The Properes of Nuclei Z N Nucleons • The nucleus is made of neutrons and protons. • The nucleons have spin ½ and (individually) obey the Pauli exclusion principle. Protons p 938.3 MeV 2.79µN Neutrons n 939.6 MeV -1.91µN 1 Nuclides • A nuclide is a nucleus with a specific number of neutrons and a specific number of protons. • Z = # of protons N = # of neutrons • A = mass number (N+Z) X = chemical symbol • Isotopes = same Z, different N • Isotones = same N, different Z Example: Carbon-14 or Nuclear ProperHes Forces, Masses and Shapes 2 Nuclear Forces • The Strong Nucleon-Nucleon Force – Binds nucleons together – Short range (≈ 3 fm) – APracHve over most distances – Repulsive “hard” core – Spin dependent – Charge Independent (n-n, n-p, and p-p forces the same aer electrostac repulsion removed) – The potenHal “well” that two nucleons experience is about 40 MeV deep and a few fermi wide. n-p and p-p interacHons 3 Nuclear PotenHal • Each proton or neutron in the nucleus feels an average force from the other nucleons. • This force can be modeled as a potenHal well. SiZe and Shape • Describing the nucleus : – Charge density – Mass density – Force density • Most (but not all) spherical 1/3 -15 R = r0A r0 = 1.2 x 10 m =1.2 fm 4 SiZe and Shape The surface thickness, t, is roughly constant across nuclei at approximately 2.3 fm. SiZe and Shape • Isomers (shape) • Most nuclei are spherical. • Some deformed nuclei in Z= 57 – 71 region 5 Determining the siZe of the nucleus • The siZe of nuclei can be determined in a variety of ways – Electron Scaering – Mirror Nuclei Decays – Alpha Scaering – APenuaon of neutron beams Determining the siZe of the nucleus: Electron Scaering • Stanford Linear Accelerator 500 MeV electrons, λ = 2.5 fm Electrons scatter from nuclear charge density 6 Determining the siZe of the nucleus: Electron Scaering • DiffracHon paerns from high energy electrons scaered from 16O and 12C. • The minimum occurs at sinθ = 0.61 λ/R • R = (1.23 ±0.01)A1/3 fm Determining the siZe of the nucleus: Electron Scaering • Radius from electron scaering experiments. • R = (1.23 ±0.01)A1/3 fm 7 Determining the siZe of the nucleus: Mirror Nuclei • Mirror nuclei are isobars with the proton and neutron numbers reversed. • As the strong force is independent of charge the mirror nuclear should differ only in terms of electrostac energy. Determining the siZe of the nucleus: Mirror Nuclei • 15N and 15O are mirror nuclei. • The decay energy should be • The 15O nucleus can equal to the electrostac spontaneously emit a positron energy difference. and neutrino decaying to 15N. • Analysis gives R = 1.2 A1/3 fm 8 Determining the siZe of the nucleus: Alpha Scaering Alpha particles are close enough to feel the strong nuclear force. • To probe maer distribuHon strongly interacHng parHcles are used. Determining the siZe of the nucleus: Neutron APenuaon • A beam of fast neutrons is aenuated by interacHng via the strong force with the nuclei (i.e. neutrons are absorbed or scaered). • R = 1.4 A1/3 fm 9 Mass of Nuclei • Mass Spectrometers can separate isotopes by their mass. Mass and Binding Energy • When two or more objects come together under the influence of an aracHve force, they become bound and the system loses mass. This lost mass leaves the system as energy: E = mc2 10 Binding energies • The nuclear binding energy is the difference between isolated neutrons and protons and the bound nucleus. • It is tradiHonal to use atomic masses in the binding energy formula. Binding Energies - Example 11 Curve of the Binding Energy Binding Energies and the Liquid Drop Model • The liquid drop model provides a theoreHcal framework to develop a general formula for binding energies: • Energies associated with – Volume – Surface area – Coulomb repulsion between protons 12 Semi-Empirical Binding Energy Volume Electrostatic Pairing Surface Area Symmetry Paring: Symmetry and Pairing • The Pauli principle requires we fill up the energy levels with one proton or neutron per state. • Fill up levels with n and p individually • Spin pairing • Proton potenHal is slightly higher in energy 13 Chart of the Nuclides • Stable Nuclei: – Light Nuclei: N ~ Z – Heavy Nuclei: N > Z Pairing InteracHon • Stable Nuclei: – 266 stable nuclides – 159 – even Z, even N – 50 – odd Z, even N – 53 – even Z, odd N – 4 – odd Z, odd N 14 Neutron and Proton Separaon Energies • Energy needed to remove one neutron from a nucleus • Energy needed to remove one proton from a nucleus Properes of Nuclei MagneHc and Electric 15 MagneHc moment of electron Nuclear MagneHc Moments • Proton: g = 5.5856912 +/- 0.0000022 • Neutron: g = -3.8260837 +/- 0.0000018 • Nuclei have magneHc moments that are combinaons of intrinsic nucleon moments and “orbital magnet moments” 16 Electric Quadrupole Moments • The charge distribuHon of the nucleus can be represented as a sum in terms of mulHpole moments. Hyperfine Spling • The magneHc and electric properHes of nuclei can be detected through the coupling of the EM fields of the nuclei with atomic electrons. • MagneHc hyperfine spling arises from the dipole-dipole interacHon between the nucleus and the atom. 21cm line of Hydrogen 17 Models of Nuclei Models of Nuclei • Independent ParHcle Models – “Shell Models” – Based on an analogy to harmonic oscillators and square wells. – Assume nucleus has some stac average potenHal made from the individual N-N interacHons (mean field approximaon.) • CollecHve Models – Based on analogies to fluid dynamics – Nucleus made of a “nuclear maer” – Allows non-spherical shapes and collecHve moHons 18 The Shell Model • Model the nucleus by a potenHal well of some shape. • Protons and neutrons fill the well according to the Pauli Principle. • Harmonic oscillator? • Square well? The Shell Model • The large gaps between levels we associate with closed shells. • Notaon: nl • n counts the number of levels with a given angular momentum l 19 Wood-Saxon PotenHal • An intermediate form based on nuclear maer distribuHon. Shell Structure with Wood-Saxon Spling differs somewhat at higher levels. 20 Do Nuclei have shell structure? • Binding energy of last neutron Do Nuclei have shell structure? • Neutron absorpHon cross-secHons 21 Do Nuclei have shell structure? • Binding energy vs. formula Do Nuclei have shell structure? • Quadrupole Moments 22 Magic Numbers • There are numbers of neutrons and protons that yield parHcularly Hghtly bound nuclei. • The magic numbers: 2, 8, 20, 28, 50, 82, 126 • 15N – magic neutron number • 58Ni – magic proton number • 40Ca – doubly magic Magic Numbers • 2, 8, 20, 28, 50, 82, 126 23 Spin Orbit InteracHon Goeppert-Mayer and Jensen Nobel Prize1963 Spin-Orbit InteracHon • The total angular momentum of a nucleon is labeled by j. • As a single nucleon has s= ½ the possible j values are 24 Spin Orbit InteracHon • The expectaon value is • And the jth level has a degeneracy of Spin Orbit InteracHon • Consider the 1f level which has a degeneracy of 14. • Possible j values are 5/2 and 7/2 thus the levels are 1f5/2 and 1f7/2 . • The degeneracy of these levels (2j+1) are 6 and 8 respecHvely. • The spin-orbit potenHal causes an energy difference proporHonal to the difference in the expectaon value of l⋅s for each state. • The energy spling increases with increasing orbital angular momentum l. If fSO(r) is negave then the states with the higher j will be pushed down in energy. 25 Spin-Orbit InteracHon 2, 8, 20, 28, 50, 82, 126, 184 PredicHng ProperHes of Nuclei from the Shell Model • Ground State spin and parity • MagneHc Dipole Moments • Electric Quadrupole Moments • Excited States 26 Ground State Properes • The group state nuclear spin and parity is determined primarily by the valance nucleon. • Closed shells have j = 0 and + parity. • Like nucleons pair up to give j = 0. • The unpaired nucleon(s) outside the closed shell determine spin and parity. • By this model all even-even nuclei should have a ground state spin of 0. • Parity is determined by the orbital angular momentum of the valance nucleons: (-1)l Example: Oxygen and Nitrogen • 16O – Z=8, N=8 (doubly Magic) – ground state is 0+ • 17O – Z=8, N=9 (1 neutron outside closed shell) – ground state is 5/2+ • 15N – Z=7, N=8 (neutron shell closed, one unpaired proton) – ground state is 1/2- 27 MagneHc Moments • In the simplest model the magneHc moment of the nucleus is determined by the valence nucleon(s). • The magneHc moment is a combinaon of the spin magneHc moment and the orbital magneHc moment of the nucleon. • But lZ and sZ do not have well defined value in a system of well-defined jZ. • There are two possible j states: MagneHc Moments • The comparison between the computed (lines) and measured values (dots) is shown. The lines are known as “Schmidt” lines. odd Z odd N 28 Excited States • We should be able to explain the spin and parity of the excited states of nuclei by using the shell model. • As energy is added to the nucleus the nucleons are promoted to higher levels in the shell model. Excited states ?? (8) 1/2 - ?? 3/2 - mirror nuclides Pairing energy increases with Independent particle shell model states 29 All have 1 unpaired nucleon in f5/2 level. Multiple nucleon excited states 2p3/2 1f5/2 1 nucleon beyond 20 CollecHve Models • Nuclei made of strongly interacHng nuclear maer. • Deformed (non-spherical nuclei). • Rotaonal, vibraonal moHons of nuclei. • Liquid-drop Model – Binding Energies 30 Even-Even Nuclei Low Lying Energy Levels of 130Sn Even-Even Nuclei Energies of the lowest 2+ states of even-even nuclei 31 Nuclear Vibraons • Constant density shape deformaons quadrupole sextupole octopole phonons: 1 2 3 2+ 0+,2+,4+ 0+,2+,3+,4+,6++ Nuclear Rotaons • Some nuclei have non-spherical shapes.