Projected Shell Model Description for Nuclear Isomers

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Model Shell discus paper Projected present a The needed. much st is isomeric effort to theoretical access experimental of possibility creasing lioetereitnet h prxmt osrainof conservation approximate the to existence their owe clei ulu a h osqec htteatohsclhalf-li astrophysical the that for consequence the has the nucleus between communication of state The pro ground abundances nucleosynthesis. elemental excitations in the thermal duced significantly through alter state could This ground its co can with state municate isomeric an [12 environments, universe astrophysical the hot in In elements the of abundances the termining [11]. stability’ of ‘island anticipated lc n edt ifrn e feeetlaudne [14 abundances elemental of set different taking a reactions to lead of in and paths place cases the change are long may (probably There microseconds) lifetime than long sufficiently stars. of the isomer neutron on an which binary place take of to disks thought iso- accretion process nuclear capture a proton that such rapid impact processes the nucleosynthesis the other at various on look have to mers beginning just is One and nvr ev ulicudsrea tpigsoe toward stones stepping the as beyond structure serve single-particle could the understanding nuclei isom heavy the that very expects One in nuclei. these for stability hanced 1]ta nsprev uli h smrcsae decreas and states fission isomeric both the for Xu probability by nuclei, out the superheavy pointed in been that has It [10] nuclei. unstable those in rdc railt noteselmdlbss otreat To basis. model shell the into triaxiality troduce 1 ihrpdygoigitrs nteioe td n in- and study isomer the in interest growing rapidly With ayln-ie,hgl-xie smr ndfre nu- deformed in isomers highly-excited long-lived, Many oevr ula smr a lyasgicn oei de- in role significant a play may isomers nuclear Moreover, , 2 ofiuain eogn odfeetshapes. different to belonging configurations e 26 r ae or ae nin 64,USA 46545, Indiana Dame, Notre Dame, tre N r,w rsn ehdbsdo h Projected the on based method a present we ers, K lcnb uhsotrta h aoaoyvle[13]. value laboratory the than shorter much be can Al = MXN NTEPOETDSELMODEL SHELL PROJECTED THE IN -MIXING vrta nodrt ecietestrong the describe to order in that ever 2 hl lsrs hc stekyt oaigthe locating to key the is which closures, shell 126 K mxn sepaie.W on u,however, out, point We emphasized. is -mixing hi204,P .China R. P. 200240, ghai 26 ladtefis xie smrcsaei this in state isomeric excited first the and Al K ioe treatment, -isomer K voain nscinIII, section In -violation. α dcy eutn nen- in resulting -decay, K - K -isomers, Z tal. et = ates, ec- ses ers m- 82 er fe d. ]. ]. e s - . 2 the K quantum number. The selection rule for an electro- The index σ denotes states with same angular momentum κ ˆ I magnetic transition would require that the multipolarity of and labels the basis states. PMK is the angular-momentum- I,σ the decay radiation, λ, be at least as large as the change in projection operator [15] and the coefficients fκ are weights I,σ the K-value (λ ≥ ∆K). However, symmetry-breaking pro- of the basis states. The weights fκ are determined by diag- cesses make possible transitions that violate the K-selection onalization of the Hamiltonian in the projected spaces, which rule. A microscopic description of K-violation is through the leads to the eigenvalue equation (for a given I) so-called K-mixing in the configuration space. A theoretical σ model that can treat K-mixing has preferably the basis states ∑(Hκκ′ − Eσ Nκκ′ ) fκ′ = 0. (2) that are eigenstates of angular momentum I but labeled by K. κ′ Diagonalization of two-body interactions mixes these states The Hamiltonian and the norm matrix elements in Eq. (2) are and the resulting wavefunctionscontain the information on the given by degree of K-mixing. In this kind of approach, the mixing and I I Hκκ′ = hφκ |Hˆ Pˆ ′ |φκ′ i, Nκκ′ = hφκ |Pˆ ′ |φκ′ i. (3) its consequences are discussed in the laboratory frame rather Kκ Kκ′ Kκ Kκ′ than in a body-fixed frame in which K is originally defined. Angular-momentum-projectionon a multi-qp state |φκ i with a sequence of I generates a band. One may define the rotational energy of a band (band energy) using the expectation values The model of the Hamiltonian with respect to the projected |φκ i

φκ ˆ Î φκ The projected shell model (PSM) [15, 16] seems to fulfil I Hκκ h |HPKκ Kκ | i Eκ = = . (4) κκ φ Î φ the requirement. It is a shell model that starts from a deformed N h κ |PKκ Kκ | κ i basis. In the PSM, the shell-modelbasis is constructedby considering a few quasiparticle (qp) orbitals near the Fermi sur- In a usual approximation with independent quasiparticle faces and performing angular momentum projection (if nec- motion, the energy for a multi-qp state is simply taken as the essary, also particle-number projection) on the chosen con- sum of those of single quasiparticles. This is the dominant figurations. With projected multi-qp states as the basis states term. The present theory modifies this quantity in the fol- of the model, the PSM is designed to describe the rotational lowing two steps. First, the band energy defined in Eq. (4) bands built upon qp excitations. introduces the correction brought by angular momentum projection and the two-body interactions, which accounts for the Suppose that a PSM calculation begins with axially de- couplings between the rotating body and the quasiparticles in formed Nilsson single-particle states, with pairing correla- a quantum-mechanical way. Second, the corresponding rota- tions incorporated into these states by a BCS calculation. This † † tional states (labeled by K) are mixed in the subsequent pro- defines a set of deformed qp states (with aν and aπ being the cedure of solving the eigenvalue equation (2). The energies creation operator for neutrons and protons, respectively) with are thus further modified by configuration mixing. respect to the qp vacuum |0i. The PSM basis is then con- For deformed states with axial symmetry, each of the basis structed in the multi-qp states with the following forms states in (1), i.e. the projected |φκ i,isa K-state. For example, n−1 † † † † † † † † an n-qp configuration gives rise to a multiplet of 2 states, e − e nuclei : {|0i,aν aν |0i,aπ aπ |0i,aν aν aπ aπ |0i, † † † † † † † † with the total K expressed by K = |K1 ± K2 ±···± Kn|, where aν aν aν aν |0i,aπ aπ aπ aπ |0i,...} Ki is for an individual neutron or proton. In this case, shell † † † † † † † † † † o − o nuclei : {aν aπ |0i,aν aν aν aπ |0i,aν aπ aπ aπ |0i, model diagonalization, i.e. solving the eigenvalue equation † † † † † † aν aν aν aπ aπ aπ |0i,...} (2), is equivalent to K-mixing. The degree of K-mixing can † † † † † † † be read from the resulting wavefunctions. odd − ν nuclei : {aν |0i,aν aν aν |0i,aν aπ aπ |0i, † † † † † The above discussion is independent of the choice of the aν aν aν aπ aπ |0i,...} two-body interactions in the Hamiltonian. In practical calcu- † † † † † † † odd − π nuclei : {aπ |0i,aν aν aπ |0i,aπ aπ aπ |0i, lations, the PSM uses the separable form of Hamiltonian with † † † † † aν aν aπ aπ aπ |0i,...} pairing plus quadrupole-quadrupole terms (these have been known to be essential in nuclear structure calculations [20]), The omitted index for each creation operator contains labels with inclusion of the quadrupole-pairing term for the Nilsson orbitals. In fact, this is the usual way of build- 1 † † † ing multi-qp states [10, 17, 18, 19]. Hˆ = Hˆ0 − χ ∑Qˆ µ Qˆ µ − GMPˆ Pˆ − GQ ∑Pˆµ Pˆµ . (5) 2 µ µ The angular-momentum-projectedmulti-qp states, each being labeled by a K quantum number, are thus the building The strength of the quadrupole-quadrupole force χ is deter- blocks in the PSM wavefunction, which can be generally writ- mined in such a way that it holds a self-consistent relation ten as with the quadrupole deformation ε2. The monopole-pairing force constants GM are ψI,σ I,σ ˆ I φ I,σ ˆ I φ | M i = ∑ fκ PMK | κ i = ∑ fκ PMKκ | κ i. (1) κ N−Z −1 κ,K≤I GM = G1 ∓ G2 A A , (6) 3 with “−" for neutrons and “+" for protons, which roughly reproduces the observed odd–even mass differences in a given mass region when G1 and G2 are properly chosen. Finally, the strength GQ for quadrupole pairing was simply assumed to be proportional to GM, with a proportionality constant fixed in a nucleus, choosing from the range 0.14 – 0.18.

The 178Hf example

The nucleus 178Hf has become a discussion focus because (MeV) Energy Excitation of the possibility to trigger the 2.45MeV, 31-year 16+-isomer decay. The triggering could be made by applying external 178Hf electromagnetic radiation which, if successful, will lead to the controlled release of nuclear energy [9]. Information on FIG. 1: Comparison of the PSM calculation with data for the rota- the detailed structure as well as the transition of this and the tional bands in 178Hf . This figure is adopted from Ref. [8]. surrounding states thus becomes a crucial issue. In the PSM calculation for 178Hf [8], the model basis was built with the deformation parameters ε2 = 0.251 and ε4 = 0.056. Fig. 1 TABLE I: Comparison of calculated 174Yb ground band with data. shows the calculated energy levels in 178Hf, which are com- E(I) are in keV and B(E2,I → I − 2) in W.u.. pared with the known data [21]. Satisfactory agreement is Spin I E(I), Exp E(I), PSM B(E2), Exp B(E2), PSM achieved for most of the states, except that for the bandhead of the first 8− band and the 14− band, the theoretical values 2 76.5 71.1 201(7) 195.18 are too low. 4 253.1 236.7 280(9) 279.01 6 526.0 496.1 370(50) 307.59 It was found that the obtained states are generally K- 8 889.9 848.0 388(21) 322.31 mixed. If the mixing is not strong, one may still talk about 10 1336 1290.3 325(22) 331.28 the dominant structure of each band by studying the wave- 12 1861 1820 369(23) 337.13 functions. We found that the 6+ band has mainly a 2-qp 14 2457 2433 320 340.91 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 PSM calculations are performed for 174Yb. The deformed − + − {ν[514]7/2 ⊕ ν[624]9/2 }, the second (higher) 8 band basis is constructed with deformation parameters ε2 = 0.275 + − has a 2-qp structure {π[404]7/2 ⊕ π[514]9/2 }, and the and ε4 = 0.042. In Tables I, II, and III, three groups of results 14− band has a 4-qp structure {ν[512]5/2− ⊕ ν[514]7/2− ⊕ are listed, for the K = 0 ground band (Table I) and K = 6 iso- π[404]7/2+ ⊕π[514]9/2−}. These states, together with many mer band (Table II) with in-band transitions, and inter-band other states (not shown in Fig. 1) obtained from a single diag- transitions (Table III) between the ground band and the K = 6 onalization, form a complete spectrum including the high-K isomer band. These results suggest that while the energy lev- isomeric states. els for both groundand isomer bands are reproduced,the E(2) As far as energy levels are concerned, the PSM can give transition probabilities are also correctly obtained. In par- a reasonable description simultaneously for multiple bands. ticular, the calculation yields a reasonable value of the very The next question is how electromagnetic transitions are de- small inter-band B(E2) as what was observed in 174Yb [23] scribed. The electromagnetic transition between any two of (see Table III). Note that without mixing configurations in the these states can be directly calculated [22] by using the wavefunctions. This is a crucial test for the model wavefunctions. TABLE II: Comparison of calculated 174Yb 6+ isomer band with data. E(I) are in keV and B(E2,I → I − 2) in W.u.. The N = 104 isotones Spin I E(I), Exp E(I), PSM B(E2), PSM 6 1518.0 1503 + There have been detailed experimental studies on the 6 7 1671.1 1683 isomer in some N = 104 isotones [23, 24, 25]. These data 8 1844.7 1886 36.76 show that along the N = 104 isotones, lifetime of the 6+ iso- 9 2038.3 2117 78.18 mer can vary very much, differing by several orders of mag- 10 2251.5 2372 115.81 nitude. Understanding the underlying physics is a challenging 11 2483.7 2652 147.96 problem: what is the microscopic mechanism for such a dras- 12 2734.4 2956 174.91 13 3003.1 3283 197.41 tic change in the neighboring isotones? 4

spontaneous symmetry breaking helps in realizing larger elec- 174 TABLE III: Comparison of calculated inter-band transition of Yb tromagnetic transitions which would otherwise be impossible 6+ isomer to ground band. B E2 is in e2 fm4 ( ) due to the selection rule. In this way, the authors in [26] were Transition B(E2), Exp B(E2), PSM able to describe the observed large inter-band B(E2) in 176Hf 178 6 → 4 4.3(8) × 10−9 8.49 × 10−8 and W rather successfully. However, their model could not i g + + give the above-discussed small 6i → 4g inter-band B(E2) in 174Yb. 2.5 Both methods, the configuration mixing implemented by Yb the PSM and the γ-tunneling by Narimatsu et al., introduce a Hf 2.0 W mechanism to break the axial symmetry; however the degree of symmetry breaking is different. If the physical process is a perturbation in the K space, then it is better described by 1.5 the PSM based on the axially symmetric mean field. If it is not, axial symmetry in the mean field must be broken as in Energy (MeV) the γ-tunneling model. The two models may be viewed as 1.0 two different simplifications of the complicated many-body problem; each emphasizes one aspect. It is desired that one 0.5 can have one unified microscopic description for all cases. 96 100 104 108 112 To efficiently introduce γ degree of freedom within the Neutron number PSM, one can break the axial symmetry of the single-particle π basis and carry out three-dimensionalangular momentum pro- FIG. 2: Experimental data for excitation energy of I = 6+ isomer (filled symbols) and 2+ γ vibrational state (open symbols). jection. The shell model diagonalization is then performed in the projected multi-quasiparticle configurations based on γ deformed basis. One example is the description of γ vibrational states within the PSM. It was shown [27] that by using pro- wavefunction, a direct transition from the 6+ isomer to ground jected triaxially-deformed basis, it is possible to describe the band would be forbidden. The obtained amount of inter-band ground band and γ band simultaneously. Thus, an extended B(E2), though small, is the consequence of K-mixing con- PSM that introduces triaxiality in the model would be use- tained in the PSM. ful for cases with large K-violation. Such an extension has On the other hand, in its isotones 176Hf and 178W, much en- + recently be developed for odd-odd nuclei [28] and even-even hanced B(E2) from the 6 isomer to ground band has been nuclei [29], and will be applied to the isomer study. obtained experimentally. The values are 1.8 × 10−5 e2fm4 for 176Hf and 2.6 × 10−2 e2fm4 for 178W. If a PSM calculation is performed for these two isotones, one gets similar small numbers for the inter-band B(E2) as in 174Yb, which disagree with SHAPE ISOMER AND CONFIGURATION MIXING data. We have to conclude that although the current PSM has K-mixing mechanism in the model, which effectively intro- Coexistence of two or more well-developed shapes at com- duces γ, the mixing within the truncated space is apparently parable excitation energies is a well-known phenomenon. The too weak. expected nuclear shapes include, among others, prolate and + In Fig. 2, we plot experimental excitation energies of the oblate deformations. In even-even nuclei, an excited 0 state + 6+ isomer states together with 2+ state of γ vibration for Yb, may decay to the ground 0 state via an electric monopole Hf, and W isotopes. There seems to be an correlation between (E0) transition. For lower excitation energies, the E0 transi- + the two plotted quantities. The correlation is such that to com- tion is usually very slow, and thus the excited 0 state be- pare with the 2+ γ states, energy of the 6+ isomers shows an comes a shape isomer. opposite variation trend with neutron number. At N = 104, nuclei have the highest excitation of γ vibrational states while they show a minimum in 6+ isomer energy. In Fig. 2, 174Yb Shape isomer in 68Se and 72Kr and the impact on isotopic appears to be the only nucleus in the collection that has the abundance in X-ray bursts 6+ isomer lower than the γ states. Therefore, below the 6+ isomer in 174Yb, there are no γ states carrying finite K to be Fig. 3 shows calculated projected energies as a function of mixed in the wavefunction. This may have naively explained deformation variable ε2 for different spin states in the N = Z why the 174Yb isomer decay is so exceptionally hindered. nuclei 68Se and 72Kr. The configuration space and the inter- In Ref. [26], a γ-tunneling model was introduced by Nari- action strengths in the Hamiltonian are the same as those em- matsu, Shimizu, and Shizuma to describe the enhanced B(E2) ployed in the previous calculations for the same mass region values. In their model, the γ degree of freedom is taken into [30]. It is found that in both nuclei, the ground state takes an account, which breaks the axial symmetry explicitly. The oblate shape with ε2 ≈−0.25. As spin increases, the oblate 5

6 the method further, it can describe those transitional nuclei 5 where energy surfaces are typically ﬂat. 68 72 4 I=8 Se Kr The heart of the present consideration is the evaluation 3 Oblate I=6 of overlapping matrix element in the angular-momentum- 2 I=8 Prolate I=4 Oblate projected bases. Let us start with the PSM wave function in 1 I=2 I=6 Prolate Eq. (1) I=0 0 I=4

Energy (MeV) Energy I,σ I,σ I -1 I=2 ψ ˆ φκ I=0 | M i = ∑ fκ PMKκ | i. -2 κ

-3 π + + K = 0 isomer Kπ = 0 isomer For an overlapping matrix element, states in the left and -4 -0.8 -0.6 -0 .4 -0.2 0 0 .2 0 .4 0.6 0.8 -0.6 -0 .4 -0.2 0 0.2 0.4 0.6 0.8 right hand side must correspond to different deformed shapes. ε Deformation 2 Therefore, two different sets of quasiparticle generated at different deformations are generally involved. Let us denote |φκ i 68 72 FIG. 3: Energy surfaces for various spin states in Se and Kr as explicitly as |φκ (a)i and |φκ (b)i, for which we define two sets ε functions of deformation variable 2. This figure is adopted from of quasiparticle operators {a†} and {b†} associated with the Ref. [14]. quasiparticle vacua |ai and |bi, respectively. For simplicity, we assume axial symmetry. The general three-dimensional angular momentum projection is reduced minimum moves gradually to ε ≈−0.3. Another local min- 2 to a problem of one-dimensional projection, with the projec- imum with a prolate shape (ε ≈ 0.4) is found to be 1.1 MeV 2 tor having the following form (68Se) and 0.7 MeV (72Kr) high in excitation. Bouchez et al. [6] observed the 671 keV shape-isomer in 72Kr with half-life π I 1 I τ = 38 ± 3 ns. The one in 68Se is the prediction, awaiting Pˆ = I + dβ sinβ d (β) Rˆy(β) (7) MK 2 MK experimental confirmation. Isomers in these nuclei have also Z0 been predicted by Kaneko, Hasegawa, and Mizusaki [31]. with The existence of low energy 0+ shape isomer along the −iβ Jˆy N = Z nuclei has opened new possibilities for the rp-process Rˆy(β)= e . (8) [32] reaction path occurring in X-ray burst. Since the ground 73 69 I β β states of Rb and Br are bound with respect to these iso- In Eq. (7), dMK ( ) is the small-d function and is one of mers, proton capture on these isomers may lead to additional the Euler angels. The evaluation of the overlapping matrix strong feeding of the 73Rb(p,γ)74Sr and 69Br(p,γ)70Kr reac- element is eventually reduced to the problem tions. However, the lifetime of the isomeric states must be Φ ˆ ˆ β Φ sufficiently long to allow proton capture to take place. No in- h κ′ (b)|ORy( )| κ (a)i, (9) formation is available about the lifetime of the 68Se isomer which is the problem of calculating the Oˆ operator sand- while the 55 ns lifetime of the isomer in 72Kr is reported [6]. wiched by a multi-qp state |Φκ′ (b)i and a rotated multi-qp Based on Hauser Feshbach estimates [32] the lifetime against state Rˆ (β)|Φκ (a)i, with a and b characterizing different qp proton capture is in the range of ≈100 ns to 10 µs depend- y sets. In Eq. (9), Oˆ stands for Hˆ or 1. ing on the density in the environment. Considering the uncer- To calculate hΦκ′ (b)|OˆRˆ (β)|Φκ (a)i, one must compute tainties in the present estimates a fair fraction may be leaking y the following types of contractions for the Fermion operators out of the 68Se, 72Kr equilibrium abundances towards higher masses. β † † β −1 β Ai j = hb|[ ]ai a j |ai = [V( )U ( )]i j, −1 Bi j = hb|bib j[β]|ai = [U (β)V(β)]i j, (10) β † −1 β Configuration mixing with different shapes Ci j = hb|bi[ ]a j |ai = [U ( )]i j,

To calculate isomer lifetime, decay probability is needed. where we have defined This involves transitions from the shape isomer to the ground Rˆy(β) state, which belong to different shapes or deformation min- [β]= , hb|Rˆ (β)|ai ima. If the energy barrier between the minima is not very y high, configuration mixing of the two shapes must be taken and into account. In the following, we outline a scheme to con- 1/2 sider such a mixing. The discussion is general; the shapes can hb|Rˆy(β)|ai = [det U(β)] . (11) be any kinds of two deformed ones in a nucleus. For example, one of them can be a prolately-deformedand the other an Eqs. (10) and (11) are written in a compact form of N × N oblately-deformed shape, or one of them can be a normally- matrix, with N being the number of total single particles. The deformed and the other a superdeformed shape. Generalizing general principle of finding U(β) and V(β) is given by the 6

Thouless theorem [33], and a well worked-out scheme can be has been developed which allows calculations for transition found in the work of Tanabe et al. [34] (see also Ref. [35]). between a shape isomer and the ground state. To write the matrices U(β) and V(β) explicitly, we con- The author is grateful to J. Hirsch and V. Velazquez for † † sider the fact that {ai,ai } and {bi,bi } can both be expressed warm hospitality during the Cocoyoc 2008 meeting. He ac- † by the spherical representation {ci,ci } through the HFB trans- knowledges helpful discussions with A. Aprahamian, Y. R. formation Shimizu, P. M. Walker, and M. Wiescher. This work is sup- ported by the Chinese Major State Basic Research Develop- c Ua Va a ment Program through grant 2007CB815005 and by the U. S. † = † c Va Ua a National Science Foundation through grant PHY-0216783. (12) c Ub Vb b † = † . c Vb Ub b U ,V ,U and V in above equations, which define the HFB a a b b [1] P. M. Walker and R. C. Johnson, Nature Phys. 3, 836 (2007). transformation, are obtained from the Nilsson-BCS calcula- [2] I. Mukha et al., Phys. Rev. Lett. 95, 022501 (2005). tion. A rotation of the spherical basis can be written in a ma- [3] I. Mukha et al., Nature 439, 298 (2006). trix form as [4] P. M. Walker and G. D. Dracoulis, Nature 399, 35 (1999). [5] M. Hasegawa, Y. Sun, K. Kaneko, and T. Mizusaki, Phys. Lett. β c † d( ) 0 c B 617, 150 (2005). Rˆy(β) Rˆ (β) = . (13) c† y 0 d(β) c† [6] E. Bouchez, et al., Phys. Rev. Lett. 90, 082502 (2003). [7] A. Blazhev, et al., Phys. Rev. C 69, 064304 (2004). Combining Eqs. (12) and (13) and noting the unitarity of the [8] Y. Sun, X.-R. Zhou, G.-L. Long, E.-G. Zhao, and P. M. Walker, HFB transformation, one obtains Phys. Lett. B 589, 83 (2004). [9] P. M. Walker and J. J. Carroll, Physics Today (June issue) 39 T β (2005). ˆ β b ˆ† β Ub Vb d( ) 0 Ry( ) † Ry( ) = [10] F.-R. Xu, E.-G. Zhao, R. Wyss, and P. M. Walker, Phys. Rev. b Vb Ub 0 d(β) Lett. 92, 252501 (2004). Ua Va a [11] R.-D. Herzberg et al., Nature 442, 896 (2006). × † . (14) Va Ua a [12] A. Aprahamian and Y. Sun, Nature Phys. 1, 81 (2005). [13] R. C. Runkle, A. E. Champagne, J. Engel, Astrophys. J. 556, U(β) and V(β) can finally be obtained from the following 970 (2001). equation [14] Y. Sun, M. Wiescher, A. Aprahamian, and J. Fisker, Nucl. Phys. A 758, 765 (2005). β β T T β [15] K. Hara and Y. Sun, Int. J. Mod. Phys. E 4, 637 (1995). U( ) V ( ) Ub Vb d( ) 0 Ua Va = T T [16] Y. Sun and K. Hara, Comp. Phys. Commun. 104, 245 (1997). V(β) U(β) V U 0 d(β) Va Ua b b [17] K. Jain et al., Nucl. Phys. A 591, 61 (1995). [18] V. G. Soloviev, Nucl. Phys. A 633, 247 (1998). T β T β T β T β Ub d( )Ua +Vb d( )Va Ub d( )Va +Vb d( )Ua [19] J.-Y. Zeng, S.-X. Liu, L.-X. Gong, and H.-B. Zhu, Phys. Rev. C = . T T T T 65, 044307 (2002). V d(β)Ua +U d(β)Va U d(β)Ua +V d(β)Va ! b b b b [20] M. Dufour and A. P. Zuker, Phys. Rev. C 54, 1641 (1996). With the overlapping matrix elements that connect config- [21] S. M. Mullins et al., Phys. Lett. B 393, 279 (1997); Phys. Lett. B 400, 401 (1997). urations belonging to different shapes, one obtains wavefunc- [22] Y. Sun and J. L. Egido, Nucl. Phys. A 580, 1 (1994). tions containing configuration mixing. Using these wavefunc- [23] G. D. Dracoulis et al., Phys. Rev. C 71, 044326 (2005). tions, one can further calculate inter-transition probabilities [24] G. D. Dracoulis et al., Phys. Lett. B 635, 200 (2006). from a shape isomer to the ground state. [25] C. S. Purry et al., Nucl. Phys. A 632, 229 (1998). [26] K. Narimatsu, Y. R. Shimizu, and T. Shizuma, Nucl. Phys. A 601, 69 (1996). SUMMARY [27] Y. Sun et al., Phys. Rev. C 61, 064323 (2000). [28] Z.-C. Gao, Y.-S. Chen, and Y. Sun, Phys. Lett. B 634, 195 (2006). We have introduced the Projected Shell Model description [29] J. A. Sheihk, G. H. Bhat, Y. Sun, G. B. Vakil, and R. Palit, Phys. for two kinds of isomers, K-isomers and shape isomers. We Rev. C, in press (arXiv:0801.0296). have shown that the physics of K-mixing in multi-qp states [30] Y. Sun, Eur. Phys. J. A 20, 133 (2004). is well incorporated in the model with the basis states hav- [31] K. Kaneko, M. Hasegawa, and T. Mizusaki, Phys. Rev. C 70, ing axial symmetry. Diagonalization mixes configurations of 051301(R) (2004). different K, which effectively introduces triaxiality. For K- [32] H. Schatz et. al., Phys. Rep. 294, 167 (1998). [33] D. J. Thouless, Nucl. Phys. 21, 225 (1960). isomers with much enhanced decay probability to the ground γ [34] K. Tanabe, K. Enami, and N. Yoshinaga, Phys. Rev. C 59, 2492 state, a triaxial PSM is needed, which employs deformed (1999). basis states. On the other hand, projected energy surface cal- [35] Z.-C. Gao, Y. Sun, and Y.-S. Chen, Phys. Rev. C. 74, 054303 culations have led to a picture of shape coexistence. A scheme (2006).