non-local nuclear dynamics
DESCRIPTION
Non-Local Nuclear Dynamics. Martin Čížek Charles University, Prague. 1348. Dedicated to Wolfgang Domcke and J iří Horáček. Studied processes:. A B(v) + e - AB(v’ v) + e - (VE) AB(v) + e - A + B - (DA) - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/1.jpg)
Martin ČížekCharles University, Prague
Non-Local Nuclear Dynamics
Dedicated to Wolfgang Domcke and Jiří Horáček
1348
![Page 2: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/2.jpg)
Studied processes:
AB(v) + e- AB(v’ v) + e- (VE) AB(v) + e- A + B- (DA)
A + B- AB(v) + e- (AD)
AB(v) + e- (AB)- A + B-
![Page 3: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/3.jpg)
Outline of Theory
Review: W. Domcke, Phys. Rep. 208 (1991) 97
http://utf.mff.cuni.cz/~cizek/
• Fixed nuclei calculation as a first step.
• Fano-Feshbach projection to get the electronic basis.
• Known analytic properties of matrix elements (threshold expansions) used to construct proper model.
• Nuclear dynamics solved assuming diabaticity of basis.
![Page 4: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/4.jpg)
Electronic structure for fixed-R
A+ + B-
A + B
Negative ion system (HCl)- Two state Landau-Zener model
H + Cl-
HCl + e-
Main idea behind the theoretical approach (O’Malley 1966):
Selection of proper diabatic electronic basis set consisting of anionic discrete state and (modified) electron scattering continuum
![Page 5: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/5.jpg)
Extraction of resonance from the continuum
Essence of the method:
Selection of a square integrable function (discrete state) describing approximately the resonance and solution of scattering problem with additional constraint (orthogonality to the discrete state)
It is show that sharp resonance structures are removed from continuum with sensitive choice of discrete state
Example:
Scattering of particle from spherical delta-shell. Discrete states – bound states in box with the same size as the shell.
![Page 6: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/6.jpg)
Discrete state … Continuum …
Coupling
Diabaticity of the basis:
Hamiltonian in the basis:
Final diabatic basis set),( rRd
),( rR
)(RVH ddeld
)'())(( 0' RVH el
)(RVH deld
0),( ,0),(
rR
RrR
R d
dVVVVTH dddddddN )( *0
00)(
0)(0
0)(0
00)(
*0
0 RV
RV
TRV
RVT
VHd
d
N
dN
![Page 7: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/7.jpg)
Nonlocal vibrational dynamics in (AB)- state
• Expansion of wave function),()( ),()(),( rRRdrRRrR d
• Projection Schrödinger equation on basis
)( )()( )(
)( )( )( )( 0),()(*
0 RRVRRVTERRVdRRVTErRHE
dN
ddN
• Formal solution of second line for (R) into first line
)'(')( )',,( where
0)( )',,(')( *1
0 RVRiVTERRVdRREF
RRREFdRRVTE
dNd
dN
• The similar procedure for Lippmann-Schwinger equation yields:
)AB(vefor )(BAfor
e wher,)( -0
-1
ivd
iKR
dNii
VEGe
FViTE
![Page 8: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/8.jpg)
wherewhere
Threshold behaviorThreshold behavior
)()(2
)1(21
22
2
RERVR
JJR dd
0)'()'()',,(')( RRRREEfdRR dJv
Jv
v
Jv
Equation of motion for nuclei
)'()()0'(')',,( *''
1 RVRVidRRf dd
)(~)(RVd
210 :scattering dipole
21 :scattering wave-s
23 :scattering wave-p
![Page 9: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/9.jpg)
Nonlocal resonance modelDynamics is fully determined by knowledge of the functions
V0(R), Vd(R), Vd(R)
)'/(),'( '..1),( |)(|2),( 2
RdpvRRVR d
')(2
)()',,(0
RiRRREF VTE N
It is convenient to define:
Then it is)(
2)( RiR
![Page 10: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/10.jpg)
Summary – our procedure
• Model parameters V0(R), Vd(R) and Vdε(R) found from Fano-Feshbach or fit for fixed-nuclei
• Analytic fit made for R and e-dependencies in Vdε(R) to be able to perform the transform
and efficient potential evaluation• Nuclear dynamics is solved for ψd(R) component• Cross sections or other interesting quantities are
evaluated
)'()()0'(')',,( *''
1 RVRVidRRf dd
![Page 11: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/11.jpg)
Results HCl(v) + e- HCl(v’) + e- (VE)
![Page 12: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/12.jpg)
Results – vibrational excitation in e- + HCl
Integral cross section. Theory versus measurement of Rohr, Linder (1975) and Ehrhardt (1989)
Differential cross section. Measurement of Schafer and Allan (1991)
![Page 13: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/13.jpg)
Results – vibrational excitation in e- + HCl
Elastic cross section. Theory -- resonant contribution (top) versus measurement of Allan 2000 (bottom)
Vibrational excitation 0->1. Theory (top) versus measurement of Allan 2000 (bottom)
![Page 14: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/14.jpg)
VE in e-+H2
![Page 15: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/15.jpg)
Interpretation of boomerang oscillations
• Dashed line = neutral molecule potential
• Solid line = negative ion – discrete state potential
• Circles = ab initio data for molecular anion
Boomerang oscillations:interference of direct process and reflection from long rangepart of anion potential
![Page 16: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/16.jpg)
Results HBr(v) + e- H + Br- (DA)
![Page 17: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/17.jpg)
Results – DA to HBr and DBr
Comparison with measurement of Sergenton and Allan 2001
![Page 18: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/18.jpg)
Results
H2 + e- ↔ H2- ↔ H-+H
![Page 19: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/19.jpg)
M. Čížek, J. Horáček, W. Domcke, J. Phys. B 31 (1998) 2571
H+H- → e- + H2
![Page 20: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/20.jpg)
M. Čížek, J. Horáček, W. Domcke, J. Phys. B 31 (1998) 2571
![Page 21: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/21.jpg)
Potentials for J=0 Potential Vad(R) for nonzero J
The Origin of the Resonances
![Page 22: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/22.jpg)
Cross section
AU
TOD
ETA
CH
MET
Resonant tunneling wave function En
ergy
)(Rd
Vad(R) + J(J+1)/2μR2
![Page 23: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/23.jpg)
Elastic cross section for e- + H2 (J=21, v=2)
![Page 24: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/24.jpg)
Γ0=2.7×10-4eV
![Page 25: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/25.jpg)
Elastic cross section for e- + H2 (J=25, v=1)
![Page 26: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/26.jpg)
Γ0=2.7×10-9eV
Γ1=1.9×10-6eV
![Page 27: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/27.jpg)
Table I: Parameters of H2- states
J Eres (relative to DA) τ
21 -136 meV 2.4 ps
22 -105 meV 12 ps
23 -75 meV 0.11 ns
24 -47 meV 0.9 ns
25 -20 meV 12 ns
26 5 meV 0.52 μs
27 28 meV 2 ns
![Page 28: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/28.jpg)
Table II: Parameters of D2- states
J Eres(relative to DA) τ
31 -118 meV 0.13 ns
32 -97 meV 0.70 ns
33 -76 meV 6 ns
34 -55 meV 39 ns
35 -35 meV 0.51 μs
36 -16 meV 5.7 μs
37 2 meV 14 μs
38 19 meV 7.2 μs
39 34 meV 41 ps
![Page 29: Non-Local Nuclear Dynamics](https://reader036.vdocument.in/reader036/viewer/2022062310/5681602c550346895dcf380f/html5/thumbnails/29.jpg)