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Physics Letters B 589 (2004) 83–88

www.elsevier.com/locate/physlet

Nuclear structure of178Hf related to the spin-16, 31-year isome

Yang Suna,b, Xian-Rong Zhouc, Gui-Lu Longc, En-Guang Zhaod, Philip M. Walkere,∗

a Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USAb Department of Physics, Xuzhou Normal University, Xuzhou, Jiangsu 221009, PR China

c Department of Physics, Tsinghua University, Beijing 100084, PR Chinad Institute of Theoretical Physics, Chinese Academy of Sciences, PO Box 2735, Beijing 100080, PR China

e Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom

Received 18 November 2003; accepted 16 March 2004

Available online 23 April 2004

Editor: J.-P. Blaizot

Abstract

The projected shell model is used to study the multi-quasiparticle and collective excitations of178Hf. With an axiallysymmetric basis, the spin-16 isomer at 2.4 MeV appears to be well separated in energy/spin space from other configuratiHowever, projected energy surface calculations suggest that178Hf has significant softness to axially asymmetric shapes, wcan strongly modify the level distribution. The implications for photodeexcitation of the isomer are discussed. 2004 Elsevier B.V. All rights reserved.

PACS: 21.60.Cs; 23.20.Lv; 23.90.+w; 27.70.+q

Keywords: Nuclear structure; Isomers; Projected shell model

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Long-lived isomers—excited nuclear states withhibited electromagnetic decay—may be consideredoffer a form of energy storage [1,2]. The possibity to trigger the decay by the application of extenal electromagnetic radiation has attracted muchterest and potentially could lead to the controlledlease of nuclear energy. First reports of the trigging of the spin-16, 2.4 MeV isomer in178Hf withlow-energy (< 100 keV) photons [3–5] have, howevebeen refuted [6,7]. Meanwhile, the threshold phoenergy for triggering the isomer remains unknow

* Corresponding author. Phone: +44-1483-686807; fax: +1483-686781.

E-mail address: p.walker@surrey.ac.uk (P.M. Walker).

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2004.03.066

This situation contrasts with the recent determinatioof a threshold triggering energy of 1.01 MeV for tspin-9, 75 keV isomer in180Ta [8], and subsequentlthe discrete intermediate states involved in the etation/deexcitation pathway have been identifiedThe significantly higher energy of the178Hf isomerleads to the expectation of a lower triggering threold, on account of the higher level density, even alloing for the pairing difference between the even–eand odd–odd nucleon numbers in178Hf and 180Ta,respectively. The higher isomer energy in178Hf alsoleads to greater potential with regard to the utilitytriggered energy release.

The two-step process of triggered gamma-ray emsion is depicted schematically in Fig. 1. A nucle

.

84 Y. Sun et al. / Physics Letters B 589 (2004) 83–88

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andusu-cur.r-

es-bletro-eri-

tureld:M)in

ten-

viaate

Vhe,masns

.

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re

m).

se-16].ede

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9]theugh

heme

hells

Fig. 1. Schematic diagram illustrating the process of triggegamma-ray emission from a nuclear isomer.

in an isomeric state is first excited to an intermeate state by absorption of an incident photon. Antermediate state may serve as a “gateway” that cnects the isomer to the ground state (g.s.). If thelection rules for transitions between the gatewaythe g.s. are fulfilled, then enhanced gamma-decay,ally a multi-step gamma-cascade, is expected to ocIn determining the favourable conditions for triggeing gamma-ray emission from the isomer, it is necsary to have information on the structure of possigateway states, as well as possible paths of elecmagnetic transitions to and from these states. Expmentally, only a few states close to the178Hf isomerhave been observed and documented in the litera[10,11]. The purpose of the present Letter is twofoto demonstrate that the projected shell model (PS[12] is an appropriate theory for studying high-spisomers and associated excitations, including potial gateway states; and to show that in the178Hf case,in order to trigger emission from the 16+ isomer withlow-energy photons, a series of external excitationsgateway states may be necessary. In addition, the gway states should be sufficiently mixed with low-K

components.The long half-life (31 years) of the 2.4 Me

isomer in178Hf is to a large extent dependent on tapproximate conservation of theK quantum numberwhereK is the projection of the angular momentuon the body-fixed symmetry axis. The isomer hKπ = 16+. Spontaneous electromagnetic transitiofrom a high-K state to a lower energy, low-K stateare strongly hindered by theK selection rule [13],|�K| � λ, with λ being the transition multipolarityThe spontaneous decay of the178Hf isomer involves

-

inhibition factors of close to 100 per degree ofK

forbiddenness,ν = �K − λ, with total inhibition≈ 100ν [14]. This strong inhibition supports the viethat178Hf has an axially symmetric intrinsic shape.

In a deformed, axially symmetric nucleus, a higK state is made by summing the contributions frseveral unpaired quasiparticles. Ann-quasiparticle(n-qp) configuration gives rise to a multiplet of 2n−1

states, with the totalK expressed byK = |K1 ± K2 ±· · · ± Kn|, whereKi is for an individual neutron oproton. Among the 2n−1 states, the state with thhighestK value,K = ∑

i |Ki |, is usually energeticallyfavoured and the most likely to form anyrast state(lowest energy state for a given angular momentuDeformed nuclei withA ≈ 180 have several high-K

single-particle (both neutron and proton) orbitals cloto the Fermi surface, and several multi-qp highK

states in this mass region are found to be yrast [15,The physics of multi-qp states is well incorporat

in the PSM framework. The PSM follows closely thshell-model philosophy, and in fact is a shell moconstructed in a deformed multi-qp basis. More pcisely, the basis is first built in the qp basis with rspect to the deformed-BCS vacuum; then rotatiosymmetry, violated in the deformed basis, isstored by angular-momentum projection [17] to foa rotational-invariant basis in the laboratory frame;nally, a two-body Hamiltonian is diagonalised in tprojected basis. In contrast to mean-field methodsemployed in the study of multi-qp states, the PSMproduce fully correlated shell-model states, andgenerate well-defined wave functions, allowing uscompute, without further approximations, the quanties such as transition probabilities.

Starting from the deformed Nilsson scheme [1plus a subsequent BCS calculation, one buildsshell-model space (for even–even systems) thromulti-qp states:

|φκ〉 = {|0〉, α†ni

α†nj

|0〉, α†pk

α†pl

|0〉,(1)α†

niα†

njα†

pkα†

pl|0〉 , . . .

},

where α† is the creation operator for a qp and tindexn (p) denotes neutron (proton) Nilsson quantunumbers which run over the low-lying orbitals. Thqp states are defined in a space with three major s(N = 4,5, 6 for neutrons andN = 3,4, 5 for protons).The corresponding qp vacuum is|0〉. The indicesn

Y. Sun et al. / Physics Letters B 589 (2004) 83–88 85

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. If

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andp in (1) are general; for example, a 2-qp state cbe of positive (or negative) parity if both quasipartici and j are from the same (or two neighbourinmajor shell(s). Positive and negative parity states sthe entire configuration space with the correspondmatrix in a block-diagonal form classified by pariAlthough in the present work we truncate the mospace beyond 4-qp’s, the bases (1) can be eaenlarged by including higher-order multi-qp statesthe configurations denoted by “· · ·” are completelyincluded, one recovers the full shell-model spawritten in the representation of qp excitations.

In the case that the nuclear potential is axiasymmetric, the basis states in (1) are labelled byK.Thus, theprojected multi-qpK-states are thebuildingblocks in our shell-model wave function

(2)∣∣Ψ Iσ

M

⟩ = ∑κ

f Iσκ P I

MKκ|φκ〉.

In Eq. (2), κ labels the basis states andσ thestates with the same angular momentum.P I

MK is theangular momentum projector [17]. The coefficief Iσ

κ in Eq. (2) are determined by diagonalisatiof the Hamiltonian. Diagonalisation is the proceof configuration mixing (here,K-mixing). Thus, therequired physics, i.e., constructing multi-qp sta(with good angular momentum and parity) and mixithese states through residual interactions, is propincorporated in the model. Electromagnetic transitprobabilities can be directly computed by usingresulting wave functions.

The Hamiltonian employed in the PSM [12] is

H = H0 − 1

2χ

∑µ

Q†µQµ − GMP †P

(3)− GQ

∑µ

P †µPµ,

whereH0 is the spherical single-particle Hamiltoniawhich contains a proper spin–orbit force [19]. Tother terms in Eq. (3) are quadrupole–quadrupole,monopole- and quadrupole-pairing interactions,spectively. The strength of the quadrupole–quadruforce χ is determined in such a way that it hasself-consistent relation with the quadrupole deformtion ε2. The monopole-pairing force constantsGM are

(4)GM =[20.12∓ 13.13

N − Z

A

]A−1,

with “−” for neutrons and “+” for protons, whichreproduces the observed odd–even mass differein the mass region. Finally, the strength parameGQ for quadrupole pairing was simply assumed toproportional toGM , with a proportionality constan0.16, as commonly used in PSM calculations [12].

For 178Hf, the model basis is built with the deformation parametersε2 = 0.251 and ε4 = 0.056.These values are taken from the literature [20]. Figshows the calculated energy levels in178Hf, comparedwith the known data [10]. Satisfactory agreemenachieved for most of the states, except that forbandhead of the first 8− band and the 14− band, thetheoretical values are too low. The dominant structof each band can be read from the wave functions.found that the 6+ band has a 2-qp structure

{ν[512]5

2− ⊕ ν[514]7

2−}

,

the 16+ band has a 4-qp structure

{ν[514]7

2− ⊕ ν[624]9

2+ ⊕ π[404]7

2+ ⊕ π[514]9

2−}

,

the first (lower) 8− band has a 2-qp structure

{ν[514]7

2− ⊕ ν[624]9

2+}

,

the second (higher) 8− band has a 2-qp structure

{π[404]7

2+ ⊕ π[514]9

2−}

,

and the 14− band has a 4-qp structure

{ν[512]5

2− ⊕ ν[514]7

2− ⊕ π[404]7

2+ ⊕ π[514]9

2−}

.

These states, together with many other statesshown in Fig. 2) obtained from the same diagonaltion process, form a complete spectrum includinghigh-K isomeric states and candidate gateway stat

In order to study the possible gateway stathrough which triggered isomeric emissions mightcur, in Fig. 3 we plot the high-K bands lying close inenergy to the 16+ isomer. We plot also the calculatelower 8− band via which the gamma-cascade coreach the g.s. The 16+ isomeric band is found to byrast, above which there are many other 4-qp highK

bands with either positive or negative parity. It canseen from Fig. 3 that the 16+ isomeric band is welseparated from the other bands, leaving an energyof nearly 500 keV. However, the bandhead energiethe other high-K bands are not much higher than t

86 Y. Sun et al. / Physics Letters B 589 (2004) 83–88

Fig. 2. Comparison of calculated energy levels in178Hf with known data [10].

atf

ayturnedveserto

tepbe

ulte

byed

heis

Mwo

Fig. 3. Calculated high-K bands in178Hf. The bands are plottedrelative to the bandhead energy of the 16+ isomer.

16+ isomer, and an external energy of<100 keV maybe able to excite the 16+ isomer.

A close look at Fig. 3 suggests, however, thin order to excite the 16+ isomer to the states o

the 8− band, a series of external excitations mbe necessary, otherwise the deexcitation would reto the original isomer. The excitation may procestepwise among the gateway states, until it arriin a low-K state (a possible state is a membof the 8− band) from which spontaneous decaythe g.s. is possible. Of course, if such a multi-sexcitation is required, then the probability wouldvery low. In addition, in order to link to the 8− band,the gateway states should contain sufficientK ≈ 8components. The latter requirement is usually difficto fulfill in an axially symmetric nucleus; however, thdiscussion below may open a possibility, through non-axial distortions.

The picture shown in Fig. 3 has been obtainedassuming an axial basis, and it could be modifiif the nucleus exhibits significant deviation from taxial shape. The amount of triaxial deformationgiven by an additional deformation parameter,γ . Tosee if theγ degree of freedom plays a role in178Hf,we have newly extended the capability of the PSto study energy surfaces as a function of the t

Y. Sun et al. / Physics Letters B 589 (2004) 83–88 87

nallar-

ares a

tumrt

-

hasleral-

n-

er,nt

d

uchro-

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oint

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un-

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quadrupole deformation parameters,ε2 andγ , basedon exact projection calculations in three-dimensioEuler space [21,22]. Here, we calculate the angumomentum-projected energies having the form

(5)EI (ε2, γ ) = 〈φ(ε2, γ )|H P I |φ(ε2, γ )〉〈φ(ε2, γ )|P I |φ(ε2, γ )〉 .

For comparison, the unprojected energy surfacesalso calculated. The unprojected energy containmixture of states with good angular momenta.

Energy surfaces for states with angular momenI = 0,2, . . . ,10 are plotted in Fig. 4. The left paof Fig. 4 shows the energies as a function ofε2 withγ = 0. A minimum atε2 = 0.25 is seen, which is consistent with the use ofε2 = 0.251 in the calculationsof Figs. 2 and 3. The unprojected energy surfacea similar form, but the minimum has a slightly smalε2 value. In the right part of Fig. 4, energies are cculated as a function ofγ , with the minimum valueof ε2 = 0.25. It is interesting to observe that the uprojected surface has a minimum atγ = 0, suggest-ing that this nucleus is axially symmetric. Howevthe projected surfaces exhibit a qualitatively differepattern: for all the spin states withI = 0,2, . . . ,10,the energies with smallγ are almost constant, anthey start to rise only forγ > 15◦. This suggests asoftness to axially asymmetric shapes. Note that squalitative differences between projected and unpjected calculations have been discussed before in aother context [23].

Nuclei having flat energy surfaces in theγ directionhave recently been suggested [24] to be critical p

Fig. 4. Energy surfaces for states with angular momenI = 0,2, . . . ,10. The energies are calculated as function(a) quadrupole deformationε2 with γ = 0, and (b)γ deformationwith ε2 = 0.25. Full (dashed) curves correspond to projected (projected) calculations.

nuclei sitting in the transitional region between taxially deformed and triaxially deformed phases. Tenergy surfaces in Fig. 4 indicate that178Hf has acharacter close to that of critical point nuclei wiconsiderableγ softness. Therefore, theγ degree offreedom may play a significant role in this nucleus.et al. [25] studied the importance ofγ deformation forhigh-K bandheads in178W and182Os, and for somehafnium isotopes [26]. While axial symmetry wpredicted for the hafnium cases, the calculationsmulti-qp states were restricted to bandhead shapesenergies, and not the associated rotational excitatiOur calculations now suggest that178Hf exhibits asignificantγ softness already near its ground staTheγ softness may strongly enhance excitation ofgateway states. However, an extension of the PSMthe description of multi-qp states, with theγ degree offreedom explicitly included, does not yet exist.

To summarise, in the PSM the multi-qp states wgood angular momentum serve as the building bloand mixing of these states is incorporated through tbody residual interactions. Our calculations for178Hfshow that the 16+ isomer lies well separated in enerfrom other states, suggesting that a series of exteexcitations may be necessary to trigger isomeric emsion. However, axially asymmetric energy-surface cculations show that in178Hf there is considerableγsoftness. This fact, in the context of recent discussof critical point nuclei [24], indicates that not only thparticular example, but also other nuclei in this mregion, may be significantly influenced by theγ de-gree of freedom. Theγ -softness may strongly enhanexcitation of the gateway states, in favour of trigging isomer decay. To describe multi-qp states propeand to further guide experiments in finding candidnuclei for triggered gamma-ray emission, an advantheory following the projected shell-model concewith theγ degree of freedom included, is very mudesired. Work along these lines is in progress.

Acknowledgements

Y.S. thanks Professor D.-H. Feng for commucations. Research at University of Notre Damesupported by the NSF under contract number PH0140324.

88 Y. Sun et al. / Physics Letters B 589 (2004) 83–88

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