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ORIGINAL RESEARCH article

Front. Phys., 25 September 2025

Sec. Nuclear Physics​

Volume 13 - 2025 | https://doi.org/10.3389/fphy.2025.1653635

This article is part of the Research TopicBeta Decay: Current Theoretical and Experimental ChallengesView all 8 articles

The pn interaction and isospin symmetry

R. B. Cakirli
R. B. Cakirli1*K. BlaumK. Blaum1R. F. CastenR. F. Casten2
  • 1Max-Planck-Institut für Kernphysik, Heidelberg, Germany
  • 2Wright Lab, Yale University, New Haven, CT, United States

A possible correlation between isospin symmetry/breaking and the average proton-neutron interaction of the last particles, δVpn, is discussed. This correlation is tested for Tz = ±1/2 mirror nuclei in terms of a differential of δVpn, Δ(δVpn), and their low-lying excited levels. Some nuclei, whose mass measurements will be useful for future studies, are suggested.

1 Introduction

The strong nuclear force is considered charge-independent and has charge symmetry. The latter means that the interaction strength between protons and neutrons is the same, and being independent of charge means that the sum of proton-proton (pp) and neutron-neutron (nn) interaction strengths is two times proton-neutron (pn) interaction strength. If the Coulomb interaction is ignored, charge independence and charge symmetry will have the same meaning for isobaric nuclei which have the same mass number with different proton and neutron numbers.

Mirror nuclei are pairs of atomic nuclei in which the number of protons in one nucleus equals the number of neutrons in the other, and vice versa (e.g., 25Mg and 25Al). In such mirror isobaric nucleus pairs, we expect similar nuclear structures [1]. We can easily see this from similar level schemes.

To understand this, it is useful to define the concept of isospin, T. Both protons and neutrons are assigned the same isospin value of T = 1/2, but differ in their isospin z-projection. Protons have Tz = -1/2, while neutrons have Tz = 1/2. Isospin symmetry is related to similar behavior of nucleons (protons and neutrons). Since some configurations such as pp and nn with T = 0 are forbidden, the Pauli principle should not be forgotten at this point. That is, the isospin symmetry only connects to T = 1 in the pp and nn interactions. For a given nucleus, the isospin projection is given by Tz = (N-Z)/2 where Z and N are, respectively, the number of protons and neutrons. While the low-lying states of a nucleus with given Tz, which we focus on here, generally have T = |Tz|, higher states can have higher T values, being part of more extended multi-isobar isospin multiplets.

Mirror nuclei have different Tz. The similar nuclear structure in such nuclei means that their excited states are (almost) identical, in terms of both their energies and spin-parity values. For example, the low-lying states of the A = 23 isobaric nuclei, 23Na with Tz = 1/2 and 23Mg with Tz = -1/2, are shown in Figure 1. As can be seen from the figure, the level schemes of the two nuclei are almost identical, so their nuclear structures are expected to be very similar. For these states, these nuclei exhibit good isospin symmetry. The assumption of perfect isospin symmetry implies that the difference between the binding energies of the mirror nuclei is zero if the differences in the Coulomb interaction in the two nuclei are ignored. Isospin symmetry breaking can occur due to increases in parts of the Coulomb interaction, especially as the mass number increases. Isospin breaking can also occur for other reasons beside the Coulomb interaction (e.g. [3, 4]). By taking these isospin symmetry breaking effects into account, the isospin concept can provide a tool for understanding the excitation energies and binding energies of exotic nuclei that are difficult to reach experimentally. In addition, the study of isospin symmetry breaking plays an important role not only in nuclear physics but also in particle physics, especially in testing the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix [59].

Figure 1
Energy level diagram comparing \(^{23}\text{Na}_{12}\) and \(^{23}\text{Mg}_{11}\). Energy levels are labeled with \(3/2^+\), \(5/2^+\), \(7/2^+\), \(1/2^+\), \(1/2^-\), and \(9/2^+\). Levels for \(^{23}\text{Na}_{12}\) are in gray, while \(^{23}\text{Mg}_{11}\) levels are in red. The y-axis represents energy in MeV.

Figure 1. (Color online) Low-lying levels and spin-parity assignments for A = 23 [2], T = 1/2 mirror nuclei are shown.

Many isobaric nuclei with different isospin projections, such as Tz = ±1, Tz = ±2 have been investigated by experimental charge-exchange reactions [10, 11] and β-decay studies (e.g. [12]). In such studies, the B(GT) values from isobaric Tz = ±1 nuclei to a Tz = 0 nucleus can be compared using both experimental techniques. If the experimental values of B(GT) are similar, isospin symmetry between mirror, Tz = ±1, nuclei can be confirmed. If the values are different, the isospin symmetry may be broken.

2 Approach and methods

In this paper, we explore another observable as a possible indicator or signature of isospin symmetry or its breaking. Since the valence proton-neutron interaction plays an important role in the evolution of nuclear structure [1315], we will investigate whether an empirical measure of those strengths correlates with isospin symmetry or its breaking. This measure is called δVpn [16; 17; 18] and is the average interaction strength of the last proton(s) and neutron(s). It reflects the spatial overlap of their respective wave functions. We will examine values of δVpn for nuclei near Z = N and will also discuss a related quantity obtained from adjacent δVpn values. We can extract the strengths of these interactions for the last valence proton(s) and neutron(s) from the following expressions in terms of binding energies [16, 17]:

δVpnoeZ,N=12BZ,NBZ,N2BZ1,NBZ1,N2(1)
δVpneoZ,N=12BZ,NBZ,N1BZ2,NBZ2,N1(2)

where B is the nuclear binding energy

BZ,N=Zmp+NmnMc2(3)

and M in Equation 3 is the nuclear mass. Equations 1, 2 are given for odd-A. More detailed information can be found in Ref. [18]. Here we look at other applications of δVpn to understand nuclear structure and its trends.

3 Results and discussion

In recent years, many light nuclei have been studied especially in such contexts as of the island of inversion, appearance, and disappearance of closed shells, etc. [19]. In addition, such nuclei have been studied in terms of δVpn, in particular for the case where the values of δVpn have obvious spikes at Z = N. This has been explained by Wigner’s SU(4) symmetry [20, 21]. In these Z = N nuclei, since protons and neutrons fill the same nuclear shell model orbitals, there can be a large spatial overlap between the proton and neutron wave functions and therefore we expect a large interaction between protons and neutrons, δVpn. As the mass number increases, the values of δVpn decrease presumably due to the Coulomb and spin-orbit interactions, and perhaps due to the greater average spacing of the last protons and neutrons.

Turning now to isobaric mirror nuclei, Figure 2 shows the experimental δVpn values of odd-AT = ±1/2 mirror nuclei versus their mass numbers. There are two δVpn values in each mass number shown with vertical bars for Tz = 1/2 (orange) and Tz = -1/2 (blue). The A = 61, 65, and 69 nuclei have only Tz = 1/2 data due to missing experimental values for the masses of the involved nuclei.

Figure 2
Bar chart comparing δVₚₙ (in MeV) versus mass number for Tz = 1/2 and Tz = -1/2. The x-axis shows mass numbers from 7 to 71, while the y-axis shows δVₚₙ values ranging from 0 to 6 MeV. Orange bars represent Tz = 1/2, and blue bars represent Tz = -1/2. Values decrease as mass number increases.

Figure 2. (Color online) Experimental δVpn values as a function of mass number for mirror Tz = ±1/2 nuclei. For each mass, there are two δVpn values shown with different colors, namely, Tz = 1/2 with orange and Tz = -1/2 with blue. There is no Tz = -1/2 data at A = 61, 65, 69 due to the lack of direct mass measurements. Masses are based on Refs. [22, 23].

Perhaps a simple way of stating the systematics in Figure 2 is that δVpn is large for nuclei with A = 4k - 1 and small for nuclei with A = 4k + 1. Interestingly, large and small δVpn values involve different sets of Tz values (see Equations 1, 2), the large bars contain |Tz| equals 0, 1/2 and 1; small bars contain |Tz| values 0, 1/2, 1 and 3/2.

There is another systematic effect in Figure 2. For mass numbers where δVpn is large (e.g., A = 7, 11, 15, 19, etc.), δVpn is always higher for Tz = -1/2 for even-Z and odd-N except for A = 7, 27, 39, 55, 57, 59 and 67. For mass numbers where δVpn is small (e.g., A = 9, 13, 17, 21, etc.), δVpn is again always higher for cases of even-Z and odd-N but now for Tz = 1/2. That is, except for a few mass numbers and regardless of what Tz is, δVpn is always higher in the case of even-Z and odd-N compared to odd-Z and even-N. This effect is even more visible in Table 1 which shows the data on which Figures 2, 3 are based on. Bold face is used for the cases of even-Z and high δVpn values for each mirror pair.

Table 1
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Table 1. A list of the nuclei discussed in this study with the Tz, δVpn and Δ(δVpn) values. While the table has data up to A = 77, Figures 2, 3 have data up to A = 71, which is the largest mass number in which a pair of δVpn is experimentally known. Bold face is used for high values of δVpn with even-Z to draw attention to these nuclei; see the text for details.

Figure 3
Bar graph depicting the change in delta Vpn (keV) against mass number, with values ranging from -550 to 100 keV. Red bars represent odd-Z Tz=1/2, and blue bars represent even-Z Tz=-1/2. Error bars are included for each data point.

Figure 3. (Color online) Experimental Δ(δVpn) values as a function of mass number for mirror Tz = ±1/2 nuclei. Shadowing is used to point out a 50 keV band around zero. See also Table 1.

When we look at the trends of the large values of δVpn in Figure 2, we see a smooth decrease except at A = 39 and 55 in which δVpn increases a little compared to the general downward trend. For A = 39, the δVpn values of (Z,N) = (19, 20) and (20, 19) are very close to each other within their error bars. A small increase is seen because both Z and N contain the magic number 20. Similarly, in A = 55, the effect of the magic number 28 is observed in (Z,N) = (27, 28) and (28, 27). For the smaller pairs of bars, in the case of A = 17, the effect of the magic number eight should also be considered for (Z,N) = (8, 9) and (9, 8). After the decrease in A = 9, there is an increase in A = 13. The question here is whether A = 9 is exceptionally low or A = 13 high.

At this point, it is useful to introduce an empirical quantity related to δVpn, but which is more sensitive to details of the p-n interactions. It is basically a differential of δVpn. If we expect the nuclear structures of the mirror isobaric nuclei to be nearly identical, then we expect the δVpn values of these nuclei to be quite close to each other. Although the δVpn values of these mirror isobaric nuclei appear to be close to each other in Figure 2, the difference between two experimental δVpn values of Tz = -1/2 and Tz = 1/2 is quite interesting. This quantity, Δ(δVpn), is defined as follows:

ΔδVpnZ,N=δVpnTz=1/2δVpnTz=1/2.(4)

Equation 4 and a similar approach as presented here have recently been discussed in Refs. [2426]. Here, however, we investigate which nuclei have isospin symmetry by looking at both the Δ(δVpn) values and some of the lowest excited states in mirror pairs.

The Δ(δVpn) results are shown in Figure 3. There are some clear trends in the results. Except for A = 7, which seems highly anomalous, the pink bars (for odd-Z, Tz = 1/2) are always positive (in some cases the values are very close to zero where uncertainties generally overlap with zero). The blue bars (for even-ZTz = -1/2) are always negative, except in a few cases above A = 50 where the data has large uncertainties that again overlap with zero. There are also some other interesting features. As can be seen in the figure, there are quite high negative Δ(δVpn) values for a few mass numbers such as -600 keV for A=13. The largest differences are seen at A = 7, 9, 13 and 17. A 50 keV band around zero is shaded as a reference to guide the eye. Most of the bars are within this 50 keV band. Note that the largest errors are at A = 7, 57, 63, 67 and 71. We see results for Δ(δVpn) closest to zero in many cases such as A = 15, 25, 37, 59, etc. Due to the lack of experimental mass values, there are no Δ(δVpn) values at A = 61, 65 and 69 (see also Figure 2).

What can we learn about the nuclear structure of mirror nuclei from these Δ(δVpn) values? Does a small value hint to a similar structure between mirror pairs? In other words, can Δ(δVpn) be used as a measure of isospin symmetry and/or its breaking? For example, in Figure 3, the Δ(δVpn) value of the mirror nuclei A=25, 25Mg and 25Al, is approximately zero, while the Δ(δVpn) value of mirror nuclei A = 13, 13C and 13N, is approximately −600 keV. In this case, is the nuclear structure of A = 25 Tz = ±1/2 mirror nuclei more similar to each other compared to the nuclear structure of A = 13 Tz = ±1/2 mirror nuclei? The rest of this paper looks at this possibility in greater detail.

Each panel of Figure 4 shows some low-lying excited levels of a pair of mirror nuclei A=25 (left) and A = 13 (right). As can be clearly seen, there is almost perfect similarity between the level schemes of 25Mg and 25Al, while there is very little similarity between 13C and 13N. In fact, the isospin symmetry between 25Mg and 25Al has been experimentally demonstrated [27]. This correlates very well with the Δ(δVpn) result. On the other hand, Ref. [3] shows isospin breaking in 13C using pion inelastic scattering. The A = 13 spectra are very dissimilar and Δ(δVpn) is large. This pair of examples suggests that Δ(δVpn) may be useful as a filter or signature for the goodness of isospin, or its breaking. To study if this approach is accidental or not, one should look at each example of Δ(δVpn) shown above in Figure 3. Of course, the absolute binding energies of the two mirror nuclei are different because of the Coulomb interaction. But, this does not play a role in the figure since we normalize the ground state energies to zero.

Figure 5 shows all pairs of mirror nuclei with level schemes that are very similar, including A = 23 from Figure 1 but not A = 25 just shown in Figure 4. Here, similarity in the level schemes, the energy difference between the excited states (level spacing) and the fact that these similar states have the same spin-parity are used as criteria. Besides the fact that the level schemes of these nuclei are very similar, their Δ(δVpn) values are quite small. The nuclei with the largest Δ(δVpn) in Figure 5 are the A = 19 mirror nuclei with 50.0 (3) keV and the A = 29 mirror nuclei with 44 (5) keV. The others have maximum Δ(δVpn) values of 25 keV.

Figure 4
Two energy level diagrams compare isotopes: \(^{25}\text{Mg}\) and \(^{25}\text{Al}\) on the left, and \(^{13}\text{C}\) and \(^{13}\text{N}\) on the right. Each panel shows energy levels indicated by horizontal lines with spin-parity values. Black lines represent \(^{25}\text{Mg}\) and \(^{13}\text{C}\), while red lines represent \(^{25}\text{Al}\) and \(^{13}\text{N}\). Energy values are measured in MeV. Δ(δVpn) values are 0.34(32) for \(^{25}\text{Mg}\)–\(^{25}\text{Al}\) and -562(3) for \(^{13}\text{C}\)–\(^{13}\text{N}\).

Figure 4. (Color online) Low-lying levels and spin-parity assignments [2] for A = 25 (left) and A = 13 (right), Tz = ±1/2 mirror nuclei are shown. Δ(δVpn) values are also given in keV.

Figure 5
The image contains a series of energy level diagrams for various isotopes. Each panel shows energy levels in mega-electron volts (MeV) for isotopes such as \(^{11}\text{B}\), \(^{19}\text{F}\), \(^{21}\text{Ne}\), among others. The energy levels are labeled with quantum numbers like \(1/2^+\), \(3/2^-\), and in some cases, red lines indicate specific states. Each diagram is labeled with isotopic and energy difference information, e.g., \(\Delta V_{pn}\) or specific values.

Figure 5. (Color online) Similar to Figure 4 for more cases of consistency between small values of ΔδVpn and level schemes that are very similar.

These results confirm that small Δ(δVpn) values might be a useful filter for mirror nuclei with small isospin symmetry breaking. We will see below that there are some exceptions to this that need to be studied further. In some cases, like A = 55 and 59, further study of experimental spectra would be useful.

This idea can be tested in an inverse way. The A = 9 and 17 cases are shown in Figure 6 and have both incompatible level schemes and Δ(δVpn) values that are rather large. At first glance, there seems to be no serious difference between the two level schemes in each pair but, for example, if we look at the level spacing in 17O and 17F carefully, there is about a factor of two difference in the energies of their first excited levels. The large Δ(δVpn), -500 keV, also points to this disagreement. Indeed, in Ref. [4] A=17 isospin breaking has been discussed on the basis of quark-meson coupling. Thus, we again see the use of Δ(δVpn) values as a signature, in this case of symmetry breaking. Note that 9B has an unbound proton, therefore a large Δ(δVpn) may be expected. However, the mass 9B is used not only for δVpn(9B) but also for δVpn(11B). In Figure 5, a small Δ(δVpn) value is given together with nice agreement on the level schemes of 11B and 11C. Clearly, the effects of extended proton radial distributions in proton unbound nuclei need further study.

Figure 6
Two side-by-side energy level diagrams illustrate nuclear states. The left graph (for \( ^9\text{Be} \) and \( ^9\text{B} \)) displays energy levels from 0 to 6 MeV, with various nuclear spin states labeled 3/2 to 7/2, indicated by black and red lines. The right graph (for \( ^{17}\text{O} \) and \( ^{17}\text{F} \)) ranges from 0.2 to 5 MeV, showing states labeled 1/2 to 5/2 with similar colored lines.

Figure 6. (Color online) Similar to Figure 4 but with cases of consistency between dissimilar excitation spectra and a large Δ(δVpn) value.

While this correlation of Δ(δVpn) and the degree of similarity in mirror pair level schemes is suggestive of a new tool to assess isospin symmetry, however, there are also a few counter examples that may hint to its limitations. Figure 7 shows one case of similar level schemes but a large Δ(δVpn) for A = 7, −185 (35) keV, and a number of Tz = ±1/2 mirror nuclei with dissimilar level schemes but low Δ(δVpn) values. There is no noticeable anomaly in the δVpn results for the 7Li and 7Be nuclei, except for mass error of about 50 keV for both 5Li and 5He. If Δ(δVpn) is a reliable filter for isospin breaking, one would expect more consistency of spectra and Δ(δVpn) values. This needs further investigation.

Figure 7
Energy level diagrams for various isotopes (7Li, 7Be, 15N, 15O, 41Ca, 41Sc, 43Sc, 43Ti, 47V, 47Cr, 49Cr, 49Mn, 53Fe, and 53Co) are presented. Black and red lines represent different energy states and transitions, with energy values in MeV on the vertical axis. The labels indicate nuclear states with various spins and parities.

Figure 7. (Color online) Similar to Figure 4 but with cases of disagreement between the value of Δ(δVpn) and the excitation spectra. A = 7 shows similar level schemes and a large Δ(δVpn) value. All the other panels show small (<50 keV) Δ(δVpn) values and spectra of mirror nuclei that either disagree with each other or where further data on levels and Jπ assignments are needed to evaluate the level of agreement.

The rest of Figure 7 shows cases of dissimilar level schemes. Most of these are in heavier nuclei compared to the nuclei in Figure 5. As the mass number increases, the Δ(δVpn) filter may simply break down. As mentioned in the beginning of this paper, isospin breaking occurs when the mass number increases due to Coulomb force among protons. Also, especially in heavier nuclei, there can be states of higher T(>1/2) at higher energies, which are part of extended isospin multiplets, and there can also be isospin mixing in complex states. This could lead to some differences in spectra.

Finally, there are a number of nuclei with insufficient data to assess the correlations. In these cases, either further spectroscopic or mass data would be highly useful. We first consider cases of insufficient level scheme information. In some nuclei, spin-parity of the excited levels is unknown or not fully known, and their Δ(δVpn) values are small. Such nuclei are shown in Figure 8. These nuclei should be studied by γ-ray spectroscopy. If the experimental data of these nuclei are clarified, further tests of the usefulness of Δ(δVpn) as a signature of isospin symmetry may emerge.

Figure 8
Energy level diagrams for different isotopes: \(^{37}\text{K}\) vs. \(^{37}\text{Ar}\), \(^{43}\text{Ti}\) vs. \(^{43}\text{Sc}\), \(^{47}\text{Cr}\) vs. \(^{47}\text{V}\), \(^{51}\text{Fe}\) vs. \(^{51}\text{Mn}\), and \(^{57}\text{Cu}\) vs. \(^{57}\text{Ni}\). Each chart shows energy levels (in MeV) with corresponding spin-parity notations, and a mix of black and red lines representing different transitions.

Figure 8. (Color online) Cases in which mirror level schemes and Δ(δVpn) values cannot be compared due to the need for more fully known level schemes (both levels and Jπ) values.

A recent γ-ray spectroscopic study focusing on isospin symmetry breaking is Ref. [28]. The study finds evidence for the breaking of isospin symmetry in the mirror system 71Kr and 71Br by β-decay. As seen in Figure 3, Δ(δVpn) of A = 71 has a large error. Therefore, in order to test our approach here with Δ(δVpn), 70Kr and even maybe 71Kr mass excess values should be improved. There are a number of other cases where additional mass measurements would be helpful to further study the use of δVpn to assess the degree of isospin symmetry breaking. These are listed in Table 2 and provide motivation for further experimental mass measurements.

Table 2
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Table 2. The successive columns of the table show δVpn values for experimentally known nuclei with large errors, the mass excess errors for those nuclei contributing the largest uncertainties to δVpn, and half-lives. The nuclei with unknown δVpn are also listed in Unknown δVpn column. Experimental masses are taken from Refs. [22, 23].

As seen in Figure 2, there are no δVpn values at A = 61 Tz = -1/2, 61Ga, A = 65 Tz = -1/2, 65As, and A = 69 Tz = -1/2, 69Br. Since the half-lives of 59Ga, 63As and 67Br are in the order of nanoseconds, it is impossible to measure the masses of these nuclei today. Finally, the δVpn values for 75Sr and 77Y are experimentally not known due to missing masses, as seen in Table 2. They are the heaviest nuclei suggested here where we can possibly test isospin symmetry/breaking with Δ(δVpn). The other nuclei in the table have δVpn values but their errors can be improved. The masses needed for this purpose are also listed. The A = 79 T = 1/2 mirror nuclei do not have any δVpn value for either Tz = 1/2 or Tz = -1/2 nuclei.

4 Conclusion

We have discussed a possible correlation between isospin symmetry in mirror nuclei and its breaking and empirical measures of the average proton-neutron interaction. The correlation is suggestive but not perfect, and breakdowns in it need to be further investigated by both γ-ray spectroscopy and mass spectrometry. For the latter, possible nuclides of interest are listed in Table 2.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

RBC: Writing – original draft, Writing – review and editing. KB: Writing – review and editing. RC: Writing – review and editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. KB and RBC express gratitude for the financial support from the Max Planck Society. RBC also thanks the ExtreMe Matter Institute EMMI at the GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany.

Acknowledgments

We thank Y. Litvinov, Y.H. Zhang and X. Yan for pointing out additional data on recent mass measurements.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: proton-neutron interaction, isospin, mirror nuclei, mass measurements, γ-ray spectroscopy

Citation: Cakirli RB, Blaum K and Casten RF (2025) The pn interaction and isospin symmetry. Front. Phys. 13:1653635. doi: 10.3389/fphy.2025.1653635

Received: 25 June 2025; Accepted: 21 July 2025;
Published: 25 September 2025.

Edited by:

Pedro Sarriguren, Spanish National Research Council (CSIC), Spain

Reviewed by:

Roelof Bijker, National Autonomous University of Mexico, Mexico
Piet Van Isacker, UPR3266 Grand accélérateur national d’ions lourds (GANIL), France

Copyright © 2025 Cakirli, Blaum and Casten. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: R. B. Cakirli, YnVyY3UuY2FraXJsaUBtcGktaGQubXBnLmRl

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