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II. Nuclear Phenomenology

Topics to be covered include: Nuclear sizes Binding energies Semi-empirical mass formula Shapes and EM moments Valley of stability Decay modes. II. Nuclear Phenomenology. References: Theoretical Nuclear Physics, Vol I, Nuclear Structure

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II. Nuclear Phenomenology

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  1. Topics to be covered include: Nuclear sizes Binding energies Semi-empirical mass formula Shapes and EM moments Valley of stability Decay modes II. Nuclear Phenomenology

  2. References: Theoretical Nuclear Physics, Vol I, Nuclear Structure Amos deShalit and Herman Feshbach Nuclear Physics, Roy and Nigam Physics of the Nucleus, Preston

  3. Nuclear Sizes: The atomic nucleus was discovered via back scattering of low energy alpha particles from a gold foil (Ernest Rutherford, 1911). Quantitative information about the sizes of nuclei are best obtained by: (1) electron scattering when the de Broglie wave length is of order the nuclear size, or electron beam energies of ~ 100 MeV (2) muonic atoms where the lower energy levels are shifted due to the overlap between the L = 0 orbitals and the nuclear charge distribution. Here I only discuss electron scattering. The electron + nucleus interaction is sufficiently weak that the first-order Born approximation is accurate. Robert Hofstadter pioneered the experimental measurement of nuclear charge distributions using electron scattering and received the 1961 Nobel Prize in physics. See the 1961 Nobel lecture at: http://www.nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf

  4. electron rchrg Nuclear Sizes: electron + nucleus elastic scattering

  5. Nuclear Sizes: electron + nucleus elastic scattering

  6. Nuclear Sizes: electron + nucleus elastic scattering The proton matter density can be obtained from the charge density by unfolding the finite spatial charge distribution of the proton where Show Including the neutron electric form factor and a magnetic form factor contribution gives the complete relation from which the proton distribution in nuclei are obtained: From Bertozzi, Friar, Heisenberg, Negele, Phys. Lett. 41B,408 (1972).

  7. Nuclear Sizes: electron + nucleus elastic scattering The neutron and magnetic corrections are at the 0.01 fm level

  8. Nuclear Sizes: electron + nucleus elastic scattering Phys. Rev. Lett. 38, 152 (1977)

  9. Nuclear Sizes: electron + nucleus elastic scattering PRL 36, 129 (1976). r(fm)

  10. Nuclear Sizes: electron + nucleus elastic scattering Phys. Rev. C 7, 1930 (1973)

  11. Nuclear Sizes: electron + nucleus elastic scattering Nuclear charge densities from Batty, Friedman, Gils, Rebel, Karlsruhe, (1987).

  12. Nuclear Sizes: electron + nucleus elastic scattering Nuclear charge- and magnetization-density-distribution parameters from elastic electron scattering Review Article Atomic Data and Nuclear Data Tables, Volume 14, Issues 5–6, November–December 1974, Pages 479-508 C.W. De Jager, H. De Vries, C. De Vries

  13. Nuclear Sizes: matter densities from elastic scattering of strong interaction probes: protons, a, p+ and p-, K+ and K- Protons and alpha-particles are mainly sensitive to the average proton and neutron densities and cannot independently measure the proton and neutron distributions. If we know the proton densities from electron scattering analysis then we can infer the neutron densities. The results show that the neutrons and protons follow approximately the same distributions. This is not too surprising if we recall that the lowest energy (L=0) I = 0 interaction (for p+n) is more attractive than the L=0, I = 1 state for pp and nn. The strong p+n interaction keeps these constituents close together. Heavy nuclei with large neutron excess develop a “neutron skin” which is typically about 0.1 to 0.2 fm thick. One goal of the rare isotope accelerator (FRIB) is to study the structure of unstable nuclei which may develop thicker neutron skins. The K+ + nucleus interaction is significantly weaker than the above and affords the opportunity to probe matter densities deep inside the nucleus. However, all hadronic probes are vulnerable to strong interaction effects which make the theoretical link between the data and the densities susceptible to theoretical uncertainty.

  14. Nuclear Sizes: matter densities from elastic scattering of strong interaction probes: protons, a, p+ and p-, K+ and K- Pions (p+ and p-) + nucleus scattering near the spin 3/2, isospin 3/2 resonance in the p+N system (at the D(1232) resonance) interact with protons and neutrons with very different strengths. The p+ + proton and p- + neutron interaction is about 3x that for p+ + neutron and p- + proton. Simultaneous analysis of p+ and p- + nucleus data can determine both proton and neutron densities, where the proton densities can be checked against the electron results. The factor of 3 near the D resonances is understood as follows. 0

  15. Nuclear Sizes: matter densities p + A at 800 MeV (lab, fixed target) G. W. Hoffmann et al PRL, 40, 1256 (1978)

  16. Nuclear Sizes: matter densities K+ + 40Ca Data: Marlow et al. Phys Rev. C 25, 2619 (1982) Gils et al., Phys. Rev. C 21, 1245 (1980)

  17. Nuclear Sizes: matter densities Nuclear matter radii from Batty, Friedman, Gils, Rebel at Karlsruhe, (1987). a+Nucleus elastic scattering

  18. Nuclear Sizes: matter densities Alpha scattering Nuclear matter densities from Batty, Friedman, Gils, Rebel at Karlsruhe, (1987).

  19. Nuclear Sizes: matter densities From proton+nucleus elastic scattering: L.R., Phys. Rev. C19, 1855 (1979).

  20. ~4.5z r0 c r Nuclear Sizes: Summary We see that the nuclear radius scales with the number of nucleons to the one-third, or the nuclear volume is proportional to the number of nucleons. As a result the density in the centers of nuclei are approximately independent of the number of nucleons and equals about 0.2 nucleons/fm3. Furthermore the surface thickness is roughly constant for most nuclei. All of this points to a strong, short-range interaction.

  21. Classical visualization of the nucleus: Typical illustration in text books: There is nowhere for the nucleons to go. Fermi motion ? This one is better; the inter-nucleon spacing is about right “Picture a nucleus as a swarm of bees” – W.R.Coker circa 1975

  22. 12C

  23. r 0 c V(r) ~4.5z Nuclear Binding Energies The short-range interaction means that within the interior of the nucleus the net force on a typical nucleon is zero, and the potential V(r) that it sees is constant. In the outer regions a nucleon experiences a net inward force, similar to surface tension in a liquid. Therefore the potential has positive slope at larger radius. Nucleons at still larger distances will see only the long-range, exponentially decreasing tails of the interactions with the outer-most nucleons. A nucleon will therefore experience a binding potential of the form: Woods-Saxon Potential A typical bound state sits at about -25 MeV. The force required to pull a nucleon out of this potential gives further meaning to the name “strong force”! Calculate the force = (25 MeV)/2 fm in the same units you use to weigh yourself

  24. r 0 V(r) r 0 ~4.5z c V(r) ~4.5z r 0 more bound states V(r) Nuclear Binding Energies As more and more nucleons are added the potential does not get any deeper because the density saturates, but the range increases as A1/3, both following & determining the density. The number of energy levels bound within the potential therefore increases as the half-radius increases; the surface thickness remains approximately constant. (smaller A) (larger A) r 0 V(r)

  25. Nuclear Binding Energies Why does the density saturate? Recall the form of the N+N interaction at low energy and low angular momentum: The strong, repulsive core prevents the nucleons from overlapping, keeping them away from each other by about 0.7 fm. The repulsive core prevents nuclei from collapsing: i.e. without the repulsive core a fluctuation of rNN to smaller distance is energetically favorable because it would decrease the energy. Such change would increase the density which in turn would increase the overall binding potential depth, thus allowing even more nucleons in the potential well, which would increase the density and so on in a run-away collapse to infinite energy density! s-wave function

  26. Nuclear Binding Energies In addition, protons interact via the repulsive Coulomb potential which raises their energy levels up. As Ze2/R increases there are fewer available energy levels for protons to occupy compared to neutrons; hence in most stable nuclei N > Z. proton energy levels neutron energy levels

  27. Maria Goeppert Mayer and Hans Jensen Nobel Prize in Physics (1963) for Nuclear Shell model r 0 more bound states V(r) Nuclear Binding Energies It is rather remarkable that something as complex as a collection of strongly interacting particles can, to first-order, be so simply described. Eigenstates of the Woods-Saxon potential provide very good basis-states for quantitative models of nuclear structure.

  28. Nuclear Binding Energies

  29. Nuclear Binding Energies: Weizsacker’s* Semi-Empirical Mass Formula This is the interaction term responsible for blocking single b-decay transitions in some isobar series, allowing access to double b-decay transitions. *C.F. von Weizsacker, Z. Phys. 96, 431 (1935). H. A. Bethe, Rev. Mod. Phys. 8, 82 (1936) E. P. Wigner, Phys. Rev. 51, 106, 947 (1937).

  30. Nuclear Binding Energies: Symmetry and Pairing effects Parabolic shape due to symmetry energy Additional shift between odd-odd and even-even nuclei: pairing energy b+b+

  31. Nuclear Binding Energies: Symmetry and Pairing effects

  32. Nuclear Binding Energies: Symmetry and Pairing effects Decay chains relevant for bb-decay experiments, as used by NEMO Insufficient energy difference for b- decay to 48Sc bb bb

  33. Nuclear Binding Energies: Symmetry and Pairing effects Decay chains relevant for bb-decay experiments, as used by NEMO A=150 bb Insufficient energy difference for b- decay to 96Nb bb

  34. A=82 Nuclear Binding Energies: Symmetry and Pairing effects Decay chains relevant for bb-decay experiments, as used by NEMO bb A=130 bb

  35. Nuclear Binding Energies: Symmetry and Pairing effects Decay chains relevant for bb-decay experiments, as used by NEMO bb

  36. ky V(x) dk k kx x 0 a Nuclear Binding Energies: Weizsacker’s Semi-Empirical Mass Formula Some of the terms in the mass formula are a consequence of the Pauli principle. Let’s consider the nucleus to be a non-interacting Fermi gas. For a particle in a box with infinite walls: Count the available states:

  37. (N,Z) p n Nuclear Binding Energies: Non-interacting Fermi Gas Model Let’s examine the neutron-proton symmetry energy in terms of the Fermi gas model:

  38. (N,Z) p n p n Nuclear Binding Energies: Symmetry Energy in Fermi Gas Model Calculate the energy required for this transition Show

  39. (N,Z) p n Nuclear Binding Energies: Symmetry Energy in Fermi Gas Model The remainder comes from additional effects of the N+N interaction, e.g. the stronger 3S1 -3D1 attraction for p+n relative to the 1S0 for p+p and n+n, i.e. binding energy is increased by maximizing the number of p+n pairs.

  40. ky kx kz Nuclear Binding Energies: Surface Energy in Fermi Gas Model The surface energy term may also be understood in terms of the Fermi gas model. In the preceding we counted some states we shouldn’t have. These correspond to states with kx, ky, or kz = 0. Refer to the following state diagram: The 3rd dimension (z-axis) is added and we can now see that the blue dots (states) corresponding to kz = 0 should not be counted in the volume of the 3D shell 4pk2dk. Only the magenta dots (states), all of which have their k-values > 0 should be counted. The k = 0 states correspond to the longest-range, surface modes. Subtracting those states estimates the energy associated with the surface nucleons. (The magenta dots are in front of the blue dots at kz = p/a.)

  41. Nuclear Binding Energies: Surface Energy in Fermi Gas Model Show Show

  42. Nuclear Binding Energies: Pairing Energy It is important to keep in mind that the Woods-Saxon binding potential represents only the average, or mean interaction that an arbitrary nucleon sees in the nucleus, and that the energy levels describe nucleons which are otherwise non-interacting. The short-range nature of the strong N+N interaction produces residual effects which significantly alter the energy levels and eigenstates in real nuclei. We have argued that the residual p+n interaction in the strongly attractive 3S1-3D1 channel increases the binding energy in nuclei when N ~ Z. Short-range interactions among p+p and n+n pairs also affect the binding energies. These interactions produce the pairing energy term in the semi-empirical mass formula. The complete understanding of this pairing energy remains the subject of on-going research which uses BCS theory from condensed matter. The p+p and n+n pairs near the Fermi surface form quasi-particles (particle-hole combinations) and the two-particle coupling between particle-hole configurations, being attractive in relative s-states, produces a coherent ground state with lowered energy analogous to Cooper pairs in conventional super-conductivity. We will return to this in the chapter on Nuclear Structure. Refs: deShalit and Feshbach, Chapter V Bohr, Mottleson, Pines, Phys. Rev. 110, 936 (1958). Moller and Nix, Nucl. Phys. A 536, 20 (1992).

  43. Nuclear Binding Energies: Pairing Energy 18F 18O 1d3/2 2s1/2 1d5/2 1p1/2 1p 3/2 1s1/2 1d3/2 2s1/2 1d5/2 1p1/2 1p 3/2 1s1/2 Filled orbitals, maximum pairing – 16O core p n p n Pairing energy refers to the additional binding that occurs for p+p and/or n+n pairs near the Fermi surface. odd-odd no additional pairing energy even-even one additional n+n pair; increased binding energy The pairing energy is defined differently in other parametrizations, e.g. odd-odd or even-even may be set to zero. Here, d(A) is defined as zero for odd-A nuclei and for even-A nuclei the B.E. difference is split between odd-odd and even-even nuclei.

  44. 18O 1d3/2 2s1/2 1d5/2 1p1/2 1p 3/2 1s1/2 p n Nuclear Binding Energies: Pairing Energy The maximum overlap, and hence the maximum energy reduction occurs when the 2 neutrons are in a relative 1S0 state. Similarly, nucleons in filled energy levels couple to Jp = 0+. Filled orbitals, maximum pairing – 16O core No.1 rule of nuclear structure: The ground state spin & parity (Jp) of ALL even – even nuclei is 0+

  45. Nuclear Binding Energies: Weizsacker’s Semi-Empirical Mass Formula The volume binding energy is about 16 MeV per nucleon. The actual B.E./A of about 8 MeV is from the combined effects of all terms: Maximum B.E./A at Fe, Ni b-stable nuclei symmetry Coulomb surface volume

  46. Nuclear Binding Energies: Shell Model Basis We have already argued that the effective binding potential for the nucleus should be approximated by the Woods-Saxon potential. But let’s backup for a moment to a more familiar form, the 3D harmonic oscillator potential. 3s, 2d, 1g 2p, 1f 2s, 1d 1p 1s r This is roughly like the nuclear binding potential and in fact many shell model calculations use H.O. basis states for numerical convenience and cpu speed. As the potential morphs from H.O. to Woods-Saxon we should expect the n,l degeneracy to break. Also, the spin-orbit potential, which originates from the N+N spin-orbit interaction, causes each orbital angular momentum state to split into 2 states corresponding to

  47. Nuclear Binding Energies: Shell Model Basis 3D harmonic oscillator states There are energy gaps between closed (filled) nlj-states. What experimental evidence is there for these gaps? 20

  48. Nuclear Binding Energies: Shell Model Basis Extra binding when protons and/or neutrons fill the levels up to these energy gaps. (amu is atomic mass unit = 931.481 MeV)

  49. Nuclear Binding Energies: Shell Model Basis Shell effects cause extra, localized binding relative to the smooth trends of the semi-empirical mass formula. For example (Garvey and Kelson, Phys. Rev. Lett. 16, 1967 (1966); see deShalit and Feshbach, p.22) fit the binding energies of many nuclei with a second-order expansion in N, Z and (N-Z):

  50. Nuclear Binding Energies: Shell Model Basis Shell effects cause extra, localized binding relative to the smooth trends of the empirical mass formula. From f1(Z) data compiled by Garvey et al., Rev. Mod. Phys. 41, S1 (1969) the residuals are: Z= 8 14(?) 28 50 82

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