Description of shape coexistence in $^{96}$Zr based on the collective quadrupole Bohr Hamiltonian

TL;DR

This paper models the shape coexistence in $^{96}$Zr using the collective quadrupole Bohr Hamiltonian, achieving good agreement with experimental data by adjusting the inertia coefficients and considering shell effects.

Contribution

It introduces a detailed geometrical model with a double-minimum potential to describe shape coexistence in $^{96}$Zr, highlighting the importance of inertia parameters and shell effects.

Findings

Good agreement with experimental energies and transition probabilities.

Shape coexistence is well reproduced with a tailored potential.

Shell effects and pairing vibrations influence specific transition rates.

Abstract

Experimental data on $^{96}$Zr indicate coexisting spherical and deformed structures with small mixing amplitudes. We investigate the properties of the low-lying collective states of $^{96}$Zr based on the collective quadrupole Bohr Hamiltonian. The $\beta$-dependent collective potential having two minima -- spherical and deformed, is fixed so to describe experimental data in the best way.Good agreement with the experimental data on the excitation energies, $B(E2)$ and $B(M1)$ reduced transition probabilities is obtained. It is shown that the low-energy structure of $^{96}$Zr can be reproduced in a satisfactory way in the geometrical model with a potential function supporting shape coexistence. However, the excitation energy of the $2^+_2$ state can be reproduced only if the rotation inertia coefficient is taken five times smaller then the vibrational one in the region of the deformed well. It is shown also that shell effects are important for the description of the $B(M1;2^+_2 \rightarrow 2^+_1)$ value. An indication on the influence of the pairing vibrational mode on the $\rho^2 (0^+_2 \rightarrow 0^+_1)$ value is obtained.

Tables

Table 1. Table 1: The results of calculations of the energies and electromagnetic reduced transition probabilities for 96 Zr. The value of b rot subscript 𝑏 rot b_{\rm rot} is taken as 0.2. The values of B ​ ( E ​ 2 ) 𝐵 𝐸 2 B(E2) are given in Weisskopf units and those of B ​ ( M ​ 1 ) 𝐵 𝑀 1 B(M1) in nuclear magnetons. The value of Q ​ ( 2 2 + ) 𝑄 subscript superscript 2 2 Q(2^{+}_{2}) is given in e ⋅ b ​ a ​ r ​ n ⋅ 𝑒 𝑏 𝑎 𝑟 𝑛 e\cdot barn . The excitation energies are given in keV. The experimental energy of the 0 2 + subscript superscript 0 2 0^{+}_{2} state is used to fix the value of B 0 subscript 𝐵 0 B_{0} . Experimental data are taken from Refs.  Kremer ; Witt

Energies and transitions	calc	exp
$E\left(2_1^{+}\right)$	1748	1750
$E\left(2_2^{+}\right)$	2268	2226
$E\left(0_2^{+}\right)$	1582∗	1582
$B\left(E2;2_2^{+}\to 0_2^{+}\right)$	26.1	36(11)
$B\left(E2;2_1^{+}\to 0_1^{+}\right)$	3.6	2.3(3)
$B\left(E2;2_2^{+}\to 0_1^{+}\right)$	0.26	0.26(8)
${\rho }^2\left(0_2^{+}\to 0_1^{+}\right)$	0.0013	0.0075
$B\left(E2;2_2^{+}\to 2_1^{+}\right)$	2.25	${2.8}_{-1.0}^{+1.5}$
$B\left(E2;2_1^{+}\to 0_2^{+}\right)$	6.8	—
$B\left(M1;2_2^{+}\to 2_1^{+}\right)$	0.11	0.14(5)
$Q\left(2_2^{+}\right)$	-0.51	—

Table 2. Table 2: Distribution of the wave function of the 0 1 + subscript superscript 0 1 0^{+}_{1} , 0 2 + subscript superscript 0 2 0^{+}_{2} , 2 1 + subscript superscript 2 1 2^{+}_{1} , and 2 2 + subscript superscript 2 2 2^{+}_{2} states between the spherical and deformed parts of the total potential.

Potential well	$0_1^{+}$	$0_2^{+}$	$2_1^{+}$	$2_2^{+}$
spherical	98.9%	3.2%	77.3%	23.7%
deformed	1.1 %	96.8 %	22.7 %	76.3 %

Equations

$$H=-\frac{{\hslash }^2}{2B_0}\frac{1}{\sqrt{wr}}\frac{1}{{\beta }^4}\frac{\partial }{\partial \beta }{\beta }^4\sqrt{\frac{r}{w}}{b}_{\gamma \gamma }\frac{\partial }{\partial \beta }$$

$$+{\hat{T}}_{\beta \gamma }+{\hat{T}}_{\gamma }+\frac{\hslash }{2B_0}\sum _{\kappa }\frac{{\hat{I}}_{\kappa }^2}{{\Im }_{\kappa }}+V(\beta ).$$

$$w$$

$$r$$

$${\Im }_{\kappa }$$

$$H$$

$$d{\tau }_{coll}=d\beta {\beta }^4\sqrt{b_{\beta \beta }{b}_{\mathrm{rot}}^3}d\Omega .$$

$$d\alpha (\beta )\equiv b_{\beta \beta }^{1/2}(\beta )d\beta ,{\tau }^{1/2}(\beta )\equiv {\beta }^4{b}_{\text{mbox}\mathrm{rot}}^{3/2},$$

$$\Psi (\beta )=g(\beta )\Phi (\beta )$$

$$g(\beta )={\tau }^{-1/4}(\beta )$$

$$\{-\frac{{\hslash }^2}{2B_0}\frac{d^2}{d{\beta }^2}+\frac{{\hslash }^2}{2B_0}\frac{\hat{\vec{I}}{}^2-{\hat{I}}_3^2}{3b_{\mathrm{rot}}{\beta }^2}+V(\beta )$$

$$+\frac{{\hslash }^2}{2B_0}\left[\frac{1}{4\tau }\frac{d^2\tau }{d{\beta }^2}-\frac{3}{16}{\left(\frac{1}{\tau }\frac{d\tau }{d\beta }\right)}^2\right]\}\Phi =E\Phi .$$

$$⟨i|\hat{Q}|j⟩={\int }_0^{\infty }d\beta {\Phi }_i^{\ast }\hat{Q}{\Phi }_j.$$

$$\hat{T}(E0)=\frac{3}{4\pi }Ze{R}^2{\beta }^2,$$

$${\rho }^2(0_2^{+}\to 0_1^{+})={\left(\frac{3}{4\pi }Z\right)}^2|⟨0_2^{+}|{\beta }^2|0_1^{+}⟩{|}^2.$$

$$\left[\left[H,{\beta }^2\right],{\beta }^2\right]=\frac{4{\hslash }^2}{B_0}{\beta }^2.$$

$$|⟨0_2^{+}|{\beta }^2|0_1^{+}⟩{|}^2\le \frac{2{\hslash }^2}{B_0}⟨0_1^{+}|{\beta }^2|0_1^{+}⟩\frac{1}{E(0_2^{+})},$$

$$⟨0_1^{+}|{\beta }^2|0_1^{+}⟩=\frac{5B(E2;2_1^{+}\to 0_1^{+})}{{\left(\frac{3}{4\pi }Ze{R}^2\right)}^2}.$$

$${\rho }^2(0_2^{+}\to 0_1^{+})\le \frac{{\hslash }^2}{B_0}\frac{1}{E(0_2^{+})}\frac{10B(E2;2_1^{+}\to 0_1^{+})}{e^2{R}^4}.$$

$${\rho }^2(0_2^{+}\to 0_1^{+})\le 0.0035$$

$$(M1{)}_{\mu }=\sqrt{\frac{3}{4\pi }}\frac{e\hslash }{2Mc}\frac{Z}{A}\frac{10}{7\sqrt{\pi }}(I{\alpha }_2{)}_{1\mu }$$

$$B(M1;2_2^{+}\to 2_1^{+})\approx 0.03\cdot 10^{-3}{\mu }_N^2,$$

$$⟨2_2^{+}||M1||2_1^{+}⟩={\mu }_N\frac{Z}{A}\frac{15}{7\pi }\int d\beta {\Phi }_{2_2^{+}}(\beta )\beta {\Phi }_{2_1^{+}}.$$

$$(M1{)}_{\mu }={\mu }_N\sqrt{\frac{3}{4\pi }}{g}_R(\beta )I_{\mu },$$

Full text

Experimental data on 96Zr indicate coexisting spherical and deformed structures with small mixing amplitudes.

Purpose:

To investigate the properties of the low-lying collective states of 96Zr based on the collective quadrupole Bohr Hamiltonian.

Method:

The $\beta$-dependent collective potential having two minima – spherical and deformed, is fixed so to describe experimental data in the best way.

Results:

Good agreement with the experimental data on the excitation energies, $B(E2)$ and $B(M1)$ reduced transition probabilities is obtained.

Conclusion:

It is shown that the low-energy structure of 96Zr can be reproduced in a satisfactory way in the geometrical model with a potential function supporting shape coexistence. However, the excitation energy of the $2^{+}_{2}$ state can be reproduced only if the rotation inertia coefficient is taken five times smaller then the vibrational one in the region of the deformed well. It is shown also that shell effects are important for the description of the $B(M1;2^{+}_{2}\rightarrow 2^{+}_{1})$ value. An indication on the influence of the pairing vibrational mode on the $\rho^{2}(0^{+}_{2}\rightarrow 0^{+}_{1})$ value is obtained.

I Introduction

Shape coexistence in nuclei is a remarkable phenomenon which became a widespreaded feature that may occur in many nuclei. The occurrence of different shapes has its origin in the evolution of shell structure with excitation energy and varying occupation of nucleonic orbitals Wood92 ; Casten ; Heyde11 . A theoretical approach suitable for consideration of the phenomena related to nuclear shape dynamics is a collective nuclear model. The basic idea of the nuclear collective model is that even though such a quantum many body system as the atomic nucleus is characterized by a huge amount of microscopic degrees of freedom they are organized in collective modes playing a crucial role in determining nuclear structure. As a next step, a collective Hamiltonian can be constructed which includes as the basic ingredients the deformation-dependent collective potential and the tensor of inertia  Li . This Hamiltonian determines nuclear collective dynamics.

A shape evolution of the nuclear states can happen as a function of excitation energy, angular momentum and the number of nucleons. The shape transition with the number of nucleons has been noticed in many chains of isotopes and isotones but appears to be rather gradual in many cases. The abrupt change of shape in Zr isotopes Cheifetz ; Federman78 ; Federman79 is exceptional. This is a notable feature of nuclear structure in $A\sim 100$ nuclei. The structure of Zr isotopes has been studied within the framework of many different models  Sieja ; Garcia ; Boyukata ; Liu ; Petrovici11 ; Petrovici12 ; Rodrigues ; Skalski ; Xiang ; Mei ; Skalski97 ; Ozen ; Fortune . Strongly deformed states coexisting with a nearly spherical ground state have been reported for Sr and Zr isotopes ISOLDE ; Buscher ; Lherson . In parallel with a smooth or abrupt establishment of a deformed shape a significant mixing between the configurations of a different shape or a suppressed mixing of two such configurations can be considered. Such information can be obtained from electromagnetic transition probabilities. By measuring the electromagnetic decay properties of the collective $2^{+}_{2}$ state of 96Zr Kremer the high purity of the coexisting states has recently been established.

It was discussed in  Chakreborty that the occurrence of the shape coexistence can be related to the existence of a sufficiently large energy gap between subshells. Closely spaced subshells lose their individuality due to pairing correlations and behave as a single, large subshell supporting deformations. In the presence of well-defined subshells in the spherical single particle level scheme a strong redistribution of nucleons over the single particle levels can take place by particle-hole transitions with increasing excitation energy. This helps to stabilize the deformation of the excited states when the ground state is spherical Togashi . Substantial change of a configuration of nucleons leads to a large difference in microscopic structure between the spherical states and the states with deformation-optimized shell structure. This property can decrease mixing of two configurations of different shape.

Some low-lying excited states of 96Zr are related by strong $E2$ transitions. Therefore, it is natural to consider them in the framework of the collective quadrupole model with the Bohr Hamiltonian. However, the appearance of shape coexistence is due to shell effects. Therefore, shell effects are important for the consideration of the structure of 96Zr and it is interesting to study to what extent the structure of 96Zr can be described in the framework of the collective geometrical model with phenomenologically determined potential, and how the impact of shell effects are manifested in such a description.

The aim of the present paper is to investigate an option to explain the properties of the low-lying collective $0^{+}_{1,2}$ and $2^{+}_{1,2}$ states of 96Zr and a weak mixing of the configurations characterized by spherical and deformed shapes based on the Bohr collective quadrupole Hamiltonian. At the end of the paper we also address briefly the characteristics of the 4${}^{+}_{1}$ and 0${}^{+}_{3}$ states.

II Model

The general Bohr collective quadrupole Hamiltonian takes the form Niksic

[TABLE]

The first term in (1) presents the kinetic energy of $\beta$-vibrations. The second and the third terms are connected to the $\gamma$-degree of freedom. The last two terms in (1) are the rotational and potential energies, correspondingly. Above

[TABLE]

The parameter $B_{0}$ is a common dimensional scaling factor for the components of the inertia tensor. Thus, the coefficients $b_{\beta\beta}$, $b_{\gamma\gamma}$, $b_{\beta\gamma}$ and $b_{\kappa}$ are dimensionless inertia coefficients for the $\beta$- and $\gamma$- vibrations, and the rotational motion. The collective potential is determined below.

For keeping out task at a manageable size we assume in this paper that the $\gamma$-degree of freedom can be separated from $\beta$ in the case of the low-lying states of ${}^{96}Zr$. It means that a schematic form of the potential $V(\beta,\gamma)=V(\beta)+V_{\gamma}(\gamma)$ is used. However, it is not enough for complete separation of $\beta$ and $\gamma$ degrees of freedom since they are coupled through the kinetic energy term. As it was shown in Caprio this coupling provides a shift of the probability distributions in $\beta$ of the various states toward larger values of $\beta$. This evolution of the probability distribution can be understood qualitatively as a five-dimensional analog of the centrifugal stretching which effectively produces an additional $\sim 1/\beta^{2}$ type term in $V(\beta)$. Strictly speaking, this term is different for different states. However, this difference can be imitated by the anharmonic correction to the spherical part of the total collective potential as it influences differently the wave functions of different states. Thus, the potential obtained below fitting the experimental data includes, in fact, the effect of the $\beta-\gamma$ coupling through the kinetic energy term. Thus, we put below the coefficient $b_{\beta\gamma}=0$ and consider $b_{\gamma\gamma}$ as a constant. We assume also, for simplicity, that $b_{1}=b_{2}=b_{3}\equiv b_{\rm rot}$. Additionally we assume that $V_{\gamma}$ provides stabilization around $\gamma=0$. We mention, however, that the results for the potential energy surface of 96Zr presented in Kremer ; Togashi , rather indicate a triaxial shape of 96Zr in its $2^{+}_{2}$ and $0^{+}_{2}$ states. Triaxiality should decrease significantly the results presented in Table I for the B(E2;$2^{+}_{2}\rightarrow 2^{+}_{1}$) value and the quadrupole moment $Q(2^{+}_{2})$. Thus, the experimental information on these quantities will give us information on the shape of 96Zr.

Separating the $\beta$-degree of freedom, we obtain as a result the following Hamiltonian which will be used to analyze the apparent shape coexistence phenomena in 96Zr:

[TABLE]

The volume element in the collective space is

[TABLE]

In cases when the $\beta$-dependence of $b_{\beta\beta}$ and $b_{\rm rot}$ is important it is convenient to introduce the following notations

[TABLE]

where $\alpha(\beta)$ is a modified collective variable. However, as it is found in our calculations the interpretation of the existing experimental data on $0^{+}_{1,2}$ and $2^{+}_{1,2}$ states does not require an introduction of a $\beta$-dependent $b_{\beta\beta}$ coefficient. Thus, we put $b_{\beta\beta}$=1 and do not use below the modified collective variable $\alpha$.

To exclude from the Hamiltonian a term with the first derivative over $\beta$ let us present the collective wave function $\Psi$ as

[TABLE]

and determine $g(\beta)$ so that the Schrödinger equation for $\Phi(\beta)$ will not include a first derivative of $\Phi$. Then

[TABLE]

and the Schrödinger equation for $\Phi$ takes the form:

[TABLE]

In these notations the matrix elements of an arbitrary operator $\hat{Q}$ are calculated as

[TABLE]

Analyzing the experimental data, it was found in Jolos07 ; Jolos08 , that in the case of well deformed axially symmetric nuclei the inertia coefficient for the rotational motion is several times smaller than the inertia coefficient for the vibrational motion. In a complete correspondence with this result it is shown below that in order to explain the excitation energy of the $2^{+}_{2}$ state it is necessary to take $b_{\rm rot}$ several times less than unity.

III Results of calculations

The Hamiltonian (8) contains two important ingredients which determine the results of our calculations: the potential energy as a function of $\beta$ and the inertia coefficient for the rotational motion. At first, we have tried to determine the shape of the collective potential which allows to get a satisfactory description of the experimental data. Then, the parameter $B_{0}$ has been varied to fix the energy scale in agreement with the experimental energies. We have initially assumed that the rotational inertia coefficient $b_{\rm rot}$=1. It was found in this case that the main problem for describing of the experimental data is related to the reproduction of the excitation energy of the second $2^{+}$ state together with the $0^{+}_{2}$ state. This energy spacing was obtained several hundreds keV lower than the experimental value. As it is shown in Fig. 1. the wave function of the second $2^{+}$ state is located in the deformed well of the potential. Therefore, this state can be interpreted qualitatively as a rotational state based on the $0^{+}_{2}$ state, and its energy is determined by the deformation at the minimum of the deformed well and the rotational inertia coefficient. A deformation at the minimum of the deformed well is related to the $B(E2;2^{+}_{2}\rightarrow 0^{+}_{2}$) value. Therefore, the calculated value of the excitation energy of the $2^{+}_{2}$ state can be improved only if we assume that the rotational inertia coefficient $b_{\rm rot}<1=b_{\beta\beta}$, i.e. smaller than the vibrational inertia coefficient. This assumption just corresponds to the conclusion obtained in Jolos08P ; Jolos08 ; Jolos06 ; Jolos07 as a result of the analysis of the experimental data on well deformed nuclei. This analysis has indicated that the vibrational inertia coefficient can exceed the rotational one by 4 – 10 times. The results for the energies of the low-lying collective states and the electromagnetic transition probabilities obtained, assuming that the rotational inertia coefficient is five times smaller than the vibrational one (i.e. $b_{\rm rot}=0.2$) are shown in Table 1 and in Fig. 2.

The results presented in Table 1 are obtained assuming $\gamma=0$. This assumption does not influence the results for $B(E2;2^{+}_{i}\rightarrow 0^{+}_{j}$). However, in the case of $\gamma=30^{\circ}$ $Q(2^{+}_{2})=0$. As it is seen from the results presented in Table 1 the agreement between the calculated results and the experimental data is satisfactory. The results for $\rho^{2}(0^{+}_{2}\rightarrow 0^{+}_{1})$ and $B(M1;2^{+}_{2}\rightarrow 2^{+}_{1})$ are discussed below. These quantitites can be calculated using the wave functions shown in Fig. 1. The value of $\rho^{2}(0^{+}_{2}\rightarrow 0^{+}_{1})$ is a factor 3.6 smaller than the experimental value. But both, the experimental and the calculated $\rho^{2}(0^{+}_{2}\rightarrow 0^{+}_{1})$ values are small in comparison to the corresponding quantitites in other nuclei Wood99 . Using the standard definition of the $E0$ transition operator in the collective model

[TABLE]

where $R$ is the nuclear radius, we obtain

[TABLE]

In order to derive an alternative expression for $\langle 0^{+}_{2}|\beta^{2}|0_{1}\rangle$ in terms of other quantities whose values are known from the experiment let us calculate the double commutator $[[H,\beta^{2}],\beta^{2}]$ using the Hamiltonian (8). The result is

[TABLE]

Taking the average of (12) over the ground state $0^{+}_{1}$ and assuming that the ground state is mainly related by $E0$ transition to the $0^{+}_{2}$ state we obtain

[TABLE]

where $E(0^{+}_{2})$ is the excitation energy of the second $0^{+}$ state. The inequality in (13) appears because we neglect the contribution to the value of $\langle 0^{+}_{1}|\beta^{2}H\beta^{2}|0^{+}_{1}\rangle$ of the $0^{+}$ states higher in energy than $0^{+}_{2}$. The quantity $\langle 0^{+}_{1}|\beta^{2}|0^{+}_{1}\rangle$ can be expressed with a good accuracy through $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ using the collective model definition of the $E2$ transition operator $Q_{2\mu}=\dfrac{3}{4}{\pi}eZR^{2}{\alpha}_{2\mu}$, where assuming axial symmetry we use $\alpha_{2\mu}=D^{2}_{\mu 0}\beta$ JvonBr :

[TABLE]

Substituting (13) and (14) into (11) we obtain

[TABLE]

In our calculations the value of $\dfrac{\hbar^{2}}{B_{0}}$ was previously fixed as $8.062$ keV in order to reproduce the experimental value of $E(0^{+}_{2})$. Substituting this value and the experimental values of $E(0^{+}_{2})$ and $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ into (15) we obtain that

[TABLE]

in satisfactory correspondence to the result given in Table 1.

This result means that the experimental value of $\rho^{2}(0^{+}_{2}\rightarrow 0^{+}_{1})$ which is $0.0075$ we can reproduce using the collective model with quadrupole degree of freedom only within a factor of two. We cannot exclude that the pairing vibrational mode plays an important role in the description of the $E0$ transitions Iwasaki .

Consider now the result obtained for the $B(M1;2^{+}_{2}\rightarrow 2^{+}_{1})$ value. In the collective quadrupole model the part of the $M1$ operator which generates transitions between the states has the form Grechukhin

[TABLE]

where $I$ is the angular momentum operator. Using the operator (17) we obtain that

[TABLE]

where $\mu_{N}$ is a nuclear magneton. This value is more than three orders of magnitude smaller than the experimental value. The reason for this result is the following. The reduced matrix element of the operator (17) is equal to

[TABLE]

As it is seen from Fig. 1 the wave function of the $2^{+}_{2}$ state has different signs in the spherical and deformed parts of the potential, whereas the wave function of the $2^{+}_{1}$ state does not change sign. As a result the integral in (19) takes a small value. To solve the problem it was suggested in Kremer to use the following $M1$ transition operator:

[TABLE]

where the $g_{R}$-factor varies from the shell model value $-0.26$ in the spherical well to $\dfrac{Z}{A}$ in the deformed well with a narrow transition region in $\beta$ where the wave function of the $2^{+}_{2}$ state changes its sign. With this transition operator we obtain $B(M1;2^{+}_{2}\rightarrow 2^{+}_{1})=0.11\mu_{N}^{2}$ which coincides in the limit of the experimental errors with the experimental value $0.14\mu_{N}^{2}$ Kremer . This result stresses the importance of the shell effects for the description of the shape coexistence phenomena, at least, in the case of 96Zr.

The collective potential which has been fixed phenomenologically so to describe the experimental data in the best way is shown in Fig. 3. The height of the barrier calculated from the position of the ground state is equal to $2.45$ MeV.

The wave functions of the ground and excited states are shown in Fig. 1. Their distribution between the spherical and deformed parts of the total potential is characterized by the values obtained by integration of the squares of the wave functions over the regions inside spherical or deformed wells. These values are presented in Table 2.

As the value of $\beta$ separating the spherical and deformed wells we have considered several points: the zeroes of the $2^{+}_{2}$ and $0^{+}_{2}$ wave functions, and the middle of the barrier. The results obtained are not very sensitive to this choice. As we see from Table 2 the wave function of the $I^{\pi}=0^{+}_{1,2}$ states are practically concentrated in one well: spherical for the $0^{+}_{1}$ state and deformed for the $0^{+}_{2}$ state. Qualitatively the same situation is realized in the case of the $I^{\pi}=2^{+}_{1,2}$ states. However, in this case the distribution of the wave functions between the spherical and deformed wells is less asymmetric. As it is seen from Table 2 the wave function of the $2^{+}_{1}$ state is also located mainly in the spherical well like the $0^{+}_{1}$ state. However, its weight in the deformed minimum is equal to 22.7 %, i.e. significantly larger than for the $0^{+}_{1}$ state.

In addition to the states considered above there are the known 4${}^{+}_{1}$ state which decays by a strong E2 transition to the 2${}^{+}_{2}$ state and the 0${}^{+}_{3}$ state also decaying to the 2${}^{+}_{2}$ state by a strong E2 transition Witt . This means that both 4${}^{+}_{1}$ and 0${}^{+}_{3}$ states can be considered in the quadrupole-collective model presented in this paper. Our preliminary calculations with the potential fitted above have shown that the wave function of the 4${}^{+}_{1}$ state is concentrated in the deformed well. Therefore, this state can be considered as a member of the band including $0^{+}_{2}$ and 2${}^{+}_{2}$ states. However, we have obtained that the ratio $[E(4^{+}_{1})-E(0^{+}_{2})]/[E(2^{+}_{2})-E(0^{+}_{2})]$ is equal to  2.5. This value is much lower than the typical rotational ratio 3.33, however, is larger than the experimental value. The reason is a closeness of the energy of the $4^{+}_{1}$ state to the barrier height.

For the energy of the $0^{+}_{3}$ state we obtained a value which is much larger than the experimental one. Probably, this means that the structure of the $0^{+}_{3}$ state is related to an excitation of the $\gamma$-mode.

IV Conclusion

We have studied a possibility to describe the properties of the low-lying collective states of 96Zr basing on the Bohr collective quadrupole Hamiltonian in terms of axially symmetric shape coexistence. The potential energy of this Hamiltonian is fixed to describe the experimental data in the best way. This potential has two minima - spherical and deformed separated by a barrier. Good agreement with the experimental data is obtained for excitation energies, $B(E2)$, and $B(M1)$ values. It is shown that the experimental value of the excitation energy of the $2^{+}_{2}$ state can be reproduced only if the rotational inertia coefficient in the region of the deformed well is taken five times smaller than in the region of the spherical well.

The calculated value of the $\rho^{2}(0^{+}_{2}\rightarrow 0^{+}_{1})$ is six times smaller than the measured value. This indicates, possible influence of the pairing vibrational mode which is not included in the present consideration.

The calculated value of the $B(M1;2^{+}_{2}\rightarrow 2^{+}_{1}$) strength demonstrates an excellent agreement with the experimental value. However, this result has been obtained due to modification of the $M1$ transition operator which takes into account the result of the shell model for the g-factor of the spherical configuration.

V Acknowledgements

Acknowledgements.

We thank A. Leviatan, V. Werner, and T. Beck for discussions. This work was supported by the German DFG under grant No. SFB 1245 and by the BMBF under grant No. 05P18RDFN1.