15. Nuclear Decay. Particle and Nuclear Physics. Dr. Tina Potter. Dr. Tina Potter 15. Nuclear Decay 1

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1 15. Nuclear Decay Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 15. Nuclear Decay 1

2 In this section... Radioactive decays Radioactive dating α decay β decay γ decay Dr. Tina Potter 15. Nuclear Decay 2

3 Radioactivity Natural radioactivity: three main types α, β, γ, and in a few cases, spontaneous fission. α decay 4 2 He nucleus emitted. A Z X A 4 Z 2 Y + 4 2He Occurs for A 210 For decay to occur, energy must be released Q > 0 Q = m X m Y m He = B Y + B He B X β decay emission of electron e or positron e + n p + e + ν e A Z X A Z+1 Y + e + ν e β decay p n + e + + ν e A Z X A Z 1 Y + e+ + ν e β + decay p + e n + ν e A Z X + e A Z 1 Y + ν e Electron capture n.b. of these processes, only n peν can occur outside a nucleus. Dr. Tina Potter 15. Nuclear Decay 3

4 Radioactivity γ decay Nuclei in excited states can decay by emission of a photon γ. Often follows α or β decay. Excited states Photons emitted Ground state ΔE E Atom 10 ev 10 7 m optical Nucleus MeV λ 10 kev m X-ray m γ-ray A variant of γ decay is: Internal Conversion an excited nucleus loses energy by emitting a virtual photon. The photon is absorbed by an atomic e, which is then ejected.(n.b. not β decay, as e not from nucleus) Dr. Tina Potter 15. Nuclear Decay 4

Natural Radioactivity The half-life, τ 1/2, is the time over which 50% of the nuclei decay τ 1/2 = ln 2 λ = 0.693τ Some τ 1/2 values may be long compared to the age of the Earth.

5 Natural Radioactivity The half-life, τ 1/2, is the time over which 50% of the nuclei decay τ 1/2 = ln 2 λ = 0.693τ Some τ 1/2 values may be long compared to the age of the Earth. 4n series λ Transition rate τ Average lifetime Series Name Type Final Nucleus (stable) Longestlived Nucleus τ 1/2 (years) Thorium 4n 208 Pb 232 Th Neptunium 4n Bi 237 Np Uranium 4n Pb 238 U Actinium 4n Pb 235 U n is an integer Dr. Tina Potter 15. Nuclear Decay 5

6 Radioactive Dating Geological Dating Can use β decay 87 Rb 87 Sr (τ 1/2 = years) N 1 N 2 87 Sr is stable λ 2 = 0 So in this case, we have (using expressions from Chapter 2) N 2 (t) = N 1 (0) [ 1 e λ 1t ] + N 2 (0) = N 1 (t) [ e λ 1t 1 ] + N 2 (0) Assume we know λ 1, and can measure N 1 (t) and N 2 (t) e.g. chemically. But we don t know N 2 (0). Solution is to normalise to another (stable) isotope 86 Sr for which number is N 0 (t) = N 0 (0). N 2 (t) = N 1(t) [ e λ 1 t 1 ] + N 2(0) N 0 N 0 N 0 Method: plot N 2 (t)/n 0 vs N 1 (t)/n 0 for lots of minerals. Gradient gives [ e λ 1t 1 ] and hence t. Intercept = N 2 (0)/N 0, which should be the same for all minerals (determined by chemistry of formation). Dr. Tina Potter 15. Nuclear Decay 6

7 Radioactive Dating Dating the Earth Using minerals from the Earth, Moon and meteorites. Plot N 2(t) N 0 vs N 1(t) N 0. Slope gives the age of the Earth = yrs Intercept gives N 2 (0)/N 0 = 0.70 Dr. Tina Potter 15. Nuclear Decay 7

8 Radioactive Dating Radio-Carbon Dating For recent organic matter, use 14 C dating 14 C is continuously formed in Earth s upper atmosphere 14 N + n 14 C + p Production rate of 14 C is approx. constant The carbon in living organisms is continuously exchanged with atmospheric carbon C 14 7 N + e + ν e Equilibrium: 1 atom of 14 C to every atoms of other carbon isotopes (98.9% 12 C, 1.1% 13 C). 14 C decays in dead organisms. No more 14 C absorbed from atmosphere. 14 C undergoes β decay, τ 1/2 = 5730 yrs in fresh organic material observe 15 decays/minute/gram of carbon Measure the specific activity to obtain age. (i.e. number of decays per second per unit mass) n.b. complications for future generations will result from burning of fossil fuels (increases 12 C in atmosphere), and nuclear bomb tests (which added 14 C). Dr. Tina Potter 15. Nuclear Decay 8

9 α Decay α decay is due to the emmission of a 4 2He nucleus. 4 2He is doubly magic and very tightly bound. α decay is energetically favourable for almost all with A 190 and for many A 150. Why α rather than any other nucleus? Consider energy release (Q) in various possible decays of 232 U n p 2 H 3 H 3 He 4 He 5 He 6 Li 7 Li Q/MeV α is easy to form inside a nucleus 2p + 2n (though the extent to which α particles really exist inside a nucleus is still debatable) Dr. Tina Potter 15. Nuclear Decay 9

10 α Decay Dependence of τ 1/2 on E 0 (Geiger and Nuttall 1911) A very striking feature of α decay is the strong dependence of lifetime on E 0 Example 232 Th E 0 = 4.08 MeV τ 1/2 = yrs 218 Th E 0 = 9.85 MeV τ 1/2 = s A factor of 2.5 in E 0 factor in τ 1/2! e.g. even N, even Z nuclei for a given Z see smooth trend (τ 1/2 increases as Z does) Dr. Tina Potter 15. Nuclear Decay 10

11 α Decay Quantum Mechanical Tunnelling The nuclear potential for the α particle due to the daughter nucleus includes a Coulomb barrier which inhibits the decay. V (r) Coulomb ~ 1/r E 0 0 R R ' r V 0 Total energy of α = E 0 + V 0 V 0 K.E. P.E. Classically, α particle cannot enter or escape from nucleus. Quantum mechanically, α particle can penetrate the Coulomb barrier Quantum Mechanical Tunnelling Dr. Tina Potter 15. Nuclear Decay 11

12 α Decay Simple Theory (Gamow, Gurney, Condon 1928) Assume α exists inside the nucleus and hits the barrier. α decay rate, λ = fp f = escape trial frequency, P = probability of tunnelling through barrier semi classically, f v/2r v= velocity of a particle inside nucleus, given by: v 2 = (2E α /m α ) and R = radius of nucleus Typical values: V 0 35 MeV, E 0 5 MeV E α = 40 MeV inside nucleus f v 2R = 1 2R 2Eα m α s 1 m α = 3.7 GeV R 2.1 fm Obtain tunnelling probability, P, by solving Schrödinger equation in three regions and using boundary conditions Dr. Tina Potter 15. Nuclear Decay 12

13 α Decay Simple Theory (Gamow, Gurney, Condon 1928) Transmission probability (1D square barrier): P = [ 1 + V0 2 ] 1 4(V 0 E)E sinh2 ka E V 0 1-D 2 k 2 2m = V 0 E m = reduced mass 0 r For ka 1, P is dominated by the exponential decay within barrier P e 2ka Coulomb potential, V 1/r, and thus k varies with r. Divide into rectangular pieces and multiply together exponentials, i.e. sum exponents. V (r) Δ r r Dr. Tina Potter 15. Nuclear Decay 13

14 α Decay Simple Theory (Gamow, Gurney, Condon 1928) Probability to tunnel through Coulomb barrier P = i e 2k i R = e 2G k = [2m α(v (r) E 0 )] 1/2 The Gamow Factor G = R R [2m α (V (r) E 0 )] 1/2 dr = R R k(r) dr For r > R, V (r) = Z αz e 2 4πɛ 0 r = B r Z = Z Z α (Z α = 2) α-particle escapes at r = R, V (R ) = E 0 R = B/E 0 G = R R G = ( 2mα 2 ( 2mα E 0 ) 1/2 [ ] 1/2 B r E 0 dr = ) 1/2 B [ cos 1 ( R R ) 1/2 ( ) 1/2 2mα B R 2 R [ 1 r 1 ] 1/2 R dr {(1 RR ) ( RR )} 1/2 ] See Appendix H To perform integration, substitute r = R cos 2 θ Dr. Tina Potter 15. Nuclear Decay 14

15 α Decay Simple Theory (Gamow, Gurney, Condon 1928) In most practical cases R R, so term in [...] π/2 G ( 2mα E 0 ) 1/2 B π 2 B = Z αz e 2 4πɛ 0 e.g. typical values: Z = 90, E 0 6 MeV R 40 fm R ( ) 1/2 3.9 MeV G Z E 0 Lifetime τ = 1 λ = 1 fp 2R v e2g ln τ 2G + ln 2R v ln λ Z E 1/2 0 + constant Geiger-Nuttall Law Not perfect, but provides an explanation of the dominant trend of the data Dr. Tina Potter 15. Nuclear Decay 15

16 α Decay Simple Theory (Gamow, Gurney, Condon 1928) Simple tunnelling model accounts for strong dependence of τ 1/2 on E 0, and also why τ 1/2 increases with Z. Also explains why decay to heavier fragments e.g. 12 C disfavoured G m 1/2 and G charge of fragment Deficiencies/complications Assumed existence of a single α particle in nucleus and have taken no account of probability of formation. Assumed semi-classical approach to estimate escape trial frequency, f v/2r, and make absolute prediction of decay rate. If α is emitted with some angular momentum, l, the radial wave equation must include a centrifugal barrier term in Schrödinger equation V l(l + 1) 2 l = relative a.m. of α and daughter nucleus = 2µr 2 µ = reduced mass which raises the barrier and suppresses emission of α in in high l states. Dr. Tina Potter 15. Nuclear Decay 16

17 α Decay Selection rules Nuclear Shell Model: α has J P = 0 + Angular momentum e.g. X Y + α Conserve J: J X = J Y J α = J Y l α l can take values from J X + J Y to J X J Y Parity Parity is conserved in α decay (strong force). Orbital wavefunction has P = ( 1) l X, Y same parity l must be even X, Y opposite parity l must be odd e.g. if X, Y are both even-even nuclei in their ground states, shell model predicts both have J P = 0 + l = 0. More generally, if X has J P = 0 +, the states of Y which can be formed in α decay are J P = 0 +, 1, 2 +, 3, 4 +,... Dr. Tina Potter 15. Nuclear Decay 17

18 β Decay β n p + e + ν e A Z X A Z+1 Y +e + ν e β + p n + e + + ν e A Z X A Z 1 Y +e+ + ν e electron capture p + e n + ν e A Z X +e A Z 1 Y +ν e β decay is a weak interaction mediated by the W boson Parity is violated in β decay Kinematics: Decay is possible if energy release E 0 > 0 Nuclear Masses Atomic Masses β E 0 = m X m Y m e m ν E 0 = M X M Y m ν β + E 0 = m X m Y m e m ν E 0 = M X M Y 2m e m ν e.c. E 0 = m X m Y + m e m ν E 0 = M X M Y m ν (and note that m ν 0) using M(A, Z) = m(a, Z) + Zm e n.b. electron capture may be possible even if β + not allowed Dr. Tina Potter 15. Nuclear Decay 18

19 β Decay Nuclear stability against β decay Consider nuclear mass as a function of N and Z m(a, Z) = Zm p + (A Z)m n a V A + a S A 2/3 + a CZ 2 A + a (N Z) 2 1/3 A δ(a) A using SEMF β decay, A is constant, Z changes by ±1 and m(a, Z) is quadratic in Z [ ] Most stable nuclide when m(a, Z) = 0 Z A Dr. Tina Potter 15. Nuclear Decay 19

20 β Decay Typical situation at constant A Usually only one isotope table against β-decay; occasionally two. Typically two even-even nuclides are stable against β-decay; almost no odd-odd ones. Dr. Tina Potter 15. Nuclear Decay 20

21 Fermi Theory of β-decay In nuclear decay, weak interaction taken to be a 4-fermion contact interaction No propagator. Absorb the effect of the exchanged W boson into an effective coupling strength given by the Fermi constant G F G F = GeV 2 β decay in Fermi Theory Y X Ye ν e X GF e Use Fermi s Golden Rule to get the transition rate Γ = 2π M fi 2 ρ(e f ) ν e ρ(e f ) = dn de f where M fi is the matrix element and ρ(e f ) is the density of final states. Dr. Tina Potter 15. Nuclear Decay 21

22 Fermi Theory of β-decay Total decay rate given by Γ = G 2 F M nuclear 2 2π 3 Γ E 5 0 β decay spectrum described by d Γ 1 2 d p e p e E0 0 (E 0 E e ) 2 E 2 e de e Sargent s Rule dγ 1 dp e pe 2 (E 0 E e ) 3 H 3 He+e + ν e Kurie Plot Endpoint E 0 E e (kev) Dr. Tina Potter 15. Nuclear Decay 22

23 Fermi Theory of β-decay BUT, the momentum of the electron is modified by the Coulomb interaction as it moves away from the nucleus (different for e and e + ). Multiply spectrum by Fermi function F (Z Y, E e ) Γ = G 2 F M nuclear 2 2π 3 E0 0 (E 0 E e ) 2 E 2 e F (Z Y, E e ) de e All the information about the nuclear wavefunctions is contained in the matrix element. Values for the complicated Fermi Integral are tabulated. f (Z Y, E 0 ) = 1 m 5 e E0 Mean lifetime τ = 1/Γ, half-life τ 1/2 = ln 2 Γ 2π 3 f τ 1/2 = ln 2 meg 5 F 2 M nuclear 2 0 (E 0 E e ) 2 E 2 e F (Z Y, E e ) de e Comparative half-life this is rather useful because it depends only on the nuclear matrix element Dr. Tina Potter 15. Nuclear Decay 23

24 Fermi Theory of β-decay Comparative half-lives In rough terms, decays with log 10 f τ 1/2 3 4 known as super-allowed 4 7 known as allowed 6 known as forbidden (i.e. supressed, small M if ) Dr. Tina Potter 15. Nuclear Decay 24

25 Fermi Theory of β-decay Selection rules in β decay Fermi theory M fi = G F ψ pe i( p e+ p ν ). r ψ n d 3 r e, ν wavefunctions Allowed Transitions Angular momentum of eν pair relative to nucleus, l = 0. Equivalent to: e i( p e+ p ν ). r 1 log 10 f τ 1/2 4 7 Superallowed Transitions (subset of Allowed transitions: often mirror nuclei in which p and n have approximately the same wavefunction) M nuclear ψ pψ n d 3 r 1 log 10 f τ 1/2 3 4 e, ν both have spin 1/2 Total spin of eν system can be S eν = 0 or 1 There are two types of allowed/superallowed transistions depending on the relative spin states of the emitted e and ν Dr. Tina Potter 15. Nuclear Decay 25

26 Fermi Theory of β-decay Fermi transitions S eν = 0 X Y + e + ν J X = J Y S eν l n p [( e ν e ) ( e ν e )] J = 0 J X = J Y S eν = 0, m s = 0 Gamow-Teller transitions S eν = 1 n p [( e ν e ) + ( e ν e )] J = 0 J X = J Y S eν = 1, m s = forbidden n p + e + ν e J = ±1 J X = J Y ± 1 S eν = 1, m s = ±1 Total number of spin states of eν = 4 (3 G-T, 1 Fermi) No change in angular momentum of the eν pair relative to the nucleus, l = 0 Parity of nucleus unchanged Dr. Tina Potter 15. Nuclear Decay 26

27 Fermi Theory of β-decay Forbidden Transitions log 10 f τ 1/2 6 typically Angular momentum of eν pair relative to nucleus, l > 0. e i( p e+ p ν ). r = 1 i( p e + p ν ). r [( p e + p ν ). r] 2... l = P = ( 1) l even odd even Allowed 1 st forbidden 2 nd forbidden Transition probabilities for l > 0 are small forbidden transitions (really means suppressed ). Forbidden transitions are only competitive if an allowed transition cannot occur (selection rules). Then the lowest permitted order of forbiddeness will dominate. In general, n th forbidden eν system carries orbital angular momentum l = n, and S eν = 0 (Fermi) or 1 (G-T). Parity change if l is odd. Dr. Tina Potter 15. Nuclear Decay 27

28 Fermi Theory of β-decay Examples 34 Cl(0 + ) 34 S(0 + ) 14 C(0 + ) 14 N(1 + ) n(1/2 + ) p(1/2 + ) 39 Ar(7/2 ) 39 K(3/2 + ) 87 Rb(3/2 ) 87 Sr(9/2 + ) Dr. Tina Potter 15. Nuclear Decay 28

29 γ Decay Emission of γ-rays (electromagnetic radiation) occurs when a nucleus is created in an excited state (e.g. following α, β decay or collision). The treatment of radiative transitions in nuclei is generally the same as for atoms, except Atom E γ ev ; λ 10 8 fm 10 3 r atom ; Γ 10 9 s 1 Only dipole transitions are important. Nuclei E γ MeV ; λ 10 2 fm 25 r nucl ; Γ s 1 Higher order transitions are also important. Collective motion of many protons lead to higher transition rates. Two types of transitions: Electric (E) transitions arise from an oscillating charge which causes an oscillation in the external electric field. Magnetic (M) transitions arise from a varying current or magnetic moment which sets up a varying magnetic field. Dr. Tina Potter 15. Nuclear Decay 29

30 γ Decay initial final γ J i l J f The photon carrys away net angular momentum l when a proton in the nucleus makes a transition from its initial a.m. state J i to its final a.m. state J f. Ji = l J f and J i J f l J i + J f The photon carries J P = 1 l 1. Single γ emission is forbidden for a transition between two J = 0 states.(0 0 transitions can only occur via internal conversion (emitting an electron) or via the emission of more than one γ.) The transition probabilities may be obtained using Γ = 2π M if 2 ρ(e f ) Fermi Golden Rule Dr. Tina Potter 15. Nuclear Decay 30

31 γ Decay Electric Dipole Transitions (E1) l = 1 Insert dipole matrix element into FGR Γ i f = ω3 3πɛ 0 c 3 ψ f e r ψ i 2 Order of magnitude estimate of this rate, see Adv. Quantum Physics; after averaging over initial and summing over final states ψ f e r ψ i 2 er 2 Γ = 4 3 αe 3 γr 2 R = radius of nucleus, Example E γ = 1 MeV, R = 5 fm Γ(E1) = 0.24 MeV 3 fm 2 = α = e2 4πɛ 0 c, E γ = ω, = c = 1. ( c = 197 MeVfm, = MeVs) 0.24 (197) s 1 = s 1 (c.f. atoms Γ 10 9 s 1 ) As nuclear wavefunctions have definite parity, the matrix element can only be non-zero if the initial and final states have opposite parity. ˆP e r e r E1 transition parity change of nucleus Dr. Tina Potter 15. Nuclear Decay 31

32 γ Decay Magnetic Dipole Transitions (M1) l = 1 Matrix Element ψ f µ σ ψ i 2 µ = magnetic moment, σ = Pauli spin matrices Typically µσ e 2m p = µ N Nuclear magneton Proton m p 0.2fm R 25 for R = 5 fm Γ(M1) Γ(E1) = ( e 2m p ) 2 1 (er) 2 = 10 3 Magnetic moment transforms the same way as angular momentum e r p ˆP e( r) ( p) = e r p EVEN M1 transition no parity change of nucleus Dr. Tina Potter 15. Nuclear Decay 32

33 γ Decay Higher Order Transitions (El, Ml where l > 1) If the initial and final nuclear states differ by more than 1 unit of angular momentum higher multipole radiation The perturbing Hamiltonian is a function of electric and magnetic fields and hence of the vector potential ψ f H ( A) ψ i A for a photon is taken to have the form of a plane wave Ae i p. r = 1 i p. r ( p. r) ( i p. r)n n! Dipole Quadrupole Octupole l = E1,M1 E2,M2 E3,M3 Each successive term in the expansion of A is reduced from the previous one by a factor of roughly p. r. e.g. p 1 MeV, R 5fm pr 5 MeVfm 0.025, pr Γ(E2) Γ(E1) 10 3 Γ(M1) Γ(E1) Dr. Tina Potter 15. Nuclear Decay 33

34 γ Decay Higher Order Transitions (El, Ml where l > 1) The matrix element for E2 transitions r 2 i.e. even under a parity transformation In general, El transitions Parity = ( 1) l Ml transitions Parity = ( 1) l+1 Rate E1 E2 E3 E4... M1 M2 M3... Parity change J P of γ E: M: In general, a decay will proceed dominantly by the lowest order (i.e. fastest) process permitted by angular momentum and parity. Example: if a process has J = 2, no parity change, it will go by the E2, even though M3, E4 are also allowed. Dr. Tina Potter 15. Nuclear Decay 34

35 γ Decay Higher Order Transitions (El, Ml where l > 1) Example: Sn E2 7/2 + 3/2 + 1/2 + M1 (E2 also allowed) M1 M4 11/2 3/2 + 1/2 + J P 11/2-3/2 + M4 More likely than 11/2-1/2 + (E5) 7/2 + 3/2 + E2 M2 less likely 7/2 + 11/2 - M3 7/2 + 1/2 + Information about the nature of transitions (based on rates and angular distributions) is very useful in inferring the J P values of states. This discussion of rates is fairly naïve. More complete formulae can be found in textbooks. Also collective effects may be important many nucleons participate in transitions. If nucleus has a large electric quadrupole moment, Q, rotational excited states enhance E2 transitions Dr. Tina Potter 15. Nuclear Decay 35

36 Summary Radioactive decays and dating. α-decay β-decay γ-decay Strong dependence on E, Z Tunnelling model (Gamow) Geiger-Nuttall law ln τ 1/2 Z E 1/2 0 β +, β, electron capture; energetics, stability Fermi theory 4-fermion interaction plus 3-body phase space. Γ = G F 2 M nuclear 2 E0 2π 3 (E 0 E e ) 2 pe 2 dp e 0 Electron energy spectrum; Kurie plot. Comparative half-lives. Selection rules; Fermi, Gamow-Teller; allowed, forbidden. Dipole, quadrupole; electric, magnetic transitions. Selection rules. + const. Up next... Section 16: Fission and Fusion Dr. Tina Potter 15. Nuclear Decay 36

α particles, β particles, and γ rays. Measurements of the energy of the nuclear

α particles, β particles, and γ rays. Measurements of the energy of the nuclear .101 Applied Nuclear Physics (Fall 004) Lecture (1/1/04) Nuclear ecays References: W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), Chap 4. A nucleus in an excited state is unstable