Shell Model A Uniﬁed View of Nuclear Structure · 2015-09-02 · The nuclear shell model Heyde K. Springer-Verlag 1994 The nuclear shell model A. Poves and F. Nowacki Lecture Notes

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 for SM calculations

Lecture 3: Shell model applications to nuclear

spectroscopy

Lecture 1

Introduction and basic notions

Basic Notions

First insights on nuclear structure:

liquid-drop model (Bethe and Bacher, 1936; von

Weizsacker, 1935): drops of charged, incompressible,

liquid nuclear matter

compound nucleus model of nuclear reactions (Bohr,

1936): incident neutron’s energy dissipate totally via

collisions

Basic Notions

Experimental evidences for shell structure in nuclei:

magic numbers

α lines systematics

15

20

25

30

35

40

45

0 20 40 60 80 100 120 140 160

BE

(Z+

2,N

+2)

-BE

(Z,N

)

N

α lines


15

20

25

30

35

40

45

0 20 40 60 80 100 120 140 160

BE

(Z+

2,N

+2)

-BE

(Z,N

)

N

α lines


15

20

25

30

35

40

45

0 20 40 60 80 100 120 140 160

BE

(Z+

2,N

+2)

-BE

(Z,N

)

N

α lines

Basic Notions

Experimental evidences for shell structure in nuclei:

magic numbers

single particle states

magnetic moments

BUT strong successes of liquid-drop and compound-nucleus

models as evidence against collisionless single-particle motion

assumed in the shell model

Basic Notions

Soon it was realized that for fermions:

compound-nucleus reactions occur at relatively high

excitations energies where many collisions are not Pauli

blocked

at low energy, suppression of collisions by Pauli exclusion

Single Particle Motion can persist

Independant Particle Model

Interaction of a nucleon with ALL the other particles is

approximated by a central potential:

Basic Notions

Empirical construction by M. Goeppert Mayer and H. Jensen

of a harmonic oscillator mean field plus a spin-orbit term to

reproduce the magic numbers:

U(r) =1

2mω2r2 + D~l2 - C~l .~s

Such a term does not commute with Lz and sz but DOES

commute with~j2= (~l + ~s)2 and jz = lz + sz ,~j2, ~s2:

-~l .~s = -1

2(~j2 −~l2 − ~s2) = -

1

2(j(j + 1)− l(l + 1)−

3

4)

=l + 1 for j=l-1

2

−l for j=l+12

Basic Notions

With spin-orbit coupling, the solutions are:

φnljm(r , σ) = Rnl(r) [Yl(θ, φ) χ 12(σ)]mj

where the orbital and spin wave functions coupling is

φnljm(r , σ) = Rnl(r)∑

ml ,ms

〈lml1

2ms|jm〉Y ml

l (θ, φ)χms12

(σ)

the corresponding energies being

ǫnljm = ~ω

[

N +3

2+ Dl(l + 1) + C

l + 1 j = l − 1

2

−l j = l + 12

]

with N = 2(n − 1) + l

Basic Notions

0

1s (2) 2

1p(8)(6)

8

1g7/2

1f2p

1d2s 2

3s

(38)(40)(50)

1g

1h

1h11/2

2d

3p

2d3/2

2p1/2

2s

1p3/21p1/2

1s

2p3/2

1h9/2

3s1/2

2d5/2

2f7/22f5/2 3p3/23p1/2

(64)

(82)

2f

1g9/2

1f5/2

1f7/2

1d3/2

1d5/2

(8)

(6)(4)(2)

(10)

(8)(6)

(4)(2)

(12)

(2)(6)(4)

(12)

(8)

(6)(2)(4)

(2)

(2)

(4)

(126)

(20)(14)

1i13/2 (14)

N=5

N=4

N=3

N=2

N=1

N=0

(28) 28

82

126

50

Hartree Fock

Link between nucleon-nucleon effective interaction and

mean-field : Hartree-Fock approximation

H =

A∑

i=1

ti +1

2

A∑

i 6= j

j = 1

vij

two body term replaced by a one body potential (mean field) U

H(0) =

A∑

i=1

ti + Ui

Single Particle in potential:

h(0)φa(r) = T (k) + U(r)φa(r) = ǫaφa(r)

System of A independant particles:

H(0) =A∑

k=1

T (k) + U(r(k))

The eigenfunctions of H(0) are

Φa1a2...aA(1,2, ...,A) =

A∏

k=1

φak(r(k))

with the eigenvalues E (0) =A∑

k=1

ǫak.

System of identical particles

For a system of identical particles, one needs to take into

account that the particles are indistinguishable

total wave function is (anti)symmetric

with exchange of two particles


in quantum mechanics, particle exchange degeneracy:

(1)

(2)

(1)

(2)

D

(1)

(2)

D

(1)

(2)

there exist two possible distinct final states (orthogonal) but

associated to a single physical state (no possible measurement

to distinguish them)


symmetrisation postulate

For a system of identical particles, only some (N-body)

eigenfunctions describe physical states: they are antisymmetric

(with respect to permutations of particles) for fermions and

symmetric for bosons

If |u〉 is a physical ket, Pα|u〉 is also a physical ket

For fermions, the physical kets are those obtained by

antisymmetrization :

A|u〉 avec A=√

1N!

∑

αǫαPα


example with two particles:

|u〉 = φa(1)φb(2) and Eu = φa(1)φb(2), φa(2)φb(1)

A|u〉 =√

12(φa(1)φb(2) − φa(2)φb(1))

Pauli principle: if φa = φb, A|u〉 = 0

antisymmetry in all particle coordinates

Generalized Pauli principle (in nuclear physics): the wave

function reverses its sign upon odd permutation of all

coordinates space, spin and isospin

For two particles in the same orbit:symmetric in space-spin and antisymmetric in isospin

antisymmetric in space-spin and symmetric in isospin

J + T = odd


example with three particles:

|u〉 = φa(1)φb(2)φc(3) and

Eu = φa(1)φb(2)φc(3), φa(2)φb(3)φc(1), φa(2)φb(1)φc(3),φa(3)φb(1)φc(2), φa(3)φb(2)φc(1), φa(1)φb(3)φc(2)

A|u〉 =√

16 ( φa(1)φb(2)φc(3) + φa(2)φb(3)φc(1) + φa(3)φb(1)φc(2)

−φa(2)φb(1)φc(3)− φa(3)φb(2)φc(1)− φa(1)φb(3)φc(2))


two particles case:

Φab(1, 2) =

√

1

2φa(1)φb(2) − φa(2)φb(1) =

√

1

2

∣∣∣∣

φa(1) φa(2)φb(1) φb(2)

∣∣∣∣

i. e. a Slater Determinant

three particles case:

Φabc(1,2,3) =

√

1

6

∣∣∣∣∣∣

φa(1) φa(2) φa(3)φb(1) φb(2) φb(3)φc(1) φc(2) φc(3)

∣∣∣∣∣∣

developped with the Sarrus rule

Φabc(1, 2, 3) =√

16

φa(1)φb(2)φc(3) + φa(3)φb(1)φc(2) + φa(2)φb(3)φc(1)

−φa(3)φb(2)φc(1) − φa(2)φb(1)φc(3)− φa(1)φb(3)φc(2)


A particles case:

Φaα1a

α2...aαA(1, 2, ...,A) =

√

1

A!

∣∣∣∣∣∣∣

φaα1

(r(1)) φaα1

(r(2)) ... φaα1

(r(A))φa

α2(r(1)) φa

α2(r(2)) ... φa

α2(r(A))

: : : :φa

αA(r(1)) φa

αA(r(2)) ... φa

αA(r(A))

∣∣∣∣∣∣∣

The global phase is determined by the order of the indices: α1,α2, ...αA withαi ≡ ni li jimi

Occupation number formalism to simplify such expressions:

Φaα1a

α2...aαA(1, 2, ..., A) = a

†aα1...a

†aαA|0〉

only occupation numbers of the single particle orbits are necessary (no

labelling of the particles)

Second quantization

Creation and annihilation operators:

a†i |0〉 = |i〉 ai |i〉 = |0〉 vacuum |0〉 such ai |0〉 = 0

For fermions, antisymmetry ensured by anti-commutation rules:

a†i ,a

†j = ai ,aj= 0

a†i ,aj=δi ,j

One body operators:

O(1) =A∑

i=1

O(r(i)),

whose matrix elements are 〈i |O|j〉 =∫

φ∗i (r)Oφj (r)dr

will write in second quantization as

O =∑

i ,j

〈i |o|j〉a†i aj

ex: n =∑

i

ni =∑

i

a†i ai

Second quantization

Two body operators:

O(2) =A∑

1=j<k

O(r(i), r(j)),

whose matrix elements are 〈ij |O|kl〉 =

∫

φ∗i (r(1))φ

∗j (r(2))(1 − P12)Oφk (r(1))φl (r(2))dr(1)dr(2)

will write in second quantization as

O = 14

∑

i ,j ,k ,l

〈ij |O|kl〉a†ia†jalak

ex: H, ββ operator

Hartree Fock

H(0) eigenfunctions are:

Ψa1a2...aA(1,2, ...,A) = det(

A∏

k=1

φak(r(k)))

=A∏

k=1

a†k |0〉

φak(r(k)) obtained by minimisation of the total energy

E =〈Ψ|H|Ψ〉

〈Ψ|Ψ〉

masses, radii, charge density distribution ...

magic numbers, single particle energies, individual wave

functions ...

Correlations in nuclei

for the description of nuclei, mean field is only the starting

point

the two body residual interaction (correlations) is

reponsable for the detailled structure of nuclei

in particular, correlations can induce coherent phenomena

i. e. collectivity


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

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


Example in 160:

νΠ

h ω

h ω

h ω

h ω

1d3/22s1/2

1d5/2

1s1/2

1p3/2

1p1/2

12 MeV

0

0 ...4

0 ...

+

+

− −


Example in 160:


Example in 160:

νΠ

1p3/2

1p1/2

1d5/22s1/2

1d3/2

νΠ

1p3/2

1p1/2

1d5/22s1/2

1d3/2


In order to incorporate the correlations, one has to go beyond

mean-field

Spherical mean-field

breaking symmetries

of the system

mixing

different mean-field configurations

• Hartree-Fock Bogoliubov

• Nilsson

• Deformed Hartree-Fock

• Tamm-Dancoff

• RPA

• Interacting shell-model

Basic notions

The succes of the independent particle model strongly suggest

that the very singular free NN interaction can be regularized in

the nuclear medium.

For a given number of protons and neutrons the mean field

orbitals can be grouped in three blocks.

Inert core: orbits that are always full.

Valence space: orbits that contain the physical degrees of

freedom relevants to a given property. The distribution of

the valence particles among these orbitals is governed by

the interaction.

External space: all the remaining orbits that are always

empty.

Basic notions

Starting with a regularized interaction, the exact solution of the

secular problem, in the (infinite) Hilbert space built on the mean

field orbits, is approximated in the large scale shell model

calculations by the solution of the Schrodinger equation in the

valence space, using an effective interaction such that:

HΨ = EΨ −→ Heff .Ψeff . = EΨeff .

In general, effective operators have to be introduced to account

for the restrictions of the Hilbert space

〈Ψ|O|Ψ〉 = 〈Ψeff .|Oeff .|Ψeff .〉

Basic notions

The microscopic description of the nucleus we adopt is that of a

non-(explicitely)-relativistic quantum many body system.

Therefore we assume:

nucleon velocities small enough to justify the use of

non-relativistic kinematics

hidden meson and quark-gluon degrees of freedom

two body interactions

Shell model

A shell model calculation needs the following ingredients:

A valence space

An effective interaction

A code to build and diagonalize the secular matrix

Obviously the last two points limit the choice of the valence

space

Valence space

The choice of the valence space:

In light nuclei the harmonic oscillator closures determine

the natural valence spaces.

4He −→ 16O −→ 40Ca −→ 80Zrp shell sd shell pf shell ↑Cohen/ Brown/ DeformedKurath Wildenthal

Valence space

0~ω

1~ω

2~ω

3~ω

4~ω

5~ω

...

1s

1p

1d2s

1f

2p

1g

2d

3s

1h

2f

3p

1s1/2 (2) 2

1p3/2 (6)1p1/2 (8) 8

1d5/2 (14)2s1/2 (16)

1d3/2 (20) 20

1f7/2 (28) 28

2p3/21f5/2 (38)

2p1/2 (40)1g9/2 (50) 50

1g7/22d5/2 (64)

2d3/23s1/2

1h11/2 (82) 82

1h9/22f7/22f5/2

3p3/23p1/2

1i13/2 (126) 126

Valence space

Valence space

0~ω

1~ω

2~ω

3~ω

4~ω

5~ω

...

1s

1p

1d2s

1f

2p

1g

2d

3s

1h

2f

3p

1s1/2 (2) 2

1p3/2 (6)1p1/2 (8) 8

1d5/2 (14)2s1/2 (16)

1d3/2 (20) 20

1f7/2 (28) 28

2p3/21f5/2 (38)

2p1/2 (40)1g9/2 (50) 50

1g7/22d5/2 (64)

2d3/23s1/2

1h11/2 (82) 82

1h9/22f7/22f5/2

3p3/23p1/2

1i13/2 (126) 126

Valence space

4 ≤ A ≤ 16 p shell

Cohen-Kurath interaction

Valence space

0~ω

1~ω

2~ω

3~ω

4~ω

5~ω

...

1s

1p

1d2s

1f

2p

1g

2d

3s

1h

2f

3p

1s1/2 (2) 2

1p3/2 (6)1p1/2 (8) 8

1d5/2 (14)2s1/2 (16)

1d3/2 (20) 20

1f7/2 (28) 28

2p3/21f5/2 (38)

2p1/2 (40)1g9/2 (50) 50

1g7/22d5/2 (64)

2d3/23s1/2

1h11/2 (82) 82

1h9/22f7/22f5/2

3p3/23p1/2

1i13/2 (126) 126

Valence space

4 ≤ A ≤ 16 p shell


16 ≤ A ≤ 40 sd shell

USD interaction

Valence space

0~ω

1~ω

2~ω

3~ω

4~ω

5~ω

...

1s

1p

1d2s

1f

2p

1g

2d

3s

1h

2f

3p

1s1/2 (2) 2

1p3/2 (6)1p1/2 (8) 8

1d5/2 (14)2s1/2 (16)

1d3/2 (20) 20

1f7/2 (28) 28

2p3/21f5/2 (38)

2p1/2 (40)1g9/2 (50) 50

1g7/22d5/2 (64)

2d3/23s1/2

1h11/2 (82) 82

1h9/22f7/22f5/2

3p3/23p1/2

1i13/2 (126) 126

Valence space

4 ≤ A ≤ 16 p shell



USD interaction

40 ≤ A ≤ 80 pf shell

KB3, GXPF1 interactions

Valence space

In heavier nuclei:

−→ jj closures due to the spin-orbit term show up

N=28, 50, 82, 126

Valence space

In heavier nuclei:

−→ jj closures due to the spin-orbit term show up

N=28, 50, 82, 126

the transition HO −→ jj : occurs between 40Ca and 100Sn

where the protagonism shifts from the 1f7/2 to the 1g9/2

Valence space

0~ω

1~ω

2~ω

3~ω

4~ω

5~ω

...

1s

1p

1d2s

1f

2p

1g

2d

3s

1h

2f

3p

1s1/2 (2) 2

1p3/2 (6)1p1/2 (8) 8

1d5/2 (14)2s1/2 (16)

1d3/2 (20) 20

1f7/2 (28) 28

2p3/21f5/2 (38)

2p1/2 (40)1g9/2 (50) 50

1g7/22d5/2 (64)

2d3/23s1/2

1h11/2 (82) 82

1h9/22f7/22f5/2

3p3/23p1/2

1i13/2 (126) 126

Valence space

4 ≤ A ≤ 16 p shell



USD interaction



Heavier nuclei :Spin-orbite shell closures

28, 50, 82, 126

Transition between 40Ca and 100Sn

Valence space

0~ω

1~ω

2~ω

3~ω

4~ω

5~ω

...

1s

1p

1d2s

1f

2p

1g

2d

3s

1h

2f

3p

1s1/2 (2) 2

1p3/2 (6)1p1/2 (8) 8

1d5/2 (14)2s1/2 (16)

1d3/2 (20) 20

1f7/2 (28) 28

2p3/21f5/2 (38)

2p1/2 (40)1g9/2 (50) 50

1g7/22d5/2 (64)

2d3/23s1/2

1h11/2 (82) 82

1h9/22f7/22f5/2

3p3/23p1/2

1i13/2 (126) 126

Valence space

4 ≤ A ≤ 16 p shell



USD interaction




28, 50, 82, 126


Valence space

0~ω

1~ω

2~ω

3~ω

4~ω

5~ω

...

1s

1p

1d2s

1f

2p

1g

2d

3s

1h

2f

3p

1s1/2 (2) 2

1p3/2 (6)1p1/2 (8) 8

1d5/2 (14)2s1/2 (16)

1d3/2 (20) 20

1f7/2 (28) 28

2p3/21f5/2 (38)

2p1/2 (40)1g9/2 (50) 50

1g7/22d5/2 (64)

2d3/23s1/2

1h11/2 (82) 82

1h9/22f7/22f5/2

3p3/23p1/2

1i13/2 (126) 126

Valence space

4 ≤ A ≤ 16 p shell



USD interaction




28, 50, 82, 126


Valence space

A valence space can be adequate to describe some

properties and completely wrong for others

48Cr (f 72)8 (f 7

2p 3

2)8 (fp)8

Q(2+) (e.fm2) 0.0 -23.3 -23.8E(2+) (MeV ) 0.63 0.44 0.80E(4+)/E(2+) 1.94 2.52 2.26

BE2(2+ → 0+) (e2.fm4) 77 150 216

B(GT) 0.90 0.95 3.88

Valence space

For the quadrupole properties f 72p 3

2

is a good space whereas for magnetic and Gamow-Teller

processes the presence of the spin orbit partners is

compulsory.

In the tin isotopes

the natural valence space consists in a 100Sn core and

valence orbits:

(d 52g 7

2s 1

2d 3

2h 11

2)ν

However, numerous E1 transitions have been measured

that are forbidden in this space because:

(a†11/2

.aj)λ6=1,

it is then necessary to incorporate the 1g9/2 orbit.

Intruder states

normal states intruder states

Ca42

νΠ

2s1/21d5/2

1d3/2

1f7/2

2p1/21f5/2

2p3/2

Ca42

νΠ

1f7/22p3/2

2p1/21f5/2

1d5/22s1/2

1d3/2

Physics Goals

Precision Spectroscopy towards larger masses

Description of the nuclear correlations in the laboratory frame

Changing Magic Numbers far from Stability: The competing

roles of spherical mean field and correlations

Double β decay, the key to the nature of the neutrinos, the

absolute scale of their masses and their hierarchy

No core shell model for light nuclei. Ab initio description of the

low-lying intruder states and of the origin of the Gamow-Teller

quenching

Nuclear Structure and Nuclear Astrophysics

The most popular “flaws” of the standard SM

description

Not all the regions of the nuclear chart are amenable to a

SM description yet

Quadrupole effective charges are needed (But their value

is universal and rather well understood)

Spin operators are quenched by another universal factor

which relates to the regularization of the interaction (also

known as short range correlation). Indeed, BMF

approaches share this shortcoming

Accessible Regions

Ν→

Ζ↑

*

* *

*

*

*

* *

*

*

* *

* ** **

* * *

**

*

* **

*

**

**

** ***

**

*

*** **

***

**

*

**

***

*

** ***** *

**

***

*

* **** **

*******

***** **** *

*

****

*

*** **** *

****

*****

*** ***** ** ***** *

**

***

*

*** *** *

**

***

*

**

****

*

*** *** *

***

*

***

**

***

*

***

**

****** ****** ****** ******* *

*****

*****

*

***** ***

**

****

*****

***

*****

**** *******

*

*

*****

****** *

*****

*****

**

*** *****

*

**

**

****

****

****

**

****

*

*** ******

*

**** ***

****

*

*** **** ****

********

**

****

***

*

***** **** **

***

*** *

****

*

**** ** * * *

n 0 H 1

He 2 Li 3

Be 4 B 5 C 6

N 7 O 8

F 9 Ne 10

Na 11 Mg 12 Al 13 Si 14

P 15 S 16

Cl 17 Ar 18

K 19 Ca 20

Sc 21 Ti 22

V 23 Cr 24

Mn 25 Fe 26

Co 27 Ni 28

Cu 29 Zn 30

Ga 31 Ge 32

As 33 Se 34

Br 35 Kr 36

Rb 37 Sr 38

Y 39 Zr 40

Nb 41 Mo 42

Tc 43 Ru 44

Rh 45 Pd 46

Ag 47 Cd 48

In 49 Sn 50

Sb 51 Te 52

I 53 Xe 54

Cs 55 Ba 56

La 57 Ce 58

Pr 59 Nd 60

Pm 61 Sm 62

Eu 63 Gd 64

Tb 65 Dy 66

2 4 6

8 10

12 14 16

18 20

22 24

26

28 30 32

34

36 38

40 42

44 46 48 50

52

54

56 58

60 62 64

66

68 70

72

74 76 78 80 82

84

86 88

90 92 94 96

98

100

*** **

**

** **

*

****

* ** *** * *

*

*

*

*** *** ***** ****** *

*

*

*****

* ***** *

**

****

*

** ****** **** ******** ***

****

*******

***

****

**********

****** ******** **

*****

*******

*

***** ******* ***** ***** **

*****

*******

***

** *** **

***********

*

*** ********************

Ho 67 Er 68

Tm 69 Yb 70

Lu 71 Hf 72

Ta 73 W 74

Re 75 Os 76

Ir 77 Pt 78

Au 79 Hg 80

Tl 81 Pb 82

Bi 83 Po 84

At 85 Rn 86

Fr 87 Ra 88

Ac 89 Th 90

Pa 91 U 92

Np 93 Pu 94

Am 95 Cm 96 Bk 97

Cf 98 Es 99

Fm 100Md 101

No 102Lr 103Rf 104

Ha 105Sg 106Ns 107

Hs 108Mt 109

10 11011 111

80 82 84 86 88 90 92 94 96 98 100 102 104106

108110

112

114 116

118 120

122124

126128

130 132

134136 138

140142 144

146

148 150

152

154 156

158

160

Shell Model Problem

CORE

Define a valence space

Shell Model Problem

CORE


Derive an effective interaction

HΨ = EΨ → HeffΨeff = EΨeff

Shell Model Problem

CORE




Build and diagonalize the

Hamiltonian matrix.

Shell Model Problem

CORE




Build and diagonalize the

Hamiltonian matrix.

In principle, all the spectroscopic properties are described

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

The interaction in second quantization

The hamiltonian can be written as:

H =A∑

k=1

T (k) +A∑

k<l

W (k , l)

and in second quantization

H =A∑

ij

〈i |T |j〉a†i aj +1

4

A∑

ijkl

〈ij |W |kl〉a†i a

†j alak

Introducing a mean fieldA∑

k

U(k), it can be written as:

H =

A∑

k=1

T (k) + U(k)

︸︷︷︸

H(0)

+ A∑

k<l

W (k , l) −A∑

k=1

U(k)

︸︷︷︸

V

The basis states are eigenvectors of the mean field and the two

body hamiltonian is diagonalized in this basis.

m scheme

Choice of the basis:

a) m-scheme : The symmetries of the hamiltonian are not

explicit. The basis is composed of Slater determinants made

from the valence orbits a†i|0〉

|Φα〉 =∏

i=nljmτ

a†i |0〉 = a

†i1...a

†iA|0〉

The physical states come out of the diagonalization of the

Hamiltonian matrix

〈Φα|Heff .|Φα′〉

The drawback is that the size of the matrices is maximal:

D ∼(

dπ

p

).(

dν

n

)

The advantage is that the Hamiltonian matrix is sparse and ...

Computing the matrix elements

The matrix elements are very easy to compute. We represent a

Slater determinant by a machine word, where each state is a bit

(0 empty 1 occupied)

Example : 12C in the p-shell

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〉

In this example the Slater determinant is a 12 bits word.

The action of the Hamiltonian on such an object is very simple:


Let |I〉 be a basis function; the action of a two body term |I〉,

a†i a

†j akal |I〉 carries

an amplitude ±Vijkl ,

if k and l are occupied and i and j are empty or equal to k , l

or zero amplitude otherwise

If the result is not zero it produces another state |J〉


Back to 12C

a†12b

†6a9b1 0 0 1 1 1 1 0 0 1 1 1 1

= 1 0 1 0 1 1 1 0 1 1 1 0

= a†12a

†10a

†8a

†7 b

†6b

†4b

†3b

†2 |0〉

In practice,

all the machine words corresponding to all the basis states

are generated |I〉 I=1, ...N

and a loop is made on all the two body operators a†ia†jakal

Computing matrix elements

The resulting states |J〉 are clasified (identified) in the list of

Slater determinants and the matrix element HIJ is

〈J|H|I〉 = ±Vijkl

This is the Glasgow method (Whitehead, (1977))


The search of |J〉 can be done by the bisection method:



1



1

2



1

2

3



1

2

3

nop ∼ ln(N)

Phases

Example

|I〉 ≡ a†6a

†5a

†4a

†2 |0〉8 ≡ 1 1 1 0 1 0

Action of :

a†3a

†1a4a2 |I〉 ≡ 1 1 0 1 0 1

Phase:

a†3a

†1a4a2 (a†

6a†5a

†4a

†2) |0〉

= −a†3a

†1a4 a

†6a

†5a

†4 |0〉

= −a†3a

†1 a

†6a

†5 |0〉

= −a†6a

†5 a

†3a

†1 |0〉

Coupled scheme

Choice of the basis:

b) coupled basis (J or JT ):

The wave function is written as successive coupling of one shell

wave functions (c. f. p. ’s) defined by |(ji )ni viγixi〉 :

[ [|(j1)

n1v1γ1x1〉 |(j2)n2v2γ2x2〉

]Γ2 ... |(jk )nk vkγkxk 〉

]Γk

~Γk = ~Γk−1 + ~γk

vi ≡ seniority i. e. number of particles non coupled by pairs

to J = 0

The interaction in second quantization

H =

A∑

ij

ǫia†i aj +

1

4

A∑

ijkl

〈ij|V res.|kl〉a†i a

†i al ak

= H(0) + V

Introducing the operators: ajm = (−1)j+m aj−m and defining thecoupling of tensors as:

[a†j1a†j2]JM =

∑

m1m2

〈j1m1j2m2|JM〉a†j1m1

a†j2m2

we obtain:

V = − 14

∑

j1 j2 j3 j4

〈j1j2|V |j3 j4〉JT ×

√

(2J + 1)(2T + 1)(1 + δ12)(1 + δ34)

×[

[a†

j1a†

j2]JT × [aj3 aj4]

JT]00

Matrix elements in coupled basis

Two body interaction can always be recoupled as:

V =[

[ Oω1Oω2

]Ω2 ... Oωk

]0

with

Oωi = (1,a†j , aj , (a

†j a

†j )

λ, (aj aj)λ, (a†

j aj)λ

[

(a†j a

†j )

λaj

]µ,[

a†j (aj aj)

λ]µ

,[

(a†j a

†j )

λ(aj aj)λ]0)

Matrix elements in coupled basis

〈nγΓx‖V ||n′γ′Γ′x ′〉 = (−1)

∑

i=2

ni (Ni−1−N′

i−1)

〈n1v1γ1x1||Oω1 ||n′

1v ′1γ

′1x ′

1〉

×∏

i=2

√(2Γi + 1).(2Γ′

i + 1).(2Ωi + 1)× 〈niγixi ||Oωi ||n′

i γ′i x

′i 〉

Γi−1 γi Γi

Γ′i−1 γ

′i Γ′

iΩi−1 ωi Ωi

n = (n1, ..., nk): number of particles in each orbital ji ,

γ = (γ1, ..., γk ): angular momentum of each state (ji)ni ,

x = (x1, ..., xk ): additional quantum numbers (e. g. seniority) for eachstate (ji)

ni ,

Γ = (Γ1, ..., Γk ): successive intermediate couplings

Ni =i∑

r=1

nr : total number of particules in the orbits j1, ..., jk

42Ca normal states

For “normal” 0+ states, one needs to

diagonalize the 4×4 matrix:

(f 72

)2 (p 32

)2 (p 12

)2 (f 52

)2

0.0 × 2 −0.783 −0.714 −2.788+(−1.920)

2.0 × 2 −1.465 −0.777+(−1.206)

4.0 × 2 −0.392+(−0.249)

6.5 × 2+(−1.687)

42Ca

0+ 0

2+ 1386

4+ 2384

2+ 4071

0+ 5292

0+ 0

2+ 1524

0+ 1837

2+ 2424

4+ 2752

3– 3447

shell model exp.

Diagonalization of the Hamiltonian

pf -shell valence space: 1f7/2,2p3/2,2p1/2,1f5/2

nucleus m-scheme jj-scheme(ANTOINE) (NATHAN)

48Cr 1,963,461 41,35554Fe 345,400,174 5,220,62156Fe 501,113,392 7,413,48856Ni 1,087,455,228 15,443,684

Impossible to store Hamiltonian matrix!

Still possible to compute HΨ.

Diagonalization using an iterative algorithm.

Choice of the basis

• m scheme

|Φα〉 =∏

i=nljmτ

a†i|0〉 = a

†i1...a†

iA|0〉

Simple HIJ but Maximal size:

D ∼(

dπp

).(

dνn

)

Huge dimensions of the matrices (109)

storage of Lanczos vectors on diskAMD Opteron 64bits 2.2 GHz / 8 Gb RAM

56Ni: D=109 1h30/it.

Very large cases:splitting of the initial and final vectors

Ψi,f =⋃

mΨm

i,f

Ψ(m)f

=∑

nH(m,n)Ψ

(n)i

• Coupled scheme

[ [

|(j1)n1 v1γ1x1〉 |(j2)

n2 v2γ2x2〉]Γ2

...|(jk )nk vkγk xk 〉

]Γk

Reduced dimensions BUT complicated andmuch more non zero termsSmall dimensions of the matrices (107)Parallelization : each processor has the ini-tial and a final vector and final vectors areadded

Ψ(k)f

= H(k)Ψi ; Ψf =∑

k

Ψ(k)f

ImaBIO Cluster 24 nodes Xeon 2.8 GHzNuc Theo Cluster 10 nodes Xeon 2.7 GHz

128Xe GS: D=70 106 (DM=0=1010)

400 Gb precalculation storage

1014 non zero terms !!! 8h/it.(34 procs)

2+ out of reach

Choice of the basis

• m scheme

|Φα〉 =∏

i=nljmτ

a†i|0〉 = a

†i1...a†

iA|0〉

Simple HIJ but Maximal size:

D ∼(

dπp

).(

dνn

)

Huge dimensions of the matrices (109)

storage of Lanczos vectors on diskAMD Opteron 64bits 2.2 GHz / 8 Gb RAM

56Ni: D=109 1h30/it.

Very large cases:splitting of the initial and final vectors

Ψi,f =⋃

mΨm

i,f

Ψ(m)f

=∑

nH(m,n)Ψ

(n)i

• Coupled scheme

[ [

|(j1)n1 v1γ1x1〉 |(j2)

n2 v2γ2x2〉]Γ2

...|(jk )nk vkγk xk 〉

]Γk

Reduced dimensions BUT complicated andmuch more non zero termsSmall dimensions of the matrices (107)Parallelization : each processor has the ini-tial and a final vector and final vectors areadded

Ψ(k)f

= H(k)Ψi ; Ψf =∑

k

Ψ(k)f

ImaBIO Cluster 24 nodes Xeon 2.8 GHzNuc Theo Cluster 10 nodes Xeon 2.7 GHz

128Xe GS: D=70 106 (DM=0=1010)

400 Gb precalculation storage

1014 non zero terms !!! 8h/it.(34 procs)

2+ out of reach

Comparison m-scheme/coupled scheme

Coupled scheme is preferable for:

the calculation of 0+ states

the calculation of a large number of states (structure

function)

seniority truncation

In all other cases, m-scheme, because more simple, is more

efficient:

Yrast band calculation

Limitations

For m-scheme, the size of the basis is THE limitation

For coupled scheme, the number of non-zero terms is THE

limitation

Dimensions HIJ 6= 0

M=0(M=0)(J=0)

(M=0)(J=4)

M=0(J=0)(M=0)

(J=4)(M=0)

48Cr 1.9 106 47.5 7.9 0.8 109 0.6 16.752Fe 1.1 108 61.9 9.4 7.4 1010 2.4 83.256Ni 1.1 109 70.4 10.3 9.6 1011 5.3 19460Zn 2.3 109 73.3 10.6 2.2 1012 6.8 254

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


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

EN−1N = ENN−1, ENN = 〈N|H|N〉

Lanczos Algorithm


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

EN−1N = ENN−1, ENN = 〈N|H|N〉

and ENN+1|N + 1〉 = (H − 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 VECTOR3

-6.345165 11.335118 29.1206876

-21.344259 -7.802025 4.637278 16.927858 29.3083099

-30.092574 -19.653950 -9.343311 0.467972 10.26573112

-32.722076 -24.462806 -17.104890 -9.353111 -1.62885715

-32.930624 -26.709841 -22.335011 -15.957805 -9.40164518

-32.952147 -28.028244 -24.233122 -19.625844 -14.77267921

-32.953570 -28.413699 -25.350732 -22.676041 -18.18035624

-32.953655 -28.537584 -26.244093 -23.883982 -20.53429827

-32.953658 -28.559930 -26.542899 -24.362551 -22.19786630

-32.953658 -28.563001 -26.646165 -24.887184 -23.55979933

-32.953658 -28.564277 -26.912739 -26.199181 -24.29916536

-32.953658 -28.564535 -27.102898 -26.382496 -24.40935739

-32.953658 -28.564567 -27.148522 -26.416873 -24.52905542

-32.953658 -28.564570 -27.156735 -26.425250 -24.72407845

-32.953658 -28.564570 -27.158085 -26.427319 -24.91091548

-32.953658 -28.564570 -27.158371 -26.428021 -25.107898

Lanczos convergence


-6.345165 11.335118 29.1206876

-21.344259 -7.802025 4.637278 16.927858 29.3083099

-30.092574 -19.653950 -9.343311 0.467972 10.26573112

-32.722076 -24.462806 -17.104890 -9.353111 -1.62885715

-32.930624 -26.709841 -22.335011 -15.957805 -9.40164518

-32.952147 -28.028244 -24.233122 -19.625844 -14.77267921

-32.953570 -28.413699 -25.350732 -22.676041 -18.18035624

-32.953655 -28.537584 -26.244093 -23.883982 -20.53429827

-32.953658 -28.559930 -26.542899 -24.362551 -22.19786630

-32.953658 -28.563001 -26.646165 -24.887184 -23.55979933

-32.953658 -28.564277 -26.912739 -26.199181 -24.29916536

-32.953658 -28.564535 -27.102898 -26.382496 -24.40935739

-32.953658 -28.564567 -27.148522 -26.416873 -24.52905542

-32.953658 -28.564570 -27.156735 -26.425250 -24.72407845

-32.953658 -28.564570 -27.158085 -26.427319 -24.91091548

-32.953658 -28.564570 -27.158371 -26.428021 -25.107898

Lanczos convergence


-6.345165 11.335118 29.1206876

-21.344259 -7.802025 4.637278 16.927858 29.3083099

-30.092574 -19.653950 -9.343311 0.467972 10.26573112

-32.722076 -24.462806 -17.104890 -9.353111 -1.62885715

-32.930624 -26.709841 -22.335011 -15.957805 -9.40164518

-32.952147 -28.028244 -24.233122 -19.625844 -14.77267921

-32.953570 -28.413699 -25.350732 -22.676041 -18.18035624

-32.953655 -28.537584 -26.244093 -23.883982 -20.53429827

-32.953658 -28.559930 -26.542899 -24.362551 -22.19786630

-32.953658 -28.563001 -26.646165 -24.887184 -23.55979933

-32.953658 -28.564277 -26.912739 -26.199181 -24.29916536

-32.953658 -28.564535 -27.102898 -26.382496 -24.40935739

-32.953658 -28.564567 -27.148522 -26.416873 -24.52905542

-32.953658 -28.564570 -27.156735 -26.425250 -24.72407845

-32.953658 -28.564570 -27.158085 -26.427319 -24.91091548

-32.953658 -28.564570 -27.158371 -26.428021 -25.107898

Lanczos convergence


-6.345165 11.335118 29.1206876

-21.344259 -7.802025 4.637278 16.927858 29.3083099

-30.092574 -19.653950 -9.343311 0.467972 10.26573112

-32.722076 -24.462806 -17.104890 -9.353111 -1.62885715

-32.930624 -26.709841 -22.335011 -15.957805 -9.40164518

-32.952147 -28.028244 -24.233122 -19.625844 -14.77267921

-32.953570 -28.413699 -25.350732 -22.676041 -18.18035624

-32.953655 -28.537584 -26.244093 -23.883982 -20.53429827

-32.953658 -28.559930 -26.542899 -24.362551 -22.19786630

-32.953658 -28.563001 -26.646165 -24.887184 -23.55979933

-32.953658 -28.564277 -26.912739 -26.199181 -24.29916536

-32.953658 -28.564535 -27.102898 -26.382496 -24.40935739

-32.953658 -28.564567 -27.148522 -26.416873 -24.52905542

-32.953658 -28.564570 -27.156735 -26.425250 -24.72407845

-32.953658 -28.564570 -27.158085 -26.427319 -24.91091548

-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 SPACEITER= 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

Numerical instabilities

STATE 2*J= 4 LANCZOS UNTIL CONVERGENCE OF THE EIGENVALUE N= 6

21 -18.887126 -16.682989 -15.490861 -13.688336 -12.096178 -9.739625CONVERGENCE DELTA E= 0.001230 0.303215 0.650853 0.985329 1.914066 2.536821



57 -18.887244 -18.322405 -17.161909 -16.549748 -15.522805 -15.125042-14.292844 -14.099706 -13.401536 -13.052378 -12.605377 -11.799747

CONVERGENCE DELTA E= 0.000000 1.160496 0.612161 1.026942 0.392340 0.831005


6666 -19.877245 -18.887244 -17.161909 -16.549748 -15.522805 -15.136157

CONVERGENCE DELTA E= 0.000144 0.000000 0.000000 0.000000 0.000000 0.000082

GROUND-STATE (AMONG THE READ STATES) ENERGY= -19.87724

N,Z= 4 2 2*J= 0 P=0 N= 1 2*M= 0 C= 4 EXC = 0.00000 E= -19.87724N,Z= 4 2 2*J= 4 P=0 N= 1 2*M= 0 C= 4 EXC = 0.00000 E= -19.87724N,Z= 4 2 2*J= 4 P=0 N= 2 2*M= 0 C= 4 EXC = 0.99000 E= -18.88724

Computing session

This afternoon:

you will calcule m-scheme dimensions with paper and

pencil

you will start to use Antoine code to set up basis, valence

space etc ...

you will check your basis calculations with the code

you will learn start to perform the first diagonalisations

Documents

Shell Model A Uniﬁed View of Nuclear Structure · 2015-09-02 · The nuclear shell model Heyde K. Springer-Verlag 1994 The nuclear shell model A. Poves and F. Nowacki Lecture Notes