Cohen & Fano (CF) Model
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Cohen & Fano (CF) Model 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 Model Basic Hypothesis
description
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. - PowerPoint PPT Presentation
Transcript of Cohen & Fano (CF) Model
Cohen & Fano (CF) ModelCohen & 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 HypothesisCF Model Basic Hypothesis
CF Cross SectionsCF Cross Sections
Interference factor
σH2 = A σH [1+Sin(ke R)/(keR)]
σH: effective H
cross section
ke: ejected
electron momentum
R: internuclear distance
σH2 / σH = A [1+Sin(ke R)/(keR)]
Interference for Interference for KrKr34+34+ /H /H22
Stolterfoht et al, PRL 87 (2001) 023201
(e,2e) Theoretical Model(e,2e) Theoretical Model
The reaction of interest is
)()( 2
2
1
2
gg HeeHe
• 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.
),,,,(),()2(
2122/3 rrRkkrρ.Rk
seCf
i
f
se
Final wave function:
),,(),,(),,()2(
),,,,( 1112/321
1
epppsjseje
i
seC CCCe e
rkRkrkrrRkk.rk
with
and
)()(),( 22
02ba rr
ff eeN r
= 1.3918, 0 = 1.406)
3C Molecular Model3C Molecular Model
To approximate the final wave function, the molecular 3C model is employed (Stia et al., 2002 PRA 66, 052709):
( j = a,b)
Coordinates used in the description.
(e,2e) Transition Matrix Element(e,2e) Transition Matrix Element
Hif
eif tt )2/cos(2)( χ.RR
= ke - K and K = ki – ks
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.
Hift
R is the internuclear vector
Ratio as a function of both the ejection angle and energy. Ei = 4087 eV (Stia et al., 2003 JPB 36 L257)
)3(2
3)3(
2
)sin(12
)2/( Hese
H RχRχ
kdddd
s = 1° s = 8°
)3(
)3(2
2 H
HRatio
• The ratios show oscillations around unity.
• Maximum interference values around e = 270° where the binary encounter condition is satisfied (ke ≈ K).
= ke - K)
3C Triple Differential Cross Section Ratios3C Triple Differential Cross Section Ratios
Ratio corresponding to D2 targets as a function of the ejection velocity. Ei = 2400 eV (Kamalou et al. PRA)
)3(2
2)2(
2
)sin(12
)2/( Hsee
H RχRχ
dkdd
d
)2(
)2(2
2 H
HRatio
Coloured lines: B1 ratios
Open circles: Experimental DDCS
2 x theor. DDCS effective H
BE region
= ke - K)
B1 Double Differential Cross Section RatiosB1 Double Differential Cross Section Ratios
B-splinesB-splines
Photoionization Matrix ElementPhotoionization Matrix ElementF. Martín, J. Phys. B 32 (1999) R197
22
||3
4 p
Emlgml
Dec
22
),(||),(3
4 p
Emlpgml
RrDeRrRdc
Fixed Nuclei Approximation (FNA):
D is the dipole momentum operatorΨg is obtained from a CI calculation
Ψ+ results from a CC calculation ep is the polarization vector
HH22 FNA FNA ((1sσ1sσgg)) ResultsResults
Total Total
l=1 l=1
l=3 l=3
l=5l=5
Fojon et al, J Phys B 37 (2004) 1
HH22 FNA 4-channel FNA 4-channel ResultsResults
Cross sections are dominated by the first ionization limit (1sσg)
CF-I is right
Present 4-channel resultsincluding the first four ionization limits 1sσg,
2pσu, 2pπu, 2sσg
Fojon et al, J Phys B 37 (2004) 1
HH22 FNA R FNA Resultsesults
HH22 /2H Ratios/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 FreedomNuclear Degrees of Freedom
Non Franck-Condon (FC) Non Franck-Condon (FC) TransitionsTransitions
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.
At Eph= 3 a.u., the
vibrational states υ=2,3 are not
dominant!
Interferences & Non FC Interferences & Non FC TransitionsTransitions
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.
ConclusionsConclusions
•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
