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LECTURE 31 SHELL MODEL PHY492 Nuclear and Elementary Particle Physics Fermi Energy and Momentum (last lecture) n(p) dp = 4πV (2πh)3 If N = Z = A/2 assumed; A/2 = N (Z) = PF p2 dp n = 2 = N(P) VPF 3 3π2h3 n(p) dp ( 2: spin ) 0 VPF3 3π2h3 N(P) PF 3 4 = πR03A × 2 3 3 3π h V h 9π 1/3 PF = PF = R0 ( 8 ) ≈ 250MeV/c N(P) EF = PF2/2M ≈ 33 MeV V0 = EF + B/A ≈ 40 MeV V0 Coulomb force March 31, 2014 PHY492, Lecture 31 2 Relation with the Mass Formula The average kinetic energy per nucleon is given, The total kinetic energy of one nucleus is given or With the parameterization of the total energy is approximated by relation to the mass formula : (contribute to Volume term) (Asymmetry term) March 31, 2014 PHY492, Lecture 31 3 Shell Model Atomic theory based on the shell model has provided remarkable clarification of the complicated details of atomic structure (magic number, energy levels…) Central potential is given by Coulomb force analogy Atom eNuclei Nuclei Nuclear shell model can also provide detailed predictions of nuclear properties Central potential? Where does it come from? the nucleons move in a potential that they themselves create March 31, 2014 PHY492, Lecture 31 4 Shell structure of atoms Atomic shell structure is based on the central Coulomb potential The energy levels are characterized by principal quantum number ( n ) n = 1, 2, 3, 4, … orbital angular momentum ( l ) l = 0, 1, 2, 3, 4, …, (n-1) (s) (p) (d) (f) (g) …. magnetic quantum number ( m ) m = -l, -l+1, .., 0, 1, …, l-1, l spin quantum number ( ms ) ms = +1/2, -1/2 (up,down) For any n, there are degenerate energy states with (l,ml,ms), number of such states is given nd = 2 × ∑ (2l+1) = 2n2 magic numbers 2 (He), 10 (Ne), 18 (Ar), 36 (Kr), … March 31, 2014 PHY492, Lecture 31 5 Magic numbers of nuclei In nuclei, one can also find magic numbers both for proton number (Z) and neutron number (N), from the enhanced values of the binging energies With respect to the semi-empirical mass formula. N = 20 28 50 Z = 20 28 82 50 126 82 Magic numbers of nuclei N = 2, 8, 20, 28, 50, 82, 126 Z = 2, 8, 20, 28, 50, 82 Magic numbers of atoms 2, 10, 18, 36, 54 … March 31, 2014 PHY492, Lecture 31 6 Evidence for magicity Other phenomena Magic nuclei have (1) more stable isotopes than other nuclei (2) small electric quarupole moment, the first excited states at high energies ( spherical ) (3) hindered neutron capture cross sections (4) sharp changes in separation energies 2 4 3 Magic numbers N = 2, 8, 20, 28, 50, 82, 126 Z = 2, 8, 20, 28, 50, 82 March 31, 2014 PHY492, Lecture 31 7 Woods-Saxon Potential atomic shell structure … Coulomb potential VCoulomb = e/4πε0r nuclear shell structure … Woods-Saxon potential (density distribution) R - V0 V(r) = 1 + e (r-R)/a 0 a - V0 shell structure with different potential shapes magic numbers of nuclei can be explained only up to 20 harmonic oscillator March 31, 2014 Woods-Saxon pot. PHY492, Lecture 31 Infinite well 8 LS force To explain magic numbers of nuclei, in 1949, Mayer and Jensen proposed a spin-orbit term ( ls force ) in the potential; V = Vcenter + Vls(r) (L·S) l-1/2 ΔEls the ls splitting between is given 2l + 1 ΔEls = l+1/2 March 31, 2014 PHY492, Lecture 31 2 l-1/2 l+1/2 states h2 < Vls > With the LS term, shell model can successfully explain nuclear magic numbers 9 Shell model configuration The shell model configuration is described using the notation (nlj)k for each sub-shell, where k is the occupancy of the sub-shell. 2s1/2 For example, configuration of 17O is written as 1d5/2 1p1/2 for protons 1p3/2 (1s1/2)2(1p3/2)4(1p1/2)2 1s1/2 for neutrons (1s1/2)2(1p3/2)4(1p1/2)2(1d5/2)1 March 31, 2014 PHY492, Lecture 31 π (proton) ν (neutron) 10 Spins and parities of the ground states The shell model can predict spins and parities of nuclear states Spins even-even nuclei : ground states have always spin J = 0 even-odd nuclei : spin of the ground states is determined by a spin of valence proton (or neutron) in case of 17O, J = 5/2 odd-odd nuclei : coupling of valence proton and neutron Parities determined by the “l” value of the valence nucleon (s) 17O 2s1/2 1d5/2 1p1/2 Parity transformation for the spherical harmonic function 1p3/2 P Ylm(θ,φ) = (-1)l Ylm(θ,φ) (HW1-2) in case of 17O one particle in 1d5/2 (l=2) … parity = 1 (+) March 31, 2014 PHY492, Lecture 31 1s1/2 π ν 11 Excited states The shell model can also predict spins and parities of excited states 17O ground state Jπ = 5/2+ 2s1/2 1d5/2 1p1/2 1p3/2 1s1/2 17O Excitation energies 3 < 2 < 1 excited states (1) π(1d5/2)1(1p1/2)-1 ×ν(1d5/2)1 March 31, 2014 (2) Jπ = 1/2- ν(1p1/2)-1 (3) Jπ = 1/2+ or 3/2+ ν(2s1/2)1 or ν(1d3/2)1 2s1/2 or 1d3/2 PHY492, Lecture 31 12 Magnetic moments Nuclei have magnetic moments µ = gj j [µN] (if spin not 0) gj is given by j(j+1) + l(l+1) – s(s+1) 2j(j+1) gl + (l:orbital) j(j+1) – l(l+1) + s(s+1) 2j(j+1) gs (s:intrinsic spin) and simplified as (l=1±1/2) With the values of gl = 1, gs = +5.6 for protons gl = 0, gs = -3.8 for neutrons, one can obtain the Schmidt values (valid for simplest configuration) proton neutron j + 2.3 (l+1/2) -1.9 (l+1/2) j – 2.3j/(j+1) (l-1/2) 1.9j/(j+1) (l-1/2) March 31, 2014 PHY492, Lecture 31 13