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Chapter 6: Nuclear Structure

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  1. Chapter 6: Nuclear Structure Abby Bickley University of Colorado NCSS ‘99 Additional References: Choppin (CLR), Radiochemistry and Nuclear Chemistry, 2nd Edition, Chapter 11 Friedlander (FKMM), Nuclear and Radiochemistry, 3rd Edition, Chapter 10

  2. The Nucleus • As chemist’s what do we already know about the nucleus of an atom? • Composed of protons and neutrons • Carries an electric charge equivalent to the number of protons & atomic number of the element • Protons and neutrons within nucleus held together by the strong force • Any model of nuclear structure must account for both Coulombic repulsion of protons and Strong force attraction between nucleons

  3. Empirical Observations • Chart of the nuclides: • 275 stable nuclei • 60% even-even • 40% even-odd or odd-even • Only 5 stable odd-odd nuclei 21H, 63Li, 105B, 147N, 5023Va (could have large t1/2) • Nuclei with an even number of protons have a large number of stable isotopes Even # protons Odd # protons 50Sn:10 (isotopes) 47Ag: 2 (isotopes) 48Cd: 8 51Sb:2 52Te: 8 45Rh:1 49In:1 53I: 1 • Roughly equal numbers of stable even-odd and odd-even nuclei

  4. Implications for Nuclear Models I • Proton-proton and neutron-neutron pairing must result in energy stabilization of bound state nuclei • Pairing of protons with protons and neutrons with neutrons results in the same degree of stabilization • Pairing of protons with neutrons does not occur (nor translate into stabilization)

  5. Chart of the nuclides • Light elements: N/Z = 1 • Heavy elements: N/Z  1.6 • Implies simple pairing not sufficient for stability • Neutron Rich: (N>Z) • N>Z: nucleus will - decay to stability • N>>Z: neutron drip line • Proton Rich: (N<Z) • N<Z: nucleus will + decay or electron capture to achieve stability • N<<Z: proton drip line (very rare) Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.

  6. Implications for Nuclear Models II 4. Pairing not sufficient to achieve stability Why? Coulomb repulsion of protons grows with Z2: Nuclear attractive force must compensate  all stable nuclei with Z > 20 contain more neutrons than protons Eq. 1

  7. General Nuclear Properties I 8.9 • Binding energy per nucleon approximately constant for all stable nuclei 7.4 Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.

  8. General Nuclear Properties II • Nuclear radius is proportional to the cube root of the mass r = r0 A1/3 Eq. 2 • Experimental studies show ~uniform distribution of the charge and mass throughout the volume of the nucleus dl = skin thickness Rl = Half density radius Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.

  9. Liquid Drop Model (1935) • Treats nucleus as a statistical assembly of neutrons and protons with an effective surface tension - similar to a drop of liquid • Rationale: • Volume of nucleus  number of nucleons • Implies nuclear matter is incompressible • Binding energy of nucleus  number of nucleons • Implies nuclear force must have a saturation character, ie each nucleon only interacts with nearest neighbors • Mathematical Representation: • Treats binding energy as sum of volume, surface and Coulomb energies: Eq. 3

  10. Volume Energy Surface Energy Liquid Drop Model Components I • Volume Energy: • Binding energy of nucleus  number of nucleons • Correction factor accounts for symmetry energy (for a given A the binding energy due to only nuclear forces is greatest for nuclei with equal numbers of protons and neutrons) • Surface Energy: • Nucleon at surface are unsaturated  reduce binding energy  surface area • Surface-to-volume ratio decreases with increasing nuclear size  term is less important for large nuclei c1 = 15.677 MeV, c2 = 18.56 MeV, c3 = 0.717 MeV, c4 = 1.211 MeV, k = 1.79

  11. Liquid Drop Model Components II • Coulomb Energy: • Electrostatic energy due to Coulomb repulsion between protons • Correction factor accounts for diffuse boundary of nucleus (accounts for skin thickness of nucleus) • Pairing Energy: • Accounts for added stability due to nucleon pairing • Even-even:  = +11/A1/2 • Even-odd & odd-even:  = 0 • Odd-odd:  = -11/A1/2 Coulomb Energy Pairing Energy c1 = 15.677 MeV, c2 = 18.56 MeV, c3 = 0.717 MeV, c4 = 1.211 MeV, k = 1.79

  12. Problem 1 • Using the binding energy equation for the liquid drop model, calculate the binding energy per nucleon for 15N and 148Gd. • Compare these results with those obtained by calculating the binding energy per nucleon from the atomic mass and the masses of the constituent nucleons.

  13. Problem 1: Answers • 15N = 6.87 MeV/nucleon • 148Gd = 8.88 MeV/nucleon • 15N = 7.699 MeV/nucleon • 148Gd = 8.25 MeV/nucleon

  14. Mass Parabolas Eq. 4 • Represent mass of atom as difference between sum of constituents and total binding energy: • Substitute binding energy equation for EB and group terms by power of Z: • For a given number of nucleons (A) f1, f2 and f3 are constants • Functional form represents a mass-energy parabola • Single parabola for odd A nuclei ( = 0) • Double parabola for even A nuclei ( = ±11/A1/2) Eq. 5

  15. Mass Parabolas Example 1A = 75 or 157 • Parabola Vertex: • ZA=[-f2/ 2f1] Eq. 6 • Minimum mass & Maximum EB • Used to find mass and EB difference between isobars • Nuclear charge of minimum mass is derivative of Eq. 5 => not necessarily integral • Comparison of Z = 75 and Z = 157 • Valley of stability broadens with increasing A • For a given value of odd-A only one stable nuclide exists • In odd-A isobaric decay chains the -decay energy increases monotonically Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.

  16. Mass Parabolas Example 2A = 156 • Eq. 5 results in two mass parabola for a given even value of A • For a given value of even-A their exist 2 (or 3) stable nuclides • In this figure both 156Gd and 156Dy are stable • In even-A isobaric decay chains the -decay energies alternate between small and large values • This model successfully reproduces experimentally observed energy levels • BUT……. Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.

  17. Problem 2 • Find the nuclear charge (ZA) corresponding to the maximum binding energy for: A = 157, 156 and 75 • To which isotopes do these values correspond? • Compare your results with the mass parabolas on slides 15 & 16.

  18. Problem 2: Answers • Find the nuclear charge (ZA) corresponding to the maximum binding energy for: A = 157, 156 and 75 ZA = 64.69, 64.32, 33.13 • To which isotopes do these values most closely correspond? 15765Tb, 15664Gd, 7533As • Compare your results with the mass parabolas on slides 15 & 16.

  19. Magic Numbers • Nuclides with “magic numbers” of protons and/or neutrons exhibit an unusual degree of stability • 2, 8, 20, 28, 50, 82, 126 • Suggestive of closed shells as observed in atomic orbitals • Analogous to noble gases • Much empirical evidence was amassed before a model capable of explaining this phenomenon was proposed • Result = Shell Model Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.

  20. Atomic Orbitals History • Plum Pudding Model: (Thomson, 1897) • Each atom has an integral number of electrons whose charge is exactly balanced by a jelly-like fluid of positive charge • Nuclear Model: (Rutherford, 1911) • Electrons arranged around a small massive core of protons and neutrons* (added later) • Planetary Atomic Orbitals: (Bohr, 1913) • Assume electrons move in a circular orbit of a given radius around a fixed nucleus • Assume quantized energy levels to account for observed atomic spectra • Fails for multi-electron systems • Schrodinger Equation: (1925) • Express electron as a probability distribution in the form of a standing wave function

  21. 3s 2p 2s Atomic Orbitals • Schrodinger equation solution reveals quantum numbers • n = principal, describes energy level • l = angular momentum, 0n-1 (s,p,d,f,g,h…) • m = magnetic, - l l, describes behavior of atom in external B field • ms = spin, -1/2 or 1/2 • Pauli Exclusion Principle: e-’s are fermions  no two e-’s can have the same set of quantum numbers • Hund’s Rule: when electrons are added to orbitals of equal energy a single electron enters each orbital before a second enters any orbital; the spins remain parallel if possible. • Example: C = 1s22s22px12py1 1s

  22. Shell Structure of NucleusHistorical Evolution • Throughout 1930’s and early 1940’s evidence of deviation from liquid drop model accumulates • 1949: Mayer & Jenson • Independently propose single-particle orbits • Long mean free path of nucleons within nucleus supports model of independent movement of nucleons • Using harmonic oscillator model can fill first three levels before results deviate from experiment (2,8,20 only) • Include spin-orbit coupling to account for magic numbers • Orbital angular momentum (l) and nucleon spin (±1/2) interact • Total angular momentum must be considered • (l+1/2) state lies at significantly lower energy than (l-1/2) state • Large energy gaps appear above 28, 50, 82 & 126

  23. Single Particle Shell Model • Assumes nucleons are distributed in a series of discrete energy levels that satisfy quantum mechanics (analogous to atomic electrons) • As each energy level is filled a closed shell forms • Protons and neutrons fill shells and energy levels independently • Mainly applicable to ground state nuclei • Only considers motion of individual nucleons

  24. Shell Model and Magic Numbers • Magic numbers represent closed shells • Elements in periodic table exhibit trends in chemical properties based on number of valence electrons (Noble gases:2,10,18,36..) • Nuclear properties also vary periodically based on outer shell nucleons

  25. Pairing • Just as electrons tend to pair up to form a stable bond, so do like-nucleons; pairing results in increased stability • Even-Z and even=N nuclides are the most abundant stable nuclides in nature (165/275) • From 15O to 35Cl all odd-Z elements have one stable isotope while all even-Z elements have three • The heaviest stable natural nuclide is 20983Bi (N=126) • The stable end product of all naturally occurring radioactive series of elements is Pb with Z=82

  26. Shell Model Evidence - Abundances • The most abundantly occurring nuclides in the universe (terrestrial and cosmogenic) have a magic number of protons and/or neutrons • Large fluctuations in natural abundances of elements below 19F are attributed to their use in thermonuclear reactions in the prestellar stage

  27. Shell Model Evidence - Abundances

  28. Shell Model Evidence - Stable Isotopes Stable Isobars • The number of stable isotopes of a given element is a reflection of the relative stability of that element. Plot of number of isotopes vs N shows peaks at • N = 20, 28, 50, 82 • A similar effect is observed as a function of Z # of Isobars # of Neutrons

  29. Shell Model Evidence - Alpha Decay • Shell Model predictions: • Nuclides with 128 neutrons => • short half life • Emit energetic  • Nuclides with 126 neutrons => • Long half life • Emit low energy 

  30. Shell Model Evidence - Beta Decay • If product contains a magic number of protons or neutrons the half-life will be short and the energy of the emitted  will be high N = 19 N = 20 N = 21 Z = 21 Z = 20 Z = 19

  31. Shell Model Evidence - Neutrons • Neutrons do not experience Coulomb barrier  even thermal neutrons (low kinetic energy) can penetrate the nucleus • Inside nucleus neutron experiences attractive strong force and becomes bound • To escape the nucleus a neutron’s KE must be greater than or equal to the nuclear potential at the surface of the nucleus • Observation: the absorption cross section for 1.0MeV neutrons is much lower for nuclides containing 20, 50, 82, 126 neutrons compared to those containing 19, 49, 81, 125 neutrons

  32. Shell Model Evidence - Energy • The energy needed to extract the last neutron from a nucleus is much higher if it happens to be a magic number neutron • Energy needed to remove a neutron • 126th neutron from 208Pb = 7.38 MeV • 127th neutron from 209Pb = 3.87 MeV

  33. Shell Model Evidence - Nucleon Interactions • Every nucleon is assumed to move in its own orbit independent of the other nucleons, but governed by a common potential due to the interaction of all of the nucleons • Implication: in ground state nucleus nucleon-nucleon interactions are negligible • Implication: mean free path of ground state nucleon is approximately equal to the nuclear diameter • Experimental data does not support this conclusion!!!

  34. Shell Model Evidence - Nucleon Interactions • Scattering experiments show frequent elastic collisions • Implication: mean free path << nuclear radius • Explanation: Pauli exclusion principle prohibits more than two protons or neutrons from occupying the same orbit (protons and neutrons are fermions) • Why Pauli?: nucleon-nucleon collisions result in momentum transfer between the participants BUT all lower energy quantum states are filled  occurrence forbidden • Severely limits nucleon-nucleon collision rate

  35. Nuclear Potential Well • Nucleon orbit = nucleon quantum state • Similar to quantum state of valence electron BUT • Nucleon “feels” total effect of interactions of all nucleons • Implication: nuclear potential is the same for all nucleons • Strong Force • All nucleons (regardless of their electrical charge) attract one another • Attractive force is short range and falls rapidly to zero outside of the nuclear boundary (~1 fm)

  36. Nuclear Potential - Protons • Protons do experience a Coulomb barrier  a proton must have kinetic energy equal or greater than ECoul to penetrate the nucleus • If Eproton< ECoul proton will back scatter • Inside nucleus proton experiences attractive strong force and becomes bound • To escape the nucleus a proton’s kinetic energy must be greater than or equal to ECoul (in the absence of quantum tunneling)

  37. Nuclear Potential - Neutrons • Neutrons do not experience Coulomb barrier  even thermal neutrons (low kinetic energy) can penetrate the nucleus • Inside nucleus neutron experiences attractive strong force and becomes bound • To escape the nucleus a neutron’s kinetic energy must be greater than or equal to the nuclear potential at the surface of the nucleus

  38. Nuclear Potential Well Depth of well represents binding energy

  39. Nuclear Potential Functions • Square Well Potential • Harmonic Oscillator Potential • Woods-Saxon • Exponential Potential • Gaussian Potential • Yukawa Potential: Note: R = nuclear radius r = distance from center of nucleus

  40. Nuclear Potential Functions Exact shape of well is uncertain and depends on mathematical function assumed for the interaction Yukawa Exponential Gaussian Square Well

  41. Neutron vs Proton Potential Wells Coulomb repulsion prevents potential well from being as deep for protons as for neutrons

  42. Quantized Energy Levels • Schrodinger Equation: developed to find wave functions and energies of molecules; also can be applied to the nucleus • Choose functional form of nuclear potential well and solve Schrodinger Equation: H = E  • Wave equation allows only certain energy states defined by quantum numbers • n = principal quantum number, related to total energy of the system • l = azimuthal (radial) quantum number, related to rotational motion of nucleus • ms = spin quantum number, intrinsic rotation of a body around its own axis

  43. Angular Momentum • Associated with the rotational motion of an object • Like linear motion, rotational motion also has an associated momentum • Orbital angular momentum: pl = mvrr • Spin angular momentum: ps = Irot • A vector quantity  always has a distinct orientation in space

  44. Magnetic Quantum Effects - Spin • A rotating charge gives rise to a magnetic moment (s). • Electrons and protons can  be conceptualized as small magnets • Neutrons have internal charge structure and can also be treated as magnets • In the absence of a B-field magnets are disoriented in space (can point any direction) • In the presence of a B-field the electron, proton and neutron spins are oriented in specific directions based upon quantum mechanical rules

  45. Spin Angular Momentum No External B-field Applied External B-field { Project spin angular momentum onto the field axes Allowed values are units of hbar ps(z) = hbar ms

  46. Spin Angular Momentum • Quantum mechanics requires that the spin angular momentum of electrons, protons and neutrons must have the magnitude • s is the spin quantum number • For protons and neutrons (just like for electrons) spin is always 1/2

  47. Magnetic Quantum Effects - Orbitals • The orbital movement of an atomic electron or a nucleon gives rise to another magnetic moment (l) • This magnetic moment also interacts with an external B-field in a similar manner to the spin magnetic moment • Quantum mechanics governs how the orbital plane may be oriented in relation to the external field • The orbital angular momentum vector (pl) can only be oriented such that its projection onto the z-axis (field axis) has values pl (z) = hbar ml • Where ml = magnetic orbital quantum number ml = -l, -l+1, -l+2….0…. l-2, l-1, l

  48. B-field axis pl pl(z) 3 h 2 h 1 h 0 h -1 h -2 h -3 h Orientations of ml in a magnetic field Orbital Angular Momentum • Project orbital angular momentum onto the field axes • Allowed values are units of hbar • pl(z) = hbar ml

  49. Orbital Angular Momentum • Quantum mechanics requires that the orbital angular momentum of electrons, protons and neutrons must have the magnitude • l is the orbital quantum number • Allowed values of l: • Nucleons: 0  l • Electrons: 0  l< (n-1) • For nucleons (but not electrons) l can exceed n

  50. Orbital Angular Momentum • The numerical values of the orbital angular momentum quantum number (l) are designated by the familiar spectroscopic notation • Remember: l can only have positive integral values (including 0)