MATR316, Nuclear Physics, 10 cr

Transcription

1 MATR316, Nuclear Physics, 10 cr Fall 2017, Period II Pertti O. Tikkanen Lecture Notes of Tuesday, Nov. 7th and Thursday, Nov. 9th Department of Physics 1

2 Beta decay Brief history Fermi Theory of Beta Decay Shape of beta spectrum Density of states Total half-life in beta decay Electron capture (EC) Classification of beta decay Recent research on beta decay Electromagnetic transitions and excitations Summary of classical theory Quantization of the radiation field Interaction between nucleus and radiation field Decay of excited states via EM interaction, γ emission, IC, IPF Excitation mechanisms, Mössbauer effect, Coulex 1

3 Beta decay

4 Some history in s Without the requirement of an extra particle, the following could not all be true simultaneously: (i) Continuous energy spectrum of β-particles (ii) Angular momenta involved (iii) Fermi-Dirac statistics (spin-1/2). W. Pauli, in an open letter, dated Dec. 4 th 1930, to L. Meitner suggested the existence of light neutrons 1, but not until June 16 th, 1931, in the first APS meeting in Pasadena, CA, Pauli was brave enough to publicly speak for the first time of his hypothesis. Unfortunately, he was not sufficiently confident to let his lectures to be printed. E. Fermi gave the hypothetical particle the name neutrino ( little neutron ), after discussing privately with Pauli during the nuclear physics conference in Rome (October 1931). Pauli still refused to give a public talk on the subject. In the Solvay Conference in 1933, the name neutrino was already in wide use. The experiments with the real neutron, discovered by Chadwick in 1932, however, kept Fermi busy for some time, until he was ready to publish, in 1934, what we now know as The Fermi theory of beta decay die Möglichkeit, es könnten elektrisch neutrale Teilchen, die ich Neutronen nennen will, in den Kernen existieren, welche Spin 1/2 haben und das Ausschliesungsprinzip befolgen und sich von Lichtquanten ausserdem noch dadurch unterschieden, daß sie nicht mit Lichtgeschwingichkeit laufen. 2

5 Historical experiments of beta decay Figure 19: One of the first published continuous spectra of electrons from the β decay of RadiumE ( 210 Bi), as measured by Ellis and Wooster (1927). 3

6 Figure 20: Cloud chamber track of a recoiling 6 Li from the 6 He β decay. The curved track is the electron. Decay energy Q β = 3505 kev. 4

7 Energy relations and Q values in beta decay Notations: T kinetic energy E total energy M nuclear mass M atomic mass A mass number mass defect = (M A) uc 2 The Q value for the β decay: A Z X N Z +1 A Y N 1 + e + ν e Q β = T e + T νe (+T D ) = M P c2 M D c2 m 0 c 2 = M P c 2 M D c 2, neglecting the small differences in electron binding energies B: M P c 2 = M P c2 + Zm 0 c 2 Z B i M D c 2 = Z +1 M D c2 + (Z + 1)m 0 c 2 B i i Thus β decay is energetically possible if M P > M D. i 5

8 Q value of β + decay, Q β + A Z X N Z 1 A Y N+1 + e + + ν e Q β + = T e + + T νe (+T D ) = M P c 2 M D c 2 2m 0 c 2, a threshold of 2m 0 c kev appears! Note: Q value of the EC process, Q EC : (EC = electron capture) A Z X N + e Z 1 A Y N+1 + ν e Q EC = T νe + T D = M P c 2 M D c 2 B n, a threshold equal to binding energy B n of electron in the n th shell Often (also in nndc database!) only Q EC and Q β are given, for apparent reasons. The β-decay half-life may depend on the chemical environment! (No electrons no EC) 6

9 Figure 21: Calculated ionization potentials for all charge states and for all elements (Handbook of Ion Sources, c CRC Press, Boca Raton ). The charge state is denoted by the number in front of each curve (n = 0 corresponds to a neutral atom). You can compare with the formula given in P2_LN1. 3 Data from T.A. Carlson et al, At. Data 2 (1970) 63. 7

10 Energy relations in beta decay Figure 22: Energy relations in β-decay processes (Heyde 2004) Table 6: Some typical beta decay energies (Note: Here the β + decay Q-values do not include 2m 0 c 2. Decay Type Q (MeV) Half-life 23 Ne 23 Na β s 99 Tc 99 Ru β years 25 Al 25 Mg β s 124 I 124 Te β s 15 O 15 N EC s 41 Ca 41 K EC years 8

11 Fermi Theory of Beta Decay Fermi Theory of Beta Decay Fermi s theory (1934) solved the (old) puzzles of beta decay: Continuous energy spectrum Emission of particles not present inside nucleus Shape of the energy spectrum Decay probability Not completely, though: Parity non-conservation Spin not included 9

12 Shape of beta spectrum and total decay probability Starting point: Golden rule (Fermi) to calculate transition rate λ fi = 2π Ψ f H int Ψ i 2 dn de (88) Need to specify the wavefunctions, interaction, and the density of states Initial state: Ψ i = Ψ P ( r 1, r 2,..., r A ), the parent nucleus (P) Final state: Ψ f product wavefunction for electron (e), daughter nucleus (D), and antineutrino ( ν e), i.e. Ψ f = Ψ D ( r 1, r 2,..., r A )Ψ e( r e, Z )Ψ νe ( r ν) Equal number A of nucleons in P and D, but one of neutrons transformed into a proton via the interaction Hamiltonian H int and, simultaneously, electron and antineutrino emitted 10

13 Ψ e( r e, Z ) is a Coulomb function in non-relativistic case Ψ νe ( r ν) described by plane wave Point-like interaction with strength g g δ( r p r n) δ( r n r e) δ( r n r ν) Rewrite transition rate as λ = 2π M fi 2 dn de (e, ν e), (89) evaluate both M fi 2 and density of states dn/de using a quantization volume V 11

14 Density of states A particle confined to move in one-dimensional box 0 < x < L with momentum between (0, p) has a number of states n determined by kl = pl = n2π. Generalizing to three-dimensions (6-dimensional phase space): 1 n = (2π ) 3 d 3 x d 3 p V = (2π ) 3 d 3 p, (90) which results in the density of states dn de = = d de [ V (2π ) 3 V dp 4πp2 (2π ) 3 de dωdp p 2 ] (91) 12

15 Relativistic relation E 2 = m 2 c 4 + p 2 c 2 yields EdE = c 2 p dp: dn de = = V p2 E 4π (2π ) 3 c 2 p 4πV (2π ) 3 pe or dn de = 1 V 2π 2 c 2 3 E E 2 m 2 c 4. (92) In β ± decay, only two of the three (electron, neutrino, recoiling nucleus D) are independent: 0 = p e + p νe + p D E = E e + E νe + E D 13

16 The resulting density of states is a product of positron (electron) and (anti)neutrino densities: dn de (e, ν e) = V 2 d (2π ) 6 de p 2 e dpedωep2 ν dpνdων. (93) = V 2 4π 4 6 c 3 p2 e (E E e ) 2 Here we explicitly allow for a possibly non-zero neutrino mass. 1 m2 ν e c 4 (E E e ) 2 Multiply with 2π M fi 2 and we have the probability per unit time that an electron is emitted with energy in [E e, E e + de e] and momentum in [p e, p e + dp e] 14

17 Shape of beta spectrum Calculation of the matrix element M fi needs Ψ ν( r ν) = 1 V exp (i k ν r ν), a plane wave, p ν = k ν, Ψ e( r e, Z ) = 1 V exp (i k e r e) F (Z, p e) a plane wave distorted by the Coulomb field of the daughter nucleus. 2πη Quantitatively F (Z, p e) 1 exp ( 2πη) Fermi function where η = ± Ze2 4πɛ 0 v e v e is the velocity of electron (positron) use η for β decay and η + for β + After combining all contributions, the momentum spectrum Λ(p e)dp e = M fi 2 F (Z D, p e) 2π 3 7 c 3 p 2 e (E E e ) 2 1 m2 ν e c 4 (E E e ) 2 (94) where the prime indicates that the proper normalisation and Coulomb distortion are included 15

18 Eq. (94) gives the partial decay probability and the distribution can be compared with measured β-ray spectra When E = E e (and p = p max) electron carries all the energy (end-point energy) Analysis of measured β spectra with the help of Eq. (94) is made using Fermi-Kurie plot: Plot the values of (but wait: put the neutrino mass 0 first!) Λ(p e) pef 2 vs. (E E e) (Z, p e) and fit a straight line. The intercept with energy axis gives end-point energy (and Q value). Works only for superallowed transitions, i.e., , must be modified when L 0 Upper limit for neutrino mass from β decay of 3 H: m νe < 9 ev/c 2 16

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20 Fermi-Kurie plot of 66 Ga β + decay. Total energies w in units of m 0 c 2 18

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22 Total half-life in beta decay How to get the total decay probability (decay constant) λ or, equivalently, the half-life T 1/2? Easy? Just integrate over the distribution, with respect to p e (or E e) Set m νe = 0 (it s of minor importance anyway!) pmax λ β = 0 M fi 2 2π 3 7 c 3 F (Z D, p e) p 2 e (E E e ) 2 dp e, (95) define u = p e/m 0 c and w = E e/m 0 c 2 = T e/m 0 c so that w 2 = u and wdw = udu, and write M 2 = g 2 M fi 2 : λ β = g2 M fi 2 m 5 0 c4 2π 3 7 w0 0 ( ) F Z D, w 2 1 w 2 1 (w 0 w) 2 wdw. (96) 20

23 The Fermi integral contains three-body phase-space information: f (Z, w 0 ) = Z =0 w0 ( ) F Z, w 2 1 w 2 1 (w 0 w) 2 wdw 0 1 ) (2w w u w 0 ln(w 0 + u 0 ) w w 5 0 (97) The final result 2π 3 7 c f T 1/2 = ln(2) g M fi 2 m0 5 (98) c4 applied to superallowed Fermi transitions (M fi = 2) gives the coupling constant of β decay as g = g V = MeV fm 3 21

24 a ue o is egect ma e correlated e i y predicted red' lue ot' ix ZI. Th' ay ' e n particle density I: 0 I o 0 ii/ /I' 3 N, (eoi) 22

25 After more than 115 years, interest in β decay continues: Nobel Price in Physics 2015: neutrino oscillations Neutrino mass 0? Majorana or Dirac neutrinos? Precise measurements of the β decay between nuclear analog states of spin, J π = 0 +, and isospin, T = 1, provide demanding and fundamental tests of the properties of the electroweak interaction. Collectively, these transitions sensitively probe the conservation of the vector weak current, set tight limits on the presence of scalar currents, and provide the most precise value for V ud, the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This latter result has become a linchpin in the most demanding available test of the unitarity of the CKM matrix, a property which is fundamental to the electroweak standard model. (J.C. Hardy and I.S. Towner: Phys. Rev. C91, 2015) 23

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27 Electron capture Only 2 particles in the final state makes all easy! Phase-space factor for neutrino final states: dn ν de = V p2 ν dp ν 2π 2 3 de = = V E ν 2 2π 2 c 3 3, assuming massless neutrino and neglecting the recoil energy. Transition matrix element M fi = g Ψ ν(0)ψ e(0)ψ n( r)ψ p( r)d 3 r ( ) 3/2 Ψ e (0) = 1 Zm 0 e 2 π 4πɛ 0 2 Transition probability for EC from the K shell: λ EC = E 2 ν π 2 c 3 4 g2 M fi 2 ( ) 3 Zm 0 e 2 4πɛ 0 2 (99) 25

28 The function f K (Z, w 0 )/f +(Z, w 0 ), giving K-capture to positron ratio for allowed transitions. 26

29 Classification of beta decay Spinless non-relativistic model (Fermi transitions) Introduction of intrinsic spin S β Fermi (F) and Gamow-Teller (GT) transitions Selection rules: allowed and forbidden transitions 27

30 Spinless non-relativistic model We have seen that in the superallowed transitions ft 1/ M fi 2 More insight can be provided by the study of the transition matrix element M fi = ψd ( r p1,..., r p,z +1 ; r n2,..., r n,n )ψe ( re)ψ ν ( rν) where the operator H int ψ P ( r p1,..., r p,z ; r n1,..., r n,n )d r ed r νd r i (100) H int = gδ( r n r p)δ( r n r e)δ( r n r ν)t, (101) ( ) 1 and T is the isospin lowering operator changing a proton p = into a 0 ( ) neutron (β + 0 decay) n =, or vise versa (β decay). Here Z = 1/2 + T z, and 1 28

31 the integration over all the nucleonic coordinates is denoted by d r i. By using simple single-particle product nuclear wavefunctions and plane waves to describe the electron (positron) and the (anti)neutrino, we have M fi = g φ D ( r)pei( k e+ k ν ) r T φ P ( r) nd r. (102) Typically, the debroglie wavelengths of the electron and neutrino are much smaller than the nuclear radius (check!), implying that the usual series expansion of the exponential converges fast: ( M fi = g φ D ( r)pt φ P ( r) nd r + i( k e + k ν) ) φ D ( r)p r T φ P ( r) nd r (103) First term 0 only if parent and daughter single-particle states have same parity, second term 0 if parities are opposite etc. 29

32 Alternatively, the use of e i( k e+ k ν ) r = L,M (4π)i L j L (kr)y M L (k e + k ν)y M L (ˆr) where k = k e + k ν and ˆr (θ r, ϕ r ) denote the angular variable, gives rise to matrix elements φ D ( r)pt Y M L (ˆr)φP ( r) n j L (kr)d r. (104) The single-particle wavefunctions separate into radial and angular parts (neglect the spin for a moment): φ D ( r) p = R D (r)y M D L D (ˆr) and φ P ( r) p = R P (r)y M P L P (ˆr) This reduces the particular L matrix element into a 30

33 product of a pure radial part and an angular part, i.e. R D (r) j L (kr)r P (r)r 2 dr Y M D L (ˆr)Y M M D L (ˆr)Y P L (ˆr) dω (105) P The second part constraints the angular momentum L β carried by the electron-antineutrino pair to L P L D L β L P + L D (on the basis of the Gaunt formula or the Wigner-Eckart theorem). Combining all the above, results finally to M fi 2 = L M fi L 2, (106) limited by the parity and angular momentum selection rules π P = ( 1) L π D and L P L D L L P + L D. (107) 31

34 The spherical Bessel functions,j L (kr) (kr)l for kr L, through their presence in (2L+1)!! the radial integral, effectively suppress the higher multipoles. Usually only the lowest multipoles (L = 0, 1) contribute significantly, except in cases where the spins of the initial and final nuclear states differ by several units. Furthermore, the parity is not completely conserved in weak interactions, as Wu et al confirmed experimentally in 1956 (see the course literature). 32

35 Introducing intrinsic spin The original Fermi theory doesn t include spin. It corresponds to the case where the spins of the emitted leptons are antiparallel, i.e., they are coupled to a total spin of S β = 0. As such, it applies only to the superallowed transitions. Improvements In non-relativistic description wavefunctions are 2-component spinors The interaction operator H int contains terms that may change the intrinsic spin The matrix element has the form M fi = ψ DˆσΨ P Ψ e ˆσΨ ν d r (108) where integral over nuclear coordinates and sum over spin orientations are implied. 33

36 The selection rules then become (J P J D J β ), (J β L β S β ), π P = ( 1) L β π D (109) where the triangle inequality is a shorthand notation for J P J D J β J P + J D and L β and S β are the angular momenta carried away by the orbital and intrinsic spin. The intrinsic spins of the electron and neutrino are naturally coupled to either S β = 0 or S β = 1. The total matrix element is then M fi 2 = M Lβ,Sβ 2. (110) fi L β,s β In a fully relativistic treatment all wavefunctions are 4-component spinors and the introduction of tensor, pseudoscalar, and pseudovector terms generate a small parity-violating contribution. 34

37 Fermi and Gamow-Teller transitions The rate of beta decay depends on whether S β = 0 or S β = 1: S β = 0, Fermi transition with strength g F S β = 1, Gamow-Teller transition, with g GT Usually in a transition both contribute, and to lowest order M fi 2 = M fi (F) 2 + g2 GT gf 2 M fi (GT) 2. (111) At quark level, g GT = g F, but in nuclear structure the value g GT /g F = ± from the neutron β decay is used (See also B.A. Brown, Lecture notes 2011) For a pure Fermi transition (J f = J i, π f = π i ), M fi 2 (F) = 1, for a pure Gamow-Teller transition (J f = J i ± 1, π f = π i ), M fi (GT) 2 = 3, as shown easily. 35

38 Selection rules for beta decay In allowed F and GT max spin change is one and no parity change. Example of pure GT decay is 6 He β 6 Li Transitions with parity change and larger spin changes occur and are called forbidden, the order of forbidness L β Reason is the j L (kr) at small values See the figure and table 36

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41 Neutrino in beta decay Several puzzles in weak interaction related to neutrino(s) Parity non-conservation, first observed in 60 Co decay Neutrino mass 0? Neutrino families, (ν e, ν µ, ν τ ), why they exist? Double beta decay processes 1) ννββ, double-neutrino double-beta, 2n 2p + 2e + 2 ν e and 2p 2n + 2e + + 2ν e experimentally well established and 2) 00ββ, neutrino-less double-beta, under very active research 2β-decay is a 2 nd -order process competing with forbidden decay. Helicity of neutrino, well defined only if massless ν. The helicity operator ĥ = ˆσ has eigenvalues of ±1, depending whether the intrinsic spin points in the same or opposite direction as p Neutrino oscillations (Nobel prize 2015) ˆ p p 39

42 MV94, M. Moe and P. Vogel, Annu. Rev. Nucl. Part. Sci. 44, (1994) V.I. Tretyak and Y.D. Zdesenko, At. Data Nucl. Data Tables 80, (2002) 40

43 Secondary (atomic or nuclear processes) related to β decay Electron capture creates a vacancy in the atomic K, (LI, LII) shell X-rays, Auger electrons Many nuclei may decay by all three processes ( 64 Cu) Nuclear charge changes by ±e perturbation in the atomic orbitals (shake-off ionisation). Effect even stronger in α or cluster decay. In α decay (or cluster decay and fission), the large change in the proton number induces naturally a multiple ionization in the daughter atom. The effect is largest for the inner-shell electron orbitals. If the β decay goes to proton- or neutron-unbound states in the daughter nucleus, those states can decay via the process of beta-delayed nucleon emission 41

44 Figure 23: Beta-delayed neutron decay of 17 N [Fig.9-15 from Krane c ] 42

45 Figure 24: Beta-decay of 126 I [Fig.9-32 from Krane c ] 43

46 Electromagnetic transitions and excitations

47 In the electromagnetic transitions and excitations, the identity of atomic nucleus does not change, as the number of protons and neutrons remains constant during the process. Instead, the configuration of the nucleons, i.e. the quantum-mechanical state described by the wavefunction changes as a result of the interaction with the quantized EM field. Our tasks 1. Construct electromagnetic interaction Hamiltonian, H int 2. Quantize the EM field 3. Understand multipoles, selection rules, transition rates in terms of W.u. 44

48 Electromagnetic interaction Starting point is Maxwell s equations (SI units): together with the equation of motion of a charged particle E = ϱ ɛ 0 (112) B = 0 (113) E = B t (114) B = E t (115) dp dt = q(e + v B). (116) For the quantisation of the EM field and the interaction, introduce the scalar potential Φ and the vector potential A: E = Φ A (117) t B = A (118) 45

49 It is always possible to choose (i.e. the radiation gauge) { Φ = 0 A = 0 The interaction Hamiltonian, H int, is usually obtained from the Hamiltonian of the unperturbed system by minimal coupling by replacing p p qa 1 H = (p qa)2 +V (r) 2m }{{} = p 2 q(a p + p A) + q 2 A 2 Thus in the radiation gauge and keeping the term linear in A: H int = q A(r, t) p. (119) m In quantum mechanics p i k and E = ω. 46

50 In the quantization, the vector potential is replaced with a suitable vector operator that (i) creates and annihilates quanta of the EM field and also (ii) describes the interaction with the system. The total Hamiltonian is then H = H 0 + H field + H int after we quantised the EM field, we redefine H0 old H0 new = H0 old + H field to describe the combined system of nucleus+field without the mutual interaction. The interaction term H int, when operating on the state vector of the system, creates/annihilates field quanta (photons) and, simultaneously, makes a nuclear transition (decay/excite) to happen. 47

51 Normalization of A The vector potential of the EM field satisfies the vector Helmholtz equation 2 A 1 2 A c 2 = 0. (120) t2 A plane-wave with a wave vector k is a solution (= normal mode in the normalization volume V ): A k (r, t) = A 0k e i(k r ωt) + (... ) (121) provided that ω = ω k c k ck. The gauge condition A = 0 requires that the amplitude vector A 0k k which guarantees transversality. The polarisation of the wave is described by two unit vectors so that A 0k = A 0 ε kµ. If the direction of z-axis is parallel to k, there are two possibilities, plane polarization (ê x, ê y ) and circular polarization ε k± = 1 (ê x ± iê y ). 2 48

52 The partial wave with fixed k is a superposition of the two polarisations A k = [ A kµ ε kµ e i(k r ωt) + A kµ ε e i(k r ωt)] kµ (122) µ The general solution is then a superposition of partial waves A(r, t) = [ A kµ ε kµ e i(k r ωt) + A kµ ε e i(k r ωt)] kµ (123) k,µ The physical fields are E = A and B = A, when we choose the radiation t gauge. 49

53 The (classical!) energy density of the EM field is 1 2 ɛ 0E B 2. 2µ 0 Its integral over the volume V should be equal to the total energy of the field k N k ω k, when there are N k photons with frequencies ω k. This implies that N [ k A(r, t) = ε kµ e i(k r ωt) + ε e i(k r ωt)] kµ (124) 2ɛ 0 V ω kµ k describes a properly normalised classical radiation field in the free space of volume V. 50

54 Quantization of EM field, photons Quantization for a harmonic oscillator was done via where H = p2 2m mω2 x 2 = 1 2 ω + ωa A, A = A = mω 2 x + i p a 2mω mω 2 x i p a 2mω Thus ( H = ω a a + 1 ) 2 and E = ω(n + 1/2) and a a is the operator that gives the number of quanta. 51

55 By analogue: the energy of the photon gas of N identical photons is N ω (plus a zero-point energy of ω/2) if we define similar operators a and a for annihilation and creation of photons: a N = N N 1 (125) a N = N + 1 N + 1 (126) Classically, A(r, t) e iωt + (c.c.), so let s try to define the operator by analogue: Â = 2ɛ 0 ωv ( εe ik r a + ε e ik r a ) (127) Check by calculating N 1 Â N = () Nε e ik r, between the states of the EM field. Note that the exponential term operates on the nuclear wave function only. 52

56 For N = 1, absorption of a photon, as the initial state is one photon with k, ε: 0 Â 1 = () 1 ε e ik r Emission of a photon, initial state with no photons, in final state one photon with k, ε: 1 Â 0 = () 1 ε e ik r Thus it seems that the recipe works OK! Complete description of the radiation field: 1. enumerate the occupation numbers n kµ, 2. eigenstate of the total field is the product of all the eigenstates n kµ. 3. Denoting n ki µ i n i, the many-photon state is written as n 1, n 2,..., n i,... 53

57 The final quantised form of the vector potential is then 1 Â H (r, t) = [a kµ ε kµ e i(k r ωt) + a kµ 2ɛ 0 V ε e i(k r ωt)] kµ (128) ωk kµ Because of the explicit time dependence, this form is called the Heisenberg representation. The annihilation and creation operators are in the Schrödinger representation. The vector potential in the latter representation is 1 Â(r, t) = [a kµ ε kµ e ik r + a kµ 2ɛ 0 V ε e ik r] kµ (129) ωk kµ 54

58 Let s collect the pieces we have: Hamiltonian of the nuclear system (in the mean-field approximation!) H nucl = Hamiltonian of the radiation field H rad = kµ A i [ ] p 2 i + U(r i ) 2m i ( ω k a kµ a kµ + 1 ) 2 Together they form the total Hamiltonian of the system nucleus+radiation field without interaction: H 0 = H nucl + H rad 55

59 Figure 25: Absorption (a) and emission (b) of photons. The zero-point energy has been omitted. If it s included A = a, n k,ε, E A = E a + (n k,ε ) ω k, E B E A = E b E a ω k = (ω ba ω k ) The number of modes per unit energy interval (density of states of EM field) that is used in the Golden Rule is 1 de dn = V (2π) 3 c 3 ω2 kdω (130) 56

60 The form of the interaction Hamiltonian (non-relativistic version, correct up to 1 st order in A): A [ qi H int = p i 2m  q ] i σ i i 2m  i The relativistic form of the interaction is given by H = d 3 r j(r) Â(r) where the total current operator j includes the convection, spin, and magnetization currents of all the nucleons. In most cases, the polarisation of the emitted photons is not measured, and the decaying nuclear state is unaligned. The experimental transition probability should then be compared with λ fi = 2π 4 ( ) L + 1 2L+1 Eγ 2J i + 1 L[(2L + 1)!!] 2 c L,ν J f ˆM(νL) J 2 i (131) where the sum is over all multipoles (or 2 L -poles), ν =E (electric) and ν =M (magnetic) that are compatible with the 57

61 Selection rules and multipole operators Parity conservation: π i π f = { ( 1) L ( 1) L+1 for electric 2 L -pole for magnetic 2 L -pole (132) Angular momentum J i J f L J i + J f. (133) In the long wavelength limit, kr 1 the multipole operators are ˆM(EL; M) = ˆM(ML; M) = A e i ri L Y LM (ˆr i ) = d 3 rρ ch (r)r L Y LM (n) i=1 A µ N (gi S Ŝ i + 2 ) L + 1 gl i ˆL i i=1 ( ) ri L Y LM (ˆr i ). (134) 58

62 In general form, they are given by M(ML, M) = M(EL, M) = d 3 r(j l)φ LM [ d 3 r ( j) 1 ] r k(r j) Φ LM k r where L = l = i [r ] is the angular momentum operator and 4 (2L + 1)!! 1 Φ LM = i L + 1 k L j L(kr)Y LM (n) Note that the multipole operators are defined with the allowance of the effective charges and effective g-factors that may differ from free-nucleon values. Therefore the sums run over all the A nucleons. It is easy to show that both electric and magnetic operators can be put in a form that are useful in testing the conservation of isospin symmetry in nuclei. Both operators have only isoscalar and isovector parts which yields the further (approximate) selection rules Only transition with T = 0, ±1 are allowed In the mirror nuclei, transitions with T = ±1 are identical in all respects. 4 The function ΦLM arises from the partial wave expansion of the plane wave 59

63 De-excitation processes The excited nuclear state may decay to an energetically lower state (in the same nucleus) via competing, mutually exclusive processes where a real or virtual photon is involved: In the first process, a γ ray photon is emitted from the nucleus and is the messenger from the decay process. The two other processes involve a virtual photon mediating the EM interaction. In the internal electron conversion (IEC), the excess energy E i E f is given to one of the atomic electrons (binding energy B n), which is emitted with kinetic energy of E e,n = E i E f B n ( E R ). The recoil energy E R of the nucleus can be usually neglected. If the decay energy is larger than 2m ec 2, an electron-positron pair (IPF, internal pair formation) may be created with a total kinetic energy of E pair = E i E f 2m ec 2. 60

64 In all three de-excitation processes, the transition probability depends on the proton number Z, on the transition energy E γ, and on the transition multipole. The probability of IEC further depends on the atomic (sub)shell. Both the internal electron conversion and the internal pair formation (IPF) can be modelled very accurately in a rather (nuclear-)model-independent way to yield theoretical electron conversion coefficients, ICC (or pair conversion coefficients), that are necessary in the analysis of the γ decay data. In some cases, the emission of a real photon is forbidden (in first order) but IEC and IPF are possible. That s the case of electric monopole transitions (E0). The total conversion coefficient (of a transition) is defined as the ratio of total number of electrons to total number of photons: α total = Ne = 1 α n(νl). (135) N γ N γ n,νl 61

65 Calculation of internal conversion coefficients (ICC) Multipole expansion of the electrostatic potential outside of the nuclear volume where n = r/r. H C = LM Ze 1 4πɛ0 2L r L+1 Y LM (n)m(el, M) (136) Monopole term (L = M = 0) normal Coulomb field binding the electrons Higher multipoles, L > 0, induce syncronized transitions in between nuclear states and the atomic shells Estimate the transition probability of internal conversion with the K electron K > emitted into continuum c > for the EL transition: dw IC fi = 2 2π H C 2 fi δ(ee E 0 e E fi )dρ ede (137) = 16π2 ɛ 0 ( ) Ze 2 2L + 1 c 1 r L+1 Y LM 2 K M(EL, M) fi dρ e M 62

66 For a nonrelativistic electron, the level density in the continuum is given by dρ e = Vm e qdω/(2π ) 3 The electron wavefunction in the 1s orbital (K shell) is ψ K = 1 e r/a 0 πa0 3 The continuum state can be expanded in terms of spherical harmonics ψ c = 1 V e iq r = 4π V l l=0 m= l i l j l (qr)y lm (ˆq)Y lm(ˆr) (138) to yield for the electron matrix element c 1 r L+1 Y LM π K = ( i) L 4 Va0 3 YLM (q) dr 0 r L 1 j L(qr)e r/a 0 }{{} q L 2 0 dx x L 1 j L(x)= ql 2 (2L 1)!! (139) after the angular integration over directions of r. 63

67 Substituting the expressions for the various contributions and integrating over the directions of the emitted conversion electron s momentum ( dω qylm (q)y LM (q) = δ MM ) we finally get the total probability of the internal conversion for a given multipole transition w IC fi = 128π mec2 3 a 3 0 q 2L 3 [(2L + 1)!!] 2 M(EL, M) 2 (140) M f M When the internal conversion coefficient (ICC) is defined as the ratio ( ) α ICC w IC K (EL) = fi w γ L L+5/2 L + 1 Z 3 α 4 2m ec 2, (141) ω fi because the nuclear part of the transition matrix element is the same in both cases (see Eq. (131)). Here α = e 2 /(4πɛ 0 c) 1/ is the fine-structure constant The conversion coefficients for an atomic shell n can be estimated from ( ) L+5/2 L Z 3 2m ec 2 α(el) L + 1 n 3 α4 ω ( ) α(ml) Z L+3/2 3 2m ec 2 n 3 α4 ω The measurement of the conversion coefficients (total for each shell or subshell ratios) together with the selections rules give detailed information on the spins and parities of the nuclear states involved in the transitions. 64

68 Figure 26: Gamma-ray (bottom) and conversion electron (top) spectra following the 108m Ag decay 65

69 Internal Pair Formation If the transition energy E fi = E i E f exceeds 2m ec 2 an electron-positron pair may be formed with the total kinetic energy T e e+ = E fi 2m ec 2. The pair-conversion coefficient β is defined similarly to the ICC. 66

70 E0 Transitions In the lowest-order radiative decay process, a single γ-ray photon is emitted, carrying an orbital angular momentum L 1. This means that transitions where J π i = J π f = 0 + can only proceed by IEC or IPF. Transitions with L = 0 are called E0- or monopole transitions, and give very specific information on nuclear charge distribution and deformation changes, related to the breathing mode of the nucleus. They may also compete with the E2 and M1 transitions in cases where J 0 The transition matrix element i f can be written as M fi exp( i k e r e)ψf ( r Z 2 e 2 i ) 4πɛ i 0 r e r i a 3/2 0 exp( r e/a 0 )ψ i ( r i )d r i d r e (142) where the sum is over all protons of the nucleus, p e = k e is the momentum of the outgoing electron and ψ f ( r i ) and ψ i ( r i ) denote the nuclear final and initial wave functions. The Bohr radius for the 1s electron is a 0 = 2 4πɛ 0 /(me 2 Z ). 67

71 The transition probability goes down with increasing the distance r e r i, implying that the K-conversion is the most important (if energetically possible). In the usual way, we can use the expansion (with r e r i = r er i cos θ) 1 1 ( ) ri l r e r i i = r e i L PL (cos θ) r r e i < r e L 1 r i i ( re ri ) l PL (cos θ) r i > r e (143) The monopole part L = 0 becomes naturally 1/r i and the matrix element to be evaluated: M fi R ( ri ) ψf ψ i ϕ 1 f ϕ i d 3 r e d 3 r i (144) i 0 0 r i where R is the nuclear radius, ψ i,f the nuclear initial and final state wavefunctions. The electron is initially assumed to be in the 1s state, and the outgoing electron is a pure s wave (why?) To a very good approximation, ϕ f (re) = ϕ f (0) and ϕ i (r e) = ϕ i (0), implying that M fi i ψ f r 2 i ψ i d 3 r i (145) a measure of the nuclear radius when ψ f ψ i. 68

72 Figure 27: Values of 10 3 ρ 2 f M(E0) i (E0) where ρ(e0) = er 2 69

73 The usefulness of EM transitions The main reasons for the usefulness of the electromagnetic interaction can be seen from Eq. (131): Measurement of mean lifetimes τ i = 1/λ i (or equivalently, the natural level widths Γ i = /τ i ) of excited nuclear states, together with the measurement of branching ratios for various possible final states, gives information that is directly related to the matrix elements connecting the initial and final nuclear states. The latter matrix elements are quantities that are theoretically modelled in the framework of the various nuclear models. 70

74 Interlude on nuclear lifetimes Mean lifetime (or level width) Transition probability Excitation energy E ex and lifetime τ are the most important parameters of nuclear levels Energies Hamiltonian of the system Lifetimes dynamics Experimental lifetimes provide stringent tests of nuclear models. Direct methods τ Indirect methods Γ Consider one powerful direct method in more detail 71

75 Nuclear lifetimes from Doppler methods The basic non-relativistic formula for the observed energy of the γ ray: ( E γ(θ(t), v(t)) = Eγ v(t) ) cos θ(t), (146) c where Eγ 0 is the energy of the γ ray in the coordinate system of the nucleus (v = 0) and θ(t) is the angle between the direction of observation and the momentum vector of the recoil nucleus. The Doppler effect can be seen in the observed energy of a gamma ray in several ways: 1) broadening of the peak, 2) shifting of peaks, and 3) both. The initial recoil velocity v(t = 0) of a nuclear reaction product the nucleus of which level lifetimes are studied is determined by the bombarding energy and the reaction Q value, Nuclei that recoil freely into vacuum from an infinitely thin target (no slowing down), v(t) = v(0) and all observed γ rays show the full Doppler shift, if the decay is fast enough: The excited levels decay at the rate e t/τ, depending on the lifetime of the level. 72

76 There are two varieties of the Doppler method: 1. In the recoil-distance method (RDM), the recoils are stopped in a movable plunger (piston). The γ rays observed after stopping show no doppler effect. Measuring the intensity ratio I stop /I flight, the mean lifetime of the nuclear level can be determined, if the recoil velocity and the recoil distance are known. 2. In the Doppler-shift-attenuation method (DSA), the excited nuclei produced in the reaction are recoiling in a suitable slowing-down material (usually solid), and the velocity of a γ emitting nucleus diminishes continuously. The energy of the observed γ ray can thus vary from Eγ 0 to the full Doppler shift, depending on the mean lifetime, τ, the stopping power of the target material, S(v(t)), and the initial recoil velocity. Provided that the stopping power and the slowing-down process known accurately enough, the lifetime can thus be determined: 73

77 Doppler-broadened line shapes The figures show how the shape of an observed gamma-ray line is affected by the 1) Level lifetime, 2) stopping power, and 3) recoil given by the projectile. 74