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Chengkui
Xiahou
and
J. N. L.
Connor
*

School of Chemistry, The University of Manchester, Manchester M13 9PL, England, UK. E-mail: j.n.l.connor@manchester.ac.uk

Received
29th October 2013
, Accepted 31st January 2014

First published on 31st January 2014

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.

More recently (in 2008), state-of-the-art DCS measurements for the F + H_{2} reaction were reported by Wang et al.^{5} using quantum-state-selected crossed molecular beams. An analysis of the angular scattering by ourselves and Zhang^{6} 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 theory^{8} 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 Miller^{9} 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 approximation^{3} 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.

(1) |

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 = α_{0}. The two cases have the following properties, which we use later:

Case A – see Fig. 1(a)

For x_{1} < x_{2} so that α ≠ α_{0}

f_{1} > f_{2}, f_{1}′ = f_{2}′ = 0, |

f_{1}′′ < 0, f_{2}′′ > 0, f_{1}′′′ > 0, f_{2}′′′ > 0 |

For x_{1} = x_{2} = x_{0} so that α = α_{0}

f_{0}′ = 0, f_{0}′′ = 0, f_{0}′′′ > 0 |

Case B – see Fig. 1(b)

For x_{1} < x_{2} so that α ≠ α_{0}

f_{1} < f_{2}, f_{1}′ = f_{2}′ = 0, |

f_{1}′′ > 0, f_{2}′′ < 0, f_{1}′′′ < 0, f_{2}′′′ < 0 |

For x_{1} = x_{2} = x_{0} so that α = α_{0}

f_{0}′ = 0, f_{0}′′ = 0, f_{0}′′′ < 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 x_{1} = −x_{2}.

• 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.

(2) |

The second limiting case is when α ≈ α_{0} so that x_{1} ≈ x_{0} ≈ x_{2}. We make a third-order Taylor series expansion of f(α;x) at the point x_{0}

f(α;x) ≈ f(α;x_{0}) + f′(α;x_{0})(x − x_{0}) + ⅙f′′′(α;x_{0})(x − x_{0})^{3} | (3) |

(4) |

(5) |

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_{x0}^{3}f(α;x) = f(α;x_{0}) + f′(α;x_{0})(x − x_{0}) + ⅙f′′′(α;x_{0})(x − x_{0})^{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.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 x_{1} and x_{2}, similar to eqn (3) or (6) for the point x_{0}. This seems straightforward and physically reasonable, but Fig. 1(a) shows there is a serious problem. We see that the 3-jet at x_{2} is accurate close to x_{2}, but it possesses another stationary point, say u_{2} = u_{2}(α), which is quite different from x_{1}. This implies that the approximation to I(α) using the 3-jet at x_{2}, denoted I_{x2}(α), will be quite different from I(α), unless f(α;x) and j_{x2}^{3}f(α;x) are very similar as functions of x. Evidently we must eliminate the contribution from the additional stationary point u_{2}.

Similar remarks apply if the 3-jet at x_{1} is used – see Fig. 1(a); we must eliminate the contribution from the additional stationary point u_{1}. And similarly for Case B in Fig. 1(b). Eliminating the unwanted contributions from u_{1} and u_{2} is the essence of our derivation.

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

using f_{2}′ = 0. Then assuming g(x) ≈ g(x_{2}), we obtain an approximation denoted I_{x2}(α) to I(α), namely

where the integration limits have been kept at −∞ to +∞. Denoting the stationary phase points of the integrand in eqn (8) by x_{2} and u_{2} as in Fig. 1(a) with u_{2} ≤ x_{2}, we find

where

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 −b^{2}/(2a^{2})^{2/3}. Applying eqn (11) to eqn (8) gives

where

The result (12) evidently contains contributions from both the stationary points, x_{2} and u_{2}.

where

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

where

and

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 x_{2}, 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 u_{2}. We now confirm this suggestion by expressing f_{2}, A_{2} and f_{2}′′ in eqn (20) in terms of u_{2} rather than x_{2}.

which simplifies to

when eqn (10) is used and A_{2} is defined by eqn (13). Also from eqn (7) we have upon differentiation

and so for x = u_{2}

which simplifies to

upon using eqn (10). With the help of eqn (22) and (23), we can now write eqn (20) in the form

Eqn (24) is the simple stationary phase result at u_{2} for j_{x2}^{3}f(α;x), except for the presence of g_{2} ≡ g(x_{2}) rather than g(u_{2}). 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 x_{1} for I(α), as given by the first term on the rhs of eqn (2), namely

j_{x2}^{3}f(α;x) = f_{2} + ½f_{2}′′(x − x_{2})^{2} + ⅙f_{2}′′′(x − x_{2})^{3} | (7) |

(8) |

x_{2} (by construction) | (9) |

u_{2} = x_{2} − (2f_{2}′′/f_{2}′′′) | (10) |

Next we use the identity

(11) |

ς = b^{3}/(3a^{2}) |

(12) |

(13) |

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}

(14) |

ξ = ⅔z^{3/2} or equivalently, z = (ξ)^{2/3} | (15) |

where

(16) |

(17) |

Next we consider the limit where x_{2} and u_{2} are well separated, so we can use the following asymptotic approximations for the Hankel functions^{43} as x → ∞

(18) |

(19) |

(20) |

(21) |

From eqn (7), we have for x = u_{2}

j_{x2}^{3}f(α;u_{2}) = f_{2} + ½f_{2}′′(u_{2} − x_{2})^{2} + ⅙f_{2}′′′(u_{2} − x_{2})^{3} |

j_{x2}^{3}f(α;u_{2}) = f_{2} + 2A_{2} | (22) |

and so for x = u

which simplifies to

(23) |

(24) |

Note: The fact that f(α;x) has a maximum at x = x_{1} did not play an essential role in the argument that should be retained and discarded when f(α;x) is approximated by j_{x2}^{3}f(α;x). Thus if f(α;x) increases monotonically for x < x_{2}, we would still retain and discard when using j_{x2}^{3}f(α;x). Also, if in eqn (1), g(x) = constant, then trivially g(x_{2}) = g(u_{2}).

In the next section, we repeat the above analysis using the 3-jet at x_{1} rather than x_{2}.

IIC.3 Contribution to I(α) from a third-order Taylor expansion at the stationary point x_{1}.
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.

using f_{1}′ = 0. Then assuming g(x) ≈ g(x_{1}), we get for the approximation, I_{x1}(α) to I(α), the integral

with the integration limits kept at −∞ to +∞. Denoting the stationary phase points for the integrand of the integral (26) by x_{1} and u_{1} with x_{1} ≤ u_{1}[see Fig. 1(a)], we have

where

The result (27) evidently contains contributions from both x_{1} and u_{1}.

and

Inspection of eqn (31) and (32) shows that is the simple stationary phase result at x_{1}, given by the first term on the rhs of eqn (2). Hence should be associated with the simple stationary phase result at u_{1}. We confirm this suggestion by manipulations in which f_{1}, A_{1} and f_{1}′′ in eqn (32) are expressed in terms of u_{1} rather than x_{1} –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_{x1}^{3}f(α;x) at u_{1}, except for the presence of g_{1} ≡ g(x_{1}) rather than g(u_{1}). 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 x_{2} for I(α), which is given by the second term on the rhs of eqn (2), namely

The 3-jet at x_{1} is given by

j_{x1}^{3}f(α;x) = f_{1} − ½|f_{1}′′|(x − x_{1})^{2} + ⅙f_{1}′′′(x − x_{1})^{3} | (25) |

(26) |

u
_{1} = x_{1} + (2|f_{1}′′|/f_{1}′′′) and x_{1} (by construction). Next we apply the identity (11) to eqn (26) obtaining

(27) |

(28) |

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

(29) |

(30) |

Next we consider the limit where x_{1} and u_{1} 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

(31) |

(32) |

(33) |

Note: The fact that f(α;x) has a minimum at x = x_{2} did not play an essential role in the argument that should be retained and discarded when f(α;x) is approximated by j_{x1}^{3}f(α;x). Thus if f(α;x) decreases monotonically for x > x_{1}, we would still retain and discard when using j_{x1}^{3}f(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(x_{1}) = g(u_{1}).

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 u_{1} and u_{2} respectively, which arise when f(α;x) is approximated by its 3-jet at x_{1} and x_{2} respectively.

where

with

and

with

Remarks:

where Γ(•) is a gamma function. These results imply that the 2Hankel expression (34) becomes numerically unstable as A_{1} and A_{2} 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 A_{1} → 0 and A_{2} → 0.

The 2Hankel approximation, denoted I_{2H}(α), is thus given by

(34) |

(35) |

(36) |

(37) |

(38) |

(39) |

(40) |

• The 2Hankel approximation has the advantage that the two stationary phase points, x_{1} and x_{2}, appear separately in eqn (34)–(40).

• The 2Hankel approximation has the disadvantage that it involves the third derivatives, f_{1}′′′ and f_{2}′′′. 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 x_{1} and x_{2} 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)

(41) |

• In practical applications of the 2Hankel approximation, the third derivatives can change sign as α varies; in particular, as x_{1} and x_{2} 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 → α_{0}.
In the derivation of the 2Hankel approximation, the case when x_{1}and x_{2} 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 u_{1} and u_{2}.

Now from the cubic approximation (3), we obtain

_{2}

where eqn (43) and (44) have been used. We then find from the definitions (37) and (40) that

To obtain f_{1} and f_{2} we again use eqn (3) together with eqn (43) and (44). The results are

Finally, we make the approximation, g_{1} ≈ g_{0} ≈ g_{2}. Then we can substitute the above results for g_{1}, f_{1}, f_{1}′′, f_{2}′′′, A_{1} and g_{2}, f_{2}, f_{2}′′, f_{2}′′′, A_{2} 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 simplification

Eqn (45) is the transitional Airy approximation mentioned in Section IIB. Although it has been obtained from the 2Hankel formula assuming f_{0}′ ≤ 0 (two real stationary phase points), it is also valid for f_{0}′ > 0, provided −|f_{0}′| is replaced by f_{0}′ in eqn (45) – see eqn (4).

In this section, we examine the 2Hankel approximation for the limiting case α → α_{0} 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 g_{1}, f_{1}, f_{1}′′, f_{1}′′′, A_{1} and g_{2}, f_{2}, f_{2}′′, f_{2}′′′, A_{2} in terms of their values at x_{0} rather than at x_{1} and x_{2}. Now for α → α_{0}, we have x_{1} ≈ x_{0} ≈ x_{2} and to extract the limiting behaviour we approximate f(α;x) by its 3-jet at x_{0}, as given by eqn (3) or (6).

When the approximation (3) is valid for f(α;x), the two real stationary phase points, x_{1} ≤ x_{2}, are the roots of

(x − x_{0})^{2} = −2f_{0}′/f_{0}′′′ | (42) |

Since x_{1} and x_{2} are assumed to be real in eqn (42), we must have f_{0}′ ≤ 0 (also recall from Section IIA that f_{0}′′′ > 0). We can write for the two roots

(43) |

(44) |

f′′(α;x) = f_{0}′′′(x − x_{0}) |

f′′′(α;x) = f_{0}′′′ |

It follows that for x = x_{1}

where eqn (43) and (44) have been used. We then find from the definitions (37) and (40) that

To obtain f

f_{1} = f_{0} + A_{1} |

f_{2} = f_{0} − A_{2} |

(45) |

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

(46) |

We now introduce standard semiclassical approximations into the PWS (46) to convert it into an oscillating integral.^{3,4}Firstly, 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

(47) |

The Hilb approximation has an error, O(J

Thirdly, we express the Bessel function as a sum of two Hankel functions

J_{0}(x) = ½[H^{(1)}_{0}(x) + H^{(2)}_{0}(x)] | (48) |

f(θ_{R}) = f^{(−)}(θ_{R}) + f^{(+)}(θ_{R}) |

(49) |

(50) |

(51) |

(52) |

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

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 (J) for our application. This is done next.

For eqn (50) and (52), the stationary phase points are the roots of

(J) ∓ θ_{R} = 0 |

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

x = J, x_{1} = J_{1}, α = θ_{R} |

f(α;x) = arg (J) − (J + ½)θ_{R} + π/4 |

f′(α;x) = (J) − θ_{R} |

f′′(α;x) = ′(J) |

f′′′(α;x) = ′′(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

(53) |

(54) |

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

x = J, x_{1} = J_{2}, x_{2} = J_{3}, α = θ_{R} |

f(α;x) = arg (J) + (J + ½)θ_{R} − π/4 |

f′(α;x) = (J) + θ_{R} |

f′′(α;x) = ′(J) |

f′′′(α;x) = ′′(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

(55) |

(56) |

(57) |

(58) |

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 (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):

(61) |

(62) |

(63) |

x = J, x_{0} = J_{r}, α = θ_{R}, α_{0} = θ^{r}_{R} |

(64) |

The approximation (64) is equivalent to replacing (J) by its 2-jet at J_{r}, namely

j_{Jr}^{2}(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,^{4,6}eqn (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.

J_{i}(θ_{R}) → J_{i}(0) ≡ J_{g}, |

′(J_{i}(θ_{R})) → ′(J_{i}(0)) = ′(J_{g}) ≡ _{g}′ |

′′(J_{i}(θ_{R})) → ′′(J_{i}(0)) = ′′(J_{g}) ≡ _{g}′′ |

B_{i}(θ_{R}) → B_{i}(0) ≡ B_{g} |

(J_{i}(θ_{R})) → (J_{i}(0)) = (J_{g}) ≡ _{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}([J_{1}(θ_{R}) + ½]θ_{R}) + H^{(1)}_{0}([J_{2}(θ_{R}) + ½]θ_{R}) → 2J_{0}(0) = 2 |

We then obtain

(65) |

(66) |

which is recognized as the semiclassical transitional approximation (STA) at θ

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.

• 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 J_{1}, J_{2}, J_{3} 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, J_{2} and J_{3}; and the uniform glory theory (uBessel),^{3,7} which involves the pair J_{1} and J_{2}.

• The 6Hankel approximation (59) has the disadvantage that it involves third derivatives of arg (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 {_{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), |(J)| → 1 and arg (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.

where

(J) = aJ^{2} + bJ |

In addition

• N = scaling factor (dimensionless).

• (dimensionless because of the following definition).

• ε = ℏ^{2}/(2μs_{0}^{2}) (dimensions of energy).

• (dimensionless). In this equation, E is the total energy and B is the rotational constant for the triatomic complex. The term BJ(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 − BJ(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.

(67) |

Remark: the parameter values (67) are based on the “standard values” employed for the H + D_{2} reaction,^{10,11} which have B/ε = 0.12247 and W_{0}/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 W_{0}/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}.

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}

f(θ_{R}) = f^{(N)}(θ_{R}) + f^{(F)}(θ_{R}) | (68) |

(69) |

(70) |

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.

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

(72) |

(73) |

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,^{23,24}i.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 J_{0} + 1/2 = 34.9 |

For a broad rainbow, we also expect J_{r} ≈ Re J_{0},^{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).

• 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, x_{1}and x_{2}, followed by a discardation of the contributions from the unwanted stationary phase points, u_{2} and u_{1}. 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, J_{1}, J_{2}, J_{3}, appears separately in the 6Hankel expression, but has the disadvantage that third derivatives of arg (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 found^{6} the 6Hankel DCS to be less accurate than the uBessel DCS at small angles – probably because of the difficulty of accurately calculating the ′′(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}

(A1) |

A(α) = ½(f_{1} + f_{2}) |

ς(α) = [¾(f_{1} − f_{2})]^{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.

(A2) |

(A3) |

where the oddoid integral is defined in ref. 29

with b

Fig. 5(a) plots I_{2}(α) versus α for the range 0 ≤ α ≤ 15 with phase parameters of a_{3} = a_{5} = 5. The oscillatory nature of I_{2}(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 I_{2}(α) for α ≲ 3, until α ≈ 0.15, when the two curves cross. The uAiry approximation agrees closely with I_{2}(α) over the whole range of α.

It was mentioned in Section IIB that the tAiry approximation is also valid for negative α. Fig. 5(b) compares I_{tAiry}(α) with I_{2}(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 I_{2}(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 a_{5} = 0 in the phase (A3). Also from eqn (5) we have

The accurate value for I_{2}(α = 0) is 1.251, so the percentage error in I_{tAiry}(α = 0) is 4.3%.

with a

is an example of a swallowtail integral,

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