Spectroscopic factors and the physics of the single-particle strength distribution in nuclei

Wim DickhoffWashington University in St. Louis

NSCL 7/19/05

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

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• Identify solvable problem by considering

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ˆ H 0 = ˆ T + ˆ U where U is a suitable auxiliary potential.

• Develop expansion in

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ˆ H 1 = ˆ V − ˆ 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?Rearrange Hamiltonian

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ˆ H = ˆ T + ˆ V = ˆ T + ˆ U ( ) + ˆ V − ˆ U ( ) = ˆ H 0 + ˆ H 1Many-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)

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G(0) α,β;E( ) =Φ0

N aα ΦmN +1 Φm

N +1 aβ† Φ0

N

E − EmA +1 − E

Φ0N( ) + iη

+Φ0

N aβ† Φn

N−1 ΦnN−1 aα Φ0

N

E − EΦ0N − En

A−1( ) − iηn∑

m∑

= δα ,βθ α − F( )E −εα + iη

+θ F −α( )E −εα − iη

using the sp basis associated with H0. Note that

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ˆ H 0aα+ Φ0

N = EΦ0

N + εα( )aα+ Φ0

N

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ˆ H 0aα Φ0N = E

Φ0N −εα( )aα Φ0

N

So that e.g.

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Sh(0)(α;E) =

1πImG(0) α,α;E( ) = δ E −εα( )θ F −α( )

and

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n(0) α( ) = dEδ E −εα( )θ F −α( )−∞

ε F( 0)−

∫ = θ F −α( ) ~ like in atoms

Perturbation expansion using G(0) and H1

Use “interaction picture”

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ˆ H 1 t( ) = eih

ˆ H 0t ˆ H 1e−

ih

ˆ H 0t

then ……

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G α,β;t − t'( ) = −ih

−ih

m 1m!

dt1L dtm∫ Φ0N T ˆ H 1 t1( )L ˆ H 1 tm( )aα (t)aβ

+ t'( )[ ]Φ0N

connected∫∑

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 organizationSum of all diagrams can be written as

Introducing some self-energy diagramsFirst order

One of the second order diagrams

The irreducible self-energyThe following self-energy diagram is reducible (previous twowere irreducible), i.e. can be obtained from lower order self-energyterms by iterating with G(0)

Sum of all irreducible diagrams is denoted by Σ*.All diagrams can then be obtained by summing

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G α,β;E( ) =G(0) α,β;E( ) + G(0) α,γ;E( )Σ* γ,δ;E( )G(0) δ,β;E( ) +Lγ ,δ∑

diagrammatically …

Towards the Dyson equation

Can be summed by

Dyson equation

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G α,β;E( ) =G(0) α,β;E( ) + G(0) α,γ;E( )Σ* γ,δ;E( )G δ,β;E( )γ ,δ∑

Looks like the propagator equation for a single particle

with the irreducible self-energy acting as the in-medium (complex) potential.

Link with two-particle propagatorEquation of motion for G

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ih ∂∂tG α,β;t − t '( ) = δ t − t '( )δα,β + εαG α,β;t − t '( ) − α U δ G δ,β;t − t '( )

δ

∑

+12

αδ V ϑζ −ihΨ0

N T aδ H

+ t( )aζ Ht( )aϑ H

t( )aβ H

+ t '( )[ ] Ψ0N

δζϑ

∑

Diagrammatic analysis of GII yields

Γ is the effective interaction (vertex function) between correlated particles in the medium.

Dyson equation and vertex functionFourier transform of equation of motion for G yields again theDyson equation with the self-energy

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Σ* γ,δ;E( ) = − γ U δ − iC↑

dE '2π

γµV δν G ν,µ;E '( )µν

∑∫

+12

dE12π∫ dE2

2πγµV εν G ε,ζ;E1( )G ν,ρ;E2( )G σ,µ;E1 + E2 − E( ) ζρ Γ E1,E2;E( ) δσ

εµνζρσ

∑∫

In diagram form

Dyson Equation and “experiment”

Equivalent to ….!!

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−h2∇2

2mΨn

N−1 ar r m Ψ0N + dr r ∫

m '∑ 'Σ'* (r r m,r r 'm';En

−) ΨnN−1 ar r 'm ' Ψ0

N = En− Ψn

N−1 ar r m Ψ0N

Self-energy: non-local, energy-dependent potential (no U)With energy dependence: spectroscopic factors < 1

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En− = E0

N − EnN−1Schrödinger-like equation with:

Physics is in the choice of the approximation to the self-energy€

S = ΨnN−1 aαqh

Ψ0N 2

=1

1−∂Σ'* αqh ,αqh;E( )

∂EEn−

αqh solution of DE at En-

Hartree-FockFor weakly interacting particles: independent propagation dominates⇒ neglect vertex function in self-energy

Democracy in action⇔ self-consistency

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ΣHF γ,δ( ) = − γ U δ − iC↑

dE '2π

γµV δν GHF ν,µ;E '( )∑∫No energy dependence ⇒ static mean fieldNot a valid strategy for realistic NN interactionsWith “effective” interactions can yield good quasihole wave functionsHF levels full or empty; spectroscopic factors 1 or 0 accordingly

HF for “closed”-shell atoms

HF good startingpoint for atomsbut total energy dominated by coreelectrons.

Description ofvalence electronsnot good enoughto do chemistry.

Spectroscopicfactors not OK.Wave functions

Energies in atomic units (Hartree)

Beyond HF ⇒ dynamical self-energy

Approximatevertex function byΓ = V

Use HF propagator to initiate self-consistent solution

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Σ(2) γ,δ;E( ) =12

γh3 V p1p2 p1p2 V δh3E − εp1 + εp2 −εh3( ) + iη

+γp3 V h1h2 h1h2 V δp3E − εh1 + εh2 −εp3( ) − iηh1h2p3

∑p1p2h3

∑

Poles at 2p1h and 2h1p energiesInteresting consequences for solution of Dyson equation

Diagonal approximationFurther simplification: assume no mixing between major shells

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Σ(2) α;E( ) =12

αh3 V p1p22

E − εp1 + εp2 −εh3( ) + iη+

αp3 V h1h22

E − εh1 + εh2 −εp3( ) − iηh1h2p3

∑p1p2h3

∑

Corresponding Dyson equation

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G α;E( ) =GHF α;E( ) +G α;E( )Σ(2) α;E( )GHF α;E( ) =1

E −εα − Σ(2) α;E( )

Assume discrete poles in Σ, then discrete solution (poles of G) for

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Enα = εα + Σ(2) α;Enα( )

With residue (spectroscopic factor)

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Rnα =1

1− ∂Σ(2) α;E( )∂E

Enα

Solutions

Explains all qualitative features of sp strength distribution in nuclei!

Self-consistent calculation with Skyrme force

Van Neck et al. NPA530,347(1991)

Data: 48Ca(e,e´p)Kramer NIKHEF(1990)

Qualitatively OKNo relation withrealistic V yet!

Self-consistent Green’s functionsand the energy of the ground state of atoms

Atoms : total ground state energies (a.u.)

Method He Be Ne Mg ArDFT -2.913 -14.671 -128.951 -200.093 -527.553HF -2.862 -14.573 -128.549 -199.617 -526.826CI -2.891 -14.617 -128.733 -199.635 -526.807Dyson(2) -2.899 -14.647 -128.939 -200.027 -527.511

Exp. -2.904 -14.667 -128.928 -200.043 -527.549

Dyson(2)

Van Neck, Peirs,WaroquierJ. Chem. Phys. 115, 15 (2001)Dahlen & von BarthJ. Chem. Phys. 120,6826 (2004)