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Quantitative interpretation: potential and deflection function =A08-m5bi

We determine the potential parameters that govern the chemi-ionization reaction Na + I Na+ + I according to the atom-atom model for ion-pair formation in molecular collisions [The model is given in Theoretical methods (link type: 'depends on/is detailed in'; to: A08-m3ci)], [More details in the mesoscopic Theoretical methods (link type: 'project/wider range/depends on/is detailed in'; to: MESO-m3c-mod)]. From the potential parameters, the theoretical cross section of the chemi-ionization reaction can be calculated via the deflection function. We determine the potential parameters, and simultaneously the deflection curve, by fitting the calculated cross section (that is based on assumed potential parameters) with the experimental cross sections.

We now restrict ourselves as much as possible to the determination of some potential parameters and to the comparison of measurements and calculations used for that purpose. The calculated differential cross section and its discrepancies with the measurements will be treated in module Quantitative interpretation A08-m5bii [theoretical cross section given and discussed in Quantitative interpretation (link type: 'is detailed in/used in'; to A09-m5bii1)].

[To the FULL figure] Figure A08-m5bi-F1: Polar differential cross section for chemi-ionization (CM system) at E=13.1 eV. (b) Differential cross section calculated with the stationary-phase approximation and uniform rainbow approximation showing separated the long-wavelength interference structures due to a + c ($\tau=0\to65$, full curve), b + c ($\tau=65\to250$, full curve) and d + e ($\tau=0\to 2300$, dashed curve) interferences. (c) Full bars indicate the measured maxima of the interference structure on the differential cross section due to net-attractive scattering. Dashed bars indicate the maxima due to net-repulsive scattering
Fig. A08-m5bi-F1b [figure given in Quantitative interpretation (link type: 'input from'; to A09-m5bii1)] shows the differential cross section with simplified interference structure. An additional simplification in Fig. A08-m5bi-F1b is the separate reproduction of the attractive and repulsive scattering contribution as though they could be distinguished. Because the origin of the oscillatory features of the calculated cross sections can be seen easily from Fig. A08-m5bi-F1b and the latter figure is more easily related to the deflection function and the potential curves, Fig. A08-m5bi-F1b is more suitable to fit the interference structure of the calculated and measured differential cross section by adjustment of the potential parameters. Moreover, convolution of the differential cross section of the complete calculated cross section, which is given in Fig. A08-m5bi-F1a [figure given in Quantitative interpretation (link type: 'input from'; to: A08-m5bii1)] , and Fig. A08-m5bi-F1b even with the smallest angular resolution of the detector of 0.3º fwhm will lead to equal results.

By comparison of measurements and calculations, the missing parameters of the covalent potential curve and the value of H12 can be estimated. For that purpose are very important particularly the wavelengths of the rainbow and Stueckelberg oscillations [oscillations are identified in Qualitative interpretation (link type: 'is detailed in'; to A08-m5a)] , which result from the semi-classical interference of different contributions to the scattering angle [semi-classical model given in Theoretical methods (link type: 'depends on'/is detailed in'; to: A08-m3cii)].

For a fitting procedure of the potential curves to the measurements, it is very helpful that the interference wavelengths can be estimated directly from the deflection curves. The oscillations are generated by the cosine of the Eqs. (11) [equation given in Qualitative interpretation (link type:'input from'; to A08-m5a)]    
 \begin{displaymath}
I(\MT<0)=A^2+C^2+2AC\cos(\alpha-\gamma){\rm
}
\end{displaymath} (-2)
or  
 \begin{displaymath}
I(\MT<0)=B^2+C^2+2BC\cos(\beta-\gamma){\rm
}
\end{displaymath} (-1)
and  
 \begin{displaymath}
I(\MT\gt)=D^2+E^2+2DE\cos(\delta-\vare).

\end{displaymath} (0)
[To the FULL figure] Figure A08-m5bi-F2: Deflection curves for chemi-ionization scattering (CM system) at Ei=13.1 eV. [Copied to the Treated results (link type: 'output to'; target: A08-m4bii1)]
The difference in the cosine argument for two neighbouring scattering angles $\theta_1$ and $\theta_2$ can be shown easily, with the help of
Fig. A08-m5bi-F2 and Eqs. ([*]) for the contributions to the scattering angle in lowest-order stationary-phase approximation .
     
 
$\displaystyle f_\ar(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\ar}{\dr{\mit\Theta}}\Big\vert b_\ar P_{b_\ar}(1-P_{b_\ar})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\ar}+kb_\ar\theta-\pi) \big]\equiv A\er^{\ir\alpha},{\rm }$
$\displaystyle f_\br(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert b_\br P_{b_\br}(1-P_{b_\br})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\br}+kb_\br\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
B\er^{\ir\beta},{\rm }$
$\displaystyle f_\mathrm{c}(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\mathrm{c}}{\dr{\mit\Theta}}\Big\vert b_\mathrm{c}
P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\mathrm{c}}+kb_\mathrm{c}\theta-\pi) \big]\equiv
C\er^{\ir\gamma},{\rm }$
$\displaystyle f_\dr(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\dr}{\dr{\mit\Theta}}\Big\vert b_\dr
P_{b_\dr}(1-P_{b_\dr})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\dr}+kb_\dr\theta-\frac{1}{2}\pi) \big]\equiv
D\er^{\ir\delta},{\rm }$
$\displaystyle f_\er(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\er}{\dr{\mit\Theta}}\Big\vert b_\er
P_{b_\er}(1-P_{b_\er})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\er}+kb_\er\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
E\er^{\ir\varepsilon},$

where $\eta_\br$ indicates the phase shift and k is given by $k=(2\muE_\ir)^{1/2}$.
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For interferences as indicated in Eqs. (11) this difference is exactly the part of the deflection curve enclosed by $\theta_1$ and $\theta_2$ and multiplied by a factor of $k/2\pi$. Then the local oscillation frequency is linearly proportional to the distance along the b-scale of the relevant branches of the deflection curve. For net-attractive and net-repulsive scattering Fig. A08-m5bi-F2 shows two slices with equal areas. It is clearly shown that in the chosen angular ranges the repulsive-scattering wavelength is much larger compared to the attractive one.
[To the FULL figure] Figure A08-m5bi-F4: Na_I adiabatic potential curves. The pseudo-crossing potentials are all of the same species $^1{\mit\Sigma}^+$. [Copied to the Treated results (link type: 'output to'; target: A08-m4bii1)]

 Repulsive scattering oscillation Acov and $\rho_\cov$

By fitting the repulsive scattering oscillations of the experimental and the theoretical cross section curves, we found Acov=3150 eV and the repulsive steepness to be $\rho_\cov$=0.435 Å. Then the calculated repulsive oscillation is in perfect agreement with the experimental one.
  
[To the FULL figure] Figure A08-m5bi-F3: Wavelength of the oscillatory differential cross section (CM system) due to net-repulsive scattering, versus the colliding energy. The measurement give the wavelength averaged over ten oscillations just beyond the rainbow angle. The error bars only indicate the error in the relative position of first and tenth oscillation. The curve gives the corresponding calculated wavelengths

 
A08-m5bi-F3 [experimental part copied from the Results(link type: `input from'; to: A08-m4bi2] shows the measured oscillation wavelength due to interference of the ionic and covalent repulsive-scattering contributions just beyond the rainbow angle. The corresponding branches of the deflection curve are d and e, where the e-branch is calculated from the known ionic potential. Because the attractive Van der Waals term of the covalent potential is not important at our kinetic-energy range, the repulsive oscillation wavelength leads to a determination of the repulsive part of the covalent potential: Acov=3150 eV and $\rho_\cov=0.435 Å.

Reliability

With the repulsive-potential parameters Acov=3150  eV and $\rho_\cov=0.435 Å the measured wavelength as a function of the kinetic energy can be fitted perfectly with the calculations. Although in first-order approximation the $\tau = constant rule predicts the wavelength being proportional to E-1/2, the measured as well as the calculated wavelengths increase more rapidly at decreasing kinetic energy. Another date from the measurements is the fact that no angular-dependent wavelength beyond the rainbow angle could be detected. The values of A and $\rho$ have been chosen in such a way that the calculated wavelength too is rather angular independent. However, it is difficult to separate the effects of varying A or $\rho$. Consequently the given set of values of Acov and $\rho_\cov$ is more reliable then the separate values. Moreover, it should be noted that the steepness of the covalent repulsive potential has been determined relative to the steepness of the ionic repulsive potential that is supposed to be known.
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Inelastic energy $\Delta E$

The general shape of the polar differential cross section is the cross section averaged over the quantal oscillations. The angular positions of special features at different collision energies agree very well: namely the maximum of the peak at $\tau\approx35$ due to covalent scattering, the minimum at $\tau\approx65$ due to scattering with maximal impact parameter and the maximum of the primary rainbow. These agreements determine that $\Delta E$, the potential-energy difference of the ionic and covalent state at infinite internuclear separation, is $\Delta=2.075eV.
Due to the flat long-range character of the covalent potential curve the chemi-ionization collision with maximal collision parameters $b\approx R_\mathrm{c}$ is affected only by the coulombic outgoing potential branch. Therefore the correct position of the minimum in the differential cross section establishes the inelastic energy $\Delta E$ (see Fig. A08-m5bi-F4).
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Potential well-depth $\vare$

The agreement of the angular positions of the special features of the general shape also establishes the well-depth of the ionic potential curve: $\vare=-3.1$\pm 0.2 eV.
The collision process with a distance of closest approach at the inflection point of the ionic potential curve causes the classical rainbow angle where $\vert\dr b/\dr\MT\vert\to\infty$. The inflection point is related to the minimum of the potential well when the shape of the potential curve is qualitatively known. Moreover, the classical rainbow angle is related to the position of the maximum of the rainbow by Eq. (10) and is situated on the ``dark side'' slope of the rainbow. Then the measured positions of the primary rainbows lead to the position of the minimum of the ionic potential well at $\vare=-3.1$\pm 0.2 eV., enclosing the values of 3.0, 3.07, 3.11 and 3.16 as tabulated by Herzberg [Value are input from another article (link type: `input from/external'; target: Rf(A08)24-m*)].
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  Coupling H12

The relative intensities of several parts of the general shape of the cross section lead to an estimation of H12 of 0.05 eV for energies in the range of 30-55 eV. However, for the 13.1 eV curve the estimation is H12=0.065.
Reliability
Using H12=0.05 eV in the calculations, the relative intensities of several parts of the calculated and measured differential cross sections are in good agreement only for the curves of 29.7, 38.7 and 55.0 eV shown in Figs. A08-m5bi1-F3c, d, e. This is in agreement with the equal value of H12 being 0.05 eV estimated from total cross section measurements on the energy range 2-20 eV [Value are input from another article (link type: `input from/external'; target: Rf(A08)9-m*])></a>. 

The sensitivity of the relative and absolute differential cross sections 
to the value of <I>H</I><SUB>12</SUB> has been indicated by three curves in <A HREF=Fig. A08-m5bi1-F3f, using H12=0.04, 0.05 and 0.07 eV. Because an obvious disagreement occurs for the curves of 13.1 and 20.7 eV shown in Figs. A08-m5bi1-F3a, b, it is not possible to give only one value of H12 resulting in an overall good fit. A consideration of the 13.1 eV curve only should lead to a value of 0.065 eV for H12 (see Fig. A08-m5bi-F1) while the 20.7 eV curve needs a coupling constant of 0.060 eV.

The determination of H12 especially requires the proper transformation of detector signal to differential cross section. A priori there are doubts on the normalization of the detector signal to an angular-independent scattering volume viewed by the detector. The calculated angular-dependent normalization factor seems to be reliable by observing the result that the total cross section due to covalent scattering is about equal to the total cross section due to ionic scattering, independent of the kinetic energy and thus independent of the different angular ranges. This requirement is postulated by the equal Landau-Zener probability Pb(1 - Pb) for both chemi-ionization trajectories and means about equal areas enclosed by the relevant parts of the differential cross section.

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Now we have determined the potential parameters of the ionic and the covalent system Na - I. The parameters are summarized in Table A08-m5bi-T1.
  
Table A08-m4bii1-T1 [Table copied to the <i>Treated results</i> (link type: `output to'; target: A08-m4bii1)]
Ionic-potential parameters Covalent-potential parameters
\alpha_Na^+ = 0.4083.a \alpha_Na = 273.h
\alpha_cl- = 6.4313.a \alpha_cl = 73.i
Cion = 11.3 eV6.b Ccov =1000 eV6.j
Aion = 1913.6 (2760l) eV.c Acov =3150 eV .j
rho_ion = 0.3489.d rho_cov =0.435 .j
epsilon = 3.11 + DeltaE_cov eV.e
re = 2.71143 (2.664l).f Coupling parameters
DeltaE_cov = 2.075eV.g H12 = 0.065 eV (0.0024 a.u. k)
Hrot = 3 x 10-17 (0.04 a.u. k)
a Dipole polarizability, [Value are input from another article (link type: `input from/external'; target: Rf(A08)14-m*]. b Van der Waals coefficient, from the London formula: , where I2 is the second ionization potential of Na and A is the electron affinity of I. c [Value are input from another article (link type: `input from/external'; target: Rf(A08)15-m*]. d [Value are input from another article (link type: `input from/external'; target: Rf(A08)16-m*]. e Potential well depth, [Value are input from another article (link type: `input from/external'; target: Rf(A08)17-m*]). f Internuclear equilibrium distance, [Value are input from another article (link type: `input from/external'; target: Rf(A08)18-m*)]. g From INa-AI. h [Value are input from another article (link type: `input from/external'; target: Rf(A08)19-m*)]. i Arbitrary value. j From the London formula: , where I is the first ionization potential. k Present work. l Alternative value due to overdefinition of the potential curve.
The ionic ground state is well known. It is described by the Rittner potential [The formulae are input from elsewhere (link type: `input/external) Rf(A08)13-m*]):
Uion(R) = $\displaystyle -\frac{e^2}{R}-\frac{e^2(\alpha_{\Na^+}+\alpha_{\I^-})}{2R^4}$  
    $\displaystyle -\frac{2e^2\alpha_{\Na^+}\alpha_{\I^-}}{R^7}-\frac{C_\ion}{R^6}$  
    $\displaystyle +A_\ion\, \er^{-R/\rho_\ion}+\Delta E.$ (3)
We describe the covalent potential only by two terms:
\begin{displaymath}U_\cov(R)=-\frac{C_\cov}{R^6}+A_\cov\,\er^{-R/\rho_\cov}
We have determined for the covalent Na-I system most of the potential parameters; the missing parameters have been chosen to construct the potential curves in Fig. A08-m5bi-F4 and the deflection functions in Fig. A08-m5bi-F2.

The ionic potential curve is overdefined by the given parameters. That is why the values for A and re have not been used but the other parameters give rise to the values given in parentheses.

For small values of the internuclear distance R the ionic and covalent potential curves bend over to negative values, leading to $\lim_{R\to 0} U(R)=-\infty$. This is due only to the mathematical form of the potential-energy expressions [Eqs. (0.3) and (0.4)]. Using the parameters of Table A08-m5bi-T1, the maxima of the potential curves are Uion=29.7 eV and Ucov=21.4 eV for R=1.16 Å and R=1.70 Å, respectively. However, this effect does not handicap the calculations. Even for the smallest impact parameter considered, the distances of closest approach are R0=2.02 Å and R0= 2.84 Å corresponding to the potential energies Ucov=-0.3 eV and Ucov=2.7 eV, respectively.