PEDAGOGÍA DE LA ERE

The scattering matrix representation is not determined by directly applying Eq. (2.164). Instead the reactance matrix of scattering theory is first calculated, according to,

K =BA−1. (2.171)

Using a standard linear-algebra diagonalisation technique, this is then decomposed into a product of two matrices according to

K=MtanξM†. (2.172)

Here, the columns of M are orthogonal and comprise the eigenvectors of K, and the nonzero elements of the diagonal matrix tanξ are its corresponding eigenvalues. Further- more, ξ are the phase shifts of scattering theory. An equivalent version of Eq. (2.164) in terms of the phase shift and eigenvectors of K is given by,

U =A−12−12iMcosξe−iξM†. (2.173)

Finally, the coupled-channels solution must be transformed to the correct asymptotic form,

χS_n =χnU. (2.174)

2.12

Coupled channels cross sections

The numerical wavefunctions described in Sec. 2.11.3 and Sec. 2.11.5 must be combined with operators corresponding to observable quantities before they can be compared with an experiment. The cross section representing absorption from an unmixed lower-state to a coupled upper-state is one such observable, and may be computed by means of Eq. (2.101) and Eq. (2.74), according to

σij(ν) = πν 3~_ǫ0 X k Z ∞ 0 χ†_ik(R)Re_kj(R)χj(R)dR·S1_J/_k2_JjΩkΩj 2 . (2.175)

Here,Re_kj is theR-dependent electric-dipole transition moment between the lower-state, j, and channel statek. Here,ν= (Ei−Ej)/2π~crepresents the wavenumber of an absorption transition, and Ei and Ej are the energies of state i and coupled-state j, respectively. Simpson’s method is used to approximate the integral between model grid points once the various quantities appearing in Eq. (2.175) have been specified in numerical form.

If the set of coupled-channels are all closed, and χij(R) has been calculated with

E corresponding to a bound-state energy eigenvalue, then σij and Eq. (2.75) may be used to calculate a discrete transition f-value. In an absorption experiment, this may be compared with the linef-value of an observed transition. If χij(R) contains at least one open channel, and the asymptotic boundary conditions of Sec. 2.11.4 have been applied, then those iwhich correspond to open channels may be used to calculate an absorption cross-section, σij(ν), which is continuous with respect to ν. In this case, transitions to closed-channel solutions are disregarded as nonphysical but nonetheless induce pseudo- resonant structure in the cross-sections calculated for open channels to which they are coupled. An example of a calculated cross section showing resonant behaviour is plotted in Fig. 2.5.

It is not necessary that all coupled states have nonzero transition moments, Re kj(R).

113200 113400 113600 113800

Transition energy (cm

)

10

10

10

10

10

Cross section (cm

)

Figure 2.5: An example absorption cross section calculated by the coupled-channels method and

showing a resonant feature with Fano lineshape and a varying background continuum.

Even nominally forbidden transitions may appear as resonances in the coupled-channels cross section if an inaccessible excited state obtains an admixture of nonzero transition moment because of off-diagonal elements in the interaction matrix.

Actual measurements of absorption are insensitive to the the final state of the dis- sociated atoms. The total absorption cross section is then found by summing over all open-channels, according to σj(ν) = NO X i=1 σij(ν). (2.176)

The quantity σij(ν)/~ν corresponds precisely to a partial photodissociation cross sec- tion. This is defined to be the rate of production of pairs of atomic dissociation products in state i, from a unit density sample of molecules in state j, exposed to a unit intensity photon beam of energy ν. This quantity may be directly compared with an experiment that detects the scattered atoms and analyses their kinetic energies. A dissociation cross section calculated in this way will only correspond to reality when the upper-state is certain to dissociate. If there exist alternative decay pathways, principally spontaneous emission, this calculation will overestimate the true photodissociation cross section; and the calcu- lated pseudo-resonance lineshapes will not include the effects of emission broadening and appear narrower than reality.

The resonant feature in Fig. 2.5 dominates the surrounding continuum in terms of its peak cross section and the integrated cross section in its neighbourhood. Such resonances then comprise much of the useful information computed by the coupled-channels model. A calculated cross section may be summarised by parameterising the transition energies, strengths, and widths of the resonances that appear within it. In most cases a Lorentzian or Fano lineshape will be well-matched to the calculated resonances.

Regardless of the treatment of dissociative or emissive decay, the strength parameter of a Fano lineshape fitted to each model resonance is directly indicative of electric-dipole transition intensity and may be directly employed by Eq. (2.70) to calculate absorption

f-values. These may be directly compared with experimental f-values. Additionally, the width of each resonance may be compared with experimental linewidths, or used to calculate the dissociative-decay lifetime according to Eq. (2.79).

2.12.1 Band models

In absorption experiments the predissociation rate of an excited state may be very fast and the observed rotational lines so broadened as to be indistinguishable. Alternatively,