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Quantitative interpretation: theoretical differential cross section with LZ =A08-m5bii1

By comparison of measurements and calculations the missing parameters of the covalent potential curve and the value of H12 can be estimated. Moreover, the suitability of the applied stationary-phase approximation and Landau-Zener theory can be rated at its true value because the ionic potential curve is well known.

Fig. A08-m5bii1-F1 a shows the polar differential cross section, defined by $I(\theta)$ $\sin (\theta)$, for the chemi-ionization process of sodium on iodine and calculated by the lowest-order stationary-phase approximation [More on the approximation in the Theoretical methods (link type:'DEPENDS ON/clarified in/is detailed in'; target: A08-m3cii)].
Use has been made of the Landau-Zener formula [details on Landau-Zener in the <i>Theoretical methods</i> (link type: 'IS DETAILED IN/input from'; target: A08-m3ci1)] Eqs. (E1)
P_{b,rot} (E1)
The scattering amplitude (E2) and the contibutions to the semi-classical differential cross section (A08-m3cii-E4)
f(th)= (E2)
leading to the differential cross section
[details on the cross section in the <i>Theoretical methods</i> (link type: 'IS DETAILED IN/input from'; target: A08-m3cii)]
I(th)=f(\theta)^2 (E3)
the JWKB approximation (E4)
\begin{displaymath}\eta_b=-\frac{1}{2}k\int_b^\infty{\mit\Theta}\,\dr b
\end{displaymath} (E4)
and the classical deflection function. The region of the classical rainbow angle has been omitted because the lowest-order approximation leads to a wrong result.
[discussion on the applicability of the approximation in the <i>Theoretical methods</i> (link type: 'reasons in'; target: A08-m3cii)]
[Hide the details]
With the potential parameters and H12 of Table 1 [The potential parameters and the deflection function are determined in another Quantitative interpretation module (link type: 'INPUT FROM'; target: A08-m5bi)] and taking into account a weight factor 1/8, the calculation leads to absolute values of the differential cross section [The potential parameters and the deflection function are determined in another Quantitative interpretation module (link type: 'INPUT FROM'; target: A08-m5bi)].
[To the FULL figure] Figure: A08-m5bii1-F1: Polar differential cross section for chemi-ionization (CM system) at Ei=13.1 eV, calculated in semi-classical approximation with the potential parameters of Table A08-m5bi-T1 and the coupling parameters H12= 0.065 eV and Hrot=0 eV.[The calculation is performed in the Quantitative interpretation (link type: 'INPUT FROM'; target: A08-m5bi)]
(a) Differential cross section with complete interference structure, calculated with the lowest-order stationary-phase approximation. The region of the classical rainbow angle cl,rb has been omitted.
(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 ( 0  65, full curve), b + c ( 65 250, full curve) and d + e ( 0  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. The complete measured cross section curve at Er =13.1 eV is given in Fig. 

Fig. A08-m5bii1-F1b shows the differential cross section with simplified interference structure. The major part has been calculated by the lowest-order stationary-phase approximation
The lowest-order stationary-phase approximation is used in the form of Eq. (E5)
 
$\displaystyle f_\ar(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\ar}{\dr{\mit\Theta}}\Big\vert b_\ar P_{b_\ar}(1-P_{b_\ar})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\ar}+kb_\ar\theta-\pi) \big]\equiv A\er^{\ir\alpha},{\rm }$
$\displaystyle f_\br(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert b_\br P_{b_\br}(1-P_{b_\br})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\br}+kb_\br\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
B\er^{\ir\beta},{\rm }$
$\displaystyle f_\mathrm{c}(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\mathrm{c}}{\dr{\mit\Theta}}\Big\vert b_\mathrm{c}
P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\mathrm{c}}+kb_\mathrm{c}\theta-\pi) \big]\equiv
C\er^{\ir\gamma},{\rm }$
$\displaystyle f_\dr(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\dr}{\dr{\mit\Theta}}\Big\vert b_\dr
P_{b_\dr}(1-P_{b_\dr})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\dr}+kb_\dr\theta-\frac{1}{2}\pi) \big]\equiv
D\er^{\ir\delta},{\rm }$
$\displaystyle f_\er(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\er}{\dr{\mit\Theta}}\Big\vert b_\er
P_{b_\er}(1-P_{b_\er})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\er}+kb_\er\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
E\er^{\ir\varepsilon},$ (5)
(where $\eta_\br$ indicates the phase shift and k is given by $k=(2\muE_\ir)^{1/2}$) and Eq. (6).
$\displaystyle I(\theta)$ = $\displaystyle \big\vert A\er^{\ir\alpha}+C\er^{\ir\gamma}+D\er^{\ir\delta}+
E\er^{\ir\varepsilon}\big\vert^2$  
  = $\displaystyle A^2+C^2+D^2+E^2+2AC\cos(\alpha-\gamma)$  
    $\displaystyle +2AD\cos(\alpha-\delta)+2AE\cos(\alpha-\varepsilon)$  
    $\displaystyle +2CD\cos(\gamma-\delta)+2CE\cos(\gamma-\vare)$  
    $\displaystyle +2DE\cos(\delta-\vare).$ (E6)
[Hide the details]
The rainbow region has been calculated in the uniform approximation
The rainbow has been calculated using Eq. (E7).
$\displaystyle {I(\theta)\sin(\theta)=\pi\bigg[ \bigg( b_\br\timesP_{b_\br}(1-P_{b_\br})\times\Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2}}$
    $\displaystyle +\bigg( b_\mathrm{c}\times
P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})\t...
...{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2} \bigg]^2\vert\xi\vert^{1/2}\A\ir(\xi)^2$  
    $\displaystyle +\pi\bigg[ \bigg( b_\br\times
P_{b_\br}(1-P_{b_\br})\times\Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2}$  
    $\displaystyle -\bigg( b_\mathrm{c}\times
P_{b_\mathrm{c}}(1-P_{b_\mathrm{c}})\t...
...Big\vert\frac{\dr b_\mathrm{c}}{\dr{\mit\Theta}}\Big\vert
\bigg)^{1/2} \bigg]^2$  
    $\displaystyle \times\vert\xi\vert^{-1/2}\A\ir'(\xi)^2,$ (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} (E7)
[Hide the details]
An additional simplification in Fig. A08-m5bii1-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 Fig. A08-m5bii1-F1a can be seen easily from Fig. A08-m5bii1-F1b and the latter figure is more easily related to the deflection function and the potential curves, Fig. A08-m5bii1-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 Fig. A08-m5bii1-F1a and Fig. A08-m5bii1-F1b even with the smallest angular resolution of the detector of 0.3º fwhm will lead to equal results.
[To the FULL figure] Figure A08-m5bii1-F2: Relative polar differential cross section for chemi-ionization (CM system). The cross sections, measured at a colliding energy of 13.1 and 18.2 eV, have been set in scales with shifted zero points [Copied from the experimental Treated results (link type: 'INPUT FROM'; target: A08-m4bi1)]

Reliability and applicability of the interpretation with Landau-Zener

Rainbow oscillation

The rainbow structure of the experimental cross sections is explained nearly perfectly by the theoretical description.
[To the FULL figure] Figure A08-m5bii1-F3: Rainbow structure versus initial colliding energy (CM system). The dashed curves show the calculated positions of the maxima of rainbow and supernumeraries
The ionic potential curve Na+Cl- is well known. Moreover, the energy difference $\Delta E$ of ionic and covalent ground state at infinite separation is known and the covalent potential must be about zero for $R\geq R_\crm$. So the ionic scattering takes place via well-known potential curves leading to a complete knowledge of the relevant deflection curve (branches b, c and e of Fig. A08-m5bi-F1).

By applying the stationary-phase approximation and the uniform rainbow approximation, the positions of primary and supernumerary rainbows from interference of the b and c contributions are calculated. Fig. A08-m5bii1-F3 compares the measured and calculated relative positions of the rainbow maxima, where the measured as well as the calculated third supernumeraries have been placed on the same straight line.

Reliability

From the primary rainbows up to the third supernumeraries the agreement is nearly perfect. Further up an increasing deviation is observed of measured and calculated spacings of the successive supernumeraries. Moreover, the deviation increases with increasing collision energy. For 13.1 eV the deviation increases from 5 up to 20 percent while for the rainbow structure at 38.7 eV the corresponding quantities are 5 and 40 percent. A look at the potential and deflection curves shows that even for the 13.1 eV deviation, a reasonable adjustment of the potential parameters is not able to decrease the distance between the b and c branches of the deflection curve enough to fit the measured and calculated rainbow structures. Moreover, an energy-dependent deviation cannot be eliminated in this way.

[Hide the details]

Small-angle oscillations

The cross sections exhibit Stueckelberg oscillations 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.

The Stueckelberg oscillations in the experimental cross sections shown in Fig. A08-m5bii1_F2 for $\tau$ < 65 eV degree are due to the interference of the differential cross-section contributions related to the a and c branches of the deflection curves. The measured maxima of the 13.1 eV curve have been indicated in Fig. A08-m5bii1-F1 c together with the calculated structure in Fig. b.

Reliability

For Stueckelberg oscillations just beyond the rainbow angle, the measured wavelength as a function of the kinetic energy can be fitted perfectly with the calculations.

However, on the range $\tau=40 \to 65 obviously the distances between the calculated maxima are too small. The relative wavelength discrepancy is the largest at $\tau\approx60$.

For decreasing scattering angles the measured and calculated oscillation wavelengths increase while the deviation decreases from twenty percent at $\tau\approx60$ to zero percent at $\tau\approx40$ and ends up in an opposite deviation meaning a calculated wavelength too large compared to the measurements. However, the latter deviation is not precarious because it can be improved by a careful adjustment of the onset of the covalent repulsive potential as can be seen from Fig. A08-m5bi-F4 of the potential and Fig. A08-m5bi-F4 of the deflection function.

More serious is the deviation on the $\tau=40 \to 65 range because this part of the differential cross section deals only with scattering from the known ionic potential and the flat part of the covalent potential curve. It is very remarkable that the deviation on this range is the same as the deviation of measured and calculated maxima just on the other side of the minimum in the differential cross section (the range of the high-number supernumeraries). The completely resolved differential cross section curve with Ei=18.2 eV gives the same trend with equal deviations of measured and calculated wavelengths around $\tau = 65 eV degree.

Some evidence of repulsive interference at small scattering angles has been indicated also in Fig. A08-m5bii1-F1c. This structure gives a check on the estimated repulsive part of the covalent potential. Because Fig. A08-m5bii1-F1 b clearly shows that at small angles the repulsive oscillation is the straight continuation of the attractive oscillation it is not very surprising that the calculated wavelength is somewhat too large compared to the measurements. The same applies to the 18.2 eV curve.

General shape

The general shape of the polar differential cross section is the cross section averaged over the quantal oscillations. This shape will nearly correspond to the classical differential cross section except at the classical rainbow angle, and thus is given by the relevant values of b, $\vert\dr b/\dr\MT\vert$ and the Landau-Zener transition probability Pb [The dependency of the cross section on these values in clarified in a mesoscopic module (link type: 'IS CLARIFIED IN/is detailed in/project/wider range'; target: MESO-m3c-defl,I(th))].
[To the FULL figure] Figure A08-m5bii1-F4: Smoothed differential cross sections for five different collision energies. (a), (b), (c), (d), (e): Full curves show the measured relative differential cross section, averaged over the interference structures. Absolute differential cross sections, calculated semi-classically and also averaged over the interference structure, have been given by the dashed curves. Use has been made of H12=0.05 eV. (f): At Ei=29.7 eV the curves show the calculated differential cross section for H12=0.04, 0.05 and 0.07 eV

The general shape of the differential cross section has been measured and calculated at collision energies of 13.1, 20.7, 29.7, 38.7 and 55.0 eV, as shown in Figs. A08-m5bii1-F4a, b, c, d, e. The calculated values have been given on absolute scales, the measurements are only relative and have been given for the different energies on arbitrary, non-related scales.

The angular positions of special features agree very well: namely the maximum of the peak at $\tau\approx35$ due to covalent scattering, the minimum at $\tau\approx 65$ due to scattering with maximal impact parameter and the maximum of the primary rainbow.

A permanent local disagreement is observed at the minimum of the differential cross section at $\tau\approx 65$, showing that the calculated cross section is too small continuously. The deviation increases at increasing energy. At least the major part cannot be caused by convolution effects on the measurements.

It has been shown that scattering from well-known parts of the internuclear potentials gives rise sometimes to a discrepancy of the calculated and measured interference structure. In spite of that, we have determined the repulsive parameters of the covalent potential curve from the repulsive interference structure.

However, the reliability is enlarged by the very good agreement of the kinetic-energy behaviour of the wavelength [This argument in favour of the reliability is given in another Quantitative interpretation module (link type: 'IS ARGUED IN/is detailed in'; target: A08-m5bi)], while the disagreement of the interference wavelength on the range $\tau 0 to 150 shows an energy-dependent discrepancy. Perhaps the collisions with large impact parameters cause this discrepancy. The comparable calculated and measured interference structures due to collisions with smaller impact parameters consist of the primary rainbows and low-number supernumeraries that are in very good agreement.

Serious discrepancies

In the preceding we have seen that there are only a few serious deviations of measurements and calculations that cannot be ascribed to measuring faults or doubts as to the correctness of the potentials used. Summarizing, these discrepancies are:

a)
disagreement of the oscillation wavelength around $\tau=65$ = 65. This discrepancy increases for increasing collision energy;
b1)
large intensity deviation at the very minimum of the cross section at $\tau\approx65$. Also this discrepancy increases for increasing energy;
b2)
rather small intensity deviation separated from $\tau=65$ = 65 somewhat more and preventing an estimation of H12 that gives an overall good fit. This discrepancy is dependent on the energy.
(A priori it is not obvious whether the deviations b1) and b2) have the same origin.)

The deviations must be viewed in the light of the used approximations, i.e. in the light of the restrictions of the theoretical methods we used [these restrictions are given in the module <i>Theoretical methods</i> (link type: `depends on/detailed in'; target: A08-m3c)]) ,the most important of which are:

A.
The stationary-phase approximation.
B.
The uniform rainbow approximation.
C.
The Landau-Zener transition-probability formula.
D.
The use of the Landau-Zener formula to collisions where the distance of closest approach R0 and the distance of pseudo-crossing Rc are not well separated [arguments against the applicability of LZ in such a case given in Dikitin (link type: `external/argument'; to: Rf(A08)25-m*)] .
E.
The use of the diabatic potentials [This restriction depends on the choice made in another Quantitative interpretation module (link type: 'DEPENDS ON/is detailed in'; target: A08-m5bi)] in the classical deflection function in spite of small deformation of the curves at the pseudo-crossing.
F.
The use of a transition point in spite of a transition region around the pseudo-crossing predicted by the Landau-Zener theory.
G.
The neglect of rotational coupling so far.

Comparing figures A08-m4bi1-F1 and  A08-m5bii1-F2, we see that the deviations between measurements and calculations occur only near $\tau=65$ = 65 eV degree, for covalent as well as ionic scattering. The deviations are energy dependent and deal with the oscillation wavelengths and relative intensities. It is very remarkable that the approximations C, D, E, F and G mostly violate the real differential cross sections at the region around $\tau$ = 65.

Those approximations don't change the oscillatory structure but only the relative intensities. Some estimates of corrections to the approximations have been made, especially in view of the influence on the differential cross section at $\tau\approx65$.

A correction to approximation D Unfold approximation(The use of the Landau-Zener formula to collisions where the distance of closest approach R0 and the distance of pseudo-crossing Rc are not well separated.)Hide details reduces Pb by passing an incomplete transition region. Because then in our energy range the product Pb(1 - Pb) decreases too, the differential cross section even decreases in the region of interest.

However, a correction to E Unfold approximation(The use of the diabatic potentials [This restriction depends on the choice made in another Quantitative interpretation module (link type: 'DEPENDS ON/is detailed in'; target: A08-m5bi)] in the classical deflection function in spite of small deformation of the curves at the pseudo-crossing.)Hide details indeed predicts a somewhat larger value of the differential cross section close to $\tau$ = 65, but there still exists the feature of $\dr b/\dr\theta\to0$ for $b\to b_{\max}$.

It is very difficult to introduce a correction to approximation F Unfold approximation(The use of a transition point in spite of a transition region around the pseudo-crossing predicted by the Landau-Zener theory].Hide details in the impact-parameter method. It will lead to a collection of deflection curves with the greatest differences of the relative shapes near $\theta_{\br_{\max}}$. Indeed, a summation over the deflection curves never gives a zero value of the differential cross section due to collisions with large impact parameters. However, this averaging effect does not cause an important rise of the cross section on the whole region around $\tau$ = 65.

The remaining corrections to the intensity at $\tau\approx65$ are C and G dealing with the Landau-Zener coupling and rotation coupling. It is not expected [arguments for the applicability of the LZ formula given in Child (link type: `external/argument/detailed in'; to: Rf(A08)25-m*)] that the Landau-Zener transition formula gives such wrong results, although this formula has been derived using the approximations that Uion and Ucov are linear in the region of the crossing and H12 is an essentially constant coupling element.

We have greater expectations for the rotation coupling to explain the intensity deviation. Rotational coupling will be treated in more detail in module A08-m5bii2 Quantitative interpretation [(link type: `used in'; target: A08-m5bii2)].

If the rotation coupling is taken into account as well, the only one of the approximations A-G that can give a wrong result for the oscillatory wavelength of the differential cross section at $\tau\approx65$ is the lowest-order stationary-phase approximation. Delos and Thorson [an argument for the application of the approximation is given in Delos (link type: `external/input from/argument'; to: Rf(A08)6-m*)] have given four statements that justify the application of this approximation on two-state collisions. One of them should not be satisfied in our case, namely the requirement of a collisional energy large compared to the potential-energy differences of the two states. This statement requires about equal classical trajectories along the two states. The deflection curves of Fig. A08-m5bi-F2[This figure is given in another Quantitative intepretation module (link type: 'DEPENDS ON'; target: A08-m5bi)] show that this requirement is not fulfilled.