the standard nuclear (shell) model a primer · 2010. 10. 19. · hartree-fock approximation,...

The Standard Nuclear (Shell) ModelA Primer

ALFREDO POVESDepartamento de Fı́sica Teórica and IFT, UAM-CSIC

International School on Exotic BeamsSantiago de Compostela, September 4-11 2010

Alfredo Poves The Standard Nuclear (Shell) Model A Primer

OUTLINE

Introduction

Basics on the ISM (Interacting Shell Model)

The physics of the neutron rich nuclei around N=20, 28and 40

Deformation and Superdeformation at N=Z

Triaxiality; The light Xenon isotopes

Quadrupole Collectivity; SU3 and its variants

Glossary


The Nuclear A-body Problem

In the Standard Nuclear Model the elementary componentsare nucleons (neutrons N and protons Z, N+Z=A). Themesonic and quark degrees of freedom are integrated out

In most cases non-relativistic kinematics is used

The bare nucleon-nucleon (or nucleon-nucleon-nucleon)interactions are inspired by meson exchange theories ormore recently by chiral perturbation theory, and mustreproduce the nn phase shifts, and the properties of thedeuteron and other few body systems



The challenge is to find Ψ(~r1,~r2,~r3, . . .~rA) such that

HΨ=EΨ, with

H=A∑

i

Ti +A∑

i ,j

V2b(~ri ,~rj) +A∑

i ,j ,k

V3b(~ri ,~rj ,~rk )

The knowledge of the eigenvectors Ψ and the eigenvaluesE make it possible to obtain electromagnetic moments,transition rates, weak decays, cross sections,spectroscopic factors, etc.



The task is indeed formidable

Only very recently and only for very light nuclei A≤10 theproblem has been solved ”exactly”,

thanks to the pioneer work of Pandharipande, Wiringa andPieper, which used variational methods (Green Function)solved by Monte Carlo techniques, GFMC

More recently, the perturbative approach has beenimplemented in the framework of the No Core Shell Model(NCSM) by Barrett, Navratil, and Vary


”Ab Initio” Approaches; GFMC

-70

-65

-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Ene

rgy

(MeV

)

α+d

α+t

α+α

α+2n 6He+2n

7Li+n

8Li+n

8Be+n

9Li+n 9Be+n

6Li+α

AV18

IL2 Exp

1+

2H1/2+

3H

0+

4He

1+

0+2+

6He 1+

3+2+

6Li3/2−1/2−

7/2−5/2−

7Li

1+

0+

2+

8He0+

2+1+3+

4+

(0,1)+

8Li

0+2+

4+

1+3+

8Be

5/2−

7/2−

3/2−

1/2−

(5/2−)

9Li9/2−

3/2−5/2−1/2−7/2−

9Be

1+

2+

?

10Li

3+

4+

0+

2+

(2+,3+)

10Be

3+1+2+4+

10B

3+

1+

2+4+

α+d

α+t

α+α

α+2n 6He+2n

7Li+n

8Li+n

8Be+n

9Li+n 9Be+n

6Li+α

Argonne v18With Illinois-2

GFMC Calculations 6 November 2002


”Ab Initio” Approaches; NCSM

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 13C


”Ab Initio” Approaches

A very important outcome of these calculations is that it iscompulsory to include three body forces in order to getcorrect solutions of the nuclear many body problem

The GFMC and the NCSM are severely limited by the hugesize of the calculations when A becomes larger than twelve

For the rest of the chart of nuclides, approximate methodshave to be used. Except for the semiclasical ones (liquiddrop) and the α-cluster models, all are based on theIndependent Particle Approximation

Mid-way between the ”ab initio” and the conventional ShellModel and Energy Density Functional ones are new (orrenewed) methods like the Coupled Cluster Expansion, theIn-medium Similarity Renormalization Group and otherMany Body Perturbation Methods


The Independent Particle Model

The basic idea of the IPM is to assume that, at zeroth order, theresult of the complicated two body interactions among thenucleons is to produce an average self-binding potential. Mayerand Jensen (1949) proposed an spherical mean field consistingin an isotropic harmonic oscillator plus a strongly attractivespin-orbit potential and an orbit-orbit term. Later, otherfunctional forms were adopted, e.g. the Woods-Saxon well

The usual procedure to generate a mean field in a system of Ninteracting fermions, starting from their free interaction, is theHartree-Fock approximation, extremely successful in atomicphysics. Whatever the origin of the mean field, the eigenstatesof the N-body problem are Slater determinants i .e.anti-symmetrized products of N single particle wave functions.



In the nucleus, there is a catch, because the very strong shortrange repulsion and the tensor force make the HFapproximation based upon the bare nucleon-nucleon forceimpracticable.

However, at low energy, the nucleus do manifest itself as asystem of independent particles in many cases, and when itdoes not, it is due to the medium range correlations thatproduce strong configuration mixing and not to the short rangerepulsion.



Does the success of the shell model really “prove” that nucleonsmove independently in a fully occupied Fermi sea as assumedin HF approaches? In fact, the single particle motion canpersist at energies in fermion systems due to the suppressionof collisions by Pauli exclusion (Pandharipande et al., RMP69)

Brueckner theory takes advantage of the Pauli blocking toregularize the bare nucleon- nucleon interaction, in the form ofdensity dependent effective interactions of use in HFcalculations or G-matrices for large scale shell modelcalculations.



The wave function of the ground state of a nucleus in the IPM isthe product of an Slater determinant for the Z protons thatoccupy the Z lowest states in the mean field and another Slaterdeterminant for the N neutrons in the N lowest states of themean fieldIn second quantization, this state can be written as:

|N〉 · |Z 〉

with

|N〉 = n†1n†2 . . . n

†N |0〉

|Z 〉 = z†1z†2 . . . z

†Z |0〉

It is obvious that the occupied states have occupation number 1and the empty ones occupation number 0


The Meaning of the Shell Model

Dilution of the Spectroscopic strength by the bare N-Ninteraction. Results for nuclear matter.

If we had a system of non interacting fermions, the figure would be astep function with occupation 1 below the Fermi level and 0 above



In spite of that, the nuclear quasi-particles resemble extraordinarily tothe mean field solutions of the IPM, as can be seen in the classicalexample of the charge density difference between 206Pb and 205Tl,fromthe electron scattering experiments of Cavedon et al, 1982



The shape of the 3s1/2 orbit is very well given by the mean fieldcalculation. To make the agreement quantitative the calculateddensity has to be scaled down by the occupation numberTo know more, Read the article “ Independent particle motion and correlations in

fermion systems” V. R. Pandharipande, et al., RMP 69 (1997) 981.


Vocabulary

state: a solution of the Schroedinger equation with a onebody potential; e.g. the H.O. or the W.S. It is characterizedby the quantum numbers nljm and the projection of theisospin tzorbit: the ensemble of states with the same nlj , e.g. the0d5/2 orbit

shell: an ensemble of orbits quasi-degenerated in energy,e.g. the pf shellmagic numbers: the numbers of protons or neutrons thatfill orderly a certain number of shellsgap: the energy difference between two shellsSPE, single particle energies, the eigenvalues of the IPMhamiltonianESPE, effective single particle energies, the eigenvalues ofthe monopole hamiltonian


The Interacting Shell Model (ISM)

Is an approximation to the exact solution of the nuclear A-bodyproblem using effective interactions in restricted spaces

The effective interactions are obtained from the barenucleon-nucleon interaction by means of a regularizationprocedure aimed to soften the short range repulsion. In otherwords, using effective interactions we can treat the A-nucleonsystem in a basis of independent quasi-particles

A Shell Model calculation amounts to diagonalizing the nuclearhamiltonian in the basis of all the Slater determinants that canbe built distributing the valence particles in a set of orbits whichis called valence space. The orbits that are always full form thecore.


The three pillars of the shell model

The Effective Interaction

Valence Spaces

Algorithms and Codes

E. Caurier, G. Martı́nez-Pinedo, F. Nowacki, A. Poves and A. P. Zuker. “The Shell

Model as a Unified View of Nuclear Structure”, RMP 77 (2005) 427.


Making the Effective Interaction Simple

The effective shell model interactions appears sometimes as along list of meanigless numbers; the two body matrix elementsof the Hamiltonian.

Without loosing the simplicity of the Fock space representation,we can recast these numbers in a way full of physical insight,following Dufour-Zuker rules

Any effective interaction can be split in two parts:H = Hm(monopole) + HM (multipole).Hm contains all the terms that are affected by a sphericalHartree-Fock variation, hence it is responsible for the globalsaturation properties and for the evolution of the sphericalsingle particle energies


The Monopole Hamiltonian

Hm = Hsp +∑

[

1(1 + δij)

aij ni(nj − δij)

+12

bij

(

Ti · Tj −3ni4

δij

)]

.

The coefficients a and b are defined in terms of the centroı̈ds:

V Tij =

∑

J VJTijij [J]

∑

J [J]

as: aij = 14(3V1ij + V

0ij ), bij = V

1ij − V

0ij , the sums run over Pauli

allowed values.


The Monopole Hamiltonian

The evolution of effective spherical single particle energies withthe number of particles in the valence space is dictated by Hm.In the case of identical particles the expression reads:

ǫj(n) = ǫj(n = 1) +∑

i

V 1ij ni

Even small defects in the centroids can produce large changesin the relative position of the different configurations due to theappearance of quadratic terms involving the number ofparticles in the different orbits.


The Drift of the Single Particle Energies

-20

-10

0

10

2016148

ES

PE

(M

eV

)

Proton number

d5/2

s1/2

d3/2

f7/2

p3/2

p1/2

f5/2


The Drift of the Single Particle Energies

Sometimes, the two body interaction is analyzed in terms of itscentral, spin-orbit and tensor components. Indeed, only themonopole parts of these terms can contribute to the evolutionof the single particle energies. Nowacki and Sieja have shownrecently that the main drivers of this evolution can be of any ofthe three types depending on which orbits are involved. Holt,Otsuka and Schweck have also shown that the monopolecomponent of the three body force is responsible for the shellstructure around 28O.

So please refrain from putting always the blame on the tensorforce. It is only in very few cases that it happens to be so


The Multipole Hamiltonian

HM can be written in two representations, particle-particle andparticle-hole:

HM =∑

r≤s,t≤u,Γ

W ΓrstuZ+

rsΓ · ZtuΓ,

HM =∑

rstuΓ

[γ]1/2(1 + δrs)1/2(1 + δtu)1/2

4ωγrtsu(S

γrtS

γsu)

0,

where Z+Γ

( ZΓ) is the coupled product of two creation(annihilation) operators and Sγ is the coupled product of onecreation and one annihilation operator.

Z+rsΓ = [a†r a

†s]Γ and Sγrs = [a

†r as]

γ



The W and ω matrix elements are related by a Racahtransformation:

ωγrtsu =∑

Γ

(−)s+t−γ−Γ{

r s Γu t γ

}

W Γrstu [Γ],

W Γrstu =∑

γ

(−)s+t−γ−Γ{

r s Γu t γ

}

ωγrtsu[γ].

The operators Sγ=0rr are just the number operators for orbits rand Sγ=0rr ′ the spherical HF particle hole vertices. Both musthave null coefficients if the monopole hamiltonian satisfies HFself-consistency.



The operator Z+rrΓ=0 creates a pair of particle coupled to J=0 (orcoupled to L=0 and S=0, or in a state of zero total momentum).Therefore the terms

Z+rrΓ=0 · ZssΓ=0

represent different pairing hamiltonians, whose specificitiesdetermine the values of the matrix elements W Γ=0rrss



The operators Sγrs are typical vertices of multipolarity γ. Forinstance, γ=(J=1,L=0,T=1) produces a (~σ · ~σ) (~τ · ~τ ) term whichis nothing but the Gamow-Teller component of the nuclearinteraction

The terms Sγrs γ=(J=2,T=0), that appear in interaction are ofquadrupole type r2Y2. They are responsible for the existence ofdeformed nuclei, and are specially large and attractive whenjr − js=2 and lr − ls=2.


Universality of the Multipole Hamiltonian

Indeed, a careful analysis of the effective nucleon-nucleoninteraction in the nucleus, reveals that the multipole hamiltonianis universal and dominated by BCS-like isovector and isoscalarpairing plus quadrupole-quadrupole and octupole-octupoleterms of very simple nature (rλYλ · rλYλ)

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 +2.46GXPF1 -4.18 -5.07 -2.92 -1.39 +2.47BONNC -4.20 -5.60 -3.33 -1.29 +2.70


The Effective Interaction

The evolution of the spherical mean field in the valence spacesremains a key issue, because we know since long thatsomething is missing in the monopole hamiltonian derived fromthe realistic NN interactions, be it through a G-matrix, Vlow−k orother options.

The need for three body forces is now confirmed. Would theybe reducible to simple monopole forms? Would they solve themonopole puzzle of the ISM calculations? The preliminaryresults seem to point in this direction

The multipole hamiltonian does not seem to demand majorchanges with respect to the one derived from the realisticnucleon-nucleon potentials


The Valence Space(s)

An ideal valence space should incorporate the most relevantdegrees of freedom AND be computationally tractableClassical 0~ω valence spaces are provided by the majoroscillator shells p, sd and pf shellsOther physically sound and computationally accessible valencespaces are proposed below.(note: in a major HO shell of principal quantum number p the orbitj=p+1/2 is called intruder and the remaining ones are denoted by rp)


Valence Space(s)

Miscellanea of computationally accessible (and physicallysound) valence spaces:

For very neutron rich nuclei around N=28, a good choice isto take the sd shell for protons and the pf shell forneutrons.

r2-pf : intruders around N and/or Z=20

r3-g9/2(d5/2): 76Ge, 82Se, and the region around 80Zr

r3-g9/2(d5/2) for the neutrons and pf for protons: neutronrich Cr, Fe, Ni, Zn

r4-h11/2 for neutrons and p1/2 − g9/2-r4 for protons: 96Zr,100Mo,110Pd, 116Cd

r4-h11/2 for neutrons and protons: 124Sn, 128−130Te, 136Xe


Algorithms and Codes

Algorithms include Direct Diagonalisation, Lanczos, MonteCarlo Shell Model, Quantum Monte Carlo Diagonalization,DMRG etc. There are also a number of different extrapolationansatzs

The Strasbourg-Madrid codes (Antoine, Nathan), can deal withproblems involving basis of 1010 Slater determinants, usingrelatively modest computational resources. Other competitiveones in the market are OXBACH, NUSHELL and MSHELL


Collectivity in Nuclei

The widespread presence of nuclei with deformed shapes is aconspicuous manifestation of the importance of thequadrupole-quadrupole terms in the nuclear multipolehamiltonian. Nuclear superfluidity (and the shift of the massparabolas in even isobaric multiplets, and many other effects)signal also the importance of the pairing terms.

For a given interaction, a many body system would or would notdisplay coherent features at low energy depending on thestructure of the mean field around the Fermi level.


Nuclear Needles and Superfluids

If the spherical mean field around the Fermi surface makes thethe pairing interaction dominant, the nucleus becomessuperfluidIf the quadrupole-quadrupole interaction is dominant thenucleus acquires permanent deformationIn the extreme limit in which the monopole hamiltonian isnegligible, the multipole interaction would produce superfluidnuclear needles.Magic nuclei are spherical despite the strong multipoleinteraction, because the large gaps in the nuclear mean field atthe Fermi surface block the correlations


The Symmetries of the Quadrupole Interaction

The isotropic harmonic oscillator has SU(3) symmetry. Thequadrupole operators are generators of this group and theCasimir of the group contains the quadrupole-quadrupoleinteraction. Therefore the states of lower energy are those withmaximal deformation compatible with the Pauli principle.The spin orbit interaction breaks the SU(3) symmetry, but otherSU3 variants emerge when there are favorable orbits aroundthe Fermi level, like Pseudo-SU3 or Quasi-SU3.


The most popular “flaws” of the standard SMdescription

Quadrupole effective charges are needed (But their valueis universal and rather well understood)

Spin operators are quenched by another universal factorwhich relates to the regularization of the interaction (alsoknown as short range correlation). Indeed, BMFapproaches share this shortcoming

Not all the regions of the nuclear chart are amenable to aSM description yet


The mean field . . . and beyond

HF-based approaches rely on the use of densitydependent interactions of different sort; Skyrme, Gogny,RMF parametrizations

The correlations are taken into account via symmetrybreaking in the mean field

Projections before (VAP) or after (PAV) variation areenforced to restore the conserved quantum numbers

Ideally, configuration mixing is also implemented throughthe GCM


Physics Goals

The precision Spectroscopy towards larger masses

The description of the nuclear correlations in the laboratoryframe

The change of the Magic Numbers far from Stability: Thecompeting roles of spherical mean field and correlations

The precision descriptions of the weak nuclear processes. Inparticular the double β decay, which has the key to the natureof the neutrinos, the absolute scale of their masses and theirhierarchy

The Nuclear Structure inputs for Nuclear Astrophysics


the standard nuclear (shell) model a primer · 2010. 10. 19. · hartree-fock approximation,...

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