A brief introduction

D S Judson

Kinetic Energy Interactions between of nucleons ith and jth nucleons

The wavefunction of a nucleus composed of A nucleons can be described using the Non-relativistic Schrödinger equation

The nuclear wavefunction is then simply the product of the individual single particle wavefunctions

In practice, the Schrödinger equation is not much use for describing the nuclei we are typically interested in looking at

The nucleus is a finite, many body problem

Such a calculation is far too computationally intensive

Interactions between nucleons are not fully understood

Analytical solution not possible for A >~ 16 - full scale sd shell calculations are possible

Even if the calculations could be performed, the results would be so complex they would be difficult to interpret / describe

To allow a useful description of the nuclear wavefunction to be developed, a number of simplifications / assumptions have to be made.

•Assume a spherical inert closed shell core which plays no role in low energy excitations

•Assume higher lying orbitals play no role either

•The low energy properties of the nucleus are then determined by the valence nucleons

•Reduce the multi-nucleon interactions to an average, attractive, central potentialE.g. Woods-Saxon potential

•Assume nucleons undergo independent motion within this potential

Solution of the Schrodinger equation for the Woods-Saxon potential (with spin-orbit term)reproduces the experimentally observed shell-gaps

g7/2

d5/2

d3/2

s1/2

h11/2

50

82

Inert core

Play no role

Model Space

102Sn

These truncations perturb the spherical shell model Hamiltonian- Effective residual interaction must be added

Effective residualinteractions

Spherical one body Shell Model Hamiltonian

This can now be solved analytically, typically using matrix formalism

These truncations perturb the spherical shell model HamiltonianEffective residual interaction must be added

Effective residualinteractions

Spherical one body Shell Model Hamiltonian

Є1 and Є2 are single particle energies given in solution to H or from experiment

Diagonal matrix elements < ψx|Hres|ψx> are expectation values of Hres on |ψi>

Non-diagonal matrix elements <ψx|Hres|ψy> describe configuration mixing

The resultant matrix is diagonalised to determine eigenvalues / eigenvectors•Eigenvalue give the energy of the state•Eigenvectors describe the wavefunction of the state

A numerical example Calculate the energy of the first two 0+ states in 42Ca •Assume can be described as a closed core of 40Ca + 2 valence neutrons•Assuming a restricted model space of 1f7/2 and 2p3/2 orbitals

The (ν2 f7/2) and (ν2 p3/2) Jπ = 0+ states are the basis vectors |ψi>

f7/2

p3/2

f5/2

p1/2

20

50

g9/2FullModel SpaceRestrict

edModel Space

The resultant matrix is diagonalised to determine eigenvalues / eigenvectors•Eigenvalue give the energy of the state•Eigenvectors describe the wavefunction

A numerical example...Calculate first two 0+ states in 42Ca •Assume can be described as a closed core of 40Ca + 2 valence neutrons•Assuming a model space of 1f7/2 and 2p3/2 orbitals

The (ν2 f7/2) and (ν2 p3/2) Jπ = 0+ are the basis vectors |ψi>S.P.Es and matrix elements are taken from fpd6 interaction

Shell Model Hamiltonian(Single particle energies)

Effective interactions(matrix elements) Diagonalisation gives Elevel

and wavefunctions

0+

0+

3.118

-2.521

5.639

42Ca

NuShell uses the Lanczos method of diagonalisation which isslightly quicker!

Difficulties

•Effective interactions / matrix elements are derived for nuclei near closed shells

•Nuclei far from closed shell exhibit structure effects not accounted for in model

•Single particle energies are not well known away from closed shells

•The more valence nuclei, the larger the matrix to be diagonalised, the harder the calculation computationally

•The larger the model space the larger the matrix also

The ‘three pillars’ of the shell model1)A ‘good’ (realistic) model space2)Effective interactions adapted to the model space3)A code that makes it possible to solve these equations

Matrix size as a function of number of valence nuclei

The size of the Hamiltonian matrix can be reduced by reducing the model spaceI.e. reducing the number of orbitals that the nucleons can occupy and / or reducing the number of nucleons that can occupy a given orbital.

HOWEVER - Non physical restrictions will give non physical results!

Just because the computer gives a result does not mean the calculation is a success!