Description of Hf in the constrained relativistic mean field theory ^{1}^{1}1Supported by the National Natural Science Foundation of China under Grant Nos 10605004 and 10705004, the Natural Science Foundation of He’nan Educational Committee under Grant No 200614003, and the Young Backbone Teacher Support Program of He’nan Polytechnic University
Abstract
The properties of the ground state of Hf and the isomeric state Hf are studied within the adiabatic and diabatic constrained relativistic mean field (RMF) approaches. The RMF calculations reproduce well the binding energy and the deformation for the ground state of Hf. Using the ground state singleparticle eigenvalues obtained in the present calculation, the lowest excitation configuration with is found to be . Its excitation energy calculated by the RMF theory with timeodd fields taken into account is equal to 2.801 MeV, i.e., close to the Hf experimental excitation energy 2.446 MeV. The selfconsistent procedure accounting for the timeodd component of the meson fields is the most important aspect of the present calculation.
pacs:
21.10.k, 21.60.n, 27.70.+qThe relativistic meanfield (RMF) theory is one of the most successful microscopic models in nuclear physics. ^{Serot1986 (); Reinhard1989 (); Ring1996 ()} From the very beginning, it incorporates the important relativistic effects, and it has achieved success in describing many nuclear phenomena related to stable nuclei ^{Reinhard1989 (); Ring1996 ()}, exotic nuclei ^{Meng1996 (); Meng1998 ()} as well as supernova and neutron stars ^{Glendenning2000 ()}. The RMF theory provides a new explanation for the identical bands in superdeformed nuclei^{Konig1993 ()} and for the neutron halo in heavy nuclei ^{Meng1996 ()}, it predicts giant neutron halos, a new phenomenon in heavy nuclei close to the neutron drip line ^{Meng1998 (); Meng2002 ()}, it naturally generates the spinorbit potential, explains the origin of the pseudospin symmetry as a relativistic symmetry ^{Ginocchio1997 (); Meng19982 (); Meng1999 ()}, and spin symmetry in the antinucleon spectrum ^{Zhou2003 ()}, and also describes well the magnetic rotation ^{Madokoro2000 ()}, the collective multipole excitations ^{Ma2002 ()} as well as the properties of hypernuclei^{Mares1994 ()}, etc. Lately, the ground state properties of about 7000 nuclei have been calculated in the RMF+BCS model and good agreements with existing experimental data were obtained ^{Geng2005 ()}. Recent and more complete reviews of the applications of the RMF model, particularly, those to exotic nuclei, can be found in Refs. Vretenar2005 (); Meng2006 () .
Recently the 31yr isomer of Hf (also called Hf, = , =2.446MeV) has attracted extensive attention ^{Hayes2002 (); Cline2005 (); Hayes2007 (); Sun2004 (); Tuya2006 ()} for its potential to be a good medium of energy storage ^{Collins1999 ()}. The long halflife of Hf is connected with the strong inhibition of spontaneous electromagnetic transitions restricted by the selection rule, and it supports the point of view that this high state has an axially symmetric intrinsic shape ^{Sun2004 ()}. The configuration originally suggested for the isomer Hf is ^{Helmer1968 ()} , which was further supported by the view of alignment and by the factor of the corresponding rotation bands. ^{Mullins1997 ()}
In this Letter, we will study the properties of the ground state of Hf and also investigate the possible configuration of Hf within the selfconsistent axially symmetric RMF theory. From the point of views of the adiabatic constrained calculation, nuclear ground state presents the global minimum of the potential energy surface (PES). To study the excited states, like isomers, the diabatic (configurationfixed) constrained approach can be applied as an effective method. ^{Lv2007 ()} Within the adiabatic constrained approach, nucleons always occupy the lowest levels, while in the diabatic constrained approach, the configuration is kept fixed by the socalled concept of “parallel transport”. In the present paper, both the adiabatic and diabatic constrained RMF approaches are applied in our investigation.
The basic ansatz of the RMF theory is a Lagrangian density where nucleons are described as Dirac particles which interact via the exchange of various mesons and the photon. The mesons considered are the isoscalarscalar , the isoscalarvector and the isovectorvector . The effective Lagrangian density reads ^{Serot1986 ()}
(1)  
in which the field tensors for the vector mesons and the photon are, respectively, defined as
(2) 
From the Lagrangian, the equation of motion for the nucleon is
(3) 
with the attractive scalar potential , the usual repulsive vector potential , and the nuclear magnetic potential . The KleinGordon equations for the mesons and electromagnetic fields are
(4) 
where is the source term and all other notations are the same as in Ref. Meng2006 ().
In the RMF approaches which are widely used, only the timeeven fields are essential for the physical observables, since the timeodd components of vector fields do not exist because of the time reversal symmetry for the ground state of an eveneven nucleus. For an odd or oddodd nucleus, the unpaired valence nucleon will give nonvanishing contribution to the nuclear current which provides the timeodd component of vector fields, i.e., the nuclear magnetic potential. It is found that the nuclear magnetic potential has small influence on the rootmeansquare radii and quadrupole moments while it plays an important role in the singleparticle properties and magnetic moments in odd or oddodd nuclei.^{Warrier1994 (); Yao2006 ()} One should keep in mind that for the excited states in the eveneven nuclei, there may also exist the unpaired nucleons, which result in nonvanishing timeodd field. Therefore the timeodd fields should also be treated carefully for some isomeric states in the eveneven nuclei, as in the odd or oddodd nuclei. In the calculation without current, the nuclear magnetic potential will be neglected.
For the adiabatic constrained approach, the binding energy at a certain deformation is obtained by constraining the mass quadruple moment to a given value , i.e.
(5) 
where is the curvature constant parameter, and is the given quadrupole moment. The expectation value of is , where . The deformation parameter is related to by , ( fm) and is the mass number. By varying , the binding energy at different deformations can be obtained ^{Ring1980 ()}.
For the adiabatic constrained approach, the occupied levels are determined by the socalled “parallel transport” ^{Lv2007 ()}, i.e.,
(6) 
where and enumerate all the singleparticle levels of two adjacent configurations. In such a way, the original configuration at can be traced and the corresponding PES can be obtained as a function of the deformation ^{Lv2007 ()}. In principle, if is small enough, the configurations at and at should be the same. In the calculation, the twostep procedure is adopted: first, the wave functions and the configuration at the initial are recorded. Second, the wave functions are mapped to one by one by searching the largest overlap in with the same quantum number . The configuration is transferred by copying the occupation number from to the mapped . The wave functions and the configuration at this are also recorded. The second step is repeated until enough points on the diabatic PES are obtained.
The constrained RMF calculations are carried out with parameter set PK1 ^{Long2004 ()}. The full = 20 deformed harmonicoscillator shells for fermions and bosons are taken into account as the basis. This basis is large enough to produce a converged binding energy at certain deformation.
The PES of Hf obtained in adiabatic (open circles) and diabatic (lines) constrained RMF calculations are plotted in Fig. 1. The calculated energy =1434.0 MeV and the deformation =0.283 of the ground state (denoted as a black asterisk in Fig. 1) are in good agreement with the experimental energy 1432.8 MeV ^{Audi2003 ()}and deformation 0.280^{Raman2001 ()}. In this figure, the adiabatic PES can be decomposed into three regions by the discontinuity, i.e., =0.22 0.32 (region 1), = 0.35 0.40 (region 2), = 0.43 0.50 (region 3). It is known that the discontinuity originates from the change of the configurations, and the configuration for all points in one region is the same. This is confirmed by the diabatic constrained calculation in which the configuration is kept fixed during the constraint procedure. It can be seen in Fig. 1 that for every region the diabatic calculation (solid curves) coincides with the adiabatic calculation (open circles) and extends the region much wider. The crossover of the solid curves announces the change of the configurations. For example, from region 1 to region 2, the configuration changes from the ground state to , where microscopically the proton level below the Fermi surface in region 1 becomes unoccupied while the proton level above the Fermi surface in region 1 becomes occupied. Since a pair of protons change the levels at the same time, the values remain zero for region 2. Similarly, from region 2 to region 3, the configuration changes from to and the values also remain zero for region 3.
Based on the singleparticle spectra of the ground state, one can construct excited states with high values. In a deformed, axially symmetric nucleus, a high state is made by summing the contributions from several unpaired quasiparticles. To form lowlying high states, several high singleparticle (both neutron and proton) levels lying close to the Fermi surface are necessary. The welldeformed nuclei with , including Hf, satisfy this requirement very well. With the restriction of the total and parity as , the candidate configurations of Hf will be constructed.
For the ground state of Hf, the neutron (proton) singleparticle levels close to the Fermi surface are shown in the first column of the left (right) panel in Fig. 2. Each level is labeled by the Nilsson notation of its main component. It can be seen that in Fig. 2, the energy required to excite one neutron or one proton is not less than 0.63 or 1.84 MeV. As the experimental excitation energy of Hf equals 2.446 MeV, it is sufficient to consider one or twoneutron excitations, together with one or twoproton excitations. For twoneutron (twoproton) excitations, the following cases are considered: in the first case, a pair of particles below the Fermi surface are excited to two different levels above the Fermi surface; in the other case, two particles occupying different levels below the Fermi surface are excited to form a new pair. Those cases which involve four or more singleparticle levels are not included for simplicity. All possible configurations are constructed by restricting the (total ) value to 16 and the nuclear parity to . Thus the configurations with the lowest excitation energies are obtained and labeled by ,, , etc.
The detailed configurations of the five lowest states of Hf are listed in column 2 of Table 1. In this table, the first 4 states are oneneutron plus oneproton excitation, while the last one is a pair of neutrons excited to high levels combined with the oneproton excitation. The configuration of is with respect to the ground state, i.e., one formerly paired neutron in the level becomes unpaired and excited to the level , and another formerly paired proton in level is excited to the level . This configuration is consistent with the former assignment of Hf ^{Helmer1968 (); Mullins1997 ()}. Note that the excitation energy of given by the sum of single particle excitations is 3.954 MeV which is about 1.5 MeV higher than the experimental value 2.446 MeV.
In order to obtain the selfconsistent excitation energy for state microscopically, the diabatic constrained RMF calculations are carried out with the respective configuration information. As discussed before, the timeodd fields ^{Yao2006 ()} caused by the unpaired nucleons should be taken into account carefully for the high state. The timeeven calculation is also done for comparison. The PES of state with (without) current is plotted as a solid (dashed) red curve in Fig. 1, together with the PES of state . The local minima of () PES are denoted as up (down) triangles in Fig. 1. For state , the excitation energy according to the calculations without and with current is, respectively 3.579 MeV and 2.801 MeV, which clearly shows that both the selfconsistent calculation and the consideration of the timeodd fields are crucial effects to obtain the reasonable excitation energy. The excitation energy of ( MeV) is close to the experimental excitation energy ( MeV) of Hf. The deformation () obtained for this isomer is similar to that of the ground state. In Table 1, all the excitation energies from calculations without and with current as well as the deformation calculated for the states to are also given. For all five states, the excitation energies calculated with current are 0.6 0.8MeV smaller than those without current. By the way, the same approaches are also applied to the isomeric state Hf (experimentally =,=1.147MeV). The configuration obtained for is , and the excitation energy without and with current is, respectively 1.552 MeV and 1.315 MeV.
In Fig. 2, the neutron and proton singleparticle levels of state from the RMF without and with current have been plotted in columns 2, 3 in both panels. In the timeodd calculation each singleparticle level splits into two levels due to the breaking of the timereversal symmetry, the level with positive being energetically favored. Such splittings will change straightforward the energy gap between two singleparticle levels. In particular, for oneneutron and oneproton excitation of the state discussed here, the neutron energy gap between and in the calculation without current decreases from 2.00 MeV to 1.25 MeV , which is the gap between and in the calculation with current. Correspondingly the energy gap for the proton excitation concerned decreases from 1.34 MeV to 0.54 MeV. As a result, the calculation with current decreases the excitation energy considerably.
In summary, the properties of the ground state of Hf and the isomeric state Hf are investigated by the adiabatic and diabatic constrained RMF approaches. The constrained RMF theory reproduces well the binding energy and deformation for the ground state of Hf. With the singleparticle levels of the ground state obtained in the adiabatic constrained RMF theory, by restricting , the configuration with the lowest excitation energy is found to be , which is consistent with the former configuration assignment. The excitation energy based on the singleparticle spectra is 3.954 MeV, which is much higher than the experimental excitation energy 2.446 MeV of Hf. By applying the selfconsistent timeeven and timeodd RMF calculation, the excitation energy of this configuration is decreased to 3.579 MeV and 2.801 MeV, respectively. Therefore both the selfconsistency and the consideration of current are important factors for studying nuclear isomers.
The authors gratefully acknowledge Professor Jie Meng, Professor L. N. Savushkin, and Dr. Jiangming Yao for their helpful suggestions and discussions.
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State 
Configuration  

(no current)  (with current)  (with current)  

2.119  3.954  3.579  2.801  0.302  
1.835  
1.549  4.146  4.025  3.342  0.272  
2.597  
3.204  5.039  4.749  4.129  0.291  
1.835  
3.473  5.308  5.380  4.601  0.273  
1.835  
4.113  5.948  5.879  5.302  0.290  
1.835 