1. Introduction
In the case of pf shell, numerous interesting phenomena have been investigated and revealed, such as shell structure evolution [1–4], competition between single-particle and collective excitation [5–8], the properties of dense matter [9] and so on. With a proton number Z = 25, Manganese (Mn) has fruitful structural information and raised lots of concern as the nuclear research goes on. In the past few decades, studies on Mn mainly focused on neutron-rich isotopes, including their lifetime [10], collective characteristics [11], electric and magnetic properties [12–14], spectroscopy properties [15, 16] as well as the influence from the ν1g9/2 orbital [13, 17, 18].
For the Mn isotope chain around N = 28 region, the most remarkable phenomenon is the abrupt changes caused by shell closure, such as charge radii and quadrupole moment [19]. The ground-state inversion is another noticeable phenomenon, that is, for all the odd-A Mn isotopes with known experimental data, the ground-state spin is 5/2−, the only exception is 53Mn with magic neutron number (N = 28), whose ground-state spin is 7/2−. Early simple shell model calculations [20] proved that permitting the configuration of $\pi {({f}_{7/2})}^{-3}$ with modified surface δ interaction fitted gives the same ground state 7/2− whether extending the valence nucleon space from π1f5/2 to π2p3/2 subshells or not. For other isotopes such as 51Mn and 55Mn, the ground state 5/2− can be explained by the configuration including the $\pi {({f}_{7/2})}^{-3}$ protons and the p1/2, p3/2, f5/2 neutrons could push the 5/2− down to be the ground state by using a proper proton-neutron interaction. Apart from the distinctive ground states of Mn, rotational phenomena in negative parity level sequences based on the ground state of 49,51 such as I(I + 1) dependence, strong E2 transitions, band crossing and signature splitting have been discussed [21–23]. However, a systematic study for low-lying states of Mn isotopes is still absent for the moment, and a general, longitudinal observation for the N = 28 region is the theme of this present work.
To further study the low-lying structure of Mn isotopes around N = 28 region systematically, theoretical calculations are vital. Shell model and density functional theory are the two most widely used microscopic nuclear structure models. The shell model is a fundamental tool for the structure research of light nuclei and those near the shell closures. For a certain model space, the Hamiltonian can be diagonalized with the effective interaction by spherical basis. Therefore, under the frame of the shell model, the collective and single-particle properties can be both described in the present pf-shell region. Considering most of the previous calculations were conducted with the shell model, and sophisticated covariant density functional theory (CDFT) [24–26] has also been adopted. The CDFT has been proved to be a powerful theory in nuclear physics by its successful description of many nuclear phenomena [24–29]. The paper is organized as follows. Section 2 and section 3 present the calculation results from the shell model and CDFT respectively. Conclusions and outlook for possible future studies are given in section 4 .
2. Shell-model calculations
With the motivation of getting a deeper understanding of the microstructure of Mn isotopes around N = 28 region, shell-model calculations have been performed to obtain the low-lying energy levels, magnetic moments and electric quadrupole moments, as well as configurations. The calculation adopts ANTOINE code [30] with GXPF1A interaction [31]. The GXPF1A effective interaction considers the nucleus to be calculated as a 40Ca core surrounded by valence nucleons, restricted to the pf space, i.e. f7/2, f5/2, p3/2 and p1/2. The present work uses standard effective charge (ep = 1.5, en = 0.5) and free g factor (${g}_{s}^{\pi }=5.586$, ${g}_{l}^{\pi }=1$, ${g}_{s}^{\nu }=-3.826$, ${g}_{l}^{\nu }=0$), with no truncation. Considering the existing experimental data, the calculations are performed for the low-lying 3/2−, 5/2−, 7/2−, 9/2− and 11/2− states.
Figure 1 presents the comparison between shell-model calculation results and the corresponding experimental energies in odd-A 49−57Mn isotopes. The spins and parities are labeled in the left accordingly. The energy levels of the 2+ states from the doubly-even core 50−58Fe isotopes are marked out in dots. From figure 1, one can easily see that the GXPF1A interaction has well reproduced the excitation energy levels of the low-lying states in odd-A 49–57Mn isotopes, including both the values and variation trend, with the deviation less than 50 keV for the first excitation energy and less than 100 keV for the rest states. The remained little deviations may be from the contribution of νg9/2 excitations [17, 18, 13]. These favorable results indicate that GXPF1A interaction is capable of giving a convincible configuration in the presented Mn isotopes. KB3 [33] effective interaction has also been calculated and it can reproduce the variation trend of the spin order as well, especially for the isotopes with fewer neutrons.
Figure 1. Comparison of energy levels between experimental data (filled circles) and large-scale shell-model calculation results (thicker bars) of odd-A 49–57Mn isotopes and 49Cr. The shell-model code Antoine [30] are adopted using GXPF1A interaction [31]. Energy levels of the 2+ states in 50–58Fe, i.e. even–even core of Mn isotopes, and 48Cr are also presented. The corresponding experimental levels are taken from ENSDF [32]. |
To further discuss the micro-forming mechanism of the energy levels and spin orders, the configuration and occupation of both proton and neutron obtained from shell-model calculations are listed in table 1. The calculations cover a range of odd-A isotopes from 49Mn to 59Mn, giving two groups of configuration with the highest probability and the average occupation for 3/2−, 5/2−, 7/2−, 9/2− and 11/2− states. The excited energies from both experiments and calculations are also listed for necessary analysis. The configurations of the mirror nucleus, 49Cr have also been given, which shows good isospin symmetry compared to 49Mn.
Table 1. The calculated excitation energy, configuration and average occupation of both proton (π) and neutron (ν) for the ground state 5/2− (with the exception of 7/2− for 53Mn), first excited state 7/2−, 9/2−, 11/2− and the first 3/2− states of odd-A 49–57Mn isotopes. The configurations are shown in the sequence of 1f7/2, 2p3/2, 1f5/2 and 2p1/2 orbitals, from left to right. For instance, the configuration of (4000)π(5000)ν stands for four protons and five neutrons on the 1f7/2 orbitals. The experimental data are also listed for necessary analysis. |
| Nuclide | J | ${E}_{\exp .}$ | Ecal. | Configuration | Average Occupation |
|---|---|---|---|---|---|
| 49Cr | 5/2− | 0 | 0 | 39% (4000)π(5000)ν+11% (4000)π(4100)ν | (3.4, 0.3, 0.2, 0.0; 4.4, 0.4, 0.2, 0.0) |
| 7/2− | 271.72 | 218.57 | 44% (4000)π(5000)ν+8% (4000)π(4100)ν | (3.5, 0.3, 0.2, 0.0; 4.5, 0.3, 0.2, 0.0) | |
| 9/2− | 1083.6 | 1099.43 | 35% (4000)π(5000)ν+12% (4000)π(4100)ν | (3.4, 0.4, 0.1, 0.0; 4.4, 0.4, 0.2, 0.0) | |
| 11/2− | 1562.1 | 1438.35 | 43% (4000)π(5000)ν+9% (4000)π(4100)ν | (3.5, 0.3, 0.2, 0.0; 4.5, 0.3, 0.2, 0.0) | |
| 3/2− | 1741.4 | 1675.29 | 27% (4000)π(4100)ν+7% (3100)π(4100)ν | (3.3, 0.4, 0.2, 0.1; 3.6, 0.9, 0.3, 0.2) | |
| 49Mn | 5/2− | 0 | 0 | 40% (5000)π(4000)ν+11% (4100)π(4000)ν | (4.4, 0.4, 0.2, 0.0; 3.5, 0.3, 0.2, 0.0) |
| 7/2− | 261.38 | 218.52 | 44% (5000)π(4000)ν+8% (4100)π(4000)ν | (4.5, 0.3, 0.2, 0.0; 3.5, 0.3, 0.2, 0.0) | |
| 9/2− | 1059.18 | 1099.42 | 35% (5000)π(4000)ν+12% (4100)π(4000)ν | (4.4, 0.4, 0.2, 0.0; 3.4, 0.4, 0.1, 0.0) | |
| 11/2− | 1541.31 | 1438.40 | 42% (5000)π(4000)ν+10% (4100)π(4000)ν | (4.5, 0.3, 0.2, 0.0; 3.5, 0.3, 0.2, 0.0) | |
| 3/2− | 1675.27 | 27% (4100)π(4000)ν+7% (4100)π(3100)ν | (3.6, 0.9, 0.3, 0.2; 3.3, 0.4, 0.2, 0.1) | ||
| 51Mn | 5/2− | 0 | 0 | 43% (5000)π(6000)ν+9% (4100)π(6000)ν | (4.5, 0.3, 0.1, 0.0; 5.4, 0.4, 0.2, 0.0) |
| 7/2− | 237.30 | 215.25 | 47% (5000)π(6000)ν+7% (5000)π(5100)ν | (4.5, 0.3, 0.1, 0.0; 5.4, 0.3, 0.2, 0.0) | |
| 9/2− | 1139.8 | 1165.49 | 41% (5000)π(6000)ν+11% (5000)π(5100)ν | (4.5, 0.3, 0.1, 0.0; 5.4, 0.4, 0.2, 0.0) | |
| 11/2− | 1488.5 | 1431.04 | 46% (5000)π(6000)ν+9% (5000)π(5100)ν | (4.6, 0.3, 0.1, 0.0; 5.4, 0.3, 0.2, 0.0) | |
| 3/2− | 1817.1 | 1721.81 | 16% (4100)π(6000)ν+12% (5000)π(5100)ν | (4.0, 0.6, 0.3, 0.1; 4.9, 0.7, 0.3, 0.1) | |
| 53Mn | 5/2− | 377.89 | 422.75 | 50% (5000)π(8000)ν+8% (5000)π(7100)ν | (4.7, 0.2, 0.1, 0.0; 7.3, 0.4, 0.2, 0.1) |
| 7/2− | 0 | 0 | 60% (5000)π(8000)ν+5% (5000)π(7100)ν | (4.8, 0.1, 0.1, 0.0; 7.5, 0.3, 0.2, 0.0) | |
| 3/2− | 1289.83 | 1201.83 | 55% (5000)π(8000)ν+9% (5000)π(7100)ν | (4.7, 0.2, 0.1, 0.0; 7.4, 0.3, 0.2, 0.1) | |
| 11/2− | 1441.15 | 1486.48 | 53% (5000)π(8000)ν+7% (5000)π(7100)ν | (4.7, 0.2, 0.1, 0.0; 7.4, 0.3, 0.2, 0.1) | |
| 9/2− | 1620.12 | 1562.31 | 52% (5000)π(8000)ν+7% (5000)π(7100)ν | (4.7, 0.2, 0.1, 0.0; 7.4, 0.3, 0.2, 0.1) | |
| 55Mn | 5/2− | 0 | 0 | 27% (5000)π(8200)ν+9% (5000)π(8101)ν | (4.5, 0.3, 0.1, 0.0; 7.6, 1.4, 0.6, 0.4) |
| 7/2− | 125.95 | 103.73 | 33% (5000)π(8200)ν+7% (5000)π(8020)ν | (4.6, 0.3, 0.1, 0.0; 7.7, 1.4, 0.6, 0.3) | |
| 9/2− | 984.26 | 1048.58 | 19% (5000)π(8200)ν+11% (5000)π(8101)ν | (4.5, 0.4, 0.1, 0.0; 7.7, 1.3, 0.6, 0.4) | |
| 11/2− | 1292.11 | 1280.03 | 29% (5000)π(8200)ν+9% (5000)π(8110)ν | (4.6, 0.3, 0.1, 0.0; 7.7, 1.4, 0.6, 0.3) | |
| 3/2− | 1528.35 | 1428.85 | 29%(5000)π(8200)ν+9% (4100)π(8200)ν | (4.5, 0.3, 0.1, 0.0; 7.5, 1.7, 0.5, 0.3) | |
| 57Mn | 5/2− | 0 | 0 | 19% (5000)π(8220)ν+9% (5000)π(8400)ν | (4.5, 0.4, 0.1, 0.0; 7.7, 2.2, 1.5, 0.5) |
| 7/2− | 83.19 | 106.47 | 20% (5000)π(8220)ν+17% (5000)π(8400)ν | (4.6, 0.3, 0.1, 0.0; 7.8, 2.4, 1.3, 0.5) | |
| 3/2− | 850.07 | 879.03 | 13% (5000)π(8220)ν+7% (4100)π(8220)ν | (4.4, 0.5, 0.1, 0.1; 7.7, 2.2, 1.6, 0.6) | |
| 9/2− | 1074 | 1206.19 | 14% (5000)π(8220)ν+14% (5000)π(8211)ν | (4.4, 0.4, 0.1, 0.0; 7.7, 2.1, 1.5, 0.6) | |
| 11/2− | 1227 | 1295.88 | 19% (5000)π(8220)ν+9% (5000)π(8400)ν | (4.5, 0.3, 0.1, 0.0; 7.7, 2.3, 1.5, 0.5) |
As shown in table 1, the occupation number for the proton 1f7/2 orbital in 53Mn is 4.7–4.8, which is closer to 5 compared to the 4.4–4.6 proton occupation in other isotopes, implying that the states in 53Mn are purer $\pi {({f}_{7/2})}^{-3}$ multiplets, similar results have been pointed out for 57,59,61Mn in [18]. Besides, for the ground state 7/2−, the dominant configuration of $[\pi {(1{{\rm{f}}}_{7/2})}^{5}\nu {(1{{\rm{f}}}_{7/2})}^{8}]$ accounts for 60%, while those of the ground state 5/2− in other isotopes are less than 50%, such as 40% $[\pi {(1{{\rm{f}}}_{7/2})}^{5}\nu {(1{{\rm{f}}}_{7/2})}^{4}]$ in 49Mn, 43% $[\pi {(1{{\rm{f}}}_{7/2})}^{5}\nu {(1{{\rm{f}}}_{7/2})}^{6}]$ in 51Mn and so on. It can be further obtained that distinct deformation in 53Mn is not favorable or it is a nearly spherical nucleus with N = 28 shell closure [19], which can be also observed from the sudden increase in excitation of the first 2+ of the doubly even core 54Fe in figure 1.
From table 1, one can see that the average occupation numbers of the 5/2−, 7/2−, 9/2− and 11/2− states are close to each other, indicating similar intrinsic structure, this might have a certain relationship with the rotational characteristics shown in figure 1. For the 3/2− states, 49Mn and 51Mn show more features of single-particle excitation, with proton occupations of 0.9 and 0.6 on the 2p3/2 orbital respectively. In contrast, the proton occupations on the 2p3/2 orbital in 53Mn, 55Mn and 57Mn are smaller, less than 0.5, showing a relatively stronger configuration mixing, which has also been pointed out by Mateja et al [34]. This may be the reason why the excitation energy of the 3/2− levels exhibits more variation from one isotope to another.
Figure 2 presents the calculation results of magnetic moments and electric quadrupole moments for 5/2− and 7/2− states, the experimental data are marked out by asterisks for reference, and the results are in good agreement with the experimental data, especially for the magnetic moments. As one can see, the experimental electric quadrupole moment of the ground states suddenly drops at 53Mn, showing an obvious spheric-inclined feature, which bears out the speculations of deformation and is consistent with the shell-model calculations. Furthermore, with K = 7/2−, the estimated quadrupole deformation β2 of 53Mn extracted from the ground-state experimental electric quadrupole moment [35] is 0.11, which indicates the smaller deformation of 53Mn and is in good agreement with the CDFT result in figure 4.
Figure 2. Comparison of magnetic moment (a) and electric quadrupole moment (b) of the ground state and first excited state, i.e. 5/2− and 7/2−, in odd-A 49–57Mn between experimental data (filled circle) and shell-model calculation results (empty circle). The Schmidt magnetic moment value for valence proton orbit f7/2, f5/2, p3/2 and p1/2 are given by dot lines. |
In [22], the I(I + 1) dependence of the low-lying negative parity states and corresponding typical collective behavior with Kπ = 5/2− band expected at low excitation energy in 49V are presented. The experimental levels of low-lying negative parity states as a function of spin in odd-A 49−57Mn and 49Cr have been demonstrated in figure 3. As 49Cr has been widely recognized as a nucleus with good rotational bands, it is presented for comparison as well. It is obvious that the levels of 49Mn, from 5/2− to 15/2−, follow a similar trend to levels of 49Cr, including the small energy staggering. Around I = 17/2 in these mirror nuclei, the staggering changes in amplitude. In [23], it is pointed out that a 3 − qp band crosses sharply the 1 − qp band with sudden changes of components in the wave function. As the neutron number increases, the levels in 51,55,57Mn begin to deviate from the trend in 49Cr and 49Mn, with relatively larger energy staggering. However, the levels in 53Mn fail to follow the behavior in 49Cr, indicating a different structure among these two nuclei. As the nuclear magnetic moment is sensitive to the wave function, it will be very helpful to understand the rotational structure and mechanism in the odd-A Mn isotopes. Next, the nuclear magnetic moments of 49,51,53,55Mn will be investigated in the CDFT.
Figure 3. Variation trend of experimental excitation energy with spin (from 5/2− to 17/2−) in odd-A 49–57Mn isotopes. The same trend of the mirror nucleus 49Cr is also presented for comparison. In order to show the evolution trend clearer, excitation energies are shifted accordingly in the vertical direction to distinguish the curves, and the specific translation distances are marked on the left. |
3. Covariant density functional theory calculations
The CDFT, which relies on the basic ideas of the effective field theory and the DFT, has achieved great success in the description of nuclear properties all over the nuclear chart [24–29] and neutron star [36]. As a microscopic and covariant theory, the CDFT has attracted a lot of attention in recent years for many attractive advantages, such as the automatic inclusion of the nucleonic spin degree of freedom, explaining naturally the pseudospin symmetry in the nucleon spectrum [37, 38] and the spin symmetry in antinucleon spectrum [38, 39], and the natural inclusion of nuclear magnetism [40], which plays an important role in nuclear magnetic moments [41–44] and nuclear rotations [27, 45].
In the present CDFT calculations, the effective meson-exchange interaction parameter PK1 [46] is adopted. Both the Dirac equation for nucleons and the Klein–Gordon equations for mesons are solved in an isotropic harmonic oscillator basis [47, 48] and a basis of 16 major oscillator shells is adopted. The oscillator frequency is given by ℏω0 = 41A−1/3 MeV. To investigate the mean-field minima, the adiabatic and the configuration-fixed quadrupole deformation-constrained calculations [49–51] are performed to obtain the energy curve, i.e. the total energies as a function of quadrupole deformation.
In figure 4(a), taking 53Mn as an example, the energy curves, i.e. the total energies as a function of the quadrupole deformation parameter β2, calculated by adiabatic (shown as open circles) and configuration-fixed (shown as solid lines) deformation constrained CDFT approach with time-odd fields, are presented. The local minima in the energy curves for two different configurations are represented by stars and labeled as A and B, respectively. A is the mean-field ground state and found to be prolate deformed, β2 = 0.13, while the odd proton occupies the π5/2−[312] orbital. For the minima of state B, the odd proton occupies the π7/2−[303] orbital, and the corresponding deformation becomes smaller with 0.09. It is easy to see that the total energy of mean-field state A with Kπ = 5/2− is around 0.9 MeV smaller than state B with 7/2−, which is not consistent with the corresponding experimental energy levels 7/2− and 5/2−.
Figure 4. The total energies (a) and magnetic moments (b) for 53Mn as functions of quadrupole deformation β2 by adiabatic and configuration-fixed (indicated by open circles and solid lines respectively) deformation constrained CDFT approach with time-odd fields using PK1 parameter set. The minima in the energy curves for different configurations are indicated by stars and marked as A and B respectively. In panel (b), the final total magnetic moment with rotational coupling is labelled as short dash lines. Experimental magnetic moment μ = 3.3 μN for 5/2− state and μ = 5.033 μN for 7/2− [52] state (solid lines) as well as the Schmidt magnetic moments of π2p1/2, π2p3/2, π1f5/2, and π1f7/2 orbitals (dotted lines) are also shown for comparison. |
By self-consistently including the nuclear superfluidity, deformation, and continuum effects, the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) has turned out to be successful in describing both stable and exotic nuclei [53–58]. To check the pairing correlation effects, the DRHBc theory calculations with point-coupling density functionals PC-PK1 [59] are performed, and the odd proton orbital with the third component of the total spin and parity 5/2− and 7/2− are blocked respectively. By blocking the 5/2− orbital, the quadrupole deformation is 0.14 and pairing energy vanishes. The final calculated total energy with 5/2− blocked orbital is −462.03 MeV. After blocking the 7/2− orbital, the quadrupole deformation is close to zero with β2 = 0.03 and the corresponding total energy is −461.66 MeV. Although the experimental ground state 7/2− still has not been reproduced with consideration of the pairing correlation, the energy difference between the two blocking case is 0.37 MeV and stiff competition of ground-state spin exists between 5/2− and 7/2− in 53Mn. Thus it is interesting to find the effect of the rotational energy correction [60] and further investigate symmetry-conserving configuration mixing approaches [61–63] that could explain the ground state 7/2− in 53Mn.
Compared with the energy, it is easy to investigate the magnetic moment with rotational symmetry. For deformed odd-A nuclei, the valence nucleon approximation is invalid and there is a strong coupling between the core and the valence nucleon. Therefore, the total magnetic moment consists of two parts, i.e. the intrinsic nucleonic motion and the collective rotational motion. In the axially deformed case with intrinsic angular momentum K, the magnetic moment can be written as [64]
$\begin{eqnarray}\mu ={g}_{R}I+({g}_{K}-{g}_{R})\displaystyle \frac{{K}^{2}}{I+1},\quad \quad ({\rm{for}}\ K\gt 1/2),\end{eqnarray}$
gR is the corresponding effective g factor, with gR ∼ Z/A. The intrinsic g factor reads gK = μintri./K and intrinsic magnetic moment μintri. is obtained from self-consistent CDFT calculations with electromagnetic current [43], i.e. $\begin{eqnarray}{\boldsymbol{\mu }}=\displaystyle \sum _{i=1}^{A}\int {\rm{d}}{\boldsymbol{r}}\,\left[\frac{{{Mc}}^{2}}{\hslash c}Q{\psi }_{i}^{+}{\boldsymbol{r}}\times {\boldsymbol{\alpha }}{\psi }_{i}+\kappa {\psi }_{i}^{+}\beta {\boldsymbol{\Sigma }}{\psi }_{i}\right],\end{eqnarray}$
in the unit of the nuclear magneton μN = eℏ/2Mc. The sum over i runs over the occupied orbits with Dirac spinor ψi. α, β and Σ are the Dirac matrices. The nucleon charge $Q\equiv \displaystyle \frac{e}{2}(1-{\tau }_{3})$, and κ the free anomalous gyromagnetic ratio of the nucleon with κp = 1.793 and κn = −1.913 for proton and neutron respectively. The first term in the above equation gives the Dirac magnetic moment, and the second term gives the anomalous magnetic moment.Using equation (2 ), the effective electromagnetic current gives the intrinsic magnetic moment of valence nucleons with assigned configurations in figure 4(b). The calculated intrinsic magnetic moments for the minima of configuration A and B are 4.71 μN and 6.28 μN respectively, and both larger than the corresponding experimental data for 5/2− and 7/2−. The results are also close to the Schmidt magnetic moment of f7/2 shown in figure 4(b), as both deformed orbitals π5/2−[312] and π7/2−[303] belong to the single-particle shell f7/2. It could be seen that the intrinsic magnetic moment is sensitive to configuration, but not so much to β2. For example, the magnetic moments of configuration B do not change much with β2, as the odd proton occupies the orbital π7/2−[303−] with the third component K = 7/2 and the configuration mixing is very weak [52].
After including the coupling of collective rotation and intrinsic single-particle motion, the calculations have been greatly improved and both the magnetic moments of 5/2− and 7/2− states in 53Mn are well reproduced as shown in figure 4(b), with relative deviation from the available data about 10% and 5% respectively. Taking 7/2− state as an example, the intrinsic magnetic moment is 6.28 μN and the final magnetic moment with collective motion is 5.25 μN, excellently reproducing the experimental value 5.033 μN. On the whole, the spin and magnetic moment for 5/2− and 7/2− states in 53Mn can be explained by the odd proton occupying the orbital π5/2−[312] and π7/2−[303] respectively in the mean-field picture of CDFT, inconsistent with the 4.7–4.8 proton occupation in the f7/2 orbital in the shell-model calculation. The weakly deformed configuration B (π7/2−[303]) and relatively large deformation for configuration A (π5/2−[312]) probably reflects the competition between the decoupling and strong coupling as shown in odd magnesium isotopes [63]. As the angular-momentum symmetry is broken in the mean-field approximation, the symmetry conserving configuration mixing (SCCM) approaches [65, 66, 61, 63, 67–69] are needed for obtaining the full spectroscopy properties including the spin, parity, energy and moments.
On the other hand, if the ground state 7/2− in 53Mn is based on the same intrinsic configuration π5/2−[312] as I = 5/2−, then the corresponding magnetic moment using equation (1 ) will be 2.86 μN, deviating much from the experimental magnetic moment of 7/2−. This further indicates that the spin sequences 7/2− and 5/2− in 53Mn originate from a different intrinsic configuration, different from the Kπ = 5/2 − band in 49Cr and other odd-A Mn isotopes. It is expected that the E2 lifetime measurement will provide strong evidence in the future.
In table 2, the calculated quadrupole deformation (β2), valence proton configuration, intrinsic mean field magnetic moment (μintri.), and final magnetic moment (μtot.) of odd-A 49−55Mn isotopes in CDFT approach with PK1 set are given, in comparison with the corresponding experimental spin, parity, energy and magnetic moment as well as electric quadrupole moment. For 49,51,55Mn, the ground state deformation is around 0.25, larger than the deformation of 53Mn and is in agreement with the relatively large experimental electric quadrupole moment in 51,55Mn. Furthermore, the relatively large deformation in 49,51,55Mn will lead to the stronger mixing for f7/2, f5/2, p3/2 and p1/2, and finally a relatively small occupation in πf7/2 compared with 53Mn as shown in table 1. The obtained deformation β2 is also in agreement with the Hartree–Fock-Bogoliubov calculations based on the D1S Gongy effective nucleon-nucleon interaction [70]. The odd proton in 49,51,55Mn occupies the π5/2−[312] orbital and the corresponding ground state spin and parity is 5/2−. Expect for 53Mn, the experimental magnetic moments of 51,55Mn are also well reproduced, after including the coupling of collective rotation and intrinsic single-particle motion.
Table 2. The calculated quadrupole deformation (β2), valence proton configuration, intrinsic mean field magnetic moment (μintri.), and final magnetic moment (μtot.) of odd-A 49–55Mn isotopes in CDFT approach with PK1 set, in comparison with the corresponding experimental spin, parity, energy and magnetic moment as well as electric quadrupole moment. |
| 49Mn | 51Mn | 53Mn | 55Mn | |||
|---|---|---|---|---|---|---|
| Exp. | Jπ | 5/2− | 5/2− | 7/2− | 5/2− | 5/2− |
| ${\mu }_{\exp .}$ | 3.577 | 5.033(5) | 3.3(3) | 3.468 | ||
| ${{\rm{Q}}}_{\exp .}$ | 0.41 | 0.17 | 0.33 | |||
| CDFT | β2 | 0.28 | 0.24 | 0.09 | 0.13 | 0.24 |
| Configuration | π5/2−[312] | π5/2−[312] | π7/2−[303] | π5/2−[312] | π5/2−[312] | |
| μintri. | 4.74 | 4.86 | 6.28 | 4.71 | 4.80 | |
| μtot. | 3.75 | 3.82 | 5.25 | 3.69 | 3.75 | |
4. Summary and prospective
In summary, systematic studies are performed upon the odd-A 49–57Mn isotope chain to reveal peculiar phenomena around N = 28 region. The remarkable features of ground-state inversion and the collectivity of the sequence based on the ground state are pointed out. Shell-model calculations with the GXPF1A effective interaction reproduce the energy levels of low-lying states well, as well as the electromagnetic moments in odd-A 49–57Mn isotopes, showing that the ground-state spin 5/2− (with 7/2− in 53Mn as an exception) may be mainly from the configuration of $\pi {({f}_{7/2})}^{-3}$, and 53Mn is a nearly spherical or weakly deformed nucleus with N = 28 shell closure. The average proton occupation reveals more single-particle characteristics for 49Mn and 51Mn, indicating their first 3/2− states mainly come from cross-shell excitation. As for 53Mn, 55Mn and 57Mn, the low average proton occupation on the π2p3/2 orbital shows that their first 3/2− states have more components of configuration mixing of π1f7/2 particles or holes coupling with the 2+ state from the doubly even core Fe.
The CDFT calculations are also performed for 49,51,53,55Mn. Using the configuration-fixed deformation-constrained calculations, the potential energy and intrinsic magnetic moment as a function of quadrupole deformation for 53Mn are given. The mean-field ground state of 53Mn has been found to be prolate deformed, β2 = 0.13, with the odd proton in the π5/2−[312] orbital, and the β2 = 0.09 for the excited state with the odd proton in the π7/2−[303] orbital. After including the coupling of collective rotation and intrinsic single-particle motion with odd proton occupying the 7/2−[303] and 5/2−[312] orbitals, the experimental magnetic moments of 5/2− and 7/2− states in 53Mn can be reproduced well. The weakly deformed configuration and relatively large deformation for another configuration probably reflect the competition between the decoupling and strong coupling, and the SCCM approaches are required for full spectroscopy property in the future. With the odd proton in 49,51,55Mn occupying the π5/2−[312] orbital, the ground-state magnetic moments for 5/2− are also well reproduced. Furthermore, the relatively large deformation in 49,51,55Mn and small deformation in 53Mn obtained from CDFT calculations are in agreement with the different proton occupations in πf7/2 between 49,51,55Mn and 53Mn from shell-model calculations. Based on the levels 7/2− and 5/2− in 53Mn originating from a different intrinsic odd proton configuration and different trend behavior of levels from 49Cr and other odd-A Mn isotopes, the low-lying levels in 53Mn do not favor the K = 5/2 band.