Unstable Nuclei; Radioactivity

1.  particle • Charged particle emitted (from nucleus) • -:np + e- + νeZ = Z+1, A = A • +: pn + e+ + νe Z = Z-1, A = A • EC: p + e- n + νe Z = Z-1, A = A Z,A Z,A e- e+,EC Z+1,A Z-1,A Unstable Nuclei; Radioactivity Radioactivity: The spontaneous emission of radiation and/or particles from a nucleus – a pure quantum physics phenomenon. •  particle • He2+ emitted • A = A–4 • Z = Z-2 Z,A •  rays • photon emitted • A = A • Z = Z Z,A α γ Z,A Z-2,A-4 Dr YA Ramachers

2. The Decay Law • The Transition Probability or Decay Constant, λ, depends on • the specific interaction process and (b) nuclear wave functions • for the initial and final state. No macroscopic effects like Pressure, • Temperature or electrodynamic fields can change λ. A single decay is a completely independent quantum process, i.e. can not be predicted. Only an ensemble of decays displays the statistical behaviour determined by the constant decay probability. (Analogy: single photon in front of double-slit experiment – many photons exhibit interference pattern; hit-location on screen for single photon is unpredictable) Dr YA Ramachers

3. The Decay Law Radioactive Decay – rate of depletion of an original quantity of nuclei is (a) constant and (b) depends on the present amount. reduction of present quantity Decay Constant Integration gives the Decay Law Why (b)? Independent alternatives with probability p for a statistical process add up, i.e. each single nucleus has λ to decay in dt, so for N possibilities its N λ=N λ. Dr YA Ramachers

4. Decay Characteristics N(t)/N0 t1/2 τ=1/λ e-1 t ln2 Half life of a decay: t1/2=ln2/λ Mean lifetime: τ=1/λ Activity: A=λN Activity Unit: 1Bq = 1 s-1 Older Unit: 1 Ci = 37 GBq Dr YA Ramachers

5. Decay Chains For multiple decays, A  B  C  etc., each with different decay constant, get a set of differential equations: where i(=A,B,C,...) labels the nucleus in the chain; q is the growth rate of i by the ‘mother’, i-1, when the daughter nucleus, i, decays. The nucleus Ni does not need to originate from another decaying nucleus but might be produced artificially, for example in a reactor. Dr YA Ramachers

6. Decay Chains The general solution of the differential equation is: using Example: radioactive daughter, N2, of a radioactive mother, N1 Put: Get: Dr YA Ramachers

7.  particle • He2+ emitted • A = A–4 • Z = Z-2 Z,A α Z-2,A-4 Decay Processes: α-Decay AZXN  A-4Z-2YN-2 + 42He2 Strong interaction process: Form a Helium nucleus inside a heavy nucleus Tunnelling through barrier: Gamow Factor  explains Geiger-Nuttall rule Dr YA Ramachers

8. Decay Processes: α-Decay Energetics: mXc2 = mYc2 + TY + mαc2 + Tα Q = TY + Tα = (mX – mY – mα)c2 kinetic Energies: TY ,Tα ;use momentum conservation: pY=pα Get: Dr YA Ramachers

9. Decay Processes: α-Decay Estimate for Q-value using SEMF [KR]: page 250 Q = [M(A,Z) – M(A-4,Z-2) – M(4,2)]c2 = -B(Z,A) + B(Z-2,A-4) + B(4,2) Simplify using 4/A  1 and 2/Z  1 and ignoring pairing term Dr YA Ramachers

10. Coulomb Barrier Nuclear Potential Binding Energy α-Decay Mechanism Tunnel through the Coulomb Barrier: Gamow Factor Barrier Height above α-particle energy Q: Dr YA Ramachers

11. α-Decay Mechanismα-Decay Spin and Parity Geiger-Nuttall rule is really just a ‘rule’ – Gamow derives ln t1/2 -1/Q as the general ‘trend’. Exact computations are far beyond syllabus here. Dr YA Ramachers

12.  particle • Charged particle emitted (from nucleus) • -:np + e- + νeZ = Z+1, A = A • +: pn + e+ + νe Z = Z-1, A = A • EC: p + e- n + νe Z = Z-1, A = A Z,A Z,A e- e+,EC Z+1,A Z-1,A Decay Processes: β-Decay AZXN  AZ+1YN-1 + e- + νe AZXN  AZ-1YN+1 + e+ + νe AZXN + e- AZ-1YN+1 + νe e- Weak interaction process: νe W- d u neutron proton u u d d Dr YA Ramachers

13. Decay Processes: β-Decay Energetics: Q = [M(A,Z) – M(A,Z+1)]c2, for β- Q = [M(A,Z) – M(A,Z-1) – 2me]c2, for β+ neglecting electron binding energy Tricky bit for Q: Three-body Decay for β Q = EY + Ee + Eν where E: total energy (rest mass + kinetic) for Y: final nucleus; e: electron(positron), ν:(anti)neutrino Result for maximum kin. energy of, e.g. the recoiling final nucleus Y m: nuclear masses; evaluated in rest frame of nucleus X Dr YA Ramachers

14. Decay Processes: β-Decay Numeric example: Free neutron decay mnc2=939.573 MeV; mpc2=938.28 MeV; mec2=0.511 MeV Emaxp - mpc20.75 keV Tiny proton recoil energy ! Combined with small upper limit for the neutrino mass (<2.3eV/c2) Electron-neutrino system practically shares the total Energy Q available. Dr YA Ramachers

15. β-Decay Energy Spectrum Dr YA Ramachers

16. Mini-Introduction to theoretical Nuclear Physics Fermi Theory of β-Decay • Treat the interaction causing the Decay as weak perturbation • can use Fermi’s Golden Rule (originating from Dirac, though). (b) Determine the density of final states. (c) Determine the Transition Matrix element. Comments: (a) is quite accurate for β-Decays of any sort – good starting point. For (b) it turns out experimentally that many Energy Spectra can be explained with a constant Matrix element, hence (b) determines the shape of many β-Spectra quite accurately. Point (c) is well beyond syllabus. Dr YA Ramachers

17. Fermi’s Theory of β-Decay Assuming point (a): N(p)dp: number of electrons in momentum interval [p,p+dp] per unit time. dn/dEf: density of final states per energy interval Vfi: Transition Matrix element Assume Vfi constant and work out dn/dEf, gives: Dr YA Ramachers

18. Selection Rules Physics in the Transition Matrix Element: Forbidden and Allowed transitions Assumption made: Zero-order approximation for particle wavefunctions (“as constants”) Hence, no orbital angular momentum for particles  Particles carry only Spin angular momentum ! Fermi-Decay Gamow-Teller-Decay Electron-Neutrino Pair in Singlet Electron-Neutrino Pair in Triplet ΔI = |If – Ii|= 0,1 (no 0  0) ΔI = |If – Ii|= 0 - + Dr YA Ramachers

19. Selection Rules The Parity Quantum Number Symmetries and conserved quantities: C: Charge conjugation q  -q P: Parity or reflection r -r T: Time reversal t  -t A Physical System invariant to the above discreet transformations C, P and T has ‘good’ quantum numbers, i.e. conserved in interactions.  = (-1)l Here (for nuclear beta decays): Additional selection rule:  = 0, no parity change Dr YA Ramachers

20. Another Quantum Number The last one for β-decay: Isospin Assumption: Complete Charge-independence of Strong Force (the symmetry) Consequence: No difference between Neutron and Proton Assign a new Quantum Number to the 2 possibilities for each Nucleon: Isospin T=½ Then a Proton gets projection value Tz=+1/2 A Neutron gets Tz=–1/2 A Nucleus then has Tz=1/2 (Z-N) Nucleus with T=TzMax can form (2T+1) states  Isospin Multiplet Dr YA Ramachers

21. Notation Quantum States of Nuclei T=1 0+ 2mec2 T=0 Notation: SpinParity or Iπ 1+ 99.4% T=1 2.312 0+ β+ γ 0.6% T=1 0+ 0.156 1+ β- T=0 14C Tz=-1 14N Tz=0 14O Tz=1 Note: 99.4% for Fermi transition only 0.6% for G-T ! Isospin Triplet T=1 and Singlet T=0 for A=14 Dr YA Ramachers

22. Nuclear Fission Similar to α-Decay: Evaporation of a part of the original nucleus; here, a big part though  Nucleus splits up ! Dr YA Ramachers

23. Nuclear Fission Liquid Drop Model Picture: Condition from stretched nuclei Coulomb Barrier Height Spontaneous fission Dr YA Ramachers

24. Nuclear Fission Typical reaction: 235U + n  93Rb + 141Cs + 2n • Observation: • Asymmetric Fragment Mass Distribution • 9337Rb58 and 14155Cs85 extremely neutron-rich • Reaction Products highly radioactive β-Decay • Get Prompt and Delayed Neutrons (Fission chain possible) • Induced Fission – getting over the fission barrier: • purely Binding Energy – activation with thermal neutrons (on 235U) • Binding Energy+Kinetic Energy – fast neutrons (e.g. on 238U) Dr YA Ramachers