Lecture 10: Nuclear Potentials and Radioactive Decay I. Nuclear Stability and Basic Decay Modes A. Schematic Representation: Synthesis Equilibration Decay X + Y + Energy A Z * Z ( 10 20 s) ( ~ 10 16 10 20 s Composite Nucleus (Activated Complex) B. Stable Nuclei 1. N/Z composition: Does not change with time peak of <BE> curve Kinetic vs. Thermodynamic stability; detection limit 10 20 y 2. Total: 266 At least one stable nucleus for all Z=183 EXCEPT 43 Tc and 61 Pm C. Radioactive Nuclei 1. Definition: A nucleus that SPONTANEOUSLY alters its neutron/proton composition or energy state FIRST-ORDER RATE PROCESS RADIOACTIVE DECAY IS IDENTICAL WITH AN ELEMENTARY UNIMOLECULAR DISSOCIATION IN CHEMISTRY. ( A B + C) Contrast with: nuclear reactions n/p changes induced by collisions, 2 nd order NUCLEAR REACTIONS HAVE THE SAME FORM AS AN ELEMENTARY BIMOLECULAR CHEMICAL REACTION (A + B C + D) 2. Half-life: t 1/2 Definition: The length of time required for one-half the nuclei in a sample to disintegrate (decay): N = N 0 e t ; = 0.693 t 1/2
3. Primary Decay Modes a. Alpha Decay: b. Beta Decay: 0 e ; -1 4 He emission 2 neutron proton conversion specifies nuclear origin SAME PARTICLE e specifies atomic origin c. Gamma Decay: 0, photon emission 0 = nuclear origin ; x-ray, uv, visible, ir = atomic/molecular origin d. Exotic decay modes: fission, protons, neutrons, 14C, etc. 4. Radioactivity in Nature (t1/2 108y) a. U Th Decay series EXTINCT: 238 U (4.5 109y) 8 92 6 206 Pb 82 (24.1%) A = 4n + 2 235 U (7.1 108y) 7 92 4 207 Pb 82 (22.1%) 232 Th (1.4 109y) 6 90 4 208 Pb 82 (52.3%) A = 4n 237 Np (2 106y) 7 93 4 209 Bi (100%) A = 4n + 1 A = 4n + 3 where n is an integer TOTAL: 45 NUCLEI (t1/2 of all daughters < t1/2 of parents.)
b. Lighter Radionuclides in Nature (1) Survivors of Nucleosynthesis ; esp. 40 K, 87 Rb, 147 Sm 19 37 62 TOTAL = 15 (2) Cosmic-Ray-Induced Activity 3 H(12y), 14 C(5280y), 7 Be(52d), 10 Be(~10 6 y), c. Natural radioactivities carry history of solar system and its evolution 5. Synthetic Nuclei (t 1/2 10 8 y) Isotopes of all elements: Z = 0 112 + [114 (??) 6. Grand Total: 3500 nuclei and still counting
Factors that Govern Decay Rate 1. Energetics large Q rapid decay (short half-life) A B + C + Q 2. Quantum Structure Spin and Parity: Changes in I between parent and daughter. Slow down decay rate e.g. 3 s 1/2 2 d 5/2 1 p 3/2 I = 1/2 + 5/2 + 3/2 even even odd II. Alpha Decay A. Mechanism: A Z 4 He + A-4 2 Z-2 X Y + Q He 2+ Y 2 Atomic Ionization State Alpha Recoil B. Energetics 1. Spectra: Discrete energies 2. Q = (x) (y) () 3. Energy systematics Range of values Q 1.5-12 MeV measured 4. 228 Th Example 228 Th 224 Ra + 4 He + Q 90 2 Q = ( 228 Th) ( 224 Ra) () = 26.758 18.313 2.425 = 5.520 MeV Measure: E = 5.423 MeV WHY?
5. Disposition of Q (1) Kinetic energy of + recoil : E + E R (2) Internal excitation energy ( heat) of recoil nucleus, E* ( has no stable excited states) a. Case I: Q Kinetic Energy Only X Y X, Y all in lowest (ground) energy state i.e., E* = 0 (T = 0) Energy Conservation: Q = E + E R (E = 1/2 Mv 2 ) Linear Momentum Conservation 0 = p + p p = 2ME R RESULT: E E R M R M M Q R M M M Q R A R Q A A R A Q A A R True for all 2-body breakup processes SPECTRA MUST BE DISCRETE, since A, Q, M are all constants Tag for nucleus ID b. Case II: Decay to Excited States X Y* E a E R + E* E* E i.e., system then undergoes -decay Energy Conservation: Q = E + E R + E* = E + E R + E Q E since M = 0
Momentum Conservation 0 = p + p R + p p 0 since ; neglect 0 = p + p R RESULT: A E R A A R ( Q E ) ; E R A A A R ( Q E ) NOTE: TOTAL ENERGY MUST BE THE SAME, REGARDLESS OF PATHWAY O E E E Q C. Alpha Decay Probability 1. Energetics: Q positive for all A>140 nuclei 2. Range of Measured Half-Lives (~10 44 ) 10 16 y > t 1/2 > 10 21 s 3. Why? a. Proton & Neutron Emission: Q p, Q n are negative near valley of beta stability (peak of peninsula); Thermodynamically forbidden b. Other Nuclei ; e.g. 12 C, 16 O Q( 12 C), Q( 16 O) positive ; possible Probability is low (i.e., t 1/2 ) is long) P( 14 C)/P ~ P 10 232 Th 14 C 218 Po (Exotic decay mode) 90 6 84
11.7 MeV particles from 212m Po are the highest energy alphas from a radioactive source 2.0 MeV alphas from Sm are the among lowest energy alphas from a radioactive source Most alpha particles from radioactive sources fall in the range of 4-8 MeV. Associated with this narrow range in energy is the enormous range in halflife noted above. 4. Coulomb Barrier Penetration What is the relevant potential for the nucleus? We can consider this by examining the approach of an alpha particle from infinity (microscopic reversibility). If the collision is head-on (dead center) there are two parts to consider the attractive nuclear potential and the repulsive Coulomb potential. V(r) = V nuclear + V Coulomb a. Coulomb Barrier V coul
V(r) o Q V coul escapes via quantummechanical tunneling through the Coulomb barrier Q p b. Electrostatic Repulsion Energy for Two Charged Spheres: V Coul = Z 1 ez 2 e R = 1.44 Z 1 Z 2 R MeV fm, where Z 1 = 2 Z 2 = Z recoil R= R 1 + R 2 = r 0 (A 1 1/3 + 4 1/3 ) c. Example: V coul = (1.44)(2)(100) (1.4)(41 / 3 216 220 No ; let r0 = 1.4 fm 102 1/ 3 ) = 25.5 MeV Typical of most heavy nuclei d. Q 10 MeV ; V coul > > Q and can't escape classically e. Relative Barriers (approximate) V coul (): V coul ( 12 C): V coul ( 16 O) 2:6:8 (since Z & R of recoil constant) heavier fragments have much higher (and thicker) hill to punch through
5. Probability P Q a. Tunneling Probability: P P formation e V coul ; P 1 t 1/2 ( ) b. P favored by: (1) large Q and low V coul or small Q V c t 1/2 () 1/P() e (Q V c ) or log t 1/2 V c Q (Figure above) D. Applications/Environment 1. 241 Am (458y) smoke detectors 2. 238 Pu (88y) remote sensing devices; power sources 3. 222 Rn (3.8d) natural radioactivity health effects 4. 226 Ra (1620y) cancer therapy 5. ( 210 Po) (138d) power sources 6. Dating tags U,Th, Sm