Mean-Field Description of Heavy Neutron-Rich Nuclei P. D. Stevenson University of Surrey NUSTAR Neutron-Rich Minischool Surrey, 2005

What’s a meanfield? Treat nucleons as if they move independently of each other in a potential caused by average of all nucleons This works (more or less) despite the strong short range repulsion of the nuclear force since Pauli blocking essentially restricts collisions Many-body theorists often then talk of quasi-particles moving in a medium. Nuclear physicists usually reserve this term for dealing with pairing.

Meanfield Evidence Classic “proof” of single-particle mean-field was Shell-Model and magic numbers

Potential Originally a simple potential such as oscillator or Woods Saxon Another method is to obtain a potential self-consistently

Hartree-Fock I Aim to solve full Schroedinger equation with two (or even more) body force by re-writing as mean-field residual interaction

Hartree-Fock II Can make any choice for u(x): Oscillator, woods-saxon etc Modern Shell Model picks oscillator, and diagonalises full H in basis of eigenstates of one-body part. HF motivation: Pick “optimum” u(x) so that residual part can be discarded. Choose u(x) such that total energy is minimised when nuclear wavefunction is approximated as a Slater Determinant.

Slater Determinants Simplest many-body wavefunction that satisfies antisymmetry choose single particle wavefunctions to be those which minimize energy

Variational Principle Expectation value of H in Slater Determinant is functional variation: We are minimizing the total energy varying the single particle wavefunctions in the Slater Determinant

Variational Principle

Derivation Continued These two terms equal so are these Ignore these for density-independent forces

Density matrix notation Simplify notation by defining one and two-body density matrices so like a usual one-body SE A slight complication

HF equation Hartree-Fock equation: the Hartree Hamiltonian Fock term

Realistic Interaction Once you have the HF equation, it is just a matter of solving it (of which more later). But what about those terms we neglected? It turns out that for a sensible reproduction of data, density- dependent forces are needed then V(r 1,r 2 ) depends on  and hence on  s, so we need to consider the effect of the variation of V(r 1,r 2 ) Simplest practical case: Hartree approximation for V(r 1,r 2 ) = a  (r 1 -r 2 ) + b  ((r 1 +r 2 )/2)  (r 1 -r 2 )

BKN Energy Expectation value of potential Integrate out delta function

BKN: Variation Hartree Mean Field Now, to solve the equations… Mean field depends on density, and hence on wavefunctions, which are the solutions of the equation. Need some kind of iterative process

Solving HF in general Make initial guess of  construct  =  i  i *  I hence construct mean field (hf) hamiltonian solve hf equation for  s if new  s = old  s, we have found self-consistent solution otherwise take new  s and go back to beginning

In practice Previous slide said “solve hf equation for  s”, but how? starting with initial guess  o we can expand it in terms of the wanted hf eigenstates as  o =  n  n then act with exponential of hf Hamiltonian: e -  (h-  o )  o =  n e -  (  n -  o )  n This will tend to damp out components in the wavefunctions with high positive energies, and leave you with the hf ground state

Imaginary time evolution A couple of problems here; we don’t know what the real HF Hamiltonian is, because we are not using the selfconsistent density until we have finished iterating Exponentiating an operator is a bit nasty also don’t know what  0 is we just use the current best guess Hamiltonian, and expectation value of h with current wf as  0 approximate exponential as Taylor series to first order:  new =  old - (h-  0 )  old This works for sensible initial guesses and small parameters 

Simplest case in practice Spherically symmetric system: 3D problem reduces to 1D 4 He: 4 single particle wavefunctions, all the same if we ignore coulomb see 50 lines of code: solves HF equations for BKN force in 4 He

A heavy nucleus: 208 Pb

Beyond To make practical calculations of heavy Neutron-Rich nuclei, need more than this: Ground states are deformed; not enough to assume spherical symmetry of wavefunctions map out potential energies as a function of deformation: Need to constrain  2,  4,  etc.. Need more sophisticated effective interaction, and pairing.

Some calculations Let’s look at some realistic calculations of heavy neutron-rich nuclei. Can we explain this? Zs. Podolyák et al., Phys. Lett. B491, 225(2000)

Shape Evolution