Annals of Physics 265, 101133 (1998) Article No. PH975766 Spectra and E2 Transition Strengths for N=Z EvenEven Nuclei in IBM-3 Dynamical Symmetry Limits with Good s and d Boson Isospins V. K. B. Kota Physical Research Laboratory, Ahmedabad 380 009, India E-mail: kotaprl.ernet.in Received June 12, 1997; revised October 3, 1997 Dynamical symmetries of IBM-3 (interacting boson model including isospin (T)) with not only good T but also good s and d boson isospins (T s, T d ) carry signatures typical of some heavy (proton rich) NtZ eveneven nuclei and therefore complete study of these symmetries is important for protorip line nuclei. With good (T s, T d ) there are three dynamical symmetries in IBM-3 and they are called [U d (5)U Td (3)]U Ts (3), [U d (15)#O d (15)]U Ts (3), and O sd (18)#O d (15)O Ts (3) limits; the last two symmetry limits carry :-particle-like correlations. For these three symmetry limits group chains, irrep labels, group generators, linear and quadratic Casimir operators, and energy formulas are given. For T=0 eveneven nuclei, in the three symmetry limits analytical formulas are given for B(E2)'s involving low-lying levels and also for the yrast band members. There are clear signatures of isospin both in spectra and B(E2)'s which can be subjected to experimental tests in the future. 1998 Academic Press 1. INTRODUCTION The U sd (6) interacting boson model (IBM-1) of Arima and Iachello, with its three dynamical U d (5), SU sd (3), and O sd (6) symmetries is established to be a standard model for describing low-lying quadrupole collective states in heavy (A>100) eveneven nuclei with active protons and neutrons occupying different oscillator shells [1, 2]. In practice usually the protonneutron IBM (pn-ibm or IBM-2) is used for numerical analysis of experimental data (see Refs. 2, 3 and references therein). With IBM-2 distinguishing proton and neutron bosons, the U sd (6) of IBM-1 goes to U sd (12) group for IBM-2. A major success of IBM-2 dynamical symmetries is iescribing scissors states in heavy eveneven nuclei [2, 4]. One has to go beyond IBM-2 for nuclei with protons and neutrons occupying the same oscillator shells, as isospiegrees of freedom cannot be ignored for these nuclei. The IBM-3 model that include s and d bosons with isospin T=1 and the IBM-4 model with s and d bosons carrying S=0, T=1 and S=1, T=0 degrees of freedom are appropriate for these nuclei; note that S is intrinsic spin. These models were introduced many years ago by Elliott and his collaborators 101 0003-491698 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
102 V. K. B. KOTA [57]. In the last two years, the possibilities with radioactive ion beam (RIB) facilities [8] led to the resurgence of interest in investigating and applying IBM-3 and IBM-4 models ietail [917]. In the near future the RIB facilities will allow us to study heavy nuclei with protons and neutrons filling the same major shells like proton rich nuclei with NtZt40 and lighter isotopes of tellurium (Z=52, A=110&134), xenon (Z=54, A=122&136), and barium (Z=56, A=130&138). Before data on such exotic nuclei become available, it is clearly desirable to have model predictions for these nuclei as they may provide guidance for new experiments to come. The IBM-3,4 models are appropriate for this purpose (other methodsmodels that are being used [18] are not discussed in this paper). The IBM-3 model which includes isospin in IBM and generates quadrupole collective states will suffice for heavy eveneven protorip line nuclei while for oddodd proton rich nuclei it is essential to have IBM-4. Restricting the present study to heavy N=Z eveneven nuclei and following the fact that major success of IBM is due to the dynamical symmetries admitted by the model [19], this paper deals with IBM-3 dynamical symmetries. The IBM-3 model includes s and d bosons with no intrinsic spin (S) as in IBM-1, but they are allowed to carry isospin T=1. The three T z values +1, &1, 0 correspond to protonproton, neutronneutron, and protonneutron pairs. With the three T z values, the U sd (6) of IBM-1 goes to U sd (18) group for IBM-3. To ensure symmetry among bosons, the symmetric irrep [N] of U(18) is taken; usually for one boson system isospin is labelled by t and for the N boson system, by T. For states with good angular momentum and isospin the U(18) model allows for two classes of symmetry schemes and they correspond to the basic associations (i) 1 U(18) [l=0(s)l=2(d)] O(3) [t=1] O(3) and (ii) [1] U(18) [[l=2(d)] O(3) [t=1] O(3) ] [[l=0(s)] O(3) [t=1] O(3) ][l=0(s)l=2(d)] O(3) [t=1] O(3). With the association (i) (i.e., with T a good quantum number) broadly speaking there are two dynamical symmetries in the U(18) model and they are: (i.a) U sd (18) #[U sd (6)#G#O d (3)] [U T (3)#O T (3)]; (i.b) U sd (18)#O sd (18)#[O sd (6)# O d (5)#O d (3)] [O T (3)]. The subgroups G in chain (i.a) correspond to the three IBM-1 symmetry limits (U(5), SU(3), O(6)), but the irreps involved are different. The above two chains are addressed in [7, 12, 20, 21] and they are not considered any further in this paper. With the association (ii), i.e. with not only T but also T s and T d good quantum numbers, there are three dynamical symmetry limits [14, 15] and they are hereafter, as in Ref. 15, called [U d (5)U Td (3)]U Ts (3), [U d (15)# O d (15)]U Ts (3), and O sd (18)#O d (15)O Ts (3) limits. A major significance of these symmetry limits is that they appear to be important for heavy NtZ proton rich nuclei and therefore they are studied ietail. Preliminary report on these symmetry limits is given in [1416] and the purpose of this paper is to give detailed results for spectra and E 2 transition strengths (B(E 2)'s) for T=0 nuclei in the three symmetry limits. Now we will give a preview. In Section 2 the three symmetry limits of IBM-3 with good (T s, T d ) are introduced. The corresponding group chains, group generators, irrep (state) labels,
N=Z EVENEVEN NUCLEI IN IBM-3 103 linear and quadratic Casimir operators for the various groups, energy formulas and structure of wavefunctions are given in Sections 2.12.3, respectively. Formulation for deriving analytical results for B(E 2)'s in T=0 nuclei is given for the three symmetry limits in Sections 3.13.3 respectively, together with explicit formulas for E2 strengths involving low-lying states. Given in Section 3.4 are results for yrast bands in the three symmetry limits. Discussed in Sections 2 and 3 are also the relationships between the three symmetry limits and the signatures that can be subjected to experimental tests. Finally, Section 4 gives some concluding remarks. 2. IBM-3 SYMMETRY LIMITS WITH GOOD (T s, T d ): SPECTRA Let us begin by introducing the single boson creation operators b - l, m l ; t, m t, where l=0 corresponds to the s boson and l=2 corresponds to the d boson; note that t=1. Similarly defined are the single bosoestruction operator b l, ml ; t, m t and b l, m l ; t, m t =(&1) 1+m l +m t bl, &ml ; t, &m t is a proper tensor, just as b - l, m l ; t, m t, with respect to angular momentum and isospin groups. The 324 double tensors (b - b l,1 l$, 1) L, T M, M T, generate the parent group U sd (18) of IBM-3. The scalar product with respect to both angular momentum and isospin is denoted as `` : '' and given two double tensors U k 1, k 2 and V k 1, k 2, their scalar product is defined by U k 1, k 2 : V k 1, k 2=(&1) k 1 +k 2 - (2k1 +1)(2k 2 +1) (U k 1, k 2 V k 1, k 2) 00. (1) The number operators n^ l and isospin operators T 1 l; +, for s and d bosons and the angular momentum operator L$ + follow from their elementary properties, n^ s=&s - : s~; n^ d=&d - : d ; N =n^ s+n^ s T 1 s; +=- 2(s - s~) 0, 1 0, +; T 1 d; +=- 10 (d - d ) 0, 1 0, +; T 1 +=T 1 s; ++T 1 d; + (2) L 1 + =- 30 (d - d ) 1, 0 +, 0. In (2) N is total boson number operator (the ``N'' used for denoting the boson number should not be confused with the ``N'' used for neutron number) and T 1 are + the generators of total isospin. The angular momentum and total isospin groups are O d (3) and O T (3), respectively, and they are generated by L 1 and T 1, respectively. + + The IBM-3 U sd (18) group admits U d (15)U Ts (3) as a subgroup with the U d (15) group generating the d-boson number and the U Ts (3) group generating the s-boson number n s. The O Ts (3) subgroup of U Ts (3) generates T s. The U d (15) group admits two subgroups. One is the U d (5)U Td (3) (first mentioned in Ref. 22) which corresponds to the decomposition of d-boson space into orbital and isospin parts and the other is the pairing subgroup O d (15) in U d (15) space. In the first chain the O Td (3) in U Td (3) generates T d while in the second chain the O Td (3) in
104 V. K. B. KOTA O d (15)#O d (5)O Td (3) generates T d. Thus these two chains, here after called [U d (5)U Td (3)]U Ts (3) and [U d (15)#O d (15)]U Ts (3) limits or limits I and II, respectively, preserve not only (T s, T d ) but also (n s, ). The third symmetry limit with good (T s, T d ) is generated via the U sd (18) space pairing group O sd (18) with O d (15)O Ts (3) subgroup. This limit is called O sd (18)#O d (15)O Ts (3) limit (or limit III). The ordering of the symmetry limits I, II, and III will be clear from the results in Sections 2, 3. It is important to remark that the chains II and III belong to a general class of groupsubgroup structures, i.e. U(N)#U(N a )U(N b ) #O(N a )O(N b ) and U(N)#O(N)#O(N a )O(N b ) with N=N a +N b. Similarly chains I and II contain the class U(NM)#U(N) U(M)#O(N) O(M) and U(NM)#O(NM)#O(N) O(M). We will return to these general structures in Sections 3 and 4. 2.1. The [U d (5)U Td (3)]U Ts (3) Limit I The group chain and irrep labels for the [U d (5)U Td (3)]U Ts (3) limit I are U sd (18)#{U d (15) d(5) # O d (5) #O d (3) # _U [f] [v 1 v 2 ] L & N # T _U d (3)#O Td (3) [f] T d &= { U T s (3)#O Ts (3) n s T s = O d (3) O T (3) L T. (3) The states corresponding to of U d (15) are totally symmetric and therefore in (3) the U d (5) and U Td (3) irreps are labeled by the same [f], and [f] contains a maximum of three rows: [f]=[f 1 f 2 f 3 ]; f 1 + f 2 + f 3 =. The [f 1 f 2 f 3 ] of U Td (3) correspond to SU(3) irrep (*+)=(f 1 & f 2, f 2 & f 3 ) and, therefore, the reduction of [f] T d follows directly from Elliott's result [23], (*+) T d : K=min(*, +), min(*, +)&2,..., 0 or 1, T d =K, K+1, K+2,..., K+max(*, +) for K{0, (4) T d =max(*, +), max(*, +)&2,..., 0 or 1 for K=0. Similarly n s of U Ts (3) correspond to (*+)=(n s 0) and, therefore, T s =n s, n s &2,..., 0 or 1. The [f 1 f 2 f 3 ] [v 1 v 2 ] L follow from the results worked out in the context of the SU sdg (5) limit of sdgibm [24, 25]. For example, some [f 1 f 2 f 3 ] [v 1 v 2 ] (U(5)#O(5)) reduction formulas are
N=Z EVENEVEN NUCLEI IN IBM-3 105 [f 1 ] : [v], v= f 1, f 1 &2,..., 0 or 1 [( f 1 &1)2] [f 1,1] [v 1 v 2 ]= : r=0 [ f 1 2]&1 [f 1,2] [v 1 v 2 ]= : r=0 [ f 1 2]&1 [ f 1 &2r, 1] : r=0 [( f 1 &1)2]&1 [ f 1 &2r, 2] : r=0 [( f 1 &4)2] [ f 1 ][ f 1 &2[ f 1 2]] : [f 1,1,1] [v 1 v 2 ]=[11][21] }}} [ f 1,1]. [ f 1 &2r&1] r=0 [ f 1 &2r&1, 1] (5) [ f 1 &2r&2] 2 In these equations [X2] is the integer part of X2 and there is multiplicity in the reductions in the case of [f 1,2]. Some numerical results that are sufficient for 6 are [221] [22][21][1], [33] [33][31][11], and [321] [32][31][22][21][2][11]. The [v 1 v 2 ] L is obtained from the known result [1] for [v] L and the reduction for the Kronecker product [m][n], n [m][n]= : k=0 n&k : r=0 [m&n+k+2r, k] for mn. (6) The quantum numbers in [U d (5)U Td (3)]U Ts (3) limit for 4 are given in Table I. The reductions (5, 6) and the results in Table I can be verified using the dimension formulas d([m])=(m+4)(m+3)(m+2)(m+1)24, d([v 1 v 2 ])= (v 1 &v 2 +1)(v 1 +v 2 +2)(2v 1 +3)(2v 2 +1)6, and d(l)=(2l+1). As Table I shows, an extra label is required for L for 4. Similarly, another label is required for denoting the multiple occurrences of [v 1 v 2 ] for 6. For example, for =6 the O(5) irrep [2] occur twice in the U(5) irrep [42] (see (5)). Leaving these multiplicity labels aside, the basis states are labelled by N; [f][v 1 v 2 ]L; (T s T d ) T). The generators and quadratic Casimir operators for various groups in (3) are identified and they are, 1 1 Normal order form for (d - d ) l, t :(d - d ) l, t provides useful checks for the formulas in Eq. (7), (d - d ) l, t :(d - d ) l, t = (2l+1)(2t+1) n^ d + : 2(2l+1)(2t+1) 15 { 2 2 l 2 2 L={ 1 1 t L, T, L+T=even _ - (2L+1)(2T+1) [(d - d - ) L, T (d d ) L, T ] 0, 0. 2 1 1 T=
106 V. K. B. KOTA TABLE I State Labels for [U d (5)U Td (3)]U Ts (3) Limit with 4 U d (15) U d (5) O d (5) O d (3) O Td (3) O Ts (3) [f] [v 1 v 2 ] L T d T s 0 [0] [0] 0 0 N, N&2,..., 0 or 1 1 [1] [1] 2 1 N&1, N&3,..., 0 or 1 2 [2] [2] 2,4 0,2 N&2, N&4,..., 0 or 1 [0] 0 0,2 N&2, N&4,..., 0 or 1 [11] [11] 1,3 1 N&2, N&4,..., 0 or 1 3 [3] [3] 0,3,4,6 1,3 N&3, N&5,..., 0 or 1 [1] 2 1,3 N&3, N&5,..., 0 or 1 [21] [21] 1,2,3,4,5 1,2 N&3, N&5,..., 0 or 1 [1] 2 1,2 N&3, N&5,..., 0 or 1 [111] [11] 1,3 0 N&3, N&5,..., 0 or 1 4 [4] [4] 2,4,5,6,8 0,2,4 N&4, N&6,..., 0 or 1 [2] 2,4 0,2,4 N&4, N&6,..., 0 or 1 [0] 0 0,2,4 N&4, N&6,..., 0 or 1 [31] [31] 1,2,3 2,4,5 2,6,7 1,2,3 N&4, N&6,..., 0 or 1 [11] 1,3 1,2,3 N&4, N&6,..., 0 or 1 [2] 2,4 1,2,3 N&4, N&6,..., 0 or 1 [22] [22] 0,2,3,4,6 0,2 N&4, N&6,..., 0 or 1 [2] 2,4 0,2 N&4, N&6,..., 0 or 1 [0] 0 0,2 N&4, N&6,..., 0 or 1 [211] [21] 1,2,3,4,5 1 N&4, N&6,..., 0 or 1 [11] 1,3 1 N&4, N&6,..., 0 or 1 Note. Given N, the s boson number is n s =N&. For a given (T s, T d ), T takes values T s &T d T T s +T d. Results in the text and the following reductions for [v 1 v 2 ] L complete the table for =5,6 cases. [5] L=2,4,5,6,7,8,10 [41] L=1,2,3 2,4 2,5 2,6 2,7 2,8,9 [32] L=1,2 2,3,4 2,5 2,6,7,8 [6] L=0,3,4,6 2,7,8,9,10,12 [51] L=1,2,3 2,4 2,5 3,6 2,7 3,8 2,9 2,10,11 [42] L=0,1,2 2,3 2,4 3,5 2,6 3,7 2,8 2,9,10 [33] L=1,3 2,4,5,6,7,9.
N=Z EVENEVEN NUCLEI IN IBM-3 107 U d (15) : (d - d ) l, t m l, m t ; l=0&4, t=0&2 C 2 (U d (15))=: (d - d ) l, t :(d - d ) l, t l, t (C 2 (U d (15))) =nd ( +14) U d (5) : (d - d ) l,0 m l,0 ; l=0&4 C 2 (U d (5))=3 : (d - d ) l,0 :(d - d ) l,0 l (C 2 (U d (5))) [f] =[ f 1 ( f 1 +4)+ f 2 ( f 2 +2)+ f 2 3 ] O d (5) : (d - d ) l,0 m l,0 ; l=1, 3 C 2 (O d (5))=6 : l=odd (C 2 (O d (5))) [v 1 v 2 ] =[v 1 (v 1 +3)+v 2 (v 2 +1)] (d - d ) l,0 :(d - d ) l,0 (7) U Td (3) : (d - d ) 0, t 0, m t ; t=0, 1, 2 C 2 (U Td (3))=5 : (d - d ) 0, t :(d - d ) 0, t t (C 2 (U Td (3))) [f] =[ f 1 ( f 1 +2)+ f 2 2 + f 3( f 3 &2)] U Ts (3) : (s - s~) 0, t 0, m t ; t=0,1,2 C 2 (U Ts (3))=: (s - s~) 0, t :(s - s~) 0, t (C 2 (U Ts (3))) n s =ns (n s +2). The linear Casimir invariants of U d (15), U d (5), U Td (3), and U Ts (3) groups are n^ d, n^ d, n^ d, n^ s, respectively. Similarly, the Casimir operators of O d (3), O Td (3), O Ts (3), and O T (3) are L 1 : L 1, T 1 d : T 1 d, T 1 s : T 1 s, and T 1 : T 1 with eigenvalues L(L+1), T d (T d +1), T s (T s +1), and T(T+1), respectively. These and the results in (7) give the following energy formula (with nine free parameters) in the [U d (5)U Td (3)] U Ts (3) limit, E [[Ud (5)U Td (3)]U Ts (3)](N; [f][v 1 v 2 ] L; (T s T d ) T) t =E 0 += d +: ( +14)+;[ f 1 ( f 1 +4)+ f 2 ( f 2 +2)+ f 2 3 ] +#[v 1 (v 1 +3)+v 2 (v 2 +1)]+# 1 L(L+1)+# 2 T d (T d +1) +# 3 T s (T s +1)+# 4 T(T+1). (8)
108 V. K. B. KOTA Using the results in Table I and the energy formula (8), typical spectra for T=0 and T=N cases with N=8 are constructed and the results are shown in Fig. 1. For T=0 necessarily T s =T d while T s =n s, T d = for T=N. The T=N spectrum, as expected, is same as IBM-1 U(5) limit spectrum with one phonon 2 + state, two phonon 0 +,2 +,4 + triplet etc. On the other hand, N=Z eveneven nuclei with T=0 will have the two phonon 0 +,2 +,4 + triplet occurring twice and they will have T s =0 and 2, respectively. Similarly, for the three phonon states, besides the IBM-1 states appearing twice with T s =1 and 3, respectively, there are states with L? =1 +,3 +,4 +,5 + occurring with T s =1 and 2 + with T s =1 twice, respectively. These extra states carry signatures of isospin and this feature is clearly seen in B(E2)'s also as discussed below. Using the results in Table I it is seen that, in general, there will be many more extra states for T=1, 2,... cases. Finally it is useful to point out that, just as the case with U(5) limits of IBM-1,2, in general the states defined by limit I can be used as a basis for detailed numerical IBM-3 calculations with a general IBM-3 hamiltonian (H IBM-3 ) and transition operators (see Appendix A for the form of H IBM-3 ). 2.2. The [U d (15)#O d (15)]U Ts (3) Limit II The group chain and irrep labels for the [U d (15)#O d (15)]U Ts (3) limit (II) are d (15)#O d (15) U sd (18)#{U N $ # _ O d(5) #O d (3) [v 1 v 2 ] L & _ O T d (3) T d & = T {U s (3)#O Ts (3) n s T s = # O d (3) O T (3) L T. (9) The generators and Casimir operators for all the groups in (9), except for the O d (15) group, are already given and for the O d (15) group they are O d (15) : (d - d ) l, t m l, m t ; l=0&4, t=0&2, l+t odd C 2 (O d (15))=2 : l, t, l+t=odd (C 2 (O d (15))) [$] =$($+13). (d - d ) l, t :(d - d ) l, t (10)
N=Z EVENEVEN NUCLEI IN IBM-3 109 Fig. 1. Typical spectrum in [U d (5)U Td (3)]U Ts (3) limit for T=0 and T=N with N=8. By the side of L? the (T d, T s ) values are shown. The parameters in (8) are chosen to be = d =1 MeV, ;=&20 kev, #=&25 kev, and E 0 =:=# 1 =# 2 =# 3 =# 4 =0. The figure is from Ref. 15. Reprinted from Phys. Lett. B 399, V. K. B. Kota, p. 185, Copyright 1997, with permission from Elsevier Science. As the states corresponding to of U d (15) are totally symmetric, the O d (15) irreps are labelled by ``seniority'' quantum number $ (i.e., by the symmetric irrep [$] of O d (15) group) and therefore in (9), $=, &2, &4,..., 0 or 1. (11) The results of limit I give the reduction [v 1 v 2 ]T d and by successive substraction, starting with =0 and using (11), produce $ [v 1 v 2 ]T d reductions. All other quantum numbers are already worked out in the context of limit I. Thus, the results of limit I give the results for limit II. The quantum numbers in [U d (15)#O d (15)] U Ts (3) limit for 4 are given in Table II. Ignoring the extra labels that are required for 4, the basis states are labelled by N; $[v 1 v 2 ]L; (T s T d ) T). A significant result here is that these states have :-particle like correlations as the O d (15) group leaves invariant the spinisospin scalar two boson system d - : d - (besides the trivial s - : s - correlation which is present also in [U d (5)U Td (3)] U Ts (3) limit.) This result follows from (B1) of Appendix B, N; $[v 1 v 2 ]L; (T s T d )T) =_ 1($+(152)) 1(T s +(32)) 2 N&$&T s 1((nd &$+2)2) 1((N& &T s +2)2) _1(( +$+15)2) 1((N& +T s +3)2)&12 _(s - : s - ) (N& &T s )2 (d - : d - ) ( &$)2 $+T s ; $$[v 1 v 2 ]L; (T s T d ) T). (12)
110 V. K. B. KOTA TABLE II State Labels for [U d (5)#O d (15)]U Ts (3) with 4 U d (15) O d (15) O d (5) O d (3) O Td (3) O Ts (3) $ [v 1 v 2 ] L T d T s 0 0 [0] 0 0 N, N&2,..., 0 or 1 1 1 [1] 2 1 N&1, N&3,..., 0 or 1 2 2 [2] 2,4 0,2 N&2, N&4,..., 0 or 1 [0] 0 2 N&2, N&4,..., 0 or 1 [11] 1,3 1 N&2, N&4,..., 0 or 1 0 [0] 0 0 N&2, N&4,..., 0 or 1 3 3 [3] 0,3,4,6 1,3 N&3, N&5,..., 0 or 1 [21] 1,2,3,4,5 1,2 N&3, N&5,..., 0 or 1 [11] 1,3 0 N&3, N&5,..., 0 or 1 [1] 2 1,2,3 N&3, N&5,..., 0 or 1 1 [1] 2 1 N&3, N&5,..., 0 or 1 4 4 [4] 2,4,5,6,8 0,2,4 N&4, N&6,..., 0 or 1 [31] 1,2,3 2,4,5 2,6,7 1,2,3 N&4, N&6,..., 0 or 1 [22] 0,2,3,4,6 0,2 N&4, N&6,..., 0 or 1 [21] 1,2,3,4,5 1 N&4, N&6,..., 0 or 1 [2] 2,4 0,2,4 N&4, N&6,..., 0 or 1 [11] 1,3 1,2,3 N&4, N&6,..., 0 or 1 [0] 0 0,2,4 N&4, N&6,..., 0 or 1 2 [2] 2,4 0,2 N&4, N&6,..., 0 or 1 [11] 1,3 1 N&4, N&6,..., 0 or 1 [0] 0 2 N&4, N&6,..., 0 or 1 0 [0] 0 0 N&4, N&6,..., 0 or 1 Note. See footnote to Table I. Using the results of Section 2.1, the table can be completed for any and $6. The pair structure in (12) leads to, as seen below, some observable effects in the spectra and B(E2)'s. Equations (7), (10) give the energy formula (with nine free parameters) in the [U d (15)#O d (15)]U Ts (3) limit, E [[Ud (15)#O d (15)]U Ts (3)](N; $[v 1 v 2 ]L; (T s T d ) T) =E$ 0 +=$ d +:$ ( +14)+;$ $($+13) +#$[v 1 (v 1 +3)+v 2 (v 2 +1)]+#$ 1 L(L+1) +#$ 2 T d (T d +1)+#$ 3 T s (T s +1)+#$ 4 T(T+1) (13) Using the results in Table II and the energy formula (13), typical spectra in limit II for T=0 and T=N cases with N=8 are constructed as shown in Fig. 2. In this
N=Z EVENEVEN NUCLEI IN IBM-3 111 Fig. 2. Typical spectrum in [U d (15)#O d (15)]U Ts (3) limit for T=0 and T=N with N=8. By the side of L? the (T d, T s ) values are shown. The parameters in (13) are chosen to be =$ d =850 kev, ;$=10 kev, #$=&20 kev, and E$ 0 =:$=#$ 1 =#$ 2 =#$ 3 =#$ 4 =0. limit II also, just as in the case of limit I, the T=N spectrum is the same as the IBM-1 U(5) limit spectrum with $. The two, three, and higher phonon 2 + states carry isospin signatures, as can be seen from the T=0 spectrum in Fig. 2. They also differentiate between the limits I and II. For example, the two 2-phonon 0 + states have T s =0 and 2, respectively, but all other quantum numbers are the same in limit I, while the two 0 + states are further distinguished by the $ quantum number in limit II. There are similar differences in 3-phonon states and they may serve as a signature for distinguishing the two limits. In the situation that the :-particle-like correlation given by (12) is present in real nuclei, then ;$ in (13) takes a positive value as in Fig. 2. Finally, it is important to mention that the groupsubgroup structure common to limits I and II is U(NM)#U(N)U(M)#O(N)O(M) and U(NM) #O(NM)#O(N)O(M) with N=5 and M=3. 2.3. The O sd (18)#O d (15)O Ts (3) Limit III The seniority subgroup O sd (18) of U sd (18) leads to a symmetry limit with good (T s, T d )aso sd (18) contains O d (15)O Ts (3) as a subgroup and O d (15) contains, as in limit II, O d (5)O Td (3) as a subgroup. The limits II and III correspond to the general group subgroup structure U(N)#U(N a )U(N b )#O(N a )O(N b ) and U(N)#O(N)#O(N a )O(N b ) with N a =15 and N b =3; see Appendix C. Thus limit II leads to the identification of limit III (Ginocchio [14] did not follow this route). The group chain and irrep labels for O sd (18)#O d (15)O Ts (3) limit III are,
112 V. K. B. KOTA d(15) U sd (18)#O sd (18)# {O $ N # _ O d(5) #O d (3) [v 1 v 2 ] L & T _O d (3) T d &= { O T s (3) T s = # O d(3) O T (3) L T. (14) As the states that correspond to a given N of U sd (18) are totally symmetric, the O d (18) irreps are labelled by ``seniority'' quantum number (corresponding to symmetric irrep [ ] ofo sd (18) group) and therefore, =N, N&2, N&4,..., 0 or 1. (15) The results of [U d (15)#O d (15)]U Ts (3) limit give the reduction N ($, T s ) and by successive substraction starting with N=0 and using (15) produce the ($, T s ) reductions (see also Appendix C), $=, &1,..., 0 T s = &$, &$&2,..., 0 or 1; (16) =2r ab +$+T s, r ab =0,1,...,[ 2]. All other quantum numbers appearing in the group chain (14) are already worked out in the context of limit II. The quantum numbers for =N and $4 are given in Table III. Ignoring the extra labels that are required for $4, the basis states are labelled by N; $[v 1 v 2 ] L; (T s T d ) T). Except for the O sd (18) group, for other groups in (14) the generators and quadratic Casimir operators are already given in (7), (10). For the O sd (18) group we have (2, t) O sd (18) : P (m 2, m t ) =(s- d +(&1) t d - s~) 2, t m 2, m t, t=0&2, (d - d ) l, t m l, m t, l=0&4, t=0&2, l+t odd, (s - s~) 0, t 0, m t, t=1, (17) C 2 (O sd (18))=: (&1) t P (2, t) : P (2, t) +C 2 (O d (15))+C 2 (O Ts (3)) (C 2 (O sd (18))) = ( +16). t The results given in Appendix C allow one to write O sd (18)#O d (15)O Ts (3) states in terms of [U d (15)#O d (15)]U Ts (3) states, N; $[v 1 v 2 ] L; (T s T d ) T)=: C N,, $, Ts (15, 3) N;, $[v 1 v 2 ] L; (T s T d )T). (18)
N=Z EVENEVEN NUCLEI IN IBM-3 113 The expansion coefficients C N,, $, T s (15, 3) for any N and are derived in a more general context in Ref. 26, 27. Formulas for C's with =N and =N&2 are given by (C2) and (C3). Just as in the case of limit II (see (12)), the states in limit III will have :-particle-like correlations. Using the O sd (18) generators it is easily seen that the O sd (18) group leaves invariant the spinisospin scalar two boson system I - = (s - : s - &d - : d - ). Using the results in Appendices B and C, the O sd (18)#O d (15) O Ts (3) states with any can be written in terms of [U d (15)#O d (15)]U Ts (3) states with N= as N; $[v 1 v 2 ] L; (T s T d ) T) = _1( +9)2N& 1 \N& +2 2 + 1 \ N+ 2 +9 +& 12 (I - ) (N& )2 _: C,, $, T s (15, 3) ;, $[v 1 v 2 ] L; (T s T d )T). (19) It is possible is to further apply (12) to Eq. (19) as done in Ref. 14. The energy formula in limit III is (with eight free parameters), E [0sd (18)#O d (15)O Ts (3)](N; $[v 1 v 2 ] L; (T s T d ) T) =E" 0 +:" ( +16)+;" $($+13)+#"[v 1 (v 1 +3)+v 2 (v 2 +1)]+#" 1 L(L+1) +#" 2 T d (T d +1)+#" 3 T s (T s +1)+#" 4 T(T+1) (20) Using the results in Table III and the energy formula (20), typical spectrum for T=0, N=8 is constructed as shown in Fig. 3. The T=0 spectrum clearly exhibits Fig. 3. Typical spectrum in O sd (18)#O d (15)O Ts (3) limit for T=0, N=8. By the side of L? the (T d, T s ) values are shown. The parameters in (20) are chosen to be :"=&30 kev, ;"=50 kev, #"=&25 kev, E" 0 =&192:", and #" 1 =#" 2 =#" 3 =#" 4 =0. The T=N spectrum in this limit, just as in Figs. 1 and 2, is identical to IBM-1 U(5) limit spectrum.
114 V. K. B. KOTA TABLE III State Labels for the O sd (18)#O d (15)O Ts (3) Limit with =N, N&2 and $4 O sd (18) O d (15) O d (5) O d (3) O Td (3) O Ts (3) $ [v 1 v 2 ] L T d T s N 0 [0] 0 0 N, N&2,..., 0 or 1 1 [1] 2 1 N&1, N&3,..., 0 or 1 2 [2] 2,4 0,2 N&2, N&4,..., 0 or 1 [11] 1,3 1 N&2, N&4,..., 0 or 1 [0] 0 2 N&2, N&4,..., 0 or 1 3 [3] 0,3,4,6 1,3 N&3, N&5,..., 0 or 1 [21] 1,2,3,4,5 1,2 N&3, N&5,..., 0 or 1 [11] 1,3 0 N&3, N&5,..., 0 or 1 [1] 2 1,2,3 N&3, N&5,..., 0 or 1 4 [4] 2,4,5,6,8 0,2,4 N&4, N&6,..., 0 or 1 [31] 1,2,3 2,4,5 2,6,7 1,2,3 N&4, N&6,..., 0 or 1 [22] 0,2,3,4,6 0,2 N&4, N&6,..., 0 or 1 [21] 1,2,3,4,5 1 N&4, N&6,..., 0 or 1 [2] 2,4 0,2,4 N&4, N&6,..., 0 or 1 [11] 1,3 1,2,3 N&4, N&6,..., 0 or 1 [0] 0 0,2,4 N&4, N&6,..., 0 or 1 N&2 0 [0] 0 0 N&2, N&4,..., 0 or 1 1 [1] 2 1 N&3, N&5,..., 0 or 1 2 [2] 2,4 0,2 N&4, N&6,..., 0 or 1 [11] 1,3 1 N&4, N&6,..., 0 or 1 [0] 0 2 N&4, N&6,..., 0 or 1 Note. See footnotes to Tables I and II and Eq. (16) for completing the table for any and $6. both IBM-1 U(5) and O(6) features. However, for T=N, just as in the case of limits I and II, the spectrum is same as IBM-1 U(5) limit spectrum with =N and $. The third equality in (16) shows that <N is not possible for T=N. There are clear differences in T=0 spectra between the limits I, II, and III. It is easily seen that the limit III spectrum beyond 2 + 1 state is different from the spectra generated by limits I and II. For example in the limits I and II, the 0 + 2 states with T s =0, 2 belong to =2 while in limit III the state with T s =0 is moved high in energy as it belongs to =N&2. As Ginocchio pointed out, some of the features given by the limit III spectra are similar to those observed in 64 Ge. The observed E 4 +E + 1 2 1 ratio is 2.1 and for the two T s =0, 2 states, (20) gives 1.88 and 2.25 and their average reproduces the observed value. The key point here is that the NtZ nuclei
N=Z EVENEVEN NUCLEI IN IBM-3 115 seem to produce E 4 +E + 1 2 1 ratio close to O(6) or #-soft limit (in IBM-1 U(5) limit gives, O(6) limit gives {({+3) and SU(3) limit gives L(L+1) spectrum). Finally, if :-particle type correlations given by I - is present in real nuclei, then =N&2, $=0, 0 + state should not be too high in energy. Therefore study of excited 0 + states will give information about these correlations. 3. E2 TRANSITION STRENGTHS IN THE THREE SYMMETRY LIMITS 3.1. [U d (5)U Td (3)]U Ts (3) Limit I For T=0 nuclei, the E2 transitions in [U d (5)U Td (3)]U Ts (3) limit (in fact, in the other two limits also), to lowest order, are generated by the operator T E 2 =eeff (d - s~+s - d ) 2, 0 +, 0, (21) where e eff is effective charge. In this symmetry limit, the reduced matrix elements of (s - d ) 2, 0 are given by ( &1, [f f 1, f f 2, f f 3 ][v f 1, v f 2 ] L f ;(T f d, T f s ) T=0 &(s- d ) 2.0 & _, [f i 1, f i 2, f i 3 ][vi 1, vi 2 ] Li ;(T i d, T i s ) T=0) O d (3) 1 = - 3(2T i +1)(2T f+1) (N&+1, T f s &s- & N&, T i ) s O Ts (3) s s _( &1, [f f 1, f f 2, f f 3 ][v f 1, v f 2 ] L f ; T f d &d & _, [f i 1, f i 2, f i 3 ][vi 1, vi 2 ] Li ; T i d ) O d (3), O Td (3). (22) In (22) (& &) Od (3), O Td (3) is defined by applying WignerEckart theorem with respect to both O d (3) and O Td (3) groups. Given a double tensor V k 1, k 2, (L f M f T f M Tf V k l, k t L m l, m i M i T i M Ti ) t = (L im i k l m l L f M f ) (T i M Ti k t m t T f M Tf ) - 2L f +1-2T f +1 (L f T f & V k l, k t &Li T i ) Od (3), O Td (3). Using the result that (N& +1, T f s & s- &N&, T i s ) O Ts (3)=- (2T f s +1) ((N&,0) T i (10) 1 &(N&n s d+1, 0) T f) s SU(3)#O(3) (N& +1_ s - _N& ) and evaluating (N& +1_ s - _N& ) via s - : s~=&n^ s, gives (N& +1, T f s & s- &N&, T i s ) O Ts (3) =- (N& +1)(2T f +1) ((N&n s d,0)t i (10) 1 &(N&n s d+1, 0) T f ) s SU(3)#O(3). (23)
116 V. K. B. KOTA Note that ( & ) SU(3)#O(3) is SU(3)#O(3) reduced Wigner coefficient [28]. The d reduced matrix element in (22) is related to d - reduced matrix element by a phase factor. Using the fact that the tensorial nature of d - with respect to the chain [U d (5)#O d (5)#O d (3)][U Td (3)#O Td (3)] is T [1][1] 2; [1]1 and applying Racach's factorization lemma [29] gives (, [f f 1, f f 2, f f 3 ][v f 1, v f 2 ] L f ; T f d & d - & &1, [f i 1, f i 2, f i 3][v i 1, v i 2] L i ; T i d) Od (3), O Td (3) =- (2L f +1)(2T f d +1) ([f f 1, f f 2, f f 2 ] _d - _ &1, [f i 1, f i 2, f i 3 ]) _, f i, f i ] 1 2 3 [fi [v i, 1 vi ] 2 [1]" [f f, f f, f f ] 1 2 3 [v f, v f] 1 2 U(5)#O(5) _, 1 vi ] 2 [1] [vi L i 2 " [v f, v f ] 1 2 L f O(5)#O(3) _(( f i 1 & f i 2, f i 2 & f i 3 ) T i d (10) 1& ( f f 1 & f f 2, f f 2& f f 3 ) T f d ) SU(3)#O(3). (24) The evaluation of the triple barred and the various double barred coefficients in (24) is discussed below. Combining (22), (23), and (24) gives the expression for B(E 2)'s, B(E2;, [f i 1, f i 2, f i 3 ][vi 1, vi 2 ] Li ;(T i d, T i s ) T=0 &1, [f f 1, f f 2, f f 3 ][v f 1, v f 2 ] L f ;(T f d, T f s ) T=0) =(e eff ) 2 \ N&+1 3 + _ (, [f i 1, f i 2, f i 3 ] _d - _ &1, [f f 1, f f 2, f f 3 ]) 2 f _, f f, f f ] 1 2 3 }[f [v f, v f ] 1 2 [1]}} [fi, f i, f i ] 1 2 3 f _, v f ] 1 2 [1] }[v L f 2 }} [vi, 1 vi ] 2 2 L } i } 2 [v i, 1 vi ] 2 O(5)#O(3) U(5)#O(5) _ (( f f 1 & f f 2, f f 2 & f f 3 ) T f d (10) 1& ( f i 1 & f i 2, f i 2& f i 3 ) T i d ) 2 SU(3)#O(3) _ ((N&,0)T i s (10) 1& (N&+1, 0) T f s ) 2 SU(3)#O(3). (25) The U(5)#O(5) and O(5)#O(3) reduced Wigner coefficients appearing in (24, 25) for all cases with 3 are available from the tabulations generated for applying the U(6) U(20) symmetry schemes of IBFM [30]. Similarly the needed SU(3)# O(3) reduced Wigner coefficients in (25) are available in [28]; see (31) below. Following closely the method due to Arima and Iachello [31] the triple barred coefficients in (24, 25) are evaluated for all cases with 3. To this end =3
N=Z EVENEVEN NUCLEI IN IBM-3 117 states (states with =1, 2 are simple to write 2 ) with L d =6, T=3 and L d =5, T d =2 are constructed. Note that these two states (see Table I) correspond uniquely to the U d (5) irreps [3] and [21], respectively. Ignoring the m-quantum numbers, the states are =3, L d =6, T d =3)= 1-6 [d - (d - d - ) L$ d=4, T$ d=2 ] Ld=6, T d=3 0) =3, L d =5, T d =2)= 1-3 [d - (d - d - ) L$ d=4, T$ d=2 ] Ld=5, T d=2 0) (26) The normalization factors in (26) follow from the commutation relations for single boson creation and destruction operators. Evaluating (n f =3 d - d n i d =2 ) for the =3 states in (26) and applying (24), results given in Table IV are obtained. Let us first consider the selection rules for B(E2)'s. The E2 operator in (21) gives the trivial selection rules 2 =\1, L 2 and T d 1. In addition, it is seen from (25) that the selection rules for the U d (5) and O d (5) irreps are [f i, f i, f i ] # 1 2 3 [f f, f f, f f ][1] and 1 2 3 [vi, 1 vi ]#[v f, v f ][1] for n 2 1 2 d &1 transitions. These Kronecker products are given by the rules [25]: [f 1, f 2, f 3 ] [1] [f 1 +1, f 2, f 3 ][f 1, f 2 +1, f 3 ] (for f 1 > f 2 ) [f 1, f 2, f 3 +1] (for f 2 > f 3 ) [v 1, v 2 ][1] [v 1 +1, v 2 ] (27) [v 1, v 2 +1][v 1 &1, v 2 ] (for v 1 >v 2 )] [v 1, v 2 ][v 1, v 2 &1] (for v 2 >0)]. For example, because of the O d (5) selection rules the 2 + of [21] irrep with =3 (see Fig. 1) cannot decay to the =2, 0 + state. Using (25), results in Table IV and the tabulations in Ref. 30 analytical results for all cases with 3 are derived. For =1 =0 and =2 =1 cases we have 2 For =1, 2, =1, L d =2, M; T d =1, M T )=d - M, M T 0) and =2, L d, M; T d, M T )=(1-2) _(d - d - ) L d, T d 0); for n M, M d =2, L d +T d is a even integer. The L d, T d values uniquely define the U d (5) and T O d (5) quantum numbers (see Table I). Evaluating the matrix elements (, d - M, M n T d &1, ) for =1, 2 and applying (24) directly gives (n f d =1, L f d =2; T f d =1& d - &n i d =0, L i d =0; T i d =0) O d (3), O Td (3)=- 15 (n f d =2, L f d ; T f d & d - &n i d =1, Li d =2; T i d =1) O d (3), O Td (3)=- 2(2L f d +1)(2T f d +1)
118 V. K. B. KOTA TABLE IV (, [f i 1, f i 2, f i 3]_ d - _ &1, [f f 1, f f 2, f f 3 ]) ( =1[1]_ d - _ =0[0])=1 ( =2[2]_ d - _ =1[1])=- 2 ( =3[3]_ d - _ =2[2])=- 3 ( =3[21]_ d - _ =2[2])=- 32 TABLE V B(E2)'s for =3 =2 Transitions in Limit I [f i ] [v i 1 vi ] 2 Li T i d [f f ] [v f v f ] 1 2 L f T f d B(E2;i f ) RED [3] [3] 6 1 [2] [2] 4 0 (53)(N&2) - 65 6 1 4 2 (815)(N+1) &2-13 6 3 4 2 (95)(N&4) - 273 4 1 4 0 (5063)(N&2) - 1507 4 1 4 2 (1663)(N+1) &2-307 4 3 4 2 (67)(N&4) 3-10 4 1 2 0 (5563)(N&2) - 1657 4 1 2 2 (88315)(N+1) &2-337 4 3 2 2 (3335)(N&4) 3-11 3 1 4 0 (1021)(N&2) &- 10 3 1 4 2 (16105)(N+1) - 8 3 3 4 2 (1835)(N&4) &- 42 3 1 2 0 (2521)(N&2) 5 3 1 2 2 (821)(N+1) &2-5 3 3 2 2 (97)(N&4) - 105 0 1 2 0 (53)(N&2) - 5 0 1 2 2 (815)(N+1) &2 0 3 2 2 (95)(N&4) - 21 [3] [1] 2 1 [2] [2] 4 0 (47)(N&2) 2-157 2 1 4 2 (32175)(N+1) &4-37 2 3 4 2 (108175)(N&4) 6 2 1 2 0 (2063)(N&2) 10-21 2 1 2 2 (32315)(N+1) &(43) - 157 2 3 2 2 (1235)(N&4) 2-5
N=Z EVENEVEN NUCLEI IN IBM-3 119 TABLE VContinued [f i ] [v i 1 vi 2 ] Li T i d [f f ] [v f 1 v f 2 ] L f T f d B(E2;i f ) RED [3] [1] 2 1 [2] [0] 0 0 (79)(N&2) - 353 2 1 0 2 (56225)(N+1) &2-73 2 3 0 2 (2125)(N&4) 7 [21] [21] 1 1 [2] [2] 2 0 (23)(N&2) - 6 1 1 2 2 (13)(N+1) - 152 2 1 2 0 (37)(N&2) 3-57 2 1 2 2 (314)(N+1) 152-7 2 1 4 0 (521)(N&2) &5-7 2 1 4 2 (542)(N+1) &(52) - 57 3 1 2 0 (421)(N&2) 2 3 1 2 2 (221)(N+1) - 5 3 1 4 0 (1021)(N&2) - 10 3 1 4 2 (521)(N+1) 5-2 4 1 2 0 (2063)(N&2) &2-157 4 1 2 2 (1063)(N+1) &5-37 4 1 4 0 (2263)(N&2) - 667 4 1 4 2 (1163)(N+1) - 16514 5 1 4 0 (23)(N&2) - 22 5 1 4 2 (13)(N+1) - 552 [21] [1] 2 1 [2] [2] 2 0 (19)(N&2) &- 53 2 1 2 2 (118)(N+1) &(56) - 3 2 1 4 0 (15)(N&2) &- 3 2 1 4 2 (110)(N+1) &- 152 [21] [1] 2 1 [2] [0] 0 0 (1645)(N&2) 4-3 2 1 0 2 (845)(N+1) (23) - 15 Note. The last column labelled ``RED'' corresponds to (n i d =3, [fi ][v i 1, vi 2 ] Li ; T i d & d - &n f d =2[ff ][v f 1, v f 2 ]L f ;T f d ) O d (3), O Td (3), T s =T d, n s =N&, and B(E2)'s are in (e 2 eff3) units.
120 V. K. B. KOTA B(E2; =1, 2 +, T s =1, T=0 =0, 0 +, T s =0, T=0)= N 3 (e eff) 2 B(E2; =2, L i, T s =0, T=0 =1, 2 +, T s =1, T=0)= 2(N+1) 9 B(E2; =2, L i, T s =2, T=0 =1, 2 +, T s =1, T=0)= 4(N&2) 9 (e eff ) 2 (28) (e eff ) 2 where L i =0, 2, 4. The important result then is that the B(E2) ratio B(E2; =2 =1)B(E2; =1 =0)=2 in the case of IBM-1 U(5) limit, which is same as T=N case in IBM-3, splits into two branches with the ratio for T s =2 =2 triplet being for large N, 43 and for the T s =0 =2 triplet, it is 23. Thus B(E2) ratios carry isospin signatures. The B(E2)'s for =3 =2 are given in Table V. In order to go beyond =3 a general framework for obtaining U(5)#O(5) and O(5)# O(3) reduced Wigner coefficients is needed. Rowe and Hecht [32] are developing, using vector coherent states, a formalism for the same. However, for certain states extention of the present work is straightforward and examples here are states with L=2, 2 &1 for any given ; see Section 3.4 below. 3.2. [U d (15)#O d (15)]U Ts (3) Limit II In the [U d (15)#O d (15)]U Ts (3) limit, the pair structure given by (12) allows one to derive seniority reduction (, $, :> $, $, :>) formulas so that B(E2)'s involving, $, :> can be reduced to those of $, $, :>. Using the E2 transition operator (21), the seniority reduced expression for B(E 2)'s is B(E2; N,, $ i [v i, 1 vi ] 2 Li ;(T i, T i ) T=0 d s N, &1, $ f [v f, v f ] L f 1 2 ;(T f, T f ) T=0) d s =(e eff ) 2 \ N&+1 3 + [(2Li +1)(2T i d +1)]&1 _ ((N&,0)T i (10)1 & (N&n s d+1, 0) T f s ) 2 SU(3)#O(3) $ $ f, $ i &1\ +$ i +13 2$ i +13 + _ ($ i $ i [v i 1, vi 2 ] Li ; T i d & d - &$ i &1, $ i &1, [v f 1, v f 2 ] L f ; T f d ) O d (3), O Td (3) 2 +$ $f, $ i +1\ &$ i 2$ i +15+ _ ($ i +1, $ i +1, [v f, v f ] L f 1 2 ; T f & d - d &$ i $ i [v i, 1 vi ] 2 Li ; T i ) d O d (3), O Td (3) 2 ]. (29) It should be noted that the selection rule with respect to $ quantum number is 2$=\1 and the selection rules with respect to O d (5) quantum numbers are same as in the case of [U d (5)U Td (3)]U Ts (3) limit. The d - reduced matrix elements
N=Z EVENEVEN NUCLEI IN IBM-3 121 appearing in the r.h.s. of (29) follow from (24) in many situations as we shall see now. The states with small ``$'' will lie low in energy if the :-particle-like correlation given in (12) is favoured. Thus E2's involving $3 will be most important. For these cases the correspondence between the states for 3 in limits I and II as given in Table VI can be exploited; except for two states, there is one-to-one correspondence. The two states N; =3$=1[1]2; (11)0) and N; =3$=3[1] 2; (11) 0) are mixtures of the [U d (5)U Td (3)]U Ts (3) limit states N; =3[3][1] 2; (11) 0) and N; =3[21][1] 2; (11) 0). For these states, symbolically we can write $=3)= sin % [3]) +cos % [21]) and $=1) =cos % [3])&sin % [21]) and the mixing angle % is determined by using the $ selection rule. The selection rule gives ( =3 $=3[1] 2; (11) 0& d - & =2 $=0[0] 0; (00) 0)=0. The numerical values in the last column of Table V immediately give sin %=&- 1651 and cos %=- 3551. With this we have the completed the determination of transformation brackets between [U d (5)U Td (3)]U Ts (3) and [U d (15)#O d (15)] U Ts (3) states for 3. Beyond =3, the general problem of transformation brackets between U(NM)#U(N) U(M)#O(N)#O(M) and U(NM)# O(NM)#O(N) O(M) for symmetric U(NM) irreps has to be addressed and this problem is not yet solved. However, it is possible to write the correspondence for certain special states such as those with L=2,2 &1, etc. for any given. The results of Section 3.1 for 3, Table VI, (24), (29), and (31) below give formulas for B(E2)'s for any but $3 in [U d (15)#O d (15)]#U Ts (3) limit. For $2 the B(E2)'s are TABLE VI Correspondence between [U d (15)#O d (15)]U Ts (3) and [U d (5)U Td (3)]U Ts (3) States for 3 N; $[v 1 v 2 ] L;(T s, T d ) T=0) N; [f][v 1 v 2 ]L; (T s, T d ) T=0) N; =0 $=0[0] 0; (00) 0) N; =0[0][0] 0; (00) 0) N; =1 $=1[1] 2; (11) 0) N; =1[1][1] 2; (11) 0) N; =2 $=2[2]L; (T s, T d )0) N; =2[2][2]L; (T s, T d )0) N; =2 $=0[0] 0; (00) 0) N; =2[2][0] 0; (00) 0) N; =2 $=2[0] 0; (22) 0) N; =2[2][0] 0; (22) 0) N; =3 $=3[3]L; (T s, T d )0) N; =3[3][3]L; (T s, T d )0) N; =3 $=3[21]L; (T s, T d )0) N; =3[21][21]L; (T s, T d )0) N; =3 $=3[1] 2; (33) 0) N; =3[3][1] 2; (33) 0) N; =3 $=1[1] 2; (11) 0) - 3551 N; =3[3][1] 2; (11) 0) +- 1651 N; =3[21][1] 2; (11) 0) N; =3 $=3[1] 2; (11) 0) &- 1651 N; =3[3][1] 2; (11) 0) +- 3551 N; =3[21][1] 2; (11) 0)
122 V. K. B. KOTA B(E2; $ i =1[1] 2; (11) 0 &1, $ f =0[0]0;(00)0) =e 2 eff (N&+1)( +14)45 B(E2; $ i =2[2] L i ;(00)0 &1, $ f =1[1] 2; (11) 0) =e 2 eff 2(N&+3)( +15)153 B(E2; $ i =2[2] L i ;(22)0 &1, $ f =1[1] 2; (11) 0) =e 2 eff 4(N&)( +15)153 B(E2; $ i =2[0] 0; (22) 0 &1, $ f =1[1]2;(11)0) =e 2 eff 4(N&)( +15)153 B(E2; $ i =0[0] 0; (00) 0 &1, $ f =1[1]2;(11)0) =e 2 eff (N&+3)( )9 B(E2; $ i =1[1] 2; (11) 0 &1, $ f =2[2] L f ; (00) 0) =e 2 eff2(n& +1)( &1)(2L f +1)765 B(E2; $ i =1[1] 2; (11) 0 &1, $ f =2[2] L f ; (22) 0) =e 2 eff 4(N&+4)( &1)(2L f +1)765 B(E2; $ i =1[1] 2; (11) 0 &1, $ f =2[0]0;(22)0) =e 2 eff 4(N&+4)( &1)765, (30) where L i =2, 4 and L f =2, 4. Explicit formulas for $=3 $=2 and $=2 $=3 transitions follow from (29), the last column of Table V and the formulas for SU(3)#O(3)-reduced Wigner coefficients [28], ((*, 0)T&1 (10) 1&(*+1, 0) T)= (*+T+2)T (*+1)(2T+1) ((*, 0)T+1 (10) 1&(*+1, 0) T)=& (*&T+1)(T+1). (*+1)(2T+1) (31) From (30) it is clear that by studying the exited 0 +,2 + levels (that belong to 3) that it is possible to identify the existence of the [U d (15)#O d (15)]U Ts (3) limit and, therefore, :-particle-like correlations in vibrational type nuclei with N=Z. 3.3. O sd (18)#O d (15)O Ts (3) Limit III The expansion of O sd (18)#O d (15)O Ts (3) states in terms of [U d (15)#O d (15)] U Ts (3) states as given by (18), (19) and the results of Section 3.2 provide the
N=Z EVENEVEN NUCLEI IN IBM-3 123 formulation for deriving B(E2)'s in the O sd (18)#O d (15)O Ts (3) limit. First as the E2 transition operator (21) is a generator of O sd (18) group (see (17)), one has the selection rule 2 =0. Selection rules with respect to other quantum numbers are already given; for example, the $ selection rule is 2$=\1. Combining (18), (19), (22), (23), (29), the expression for B(E2; i f )'s with $ f =$ i +1 is B(E2; N,, $ i [v i 1, vi 2 ] Li ;(T i d, T i s ) T=0 N,, $ f =$ i +1, [v f 1, v f 2 ] L f ;(T f d, T f s ) T=0) =(e eff ) 2 (2L i +1) &1 (N,, $ f =$ i +1, [v f 1, v f 2 ] L f ;(T f d, T f s ) T=0 &(d - s~+s - d ) 2, 0 & N,, $ i [v i 1, vi 2 ] Li ;(T i d, T i s ) T=0) O d (3) 2 ; (N,, $ f =$ i +1, [v f 1, v f 2 ] L f ;(T f d, T f s ) T=0 &(d - s~+s - d ) 2, 0 & N,, $ i [v i 1, vi 2 ] Li ;(T i d, T i s ) T=0) O d (3) =($ i +1, $ i +1, [v f 1, v f 2 ] L f ; T f d& d - &$ i, $ i,[v i 1, v i 2] L i ; T i d ) Od (3); O Td (3) _ { : (n s ) C N,, $ i +1, T f s(15, 3) C N,, $ i, T i s(15, 3) d+$ i +15 +1 n 2$ i +15 _(&1) T i s +T f s +1 N& 3(T f s +1) ((N&&1, 0) T f s (10) 1&(N&,0)T i s ) SU(3)#O(3) + : (n s ) C N,, $ i +1, T f s &1 (15, 3) C N,, $ i, T i s(15, 3) &$ i 2$ i +15 _ N&+1 3(T i s +1) ((N&,0)T i s (10) 1 & (N&+1, 0) T f s ) SU(3)#O(3)=. (32) Let us first consider =N cases. In order to produce formulas for B(E2)'s for $ i =0, 1, the relationships between C's in (32) are derived using (C2), C C N, N, $, $ N, N, $+1, $+1&2r (15, 3) C +1 (15, 3) =(&1) r+1 N, N, $, $ C (15, 3) 2 N, N, $, $ N, N, $+1, $+1&2r (15, 3) C &1 (15, 3) 2_ =(&1) r N, N, $, $ C (15, 3) _ (N+2$&3r+16)(N&n 12 d&$+3r) (N&2$+3r)( +$+15) & (33) (N+2$&3r+16)( &$) (N&2$+3r)(N& +$&3r+3)& where ($, r)=($, 0) or (1, 1). Substituting these relations and (31) in (32) and applying the sum rule (C1) generate compact formulas for B(E2)'s, 12,
124 V. K. B. KOTA B(E2; N, N, $ i =0[0]0;(00)0N, N, $ f =1[1] 2; (11) 0)=e 2 N(N+16)9 eff B(E2; N, N, $ i =1[1]2;(11)0N, N, $ f =2[v1, f v2] f L f ; (22) 0) =e 2 4(2L f eff +1)(N&2)(N+18)765 (34) B(E2; N, N, $ i =1[1]2;(11)0N, N, $ f =2[v f, v f ] L f 1 2 ; (00) 0) =e 2 2(2L f eff +1)(N+1)(N+15)765. In (34), the O d (5) irreps [v f, v f ] are uniquely specified by (L f 1 2, T f d ) (see Table III). It is important to mention that the first formula in (34) is also given in Ref. [14]. Equations (31), (32), (C2) and the reduced matrix elements in the last column of Table V give all the results for any N and $ i =2 $ f =3 transitions. Similarly, the general formula for C's given in [26] and the results in Section 2.1, Table V and (19), (32) allow us to calculate for any N,, and $ f 3, the $ i $ f =$ i +1 E2 transition strengths. For example, B(E2; N, N&2, $ i =0[0] 0; (00) 0 N, N&2, $ f =1[1]2;(11)0) =e 2 eff (N&2)(N+14)9. (35) One important feature of B(E2)'s in limit III is that they depend quadratically on N (as in the O(6) limit of IBM-1) for T=0 and linearly on N (as in IBM-1 U(5) limit) for T=N. Thus for T=0 nuclei, the limit III case and the spherical IBM-1 U(5) limit case have clearly quite different mass dependence for B(E 2)'s. It is worth recalling that for T=0 the B(E2)'s depend linearly on N in limits I and II. At present experimental data for B(E 2)'s for T=0 nuclei is scarce for any meaningful analysis. If we consider, for example, 48 Cr and 44 Ti (with N=4, 2) nuclei, the experimental values for B(E2; 0 + 1 2+ ) are [33] 0.133\0.020 1 (eb)2 and 0.061\ 0.015 (eb) 2, respectively. Then the ratio of B(E2)'s for 48 Cr and 44 Ti is between 3.3 and 1.5 with average value 2.2. This ratio for limits I, II, and III (using (28), (30), (34)) is 2, 2, and 2.2, respectively. Thus data do not distinguish the three limits. Now we will turn to yrast bands in the three symmetry limits. 3.4. Yrast Bands The formulation given in Sections 3.13.3 is applied to E 2 transition strengths along the yrast line in the three symmetry limits for T=0 nuclei. Given the boson number N, the cutoff angular momentum for the yrast L=0, 2, 4... band is 2N. As Tables I, II, and III, show, for a given yrast L-level several T d values are possible. Therefore, for a unique definition of the yrast bands in the symmetry limits, T d values for the band members should be chosen in a consistant manner. With this criterion, the yrast band in limit I (similarly in limits II and III as given below) is N; L) yrast&i = N; [ ][ ] L=2 ;(T s, T d ) T=0) T s =T d =0 for even (36) T s =T d =1 for odd.
N=Z EVENEVEN NUCLEI IN IBM-3 125 As N is even for eveneven nuclei, n s =N& is even or odd as is even or odd and with this the cutoff L for the bands (36) is seen to be 2N. The choice L=2 and T s =T d = also gives a band structure but here the cutoff L=N (see Appendix D). It is important to point out that the parameters in the energy formula (8) (similarly in (13), (20) for limits II and III) can be chosen for any N such that the levels (36) are indeed the yrast levels in limit I. Following the procedure that led to (26), gives ( [ ]_ d - _ &1, [ &1])=- (, L f =2, T f d & d - & &1, L i =2 &2, T i d ) O d (3), O Td (3) =(&1) +1 [( +1&(&1) )(4nd +1)] 12, (37) where T f =0, T i =1 for even and T f =1, T i =0 for odd. Applying (25) and using (31), (37), together with the fact that the U(5)#O(5) and O(5)#O(3) reduced Wigner coefficients in (25) are unity for the yrast band members give for B(E2)'s, B(E2; N, L N, L&2) yrast&i = e2 eff 36 (L+2&2(&1)L2 )(2N&L+4+2(&1) L2 ). (38) Then the B(E2) ratio in limit I is R(N, L)=B(E2; N, L N, L&2)B(E2; N, 2 + 1 N, 0+ 1 ) (39) R(N, L)=(L+2&2(&1) L2 )(2N&L+4+2(&1) L2 )(12N) N = (L+2&2(&1) L2 )6. (40) The alternating (from 0 to 1) values of T d for the yrast band members generate a 2L=4 staggering in B(E2)'s; see the (&1) L2 factor in (38), (40). This is a striking feature of limit I yrast band and it is absent in IBM-1. In the U(5) limit of IBM-1 [1, 31] R(N, L)=L(2N&L+2)4N and R(, L)=L2. For example, as N, for L=2,4,6,8,10,12,R takes values 1, 23, 53, 43, 73, 2 in limit I and 1, 2, 3, 4, 5, 6 in the U(5) limit, respectively. It is clear that by measuring the ratio R along the yrast line it is possible to identify limit I if it is present in the observed spectrum. Following (36), in limit II the yrast band is defined by N, L) yrast&ii = N; $= [ ] L=2 ;(T s, T d ) T=0) T s =T d =0 for even (41) T s =T d =1 for odd and this is identical to the yrast band in limit I as L=2 implies [f]=[ ], $=, and [v 1 v 2 ]=[,0].
126 V. K. B. KOTA For T=0 nuclei in limit III, the yrast band with cutoff L=2N is N; L) yrast&ii = N; =N, $[$] L=2$; (T s, T d ) T=0) T s =T d =0 for $ even (42) T s =T d =1 for $ odd. The general expression (32) and the relations N, N, $, 0 C N, N, $+1, 1 (15, 3) C +1 (15, 3) $ = even C N, N, $, 0 C N, N, $, 0 (15, 3) 2 N, N, $+1, 1 (15, 3) C &1 (15, 3) N, N, $, 1 C $ = even & C N, N, $, 0 (15, 3) 2 N, N, $+1, 0 (15, 3) C +1 (15, 3) $ = odd C N, N, $, 1 C N, N, $, 1 (15, 3) 2 N, N, $+1, 0 (15, 3) C &1 (15, 3) $ = odd & C N, N, $, 1 (15, 3) 2 together with (31), (37) give for the yrast band _ (N+$+16)(N&n 12 d) (N&$)( +$+15)& _ (N+$+16)(n 12 d&$) (N&$)(N& +3)& _ (N+$+14)(N&n 12 d+2) (N&$+2)( +$+15)& _ (N+$+14)(n 12 d&$), (N&$+2)(N& +1)& (43) B(E2; N, L N, L&2) yrast&iii = (e2 eff) 9 (L+2&2(&1) L2 )(2N+L+28&2(&1) L2 )(2N&L+4+2(&1) L2 ). 8(L+13) (44) Then the B(E2) ratio R(N, L) is R(N, L)= 5(L+2&2(&1)L2 )(2N+L+28&2(&1) L2 )(2N&L+4+2(&1) L2 ) 8N(N+16)(L+13) N 5(L+2&2(&1) L2 ) =. (45) 2(L+13) The results in (45) should be compared with the IBM-1 O(6) limit results [1, 34] R(N, L)=5L(2N&L+2)(2N+L+6)8(L+3) N(N+4) and R(, L)=5L2(L+3), respectively. Once again, as in limit I, there is 2L=4 staggering in the B(E2)'s and this feature is not present in the O(6) limit. For example, as N, for
N=Z EVENEVEN NUCLEI IN IBM-3 127 L=2,4,6,8,10,12,14,16,R takes values 1, 0.59, 1.32, 0.95, 1.52, 1.20, 1.67, 1.38 in limit III and 1, 1.43, 1.67, 1.82, 1.92, 2, 2.06, 2.11 in the O(6) limit, respectively. Thus, by measuring R along the yrast line one can clearly distinguish the limit III and O(6) descriptions. Finally it is important to mention that the isospin generated 2L=4 staggering in yrast band B(E2)'s in limits I, II, and III is different from the 2L=4 staggering in superdeformed bands which is due to a C 4 symmetry [35]. 4. CONCLUSIONS The IBM-3 model admits three dynamical symmetry limits, [U d (5)U Td (3)] U Ts (3), [U d (15)#O d (15)]U Ts (3), and O sd (18)#O d (15)O Ts (3), with good s and d boson isospins. The low-lying phonon (or $) multiplets and the yrast band members generated by the three symmetry limits carry, for heavy N=Z eveneven nuclei, clear signatures of isospin both in spectra and B(E2)'s. Some of the main signatures for T=0 states that can be looked for using future RIB facilities are as follows: 1. In the limits I and II the =2 two phonon 0 +,2 +,4 + triplet appears twice and they will have T s =0 and 2, respectively. For the triplet members for large N the B(E2) ratio B(E2; =2 =1)B(E2; =1 =0) is 23 and 43, respectively; the ratio is 2 for the vibrational (IBM-1) U(5) limit states. 2. In the limits I and II for the =3 three phonon states there are: (i) IBM-1 U(5) states [3] 0 +,3 +,4 +,6 + appearing twice with T s =1 and 3, respectively; (ii) states with L? =1 +,2 +,3 +,4 +,5 + occurring with O d (5) irrep [21] and T s =1; (iii) three more 2 + states, two with T s =1 and one with T s =3. Besides looking for these states in experiments, the B(E2) results in Table V and Eq. (30) can be used to establish the structure of these states in the two limits. A good test comes from the selection rules for the four =3, 2 + states. Because of the O d (5) selection rules (27), the 2 + state belonging to [21] irrep cannot decay to the =2, 0 + states in both limits. The 2 + states with O d (5) irrep [1] and T s =1 have quite different structure in the two limits. The state with $=3 in limit II cannot decay to =2, $=0, 0 + state due to $ selection rule and there is no such selection rule in limit I. Finally, in the two limits the 2 + state with T s =3 cannot decay to =2 states with T s =0. 3. In the limit III the spectrum exhibits both IBM-1 U(5) and O(6) features and beyond 2 + 1 state, it is different from the spectra generated by limits I and II. For example, the 0 + 2 state with T s =0 is moved high in energy in limit III as it belongs to =N&2 (therefore its E2 decay to 2 + 1 state is forbidden). With the quadrupolequadrupole force given by the P 2 operator in (17), the E(L? )E(2 + ) 1 ratio for L? states with =N, $=L2 is given by [$($+13)+T s (T s +1)]16. An important feature of B(E2)'s in limit III is that they depend quadratically on N while they depend linearly on N in limits I and II.