Quantitative interpretation: potential and deflection function =A08m5bi
We determine the potential parameters that govern the chemiionization reaction Na + I Na^{+} + I according to the atomatom model for ionpair formation in molecular collisions
,
.
From the potential parameters, the theoretical cross section of the chemiionization 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 A08m5bii
.

Figure A08m5biF1: Polar differential cross section for chemiionization (CM
system) at E=13.1 eV. (b) Differential cross section calculated with the stationaryphase
approximation and uniform rainbow approximation showing separated the
longwavelength interference structures due to a + c (,
full curve), b + c (, full curve) and d + e (,
dashed curve) interferences. (c) Full bars indicate the
measured maxima of the interference structure on the differential
cross section due to netattractive scattering. Dashed bars indicate
the maxima due to netrepulsive scattering 
Fig. A08m5biF1b
shows the differential cross section with simplified
interference structure. An additional simplification in Fig. A08m5biF1b 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. A08m5biF1b and the latter
figure is more easily related to the deflection function and the
potential curves, Fig. A08m5biF1b 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. A08m5biF1a
, and Fig. A08m5biF1b 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 H_{12} can be estimated.
For that purpose are very important particularly the
wavelengths of the rainbow and Stueckelberg oscillations
, which result from the semiclassical interference of different contributions to the scattering angle
.
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)

(2) 
or

(1) 
and

(0) 

Figure A08m5biF2: Deflection curves for chemiionization scattering (CM system) at E_{i}=13.1 eV. 
The
difference in the cosine argument for two neighbouring scattering
angles and can be shown easily, with the help of Fig. A08m5biF2 and
Eqs. () for the contributions to the scattering angle in lowestorder stationaryphase
approximation
.

where
indicates the phase shift and k is given by .



For interferences as indicated
in Eqs. (11) this difference is exactly the part of the deflection
curve enclosed by and and multiplied by a factor of
. Then
the local oscillation frequency is linearly proportional to the
distance along the bscale of the relevant branches of the deflection
curve. For netattractive and netrepulsive scattering Fig. A08m5biF2 shows
two slices with equal areas. It is clearly shown that in the chosen
angular ranges the repulsivescattering wavelength is much larger
compared to the attractive one.
Repulsive scattering oscillation A_{cov} and
By fitting the repulsive scattering oscillations of the experimental and the theoretical cross section curves, we found A_{cov}=3150 eV and the repulsive steepness to be =0.435 Å. Then the calculated repulsive oscillation is in perfect agreement with the experimental one.

Figure A08m5biF3:
Wavelength of the oscillatory differential cross section (CM
system) due to netrepulsive 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 

A08m5biF3
shows the measured
oscillation wavelength due to interference of the ionic and covalent
repulsivescattering contributions just beyond the rainbow angle. The
corresponding branches of the deflection curve are d and e, where the
ebranch is calculated from the known ionic potential. Because the
attractive Van der Waals term of the covalent potential is not
important at our kineticenergy range, the repulsive oscillation
wavelength leads to a determination of the repulsive part of the
covalent potential:
A_{cov}=3150 eV and
=0.435 Å.
Reliability
With the repulsivepotential parameters A_{cov}=3150
eV and
=0.435 Å the measured wavelength as a
function of the kinetic energy can be fitted perfectly with the
calculations.
Although in firstorder approximation the = 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 angulardependent wavelength beyond the rainbow
angle could be detected. The values of A and
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 .
Consequently the given set of values of A_{cov} and
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.



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
due to covalent scattering, the minimum at
due to
scattering with maximal impact parameter and the maximum of the
primary rainbow.
These agreements
determine that ,
the potentialenergy difference of the ionic and covalent state at infinite internuclear separation, is
=2.075eV.

Due to the flat longrange character of the covalent potential
curve the chemiionization collision with maximal collision parameters
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
(see Fig. A08m5biF4).



The agreement of the angular positions of the special features of the general shape also establishes the welldepth of the ionic potential curve:
=3.1 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
.
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
=3.1 0.2 eV., enclosing
the values of 3.0, 3.07, 3.11 and 3.16 as tabulated by Herzberg
.



Coupling H_{12}
The relative intensities of several parts of the general shape of the cross section lead to an estimation of H_{12} of 0.05 eV for energies in the range of 3055 eV. However, for the 13.1 eV curve the estimation is H_{12}=0.065.

Reliability
Using
H_{12}=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. A08m5bi1F3c, d, e. This is in
agreement with the equal value of H_{12} being 0.05 eV estimated from
total cross section measurements on the energy range 220 eV
Fig. A08m5bi1F3f, using H_{12}=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. A08m5bi1F3a, b, it is not possible to give only one value of H_{12} 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 H_{12} (see Fig. A08m5biF1) while the 20.7 eV curve needs a coupling constant of 0.060 eV.
The determination of H_{12} 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 angularindependent scattering volume viewed by the detector.
The calculated angulardependent 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 LandauZener probability P_{b}(1  P_{b})
for both
chemiionization trajectories and means about equal areas enclosed by
the relevant parts of the differential cross section.



Now we have determined the potential parameters of the ionic and the covalent system Na  I. The parameters are summarized in Table A08m5biT1.
Table A08m4bii1T1
Ionicpotential parameters 
Covalentpotential parameters 
= 0.408^{3}.^{a} 
= 27^{3}.^{h} 
= 6.431^{3}.^{a} 
= 7^{3}.^{i} 
C_{ion} = 11.3 eV^{6}.^{b} 
C_{cov} =1000 eV^{6}.^{j} 
A_{ion} = 1913.6 (2760^{l}) eV.^{c} 
A_{cov} =3150 eV .^{j} 
= 0.3489.^{d} 
=0.435 .^{j} 
= 3.11 + eV.^{e} 
r_{e} = 2.71143 (2.664^{l}).^{f} 
Coupling parameters 
= 2.075eV.^{g} 
H_{12} = 0.065 eV (0.0024 a.u. ^{k}) 

H_{rot} = 3 x 10^{17} (0.04 a.u. ^{k}) 
^{a} Dipole polarizability, . ^{b} Van der Waals coefficient,
from the London formula: , where I_{2} is the second ionization potential of Na and A is the electron affinity of I. ^{c}
. ^{d} .
^{e} Potential well depth, . ^{f} Internuclear equilibrium distance, . ^{g} From I_{Na}A_{I}. ^{h} . ^{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
:
We describe the covalent potential only by two terms:
We have determined for the covalent
NaI system most of the potential parameters; the missing parameters have been chosen
to construct the potential curves in Fig. A08m5biF4 and the
deflection functions in Fig. A08m5biF2.
The ionic potential curve is
overdefined by the given parameters. That is why the values for A and r_{e}
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
.
This is due only to the mathematical form of
the potentialenergy expressions [Eqs. (0.3) and (0.4)]. Using the
parameters of Table A08m5biT1, the maxima of the potential curves are
U_{ion}=29.7 eV and
U_{cov}=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 R_{0}=2.02 Å and R_{0}=
2.84 Å
corresponding to the potential energies
U_{cov}=0.3 eV and
U_{cov}=2.7 eV, respectively.