EVIDENCE FOR K MIXING IN Hf-178
C. B. Collins
The University of Texas at Dallas, Center for Quantum Electronics P. O. Box 830688, Richardson, Texas 75083-0688
J. J. Carroll
Department of Physics and Astronomy, Youngstown State University Youngstown, Ohio 44555
Yu. Ts. Oganessian and S. A. Karamian
Joint Institute for Nuclear Research, Flerov Laboratory of Nuclear Reactions Dubna, P.O.Box 79, 101000 Moscow, Russia
Systematics for the appearance of K-mixing levels for the pumping or spontaneous decay of multi-quasiparticle isomers in Hf isotopes are detailed in this letter. The possible location of such a level in the nuclide Hf-178 is discussed and an experiment is proposed to investigate its existence.
Multi-quasiparticle states in the isotopes of Hf appear as high-spin isomers that are distinguished by the combination of MeV excitation energies and long lifetimes . As is typical for mid-shell isotopes, Hf nuclei have marked prolate deformations that provide a body axis upon which the total angular momentum, J can be quantized in units of K. Thus multi- quasiparticle states serve as the heads of rotational bands corresponding to large values of projection quantum number, K.
The Hf isomers also have angular momenta which are quite different from those of other intrinsic nuclear states. Since strong electromagnetic transitions are generally characterized by low orders of multipolarity, L and selection rules require J < L, the apparent necessity for large changes in angular momentum would seem to be the cause of the long isomeric lifetimes. However, large J transitions need not be required. The excitation energies of multi- quasiparticle states are so great that levels with similar angular momenta can be found, built upon the low-spin ground state from as many as 6 - 8 quanta of rotation, while still remaining below the energy of the isomers. Thus the selection rule upon J alone cannot be the cause of the isomerism.
The long lifetimes of multi-quasiparticle states of Hf result from the analogous selection rule upon K, generally accepted to be K < L for transitions between states in different rotational bands. This is the restriction which severely hinders electromagnetic decay of the isomers to yrast levels of similar J, but dissimilar K. While providing storage of great energy densities, the selection rule on K would also seem to vitiate otherwise attractive proposals to use concentrations of high-spin isomers for superelastic particle beam applications, (gamma, gamma') frequency upconversion, and the nuclear analog to the ruby laser. It is very difficult a priori to conceive of a trigger process or reaction which would transfer so much K as would be needed to deexcite populations of high-spin isomers to freely-radiating states in an efficient, controlled manner.
The key to transitions between high-spin isomers of Hf and yrast states may lie in the existence of mediating levels having mixed values of K. Those "K-mixing" levels would be described by superpositions of eigenfunctions for several different projection quanta corresponding to comparable values of J and could be reached by transitions of low multipolarity both from isomers and from members of bands built upon much lower values of K. A discussion of what type of perturbation could provide the interaction energy necessary for such a fortunate K-mixing is left for later.
The first evidence for the deexcitation, or "dumping," of a multi-quasiparticle isomer through a K-mixing level seems to have been reported in 1987 and published  in 1988. Populations of the 10^15 year, two-quasiparticle isomer Ta-180m were dumped to the ground state in the reaction Ta-180m(gamma, gamma')Ta-180, despite a total change of K = 8. Identified  in 1990 as proceeding directly through a previously unobserved state at 2.8 MeV, the reaction was found to be excited with a surprisingly large integrated cross section of 1.2 x 10^-25 cm^2 keV. Independently, the same year the deexcitation of the 3.7-microsec., four-quasiparticle isomer Hf-174m was reported  to occur as a result of spontaneous emission through a K-mixing level lying below the metastable state at 2.685 MeV and providing K = 14. The similarity of the excitation energies of the mediating levels in Ta-180 and Hf-174, on the order of a pairing interaction, suggested an examination of the systematics of other (gamma, gamma') reactions that spanned large K. While those reactions might have proceeded through complex cascades, they could have also benefitted from more direct transitions through K-mixing states.
Figure 1 summarizes the results reported earlier for a systematic investigation [5,6] of integrated cross sections and excitation energies measured for (gamma, gamma') processes mediating large changes of K. The only known reactions starting on multi-quasiparticle levels seem to be those of Hf-174m and Ta-180m, so the others shown are for the excitation of one- or two-quasiparticle isomers from ground-state targets. Nevertheless, the trend is compelling and suggests the pervasive existence of a K-mixing level between 2.5 - 2.8 MeV in many mid-shell nuclides below p = 82. Moreover, the values of integrated cross section for (gamma, gamma') reactions occurring through such levels seem to peak in this region, notwithstanding the large K between initial and final states. Such magnitudes would suggest strong transitions of low multipolarity, requiring the change in K to occur in the mediating state.
The persuasive nature of the data plotted in Fig. 1 encourages the tentative conclusion that a similar K-mixing level can be expected in the Hf-178 system. This would be particularly important for the 31-year, four-quasiparticle isomer Hf-178m2 since such an intermediate state could connect it to yrast states, providing the means for realizing an induced release of the stored energies. Unfortunately, few studies have been conducted for reactions involving Hf-178m2 due to great difficulties in obtaining appropriate targets. The present objective is to attempt to estimate the parameters that would describe such a reaction, Hf-178m2(gamma, gamma')Hf-178 from the data which has recently become available .
Figure 2 shows an expanded view of the data of Fig. 1 for the mass island immediately below p = 82. Measured excitation energies of apparent K-mixing levels mediating those reactions are plotted as X and + symbols by the right-hand ordinate and fall between 2.5 - 2.8 MeV, defining an interval Delta(EK) within which such levels may be reasonably expected. The green circles give the energies of the four- and five-quasiparticle isomers Hf-174m, Hf-175m, Hf-176m3, Hf-177m2 and Hf-178m2, which trend to lower values with increasing mass numbers. The important detail in Fig. 2 is the position of those isomers relative to the possible excitation energies of K-mixing levels. At present no direct observations of (gamma, gamma') reactions are available for those isotopes and thus the discussion is based on Delta(EK) which bounds the likely energies of mediating states.
For Hf-174m and Hf-175m the isomers lie well above the upper bound of Delta(EK) and it can be reasonably expected from Fig. 2 that those multi-quasiparticle states would be relatively short- lived due to the availability of K-mixing states which could mediate spontaneous decay. As mentioned before, this is indeed the case for Hf-174m with a halflife of 3.7 microsec. Similarly, Hf-175m has a 1.21-microsec. halflife. In the case of Hf-176m3, the isomer lies at 2.866 MeV, slightly above the upper bound of Delta(EK). Thus, it could be expected to have a short lifetime somewhat lengthened by a small transition energy to a K-mixing level. This is supported by a measured halflife of 401 microsec. The isomer Hf-177m2 lies just below the upper bound of Delta(EK), possibly at a lower energy than any mediating state. In that event a long lifetime would be suggested since spontaneous decay could not occur, but this could also be the result if a K-mixing level were located little below the isomer. The measured halflife of 51 min agrees well with either possibility. Of most importance to the present discussion, the isomer Hf-178m2 appears to lie below the likely position of a mediating state. This points to a long lifetime and indeed the measured value is 31 years. Clearly the known halflives of these multi-quasiparticle Hf isomers agree well with the speculation that K-mixing levels are prevalent and lie between 2.5 - 2.8 MeV.
The strong dependence of isomer lifetime on the possible excitation energy of a mediating state is shown directly in Fig. 3. Again, Delta(EK) indicates the interval within which all measured values for possible K-mixing levels lie. This interval can be further constrained by comparing the data to lifetimes derived from single-particle widths. Since low-multipolarities would be indicated for transitions between the isomers and states of mixed K, dipole radiation is considered here. Derived using the typical 1/(Ei -Delta(EK))^3 dependence for dipole transition widths where Ei is the isomer energy and EK is the energy of the mediating level, the curves in Fig. 3 show halflives in good agreement with the measured values. It is doubtful that a K-mixing state could lie much below the isomer Hf-177m2 without affecting its 51 min lifetime. Thus, the value EK = 2.7 MeV used to generate the curve passing just below that isomer represents a lower bound on the likely energy of an intermediate state of mixed K. Likewise, EK = 2.8 MeV provides an upper bound for that energy since larger values produce curves which do not reproduce the lifetimes of the isomers Hf-174m, Hf-175m and Hf-176m3. The excellent agreement seen between halflives predicted for dipole transitions and those measured provides further evidence for a K-mixing level in Hf-178m2 lying no more than about 300 keV above the isomer.
The presence of such a K-mixing level could in principle be identified in a (gamma, gamma') reaction study similar to those conducted previously [2,3,6], but might be most easily detected in the proposed experiment shown schematically in Fig. 4a. A beam of protons of energy E0 would be incident on Hf-178m2 nuclei in a sample located within a 4-pi granular detector array, or "gamma-ball." This arrangement would provide for inelastic scattering of protons in (p,p'gamma) reactions where the gamma refers to photons emitted in the decay of the excited nucleus. Many scattering events would merely excite low-lying members of the ground-state or isomeric rotational bands with a low multiplicity, n of accompanying gamma-rays. However, the excitation of a K-mixing state from the isomer would result in a transfer of the isomer to the yrast band with an accompanying large multiplicity of detected photons (n > 4), since the gamma-cascade multiplicity, M(gamma) is expected to be more than 6 - 7. Due to the detection efficiency, n < M(gamma) . Thus the important aspect in the observation of a K-mixing level would be the measurement of the spectrum of inelastically scattered protons in coincidence with a high multiplicity of photons detected by the ball. By selecting only events satisfying this criterion it would be possible to confirm the existence of the K-mixing level in an excitation function like that depicted in Fig. 4b. The location of the mediating level would be clearly marked by a threshold at an energy Et = E0 - EK.
This letter makes no attempt to estimate the probability for the actual existence of a K- mixing state. However, systematics strongly suggest that such a level may exist in Hf-178m2, and at a sufficiently low excitation energy to be of great importance to proposals for the use of this isomer in superelastic particle-beam studies, gamma-ray upconversion and the nuclear analog to the ruby laser. At this point further study is needed for a definitive resolution.