The Proper)es 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 2.79µ N Neutrons n 939.6 MeV -1.91µ N 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 auer electrostahc 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 : R = r 0 A 1/3 r 0 = 1.2 x 10-15 m =1.2 fm Charge density Mass density Force density Most (but not all) spherical 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 ScaPering Mirror Nuclei Decays Alpha ScaPering APenuaHon of neutron beams Determining the size of the nucleus: Electron ScaPering Stanford Linear Accelerator 500 MeV electrons, λ = 2.5 fm Electrons scatter from nuclear charge density 6
Determining the size of the nucleus: Electron ScaPering DiffracHon paperns from high energy electrons scapered from 16 O and 12 C. The minimum occurs at sinθ = 0.61 λ/r R = (1.23 ±0.01)A 1/3 fm Determining the size of the nucleus: Electron ScaPering Radius from electron scapering experiments. R = (1.23 ±0.01)A 1/3 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 electrostahc energy. Determining the size of the nucleus: Mirror Nuclei 15 N and 15 O are mirror nuclei. The 15 O nucleus can spontaneously emit a positron and neutrino decaying to 15 N. The decay energy should be equal to the electrostahc energy difference. Analysis gives R = 1.2 A 1/3 fm 8
Determining the size of the nucleus: Alpha ScaPering Alpha particles are close enough to feel the strong nuclear force. To probe maper distribuhon strongly interachng parhcles are used. Determining the size of the nucleus: Neutron APenuaHon A beam of fast neutrons is apenuated by interachng via the strong force with the nuclei (i.e. neutrons are absorbed or scapered). R = 1.4 A 1/3 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 aprachve force, they become bound and the system loses mass. This lost mass leaves the system as energy: E = mc 2 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 Stable Nuclei: 266 stable nuclides Pairing InteracHon 159 even Z, even N 50 odd Z, even N 53 even Z, odd N 4 odd Z, odd N 14
Neutron and Proton SeparaHon Energies Energy needed to remove one neutron from a nucleus Energy needed to remove one proton from a nucleus ProperHes 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 combinahons 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 Spliqng 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 spliqng 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 stahc average potenhal made from the individual N- N interachons (mean field approximahon.) CollecHve Models Based on analogies to fluid dynamics Nucleus made of a nuclear maper 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. NotaHon: nl n counts the number of levels with a given angular momentum l 19
Wood- Saxon PotenHal An intermediate form based on nuclear maper distribuhon. Shell Structure with Wood- Saxon Spliqng 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 15 N magic neutron number 58 Ni magic proton number 40 Ca 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 expectahon value is And the j th 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 1f 5/2 and 1f 7/2. 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 expectahon value of l s for each state. The energy spliqng increases with increasing orbital angular momentum l. If f SO (r) is negahve 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 ProperHes 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 16 O Z=8, N=8 (doubly Magic) ground state is 0 + 17 O Z=8, N=9 (1 neutron outside closed shell) ground state is 5/2 + 15 N 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 combinahon of the spin magnehc moment and the orbital magnehc moment of the nucleon. But l z and s z do not have well defined value in a system of well- defined j z. 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 Independent particle shell model states Pairing energy increases with 29
All have 1 unpaired nucleon in f 5/2 level. Multiple nucleon excited states 2p 3/2 1f 5/2 1 nucleon beyond 20 CollecHve Models Nuclei made of strongly interachng nuclear maper. Deformed (non- spherical nuclei). RotaHonal, vibrahonal mohons of nuclei. Liquid- drop Model Binding Energies 30
Even- Even Nuclei Low Lying Energy Levels of 130 Sn Even- Even Nuclei Energies of the lowest 2 + states of even-even nuclei 31
Nuclear VibraHons Constant density shape deformahons quadrupole sextupole octopole phonons: 1 2 3 2 + 0 +,2 +,4 + 0 +,2 +,3 +,4 +,6 ++ Nuclear RotaHons Some nuclei have non- spherical shapes. These deformed nuclei can undergo rotahon yielding a set of rotahonal bands built on top of shell model states. 32