ICD - THE DECAY WIDTHS I

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1 CD - THE DECAY WDTHS 1 st ORDER THEORY AND GENERAL CHARACTERSTCS Přemysl Kolorenč nstitute of Theoretical Physics, Charles University in Prague CD Summer School Bad Honnef September 1 st 5 th P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

2 1 Wigner-Weisskopf theory Evolution of the state vector CD the mechanism Asymptotic behavior of the decay width 2 CD widths examples Short-range and asymptotic behavior Polarization dependence 3 Summary P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

3 Resonant scattering Siegert states Regular solution of Schrödinger equation [l = 0, V (r 0) = 0] Ψ E (r) = Ae ikr + Be ikr, E = k 2 2 bound states: k = iκ, κ > 0 E < 0, Ψ E (r) L 2 Kukulin, Krasnopol sky and Horáček, Theory of Resonances, Kluwer (1989) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

4 Resonant scattering Siegert states Regular solution of Schrödinger equation [l = 0, V (r 0) = 0] Ψ E (r) = Ae ikr + Be ikr, E = k 2 2 bound states: k = iκ, κ > 0 E < 0, Ψ E (r) L 2 continuum states: k R, k > 0 E > 0, Ψ E Ψ E = δ(e E ) Kukulin, Krasnopol sky and Horáček, Theory of Resonances, Kluwer (1989) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

5 Resonant scattering Siegert states Regular solution of Schrödinger equation [l = 0, V (r 0) = 0] Ψ E (r) = Ae ikr + Be ikr, E = k 2 2 bound states: k = iκ, κ > 0 E < 0, Ψ E (r) L 2 continuum states: k R, k > 0 E > 0, Ψ E Ψ E = δ(e E ) resonance states: k = α iβ, α > 0, β > 0 E = E R iγ/2, Ψ E (r) divergent (Siegert state) cannot be represented in an L 2 basis Kukulin, Krasnopol sky and Horáček, Theory of Resonances, Kluwer (1989) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

6 Resonant scattering Siegert states Regular solution of Schrödinger equation [l = 0, V (r 0) = 0] Ψ E (r) = Ae ikr + Be ikr, E = k 2 2 bound states: k = iκ, κ > 0 E < 0, Ψ E (r) L 2 continuum states: k R, k > 0 E > 0, Ψ E Ψ E = δ(e E ) resonance states: k = α iβ, α > 0, β > 0 E = E R iγ/2, Ψ E (r) divergent (Siegert state) cannot be represented in an L 2 basis virtual states: k = iβ, β > 0 E < 0, Ψ E (r) divergent Kukulin, Krasnopol sky and Horáček, Theory of Resonances, Kluwer (1989) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

7 Complex energy plane F. Nunes, PHY982 course, NSCL ( nunes/phy982/phy982prog2014.htm) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

8 Example potential scattering V (r) = 2r 2 e r E res = i Solution of SE at real and complex energy (H E) Ψ E = 0 E R/C Discrete state L 2, not Hamiltonian eigenstate P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

9 Wigner-Weisskopf theory Wigner-Weisskopf theory ntroduction First order time-dependent perturbation theory Semiquantitative, but offers useful qualitative insights Applicable for metastable states well described in single-particle picture: Φ (N 1) = c iv Φ N 0 Detailed derivation Santra and Cederbaum, Phys. Rep. 368 (2002) Weisskopf and Wigner, Z. Phys. 63, 54 (1930) Sakurai, Modern Quantum Mechanics, Addison-Wesley, 1994 P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

10 Wigner-Weisskopf theory Wigner-Weisskopf The Goal Prepare system in the metastable state Ψ (N 1) (t = 0) = Φ (N 1) = c iv Φ N 0 Solve TDSE for Ψ (N 1) (t) Evaluate the survival probability amplitude C (t) = Φ (N 1) Ψ (N 1) (t) Show that the survival probability decreases exponentially P (t) = P (0)e t τ τ = Γ P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

11 Outline Wigner-Weisskopf theory Evolution of the state vector 1 Wigner-Weisskopf theory Evolution of the state vector CD the mechanism Asymptotic behavior of the decay width 2 CD widths examples Short-range and asymptotic behavior Polarization dependence 3 Summary P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

12 Wigner-Weisskopf theory Evolution of the state vector Wigner-Weisskopf theory Hamiltonian Ĥ (N 1) 0 = J Ĥ (N 1) = Ĥ(N 1) 0 + Ĥ(N 1) Φ (N 1) J Ĥ Φ(N 1) J Φ (N 1) J Φ (N 1) J Ĥ (N 1) = J K J Φ (N 1) J Ĥ Φ(N 1) K Φ (N 1) J Φ (N 1) K Unperturbed part Ĥ(N 1) 0 diagonal in the chosen basis nitial metastable state Φ (N 1) discrete eigenvector of Ĥ(N 1) 0 Decay of Φ (N 1) is induced by the off-diagonal part Ĥ(N 1) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

13 Wigner-Weisskopf theory Evolution of the state vector Wigner-Weisskopf theory TDSE Time-dependent Hamiltonian η 0 + gives physical Ĥ Ĥ (N 1) (t) Ĥ(N 1) 0 + e ηt Ĥ (N 1), η > 0 t gives unperturbed Ĥ0 TDSE in interaction picture i t Ψ(N 1) (t) int = ˆV int (t) Ψ (N 1) (t) int Ψ (N 1) (t) int = e iĥ(n 1) 0 t/ Ψ (N 1) (t) ˆV int (t) e ηt e iĥ(n 1) 0 t/ Ĥ (N 1) e iĥ(n 1) t/ 0 P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

14 Solution of TDSE Wigner-Weisskopf theory Evolution of the state vector nitial condition Ψ (N 1) (t) = Φ (N 1) Formal integration of TDSE yields Ψ (N 1) (t) int = Φ (N 1) i t ˆV int (t ) Ψ (N 1) (t ) int dt Perturbation expansion Ψ (N 1) (t) int = Φ (N 1) i t dt ˆVint (t ) Φ (N 1) 1 2 t dt ˆVint (t ) t dt ˆVint (t ) Φ (N 1) +... P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

15 Wigner-Weisskopf theory Evolution of the state vector Decay of the initial state Exploiting simple time-dependence of ˆV int(t) the off-diagonal character of Ĥ yields the probability amplitude C (t) = F Derivative of the probability amplitude Ċ (t) = 1 2 F Φ (N 1) F Ĥ(N 1) Φ (N 1) 2 e 2ηt 2η[η + i(ω F ω )] Φ (N 1) F Ĥ(N 1) Φ (N 1) 2 e 2ηt η + i(ω F ω ) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

16 Wigner-Weisskopf theory Decay of the initial state Evolution of the state vector Ratio Ċ(t)/C (t) through lowest order in interaction Ċ (t) C (t) = i e2ηt/ F Limit η 0 + now possible with the aid of Ċ (t) C (t) = P i + π F Φ (N 1) F Ĥ(N 1) Φ (N 1) 2 (ω ω F ) + iη 1 lim η 0 + x + iη = P 1 x iπδ(x) Φ (N 1) F Ĥ(N 1) Φ (N 1) 2 (ω ω F ) F Φ (N 1) F Ĥ(N 1) Φ (N 1) 2 δ( [ω ω F ]) i ( i ) 2 Γ P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

17 Wigner-Weisskopf theory Evolution of the state vector Probability amplitude in Schrödinger picture ntegration of the above differential equation C (t) = C (0)e i( iγ /2)t/ Probability amplitude in Schrödinger picture Φ (N 1) Ψ (N 1) (t) = Φ (N 1) e iĥ0t/ Ψ (N 1) (t) int = C (0)e i( ω + iγ /2)t/ C (0)e ierest/ nteraction with continuum resonance energy shifted from ω by level shift (real) Γ decay width (imaginary) Probability of finding system in the initial state Φ (N 1) P (t) = C (t) 2 = P (0)e t τ τ = Γ P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

18 Wigner-Weisskopf theory Evolution of the state vector Fourier decomposition resonance profile Fourier transformation e i( ω + iγ /2)t/ = f (ω)e iωt dω gives f (ω) 2 1 ( ω ω ) 2 + Γ 2 /4 f (ω) distribution of the discrete state over the continuum of eigenstates of full Hamiltonian (cf. Fano theory tomorrow) Γ has the meaning of full width at half maximum of the resonance profile Sakurai, Modern Quantum Mechanics, Addison-Wesley, 1994 P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

19 Outline Wigner-Weisskopf theory CD the mechanism 1 Wigner-Weisskopf theory Evolution of the state vector CD the mechanism Asymptotic behavior of the decay width 2 CD widths examples Short-range and asymptotic behavior Polarization dependence 3 Summary P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

20 Wigner-Weisskopf theory CD the mechanism CD decay of an inner-valence vacancy nitial state inner-valence vacancy Φ (N 1) = c iv Φ N 0 Golden-rule formula for the decay width Γ = 2π F Φ (N 1) F Ĥ Φ(N 1) 2 δ(e F E ) Hamiltonian in HF quasi-particle picture: Ĥ = ε p c pc p V pi[qi] c pc q + 1 V pqrs c 2 pc qc s c r p pqi pqrs V pqrs = ϕ e 2 p(x 1 )ϕ r (x 1 ) x 1 x 2 ϕ q(x 2 )ϕ s (x 2 )dx 1 dx 2 P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

21 Wigner-Weisskopf theory CD the mechanism Decay mechanism NeAr example nitial state: Ne + (2s 1 ) Φ (N 1) = c Ne(2s) Φ N 0 Final state: Ne + (2p 1 ) + Ar + (3p 1 ) + e Φ (N 1) F = 1 ( ) c 2 k c Ne(2p) c Ar(3p) c k c Ne(2p) c Ar(3p) Φ N 0 P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

22 Wigner-Weisskopf theory CD the mechanism Decay mechanism NeAr example nitial state: Ne + (2s 1 ) Φ (N 1) = c Ne(2s) Φ N 0 Final state: Ne + (2p 1 ) + Ar + (3p 1 ) + e Φ (N 1) F Ĥ Φ(N 1) = Φ (N 1) F = 1 ( ) c 2 k c Ne(2p) c Ar(3p) c k c Ne(2p) c Ar(3p) Φ N 0 ϕ Ne 2s (x 1)ϕ Ne 2p (x 1) + e 2 x 1 x 2 ϕar 3p (x 2)ϕ k (x 2 )dx 1 dx 2 ϕ Ne 2s (x 1)ϕ Ar 3p (x 1) e 2 x 1 x 2 ϕne 2p (x 2)ϕ k (x 2 )dx 1 dx 2 P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

23 Outline Wigner-Weisskopf theory Asymptotic behavior of the decay width 1 Wigner-Weisskopf theory Evolution of the state vector CD the mechanism Asymptotic behavior of the decay width 2 CD widths examples Short-range and asymptotic behavior Polarization dependence 3 Summary P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

24 Wigner-Weisskopf theory Asymptotic behavior of the decay width Asymptotic dependence on intermolecular distance 1 x 1 x 2 = 1 R u R (r 1 r 2 ) R 2 + 3(u R (r 1 r 2 )) 2 (r 1 r 2 ) 2 ( ) 1 2R 3 + O R 4 ϕ ov1 ϕ iv1 = 0 ϕ ov2 ϕ k = 0 Santra and Cederbaum, Phys. Rep. 368, 1 (2002) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

25 Wigner-Weisskopf theory Asymptotic behavior of the decay width Asymptotic dependence on intermolecular distance 1 x 1 x 2 = 1 R u R (r 1 r 2 ) R 2 + 3(u R (r 1 r 2 )) 2 (r 1 r 2 ) 2 ( ) 1 2R 3 + O R 4 ϕ ov1 ϕ iv1 = 0 ϕ ov2 ϕ k = 0 e 2 V ov1,ov 2,iv 1,k = ϕ iv1 (x 1 )ϕ ov1 (x 1 ) x 1 x 2 ϕ ov 2 (x 2 )ϕ k (x 2 )dx 1 dx 2 e2 R 3 ϕ ov 1 r 1 ϕ iv1 ϕ ov2 r 2 ϕ k 3e2 R 3 ϕ ov 1 r 1 u R ϕ iv1 ϕ ov2 r 2 u R ϕ k Santra and Cederbaum, Phys. Rep. 368, 1 (2002) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

26 Wigner-Weisskopf theory Asymptotic behavior of the decay width Γ(R) Asymptotic behavior Asymptotic behavior of the decay width F ˆV 1 R 3 ( 1 + P(R)e αr ) = lim R Γ(R) 1 R 6 Leading term dipole-dipole interaction virtual photon transfer model Γ = 3 4π ( c ω ) 4 τ 1 rad σ R 6 τ rad radiative lifetime of the inner-valence vacancy state σ total photoionization cross section of the neigbor at the virtual photon energy ω Matthew and Komninos, Surf. Sci. 53, 716 (1975) Averkukh, Müller and Cederbaum, PRL 93, (2004) Averbukh and Cederbaum, JCP 125, (2006) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

27 Wigner-Weisskopf theory Asymptotic behavior of the decay width Γ(R) Deviations from the R 6 rule ETMD electron transfer process dominated by orbital overlap, e αr dependence Dipole forbidden recombination transition example: Zn(3d) vacancy in BaZn dimer quadrupole-dipole interaction Γ 0,±1 = π ( c ω ) 6 τ 1 Zn σba R 8 CD in He dimer: He + (2s 1 )He He + + He + + e 2s 1s single-photon recombination forbidden e αr dominates at small distances R 8 asymptotics 3 rd order decay pathway Averkukh, Müller and Cederbaum, PRL 93, (2004) Kolorenč et al., PRA 82, (2010) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

28 Outline CD widths examples Short-range and asymptotic behavior 1 Wigner-Weisskopf theory Evolution of the state vector CD the mechanism Asymptotic behavior of the decay width 2 CD widths examples Short-range and asymptotic behavior Polarization dependence 3 Summary P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

29 CD widths examples Short-range and asymptotic behavior CD & ETMD widths in NeAr 10 2 αr 6 Ne + (2s 1 )Ar Ne 2+ (2s 1 2p 1 )Ar ( 1 Σ + ) Ne 2+ (2s 1 2p 1 )Ar ( 1 Π) 10 1 Γ [mev] ETMD 10 0 R eq 10 1 τ = 60 fs R [Å] CD : Ne + (2s 1 )Ar Ne + (2p 1 ) + Ar + (3p 1 ) + e ETMD : Ne + (2s 1 )Ar NeAr 2+ (3p 2 ) + e P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

30 CD widths examples Short-range and asymptotic behavior CD & ETMD widths in MgNe αr 6 MgNe + (2s 1 ) MgNe 2+ (2s 1 2p 1 ) ( 1 Σ) MgNe 2+ (2s 1 2p 1 ) ( 1 Π) Γ [mev] ETMD R eq τ = 820 fs R [Å] Ne + (2s 1 )Mg Ne + (2p 1 ) + Mg + (3s 1 ) + e P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

31 CD widths examples Short-range and asymptotic behavior CD in (He + 2 ) Γ [mev] 10 4 He + (2p 1 ) 2 + Σ g R 8 (3 rd order pathway) 10 6 He + (2p 1 ) 2 + Σ u He + (2p 1 ) 2 + Π u 10 8 He + (2p 1 ) 2 + Π g He + (2s 1 ) 2 + Σ g He + (2s 1 ) 2 + Σ u R [Å] (He + ) He He + + He + + e P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

32 CD widths examples Short-range and asymptotic behavior CD in He 3 selection rules on the emssion site He 2 + (2py 1 ) 2 Πy,g He 2 + (2py 1 ) 2 Πy,u Γ p [mev] (1s 1 1σ 1 g ) 1 (1s 1 1σ 1 g ) Γ p [mev] (1s 1 1σ 1 g ) 1 (1s 1 1σ 1 g ) 1e 05 1 (1σg 2 1σu 2 ) 3 (1s 1 1σ 1 u ) 1e 05 1 (1σg 2 1σu 2 ) 3 (1s 1 1σ 1 u ) 1e 06 1 (1s 1 1σ 1 u ) 1e 06 1 (1s 1 1σ 1 u ) 3 (1σ 1 g 1σ 1 u ) 1e R [Å] 3 (1σ 1 g 1σ 1 u ) 1e R [Å] Γ R 6 : symmetry of CD e match the polarization of the virtual photon 2 Π y,g : (He + 2 ) (2p 1 y) He He + 2 (1σ 1 u ) + He + (1s) + e (p y ) Γ R 8 : other channels: 2 Π y,g : (He + 2 ) (2py) 1 He He + 2 (1σ 1 g ) + He + (1s) + e (d) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

33 Outline CD widths examples Polarization dependence 1 Wigner-Weisskopf theory Evolution of the state vector CD the mechanism Asymptotic behavior of the decay width 2 CD widths examples Short-range and asymptotic behavior Polarization dependence 3 Summary P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

34 CD widths examples Polarization dependence Symmetry dependence of the CD width theoretical ratio Γ Σg /Γ Πg Γ [mev] He + (2p 1 ) 2 Σ g + He + (2p 1 ) 2 Σ u + He + (2p 1 ) 2 Π u + He + (2p 1 ) 2 Π g + He + (2s 1 ) 2 Σ g + He + (2s 1 ) 2 Σ u + R 8 (3 rd order pathway) R [Å] interaction energy between two classical dipoles Γ Σ /Γ Π W = p 1 p 2 3(n p 1 )(n p 2 ) x 1 x 2 3 dipoles parallel vs. perpendicular to molecular axis W /W = 2 Γ2 Σ/Γ2 Π = R [Å] Gokhberg et al., PRA 81, (2010) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

35 CD widths examples Polarization dependence Symmetry dependence of the CD width 10 1 αr theoretical ratio 10 0 MgNe + (2s 1 ) MgNe 2+ (2s 1 2p 1 ) ( 3 Σ) 1 Γ3 Σ /Γ3 Π 10 1 MgNe 2+ (2s 1 2p 1 ) ( 3 Π) 0.8 Γ [mev] R eq Γ Σ /Γ Π ETMD τ = 820 fs R [Å] R [Å] CD after Auger in MgNe, triplet metastable states more complicated case summation over all partial rates gives asymptotic ratio Γ3 Σ Γ3 Π = 2 5 Gokhberg et al., PRA 81, (2010) P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31

36 Summary Summary Survival probability of finding system in the initial state ( P (t) = P (0) exp Γ ) t Wigner-Weisskopf golden-rule formula for decay width Γ = 2π F Φ (N 1) F Ĥ Φ(N 1) 2 δ(e F E ) Typical CD mechanism dipole-dipole interaction, virtual photon transfer Γ R 6 R Deviations from the R 6 rule electron transfer or orbital overlap contribution (at short distances) dipole-forbidden transitions Beyond 1 st order CAP, complex scaling, Fano-Feshbach,... P. Kolorenč (Charles University in Prague) CD widths September 3rd, / 31