Quasiparticle-phonon model and quadrupole mixed-symmetry states of 96 Ru

The structure of low-lying quadrupole states of 96Ru was calculated within the Quasiparticle-Phonon Model. It is shown that symmetric and mixed-symmetry properties manifest themselves via the structure of the excited states. The first 2 state is collective and neutron and proton transition matrix elements Mn and Mp are in-phase, while the neutron and proton transition matrix elements Mn and Mp have opposite signs for the third 2 state. This property of the third 2 state leads to a large M1 transition between the first and third 2 states. It is an unambigous demonstration of the mixed-symmetry nature of the third 2 state. The structure of the first 1 state is calculated. The state is a member of the two-phonon multiplet generated by the coupling of the [2+1 ]QRPA and the [2 + 2 ]QRPA states.


Introduction
Recent experiments have discovered quite complex structures of low-lying states of nearly spherical nuclei.It originates from proton-neutron oscillations and generates mixed-symmetry states (MS S ).This phenomenon was predicted within the proton-neutron version of the interacting boson model (IBM − 2) [1].Many years after the predictions unambiguous and comprehensive evidence of mixed-symmetry states based on absolute M1 transition rates was found in experiments on 94 Mo [2,3].A review of the experimental aspects on mixed-symmetry states in vibrational and weakly deformed transitional nuclei is given in [4].A detailed explanation of the structure of mixed symmetry states was given within a microscopic approach based on the Quasiparticle-Phonon Model (QPM) [5].The model treats the excitation as superposition of multiphonon states and was successfully applied in the domains around semi-magic numbers N = 50 and N = 82.These two domains reveal different behavior of mixed-symmetry excitations.The mixed-symmetry properties are concentrated predominantly in a single state [6,7] of 94 Mo while around N = 82 there is fragmentation of mixed-symmetry strength [8][9][10].
The existence of the quadrupole mixed-symmetry state of 96 Ru has been announced in Ref. [11].Its identification is based on the large M1 transition rate between the first and third 2 + states.Later on two-phonon mixed-symmetry states, 3 + 1,ms and 2 + 2,ms , of 96 Ru have been proposed [12].Recently, new experimental information concerning mixed symmetry states of N = 52 isotones has been puba e-mail: stoyanov@inrne.bas.bgb e-mail: pietralla@ikp.tu-darmstadt.delished.The mixed-symmetry excitations of 96 Ru have been studied via inelastic proton-scattering [13,14].
Calculations of the structure of quadrupole mixedsymmetry states of 96 Ru within the QPM are presented in this paper.

The model
Following [15] the main building blocks of the QPM are QRPA phonons.The phonon operator reads: where jm denote a single-particle level of the average field for neutrons (or protons) and the notation [• • • ] λμ means coupling to the total angular momentum λ with projection μ; [α + j α + j ] λμ = mm C λμ jm j m α + jm α + j m ; the quantity C λμ jm j m is the Clebsch-Gordon coefficient.Quasiparticles themselves are the result of the Bogoliubov transformation.In the QPM, quasiparticle energies and Bogoliubov's coefficients u j and v j are obtained by solving the BCS equations.A phonon basis is constructed by diagonalizing the QPM Hamiltonian on the set of one-phonon states [15].The procedure yields the QRPA equations, and solving these equations one obtains the phonon spectrum and the internal phonon structure, i.e., the coefficients ψ λi j j and ϕ λi j j of Eq. ( 1) for any multipolarity λ under consideration.The index i in the definition of the phonon operator (1) gets the meaning of the QRPA root number.The phonons are of different degree of collectivity, from collective ones (e.g.In the case of even-even nuclei the QPM Hamiltonian is diagonalized in a basis of wave functions constructed as a superposition of one-, two-, and three-phonon components [16,17].
where Ψ 0 represents the phonon vacuum state and R, P, and T are unknown amplitudes.The index ν specifies the particular excited state.
The foregoing formalism was applied to study the lowlying excited states of even-even nuclei having neutron numbers N = 80 and N = 84 and the domain around neutron number N = 50.The results are published in [6,7,9,10].

Results
In the calculation the parameters of the QPM Hamiltonian are the same as used in [6,7] for 94 Mo.The corresponding single-particle spectra for the A = 90 region can be found in [18].The strengths of the quadrupole-quadrupole and octupole-octupole interactions were fixed by a fit of the lowest 2 + 1 and 3 − 1 levels of 96 Ru.The strengths of the other multipole terms are adjusted to keep unchanged the energy of the computed two-quasiparticle states [18].This set of parameters was widely used and gave an overall description of the low-lying as well as the high-lying states of nuclei in this mass region [18].
The structure of the 2 + states of 96 Ru calculated in QRPA reveals that the [2 +  1 ] QRPA state is symmetric and the [2 +  2 ] QRPA state shows mixed symmetry.The total contribution of neutrons and protons to the structure of the 2 + 1 state is in-phase (the sum of neutron and proton transition matrix elements M n and M p is large and both have positive sign).The contribution of neutrons and protons in the structure of the [2 Table 3. Results of the QPM calculations for 96 Ru in comparison to the experimental data [13,14].M1 strengths are in units of μ 2 N , E2 strengths are given in units of W.u.

State Transition
Transition strength

State Transition
Transition strength state is almost an isovector one-phonon state and therefore it is the one-phonon mixed-symmetry state.
The structure of the first 1 + state is mainly a twophonon mixed-symmetry state.The main component is of two-phonon character, coupling the symmetric [2 +  1 ] QRPA state with the mixed symmetry [2 +  2 ] QRPA state.The corresponding transition probabilities between the low-lying quadrupole states of 96 Ru are given in Table 3.The comparison with IBM results is presented in Table 4.

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The experimental data as well as the IBM results are taken from [13,14].
It is seen from Table 3 that the QPM reproduces the main features of the low-lying quadrupole states in 96 The state is a member of the two-phonon multiplet generated by symmetric and mixed-symmetry quadrupole QRPA states.The corresponding M1 transition connecting the 1 + state with the ground state is less than the experimental value but nevertheless appears to be quite large that confirms the twophonon mixed-symmetry nature of the first 1 + state qualitatively.It is seen from Table 4 that the calculated values within the QPM are very similar to those within the IBM-2.The results of both the models on the M1 transition rate be-tween the 2 + 3 state and the 2 + 1 state are in good agreement.This confirms the mixed-symmetry nature of the 2 + 3 state.It is interesting to compare the structure of low-lying quadrupole states of 94 Mo with those of 96 Ru.The structure of the states of 94 Mo is published in [6,7] and is shown in Table 5.The contribution of the main components in the structure of the lowest 2 + states is very similar to those of 96 Ru.The calculated values of B(M1; 2 + i → 2 + 1 ) for both the nuclei are shown in Fig. 1.The behaviour of both distributions is very similar.The M1 strength is concentrated predominantly in a single state.In both nuclei the 2 + 3 state is the quadrupole one-phonon mixedsymmetry state.In 96 Ru its excitation is higher than in 94 Mo.This trend is also well reproduced in the QPM calculations.

Conclusion
The presented results on the structure of low-lying quadrupole states within the QPM confirm the complicated nature of low-lying quadrupole states of 96 Ru.The spectrum consists of symmetric as well as mixed symmetry states.The regularities of E2 and M1 transitions correspond to the idea about the existing of both symmetries in the low-energy sector of excited states.

Table 1 .
+ 2 ] QRPA state is out-of-phase (the neutron and proton transition matrix elements M n and M p have opposite signs).It means that isoscalar correlations dominate in the first [2 + ] QRPA state, while the structure of the second [2 + ] QRPA state is dominated by isovector correlations.The main two-quasiparticle components contributing to the structure of the low-lying [2 + ] QRPA states are given in Table 1.The first [2 + ] QRPA state is collective.The state is connected with the second [2 + ] QRPA with a large M1 transition.The B(M1, 2 + 2 → 2 + 1 ) value is 1.37 μ 2 N .The energies and the structure of the low-lying QPM states are given in Table 2.The first 2 + state is dominated by the isoscalar [2 + 1 ] QRPA component and therefore it is a symmetric state.The second 2 + state is mainly a twophonon state dominated by isoscalar phonons and, therefore it is a two-phonon symmetric 2 + state.The third 2 + Contributions of the main components in the structure of the low-lying QRPA 2 + states of 96 Ru.The corresponding E2 transitions are shown.

Table 2 .
Structure of the low-lying quadrupole QPM states of 96 Ru.

Table 5 .
Structure of the low-lying quadrupole QPM states of 94 Mo.Values of the largest components are given.
Ru.The agreement with experimental data is quite well.The M1 transition strength between the 2 + 3 state and the 2 + 1 state is in good agreement with the data.Its relatively large value confirms the suggestion that the structure of the 2 + 3 state corresponds to a mixed-symmetric state.The structure of the first 1 + state is dominated by the two-phonon (