Prediction of two-neutron halos in the N = 28 isotones 40 Mg and 39 Na

,

Above N = 20, an explanation of the magic numbers requires a spin-orbit force [14].Thus, N = 28 is the smallest magic number that is sensitive to the spin-orbit force.As shown in the lower panel of Fig. 1, for 20 ≤ N ≤ 28 the 1 f 7/2 orbit is filled, while in the absence of spin-orbit splitting the 1 f 7/2 and 2p 3/2 orbits are almost degenerate.With modifications to the spin-orbit force, the spacing between these two orbits may change, and this can lead to an "inverted" order, where the 2p 3/2 lies below the 1 f 7/2 .The same result may also be caused by deformation [1,15].Such islands of inversion (IOI) have been found in isolated cases.It has also been shown that for N = 20 and 28 the IOIs merge in the neon, sodium, and magnesium isotopic chains, creating a "big island of inversion", called the B-IOI [16,17].
The recent observation of the disappearance of the N = 20 shell gap at the low-Z side of the N = 20 chain (shown in the red circle in the upper panel of Fig. 1), has led to the identification of the 29 F system as the heaviest known two-neutron Borromean-halo nucleus [13,[30][31][32].Motivated by this observation, it is interesting to explore the low-Z side of the N = 28 shell closure for potential two-neutron Borromean halos in the Na and Mg isotopes (as shown in the blue circles in the upper panel of Fig. 1).The magnesium isotopic chain also provides interesting candidates to explore the transition from stronglybound to weakly-bound nuclei and the form of n-n correlations at the limits of stability.Deformation will also play a role: e.g., the nucleus 32 Mg, with N = 20, can be described as a prolatedeformed rotor [8,33].There is only limited knowledge of the properties of heavier neutron-rich Mg isotopes with N ≥ 25.The weakly-bound nucleus 37 Mg (N = 25) has been observed, and its ground state can be interpreted as a one-neutron halo with the valence neutron occupying 2p 3/2 orbit [34][35][36][37][38].The neighbouring odd-A nucleus 39 Mg was confirmed as unbound, indicating the important role of pair correlations.
For the present study, we start from the medium-mass open shell nucleus 40 Mg, which is the next two-neutron halo candi-  date after 29 F according to various theoretical models [16, [39][40][41].The evaluated (i.e., not measured but estimated based on systematics) two-neutron separation energy of 40 Mg is s 2n = 0.670 ± 0.710 MeV [42].This nucleus was first observed experimentally in 2007 [43] and more recently low-lying excited states of 40 Mg have been studied using a one-proton removal reaction from 41 Al [12].Recent ab initio calculations suggest large deformation in 40 Mg [44], though no connection to halo physics is made.Because 39 Mg is neutron unbound, it is reasonable to assume that the correlation between the two valence neutrons in 40 Mg plays an important role in binding this system.Thus, the 40 Mg nucleus, when interpreted as a 38 Mg core plus two valence neutrons, provides an example of a Borromean system, similar to the well-studied two-neutron halo nuclei 6 He [45,46] or 11 Li [47].Because three-body models give a good description of the structural properties of many Borromean nuclei, we apply the same technique, using effective three-body calculations to study the configuration mixing in and matter radius of the ground state of 40 Mg.Given the uncertainties in the two-neutron separation energy, we explore how the allowed values influence the configuration mixing and matter radius, by analysing different scenarios.This allows us to explore the possibility of halo formation in this region of the nuclear chart, well beyond the heaviest observed two-neutron halo in 29 F [13,[30][31][32].The most recent atomic mass evaluation [42] indicates that 39 Na is unbound, contradicting Ahn et al. [26] recent experimental discovery; our calculations within same three-body frame-work aim to predict the two-neutron separation energy (s 2n ) and matter radii for 39 Na systematically.Finally, we also provide Glauber-model predictions for the total reaction cross sections of 40 Mg and 39 Na on a carbon target, which will further support the finding of halo formation in these systems.

Model Formulation
To focus on the effective core + n + n system, we employ the hyperspherical formalism [48,49].The eigenstates of the three-body Hamiltonian can be written as where ρ 2 = x 2 + y 2 is the hyperradius defined from the usual mass-scaled Jacobi coordinates {x, y}, and the hyperangle α = arctan (x/y) is contained in the angular variables Ω = α, x, y .
For simplicity, we choose x as the coordinate describing the neutron-neutron relative motion, and y as the coordinate between the core and the two neutrons.This is typically referred to as the Jacobi-T representation.In Eq. ( 1), the index β ≡ {K, l x , l y , l, S x } j labels the different components of the wave function, with Y β (Ω) representing states of total angular momentum j which follow the coupling order Here, Υ l x l y Klm l (Ω) are the hyperspherical harmonics [48], eigenstates of the hypermomentum operator K, and κ S x is a spin function.This equation implies that l = l x + l y is the total orbital angular momentum, S x is the coupled spin of the two neutrons, and the total j results from j = l + S x .Note that we assume the core to be inert and its spin to be zero.More details can be found, for instance, in Ref. [50].
To determine the hyperradial functions U β (ρ) for bound states, we diagonalize the Hamiltonian in a suitable basis and focus on the negative-energy solutions.Thus, we expand where C iβ are the diagonalization coefficients to be determined numerically.For this purpose, different bases have been used in various works [51][52][53], and in the present work, we employ the analytical transformed harmonic oscillator (THO) basis from Ref.
[54].The THO functions are built by performing a local scale transformation to the harmonic oscillator functions, changing their Gaussian asymptotic behaviour into an exponential decay, which is better suited to accurately describe bound nuclear states with a relatively small set of basis functions.This procedure was also adopted, for instance, in Refs.[30,32].The diagonalization of the three-body Hamiltonian requires the computation of the corresponding kinetic energy and potential matrix elements.With the above definition in hyperspherical coordinates, we can write [55, 56] for the kinetic energy operator, where m is the mass of the nucleon.The coupling potentials are taken to be of the form In this expression, V i j are the two-body interactions within the three-body composite system, to be fixed from the known information on the binary subsystems, and V 3b (ρ) is a phenomenological three-body force, to account for effects not included in the two-body interactions alone [53,54,56,57].This term contains the only free parameter in this model and is used to adjust the energy of the three-body states to the experimental values if known.

Two-body (core + n) subsystems
The spectral properties of the core+n subsystems play a key role in the study of the structure of Borromean three-body nuclei, as the two-body potentials are contained in the three-body Hamiltonian.In the present case, this amounts to fixing a phenomenological 38 Mg + n and 37 Na + n potential to describe the low-lying continuum spectrum of 39 Mg and 38 Na, respectively.
We will start with 38 Mg + n system.Though N = 26 has a partially filled 1 f 7/2 subshell, we model 39 Mg as a 38 Mg core surrounded by an unbound neutron moving in the f 7/2 , p 3/2 , f 5/2 , and p 1/2 orbitals in a simple independent-particle shellmodel picture.Even though the distinction between the core and valence neutrons is not completely obvious, for simplicity, we nevertheless make an inert-core approximation.This means that the effects from internal rearrangement or corevalence exchange must be contained in effective potential parameters.Note that a similar approach has been successfully followed in other three-body calculations, such as the 27 F + n and the 29 F + n subsystems for the description of 29 F [30][31][32] and 31 F [58].Core excitations could be incorporated by using, for instance, a particle-rotor model [41] with an effective core deformation parameter.
The only theoretical study on the spectrum of the neutronunbound nucleus 39 Mg reports an indication of a low-lying resonance at 0.129 MeV with J π either 7/2 − or 3/2 − [59].However, due to the unavailability of experimental data, this does not rule out the existence of other possibilities such as a single resonant structure or more than two states, and in any case, the spin-parity assignment is not unambiguous.
Considering the limited information available, we model the 38 Mg + n interaction as a Woods-Saxon potential including only central and spin-orbit terms, where V l 0 is in general l-dependent, and R c = r 0 A 1/3 c with A c the mass number of the core.The spin-orbit interaction is written in terms of the Compton wavelength λ π = 1.414 fm.Following Ref.
[60], the spin-orbit strength is taken to follow the systematic trend [61] and has the value V ls = 16.842MeV for 38 Mg + n and 16.324 MeV for 37 Na + n.The value r 0 = 1.25 fm  38 Mg + n (upper-panel) and 37 Na + n (lowerpanel) Woods-Saxon interactions, Eq. ( 6).Here a is diffuseness, V (l)  0 is the potential depth and E R is the position of the resonances.Note that r 0 = 1.25 fm, and V ls = 16.842MeV (for 38 Mg + n) and 16.324 MeV (for 37 Na + n) are fixed. 38 is as originally suggested for 31 Ne [30,60], and we examine three scenarios, see Table 1.In Set 1 and 2, V 0 is chosen to be l-independent.In set 1 it is adjusted to fix the f 7/2 ground-state resonance of 39 Mg at 0.129 MeV, corresponding to the prediction of Ref.
To explore the competition of the p-wave with the f -wave ground state, we consider two additional scenarios defined by Sets 2 and 3, respectively.In the "degenerate" case, we tune the f -wave strength and diffuseness parameter a so that the f 7/2 and p 3/2 resonances become nearly degenerate.The "inverted" scenario assumes the ground state of 39 Mg to be the p-wave resonance.Here we need to allow orbital dependence, and we have different potential depths for l = 1 and l = 3.
These potentials generate Pauli forbidden single-particle states, which would result in unphysical eigenstates of the three-body Hamiltonian, and need to be removed.In this work, this is achieved through a supersymmetric transformation [62,63].
As we do not have either theoretical or experimental predictions and data for 38 Na, we use the same potential parameters as for 39 Mg.The only changes are R c and the spin-orbit strength, as explained above.The latter takes into account effectively the tensor force between the unpaired proton and valence neutron [64,65].Indeed, since 39 Na has an odd proton number, its ground state angular momentum is not zero.However, we have found before [30,58,66] that approximating the ground state of these light systems as a 0 + state gives reasonable results.The lower half of Table 1 shows the parameter sets for the 37 Na + n potential.Our predictions for the low-lying spectrum of 38 Na show no "normal" ordering of the single-particle states; most scenarios present an inversion, indicating a shell-evolution effect, that we describe effectively by just changing the spin-orbit strength.

Three-body (core
Using the core+n potentials outlined in the preceding section, we compute the three-body ground-state of the 40 Mg and 39 Na nuclei assuming a simple description in which two valence neutrons couple to J π = 0 + .To provide a more comprehensive description, it is essential to have precise experimental data on the core+n spectrum and a thorough understanding of the intricate core+n interactions that can result in the splitting of singleparticle energy levels.In the present calculation, we also need a prescription for the nn interaction.We use the Gogny-Pires-Tourreil (GPT) potential [67] as in previous works [54,56,68].It is important to emphasize that when it comes to the groundstate properties of core + n + n systems, the specific selection of the nn interaction is not particularly critical, as long as the chosen interaction adequately represents nn scattering data, as discussed in Ref. [48].As previously introduced in Eq. ( 5), we incorporate a phenomenological three-body force to obtain the energy of the three-body ground state, following a methodology similar to that employed in our earlier studies [30,54,58].The three-body problem is solved by diagonalizing the Hamiltonian in a THO basis including hyperspherical harmonics up to a maximum hypermomentum K max in the wave-function expansion (Eq.( 1)), and i = 0, . . ., N excitations for the hyperradial functions (Eq.(3)).Note that fixing K max implies that the orbital angular momenta associated with each Jacobi coordinate are restricted to l x + l y ≤ K [48].In this work, K max = 46 and N = 30 are found to provide converged results.
In order to compute the 40 Mg and 39 Na matter radius within the present three-body model [55] using R m = 1 A A c R 2 c + ⟨r 2 nn ⟩ , the size of the 38 Mg and 37 Na cores are required as input.We have used R c = R m ( 38 Mg, 37 Na) = 3.600, 3.637 fm, respectively, adopted from Ref. [69,70].These are somewhat uncertain, so it is more precise to look at a comparison of the differences across various potential models as well as the relative increase between core and three-body systems.
With all these ingredients, our three-body results for the ground state of 40 Mg and 39 Na using the scenarios for the core + n potentials are shown in Fig 2 .As already discussed, there are large uncertainties in the evaluated s 2n value [42] of 40 Mg and no information is available for the s 2n value of 39 Na.Given these uncertainties, we explore the sensitivity of the configuration mixing and matter radius of the ground state with s 2n , using different values of the three-body potential strength.For 40 Mg, V 3b = 0 leads to an overbinding of the system, so we need to choose V 3b > 0 to get s 2n corresponding to the lower, central and upper limits of the evaluation (0.010, 0.670 and 1.380 MeV).Note that the lower limit would result in an unbound state.However, the experimental findings support a for 40 Mg and 39 Na with s 2n .The black dotted line shows the core radius, 38 Mg and 37 Na, respectively.It should be noted that the labels Inverted, Degenerate and Normal pictures here refer to the outcome of full three-body calculations and should not be confused with the single-particle scenarios of Table 1.
bound ground state, so we have considered a barely bound case as well.For 39 Na, due to the lack of experimental information and systematic information on s 2n , it seems reasonable to keep the range of V 3b values determined for 40 Mg and make predictions.With this prescription, the shallowest case becomes unbound for all sets.To restrict the situation to bound states, we modify V 3b in that case so that s 2n is 0.010 MeV, since we are interested in giving predictions for a bound halo nucleus, which could be tested experimentally.We predict s 2n for 39 Na between 0.010-0.824(1.828) MeV without (with) V 3b = 0 cases.
The computed partial wave content and radii for each s 2n value are shown in the upper panel and lower panel, respectively, of Fig 2 .As expected, shallower (bound) ground states yield larger p-wave occupancy which in turn leads to an "inverted" picture, in contrast to the deeply-bound ground states which lead to the "degenerate" and "normal" pictures for all sets.As can be seen from the upper panel of Fig. 2, when moving from Set 1 to Set 3 the p-wave content in the ground state increases, whereas the f -wave content decreases for each choice of s 2n value.We remark that the similar behavior was found in Ref. [71].
Similarly, we see in the lower panel of Fig. 2 that, as expected, the shallowest cases give larger matter radii in contrast with more deeply bound ground states, which result in smaller matter radii for all sets in both systems considered.More interestingly, we note that the effect of the value of s 2n on the change in radius is most substantial when we have a shell inversion.However, due to the rather heavy cores, the calculated total radii are similar for the scenarios considered.Even with a substantial neutron halo, for a fixed s 2n the variations amount up to a maximum of 5 % difference for both systems.Differentiating scenarios solely by radius could pose experimental difficulties.Additionally, data on knockout or transfer reactions [72], [73], and [34], respectively.The dotted black line corresponds to the R 0 A 1/3 fit for A = 24-37 for Mgisotopes and A = 23-32 for Na-isotopes.It should be noted that each colour band associated with a specific s 2n corresponds to the range of values for Sets 1, 2, and 3, and the same applies to Fig. 5.
that can reveal the partial wave content in the ground state of these nuclei would be very useful to tighten the constraints on theoretical models and enable us to distinguish between various wave functions that have been presented in this text.
In the lower panel of Fig 2 , we show the radius of the 38 Mg and 37 Na cores as a horizontal dotted black line which makes it easy to extract the relative change in matter radius of 40 Mg and 39 Na relative to the radius of the appropriate core, ∆R = R m − R core .With the current uncertainties in s 2n and the low-lying spectrum of subsystems, the difference of the matter radius of these nuclei and their cores ranges between 0.1-0.5 fm for different choices of potential sets and s 2n values.This number is notably smaller than those for well-established halo nuclei such as 6 He or 11 Li, but is close to the heaviest known two-neutron halo, 29 F, which is only 0.35 fm larger than the 27 F core [13,30,32].It is worth mentioning that, recently, we have shown that the halo formation in 29 F is connected to the weakening of the N = 20 shell gap, which leads to the intrusion of the p 3/2 orbital and reduced binding [30][31][32], which leads to halo formation.The same situation is observed when N = 28 melts, with the difference that the p 3/2 orbital is within the same major shell.As depicted in Fig. 2, our three-body findings reveal a substantial increase in ∆R for the least deeply bound state, with a substantial p 3/2 component.This suggests that our study provides evidence for a halo structure in the ground states of both 40 Mg and 39 Na.However, experimental cross section measurements are necessary to confirm this assertion.
In Fig. 3 we show the variation of matter radii for the Mg and Na isotopes with increasing mass numbers.The experimental derived values for the radii for 20−32 Mg and 20−32 Na (shown in black circles) are adopted from Ref.
[72], while those for 32−25 Mg are taken from Ref. [73] (shown in blue circles).Data for the one-neutron halo 37 Mg (shown as a red circle) is taken from Ref. [34].The dotted black lines in the figure correspond to a weighted fit of the experimental data points of the form R 0 A 1/3 for A = 24-37 for Mg isotopes and A = 23-32 for Na isotopes, respectively.It must be noted that the sharp rise of radius in 37 Mg compared to the nearest lighter isotope can be attributed to the fact that 37 Mg is itself a one-neutron halo nucleus [34].It can be easily seen from Fig. 3 that the radii of 40 Mg and 39 Na are larger than the standard R 0 A 1/3 fitted value for all choices of s 2n .The present calculations provide evidence of a modest halo in all instances, with a more pronounced effect observed as the occupation of the p 3/2 intruder orbital increases, akin to the case of 29 F. Thus, this observation implies a likely two-neutron halo structure in the ground state of 40 Mg and 39 Na, and the corresponding melting of the traditional N = 28 shell gap is due to the intrusion of the p 3/2 orbital.
We can gain a more in-depth understanding of the wave function by examining the probability densities in the Jacobi-T system, scaling x and y to the distance between the valence neutrons (r nn ) and that between the center of mass of the neutrons and the core (r c−nn ). Figure 4 illustrates density distributions for 40 Mg in the inverted scenario, considering both lower and upper limits of s 2n , which encourages halo formation in its ground state.The density distributions for 39 Na closely resemble those of 40 Mg and are consequently not shown.Both selections of the s 2n result in a peak where two neutrons are situated near each other at a distance from the core, the di-neutron peak.However, the latter choice leads to a more concentrated wave function, which is a consequence of the larger f -wave component (evidenced by the presence of four peaks in Fig. 4b).It is also evident that the valence neutrons explore shorter relative distances.These characteristics are reflected in the root-meansquare (rms) values of the coordinates, with values of 11.35 and 8.06 fm for r nn and r c−nn , respectively, in the shallow case, and 6.43 and 4.55 fm in the deeper case.
In summary, the results presented in this section provide evidence for the formation of halos in the ground state of both 40 Mg and 39 Na systems.The formation of these halos is attributed to the dissolution of the N = 28 shell gap, leading to the intrusion of the p 3/2 orbital and subsequent weak binding.(c) 39 Na+ 12 C at 1000 MeV/A

Total reaction cross section
Experimentally, a very obvious way to determine whether a nucleus is a halo nucleus, is to look for an enhanced reaction cross section.Thus we examine the total reaction cross section (σ R ) by employing the conventional Glauber theory [74].We utilize the nucleon-target formalism [75] and apply the nucleonnucleon profile function from Refs.[76][77][78].This approach has proven effective in various high-energy nucleus-nucleus collision reactions, particularly those with unstable nuclei, and it successfully replicated isotope dependence of the total reaction cross sections with appropriate density distributions [60, 79-84].
The additional theoretical inputs required for this reaction model include the density distributions of both the projectile and target nuclei.In the case of 40 Mg, we first generate a harmonic-oscillator (HO) type density distribution for 38 Mg that reproduces the measured total reaction cross section of 38 Mg+ 12 C by Takechi et al., [34] at 240MeV/nucleon (1535±21 mb) by assuming a simple shell model state (1 f 7/2 ) 8 ν ⊗ (1d 5/2 ) 4  π .The density for 40 Mg was constructed by simply adding the 2n densities calculated in the previous section to the 38 Mg density.No center-of-mass correction was applied in these calculations, but we believe this is a safe strategy for a system with such a heavy core.Using this prescription, we predict the σ R for 40 Mg at different incident energies, 240 MeV/nucleon (lies between 1601-1807 mb, shown in Fig. 5(a)) and 1000 MeV/A (lies between 1695-1944 mb, shown in the Fig. 5(b)).As can be seen from Fig. 5(a) and (b), the predicted values of σ R for 40 Mg show significant enhancement with respect to the observed σ R in the lower-A isotopes for both choices of energy.This shows that we can use these reaction cross sections as an indication of the melting of the N = 28 shell closure.
We use the same HO parameterization established for 38 Mg for 37 Na.Hence, we forecast the σ R for 39 Na at an energy of 1000 MeV/nucleon, as depicted in Fig. 5(c) (lies between 1710-1894 mb).Our computed estimates of σ R for 39 Na reveal a noteworthy increase compared to the observed σ R in its lower-mass isotopes.This reinforces our earlier finding in the 40 Mg, which points to the formation of a two-neutron halo in the ground state of 39 Na, leading to the disruption of the neutron shell at N = 28.

Conclusions
We have analysed the configuration mixing and matter radius of the ground states of the nuclei 40 Mg and 39 Na at the potential N = 28 shell closure.Since we are using an effective three-body model, we can study two-neutron halo formation.Unfortunately, there is very little information on the properties of 39 Mg, which we would need to constrain our models.We have instead analysed three scenarios that span the range of allowed parameters.We have then used the same parameters for both 40 Mg and 39 Na.We have thus mapped out the whole range of the expected s 2n values in 40 Mg and 39 Na.
Our results indicate a picture with mild to strong mixing of the intruder p 3/2 orbit with the normal f 7/2 orbit due to the magnitude of the core + n potentials in our three scenarios.This seems difficult to avoid in 39 Na, where most scenarios lead to a shell inversion.It seems only plausible for 40 Mg, where a normal ordering is also possible.The mixing enhances the dineutron configuration in the ground-state density.Due to the heavy masses of these nuclei, the computed matter radii vary only up to 5% for a fixed s 2n and considering the different choices of the core+n interaction.This makes it challenging to verify this experimentally, since one must make highly accurate mass measurements.On the theory side, we need a thorough understanding of the low-energy continuum spectrum of core+n subsystems.Additionally, the inclusion of transfer or knockout data, capable of probing the partial-wave characteristics, would be highly valuable in distinguishing between the scenarios used in this study.
The matter radii of 40 Mg and 39 Na show an increase of 0.1 to 0.5 fm compared to their respective cores 38 Mg and 37 Na, lending support to the idea of a potential halo structure in their ground state.This assertion gains further backing from the elevated predicted reaction cross sections for these nuclei in comparison to the lower mass isotopes.Nevertheless, it is imperative to verify this conclusion through experimental measurements of interaction cross sections.Future perspectives include incorporating core deformation within the effective three-body description explicitly, to assess its influence on the halo structure in this region of the nuclear chart.

Figure 1 :
Figure1: (upper panel) A selected area of the nuclear Segrè chart for the 9 ≤ Z ≤ 14 isotopes.The green and orange blocks correspond to neutron numbers N = 20 and 28, respectively.(lower panel) A sketch of the scenarios for the single-particle levels that lead to shell inversion.On the left, we show the shell gaps for stable nuclei (normal shell model), and on the right, we show the active shells at N = 20 and N = 28 inside the coloured blocks.A change in the order of the shells leads to shell inversion.

Figure 2 :
Figure2: (upper panel) The occupancy of the ( f 7/2 ) 2 (solid lines) and (p 3/2 ) 2 orbits (dashed lines) for different parameter sets in 40 Mg (left) and 39 Na (right), respectively, as a function of s 2n .(lower panel) Variation of matter radii (R m ) for40 Mg and39 Na with s 2n .The black dotted line shows the core radius,38 Mg and37 Na, respectively.It should be noted that the labels Inverted, Degenerate and Normal pictures here refer to the outcome of full three-body calculations and should not be confused with the single-particle scenarios of Table1.

Figure 3 :
Figure 3: The Variation of matter radii (R m ) for Mg (upper panel) and Na (lower panel) isotopes with mass number A. Experimental values, shown in black, blue, and red circles, were taken from Refs.[72],[73], and[34], respectively.The dotted black line corresponds to the R 0 A 1/3 fit for A = 24-37 for Mgisotopes and A = 23-32 for Na-isotopes.It should be noted that each colour band associated with a specific s 2n corresponds to the range of values for Sets 1, 2, and 3, and the same applies to Fig.5.

Figure 4 :
Figure 4: The ground state probability density distribution (in units of fm −2 ) for 40 Mg for Set-3 (a) with s 2n = 0.010 MeV and (b) with s 2n = 1.380MeV.The contour lines in (b) show the shape of the peak above a probability of 0.04, which is twice as high as that in (a).

Figure 5 :
Figure 5: Measured and calculated total reaction cross section for different isotopes of Mg and Na at different energies.Theoretical results (shown with green, magenta, and indigo bars) for 40 Mg+ 12 C are shown at (a) 240 MeV/A and (b) 1000 MeV/A.Experimental data shown with red, blue, and black dots are taken from[34] (at 240 MeV/A),[73] (at 900 MeV/A), and [72] (at 950 MeV/A), respectively.(c) Shows the results for39 Na+ 12 C at 1000 MeV/A and experimental data shown with black dots is taken from [72] (at 950 MeV/A).The dotted black line corresponds to a σ 0 A 2/3 fit to guide the eye of the reader.

Table 1 :
Parameter sets for the