Presentation on theme: "Cohen & Fano (CF) Model CF-I: Monoelectronic Process CF-II: LCAO for the bound molecular state CF-III: Free Wave for the ejected electron H H Interferences."— Presentation transcript:
Cohen & Fano (CF) Model CF-I: Monoelectronic Process CF-II: LCAO for the bound molecular state CF-III: Free Wave for the ejected electron H H Interferences come from the coherent emission from both nuclei of the molecule CF Model Basic Hypothesis
CF Cross Sections Interference factor σ H2 = A σ H [1+Sin(k e R)/(k e R)] σ H : effective H cross section k e : ejected electron momentum R: internuclear distance σ H2 / σ H = A [1+Sin(k e R)/(k e R)]
Interference for Kr 34+ /H 2 Stolterfoht et al, PRL 87 (2001)
(e,2e) Theoretical Model The reaction of interest is The ionization process may be treated as a pure electronic transition. We consider only asymmetric arrangements and coplanar geometries at high incident energies. The following approximations are made: Only vertical transitions at the fixed equilibrium distance are considered (Fixed Nuclei Approximation, FNA). Exchange effects are neglected.
Final wave function: with and = , 0 = 1.406) 3C Molecular Model To approximate the final wave function, the molecular 3C model is employed (Stia et al., 2002 PRA 66, ): ( j = a,b) Coordinates used in the description.
(e,2e) Transition Matrix Element = k e - K and K = k i – k s The T-matrix element for 3C is approximately given by, Analogous results are obtained for a First Born approximation. Transition matrix element for an effective H atom placed at either molecular nuclei. R is the internuclear vector
Ratio as a function of both the ejection angle and energy. E i = 4087 eV (Stia et al., 2003 JPB 36 L257) s = 1° s = 8° The ratios show oscillations around unity. Maximum interference values around e = 270° where the binary encounter condition is satisfied (k e ≈ K). = k e - K) 3C Triple Differential Cross Section Ratios
Ratio corresponding to D 2 targets as a function of the ejection velocity. E i = 2400 eV (Kamalou et al. PRA) Coloured lines: B1 ratios Open circles: Experimental DDCS 2 x theor. DDCS effective H BE region = k e - K) B1 Double Differential Cross Section Ratios
Photoionization Matrix Element Photoionization Matrix Element F. Martín, J. Phys. B 32 (1999) R197 Fixed Nuclei Approximation (FNA): D is the dipole momentum operator Ψ g is obtained from a CI calculation Ψ + results from a CC calculation e p is the polarization vector
H 2 FNA (1sσ g ) Results Total l=1 l=3 l=5 Fojon et al, J Phys B 37 (2004) 1
H 2 FNA 4-channel Results Cross sections are dominated by the first ionization limit (1sσ g ) CF-I is right Present 4-channel results including the first four ionization limits 1sσ g, 2pσ u, 2pπ u, 2sσ g Fojon et al, J Phys B 37 (2004) 1
H 2 FNA Results
H 2 /2H Ratios CF Bad Behaviour at low energies Present 4-channel and 10-channel results are in good agreement Model calculations show that failure of CF is related with screening and electronic correlation Model Fojon et al, J Phys B 37 (2004) 1
Nuclear Degrees of Freedom
Non Franck-Condon (FC) Transitions At Eph= 1, 8 and 14 a.u., the most probable final vibrational states are the ones with ν=2,3 which is in agreement with previous results (Martín, 1999) and with a FC transition. However, at Eph= 3 a.u., the contribution of these states is “swallowed up” by the interference effect affecting mainly l=1 partial waves in the 1 Σ u + states.
At E ph = 3 a.u., the vibrational states υ=2,3 are not dominant!
Interferences & Non FC Transitions Non FC transitions are put in evidence in the vibrational analysis of the cross sections ratio. A FC process corresponds to a vertical transition in which the internuclear distance is constant. In this way, the interference pattern coming from the coherent emission from the two nuclei of the molecule is closer to the one corresponding to ν=2,3.
Interferences effects were identified in (e,2e) and Photoionization processes. Conclusions These effects are due to the two-centre nature of the molecular target. Experimental evidence of the interferences was presented for electrons impacting on D 2 molecules. Fairly good agreement between first-order calculations and measurements data is found. Failure of the CF model has been detected at low photon energies for H 2 Non Franck-Condon transitions related to the studied interference effects have been predicted for H 2