Investigation of the (cid:2) n = 0 selection rule in Gamow-Teller transitions: The β -decay of 207 Hg

decay is forbidden if the number of nodes in the radial wave functions of the initial and ﬁnal states is different. This (cid:2) n requirement plays a major role in the β decay of heavy neutron- rich nuclei, affecting the nucleosynthesis through the increased half-lives of nuclei on the astrophysical r -process pathway below both Z = 50 (for N > 82) and Z = 82 (for N > 126). The level of forbiddenness of the (cid:2) n / 7 / 2 transition has been investigated from the β − decay of the ground state of 207 Hg into the single-proton-hole nucleus 207 Tl in an experiment at the ISOLDE Decay Station. From statistical observational limits on possible γ -ray transitions depopulating the π 0g − 1 7 / 2 state in 207 Tl, an

upper limit of 3.9 × 10 −3 % was obtained for the probability of this decay, corresponding to log ft > 8.8 within a 95% confidence limit. This is the most stringent test of the n = 0 selection rule to date. The theory of β decay was developed by Fermi in 1934 [1]. In its modern form it is derived from the standard model of the electroweak interaction [2,3]. Based on the orbital angular momentum L carried by the β particle and neutrino, the β decay is classified as allowed (L = 0), first-forbidden (L = 1) and so on. The so-called forbidden transitions are hindered, but not completely suppressed.
The selection rules for allowed β decay are total angular momentum change I = 0, ±1 and no parity change between the initial (decaying) and final (populated) states. In contrast, first-forbidden transitions have I = 0, ±1, ±2 and a change in parity. These are the selection rules which can be found in textbooks. However, in the case of allowed β transitions, there is one additional rule: the number of nodes, n, in the radial wave functions of the decaying and populated states must be equal.
The n = 0 selection rule plays a major role in heavy neutronrich nuclei. The single-particle shell-model orbitals for the N > 126, Z < 82 region ('south-east' of 208 Pb) are shown in Fig. 1. The several pairs of n = 1, l = 0o r b i t a l s are indicated. If the n = 0 selection rule is strictly obeyed, β decay between them is forbidden, resulting in longer lifetimes. The greatest impact is on nuclei where the Fermi level lies high above N = 126 and/or much below Z = 82, e.g. nuclei on the astrophysical r-process pathway, influencing the nucleosynthesis of heavy elements. On the other hand, this selection rule has little effect on isotopes which are protonrich or close to the stability line, making the experimental investigation of its validity difficult. We explored the ν1g 9/2 → π0g 7/2 transition in the β decay of 207 Hg in a high-statistics experiment performed at CERN-ISOLDE. In this letter we provide the most stringent test of the n = 0 selection rule to date.
The β-decay probabilities are characterised by log ft values, where f is the Fermi function and t is the partial half-life. For allowed (L = 0) decay ft can be written as [5]: where B F and B GT are the Fermi (total spin of the electron and antineutrino is S = 0) and the Gamow-Teller (S = 1) reduced transition probabilities, and are proportional to the |M F fi | 2 and |M GT fi | 2 matrix elements respectively. These in turn are summed over the single-particle matrix elements. These single-particle matrix elements are equal to zero if the number of radial nodes in the wave function, n, changes between the initial and final states. The n = 0 selection rule applies for both Fermi and Gamow-Teller transitions. For details see ref. [6].
Far from the N = Z line, Fermi decays are isospin-forbidden ( T = 0), so considering Gamow-Teller transitions only one obtains [5][6][7]: ft= 6147 where τ ± is the isospin step operator converting either a proton to a neutron or a neutron to a proton respectively, σ is the Pauli spin operator, and i, f are the initial and final total nuclear wave functions. From the form of the operators the selection rules of I = 0, ±1 with no parity change follow. The matrix element is related to the overlap L of the initial and final state wave functions: For n = 0, if the Hamiltonians are identical (i.e. spin-orbit and Coulomb effects are ignored) then the radial wave functions are orthogonal, and L = 0. The assessment of the n = 0 selection rule requires a good understanding of both initial and final states. This is achieved in the vicinity of the doubly magic nuclei, when the wave functions can be given in terms of a few single-particle components.
According to our knowledge, the n = 0 selection rule was verified for a single case: that of the β decay of 209 Tl into 209 Pb [8]. The present case of the 207 Hg → 207 Tl decay provides a more stringent test, due to larger spin-coupling coefficients and the stronger contributions of the relevant single-particle orbitals to the overall wave functions.
The 207 Hg ground state has an expected spin-parity of 9/2 + , corresponding to a neutron in the ν1g 9/2 orbital above the N = 126 magic number. The daughter nucleus 207 Tl is one proton hole away from doubly magic 208 Pb. It exhibits clear single-particle behaviour with the 7/2 + π0g −1 7/2 hole state at an energy of 3474(6) keV [9]. This state has been populated in a number of particle transfer experiments [10][11][12][13][14]. In the present letter we examine the ν1g 9/2 → π0g 7/2 β decay, allowed in terms of spin-parities but forbidden by the n = 0 selection rule.
The β decay of 207 Hg into 207 Tl has been studied at the ISOLDE Decay Station (IDS). A molten lead target was bombarded with 1.4 GeV protons. 207 Hg ions were extracted at 30 kV from the VD5 FEBIAD source [15], separated by the General Purpose Separator (GPS) and collected on the tape at IDS. The average implantation yield was 4.8(2) × 10 4 pps. The dominant beam contaminant was 206 Hg, with a presence of around 0.6 times that of 207 Hg. However, with a Q β = 1.31(2) MeV and few γ transitions [16]i t does not impact the current results. β decay and γ rays were detected by three plastic scintillators in a close configuration and an array of high-purity germanium (HPGe) detectors, respectively. The HPGe array consisted of four Canberra Clover detectors, arranged at equal angular separation at a backward angle, and a Miniball [17]c l u s t e r detector along the beam axis. The full detector set-up had a gamma-ray add-back efficiency of 22% at 100 keV and 4% at 2.6 MeV, and a β-particle efficiency of ∼30%. Extension of the efficiency calibration up to 2.6 MeV was performed by using the relative intensities of the 583 keV and 2614 keV peaks from the β decay of 208 Tl [18], measured on a separate GPS mass setting during the same experiment. The analysis of this experiment revealed the level scheme of 207 Tl up to an energy of 3.94 MeV [19]. It is in good agreement with the previously established scheme [20], and contains a number of newly-observed states and transitions. The level scheme is now complete and balanced. A partial level scheme, showing levels and transitions used in this analysis, is shown in Fig. 2.
No transitions into or out of the π0g −1 7/2 proton-hole state at 3474(6) keV were observed. As the decay of this state was never observed [9], we used theoretical considerations to evaluate which states it would populate (see Fig. 2). The M3 and M2 transitions to the 1/2 + ground state and 11/2 − isomeric state, respectively, would be extremely weak and so are ignored. The high-energy 3123(6) keV E2 transition to the 3/2 + first excited state is expected to be dominant. This is a g 7/2 → d 3/2 transition connecting j = l = 2s t a t e s . The 1791(6) keV M1+E2 transition to the 5/2 + state may be of comparable strength, although the M1 decay in this g 7/2 → d 5/2 transition is l-forbidden. The next two excited states at 2676 keV and 2709 keV have spin-parity (7/2 − ) and (5/2 − ), respectively [20]. Many E1 transitions, some connecting equivalent single-particle and octupole states, have been observed in nearby nuclei [16,[21][22][23][24]. Measured B(E1) transition strengths are in the range of 10 −3 -10 −5 W.u. Therefore we adopt an up-  Table 1. The latter are based on shell-model values of B(M1) and B(E2) transition strengths. These calculations assume pure (0g 7/2 , 1d, 2s) proton hole states and standard effective E2 proton charge of 1.5e [25][26][27]. In order to obtain the tensor term into the effective operator [28,29]w a s introduced. Adopting the 0g 7/2 -1d 5/2 vs. 1d 3/2 -2s 1/2 scaling calculated for a 132 Sn core in the same model space [29], and considering the radial overlaps of the involved orbitals, the B(M1) value of 0.0041 W.u. was obtained.
The most stringent non-observation limits were obtained from βγγ coincidence data. Where a transition is not observed, an upper limit on the intensity can be deduced from the uncertainty in the relevant background area, I γ < N √ 2 A. N is the number of standard deviations used, and A is the background area below the expected peak. The γγ coincidences used are indicated in Fig. 2, and the resulting coincidence spectra for the four different gating transitions are shown in Fig. 3. The extracted intensity limits, for N = 2 (95% confidence limit), are given in Table 1. The limits cover all energies within 2σ (12 keV) of the central energy. As shown in Table 1, the highest-energy transition is expected to dominate. Assuming a lower limit of 90% for the branching ratio of the 3123 keV transition, we would obtain a maximum possible population of the 3474 keV 7/2 + state of 1.2 × 10 −3 % leading to the final result log ft> 9.3. However, we can reduce our reliance on the shell model calculations by considering any branching ratios between all four transitions. In this case, the maximum possible population of the 3474 keV 7/2 + state is 3.9 × 10 −3 % leading to the final result log ft> 8.8.
The initial and final wave functions are: where φ C is the 208 Pb core and j α j (π j −2 ) 0 + is the wave function of two proton holes outside the core. The resulting reduced Gamow-Teller matrix element is given by the equation where the notation ˆj = √ 2 j + 1i s used. The Racah coefficient W is equal to 0.19245 for this decay. α 2 7/2 is the probability that there are two holes in the π0g 7/2 orbital in the ground state of 207 Hg. The shell model calculation in the πh-ν p space yields α 7/2 = 0.095. L = ν1g 9/2 |π 0g 7/2 is the overlap integral between the specified single particle states.
From this information we obtain a limit |L| < 0.019. This is a stricter limit than that obtained from the β decay of 209 Tl for the overlap of the ν3s 1/2 and π2s 1/2 orbitals [8]. The reasons are the lower experimental log ft> 8.35 limit in the case of 209 Tl, and the lower ν3s −2 1/2 contribution to the 209 Tl ground state (α 1/2 = 0.05 using the same shell model calculation).
The overlap integral L is very sensitive to the difference between the neutron and proton well radii. By calculating the value of the integral using a Woods-Saxon potential, using known proton and neutron separation energies and fixing the surface thickness and neutron radius parameter to a = 0.65 fm and r ν = 1.25 × A 1/3 fm respectively, the result suggests that the 0g 7/2 proton well radius parameter is between 0% and 1.5% larger than the 1g 9/2 neutron orbital parameter. Knowledge of these parameters is important in models of transfer reactions and neutron skin calculations.
In conclusion, the theoretically predicted population of the yrast 7/2 + state is around twice the experimentally determined upper limit for the excited state at 3474 (6) keV. An uncertainty of 50% in the population predicted by the shell model would reconcile the observed and predicted values.
While no peak was observed in the 351 keV γγ coincidence spectrum at the expected energy of 3123 keV, a small peak at 3142 keV (shown in Fig. 3) implies the existence of a state with an energy of 3494 keV. This lies three standard deviations away from 3474(6) keV, and with no other supporting transitions a spin-parity assignment is not possible. Nevertheless, in the case that the state at 3494 keV is the 7/2 + 1 state previously placed at 3474(6) keV, this would imply an observed β-feeding intensity of 8(1) × 10 −3 % and log ft = 8.53 (9). This is exactly the value predicted by the shell model for the population of the yrast π 0g 7/2 -dominated 7/2 + state. It would still imply that the n = 0 selection rule is observed, as the observed population of this state from the β decay of 207 Hg would be fully accounted for by the 2p − 3h mixing.
In the absence of the n = 0 selection rule, the lifetimes of the 207 Hg and 209 Tl nuclei would be shorter. Assuming log ft equal to the modal average of the empirical distribution for allowed transitions [39], the Gamow-Teller transitions would take approximately 5% and 20% of the decay intensity into 207 Tl and 209 Pb respectively, shortening the half-life of 207 Hg by around 7% and that of 209 Tl by around 20%. The effects are much larger for extremely neutron-rich heavy nuclei. Close to 208 Pb the contributions of the relevant wave functions in the mother/daughter nuclei are very low, with amplitudes α 1/2 = 0.05 for 209 Tl and α 7/2 = 0.095 for 207 Hg. These increase dramatically for extremely neutron-rich nuclei, when either exploring deeper into the proton shell below Z = 82 or extending further into N > 126 (see the ordering of orbitals on Fig. 1). For example, the r-process path is predicted to cross the Z = 82 line at around neutron number N∼160-170 [40]. In this region, just below Z = 82, the n = 0 selection rule will have an impact on the lifetimes by forbidding the ν3s 1/2 → π2s 1/2 and ν2d 3/2 → π1d 3/2 Gamow-Teller transitions. The effect of this selection rule for the decay of N = 126 r-process waiting-point nuclei is negligible because of the minimal contribution from N > 126 orbitals. The same selection rule also affects nuclei 'south-east' of 132 Sn [41]d u e to the existence of n = 1, l = 0 neutron-proton orbital pairs in the region of N > 82 and Z < 50. Experimental investigation of the forbiddenness in this mass region is an interesting possibility but remains challenging due to the large Q β values.
The β decay of 207 Hg was studied and a search was made for γ -rays following the n-forbidden ν1g 9/2 → π0g 7/2 Gamow-Teller β decay. From the non-observation of this decay, a log ft > 8.9 limit at 95% confidence level was deduced. This is the most stringent test of the n = 0r u l e to date, and suggests that the selection rule is indeed obeyed. The rule has implications for the decays of the neutron-rich nuclei in both the N > 82, Z < 50 and N > 126, Z < 82 regions. In the latter region, the lifetimes of nuclei on the astrophysical r-process path are considerably increased, affecting the nucleosynthesis of heavy elements.