Northeastern Illinois University
Nuclear Physics
Greg Anderson Department of Physics & Astronomy Northeastern Illinois University
2020
c 2004-2020 G. Anderson Modern Physics – slide 1 / 78 Northeastern Illinois Overview University
Introduction Nuclei Nuclear Forces Basic Properties Spin & Nuclear Magnetic Moments Nuclear Binding Energy Radioactivity α, β, γ Decay Weak Interaction Decays β±, EC
c 2004-2020 G. Anderson Modern Physics – slide 2 / 78 Northeastern Illinois University
Introduction Historical Milestones Rutherford’s Atom The neutron Nuclear electrons n Discovery
Nuclei
Nuclear Forces Basic Properties Introduction Spin & Nuclear Magnetic Moments Nuclear Binding Energy
Radioactivity
α, β, γ Decay Weak Interaction ± Decays β , EC
c 2004-2020 G. Anderson Modern Physics – slide 3 / 78 Northeastern Illinois Historical Milestones University
1895 Roentgen discovers X-rays. 1896 A.H. Becquerel discovers radioactivity. 1898 Curies study radioactive substances and coin the term radioactivity. 1902 Rutherford’s classifications α,β,γ rays. 1911 Rutherford proposes positive charge in atom exists in a nucleus. 1928 Gamow & Gurney explain radiation as quantum mechanical barrier tunneling. 1930 Pauli postulates the neutrino. 1932 Chadwick discovers the neutron. 1934 Fermi’s theory of beta decay. 1938 Discovery of nuclear fission by Otto Hahn, Lise Meitner and Fritz Strassman. 1949 Nuclear shell model, Maria Geoppert Mayer and Hans Jensen. 1956 Neutrino discovered, Reines and Cowan.
c 2004-2020 G. Anderson Modern Physics – slide 4 / 78 Northeastern Illinois Rutherford’s Atom University
Atom: 10−10 m
Nucleus: 10−15 m.
c 2004-2020 G. Anderson Modern Physics – slide 5 / 78 Northeastern Illinois Rutherford’s Atom University
Atom: 10−10 m
Nucleus: 10−15 m.
After 1920 (1932)
c 2004-2020 G. Anderson Modern Physics – slide 5 / 78 Northeastern Illinois The neutron University
The nucleus has a neutral constituent: • Geiger and Marsden’s 1911 α-particle scattering experiment leads to the view of nuclei (1 fm = 10−15 m) surrounded by a cloud of electrons (10−10 m). • Radiochemist Frederick Soddy discoverd isotopes by 1913. • Francis Aston builds the first mass spectrograph in 1919 to study isotopes and discovers the “whole number rule”
mN = AmH ,A =1, 2, 3,...
• Protons don’t account for the entire mass of a nuclei.
– Except for hydrogen, MN ≈ Amp =6 Zmp.
• View in the 1920’s: Nucleous is composed of protons and electrons. c 2004-2020 G. Anderson Modern Physics – slide 6 / 78 Northeastern Illinois Nuclear electrons are problematic University
The neutron can’t be a combination n =(e−,p): • Electrons confined to 1 fm would have too much energy: hc 1.24 MeV · nm E ∼ pc ∼ ∼ ∼ 1 TeV! λ 10−6 nm
• Klein paradox: electrons confined to the nucleus would escape through quantum tunneling. 2 − • Spin: the deuteron 1H, has s = 1, but p +(e + p) has s =(n +1/2). • Nuclear magnetic moments are too small to accommodate electrons: e~ e~ µN = ≪ µB = 2mp 2me
c 2004-2020 G. Anderson Modern Physics – slide 7 / 78 Northeastern Illinois Discovery of the Neutron University
• Bothe and Becker (1930) bombarded light elements with α-particles and produced extremely pennetrating radation (neutrons). 9 4 1 12 4Be + 2He −→ 0n+ 6 C • 1932 Curie and Joliot study effect of this new penetrating radiation on paraffin (CnH2n+2) and observe 5 MeV protons. • 1932 Chadwick shows γ-ray hypothesis is untenable. Proposes a neutral particle (neutron) and estimates:
1.005 u compare to mn =1.0087 u. c 2004-2020 G. Anderson Modern Physics – slide 8 / 78 Northeastern Illinois University Introduction Nuclei Composition Units Particle Numbers Isotopes & Isotones H Isotopes The Deuteron Deuteron Reactions Nuclei Chart of Nuclides Chart of Nuclides Nuclear Forces Basic Properties Spin & Nuclear Magnetic Moments Nuclear Binding Energy Radioactivity α, β, γ Decay Weak Interaction ± Decays β , EC c 2004-2020 G. Anderson Modern Physics – slide 9 / 78 Northeastern Illinois Composition of the Nucleus University • A nucleus is composed of neutrons and protons. • Protons and neutrons are known collectively as nucleons. Nucleon Symbol Charge Spin (s) Mass proton p +e 1/2 1.673 × 10−27 kg 938.272 MeV/c2 1.007276 u neutron n 0 1/2 1.675 × 10−27 kg 939.565 MeV/c2 1.008665 u 2 2 Note that mn − mp =1.293 MeV/c >me =0.511 MeV/c c 2004-2020 G. Anderson Modern Physics – slide 10 / 78 Northeastern Illinois Nuclear sized units University Unit for nuclear lengths 1 femtometer = 1 fermi = 1 fm = 10−15 m There is more empty space in an atom than in the solar system. Units for nuclear masses 1 u = 1 unified atomic mass unit = 1.661×10−27 kg Defined for neutral carbon-12 1 1u= M12 12 C c 2004-2020 G. Anderson Modern Physics – slide 11 / 78 Northeastern Illinois Particle Numbers University Z Atomic number. The number of protons in the nucleus. Charge of a nucleus: + + + + + + + + Q = Ze + N Neutron number. The number of neutrons in the nucleus. A Mass number. The number of nucleons in the nucleus. A = Z + N, MN ∼ Amp A 13 13 12 Notation for nuclei: Z(symbol)N . e.g., 7 N, 6 C, 6 C. c 2004-2020 G. Anderson Modern Physics – slide 12 / 78 Northeastern Illinois Nuclides: Isotopes, Isotones, & Isomers University • Nucleus (Nuclei): Bound state(s) of n’s and p’s. • Nuclide: A particular nuclear species. • Isotope: Nuclides with the same Z. • Isotone: Nuclides with the same N. • Isomer: A metastable state of an nucleus caused by the excitation of a proton or neutron so that it requires a change in spin before the nucleus can release its extra energy. c 2004-2020 G. Anderson Modern Physics – slide 13 / 78 Northeastern Illinois Hydrogen Isotopes University Three isotopes of hydrogen: 1 • Hydrogen: 1H. 2 • Deuterium: 1H, or the symbol D. The nucleus of deuterium is called a deuteron. Deuterium is a stable isotope of hydrogen. Heavy water is deuterium water. 3 • Tritium: 1H, or the symbol T . Tritium is radioactive. It is a pure β emitter with a half-life of 12.43 years. + + + H D T c 2004-2020 G. Anderson Modern Physics – slide 14 / 78 Northeastern 2 + Illinois The Deuteron d= H University Deuterium D = 2H aka, heavy hydrogen: stable isotope, natural abundance in the oceans ≈ 1/6500. + 0 mass (u) mass (u) d n 1.008665 n 1.008665 p 1.007276 1H 1.007825 1 + 0 n+p 2.015941 n+ H 2.016490 p n d 2.013553 2H 2.014102 Binding Energy 2 2 Bd/c = mn + M1H − M2H ≈ 0.002388 u ≈ 2.224 MeV/c The deuteron is a boson s = 1. Magnetic moment: nuclide spin (s) µz n 1/2 −1.91µN p 1/2 2.79µN d 1 0.86µN c 2004-2020 G. Anderson Modern Physics – slide 15 / 78 Northeastern Illinois Deuteron Reactions University Neutron-proton fusion: p + n −→ d + γ Photo-disintegration: d + γ −→ p + n Conservation of energy 2 2 2 hf + M2Hc = mnc + M1Hc + Kn + Kp 2 Binding energy: Bd/c = mn + M1H − M1H, photon energy: Bd hfmin = Bd 1+ 2 ≈ 2.2 MeV 2M2Hc c 2004-2020 G. Anderson Modern Physics – slide 16 / 78 Northeastern Illinois Chart of Nuclides University 30 30 29 28 Stable τ > 1 Gyr 27 τ = 1Myr–1Gyr 26 τ = 1 kyr–1 Myr 25 FeFeFe τ = 1 yr–1 yr 24 25 τ = 1 day–1 yr 23 1 min–1 day 22 1 s–1 day 21 20 1 ms –1 s < 1 ms 20 19 18 17 Z 16 15 15 14 13 12 11 10 10 9 8 7 O O O 6 N N 5 C 5 4 3 2 Li 1 α p 0 D 0 0 n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 0 10 20 30 40 N c 2004-2020 G. Anderson Modern Physics – slide 17 / 78 Northeastern Illinois Chart of Nuclides University Image Credit: http://www-nds.iaea.org c 2004-2020 G. Anderson Modern Physics – slide 18 / 78 Northeastern Illinois University Introduction Nuclei Nuclear Forces Nucleon-Nucleon Potential QED vs. Yukawa Potential Basic Properties Spin & Nuclear Magnetic Moments Nuclear Forces Nuclear Binding Energy Radioactivity α, β, γ Decay Weak Interaction ± Decays β , EC c 2004-2020 G. Anderson Modern Physics – slide 19 / 78 Northeastern Illinois Nucleon-Nucleon Potential University U(r) U(r) neutron-neutron neutron-proton proton-proton • Width: ∼ 2 − 3 fm • Depth: ∼ 40 MeV • Barrier ∼ 20 MeV. c 2004-2020 G. Anderson Modern Physics – slide 20 / 78 Northeastern Illinois QED vs. Yukawa Potential University γ π Range: Range: c~ c~ λ ~ c~ R = c∆t & ≈ = R = c∆t & ≈ ∆E hf 2π ∆E mc2 Potential: Potential: 2 q1q2 λ − U(r)= U(r)= e mπr 4πǫ0r 4πr For pi-meson exchange: R ≈ ~/mπc ≈ 1.4fm c 2004-2020 G. Anderson Modern Physics – slide 21 / 78 Northeastern Illinois University Introduction Nuclei Nuclear Forces Basic Properties Nuclear Radii Nuclear Charge Density Mirror Nuclides Nuclear Shape Spin & Nuclear Basic Properties Magnetic Moments Nuclear Binding Energy Radioactivity α, β, γ Decay Weak Interaction ± Decays β , EC c 2004-2020 G. Anderson Modern Physics – slide 22 / 78 Northeastern Illinois Nuclear Radii University 4 3 Nuclear densities are fairly constant: V = 3 πR ∝ A. 1/3 R = R0A β-decay of mirror nuclides: R0 =1.2 ± 0.2 fm Electron diffraction, Hofstadter et al. R0 = 1.07 ± 0.02 fm, t = 2.4 ± 0.3 fm λ 2 Attenuation of fast neutrons, σ =2π R + 2π : R0 =1.4 fm c 2004-2020 G. Anderson Modern Physics – slide 23 / 78 Northeastern Illinois Nuclear Charge Density University Hofstadter et.al. in 1950s, high energy electron scattering approximately described by Fermi distribution for nuclear charge density: ρ/ρ0 1 ρ(r)= ρ0 1+e(r−R)/a t ≈ 4.4a Relative Density 0 0 1 2 3 4 5 6 7 8 9 10 Radial Distance (fm) Charge distribution: Matter distribution: R ≈ 1.07 A1/3 fm, R ≈ 1.25 A1/3 fm, a ≈ 0.55 fm a ≈ 0.65 fm c 2004-2020 G. Anderson Modern Physics – slide 24 / 78 Northeastern Illinois Mirror Nuclides University Electrostatic energy of a uniform charge 3 1 q2 U = 5 4πǫ0 R Energy difference between 15O and 15N 3 1 e2 ∆U = Z2 − (Z − 1)2 5 4πǫ R 0 15N 15O Measure energy in β decay 15 15 + 8 O −→ 7 N+ β + ν 1/3 R = R0A with R0 =1.2 ± 0.2 fm c 2004-2020 G. Anderson Modern Physics – slide 25 / 78 Northeastern Illinois Nuclear Shape University Quadrupole Moment (Nuclei mostly spherical, exceptions 57 hQi = Z ψ∗ 3z2 − (x2 + y2 + z2) ψdV Z Q> 0 Q =0 Q< 0 z2 >x2,y2 z2 = x2,y2 z2 c 2004-2020 G. Anderson Modern Physics – slide 26 / 78 Northeastern Illinois University Introduction Nuclei Nuclear Forces Basic Properties Spin & Nuclear Magnetic Moments Magnetic Moments Spin & Nuclear Anomalous Magnetic Moments Nuclear Magnetic Moments Magnetic Moments Hyperfine Structure Nuclear Binding Energy Radioactivity α, β, γ Decay Weak Interaction ± Decays β , EC c 2004-2020 G. Anderson Modern Physics – slide 27 / 78 Northeastern Illinois Magnetic Moments University Orbital magnetic moment ˆ B = Bk L L µ z = −µB ~ , µz = −µB ~ = −µBmℓ µ Spin magnetic moment (anomalous) ~ µ = −gsµBS/ Potential energy ~ µz = −gsµBSz/ = −gsµBms U = −µ · B G-factor g: = −µzB α 2 = mℓµBB gs = 2 1+ π + Oα Schwinger 1948 = 2.002319 experiment c 2004-2020 G. Anderson Modern Physics – slide 28 / 78 Northeastern Illinois Anomalous Magnetic Moments University The magnetic moment due to the intrinsic spin S is anomalous. e µ = g S 2m Classically, the g-factor, or gyromagnetic, ratio is | g |= 1. ge = −2.0023193043718 gp = +5.585694701 gn = −3.82608546 In quantum electrodynamics (QED) α ge =2 1+ + ··· 2π In NMR (MRI) the dimensionless quantity γ = ω/B is refered to as the gyromagnetic ratio c 2004-2020 G. Anderson Modern Physics – slide 29 / 78 Northeastern Illinois Nuclear Magnetic Moments University Nuclear spin: I, total angular momentum: F: F = I + J, J = L + S Possible quantum numbers f =(i + j), (i + j − 1),... | i − j| Hyperfine splitting ≈ 10−3 fine structure. ∆E = gN miµN B, µN = e~/2mp Moments of nucleons: (µp)z =2.79285µN , (µn)z = −1.91304µN c 2004-2020 G. Anderson Modern Physics – slide 30 / 78 Northeastern Illinois Hyperfine Structure University Using radio telescopes, the hydrogen gas in our galaxy can be mapped by observation of the 21 cm wavelength line (1420 MHz). nuclear electron spin spin ↑ ↑ 1S ∆U ≈ 5.9 × 10−6 eV ↑ ↓ λ = 21 cm c 2004-2020 G. Anderson Modern Physics – slide 31 / 78 Northeastern Illinois University Introduction Nuclei Nuclear Forces Basic Properties Spin & Nuclear Magnetic Moments Nuclear Binding Energy Nuclear Stability Nuclear Binding Energy Hydrogen and constituents Deuterium to Helium Fusion Masses & Binding Energies Curve of Binding Energy Mass per Nucleon Chart of Nuclides Chart of Nuclides Weizs¨acker semi-empirical formula Weizs¨acker semi-empirical formula Weizs¨acker c 2004-2020 G. Anderson Modern Physics – slide 32 / 78 semi-empirical Northeastern Illinois Nuclear Stability University n En 266 of > 3000 nuclides are stable 4 16E1 Z = 3 9E1 N 2 + + 4E1 (proton number) stability line Z + + E 1 1 N (neutron number) Particle in Box Electromagnetic Repulsion 2 2 En = n E1 ∆U ∝ Z c 2004-2020 G. Anderson Modern Physics – slide 33 / 78 Northeastern Illinois Hydrogen and constituents University Hydrogen is composed of an electron and a proton. The mass of hydrogen is slightly less than the sum of the mass of its constituents. Hydrogen Mass 2 mH = 938.78 MeV/c Mass defect 2 e,p H mp + me − mH = 13.6 eV/c ∆m is one part in 108, but e− +p → H + energy = m c 2004-2020 G. Anderson Modern Physics – slide 34 / 78 Northeastern Illinois Deuterium to Helium Fusion University M (u) M (MeV/c2) D 2.014102 1876.12 D,D He 2D 4.028204 3752.25 Deuterium fusion: He 4.002602 3728.40 1 u = 1 unified mass unit = 931.494 MeV/c2 D+D → 4He + energy Mass defect: 2 ∆m =2mD − mHe =0.0256 u = 23.8 MeV/c ∆m is 6.4 parts in 103. c 2004-2020 G. Anderson Modern Physics – slide 35 / 78 Northeastern Illinois Masses & Binding Energies University Atomic mass deficit: 2 2 2 2 Batomic = Mnucc + Zmec − Matomc =∆mc Nuclear mass deficit: 2 2 2 Bnuclear = Zmpc + Nmnc − Mnucc 2 2 2 ≃ ZMH c + Nmnc − Matomc Mean value per nucleon: B/A =8.3 MeV/nucleon c 2004-2020 G. Anderson Modern Physics – slide 36 / 78 Northeastern Illinois The Curve of Binding Energy University 10 62 56Fe Ni b b b b b b b b b b b b b b b b b 9 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 4 b b b b b b b b b b b b b b b b b b b 238 He b b b 8 b b b b b b U b b b b b b b b b b b b b b 7 b b b 6 b b 6Li (MeV) 5 4 B/A 3 b b 3He 2 1 b 2H 1 0 b H 0 50 100 150 200 250 A c 2004-2020 G. Anderson Modern Physics – slide 37 / 78 Northeastern Illinois Mass Excess per Nucleon University b 1H bb 2H 7 b b b b 6 b bb 5 b b b b b b b b 4 b b b b b b b b b b b b (MeV) 3 b b b b b bb b b bb b b bb b b b b b 2 b b bb b b b b b b b bb b b b b b b b b b b M/A b b b b b b b bb b b b b 1 b b b b b bb bb b b b ∆ b b b b b b b b b b b b b b b b bb bb bb bb 4 bb b b b bb bb bb bb bb bb bb b bb bb bb bb bb bb b b b b b b b b b b b bb bb bb bb bb bb bb He b b b b b b bb bb bb bb bb bb b b bb bb bb bb bb bb bb bb b b b b b b b b b b b b b bb bb bb bb bb bb bb b b b b b b b b b b b b bb bb bb bb b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bbb bb bb b b b b b b b b b b b b b b bb bb bb bb bb bb bb 0 bb bb b b b b b b b b b b b b b b b b b b bb bb bb bb bb b b b b b b b b b b b b b bb bb bb bb bb b b b b b b b b b b b b b b b bb bb bb bb bb bb b b b b b b b b b b b b bb bb bb bb bbbb b bb bb bb b b b b b b b b b b b b b b b b bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb 238 bb bb bb bb bb b b b b b b b b b b b b b b b b b b bb b bb bb bb bb bb bb bb bb bb bb b b b b b b b b b b b bb b b b b b b b b b b b b bb bb bb bb bb bb bb bb b b b b b b b b b b b bb b bb b bb b bb b bb bb bb bb bb bb bb bb b b b bb bb bb bb bb bb bb bb b b b b b b b b bb b bb bb bb bb b bb bb bb bb bb bb bb −1 bb bb bb bb bb bb bb bb bb bb bb bb bb b bb U 56 −2 Fe 0 20 40 60 80 100 120 140 160 180 200 220 240 Mass Number (A) c 2004-2020 G. Anderson Modern Physics – slide 38 / 78 Northeastern Illinois Chart of Nuclides University 30 30 29 28 Stable τ > 1 Gyr 27 τ = 1Myr–1Gyr 26 τ = 1 kyr–1 Myr 25 FeFeFe τ = 1 yr–1 yr 24 25 τ = 1 day–1 yr 23 1 min–1 day 22 1 s–1 day 21 20 1 ms –1 s < 1 ms 20 19 18 17 Z 16 15 15 14 13 12 11 10 10 9 8 7 O O O 6 N N 5 C 5 4 3 2 Li 1 α p 0 D 0 0 n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 0 10 20 30 40 N c 2004-2020 G. Anderson Modern Physics – slide 39 / 78 Northeastern Illinois Chart of Nuclides University Image Credit: http://www-nds.iaea.org c 2004-2020 G. Anderson Modern Physics – slide 40 / 78 Northeastern Illinois Weizs¨acker semi-empirical formula University 2 B 2/3 Z(Z − 1) (A − 2Z) −1/2 = c1A − c2A − c3 − c4 + c5A c2 A1/3 A • c1 Volume term. Number of interactions ∝ A 2 2/3 • c2 Surface term. Surface Area: R ∝ A 2 −1/3 • c3 Coulomb term. Electrostatic repulsion ∝ Z(Z − 1)/R ∼ Z A • c4 Symmetry Term. Exclusion principle N 6= Z. • c5 Empirical pairing 2 Best fit, in (MeV/c ) : c1 = 15.8, c2 = 18.3, c3 = 0.714, c4 = 23.2, c5(N,Z) = 12 (even-even), = −12 (odd-odd) =0 (odd-even) c 2004-2020 G. Anderson Modern Physics – slide 41 / 78 Northeastern Illinois Weizs¨acker semi-empirical formula University 2 B 2/3 Z(Z − 1) (N − Z) −1/2 = c1A − c2A − c3 − c4 + c5A c2 A1/3 A Liquid Drop Model: Nucleons on surface have fewer neighbor interactions. Volume and surface terms B ≈ a1V − a2S 3 2 = b1R − b2R 2/3 = c1A − c2A R3 ∝ A c 2004-2020 G. Anderson Modern Physics – slide 42 / 78 Northeastern Illinois Weizs¨acker semi-empirical formula University 2 B 2/3 Z(Z − 1) (N − Z) −1/2 = c A − c A − c3 − c + c A c2 1 2 A1/3 4 A 5 Coulomb term: empirical motivation from liquid drop model with electrostatic repulsion. Coulomb repulsion of protons 6 e2 Z(Z−1) ∆B ≈ 5 k R 2 Z(Z−1) ∝ R ∝ Z(Z − 1)A−1/3 R3 ∝ A c 2004-2020 G. Anderson Modern Physics – slide 43 / 78 Northeastern Illinois Weizs¨acker semi-empirical formula University 2 B 2/3 Z(Z − 1) (N − Z) −1/2 = c1A − c2A − c3 − c4 + c5A c2 A1/3 A Asymmetry term: empirical motivation from exclusion principle and shell model. Favors nuclei with N ≈ Z. | N − Z |=0 | N − Z |=4 c 2004-2020 G. Anderson Modern Physics – slide 44 / 78 Northeastern Illinois Weizs¨acker semi-empirical formula University 2 B 2/3 Z(Z − 1) (N − Z) −1/2 = c1A − c2A − c3 − c4 + c5A c2 A1/3 A Empirical Pairing: Odd+1 −→ Even, with no need to use a higher energy level. More nucleons increases binding. Number of Stable Nuclei Z odd even odd few some N even some many c 2004-2020 G. Anderson Modern Physics – slide 45 / 78 Northeastern Illinois 3D Infinite Spherical Well University Spherical potential: 0 r jℓ(kℓnR)=0, k0,n = nπ Quantum energies: ~2 E = k2 n,ℓ n,ℓ 2mR2 c 2004-2020 G. Anderson Modern Physics – slide 46 / 78 Northeastern Illinois Spherical Besssel functions University sin x j0(x) = x sin x cos x j1(x) = x2 − x 3 sin x 3 cos x j2(x) = x2 − 1 x − x2 15 6 sin x 15 3 cos x j3(x) = x3 − x x − x2 − 1 x2 Selected zeros of jℓ(z) n =1 n =2 n =3 n =4 l =0 3.142 6.283 9.425 12.566 l =1 4.493 7.725 10.904 14.066 l =2 5.763 9.095 12.323 15.515 l =3 6.988 10.417 13.698 16.924 l =4 8.183 11.705 15.040 18.301 c 2004-2020 G. Anderson Modern Physics – slide 47 / 78 Northeastern Illinois Magic Numbers University Numbers of nucleons which form a complete shell within an atomic nucleus. Closed shells are indicated by “magic numbers” of nucleons. N,Z =2, 8, 20, 28, 50, 82, 126, (184?) These nuclei have a larger binding energy than predicted by the semi-empirical mass formula. c 2004-2020 G. Anderson Modern Physics – slide 48 / 78 Northeastern Illinois University Introduction Nuclei Nuclear Forces Basic Properties Spin & Nuclear Magnetic Moments Nuclear Binding Energy Radioactivity Radioactivity Radioactivity Radioactivity: Radioactive Decay Radioactivity: Mean lifetime Radioactivity: Half-life Radioactivity in Nature Conserved Quantities α, β, γ Decay Weak Interaction ± Decays β , EC c 2004-2020 G. Anderson Modern Physics – slide 49 / 78 Northeastern Illinois Radioactivity University The international radiation warning symbol (trefoil symbol), used to indicate radioactive materials. c 2004-2020 G. Anderson Modern Physics – slide 50 / 78 Northeastern Illinois Radioactivity: Radioactive Decay University Decays per unit time proportional to number of ptls. −λt dN = −λNdt, ⇒ N(t)= N0e Both number and decay-rate decay exponentially. dN R = − = λN e−λt = R e−λt dt 0 0 N0 N(t)= N e−λt SI units of radioactivity: 0 1becquerel = 1B = 1decay/s t c 2004-2020 G. Anderson Modern Physics – slide 51 / 78 Northeastern Illinois Radioactivity: Mean lifetime University Both number and decay-rate decay exponentially. −λt N(t)= N0e Fraction with lifetime between t and t + dt: N0 −dN −t/τ f(t)dt = = λe−λtdt N(t)= N0e N0 N0 Mean lifetime: e ∞ 1 τ = tf(t)dt = Z0 λ τ t c 2004-2020 G. Anderson Modern Physics – slide 52 / 78 Northeastern Illinois Radioactivity: Half-life University Mean lifetime: Half-life: 1 −t1/2/τ τ =1/λ N(t1/2)= N0e = N0 2 Half-life vs. mean-life: t1/2 = (ln2)τ ≃ 0.693τ N0 −t/τ N(t)= N0e N0/2 N0/e t1/2 τ t c 2004-2020 G. Anderson Modern Physics – slide 53 / 78 Northeastern Illinois Radioactivity in Nature University • More than 3000 nuclides are known. • Only 266 are stable. • Over 60 radionuclides are found in nature: – Primordial. Common primordial radionuclides 235U, 238U, 232Th, 226Ra, 222Rn, 40K. – Cosmogenic. formed by interactions between cosmic rays and upper atmosphere, e.g., 14C, 3H, 7Be. – Human produced or enhanced. e.g., 3H, 131I, 129I, 137Cs, 90Sr, 99Tc, 239Pu. c 2004-2020 G. Anderson Modern Physics – slide 54 / 78 Northeastern Illinois Conserved Quantities University In all nuclear reactions and decays the following quantities are conserved: • Linear Momentum: p • Total Angular Momentum: L • Total Energy: E • Electric Charge: Q • Nucleon Number: A = Np + Nn • Lepton Number: Ne− + Nν − Ne+ − Nν¯ c 2004-2020 G. Anderson Modern Physics – slide 55 / 78 Northeastern Illinois University Introduction Nuclei Nuclear Forces Basic Properties Spin & Nuclear Magnetic Moments Nuclear Binding Energy α, β, γ Decay Radioactivity α, β, γ Decay αβγ Types of Radioactive Decay α Decay α Decay Energetics: A Z P → A−4 4 Z−2D + 2α Weak Interaction ± Decays β , EC c 2004-2020 G. Anderson Modern Physics – slide 56 / 78 Northeastern Illinois α, β & γ Ray Penetration University Rutherford’s classification of rays produced by radioactive nuclei according to ability to penetrate matter and ionize air: α: 4He, least penetration, most ionization: stopped by skin or paper. β: e± penetrate a few mm of aluminum. γ: photons penetrate a few cm of lead. ν: Neutrinos can pass through several light-years of normal matter. α β γ ν Al Pb c 2004-2020 G. Anderson Modern Physics – slide 57 / 78 Northeastern Illinois Types of Radioactive Decay University • Alpha emission: 238 234 4 92 U −→ 90 Th + 2α • Beta emission: 234 234 0 − 90 Th −→ 91 Pa + −1β +¯ν • Gamma emission: 234 234 0 90 Th∗ −→ 90 Th + 0γ 82 0 − 82 37Rb + −1e −→ 36Kr + ν c 2004-2020 G. Anderson Modern Physics – slide 58 / 78 Northeastern Illinois α Decay University An α particle is a Helium-4 nucleus: 4 4 2α = 2He Generic α decay: A A−4 4 −→ + ZP → Z−2D + 2α Example decay: 232 228 4 90 Th → 88 Ra + 2α • In all decays, the total number of nucleons A and the electric charge Z are conserved. Very heavy nuclei are unstable to α decay. c 2004-2020 G. Anderson Modern Physics – slide 59 / 78 Northeastern A A−4 4 Illinois α Decay Energetics: ZP → Z−2D + 2α University P Before decay Disintegration energy Q 2 After decay Q/c = MP − (MD + Mα) D α In the parent rest frame: p2 p2 p2 M p2 4 Q = + = 1+ α = 1+ 2MD 2Mα 2Mα MD 2Mα A − 4 Kinetic energy of α particle A − 4 E = Q α A c 2004-2020 G. Anderson Modern Physics – slide 60 / 78 Northeastern Illinois University Introduction Nuclei Nuclear Forces Basic Properties Spin & Nuclear Magnetic Moments Nuclear Binding Weak Interaction Energy Radioactivity ± α, β, γ Decay Decays β , EC Weak Interaction ± Decays β , EC Neutron Decay − Nuclear β Decay − β decay: Q Nuclear Proton Decay Nuclear β+ Decay β+ decay: Q Weak Interactions and EC Electron Capture (EC) c 2004-2020 G. Anderson Modern Physics – slide 61 / 78 EC decay: Q Northeastern Illinois Neutron Decay University Weak interactions are responsible for beta decay: − n −→ p + e + νe d d p n u u d u − − e W ν¯e c 2004-2020 G. Anderson Modern Physics – slide 62 / 78 Northeastern − Illinois Nuclear β Decay University Nuclear beta decay arises from neutron beta decay: − 0 − 0 − n −→ p + β +¯νe, −1β = −1e A β− particle is an electron. Generic β− decay: e− − A A + 0 − 0 Z P → Z+1D + −1β + 0ν¯e − ⇒ Example β decay: 198 198 − 79 Au → 80 Hg + β +¯νe P D νe In all decays, the total number of nucleons A, the electric charge Z, and lepton number are conserved. c 2004-2020 G. Anderson Modern Physics – slide 63 / 78 Northeastern − Illinois β -decay Disintegration Energy University Generic β− decay A A + 0 − P ZP → Z+1D + −1e +¯νe before Daughter ion D+ lacks an e−: MD = MD+ + me ν¯ e− Disintegration energy Q: D+ Q/c2 = M − M P D after c 2004-2020 G. Anderson Modern Physics – slide 64 / 78 Northeastern Illinois Nuclear Proton Decay University Weak interactions are responsible for beta decay: + p −→ n + e + νe p d d u u n u d νe W + e+ c 2004-2020 G. Anderson Modern Physics – slide 65 / 78 Northeastern + Illinois Nuclear β Decay University Nuclear beta decay occurs when a nuclear proton decays: + 0 + 0 + p −→ n + β + νe, 1β = 1e A β+ particle is a positron. Generic β+ decay: e+ + A A − 0 + 0 Z P → Z−1D + 1β + 0νe ⇒ Example β+ decay: 13 13 + 7 N → 6 C+ β + νe P D νe In all decays, the total number of nucleons A, the electric charge Z, and lepton number are conserved. c 2004-2020 G. Anderson Modern Physics – slide 66 / 78 Northeastern + Illinois β -decay Disintegration Energy University Generic β+ decay: A A − 0 + P ZP → Z−1D + 1e + νe before Daughter ion D− has an extra e−: MD− = MD + me νe e+ Disintegration energy Q: D− Q/c2 = M − M − 2m P D e after c 2004-2020 G. Anderson Modern Physics – slide 67 / 78 Northeastern Illinois Weak Interactions and EC University Weak interactions are responsible for EC decay: − p + e −→ n + νe p d d u u n u d W + νe e− c 2004-2020 G. Anderson Modern Physics – slide 68 / 78 Northeastern Illinois Electron Capture (EC) University “Capture” of an atomic electron produces the reaction: − p + e −→ n + νe Generic EC decay: e− − AP → A D+ 0ν Z Z−1 0 e ⇒ Example EC decay: P D νe 51 51 24Cr → 23V+ νe In all decays, the total number of nucleons A, the electric charge, and lepton number are conserved. c 2004-2020 G. Anderson Modern Physics – slide 69 / 78 Northeastern Illinois EC-decay Disintegration Energy University e− − Generic electron capture: P A A ZP → Z+1D+ νe before Capture of K or L shell electron. Disintegration energy Q: νe 2 Q/c = MP − MD D after c 2004-2020 G. Anderson Modern Physics – slide 70 / 78 Northeastern Illinois Gamma Decay University Nucleon in exited energy state undergoes transition: A ∗ A ZX → ZX + γ Example sequence: 12 5 B 12 12 ∗ − e − B → C + e +¯νe 5 6 e − + ¯ ν + ¯ Followed by γ decay ν 12 ∗ 6 C γ 12 ∗ 12 6 C → 6 C+ γ 12 6 C c 2004-2020 G. Anderson Modern Physics – slide 71 / 78 Northeastern ± Illinois Decays: α, β & EC University β − D decay Z P decay β + D α EC, D N c 2004-2020 G. Anderson Modern Physics – slide 72 / 78 Northeastern 238 Illinois U Decay Series University 93 234 238 92 b U b U 70 s 234 91 b Pa 246 ky 2.41 d 90 b b 4.47 Gy 230 234 89 Th Th 75.4 ky 88 b 226 87 Ra Z 1.6 ky 86 b 222 85 Rn 210 214 3.82 d 84 Po b Po b b 20 m 218 5 d 210 214 Po 83 b Bi b Bi 22 y 27 m 138 d 82 b b 0.16 ms b 3.10 m 214 206 210 1.3 m 20 m Pb 81 Pb Pb b 210 80 Ti 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 N c 2004-2020 G. Anderson Modern Physics – slide 73 / 78 Northeastern Illinois Energy Valley Cross Section University 101 Cdl 101Zr l 100.904 A = 101 100.900 101 101Nb Agl l 100.896 101 101 Pd Mo l Mass (u) 100.892 l 101 + − Rh β , EC β 101Tc 101 l l Ru 100.888 l 100.884 40 41 42 43 44 45 46 47 48 Z c 2004-2020 G. Anderson Modern Physics – slide 74 / 78 Northeastern Illinois Nuclear Decay Type University α decay A A−4 4 Z P → Z−2D+ 2α β− decay A A + 0 − Z P → Z+1D + −1β +¯νe β+ decay A A − 0 + Z P → Z−1D + 1β + νe c 2004-2020 G. Anderson Modern Physics – slide 75 / 78 Northeastern Illinois Decay Modes of Unstable Nuclides University Chartfrom c 2004-2020 G. Anderson Modern Physics – slide 76 / 78 Northeastern Illinois Chart of Nuclides University Image Credit: http://www-nds.iaea.org c 2004-2020 G. Anderson Modern Physics – slide 77 / 78 Northeastern Illinois Chart of Nuclides University 30 30 29 28 Stable τ > 1 Gyr 27 τ = 1Myr–1Gyr 26 τ = 1 kyr–1 Myr 25 FeFeFe τ = 1 yr–1 yr 24 25 τ = 1 day–1 yr 23 1 min–1 day 22 1 s–1 day 21 20 1 ms –1 s < 1 ms 20 19 18 17 Z 16 15 15 14 13 12 11 10 10 9 8 7 O O O 6 N N 5 C 5 4 3 2 Li 1 α p 0 D 0 0 n1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 0 10 20 30 40 N c 2004-2020 G. Anderson Modern Physics – slide 78 / 78