Magic nuclei at explosive dynamo activity

Explosive nucleosynthesis at conditions of magnetorotational instabilities is considered for iron group nuclides by employing arguments of nuclear statistical equilibrium. Effects of ultra-strong nuclear magnetization are demonstrated to enhance the portion of titanium product. The results are corroborated with an excess of 44Ti revealed from the Integral mission data.


Introduction
Energetic nuclear processes, e.g., core-collapse supernovae (SNe) [1,2], heavy ion collisions [3] give rise to ultrastrong magnetic fields exceeding teratesla (TT).Magnetorotational instabilities (MRI) and/or dynamo action represent, plausibly, main respective mechanism.Then maximum magnetic field intensity at distances smaller than a radius r0 from a center of the vortex dynamo process can be taken as a constant H0.Strong magnetization of the MRI central region stabilizes magnetic fluxes Ф0 also for field decaying components.Respectively, in conditions of a constant flux at radius r > r0 the dependence of strength H on distance r can be represented as H = Ф0/Sr 2 .Comparable values for magnetic pressure gradients and gravitational force, i.e., dH 2 (r)/dr = 4H0 2 /b 2 ra ~ 8S GM n(R)/R 2 , determine the radius ra relative to the MRI center corresponding to material irruption.Here, the gravitational constant G, the star mass M inside the bifurcation radius R is related to the matter density n(R) as 4SR 2 n(R) = -dM/dR.Additional constraints on MRI parameters are given by shock wave energy Es (see [1]) where L is total length of MRI areas, b=(ra/r0) 2 .For typical SNe Type II values R ~ 30 km, Es ~ 10 51.5 ergs at H0 ~ 3 TT one obtains r0 ~ 10 -0.5 km and b ~ 10.Nuclides produced in so strong fields (i.e., larger than 0.1 TT) contain an information on matter structure and explosion mechanisms (see [1,2,[4][5][6] and refs.therein).In this contribution we analyze possibilities for using radionuclides to probe internal regions of respective sites.

Synthesis of ultramagnetized atomic nuclei
Abundances of iron group and nearby nuclides are described very successfully within nuclear statistical equilibrium (NSE) approach for over half a century, cf.[1,2,6,7], and for broader discussion of statistical models see [8] and refs.therein.At such conditions a nuclide abundance is determined mainly by its binding energy.The magnetic effects on the NSE were considered in [1,2,6] and refs.therein.Recall that at temperatures (ܶ ≤ 10 ଽ.ହ K) and field strengths (H ≥ 0.1 TT), the magnetic field dependence of relative output value ‫ݕ‬ = ‫)0(ܻ/)ܪ(ܻ‬ is determined by a change in the binding energy of nuclei in a mgnetic field and can be written in the following form : It is worthy to remind that NSE corresponds to a balance of strong and electromagnetic reactions with their inverse and meets β-equilibrium conditions with an equal number of protons and neutrons (i.e.Ye = 0.5).In this situation predominantly symmetric nuclei (N=Z) are produced.Therefore, we consider as examples 56 Ni and 44 Ti.The choice of symmetric nuclei, double-magic and anti-magic at vanishing magnetization, gives a clear picture of magnetic effects in the formation of chemical elements and leads to fundamental conclusions about transmutation and synthesis of nuclei at ultra-strong magnetization.

Structure of magnetized nuclei
Within the mean field approximation the properties of nuclei are determined by single-particle energy levels filled up to the Fermi energy ‫ܧ‬ ி , see [9].The binding energy B can be written as ‫ܤ‬ = ‫ܤ‬ LDM + ‫ܥ‬ + ‫ܥ‬ , where ‫ܥ‬ ୧ are the shell corrections for neutrons and protons, and the component ‫ܤ‬ LDM is calculated in semiclassical liquid drop model and varies only slightly in the magnetic field, according to the Bohr-van Leuven theorem, see [1,2].
Spin magnetization of Pauli type dominates for the neutron magnetic reactivity.Interaction of a field and the spin-magnetic moment corresponding to a spin projection EPJ Web of Conferences ݉ on a field vector gives rise to a linear shift of level energy Δ = ݉ ݃ ߱ , where ߱ = ߤ ‫ܪ‬ with nucleon magneton ߤ , and ݃ -neutron ݃-factor.Accordingly, the shell energy in a field H is modified as follows where the indices + and -indicate a sign of the projection of spin magnetic moment on the field direction.This leads to a phase shift in dependence of the shell energy on number of neutrons N [1,2].The proton magnetic response is represented by a superposition of the field interaction with spin and orbital magnetic moments.Great success in an understanding of many properties of stable nuclei of mass numbers A~ 10 -100 is associated with the Nilsson model (see, e.g., [9]) where the nuclear mean field is approximated by the harmonic oscillator with frequency ߱ ≈ ‫ܣ/14‬ ଵ/ଷ MeV and the spin-orbit interaction, which leads to a splitting of energy levels ߟ ௦ ≈ 0.12 (in units ߱ ).Magnetic field dependence of the total shell effect for neutron and proton contributions is shown in Fig. 1.We recall here similar consideration within the covariant density functional theory [10], which includes the interaction of the total magnetic moment of a nucleus with magnetic field, as well.As is evident from Fig. 1, at values h < 0.07, i.e., field strengths H < 20 TT, the binding energy shows nearly linear H dependence for considered nuclei, ‫ܤ‬ = ‫ܤ‬ + ݇ ‫ܪ‬ (MeV) with magnetic susceptibility parameters ݇ depending on a nucleus ܼ ே .For 44 Ti the value of this parameter is positive k(Ti) ~ 0.3 MeV/TT, whereas for 56 Ni it becomes negative k(Ni) ~ -0.3 MeV/TT.Evidently, for the anti-magic at zero field strength nucleus the shell energy always increases with the field H, and for the magic onedecreases, indicating positive and negative values of magnetic susceptibility кi , respectively.

MRI explosive nucleosynthesis
Let us consider an average relative yield of nucleosynthesis products over MRI region V (see Sec. 1) ‫〉ݕ〈‬ = ܸ ିଵ න ݀ ଷ ‫ݎ‬ ‫))(ܪ(ݕ‬ Then using linear approximation for the binding energy (sect.2.1), in a field H the average relative yield is written as follows: Where ܽ = ݇ ‫ܪ‬ /݇ܶ and the integral exponential function ^èxp Ei( ) In Fig. 2, one sees significant difference for magnetic field dependence of nuclide output, magic and anti-magic at vanishing field.For anti-magic nuclei and, therefore, increasing binding energy with the increase of the field strength (or positive magnetic susceptibility кi) relative volume of nucleosynthesis increases significantly with increasing a.At the same time, the relative production of magic nuclides, i.e., negative value a, is not substantially changed with increasing field.This behavior significantly differs from the case of a spatially uniform magnetization, see Fig. 2, which corresponds to the exponential dependence of ‫〉ݕ〈‬ or b = 1 in Eq. ( 3).In this case the coefficients of suppression and enhancement are the same at the same absolute value of a.The presence of a diffusion layer corresponding to fade-out field strength with increasing r (or b > 1) in a real MRI region leads to substantial differences of relevant factors.Significant increase in a synthesis of anti-magic nuclei is accompanied by a slight change in mass volume of magic nuclides.Model predictions in an absence of magnetic effects (see Ref. [7]) give the mass of initially synthesized 44 Ti, M(Ti) ~ 10 ି M ⨀ (in solar masses M ⨀ ).For realistic characteristics of Type II SNe explosion (see Sec. 1) enhancement factor 〈‫(〉ݕ‬Ti)∼ 30 -300 corresponds to a mass M(Ti) ~ 10 -3.5 -10 -2.5 M ⨀ .

10006-p.2 3 Probing SN by radionuclides
As was shown above magnetic effects on the nuclear binding energy lead to an increase of the Ti portion in the explosive nucleosynthesis.Consequently, the characteristic lines of respective nuclei in spectra of astrophysical objects are considerably enhanced and become noticeable allowing for an analysis of synthesized elements.The radioactive decay chain 44 Ti → 44 Sc → 44 Ca gives rise to emission of lines with energies of 67.9 keV and 78.4 keV (from 44 Sc*) and 1157 keV (from 44 Ca*) of approximately equal intensity.The respective image-mosaics from the Integral data (e.g., [1,2]) for the Cassiopeia region in various energy ranges of registered photons are presented in Fig. 3.The color (brightness) is proportional to the gamma-quanta flux: as larger the flux as lighter (brighter) the color of a pixel.As seen, the SNR CAS A gives the brightest spot for energies matching the 44 Sc lines.
Then the 44 Ti half-life, about 60 years, allows to determine this isotope initial mass in SN remnants.Table 1 shows the observational results for the total mass of 44 Ti nuclide synthesized in SNe explosions [6].These values are significantly larger as compared to the model predictions in absence of magnetic effects (see Sec. 2).However, magnetic enhancement exceeds noticeably observational data.It is worthy to notice that not all the material ejected from the central part of a star is formed in MRI areas (see [1,2]).Accordingly, consideration of magnetic effects in nucleosynthesis when accounting for the realistic MRI structures and geometry can provide consistent understanding of SNe explosion mechanisms in detail.

Conclusions
Nuclide abundances at conditions of the nuclear statistical equilibrium are investigated for developed explosive magneto-rotational instabilities and, respectively, the ultra-magnetized matter.For iron group nuclei the magnetic modification of nuclear structure shifts a maximum of the nucleosynthesis products towards smaller mass numbers approaching titanium.Magnetic effects in a nuclide production are favorable compared to observational Integral data and sensitive to the field geometry.Thermal effects can also affect structural and dynamical properties of atomic nuclei at high temperatures, see [11].Finally, we notice that similar magnetic effects in atomic clusters [12] will also lead to a shift of electronic magic numbers and change the mass distribution.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 201

Fig. 1 .
Fig. 1.Dependence of nuclear binding energy on magnetic field strength indicated by the parameter h = ωL/ω0.Results for 56 Ni -solid line and for 44 Ti -dotted line.

Fig. 2 .
Fig. 2. Relative yield of nucleosynthesis products depending on parameters a and b.