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Cohen & Fano (CF) Model

Cohen & Fano (CF) Model. H. Interferences come from the coherent emission from both nuclei of the molecule. H. CF Model Basic Hypothesis. CF-I: Monoelectronic Process. CF-II: LCAO for the bound molecular state. CF-III: Free Wave for the ejected electron. CF Cross Sections.

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Cohen & Fano (CF) Model

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  1. Cohen & Fano (CF) Model H Interferences come from the coherent emission from both nuclei of the molecule H CF Model Basic Hypothesis • CF-I: Monoelectronic Process • CF-II: LCAO for the bound molecular state • CF-III: Free Wave for the ejected electron

  2. CF Cross Sections σH2 = A σH [1+Sin(ke R)/(keR)] σH: effective H cross section ke: ejected electron momentum R: internuclear distance Interference factor σH2 / σH= A [1+Sin(ke R)/(keR)]

  3. Interference for Kr34+ /H2 Stolterfoht et al, PRL 87 (2001) 023201

  4. (e,2e) Theoretical Model The reaction of interest is We consider only asymmetric arrangements and coplanar geometries at high incident energies. The following approximations are made: • The ionization process may be treated as a pure electronic transition. • Only vertical transitions at the fixed equilibrium distance are considered (Fixed Nuclei Approximation, FNA). • Exchange effects are neglected.

  5. 3C Molecular Model To approximate the final wave function, the molecular 3C model is employed (Stia et al., 2002 PRA 66, 052709): Final wave function: with (a = 1.3918, r0 = 1.406) Coordinates used in the description. and ( j = a,b)

  6. (e,2e) Transition Matrix Element The T-matrix element for 3C is approximately given by, c = ke- K and K = ki – ks R is the internuclear vector Transition matrix element for an effective H atom placed at either molecular nuclei. Analogous results are obtained for a First Born approximation.

  7. qs = 1° qs = 8° 3C Triple Differential Cross Section Ratios (c = ke- K) Ratio as a function of both the ejection angle and energy. Ei = 4087 eV (Stia et al., 2003 JPB 36 L257) • The ratios show oscillations around unity. • Maximum interference values around qe = 270° where the binary encounter condition is satisfied (ke ≈ K).

  8. Open circles: Experimental DDCS 2 x theor. DDCS effective H B1 Double Differential Cross Section Ratios (c = ke- K) Ratio corresponding to D2 targets as a function of the ejection velocity. Ei = 2400 eV (Kamalou et al. PRA) Coloured lines: B1 ratios BE region

  9. B-splines

  10. Photoionization Matrix ElementF. Martín, J. Phys. B 32 (1999) R197 Ψg is obtained from a CI calculation D is the dipole momentum operator Ψ+ results from a CC calculation ep is the polarization vector Fixed Nuclei Approximation (FNA):

  11. H2FNA (1sσg)Results l=1 l=1 Total Total l=3 l=3 l=5 l=5 Fojon et al, J Phys B 37 (2004) 1

  12. H2 FNA 4-channel Results Present 4-channel results including the first four ionization limits 1sσg, 2pσu, 2pπu, 2sσg Cross sections are dominated by the first ionization limit (1sσg) CF-I is right Fojon et al, J Phys B 37 (2004) 1

  13. H2 FNA Results

  14. H2/2H Ratios • Present 4-channel and 10-channel results are in good agreement • CF Bad Behaviour at low energies • Model calculations show that failure of CF is related with screening and electronic correlation Model Fojon et al, J Phys B 37 (2004) 1

  15. Nuclear Degrees of Freedom

  16. 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 1Σu+ states.

  17. At Eph= 3 a.u., the vibrational states υ=2,3 are not dominant!

  18. 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.

  19. Conclusions • Interferences effects were identified in (e,2e) and Photoionization processes. • These effects are due to the two-centre nature of the molecular target. • Experimental evidence of the interferences was presented for electrons impacting on D2 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 H2 • Non Franck-Condon transitions related to the studied interference effects have been predicted for H2

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