Materials harbouring exotic quasiparticles, such as massless Dirac and Weyl fermions, have garnered much attention from physics and material science communities due to their exceptional physical properties such as ultra-high mobility and extremely large magnetoresistances. Here, we show that the highly stable, non-toxic and earth-abundant material, ZrSiS, has an electronic band structure that hosts several Dirac cones that form a Fermi surface with a diamond-shaped line of Dirac nodes. We also show that the square Si lattice in ZrSiS is an excellent template for realizing new types of two-dimensional Dirac cones recently predicted by Young and Kane. Finally, we find that the energy range of the linearly dispersed bands is as high as 2 eV above and below the Fermi level; much larger than of other known Dirac materials. This makes ZrSiS a very promising candidate to study Dirac electrons, as well as the properties of lines of Dirac nodes.

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Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS.

Constraining the neutron–proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density

The nuclear symmetry energy

The atomic nucleus is a quantum many-body system whose constituent nucleons (protons and neutrons) are subject to complex nucleon-nucleon interactions that include spin- and isospin-dependent components. For stable nuclei, several decades ago, emerging seemingly regular patterns in some observables could already be described successfully within a shell-model picture that results in particularly stable nuclei at certain magic fillings of the shells with protons and/or neutrons: N, Z=8, 20, 28, 50, 82, 126. However, in short-lived, so-called exotic nuclei or rare isotopes, characterized by a large N/Z asymmetry and located far from the valley of β stability on the nuclear chart, these magic numbers, viewed through observables, were shown to change. These changes in the regime of exotic nuclei offer an unprecedented view at the roles of the various components of the nuclear force when theoretical descriptions are confronted with experimental data on exotic nuclei where certain effects are enhanced. This article reviews the driving forces behind shell evolution from a theoretical point of view and connects this to experimental signatures.

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Evolution of shell structure in exotic nuclei

Background: Nuclear density functional theory is the only microscopical theory that can be applied throughout the entire nuclear landscape. Its key ingredient is the energy density functional.Purpose: In this work, we propose a new parametrization unedf2 of the Skyrme energy density functional.Methods: The functional optimization is carried out using the pounders optimization algorithm within the framework of the Skyrme Hartree-Fock-Bogoliubov theory. Compared to the previous parametrization unedf1, restrictions on the tensor term of the energy density have been lifted, yielding a very general form of the energy density functional up to second order in derivatives of the one-body density matrix. In order to impose constraints on all the parameters of the functional, selected data on single-particle splittings in spherical doubly-magic nuclei have been included into the experimental dataset.Results: The agreement with both bulk and spectroscopic nuclear properties achieved by the resulting unedf2 parametrization is comparable with unedf1. While there is a small improvement on single-particle spectra and binding energies of closed shell nuclei, the reproduction of fission barriers and fission isomer excitation energies has degraded. As compared to previous unedf parametrizations, the parameter confidence interval for unedf2 is narrower. In particular, our results overlap well with those obtained in previous systematic studies of the spin-orbit and tensor terms.Conclusions: unedf2 can be viewed as an all-around Skyrme EDF that performs reasonably well for both global nuclear properties and shell structure. However, after adding new data aiming to better constrain the nuclear functional, its quality has improved only marginally. These results suggest that the standard Skyrme energy density has reached its limits, and significant changes to the form of the functional are needed.

Nuclear energy density optimization: Shell structure

The $\ensuremath{\alpha}$-decay half-lives of recently synthesized superheavy nuclei (SHN) are investigated by employing a unified fission model (UFM) where a new method to calculate the assault frequency of $\ensuremath{\alpha}$ emission is used. The excellent agreement with the experimental data indicates the UFM is a useful tool to investigate these $\ensuremath{\alpha}$ decays. It is found that the $\ensuremath{\alpha}$-decay half-lives become more and more insensitive to the ${Q}_{\ensuremath{\alpha}}$ values as the atomic number increases on the whole, which is favorable for us to predict the half-lives of SHN. In addition, a formula is proposed to compute the ${Q}_{\ensuremath{\alpha}}$ values for the nuclei with $Z\ensuremath{\geqslant}92$ and $N\ensuremath{\geqslant}140$ with a good accuracy, according to which the long-lived SHN should be neutron rich. Several weeks ago, two isotopes of a new element with atomic number $Z=117$ were synthesized and their $\ensuremath{\alpha}$-decay chains have been observed. The ${Q}_{\ensuremath{\alpha}}$ formula is found to work well for these nuclei, confirming its predictive power. The experimental half-lives are well reproduced by employing the UFM with the experimental ${Q}_{\ensuremath{\alpha}}$ values. This fact that the experimental half-lives are compatible with experimental ${Q}_{\ensuremath{\alpha}}$ values supports the synthesis of a new element $117$ and the experimental measurements to a certain extent.

α -decay half-lives and Q α values of superheavy nuclei

We establish a correlation for the symmetry energy at saturation density $S_{0}$, slope parameter $L$ and curvature parameter $K_{\text{sym}}$ based on widely different mean field interactions. With the help of this correlation and available empirical and theoretical information, the density dependent behavior around the saturation density is determined. We compare the results obtained with the present approach with those by other analyses. With this obtained density dependent behavior of the symmetry energy, the neutron skin thickness of $^{208}$Pb and some properties of neutron stars are investigated. In addition, it is found that the expression $S(\rho)=S_{0}(\rho /\rho_{0})^{\gamma}$ or $S(\rho)=12.5(\rho /\rho_{0}) ^{2/3}+C_{p}(\rho /\rho_{0}) ^{\gamma}$ does not reproduce the density dependence of the symmetry energy as predicted by the mean-field approach around nuclear saturation density.

Density dependence of the nuclear symmetry energy constrained by mean-field calculations

In the framework of the Hartree-Fock-Bogoliubov (HFB) approach with Skyrme interactions SLy5+T, SLy5+T${}_{w}$ and several sets of the TIJ parametrizations, the evolution of the proton single-particle energy difference between the 2$s$${}_{1/2}$ and 1$d$${}_{3/2}$ orbits in Ca isotopes is investigated for the cases with and without the tensor force. It is shown that the energy difference evolution trends by using different Skyrme interactions without the tensor force are similar to each other. However, the energy differences are obviously influenced by the tensor force. Meanwhile, it is found that the inversion between the two orbits (2$s$${}_{1/2}$-1$d$${}_{3/2}$ inversion) induced by the tensor force occurs around ${}^{48}$Ca (by SLy5+T${}_{w}$) as well as in the nuclei very far from the stability. We understand the inversion phenomenon by analyzing the tensor contributions to the spin-orbit potentials and the radial wave functions of the two orbits. Finally, the proton density distributions of ${}^{46}$Ar and ${}^{206}$Hg are calculated by using several sets of Skyrme interactions with and without the tensor component. It is found that the parametrization SLy5+T${}_{w}$ favors the bubble structure of ${}^{46}$Ar resulting from the 2$s$${}_{1/2}$-1$d$${}_{3/2}$ inversion. However, for ${}^{206}$Hg any tensor parametrizations do not favor its bubble structure.

Tensor effects on the proton sd states in neutron-rich Ca isotopes and bubble structure of exotic nuclei

In the framework of the Hartree-Fock-Bogoliubov approach with the Skyrme interactions SLy5, SLy5$+$T, SLy5$+$T${}_{w}$, and several sets of the TIJ parametrizations (Skyrme effective interaction parametrizations including the tensor terms), the effect of the tensor force on the shell evolution at $Z,N=8$, 20, and 28 is investigated It is shown that the evolution of the gap (defined as the energy difference between the last occupied and first unoccupied single-particle orbits) with SLy5$+$T is similar to that with SLy5$+$T${}_{w}$, and the gap values with SLy5$+$T are smaller than those with SLy5$+$T${}_{w}$ in the cases of $Z,N=8$, 20 At Z, N $=$ 28, the gap values with SLy5 are smaller than those with SLy5$+$T and larger than those with SLy5$+$T${}_{w}$ To understand these features, we analyze the spin-orbit potentials with and without the tensor contributions and the radial wave functions of relevant orbits Meanwhile, we find that the deviation of the calculated gaps with SLy5$+$T${}_{w}$ from the experimental ones is larger than that with SLy5$+$T This indicates that SLy5$+$T${}_{w}$ is not suitable for investigating the properties of the ground-state gap evolution Finally, it is seen that the gap evolutions with different sets of the TIJ parametrizations are similar to each other except for the ones with T22, and there is almost no difference between the tensor effect in gap evolution with the perturbative method and the one with the complete fitting method

Systematic study of tensor effects in shell evolution

Based on the Skyrme energy density functional, the spatial distribution of the symmetry energy of a finite nucleus is derived in order to examine whether the symmetry energy of a finite nucleus originates from its interior or from its surface. It is found that the surface part of a heavy nucleus contributes dominantly to its symmetry energy compared to its inner part. The symmetry energy coefficient $a_{\text{sym}}({A})$ is then directly extracted and the ratio of the surface symmetry coefficient to the volume symmetry coefficient $\kappa$ is estimated. Meanwhile, with the help of experimental alpha decay energies, a macroscopic method is developed to determine the symmetry energy coefficient of heavy nuclei. The resultant $a_{\text{sym}}({A})$ is used to analyze the density dependence of the symmetry energy coefficient of nuclear matter around the saturation density, and furthermore, the neutron skin thickness of $^{208}\text{Pb}$ is deduced which is consistent with the pygmy dipole resonance analysis. In addition, it is shown that the ratio $\kappa$ obtained from the macroscopic method is in agreement with that from the Skyrme energy density functional. Thus the two completely different approaches may validate each other to achieve more compelling results.

Jianzhong Gu

Papers

α -decay half-lives and Q α values of superheavy nuclei

Density dependence of the nuclear symmetry energy constrained by mean-field calculations

Tensor effects on the proton sd states in neutron-rich Ca isotopes and bubble structure of exotic nuclei

Systematic study of tensor effects in shell evolution

Origin of symmetry energy in finite nuclei and density dependence of nuclear matter symmetry energy from measured $\alpha$-decay energies