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In this paper, we review the theoretical analyses and simulations of the interactions between curvature-inducing proteins and biomembranes. Laterally isotropic proteins induce spherical budding, whereas anisotropic proteins such as Bin/Amphiphysin/Rvs (BAR) superfamily proteins, induce tubulation. Both types of proteins can sense the membrane curvature. We describe the theoretical analyzes of various transitions of protein binding accompanied by a change in various properties, such as the number of buds, the radius of membrane tubes, and the nematic order of anisotropic proteins. Moreover, we explain the membrane-mediated interactions and protein assembly. Many types of membrane shape transformations (spontaneous tubulation, formation of polyhedral vesicles, polygonal tubes, periodic bumps, network structures, etc.) have been demonstrated by coarse-grained simulations. Furthermore, traveling waves and turing patterns under the coupling of reaction-diffusion dynamics and membrane deformation are described.

Thermal and rotational behaviors of neutron rich fp-shell isotopes such as ${}_{26}^{56}\mathrm{\text{Fe}}$, ${}_{26}^{58}\mathrm{\text{Fe}}$ and ${}_{26}^{60}\mathrm{\text{Fe}}$ were analyzed microscopically within the framework of statistical theory of hot rotating nuclei (STHRN) for the angular momentum range (0–15)$\hslash$ at excitation energy above 4MeV. Pair-breaking phenomenon and band-crossing phenomenon of Fe isotopes were discussed with and without the inclusion of BCS pairing. The STHRN method with BCS effect was extended to the Fe isotopes to determine the critical temperature $(T_C)$, where the pair-breaking phenomenon takes place and it is found to occur below $T=1.0$MeV. The observation of band-crossing phenomenon above the $T_C$ was explained without considering the effect of BCS pairing. The single particle energy levels were engendered from triaxially deformed Nilsson Hamiltonian. The outcomes of the present investigation on moment of inertia (MOI), back-bending phenomenon, spin cutoff parameter show a strong evidence of band-crossing phenomenon and it eventually led to a shape transition from spherical to noncollective oblate. Moreover, attention on separation energy of protons and neutrons reveals that the neutron pairs are responsible for band-crossing. STHRN method was able to reproduce results in good agreement with experiments and comparable with other theories such as projected shellmodel (PSM) and shellmodel Monte Carlo (SMMC) method.

The ground state shapes and properties of even–even ${}^{98-132}$Cd isotopes have been studied in the framework of relativistic Hartree–Bogoliubov model with two effective interactions that are density-dependent meson exchange and point coupling. The theoretical results on the binding energies, charge radii and two neutron separation energies are presented in this paper in comparison to experimental data and a reasonable agreement is found between the two. Both the interactions predict spherical to axially prolate, prolate to triaxial, triaxial to prolate and prolate to spherical transitions in cadmium isotopic mass chain. In addition, the present calculations confirm the existence of shape coexistence observed in ${}^{110,112}$Cd. The comparison of calculated ground state properties with the results based on PC-PK1 functional shows that PC-PK1 employed in the deformed relativistic Hartree–Bogoliubov theory in continuum has the lowest root mean square deviation for average binding energies and two neutron separation energies.

The shape evolution and potential energy surfaces of even–even ${}^{220-240}$Ra are investigated in the $({\beta }_2,{\beta }_3)$ plane by the covariant density functional theory. For octupole-deformed ${}^{222}$Ra, the free energy surfaces, the deformations, the pairing gaps, the excitation energy as well as the specific heat with increasing temperature are analyzed. Based on the specific heat, ${}^{222}$Ra exhibits three distinct discontinuities as the temperature rises. The first pairing transition occurs at $T=0.45$MeV, while the second one happens at $T=0.65$MeV when octupole deformation disappears, and the third one takes place at $T=0.80$MeV as quadrupole deformation approaches zero. The gaps at $N=132,136$ and $Z=88$ in the single-particle levels are responsible for the octupole global minima. The shape transitions not only occur between even–even ${}^{220-240}$Ra, but also occur with increasing temperature for ${}^{222}$Ra.

Si_n clusters in the size range n = 4–30 have been investigated using a combination of global structure optimization methods with DFT and ab initio calculations. One of the central aims is to provide explanations for the structural transition from prolate to spherical outer shapes at about n = 25, as observed in ion mobility measurements. Firstly, several existing empirical potentials for silicon and a newly generated variant of one of them were better adapted to small silicon clusters, by global optimization of their parameters. The best resulting empirical potentials were then employed in global cluster structure optimizations. The most promising structures from this stage were relaxed further at the DFT level with the hybrid B3LYP functional. For the resulting structures, single point energies have been calculated at the LMP2 level with a reasonable medium-sized basis set, cc-pVTZ. These DFT and LMP2 calculations were also carried out for the best structures proposed in the literature, including the most recent ones, to obtain the currently best and most complete overall picture of the structural preferences of silicon clusters. In agreement with recent findings, results obtained at the DFT level do support the shape transition from prolate to spherical structures, beginning with Si₂₆ (albeit not completely without problems). In stark contrast, at the LMP2 level, the dominance of spherical structures after the transition region could not be confirmed. Instead, just as below the transition region, prolate isomers are obtained as the lowest-energy structures for n ≤ 29. We conclude that higher (probably multireference) levels of theoretical treatments are needed before the puzzle of the silicon cluster shape transition at n = 25 can safely be considered as explained.

The low-lying structure of ¹³²Xe and ¹³⁴Xe has been studied using the (n, n′ γ) reaction at the University of Kentucky 7-MV Van de Graaff accelerator facility. Gamma-ray excitation function and angular distribution measurements were performed. Lifetimes were measured using the Doppler-shift attenuation method, and transition probabilities were obtained. Previous assignments of mixed-symmetry states in each nucleus are supported. Also, a tentative J^π = 0⁺ state is supported for ¹³⁴Xe, and a new one is proposed for ¹³⁴Xe.

Octupole instabilities in even-even nuclei near nucleus have been investigated by means of improved total-Routhian-surface calculations with the inclusion of reflection-asymmetric deformations. It is found that the shape evolutions of mirror nuclei are similar between Z = 56 isotopes and N = 56 isotones in the ground-states. As an example, , which has reflection-symmetric shape at low spins, is predicted to be octupole deformed at certain high rotational frequencies. The prediction is confirmed by experiment.

The shape evolution and potential energy surfaces of even{even ^220-240Ra are investigated in the (β₂,β₃) plane by the covariant density functional theory. For octupole-deformed ²²²Ra, the free energy surfaces, the deformations, the pairing gaps, the excitation energy as well as the specific heat with increasing temperature are analyzed. Based on the specific heat, ²²²Ra exhibits three distinct discontinuities as the temperature rises. The first pairing transition occurs at T = 0.45 MeV, while the second one happens at T = 0.65 MeV when octupole deformation disappears, and the third one takes place at T = 0.80 MeV as quadrupole deformation approaches zero. The gaps at N = 132; 136 and Z = 88 in the single-particle levels are responsible for the octupole global minima. The shape transitions not only occur between even–even ^220-240Ra, but also occur with increasing temperature for ²²²Ra.