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# Chapters 5-6: Strong Force & Nuclear Structure

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1. Chapters 5-6: Strong Force & 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. Particle Classifications • Fermions: • Spin 1/2 particles • Obey Pauli Exclusion Principle • No more than one fermion can occupy the same quantum state • Antisymmetric wave functions • Two classifications • Hadrons - interact via strong nuclear force; composed of quarks; neutrons, protons, etc. • Leptons - fundamental particles, ie no substructure; electron, muon, tau, neutrino. • Bosons: • Integer spins • Obey Bose-Einstein Statistics • Any number of bosons can share the same quantum state • Symmetric wave functions • Force mediators- Photons, gluons, etc.

3. Fundamental Forces • Virtual particles - exchange particles that exist for a short period of time to convey force; existence must obey Heisenberg Uncertainty Principle

4. Heisenberg’s Uncertainty Principle • Conservation of energy can only be violated by E and t as long as the uncertainty principle remains valid. • How far can an exchange particle travel without violating HUP? • Assume exchange particle moves at the speed of light, c. • Insert HUP for t and E=mc2

5. Problem • If the effective range of the weak force is 10-18 m, what is the maximum allowed mass of the exchange particle? Answer: (GeV/c2)

6. Problem • If the effective range of the weak force is 10-18 m, what is the maximum allowed mass of the exchange particle? Answer: (GeV/c2)

7. Properties of the Strong Force V(r) r • Range: less than nuclear radius, <1.4fm • Attractive: on the distance scale of 1fm, overcomes coulombic force to hold charged protons together • Repulsive: on the distance scale of <0.5fm • Charge Independent: interaction is independent of nucleon electrical charge, ie p-n = p-p, n-n

8. Charge Independence

9. The Nucleus • As chemist’s what do we already know about the nucleus of an atom?

10. 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

11. Two Nucleon Systems • Combinatorics gives us three possible states, but only one occurs in nature! • nn • Unbound and comes apart easily • Free neutrons decay on the time scale of ~10min • pp • More unstable than nn due to Coulomb repulsion • pn - deuteron • Stable and naturally occuring • Spins of n and p align parallel in ground state configuration • Non-spherical structure

12. Chart of the Nuclides

13. 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

14. 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)

15. Problem • Which nucleus is stable? p n p n p n 12B 12C 12N

16. Problem • Which nucleus is stable?

17. 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.

18. 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

19. General Nuclear Properties I • Binding energy per nucleon ~ constant for all stable nuclei • Implies all nucleons in the nucleus do not interact with one another • If they did the BE per nucleon would be proportional to the mass number 8.9 MeV/u 7.4 Mass Number Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.

20. 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.

21. Simple Nuclear Potential Well • Potential energy of nucleons as they approach the nucleus • Neutron - feels no effective force until reaches surface of nucleus; constant attractive force at interior of nucleus. • Proton - coulombic repulsion as approaches nucleus; constant attractive force at interior of nucleus; coulombic repulsion from other protons decreases depth of potential well relative to neutron well.

22. 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:

23. Liquid Drop Model Components I Volume Energy Surface Energy • 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

24. 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

25. Problem • 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.

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

27. Mass Parabolas • 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)

28. Mass Parabolas Example 1A = 75 or 157 • Parabola Vertex: • ZA=[-f2/ 2f1] • Minimum mass & Maximum EB • Used to find mass and EB difference between isobars • Nuclear charge of minimum mass is derivative of M Eqn => 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.

29. Mass Parabolas Example 2A = 156 • Eqn 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.

30. Problem • 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 28 & 29.

31. Problem: 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 28 & 29.

32. Fermi Gas Model • Model emphasizes free particle character of nucleons & allows average behavior of lg nuclei to be described by thermodynamics • Assume nucleus is composed of a degenerate Fermi gas of p & n • Degenerate - particles occupy lowest possible energy states • Fermi gas - all particles obey Pauli Principle • Fermi Wavenumber - highest state occupied by nucleons • Fermi Energy - gas is characterized by kinetic energy of highest filled state • When # n>p then must calculate Fermi energies of neutrons and protons separately

33. Fermi Gas Potential Well • EF,p - Fermi energy of proton • EF,n - Fermi energy of neutron • EC - coulombic energy • B - binding energy • U0 - depth of potential well • Fermi level - uppermost filled energy level, approximately -8MeV. Excited states

34. Problem • What is the average Fermi energy of a neutron in a 208Pb nucleus?

35. Problem • What is the average Fermi energy of a neutron in a 208Pb nucleus?

36. 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.

37. 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

38. Atomic Orbitals 3s 2p 2s • 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. Shell Model Evidence - Abundances

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

46. 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 

47. 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

48. 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

49. 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

50. 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!!!