•) is the Gamma function.

In the following, to avoid repeating phrases like “the kth-order Taylor series expansion of f(α;x) at the point x_i”, we will often use the concept of a k-jet,^40,41 which is defined as the Taylor series expansion of order k for f(α;x) at the point x_i. For example, eqn (3) written as a 3-jet is

j_x0³f(α;x) = f(α;x₀) + f′(α;x₀)(x − x₀) + ⅙f′′′(α;x₀)(x − x₀)³ (6)

Note: all k-jets in this paper are functions and not members of a polynomial ring.

The problem now is to deduce a uniform approximation which reduces to eqn (2) and (4) in the appropriate limits.

IIC. Extension of Carrier's method

IIC.1 Introduction. Carrier presents his derivation for the integral I(α) in the Appendix of his paper,¹⁸ assuming that f(α;x) is an odd function of x, and for g(x) = 1 (n.b., there are many misprints). At the end of his derivation, he extends his result assuming g(x) to be an even function of x. We need to generalize Carrier's derivation to the case when neither of these assumptions is valid.

The basic idea of Carrier is to use 3-jets at the points x₁ and x₂, similar to eqn (3) or (6) for the point x₀. This seems straightforward and physically reasonable, but Fig. 1(a) shows there is a serious problem. We see that the 3-jet at x₂ is accurate close to x₂, but it possesses another stationary point, say u₂ = u₂(α), which is quite different from x₁. This implies that the approximation to I(α) using the 3-jet at x₂, denoted I_x2(α), will be quite different from I(α), unless f(α;x) and j_x2³f(α;x) are very similar as functions of x. Evidently we must eliminate the contribution from the additional stationary point u₂.

Similar remarks apply if the 3-jet at x₁ is used – see Fig. 1(a); we must eliminate the contribution from the additional stationary point u₁. And similarly for Case B in Fig. 1(b). Eliminating the unwanted contributions from u₁ and u₂ is the essence of our derivation.

IIC.2 Contribution to I(α) from a third-order Taylor expansion at the stationary point x₂. The 3-jet at x₂ is given by

j_x2³f(α;x) = f₂ + ½f₂′′(x − x₂)² + ⅙f₂′′′(x − x₂)³ (7)

using f₂′ = 0. Then assuming g(x) ≈ g(x₂), we obtain an approximation denoted I_x2(α) to I(α), namelywhere the integration limits have been kept at −∞ to +∞. Denoting the stationary phase points of the integrand in eqn (8) by x₂ and u₂ as in Fig. 1(a) with u₂ ≤ x₂, we find

x₂ (by construction) (9)

u₂ = x₂ − (2f₂′′/f₂′′′) (10)

Next we use the identity

where

ς = b³/(3a²)

which is valid for a < 0 or a > 0 but a ≠ 0, and b ≤ 0 or b > 0. When b < 0, the argument of the Airy function is to be interpreted as −b²/(2a²)^2/3. Applying eqn (11) to eqn (8) giveswhere The result (12) evidently contains contributions from both the stationary points, x₂ and u₂.

Next we use the following identity connecting Ai(•) and Hankel functions of the first and second kinds, H⁽¹⁾_1/3(•) and H⁽²⁾_1/3(•) respectively, both of order one-third.⁴²

where

ξ = ⅔z^3/2 or equivalently, z = ([/]ξ)^2/3 (15)

We now use the identity (14) to write eqn (12) as the sum of two terms
whereand

Next we consider the limit where x₂ and u₂ are well separated, so we can use the following asymptotic approximations for the Hankel functions⁴³ as x → ∞

with ν = 1/3. We obtain for and the results Inspection of eqn (20) and (21) shows that is the simple stationary phase result for x₂, as given by the second term on the rhs of eqn (2). This suggests the other integral, , is associated with the simple stationary phase result at u₂. We now confirm this suggestion by expressing f₂, A₂ and f₂′′ in eqn (20) in terms of u₂ rather than x₂.

From eqn (7), we have for x = u₂

j_x2³f(α;u₂) = f₂ + ½f₂′′(u₂ − x₂)² + ⅙f₂′′′(u₂ − x₂)³

which simplifies to

j_x2³f(α;u₂) = f₂ + 2A₂ (22)

when eqn (10) is used and A₂ is defined by eqn (13). Also from eqn (7) we have upon differentiation
and so for x = u₂
which simplifies toupon using eqn (10). With the help of eqn (22) and (23), we can now write eqn (20) in the formEqn (24) is the simple stationary phase result at u₂ for j_x2³f(α;x), except for the presence of g₂ ≡ g(x₂) rather than g(u₂). However this small difference is of no consequence because it is that we must eliminate from our theory, since eqn (24) may be quite different from the simple stationary phase result at x₁ for I(α), as given by the first term on the rhs of eqn (2), namely

Note: The fact that f(α;x) has a maximum at x = x₁ did not play an essential role in the argument that should be retained and discarded when f(α;x) is approximated by j_x2³f(α;x). Thus if f(α;x) increases monotonically for x < x₂, we would still retain and discard when using j_x2³f(α;x). Also, if in eqn (1), g(x) = constant, then trivially g(x₂) = g(u₂).

In the next section, we repeat the above analysis using the 3-jet at x₁ rather than x₂.

IIC.3 Contribution to I(α) from a third-order Taylor expansion at the stationary point x₁. The following derivation is analogous to that presented in Section IIC.2 and is shorter because only an outline is presented. Note that eqn (27)–(30) derived below are used in Sections IIC.4 and IIIC.

The 3-jet at x₁ is given by

j_x1³f(α;x) = f₁ − ½|f₁′′|(x − x₁)² + ⅙f₁′′′(x − x₁)³ (25)

using f₁′ = 0. Then assuming g(x) ≈ g(x₁), we get for the approximation, I_x1(α) to I(α), the integralwith the integration limits kept at −∞ to +∞. Denoting the stationary phase points for the integrand of the integral (26) by x₁ and u₁ with x₁ ≤ u₁[see Fig. 1(a)], we have

u ₁ = x₁ + (2|f₁′′|/f₁′′′) and x₁ (by construction). Next we apply the identity (11) to eqn (26) obtaining

where The result (27) evidently contains contributions from both x₁ and u₁.

We again use the relations (14) and (15) connecting Ai(•) and the Hankel functions of the first and second kinds. This lets write eqn (27) as the sum of two terms

whereand

Next we consider the limit where x₁ and u₁ are sufficiently well separated that we can use the asymptotic approximations for the Hankel functions, as given by eqn (18) and (19) for x → ∞ with ν = 1/3. We obtain

Inspection of eqn (31) and (32) shows that is the simple stationary phase result at x₁, given by the first term on the rhs of eqn (2). Hence should be associated with the simple stationary phase result at u₁. We confirm this suggestion by manipulations in which f₁, A₁ and f₁′′ in eqn (32) are expressed in terms of u₁ rather than x₁ –these manipulations are analogous to those in Section IIC.2 for eqn (20). It is found that can be written in the alternative form Now eqn (33) is the simple stationary phase result for j_x1³f(α;x) at u₁, except for the presence of g₁ ≡ g(x₁) rather than g(u₁). Again this small difference is of no consequence since it is that we must eliminate from our theory. Note that eqn (33) may be quite different from the simple stationary phase result at x₂ for I(α), which is given by the second term on the rhs of eqn (2), namely

Note: The fact that f(α;x) has a minimum at x = x₂ did not play an essential role in the argument that should be retained and discarded when f(α;x) is approximated by j_x1³f(α;x). Thus if f(α;x) decreases monotonically for x > x₁, we would still retain and discard when using j_x1³f(α;x). This observation is used in Section IIIC for the nearside scattering of the J-shifted Eckart parametrization of the S matrix. Also, if in eqn (1), g(x) = constant, then trivially g(x₁) = g(u₁).

IIC.4 2Hankel approximation to I(α). The 2Hankel approximation is obtained by retaining and [i.e., eqn (29) and (17) respectively]. But we drop and [i.e., eqn (30) and (16) respectively] because they correspond to unwanted contributions from the stationary phase points u₁ and u₂ respectively, which arise when f(α;x) is approximated by its 3-jet at x₁ and x₂ respectively.

The 2Hankel approximation, denoted I_2H(α), is thus given by

wherewithandwith Remarks:

• The 2Hankel approximation has the advantage that the two stationary phase points, x₁ and x₂, appear separately in eqn (34)–(40).

• The 2Hankel approximation has the disadvantage that it involves the third derivatives, f₁′′′ and f₂′′′. For numerical input data with associated errors, these third derivatives may be difficult to determine accurately.

• Note that the asymptotic limits (36) and (39), valid when x₁ and x₂ are well separated, do not involve third derivatives. Typically, H⁽¹⁾_1/3(x) and H⁽²⁾_1/3(x) attain their asymptotic limits in eqn (18) and (19) respectively for x ≳ 1.

• As x → 0, we have H⁽¹⁾_1/3(x) → −i∞ and H⁽²⁾_1/3(x) → i∞, or in more detail (from ref. 44)

where Γ(•) is a gamma function. These results imply that the 2Hankel expression (34) becomes numerically unstable as A₁ and A₂ approach zero because of subtractive cancellation. This can be a problem for numerical input data. In Section IIC.5, we investigate analytically the 2Hankel limit when A₁ → 0 and A₂ → 0.

• In practical applications of the 2Hankel approximation, the third derivatives can change sign as α varies; in particular, as x₁ and x₂ separate and we approach the stationary phase results (36) and (39). Eqn (34)–(40) have been written so they are valid for either sign [If the third derivatives are identically zero, then eqn (34), (35), (37), (38) and (40) are ill-defined].

• If g(x) = 1 and f(α;x) is exactly a cubic polynomial in x, eqn (4), (12), (27) and the 2Hankel formula (34) are exact results for I(α) because the 3-jets are then exact representations of f(α;x). This is useful for the checking of computer programs.

Since our application of the 2Hankel approximation to reactive scattering in Section III is relatively complicated, in the Appendix we apply the 2Hankel approximation to three oscillating integrals of the cuspoid type.^28,29

IIC.5 Limiting case for the 2Hankel approximation when α → α₀. In the derivation of the 2Hankel approximation, the case when x₁and x₂ are well separated played an important role. For this limiting case, the simple stationary phase approximation (2) is valid, which enabled us to eliminate the unwanted contributions from the stationary phase points u₁ and u₂.

In this section, we examine the 2Hankel approximation for the limiting case α → α₀ and ask the question: Do we obtain the transitional Airy approximation (4) in this limit? In eqn (34), (35), (37), (38) and (40) we evidently have to express g₁, f₁, f₁′′, f₁′′′, A₁ and g₂, f₂, f₂′′, f₂′′′, A₂ in terms of their values at x₀ rather than at x₁ and x₂. Now for α → α₀, we have x₁ ≈ x₀ ≈ x₂ and to extract the limiting behaviour we approximate f(α;x) by its 3-jet at x₀, as given by eqn (3) or (6).

When the approximation (3) is valid for f(α;x), the two real stationary phase points, x₁ ≤ x₂, are the roots of

(x − x₀)² = −2f₀′/f₀′′′ (42)

Since x₁ and x₂ are assumed to be real in eqn (42), we must have f₀′ ≤ 0 (also recall from Section IIA that f₀′′′ > 0). We can write for the two roots

Now from the cubic approximation (3), we obtain

f′′(α;x) = f₀′′′(x − x₀)

f′′′(α;x) = f₀′′′

It follows that for x = x₁

and for x = x₂
where eqn (43) and (44) have been used. We then find from the definitions (37) and (40) that
To obtain f₁ and f₂ we again use eqn (3) together with eqn (43) and (44). The results are

f₁ = f₀ + A₁

f₂ = f₀ − A₂

Finally, we make the approximation, g₁ ≈ g₀ ≈ g₂. Then we can substitute the above results for g₁, f₁, f₁′′, f₂′′′, A₁ and g₂, f₂, f₂′′, f₂′′′, A₂ into the 2Hankel approximation, eqn (34), (35), (37), (38) and (40). The last step is to use the identity (14) to replace the two Hankel functions with an Airy function, yielding after simplificationEqn (45) is the transitional Airy approximation mentioned in Section IIB. Although it has been obtained from the 2Hankel formula assuming f₀′ ≤ 0 (two real stationary phase points), it is also valid for f₀′ > 0, provided −|f₀′| is replaced by f₀′ in eqn (45) – see eqn (4).

Thus we have demonstrated that the 2Hankel approximation contains both limiting cases presented in Section IIB.

III. Derivation of the 6Hankel approximation for rainbow and glory scattering

IIIA. Introduction

Our starting point is the expansion for the scattering amplitude, f(θ_R), in a basis set of Legendre polynomials. We write the resulting partial wave series (PWS) in the formwhere k is the initial translational wavenumber, J is the total angular momentum quantum number, S̃_J is the Jth modified scattering matrix element, and P_J(•) is a Legendre polynomial of degree J. In order to keep the notation simple, the label, v_i, j_i, m_i → v_f, j_f, m_f for the initial and final states, has been omitted from f(θ_R) and S̃_J, as has the label, v_i, j_i, from k. Here v, j, m are vibrational, rotational and helicity quantum numbers respectively, with m_i = m_f = 0 in our case.

We now introduce standard semiclassical approximations into the PWS (46) to convert it into an oscillating integral.^3,4Firstly, we transform the PWS into a Poisson series and retain the leading (m = 0) term; this assumes all the stationary phase points lie in (−π, +π), which is the case for our application – see Section IIIB. We have

In eqn (47), S̃(J) is the continuation of the set of values, {S̃_J}, from integer to real values of J and P_J(cos θ_R) is now a Legendre function of the first kind. Secondly, we introduce the Hilb approximation to express the Legendre function in terms of a Bessel function, J₀(•), of order zero^45,46
The Hilb approximation has an error, O(J^−3/2), which is uniform for θ_R ∈ [0, π − ε] with ε > 0, We obtain
Thirdly, we express the Bessel function as a sum of two Hankel functions⁴⁷

J₀(x) = ½[H⁽¹⁾₀(x) + H⁽²⁾₀(x)] (48)

We can then write the scattering amplitude as the sum of nearside and farside subamplitudes,^3,4,6,19,20 which are indicated by superscripts, “(−)” and “(+)”, respectively:

f(θ_R) = f⁽⁻⁾(θ_R) + f⁽⁺⁾(θ_R)

The nearside subamplitude is given byor The farside subamplitude is given byor In writing down eqn (50) and (52), we have followed Miller⁹ and introduced the factors

exp{±i[(J + ½)θ_R − π/4]}exp{∓i[(J + ½)θ_R − π/4]} = 1

This is because the asymptotic eqn (18) and (19) with ν = 0 and (J + ½)θ_R ≫ 1 show that
which implies that the expressions in double braces, , are relatively slowly varying with respect to J.

Remark: As in our previous work,^3,4,6 we use “∓” to indicate nearside/farside in the semiclassical theory and reserve “N/F” for nearside/farside decompositions in the PWS theory.

The fourth step is the asymptotic evaluation of the oscillatory integrals (50) and (52) using the theory developed in Section II. To do this we must first examine the general properties of S̃(J) for our application. This is done next.

IIIB. Properties of S̃(J)

Fig. 2 illustrates the properties of S̃(J) for the J-shifted Eckart model defined in Section IV. In particular, Fig. 2(a) and (b) display |S̃(J)| and arg S̃(J) versus J respectively, whilst Fig. 2(c) is a plot of the quantum deflection function versus J, which is defined by
For eqn (50) and (52), the stationary phase points are the roots of

[capital Theta, Greek, tilde](J) ∓ θ_R = 0

where “−” corresponds to nearside scattering and “+” to farside scattering. We see there can be a contribution from both the nearside and farside scattering at a given value of θ_R; we consider the nearside and farside cases separately.

Fig. 2 S matrix data for the J-shifted Eckart parameterization. The values of the parameters are given in Section IVB. (a) |S̃(J)| versus J. (b) arg S̃(J)/rad versus J. The maximum of the arg S̃(J)/rad curve defines the glory angular momentum variable, J_g, which is indicated by a green dashed line and arrow. (c) [capital Theta, Greek, tilde](J)/deg versus J. The red dashed lines and arrow indicate + θ_R and J₁ = J₁(θ_R) for the nearside scattering. The blue dashed lines and arrows indicate −θ_R, as well as J₂ = J₂(θ_R) and J₃ = J₃(θ_R) for the farside scattering. Also shown is the rainbow angular momentum variable, J_r, which is located at the minimum of the [capital Theta, Greek, tilde](J)/deg curve, where [capital Theta, Greek, tilde](J_r) = −θ^r_R (pink arrow and dashed line respectively), together with J_g, which satisfies the equation [capital Theta, Greek, tilde](J_g) = 0 (green arrow).

IIIC. Nearside scattering

Fig. 2(c) shows there is a single real root to [capital Theta, Greek, tilde](J) = +θ_R, which we denote by J₁ ≡ J₁(θ_R). The asymptotic theory developed in Section IIC.3 applies, in particular eqn (29).

We make the following identifications between eqn (1) and (50)

x = J, x₁ = J₁, α = θ_R

f(α;x) = arg S̃(J) − (J + ½)θ_R + π/4

so that

f′(α;x) = [capital Theta, Greek, tilde](J) − θ_R

f′′(α;x) = [capital Theta, Greek, tilde]′(J)

f′′′(α;x) = [capital Theta, Greek, tilde]′′(J)

Note that the phase has a linear dependence on θ_R, namely, −(J + ½)θ_R. Then from eqn (28), (29) and (50), we can write for the nearside subamplitude to the 6Hankel approximation

with

IIID. Farside scattering

Perusal of Fig. 2(c) shows there are two real roots to [capital Theta, Greek, tilde](J) = −θ_R, which we denote J₂ ≡ J₂(θ_R) and J₃ ≡ J₃(θ_R) with J₂ ≤ J₃. At the rainbow angle, θ_R = θ^r_R, J₂ and J₃ coalesce to J_r, the value of the rainbow angular momentum variable; we then have, [capital Theta, Greek, tilde](J_r) = −θ^r_R. For the farside scattering, the asymptotic theory developed in Section IIC.4 applies and in particular we can use the 2Hankel approximation.

We make the following identifications between eqn (1) and (52)

x = J, x₁ = J₂, x₂ = J₃, α = θ_R

f(α;x) = arg S̃(J) + (J + ½)θ_R − π/4

so that

f′(α;x) = [capital Theta, Greek, tilde](J) + θ_R

f′′(α;x) = [capital Theta, Greek, tilde]′(J)

f′′′(α;x) = [capital Theta, Greek, tilde]′′(J)

Note that the phase has a linear dependence on θ_R, namely, +(J + ½)θ_R. Then from eqn (34), (35), (37), (38), (40) and (52) we have for the two farside subamplitudes which contribute to the 6Hankel approximation

withandwith

IIIE. 6Hankel approximation

The 6Hankel approximation for the full scattering amplitude is then obtained by summing the subamplitudes (53), (55) and (57)

f_6H(θ_R) = f⁽⁻⁾_6H(1|θ_R) + f⁽⁺⁾_6H(2|θ_R) + f⁽⁺⁾_6H(3|θ_R) (59)

The corresponding DCS is

σ_6H(θ_R) = |f_6H(θ_R)|² (60)

Eqn (59) is the 6Hankel approximation that was written down without proof in our recent paper with Zhang.⁶ We see that it contains six Hankel functions, three of order zero, H^(1,2)₀(•), and three of order 1/3, H^(1,2)_1/3(•).

Note that the farside subamplitudes (55) and (57) are only valid on the bright side of the rainbow angle, where the two roots of [capital Theta, Greek, tilde](J) = −θ_R are real. Thus in our calculations in Section V, we will only apply the full 6Hankel approximation (59) for 0 < θ_R < θ^r_R.

When the arguments of the Hankel functions in eqn (53), (55) and (57) are large, we can substitute the asymptotic results (18) and (19) with ν = 0 or 1/3. We then obtain the following expressions for the subamplitudes, which constitute the primitive semiclassical approximation (PSA):

where the “classical-like” DCSs are given byand the phases are In a systematic semiclassical (SC) notation,^4,6eqn (61)–(63) are written SC/N/PSA. We call the sum of the two farside subamplitudes, SC/F/PSA.

IIIF. The limit θ_R → θ^r_R: farside rainbow scattering

When θ_R → θ^r_R, f⁽⁺⁾_6H(2|θ_R) + f⁽⁺⁾_6H(3|θ_R) becomes numerically unstable because of subtractive cancellation. This arises because [capital Theta, Greek, tilde]′(J₂(θ_R)) → 0 and [capital Theta, Greek, tilde]′(J₃(θ_R)) → 0 as the rainbow angle is approached, and the asymptotic limits (41) apply to the H⁽¹⁾_1/3(B_i(θ_R)) with i = 1 and 2. We examined this limit in Section IIC.5, and showed that the 2Hankel approximation contains the transitional Airy approximation (45) [or eqn (4)] when α → α₀. We can apply this result to the farside scattering, as given by eqn (55) and (57). Making the identifications

x = J, x₀ = J_r, α = θ_R, α₀ = θ^r_R

we can write for the farside subamplitude.where

The approximation (64) is equivalent to replacing [capital Theta, Greek, tilde](J) by its 2-jet at J_r, namely

j_Jr²[capital Theta, Greek, tilde](J) = −θ^r_R + q_r(J − J_r)²

Eqn (64) has the advantage that it can be used on both the bright side, θ_R < θ^r_R, and the dark side, θ_R > θ^r_R, of the rainbow, as well as at θ_R = θ^r_R.

Remarks:

• In a systematic semiclassical notation,^4,6eqn (64) is written as SC/F/tAiry, or tAiry for short.

• When the tAiry subamplitude (64) is used to calculate the DCS, we must also include the contribution from the nearside scattering. We use the SC/N/PSA subamplitude of eqn (61)–(63) for this purpose.

• It is known that the tAiry subamplitude is a special case of the more general uniform Airy approximation, as derived in ref. 4 and 6. In a systematic notation,^4,6 it is described as SC/F/uAiry, or uAiry for short.

IIIG. The limit θ_R → 0: glory scattering

Finally we must examine the glory limit, θ_R → 0, for f⁽⁻⁾_6H(1|θ_R) + f⁽⁺⁾_6H(2|θ_R). In eqn (53) and (55), we note the following limits as θ_R → 0, for i = 1 or 2;

J_i(θ_R) → J_i(0) ≡ J_g,

[capital Theta, Greek, tilde]′(J_i(θ_R)) → [capital Theta, Greek, tilde]′(J_i(0)) = [capital Theta, Greek, tilde]′(J_g) ≡ [capital Theta, Greek, tilde]_g′

[capital Theta, Greek, tilde]′′(J_i(θ_R)) → [capital Theta, Greek, tilde]′′(J_i(0)) = [capital Theta, Greek, tilde]′′(J_g) ≡ [capital Theta, Greek, tilde]_g′′

B_i(θ_R) → B_i(0) ≡ B_g

S̃(J_i(θ_R)) → S̃(J_i(0)) = S̃(J_g) ≡ S̃_g

H⁽¹⁾_1/3(B_i(θ_R)) → H⁽¹⁾_1/3(B_i(0)) ≡ H⁽¹⁾_1/3(B_g)

We can use the above limits and the identity (48) to deduce that in the limit θ_R → 0

H⁽²⁾₀([J₁(θ_R) + ½]θ_R) + H⁽¹⁾₀([J₂(θ_R) + ½]θ_R) → 2J₀(0) = 2

We then obtain

where Inspection of Fig. 2(c) shows that we expect [capital Theta, Greek, tilde]′′(J_g) to be very small in magnitude; hence from eqn (66), we conclude that B_g ≫ 1, and so can use the asymptotic approximation (18) with ν = 1/3 in eqn (65). We obtain for the limit θ_R → 0
which is recognized as the semiclassical transitional approximation (STA) at θ_R = 0, as given by eqn (14) of ref. 3 or eqn (29) of ref. 7.

Remarks:

• The contribution from f⁽⁺⁾_6H(3|θ_R) to the scattering at θ_R ≈ 0 is very small in our application in Section V. It has been neglected in the above derivation.

• f⁽⁻⁾_6H(1|θ_R) and f⁽⁺⁾_6H(2|θ_R) become large in magnitude as θ_R → 0, but their sum f⁽⁻⁾_6H(1|θ_R) + f⁽⁺⁾_6H(2|θ_R) is much smaller. This implies the 6Hankel approximation becomes numerically unstable as θ_R → 0.

• The STA is a special case of the uniform semiclassical approximation (USA) for forward glory scattering which was derived and discussed in ref. 3 and 7. However, in this paper we will use the more explicit name, uniform Bessel approximation, and use the abbreviation, uBessel.

IIIH. Discussion

We note the following:

• The 6Hankel approximation (59) is generic. This means it is applicable to numerous chemical reactions at numerous collision energies (in principle, an infinite number of cases) which have S matrix elements that are analogous to those in Fig. 2.

• The 6Hankel approximation (59) is uniform for both rainbow and glory scattering, with the advantage that the contributions from all three roots J₁, J₂, J₃ appear separately. In contrast, in the standard semiclassical approach, it is necessary to apply two different theories: the uniform rainbow theory (SC/F/uAiry or uAiry),^4,6 which involves the pair of roots, J₂ and J₃; and the uniform glory theory (uBessel),^3,7 which involves the pair J₁ and J₂.

• The 6Hankel approximation (59) has the disadvantage that it involves third derivatives of arg S̃(J); these can be difficult to calculate accurately for numerical input data. Moreover in practical applications, these third derivatives can change sign. In contrast, the uAiry and uBessel theories involve just the second derivatives.

• The 6Hankel approximation (59) has the disadvantage that it becomes numerically unstable for θ_R → 0 and for θ_R → θ^r_R, in particular for numerical {S̃_J} that possess errors. These problems can be overcome by using the STA for glory scattering and the tAiry approximation for rainbow scattering.

• The 6Hankel formula for the DCS is a generalization to reactive scattering of a result given by Miller for elastic collisions. In particular, if we set in eqn (53)–(60), |S̃(J)| → 1 and arg S̃(J) → 2δ(J), where δ(J) is the elastic phase shift, then, after considerable algebraic manipulations, we obtain for σ_6H(θ_R) the result quoted by Miller [eqn (6) of ref. 9]. One difference is that Miller assumes the Wentzel–Kramers–Brillouin approximation for δ(J), whereas our derivation shows that this assumption is not necessary.

IV. J-shifted Eckart parameterization and the values of its parameters

This section defines the J-shifted Eckart parameterization for S̃(J) in Section IVA, whilst Section IVB gives the values of its parameters that are used in our DCS computations in Section V.

IVA. J-shifted Eckart parameterization

Sokolovski introduced the J-shifted Eckart parameterization for S̃(J) in order to model the DCS for the state-selected H + D₂ → HD + D reaction; in particular to mimic a resonance near the reaction threshold.^10,48 It has subsequently been found to be very useful for understanding the dynamics of chemical reactions.^11,48,49 The J-shifted Eckart parameterization is based on the exact quantum solution for a particle of reduced mass μ transmitted by a symmetric Eckart potential, W₀/cosh²(s/s₀), where s is a reaction coordinate, s₀ is a reference distance and W₀ is the height of the energy barrier. The following formulae define this parameterization:
where

[small phi, Greek, tilde](J) = aJ² + bJ

is a quadratic phase, with real parameters, a and b. Also

In addition

• N = scaling factor (dimensionless).

• (dimensionless because of the following definition).

• ε = ℏ²/(2μs₀²) (dimensions of energy).

• (dimensionless). In this equation, E is the total energy and B is the rotational constant for the triatomic complex. The term B J(J + 1) represents the J-shifted approximation, because it introduces a J dependence into the mathematically one-dimensional Eckart barrier.

• K(J) has two branch points at real values of J, which are the roots of E − B J(J + 1) = 0. The larger root is located at . The integer J_max is then defined by J_max = Floor(J_c). It follows that K(J_max) is real and K(J_max + 1) is purely imaginary. In our application in Section V, we have J_max ≫ 1.

IVB. Parameter values for the J-shifted Eckart parameterization

We used the following values for the parameterswhich results in J_max = 44. Plots of |S̃(J)|, arg S̃(J) and [capital Theta, Greek, tilde](J) versus J respectively have already been shown and discussed in Fig. 2(a), (b) and (c) respectively. In Section V, we will require the position of the leading Regge pole (n = 0) in the first quadrant of the complex angular momentum plane for the J-shifted Eckart potential with the parameter values (67). It is obtained from the poles of Γ(−iK(J) + iQ + 1/2) and is given by J₀ = 34.4 + 1.21i, which corresponds to a life-angle of 1/(2Im J₀) = 0.415 rad = 23.8°.

Remark: the parameter values (67) are based on the “standard values” employed for the H + D₂ reaction,^10,11 which have B/ε = 0.12247 and W₀/ε = 150. However, the standard values result in θ^r_R = 24.0° at J_r = 25.2, which only allows a test of the 6Hankel approximation over a relatively small range of angles. Changing B/ε to 0.11247 and W₀/ε to 90 gives θ^r_R = 109.2° at J_r = 34.5, which permits the semiclassical approximations to be tested over a wider range of θ_R.

V. Results for the J-shifted Eckart parameterization

This section presents our results for the angular scattering in Section VA, whilst Section VB reports a nearside–farside (NF) analysis of the (dimensionless) DCS, as well as a NF analysis for the local angular momentum (LAM). All the semiclassical DCSs are defined as the square modulus of the corresponding scattering amplitude.

VA. Dimensionless differential cross sections

Fig. 3 plots, on a linear scale, the dimensionless differential cross section (dDCS), k²σ(θ_R), versus θ_R for the PWS, where it is compared with the uAiry + SC/N/PSA, uBessel and 6Hankel approximations for θ_R < θ^r_R. When θ_R ≥ θ^r_R, the dDCS for the tAiry + SC/N/PSA approximation is drawn. For consistency with ref. 4, the tAiry subamplitude includes the first correction term to eqn (64), even though it makes only a small contribution to the dDCS [see eqn (33) of ref. 4]. Notice that the uAiry + SC/N/PSA curve diverges as θ_R → 0°, the uBessel curve diverges as θ_R → θ^r_R, and the tAiry + SC/N/PSA curve diverges as θ_R → 180°.

Fig. 3 Linear plot of the dimensionless angular distribution, k²σ(θ_R) versus θ_R. Black solid curve: PWS. Purple solid curve: uAiry + SC/N/PSA. Purple dashed curve: tAiry + SC/N/PSA. Green solid curve: uBessel. Orange solid curve: 6Hankel. (a) 0° ≤ θ_R ≤ 30°. The uAiry + SC/N/PSA curve diverges as θ_R → 0°. (b) 30° ≤ θ_R ≤ 80°. The PWS and uAiry + SC/N/PSA curves are almost coincident for θ_R ≳ 35°. (c) 80° ≤ θ_R ≤ 180°. The pink solid arrows indicate the rainbow angle, θ^r_R = 109.2°. The uBessel curve diverges as θ_R → θ^r_R. The tAiry + SC/N/PSA approximation is used for θ_R ≥ θ^r_R; it diverges as θ_R → 180°.

Fig. 3(a) shows the range, 0° ≤ θ_R ≤ 30°. We see that the PWS, uBessel and 6Hankel dDCSs agree to graphical accuracy, as does the uAiry + SC/N/PSA dDCS for θ_R ≳ 2°. The same is mostly true for the range, 30° ≤ θ_R ≤ 80°, displayed in Fig. 3(b), except near the maxima of the diffraction oscillations.

For 80° ≤ θ_R ≤ θ^r_R, Fig. 3(c) shows that the uAiry + SC/N/PSA curve generally agrees best with the PWS dDCS, with the uBessel and 6Hankel dDCSs being less accurate. This last result can be traced back to the 2Hankel approximation, which is generally less accurate than the uAiry approximation for the cuspoid test integrals in the Appendix. For θ_R ≳ θ^r_R, the agreement between the PWS and tAiry + SC/N/PSA curves is satisfactory. As expected, the discrepancies between these two curves generally increases as we move further into the dark side of the rainbow.

Notice that the rainbow does not possess any supernumerary rainbows in the angular scattering; rather it is an example of a “broad rainbow”.^4,6,50

VB. Nearside–farside analyses

The NF decomposition of the PWS for f(θ_R) is given by ref. 3, 4, 6, 11, 13–17 and 19–24

f(θ_R) = f^(N)(θ_R) + f^(F)(θ_R) (68)

where the N, F subamplitudes are (θ_R ≠ 0, π)withEqn (68)–(70) are the Fuller decomposition for f(θ_R).⁵¹ The corresponding N, F PWS DCSs are defined by

σ^(N,F)(θ_R) = |f^(N,F)(θ_R)|² (71)

In practice, we resum the PWS (46) three times (r = 3) before carrying out the NF decomposition because it is known this helps “clean” the N and F unresummed dDCSs and LAMs of unphysical structure.^21–24 We then write PWS/N/r = 3 and PWS/F/r = 3 for the N and F resummed subamplitudes respectively. Totenhofer et al.²⁴ have presented a detailed account of resummation theory for a Legendre PWS; this theory is not repeated here.

Fig. 4(a) reports a NF analysis of the PWS dDCS. We see that the reaction is N dominant, although the PWS/F/r = 3 dDCS is always significant, becoming increasingly important upon moving to smaller scattering angles. This results in pronounced diffraction oscillations in the full dDCS, which arise from interference between the N and F subamplitudes.

Fig. 4 Nearside–farside analyses for the dimensionless logarithmic angular distribution and the Local Angular Momentum. Black solid curve: PWS. Red solid curve: PWS/N/r = 3. Red dashed curve: SC/N/PSA. Solid purple curve: SC/F/uAiry. Purple dashed curve: SC/F/tAiry. Blue solid curve: PWS/F/r = 3. Blue dashed curve: SC/F/PSA. Green solid curve: SC/N/6Hankel. Orange solid curve: SC/F/6Hankel. The pink solid arrows indicate the rainbow angle, θ^r_R = 109.2°. (a) Full and NF log k²σ(θ_R) versus θ_R. (b) Full and NF LAM(θ_R) versus θ_R.

The PWS results in Fig. 3 for the dDCS, allowed us to test the accuracy of the semiclassical approximations. This is not the case for Fig. 4, because it is known that the NF PWS decomposition of eqn (68)–(71) is not unique: there is no guarantee that it will produce physically useful results.^20–24 Rather we use the semiclassical approximations to provide a check on the physical effectiveness of the N, F PWS/r = 3 results. We also recall that the SC/F/6Hankel, SC/F/uAiry, SC/F/PSA and SC/N/6Hankel approximations are only defined for θ_R < θ^r_R.

Fig. 4(a) plots the N and F dDCSs for the PWS/r = 3, PSA and 6Hankel approximations, and the F dDCSs for the uAiry and tAiry approximations. The agreement between the semiclassical and N, F PWS/r = 3 dDCSs is seen to be generally good. The discrepancies are those expected from our earlier work.^{3,4,6,11,13–17,19–24} For example, the SC/F/PSA dDCS diverges as θ_R → θ^r_R, whilst the N and F PWS/r = 3 dDCSs exhibit unphysical oscillations at large angles.

Next we consider the NF analysis of the LAMs.^21–24 The full LAM is defined by

whilst the N, F LAMs are obtained from ref. 21–24 In the PWS calculations, we again resum f(θ_R) three times (r = 3) before making the NF decomposition.²⁴ The args in eqn (72) and (73) are not necessarily principal values in order that the derivatives be well defined.

Fig. 4(b) shows full and N, F LAMs for the PWS/r = 3 and semiclassical approximations. We observe that the LAM information in Fig. 4(b) is consistent with the dDCS information in Fig. 4(a). The discrepancies between the N, F curves are similar to those we have seen previously, e.g., the unphysical oscillations in the N, F PWS/r = 3 LAMs as we approach the backward direction.^21–24

We observe that the semiclassical N LAMs decrease in magnitude as θ_R increases, and are similar to the LAM for the repulsive scattering of two hard spheres,^23,24i.e., the N scattering is direct. Next we examine the semiclassical F LAMs. We see that the semiclassical F LAMs are slowly increasing at small θ_R; then the SC/F/tAiry LAM becomes approximately constant at large angles, where it has the value 34.9. In fact to a very good approximation we have (from ref. 4 and 6)

LAM⁽⁺⁾_tAiry(θ_R) ≈ J_r + 1/2 = 35.0

This behaviour of SC/F/tAiry corresponds in a Regge treatment to decaying (creeping) surface waves that propagate around the reaction zone (see also ref. 52.) When a single Regge pole (n = 0) dominates, we have (from ref. 6 and 27)

LAM⁽⁺⁾_Regge(θ_R) ≈ Re J₀ + 1/2 = 34.9

For a broad rainbow, we also expect J_r ≈ Re J₀,^6,27 which is what we find. If we average over the oscillations in the PWS/F/r = 3 LAM for 100° ≤ θ_R ≤ 165° we obtain a mean value of 34.5. Thus we have the result

This important result tells us that the PWS, semiclassical and Regge theories are all consistent with each other for the broad rainbow in Fig. 3 and 4(a).

VI. Conclusions

In our recent paper with Zhang,⁶ we stated without proof the 6Hankel asymptotic approximation for the scattering amplitude, which has the desirable property of being uniform for both a forward glory and a rainbow in the DCS of a chemical reaction. It is also a generic approximation. In this paper we have:

• Presented a detailed derivation of the 6Hankel approximation. Our derivation generalizes a method described by Carrier for an oscillating integral with two coalescing real stationary phase points. The generalization uses 3-jets of the phase at the stationary phase points, x₁and x₂, followed by a discardation of the contributions from the unwanted stationary phase points, u₂ and u₁. Applying the resulting 2Hankel approximation to the Legendre PWS for the scattering amplitude gives rise to the generic 6Hankel approximation. We also made a test of the accuracy of the 2Hankel approximation by applying it to three cuspoid oscillating integrals.

• Investigated some properties of the 6Hankel approximation. It has the advantage that each root contribution, J₁, J₂, J₃, appears separately in the 6Hankel expression, but has the disadvantage that third derivatives of arg S̃(J) are required. In the limit θ_R → 0, the 6Hankel approximation reduces to the STA for describing the glory. And in the limit θ_R → θ^r_R, we obtain the tAiry approximation for rainbow scattering.

• Assessed the accuracy of the 6Hankel approximation for θ_R < θ^r_R. Using a J-shifted Eckart parametrization of the S matrix, we found that both the 6Hankel and uBessel DCSs agreed well with the PWS DCS at angles close to the forward direction. However using numerical S matrix data, we earlier had found⁶ the 6Hankel DCS to be less accurate than the uBessel DCS at small angles – probably because of the difficulty of accurately calculating the [capital Theta, Greek, tilde]′′(J_i). Near the rainbow angle, the 6Hankel DCS generally exhibited greater deviations from the PWS DCS compared to the uAiry + SC/N/PSA DCS. This can be traced back to the 2Hankel approximation, which was generally less accurate than the uAiry approximation for the cuspoid test integrals.

The above trends can be understood in a more general way by remembering that the phases of the semiclassical integrands are not approximated in the uBessel and uAiry approximations – rather exact local one-to-one transformations are made – and only the pre-exponential factors are approximated.^3,4,7,8 Whereas, in the 2Hankel and 6Hankel approximations, both the phases and pre-exponential factors are approximated.

Also relevant is the following comment made by Ursell in his last published paper concerning the solution of (water) wave problems:⁵³

“Such a problem is usually solved by applying a sequence of mathematical arguments, and it would be helpful if some or all of the successive steps in this sequence could be given a physical interpretation. In the author's experience this is generally not possible.”

Ursell illustrates his comment by several examples of mathematical steps that do not have a physical interpretation, in particular the use of the exact local one-to-one transformation employed in the method of Chester et al.³⁹ for the uniform asymptotic evaluation of an oscillating integral with two coalescing saddle points. This is also the key transformation used in the derivation of the uAiry approximation.^4,8

Appendix: application of the 2Hankel approximation to cuspoid integrals

In this Appendix, we assess the accuracy of the 2Hankel approximation (34) when it is applied to three cuspoid integrals^28,29 of the Case A type. We also compare with results from the uniform Airy approximation (uAiry) derived by CM,⁸ which is based on the technique of Chester et al.³⁹ For a Case A integral, the uniform Airy approximation is given in Section IIIG of CM, namelywhere

A(α) = ½(f₁ + f₂)

and

ς(α) = [¾(f₁ − f₂)]^2/3

Here Ai′(x) means dAi(x)/dx. Note that ς(α) ≥ 0 for real roots. When ς ≫ 1, so the stationary phase points are well separated, the simple stationary phase result (2) is obtained upon replacing Ai(−ς) and Ai′(−ς) by their asymptotic approximations. When ς ≈ 0, so the stationary phase points are close together, the transitional Airy approximation (4) is obtained when f(α;x) is approximated by its 3-jet, as in eqn (3) or (6). For this situation, eqn (A1) for I_uAiry(α) becomes an exact result, provided g(x) = 1.

In the following three examples, all the phases are real and have a linear dependence on α of the type, −αx. In all three cases, g(x) = 1. The corresponding oscillating integrals have been calculated numerically by deforming the contour of integration from the real axis into the complex plane as explained in ref. 54–56.

Example 1. Cubic polynomial phase

We write the cubic phase in the formwhere α, a₂, a₃ are real numbers chosen so that f₁(α;x) has two real stationary phase points [i.e., real roots of df₁(α;x)/dx = 0], which can coalesce as α varies; for example, a₂ = 1, a₃ = 1, α ≥ −1/4. More generally, we require the discriminant, D₁, of the quadratic equation df₁(α;x)/dx = 0, namely D₁ = a₂² + 4αa₃, to satisfy D₁ > 0 for separate real roots or D₁ = 0 for one real double root. Then I_2H(α), I_uAiry(α) and I_tAiry(α) are exact results for the cubic phase (A2), which is useful for the checking of computer programs.

Example 2. Quintic polynomial phase: an oddoid integral of order two

In this example we choosewhere a₃ > 0, a₅ > 0 and α ≥ 0. (For the tAiry approximation we also let α < 0.) Since f₂(α;x) is an odd function of x, I₂(α) is purely real; it is an example of an oddoid integral of order two.²⁹ We have
where the oddoid integral is defined in ref. 29
with b₁ and b₂ real. Hobbs et al.²⁹ have discussed the properties of oddoid (and evenoid) integrals. Now Descartes' Rule of Signs⁵⁷ tells us that the quartic equation df₂(α;x)/dx = 0 for α > 0 has one positive root and one negative root, which coalesce when α = 0. For α < 0, these roots become complex conjugates.

Fig. 5(a) plots I₂(α) versus α for the range 0 ≤ α ≤ 15 with phase parameters of a₃ = a₅ = 5. The oscillatory nature of I₂(α) can be clearly seen. Also plotted are the 2Hankel and uAiry approximations as well as the SPA [these curves are also drawn in more detail for α ≤ 5 in Fig. 5(b)]. The SPA is accurate for α ≳ 2 and, as expected, it diverges as α → 0. The 2Hankel approximation becomes systematically smaller than I₂(α) for α ≲ 3, until α ≈ 0.15, when the two curves cross. The uAiry approximation agrees closely with I₂(α) over the whole range of α.

Fig. 5 (a) I₂(α) and three asymptotic approximations for 0 ≤ α ≤ 15. Black solid curve: I₂(α). Red solid curve: uAiry. Blue solid curve: 2Hankel. Green dashed curve: SPA. (b) I₂(α) and four asymptotic approximations for −5 ≤ α ≤ 5. Black solid curve: I₂(α). Red solid curve: uAiry. Blue solid curve: 2Hankel. Green dashed curve: SPA. Pink dashed curve: tAiry. The inset shows the corresponding curves for −0.6 ≤ α ≤ 0.6 (not SPA).

It was mentioned in Section IIB that the tAiry approximation is also valid for negative α. Fig. 5(b) compares I_tAiry(α) with I₂(α) for −5 ≤ α ≤ 5. It can be seen that there is good agreement between the two curves for negative α. This finding is very useful since the theory and application of the 2Hankel and uAiry approximations for negative α in practical applications is usually much more difficult than for α > 0. However Fig. 5(b) shows that the tAiry approximation quickly becomes inaccurate for positive α, in particular for the amplitude of the oscillation at α ≈ 2. And this continues to be the case for α > 5 where the positions and amplitudes of the maxima and minima in I₂(α) are not reproduced (not shown). At α = 0, the 6Hankel and uAiry approximations become equivalent to the tAiry approximation; this result can be seen visually in the inset to Fig. 5(b). Note that the tAiry approximation is obtained by putting a₅ = 0 in the phase (A3). Also from eqn (5) we have

The accurate value for I₂(α = 0) is 1.251, so the percentage error in I_tAiry(α = 0) is 4.3%.

Example 3. Quintic polynomial phase: A swallowtail integral

In our third example we choose
with a₃ > 0, a₄ > 0, a₅ > 0 and α ≥ 0. (Again we also let α < 0 for the tAiry approximation.) The corresponding oscillating integral
is an example of a swallowtail integral,^28,55 which is complex valued in general. Descartes' Rule of Signs⁵⁷ tells us that the quartic equation df₃(α;x)/dx = 0 for α > 0 has one positive root and one negative root, which coalesce at α = 0. For α < 0, these roots become complex conjugates. We have compared I_2H(α), I_uAiry(α), I_tAiry(α) with I₃(α) for many values of a₃, a₄, a₅ when 0 ≤ α ≤ 20 [and also −5 ≤ α ≤ 20 for the tAiry approximation]. In general, the accuracy of the asymptotic formulae are similar to that already discussed in Fig. 5 for Example 2, so, we do not display the corresponding graphs.

Acknowledgements

Support of this research by the UK Engineering and Physical Sciences Research Council and a UK Leverhulme Trust Emeritus Fellowship to JNLC is gratefully acknowledged.

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