16.451: Yesterday’s news!

1. 1 16.451: Yesterday’s news!

2. Solid line: fit to the semi-empirical formula almost flat, apart from Coulomb effects Most stable: 56Fe, 8.8 MeV/ nucleon gradual decrease at large A due to Coulomb repulsion some large oscillations at small mass very sharp rise at small A 2 16.451 Lecture 22: Beyond the mass formula... 25/11/2003

3. Semi-Empirical Mass (binding energy) Formula (SEMF) implications: 3 Stable nuclei have the maximum B for a given A; for constant mass number, B is quadratic in Z “mass parabolas”, e.g.: Even A: offset is the pairing term!

4. 4 Beyond the SEMF: “Magic Numbers” and the Shell Model We already noted that there were some marked deviations from the SEMF curve at small mass number, e.g. A = 4. On an enlarged scale, a systematic pattern of deviations occurs, with maxima in B occurring for certain “magic” values of N and Z, given by: N/Z = 2, 8, 20, 28, 50, 82, 126 These values of neutron and proton number are anomalously stable with respect to the average – the pattern must therefore reflect something important about the average nuclear potential V(r) that the neutrons and protons are bound in.... (NB, the most stable nucleus of all is 56Fe, which has Z = 28, N = 28, “double magic” ...)

5. Other evidence for “magic numbers” 2-n and 2-p separation energies: 5 • Energy required to remove a pair of • neutrons or protons from a given nucleus is • referred to as S2n or S2p • like the ionization energy for atoms, but • the pairing force is so strong in nuclei that • systematics are more easily seen comparing • nuclei that differ by 2 nucleons • the same pattern of “magic numbers” • appears – large separation energies • correspond to particularly stable nuclei: • N/Z = 2, 8, 20, 28, 50, 82, 126 ....

6. A periodic table of nuclei? 6 Systematics are reminiscent of the periodic structure of atoms, which results from filling independent single-particle electron states with electrons in the most efficient way consistent with the Pauli principle, but the magic numbers are different: “Magic numbers” for atoms: Z = 2, 10, 18, 30, 36, 48, 54, 70, 80, 86 ....

7. Q A Single Particle Shell Model for Nuclei: 7 Self-consistent approximation: assume the quantum state of the ith nucleon can be found by solving a Schrödinger equation for its interaction via an average nuclear potential VN(r) due to the other (A-1) nucleons: Assume a spherically symmetric potential VN(r); then the eigenstates have definite orbital angular momentum, and the standard radial and angular momentum quantum numbers (n,l,m) as indicated. (Justification: measured quadrupole moments of nuclei are relatively small, at least near the “magic numbers” that we are interested in explaining; midway between the last two magic numbers, ie around Z or N = 70, 100, the picture changes, and we will have to use a different approach, but at least for the lighter nuclei this assumption should be reasonable.)

8. 8 Shell Model continued... If we choose the right potential function VN(r), then the wave function for the whole nucleus can be written as a product of the single particle wave functions for all A nucleons, or at least schematically: oversimplification here... actually, it has to be written as an antisymmetrized product wavefunction since the nucleons are identical Fermions – the procedure is well-documented in advanced textbooks in any case! With total angular momentum given by: And parity: Always + for an even number of nucleons...

9. What to use for VN(r)? – three candidate potential functions: 9 Advantage: easy to write down Disadvantages: numerical solutions only edges unrealistically sharp Advantage: easy to write down and can be solved analytically Disadvantage: potential should not go to infinity, have to cut off the function at some finite r and adjust parameters to fit data. Advantage: same shape as measured charge density distributions of nuclei. smooth edge makes sense Disadvantage: numerical solution needed

10. Comparison: Harmonic Oscillator versus Woods-Saxon solutions: 10 • since both potentials are spherically • symmetric, the only difference is in • the radial dependence of the wave • functions • amazingly, when parameters are • adjusted to make the average • potential the same, as shown in the • top panel, there is remarkably • little difference in the radial • probability densities for these • two potential energy functions! • this being the case, the simplicity • of the harmonic oscillator potential • means that it is strongly preferred • as a model for nuclei

11. Theory: square of the harmonic oscillator wave function for the last proton in 206Pb, quantum numbers: n = 3, l = 0 --- it works !!! Evidence that this works: (Krane, Fig. 5.13) 11 electric charge density, measured via electron scattering: charge density difference between 205Tl and 206 Pb is proportional to the square of the wave function for the extra proton in 206 Pb, i.e. we can actually measure the square of the wave function for a single proton in a complex nucleus this way!

12. Various potential shapes lead to similar patterns of energy gaps, e.g.: 12 But the magic numbers are wrong  ! N/Z = 2, 8, 20, 28, 50, 82, 126 Something else is needed to explain the observed behaviour...

13. as in the calculation of magnetic moments, lecture 19, we can write: • but there are only two ways the orbital and spin angular momentum can add for a • single particle nucleon state: • a) “stretched state” j = l + ½ : • b)“jack-knife state” j = l – ½: Solution: the “spin-orbit” potential 13 • Meyer and Jensen, 1949: enormous breakthrough at the time because it was the • only explanation for the observed pattern of “magic numbers” and paved the way • for a “periodic table” of nuclei ... and the Nobel prize in physics, 1963! • (http://www.nobel.se/physics/laureates/1963/index.html • – see Maria Goeppert-Meyer’s Nobel Lecture link on this page) • simple idea:

14. E 14 Spin-orbit force, continued: • The energy shift due to the spin-orbit interaction is between states of the same • l but different j; • the splitting is proportional to land so it increases as the energy increases for the • single particle solutions to V(r) • each state can accommodate (2j+1) neutrons or protons, each with different mj • empirically, the sign of the spin-orbit term for nuclei is opposite to that for atoms • and the effect is much stronger in nuclei– the phenomenon has nothing to do with • magnetism, which is the origin of this effect in atoms, but rather it reflects a basic • feature of the strong nuclear force. • with these features, the spin-orbit potential is the “missing link” required to • correctly predict the observed sequence of magic numbers in nuclear physics 

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16. something to read.... http://www.nobel.se/physics/laureates/1963/ 16 see also!

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