The Shell Model Can we treat the nucleus in an analogous way to Atomic Physics? i.e. Electrons moving in a potential well. There is evidence for Magic Numbers, just as there are Magic Numbers in atomic physics. e.g. The Nobel Gasses appear at Z = 2, 10, 18, 36, 54, 80, 86 e.g. First ionization energies for the atoms. The Shell Model In Nuclei, the Magic Numbers are N or Z = 2, 8, 20, 28, 50, 82, 126 Evidence: • For Z = magic, there are more stable isotopes. • For N = magic, there are more stable isotones. The Shell Model In Nuclei, the Magic Numbers are N or Z = 2, 8, 20, 28, 50, 82, 126 Evidence: • Neutron separation energies are greater for N = magic. • Proton separation energies are greater for Z = magic. The Shell Model In Nuclei, the Magic Numbers are N or Z = 2, 8, 20, 28, 50, 82, 126 Evidence: • Nuclear binding energies are greater when either N or Z = magic. • They are even greater when both N and Z are magic. These are Doubly Magic nuclei. • First excited state energies are greater for nuclei with either N or Z = magic. The Shell Model First Excited state energies The Shell Model This suggests a model in which the nucleons move within a potential well. Unlike the atomic case there is no central potential. [Note: Potential = PE just as in QM, not PE/q as in E&M.] The short range force would imply that a nucleon, within a nucleus, would feel an approximately constant potential. So in this model, we can imagine each nucleon, moving in a potential well which is established by the presence of the other nucleons. The Shell Model So each nucleon moves in a potential that looks like: V(r) V(r) 0 r 0 r For Neutrons For Protons The Shell Model In the Atomic case – to first order – we ignore the interaction between the electrons. In the Nuclear case the interaction between the nucleons is strong, but this is taken into account by the potential. i.e. We can treat each nucleon as moving independently in the potential. This is known as the Independent Particle Model. We solve the Schrödinger equation for a single nucleon moving in the potential well. (We assume nucleons are non-relativistic. Can be justified using the uncertainty principle.) The Shell Model If we assume the potential is spherically symmetric, we can write the wave function for a particular nucleon as (r) R(r)Yl,m (,) where, Y l , m ( , ) are the spherical harmonics, l = orbital angular momentum quantum number = 0, 1, 2,… m = magnetic quantum number. l m l 2 So 2 L l(l 1) Lz m We will use the usual spectroscopic notation for l. i.e. l = 0, 1, 2, 3, … s p d f … The Shell Model The time-independent Schrödinger equation becomes, 2 d 2u(r) l(l 1)2 2 V (r) 2 u(r) Eu(r) 2M dr 2Mr with u(r) rR(r) V (r) (Effective potential) eff For nucleons bound within the potential V(r) the boundary conditions force us to have a discrete set of solutions u ( r ) u n , l ( r ) and E En,l with n = principal quantum number. The full wave function is u (r) (r,,) R (r)Y (,) n,l Y (,) n,l,m n,l l,m r l,m The Shell Model l(l 1)2 is the potential modified by the Veff (r) V (r) 2 2Mr switch from a 3-dim to a 1-dim problem. e.g. for a neutron. Veff(r) l = 0 Veff(r) l = 1 0 r 0 r Sometimes called the Angular Momentum barrier. The Shell Model We can choose any convenient form for V(r). We know analytical solutions for potentials such as the simple harmonic oscillator or a square well. However these simple models do not reproduce the magic numbers. The best results come from assuming a potential shape that mimics the nuclear density. i.e. V(r) V a V (r) 0 ra r 1 e d for neutrons a 1.2A1/3 fm, d 0.75 fm The Schrödinger equation must be solved numerically. The Shell Model The result is a single-particle energy level scheme that looks something like… Nucleons are fermions so the 1g 18 Pauli Exclusion principle applies. 40 # m values = 2l + 1 2p l 6 nucleon spin s = ½ 1f 14 2 fermions in each ml (ms = + ½, ½) 20 Prediction for magic numbers. 2s 2 But, the magic numbers are E 1d 10 n,l actually 2, 8, 20 28, 50… 8 1p 6 Predicting the excited states 2 of nuclei with these energy 1s 2 levels did not work well. n(l) # neutrons (or protons) The Shell Model What we have neglected is a contribution to the potential due to the spin of the neutron and protons. In the Atom, the spin ½ electron has a magnetic moment , which interacts with the magnetic field generated by its orbital motion. The result is the Spin-Orbit interaction that gives rise to the Fine-Structure splitting in atomic states. This is a very small effect in atoms. Magnetic Moments S Electrons g S Spin angular momentum s B S 2 2 1 z 1 With S s(s 1) s z gs B S 2 z 2 e B = Bohr Magneton 2me gs 2.00232 = Landé g factor for the electron. So z B opposite to S Note: This is what is quoted in tables of magnetic moments for nuclei, rather than , because this is what is measured (in a Stern-Gerlach type experiment, for example). Spin-Orbit Effect The Spin-Orbit Effect in atoms If an electron is in a state with non-zero orbital angular momentum, it has a net motion around the nucleus. In the frame of reference of the electron it would appear that the nucleus is moving around the electron. That is like a current going around B the electron so there would be magnetic field at the position of nucleus e the electron. +Ze The magnetic moment of the electron will tend to “align” with this magnetic field. The Spin-Orbit Effect in atoms B B There are two possible “alignments” of the magnetic moment with the magnetic field. z The system is in a lower energy state z when “aligned.” “Aligned” “Anti-Aligned” The potential energy is U B z B We will attempt to calculate this energy shift U for a given hydrogen-like state of the electron. To do that we will calculate U . First we need to find the magnetic field. The Spin-Orbit Effect in atoms B The magnetic field in the centre nucleus of a current loop is e r +Ze I where is the permeability of B 0 0 2r free space. Ze The current I with period T T We can relate T to the angular momentum of the electron. 2 2 For the electron going around the nucleus: L (mer ) Ze T B 0 L So 2 2r 2 mer 1 Using c and recognizing that B 0 0 is in the same direction as L 1 Ze keZe B 2 3 L 2 3 L 4 0 mec r mec r The Spin-Orbit Effect in atoms k Ze The energy shift B e L m c2r 3 S e And we had g s B e B keZe and since B So U gs 2 3 S L m c r 2me e 2 keZe U B U gs 2 2 3 S L 2me c r But actually our analysis is not entirely correct because we did this in the non-inertial frame of reference of the electron. The correct result is: 2 keZe U (gs 1) 2 2 3 S L 2me c r The Spin-Orbit Effect in atoms First we look at: L S J 2 J J (L S)(L S) L2 S 2 2LS 1 2 2 2 So LS 2 (J L S ) 2 Therefore LS (n,l, j,m ) j( j 1) l(l 1) s(s 1) (n,l, j,m ) j 2 j Then U *UdV k Ze2 1 (g 1) e * L SdV s 2 2 3 2me c r 2 2 keZe 1 (gs 1) 2 2 j( j 1) l(l 1) s(s 1) 3 2me c 2 r 1 Z 3 for We can show that 3 1 3 3 l 0 r l(l 2)(l 1)n a0 The Spin-Orbit Effect in atoms Combining these we get an estimate for the energy shift: Z 2 j( j 1) l(l 1) 3 2 2 2 Z E0 U En 1 Where En 2 2n l(l 2 )(l 1) n k e2 1 and e a dimensionless quantity. c 137 is called the Fine Structure Constant e.g. For the 2p states in Hydrogen 2 1 1 3 1 1 1 13.6 eV 2 ( 2 1) 1(11) 2 5 2 p 1 U 3.010 eV 2 state 2 1 2 2 137 2 1(1 2 )(11) 2 3 3 3 1 1 1 13.6 eV 2 ( 2 1) 1(11) 2 5 2 p3 U 1.510 eV 2 state 2 1 2 2 137 2 1(1 2 )(11) So the energy difference between these states is 5105 eV The Spin-Orbit Effect in atoms This introduces a splitting of the spectral lines which can be seen with a high resolution spectrometer.