The nuclear mean field and its symmetries W. Udo Schröder, 2011 Mean Field 1

Gross Estimates of Mean Field W. Udo Schröder, 2011 Mean Field 2 Fermi gas kinetic energy estimates: Light nuclei, N  Z  A/2: r V C (r) R (r) r E E 0 0 For stable nuclei, Fermi energies for protons and neutrons are equal. Otherwise beta decay. Coulomb Barrier Protons somewhat less bound but confined by Coulomb barrier.

The Nuclear Mean Field W. Udo Schröder, 2011 Mean Field 3 General properties of mean potential of nucleus (A=N+Z), radius R A 3D Square Well Oscill Woods Saxon  Nucleons close to center (r=0): V(r0)  V 0 =const.  NN forces are short range: V(r)  0 for r  R rapidly Range of potential > R A  NN forces have saturation character: central mass density (r0)  const. for all A, V 0 = const.  Total s.p. interaction: H = H nucl +H elm +H weak +… elm=electromagnetic, not all conservative! V C (r) R r (r) M=inertia  0 =frequency V 0 = depth a=diffuseness V 0 = depth R=range R=range 3D Square Well Oscill. Woods Saxon

The Nuclear A-Body Schrödinger problem W. Udo Schröder, 2011 Mean Field 4 Nucleus with A =N+Z nucleons 3-dimensional Schrödinger problem Symmetries  Further simplifications of 3D Schrödinger problem 3A-dim. Schrödinger problem 3D Square Well Oscill. Woods Saxon

Single-Particle Symmetries W. Udo Schröder, 2011 Mean Field 5 3D  1D Schrödinger problem Oscillator Woods Saxon Square Well 3D Square Well Oscill. Woods Saxon

Single-Particle Wave Functions See Nilsson Book W. Udo Schröder, 2011 Mean Field 6

Multi-Particle Symmetries W. Udo Schröder, 2011 Mean Field 7