Quantum phase transitions between competing equilibrium shapes of nuclei with an odd number of nucleons are explored using a microscopic framework of nuclear energy density functionals and a fermion-boson coupling model. The boson Hamiltonian for the even-even core nucleus, as well as the spherical single-particle energies and occupation probabilities of unpaired nucleons, are completely determined by a constrained self-consistent mean-field calculation for a specific choice of the energy density functional and pairing interaction. Only the strength parameters of the particle-core coupling have to be adjusted to reproduce a few empirical low-energy spectroscopic properties of the corresponding odd-mass system. The model is applied to the odd-$A$ Ba, Xe, La, and Cs isotopes with mass $A\approx 130$, for which the corresponding even-even Ba and Xe nuclei present a typical case of $\gamma$-soft nuclear potential. The theoretical results reproduce the experimental low-energy excitation spectra and electromagnetic properties, and confirm that a phase transition between nearly spherical and $\gamma$-soft nuclear shapes occurs also in the odd-$A$ systems.

Finite-temperature mean-field theories, such as the Hartree–Fock (HF) and Hartree–Fock–Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen–Cooper–Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei ${}^{162}\mathrm{Dy}, {}^{148}\mathrm{Sm}$, and ${}^{150}\mathrm{Sm}$. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.

Background: Near- and sub-barrier fusion of various Ca + Zr isotopic combinations have been widely investigated. A recent analysis of ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ data has highlighted the importance of couplings to multiphonon excitations and to both neutron and proton transfer channels. Analogous studies of ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ tend to exclude any role of transfer couplings. However, the lowest measured cross section for this system is rather high ($840\mu \mathrm{b}$). A rather complete data set is available for ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$, while no measurement of ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ fusion has been performed in the past.

Purpose: Our aim is to measure the full excitation function of ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ near the barrier and to extend downward the existing data on ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$, in order to estimate the transfer couplings that should be used in coupled-channels calculations of the fusion of these two systems and of ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$.

Methods: ${}^{40}\mathrm{Ca}$ beams from the XTU Tandem accelerator of INFN–Laboratori Nazionali di Legnaro were used, bombarding thin metallic ${}^{90}\mathrm{Zr}$ ($50\mu \mathrm{g}/{\mathrm{cm}}^2$) and ${{}^{92}\mathrm{ZrO}}_2$ targets (same thickness) enriched to $99.36\%$ and $98.06\%$ in masses 90 and 92, respectively. An electrostatic beam deflector allowed the detection of fusion evaporation residues (ER) at very forward angles, and angular distributions of ER were measured.

Results: The excitation function of ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ has been measured down to the level of $\simeq 60\mu \mathrm{b}$. Coupled-channels (CC) calculations using a standard Woods-Saxon (WS) potential and following the line of a previous analysis of ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ fusion data give a good account of the new data, as well as of the existing data for ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$. The previous excitation function of ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ has been extended down to $40\mu \mathrm{b}$.

Conclusions: Transfer couplings play an important role in explaining the fusion data for ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ and ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$. The strength of the pair-transfer coupling is deduced by applying a simple recipe based on the value obtained for ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$. The logarithmic slopes and the $S$ factors for fusion are reproduced fairly well for all three systems by the CC calculations, and there are no indications of a fusion hindrance at the lowest energies. In contrast, the new data for ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ indicate the onset of a fusion hindrance at the lowest energies.

We analyze the phenomenon of size-flow transmutation in ultrarelativistic nuclear collisions in a model where the initial size fluctuations are driven by the wounded quarks and the collectivity is provided by viscous hydrodynamics. It is found that the model properly reproduces the data for the transverse momentum fluctuations measured for $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76$ TeV by the ALICE Collaboration. The agreement holds for a remarkably wide range of centralities, from 0–5 % up to 70–80 %, and displays a departure from a simple scaling with ${(d{N}_{\mathrm{ch}}/d\eta )}^{1/2}$ in the form seen in the data. The overall agreement in the model with wounded quarks is significantly better than with nucleon participants. This feature joins the previously found wounded quark multiplicity scaling in the argumentation in favor of subnucleonic degrees of freedom in the early dynamics. We also examine in detail the correlations between measures of the initial size and final average transverse momentum of hadrons. Predictions are made for the transverse momentum fluctuations in $p+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=5.02$ TeV.

We study the single electron spectra from $D-$ and $B-$meson semileptonic decays in $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$, 62.4, and 19.2 GeV by employing the parton-hadron-string dynamics (PHSD) transport approach that has been shown to reasonably describe the charm dynamics at Relativistic Heavy Ion Collider and Large Hadron Collider energies on a microscopic level. In this approach the initial charm and bottom quarks are produced by using the PYTHIA event generator which is tuned to reproduce the fixed-order next-to-leading logarithm calculations for charm and bottom production. The produced charm and bottom quarks interact with off-shell (massive) partons in the quark-gluon plasma with scattering cross sections which are calculated in the dynamical quasiparticle model that is matched to reproduce the equation of state of the partonic system above the deconfinement temperature $T_c$. At energy densities close to the critical energy density ($\approx 0.5\mathrm{GeV}/{\mathrm{fm}}^3$) the charm and bottom quarks are hadronized into $D$ and $B$ mesons through either coalescence or fragmentation. After hadronization the $D$ and $B$ mesons interact with the light hadrons by employing the scattering cross sections from an effective Lagrangian. The final $D$ and $B$ mesons then produce single electrons through semileptonic decay. We find that the PHSD approach well describes the nuclear modification factor $R_{\mathrm{AA}}$ and elliptic flow $v_2$ of single electrons in $d+\text{Au}$ and $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{\mathrm{NN}}}=200\mathrm{GeV}$ and the elliptic flow in $\text{Au}+\text{Au}$ reactions at $\sqrt{s_{\mathrm{NN}}}=62.4\mathrm{GeV}$ from the PHENIX Collaboration, however, the large $R_{\mathrm{AA}}$ at $\sqrt{s_{\mathrm{NN}}}=62.4\mathrm{GeV}$ is not described at all. Furthermore, we make predictions for the $R_{\mathrm{AA}}$ of $D$ mesons and of single electrons at the lower energy of $\sqrt{s_{\mathrm{NN}}}=19.2\mathrm{GeV}$. Additionally, the medium modification of the azimuthal angle $\varphi$ between a heavy quark and a heavy antiquark is studied. We find that the transverse flow enhances the azimuthal angular distributions close to $\varphi =0$ because the heavy flavors strongly interact with nuclear medium in relativistic heavy-ion collisions and almost flow with the bulk matter.

A high precision lifetime measurement of the $2_1^{+}$ state in ${}^{94}\mathrm{Sr}$ was performed at TRIUMF's ISAC-II facility by coupling the recoil distance method implemented via the TIGRESS integrated plunger with unsafe Coulomb excitation in inverse kinematics. Due to limited statistics imposed by the use of a radioactive ${}^{94}\mathrm{Sr}$ beam, a likelihood ratio ${\chi }^2$ method was derived and used to compare experimental data to Geant4 simulations. The $B(E2;2_1^{+}\to 0_1^{+})$ value extracted from the lifetime measurement of $7.{80}_{-0.40}^{+0.50}\text{(stat.)}\pm 0.07\text{(sys.)}$ ps is approximately 25% larger than previously reported while the relative error has been reduced by a factor of approximately 8. A baseline deformation has been established for Sr isotopes with $N\le 58$ which is a necessary condition for the quantum phase transition interpretation of the onset of deformation in this region. A comparison to existing theoretical models is presented.

We generalize the theory of nuclear decay and capture of Gamow that is based on tunneling through the barrier and internal oscillations inside the nucleus. In our formalism an additional factor is obtained, which describes distribution of the wave function of the the $\alpha$ particle inside the nuclear region. We discover new most stable states (called quasibound states) of the compound nucleus (CN) formed during the capture of $\alpha$ particle by the nucleus. With a simple example, we explain why these states cannot appear in traditional calculations of the $\alpha$ capture cross sections based on monotonic penetrabilities of a barrier, but they appear in a complete description of the evolution of the CN. Our result is obtained by a complete description of the CN evolution, which has the advantages of (1) a clear picture of the formation of the CN and its disintegration, (2) a detailed quantum description of the CN, (3) tests of the calculated amplitudes based on quantum mechanics (not realized in other approaches), and (4) high accuracy of calculations (not achieved in other approaches). These peculiarities are shown with the capture reaction of $\alpha +{}^{44}\mathrm{Ca}$. We predict quasibound energy levels and determine fusion probabilities for this reaction. The difference between our approach and theory of quasistationary states with complex energies applied for the $\alpha$ capture is also discussed. We show (1) that theory does not provide calculations for the cross section of $\alpha$ capture (according to modern models of the $\alpha$ capture), in contrast with our formalism, and (2) these two approaches describe different states of the $\alpha$ capture (for the same $\alpha$-nucleus potential).

We estimate chemical freeze-out parameters in Hadron Resonance Gas (HRG) and Excluded Volume HRG (EVHRG) models by fitting the experimental information of net-proton and net-charge fluctuations measured in Au + Au collisions by the STAR Collaboration at the BNL Relativistic Heavy Ion Collider (RHIC). We observe that chemical freeze-out parameters obtained from lower and higher order fluctuations are almost the same for $\sqrt{s_{\mathit{NN}}}>27$ GeV, but tend to deviate from each other at lower $\sqrt{s_{\mathit{NN}}}$. Moreover, these separations increase with decrease of $\sqrt{s_{\mathit{NN}}}$, and for a fixed $\sqrt{s_{\mathit{NN}}}$ increase towards central collisions. Furthermore, we observe an approximate scaling behavior of $({\mu }_B/T)/{({\mu }_B/T)}_{\mathrm{central}}$ with $\left(N_{\mathrm{part}}\right)/{\left(N_{\mathrm{part}}\right)}_{\mathrm{central}}$ for the parameters estimated from lower order fluctuations for $11.5\le \sqrt{s_{\mathit{NN}}}\le 200$ GeV. Scaling is violated for the parameters estimated from higher order fluctuations for $\sqrt{s_{\mathit{NN}}}=11.5$ and 19.6 GeV. It is observed that the chemical freeze-out parameter, which can describe ${\sigma }^2/M$ of net protons very well in all energies and centralities, cannot describe the $s\sigma$ equally well, and vice versa.

In high-energy heavy-ion collisions, the magnetic field is very strong right after the nuclei penetrate each other and a nonequilibrium system of quarks and gluons builds up. Even though quarks might not be very abundant initially, their dynamics must necessarily be influenced by the Lorentz force. Employing the (3+1)-d partonic cascade Boltzmann approach to multiparton scatterings (BAMPS), we show that the circular Larmor movement of the quarks leads to a strong positive anisotropic flow of quarks at very soft transverse momenta. We explore the regions where the effect is visible and explicitly show how collisions damp the effect. As a possible application, we look at photon production from the flowing nonequilibrium medium.

$\gamma$-ray transitions between low-spin states of the neutron-rich ${}^{84,86,88}\mathrm{Ge}$ were measured by means of in-flight $\gamma$-ray spectroscopy at 270 MeV/u. Excited $6_1^{+},4_{1,2}^{+}$, and $2_{1,2}^{+}$ states of ${}^{84,86}\mathrm{Ge}$ and $4_1^{+}$ and $2_{1,2}^{+}$ states of ${}^{88}\mathrm{Ge}$ were observed. Furthermore, a candidate for a $3_1^{+}$ state of ${}^{86}\mathrm{Ge}$ was identified. This state plays a key role in the discussion of ground-state triaxiality of ${}^{86}\mathrm{Ge}$, along with other features of its low-energy level scheme. A new region of triaxially deformed nuclei is proposed in the Ge isotopic chain.

An experiment focused on the study of shape coexistence and new high-spin structures in ${}^{195}\mathrm{Bi}$ has been performed. The nucleus is in a transitional region of the bismuth isotope chain. A large number of new states have been found, resulting in a significant extension of the previously known level scheme. Several new collective structures have been identified. A strongly coupled rotational band built upon the $13/2^{+}$ isomeric state was extended up to $I^{\pi }=(49/2^{+})$ and an energy of 5706 keV. The $I^{\pi }=31/2^{+}$ member of the $\pi i_{13/2}$ band was also found to feed a new long-lived isomeric state with an excitation energy of 2616 keV and a spin and parity of $I^{\pi }=29/2^{+}$. The half-life of the $29/2^{+}$ isomeric state was determined to be $1.49\left(1\right)\mu \mathrm{s}$. It decays via the emission of 457-keV $E2$ and 236-keV $E1$ transitions, respectively. A low-energy 46-keV $E2$ transition has been identified to depopulate the ($29/2^{-}$) isomeric state, with a measured half-life of $T_{1/2}=614\left(5\right)\mathrm{ns}$. This transition allows the excitation energy of the isomeric state to be determined as 2381 keV. The feeding patterns of both $29/2^{+}$ and ($29/2^{-}$) isomeric states have also been described. This is the first time collective structures have also been observed up to high spins and excitation energies in the neutron-deficient ${}^{195}\mathrm{Bi}$ nucleus. Evidence for the manifestation of shape coexistence in ${}^{195}\mathrm{Bi}$ is also discussed.

Large-scale shell-model calculations for the even-even cadmium isotopes ${}^{98}\mathrm{Cd}$–${}^{108}\mathrm{Cd}$ have been performed with the antoine code in the $\pi (2p_{1/2};1g_{9/2})\nu (2d_{5/2};3s_{1/2};2d_{3/2};1g_{7/2};1h_{11/2})$ model space without further truncation. Known experimental energy levels and $B\left(E2\right)$ values could be well reproduced. Taking these calculations as a starting ground we analyze the deformation parameters predicted for the Cd isotopes as a function of neutron number $N$ and spin $J$ using the methods of model independent invariants introduced by Kumar [Phys. Rev. Lett. 28, 249 (1972)] and Cline [Annu. Rev. Nucl. Part. Sci. 36, 683 (1986)].

We explore the impact of nuclear matter saturation on the properties and systematics of finite nuclei across the nuclear chart. By using the ab initio in-medium similarity renormalization group (IM-SRG), we study ground-state energies and charge radii of closed-shell nuclei from ${}^4\mathrm{He}$ to ${}^{78}\mathrm{Ni}$ based on a set of low-resolution two- and three-nucleon interactions that predict realistic saturation properties. We first investigate in detail the convergence properties of these Hamiltonians with respect to model-space truncations for both two- and three-body interactions. We find one particular interaction that reproduces well the ground-state energies of all closed-shell nuclei studied. As expected from their saturation points relative to this interaction, the other Hamiltonians underbind nuclei but lead to a remarkably similar systematics of ground-state energies. Extending our calculations to complete isotopic chains in the $sd$ and $pf$ shells with the valence-space IM-SRG, the same interaction reproduces not only experimental ground states but two-neutron-separation energies and first-excited $2^{+}$ states. We also extend the valence-space IM-SRG to calculate radii. Since this particular interaction saturates at too high density, charge radii are still too small compared with experiment. Except for this underprediction, the radius systematics is, however, well reproduced. Our results highlight the renewed importance of nuclear matter as a theoretical benchmark for the development of next-generation chiral interactions.

Prompt-fission $\gamma$ rays are responsible for approximately 5% of the total energy released in fission, and therefore important to understand when modeling nuclear reactors. In this work we present prompt $\gamma$-ray emission characteristics in fission as a function of the nuclear excitation energy of the fissioning system. Emitted $\gamma$-ray spectra were measured, and $\gamma$-ray multiplicities and average and total $\gamma$ energies per fission were determined for the ${}^{233}\text{U}(d,pf)$ reaction for excitation energies between 4.8 and 10 MeV, and for the ${}^{239}\text{Pu}(d,pf)$ reaction between 4.5 and 9 MeV. The spectral characteristics show no significant change as a function of excitation energy above the fission barrier, despite the fact that an extra $\sim 5$ MeV of energy is potentially available in the excited fragments for $\gamma$ decay. The measured results are compared with model calculations made for prompt $\gamma$-ray emission with the fission model code gef. Further comparison with previously obtained results from thermal neutron induced fission is made to characterize possible differences arising from using the surrogate ($d,p$) reaction.

Both hydrodynamics-based models and a multiphase transport (AMPT) model can reproduce the mass splitting of azimuthal anisotropy $\left(v_n\right)$ at low transverse momentum $\left(p_{\perp }\right)$ as observed in heavy ion collisions. In the AMPT model, however, $v_n$ is mainly generated by the parton escape mechanism, not by the hydrodynamic flow. In this study we provide detailed results on the mass splitting of $v_n$ in this transport model, including $v_2$ and $v_3$ of various hadron species in $d$+Au and Au+Au collisions at the Relativistic Heavy Ion Collider and $p$+Pb collisions at the Large Hadron Collider. We show that the mass splitting of hadron $v_2$ and $v_3$ in AMPT first arises from the kinematics in the quark coalescence hadronization process, and then, more dominantly, comes from hadronic rescatterings, even though the contribution from the latter to the overall charged hadron $v_n$ is small. We further show that there is no qualitative difference between heavy ion collisions and small-system collisions or between elliptic $\left(v_2\right)$ and triangular $\left(v_3\right)$ anisotropies. Our studies thus demonstrate that the mass splitting of $v_2$ and $v_3$ at low $p_{\perp }$ is not a unique signature of hydrodynamic collective flow but can be the interplay of several physics effects.

The ${}^{242}\mathrm{Am}$ metastable isomer's neutron-induced destruction mechanisms were studied at the Los Alamos Neutron Science Center using the Detector for Advanced Neutron-Capture Experiments array with a compact parallel-plate avalanche counter. New ${}^{242\mathrm{m}}\mathrm{Am}$ neutron-capture cross sections were determined from 100 meV to 10 keV, and the absolute scale was set with respect to a concurrent measurement of the well-known ${}^{242\mathrm{m}}\mathrm{Am}$ neutron-induced-fission cross section. The new fission cross section spans an energy range from 100 meV to 1 MeV and was normalized to the ENDF/B-VII.1 evaluated cross section to set the absolute scale. Our ${}^{242\mathrm{m}}\mathrm{Am}(n,f)$ cross section agrees well with the cross section of Browne et al. [Phys. Rev. C 29, 2188 (1984)] over this large energy interval. The new neutron-capture cross section measurement complements and agrees well with our recent results reported below 1 eV in Buckner et al. [Phys. Rev. C 95, 024610 (2017)]. This new work comprises the most comprehensive study of ${}^{242\mathrm{m}}\mathrm{Am}(n,\gamma )$ above thermal energy. Neutron-induced resonance energies and parameters were deduced with the sammy $R$-matrix code for incident neutron energies up to 45 eV, and the new average ${\mathrm{\Gamma }}_{\gamma }$ is $13\%$ higher than the evaluated average $\gamma$ width.

Halo is one of the most interesting phenomena in exotic nuclei especially for ${}^{31}\mathrm{Ne}$, which is deemed to be a halo nucleus formed by a $p$-wave resonance. However, the theoretical calculations do not suggest a $p$-wave resonance using the scattering phase shift approach or complex scaling method. Here, we apply the complex momentum representation method to explore resonances in ${}^{31}\mathrm{Ne}$. We have calculated the single-particle energies for bound and resonant states together with their evolutions with deformation. The results show that the $p$-wave resonances appear clearly in the complex momentum plane accompanied by a $p\text{−}f$ inversion in the single-particle levels. Because the $p\text{−}f$ inversion happens, calculated energy, width, and occupation probabilities of major components in the level occupied by the valance neutron support a $p$-wave halo for ${}^{31}\mathrm{Ne}$.

We systematically investigate the thermodynamic properties of homogeneous nuclear matter with light clusters at low densities and finite temperatures using a generalized nonlinear relativistic mean-field (gNL-RMF) model, in which light clusters up to $\alpha \left(1\le A\le 4\right)$ are included as explicit degrees of freedom and treated as pointlike particles, with their interactions described by meson exchanges. The medium effects on the cluster binding energies are described by density- and temperature-dependent energy shifts with the parameters obtained by fitting the experimental cluster Mott densities. We find that the composition of low density nuclear matter with light clusters is essentially determined by the density and temperature dependence of the cluster binding energy shifts. Compared with the values of the conventional (second-order) symmetry energy, symmetry free energy, and symmetry entropy, their fourth-order values are found to be significant at low densities $\left(n\sim {10}^{-3}{\mathrm{fm}}^{-3}\right)$ and low temperatures $\left(T\lesssim 3\mathrm{MeV}\right)$, indicating the invalidity of the empirical parabolic law for the isospin asymmetry dependence of these nuclear matter properties. Our results indicate that, in the density region of $n\gtrsim 0.02{\mathrm{fm}}^{-3}$, the clustering effects become insignificant and the nuclear matter is dominated by the nucleon degree of freedom. In addition, we compare the gNL-RMF model predictions with the corresponding experimental data on the symmetry energy and symmetry free energy at low densities and finite temperatures extracted from heavy-ion collisions, and reasonable agreement is found.

The prompt fission neutron spectrum from spontaneous fission of ${}^{252}\mathrm{Cf}$ is an integral part of several aspects of nuclear data. Not only is the spectrum itself of interest, but neutron detectors often use the spectrum for calibration, and other prompt fission neutron spectra are measured as a ratio to ${}^{252}\mathrm{Cf}$. Therefore, reducing the uncertainties in this spectrum will allow for more accurate nuclear data to be available across a wide range of fields. The prompt fission neutron spectrum for the spontaneous fission of ${}^{252}\mathrm{Cf}$ was measured at Rensselaer Polytechnic Institute using the multiple $\gamma$ tagging method with a 18.4-ng fission sample. An EJ-301 liquid scintillator fast neutron detector was used to measure the high energy portion of the spectrum, 0.5–7 MeV, and a thin EJ-204 plastic scintillator was used to measure the low energy portion of the spectrum, from 50 keV to 2 MeV. These spectra both show good agreement with the current evaluation of ${}^{252}\mathrm{Cf}$ and have low associated uncertainties providing a new high precision measurement that helps reduce the uncertainties in the prompt fission neutron spectrum for the spontaneous fission of ${}^{252}\mathrm{Cf}$.

By employing four typical equation of states (EOSs) of nuclear matter in the inner crust, the properties of low-mass neutron stars are investigated theoretically. Based on the well-known fact that there is a big gap between the neutron stars and white dwarfs in the mass-radius sequence of compact stars, according to the mass-radius relations of the four adopted EOSs, we conclude that there is a rough forbidden region for the central density and stellar radius to form a compact star; that is, there is no compact star in nature having central density in the region from about ${10}^{12}{\mathrm{kgm}}^{-3}$ to ${10}^{17}{\mathrm{kgm}}^{-3}$, and there is also no compact star having a radius in the region from about 400 km to 2000 km. Moreover, the properties of the low-mass neutron stars are also explored. It is shown that for a stable neutron star near the minimum mass point, the stellar size (with radius $>200$ km) is much larger than that of normal neutron stars, and there is a compact “core” concentrated at about 95% of the stellar mass in the inner core with a radius of about 13 km and density higher than the neutron-drip point $(4.3\times {10}^{14}{\mathrm{kgm}}^{-3})$. This property totally differs from that of normal neutron stars and white dwarfs. Furthermore, the Keplerian period, the moment of inertia, and the surface gravitational redshift of the star near the minimum-mass point are also investigated.