**1. Introduction**

Radioactive nuclides synthesized during nuclear processes make it possible to probe active regions of nuclear reactions in respective sites, cf., e.g., [1–8]. For example, the radioactive decay of iron group isotopes (44Ti, 56Co, 57Co) isthe most plausible source of energy [5], which feeds infrared, optical, and ultraviolet radiation insupernova (SN) remnants. The contribution of44Ti dominates for SNe older than three or four years, until an interaction of ejecta with the surrounding matter increases and becomesthe dominant source. Accordingly, the light curves and spectra of infrared and ultraviolet radiation were analyzed using complex and model-dependent computer simulations [5]. An estimate ofthe initial mass of 44Ti in SNR 1987A was made, i.e.,(1 − <sup>2</sup>)· 10−<sup>4</sup> · *M*solar (in solar masses). This value significantly exceeds model predictions (see [9] and below). Neutron star mergers are another plausible source [10,11] of nucleosynthetic components of r-process nuclide enrichment in galactic chemical evolution.

Radioisotopes synthesized in SN explosions can be observed directly by recording the characteristic gamma lines accompanying their decay [1,9]. The radioactive decay chain 44Ti → 44Sc → 44 Ca leads to the emission of lines with energies of 67.9 keV and 78.4 keV (from 44Sc) and 1157 keV (from 44Ca) of approximately the same intensity. The half-life of 44Ti, i.e., about 60 years under Earth conditions, allows us to estimate the mass of this isotope in the remnant. The obtained observational values for the total mass of 44Ti nuclides synthesized in SN explosions significantly exceed model predictions, showing a mass of initially synthesized nuclides of 44Ti at *M*Ti ∼ 10−<sup>5</sup> · *M*solar in the absence of magnetic effects. These predictions are consistent with observational data of SN-type I, see [6] and refs. therein. Consideration of specific SN explosion scenarios leads, in some cases, c.f., e.g., [12,13], to mass values approaching those in the observational data.

Superstrong magnetic fields exceeding *teratesla* (TT, 1 TT = 10<sup>16</sup> G) arise in SN explosions [1,2], neutron star mergers [3], heavy ion collisions [4], and magnetar crusts [7], in conjunction with observations of soft-gamma repeaters and abnormal X-ray pulsars. The nuclides formed in such processes contain information about the structure of matter and the mechanisms of explosive processes. In this contribution, we analyze an e ffect of a relatively weak magnetic field on nuclear structure, and discuss the possibility of using radionuclides to probe the internal regions of these processes. The next section briefly describes the used methods of nuclear statistical equilibrium for the description and analysis of nucleosynthesis. Section 3 considers changes in the structure and properties of atomic nuclei due to Zeeman splitting of energy levels of nucleons. It is shown that such a mechanism dominates with a magnetic field strength range of 0.1–10 TT, and results in a linear nuclear magnetic response which is in agreemen<sup>t</sup> with calculations made using covariant density functional theory, cf. [14,15]. Magnetic susceptibility is a key quantity for the description and analysis of nuclear magnetization. The influence of magnetic fields on the composition of nuclei is considered in Sections 3.2 and 3.3. Conclusions are presented in Section 4.

#### **2. Abundance of Atomic Nuclei at Statistical Equilibrium**

Nuclear statistical equilibrium (NSE) approximation has been used very successfully to describe the abundance of iron group nuclei and nearby nuclides for more than half a century. Under NSE conditions, the yield of nuclides is mainly determined by the binding energy of the resulting atomic nuclei. The magnetic e ffects in NSE were considered in [1,2,8]. Recall that at temperatures ( *T* ≤ 109.5 K) and field strengths ( *H* ≥ 0.1 TT), the dependence on the magnetic field of the relative yield *y* = *Y*(*H*)/*Y*(0) is determined mainly by a change of nuclear binding energy, Δ*B*, in a field, and can be written in the following form:

$$y = \exp\left\{\Delta B / kT\right\},\tag{1}$$

The binding energy of a nucleusis givenin theform of the energy difference between noninteractingfree nucleons*E*N and the nucleus consisting of them, i.e.,*EA*, *B* = *E*N − *EA*. Under conditions of thermodynamic equilibrium at temperature *T*, the corresponding energy is expressed as follows:

$$E = \frac{kT^2}{\Sigma} \frac{\partial \Sigma}{\partial T} \tag{2}$$

in terms of a partition function Σ = - *i* exp{−*ei*/*kT*}, where *ei* denotes theenergy of nuclear particles in an *i*-state and *k* is the Boltzmann constant. Using Equation (2) for free nucleons, the energy component due to an interaction with a magnetic field can be written in the following form: *E*α = −*g*<sup>α</sup> 2 <sup>ω</sup>Lth(*g*α<sup>ω</sup>L/2*kT*), where th(*x*) is the hyperbolic tangent and the Larmor frequency ωL = μN *H*. Here, the well-known [16] spin *g*− factors *g*p ≈ 5.586 and *g*n ≈ −3.826 for protons α = p and neutrons α = n. For values of temperature ( *T* ∼ 109.5 K) and field strengths ( *H* ∼ 1 TT), here, one gets *E*α ∼ −100.5 keV.

#### **3. Synthesis of Ultramagnetized Atomic Nuclei**

The Zeeman—Paschen—Back e ffect is associated with a shift of nucleon energy levels due to an interaction of nucleon magnetic moments with a field. Dramatic change in nuclear structure occurs under conditions of nuclear level crossing [2,8]. The characteristic energy interval Δε ∼ 1 MeV determines the scale of a field strength, i.e., Δ *H*cross ∼ Δε/μN ∼ 101.5 TT, at which nonlinear e ffects dominate. Here, μN stands for nuclear magneton. In case of a small field strength, i.e., *H* 101.5 TT, a linear approximation can be used. At field intensities *H* ≥ 0.1TT, one can neglect the residual interaction [8]. Under such conditions, the total value of a nucleon spin quantum number on a subshell (and a nucleus) is the maximum possible, similar to the Hund rule, which is well known for the electrons of atoms.

#### *3.1. Zeeman Energy in Atomic Nuclei*

The self-consistent mean field is a widely used approach for obtaining realistic descriptions and analyses of the properties of atomic nuclei. The single-particle (sp) Hamiltonian Hˆ α for nuclei in a relatively weak magnetic field *H* within the linear approximation can be written as

$$\mathbf{H}\_{\alpha} = \mathbf{H}\_{\alpha}{}^{0} - \left(\mathbf{g}\_{\alpha}{}^{\alpha}\hat{\mathbf{I}} + \mathbf{g}\_{\alpha}\hat{\mathbf{s}}\right)\omega\_{\mathbf{L}} \tag{3}$$

for protons α = p and neutrons α = n. Here, Hˆ 0 α represents the sp Hamiltonian for isolated nuclei, while the orbital moment and spin operators are denoted by ˆ **l** and **s**ˆ, respectively. The interaction of dipole nucleon magnetic moments with a field is represented by terms containing the vector ωL = μN *H*, and *g*o α denotes orbital *g*− factors *g*o p = 1 and *g*o n = 0.

Thus, the binding energy decreases for magic nuclei with a closed shell, zero magnetic moment and, therefore, zero interaction energy with a magnetic field. In cases of antimagic nuclei with open shells, a significant (maximum possible under these conditions) magnetic moment leads to an additional increase in the binding energy *B* in a field. In this case, the leading component of such a magnetic contribution is represented by the sum over the filled *i* sp energy levels ε*i*, *Bm* = - *i*−*occ* ε*i*, see [8]. In the representation of angular momentum for spherical nuclei, the sp states |*i* > are conveniently characterized by quantum numbers (see [16]): *n*-radial quantum number, angular momentum *l*, total spin *j*, and spin projection on the direction of the magnetic field *mj*. Then, using sp energies <sup>ε</sup>*nljmj* and wave functions *nljmj* >, the magnetic energy change Δ*Bm* = *Bm*(*H*) − *Bm*(0) in a field *H* can be written as 

$$\begin{aligned} \Delta B^{m}\_{\alpha} &= \kappa\_{\alpha} \omega\_{\mathbf{L},\prime} \kappa\_{\alpha} = \sum\_{i-\text{occ}} \kappa^{i}\_{\alpha\prime} \\ \kappa^{i}\_{\alpha} &= \sum\_{m,s} | \widehat{\ell}^{2}(\mathcal{g}^{o}\_{\alpha}m + \mathcal{g}\_{\alpha}s) \\ &= \begin{cases} (\mathcal{g}^{o}\_{\alpha}l + \mathcal{g}\_{\alpha}/2)m\_{j}/j, & \text{for } j = l + 1/2, \\ (\mathcal{g}^{o}\_{\alpha}(l+1) - \mathcal{g}\_{\alpha}/2)m\_{j}/(j+1), & \text{for } j = l - 1/2, \end{cases} \end{aligned} \tag{4}$$

where (α = n, p), < *lm*, 1 2 *s jmj* > is the Clebsh-Gordan coe fficient. The result from Equation (4) is similar to that obtained in the Schmidt model [16]. We stress here that in this case, parameter κα is given by the combined susceptibility of all the independent nucleons spatially confined due to a mean field. The linear response regime at magnetic induction *H* < 10 TT is also confirmed by consideration within the covariant density functional theory, cf. [14,15]. The present analysis in terms of magnetic susceptibility yields transparent and clear results for nuclear magnetic reactivity with fundamental consequences for the study of nuclear structure and dynamics in strong magnetic fields.
