An analysis of E(5) shape phase transitions in Cr isotopes with covariant density functional theory

An analysis of E(5) shape phase transitions in Cr

isotopes with covariant density functional theory

Tuncay Bayram1 and Serkan Akkoyun2

1 Department of Physics, Faculty of Science and Arts, Sinop University, Sinop,

Turkey2 Department of Physics, Faculty of Science, Cumhuriyet University, Sivas, Turkey

E-mail: [email protected]

Abstract. Constrained relativistic mean field theory (RMF) has been employed for

analysis of the shape phase transitions of even-even 52−66Cr isotopes. The systematic

investigation of ground-state shape evolution between spherical U(5) and γ−unstable

O(6) for these nuclei has been carried out by using the potential energy curves

(PECs) obtained from the effective interactions NL3*, TM1, PK1 and DD-ME2. The

calculated PECs have indicated that 58Cr can be a candidate for the critical-point

nucleus with E(5) symmetry. Similar conclusion is also drawn from calculated single-

particle spectra of 58Cr and is supported by the experimental data via observed ratios

of the excitation energies. Furthermore, γ-independent character of 58Cr has been

pointed out by using its binding energy map in β-γ plane obtained from the triaxial

RMF calculations as in agreement with the condition of E(5) symmetry.

PACS numbers: 21.10.Dr, 21.10.Pc, 21.10.Re, 21.60.Fw

An analysis of E(5) shape phase transitions in Cr isotopes 2

1. Introduction

Nuclear quantum phase transitions have been the subject of experimental and theoretical

studies during the last decade. In nuclear physics, phase transitions in the equilibrium

shapes of nuclei correspond to first- and second-order quantum phase transitions (QPTs)

between competing ground-state phases induced by variation of a nonthermal control

parameter (number of nucleons) at zero temperature [1]. As is well known, atomic nu-

clei possess modes of collective motion and geometric shapes, such as spherical vibrator,

rotational ellipsoid, and deformed shapes [2, 3, 4]. Nuclei can exhibit changes of their

energy levels and electromagnetic transition rates among them when the number of

neutrons or protons is changed, resulting in shape phase transitions from one kind of

collective behavior to another. These transitions, which describe changes in equilibrium

structure, that is, the ground-state, are quantum phase transitions (QPTs) [5] and are

different from usual thermal phase transitions which occur as a function of temperature.

This refers to changes in the shape of the nucleus, and the control parameter is the num-

ber of nucleons. In the last decade, a number of theoretical studies have given insights

into the evolution of the structure (particularly, in transitional regions of rapid change).

They involve the concepts of QPTs, phase coexistence and the critical-point symmetries

(CPS) [6, 7] developed by Iachello, as well as a raft of geometrical models [8, 9, 10] that

can be mock up the changing shapes of the nuclei with number of neutrons (N ) and

protons (Z ).

A new class of symmetries E(5) [6] and X(5) [7] have been proposed to de-

scribe shape phase transitions in atomic nuclei within the collective Bohr Hamil-

tonian. These symmetries have been experimentally identified in the spectrum of134Ba [11] and 152Sm [12], respectively. The E(5) symmetry has been suggested to

describe the systems lying at the critical-point in the transition from spherical to γ-soft

shapes while X(5) symmetry is offered to define critical-points in the phase transi-

tions from spherical to axially deformed systems. Although QPTs are usually stud-

ied within the geometrical models [13] and algebraic models [14], theoretical stud-

ies [15, 16, 17, 18, 19, 20] have been conducted within the framework of the rela-

tivistic mean field (RMF) model [21, 22, 23, 24] and Hartree-Fock-Bogoliubov (HFB)

model [3, 25, 26]. 148,150,152Sm [15] and 128,130,132,134Ce [17] have been suggested to be

the possible critical-point nuclei with X(5) symmetry while 48,52,60Ti [18] have been pro-

posed to be the critical-point nuclei with E(5) symmetry within the framework of the

RMF model. Also, a series of isotopes which exhibit critical-point symmetry have been

offered in RMF model by Ref. [16]. A similar study for rare-earth region has been car-

ried out by means of the HFB model in Ref. [19]. Besides, 46,52,60Ti have been suggested

as critical-point nuclei with E(5) symmetry in HFB model [20]. In these studies, RMF

and HFB models were employed in calculation of the potential energy curves (PECs)

by considering constraint on quadrupole moment. The resulting PECs show a shape

transition from spherical to deformed. PECs appear relatively flat and are interpreted


as a sign of certain types of critical-point symmetries. However, one should go beyond

the mean field level which means that excitation energies and electromagnetic transi-

tion rates should be calculated for a quantitative analysis of QPT in nuclei. Because of

this, the generator coordinate method (GCM) was used to perform configuration mix-

ing of angular-momentum and particle-number projected relativistic wave functions in

Ref. [27]. The GCM has been extended by implementing the exact three-dimensional

angular momentum projection on triaxial states in recent years [28, 29, 30]. It should be

noted, however, that the applications of these methods with triaxiality for a systematic

study of QPT are still very time consuming at present. The evolution of the PECs ob-

tained from quadrupole constrained calculations along the isotopic and isotonic chains

can provide a qualitative understanding of the shape phase transition.

The classical shape transition regions (Si−Mg region, A=100 (Z∼40), light rare

earth nuclei (A∼=150) and actinides) are well known (see Ref. [31] and references therein).

Besides, neutron-rich nuclei (in the A≈60 mass region) constitute an ideal region for

investigations of shape evolution [32] because of the appearance of a sub-shell gap at

N=32, observed in Ca, Ti and Cr isotopes. The onset of stable deformation is expected

in heavy Cr and Fe isotopes when approaching N=40 [33, 34]. A critical-point for the

phase transition could be encountered in this spherical-to-deformed path. Also, the

recent experimental study [32] on the shape of Cr isotopes has pointed out that 58Cr

can be a possible critical-point nucleus with E(5) symmetry. Because self-consistent

RMF theory achieves considerable success in the description of various ground-state

properties of the nuclei throughout both in and far from the stability line [23, 24], we

have employed constrained RMF theory with various interactions for investigation of

the shape evolution of even-even 52−66Cr isotopes in this study.

The paper is organized as follows. In Section 2, the relativistic mean field theory

with non-linear self coupling of mesons and with density dependent meson-nucleon

coupling and the relations of the both version of the RMF theory has been presented

briefly. In section 3, some details on the numerical calculations have been given. In

Section 4, the results of the study and discussions have been presented. Finally, a

summary has been given in Section 5.

2. Theoretical framework

The relativistic mean field (RMF) model describes the nucleus as a system of Dirac

nucleons which interact via exchange of various mesons and photons. The mesons con-

sidered in the RMF model are scalar σ-, vector ω- and isovector vector ρ-mesons. The

scalar σ- and vector ω-mesons are responsible for attractive and repulsive part of the

interaction of nucleons, respectively. The isovector vector ρ-meson and photon are im-

portant for describing of isospin-dependent effects and the electromagnetic interaction

in nuclei, respectively. The mesons do not interact among themselves in the simplest


version of RMF model. However, this version of RMF model encountered a problem

for describing incompressibility for nuclear matter. Therefore, Boguta and Bodmer [35]

proposed to include a non-linear self interaction of the σ-mesons. This non-linear ver-

sion of RMF model has been widely used in describing many nuclear phenomena during

the last two decades, because it is successful for providing essential properties of the

equation of state and the surface properties such as the nuclear deformation [22] and

the incompressibility [35]. Moreover, there are models with non-linear self-interaction

of the ω-mesons [36] and ρ-mesons [37, 38]. Also, there is a version of RMF model with

density-dependent meson-nucleon couplings [39, 40, 41, 42, 43] which is an alternative

approach for describing nuclear matter and finite nuclei.

The starting point of the RMF model is a phenomenological effective Lagrangian

density. It contains terms for free nucleon, free mesons and the interaction of nucleon

with the mesons. The phenological Lagrangian density can be written as:

L = ψ[iγµ∂µ −M − gσσ − gωγµωµ − gργ

µτ .ρµ − eγµ1− τ3

2Aµ]ψ

+1

2∂µσ∂µσ − 1

2m2

σσ2 − 1

3g2σ

3 − 1

4g3σ

4

−1

4ωµνωµν +

1

2m2

ωωµωµ +

1

4c3(ω

µωµ)2 − 1

4ρµν .ρµν

+1

2m2

ρρµ.ρµ +

1

4d3(ρ

µ.ρµ)2 − 1

4AµνAµν , (1)

where field tensors of the vector mesons and of the electromagnetic field are:

ωµν = ∂µων − ∂νωµ,

ρµν = ∂µρν − ∂νρµ + gρρµ × ρν , (2)

Aµν = ∂µAν − ∂νAµ.

In the equations, the meson fields (masses) are denoted by σ (mσ), ω (mω) and ρ (mρ).

gσ, gω and gρ are the corresponding coupling constants for the mesons. The Dirac spinor

ψ denotes the nucleon with mass M . In the present study, space vectors are denoted

by bold type symbols while arrows indicate isospin vectors. Unknown meson masses

and the coupling constants are parameters. They are adjusted to reproduce ground-

state properties of finite nuclei and nuclear matter properties. The Lagrangian density

given by Eq. (1) includes four mass parameters (M , mσ, mω, mρ), three nucleon-

meson coupling constants (gσ, gω, gρ) and four self-coupling constants (g2, g3, c3, d3).

The nucleon and ρ-meson masses are fixed to their free values in general and coupling

coefficients c3 and d3 are taken into account as zeros. In addition to the masses of nucleon

and ρ-meson, the mass of ω-meson is fixed to its free value in some parametrizations.

Thus six parameters are obtained by fitting experimental binding energies and nuclear

charge radii of some spherical nuclei as in the Ref. [44]. In another version of the

RMF theory meson-nucleon interactions are considered as density dependent. In this

alternative version, the coupling constants gσ, gω, gρ are functions of the baryonic density

ρυ =√jµjµ with jµ = ψγµψ. (3)


The non-linear self-coupling constants g2, g3, c3, d3 are set to zero. The baryonic density

dependence of the coupling constants for σ and ω mesons reads

gi(ρυ) = gi(ρsat)fi(x) for i = σ, ω, (4)

where

fi(x) = ai1 + bi(x+ di)

2

1 + ci(x+ di)2(5)

is a function of x = ρ/ρsat, and ρsat indicates the baryon density at saturation in

symmetric nuclear matter.

The functional form of the density dependence of ρ-meson is given by an exponential

form

gρ = gρ(ρsat)exp[−aρ(x− 1)]. (6)

The eight parameters related to density dependence for nucleon-meson couplings

in Eq. 5 are not independent. The five constraint conditions fi(1) = 1, f′′σ (1) = f

′′ω (1),

and f′′i (0) = 0 for the functions fi(x) reduce the number of free parameters to three.

The equations of motion for the fields can be obtained from the Lagrangian density

(1) by using the classical variational principle. A set of coupled equations namely

the Dirac equation with potential terms for the nucleons and the Klein-Gordon type

equations with sources for mesons and photon are obtained by application of variational

principle. These equations can be solved by expanding the wave functions in terms of a

deformed axially symmetric harmonic oscillator potential [22, 45]. Also, they are solved

by expanding the Dirac spinors and the mesonic fields in terms of the complete set

of eigenfunctions of the three-dimensional harmonic oscillator, expressed in Cartesian

coordinates as prescribed in Ref. [46].

3. Numerical details

In the calculations, the numbers of oscillator shells taken into account are 14 and 20

for the fermionic and bosonic expansions, respectively. The oscillator basis parameter

is taken to be 41A−1/3. The PECs of Cr isotopes are obtained from constrained

calculations in such a way that the binding energy at a desired deformation is calculated

by constraining the quadrupole moment ⟨Q2⟩ to a given value µ2 in the expectation value

of the Hamiltonian

⟨H ′⟩ = ⟨H⟩+ Cµ(⟨Q2⟩ − µ2)2 (7)

where Cµ is the constraint multiplier. The relation between the expectation value

of quadrupole moment ⟨Q2⟩ and the deformation parameter β2 is taken as ⟨Q2⟩ =

(3/√5π)Ar2β2, where r = R0A

1/3 (R0 = 1.2 fm) and A is the mass number.


For pairing correlations, BCS (Barden-Cooper-Schiffer) method is used in the

present study. Generally constant gap approximation is used in the RMF calculations.

However, the constant gap approximation is not a good one for calculation of the PECs

because the gaps are kept fixed along the variation of the energy with deformation.

Because of this, we adopt constant G approximation in which the pairing strength G

is kept fixed as proposed in Ref. [48]. In RMF+BCS methods, the RMF equations

are solved self-consistently and at each iterative step the BCS occupation probabilities

ν2k are determined. In the constant gap approximation, the BCS expression for the

occupation numbers

ν2k =1

2

(1− ϵk − λ√

(ϵk − λ)2 +∆2

)(8)

is used by starting with fixed gap parameters ∆ which can be determined from

experimental odd-even mass differences or estimated formulae (i.e., ∆n = 4.8/N1/3

and ∆p = 4.8/Z1/3 for neutron and proton, respectively). In Eq. (8), ϵk and λ are the

eigenvalues of the Dirac equation and the chemical potential, respectively. However,

in the constant G method considered in the present study, the occupation numbers

are evaluated by the solution of the gap equations based on a monopole pairing force

with the strength parameters Gn for neutrons and Gp for protons. In the following

expressions, Gn and Gp are represented by G. In the method, by starting with a pairing

strength parameter G the gap equation [3] given by

1

G=∑k>0

1

2√

(ϵk − λ)2 +∆2(9)

is solved in a self consistent manner to determine the gap parameters

∆ = G∑k>0

ukνk. (10)

Pairing window determined by the cutoff energy Ecutoff is an important quantity

to put limit on the sums in Eq. (10) which shows an ultra-violet divergence. The pairing

window is taken as being Ecutoff = 60 MeV. The size of the effective pairing constants

G are adjusted in such a way that the resulting gap parameters correspond roughly to

the observed gaps in the present work. It should be noted, however, that there is an

important difference between the method of constant G and the constant gap approxi-

mation in determining the pairing window. The reader is referred to Ref. [48] for detail

about the influence of the pairing window on the size of the fission barriers.

Besides, the non-linear sigma parameter sets NL3* [44] (revised version of NL3 [47])

and TM1 [36], non-linear meson coupling (in the ρ-channel) parameter set PK1 [38]

and density-dependent covariant functional DD-ME2 [39] are taken into account in the

present study. Parameters of these effective interactions are listed in Table 1.


4. Results and discussions

The calculated binding energies per nucleon (BE/A) for even-even 52−66Cr isotopes

obtained from quadrupole constrained RMF theory with the effective interactions NL3*,

TM1, PK1 and DD-ME2, and the experimental data are shown in figure 1. The four

different interactions produce similar results for the binding energy per nucleon as in

good agrement with the experimental data [49] and the RMF calculations are found

to be very stable against changes of the parametrizations. In Table 2, we show the

differences between the calculated and experimental binding energies (BE) of 52−66Cr.

The root-mean-square deviations (defined by σ =√∑N

i=1(BEitheo −BEi

exp)2/N , where

N is the total number of experimental data) between the results of the present study

and the experimental data for binding energies of the eight Cr isotopes are 2.038, 2.334,

2.014 and 1.973 for the interactions NL3*, TM1, PK1 and DD-ME2, respectively. In

Table 2, the largest deviation between experimental data and calculated results is seen

for 52Cr obtained from the interaction TM1. After 52Cr, the maximum difference from

experimental results appears in binding energies of 66Cr obtained from the interaction

Table 1. Parameters of the effective interactions NL3*, TM1, PK1 and DD-ME2.

The masses are in units of MeV.

Parameters NL3* TM1 PK1 DD-ME2

Mn 939 938 939.5731 938.5

Mp 939 938 938.2796 938.5

mσ 502.5742 511.198 514.0891 550.1238

mω 782.600 783 784.254 783

mρ 763 770 763 763

gσ 10.0944 10.0289 10.3222 10.5396

gω 12.8065 12.6139 13.0131 13.0189

gρ 4.5748 4.6322 4.5297 3.6836

g2 −10.8093 −7.2325 −8.1688 0

g3 −30.1486 0.6183 −9.9976 0

c3 0 71.3075 55.636 0

d3 0 0 0 0

aσ 0 0 0 1.3881

bσ 0 0 0 1.0943

cσ 0 0 0 1.7057

dσ 0 0 0 0.4421

aω 0 0 0 1.3892

bω 0 0 0 0.9240

cω 0 0 0 1.4620

dω 0 0 0 0.4775

aρ 0 0 0 0.5647


PK1. However, it is less than 3 MeV.

The calculated ground-state quadrupole moments of 52−66Cr obtained from

RMF+BCS calculations are shown in figure 2. In the figure, the predicted quadrupole

moments obtained with NL3*, TM1, PK1 and DD-ME2 parameter sets are agree quite

well with each other as in the case of binding energy. Spherical-deformed-spherical path

are clearly visible for 52−66Cr in our calculations.

The E(5) critical point symmetry, which is expected to be realized in nuclei when

they undergo a shape transition from spherical [U(5)] to γ-unstable [O(6)], has been

carried out as a particular case of Bohr Hamiltonian [2] in Ref. [6]. The potential is

supposed to depend only on the collective variable β and not on γ. Then exact separation

of variables is achieved and the equation containing β can be solved exactly for an

Table 2. The differences of the total binding energies for Cr isotopes between the

experimental data [49] and calculated results obtained from the constrained RMF

theory with NL3*, TM1, PK1 and DD-ME2 interactions (in units of MeV).

Nuclei NL3* TM1 PK1 DD-ME252Cr 4.524 5.492 3.862 3.03954Cr 0.409 1.087 0.363 −1.10856Cr −1.610 −0.794 −1.109 −2.58558Cr −1.598 −0.714 −0.944 −2.53060Cr −0.606 0.198 0.104 −1.88862Cr 0.078 0.851 0.848 −1.70264Cr 1.509 2.374 2.413 −0.01166Cr 2.119 2.170 2.945 −1.061

Figure 1. The ground-state binding energies per nucleon for Cr isotopes obtained

from RMF calculations with NL3*, TM1, PK1 and DD-ME2 interactions.


infinite square well potential in β. It should be noted that the same phenomenology can

be studied with a finite number of particles in the interaction boson model (IBM).

The large-N limit of the IBM Hamiltonian at the critical point in the transition

from spherical to γ-unstable shape can be represented in the geometrical model by

a β4 potential [50]. The E(5) critical point symmetry corresponds to a flat-bottomed

potential. For better understanding of phase translational character of nuclei at the

E(5) critical point from models, the evolution of the energy surface can be used [31].

In Ref. [16], the authors used RMF theory with NL3 force to produce potential energy

curves (PECs) as a function of quadrupole deformation parameter β for isotopic chains

involving nuclei which were suggested to be the candidates of the E(5) and X(5) critical

point symmetries. They have found rather flat PECs for the nuclei suggested by the

experiments and numerical solutions using a β4 potential as good examples of E(5)

such as 100,102,108Pd, 126,128,130,134Xe, and 134,136Ba (detail can be found in Ref. [16] and

reference therein). On the other hand, the authors of Ref.[19] have examined PECs

of nuclei which were suggested as the possible critical point nuclei by using Skyrme-

Hartree-Fock+BCS approach. They have found rather flat PECs for 108,110Pd, 128,130Xe

and 130,132Ba suggested as examples of critical point nuclei with E(5) symmetry. Their

results confirm the assumed square well potential in the β degree of freedom that leads

to the critical point symmetry E(5). As mentioned in Section 1, the A≈60 mass area

is an ideal region to investigate shape evolution in neutron-rich nuclei such as Ca, Ti

and Cr isotopes. This is the reason why Cr isotopes are mainly studied in the present

research. Figure 3 shows the potential energy curves for 52−66Cr obtained in the present

work by using constrained RMF theory with NL3*, TM1, PK1 and DD-ME2 effective

interactions. The energy for the ground-state is taken as a reference in each curve. As

can be seen in the figure 3, the PECs for 52−66Cr obtained with the four parameter sets

Figure 2. Calculated quadrupole moments for the even-even 52−66Cr by using

RMF+BCS with NL3, TM1, PK1 and DD-ME2 forces.


are found similar. The ground-state of 52Cr and 54Cr are found to be spherical shape.

The calculated deformation energy curve becomes soft with increasing neutron number

from N=32 (A=56). Relatively flat PECs are obtained with the four interactions for56Cr and 58Cr nuclei. The PEC for 58Cr is quite flat and the change of the binding

energy is around 1 MeV through the deformation between β2 = −0.2 and β2 = 0.25.

Taking into account that the relatively flat PEC is the characteristic of the E(5) critical-

point symmetry [16], 58Cr can be suggested to be a possible critical-point nucleus with

E(5) symmetry. With the increase of the neutron number from N=36, the ground-

state moves toward the deformed side and 64Cr which has semi-magic neutron number

(N=40) seems to be spherical. The experimental data of 58Cr supports the conclusion.

The experimental characteristic ratio R4/2 = E(4+1 )/E(2+1 ) and the energy ratio of the

first excited 6+ state R6/2 = E(6+1 )/E(2+1 ) are 2.20 and 3.59, respectively [32]. These

values are in excellent agrement with the excitation energies of the E(5) symmetry [6].

The E(5) symmetry, associated with a second-order QPT between spherical and

γ-soft potential shapes, is a dynamical symmetry of a five-dimensional (the collective

variables β and γ and the three Euler angles) infinite well in the axial deformation

variable β (V (β) = 0 for |β| ≤ βW , and V (β) = ∞ for |β| > βW ). The potential does

not depend on γ. To search γ-dependence of even-even 52−66Cr, we have performed

constrained self-consistent RMF+BCS calculations with NL3* interaction for triaxial

shapes, that is, including both β and γ deformations. Figure 4 shows the potential

energy surfaces in the β-γ plane (0 ≤ γ ≤ 60◦). Contour lines represent a step of 0.4

MeV and the binding energy is set to zero at the minimum of each surface. In our

calculations based on NL3* functional, a very weak γ-dependence of binding energy for58Cr is obtained as can be seen in the figure 4.

One of the achievements of the microscopic models as RMF theory is that it can

provide detailed information on the single-particle levels, shell structure etc., which is

very important in investigations of nuclear structure. The single-particle levels can be

used in studying the deformation-driving effect and understanding of the physical origin

for the critical-point nuclei. In figure 5, the neutron single-particle levels for 58Cr lying

between −20 and 0 MeV are presented, as functions of the axial deformation parameter

β. The dashed-dot-dot curve denotes the position of the corresponding Fermi levels.

This figure presents results calculated with the effective interaction DD-ME2. The

other three effective interactions give similar single-particle structure, thus not presented

here. The fermi energy curve of 58Cr seems flat for the range of −0.25 ≤ β ≤ 0.25 which

means that the barrier against deformation is weak. The result is consistent with the

PECs result of 58Cr which implies that it is the possible critical-point nucleus with E(5)

symmetry.

The differences in binding energies between the spherical state and the ground-

state for Cr isotopes can be used for analyzing of how the shape of the Cr isotopes

changes with the neutron number. These differences can give a clue about how soft

the nucleus is against deformation and phase transition of the nuclear shape. The

calculated differences in the binding energies between the ground-state and the state


with the spherical shape for Cr isotopes are presented in Table 3. As can be seen in

Table 3, starting from 52Cr to 66Cr the energy differences obtained from the interactions

NL3*, TM1, PK1 and DD-ME2 change from 0 to 0.833, 0.910, 0.799 and 1.378 MeV,

respectively. For all interactions, a clear jump in the energy differences appears at 58Cr

which suggests that the shape changes from spherical to critical-point nuclei.

As a function of the control parameter (number of neutrons), how precisely can

a point of phase transition be associated with a particular isotope while the control

parameter takes only discrete integer values? The differences between squares of ground-

Figure 3. The potential energy curves (PECs) for 52−66Cr obtained by the constrained

RMF+BCS theory with NL3*, TM1, PK1 and DD-ME2 interactions where binding

energy of the ground-state is taken as a reference.


state charge radii and the isomer shift between the first 2+ state and the ground-state

can be used to determine a point of phase transition (see Ref. [1] and references therein).

In this study, we have calculated the differences between squares of ground-state charge

radii of Cr isotopes with mass number A and those of reference nucleus (A = 52) for

better understanding of shape evolution of Cr isotopes with increasing neutron number.

The calculations from the interactions NL3*, TM1, PK1 and DD-ME2 are shown in

figure 6. Similar patterns are found for all the effective interactions. As can be seen

in figure 6, after two steps rising in neutron number for A ≤ 58 an abrupt change

starts at A = 58 and then the nuclei remain in deformed shape. This is a signature of

phase-transitional behavior.

Furthermore, one can expect to find nuclei with E(5) symmetry in even-even Ti

(Z=22) and Fe (Z=26) isotopic chains which are the neighbors of even-even isotopic chain

of Cr (Z=24). In Ref. [18], constrained RMF theory with various non-linear parameter

sets has been performed to search shape evolution of even-even 42−64Ti. Relatively flat

and symmetric PECs around β = 0 have been obtained for 48,52,60Ti. In particular,

the authors of Ref. [18] have suggested that 48Ti can be a better candidate for nucleus

with E(5) symmetry by taking experimental data into account. On the other hand,

we have performed axial and triaxial RMF+BCS calculations with NL3* functional to

search shape evolution of even-even 54−68Fe isotopes as in the case of 52−66Cr. Relatively

flat PEC around β = 0 has been obtained for only 58Fe. It varies from β2 = −0.2 to

β2 = 0.25 as can be seen in the upper panel of figure 7. However, the potential energy

surface of 58Fe in β-γ plane which is shown in the lower panel of the same figure indicates

that the binding energy of 58Fe depends on γ. Because of this, 58Fe may not be indicated

as the possible candidate nucleus with E(5) symmetry in the present calculations.

Table 3. The differences of the total binding energy between the spherical-state and

the ground-state obtained from the constrained RMF theory with NL3*, TM1, PK1

and DD-ME2 interactions for even−even 52−66Cr (in units of MeV).

Nuclei NL3* TM1 PK1 DD-ME252Cr 0.000 0.001 0.000 0.00054Cr 0.001 0.001 0.000 0.00056Cr 0.008 0.223 0.074 0.55358Cr 0.670 0.910 0.653 1.37860Cr 0.833 0.833 0.799 1.36062Cr 0.315 0.142 0.258 0.41564Cr 0.000 0.000 0.000 0.00066Cr 0.281 0.000 0.295 0.154


5. Summary

Self-consistent RMF theory with effective interactions has been employed to investigate

shape evolution of even-even 52−66Cr. The calculated ground-state binding energies of Cr

isotopes have been found as in good agreement with the experimental data. As a result

of the systematic investigation of ground-state shape evolution between spherical U(5)

and γ−unstable O(6) for Cr isotopes obtained from the PECs, 58Cr has been suggested

as a possible candidate of critical-point nucleus with E(5) symmetry which is favored by

the experimental data. The potential energy surface in β-γ plane for 58Cr obtained from

triaxial RMF+BCS calculations with NL3* force has shown that 58Cr is γ-independent

nucleus as in agreement with the condition of E(5) symmetry. In addition, the same

conclusion of the transition has been supported by examining the neutron single-particle

levels of 58Cr, the differences of the ground-state energies from spherical-state of 52−66Cr

and the differences in ground-state charge radii of Cr isotopes.


Figure 4. Potential energy surfaces in the β-γ plane (0 ≤ γ ≤ 60◦) obtained from

triaxial RMF+BCS calculations with the parameter set NL3* for 52−66Cr. Contour

lines represent a step of 0.4 MeV and the binding energy is set to zero at the minimum

of each surface.


Figure 5. Neutron single-particle levels for 58Cr, as functions of the axial deformation

parameter β. The positions of the corresponding Fermi levels are shown with thick

dashed-dot-dot curve.


Figure 6. Calculated differences between squares of ground-state charge radii:

⟨r2⟩(A+2) − ⟨r2⟩(A=52) as functions of mass number in Cr isotopes.


Figure 7. The PEC for 58Fe obtained with NL3* force (upper panel). Potential

energy surfaces in the β-γ plane (0 ≤ γ ≤ 60◦) obtained from triaxial RMF+BCS

calculations with the parameter set NL3* for 58Fe (lower panel).


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An analysis of E(5) shape phase transitions in Cr isotopes with covariant density functional theory

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