Mean-Field Description of Heavy Neutron-Rich Nuclei

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

2. 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.

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

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

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

6. 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.

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

8. 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

9. Variational Principle

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

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

12. HF equation • Hartree-Fock equation: Fock term the Hartree Hamiltonian

13. 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(r1,r2) depends on  and hence on s, so we need to consider the effect of the variation of V(r1,r2) • Simplest practical case: Hartree approximation for V(r1,r2) = a(r1-r2) + b((r1+r2)/2)(r1-r2)

14. BKN Energy • Expectation value of potential • Integrate out delta function

15. 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

16. 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

17. 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

18. 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 

19. Simplest case in practice • Spherically symmetric system: 3D problem reduces to 1D • 4He: 4 single particle wavefunctions, all the same if we ignore coulomb • see http://www.ph.surrey.ac.uk/~phs3ps/simple-hf.html 50 lines of code: solves HF equations for BKN force in 4He

20. A heavy nucleus: 208Pb

21. 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.

22. 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)

24. Shape Evolution