Ergash Maxkamovich Tursunov

An effective operator is exactly equivalent to the long-wavelength form of the $M1$ operator in transition matrix elements. It allows us to analytically and numerically analyze the $M1$ contribution to the $^{4}\mathrm{He}{(d,\ensuremath{\gamma})}^{6}\mathrm{Li}$ reaction. Isoscalar $M1$ transitions from an initial $S$ wave are shown to be forbidden in radiative capture reactions when distortion is neglected in the initial state. An illustrative calculation in a three-body model with proton, neutron, and a structureless $\ensuremath{\alpha}$ interacting through effective forces leads to a negligible $M1 S$ factor at small energies. The dominant $M1$ contribution comes from transitions from an initial $S$ wave to isospin 1 components of the $^{6}\mathrm{Li}$ ground state. The importance of the orthogonality between initial and final states is emphasized. It is suggested that using this effective $M1$ operator in other models should clarify the origin of large discrepancies between $M1 S$ factors appearing in the literature.

An effective operator is exactly equivalent to the long-wavelength form of the $M1$ operator in transition matrix elements. It allows us to analytically and numerically analyze the $M1$ contribution to the $α(d,γ)^6$Li reaction. Isoscalar $M1$ transitions from an initial $S$ wave are shown to be forbidden in radiative capture reactions when distortion is neglected in the initial state. A calculation in a three-body model with proton, neutron, and a structureless $α$ interacting through effective forces leads to a negligible $M1$ $S$-factor at small energies. The dominant $M1$ contribution comes from transitions from an initial $S$ wave to isospin 1 components of the $^6$Li ground state. It is suggested that using this effective $M1$ operator in other models should clarify the origin of large discrepancies between $M1$ $S$-factors appearing in the literature.

The astrophysical direct nuclear capture reaction $^{12}{\rm C}(p, γ)^{13}{\rm N}$ is studied within the framework of a potential model. Parameters of the nuclear $p-^{12}$C interaction potentials of the Woods-Saxon form are adjusted to reproduce experimental $p-^{12}$C scattering phase shifts, as well as the binding energies and empirical values of the asymptotic normalization coefficient (ANC) for the $^{13}$N(1/2$^-$) ground state from the literature. The reaction rates are found to be very sensitive to the description of the value of the ANC of the $^{13}$N($1/2^{-}$) ground state and width of the $^{13}$N($1/2^+$) resonance at the $E_x=2.365$ MeV excitation energy. The potential model, which yields the ANC value of 1.63 fm$^{-1/2}$ for the $^{13}$N($1/2^{-}$) ground state and a value $Γ$=39 keV for the $^{13}$N($1/2^+$) resonance width, is able to reproduce the astrophysical $S$ factor in the energy interval up to 2 MeV, the empirical values of the reaction rates in the temperature region up to $T=10^{10}$ K of the LUNA Collaboration and the results of the R-matrix fit. The astrophysical factor $S(0)=1.35$ keV b is found using the asymptotic expansion method of D. Baye. The obtained value is in a good agreement with the Solar Fusion II result. At the same time, the calculated value of 1.44 keV b of the astrophysical $S$ factor at the Solar Gamow energy is consistent with the result of the R-matrix fit of $S(25~\rm{keV})=1.48 \pm 0.09$ keV b by Kettner {\it et al.}, but slightly less than the result of $S(25~\rm{keV})=1.53 \pm 0.06$ keV b the LUNA Collaboration.

The astrophysical $S$ factor and reaction rates of the direct capture process ${}^6\mathrm{Li}(p,\gamma ){}^7\mathrm{Be}$ are estimated within a two-body single-channel potential model approach. Nuclear potentials of the Gaussian form in the ${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$ waves are adjusted to reproduce the binding energies and the empirical values of the asymptotic normalization coefficients (ANC) for the ${}^7\mathrm{Be}(3/2^{-})$ ground and ${}^7\mathrm{Be}(1/2^{-})$ excited bound states, respectively. The parameters of the potential in the most important ${}^{2}S_{1/2}$ scattering channel were fitted to reproduce the empirical phase shifts from the literature and the low-energy astrophysical $S$ factor of the LUNA Collaboration. The obtained results for the astrophysical $S$ factor and the reaction rates are in very good agreement with available experimental data sets. The numerical estimates reproduce not only the absolute values, but also the energy dependence of the $S$ factor and the temperature dependence of the reaction rates of the LUNA Collaboration. The estimated ${}^7\mathrm{Li}/\mathrm{H}$ primordial abundance ratio of $(4.67\pm 0.04)\times {10}^{-10}$ is consistent with recent big bang nucleosynthesis result of $(4.72\pm 0.72)\times {10}^{-10}$ after the Planck telescope observation.

The astrophysical $S$ factor and reaction rates of the direct capture process $^{6}$Li(p,$γ)^{7}$Be are estimated within a two-body single-channel potential model approach. Central potentials of the Gaussian-form in the $^2P_{3/2}$ and $^2P_{1/2}$ waves are adjusted to reproduce the binding energies and the empirical values of the asymptotic normalization coefficients (ANC) for the $^7$Be(3/2$^-$) ground and $^7$Be(1/2$^-$) excited bound states, respectively. The parameters of the potential in the most important $^2S_{1/2}$ scattering channel were fitted to reproduce the empirical phase shifts from the literature and the low-energy astrophysical $S$ factor of the LUNA collaboration. The obtained results for the astrophysical $S$ factor and the reaction rates are in a very good agreement with available experimental data sets. The numerical estimates reproduce not only the absolute values, but also the energy and temperature dependence of the $S$ factor and reaction rates of the LUNA collaboration, respectively. The estimated $^{7}{\rm Li/H}$ primordial abundance ratio $(4.67\pm 0.04 )\times 10^{-10}$ is well consistent with recent BBN result of $(4.72\pm 0.72) \times 10^{-10}$ after the Planck observation.

The spatial structure of the lowest 0\({}_{1}^{+}\), 0\({}_{2}^{+}\), 2\({}_{1}^{+}\) and 2\({}_{2}^{+}\) states of the \({}^{12}\)C nucleus is studied within the 3\(\alpha\) model with the Buck, Friedrich and Wheatley \(\alpha\alpha\) potential with Pauli forbidden states in the \(S\) and \(D\) waves. The Pauli forbidden states in the three-body system are treated by the exact orthogonalization method. The largest contributions to the ground and excited 2\({}_{1}^{+}\) bound states energies come from the partial waves \((\lambda,\ell)=(2,2)\) and \((\lambda,\ell)=(4,4)\). In contrast to the bound states, for the Hoyle resonance 0\({}_{2}^{+}\) and its analog state 2\({}_{2}^{+}\), dominant contributions come from the \((\lambda,\ell)=(0,0)\) and \((\lambda,\ell)=(2,2)\) configurations, respectively. The estimated probability density functions for the \({}^{12}\)C(0\({}_{1}^{+}\)) ground and 2\({}_{1}^{+}\) excited bound states show mostly a triangular structure, where the \(\alpha\) particles move at a distance of about 2.5 fm from each other. However, the spatial structure of the Hoyle resonance and its analog state have a strongly different structure, like \({}^{8}\textrm{Be}+\alpha\). In the Hoyle state the last \(\alpha\) particle moves far from the doublet at the distance between \(R=3.0\) fm and \(R=5.0\) fm. In the Hoyle analog 2\({}_{2}^{+}\) state the two alpha particles move at a distance of about 15 fm, but the last \(\alpha\) particle can move far from the doublet at the distanse up to \(R=30.0\) fm.

Lowest energy spectrum of the \({}^{12}\)C nucleus is analyzed in the 3\(\alpha\)-cluster model with a deep \(\alpha\alpha\) potential of Buck, Friedrich and Wheatley with Pauli forbidden states in the \(S\) and \(D\) waves. The direct orthogonalization method is applied for the elimination of the 3\(\alpha\) Pauli forbidden states. The effects of possible first order quantum phase transition are shown in the lowest \({}^{12}\)C(\(0_{1}^{+}\)) and \({}^{12}\)C(\(2_{1}^{+}\)) states from weakly bound phase to a deep phase. The ground and lowest 2\({}^{+}\) states of the \({}^{12}\)C nucleus in the deep phase are created by the critical eigen states of the Pauli projector for the 0\({}^{+}\) and 2\({}^{+}\) three-alpha functional spaces, respectively.