Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron

Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography

Modern nuclear energy density functional methods

T. Duguet

1DSM/Irfu/SPhN, CEA Saclay, France

2National Superconducting Cyclotron Laboratory,Department of Physics and Astronomy, Michigan State University, USA

Euroschool on exotic beams, Jyvaskyla, Finland, Aug. 20th - 26th, 2011

EDF methods


Lecture series

0utline

1 Basics of the nuclear energy density functional method

■ Elements of formalism

■ Form, performances and limitations of empirical energy functionals

2 Typical applications and results

Practice session

1 Deriving the (simplified) Skyrme energy density functional

2 Computing the corresponding equation of state of infinite nuclear matter

Present lectures assume that you are familiar with

1 Basics of quantum mechanics, e.g. Dirac notation, variational method, . . .

2 Second quantization formulation of quantum mechanics, i.e. ai ;a†j

3 Basics of many-body technique, e.g. Slater determinant, Hartree-Fock, . . .

EDF methods


Lecture series

0utline

1 Basics of the nuclear energy density functional method

■ Elements of formalism

■ Form, performances and limitations of empirical energy functionals

2 Typical applications and results

Practice session

1 Deriving the (simplified) Skyrme energy density functional

2 Computing the corresponding equation of state of infinite nuclear matter

Present lectures assume that you are familiar with

1 Basics of quantum mechanics, e.g. Dirac notation, variational method, . . .

2 Second quantization formulation of quantum mechanics, i.e. ai ;a†j

3 Basics of many-body technique, e.g. Slater determinant, Hartree-Fock, . . .

EDF methods


Outline

1 The energy density functional method in a nutshell

2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes

3 Multi-reference EDF method

4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family

5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods

6 Summary and outlook

7 Bibliography

EDF methods


Outline







7 Bibliography

EDF methods


What is low-energy nuclear physics interested in?

In generic terms

■ Spectrum of H |ΨAµ 〉= EA

µ |ΨAµ 〉 for all A = N + Z

■ Observables for each state, e.g. r2 ≡ 〈ΨAµ |∑A

kr2k |Ψ

Aµ 〉/A

■ Decays between |ΨAµ 〉, i.e. nuclear, electromagnetic, electro-weak

Ground state

Mass, deformation

Spectroscopy

Excitations modes

Limits

Drip-lines, halos

Reaction properties

Fusion, transfer...

Heavy elements

Fission, fusion, SHE

Astrophysics

NS, SN, r-process

EDF methods


Towards a unified theoretical description of nucleonic matter

Extended matter ρ ∈ [0,∼ 3ρsat]

-50

0

50

100

150

200

250

300

350

0 0.16 0.32 0.48 0.64 0.8 0.96

E/A

ρ [fm-3

]

f3f4f5

SLY5’

[T. Lesinski et al., PRC74 (2006) 044315]

Theoretical methods

1 "Exact" (VMC, . . . )

2 Ab-initio (SCGF, BHF, . . . )

3 Effective (EDF)

Finite nuclei Z ∈ [0,118+]

Theoretical methods

1 "Exact" (GFMC, NCSM, . . . )

2 Ab-initio (CC, SCGF, IMSRG)

3 Effective (SM, EDF)

EDF methods


Energy density functional method in a nutshell

Basic ingredients

1 Approach NOT based on {H ; |Ψ〉} ; i.e. E = 〈Ψ|H |Ψ〉 with |Ψ〉 ≈ |Ψexact〉

2 Key object is the off-diagonal energy kernel Eqq′

Eqq′ ≡ E [〈Φq |; |Φq′〉] = E [ρqq′

ĳ ,κqq′

ĳ ,κq′q∗ĳ ]

which is a functional of one-body transition density matrices

ρqq′

ĳ ≡〈Φq |a†j ai |Φ

q′〉

〈Φq|Φq′ 〉; κqq′

ĳ ≡〈Φq |ajai |Φ

q′〉

〈Φq|Φq′ 〉; κq′q ∗

ĳ ≡〈Φq|a†i a

†j |Φ

q′ 〉

〈Φq |Φq′〉

with {a†i }= arbitrary single-particle basis; e.g. i = (~r ,σ,τ )

■ |Φq〉= product (Bogoliubov) state with collective label q

■ EDF kernel re-sums bulk correlations through functional character

■ Approach relies on symmetry breaking and restoration (J2,N ,Z ,Π, . . .)

3 Empirical param. (Gogny, Skyrme,. . . ) successful but lack predictive power

EDF methods



Basic ingredients



Eqq′ ≡ E [〈Φq |; |Φq′〉] = E [ρqq′

ĳ ,κqq′

ĳ ,κq′q∗ĳ ]


ρqq′


q′〉



q′〉



†j |Φ

q′ 〉

〈Φq |Φq′〉






EDF methods



Basic ingredients



Eqq′ ≡ E [〈Φq |; |Φq′〉] = E [ρqq′

ĳ ,κqq′

ĳ ,κq′q∗ĳ ]


ρqq′


q′〉



q′〉



†j |Φ

q′ 〉

〈Φq |Φq′〉






EDF methods



Two successive levels of implementation

1 Single-Reference ("mean-field")

■ Invokes single |Φq〉 and diagonal energy kernel ESR ≡Min{|Φq〉} E|q||q|

■ |Φq〉 may break symmetries and acquire finite order parameters |q|eiϕq

■ Provides first approx to BE, 〈r2ch〉, ρτ (~r), β2 and ESPE {ǫα}

2 Multi-Reference ("beyond mean-field")

■ Mixes off-diagonal energy kernels associated with chosen MR set {|Φq〉}

EMRk ≡ Min{f k

q }

∑

q,q′∈MR f k ∗q f k

q′ Eqq′ 〈Φq|Φq′ 〉

∑


q′〈Φq|Φq′ 〉

■ Large-amplitude quantum fluctuations

■ Treats collective vibrations (|q|)■ Restores broken symmetries (ϕq)■ Includes associated G.S. correlations■ Provides associated collective excitations

■ QRPA, Bohr-Hamiltonian are approximations

EDF methods










EMRk ≡ Min{f k

q }

∑



∑






EDF methods










EMRk ≡ Min{f k

q }

∑



∑






EDF methods


Outline







7 Bibliography

EDF methods


Outline







7 Bibliography

EDF methods


Auxiliary many-body state

Single-particle basis {|α〉}

■ Creation operators a†α|0〉 = |α〉 with

{

aα,a†β

}

= δαβ

■ Wave-functions ψα(~rστ )≡ 〈~rστ |α〉

Slater determinant

1 Standard product state

|Φ〉 ≡

N∏

µ=1

b†µ |0〉

b†µ ≡

∑

α

Uαµ a†α

2 Unitary transformation U

3 Effective "vacuum"

b†µ |Φ〉 = 0 forµ ∈ [1,N ]

bµ |Φ〉 = 0 forµ ∈ [N +1,∞]

4 Particle number N |Φ〉 = N |Φ〉

Bogoliubov state

1 Product of quasi-particle operators

|Φ〉 ≡∏

µ

βµ |0〉

βµ ≡∑

α

U∗αµ aα+ V

∗αµ a

+α

2 Unitary transformation

(

U † V †

V T U T

)

3 Vacuum of quasi-particle operators

βµ |Φ〉 = 0 ∀µ

4 Symmetry breaking N |Φ〉 6= N |Φ〉

EDF methods





{

aα,a†β

}

= δαβ


Slater determinant


|Φ〉 ≡

N∏

µ=1

b†µ |0〉

b†µ ≡

∑

α

Uαµ a†α



b†µ |Φ〉 = 0 forµ ∈ [1,N ]

bµ |Φ〉 = 0 forµ ∈ [N +1,∞]


Bogoliubov state


|Φ〉 ≡∏

µ

βµ |0〉

βµ ≡∑

α

U∗αµ aα+ V

∗αµ a

+α


(

U † V †

V T U T

)


βµ |Φ〉 = 0 ∀µ


EDF methods





{

aα,a†β

}

= δαβ


Slater determinant


|Φ〉 ≡

N∏

µ=1

b†µ |0〉

b†µ ≡

∑

α

Uαµ a†α



b†µ |Φ〉 = 0 forµ ∈ [1,N ]

bµ |Φ〉 = 0 forµ ∈ [N +1,∞]


Bogoliubov state


|Φ〉 ≡∏

µ

βµ |0〉

βµ ≡∑

α

U∗αµ aα+ V

∗αµ a

+α


(

U † V †

V T U T

)


βµ |Φ〉 = 0 ∀µ


EDF methods


Single-reference one-body density matrices

Hermitian normal density matrix

ραβ ≡〈Φ|a†βaα|Φ〉

〈Φ|Φ〉

=∑

µ

V∗αµV

Tµβ

= ρ∗βα

Antisymmetric anomalous density matrix

καβ ≡〈Φ|aβaα|Φ〉

〈Φ|Φ〉

=∑

µ

V∗αµU

Tµβ

= −κβα

EDF methods





〈Φ|Φ〉



〈Φ|Φ〉

Slater determinant

1 Fixed particle number κ= 0

2 ρ is idempotent ρ2 = ρ

3 In basis {b†µ} diagonalizing ρ

ρ=

1 0 . . . 0 . . .

0. . .

... 10 0...

. . .

with ρµµ = individual occupations

EDF methods





〈Φ|Φ〉



〈Φ|Φ〉

Slater determinant

1 Fixed particle number κ= 0

2 ρ is idempotent ρ2 = ρ

3 In basis {b†µ} diagonalizing ρ

ρ=

1 0 . . . 0 . . .

0. . .

... 10 0...

. . .

with ρµµ = individual occupations

Bogoliubov state

1 Indefinite particle number κ 6= 0

2 Generalized density matrix

R≡

(

ρ κ

−κ∗ 1−ρ∗

)

=R2

3 In basis {c†ν} diagonalizing ρ

|Φ〉 =∏

ν>0

(

uν + vν c+ν c

+ν

)

|0〉

BCS-like state with occupations ρνν = v2ν

EDF methods





〈Φ|Φ〉



〈Φ|Φ〉

Coordinate basis {|~rστ 〉}

■ Basis transformation

c~rστ =∑

α

ψα(~rστ ) aα

■ Density matrices are non-local functions in space

ρ~rστ~r ′σ′τ ′ =∑

αβ

ψ∗β(~r ′σ′τ )ψα(~rστ )ραβ δττ ′

κ~rστ~r ′σ′τ ′ =∑

αβ

ψβ(~r ′σ′τ )ψα(~rστ )καβ δττ ′

EDF methods





〈Φ|Φ〉



〈Φ|Φ〉

Physical content in coordinate basis

1 Value at ~r = ~r ′ ⇔ average nucleonic density distribution

ρτ (~r) ≡∑

σ

ρ~rστ~r ′στ∣

∣

~r=~r ′

2 One derivative at ~r = ~r ′ ⇔ average nucleonic current distribution

~jτ (~r) ≡i

2

(

~∇′− ~∇)

∑

σ


∣

~r=~r ′

3 Two derivatives at ~r = ~r ′ ⇔ average nucleonic kinetic distribution

ττ (~r) ≡ ~∇· ~∇′∑

σ


∣

~r=~r ′

EDF methods


Outline







7 Bibliography

EDF methods


Ansatz

SR-EDF ansatz

■ Diagonal energy kernel postulated as a functional E ≡ E [ρ,κ,κ∗]

■ The energy results from the minimization of the diagonal kernel

ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]

within the manyfold span by product states |Φ〉


Minimization and constraints

1 Independent variables are {ραβ , ρ∗αβ , καβ , κ

∗αβ for β ≤ α}

2 The constraint that R2 =R must be enforced

➨ Matrix Λ of Lagrange parameters (one per isospin τ )

3 Constrain 〈Φ|N |Φ〉 = Tr{ρ}= N as a free variation would lead to ESR→−∞

➨ Lagrange parameter λ (one per isospin τ )

EDF methods


Ansatz

SR-EDF ansatz

■ Diagonal energy kernel postulated as a functional E ≡ E [ρ,κ,κ∗]

■ The energy results from the minimization of the diagonal kernel

ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]

within the manyfold span by product states |Φ〉


Minimization and constraints

1 Independent variables are {ραβ , ρ∗αβ , καβ , κ

∗αβ for β ≤ α}

2 The constraint that R2 =R must be enforced

➨ Matrix Λ of Lagrange parameters (one per isospin τ )

3 Constrain 〈Φ|N |Φ〉 = Tr{ρ}= N as a free variation would lead to ESR→−∞

➨ Lagrange parameter λ (one per isospin τ )

EDF methods


Equation of motion

Minimization

1 Set δR

[

E [ρ,κ,κ∗]−λTr{ρ}−Tr{Λ(R2−R)}]

= 0 leads to [H,R] = 0

H≡

(

h−λ ∆−∆∗ −h∗+λ

)

where the effective one-body fields are defined through

hαγ ≡δE [ρ,κ,κ∗]

δργα; ∆αγ ≡

δE [ρ,κ,κ∗]

δκ∗αγ

2 This is equivalent to solving the Bogoliubov-De-Gennes eigenvalue problem

H

(

U

V

)

µ

= Eµ

(

U

V

)

µ

3 {(U ,V )µ} and {Eµ} denote quasi-particle states and energies

4 The field h drives the shell structure while ∆ drives pair scattering

5 Iterative procedure H[{(U ,V )[n]µ }]→{(U ,V )

[n+1]µ } →H[{(U ,V )

[n+1]µ }]→ . . .

EDF methods


Iterative procedure and self-consistency

In

1 E ≡ E [ρ,κ,κ∗]

2 N ,Z

3 |α〉(0)

4 (U ,V )(0)µ

E (n)

(ρ,κ)(n)(h,∆)(n)

(U ,V )(n+1)µ

δEDF

Bogo

SR

Out

■ ESR

■ (U ,V )µ

■ Eµ

■ ρ(~r),r2ch

■ . . .

EDF methods


Individual excitations

HFB spectrum Eµ with ≈√

(hµµ−λ)2 + ∆µµ and without = |hµµ−λ| pairing

Pairing gap opens at low energy + mix of hole- and particle-like excitations

Slater determinant

1 Auxiliary excited state

|Φph〉 ≡ b†p bh |Φ〉

2 Spectrum (hµν ≡ ǫµδµν)

ESRph −E

SR ≈ ǫp− ǫh≥ 0

Bogoliubov state

1 Auxiliary excited state

|Φµν〉 ≡ β†µβ†ν |Φ〉

2 Spectrum

ESRαβ −E

SR ≈ Eα+ Eβ ≥ 2∆F

EDF methods


Observable

At convergence

1 Binding energy ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]

2 Density matrices (ρ,κ) from which other observables can be computed, e.g.

Local matter density ρ0(~r) ≡∑

σ,τ

ρ~rστ~rστ

Matter square radius r2mat ≡

1

A

∫

d~r r2 ρ0(~r)

SR-EDF method

1 Correlations that vary smoothly with N and Z are mocked up

1 Through rich (enough?) functional form

2 Through the fitting of its parameters

2 Not all correlations are easily resummed into E [ρ,κ,κ∗] itself

1 Symmetry breaking captures important correlations

2 Further correlations brought by restoring symmetries = MR-EDF

EDF methods


Observable

At convergence

1 Binding energy ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]

2 Density matrices (ρ,κ) from which other observables can be computed, e.g.

Local matter density ρ0(~r) ≡∑

σ,τ

ρ~rστ~rστ

Matter square radius r2mat ≡

1

A

∫

d~r r2 ρ0(~r)

SR-EDF method

1 Correlations that vary smoothly with N and Z are mocked up

1 Through rich (enough?) functional form

2 Through the fitting of its parameters

2 Not all correlations are easily resummed into E [ρ,κ,κ∗] itself

1 Symmetry breaking captures important correlations

2 Further correlations brought by restoring symmetries = MR-EDF

EDF methods


Outline







7 Bibliography

EDF methods


Symmetries

Symmetries of H and quantum numbers

■ In nuclear physics [X ,H ] = 0 for {X}= {N , Z , ~P, J2, Jz , Π, T2}

■ Solutions |Ψxi 〉 are eigenstates of {X} and labeled by quantum numbers {x}

➨ (N ,Z) ∈ N2

➨ ~P ∈ R3

➨ 2J ∈ N and 2M ∈ Z such that −J ≤M ≤+J

➨ Π =±1➨ T 2 = (−1)N (−1)Z

■ The nuclear part of H also nearly commutes with isospin operators (T2, Tz)

In which space do we actually vary |Φ〉/{ρ,κ} when minimizing E [ρ,κ,κ∗]?

■ Natural to vary such that |Φ〉 carries the same quantum numbers {x} as |Ψxi 〉

➨ Too constraining for simple product states, e.g. Slater determinants

■ Example: ~P|Φ〉 = ~P|Φ〉 ⇒ ϕα(~r) = ϕ~k(~r) = ei~k·~r

■ Miss correlations that induce spatial localization of the internal motion■ A similar situation holds for all symmetries

➨ Need to expand our horizon. . .

EDF methods


Symmetries

Symmetries of H and quantum numbers

■ In nuclear physics [X ,H ] = 0 for {X}= {N , Z , ~P, J2, Jz , Π, T2}

■ Solutions |Ψxi 〉 are eigenstates of {X} and labeled by quantum numbers {x}

➨ (N ,Z) ∈ N2

➨ ~P ∈ R3

➨ 2J ∈ N and 2M ∈ Z such that −J ≤M ≤+J

➨ Π =±1➨ T 2 = (−1)N (−1)Z

■ The nuclear part of H also nearly commutes with isospin operators (T2, Tz)

In which space do we actually vary |Φ〉/{ρ,κ} when minimizing E [ρ,κ,κ∗]?

■ Natural to vary such that |Φ〉 carries the same quantum numbers {x} as |Ψxi 〉

➨ Too constraining for simple product states, e.g. Slater determinants

■ Example: ~P|Φ〉 = ~P|Φ〉 ⇒ ϕα(~r) = ϕ~k(~r) = ei~k·~r

■ Miss correlations that induce spatial localization of the internal motion■ A similar situation holds for all symmetries

➨ Need to expand our horizon. . .

EDF methods


Spontaneous symmetry breaking

Variation allowing for spontaneous symmetry breaking

■ Let |Φ〉 span several irreducible representations of the symmetry group G

|Φ〉 ≡∑

x

cx |Θx〉

✔ ESR lower than for a symmetry-conserving variation

■ Specific long-range correlations are included in this way

■ SM method grasps such correlations through |Ψx〉 ≈∑

jaj |Φ

xj 〉

✘ ESR does not access the ground-state energy but the one of a wave-packet

ESR =

∑

x

|cx |2

Ex

Symmetries have to be restored eventually ⇒ MR-EDF method

■ Additional ground-state correlation energy by extracting Ex

■ Compute reliably quantities dictated by selection rules, e.g. B(E2)

■ Particularly important for weakly-broken symmetries

EDF methods


Spontaneous symmetry breaking

Variation allowing for spontaneous symmetry breaking

■ Let |Φ〉 span several irreducible representations of the symmetry group G

|Φ〉 ≡∑

x

cx |Θx〉

✔ ESR lower than for a symmetry-conserving variation

■ Specific long-range correlations are included in this way

■ SM method grasps such correlations through |Ψx〉 ≈∑

jaj |Φ

xj 〉

✘ ESR does not access the ground-state energy but the one of a wave-packet

ESR =

∑

x

|cx |2

Ex

Symmetries have to be restored eventually ⇒ MR-EDF method

■ Additional ground-state correlation energy by extracting Ex

■ Compute reliably quantities dictated by selection rules, e.g. B(E2)

■ Particularly important for weakly-broken symmetries

EDF methods


Physical content and patterns

Origin and manifestation of spontaneous symmetry breaking

Group Correlations

Label Casimir V NN Internal motion

Translation ~a T (3) P Short range Spatial localization

Rotation ϕ U (1) N S-wave virtual state Pairing

Rotation α,β,γ SO(3) J2 Quadrup.-quadrup. Angular localization

Nuclei and spontaneous symmetry breaking

Nuclei Excitation pattern

Translation ~a All of them Surface vibrationsRotation ϕ All but doubly-magic ones Energy gapRotation α,β,γ All but singly-magic ones Rotational bands

■ Symmetries are enforced or relaxed according to which nucleus is studied

■ Enforcing symmetries leads to a great gain in CPU time

EDF methods


Physical content and patterns

Origin and manifestation of spontaneous symmetry breaking

Group Correlations

Label Casimir V NN Internal motion

Translation ~a T (3) P Short range Spatial localization

Rotation ϕ U (1) N S-wave virtual state Pairing

Rotation α,β,γ SO(3) J2 Quadrup.-quadrup. Angular localization

Nuclei and spontaneous symmetry breaking

Nuclei Excitation pattern

Translation ~a All of them Surface vibrationsRotation ϕ All but doubly-magic ones Energy gapRotation α,β,γ All but singly-magic ones Rotational bands



EDF methods


Symmetry breaking and densities

Symmetry, densities and collective deformation

■ Quantum numbers ~P,(L,M ),(N ,Z) . . . translate into properties of {ρ,κ}

1 Local density ρ(~r) is constant over space for state with any good ~P

2 Local density ρ(~r) is spherically symmetric for state with L = 0

3 Anomalous density καβ is a null matrix for state with fixed N ,Z

■ Symmetry breaking translates into a violation of such properties

➨ Deformation with respect to a collective variable

~P vs density localization L = 0 vs density deformation

EDF methods


Symmetry breaking and densities

Symmetry, densities and collective deformation

■ Quantum numbers ~P,(L,M ),(N ,Z) . . . translate into properties of {ρ,κ}

■ Symmetry breaking translates into a violation of such properties

■ Deformation with respect to a collective variable

➨ Characterized by a non-zero order parameter q ≡ |q|eiArg(q)

Group Order parameter"Phase" Arg(q) Magnitude |q|

SO(3) Euler angles (α,β,γ) ρλµ ≡ 〈rλYµλ 〉

{

odd λ≤ 2L and any µeven λ≤ 2L and µ 6= 0any λ > 2L and any µ

U (1) Gauge angle ϕ ‖κ‖ such that καβ ≡ 〈Φ|aβaα|Φ〉

EDF methods


Minimization and correlation energy

Order parameter q ≡ |q|eiArg(q) and energy minimization

■ ESR is independent of Arg(q) as Eqq = E|q||q|

■ The magnitude |q| may vary when minimizing Eqq

1 Variation at fixed |q|= 0 provides symmetry-conserving solution2 Free variation (|q| 6= 0) may provide a lower solution in the larger space3 Constrained variation via Lagrange (−λq 〈Φq |Q|Φq〉) to access E|q||q|

EDF methods


Minimization and correlation energy

Order parameter ”|q|eiArg(q)” and energy minimization

■ ESR is independent of Arg(q) as Eqq = E|q||q|

■ The magnitude |q| may vary when minimizing Eqq

1 Variation at fixed |q|= 0 provides symmetry-conserving solution2 Free variation (|q| 6= 0) may provide a lower solution in the larger space3 Constrained variation via Lagrange (−λq 〈Φq |Q|Φq〉) to access E|q||q|

Quadrupole (J/ = 0) correlation ∆Eρ20

[M. Bender, private communication]

Octupole (Jπ/ = 0+) correlation ∆Eρ30


EDF methods


Spatial symmetry and quantum numbers

Symmetries of quasi-particle wave-functions (U ,V )µ

■ Quantum numbers µ reflects symmetries of (h,∆) and thus of (ρ,κ)

Sym. ρ Pref. coord. Broken sym. Quant. numb. Degeneracy

ρ (x,y,z) none (kxkykzσq) ρ(E)

ρ00 (r , θ,φ) ~p (nljmq) 2j + 1

ρ2l0 (ρ,θ,z) ~p, j2 (nπmq) 2

ρ2lµ none ~p, j2, jz (nπζq) 2

ρλµ none ~p, j2, jz ,π (nζq) 2

ρλµ none ~p, j2, jz ,π,T (nq) 0

➨ Degeneracies of Eµ reflect symmetries of (h,∆)/(ρ,κ)

Practical consequences



EDF methods


Deformation and Nilsson diagram

Calculation of 250Fm (N = 150,Z = 100) as a function of β2 ∝ ρ20

■ Spectrum of hϕα = ǫαϕα as a function of β2 = Nilsson diagram

[M. Bender, P.-H. Heenen, unpublished]

■ β2,min is a compromise between N = 150 and Z ≈ 100 deformed gaps

EDF methods


Manifestations of pairing correlations in nuclei

Fingerprint at finite density of n-n virtual state in vacuum

■ Additional ground-state correlations

■ Odd-even mass staggering (OEMS)

■ Gap opens up in individual excitation spectrum

■ Moment of inertia ր with J ⇐⇒ Meissner effect in superconductors

Pairing (N/ = 70) correlation ∆E‖κ‖

Data from [M. Bender, T.D., ĲME16 (2007) 222]

Pairing correlations

■ Weak symmetry breaking

■ |∆E‖κ‖| ≪∆Eρ20

■ |∆E‖κ‖| ≥ needed accuracy

■ Sharp collapse for ‖κ‖ ≤ ‖κ‖crit.

EDF methods


Manifestations of pairing correlations in nuclei

Fingerprint at finite density of n-n virtual state in vacuum

■ Additional ground-state correlations

■ Odd-even mass staggering (OEMS)

■ Gap opens up in individual excitation spectrum

■ Moment of inertia ր with J ⇐⇒ Meissner effect in superconductors

Odd-even mass staggering

From [S. Baroni, private communication]

Gap extracted from the OEMS

Data from [T. D., T. Lesinski, arXiv:0907.1043]

EDF methods


Outline







7 Bibliography

EDF methods


SR-EDF numerical codes

Some published SR-EDF (Skyrme) codes

1 1D code with spherical symmetry HFBRAD[K. Bennaceur, J. Dobaczewski, Comp. Phys. Comm. 168 (2005) 96]

http://www.sciencedirect.com/science/article/pii/S0010465505002304

2 2D code with axial symmetry HFBTHO[M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comp. Phys. Comm. 167 (2005) 43]


3 3D code with 3 symmetry planes ev8[P. Bonche, H. Flocard, P.-H. Heenen, Comp. Phys. Comm. 171 (2005) 49]


4 3D code with no symmetry HFODD[J. Dobaczewski et al., Comp. Phys. Comm. 180 (2009) 2361]


EDF methods


Outline







7 Bibliography

EDF methods


Limitations of the single-reference method

Using a single reference state |Φq〉

■ Casimir operator A, i.e. J2, of G

A |Φqmin〉 6= a|Φqmin〉 with a≡ Irrep(a)

➨ Zero-energy mode at any |q| 6= 0

Collective landscape

EDF methods







■ What about using a fixed |q|min?

Different categories of nuclei (1) Stiff



EDF methods








➨ Soft mode as a function of |q|?

Different categories of nuclei (2) Soft



EDF methods








➨ Soft mode as a function of |q|?


How to improve on that consistently?

■ Go beyond |Φq〉 at fixed q

➨ Mixing in |q| and Arg(q)

➨ Add dynamical correlations

➨ Provide excitations

EDF methods


Multi-reference energy density functional method in a nutshell

Second level of implementation

■ Mixes off-diagonal energy kernels associated with MR set {|Φq〉}

EMRk ≡ Min{f k

q }

∑



∑



➨ Treats collective vibrations = mixing along |q|

➨ {f kq } obtained via minimization

➨ Restores broken symmetries = mixing along ϕq

➨ {f kq } fixed by symmetry-group structure


1 Includes G.S. correlations

2 Provides associated collective excitations

■ Common approximation schemes

1 QRPA = harmonic fluctuations

2 Bohr-Hamiltonian = GOA

EDF methods





EMRk ≡ Min{f k

q }

∑



∑













EDF methods





EMRk ≡ Min{f k

q }

∑



∑













EDF methods


Outline







7 Bibliography

EDF methods


Outline







7 Bibliography

EDF methods


Building Eqq′ = E [〈Φq |; |Φq′〉] = E [ρqq′ ,κqq′ ,κq′q ∗]

The mathematical form

1 It can be as elaborate and as complicated as one finds empirically necessary

➨ It can be a very non-local functional in space, i.e.

Eqq′ =∑

σ1...σn ,τ1...τn

∫

· · ·

∫

d~r1 . . .d~rn E(ρqq′

~r1σ1τ1~r2σ2τ2, . . . ,κqq′

~rn−1σn−1τn−1~rnσnτn)

➨ One tries to limit the complexity and the number of free parameters

2 The form is constrained as Eqq′ (implicitly) relates to the scalar operator H

E [〈Φq|R†(g);R(g)|Φq′〉] = E [〈Φq|; |Φq′ 〉] for any R(g) ∈ G

Fitting its free parameters (so far done at the SR level)

1 Use empirical knowledge of Infinite Nuclear Matter equation of state

➨ esat, ρsat,K∞,m∗0 , asym + various ρinst⇒see practice session

2 Use a selected set of finite nuclei properties

➨ BE , r2ch,∆ǫj=l∓1/2,∆Efission

isomer, . . .

EDF methods


Building Eqq′ = E [〈Φq |; |Φq′〉] = E [ρqq′ ,κqq′ ,κq′q ∗]

The mathematical form

1 It can be as elaborate and as complicated as one finds empirically necessary

➨ It can be a very non-local functional in space, i.e.

Eqq′ =∑

σ1...σn ,τ1...τn

∫

· · ·

∫

d~r1 . . .d~rn E(ρqq′

~r1σ1τ1~r2σ2τ2, . . . ,κqq′

~rn−1σn−1τn−1~rnσnτn)

➨ One tries to limit the complexity and the number of free parameters

2 The form is constrained as Eqq′ (implicitly) relates to the scalar operator H

E [〈Φq|R†(g);R(g)|Φq′〉] = E [〈Φq|; |Φq′ 〉] for any R(g) ∈ G

Fitting its free parameters (so far done at the SR level)

1 Use empirical knowledge of Infinite Nuclear Matter equation of state

➨ esat, ρsat,K∞,m∗0 , asym + various ρinst⇒see practice session

2 Use a selected set of finite nuclei properties

➨ BE , r2ch,∆ǫj=l∓1/2,∆Efission

isomer, . . .

EDF methods


Outline







7 Bibliography

EDF methods


Quasi-local form of E = E [ρ,κ,κ∗]

Strategy followed below

1 Diagonal kernel Eqq ≡ E [ρ,κ,κ∗]

2 Focus on the ρ dependence

Scalar and vector non-local densities

ρτ (~r ,~r′) ≡

∑

σσ′

ρ~rστ~r ′σ′τ 1σ′σ

sτ,ν(~r ,~r′) ≡

∑

σσ′

ρ~rστ~r ′σ′τ σσ′σν

Time-even local densities

ρτ (~r) ≡ ρτ (~r ,~r′)|~r=~r ′ Matter density

ττ (~r) ≡∑

µ

∇µ∇′µ ρτ (~r ,~r

′)|~r=~r ′ Kinetic density

Jτ,µν(~r) ≡i

2

(

∇′µ−∇µ)

sτ,ν(~r ,~r′)|~r=~r ′ Spin-current tensor density

Jτ,κ(~r) ≡

z∑

µ,ν=x

ǫκµν Jτ,µν(~r) Spin-orbit density

EDF methods


Quasi-local form of E = E [ρ,κ,κ∗]

Strategy followed below

1 Diagonal kernel Eqq ≡ E [ρ,κ,κ∗]

2 Focus on the ρ dependence

Scalar and vector non-local densities

ρτ (~r ,~r′) ≡

∑

σσ′

ρ~rστ~r ′σ′τ 1σ′σ

sτ,ν(~r ,~r′) ≡

∑

σσ′

ρ~rστ~r ′σ′τ σσ′σν

Time-odd local densities

sτ,ν(~r) ≡ sτ,ν(~r ,~r′)|~r=~r ′ Spin density

Tτ,ν(~r) ≡∑

µ

∇µ∇′µ sτ,ν(~r ,~r

′)|~r=~r ′ Spin-kinetic density

Fτ,µ(~r) ≡1

2

∑

ν

(

∇µ∇′ν +∇ν∇

′µ

)

sτ,ν(~r ,~r′)|~r=~r ′ Tensor-kinetic density

jτ,µ(~r) ≡i

2

(

∇′µ−∇µ)

ρτ (~r ,~r′)|~r=~r ′ Current density

EDF methods


Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]

Building procedure

1 Local bilinear form up to two gradients and two Pauli matrices

2 Form scalars

3 Couplings Cff ′

ττ ′ may further depend on ~r ≡ higher-order terms

Skyrme energy density functional

E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[

Cρρττ ′ ρτ ρτ ′ + C

ρ∆ρττ ′ ρτ∆ρτ ′ + C

ρτττ ′

(

ρτ ττ ′−~jτ ·~jτ ′)

+Cssττ ′ ~sτ ·~sτ ′ + C

s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C

JJττ ′

(

∑

µν

Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)

+CJJττ ′

(

∑

µν

Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)

]

+ terms in [κ,κ∗]

EDF methods



Building procedure


2 Form scalars with respect to (i) Galilean transformation

3 Couplings Cff ′



E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[



ρτττ ′

(



s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C

JJττ ′

(

∑

µν


+CJJττ ′

(

∑

µν


]


EDF methods



Building procedure


2 Form scalars with respect to (ii) space rotation

3 Couplings Cff ′



E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[



ρτττ ′

(



s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C

JJττ ′

(

∑

µν


+CJJττ ′

(

∑

µν


]


EDF methods



Building procedure


2 Form scalars with respect to (iii) time reversal

3 Couplings Cff ′



E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[



ρτττ ′

(



s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C

JJττ ′

(

∑

µν


+CJJττ ′

(

∑

µν


]


EDF methods



Building procedure


2 Form scalars

3 Couplings Cff ′



E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[



ρτττ ′

(



s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C

JJττ ′

(

∑

µν


+CJJττ ′

(

∑

µν


]


EDF methods



Historically: kernel derived from an effective pseudo-potential

1 Quasi zero-range two-body vertex with 9 parameters

Vskyrme ≡ Vcent + Vls + Vtens

Vcent = t0 (1 + x0Pσ)δ(~r)

+t1

2(1 + x1Pσ)

[

δ(~r)−→k

2 +←−k′2 δ(~r)

]

+ t2 (1 + x2Pσ)←−k′ · δ(~r)

−→k

Vls = iW0 ( ~σ1 + ~σ2)←−k′ ∧ δ(~r)

−→k

Vtens =te

2

{

[

3(~σ1 ·←−k′ ) (~σ2 ·

←−k′ )− (~σ1 ·~σ2)

←−k′2]

δ(~r)

+δ(~r)[

3(~σ1 ·−→k ) (~σ2 ·

−→k )− (~σ1 ·~σ2)

−→k

2]}

+ to

{

3(~σ1 ·←−k′ )δ(~r) (~σ2 ·

−→k )− (~σ1 ·~σ2)

←−k′ · δ(~r)

−→k

}

2 E [ρ,κ,κ∗]≡ 〈Φ|T + Vskyrme|Φ〉 ⇐⇒ Effective Hartree-Fock

EDF methods







+t1

2(1 + x1Pσ)

[

δ(~r)−→k

2 +←−k′2 δ(~r)

]

+ t2 (1 + x2Pσ)←−k′ · δ(~r)

−→k

Vls = iW0 ( ~σ1 + ~σ2)←−k′ ∧ δ(~r)

−→k

Vtens =te

2

{

[

3(~σ1 ·←−k′ ) (~σ2 ·

←−k′ )− (~σ1 ·~σ2)

←−k′2]

δ(~r)

+δ(~r)[

3(~σ1 ·−→k ) (~σ2 ·

−→k )− (~σ1 ·~σ2)

−→k

2]}

+ to

{

3(~σ1 ·←−k′ )δ(~r) (~σ2 ·

−→k )− (~σ1 ·~σ2)

←−k′ · δ(~r)

−→k

}


EDF methods







+t1

2(1 + x1Pσ)

[

δ(~r)−→k

2 +←−k′2 δ(~r)

]

+ t2 (1 + x2Pσ)←−k′ · δ(~r)

−→k

Vls = iW0 ( ~σ1 + ~σ2)←−k′ ∧ δ(~r)

−→k

Vtens =te

2

{

[

3(~σ1 ·←−k′ ) (~σ2 ·

←−k′ )− (~σ1 ·~σ2)

←−k′2]

δ(~r)

+δ(~r)[

3(~σ1 ·−→k ) (~σ2 ·

−→k )− (~σ1 ·~σ2)

−→k

2]}

+ to

{

3(~σ1 ·←−k′ )δ(~r) (~σ2 ·

−→k )− (~σ1 ·~σ2)

←−k′ · δ(~r)

−→k

}


EDF methods




1 〈Φ|T + Vskyrme|Φ〉 = purely bilinear Skyrme-type EDF ⇒see practice session

2 The 9 vertex parameters induce correlations among the 18 EDF couplings

E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[

Cρρττ ′ ρτρτ ′ + C


ρτττ ′

(

ρτ ττ ′ −~jτ ·~jτ ′)

+ Cssττ ′ ~sτ ·~sτ ′ + C

s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+ C∇s∇sττ ′ (∇·~sτ )(∇·~sτ ′) + C

JJττ ′

(

∑

µν


+ CJJττ ′

(

∑

µν


]


➨ Not enough (i) powers of the densities and (ii) freedom to fix the Cff ′

ττ ′

EDF methods






E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[



ρτττ ′

(



s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+ C∇s∇sττ ′ (∇·~sτ )(∇·~sτ ′) + C

JJττ ′

(

∑

µν


+ CJJττ ′

(

∑

µν


]



ττ ′

EDF methods






E [ρ,κ,κ∗] =∑

τ

∫

d~rh2

2mττ

+∑

ττ ′

∫

d~r

[



ρτττ ′

(



s∆sττ ′ ~sτ ·∆~sτ ′ + C

ρ∇Jττ ′

(

ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)

+ C∇s∇sττ ′ (∇·~sτ )(∇·~sτ ′) + C

JJττ ′

(

∑

µν


+ CJJττ ′

(

∑

µν


]



ττ ′

EDF methods



Historically: kernel derived from a density-dependent effective pseudo-potential

1 Step towards EDF philosophy by adding a density dependence to Vskyrme

”Vcent” = t0 (1 + x0Pσ)δ(~r)

+1

2t1 (1 + x1Pσ)

[

δ(~r)−→k

2 +←−k′ 2 δ(~r)

]

+ t2 (1 + x2Pσ)←−k′ · δ(~r)

−→k

+1

6t3 (1 + x3Pσ)ρα0 (~r)δ(~r)

2 Compute E [ρ,κ,κ∗]≡ 〈Φ|T + ”Vskyrme”|Φ〉

➨ Only Cρρττ ′ further depend on ~r via ρα0 (~r)

3 Modern EDF approach = exploit full freedom offered by EDF kernel

■ Bypass ”Vskyrme” entirely to parameterize E [ρ,κ,κ∗] directly

■ Freedom to use any functional form; e.g. Padé, exponential. . .

■ Relax constraints between EDF couplings

➨ Responsible for serious problems that cannot be discussed here. . .

EDF methods



Historically: kernel derived from a density-dependent effective pseudo-potential

1 Step towards EDF philosophy by adding a density dependence to Vskyrme

”Vcent” = t0 (1 + x0Pσ)δ(~r)

+1

2t1 (1 + x1Pσ)

[

δ(~r)−→k

2 +←−k′ 2 δ(~r)

]

+ t2 (1 + x2Pσ)←−k′ · δ(~r)

−→k

+1

6t3 (1 + x3Pσ)ρα0 (~r)δ(~r)

2 Compute E [ρ,κ,κ∗]≡ 〈Φ|T + ”Vskyrme”|Φ〉

➨ Only Cρρττ ′ further depend on ~r via ρα0 (~r)

3 Modern EDF approach = exploit full freedom offered by EDF kernel

■ Bypass ”Vskyrme” entirely to parameterize E [ρ,κ,κ∗] directly

■ Freedom to use any functional form; e.g. Padé, exponential. . .

■ Relax constraints between EDF couplings

➨ Responsible for serious problems that cannot be discussed here. . .

EDF methods


Single-particle field h

Local h derived from the Skyrme EDF ⇒see practice session

hĳ ≡δE

δρji=

∫

d~r ϕ†i (~r)hτ (~r)ϕj(~r)

hτ (~r) ≡ −~∇·Bτ (~r)~∇+ Uτ (~r)−i

2

z∑

µ,ν=x

[

Wτ,µν(~r)∇µ+∇µWτ,µν(~r)]

σν

−~∇·[

~Cτ (~r) ·~σ]

~∇+~Sτ (~r) ·~σ−1

2

[

~∇· ~Dτ (~r)~σ · ~∇+~σ · ~∇ ~Dτ (~r) · ~∇]

−i

2

[

~Aτ (~r) · ~∇+ ~∇· ~Aτ (~r)]

with multiplicative potentials defined through

Bτ (~r) ≡δE

δττ (~r); Uτ (~r)≡

δE

δρτ (~r); Wτ,µν(~r)≡

δE

δJτ,µν(~r)

Cτ,µ(~r) ≡δE

δTτ,µ(~r); Sτ,µ(~r)≡

δE

δsτ,µ(~r); Dτ,µ(~r)≡

δE

δFτ,µ(~r)

Aτ,µ(~r) ≡δE

δjτ,µ(~r)

EDF methods


Nuclear shell structure

SR-EDF h drives the s.p. motion

1 Observable fragmented shell structure

E+µ ≡E

A+1µ −E

A0 with spectroscopic factor S

+µ

E−µ ≡E

A0 −E

A-1µ with spectroscopic factor S

−µ

2 h provides the centroid shell structure

hτ (~r)ϕα(~r) ≡ ǫαϕα(~r)

[T. D., J. Sadoudi, (2011) unpublished]

➨ Add dynamical correlations to access E±µ

➨ Configuration mixing via MR-EDF

1 Symmetry restorations

2 Dynamical particle-vibrations coupling

EDF methods


Outline







7 Bibliography

EDF methods


Outline







7 Bibliography

EDF methods


Typical results from EDF calculations

Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Ground-state properties over the nuclear chart

■ Large-scale deformed calculations in one day

■ Masses, deformation, radii, s.p.e energies. . .

■ Tables for experimentalists/astrophysics

■ Ex: http://www-phynu.cea.fr/HFB-Gogny.htm

[S. Hilaire and M. Girod, Eur. Phys. J. A33 (2007) 237]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Neutron shell-structure in Sn and Pb isotopes

■ From SR-EDF calculation hϕα = ǫαϕα

■ ǫnlj ≡ centroid of one-nucleon separation energies

[T. D., J. Sadoudi, (2011) unpublished]

■ Too spread out to anticipate correlations

■ Current interest focuses on N−Z behavior

[M. Bender, G. F. Bertsch, P.-H. Heenen, PRC73 (2006) 034322]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Z = 50 magic gaps in Sn isotopes

■ From difference of −S2p(Z) = EZ0 −EZ+2

0

■ Importance of correlations to reproduce data

■ Current focus on evolution with N−Z


EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Systematic of S2q and magic gaps

■ Visible (e.g. N = 50) shell quenching

■ Collective correlations are essential

■ Shell effects too pronounced in heavy nuclei


EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Charge density distribution in 48Ca

■ ρch(~r)≈ ρp(~r)

■ Saturation density ρsat = 2ρp(0)≈ 0.16 fm−3

■ Stable nuclei = good reproduction of data

■ No measure yet in unstable nuclei[V. Rotival and T. D., unpublished]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Charge density distribution in 208Pb

■ ρch(~r)

■ Saturation density ρsat ≈ 2ρp(0)≈ 0.16 fm−3

■ Sensitive to shell effects

■ Sensitive to correlations[V. Rotival and T. D., unpublished]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Charge radii in Xe and Ba isotopes

■ Good reproduction of data

■ Correct N−Z dependence

■ Importance of correlations


EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Systematic of charge radii

■ Good agreement (≤ 3%)

■ More difficult to extract r2n experimentally

■ Neutron skin√

r2n−√

r2p of importance

■ New interest triggered by JLAB exp. on 208Pb

[J. P. Delaroche et al., unpublished]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Halos in medium-mass nuclei

0 4 8 12r [fm]

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

n(r

) [f

m-3

]

54Cr

80Cr

■ Nuclei with anomalous extensions

■ Predictions done over the nuclear chart

■ Can only exist very close to neutron drip-line

■ Best candidates are Cr and Fe isotopes

[V. Rotival, T. D., PRC79 (2009) 054308]

[V. Rotival, K. Bennaceur, T. D., PRC79 (2009) 054309]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Halos in medium-mass nuclei

■ Nuclei with anomalous extensions

■ Predictions done over the nuclear chart

■ Can only exist very close to neutron drip-line

■ Best candidates are Cr and Fe isotopes

[V. Rotival, T. D., PRC79 (2009) 054308]

[V. Rotival, K. Bennaceur, T. D., PRC79 (2009) 054309]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Systematic of pairing gap in semi-magic nuclei

■ The nucleus is superfluid

■ Odd-even mass staggering is a measure of ∆τ

■ First microscopic calculations

■ All low-energy nuclear properties impacted

[T. Lesinski, K. Hebeler, T. D., A. Schwenk, JPG (2011) in press]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Isoscalar giant monopole resonance in 120Sn

■ QRPA = harmonic approximation of MR-EDF

■ Strength distribution

Sλ(E) =∑

i

∣

∣

∣〈ΨA

0 |Fλ|ΨAi 〉

∣

∣

∣

2

δ(EAi −EA

0 )

■ Resonances for all λ= Lπ and T = 0,1 with L≤ 3

■ Test features of EDF (K∞, m∗τ , asym, pairing. . . )

[J. Engel, unpublished]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Low-lying collective spectroscopy of 186Pb

■ Complex nucleus displaying shape coexistence

■ Good overall picture: bands, in/out B(E2)s

■ Too spread out spectra and too large E0

■ Extensions needed

[M. Bender et al., PRC69 (2004) 064303]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Nuclear fission of 238Pu

80 100 120 140 160

Fragment Mass

10-5

10-4

10-3

10-2

10-1

Yie

ld

■ Fission fragments properties: A, Ekin distrib.

■ SR-EDF ⇒ static paths for asymmetric fission

■ Bohr Hamiltonian ⇒ fission dynamics

■ Need exp. Z distrib., T1/2, n/γ with N −Z

[H. Goutte et al., PRC71 (2005) 024316]

EDF methods



Calculations

1 Systematics

2 Shells

3 Densities

4 Radii

5 Halos

6 Superfluidity

7 Resonances

8 Spectroscopy

9 Fission

10 Reactions

Reactions from time-dependent EDF calculations

■ Ex: fusion at the Coulomb barrier (16O+208Pb)

■ Pre-transfer, barrier distribution. . .

■ Note: full EDF essential to solve old problems

[C. Simenel and B. Avez, arXiv:0711.0934]

EDF methods


Outline







7 Bibliography

EDF methods


Limitations of current EDF methods

Limitations

1 Masses

2 Nuclear EOS

3 Instabilities

4 Shells

5 Predictability

Systematics of Eth−Eexp (σmass2135 ≈ 1.5 MeV from SR)

■ Ravines at magic numbers at SR level

■ Improved by dynamical correlations

■ Miss MR correlations, terms in EDF, fit?

[F. Chappert et al., unpublished]

EDF methods



Limitations

1 Masses

2 Nuclear EOS

3 Instabilities

4 Shells

5 Predictability

(S ,T ) content of nuclear EOS

-20

-10

0

10

20

30

40

50

60

0 0.08 0.16 0.24 0.32 0.4E

/A [

MeV

]ρ [fm-3]

f-f0f+

SLy5LNS

■ Good reproduction of SNM and PNM EOS

■ Wrong splitting in (S ,T ) channels

■ Current EDF are overconstrained


EDF methods



Limitations

1 Masses

2 Nuclear EOS

3 Instabilities

4 Shells

5 Predictability

(S ,T ) content of nuclear EOS

-25-20-15-10-5 0 5

10

0 0.1 0.2 0.3

E/A

S,T

=0,

0 [M

eV]

ρ [fm-3] 0 0.1 0.2 0.3 0.4

-30-25-20-15-10-5

E/A

S,T

=0,

1 [M

eV]

ρ [fm-3]

SLy5

-30-20-10

0 10 20

E/A

S,T

=1,

0 [M

eV]

-10

-5

0

5

E/A

S,T

=1,

1 [M

eV]

f-f0f+

■ Good reproduction of SNM and PNM EOS

■ Wrong splitting in (S ,T ) channels

■ Current EDFs are overconstrained


EDF methods



Limitations

1 Masses

2 Nuclear EOS

3 Instabilities

4 Shells

5 Predictability

Spin/isospin instabilities of nuclear matter

0 0.16 0.32 0.48 0.64

0 1 2 3s.-i

sos.

ρc

[fm

-3]

q [fm-1]

f-f0f+

SLy5

0 1 2 3 4 0 0.16 0.32 0.48 0.64

v.-i

sos.

ρc

[fm

-3]

q [fm-1]

0 0.16 0.32 0.48 0.64 0.8

s.-i

sov.

ρc

[fm

-3]

0 0.16 0.32 0.48 0.64 0.8

v.-i

sov.

ρc

[fm

-3]

■ ∞/finite size instabilities of INM

■ Potentially spurious predictions in nuclei

■ Need control but EDFs are overconstrained


EDF methods



Limitations

1 Masses

2 Nuclear EOS

3 Instabilities

4 Shells

5 Predictability

∆ǫi = ǫi− ǫexpi in doubly-magic nuclei

0

5

10

15

20

-4 -2 0 2 4

SkO' (fit)

εε εε i - εε εε

iexp

(MeV)

0

5

10

15

20SkO'

εε εε i - εε εε

iexp

-4 -2 0 2 4

SkP (fit)

(MeV)

SkP

-4 -2 0 2 4

SLy5 (fit)

(MeV)

SLy5

[M. Kortelainen et al., PRC77 (2008) 064307]

■ Refit on 58 data points

■ ∆ǫrms ≥ 1.1 MeV = poor

■ Wrong ℓ-dependence of centroids/s.o. splittings

■ Wrong evolution with N−Z

■ Enriched EDF needed

EDF methods



Limitations

1 Masses

2 Nuclear EOS

3 Instabilities

4 Shells

5 Predictability

Extrapolations towards the drip-line: masses

■ Agreement within models where fitted on data

■ Divergence "in the next major shell"

■ Strong limitations of empirical EDFs


EDF methods


Outline







7 Bibliography

EDF methods


Energy Density Functional method: single-reference implementation

EDF methods


Energy Density Functional method: multi-reference implementation

EDF methods


Towards non-empirical EDF parameterization and method

Design non-empirical Energy Density Functionals

■ Bridge with ab-initio many-body techniques

■ Calculate properties of heavy/complex nuclei from NN+NNN

Empirical

Predictive?

Finite nuclei and extended nuclear matter

EDF methods


Towards non-empirical EDF parameterization and method

Design non-empirical Energy Density Functionals

■ Bridge with ab-initio many-body techniques

■ Calculate properties of heavy/complex nuclei from NN+NNN

Empirical

Predictive?

Non-empirical

Predictive...

Low-k NN+NNN QCD / χ-EFT

Finite nuclei and extended nuclear matter

EDF methods


Outline







7 Bibliography

EDF methods


Selected bibliography

P. Ring, P. Schuck,

The nuclear many-body problem, 1980, Springer-Verlag, Berlin

M. Bender, P.-H. Heenen, P.-G. Reinhard,

Rev. Mod. Phys. 75 (2003) 121

T. Duguet, K. Bennaceur, T. Lesinski, J. Meyer,

Opportunities with Exotic Beams, 2007, World Scientific, p. 21

K. Bennaceur, J. Dobaczewski,

Comp. Phys. Comm. 168 (2005) 96

M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring,

Comp. Phys. Comm. 167 (2005) 43

P. Bonche, H. Flocard, P.-H. Heenen,

Comp. Phys. Comm. 171 (2005) 49

J. Dobaczewski et al.,

Comp. Phys. Comm. 180 (2009) 2361

EDF methods

Documents

Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron