A proton pairing isomer in 174Hf: Reinterpretation of data and support for the split prolate–oblate pairing approximation for deformed nuclei

The available data on the even–even isotope 174Hf are reinterpreted to conclude that the low-lying 02+ level, usually regarded as a “β vibration”, is actually a proton (2p–2 h) excitation known as a Pairing Isomer. This is the first identification of a Pairing Isomer based on Proton Nilsson Orbitals. Its identification demonstrates that Pairing Isomers can exist anywhere in the Nuclear Chart if the positioning of the Fermi Surface is favourable with respect to relevant Nilsson orbitals.

In most Nuclear Structure textbooks, and indeed in the current Nuclear Data Tables, the first excited spin zero 0 2 + level, in the pairing gaps of even-even deformed nuclei, has usually been identified as a β vibration along the symmetry axis in the manner of the collective model of Bohr and Mottelson [1,2]. In particular the nucleus 174 72 Hf 102 has been taken to be one of the best examples of such a vibration [2, p. 168, 3]. Quantum mechanically the "vibration" is an excitation in the one-dimensional potential V (β − β 0 ) where β is the usual quadrupole axial deformation parameter. Initially V was taken to be a simple harmonic oscillator (SHO) potential with the 0 2 + level excitation energy E x (0 2 + ) èω β . More complex versions of V have been used and recently 5-Dimensional Collective Hamiltonians (5-DCH) [4] and references therein] have been used to represent the current wealth of data.
Paul Garrett, in a seminal review paper [5], analysed the then current data on 0 2 + levels which were generally accepted to be "β vibrations". He comes to the conclusion that, "While it is clear that there are very few good examples of a β vibration, the question arises as to the nature of the 0 2 + states, indeed of all low-lying 0 + states." a e-mail: johnsharpeyschafer@gmail.com(corresponding author) Recent publications [4,[6][7][8] have presented data on even-even nuclei near the neutron number N 90 demonstrating that the first excited 0 2 + levels are Pairing Isomers [9][10][11][12][13][14][15][16][17] due to pairs of neutrons being excited into the neutron high-K ν[505]11/2 − Nilsson orbital. In this contribution we point out that current data indicate that the low-lying 0 2 + level in 174 Hf is not a "β vibration" but is a Pairing Isomer formed by 2-particle-2-hole (2p-2h) excitations mainly involving the proton Nilsson orbital π[404]7/2 + .
The even-even Hafnium isotope, with proton number Z 72, 174 Hf has its first excited 0 2 + state at the unusually low excitation energy of 828.1 keV. This is a similar situation to the isotones with neutron number N 90 where the nuclei 154 Gd and 156 Dy have their first excited 0 2 + levels at 680.7 and 675.6 keV respectively. Evidence indicating that these levels were not β vibrations came from (p,t) and (t,p) reactions [8,15,18] and blocking arguments [6][7][8]. For example the 152 Gd(t,p) 154 Gd two neutron L 0 stripping reaction [18] strongly populates the 0 2 + level in 154 Gd. The 158 Dy(p,t) 156 Dy two neutron L 0 pick-up reaction [15] strongly populates the 0 2 + level in 156 Dy. These two neutron transfer reactions show that simple monopole pairing will not account for these data [19,20]. An improved solution to monopole pairing was to split the pairing into two parts [10]. The main 'normal' pairing based on summing over prolate low-Nilsson levels and a second separate pairing due to summing over oblate high-K orbitals. In the case of the nuclei near N 90 the density of low-Nilsson neutron orbitals is much higher than that of neutron high-Nilsson orbitals, so that the neutron prolate pairing is stronger than that of the oblate pairing. For the N 90 nuclei the neutron high-Nilsson level nearest the Fermi surface is the orbital ν[505]11/2 − extruded from the lower shell by the increasing deformation. Hence the identification of the 0 2 + levels in 154 Gd and 156 Dy as Pairing Isomers based mainly on the ν[505]11/2 − configuration. It was also found that the dynamic moments-of-inertia I (2) 2 of the rotational bands built on the 0 2 + levels were larger than the I 1 of the ground state bands [4]. This is due to the pairing being weaker for the 0 2 + levels, confirming the conjecture of Mottelson and Valatin [21].
In a venerable publication [22] a rotational band based on the 0 2 + level in 174 Hf was identified up to a spin of 24è. These data were subsequently modified in a conference contribution [23] which are the data reproduced in the Data Tables [24]. Ref. [22] surmised that the band crossing structures seen near spin 20 + were due to a discontinuity in the deformation. They also pointed out that that it could also be due to the aligned i 13/2 2 neutrons of the so-called S-band forming a Cranked Shell Model (CSM) AB crossing with the 0 2 + band. Such a crossing of the "β band" by the aligned S-band had been observed early on at the Argonne National Laboratory in the USA and Chalk River in Canada in both 154 Gd [25] and 156 Dy [26]. Using the data of [23,24] in Fig. 1, a comparison of the excitation energies of the ground state bands E gsb in 174 Hf, 154 Gd and 156 Dy with the energies of the 0 2 + levels and their band extensions is shown, where a rigid rotor reference E r 7.7I(I + 1) keV has been subtracted. The similarity between the experimental data for each isotope is clear.
The data implies that similar physics is involved in 174 Hf to that which is determining the structure of the N 90 isotones. The deuteron elastic scattering on 154 Gd is very weak, in comparison with the (d,d ) excitations of the K 2 + "γ band" and the K 3 − "octupole band", indicating that the assumed "β vibration" is not a simple collective state [27]. In Ref. [28] it was pointed out that a similar situation occurs in the 174 Hf(d,d ) reaction where again the (d,d ) cross-sections to the levels in the "β vibrational band" are very much smaller than those in the γ and octupole bands. We reproduce from [28] the relevant figure here as Fig. 2, which was a private communication from Peter Kleinheinz and Bent Elbeck.
Again the data supports the idea that the 0 2 + level in 174 Hf is not a classical "β vibration". The level cannot be a neutron pairing isomer as there are no high-K Nilsson orbitals available near the Fermi Surface at the N 102 neutron number. There are no 176 Hf(p,t) 174 Hf two neutron pick-up data to check this. In Fig. 3 we show a partial level scheme for 174 Hf showing the unusual decay of the 133 ns 6 iso + isomer at 1549 keV. It decays primarily to the 4 1 + and 6 1 + of the ground state band and also weakly to the 8 1 + . Its two other weak decays are to the 4 2 + and 6 2 + levels of the 0 2 + band [29]. The configuration of this isomer has been identified as overwhelmingly the coupling of the proton orbitals π[404]7/2 + and π[402]5/2 + to spin 6 + [29], see Fig. 4 for the relevant Nilsson orbitals. The π[404]7/2 + orbital is very near the Fermi surface as it is also a primary component of other isomers in 174 Hf [29]. The importance of the π[404]7/2 + Fig. 1 Excitations energies of levels, spin I, in the ground state and 0 2 + bands in 154 Gd, 156 Dy and 174 Hf minus a rigid rotor reference E r 7.7I(I + 1) keV orbital is also emphasised by reactions to neighbouring Hf isotopes. An example is that the 181 Ta(p,α) 178 Hf reaction populates the 0 2 + band at 1199 keV with a strength equal to that of the 0 1 + ground state band [30]. The odd proton in 181 Ta is in the π[404]7/2 + orbital indicating that the 0 2 + level contains a large component of two protons in the π[404]7/2 + Nilsson orbital. This is discussed fully in Ref. [30] and supports the proposition that a candidate for the configuration  It also decays to the 4 2 + and 6 2 + levels of the 0 2 + proton pairing isomer band; transitions shown in red [29] of the 0 2 + level in 174 Hf is a pair of protons primarily in this π[404]7/2 + level to form a proton pairing isomer.
It is clear from Fig. 1 that the dynamic moment-of-inertia I (2) 2 ≈ 51 MeV −1 è 2 in the 0 2 + band in 174 Hf is considerably larger than that of the ground state 0 1 + band I 1 ≈ 39 MeV −1 è 2 . Again this follows the behaviour in the N 90 nuclei indicating that the effective pairing in the 174 Hf 0 2 + band is reduced; the situation expected for Pairing Isomers [17]. The reduction of monopole pairing strength with increasing seniority of well-established isomers has been Fig. 4 Section of the Nilsson diagram for prolate deformed nuclei with proton number Z between 50 and 82 with Z 72 for 174 Hf indicated extensively studied by George Dracoulis and his co-workers at the Australian National University [31,32]. They used the Lipkin-Nogami scheme [33][34][35] to calculate the pairing for well-defined isomers in the region of A 174. They found that pairing was reduced with increasing isomer seniority. For the lowest 6 iso + isomer in 174 Hf (Fig. 3) I (2) 6+ ≈ 57 MeV −1 è 2 similar to I 2 . The value for the 3 + band is I (2) 3+ ≈ 51 MeV −1 è 2 . It would be interesting to discover the effect of using the split pairing scheme in Lipkin-Nogami calculations.
In conclusion; the isotope 174 Hf has its K π 0 2 + level as having strong evidence for being the first example of a Pairing Isomer with a two proton configuration. A way of confirming this is to carry out a two proton transfer experiment in which the 0 2 + level should be strongly populated. Currently there are no high resolution data on the 172 Yb( 3 He,n) 174 Hf reaction. This reaction could be explored using the techniques of Refs. [36][37][38][39]. Clearly Pairing Isomers are not confined to the start of deformation at the beginning of shells but can arise in mid-shell as well if the positioning of the Fermi surface is favourable with respect to relevant Nilsson orbitals. Isotopes near to 174 Hf, and in other parts of the nuclear chart, need to be examined in the light of the present proposition as do higher lying 0 + states. It is also clear that realistic calculations using pairing in their Hamiltonians should do a much better job of separating the pairing contributions, from prolate low-K and oblate high-K Nilsson orbitals, than they usually do.