Changes in rotational characters of one- and two-phonon $\gamma$-vibrational bands in $^{105}$Mo

The $\gamma$ vibration is the most typical low-lying collective motion prevailing the nuclear chart. But only few one-phonon rotational bands in odd-$A$ nuclei have been known. Furthermore, two-phonon states, even the band head, have been observed in a very limited number of nuclides not only of odd-$A$ but even-even. Among them, that in $^{105}$Mo is unique in that Coriolis effects are expected to be stronger than in $^{103}$Nb and $^{105}$Nb on which theoretical studies were reported. Then the purpose of the present work is to study $^{105}$Mo paying attention to rotational character change of the one-phonon and two-phonon bands. The particle-vibration coupling model based on the cranking model and the random-phase approximation is used to calculate the vibrational states in rotating odd-$A$ nuclei. The present model reproduces the observed yrast zero-phonon and one-phonon bands well. Emerging general features of the rotational character change from low spin to high spin are elucidated. In particular, the reason why the one-phonon band does not exhibit signature splitting is clarified. The calculated collectivity of the two-phonon states, however, is located higher than observed.


I. INTRODUCTION
Based on the long history of physics of the atomic nucleus as a finite quantum many-body system bound by the strong interaction, nowadays, on the one hand interactions between two, or among three or more, nucleons can be related to quantum chromodynamics and the interaction thus obtained can be used as an input to large-scale shell-model calculations.
On the other hand, from the mean-field picture, density-functional theories that describe ground states of nuclides of wide range in the nuclear chart are developing. For a class of excited states, the generator-coordinate method is used. Applications of such a framework to light odd-A nuclei are just begun [1].
Aside from these progress, traditional effective models are still indispensable to describe detailed properties of collective excitations. In finite many-body systems, individual particle motions and collective motions are of similar energy scales hence couple to each other. In addition, they are sensitive to the numbers of constituents or the shell filling. Collective vibrations are not necessarily harmonic oscillations and whether multiple excitations exist or not is not trivial.
In odd-A nuclei, the first observation of the two-phonon γ vibration was made in 105 Mo [11], on which any theoretical study has not been reported to the author's knowledge, and to which the present study is devoted. Then similar excitations were observed in 103 Nb [12] and 107 Tc [13]. The first realistic calculation on 103 Nb [14] was performed in terms of the triaxial projected shell model. After that the present author made a calculation on this nuclide in terms of the particle-vibration coupling model in Ref. [10].
Quite recently, two-phonon states very similar to those in 103 Nb were observed in 105 Nb [15]. In these isotopes, candidates of three-phonon states that fed two-phonon bands strongly were also indicated. If their property is confirmed, this is the first three-phonon excitation in deformed nuclei, and may indicate that there is a mechanism that makes odd-A nuclei more favorable for realizing a three-phonon γ vibration than even-even nuclei, al-though spectra of odd-A nuclei are thought to be more complex than those of even-even nuclei in general. The present author studied in Ref. [16] these three-phonon candidates with invoking a method to calculate the interband B(E2) based on the generalized intensity relation [17]. The result is promising but still not definitive.
Returning to two-phonon states, the odd particle that couples to them in 103 Nb and 105 Nb is the π[422] 5/2 + originating from the g 9/2 subshell on which Coriolis effects are not very strong. Then states at finite rotation still can be approximately classified in terms of K, the projection of the angular momentum to the z axis. In contrast, the odd particle in 105 Mo is the ν[532] 5/2 − originating from the h 11/2 subshell on which Coriolis effects are stronger.
Accordingly character of rotational band members would be different. This point is the main concern of the present work.
Throughout this paper theh = 1 unit is used.

II. THE MODEL AND PARAMETERS
The particle-vibration coupling (PVC) model based on the cranking model and the random-phase approximation (RPA) is used for calculating eigenstates of odd-A nuclei.
In the first step, the one-dimensionally cranked Nilsson plus BCS one-body Hamiltonian is diagonalized, where h is the standard Nilsson plus BCS Hamiltonian. This gives a set of quasiparticle (qp) states. Second, I apply the RPA to the residual two-body pairing plus doubly-stretched quadrupole-quadrupole interaction between qps, This gives many RPA modes. Cranking and RPA calculations are done in the five major shells, N osc = 2 -6 for the neutron and 1 -5 for the proton. Among RPA modes, I choose the γ-vibrational phonons, n = γ andγ, with signature r = exp (−iπα) = 1 and −1, respectively, which have outstandingly large K = 2 transition amplitudes. Regarding them as elementary excitations, I diagonalize the PVC Hamiltonian, These calculations are performed in terms of signature-classified bases, and all the resulting quantities are given as continuous functions of the rotational frequency ω rot . Detailed expressions of these formulations were given in Ref. [10].
The diagonalization of the PVC Hamiltonian is performed in the model space spanned by 0 -4γ basis states as in Ref. [16]. Here nγ basis states designate 1qp ⊗ (γ orγ) n states. Because there are two types of γ-vibrational phonons, γ andγ, (1qp) r=±i ⊗ γ and (1qp) r=∓i ⊗γ are possible for the 1γ states in the r = ±i sector. Similarly, there are 3 -5 types of 2 -4γ basis states, respectively, for each signature. The concrete form of eigenstates with r = −i was given by Eq. (7) in Ref. [16]. That with r = +i, obtained by interchanging µ andμ, is given here.

The bands in
The sum of the fractions of these main components defines the collectivity of calculated eigenstates.
The parameters entering into the calculation are chosen in a manner similar to the case of Refs. [10] and [16]. Concretely, the pairing gaps, ∆ n = 1.05 MeV and ∆ p = 0.85 MeV, are those widely used for both even-and odd-A nuclides in this mass region [7,12]. The quadrupole deformation, ǫ 2 = 0.3254, is the same as that adopted for the isobar 105 Nb in Ref. [16]. The triaxial deformation γ is chosen so that the calculated signature splitting ∆e ′ between the lowest PVC eigenstates reproduces the overall features of the observed one. The resulting value, γ = −10 • , is again the same as that for 105 Nb. For the quadrupole interaction strengths, examined in the first attempt were those which reproduce the observed γ-vibrational energy ω γ , 0.8121 MeV of 104 Mo [8] or 0.7104 MeV of 106 Mo [7] or their average, in the RPA calculation for the even-even core state at each γ as in previous calculations [10,16,18,19]. But these were unsuccessful because there exists the ν[541] 3/2 − qp-dominant state slightly higher than the one-phonon states in the PVC calculation and the former pushes down the latter. This contradicts the observed situation that the γ-vibrational energy of 105 Mo is higher than those of 104 Mo and 106 Mo, see Fig. 10 of Ref. [11]. To reproduce this phenomenologically, adopted are the force strengths that give ω γ = 1.0 MeV in the RPA calculation for the even-even core and consequently bring the one-phonon states higher than If the static triaxial deformation that mixes the K quantum number is ignored, the second, third, and fourth states are the ν[541] 3/2 − , the K = Ω − 2 = 1/2 γ vibration, and the K = Ω + 2 = 9/2 γ vibration, respectively, hereafter j and Ω are the single-particle angular momentum and its projection to the z axis. In the single-j approximation effective for high-j cases, the favored state (r = +i in the present case) is written as |f Ω = 1 Ref. [16]. This pattern is preserved in the whole calculated range of ω rot in the cases of 103 Nb and 105 Nb with the πg 9/2 odd particle with small signature splittings studied there.
In the present case of 105 Mo with the higher-j, νh 11/2 , odd particle, in contrast, stronger Coriolis interactions change their character as ω rot increases. Actually slopes of their Routhians are as twice as those of 103 Nb and 105 Nb. Because the signature splitting between the 1 and the1 basis states is significantly larger than that between γ andγ, the1 ⊗γ compo-  show signature splitting at ω rot ∼ 0 because of the high K as argued in Ref. [16]. And at finite ω rot , the r = +i and r = −i sequences of the band consist dominantly of the 1 ⊗γ and 1 ⊗ γ, respectively, hence the splitting between them is essentially the difference ω γ − ωγ, which is much smaller than the splitting of the yrast band. When the r = −i is favored as in the g 9/2 and i 13/2 cases, 1 and1 in the figure should be interchanged. The calculated zero-and one-phonon states are compared quantitatively with the observed ones. The reference rotating frame to which the data are converted is determined in two steps. First the Harris parameters are determined to fit the ground band up to 10 + of 106 Mo as J 0 = 18.08 MeV −1 and J 1 = 43.21 MeV −3 . In the second step the origin of this frame must be shifted in order to be compared with states in the odd-A system. Usually this overall shift is given by the pairing gap, in the present case ∆ n = 1.05 MeV, in the cranking calculation. In the present calculation, however, the cranking model is extended to the PVC model and this produces a downward shift of the Routhian of the lowest state, 0.078 MeV at ω rot = 0. Accordingly the overall shift is determined to be 0.972 MeV. The result is shown in Fig. 4. This shows that the zero-and one-phonon states are reproduced well. an anharmonicity, E 2γ /E 1γ = 1.76, was reported in contrast to the harmonic spectra in the adjacent even-even isotopes, 104 Mo [8] and 106 Mo [7]. Note here that an anharmonic E 2γ in 106 Mo taken from Ref. [20] was cited in Ref. [11] but the harmonic value was reported later in Ref. [21] again. The present argument relies on the latest data. These data suggest an unknown mechanism to produce anharmonicity proper to odd-A systems, different from the one discussed for even-even nuclei from a microscopic theoretical point of view in Ref. [22]. that the order of the collective states is ruled by the signature splitting between the 1 and 1 basis states as in the one-phonon states in Fig. 3. This indicates that their connection to the band head with fairly pure K is not trivial.
Then a natural criterion to choose the state to be identified with (the main component of) the observed one is the largest collectivity, as argued in the three-phonon-candidate cases in Ref. [16], because it is generally expected that the most collective state connected by strong E2 transitions would be observed. Actually the upper one, identified to the observed one at the band head, is the most collective at ω rot = 0 because its high-K property prevents the wave function from mixing with other states. But three sequences become to share similar collectivities as soon as rotation sets in. As in the three-phonon case, the state with the highest K effectively has the highest j and accordingly feels the strongest Coriolis force as ω rot increases. Then it is expected that band members that have large overlaps with the most collective eleventh state with the highest K at ω rot ∼ 0 would be energetically lowered with reducing K as ω rot increases and appear at lower Routhians through bandcrossing(s).
The observed transitions are located near to (r = +i) or even lower than (r = −i) the lowest calculated two-phonon states. However, lowering of the location of collectivity to these lowest states in the present calculation is insufficient in contrast to the three-phonon case although j is larger. Therefore it is difficult to establish the mapping between them.

IV. CONCLUSIONS
To conclude, the yrast (zero-phonon) ν[532] 5/2 − , the one-and two-phonon γ vibrational bands in 105 Mo have been calculated in the particle-vibration coupling model based on the cranking model and the random-phase approximation paying attention to the rotational effects on the spectra in comparison to the lower-j cases of 103 Nb and 105 Nb studied in the previous works.
The zero-and one-phonon bands have been reproduced well. In particular, the rotational character change from the K scheme to the signature scheme through the Coriolis K mixing in the one-phonon states is stressed. This naturally accounts for the reason why the observed one-phonon band does not exhibit signature splitting in contrast to the yrast zero-phonon band. A specific feature of the data is that the two-phonon states show anharmonicity in the spectra that is absent in 103 Nb, 105 Nb, 104 Mo, and 106 Mo. This fact suggests that there exists an unknown mechanism to produce anharmonicity proper to high-j odd-A nuclei. The particle-vibration coupling pushes down them but the result is still harmonic with a slightly reduced interval at ω rot = 0. A possibility applicable to finite ω rot , inspired from the previous calculation for the three-phonon-candidate states in 103 Nb and 105 Nb, is that the continuation of the highest-lying two-phonon state with the highest K would be energetically lowered by a strong Coriolis force. In the present calculation, however, its lowering is insufficient.