2. COLLISIONAL IONISATION PROCESSES
Besides the energy transfer in collisions of excited atoms and / or molecules,
also ionisation is possible. Three types of processes can be distinguished:
A* + B* 
(i)
A+ + B,
(ion pair formation)
(2.1a)
(ii)
A+ + B + e,
(Penning ionisation)
(2.1b)
(iii)
AB+ + e.
(associative ionisation)
(2.1c)
Ion pair formation takes place thanks to non-adiabatic transitions between covalent
(AB**) and ionic (A+B) terms at large internuclear distances. Penning (PI) and
associative (AI) ionisation result from autoionisation of quasimolecular continuum
states (AB**) at small internuclear distances. AI possess the smallest threshold energy
among the other ionisation processes, and it may prove itself as an important
mechanism in cluster formation. Recently, a hypothesis was proposed that dissociative
recombination of trimer ions might be responsible for the violet excimer type emission
on the 2 3  g  a 3  u transition in Na2 molecule [25,67,68]:
Na 3 + e  Na2( 23 g ) + Na,
where trimer ions could be formed in some of the molecule-atom AI processes, e.g.,
Na2( C 1  u ) + Na(3S)  Na 3 + e,
Na2( A1 u ) + Na(3P)  Na 3 + e.
(2.2)
The purpose of the present work is to prove the formation of trimer ions in collisions of
excited atoms and molecules, and to clarify the influence of initial vibrational excitation
of Na2 molecule on the efficiency of AI.
2.1. THEORETICAL INTERPRETATION OF ASSOCIATIVE IONISATION
For the sake of simplicity, let us consider a collision of two excited atoms, A*
and B*, following [21,69,70]. As the atoms approach each other, a quasimolecular
system is formed which may rich an autoionising state and emit an electron. AI can take
place when the total excitation energy is larger than the ionisation potential of an atom
(Fig. 2.1a), as well as when it is smaller (Fig. 2.1b). Usually one works in the BornOppenheimer approximation, which is valid at small (thermal) collision energies of the
particles. The dynamics of the process can be fully described by three local functions:
molecular potentials U  R  and U  R  , which correspond to the covalent entrance and
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Fig. 2.1. Schematic illustration of covalent [ U  R  ] and ionic [ U  R  ] potential curves. E0 is the initial
kinetic energy of the collision, but E(R) is the kinetic energy of nuclei at internuclear distance R.
According to the Franck-Condon principle, E(R) does not change during the electronic transition, i.e.,
autoionisation results in a transition to the so-called difference term (broken line). (a) The sum of
potential energies of A  and B  is larger than the ionisation potential of atom B. Depending on the
internuclear distance at which the transition happens, either Penning (R>RAI) or associative (R0RRAI)
ionisation will take place. (b) The sum of potential energies of A  and B  is smaller than the ionisation
potential of atom B. Autoionisation is possible only at internuclear distances R0RRc. In the displayed
case, U    E0  U   , therefore only the associative ionisation takes place.
ionic exit channels, and the autoionisation width of the covalent state (R)2. Molecule in
an autoionising state U  R  may spontaneously emit an electron to go over to the ionic
state U  R  . Franck-Condon principle anticipates that this electronic transition does not
change the kinetic energy and momentum of the relative motion of the nuclei. The total
energy of the system after collision will be


W R   E0  U     U  R   U  R  ,
where energy  R   U  R   U  R  is carried away by the free electron. Depending on
if W(R) will be smaller or larger than U+(), in the case shown in Fig. 2.1a either
associative (at R0  R RAI) or Penning (at R > RAI) ionisation will take place. In the
case shown in Fig. 2.1b only the associative ionisation is possible.
At a given initial angular momentum of nuclei, l , ionisation probability can be
expressed as [70]
2
  R  is the lifetime of the system of colliding particles with respect to autoionisation, but (R) is
determined as  R   2g 

Ĥ  
2
, where   and 

are the wavefunctions of the inital and
final states, Ĥ is the hamiltonian of the system, and g is the density of continuum states.
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
   R 

Pl   Pl R dR  1  exp  2 
dR  ,
 Rl v l R  
Rl
(2.3)
where Pl(R) accounts for the dependence of ionisation probability on R; Rl is the ldependent turning point on U  R  , but the radial velocity of the nuclei, vl, can be
determined from the initial kinetic energy E0 as
vl 
2
 2l 2 


 E0  U    U R  
,
 
2R 2 
(2.4)
where  is the reduced mass of the system A-B. The total ionisation cross section is
obtained from (2.3) by summing it over different values of l:
 tot E0  

2
 2l  1P E  .
l
0
(2.5)
l
Since the main contribution to the cross section is due to large l values, the discrete
angular momenta can be replaced by the relation   l  1 2 , where  is the impact
parameter, and   2E0 . Transforming the sum (2.5) into integral over , one
obtains:

 tot E0   2  P  d ,
(2.6)
0
where P() is calculated from (2.3) and (2.4) with l replaced in the above described
way. In the case of AI, the upper limit of integral (2.6) should be replaced by max,
which is determined by the maximal internuclear distance RAI [Fig. 2.1(a)], or Rc [Fig.
2.1(b)], at which the associative ionisation is still possible.
In order to compare with the experimental cross sections, (2.6) should be
averaged over the corresponding distributions f(E0) of energy E0 (i.e., the collision
velocity v 0  2E0  ):
 tot    tot E 0  f E 0 dE 0 .
Distribution functions for different experimental conditions (single and crossed beams,
thermal cells) are considered in [1,38].
Detailed theoretical calculations for particular reactions require precise
information on behaviour of the molecular terms at small internuclear distances, which
is by itself a complicated task of quantum chemistry. Therefore, in most cases
qualitative molecular potentials are used (see, e.g., [71]), which limits the accuracy of
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calculations. The calculations are complicated also by a fact that it is necessary to take
into consideration the energy-transfer-type processes described in part 1, which
influence the probability that the system will survive on the initial tem U  R  until the
beginning of autoionisation. Moreover, at small internuclear distances one has to
account for interactions with Rydberg terms, which not only diminish the population of
U  R  , but may also give a contribution to the ionisation (see, e.g., [72]).
Until now calculations based on accurate molecular terms have been possible
only for the well-known Na(3p) + Na(3p) AI reaction [20,72]. As about the AI in
collisions of molecules and atoms, it has not been treated theoretically until now.
2.2. MOLECULE-ATOM ASSOCIATIVE IONISATION
Contrary to the associative ionisation in atom-atom collisions, only few
qualitative experiments have been conducted until recently on molecule-atom AI, with
either rare gas atoms or N2 molecules as one of the collision partners [23,24,73,74].
Molecule-atom AI in alkalis was observed for the first time recently in collisions of
electronically excited molecules with ground state atoms [75]:
Na2( 21  g ) + Na  Na 3 .
(2.7)
The authors of [75] used the process (2.7) as an ion source in an ionisation detector
diode, and attempts to study the very AI process were not reported. In another study
[26] AI in collisions of ground state molecules and excited atoms was considered:
Na2( X 1 g ) + Na(4D)  Na 3 + e;
(2.8)
Na2( X 1 g ) + Na(5S)  Na 3 + e.
(2.9)
These experiments were performed in an effusive Na / Na2 beam, using time-of-flight
(TOF) mass spectrometry to distinguish among different ionic products. The cross
sections of processes (2.8) and (2.9) were determined relative to those of the atom-atom
AI reactions [76]:
Na(3S) + Na(4D)  Na 2 + e,
(2.10)
Na(3S) + Na(5S)  Na 2 + e.
(2.11)
The experiments showed that the cross sections of molecule-atom AI exceed those of
the corresponding atomic reactions (2.10) and (2.11) two times in the case of the 4D 5/2
state, and 10 times in the case of the 5S1/2 state. Except the above mentioned processes,
no more data are available in the literature on the molecule-atom AI.
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