[Legenda]
[Contents of the thesis]
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[Legenda]
[Contents of the thesis]
[Comments on this module]

Theoretical methods: Differential cross section=A08-m3cii

Based on the potential curves, the deflection function of the system can be calculated Starting with the deflection function, the classical differential cross section for chemi-ionization is given by :

(E1)

where $\theta=\vert{\mit\Theta}\vert$. Figure MESO-m3c-defl-F1 of the deflection function shows that up to the rainbow angle the summation is over four impact parameters and over two impact parameters for larger scattering angles.

  

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. ]

The same impact parameters are important in the semiclassical treatment extended with the stationary phase approximation . Then the scattering amplitude $f(\theta)$ is built up again either by four or by two contributions:

(E2)

leading to the differential cross section

(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 the possible contributions $f_i(\theta)$ to $f(\theta)$
are then given by

$$f_\ar(\theta) \bigg[ \Big\vert\frac{\dr b_\ar}{\dr{\mit\Theta}}\Big\vert b_\ar P_{b_\ar}(1-P_{b_\ar})/\sin\theta \bigg]^{1/2}$$ =

$$\times\exp\big[ \ir(2\eta_{b_\ar}+kb_\ar\theta-\pi) \big]\equiv A\er^{\ir\alpha},{\rm }$$ (E4a)

$$f_\br(\theta) \bigg[ \Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert b_\br P_{b_\br}(1-P_{b_\br})/\sin\theta \bigg]^{1/2}$$ =

$$\times\exp\bigg[ \ir\Big(2\eta_{b_\br}+kb_\br\theta-\frac{1}{2}\pi\Big) \bigg]\equiv B\er^{\ir\beta},{\rm }$$ (E4b)

$$f_\mathrm{c}(\theta) \bigg[ \Big\vert\frac{\dr b_\mathrm{c}}{\dr{\mit\Theta}}\Big\vert b_\mathrm{c} P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})/\sin\theta \bigg]^{1/2}$$ =

$$\times\exp\big[ \ir(2\eta_{b_\mathrm{c}}+kb_\mathrm{c}\theta-\pi) \big]\equiv C\er^{\ir\gamma},{\rm }$$ (E4c)

$$f_\dr(\theta) \bigg[ \Big\vert\frac{\dr b_\dr}{\dr{\mit\Theta}}\Big\vert b_\dr P_{b_\dr}(1-P_{b_\dr})/\sin\theta \bigg]^{1/2}$$ =

$$\times\exp\big[ \ir(2\eta_{b_\dr}+kb_\dr\theta-\frac{1}{2}\pi) \big]\equiv D\er^{\ir\delta},{\rm }$$ (E4d)

$$f_\er(\theta) \bigg[ \Big\vert\frac{\dr b_\er}{\dr{\mit\Theta}}\Big\vert b_\er P_{b_\er}(1-P_{b_\er})/\sin\theta \bigg]^{1/2}$$ =

$$\times\exp\bigg[ \ir\Big(2\eta_{b_\er}+kb_\er\theta-\frac{1}{2}\pi\Big) \bigg]\equiv E\er^{\ir\varepsilon},$$ (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:

$$I(\theta) \big\vert A\er^{\ir\alpha}+C\er^{\ir\gamma}+D\er^{\ir\delta}+ E\er^{\ir\varepsilon}\big\vert^2$$ =

$$A^2+C^2+D^2+E^2+2AC\cos(\alpha-\gamma)$$ =

$$+2AD\cos(\alpha-\delta)+2AE\cos(\alpha-\varepsilon)$$

$$+2CD\cos(\gamma-\delta)+2CE\cos(\gamma-\vare)$$

$$+2DE\cos(\delta-\vare).$$ (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 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 by

$$\eta_b=-\frac{1}{2}k\int_b^\infty{\mit\Theta}\,\dr b$$

(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 , 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. .

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:

$${I(\theta)\sin(\theta)=\pi\bigg[ \bigg( b_\br\times P_{b_\br}(1-P_{b_\br})\times\Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert \bigg)^{1/2}}$$

$$+\bigg( b_\mathrm{c}\times P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})\t... ...{\dr{\mit\Theta}}\Big\vert \bigg)^{1/2} \bigg]^2\vert\xi\vert^{1/2}\A\ir(\xi)^2$$

$$+\pi\bigg[ \bigg( b_\br\times P_{b_\br}(1-P_{b_\br})\times\Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert \bigg)^{1/2}$$

$$-\bigg( b_\mathrm{c}\times P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})\t... ...Big\vert\frac{\dr b_\mathrm{c}}{\dr{\mit\Theta}}\Big\vert \bigg)^{1/2} \bigg]^2$$

$$\times\vert\xi\vert^{-1/2}\A\ir'(\xi)^2,$$ (E7)

where

$$\xi=-\big(3/4\big\vert\{2\eta_{b_\mathrm{c}}-2\eta_{b_\br}+k[b_\mathrm{c}-b_\br]\theta\}\big\vert \big)^{2/3}.$$

(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 . 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 . The collisional process discussed now does not fulfill completely those statements.