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Application of Heisenberg’s S Matrix Programme to rainbows, supernumerary rainbows and interference effects in the angular scattering of chemical reactions
We investigate the existence of primary rainbows, supernumerary rainbows and diffraction interference effects in the product differential cross sections (DCSs) of state-to-state chemical reactions. The rainbows can be “pronounced” or “hidden”. Our theoretical approach uses a “weak” version of Heisenberg’s scattering matrix program (wHSMP) introduced by X. Shan and J. N. L. Connor, 2011, Phys. Chem. Chem. Phys. 13 8392. This wHSMP uses four general physical principles for chemical reactions to suggest simple parametrized forms for the S matrix; it does not employ a potential energy surface. We use a realistic parametrization in which the modulus of the S matrix is the sum of a smooth-step function and a gaussian function; both are functions of the total angular momentum quantum number, J. We then vary the parameters in the modulus.The phase of the S matrix is a cubic polynomial in J, which is held fixed. We demonstrate for a Legendre partial wave series (PWS) the existence of primary rainbows and supernumerary rainbows (both pronounced and hidden) as well as diffraction interference effects, in reactive DCSs. We find that reactive rainbows can be complicated in their structure. We also analyse for five examples, the angular scattering using Nearside-Farside (NF) PWS theory, including resummations of the PWS. In addition, we apply full and NF asymptotic (semiclassical) rainbow theories to the PWS − in particular, the uniform Airy and transitional Airy approximations for the farside scattering. This lets us prove that structures in the DCSs are indeed primary rainbows, supernumerary rainbows as well as diffraction interference effects.
The experimental measurement, and theoretical calculation, of differential cross sections (DCSs) for state-to-state chemical reactions is a topic of fundamental significance in atomic and molecular physics. This is because DCSs contain detailed information on the mechanism and dynamics of a reaction. However, one major difficulty is that DCSs often exhibit complicated interference structures: the interpretation of these structures is then an important and difficult task.
One powerful approach for understanding structure in DCSs is the discovery and analysis of generic dynamical phenomena. Consider, for example, the recent discovery of rainbows in the DCSs of state-to-state reactions [1-4]. Broad farside rainbows of the Airy type (fold catastrophe) have been found in the DCSs of the F + H2 → FH(vf = 3) + H reaction[1, 2] (here vf is the quantum number for the final vibrational state). A uniform asymptotic (semiclassical) analysis of the large number of interfering partial waves contributing to the scattering amplitude proved that these rainbows are a generic phenomenon. They have also been describedas “hidden” rainbows  because their appearance in a DCS is quite different from the familiar rainbows of elastic scattering.[5-7] Indeed, it took 24 years before it was realized  that the DCSs measured by Neumark et al. for the F + H2 reaction contained an attractive (or farside) rainbow with unusual properties. Rainbows are also present  in the more recent experimental measurements of DCSs for the F + H2 reaction by Wang et al. 
The discovery of broad hidden rainbows in refs. 1, 2 raised the question: Can localized and pronounced rainbows occur in the DCSs of chemical reactions? E.g., are there rainbows possessing a prominent primary rainbow of the Airy type accompanied by supernumerary rainbows? A start to answering this question was made in ref. 4, where the answer was found to be “yes”.
The theoretical approach of ref. 4 employed a simple, yet powerful, procedure based on a “weak” version of Heisenberg’s scattering matrix program (wHSMP).[10-16] In particular, it employed a parametrized form for the scattering (S) matrix, which obeyed four general physical principles relevant to state-to-state chemical reactions.
The procedure we used previously to calculate DCSs varied the parameters in the phase of the S matrix, but kept its modulus fixed. This allowed us to obtain a partial understanding of rainbow phenomena in reactive DCSs. However our goal is to obtain a more comprehensive understanding. In this paper we extend and complement our earlier findings  by using a fixed phase for the parametrized S matrix but changing its modulus.
This paper is arranged as follows. Section 2 describes the general physical principles used in wHSMP for the design of an S matrix. Our approach can be summarized by the following scheme
initial states → S matrix → final states.
It does not employ a potential energy surface. Previously wHSMP has been used to obtain worthwhile and novel insights into structured DCSs for the H + D2 and F + H2 reactions.[14-16] In this paper, rather than consider a particular reaction, we generalize our previous work and consider a whole class of reactions as defined by their S matrix elements.
The parametrized S matrix form we use is described in section 3. The phase of the S matrix is a cubic polynomial in J, which is held fixed. Here J is the total angular momentum quantum number. The modulus of the S matrix is the sum of a realistic smooth-step function and a gaussian function, both dependent on J. We then vary the parameters in the modulus to understand better the properties of reactive DCSs.
In section 4, we summarise the theoretical techniques used in our calculations. They include Legendre partial wave series (PWS), and its first order resummation, for the scattering amplitude, as well as nearside-farside (NF) theories and the uniform Airy and transitional Airy asymptotic (semiclassical) approximations for the PWS. The use of the Airy approximations is important because it lets us prove the genericity of the rainbows, both hidden and pronounced.
Section 5 describes and discusses our results for five different examples; most results are displayed graphically. Our conclusions are in section 6.
The theory presented in this paper is for the following generic state-to-state chemical reaction:
where vi, ji, mi and vf, jf, mf are vibrational, rotational and helicity quantum numbers for the initial and final states, respectively. It is assumed that the reaction occurs at a fixed total energy, or equivalently a fixed initial translational wavenumber. Note that wHSMP is also valid for reactions with non-zero helicity quantum numbers, as described in ref. .
In addition, our theoretical development applies to reactions treated by many other approximate theoretical techniques such as reduced dimensionality methods,[17, 18] e.g., the rotating-line-umbrella approximation for reactions of the type
A + BCD3(si = 0) → AB(vf, jf , mf = 0) + CD3(sf= 0)
where the labels si = 0 and sf = 0 indicate that the molecules BCD3 and CD3, respectively, are in their ground states.
2. Weak version of Heisenberg’s S matrix program
Heisenberg’s original S matrix program (HSMP) is based on his fundamental insight that all the information needed to calculate collisional observables, such as DCSs, is contained within the S matrix.[10-13] No potential energy surface(s) is(are) necessary. It was originally hoped that general physical principles satisfied by the collision system, such as unitarity, causality and analyticity, could be used to determine the S matrix.[10-13] This hope has never been realised. To avoid this difficulty, we have introduced  and applied to the H + D2 and F + H2 reactions [14-16],a weak version of HSMP. Historical remarks on HSMP can be found in ref. 14. Note that we always work with a modified S matrix element in the following[1-4] denoted, .
The wHSMP employs four general physical principles relevant to state-to-state chemical reactions. They are used in our design strategy for the construction of the S matrix. The four general physical principles are:
(1)The forces responsible for chemical reactions are short ranged, of the order of 10−10 m. This implies as . In practice, there is a maximum value of J, denoted, beyond which partial waves make a negligible numerical contribution to the PWS. N.b., this principle excludes reactions that are asymptotically Coulombic, for which the PWS is divergent.
(2) Conservation of probability holds. This implies the S matrix is unitary with .
(3) Under semiclassical conditions, namely, , we can continue the set of J-dependent S matrix elements, denoted , to a smoothly varying function,, with simple properties. In our applications, is an analytic function, i.e., one of class in the notation used for the continuity and differentiability of functions.
(4) In the classical limit, we require a head-on collision to correspond to backward (or rebound) scattering of the products.
Notice that Principle (3) was used in a weaker form in refs.  and , where was allowed to be (i) a piecewise-continuous function (of class ), with simple properties for the pieces,or (ii) a piecewise-discontinuous function (of class ), again with simple properties for the pieces.
3. Parametrization of the S matrix
Keeping the wHSMP principles of section 2 in mind, we use the same polar parametrization as ref. , and write in the form
is positive or zero with for . We must choose so that for all . Also is a real scattering phase (or argument) and we evidently have, .
3A. Parametrization of
We use the same functional form for as previously. Its parameters are held fixed in the results section 5. The quantum deflection function, , is also important for the asymptotic theory of section 4. It is defined by
Note: the arg in Eq. (2) is not necessarily the principal value in order that the derivative be well defined.
As before, we choose to be a cubic polynomial, which we can write in three equivalent forms: 
(1) The first form is an exact Taylor expansion about :
where are four real phase parameters with . The corresponding quantum defection function is
. We choose two of the phase parameters in as follows:
- The PWS DCSs and asymptotic DCSs are independent of the value of , so we choose .
- Equation (4) shows that, , i.e., in a classical picture, a head-on collision determines . So remembering Principle 4 of the wHSMP, we choose .
Figure 1 shows a plot of and , both versus J, for the parameters used in our computations of section 5. Also marked on the graphs are quantities important for the asymptotic analysis in the next section. In particular, the real stationary points, , , , as well as the rainbow angular momentum variable, , and the glory angular momentum variables, and .
(2) For the second form, we use instead of a2 and a3, the more physically meaningful variables, (i) , the rainbow angular momentum variable which satisfies, , and (ii) , the rainbow angle which satisfies . Illustrations of these quantities are given in Figure 1. We have 
The cubic phase can then be written in the representation as
(3) The third form makes an exact Taylor expansion of in Eq. (5) about the point . We have
Note that is always negative and is always positive; this is because , and are all positive in our computations.
3B. Parametrization of
We write as the sum of two contributions:
In Eq. (7), is a smooth step function given by [4, 14]
where is as a normalization factor, acts as a “diffuseness” parameter in J space, and is the “cut off” value of J. The quantities,, and are positive, although not necessarily integers.
The second term in Eq. (7) is a gaussian (or bell-shaped) function of J defined by
In Eq. (9), acts as a normalization factor, determines the width in J space and determines the position of the maximum of the gaussian curve in J space. In addition, , and are positive, although not necessarily integers.
Figure 2 (left-hand column) shows plots of versus J for the five parametrizations, A – E, used in section 5.
4. Theoretical methods
We use the same theoretical methods and working equations as in our companion paper ; so in this section we briefly summarize the techniques used, highlight important aspects of the theory and establish our notation.
4A. Partial Wave Theory
Our starting point is the Legendre PWS for the scattering amplitude,, given by
where k is the initial translational wavenumber, and is a Legendre polynomial of degree J. Also, is the reactive scattering angle, which is defined as the angle between the incoming A atom and the outgoing AB molecule in the centre-of-mass collision system. The dependence of , (and related quantities) on has been omitted for notational simplicity from Eq. (10), and below, as has the label from k. In addition, we omit the channel label “reactive” from etc., so our results are not relevant for the elastic/inelastic channel. The full DCS is then given by
We shall see that the full DCS often exhibits complicated oscillatory structures. A powerful method to help understand these structures is an exact decomposition of into the sum of nearside and farside subamplitudes.[19-21] We use the Fuller decomposition  and write
where the are travelling Legendre functions defined by (for )
and is a Legendre function of the second kind of degree J. The corresponding N, F DCSs are defined by (for )
The fundamental identity for full and N,F DCSs is [19-21]
Although the decomposition (12) – (14) is mathematically exact, there is no guarantee that it is physically meaningful. However it is known that a resummation of the PWS (10) can significantly improve the physical effectiveness of a NF decomposition; this process is known as “cleaning”. [4,14-17,23-27] A detailed account of resummation theory for a Legendre PWS has been given by Totenhofer et al.
A recent review of NF theory has been presented by Child, while shorter reviews can be found in refs.29-32. Recent applications of NF techniques have provided insights into the dynamics of: the CH4 + Cl → CH3 + HCl reaction,  the SN2 Cl¯ + CH3Br → ClCH3 + Br¯ reaction,  four complex-mode reactions  and the forward- angle scattering of the H + HD → H2 + D reaction.
4B. Asymptotic (semiclassical) rainbow representations
Here we outline the asymptotic (semiclassical ≡ SC) techniques that we use for the N and F scattering. We write the SC full scattering amplitude as
The corresponding full SC DCS is
and the N and F SC DCSs are given by
Note we use the superscripts,, to label the N and F SC subamplitudes respectively; this is to avoid confusion with the corresponding PWS subamplitudes. The SC theory uses the continuation of , with J = 0, 1, 2,…, to real values of J, which we denote by , i.e., the S matrix elements are now considered to be a continuous function of the total angular momentum variable.
Firstly, we consider the F scattering, for which the stationary phase condition is 
Figure 1(b) shows Eq. (18) has two real roots for a given value of , provided ; this corresponds to the bright side of the rainbow. The stationary points are denoted and , which coalesce at . In this angular region we use the uniform Airy approximation, which is denoted in a systematic notation by SC/F/uAiry, or for short, uAiry. It allows for the proximity of and as . When , the uAiry approximation becomes numerically indeterminate. On the dark side of the rainbow where , the roots of Eq. (18) become complex valued, which are more awkward to handle.
To avoid these difficulties, we use the transitional Airy approximation for, which is denoted SC/F/tAiry, or tAiry for short. The tAiry approximation only depends on the properties of at .
Secondly, we consider the N scattering, for which the stationary phase condition is 
Figure 1(b) shows the stationary points, and , are well separated from each other, so we can use the primitive semiclassical approximation at each point and write 
In a systematic notation, Eq. (20) is written, SC/N/PSA, or PSA for short; with PSA1 and PSA4 being used for the individual SC sub-subamplitudes. In practice, we shall see that only one of our parametrizations, (namely D) in section 5 needs the PSA at (i.e., PSA4) because the are negligible for the other parametrizations.
The full asymptotic (SC) DCS is then given by
We also note that:
- Comparing the SC N, F DCSs with the PWS N, F DCSs provides an important test of the physical reliability of the Fuller PWS N, F decomposition in Eqs. (12) – (14), or the corresponding equations starting with the resummed PWS.
- It is also useful to have the explicit formulae for the stationary points. They are 
Equations (21) and (24) are the real N roots of , whilst Eqs. (22) and (23) are the real F roots of .
- There is a simple formula for the period of the NF diffraction oscillations in the DCS when just and contribute to the scattering, as occurs in forward glory scattering. This formula is given by Eq. (50) of ref. 26, namely
where is the forward glory angular momentum variable shown in Figure 1. Notice that Eq. (25) is independent of . Since we always have in our calculations, , Eq. (25) yields, . Conversely, if we find a PWS DCS where varies with , and is much different from the simple result of Eq. (25), then this is a hint that other stationary points are contributing to the SC scattering.
5. Results and discussion
We recall from Section 3, that contains 10 parameters. They are: , Jcut, dcut in ; ,, in ; and in the cubic phase. In the calculations that follow, we always keep the parameters in fixed, with the values, , , , . These values were also used previously by us for one of our phases in the varying-the-phase calculations, but keeping the modulus fixed (see Table 1 of ref. ).
Table 1 reports the values of the six parameters in and employed in our computations. We report results for five sets of parameters: the resulting parametrizations are denoted A, B, C, D, E. Note that parametrization A contains only and the phase, whilst parametrization E comprises and the phase only. Sometimes, we will add a subscript, A, B, C, D, E, to or to distinguish different parametrizations, e.g., .
Plots of (or equivalently, ) versus J for the five parametrizations are shown in the left-hand column of Figure 2. The shapes of the curves are typical of many state-to-state chemical reactions. For example, in Figure 2a, the smooth decline in was used previously for the H + D2 reaction , whereas in Figure 2b the peak at larger J is typical of the F + H2 and CH4 + Cl reactions [16,17]. Table 1 shows that . However, we will see that this is not true for the corresponding DCSs displayed below, because of interference between the A and E subamplitudes.
The right-hand column of Figure 2 displays the corresponding graphs of versus J. These graphs are useful because they show the magnitude of the numerical contribution of the modified S matrix to the DCS; e.g., Figure 2d, shows that extends into the second nearside region.
As well as the five parametrizations A–E, we have carried calculations for many other parametrizations, F, G, H,… They show similar results to the five parametrizations we discuss. In all cases we have performed PWS calculations for resummations orders [4,14-17, 23-27] of r = 0 (no resummation, i.e., Eq. (10)), and r = 1,2,3. We find the biggest cleaning effect on the PWS N, F DCSs occurs on going from r = 0 to r = 1. Additional resummations to r = 2 and r = 3 have a smaller cleaning effect. Thus in Sections 5A-5E, we shall only show PWS N, F DCS results for r = 1.
We have also computed values of the full and N, F local angular momenta [23-25,36], , for the SC theory as well as for r = 0,1,2,3, for the PWS theory. They are not shown, because the LAM results are consistent with our DCS results.
Table 1. Values of the S matrix parameters. Every parametrization has the same cubic phase with , , and .
In the following, we shall show dimensionless DCSs (denoted dDCSs), defined as , in our logarithmic plots. In every case we have also made linear plots. We find logarithmic plots are the more meaningful since they allow the reader to see clearly the contribution of the farside scattering to the dark side of rainbows.
Our figures adopt the following colour conventions for the dDCS curves in the versus graphs:
- Full PWS: solid black.
- N PWS/r=1: solid red.
- F PWS/r=1: solid blue.
- Full SC: solid green.
- N SC/PSA: dashed yellow.
- F SC/uAiry: solid purple.
- F SC/tAiry: dashed purple.
On every graph, solid pink arrows mark the value of the rainbow angle, .
5A. dDCSs for Parametrization A
Figure 3 displays dDCSs for the smooth-step function, of Figure 2a, in the form of logarithmic plots for . The PWS curve in Figure 3a shows the characteristic shape of a primary rainbow for , accompanied by diffraction (high frequency) oscillations extending out to . The PWS N, F dDCSs show that the diffraction oscillations arise from interference of the corresponding subamplitudes using Eq. (15) generalized to r = 1.  The F contribution dominates the scattering for , with a change in mechanism at from F-dominant to N-dominant.
It is difficult to decide from the full PWS dDCS in Figure 3a whether there are supernumerary rainbows present or not? Here an examination of the PWS F dDCS is very helpful, as it is seen to possess several weak oscillations. These suggest there are indeed weak supernumerary rainbows in the full PWS dDCS.
Figure 3b compares the SC N, F dDCSs with the corresponding PWS N, F dDCSs. We see that the SC and PWS curves agree closely. This is an important result as it demonstrates that the Fuller NF decomposition of Eqs. (12) − (14) is physically meaningful. In particular, it proves that the oscillations in the PWS F dDCS correspond to a primary rainbow and weak supernumeraries. Note that our SC N analysis of Eq. (20) uses only the stationary point, since Figure 2a shows that and are negligible.
The full SC and PWS dDCSs are compared in Figure 3c. We see very good agreement between the two curves. Thus we have achieved a comprehensive physical and mathematical (generic) understanding of the rainbow and diffraction effects for parametrization A. Finally we note that the period of the full PWS interference oscillations is 5.9°, 5.3°, 4.8° close to = 30°,50°,70° respectively. This tells us that there are more stationary points than and contributing to the forward scattering, since Eq. (25) predicts 5.9°, independent of .
5B. dDCSs for Parametrization B
Next we consider parametrization B , for which and are plotted in Figure 2b, left- and right-hand columns, respectively. We see that the curve, which possesses a pronounced peak at , is qualitatively different from .
The corresponding PWS and SC dDCSs are displayed in Figure 4 as logarithmic plots for . The full PWS dDCS in Figure 4a is dominated by diffraction oscillations which arise from N-F interference (see the N and F curves in Figures 4a and 4b). Now there are no obvious rainbows or supernumerary rainbows in the full PWS dDCS. However the F PWS and F SC dDCSs reveal the presence of a weak primary rainbow accompanied by several weak supernumeraries. As in the case for parametrization A, our SC N calculation uses only the stationary point, since Figure 2b shows that and are negligible.
Our SC N and F analysis is confirmed in Figure 4c for the full dDCS, which shows very good agreement for the PWS and SC curves. In particular we have shown the presence of hidden primary and supernumerary rainbows for parametrization B.
Also note that the period of the full PWS interference oscillations is 5.8°, 5.3°, 4.8° for 30°, 50°, 70° respectively. This again tells us that there are more stationary points than and contributing to the forward scattering, since Eq. (25) predicts 5.9°, independent of .
5C. dDCSs for Parametrization C
Figure 2c shows the plot of versus J. Compared to , we see the maximum of has moved to . Figure 5 displays the PWS and SC dDCSs as logarithmic plots for . Unlike the previous two examples, the full PWS dDCS in Figure 5a displays a prominent primary rainbow, together with noticeable supernumeraries, accompanied by N-F diffraction oscillations. The rainbows are clearly visible in the F PWS and F SC plots in Figure 5b. As previously, our SC N calculation uses only the stationary point; Figure 2c shows that and are negligible.
Our SC N and F computations for the pronounced primary and supernumerary rainbows in the full dDCS are confirmed in Figure 5c; there is very good agreement between the PWS and SC curves.
Finally we note that the period of the PWS interference oscillations is 5.4°, 5.2°, 4.4° close to = 30°,50°,70° respectively. Once again, this suggests that there are more stationary points than and contributing to the forward scattering, since Eq. (25) gives 5.9°, independent of .
5D. dDCSs for Parametrization D
The graph of versus J in Figure 2d exhibits two new features. (i) There is a local minimum at , in addition to the maximum at , (ii) and are no longer negligible, i.e., the first and second N regions contribute to the scattering.
Next we examine Figure 6, which shows logarithmic plots of the dDCSs for . We see the PWS dDCS in Figure 6a possesses oscillations out to . There is visible a primary rainbow at and a supernumerary rainbow at . At smaller angles, the dDCS exhibits rapidly-varying irregular oscillations. How are we to understand this complicated dDCS?
Firstly, we examine the F PWS dDCS and F SC dDCS in Figure 6b, which are seen to agree closely. We observe that the F scattering possesses a pronounced primary rainbow and five supernumeraries, which extend down to . Secondly, we inspect the N PWS and N SC curves in Figure 6b (which are also in close agreement). They possess regular oscillations extending out to , whereas previously in Figures 3-5, the N curves are monotonic. The SC analysis reveals that the oscillations arise from the interference of the sub-subamplitudes associated with the stationary points, and , i.e., the N scattering receives contributions from the first and second N regions.
The full PWS dDCSs and full SC dDCSs are compared in Figure 6c; they are seen to agree closely. It is remarkable that the SC analysis provides a relatively simple explanation for the complicated PWS dDCS in terms of the four interfering stationary points, , , , . The simple formula (25) for the period of the oscillations no longer applies.
5E. dDCSs for Parametrization E
Finally we consider parametrization E, for which is a pure-gaussian function of J. Figure 2e shows that both and are very small at . This tells us that the sub-subamplitude for the scattering from will be tiny. We then expect that the resulting rainbow in the dDCS will be miniscule, and in practice we cannot see it, even for . This is confirmed by inspection of Figure 7a where the plot of the PWS dDCS versus just shows diffraction oscillations at small angles, .
Figure 7b shows that the F PWS dDCS and the F SC dDCS do not exhibit rainbow oscillations. Since the N PWS dDCS and N SC dDCS agree closely, and are monotonic, we deduce that the small-angle diffraction oscillations arise from N-F interference of the corresponding subamplitudes. Figure 7c shows that the full PWS and SC dDCSs are in good agreement.
We note that the period of the PWS interference oscillations is 5.8°, 5.8°, 5.8° for 10°,20°,30° respectively, which is very close to the result from Eq. (25) of 5.9° (independent of ), i.e., we expect just the stationary points and to contribute to the scattering in this angular range, which is the case.
This paper has investigated the occurrence of pronounced and hidden rainbows in the state-to-state product dDCSs of chemical reactions. We used a novel theoretical approach, wHSMP; it does not employ a potential energy surface. Rather wHSMP uses four general physical principles for chemical reactions to suggest simple parametrized forms for the S matrix.
We used a realistic parametrization in which the modulus of the S matrix is the sum of a smooth-step function and a gaussian function, both functions of J; whilst the phase of the S matrix is a cubic polynomial in J. We varied the modulus and held the phase fixed. Our calculations complement and extend earlier work  in which the modulus is held fixed and the phase varied.
We then analysed five Legendre PWS dDCSs using NF PWS theory, including resummations of the PWS for r = 0, 1, 2, 3. We also applied full and NF SC (asymptotic) rainbow theories − in particular, the uAiry and tAiry approximations for the farside scattering. In every case we proved the existence of primary rainbows, supernumerary rainbows (both hidden and pronounced) and other interference effects. We obtained a comprehensive physical and mathematical (generic) understanding of structures in the angular scattering in terms of contributions from (up to) four real stationary points, , , , . In addition, we used the SC theory to demonstrate that the Fuller NF decomposition of the PWS is physically meaningful, as well as being mathematically exact.
More generally, it is well established that rainbows occur in many different wave phenomena.[37-41] The present calculations, and our earlier work, [1-4] show that this is also true for reactive DCSs.
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Figure 1. Black solid curves with and : (a) versus J, (b) quantum deflection function versus J. In (b), The red dashed lines and arrows indicate , as well as and for the nearside scattering. The blue dashed lines and arrows indicate , as well as and , for the farside scattering. In (a) and (b), the pink dashed lines and arrows indicate the rainbow angular momentum variable,, which is located at the minimum of the curve and satisfies, , where is the rainbow angle. The orange arrows indicate the two glory angular momentum variables, and , which satisfy for i = 1, 2.
Figure 2. Plots of versus J (left hand column) and versus J (right hand column) for parametrizations A–E. The pink arrows and dashed lines indicate the rainbow angular momentum variable, . The orange arrows and dashed lines indicate the two glory angular momentum variables, and .
Figure 3. Plots of versus for parametrization A: (a) full PWS (black curve), N(r=1) PWS (red curve), F(r=1) PWS (blue curve). (b) N(r=1) PWS (red curve), F(r=1) PWS (blue curve), SC/N/PSA (dashed yellow curve), SC/F/uAiry (solid purple curve), SC/F/tAiry (dashed purple curve). (c) full PWS (black curve), full SC (green curve). The pink arrows denote the rainbow angle at .
Figure 4. Plots of versus for parametrization B: (a) full PWS (black curve), N(r=1) PWS (red curve), F(r=1) PWS (blue curve). (b) N(r=1) PWS (red curve), F(r=1) PWS (blue curve), SC/N/PSA (dashed yellow curve), SC/F/uAiry (solid purple curve), SC/F/tAiry (dashed purple curve). (c) full PWS (black curve), full SC (green curve). The pink arrows denote the rainbow angle at .
Figure 5. Plots of versus for parametrization C: (a) full PWS (black curve), N(r=1) PWS (red curve), F(r=1) PWS (blue curve). (b) N(r=1) PWS (red curve), F(r=1) PWS (blue curve), SC/N/PSA (dashed yellow curve), SC/F/uAiry (solid purple curve), SC/F/tAiry (dashed purple curve). (c) full PWS (black curve), full SC (green curve). The pink arrows denote the rainbow angle at .
Figure 6. Plots of versus for parametrization D: (a) full PWS (black curve), N(r=1) PWS (red curve), F(r=1) PWS (blue curve). (b) N(r=1) PWS (red curve), F(r=1) PWS (blue curve), SC/N/PSA for branches 1 and 4 (dashed yellow curve), SC/F/uAiry (solid purple curve), SC/F/tAiry (dashed purple curve). (c) full PWS (black curve), full SC (green curve). The pink arrows denote the rainbow angle at .
Figure 7. Plots of versus for parametrization E: (a) full PWS (black curve), N(r=1) PWS (red curve), F(r=1) PWS (blue curve). (b) N(r=1) PWS (red curve), F(r=1) PWS (blue curve), SC/N/PSA (dashed yellow curve), SC/F/uAiry (solid purple curve), SC/F/tAiry (dashed purple curve). (c) full PWS (black curve), full SC (green curve). The pink arrows denote the rainbow angle at .