The transition for the H2O (x ^Ai) + e' {Eo) = H2O’ (^Ai, ^Bi or ^8 2) reaction is zero when the beam of the incident electrons is in the plane of the water molecule and non-zero if it is perpendicular to the plane of the water molecule. In other words, the transition reaction will behave as a cosine function in ^(and such that the transition
Toe u.{OH X O H' ) , where u is the velocity unit vector of the incident electron beam and { O Hx QH ' ) represents the area vector o f the water molecule H-O-H’; $ i s the elevation angle in the horizontal scattering plane and (j) is the azimuthal angle in the vertical x-y plane.
If the incident electron energy is greater than the threshold energy for any of the resonant states o f the negative molecular ion then the molecule may undergo a dissociative attachment process. Provided that the lifetime o f the H2O' resonant state is shorter than the rotational period of the water molecule the KT ion, which is released in dissociative attachment process via OH + H~, intensity will be symmetrically distributed about ^(and = 90° {Lassettre and Huo 1974).
Azria et al (1979) have computed the angular distribution of the H' ions produced by dissociative attachment mechanism of ^Ai, ^A2, ^Bi or ^ 8 2 states o f H2S", which is very similar to the molecular structure o f the H2O’, as shown in figure 4.1.2a.
Azria uses the O'Malley and Taylor (1968) theory adapted to polyatomic molecules that is developed to describe the symmetry resonant states of the H2S' ions.
s wave d wave d W a v e f/w a v e 0.5
I
X) (5 d \w av e pwave p /w ave 0,5 (p+d)/2, d \w a v e 30 60 90 120 150 30 60 90 120 150 Angle (degrees)F ig u re 4.1.2a Calculated angular distribution for H' ions dissociated from H2S' for allowed values o f / < 3 for each o f the resonant states. H2O' has a very sim ilar m olecular structure and angular distribution to H2S' {Azria et al 1979).
4.2 Interaction Potentials in H2O
Theoretical calculations have been made to determine the potentials of interaction (K% and in an attempt to understand the forces that play a crucial role in the angular distribution of the scattered electrons.
Gianturco (1991) computed the parameter-free model interaction for the H2O
molecule in the fixed-nuclei approximation. The static exchange and
polarisation potentials are evaluated and the eigenphase sums are also examined where the number of partial waves included in the expansion of the potentials reaches
1=1.
Figure 4.2a represents the static, exchange and polarisation potentials as a function of the electron separation from the centre of the water molecule. Figure 4.2a (a) shows the spherical component of these potentials for / = 0 and it is clear that the static potential dominates the interaction process in the inner region (r < rc) of the molecular volume. However, as r increases beyond 30o the interaction becomes dominated by the polarisation potential and hence it dominates the electron scattering process in the forward direction. For / = 1 the three components of the interaction potentials are shown in Figure 4.2a (b). The polarisation and exchange potentials are ~ zero while the peak of the static potential at ~ 1 . 8 Oq indicates that the incoming electron will almost only see the static potential over the H atom and H2O molecule will appear as a nearly spherical heavy atom.
0 9 1 = 0 3 re re c 0.0 1.2 0.4 1.2 2.0 2.8 3.6 4.4 0.4 2.0 2.8 3.6 4.4 r (a.u.) /-(a.u.)
F ig u re 4.2a C om puted interaction potentials: static potential exchange potential (V ^ ) and correlation polarisation potential (P ^) for e' + H^O (^A,) in the fixed nuclei approxim ation, (a) w ith / = 0 and (b) / = 1 components. Note the potential sign is changed {Gianturco 1991).
Gianturco (1991) has also included the eigenphase sums for the two symmetries as a function of the incident electron energies. The changes in the eigenphase sum with collision energy indicate the presence of resonances over the range of changes as shown in figure 4.2b.
In figure 4.2b (b) the calculations only include the static and exchange potentials (SE) where the changes in the eigenphase sum with the collision energy are marginal. This indicates that the static and exchange potentials contribute little to the resonances. Figure 4.2b (a) where the static, exchange and polarisation (SEP) potentials are employed shows a rapid change in the 6 2 eigenphase sum with collision energy. The conclusion is that there is a 8 2 resonance over a wide range of collision
energies and that the polarisation potential is very crucial in detecting and locating shape resonances. Regarding the Ai resonant state there is a rapid change in its eigenphase sum for collision energies below 6 eV and therefore it contributes to the scattering process at low energies (< 6 eV). Jain and Thompson (1983) also see a significant contribution from the ^Ai resonant state at < 5 eV incident electron energy, which is outside the range of energies in this work, as shown in figure 4.2c.
(a) SEP e + H j O 2.0 S 0.6- M 0.4- Cl. g 0.2- 00 ^ 0.0- 0.4- -0.2- 0.0 6 18 2 10 14 22 2 6 10 14 18 22
Energ> (eV) Energy Eo (eV )
F ig u re 4.2b Com puted eigenphase sums in the energy range 2 to 20 eV. (a) All interaction potentials are included i.e. static, exchange and polarisation (SEP) and (b) only static and exchange (SE) potentials are included in the scattering resonance states and {G ianturco 1991).
8 6 I o 4 V] U 2 0 0 2 4 6 8 10
Incident Electron Energy (eV)
F ig u re 4.2c The contribution o f the individual symm etry states Bj, A% and B2 to the integral cross section o f th e vibrational excitation ( 0 0 0 1 0 0) as a function o f the incident electron energies {Jain and Thompson 1983).
Gianturco (1991) comments on previous experimental results: “Earlier suggestions of shape resonances in the vibrationally inelastic channels {Seng and
Linder 1976) had tentatively tried to assign it to a ^Ai state for the negative ion. The present, fixed-nuclei calculations indicate, however, the possible assignment o f a
broad shape resonance to a molecular state. ... one indeed finds that the dominant
angular momentum component in the ^ 8 2 channel for the eigenphase sums is the p wave with / = 1. This suggests therefore that a short-lived H2O' species can exist in the vibrationally inelastic channel.”
Further theoretical studies of the electron impact excitation of the water molecule by Morgan (1998) and Gil et al (1994) have confirmed the three Feshbach resonant states, but at slightly different resonance energies. This is because they use different computational methods, different excitation thresholds and different wave functions.
Morgan finds that the cross sections o f excitation o f the three states (’Bi, ^Ai and ^A]) and the fourth state (^Bi) are dominated by the resonant states ^ 8 2 and ^Ai, respectively, in the energy range from 7 to 20 eV. However, Gil et al have employed a trial wave function which shows insufficient resonant structure and energy position. It also leads to the Ai state dominating the electronic excitation in e' + H2O collisions. One deduces that Gil et al's trial wave function may be not adequate to describe the collision process.
4.3 Experim ental Review
Experimental electron scattering from water molecules are scarce, mainly inelastic scattering, and the available data are compared with the current results in the relevant sections. In this section the comparison will be extended to include previous experimental results on dissociatve attachment and to relate them to the current work.
As mentioned in section 4.1.1 many groups confirmed the existence o f the resonant states from dissociative attachment experiments. Figure 4.3a shows the results obtained by Belie et al (1981) for H‘ ions produced by e' + H2O via dissociative attachment processes at different scattering angles. The H" ion intensities at each scattering angle are normalised to the ion’s intensity at 90° scattering angle to give the ion intensity angular distribution. In figure 4.3a (a) the incident electron energy is 6.5 eV and the angular distribution o f the H" ions exhibit the ^8 % resonant
State dominated by a p-type-wave when compared with figure 4.3a. Similarly figure 4.3a (b) and (c), where incident electron energies are 8 . 8 and 12 eV respectively, the H ion angular distribution exhibit ^Ai and resonant state characteristics which are dominated by s+d and d waves respectively. Melton (1972) and Jungen and Vogt
(1979) have obtained similar results for H2O while Azria et al (1979) have obtained very similar results for H2S.
The angular behaviour in resonant scattering processes is determined directly by the state symmetries and the number of partial waves probed, which is limited to / < 3 due to the almost atomic nature of the resonant electron orbitals. The presence of direct potential scattering, which distorts the incident electron wave (s wave), shifts the peak position to 100® scattering angle as seen in figure 4.3a (a).
2.00 Eo= 6.5 eV 1.75
I
1.50 5 0.6 1,00 0.2 0.75 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 12 eV 1.00I
I
I 0.75 r 0.50 0.25 0 20 40 60 80 100 120 140 160 180Scattering Angle 8 (degrees)
F ig u re 4.3a Experim ental results obtained by B elie et at (1981) for e' + H2O dissociative attachm ent. The data points are the angular distribution o f H ions norm alised to the ion intensity at 90° scattering angle for different incident electron energies as shown in the top right com er o f each graph, (a) Indicates that the process is dom inated by p-wave, (b) is A], dom inated by s and d-waves and (c) is
dom inated by d-wave.
Scattering Angle 6 (degrees)