A brief introduction D S Judson

Kinetic Energy Interactions between of nucleons i th and j th 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 potential E.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 g 7/2 d 5/2 d 3/2 s 1/2 h 11/ Inert core Play no role Model Space 102 Sn

These truncations perturb the spherical shell model Hamiltonian - Effective residual interaction must be added Effective residual interactions Spherical one body Shell Model Hamiltonian This can now be solved analytically, typically using matrix formalism

These truncations perturb the spherical shell model Hamiltonian Effective residual interaction must be added Effective residual interactions Spherical one body Shell Model Hamiltonian Є 1 and Є 2 are single particle energies given in solution to H or from experiment Diagonal matrix elements are expectation values of H res on | ψ i > Non-diagonal matrix elements 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 42 Ca Assume can be described as a closed core of 40 Ca + 2 valence neutrons Assuming a restricted model space of 1f 7/2 and 2p 3/2 orbitals The (ν 2 f 7/2 ) and (ν 2 p 3/2 ) J π = 0 + states are the basis vectors | ψ i > f 7/2 p 3/2 f 5/2 p 1/ g 9/2 Full Model Space Restricted Model 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 42 Ca Assume can be described as a closed core of 40 Ca + 2 valence neutrons Assuming a model space of 1f 7/2 and 2p 3/2 orbitals The (ν 2 f 7/2 ) and (ν 2 p 3/2 ) J π = 0 + are the basis vectors | ψ i > S.P.Es and matrix elements are taken from fpd6 interaction Shell Model Hamiltonian (Single particle energies) Effective interactions (matrix elements) Diagonalisation gives E level and wavefunctions

Ca NuShell uses the Lanczos method of diagonalisation which is slightly 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 model 1)A ‘good’ (realistic) model space 2)Effective interactions adapted to the model space 3)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 space I.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!