shell model a uniﬁed view of nuclear structurebasic ideas and concepts in nuclear physics an...

Shell Model

A Unified View of Nuclear Structure

Frederic Nowacki1

18th STFC UK Postgraduate Summer School

Lancaster, August 24th-September 5th-2015

1Strasbourg-Madrid Shell-Model collaboration

Bibliography

Basic ideas and concepts in nuclear physicsan introductory approachHeyde K.IOP Publishing 1994

Shell model applications in nuclear spectroscopyBrussaard P.J., Glaudemans P.W.M.North-Holland 1977

The nuclear shell modelHeyde K.Springer-Verlag 1994

The nuclear shell modelA. Poves and F. NowackiLecture Notes in Physics 581 (2001) 70ff

The shell model as a unified view of nuclear structureE. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, A. P. ZukerRev. Mod. Phys. 77, 427 (2005)

Shell structure evolution and effective in-medium NN interactionN. SmirnovaEcole Joliot-Curie 2009

Outline

Lecture 1: Introduction, basic notions, shell model codes

and calculations

Lecture 2: Lanczos structure functions, Effective

Interactions

Lecture 3: Shell model applications to nuclear

spectroscopy

Reminder of Shell Model Problem

CORE

Define a valence space

Derive an effective interaction

HΨ = EΨ → HeffΨeff = EΨeff

Build and diagonalize the

Hamiltonian matrix.

In principle, all the spectroscopic properties are described

simultaneously (Rotational band AND β decay half-life).

Reminder of basis representation

Reminder of basis representation:

We represent a Slater determinant by a machine word, where

each state is a bit (0 empty 1 occupied)

1/2 3/2 1/2- 1/2 -1/2 -3/2 1/2 3/2 1/2- 1/2 -1/2 -3/2

0 0 1 1 1 1 1 1 1 10 0

12 11 10 9 8 7 6 5 4 3 2 1

Mn

i=

Mp

0p1/2 0p3/2 0p1/2 0p3/2

≡ a†10a

†9a

†8a

†7 b

†4b

†3b

†2b

†1 |0〉

After diagonalization, the eigenstates of the system are linear

combinations of Slater Determinants of the basis:

|Ψα〉 =∑

i

ci |Φi〉 with |Φi〉 =∏

k=nljmτ

a†k |0〉 = a

†k1...a

†kA|0〉

Diagonalisation with Lanczos algorithm

The Lanczos algorithm consist in the construction of an

orthonormal basis by orthogonalization of the states Hn|1〉,obtained by the repeated action of the hamiltonian H, on a

basis state |1〉 called pivot. From this procedure results a

tridiagonal matrix. In the first step we write:

H|1〉 = E11|1〉 + E12|2〉

where E11 is just 〈1|H|1〉 = 〈H〉,the mean value of H.

Lanczos Algorithm

E12 is obtained by normalization :

E12|2〉 = H|1〉 - E11|1〉 = (H − E11)|1〉

In the second step:

H|2〉 = E21|1〉 + E22|2〉 + E23|3〉

The hermiticity of H implies E21 = E12

E22 is just 〈2|H|2〉and E23 is obtained by normalization :

E23|3〉 = (H− E22) |2〉 - E21|1〉

Lanczos Algorithm

At rank N, the following relations hold:

H|N〉 = ENN−1|N − 1〉 + ENN|N〉 + ENN+1|N + 1〉

Lanczos Algorithm

It is explicit that we have built a tridiagonal matrix

〈I|H|J〉 = 〈J|H|I〉 = 0 if |I − J| > 1

E11 E12 0 0 0 0

E12 E22 E23 0 0 0

0 E32 E33 E34 0 0

0 0 E43 E44 E45 0

Lanczos convergence

RANDOM STARTING VECTOR

3

-6.345165 11.335118 29.120687

6

-21.344259 -7.802025 4.637278 16.927858 29.308309

9

-30.092574 -19.653950 -9.343311 0.467972 10.265731

12

-32.722076 -24.462806 -17.104890 -9.353111 -1.628857

15

-32.930624 -26.709841 -22.335011 -15.957805 -9.401645

18

-32.952147 -28.028244 -24.233122 -19.625844 -14.772679

21

-32.953570 -28.413699 -25.350732 -22.676041 -18.180356

24

-32.953655 -28.537584 -26.244093 -23.883982 -20.534298

27

-32.953658 -28.559930 -26.542899 -24.362551 -22.197866

30

-32.953658 -28.563001 -26.646165 -24.887184 -23.559799

33

-32.953658 -28.564277 -26.912739 -26.199181 -24.299165

36

-32.953658 -28.564535 -27.102898 -26.382496 -24.409357

39

-32.953658 -28.564567 -27.148522 -26.416873 -24.529055

42

-32.953658 -28.564570 -27.156735 -26.425250 -24.724078

45

-32.953658 -28.564570 -27.158085 -26.427319 -24.910915

48

-32.953658 -28.564570 -27.158371 -26.428021 -25.107898

Lanczos convergence

48Cr

Dim (t=2) = 6.105

Dim (full space) = 2.106

Ca40

f7/2 f7/2

Π ν

f5/2p1/2p3/2

f5/2p1/2p3/2

STARTING VECTOR :EIGENVECTOR OF A SMALLER SPACE

ITER= 1 DIA= -31.105920 NONDIA= 4.642871

3

-32.578285 -21.260843 5.090417

6

-32.929531 -27.208522 -16.116780 -1.200061 14.816894

9

-32.952149 -28.024347 -22.702052 -13.782511 -3.514506

12

-32.953553 -28.345536 -25.965169 -20.636169 -12.806719

15

-32.953655 -28.528301 -26.951521 -22.532438 -18.004439

Lecture 2

Lanczos structure functions,Effective Interactions

Computation of transition operators

Given a one-body transition operator O(r), how do we compute

〈Ψf |O|Ψi〉

here Ψi and Ψf are many-body wave functions obtained from

shell-model diagonalization

One body operators:

O =A∑

i=1

o(r(i)) −→ O =∑

i ,j

〈i |o|j〉a†i aj

We need to know

the value of our one body operator between single particlewave functions 〈i|o|j〉

the one body density matrix elements 〈Ψf |a†i aj |Ψi〉

Computation of transition operators

for 〈i |o|j〉, one needs (eventually) to know the radial part of

the wave function: usually harmonic oscillator, sometimes

wood-saxon.

〈i |o|j〉 =

∫

d3r φ∗i oφj

for the one body density matrix elements, same procedure

as for the hamiltonian:

a†5a2|001011〉 = |011001〉

now we know the procedure to compute :

EL gamma decay rLYL0

ML gamma decay rL−1YL0

β decay :

Fermi decay : τ±Gamow-Teller decay: στ±

Spectroscopic factors a†i,ai

Lanczos Structure Function

We can be interested in the transition matrix elements

〈Ψf |O|Ψi〉 for many final states |Ψf 〉

example: β decay half-life calculation

Determine initial state |Ψi〉

Determine all posible final states |Ψf 〉

Compute matrix elements 〈Ψf |O|Ψi〉

λf =ln 2

Kf (Z ,W f

0)[Bf (F ) + Bf (GT )]

Determine total decay rate:

λ =ln 2

T1/2

=∑

f

λf

Lanczos Strength Function

Let O be an operator acting on some initial state |Φini 〉, we obtain the state

O|Φini 〉 whose norm is the sum rule of the operator O in the initial state:

S = |O|Φini〉| =√

〈Φini |O2|Φini〉

Depending on the nature of the operator O,the state O|Φini 〉 belongs to the same nucleus (if O is a e.m transition operator) or

to another (Gamow-Teller, nucleon transfer, a†

j /aj , ββ, ...)

If the operator O does not commute with H, O|Φini 〉 is not necessarily an

eigenvector of the system BUT it can be developped in energy eigenstates:

O|Φini〉 =∑

f

S(Ef )|Ef 〉 and 〈Φini |O2|Φini〉 =

∑

f

S2(Ef )

with S(Ef ) = 〈Ef |O|Φini〉 being the strength function (or structure

function)


If we carry on the Lanczos procedure

using |O〉 = O|Φini〉 as initial pivot.

then H is again diagonalized to obtain the eigenvalues |Ef 〉

U is the unitary matrix that diagonalizes H and gives the

expression of the eigenvectors in terms of the Lanczos vectors:

U =

|E1 |E2〉 |E3〉 ... |EN〉|O〉|2〉|3〉::

|N〉

S(Ef ) = U(1, f ) = 〈Ef |O〉 = 〈Ef |O|Φini〉

How good is the Strength function S obtained at iteration N

compared to the exact one S?


Any distribution can be characterized by the moments of the

distribution.

E = 〈O|H|O〉 =∑

f

Ef S2(Ef )

mn = 〈O|(H − E)n|O〉 =∑

f

(Ef − E)nS2(Ef )

Gaussian distribution characterized by two

moments (E , σ2 = m2)

g(E) = 1

σ√

2πexp(− (E−E)2

2σ2 )

Eg(

E)

2σ


Lanczos method provides a natural way of determining a finite

number of momenta.

Initial vector |1〉 = |O〉√〈O|O〉

E12|2〉 = (H − E11)|1〉E23|3〉 = (H − E22)|2〉 − E12|1〉. . .ENN+1|N + 1〉 = (H − ENN)|N〉

−EN−1N |N − 1〉

whereENN = 〈N|H |N〉, ENN+1 = EN+1N

Each Lanczos iteration gives informa-tion about two new moments of the dis-tribution.

E11 = 〈1|H |1〉 = E

E212 = 〈O|(H − E11)

2|O〉 = m2

E22 =m3

m2

+ E

E223 =

m4

m2

−m2

3

m22

− m2

Diagonalizing Lanczos matrix after N iterations gives an approximation to

the distribution with the same lowest 2N moments.

Evolution of Strength Distribution

GT Strength on 48Sc


GT Strength on 48Sc

48Ca(p,n)48Sc Strength Function

Quenching of GT operator in the pf -shell

Nucleus Uncorrelated Correlated Expt.

Unquenched Q = 0.74

51V 5.15 2.42 1.33 1.2 ± 0.154Fe 10.19 5.98 3.27 3.3 ± 0.555Mn 7.96 3.64 1.99 1.7 ± 0.256Fe 9.44 4.38 2.40 2.8 ± 0.358Ni 11.9 7.24 3.97 3.8 ± 0.459Co 8.52 3.98 2.18 1.9 ± 0.162Ni 7.83 3.65 2.00 2.5 ± 0.1

Quenching of GT strength in the pf -shell

Quenching of M1 operator in the pf -shell

KB3 interaction

Neumann-Cosel et al.

Phys. Lett. B433 1 (1998)


GXPF1 interaction

−4 −2 0 2 4 6 8 10µfree

(µN)

−4

−2

0

2

4

6

8

10

µexp (

µ N)

−4 −2 0 2 4 6 8 10µeff

(µN)

−4

−2

0

2

4

6

8

10

µexp (

µ N)

M. Homma, T. Otsuka, B. A. Brown, T. Mizusaki

Phys. ReV. C69, 034335 (2004)


0

0.04

0.08

6 8 10 12 140.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

6 8 10 12 14Excitation Energy (MeV)

0.0

0.5

1.0

Orbital

Spin

Shell−ModelTotal

Expt.

B(M

1) (

µ N2)

52Cr

K. Langanke, G. Martinez-Pinedo,

P. Von Neumann-Cosel, and A. Richter

Phys. Rev. Lett. 93, (2004) 202501

KB3G interaction

Quenching of GT operator in the pf -shellIf we write

|i〉 = α|0~ω〉+∑

n 6=0

βn|n~ω〉,

|f 〉 = α′|0~ω〉+∑

n 6=0

β′n|n~ω〉

then

〈f ‖ T ‖ i〉2 =

αα′ T0 +∑

n 6=0

βnβ′n Tn

2

,

n 6= 0 contributions negligible

α ≈ α′

projection of the physical wavefunction in the

0~ω space is Q ≈ α2

transition quenched by Q2

GT and M1 strengths in 12C

0 50 100 150 200E (MeV)

0

0.02

0.04

0.06

0.08

GT

Str

engt

h

0

0.02

0.04

0.06

0.08

GT

Str

engt

h

4~ω

6~ω

/10

/10

67% in 0~ω states

78% in 0~ω states

0 50 100 150 200E (MeV)

0

0.1

0.2

B(M

1) (

µ N2 )

0

0.1

0.2

0.3

B(M

1) (

µ N2 )

4~ω

6~ω

/10

/10

64% in 0~ω states

75% in 0~ω states

Correlations in nuclei

V. R. Pandharipande, I. Sick and P. K. A. deWitt

Huberts, Rev. mod. Phys. 69 (1997) 981

SCGF microscopic description

Lecture 2

Effective Interactions

Symmetries of the hamiltonian

The information related to the two body interaction is fully

contained in the matrix elements:

〈(j1j2)JTMTz

|V |(j3j4)JTMTz

〉

Symmetries of the hamiltonian have the following

consequences:

at fixed J (T) values, the matrix elements corresponding to

all possible values of M (Tz) are equal

matrix elements between different J (T) values are

vanishing

The effective interaction

Different approaches are possible to determine the effective

interaction, or equivalently the set of matrix elements:

〈(j1j2)JT |VJTj1j2j3j4

|(j3j4)JT 〉

EMPIRICAL

E icalc. =

p∑

k=1

c(i)k vk

and minimize the function

Q2 =N∑

i=1

(E icalc. − E i

exp.)2 i. e.

∂Q2

∂vr= 0 = ∂

∂vr

(N∑

i=1

((p∑

k=1

c(i)k vk )− E i

exp.)2

)

for r=1, ... p

The effective interaction

N∑

i=1

((

p∑

k=1

c(i)k

vk )− E iexp.))c

(i)r = 0

from thereN∑

i=1

((p∑

k=1

c(i)k vk )− E i

exp.) =N∑

i=1

c(i)r E i

exp.

i.e. a set of p linear equations with p parameters with the

hypothesis that the wave functions do not change (c(i)k fixed)

Effective interactions

choose

initial

parameters

calculate

hamiltonian

matrix

calculate

eigenvectors

eigenvalues

no

yesconv.? end

parameters

of

variation

new

interaction

parameters

new

linear

system

Realistic Interactions

Free nucleon-nucleon interaction:

N N

V(1,2)

• Spin state S=0,1

• Charge state pp,pn,nn (isospin T)

• Spatial state L=even,odd

δ

• Deuteron properties

=⇒ Realistic potentials : Hamada-Johnston, Paris, CD-Bonn,

Argonne, Idaho-A

v(NN) = vEM (NN) + vπ(NN) + vR(NN)

Failure of realistic potentials in many-body

calculations

NN potentails with hard core are not suited for low energy nuclear physicscalculations. Before using in medium, the realistic potential is ”softened”(G-matrix, Vlowk , SRG). Effective interactions for SM take into account the coreparticles and scattering to outer space by perturbation (MBPT).

0

2.5

5

8 14 16

Ee

xc(

2+)

(Me

V)

Neutron number

O, Z=8

THE EXP 0

2.5

5

20 28 32

Neutron number

Ca, Z=20

THE EXP

Realistic effective interactions

Exp. KB KLS Bonn A Bonn B Bonn C

2+1 excitation energy

44Ca 1.16 1.45 1.43 1.31 1.25 1.2646Ca 1.35 1.45 1.42 1.26 1.22 1.2348Ca 3.83 1.80 1.60 1.23 1.30 1.4150Ca 1.03 1.41 1.35 1.27 1.10 1.17

〈(f 72)8|ΨGS〉 0.468 0.381 0.214 0.345 0.437

56Ni model space (f 72p 3

2)16

56Ni 2.70 0.39 0.31 0.43 0.42 0.42

〈(f 72)16|ΨGS〉 0.04 0.015 0.018 0.011 0.019

〈np3/2〉 4.5 5.2 5.7 5.2 5.0

Realistic effective interactions

Exp. KB KLS Bonn A Bonn B Bonn C

2+1 excitation energy

44Ca 1.16 1.45 1.43 1.31 1.25 1.2646Ca 1.35 1.45 1.42 1.26 1.22 1.2348Ca 3.83 1.80 1.60 1.23 1.30 1.4150Ca 1.03 1.41 1.35 1.27 1.10 1.17

〈(f 72)8|ΨGS〉 0.468 0.381 0.214 0.345 0.437

56Ni model space (f 72p 3

2)16

56Ni 2.70 0.39 0.31 0.43 0.42 0.42

〈(f 72)16|ΨGS〉 0.04 0.015 0.018 0.011 0.019

〈np3/2〉 4.5 5.2 5.7 5.2 5.0

The realistic interactions do not reproduce the shell closure

N or Z=28back

Modern nuclear forces: N3LO

We use pions and nucleons asdegrees of freedom.

The effective Lagrangian isclassified using a systematicexpansion based on a powercounting in terms of (Q/Λχ)ν ,where ν is called chiral order andΛχ is the hard scale (∼700MeV)

ν=0 is called leading order, ν > 1are called next-to-ν − 1 leadingorders.

Note hierarchy of nuclear forces.

Coupling constants (LEC)adjusted to phase shifts anddeuteron properties.

E. Epelbaum et al., Rev. Mod. Phys. 81 (2009) 1773.

N3LO and no-core SM calculations

0

4

8

NN+NNN Exp NN

3 + 3

+1

+

1 +

0 +; 1

0 +; 1

1 +

1 +

2 +

2 +

3 +

3 +

2 +; 1

2 +; 1

2 +

2 +

4 +

4 +

2 +; 1

2 +; 1

hΩ=14

0

4

8

12

16

NN+NNN Exp NN

3/2-

3/2-

1/2-

1/2-

5/2-

5/2-

3/2-

3/2-

7/2-

7/2-

5/2-

5/2-

5/2-

5/2-

1/2-; 3/2

1/2-; 3/2

0

4

8

12

16

NN+NNN Exp NN

0 +

0 +

2 +

2 +

1 +

1 +

4 +

4 +

1 +; 1

1 +; 1

2 +; 1

2 +; 1

0 +; 1

0 +; 1

0

4

8

12

16

NN+NNN Exp NN

1/2- 1/2

-

3/2-

3/2-

5/2-

5/2-

1/2-

1/2-

3/2-

3/2-

7/2-

7/2-

3/2-; 3/2

3/2-; 3/2

10B

11B

12C 13

C

Excitation energies (in MeV) in light nuclei in NOCSM with chiral EFT interactions.NN force at N3LO level, 3N force from N2LO.P. Navratil et al., Phys. Rev. Lett. 99 (2007) 042501.

Effective 2-body HamiltoniansImportant features

/ Effective 2-body Hamiltonians derived from realistic

potentials fail to reproduce correctly the spectroscopy of

many-body systems and bulk properties of nuclei-no right

binding, no double shell closures...

, The matrix elements depend very little on the potential used

(Argonne, Paris, Bonn, N3LO...) and the method of

regularization (G-matrix, Vlowk , ...). The link to phase shifts is

then nearly model independent.

,/ NN potentials are nowadays ”perfect”. We need a NNN

force.

What we can do ?

we have solved the problem of s.p. energies replacing

them by the experimental ones

we can treat all the matrix elements as free parameters

and fit them to the many-body data (e.g. USD, USDb

interactions)

or we can proceed in a more general manner: understand

the physics behind the set of the matrix elements and

change only some of them

Outline

Separation of the effective Hamiltonian: monopole and

multipole

3-body forces: corrections to the monopole Hamiltonian

Some useful definitions

Separation of the effective HamiltonianMonopole and multipole

From the work of M. Dufour and A. Zuker (PRC 54 1996 1641)

Separation theorem:

Any effective interaction can be split in two parts:

H = Hmonopole + Hmultipole

Hmonopole: spherical mean-field

Zresponsible for the global saturation properties and for the

evolution of the spherical single particle levels.

Hmultipole: correlator

Zpairing, quadrupole, octupole...

Important property:

〈CS ± 1|H|CS ± 1〉 = 〈CS ± 1|Hmonopole|CS ± 1〉

Effective HamiltonianMonopole and Multipole Hamiltonians

V =∑

JT

V JTijkl

[

(a+i a+

j )JT (ak al)

JT]00

In order to express the number of particles operators

ni = a+i

ai ∝ (a+i

ai)0,

particle-hole recoupling :

V =∑

λτ

ωλτikjl

[

(a+i

ak )λτ (a+

jal)

λτ]00

ωλτikjl ∝

∑

JT

V JTijkl

i k λ

j l λ

J J 0

12

12 τ

12

12

τ

T T 0

Effective HamiltonianMonopole and Multipole Hamiltonians

Hmonopole corresponds only to the terms λτ=00 and 01 which

implies that i = j and k = l and writes as

Hmonopole =∑

i

niǫi +∑

i≤j

ni .nj Vij

Hmultipole corresponds to all other combinations of λτ

Effective HamiltonianMultipole Hamiltonian

Hmultipole can be written in two representations, particle-particle

Hmultipole =∑

ik<jlΓ

WΓijkl [(a

†i a

†j )

Γ(akal)Γ]0,

where Γ = JT or in particle-hole

Hmultipole =∑

ik<jlΓ

√

(2γ + 1)

√

(1 + δij)(1 + δkl)

4ωγ

ikjl [(a†i ak)

γ(a†j al)

γ ]0,

where γ = λτThe W and ω matrix elements are related by a Racah transformation:

ωγikjl =

∑

Γ

(−)j+k−γ−Γ

i j Γl k γ

WΓijkl(2J + 1)(2T + 1),

WΓijkl =

∑

γ

(−)j+k−γ−Γ

i j Γl k γ

ωγikjl(2λ+ 1)(2τ + 1).


In the preceding expressions we can replace pairs of indices by a single one

ij ≡ x , kl ≡ y , ik ≡ a et jl ≡ b, and diagonalise the matrices W Γxy and

f γab = ωγab

√

(1 + δij)(1 + δkl)/4, by unitary transformations UΓxk , u

γak :

U−1

WU = E =⇒ WΓxy =

∑

k

UΓxk U

Γyk E

Γk

u−1

fu = e =⇒ fγab =

∑

k

uγaku

γbke

γk ,

then

Hmultipole =∑

k,Γ

EΓk

∑

x

UΓxk Z

+xΓ ·∑

y

UΓyk ZyΓ,

Hmultipole =∑

k,γ

eγk

(∑

a

uγakS

γa

∑

b

uγbkS

γb

)0

[γ]1/2,

that we call representations E and e.


E-eigenvalue density for the KLS interaction in the pf+sdg major shells ~ω = 9. Eacheigenvalue has multiplicity [Γ].


e-eigenvalue density for the KLS interaction in the pf+sdg major shells ~ω = 9. Eacheigenvalue has multiplicity [γ].

0

5000

-4 -3 -2 -1 0 1 2 3 4 5

num

ber

of s

tate

s

e-eigenvalue density

2 0+4 0+

1 0+ 3 0-

1 1+

1 0-

Energy (Mev)


E-eigenvalue density for the KLS interaction in the pf+sdg major shells ~ω = 9, afterremoval of the five largest multipole contributions. Each eigenvalue has multiplicity [Γ].

0

2000

-10 -8 -6 -4 -2 0 2 4

num

ber

of s

tate

s

1 0+ 0 1+ 2 0+

Energy (Mev)

E-eigenvalue densityepsilon=2.0

Multipole Hamiltonian

Hmultipole can be written in two representations, particle-particleand particle-hole. Both can be brought into a diagonal form.When this is done, it comes out that only a few terms arecoherent, and those are the simplest ones:

L = 0 isovector and isoscalar pairing

Elliott’s quadrupole

~σ~τ · ~σ~τ

Octupole and hexadecapole terms of the type rλYλ · rλYλ

Besides, they are universal (all the realistic interactions give

similar values) and scale simply with the mass number

Interaction particle-particle particle-holeJT = 01 JT = 10 λτ = 20 λτ = 40 λτ = 11

KB3 -4.75 -4.46 -2.79 -1.39 +2.46FPD6 -5.06 -5.08 -3.11 -1.67 +3.17

GOGNY -4.07 -5.74 -3.23 -1.77 +2.46

Multipole Hamiltonian

γ eγ

1 eγ

2 eγ

1+2 M 〈u1|M〉〈u2|M〉〈u1|u′1〉〈u2|u

′2〉 α

2

11 1.77 2.01 3.90 στ .992 .994 .999 1.000 .94

20 -1.97 -2.14 -3.88 r2Y2 .996 .997 1.000 1.000 .9510 -1.02 -0.97 -1.96 σ .880 .863 .997 .994 1.04

21 -0.75 -0.85 -1.60 r2Y2τ .991 .998 .999 .997 .94

40 -1.12 -1.24 -2.11 r4Y4

Γ EΓ1 EΓ

2 EΓ1+2 P 〈U1|P〉〈U2|P〉〈U1|U

′1〉〈U2|U

′2〉 α

2

01 -2.95 -2.65 -5.51 P01 .992 .998 1.000 .994 1.04810 -4.59 -4.78 -10.12 P10 .928 .910 .998 .997 .991

Effective HamiltonianMonopole Hamiltonian

The evolution of effective spherical single particle energies with

the number of particles in the valence space can be extracted

from Hmonopole. In the case of identical particles the expresion

is:

ǫj(n) = ǫj(n = 1) +∑

i

V 1ij ni

The monopole Hamiltonian Hmonopole also governs the relative

position of the various T-values in the same nucleus, via the

terms:

bij Ti · Tj

Even small defects in the centroids can produce large changes

in the relative position of the different configurations due to the

appearance of quadratic terms involving the number of

particles in the different orbits

Effective HamiltonianClosed shell vs 4p4h states in 56Ni

ECS = 16 ǫf + 16∗152 Vff

2p3/22p1/21f5/2Π ν

1f7/2

r

f

E4p4h = 12 ǫf + 4 ǫr

66Vff + 48Vfr + 6Vrr

2p3/22p1/21f5/2Π ν

1f7/2f

r

∆ E = 4(ǫf − ǫr ) + 48(Vff − Vfr ) + 6(Vff − Vrr )

Effective HamiltonianMonopole Hamiltonian

table

Effective HamiltonianMonopole Hamiltonian: quadratic effects

Two shells (i and j ) system:

Hmonopole = E0 + niǫi + njǫj +ni(ni − 1)

2Vii +

nj(nj − 1)

2Vjj + ninjVij

It is possible to rewrite this equation and separate a global term H0

(depending only on the total number of particles n = ni + nj ) from a linear

term H1 and a quadratic term H2 in ni et nj to get:

Hmonopole = E0 + n ǫ0 +n(n − 1)

2W0

︸︷︷︸

+ Γij [ǫ1 + (n − 1)W1]︸︷︷︸

+ Γ(2)ij W2︸︷︷︸

= H0 + H1 + H2

with

Γij =Djni − Dinj

Di + Dj

Γ(2)ij =

Di Dj

2

(2ninj

Di Dj

− ni(nj − 1)

Di(Dj − 1)− ni(nj − 1)

Dj(Di − 1)

)

Effective HamiltonianMonopole Hamiltonian: quadratic effects

νΠ

1p3/2

1p1/2

1d5/22s1/2

1d3/2

νΠ

1p3/2

1p1/2

1d5/22s1/2

1d3/2

0 1 2 3 40

5

10

15

20

25

30

35

40

45

lineairequadratique

kp-kh excitation energies for 16O

3-body forcescorrections to the monopole Hamiltonian

The 2-body potentials are now ’perfect’. Exact calculations are

possible for light systems. The bad spectroscopy has to be then

related with the lack of 3-body forces.

Let’s add 3-body term to the monopole Hamiltonian

Hmonopole =∑

i

eini

︸︷︷︸

+∑

i≤j

aijnij +∑

i≤j

bijTij

︸︷︷︸

+∑

ijk

aijknijk +∑

ijk

bijkTijk

︸︷︷︸

,

1-body 2-body 3-body

where

nijk = ninjnk , Tijk = TiTjTk , etc.

3-body forces I

One should remember that the 2-body force contributes to the

1-body piece of the effective interaction, when we make

summation over orbits in the core:

∑

c

aicninc = ni

∑

c

aicnc ≡ niei

Similarly, the 3-body force contributes to 1-body and 2-body

terms.

∑

c

aijcninjnc = ninj

∑

c

aijcnc ≡ ninjaij

∑

cc′aic′cnincncc′ = ni

∑

cc′aicc′ncnc′ ≡ niei

3-body forces II

ZDifferent studies (no-core, coupled-cluster) suggest that the

residual 3-body force is much smaller than 1-and 2-body parts

of the 3-body force. As a first step one should look to the

contributions of the 3-body terms to 1 and 2-body pieces.

Contributions from

3-body force to 4He

binding energy (from

Hagen et al. Phys.

Rev. C 76, 034302,

2007)

Shell gapDefinition

The shell gap ∆ is defined as the difference of binding energies

(positively defined)

∆ = [BE(N)− BE(N − 1)]− [BE(N + 1)− BE(N)]

= 2BE(N)− BE(N + 1)− BE(N − 1)

ZThe uncorrelated shell gap is therefore equal to the difference

of the corresponding ESPE.

Computing session

This afternoon:

you will learn what a Hamiltonian file contains

you will check yourself how calculated spectra differ

depending on the interaction used (tuned, not tuned)

you will do some mathematics on the paper, please bring

calculators!

you will learn the usage of the option 52 to

calculate/change monopoles

you will ”repare” yourself a bad interaction

you will calculate ESPE

Further reading

E. Caurier et al., The shell model as a unified view of the

nuclear structure, Rev. Mod. Phys. 77 (2005) 427.

M. Dufour, A.P. Zuker, The realistic collective nuclear

Hamiltonian, Phys. Rev. C52 (1996) 1641.

A.P. Zuker, Separation of the monopole contribution to the

nuclear Hamiltonian, nucl-th/9505012.

A.P. Zuker, Three body monopole corrections to the

realistic interactions, Phys. Rev. Lett.90 (2003) 042502.

A. Schwenk, A.P. Zuker, Shell model phenomenology of

low momentum interactions, Phys. Rev. C74 (2007)

061302.

shell model a uniﬁed view of nuclear structurebasic ideas and concepts in nuclear physics an...

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