Milestones in the history of beta decay

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1 Milestones in the history of beta decay Figure : Continuous spectrum of electrons from the β decay of RadiumE ( 210 Bi), as measured by Ellis and Wooster (1927).

2 Figure : 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.

3 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 M P c 2 = M P c2 + Zm 0 c 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: Z Z+1 M D c 2 = M D c2 + (Z + 1)m 0 c 2 Thus β decay is energetically possible if M P > M D. i B i i B i

4 Q value of β + decay, Q β + A Z X N Z 1 A 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, 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 of 2m 0 c kev appears! Note: 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)

5 Secondary (atomic 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.

6 Energy relations in beta decay Figure : Energy relations in β-decay processes (Heyde 2004) Table : 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

7 Dynamics 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

8 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 (1) 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

9 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 strenth g Rewrite transition rate as g δ( r p r n ) δ( r n r e ) δ( r n r ν ) λ = 2π M fi 2 dn de (e, ν e ) (2) and evaluate both M fi 2 and density of states dn/de using a quantization volume V

10 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, (3) which results in the density of states dn de = d [ ] V de (2π ) 3 dωdp p 2 V dp = 4πp2 (2π ) 3 de (4)

11 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. (5) 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

12 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 dp e dω e p 2 νdp ν dω ν. (6) V 2 = 4π 4 6 c 3 p2 e (E E e ) 2 1 m2 ν e c 4 (E E e ) 2 Here we explicitly allow for a possibly non-zero neutrino mass 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 ]

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

14 Eq. (7) 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. (7) is made using Fermi-Kurie plot: Plot the values of (but wait: put the neutrino mass 0 first!) Λ(p e ) pe 2 F (Z, p e ) vs. (E E 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

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

17 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!) λ β = pmax 0 M fi 2 2π 3 7 c 3 F (Z D, p e ) p 2 e (E E e ) 2 dp e, (8) 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 = g 2 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. (9)

18 The Fermi integral contains three-body phase-space information: f (Z, w 0 ) = Z=0 w F ( Z, w 2 1) w 2 1 (w 0 w) 2 wdw ( 2w 4 0 9w ) u w 0 ln(w 0 + u 0 ) w w 5 0 (10) The final result 2π 3 7 c ft = ln(2) g M fi 2 m0 5 (11) c4 applied to superallowed Fermi transitions (M fi = 2) gives the coupling constant of β decay as g = g V = MeV fm 3

19 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)

20 After more than 115 years, interest in β decay continues: Nobel Price in Physics 2015: neutrino oscillations Neutrino mass? 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)

22 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 2π 2 3 dp ν de = = V E 2 ν 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 Ψ e(0) = 1 ( Zm0 e 2 ) 3/2 π 4πɛ 0 2 Transition probability for EC from the K shell: λ EC = E 2 ν π 2 c 3 3 g 2 M fi 2 ( Zm0 e 2 ) 3 (12) 4πɛ 0 2

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

24 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

25 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( r e )ψν( r ν ) where the operator H int ψ P ( r p1,..., r p,z ; r n1,..., r n,n )d r e d r ν d r i (13) H int = gδ( r n r p )δ( r n r e )δ( r n r ν )T, (14) and T ( is the ) isospin lowering operator changing ( a proton ) 1 0 p = into a neutron (β 0 decay) n =, or vise 1 versa (β + decay). Here Z = 1/2 + T z, and

26 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) pe i( k e+ k ν) r T φ P ( r) n d r. (15) 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) n d r + i( k e + k ν ) φ D ( r) p r T φ P ( r) n d r +... ). (16) First term 0 only if parent and daughter single-particle states have same parity, second term 0 if parities are opposite etc.

27 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. (17) 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

28 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 D (ˆr)Y M L (ˆr)Y M P L P (ˆr) dω (18) 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, (19) 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. (20) The spherical Bessel functions,j L (kr) (kr)l (2L+1)!! for kr L,

29 through their presence in 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).

30 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 (21) where integral over nuclear coordinates and sum over spin orientations are implied.

31 The selection rules then become (J P J D J β ), (J β L β S β ), π P = ( 1) L β π D (22) 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 = Lβ,Sβ M fi 2. (23) 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.

32 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 + g GT 2 gf 2 M fi (GT) 2. (24) 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.

33 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 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 Double beta decay processes 1) ννββ, double-neutrino double-beta experimentally well established and 2) 00ββ, neutrino-less double-beta, under very active research Helicity of neutrino, well defined for massless only. The helicity operator ˆ p ĥ = ˆσ p has eigenvalues of ±1, depending whether the intrinsic spin points in the same or opposite direction Neutrino oscillations (Nobel prize 2015)

37 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)

Chapter VI: Beta decay

Chapter VI: Beta decay 1 Summary 1. General principles 2. Energy release in decay 3. Fermi theory of decay 4. Selections rules 5. Electron capture decay 6. Other decays 2 General principles (1) The decay