Population of mixed-symmetry states via α transfer reactions

PHYSICAL REVIEW C 78, 017301 (2008) Population of mixed-symmetry states via αtransfer reactions C. E. Alonso,1J. M. Arias,1L. Fortunato,2N. Pietralla,3and A. Vitturi2 1Departamento de F´ ısica At´ omica, Molecular y Nuclear, Facultad de F´ ısica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 2Dipartimento di Fisica Galileo Galilei and INFN, Via Marzolo 8, I-35131 Padova, Italy 3Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, D-64289 Darmstadt, Germany (Received 7 April 2008; published 8 July 2008) Within the neutron-proton interacting boson model we study the population of mixed-symmetry states via αtransfer processes. Closed expressions are deduced in the case of the limiting Uπ+ν(5) and SUπ+ν(3). We find that the population of the lowest mixed-symmetry 2+state, vanishing along the Nπ=Nνline, depends on the number of active bosons and is normally smaller than that of the lowest full symmetric 2+state. In particular, for deformed nuclei where the number of bosons is normally large, the relative population of the mixed-symmetry 2+state is of the order of a few percent. More favorable cases can be found near shell closures, as in the case of αtransfer leading to 140Ba. DOI: 10.1103/PhysRevC.78.017301 PACS number(s): 21.60.−n One of the most interesting features of the neutron-proton version of the interacting boson model (IBM-2) [1]isthe occurrence, in addition to all the states already present in the original IBM-1 version, of states that are not fully symmetric in the proton-neutron degree of freedom [2]. In spherical nuclei, the lowest of such mixed-symmetry states is expected tobea2 +state (normally indicated as 2+ M)[3], while in the well-deformed case one expects a mixed-symmetry K=1+ band head [4], with a spin quantum number that is outside the simple IBM-1. These mixed-symmetry states display particular properties, such as an electromagnetic decay by a strong M1 transition. They have been populated in many nuclear reactions such as inelastic scattering of electrons [4], photons [5], or hadrons [6]; βdecay [3]; fusion evaporation reaction [7]; and Coulomb excitation [8]. For a review article, see Ref. [9]. The possibility of populating these states by (12C,8Be) reactions, namely, via αtransfer processes, has recently been suggested [10]. The reaction mechanism associated with the αtransfer reaction is not fully understood [11]. Whether the four particles are transferred sequentially due to the action of the one-body field or, as the other extreme, are transferred as a unique cluster in a single shot is a matter for discussion. In the former case the particle correlations act via constructive interference of the different paths in the intermediate states due to the correlated features of the initial and final states. In the latter case, one assumes that the correlations have preformed an α particle in the initial and final states, with a probability given by the so-called αspectroscopic factors, and the process is then formally described as the transfer of a single object. In addition, each approach relies on the choice of a number of ingredients, such as the proper optical potentials, coupling form factors, etc. Therefore, in the end, different descriptions of the reaction mechanism yield estimates of the absolute cross sections that, in addition to the dependence on the bombarding energy, can vary even by orders of magnitude from one approach to the other. As far as relative cross sections are concerned, on the other hand, one can profitably assume that the transfer intensities will in all cases scale as the square of the matrix element of the four-particle creation (or annihilation, in the case of pickup reactions) operator. In this article we therefore calculate four-particle creation matrix elements, comparing those involving fully symmetric or mixed-symmetry final states. In the framework of the interacting boson model the operator associated with the α-particle creation does not simply acquire a two-boson creation form. A mapping procedure from the underlying fermion space will in fact map the four fermion creation operator into a sum of boson operators of increasing order. At the lowest order, however, we assume the simplest expressions for the L=0 and L=2α-particle creation operator, A† α(L=0) =c1s+ πs+ ν+c3(d+ π×d+ ν)(0) (1) and A† α(L=2) =c2[s+ π×d+ ν+d+ π×s+ ν)](2) +c4(d+ π×d+ ν)(2). (2) The parameters cican in principle be derived microscopically from the mapping procedure. In practice, and for the purpose of this article, we can treat them as phenomenological parameters to be determined from experimental data. Note also that onestep transfer processes can only populate natural-parity states [11]; so the transfer to 1+states is not allowed. On the other hand, the dipole L=1−transfer operator has vanishing matrix elements within the IBM-2 space, characterized by states with positive parity only. These simple forms, Eqs. (1) and (2), are close to the original operators introduced by Bennett and Fulbright [12] and applied by the authors to different isotopic and isotonic chains in the f-pshell. We consider as the initial state the ground state of an even-even nucleus characterized by the boson number Ni= Nπ+Nνand as the final states the lowest fully symmetric states and the first mixed-symmetry 2+states in a nucleus with boson number Nf=Nπ+Nν+2. In general, matrix elements have to be determined after the diagonalization of the IBM-2 Hamiltonian, from the explicit wave functions. 0556-2813/2008/78(1)/017301(4) 017301-1 ©2008 The American Physical Society BRIEF REPORTS PHYSICAL REVIEW C 78, 017301 (2008) Analytic expressions can, however, be obtained in the case of limiting situations. We first consider the case of the Uν+π(5) limit. The group chain in this case is Uν(6) ×Uπ(6) ⊃Uν+π(6) ⊃Uν+π(5) ⊃Oν+π(5) ⊃Oν+π(3) ⊃Oν+π(2) (3) and the U(5) wave functions are characterized by the 12 quantum numbers that classify uniquely the basis states, namely, |[Nν]×[Nπ]; [N−f,f]{n1,n2}(v1,v2)αLM,(4) where αstands for the two missing labels necessary to completely specify the O(5) ⊃O(3) reduction, omitted when the reduction is unique. With this notation the initial state is |0+ 1=|[Nν]×[Nπ]; [N,0]{0,0}(0,0)0=sNν νsNπ π;0 ,(5) where N=Nν+Nπ. The final states that can be populated via the operators (1) and (2) are the lowest fully symmetric states [2] and the lowest mixed-symmetry state. Their labels and full expressions are given in the first three columns of Table I. The associated reduced matrix elements of the α transfer operator are given in the last column. One can see from these expressions that the αtransfer intensities are expected to scale approximately as NπNνto the ground state, as NπNν/N to the one-phonon state, and as NπNν/N2to the two-phonon states. Note that the population of the mixed-symmetry state, which scales as (Nπ−Nν)2/N, will in general be much lower than that of the corresponding symmetric one. The ratio of α transfer cross sections to the 2+ Mstate and to the 2+ 1state does not depend on the parameters of the transfer operators and amounts to (Nπ−Nν)2/4(Nπ+1)(Nν+1). In particular, for Nν=Nπwe obtain the selection rule that the population of the mixed-symmetry state is completely forbidden. We consider now the case of SU(3). In this limit the group chain is Uν(6) ×Uπ(6) ⊃Uν+π(6) ⊃SUν+π(3) ⊃Oν+π(3) ⊃Oν+π(2) (6) and the SU(3) wave functions are characterized by the 12 necessary quantum numbers, namely, |[Nν]×[Nπ]; [N−f,f]ρ(λ,µ)κLM,(7) where ρ(set of three) and κare missing labels necessary to completely specify the U(6) ⊃SU(3) and SU(3) ⊃O(3) reductions, omitted when these are unique. In the following we omit the Oν+π(2) label, M. With this notation the initial state is |0+ 1=|[Nν]×[Nπ]; [N,0](2N,0)0,(8) where N=Nν+Nπ. The final states that can be populated via the operators (1) and (2) are the lowest fully symmetric states and the lowest 2+mixed-symmetry state (see Table II, first and second columns). The evaluation of the αtransfer matrix elements is more feasible in the intrinsic frame. The ground, β,γ, and mixedsymmetry bands can be associated with intrinsic states of the form [13] |(Nπ+1,N ν+1; gs) =Bπ gNπ+1Bν gNν+1|(Nπ+1,N ν+1; β) =nˆ NπBπ gNπBπ βBν gNν+1+ˆ NνBπ gNπ+1Bν gNνBν β ×|(Nπ+1,N ν+1; γ) =nˆ NπBπ gNπBπ γBν gNν+1+ˆ NνBπ gNπ+1Bν gNνBν γ ×|(Nπ+1,N ν+1; M) =nˆ NνBπ gNπBπ MBν gNν+1−ˆ NπBπ gNπ+1Bν gNνBν M, (9) where n=(N+2)−1/2and where we have used the general notations ˆ Nα=√Nα+1 and Bα iNα,i =(b† α,i )Nα,i Nα,i !|0,(10) TABLE I. Notation and explicit expressions of U(5) states (first three columns) and αtransfer reduced matrix elements. |f|[Nν+1] ×[Nπ+1]; [N− f, f ]{n1,n2}(v1,v2)αLM Explicit expression f||A† α(L)||0+ 1(Nν,N π) |0+ 1 |···;[N+2,0]{0,0}(0,0)0|sNν+1 νsNπ+1 π;0c1(Nν+1)(Nπ+1) |0+ 2 |···;[N+2,0]{2,0}(0,0)01 (N+2)(N+1) (√Nν(Nν+1)|sNν−1 νd2 νsNπ+1 π;0.+ √2(Nν+1)(Nπ+1)|sNν νdνsNπ πdπ;0.+√Nπ(Nπ+1)|sNν+1 νsNπ−1 πd2 π;0) c32(Nν+1)(Nπ+1) (N+2)(N+1) |2+ 1 |···;[N+2,0]{1,0}(1,0)21 N+2(√Nν+1|sNν νdνsNπ+1 π;2+√Nπ+1|sNν+1 νsNπ πdπ;2)2c25(Nν+1)(Nπ+1) N+2 |2+ 2 |···;[N+2,0]{2,0}(2,0)21 (N+2)(N+1) √Nν(Nν+1)|sNν−1 νd2 νsNπ+1 π;2+ √2(Nν+1)(Nπ+1)|sNν νdνsNπ πdπ;2+ √Nπ(Nπ+1)|sNν+1 νsNπ−1 πd2 π;2 c410(Nν+1)(Nπ+1) (N+2)(N+1) |2+ M |···;[N+1,1]{1,0}(1,0)21 N+2√Nπ+1|sNν νdνsNπ+1 π;2− √Nν+1|sNν+1 νsNπ πdπ;2c2 √5(Nπ−Nν) √N+2 017301-2 BRIEF REPORTS PHYSICAL REVIEW C 78, 017301 (2008) TABLE II. Explicit expressions of SU(3) states (first and second columns) and αtransfer reduced matrix elements. |f|[Nν]×[Nπ]; [N−f, f ]ρ(λ,µ)κLMf||A† α(L)||0+ 1(Nν,N π) |0+ 1|[Nν+1] ×[Nπ+1]; [N+2,0](2N+4,0)01 3(Nπ+1)(Nν+1) c1+2c3 √5 |2+ 1|[Nν+1] ×[Nπ+1]; [N+2,0](2N+4,0)22 32(Nπ+1)(Nν+1) c2−c4 √7 |0+ β|[Nν+1] ×[Nπ+1]; [N+2,0](2N,2)02 32(Nπ+1)(Nν+1) (N+2) c1−c3 √5 |2+ β|[Nν+1] ×[Nπ+1]; [N+2,0](2N,2)κ=022 3(Nπ+1)(Nν+1) (N+2) c2+2c4 √7 |2+ γ|[Nν+1] ×[Nπ+1]; [N+2,0](2N,2)κ=2222(Nπ+1)(Nν+1) 3(N+2) c2+2c4 √7 |2+ M|[Nν+1] ×[Nπ+1]; [N+1,1](2N+2,1)22 3(N+2)(Nν−Nπ)c2−c4 √7 with α=ν,π and i={g,β,γ,M}. The basis (proton or neutron) bosons bg,b β,b γ, and bMare given by b† g=1 √3(s†+√2d† 0),b † β=1 √3(−√2s†+d† 0), (11) b† γ=1 √2(d† 2+d† −2),b † M=1 √2(d† 1+d† −1). Once the matrix elements are obtained in the intrinsic frame a projection to the laboratory system is performed. After that projection, we obtain, for the reduced matrix elements to the fully symmetric states and to the mixed-symmetry 2+ Mstate, the expressions in the third column of Table II. It is worth noting that more involved expressions than Eq. (10) for the structure of the intrinsic ground, β,γ, and mixed-symmetry bands can be found in the literature [14] but they give the same results for the αtransfer intensities in leading order of an expansion in N. In the SU(3) case, therefore, one expects that the αtransfer intensities scale approximately as NπNνto the ground-state band, as NπNν/N to the one-phonon (either βor γ) state, and as (Nπ−Nν)2/N to the mixed-symmetry states. This is much lower than the values for the corresponding symmetric states. The ratio of αtransfer cross sections to the 2+ Mstate and to the 2+ 1state does not depend on the parameters of the transfer operators and amounts to 3(Nπ−Nν)2/(4(N+2)(Nπ+1) (Nν+1)). Again, the population of the mixed-symmetry state is forbidden for Nν=Nπ. For the sake of clarity, in Fig. 1 the L=2αtransfer transitions studied in this work are schematically presented. We finally mention the case of the O(6) dynamical symmetry. In this case the straightforward procedure is to express the IBM-2 O(6) states in terms of products of proton and neutron O(6) states (see Ref. [2]) and then expand each IBM-1 O(6) stateintheIBM-1U(5) basis using the transformation brackets given, for example, in Ref. [15]. The evaluation of the matrix elements of the αtransfer operator is reduced to a summation of U(5) matrix elements. A trivial simplification occurs, due to the selection rules of the boson creation operators, but the remaining sum does not seem to be easily reducible to a compact expression. As an example of the spherical limit, we discuss the case of 140Ba, considered in Ref. [10]. In this case the considered reaction is 136Xe(12C,8Be)140Ba. As already discussed in Ref. [3] one can assume a proper description of the low-lying spectrum of 140Ba within the simple U(5) limit. Assuming as cores the Z=50 and N=82 shell closure, one has in the initial system Nπ=2 and Nν=0. This leads to the relative αtransfer intensities, with respect to the cross section to the ground state, equal to c2 2 c2 1 for the first 2+ 1state, to c2 2 3c2 1 for the mixed-symmetry 2+ Mstate, to c2 3 6c2 1 for the 0+ 2state, and finally to c2 4 6c2 1 for the 2+ 2state. In this rather fortunate and promising case, due to the small values of the boson numbers Nπand Nν, the population of the first 2+ 1state is expected to be of the same order of magnitude as that of the ground state, but FIG. 1. Schematic representation of the L= 2αtransfer intensities studied in this work in the SU(3) (left) and U(5) (right) states. 017301-3 BRIEF REPORTS PHYSICAL REVIEW C 78, 017301 (2008) more importantly the transition to the mixed-symmetry 2+ M state is expected to be only a factor of three smaller than the population of the first 2+ 1state. As an example of the deformed limit, we consider 156Gd, a typical SU(3) case in the rare earth region where 1+states corresponding to the scissors mode with mixed symmetry have been observed. The corresponding α-transfer reaction is 152Sm(12C,8Be)156Gd. In this case the initial nucleus has Nπ=6 and Nν=4. These higher boson numbers, typical of the SU(3) limit, imply that the population of the mixedsymmetry 2+ Mstate relative to the population of the first 2+ 1 state is less than 1%. In summary, we have evaluated the matrix elements of αtransfer as two-boson operators in the U(5) and the SU(3) dynamical symmetries of the F-spin limit of the IBM-2 from the initial ground state with boson numbers Ni=Nπ+Nνto selected final states with boson numbers Nf=(Nπ+1) +(Nν+1) and spin quantum numbers 0 or 2. Lowest order expressions for αtransfer cross sections to full-symmetry states and to the lowest mixed-symmetry 2+ state have been derived this way. Corresponding experiments have been proposed. We stress the importance of these experiments with respect to two points. First, αtransfer reactions might turn out to be an efficient way to populate mixed-symmetry states, in particular in spherical nuclei. 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