Feshbach-Fano-R-matrix (FFR) method

Transcription

1 Feshbach-Fano-R-matrix (FFR) method Přemysl Kolorenč Institute of Theoretical Physics Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic Feshbach-Fano-R-matrix (FFR) method p.1/26

2 Outline Feshbach-Fano partitioning Motivation for the FFR method FFR construction of the discrete state FFR resonance position and width, cross section Application to potential scattering Application to low-energy electron-molecule scattering Conclusions Feshbach-Fano-R-matrix (FFR) method p.2/26

3 Feshbach-Fano projection approach Main idea decomposition of the T-matrix into the resonant and background part T (ɛ, ɛ )=T res (ɛ, ɛ )+T bg (ɛ, ɛ ) Resonant term T res corresponds to all rapid variations in the cross section Background term T bg smooth quantity in the energy region of interest, allows for the use of different approximations Corresponding separation of the Hilbert space: H = Q P Projection operator onto the resonant subspace Q: Q = ϕ d ϕ d Feshbach-Fano-R-matrix (FFR) method p.3/26

4 Feshbach-Fano projection approach Discrete state ϕ d square integrable function, ɛ d = ϕ d H Ω ϕ d Resonance parameters Γ(ɛ) and (ɛ) are determined by the discrete state continuum coupling V dɛ = ϕ d H bg ϕ (+) ɛ Energy-dependent resonance width and level shift function Γ(ɛ) = 2π V dɛ 2 (ɛ) = 1 2π Resonant part of the T-matrix T res (ɛ) = 1 2π Γ(ɛ ) ɛ ɛ dɛ Γ(ɛ) ɛ ɛ d (ɛ)+iγ(ɛ) Feshbach-Fano-R-matrix (FFR) method p.3/26

5 Motivation for the FFR method Major drawback of the FF approach definition of the discrete state is ambiguous Calculation of the coupling V dɛ is a very difficult task. Complex absorbing potential, stabilization methods Sommerfeld and Meyer, J. Phys. B 35 (2002), 1842 The coupling can be easily extracted from the R-matrix results provided ϕ d (r) can be expanded in terms of R-matrix basis Kazansky, J. Phys. B 31 (1998), L431 FFR method general procedure that defines the discrete state and determines its coupling with background continuum. Nestmann, J. Phys. B 31 (1998), 3929 Feshbach-Fano-R-matrix (FFR) method p.4/26

6 FFR method Expand the discrete state in terms of the R-matrix basis H Ω (r)φ Ω k (r) =(T + V eff + L)φ Ω k = E Ω k φ Ω k (r) To make it possible we have to restrict ϕ d (r) to be not only square integrable but completely contained within the internal region: ϕ d (r) =0for r r Ω. = ϕ d (r) = c k φ Ω k (r), ϕ d (r Ω )= c k φ Ω k (r Ω )=0 The derivative of ϕ d (r) vanish automatically at r Ω because of the properties of the basis: d dr φω k (r) =0 r=rω Feshbach-Fano-R-matrix (FFR) method p.5/26

7 Spectra comparison Feshbach-Fano-R-matrix (FFR) method p.6/26

8 Spectra comparison Feshbach-Fano-R-matrix (FFR) method p.6/26

9 Spectra comparison Feshbach-Fano-R-matrix (FFR) method p.6/26

10 Construction of the projector onto the background subspace Σ res energy region where cross section shows resonance structure. Basic idea of the FFR method similarity of the background Hamiltonian PHP and the model Hamiltonian H Existence of the unitary mapping within Σ res ( bg Ej Ω Σ res ) bg φ Ω j (r) = a jl φ Ω l (r), a }{{} }{{} ila lj = δ ij E l Σ res PH Ω P H Ω l Outside Σ res, the R-matrix spectrum is not affected by the presence of the resonance bg φ Ω j (r) =φ Ω j (r) E Ω j / Σ res Feshbach-Fano-R-matrix (FFR) method p.7/26

11 Discrete state expansion Restricted projector onto the P-subspace P Ω = φ Ω j φ Ω j + φ Ω j φ Ω j E Ω j Σ res E Ω j /Σ res Projector onto the resonant subspace Q = ϕ d ϕ d ϕ d (r) = c k φ Ω k (r) E Ω k Σ res Mutual orthogonality of the P and Q subspaces: PQ =0 c k φ Ω j φ Ω k = 0 for E Ω j Σ res E Ω k Σ res c k φ Ω k (r Ω ) = 0 E Ω k Σ res Feshbach-Fano-R-matrix (FFR) method p.8/26

12 Discrete state expansion Restricted projector onto the P-subspace P Ω = φ Ω j φ Ω j + φ Ω j φ Ω j E Ω j Σ res E Ω j /Σ res Projector onto the resonant subspace Q = ϕ d ϕ d ϕ d (r) = c k φ Ω k (r) E Ω k Σ res Mutual orthogonality of the P and Q subspaces: PQ =0 c k φ Ω j φ Ω k = 0 for E Ω j Σ res E Ω k Σ res c k φ Ω k (r Ω ) = 0 E Ω k Σ res Feshbach-Fano-R-matrix (FFR) method p.8/26

13 Number of R-matrix levels within Σ res : Equations for the expansion coefficients #(E Ω k Σ res ) : N #( E Ω j Σ res ) : N 1 Homogeneous system of N equations for N coefficients: N c k φ Ω j φ Ω k = 0 for j =1,...,N 1 k=1 N c k φ Ω k (r Ω ) = 0 k=1 Non-trivial solution exists only if determinant of the system is zero. This is very restrictive condition for the model system H. Feshbach-Fano-R-matrix (FFR) method p.9/26

14 Number of R-matrix levels within Σ res : Equations for the expansion coefficients #(E Ω k Σ res ) : N #( E Ω j Σ res ) : N 1 Homogeneous system of N equations for N coefficients: N c k φ Ω j φ Ω k = 0 for j =1,...,N 1 k=1 N c k φ Ω k (r Ω ) = 0 k=1 Non-trivial solution exists only if determinant of the system is zero. This is very restrictive condition for the model system H. Possible problem solving use of approximative values for the overlap integrals φ Ω j φω k Feshbach-Fano-R-matrix (FFR) method p.9/26

15 Improved Nestmann approximation Objective: to construct such an approximation for φ Ω j φω k that the conditions ϕ d (r Ω )=0and PQ =0will be linearly dependent. Schrödinger equation for the P -component of the wave function ( PHP + PHQ(ɛ QHQ) 1 QHP ) P ψ ɛ = ɛp ψ ɛ Ω-confined form, ɛ = Ek Ω (bg H Ω + P Ω H Ω ϕ d ϕ d H Ω P Ω ) Ek Ω ɛ P Ω φ Ω k = Ek Ω P Ω φ Ω k d Introduce residual potential: bg H Ω = H Ω + V rsd Multiply by φ Ω j from left, expand the projector P Ω Feshbach-Fano-R-matrix (FFR) method p.10/26

16 Improved Nestmann approximation Final (exact) implicit formula φ Ω j φ Ω k = v kj + B k φ Ω j HΩ ϕ d E Ω k E Ω j Interaction of the discrete state with background continuum B k = E Ω l Σ res φ Ω l φ Ω k ϕ d H Ω φ Ω l E Ω k ɛ d Contribution of the residual potential v kj = φ Ω l φ Ω k φ Ω j V rsd φ Ω l E Ω l Σ res Feshbach-Fano-R-matrix (FFR) method p.11/26

17 Improved Nestmann approximation Final implicit formula φ Ω j φ Ω k = B k φ Ω j HΩ ϕ d E Ω k E Ω j Interaction of the discrete state with background continuum B k = E Ω l Σ res φ Ω l φ Ω k ϕ d H Ω φ Ω l E Ω k ɛ d Contribution of the residual potential v kj = φ Ω l φ Ω k φ Ω j V rsd φ Ω l 0 E Ω l Σ res Feshbach-Fano-R-matrix (FFR) method p.11/26

18 Improved Nestmann approximation Coefficients B k are determined from the condition ϕ d (r) =0for r r Ω, which is equivalent to ( Pφ Ω k ) (r) =φ Ω k (r) for r r Ω, as B k = E Ω i Σ res φ Ω i HΩ ϕ d φ Ω i (r Ω) E Ω k E Ω i 1 φ Ω k (r Ω ) Iterative process, starting from ϕ (1) d (r) =φω l (r) φ Ω j H Ω ϕ d (1) = E Ω l φ Ω j φ Ω k (0) Feshbach-Fano-R-matrix (FFR) method p.12/26

19 Recapitulation of Improved Nestmann approximation Discrete state expansion is determined in a self-consistent iterative process via solving system of equations c k φ Ω j φ Ω k =0 for Σ res E Ω k Σ res E Ω j In order to ensure ϕ d (r Ω )=0the overlap integrals φ Ω j φω k are approximated as φ Ω j φ Ω k = B k φ Ω j HΩ ϕ d E Ω k E Ω j B k = E Ω i Σ res φ Ω i HΩ ϕ d φ Ω i (r Ω) E Ω k E Ω i 1 φ Ω k (r Ω ) Feshbach-Fano-R-matrix (FFR) method p.13/26

20 Advantages of Improved Nestmann approximation Influence of the model potential is reduced. Difficult and time consuming explicit numerical evaluation of multi-dimensional integrals φ Ω j φω k is avoided. The algorithm requires only the knowledge of the poles Ek Ω Ej Ω, and amplitudes φω k (r Ω) and φ Ω j (r Ω). and Possible simplification: φ Ω j HΩ ϕ d is assumed to be j-independent constant. Iterative process is avoided, but the algorithm becomes less stable and strongly dependent on unphysical parameters r Ω and Σ res. Feshbach-Fano-R-matrix (FFR) method p.14/26

21 Background R-matrix spectrum Having defined the discrete state ϕ d (r), we can construct projector onto the background subspace P Background R-matrix spectrum P Ω =1 Ω ϕ d ϕ d P Ω H Ω (r)p Ωbg φ Ω l (r) = bg H Ωbg φ Ω l (r) = bg E Ω l bg φ Ω l (r) Expansion into original R-matrix basis bg φ Ω l (r) = k b lk φ Ω k (r) Discrete state-continuum coupling (depending on discrete index l) Vdl Ω = bg φ Ω l H Ω ϕ d = b li Ei Ω c i E Ω i Σ res Feshbach-Fano-R-matrix (FFR) method p.15/26

22 Discrete state-continuum coupling Background scattering states bg ψ (+) ɛ (r) = 1 2 k bg φ Ω k (r) bg φ Ω k (r Ω) E Ω k ɛ ( d bg ψ ɛ (r)) (+) dr r=r Ω Extension of the coupling to continuous energy V dɛ = ϕ d H bg ψ (+) ɛ = 1 2 ki b ki Ei Ω bg φ Ω k c (r ( Ω) d bg i bg Ek Ω ɛ ψ ɛ (r)) (+) dr r=r Ω Resonance term of the T-matrix is determined by the coupling via ɛ d, Γ(ɛ) and (ɛ). Background term of the T-matrix is determined by the background R-matrix bg R(ɛ) = 1 bg φ Ω l (r Ω) 2 2 ɛ l bg E Ω l Feshbach-Fano-R-matrix (FFR) method p.16/26

23 Application to potential scattering 5 V(r), λ=10 0 V(r), λ0 = full phase shift model potential phase shift Energy 3 2 δ r Energy V (r) = λ 2 r2 e R V (r) = λ 0 4 e (r 3)/2 Feshbach-Fano-R-matrix (FFR) method p.17/26

24 Application to potential scattering R-matrix level closest to the resonance, r Ω = φ 9 Ω (r) 0.6 wave function amplitude r Feshbach-Fano-R-matrix (FFR) method p.18/26

25 Application to potential scattering Discrete state wave function, r Ω = ϕ d NA (r) ϕ d INA (r) wave function amplitude r Feshbach-Fano-R-matrix (FFR) method p.19/26

26 Application to potential scattering Background phase shift, r Ω = full phase shift δ bg, NA δ bg, INA δ Energy Feshbach-Fano-R-matrix (FFR) method p.20/26

27 Application to potential scattering Energy-dependent resonance width, r Ω = NA INA Γ Energy Feshbach-Fano-R-matrix (FFR) method p.21/26

28 Application to potential scattering Energy-dependent resonance width, r Ω = NA INA 0.20 Γ Energy Feshbach-Fano-R-matrix (FFR) method p.22/26

29 Application to low-energy electron-cl 2 scattering SEP level Fixed-nuclei cross section R=3.2 a.u. R=3.3 a.u. R=3.4 a.u. R=3.5 a.u. R=3.6 a.u. σ FN (E) Energy [ev] Feshbach-Fano-R-matrix (FFR) method p.23/26

30 Application to low-energy electron-cl 2 scattering SEP level Fixed-nuclei cross section R=3.2 a.u. R=3.3 a.u. R=3.4 a.u. R=3.5 a.u. R=3.6 a.u. 30 σ bg (E) Energy [ev] Feshbach-Fano-R-matrix (FFR) method p.23/26

31 Application to low-energy electron-cl 2 scattering SEP level Background fixed-nuclei cross section 6 5 R=3.2 a.u. R=3.3 a.u. R=3.4 a.u. R=3.5 a.u. R=3.6 a.u. 4 σ bg (E) Energy [ev] Feshbach-Fano-R-matrix (FFR) method p.23/26

32 Application to low-energy electron-cl 2 scattering SEP level Resonance width and level shift function 1.0 R=3.0 a.u. R=3.2 a.u. R=3.4 a.u. R=3.6 a.u. 0.5 Γ/2, [ev] Energy [ev] Feshbach-Fano-R-matrix (FFR) method p.24/26

33 Application to low-energy electron-cl 2 scattering SEP level Diabatization of the R-matrix spectrum Energy [ev] SE target SEP discrete state R-matrix poles r [a.u.] Feshbach-Fano-R-matrix (FFR) method p.25/26

34 Conclusions The FFR method provides the discrete state, its potential curve and the associated coupling terms to the background continuum. It requires a priori definition of the energy domain Σ res and the model potential. It can be applied to systems which can be investigated via the R-matrix method. Once the R-matrix calculations are done FFR is computationally very cheap method to analyze the results only the R-matrix poles E Ω k and E Ω j and amplitudes φ Ω k (r Ω) and φ Ω j (r Ω) are required to obtain the discrete state position and coupling. Feshbach-Fano-R-matrix (FFR) method p.26/26