MONTAJE DE LA TORRE ESTRUCTURAL EN CAMPO

The non-spherical and multicentred nature of the molecules and the number of degrees of freedom involved in the electron-molecule interaction make it very difficult to calculate the equations and the potentials of the interaction. Therefore, different methods have been used in order to simplify the equations that govern the electron-molecule interaction. These methods will be listed with a brief description • The Born Approximation (BA) is used by replacing the scattered spherical wave

function by a plane one for large radial distance r. This leads to a Bom expansion of the scattering amplitude as powers in the potential strength, which makes it useful in the cases of a weak potential where only the first, and may be the second, term in Bom series is important. The modified scattering amplitude becomes

/ ( ^ ) = TTT" X “ i)f;(cos 9) + 1.32 Q(^) - 2 i k lin a k /=o 1 1 . ^ sin 3 2 0 ^ /) (cos 9) 1.33 V ^ y i>L (2/ + 3X2/ - 1) tan S f = I U (,f^dr 1.34 0 9 l2/+1 » 1.35

where Cuo) represents the dipole-Bom approximation of the higher partial waves (/>L), a is the polarisability and Qq is the Bohr radius. Uq-) is the reduced spherical

symmetric potential The Bom approximation is the basis for most

scattering theories and modifications have been introduced to equation 1.32 (see

chapter 4, section 4.2) to include more complicated scattering processes and the interaction potentials.

• Close Coupling Theory (CCT) truncates the higher terms in the expansion o f the total wave function in equation 1.22. The theory limits the expansion to the terms that are close in energy between the initial and final states i.e. before and after scattering. The truncated terms, even though they are inaccessible, still contribute to the scattering process as they are responsible for polarisation effects (see section 1.3 polarisation potential). However, a large number of coupled equations are generated in this approach due to small energy spacing between rotational and vibrational states, therefore, a full ro-vibrational close coupling approach is only feasible for low energy e-Hi collision.

A result of the CCT is the convergent close coupling (CCC) method developed by Bray and Stelbovics (1992) for solving the coupled equations. Convergence is tested by including an ever increasing set of target states in the CCT formalism. It is the CCC theory that describes the electron-atom scattering processes at all non- relativistic energies.

• Fixed Nuclei Approximation (FNA) was introduced by Massey ( 1930) to simplify the electron-molecule interaction by assuming that the inter-nuclear distance and the direction o f the molecular axis are fixed during the scattering process. Also assuming that the molecule is in the ground electronic state, the expansion in equation 1.22 is reduced to one term only. The calculations are computed frame by frame allowing for the change of the molecular orientation and the inter- nuclear separation, then the scattering amplitude is averaged over all possible states. The predictions of the FNA are in good agreement with experiments mainly at high incident electron energies (>100 eV).

Massey extended the theory to calculate rotational and vibrational cross

sections using what is called Adiabatic Nuclear Approximation (ANA), where the

scattering process is solved in two steps. First the FNA is used in the molecular frame then the scattering amplitude is averaged over the initial and final states o f the nuclear wave function before transforming into the laboratory frame. This approximation is valid where the collision time ( -1 0’^^ s) is very short compared to rotational (1 0'^^ s) and vibrational (1 0'^"* s) periods, and where the number of excited states o f the target is limited. However the FNA and the ANA are found to diverge for forward scattering directions, 0< 30°, of polar molecules but still valid for scattering angles 6> 30°.

• Frame transformation Theory(FTT) was formulated by F ana (1970) and Chang

and Fano (1972) to combine the advantages o f the CCT and FNA. The theory divides the electron scattering into two regions: 1) In the inner region the incoming electron is just inside the molecular electron cloud (r^), so it accelerates and sees a frozen nucleus. Thus for r < Vc the FNA is suitable for treating the collision process. 2) In the outer region the nuclear motion is included where the degree of coupling between different states is weak. Thus for r > the CCT is suitable to describe the scattering process. Then the FFT is utilised to bring the two regions together and matched at an arbitrary value (see Burke 1979 and

Buckley et al 1984 for details).

An alternative approach, known as the Angular Frame Transformation(AFT,

Collins and Norcross 1978), divides the orbital angular momentum space into two regions having angular momentum h at a common boundary. For / < // ANA is used and for / > // a suitable lab-frame calculation is used (full details in Norcross and

Collins 1982).

A number o f computational methods have been used to solve the electron- molecule scattering processes within the various theoretical approximations mentioned above. Many of these methods predict DCSs and GTSs very close to the experimental results and within the experimental errors. The main disagreement usually lies in the forward and backward scattering regions, as they were until recently experimentally inaccessible.