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Theoretical methods: Differential cross section=A08-m3cii

Based on the potential curves, the deflection function [The definition of deflection function is given in a mesoscopic module (link type: defined in/wider range/project'; target MESO-m3c-defl#\theta(b)] of the system can be calculated [To details on the method to calculate the deflection function (link type: 'is detailed in/project'; target: A06-m*] (assuming the impact-parameter approximation) Starting with the deflection function, the classical differential cross section for chemi-ionization is given by [The formula is copied from a mesoscopic module (link type: 'input from/wider range/project'; target: MESO-m3c-defl#I(\theta)]:
I(th)= (E1)
where $\theta=\vert{\mit\Theta}\vert$. Figure MESO-m3c-defl-F1 of the deflection function
[The figure is copied from a mesoscopic module (link type: 'input from/project/wider range'; target MESO-m3c-defl#\theta(b)] shows that up to the rainbow angle [The notion of a rainbow angle is clarified by Bernstein (link type: 'is detailed in/clarified in/wider range/external'; target: Bernstein] the summation is over four impact parameters and over two impact parameters for larger scattering angles.

  
[To the FULL figure] Figure A08-m3cii-F1 = MESO-m3c-defl-F1: Typical deflection curves for chemi-ionization scattering (CM system). The two curves due to ionic and covalent scattering are connected $b\approx R_\c$. Because of the several interference features, the ionic curve is split up into b, c and e branches, the covalent curve into a and d branches. [Figure copied from MESO-m3c-defl]]

The same impact parameters are important in the semiclassical treatment extended with the stationary phase approximation [The semi-classical treatment is elucidated by Ford and Wheeler (link type: 'elucidated in/is detailed in/external '; target: R21-m3]. Then the scattering amplitude $f(\theta)$ is built up again either by four or by two contributions:
f(th)= (E2)
leading to the differential cross section
I(th)=f(\theta)^2 (E3)
The contributions to the differential cross section are to be approximated. We use the lowest order stationary-phase approximation and the uniform approximation, which in their turn both require the JWKB approximation.

Lowest-order stationary-phase approximation

In the lowest-order stationary-phase approximation [NOT AVAILABLE This approximation has to be elucidated elsewhere, in some macroscopic module Theoretical methods (link type: `IS ELUCIDATED IN/is detailed in/wider range'; target: some MACRO-m3c)] the possible contributions $f_i(\theta)$ to $f(\theta)$
are then given by
$\displaystyle f_\ar(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\ar}{\dr{\mit\Theta}}\Big\vert b_\ar P_{b_\ar}(1-P_{b_\ar})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\ar}+kb_\ar\theta-\pi) \big]\equiv A\er^{\ir\alpha},{\rm }$ (E4a)
$\displaystyle f_\br(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert b_\br P_{b_\br}(1-P_{b_\br})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\br}+kb_\br\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
B\er^{\ir\beta},{\rm }$ (E4b)
$\displaystyle f_\mathrm{c}(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\mathrm{c}}{\dr{\mit\Theta}}\Big\vert b_\mathrm{c}
P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\mathrm{c}}+kb_\mathrm{c}\theta-\pi) \big]\equiv
C\er^{\ir\gamma},{\rm }$ (E4c)
$\displaystyle f_\dr(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\dr}{\dr{\mit\Theta}}\Big\vert b_\dr
P_{b_\dr}(1-P_{b_\dr})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\dr}+kb_\dr\theta-\frac{1}{2}\pi) \big]\equiv
D\er^{\ir\delta},{\rm }$ (E4d)
$\displaystyle f_\er(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\er}{\dr{\mit\Theta}}\Big\vert b_\er
P_{b_\er}(1-P_{b_\er})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\er}+kb_\er\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
E\er^{\ir\varepsilon},$ (E4e)

where $\eta_\br$ indicates the phase shift and k is given by $k=(2\muE_\ir)^{1/2}$.

For a certain scattering angle $\theta$ where four contributions form the cross section (for instance the contributions from the branches a, c, d and e), the differential cross section is given by:
$\displaystyle I(\theta)$ = $\displaystyle \big\vert A\er^{\ir\alpha}+C\er^{\ir\gamma}+D\er^{\ir\delta}+
E\er^{\ir\varepsilon}\big\vert^2$  
  = $\displaystyle A^2+C^2+D^2+E^2+2AC\cos(\alpha-\gamma)$  
    $\displaystyle +2AD\cos(\alpha-\delta)+2AE\cos(\alpha-\varepsilon)$  
    $\displaystyle +2CD\cos(\gamma-\delta)+2CE\cos(\gamma-\vare)$  
    $\displaystyle +2DE\cos(\delta-\vare).$ (E5)

The lowest-order stationary-phase approximation is not applicable to the entire range, as it fails in giving a good description of the rainbow. Also the usual Airy description
[This description is elucidated by Airy (link type: `IS ELUCIDATED IN/ is detailed in/external'; target: Rf(A08)22-m*)] is not very suitable because the shape of the deflection curve in the rainbow-angle region deviates too much from the supposed parabolic behaviour.

JWKB approximation

The phase shift $\eta_\br$ in equations for  and  is given in the JWKB approximation [This approximation is elucidated by Bernstein link type: `IS ELUCIDATED IN/is detailed in/wider range/external'; target: Bernstein)] by
\begin{displaymath}\eta_b=-\frac{1}{2}k\int_b^\infty{\mit\Theta}\,\dr b
\end{displaymath} (E6)
so making necessary an integration over the deflection curve.

However, the first problem in applying this approximation is that in the present case there does not exist a deflection curve for collision parameters from infinity down to the largest value of b that gives rise to chemi-ionization $(b_{\max}\approx R_\mathrm{c})$. Fortunately, the equation for  shows that for the calculation of the differential cross section we only need the phase-shift differences of the relevant contributions. In our calculations therefore we have taken arbitrarily $\eta_{b_{\max}}\equiv0$.

A second point is that due to the coulombic nature of the outgoing channel, the argument of the sine of the wave function contains a term with ln(2kR) , and therefore the validity of the semiclassical approximation is questionable. However, as in the asymptotic wave functions the phase shifts contain for all deflection branches this ln(2kR) term; this term cancels out for phase-shift differences and the problem is circumvented. Thus, the JWKB approximation can be applied in the present case.

Uniform approximation

The approximation that can be applied to the area of the rainbow angle is the uniform approximation of rainbow scattering [The uniform approximation is elucidated and its applicability argued by Berry (link type: `IS ELUCIDATED IN/IS ARGUED IN/detailed in/external'; target: Rf(A08)23-*)], in which the actual shape of the deflection function is used. In our case it turns out that the uniform approximation is only necessary to describe the primary rainbow because the supernumerary rainbows calculated in this way completely coincide with the ones calculated by the stationary-phase method. [These rainbows are calculated in the Interpretation (link type: `depends on'; target: A08-m5bii1].

In the uniform approximation of rainbow scattering the resulting formula to calculate the rainbow structure for $\theta<\theta_{\mathrm{class.
rainbow}}$, being the interference of the b and c branches, reads:
$\displaystyle {I(\theta)\sin(\theta)=\pi\bigg[ \bigg( b_\br\times
P_{b_\br}(1-P_{b_\br})\times\Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2}}$
    $\displaystyle +\bigg( b_\mathrm{c}\times
P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})\t...
...{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2} \bigg]^2\vert\xi\vert^{1/2}\A\ir(\xi)^2$  
    $\displaystyle +\pi\bigg[ \bigg( b_\br\times
P_{b_\br}(1-P_{b_\br})\times\Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2}$  
    $\displaystyle -\bigg( b_\mathrm{c}\times
P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})\t...
...Big\vert\frac{\dr b_\mathrm{c}}{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2} \bigg]^2$  
    $\displaystyle \times\vert\xi\vert^{-1/2}\A\ir'(\xi)^2,$ (E7)

where
\begin{displaymath}\xi=-\big(3/4\big\vert\{2\eta_{b_\mathrm{c}}-2\eta_{b_\br}+k[b_\mathrm{c}-b_\br]\theta\}\big\vert
\big)^{2/3}.
\end{displaymath} (E8)
The Airy functions Ai and Ai' have been replaced by their asymptotic approximations.

Applicability of this theoretical method

The semiclassical differential cross section can be calculated using the stationary phase approximation, the uniform approximation of rainbow scattering and JWKB phase shifts.

The application of the semiclassical approximation on two-state collisional processes and the Landau-Zener theory have been discussed recently in a series of papers by Delos and Thorson [(link type: `IS ARGUED IN/external'; target: Rf(A08)6-m*]. As far as the applicability of the semiclassical approximation is concerned, they conclude with four statements about kinetic energy and potential states of the collision partners [To these statements (link type: 'IS ELABORATED IN/external/clarified in'; target: Rf(A08)6-m*] . The collisional process discussed now does not fulfill completely those statements.