Systematic analysis on spectral statistics of odd-A nuclei

Annals of Physics 407 (2019) 250–260

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Systematic analysis on spectral statistics of odd-A nuclei ∗

A. Jalili Majarshin a,c , , Feng Pan a,b , H. Sabri c , Jerry P. Draayer b a

Department of Physics, Liaoning Normal University, Dalian 116029, China Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA c Department of Physics, University of Tabriz, Tabriz 51664, Iran b

article

info

Article history: Received 27 January 2019 Accepted 4 May 2019 Available online 29 May 2019 Keywords: Odd-mass nuclei Spectral statistics Transition probabilities Maximum likelihood estimation (MLE)

a b s t r a c t Level statistics for given spin and parity below 10 MeV within A ≤ 71, 71 ≤ A ≤ 151, and 151 ≤ A ≤ 221 mass regions are systematically analyzed. We have taken 5316 level spacings from 266 odd-A nuclei in the light mass region, 6897 level spacings from 478 ones in the medium mass region, and 2768 level spacings from 143 ones in the heavy mass region for the statistics. In the analysis, the chaoticity parameter in the Berry–Robnik distribution is determined based on the method of maximum likelihood estimation. The results show a mass number A-dependence in the level statistics. The larger the mass number, the relatively stronger in chaoticity. In addition, with the level statistics in using data sequences classified according to the order of magnitude of half-lives, it is shown that the level energies to be more chaotic for longer lived light and heavy mass odd-A nuclei, and intermediate between regular and chaotic in medium mass odd-A nuclei. The level statistical analysis in concerning E2 transition strengths of E2 or mixed E2+M1 transitions in odd-A nuclei is also made, which shows chaos in level energies likely occurring in those with weaker E2 transition strengths. Therefore, it can be concluded that odd-A nuclei are, more often than not, chaotic. Chaos is likely emergent in level energies of stable light mass nuclei with weaker E2 transition strengths, while it is definitely emergent in level energies of heavy mass nuclei, and even stronger in comparatively stable ones and those with weaker E2 transition strengths. © 2019 Elsevier Inc. All rights reserved.

∗ Corresponding author at: Department of Physics, University of Tabriz, Tabriz 51664, Iran. E-mail address: [email protected] (A.J. Majarshin). https://doi.org/10.1016/j.aop.2019.05.002 0003-4916/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction Atomic nuclei are often considered as quantum chaotic systems due to the strong many-body interactions among nucleons bounded within a tiny region. In the past few decades, level-statistic properties of even–even nuclei were intensively studied in both theory and experiment [1,2]. These investigations are based on statistics of a set of level-energies of even–even nuclei with integer spin. It was suggested that the chaotic behavior shown in the level-statistics may be modeled by the Random Matrix Theory (RMT) [3,4]. Several effective measures in characterizing the degree of chaoticity were proposed [5,6], which were also used in elucidating chaotic behaviors in other quantum many-body systems [7,8]. There are quite a lot of investigations [9–22] on chaotic behaviors in spectra of even–even nuclei with respect to mass, quadrupole deformation, and spin, etc while collecting relevant samples. Typically, Shriner et al. had considered the spectral statistics of some nuclei classified into several mass regions [10]. The level fluctuation properties of nuclei may also be classified according to excitation energy, with which the nearest neighbor spacing distribution (NNSD) in even–even nuclei was investigated [11,14]. Level-statistics of 0+ states of even–even nuclei in the rare-earth region was studied [15], in which the degree of level mixing extracted from Brody distribution to the energy spacings of adjacent excited 0+ levels is also explored. Possible effects of shell model configurations on the spectral statistics were considered in [1], while effect of pairing correlations on the spectral statistical behavior of the spherical meanfield plus standard pairing model was studied in [16]. Very recently, the fluctuation properties in the energy spectra of the unprecedented set of bound states of both normal and abnormal parity in 208 Pb have been analyzed [18]. However, the level-statistics has only been applied to even–even nuclear systems. On the other hand, energy spectra of odd-A nuclei are much more complicated than those of even–even systems due to the fact that both collective and single-particle excitations should be considered simultaneously [23]. It is especially the case in describing spectra and decay properties of odd-A nuclei in medium- and heavy-mass regions, where the shell model calculations are still beyond reach, while interactions between the core and a single-nucleon in the Bohr–Mottelson collective model [24] or different configuration-mixing considerations in the interacting boson– fermion model (IBFM) [25] should be properly considered. For example, besides deformations and other considerations, pairing interactions among nucleon pairs and pair-broken configurations of unpaired nucleons in odd-A nuclei may be considered [26,27]. Nevertheless, according to previous investigations in even–even nuclei [28,29], similar statistical analysis should also apply for oddA cases as well to reveal whether the dynamics in odd-A nuclei is affected by an additional single-nucleon involved. In this work, inspired by the statistical analysis on even–even nuclei [29,30], fluctuation properties of level energies for given spin and parity below 10 MeV, and E2 or mixed E2+M1 transition rates among these levels in odd-A nuclei within A ≤ 71, 71 ≤ A ≤ 151, and 151 ≤ A ≤ 221 mass regions, respectively, are studied systematically, for which the NNSD and Berry–Robnik distribution (BRD) parameter determined by the maximum likelihood technique shown in [31–34] are adopted. The analysis is not restricted to the individual nucleus, but rather covers all stable odd-A nuclei with existing data in the three mass regions. In order to describe the NNSD, all level sequences of odd-A nuclei are prepared from the available experimental data [35,36] classified based on their mass, half-lives, and E2 transition rates. 2. Method of the analysis The NNSD measures the spectral behavior from adjacent levels separated by a spacing in the unfolded spectrum. In general, the NNSD requires as many level energies with the same spin and parity as possible. Since the departure of the actual level density from a local uniform density is of importance, it is essential to eliminate smooth part of in the level density, so that a mapping from actual spectrum onto a quasi-uniform spectrum with unit mean spacing should be carried out. Hence, for a set of level energies collected, it is necessary to separate the whole spectrum into

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a fluctuation part and a smoothed average part, of which the latter is non-universal and cannot be described by the RMT [1]. To do so, the number of levels below E is expressed as [10,34]

∫

E

ρ (E)dE = e

N(E) =

E −E 0 T

E0

− e− T ,

(1)

E0

for which the constant temperature is assumed in the level density ρ (E), where temperature T and E0 are treated as adjustable parameters. The unfold level energy sequence E˜ i = N(Ei ) is dimensionless and has a unit average spacing. The transformed (unfold) energies should now display on average a constant level density. The nearest neighbor level spacing is then defined as si = E˜ i+1 − E˜ i . Thus, the NNSD, P(s), gives the probability of two unfold nearest-neighbor energy levels with a spacing s. According to the results of the RMT, the Gaussian Orthogonal Ensemble (GOE) statistics characterizing a chaotic system follows the Wigner distribution [2,13,21]: 1

π se−

π s2

4 , (2) 2 while the NNSD of systems with Regular dynamics is generically represented by the Poisson distribution:

P(s) =

P(s) = e−s .

(3)

The Regular (Poisson), chaotic (GOE), and other distribution functions are often used as criterions to describe the degree of chaoticity in level energies with the same spin and parity in even–even nuclei [13,18,21]. These days there are many interesting new methods for describing the spectral statistics of nuclei [37,38]. But in order to describe the level spacing in nuclei, it might be worth to apply simple distributions proposed by Brody, Berry–Robnik, and Abul-Magd [39–45] with less fitting parameters. For example, the Abul-Magd distribution [43,44] is the best approximation for lowlying levels in even–even mass nuclei, while in contrast to the two-parameter Brody approach [39], the BRD [40,42] only has one fitting parameter. It seems that the Berry–Robnik approximation is the best procedure even for high-lying level energies. Another reason to select the Berry–Robnik method is related to energy levels when regular and chaotic orbits coexist with superposing of all sequences. In such a case, the energy interval must be small. For this reason, we need enough levels in this interval to calculate the P(S) of odd-A nuclei. The BRD is derived by assuming that, the energy level spectrum is a product of the superposition of independent sub-spectra, which are contributed respectively from localized eigenfunctions onto invariant (disjoint) phase space region. Also, the BRD also provides a physical meaning for the estimated values in different sequences. The main idea in a derivation of this distribution is that the intermediate behavior of the NNSD of low-lying nuclear levels does not necessarily imply that nuclei in the vicinity of the ground state have mixed regular-chaotic dynamics. The key ingredient of the analysis by BRD is the assumption that the deviation of the NNSD of low-lying nuclear levels from the GOE statistics is caused by neglecting of possibly existing conserved quantum numbers other than energy, spin, and parity. As have shown in [46], the BRD has been simplified by observing that P(s) is mainly determined by short-range level correlations, which reduces the number of parameters to unity, and allows us to refer to the parameter of this distribution as to the chaoticity parameter. Such parameterization is not restricted to statistically independent sequences. A system with partially broken symmetries can also be approximately represented by a superposition of independent sequences. Thus, in order to describe the NNSD of odd-A nuclei as simple and appropriate as possible, the BRD will be adopted. If there is a strong correlation in level energies with a wide range population of level energies in nuclei, the NNSD should be close to the Wigner distribution, while it is close to the Poisson distribution if the correlation is weak. Accordingly, the Berry–Robnik distribution (BRD) [39,47] defined as 1 1 (4) P(s, q) = [q + π (1 − q)s] exp(−qs − π (1 − q)s2 ) 2 4 is introduced, where the mixing of the Poisson and the GOE distribution is weighted by the parameter q. It is obvious that (4) becomes the Poisson distribution when q = 1, while it becomes

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that of the GOE when q = 0. Hence, the actual value of q may be used to characterize the chaoticity of a quantum many-body system. In studying spectral statistics of unfolding energy sequences, a comparison of a set of histograms of sequences with the BRD should be made in order to extract the best value of q via estimation techniques. To avoid drawbacks in the estimation, such as the least square fitting (LSF), in which unusual uncertainty in the estimated value of q is involved leading wrongly to chaotic dynamics, Maximum Likelihood Estimation (MLE) has been used [33,48,49]. The MLE yields a better result with less uncertainty than that obtained from other methods. In order to estimate the parameter q of the BRD, the likelihood function of q is taken as a product of the BRDs of all sequences considered with L(q) =

n ∏

P(si , q) =

i=1

n ∏

1

1

[q + π (1 − q)si ]e−qsi − 4 π (1−q)si . 2

2

i=1

(5)

The desired estimator F (q) is obtained by maximizing the likelihood function shown in (5) with F (q) =

1−

∑ q+

i

π 2

π si 2

(1 − q)si

−

∑

(si −

i

π s2i 4

),

(6)

in which the parameter q should be the isolated zero of F (q) within the closed interval q ∈ [0, 1] and can be evaluated by using various numerical methods. Moreover, the spectral rigidity of considered sequences is also studied. The spectral rigidity, that is, the departure from uniformity over a given span of levels, ∆3 (α, L), defined by [5]

∆3 (α, L) =

1 L

α+L

∫ min A, B

[

˜ − (AE˜ + B) N(E)

α

]2

dE˜ ,

(7)

˜ for which is the average of the least-square deviations between the number staircase function N(E) ˜ the unfold spectrum and its best linear fit (AE + B) over the energy interval [α, α + L]. For a given L, smaller values of ∆3 (α, L) imply stronger long-term correlations between the levels. The average of ∆3 (α, L) over na intervals [α, α + L], which overlaps by 2L successively, yields a smoother measure with 1 ∑ ¯ 3 (L) = ∆ ∆3 (α, L). (8) nα α

It is shown that

¯ 3 (L) = L/15 ∆

(9)

for a regular spectrum with the Poisson statistics, while, in the large-L limit,

¯ 3 (L) ≈ ∆

1

π2

(ln L − 0.0687)

(10)

for a chaotic system with the GOE statistics. Furthermore, half-lives of excited states of odd-A nuclei in the three mass regions may be considered in the level statistical analysis. In this case, the levels collected from odd-A nuclei in the three mass regions are regrouped according to the order of magnitude of the half-lives, from which the samples of level spacings in each group are thus prepared. In addition, it is shown that there are transition-strength fluctuations related to chaos [17,50,51]. Instead of the statistical analysis of transition-strengths, level energy statistics in concerning E2 transition strength of the levels is made, for which the level energies involved in the statistics are divided into four groups according to their E2 transition strengths ranging from 0–50 W.u. The BRD for each group in the three mass regions is also calculated. 3. Data sets and statistical results Similar to the procedure used for even–even nuclei [10], experimental data of level energies for given spin and parity of odd-A nuclei are collected from [35,36], which are then divided into

254

A.J. Majarshin, F. Pan, H. Sabri et al. / Annals of Physics 407 (2019) 250–260 Table 1 The chaoticity degree q determined from energy levels with the same spin and parity of odd-A nuclei in the three mass regions, where the value of q for each case is derived by using the MLE method, and the corresponding spin and parity J π of the levels used to determine the value of q for each region are provided. A ≤ 71 q

1/2+ 0.88

1/2− 0.91

3/2+ 0.75

3/2− 0.83

5/2+ 0.92

5/2− 0.79

71 ≤ A ≤ 151 q

1/2+ 0.69

1/2− 0.54

3/2+ 0.53

3/2− 0.50

5/2+ 0.52

5/2− 0.47

151 ≤ A ≤ 221 q

1/2+ 0.41

1/2− 0.35

3/2+ 0.36

3/2− 0.33

5/2+ 0.38

5/2− 0.42

Fig. 1. The NNSDs with the same spin and parity for light mass odd-A nuclei with A ≤ 71, where the solid broken line is the NNSD, the thick solid curve and open circles correspond to the GOE and the BRD prediction, respectively, and the dashed curve is the regular distribution.

three groups according to the mass number with A ≤ 71, 71 ≤ A ≤ 151, and 151 ≤ A ≤ 221, respectively. In order to collect level energy sequences from different nuclei with existing experimental data [35,36] within the same mass region, the procedure shown in [11,13,14,17,18] is taken. Namely, level energies in a nucleus are taken only when there are at least five consecutive ones with the same spin and parity including those with uncertain spin and parity shown in brackets. The level energy sequence of each nucleus with the same spin and parity is terminated till the highest level observed below 10 MeV. These sequences are unfolded and then analyzed via the BRD and the MLE method. The MLE predictions for the chaoticity (or regularity) degree of all sequences are presented in Table 1, in which the corresponding value of q for the levels with the same spin and parity was determined by using the corresponding level spacings. It should be stated that the total number of sample-level spacings collected and used is 5316 from 266 odd-A nuclei in A ≤ 71 mass region, 6897 from 478 ones in 71 ≤ A ≤ 151 mass region, and 2768 from 143 ones in 151 ≤ A ≤ 221 mass region with parity and the lowest three spins J π = 1/2± , 3/2± , and 5/2± . For all the three mass regions, we have used these sample level spacings to get the corresponding NNSD and BRD via the MLE method. It can be seen from the estimated values of the BRD parameter q shown in Table 1 that level energies with the same spin and parity of odd-A nuclei in the light mass region with A ≤ 71 seem more regular than those in other two mass regions, while there is no clear spin or parity

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Fig. 2. The same as Fig. 1, but for odd-A nuclei in medium mass region with 71 ≤ A ≤ 151.

Fig. 3. The same as Fig. 1, but for odd-A nuclei in heavy mass region with 151 ≤ A ≤ 221.

dependence of q can be observed. Similar to the predictions made in [9], the spectra of light odd-A nuclei seem almost regular, while those of the medium and heavy mass odd-A nuclei seem more or less chaotic in comparison to the light ones. With the increasing of the mass number A, the statistics characterized by the parameter q gradually departs from regular towards the GOE type. The NNSDs with the same spin and parity for odd-A nuclei in the three mass regions are shown in Figs. 1–3. As is clearly shown, the NNSDs of the light mass nuclei seem almost regular, and those of the medium mass ones seem intermediate between the exponential and the Wigner distribution, while those of heavy mass ones become more close to the Wigner distribution. The results shown in Figs. 1–3 are indeed consistent with the chaoticity prediction made by the BRD shown in Table 1. It

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Fig. 4. The same as Fig. 1, but for level energies with the same parity and the lowest three spins 1/2, 3/2, and 5/2 in the three mass regions, where LMN, MMN, and HMN are abbreviations for Light, Medium, and Heavy Mass Nuclei, respectively.

should be noted that the results shown in Figs. 1–3 are only of statistical meaning within the mass regions. There are quite many exceptions to the above conclusion in each region if one makes the same statistics of level energies with the same spin and parity for the individual nucleus in each region. For example, according to the above conclusion, a spectrum of a light mass nucleus with − A ≤ 71 should be often regular. However, we find the level statistics with spin and parity 21 of 41 51

K, 43 Sc, and 55 Co, that with 53

55

3+ 2

of 47 Ca, 47 V, 57 Ni, 63 Ni, 65 Ni, that with 5+ 2

47

41

53

55

57

59

3− 2 57

of

41

K, 49 Sc, 47 Ti, 49 V, 51 V,

Mn, Mn, Mn, and that with of Ca, Sc, Fe, Fe, Fe, Fe, Ni are more or less chaotic if their spectra are analyzed separately. In order to make the statistical results clearer, the NNSDs for level energies with the same parity and the lowest three spins 1/2, 3/2, and 5/2 grouped together for the three mass regions are shown in Fig. 4. It can be seen that the statistics in each mass region for the fixed parity becomes more apparent in comparison to the results shown in Figs. 1–3 in distinguishing from different spin and parity. In addition, the spectral rigidities calculated from all level energy sequences with no distinction of spin and parity in the three mass regions are shown in Fig. 5. Again, the results are shown in Fig. 5 are consistent with those shown in Fig. 4, namely, the larger the mass number, the relatively stronger in chaoticity. According to the shell model, single-particle energy of an unpaired nucleon in an odd-A system increases with the increasing of the mass number A. As the consequence, the heavier the mass, the more disturbance an unpaired nucleon in a valence orbit suffers [25], which answers why the spectra of odd-A nuclei become more chaotic with the increasing of the mass number A. Furthermore, half-lives of excited states of odd-A nuclei in the three mass regions may be considered in the level statistical analysis. In this case, the levels collected from odd-A nuclei in the three mass regions are regrouped according to the order of magnitude of the half-lives, from which the samples of level spacings in each group are prepared for the level statistics. The NNSDs thus generated are shown in Fig. 6, in which the BRD parameter q for each case is also provided. In this way, in addition to the conclusion on the level statistics observed from Fig. 4, it is shown that the spectra of relatively stable nuclei are more chaotic in light and heavy mass regions, while the spectra of odd-A nuclei are always intermediate between regular and chaotic in the medium mass region.

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Fig. 5. Spectral rigidity of the three mass regions. The parameters A and B are introduced in Eq. (7).

Fig. 6. The NNSDs for levels with the same order of magnitude of the half-lives of odd-A nuclei in the three mass regions.

On the other hand, as shown in [17] for even–even nuclei, the statistical result of transition rates varies with the magnitude of strength due to the associated collectivity of initial and final states involved. The stronger the collectivity, the higher transition rates of both E2 and mixed E2+M1 transitions. Inspired by [17], the level statistics related to E2 or mixed E2+M1 transition strengths of odd-A nuclei is also made. In our work, the level energies are classified according to their E2 transition strengths ranging from 0 to 50 (in W.u.) into four groups as shown in Table 2. Then, the BRD parameter q in each group of the three mass regions is thus calculated. If the BRD parameter q is taken as the criterion, it can be seen from Table 2 that the spectra with weaker E2 strengths in the three mass regions, which means weaker collectivity, are more chaotic. This conclusion is consistent with the level statistical results in concerning half-lives, because the weaker the transition rates, the longer the half-lives. It should be pointed out that the above conclusion is made based on the global statistical analysis. There may be quite a few exceptions due to the dynamics associated with shape phase transitions in some isotopes or isotones as analyzed in the IBM and other calculations previously [2,52–54]. Globally, it is shown that the statistics of level energies of light mass nuclei is more regular, that of medium mass nuclei is intermediate, while that of heavy mass nuclei tends towards chaotic. From the above observation, it can be concluded that an odd-A nucleus is likely to be chaotic with heavier mass, longer half-life, and weaker E2 transition rate. Fig. 7 provides contour plot of

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Table 2 The chaoticity degree q of level energies of four categories classified according to E2 transition strengths of E2 or mixed E2+M1 transitions of odd-A nuclei in the three mass regions, which is obtained according to the corresponding BRD by using the MLE method. A ≤ 71 q value of level energies with pure E2 transitions q value of level energies with E2+M1 transitions

0–5 (W.u.) 0.59 0.46

5–10 (W.u.) 0.61 0.75

10–20 (W.u.) 0.77 0.81

20–50 (W.u.) 0.82 0.87

71 ≤ A ≤ 151 q value of level energies with pure E2 transitions q value of level energies with E2+M1 transitions

0–5 (W.u.) 0.51 0.40

5–10 (W.u.) 0.46 0.56

10–20 (W.u.) 0.60 0.71

20–50 (W.u.) 0.83 0.72

151 ≤ A ≤ 221 q value of level energies with pure E2 transitions q value of level energies with E2+M1 transitions

0–5 (W.u.) 0.32 0.37

5–10 (W.u.) 0.55 0.67

10–20 (W.u.) 0.48 0.55

20–50 (W.u.) 0.66 0.73

Fig. 7. Contour plots of B(E2) values for E2 and mixed E2+M1 transitions of odd-A nuclei on the N-Z plane.

B(E2) values for E2 or mixed E2+M1 transitions of odd-A nuclei on the N-Z plane from the data collected. It is clearly shown that transitions in odd-A nuclei near to the β -stability line with more nucleons seem often chaotic because the transitions are often weak, while the regularity occurs along the verge of the stable island due to the transitions in these nuclei are often strong with shorter half-lives based on the results shown in Table 2. 4. Summary and conclusion In this work, spectral statistics of odd-A nuclei are systematically analyzed. Specifically, the NNSD and the BRD for the lowest three spins and parity below 10 MeV within A ≤ 71, 71 ≤ A ≤ 151, and 151 ≤ A ≤ 221 mass regions are calculated. In the analysis, we have taken 5316 sample spacings from 266 odd-A nuclei in the light mass region, 6897 sample spacings from 478 ones in the medium mass region, and 2768 sample spacings from 143 ones in the heavy mass region for the statistics. The chaoticity parameter in the BRD is determined based on the MLE. The level statistics for given spin and parity or for fixed parity is mass number A-dependent. Namely, the larger the mass number, the relatively stronger in chaoticity. In addition, when the order of magnitude of half-lives of these levels are taken into account, the results of the NNSD and the BRD all indicate that the spectra are chaotic for longer-lived odd-A nuclei, especially in the heavy mass region. The level statistical analysis in concerning E2 transition strengths in odd-A nuclei show that chaos often occurs in level energies with weaker E2 transition strengths. By combining the results of our analysis, we come to the conclusion that odd-A nuclei are, more often than not, chaotic. Chaos is emergent in level energies of a stable lighter mass nuclei with weaker E2 transition strength, while it is definitely emergent in level energies of heavy mass odd-A nuclei, and even stronger in comparatively stable ones.

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Since the above conclusion is made based on the global statistical analysis, there may be exceptions due to the dynamics associated with shape phase transitions in some isotopes or isotones [2,52–54]. Therefore, in order to study the actual dynamical structure in a specific odd-A nucleus or its evolution in a chain of isotopes or isotones, a shell model or IBFM based study are in demand. For example, the NNSD and the BRD analysis in the framework of the three-level and four-level bosonic pairing model [55,56] for the odd-A system may be studied. The related work is in progress. Acknowledgments Support from the National Natural Science Foundation of China (11675071, 11747318), the U. S. National Science Foundation (OIA-1738287 and ACI -1713690), U. S. Department of Energy (DESC0005248), the Southeastern Universities Research Association, the China-U. S. Theory Institute for Physics with Exotic Nuclei (CUSTIPEN) (DE-SC0009971), and the LSU–LNNU joint research program (9961) is acknowledged. Also, this work is supported by the Research Council of University of Tabriz. References [1] [2] [3] [4] [5]

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