Transition probabilities in the U(3,3) limit of the symplectic IVBM

# Transition probabilities in the U(3,3) limit of the symplectic IVBM

H. G. Ganev Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia
Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
Sofia 1784, Bulgaria
###### Abstract

The tensor properties of the algebra generators are determined in respect to the reduction chain , which defines one of the dynamical symmetry limits of the Interacting Vector Boson Model (IVBM). The symplectic basis according to the considered chain is thus constructed and the action of the generators as transition operators between the basis states is illustrated. The matrix elements of the ladder operators in the so obtained symmetry-adapted basis are given. The limit of the model is further tested on the more complicated and complex problem of reproducing the transition probabilities between the collective states of the ground band in , , , and isotopes, considered by many authors to be axially asymmetric. Additionally, the excitation energies of the ground and bands in are calculated. The theoretical predictions are compared with the experimental data and some other collective models which accommodate the rigid or soft structures. The obtained results reveal the applicability of the model for the description of the collective properties of nuclei, exhibiting axially asymmetric features.

PACS 21.60.Fw, 23.20.-g, 21.10.Re, 27.80.+w, 27.60.+j

## I Introduction

Symmetry is an important concept in physics. In finite many-body systems, it appears as time reversal, parity, and rotational invariance, but also in the form of dynamical symmetries DG ()-RW (). In the algebraic models, the use of the dynamical symmetries defined by a certain reduction chain of the group of dynamical symmetry yields exact solutions for the eigenvalues and eigenfunctions of the model Hamiltonian, which is constructed from the invariant operators of the subgroups in the chain. Many properties of atomic nuclei have been investigated using such models, in which one obtains bands of collective states which span irreducible representations of the corresponding dynamical groups.

Something more, it is very simple and straightforward to calculate the matrix elements of transition operators between the eigenstates as both - the basis states and the operators - can be defined as tensor operators in respect to the considered dynamical symmetry. Then the calculation of matrix elements is simplified by the use of a respective generalization of the Wigner-Eckart theorem, which requires the calculation of the isoscalar factors and reduced matrix elements. By definition such matrix elements give the transition probabilities between the collective states attributed to the basis states of the Hamiltonian.

The comparison of the experimental data with the calculated transition probabilities is one of the best tests of the validity of the considered algebraic model. With the aim of such applications of one of the dynamical symmetries of the symplectic Interacting Vector Boson Model (IVBM), we develop in this paper a practical mathematical approach for explicit evaluation of the matrix elements of transitional operators in the model.

The IVBM and its recent applications for the description of diverse collective phenomena in the low-lying energy spectra (see, e.g., the review article pepan ()) exploit the symplectic algebraic structures and the Sp(12,) is used as a dynamical symmetry group. Symplectic algebras and their substructures have been applied extensively in the theory of nuclear structure GL ()-Vanagas (). They are used generally to describe systems with a changing number of particles or excitation quanta and in this way provide for larger representation spaces and richer subalgebraic structures that can accommodate the more complex structural effects as realized in nuclei with nucleon numbers that lie far from the magic numbers of closed shells. In particular, the model approach was adapted to incorporate the newly observed higher collective states, both in the first positive and negative parity bands Sp12U6 () by considering the basis states as ”yrast” states for the different values of the number of bosons that built them.

In Ref.TSIVBM () a new dynamical symmetry limit of the IVBM was introduced, which seems to be appropriate for the description of deformed even-even nuclei, exhibiting triaxial features. Usually, in the geometrical approach the triaxial nuclear properties are interpreted in terms of either the -unstable rotor model of Wilets and Jean WJ () or the rigid triaxial rotor model (RTRM) of Davydov et al. DF (). An alternative description can be achieved by exploiting the properties of the algebra introduced in Ref.TSIVBM () (and appearing also in the context of IBM-2 PSDiep ()). The latter is appropriate for nuclei in which the one type of particles is particle-like and the other is hole-like. Using a schematic Hamiltonian with a perturbed dynamical symmetry, the IVBM was applied for the calculation of the low-lying energy spectrum of the nucleus Os TSIVBM (). The obtained results proved the relevance of the proposed dynamical symmetry in the description of deformed triaxial nuclei.

In this paper we develop further our theoretical approach initiated in Ref.TSIVBM () by considering the transition probabilities in the framework of the symplectic IVBM with as a group of dynamical symmetry. For this purpose we consider the tensorial properties of the algebra generators in respect to the reduction chain:

 (1)

where and are the one-fluid algebras corresponding to the two nuclear subsystems, is the combined two-fluid algebra, and is the standard angular momentum algebra. Further we classify the basis states by the quantum numbers corresponding to the irreducible representations (irreps) of different subgroups along the chain (1). In this way we are able to define the transition operators between the basis states and then to evaluate analytically their matrix elements. This will allow us further to test the model in the description of the electromagnetic properties observed in some non-axial nuclei. As a first step we will test the theory on the transitions between the states belonging to the ground state bands (GSB) in some even-even nuclei from the and mass regions.

## Ii Tensorial properties of the generators of the Sp(12,R) group

It was suggested by Bargmann and Moshinsky BargMosh () that two types of bosons are needed for the description of nuclear dynamics. It was shown there that the consideration of only two-body system consisting of two different interacting vector particles will suffice to give a complete description of three-dimensional oscillators with a quadrupole-quadrupole interaction. The latter can be considered as the underlying basis in the algebraic construction of the phenomenological IVBM.

The basic building blocks of the IVBM TSIVBM () are the creation and annihilation operators of two kinds of vector bosons and , which differ in an additional quantum number (or and )the projection of the -spin (an analogue to the -spin of IBM-2 or the -spin of the particle-hole IBM). In the present paper, we consider these two bosons just as elementary building blocks or quanta of elementary excitations (phonons) rather than real fermion pairs, which generate a given type of algebraic structures. Thus, only their tensorial structure is of importance and they are used as an auxiliary tool, generating an appropriate dynamical symmetry.

The vector bosons can be considered as components of a dimensional vector, which transform according to the fundamental irreducible representation and its conjugate (contragradient) one , respectively. These irreducible representations become reducible along the chain of subgroups (1) defining the dynamical symmetry. This means that along with the quantum number characterizing the representations of , the operators are also characterized by the quantum numbers of the subgroups of chain (1). Introducing the notations and , the components of the creation operators labeled by the chain (1) can be written as:

 p†m≡p†[1]6[1]3[0]∗3 [1]3(1)3m,   n†m≡n†[1]6[0]3[1]∗3 [1]∗3(1)3m. (2)

According to the chain (1), the fundamental irrep decomposes as

 [1]6⊃[1]3⊕[1]∗3, (3)

i.e. as a direct product sum of the and fundamental irreps. In Eq.(3) the denotes the (contragradient) irrep of which is conjugate to the of . This corresponds to the case when the one type of particles in the two-fluid nuclear system is particle-like and the other is hole-like. Note that there is an alternative decomposition of the fundamental irrep :

 [1]6⊃[1]3⊕[1]3, (4)

where the group in Eq.(1) should be replaced by the one. The decomposition (4) is appropriate for the situation when the nucleus is considered as consisting of two particle-like constituents. In our further considerations we will need also the reduction of the irrep along the chain (1). According to the decomposition rules for the fully symmetric irreps, we obtain for the content

 [2]6⊃[2]3[0]∗3+[1]3[1]∗3+[0]3[2]∗3. (5)

Thus, the generators of the symplectic group can already be defined as irreducible tensor operators according to the whole chain (1) of subgroups as follows.

The raising operators of can be expressed as

 F[χ]6LM[λ]3[0]∗3[λ]3 =C[1]6[1]6[χ]6[1]3[0]∗3[1]3[0]∗3[λ]3[0]∗3C[λ]3[1]3,[1]3 ×C[1]3[1]3[λ]3(1)3(1)3(L)3CLM1m1n ×p†[1]6[1]3[0]∗3 [1]3(1)3mp†[1]6[1]3[0]∗3 [1]3(1)3n, (6)
 F[χ]6LM[0]3[λ]∗3[λ]∗3 =C[1]6[1]6[χ]6[0]3[1]∗3[0]3[1]∗3[0]3[λ]∗3C[λ]3[−1]3,[−1]3 ×C[1]∗3[1]∗3[λ]∗3(1)3(1)3(L)3CLM1m1n ×n†[1]6[0]3[1]∗3 [1]∗3(1)3mn†[1]6[0]3[1]∗3 [1]∗3(1)3n, (7)
 F[χ]6LM[1]3[1]∗3[λ]3 =C[1]6[1]6[χ]6[1]3[0]∗3[0]3[1]∗3[0]3[λ]3C[λ]3[1]3,[−1]3 ×C[1]3[1]∗3[λ]3(1)3(1)3(L)3CLM1m1n ×p†[1]6[1]3[0]∗3 [1]3(1)3mn†[1]6[0]3[1]∗3 [1]∗3(1)3n, (8)

where, according to the lemma of Racah Racah (), the Clebsch-Gordan coefficients along the chain are factorized by means of the isoscalar factors (IF), defined for each step of decomposition (1). The lowering operators of are obtained from the rasing ones by Hermition conjugation. That is why we consider only the tensor properties of the raising operators.

The tensors (6)-(8) transform according to

 [1]6×[1]6=[2]6+[1,1]6, (9)

and their Hermition conjugate counterparts according to

 [1]∗6×[1]∗6=[−2]6+[−1,−1]6, (10)

respectively. But, since the basis states of the IVBM are fully symmetric, we consider only the fully symmetric representation and its conjugate . Hence, the tensors (6)-(8) transform according to the irrep .

The tensor (6) with respect to the subgroup transforms according to the direct product

 [1]3×[1]3=[2]3+[1,1]3, (11)

while (7) and (8) transform according to

 [1]∗3×[1]∗3=[2,2]3+[2,1,1]3=[−2]3+[1]3, (12)
 [1]3×[1]∗3=[2,1]3+[1,1,1]3=[1,−1]3+[0]3 (13)

and obviously, because of their symmetric character, (6) and (7) transform only according to the symmetric representations and , respectively. The latter follows also from the reduction (5). In this way we obtain the following set of raising generators:

 F[2]6LM[2]3[0]∗3[2]3,F[2]6LM[0]3[2]∗3[−2]3, (14)
 F[2]6LM[1]3[1]∗3[2,1,0]3,F[2]6LM[1]3[1]∗3[0]3, (15)

which together with their conjugate (lowering) operators change the number of bosons by two. The operators (15) and their conjugate counterparts are the ladder generators of algebra.

In terms of Elliott’s notations Elliott () , we have , , and . The corresponding values of from the reduction rules are in both the and irreps, in the irrep and in the .

The number preserving operators transform according to the direct product of the corresponding representations and , namely

 [1]6×[1]∗6=[1,−1]6+[0]6, (16)

where and is the scalar representation. They generate the maximal compact subgroup of .

The tensor operators

 A[1−1]6LM[λ]3[0]∗3[λ]3≃1√2∑m,kCLM1m1kp†mpk (17)
 A[1−1]6LM[0]3[λ]∗3[λ]∗3≃1√2∑m,kCLM1m1k n†mnk (18)

correspond to the generators of the and algebras, respectively. The operators with represent the angular momentum components, whereas those with correspond to the quadrupole momentum operators and together they generate the one-fluid () algebra. The tensors (17), (18) together with (15) and their conjugate counterparts, in turn, constitute the full set of generators.

The linear combination operators

 A′LM[λ]3=A[1−1]6LM[λ]3[0]∗3[λ]3−(−1)LA[1−1]6LM[0]3[λ]∗3[λ]3 (19)

generate the algebra. The algebra is obtained by excluding the operator which is the single generator of the algebra, whereas the angular momentum algebra is generated by the generators only. The operator , counting the difference between particle and holes, is also the first order Casimir of algebra and it decomposes the action space of the generators to the ladder subspaces of the boson representations of with Sp2NRbr ().

Finally, the tensors

 A[1−1]6LM[1]3[1]3[λ]3≃1√2∑m,kCLM1m1k p†mnk, (20)
 A[1−1]6LM[1]∗3[1]∗3[λ]3≃1√2∑m,kCLM1m1k n†mpk (21)

with and extend the algebra to the one.

In this way we have listed all the irreducible tensor operators in respect to the reduction chain (1) that correspond to the infinitesimal operators of the algebra.

## Iii Construction of the symplectic basis states of IVBM

Next, we can introduce the tensor products

 T([χ1]6[χ2]6)ω[χ]6[λ1]3[λ2]3[λ]3LM= ∑T[χ1]6L1M1[λ′1]3[λ′′1]∗3 [λ1]3T[χ2]6L2M2[λ′2]3[λ′′2]∗3 [λ2]3 ×C[χ1]6[χ2]6ω[χ]6[λ′1]3[λ′′1]∗3[λ′2]3[λ′′2]∗3[λ1]3[λ2]3 ×C[λ]3[λ1]3,[λ2]3C[λ1]3[λ2]3[λ]3K1L1 K2L2KLCL1L2LM1M2M (22)

of two tensor operators , which are as well tensors in respect to the considered reduction chain. We use (22) to obtain the tensorial properties of the operators in the enveloping algebra of containing the products of the algebra generators. In this particular case we are interested in the transition operators between states differing by four bosons , expressed in terms of the products of two operators . Making use of the decomposition (5) and the reduction rules in the chain (1), we list in Table 1 all the representations of the chain subgroups that define the transformation properties of the resulting tensors.

In order to clarify the role of the tensor operators introduced in previous section as transition operators and to simplify the calculation of their matrix elements, the basis for the Hilbert space must be symmetry adapted to the algebraic structure along the considered subgroup chain (1). It is evident from (14) and (15) that the basis states of the IVBM in the (even) subspace of the boson representations of can be obtained by a consecutive application of the raising operators on the boson vacuum (ground state) , annihilated by the tensor operators  and .

Thus, in general a basis for the considered dynamical symmetry of the IVBM can be constructed by applying the multiple symmetric couplings (22) of the raising tensors with itself - . The possible couplings are enumerated by the set . We note that the integers can take non-negative as well as negative values and hence correspond to mixed irreps of Flores (). The number of copies of the operator in the symmetric product tensor is , where . Each raising operator will increase the number of bosons by two. Then, the resulting infinite basis can be written as:

 |[N]6;[Np]3,[Nn]∗3;(λ,μ);KLM⟩, (23)

where , and denote the irreducible representations of the , and groups respectively, while the quantum numbers denote the basis of the irrep of . By means of these labels, the basis states can be classified in each of the two irreducible even with and odd with representations of .

The classification scheme for the boson representations obtained by applying the reduction rules for the irreps in the chain (1) for even value of the number of bosons is shown on Table 2. Each row (fixed ) of the table corresponds to a given irreducible representation of the algebra, whereas the quantum numbers define the cells of the Table 2. On the other hand, the so called ladder representation of the non-compact algebra acts in the space of the boson representation of the algebra. Thus the ladder representations of correspond to the columns (fixed value of ) of the Table 2. Note that along the columns the irreps repeat each other except the ones corresponding to the first row for each .

Now, it is clear which of the tensor operators act as transition operators between the basis states ordered in the classification scheme presented on Table 2. The operators give the transitions between two neighboring cells from one column, while the or ones change the column as well. The tensors and , acting within the rows, change a given irrep to the neighboring one on the left and right , respectively. The operators (19), which correspond to the generators do not change the representations , but can change the angular momentum inside it. The action of the tensor operators on the vectors inside a given cell or between the cells of Table 2. is also schematically presented on it with corresponding arrows, given above in parentheses.

## Iv Matrix elements of the U(3,3) ladder operators

Physical applications are based on the correspondence of sequences of vectors to sequences of collective states belonging to different bands in the nuclear spectra. The above analysis permits the definition of the appropriate transition operators as corresponding combinations of the tensor operators given in Sections II and III.

In the present work we are interesting in the calculation of the matrix elements of the generators in appropriately chosen symmetry-adapted basis. For this purpose we consider the following reduction chain:

 U(3,3)⊃Up(3)⊗¯¯¯¯¯¯¯¯¯¯¯¯¯Un(3)⊃U∗(3)⊃SO(3), (24)

which is a part of (1). The basis is

 |ν;[Np]3,[Nn]∗3;[λ]3;KLM⟩, (25)

where and the new label denotes the different ladder representations. Note that the number of bosons is not a good quantum number along the chain (24) and hence the irrep label is irrelevant and will be omitted in the further considerations.

The matrix elements of generators can be calculated using the fact that the Hilbert state space is the tensor product of the and boson representation spaces and , i.e.

 |[Np]3,[Nn]∗3;[λ]3⟩=|[Np]3⟩⊗|[Nn]∗3⟩, (26)

coupled to good total symmetry. Tensor operators in the p-n space can be constructed by coupling tensors in the separate spaces to good total symmetry.

In the preceding sections we expressed all the symplectic generators and the basis states as components of irreducible tensors in respect to the reduction chain (1). Thus, for calculating of the matrix elements of the generators (which are a subset of the symplectic generators), one can use the generalized Wigner-Eckart theorem with respect to the subgroup:

 ⟨ν;[N′p]3,[N′n]∗3;[λ′]3;K′L′M′|Tlm[σ′]3[σ′′]3[σ]3|ν;[Np]3,[Nn]∗3;[λ]3;KLM⟩ =⟨ν;[N′p]3,[N′n]∗3;[λ′]3;K′L′||Tlm[σ′]3[σ′′]3[σ]3||ν;[Np]3,[Nn]∗3;[λ]3;KL⟩CL′M′LM,lm. (27)

Note that the generators (15) act within a given ladder representation (fixed ) and change the number of bosons by two, whereas the generators (14) change the irrep as well. The double-barred reduced matrix elements in (27) are determined by the triple-barred matrix elements:

 ⟨ν;[N′p]3,[N′n]∗3;[λ′]3;K′L′||Tlm[σ′]3[σ′′]3[σ]3||ν;[Np]3,[Nn]∗3;[λ]3;KL⟩ =⟨ν;[N′p]3,[N′n]∗3;[λ′]3|||Tlm[σ′]3[σ′′]3[σ]3|||ν;[Np]3,[Nn]∗3;[λ]3⟩C[λ]3[σ]3[λ′]3KLklK′L′ (28)

where are the isoscalar factors and the triple-barred matrix elements depend only on the , and quantum numbers. Obviously, for the evaluation of the matrix elements (27) of the operators in respect to the chain (1) the knowledge of the IF as well as the reduced triple-barred matrix elements is required.

We consider the reduced matrix element of the ladder operator :

 ⟨ν;[N′p]3,[N′n]∗3;[λ′]3;K′L′||Flm[1]3[1]∗3[2,1,0]3||ν;[Np]3,[Nn]∗3;[λ]3;KL⟩ =⟨ν;[N′p]3,[N′n]∗3;[λ′]3|||Flm[1]3[1]∗3[2,1,0]3|||ν;[Np]3,[Nn]∗3;[λ]3⟩C[λ]3  [2,1,0]3[λ′]3KL klK′L′. (29)

Since the operator under consideration acts on the separate and spaces, the reduced triple-barred matrix element can be expressed as a product of the separate reduced triple-barred matrix elements tme ():

 ⟨ν;[N′p]3,[N′n]∗3;[λ′]3|||Flm[1]3[1]∗3[2,1,0]