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Spectroscopic Factors and the Physics of the Single- Particle Strength Distribution in Nuclei

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Spectroscopic Factors and the Physics of the Single- Particle Strength Distribution in Nuclei NSCL 7/19/05 Spectroscopic factors and the physics of the single- particle strength distribution in nuclei Lecture 1: 7/18/05 Propagator description of single-particle motion and the link with experimental data Lecture 2: 7/19/05 From diagrams to Hartree-Fock and spectroscopic factors < 1 Lecture 3: 7/20/05 Influence of long-range correlations and the relation to excited states Lecture 4: 7/21/05 Role of short-range and tensor correlations associated with realistic interactions. Prospects for nuclei with N very different from Z. Lecture 5: 7/22/05 Saturation problem of nuclear matter Wim Dickhoff Washington University in St. Louis Outline • Outline of perturbation theory • Diagrams and diagram rules • Self-energy and Dyson equation • Link between sp and two-particle propagator • Self-consistent Green’s functions • Hartree-Fock • Dynamical self-energy and spectroscopic factors < 1 Many-body perturbation theory for G ˆ ˆ ˆ • Identify solvable problem by considering H 0 = T + U where U is a suitable auxiliary potential. ˆ ˆ ˆ • Develop expansion in H1 = V −U • Employs time-evolution, Heisenberg,€ Schrödinger, and interaction picture of quantum mechanics. • Once established,€ this expansion (expressed in Feynman diagrams) is organized in such a way that nonperturbative results can be obtained leading to the Dyson equation. The Dyson equation describes sp motion in the medium under the influence of the self-energy which is an energy-dependent complex sp potential. • Further insight into the proper description of sp motion in the medium is obtained by studying the relation between sp and two-particle propagation. This allows the selection of appropriate choices of the relevant ingredients for the system under study. How to calculate G? Hˆ Tˆ Vˆ Tˆ Uˆ Vˆ Uˆ Hˆ Hˆ Rearrange Hamiltonian = + = ( + ) + ( − ) = 0 + 1 Many-body problem with H0 can be exactly solved when U is a one-body potential like a Woods-Saxon or HO potential. Corresponding sp propagator (replace H by H0) € N N +1 N +1 † N N † N−1 N−1 N Φ0 a Φm Φm a Φ0 Φ0 a Φn Φn a Φ0 G(0) α,β;E = α β + β α ( ) ∑ A +1 ∑ A−1 m E − E m − E N + iη n E − E N − E n − iη ( Φ 0 ) ( Φ 0 ) θ(α − F) θ(F −α) = δα,β + E −εα + iη E −εα − iη using the sp basis associated with H . Note that ˆ + N + N 0 H 0aα Φ0 = E N + εα aα Φ0 ( Φ 0 ) ˆ N N H 0aα Φ0 = E N −εα aα Φ0 € ( Φ 0 ) So that e.g. (0) 1 (0) Sh (α;E) = ImG (α,α;E) = δ(E −εα )θ(F −α) π € ( 0)− ε F and (0) € ~ like in atoms n (α) = ∫ dEδ(E −εα )θ(F −α) = θ(F −α) −∞ € € (0) Perturbation expansion using G and H1 i ˆ i ˆ H 0t − H 0t Use “interaction picture” Hˆ t = e h Hˆ e h 1( ) 1 then …… € m i −i 1 G α,β;t − t' = − dt dt ΦN T Hˆ t Hˆ t a (t)a+ t' ΦN ( ) ∑ ∫ 1L ∫ m 0 [ 1( 1)L 1( m ) α β ( )] 0 connected h h m! € Can be calculated order by order using diagrams and Wick’s theorem. Yields expressions involving G(0) and matrix elements of the two-body interaction V (and the auxiliary potential U) Simple diagram rules in time formulation. For practical calculations use energy formulation. Diagrams Diagram rules in energy formulation Examples of diagrams More diagrams Diagram organization Sum of all diagrams can be written as Introducing some self-energy diagrams First order One of the second order diagrams The irreducible self-energy The following self-energy diagram is reducible (previous two were irreducible), i.e. can be obtained from lower order self-energy terms by iterating with G(0) Sum of all irreducible diagrams is denoted by Σ*. All diagrams can then be obtained by summing (0) (0) * (0) G(α,β;E) = G (α,β;E) + ∑G (α,γ;E)Σ (γ,δ;E)G (δ,β;E) +L γ,δ diagrammatically … € Towards the Dyson equation Can be summed by Dyson equation Looks like the propagator equation for a single particle G(α,β;E) = G(0)(α,β;E) + ∑G(0)(α,γ;E)Σ*(γ,δ;E)G(δ,β;E) γ,δ with the irreducible self-energy acting as the in-medium (complex) potential. € Link with two-particle propagator Equation of motion for G ∂ ih G(α,β;t − t') = δ(t − t')δα,β + εαG(α,β;t − t') − ∑ α U δ G(δ,β;t − t') ∂t δ 1 i + αδ V ϑζ − ΨN T a+ t a t a t a+ t' ΨN ∑ 0 [ δ H ( ) ζ H ( ) ϑ H ( ) β H ( )] 0 2 δζϑ h Diagrammatic analysis of GII yields € Γ is the effective interaction (vertex function) between correlated particles in the medium. Dyson equation and vertex function Fourier transform of equation of motion for G yields again the Dyson equation with the self-energy dE' Σ*(γ,δ;E) = − γ U δ − i ∫ ∑ γµ V δν G(ν,µ;E') C↑ 2π µν 1 dE1 dE 2 + ∫ ∫ ∑ γµ V εν G(ε,ζ;E1)G(ν,ρ;E 2 )G(σ,µ;E1 + E 2 − E) ζρ Γ(E1,E 2;E) δσ 2 2π 2π εµνζρσ In diagram form € Dyson Equation and “experiment” Equivalent to ….!! − N N−1 Schrödinger-like equation with: E n = E 0 − E n 2 2 h ∇ N−1 N r '* r r − N−1 N − N−1 N − Ψ ar Ψ + dr 'Σ (r m,r 'm';E ) Ψ ar Ψ = E Ψ ar Ψ 2m n r m 0 ∑ ∫ n n r 'm' 0 n n r m 0 m' € Self-energy: non-local, energy-dependent potential (no U) With energy dependence: spectroscopic factors < 1 € 2 N−1 N 1 S = Ψn aα Ψ0 = - qh * α solution of DE at E ∂Σ' α qh ,α qh ;E qh n 1− ( ) E ∂ − En Physics is in the choice of the approximation to the self-energy € Hartree-Fock For weakly interacting particles: independent propagation dominates ⇒ neglect vertex function in self-energy Democracy in action ⇔ self-consistency dE' ΣHF (γ,δ) = − γ U δ − i ∫ ∑ γµ V δν G HF (ν,µ;E') C↑ 2π No energy dependence ⇒ static mean field Not a valid strategy for realistic NN interactions € With “effective” interactions can yield good quasihole wave functions HF levels full or empty; spectroscopic factors 1 or 0 accordingly HF for “closed”-shell atoms HF good starting point for atoms but total energy dominated by core electrons. Description of valence electrons not good enough to do chemistry. Spectroscopic factors not OK. Wave functions Energies in atomic units (Hartree) Beyond HF ⇒ dynamical self-energy Approximate vertex function by Γ = V Use HF propagator to initiate self-consistent solution 1 γh V p p p p V δh γp V h h h h V δp Σ(2)(γ,δ;E) = ∑ 3 1 2 1 2 3 + ∑ 3 1 2 1 2 3 2 E − ε + ε −ε + iη E − ε + ε −ε − iη p1 p2h3 ( p1 p2 h3 ) h1h2 p3 ( h1 h2 p3 ) Poles at 2p1h and 2h1p energies € Interesting consequences for solution of Dyson equation Diagonal approximation Further simplification: assume no mixing between major shells 2 2 1 αh3 V p1 p2 αp3 V h1h2 Σ(2)(α;E) = ∑ + ∑ 2 p p h E − ε + ε −ε + iη h h p E − ε + ε −ε − iη 1 2 3 ( p1 p2 h3 ) 1 2 3 ( h1 h2 p3 ) Corresponding Dyson equation HF (2) HF 1 € G(α;E) = G (α;E) + G(α;E)Σ (α;E)G (α;E) = (2) E −εα − Σ (α;E) Assume discrete poles in Σ, then discrete solution (poles of G) for (2) € E nα = εα + Σ (α;E nα ) 1 With residue (spectroscopic factor) Rn = α ∂Σ(2)(α;E) 1− ∂E € Enα € Solutions Explains all qualitative features of sp strength distribution in nuclei! Self-consistent calculation with Skyrme force Data: 48Ca(e,e´p) Kramer NIKHEF (1990) Qualitatively OK No relation with realistic V yet! Van Neck et al. NPA530,347(1991) Self-consistent Green’s functions and the energy of the ground state of atoms Dyson(2) Van Neck, Peirs,Waroquier J. Chem. Phys. 115, 15 (2001) Dahlen & von Barth J. Chem. Phys. 120,6826 (2004) Atoms : total ground state energies (a.u.) Method He Be Ne Mg Ar DFT -2.913 -14.671 -128.951 -200.093 -527.553 HF -2.862 -14.573 -128.549 -199.617 -526.826 CI -2.891 -14.617 -128.733 -199.635 -526.807 Dyson(2) -2.899 -14.647 -128.939 -200.027 -527.511 Exp. -2.904 -14.667 -128.928 -200.043 -527.549.
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