Kinetic energy (KE) functional is crucial to speed up density functional theory calculation. However, deriving it accurately through traditional physics reasoning is challenging. We develop a generally applicable KE functional estimator for a one-dimensional (1D) extended system using a machine learning method. Our end-to-end solution combines the dimensionality reduction method with the Gaussian process regression, and simple scaling method to adapt to various 1D lattices. In addition to reaching chemical accuracy in KE calculation, our estimator also performs well on KE functional derivative prediction. Integrating this machine learning KE functional into the current orbital free density functional theory scheme is able to provide us with expected ground state electron density.

The non-Hermitian $PT$-symmetric system can live in either unbroken or broken $PT$-symmetric phase. The separation point of the unbroken and broken $PT$-symmetric phases is called the $PT$-phase-transition point. Conventionally, given an arbitrary non-Hermitian $PT$-symmetric Hamiltonian, one has to solve the corresponding Schrödinger equation explicitly in order to determine which phase it is actually in. Here, we propose to use artificial neural network (ANN) to determine the $PT$-phase-transition points for non-Hermitian $PT$-symmetric systems with short-range potentials. The numerical results given by ANN agree well with the literature, which shows the reliability of our new method.

A unified description of finite nuclei and equation of state of neutron stars presents both a major challenge and also opportunities for understanding nuclear interactions. Inspired by the Lee–Huang–Yang formula of hard-sphere gases, we develop effective nuclear interactions with an additional high-order density dependent term. While the original Skyrme force SLy4 is widely used in studies of neutron stars, there are not satisfactory global descriptions of finite nuclei. The refitted SLy4${'}$ force can improve descriptions of finite nuclei but slightly reduces the radius of neutron star of 1.4$M_{\odot}$ with $M_{\odot}$ being the solar mass. We find that the extended SLy4 force with a higher-order density dependence can properly describe properties of both finite nuclei and GW170817 binary neutron stars, including the mass-radius relation and the tidal deformability. This demonstrates the essential role of high-order density dependence at ultrahigh densities. Our work provides a unified and predictive model for neutron stars, as well as new insights for the future development of effective interactions.

At the China Spallation Neutron Source (CSNS), we have developed a custom gas-filling station, a glassblowing workshop, and a spin-exchange optical pumping (SEOP) system for producing high-quality $^3$He-based neutron spin filter (NSF) cells. The gas-filling station is capable of routinely filling $^3$He cells made from GE180 glass of various dimensions, to be used as neutron polarizers and analyzers on beamlines at the CSNS. Performance tests on cells fabricated at our gas-filling station are conducted via neutron transmission and nuclear-magnetic-resonance measurements, revealing nominal filling pressures, and a saturated $^3$He polarization in the region of 80%, with a lifetime of approximately 240 hours. These results demonstrate our ability to produce competitive NSF cells to meet the ever-increasing research needs of the polarized neutron research community.

It has been reported that electron-rotation coupling plays a significant role in diatomic nuclear dynamics induced by intense VUV pulses [Phys. Rev. A 102 (2020) 033114; Phys. Rev. Res. 2 (2020) 043348]. As a further step, we present here investigations of the electron-rotation coupling effect in the presence of Auger decay channel for core-excited molecules, based on theoretical modeling of the total electron yield (TEY), resonant Auger scattering (RAS) and x-ray absorption spectra (XAS) for two showcases of CO and CH$^{+}$ molecules excited by resonant intense x-ray pulses. The Wigner D-functions and the universal transition dipole operators are introduced to include the electron-rotation coupling for the core-excitation process. It is shown that with the pulse intensity up to $\mathrm{10^{16}\,W/cm^{2}}$, no sufficient influence of the electron-rotation coupling on the TEY and RAS spectra can be observed. This can be explained by a suppression of the induced electron-rotation dynamics due to the fast Auger decay channel, which does not allow for effective Rabi cycling even at extreme field intensities, contrary to transitions in optical or VUV range. For the case of XAS, however, relative errors of about 10% and 30% are observed for the case of CO and CH$^{+}$, respectively, when the electron-rotation coupling is neglected. It is concluded that conventional treatment of the photoexcitation, neglecting the electron-rotation coupling, can be safely and efficiently employed to study dynamics at the x-ray transitions by means of electron emission spectroscopy, yet the approximation breaks down for nonlinear processes as stimulated emission, especially for systems with light atoms.

We study the multiphoton ionization of potassium atoms in 800 nm and 400 nm femtosecond laser fields. In the 800 nm laser field, the potassium atom absorbs three photons and emits one electron via one photon resonance with the $4p$ intermediate state with the help of the ac-Stark shift. The resonance feature is clearly shown as an Autler–Townes (AT) splitting and is mapped out in the electron kinetic energy spectrum. In a 400 nm laser field, although one photon resonance is possible with the $5p$ state, no splitting is observed. The different transition amplitudes between $4s$–$4p$ and $4s$–$5p$ explain the observed results. Due to the AT effect, an unexpected peak in the photoelectron energy spectrum that violates the dipole transition rule is observed. A preliminary explanation involving the spin-orbit interaction in the $p$ state is given to account for this component. The observed AT-splitting in the electron kinetic energy distribution can be used as an effective method to calibrate the intensity of a laser field.

We theoretically investigate the characteristics of terahertz (THz) radiation from monolayer graphene exposed to normal incident few-cycle laser pulses, by numerically solving the extended semiconductor Bloch equations. Our simulations show that the THz spectra in low frequency regions are highly dependent on the carrier envelope phase (CEP) of driving laser pulses. Using an optimal CEP of few-cycle laser pulses, we can obtain broadband strong THz waves, due to the symmetry breaking of the laser-graphene system. Our results also show that the strength of the THz spectra depend on both the intensity and central wavelength of the laser pulses. The intensity dependence of the THz wave can be described by the excitation rate of graphene, while wavelength dependence can be traced back to the band velocity and the population of graphene. We find that a near single-cycle THz pulse can be obtained from graphene driven by a mid-infrared laser pulse.

Fusion born $\alpha$ particle confinement is one of the most important issues in burning plasmas, such as ITER and CFETR. However, it is extremely complex due to the nonequilibrium characteristics, and multiple temporal and spatial scales coupling with background plasma. A numerical code using particle orbit tracing method (PTC) has been developed to study energetic particle confinement in tokamak plasmas. Both full orbit and drift orbit solvers are implemented to analyze the Larmor radius effects on $\alpha$ particle confinement. The elastic collisions between alpha particles and thermal plasma are calculated by a Monte Carlo method. A triangle mesh in poloidal section is generated for electromagnetic fields expression. Benchmark between PTC and ORBIT has been accomplished for verification. For CFETR burning plasmas, PTC code is used for $\alpha$ particle source and slowing down process calculation in 2D equilibrium. In future work, 3D field like toroidal field ripples, Alfvén and magnetohydrodynamics instabilities perturbation inducing $\alpha$ particle transport will be analyzed.

Multiple broadband Alfvénic chirping modes (CMs), with frequencies in the wide range of $f\sim35$–150 kHz and chirping down rapidly, are found in HL-2A neutral beam injection plasmas, and the CMs can even coexist. The frequency chirping down process can be completed within $\sim$1 ms, and the frequency shift can reach 30–50 kHz. The CMs propagate in ion diamagnetic drift directions poloidally. The toroidal mode number is confirmed to be $n=1$, $2$, $3$ and $4$ for the $f\sim35$–65, 55–90, 70–120 and 100–150 kHz CMs, respectively. The CMs are more like to be energetic-particle continuum modes (EPMs), since the modes almost locate on the Alfvén continuum.

This research applies experimental measurements and NUBEAM, ONETWO and TRANSP modules to investigate the shine-through (ST) loss ratio and beam heating percentage of neutral beam injection on EAST. Measurements and simulations confirm that the ST loss ratio increases linearly with beam energy, and decreases exponentially with plasma density. Moreover, using the multi-step fitting method, we present analytical quantitative expressions of ST loss ratio and beam heating percentage, which are valuable for the high parameter long-pulse experiments of EAST.

Diamond, cubic boron nitride (c-BN), silicon (Si), and germanium (Ge), as examples of typical strong covalent materials, have been extensively investigated in recent decades, owing to their fundamental importance in material science and industry. However, an in-depth analysis of the character of these materials' mechanical behaviors under harsh service environments, such as high pressure, has yet to be conducted. Based on several mechanical criteria, the effect of pressure on the mechanical properties of these materials is comprehensively investigated. It is demonstrated that, with respect to their intrinsic brittleness/ductile nature, all these materials exhibit ubiquitous pressure-enhanced ductility. By analyzing the strength variation under uniform deformation, together with the corresponding electronic structures, we reveal for the first time that the pressure-induced mechanical softening/weakening exhibits distinct characteristics between diamond and c-BN, owing to the differences in their abnormal charge-depletion evolution under applied strain, whereas a monotonous weakening phenomenon is observed in Si and Ge. Further investigation into dislocation-mediated plastic resistance indicates that the pressure-induced shuffle-set plane softening in diamond (c-BN), and weakening in Si (Ge), can be attributed to the reduction of antibonding states below the Fermi level, and an enhanced metallization, corresponding to the weakening of the bonds around the slipped plane with increasing pressure, respectively. These findings not only reveal the physical mechanism of pressure-induced softening/weakening in covalent materials, but also highlights the necessity of exploring strain-tunable electronic structures to emphasize the mechanical response in such covalent materials.

Detection of local strain at the nanometer scale with high sensitivity remains challenging. Here we report near-field infrared nano-imaging of local strains in bilayer graphene by probing strain-induced shifts of phonon frequency. As a non-polar crystal, intrinsic bilayer graphene possesses little infrared response at its transverse optical phonon frequency. The reported optical detection of local strain is enabled by applying a vertical electrical field that breaks the symmetry of the two graphene layers and introduces finite electrical dipole moment to graphene phonon. The activated phonon further interacts with continuum electronic transitions, and generates a strong Fano resonance. The resulted Fano resonance features a very sharp near-field infrared scattering peak, which leads to an extraordinary sensitivity of $\sim $0.002% for the strain detection. Our results demonstrate the first nano-scale near-field Fano resonance, provide a new way to probe local strains with high sensitivity in non-polar crystals, and open exciting possibilities for studying strain-induced rich phenomena.

We report the experimental production of degenerate Fermi gases of $^6$Li atoms in an optical dipole trap. The gray-molasses technique is carried out to decrease the atomic temperature to 57 µK, which facilitates the efficient loading of cold atoms into the optical dipole trap. The Fermi degeneracy is achieved by evaporative cooling of a two-spin mixture of $^6$Li atoms on the Feshbach resonance. The degenerate atom number per spin is $3.5\times10^4$, and the reduced temperature $T/T_{\rm F}$ is as low as 0.1, where $T_{\rm F}$ is the Fermi temperature of the non-interacting Fermi gas. We also observe the anisotropic expansion of the atom cloud in the strongly interacting regime.

The Magnus Hall effect (MHE) is a new type of linear-response Hall effect, recently proposed to appear in two-dimensional (2D) nonmagnetic systems at zero magnetic field in the ballistic limit. The MHE arises from a self-rotating Bloch electron moving under a gradient-electrostatic potential, analogous to the Magnus effect in the macrocosm. Unfortunately, the MHE is usually accompanied by a trivial transverse signal, which hinders its experimental observation. We systematically investigate the material realization and experimental measurement of the MHE, based on symmetry analysis and first-principles calculations. It is found that both the out-of-plane mirror and in-plane two-fold symmetries can neutralize the trivial transverse signal to generate clean MHE signals. We choose two representative 2D materials, monolayer MoS$_2$, and bilayer WTe$_2$, to study the quantitative dependency of MHE signals on the direction of the electric field. The results are qualitatively consistent with the symmetry analysis, and suggest that an observable MHE signal requires giant Berry curvatures. Our results provide detailed guidance for the future experimental exploration of MHE.

It has recently been demonstrated that various topological states, including Dirac, Weyl, nodal-line, and triple-point semimetal phases, can emerge in antiferromagnetic (AFM) half-Heusler compounds. However, how to determine the AFM structure and to distinguish different topological phases from transport behaviors remains unknown. We show that, due to the presence of combined time-reversal and fractional translation symmetry, the recently proposed second-order nonlinear Hall effect can be used to characterize different topological phases with various AFM configurations. Guided by the symmetry analysis, we obtain expressions of the Berry curvature dipole for different AFM configurations. Based on the effective model, we explicitly calculate the Berry curvature dipole, which is found to be vanishingly small for the triple-point semimetal phase, and large in the Weyl semimetal phase. Our results not only put forward an effective method for the identification of magnetic orders and topological phases in AFM half-Heusler materials, but also suggest these materials as a versatile platform for engineering the nonlinear Hall effect.

We report an experimental study of electron transport properties of MnSe/(Bi,Sb)$_{2}$Te$_{3}$ heterostructures, in which MnSe is an antiferromagnetic insulator, and (Bi,Sb)$_{2}$Te$_{3}$ is a three-dimensional topological insulator (TI). Strong magnetic proximity effect is manifested in the measurements of the Hall effect and longitudinal resistances. Our analysis shows that the gate voltage can substantially modify the anomalous Hall conductance, which exceeds 0.1 $e^{2}/h$ at temperature $T = 1.6$ K and magnetic field $\mu_{0}H = 5$ T, even though only the top TI surface is in proximity to MnSe. This work suggests that heterostructures based on antiferromagnetic insulators provide a promising platform for investigating a wide range of topological spintronic phenomena.

Atomically thin two-dimensional (2D) materials are the building bricks for next-generation electronics and optoelectronics, which demand plentiful functional properties in mechanics, transport, magnetism and photoresponse. For electronic devices, not only metals and high-performance semiconductors but also insulators and dielectric materials are highly desirable. Layered structures composed of 2D materials of different properties can be delicately designed as various useful heterojunction or homojunction devices, in which the designs on the same material (namely homojunction) are of special interest because preparation techniques can be greatly simplified and atomically seamless interfaces can be achieved. We demonstrate that the insulating pristine ZnPS$_{3}$, a ternary transition-metal phosphorus trichalcogenide, can be transformed into a highly conductive metal and an n-type semiconductor by intercalating Co and Cu atoms, respectively. The field-effect-transistor (FET) devices are prepared via an ultraviolet exposure lithography technique. The Co-ZnPS$_{3}$ device exhibits an electrical conductivity of $8\times10^{4}$ S/m, which is comparable to the conductivity of graphene. The Cu-ZnPS$_{3}$ FET reveals a current ON/OFF ratio of 10$^{5}$ and a mobility of $3\times10^{-2}$ cm$^{2}\cdot$V$^{-1}\cdot$s$^{-1}$. The realization of an insulator, a typical semiconductor and a metallic state in the same 2D material provides an opportunity to fabricate n-metal homojunctions and other in-plane electronic functional devices.

The modulation of electrical properties of MoS$_{2}$ has attracted extensive research interest because of its potential applications in electronic and optoelectronic devices. Herein, interfacial charge transfer induced electronic property tuning of MoS$_{2}$ are investigated by in situ ultraviolet photoelectron spectroscopy and x-ray photoelectron spectroscopy measurements. A downward band-bending of MoS$_{2}$-related electronic states along with the decreasing work function, which are induced by the electron transfer from Cs overlayers to MoS$_{2}$, is observed after the functionalization of MoS$_{2}$ with Cs, leading to n-type doping. Meanwhile, when MoS$_{2}$ is modified with 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane ($F_{4}$-TCNQ), an upward band-bending of MoS$_{2}$-related electronic states along with the increasing work function is observed at the interfaces. This is attributed to the electron depletion within MoS$_{2}$ due to the strong electron withdrawing property of $F_{4}$-TCNQ, indicating p-type doping of MoS$_{2}$. Our findings reveal that surface transfer doping is an effective approach for electronic property tuning of MoS$_{2}$ and paves the way to optimize its performance in electronic and optoelectronic devices.

Recent experiments have demonstrated the realization of the three-dimensional quantum Hall effect in highly anisotropic crystalline materials, such as ZrTe$_{5}$ and BaMnSb$_{2}$. Such a system supports chiral surface states in the presence of a strong magnetic field, which exhibit a one-dimensional metal-insulator crossover due to suppression of surface diffusion by disorder potential. We study the nontrivial surface states in a lattice model and find a wide crossover of the level-spacing distribution through a semi-Poisson distribution. We also discover a nonmonotonic evolution of the level statistics due to the disorder-induced mixture of surface and bulk states.

Twisting the stacking of layered materials leads to rich new physics. A three-dimensional topological insulator film hosts two-dimensional gapless Dirac electrons on top and bottom surfaces, which, when the film is below some critical thickness, will hybridize and open a gap in the surface state structure. The hybridization gap can be tuned by various parameters such as film thickness and inversion symmetry, according to the literature. The three-dimensional strong topological insulator Bi(Sb)Se(Te) family has layered structures composed of quintuple layers (QLs) stacked together by van der Waals interaction. Here we successfully grow twistedly stacked Sb$_{2}$Te$_{3}$ QLs and investigate the effect of twist angels on the hybridization gaps below the thickness limit. It is found that the hybridization gap can be tuned for films of three QLs, which may lead to quantum spin Hall states. Signatures of gap-closing are found in 3-QL films. The successful in situ application of this approach opens a new route to search for exotic physics in topological insulators.

The electronic and superconducting properties of Fe$_{1-\delta}$Se single-crystal flakes grown hydrothermally are studied by the transport measurements under zero and high magnetic fields up to 38.5 T. The results contrast sharply with those previously reported for nematically ordered FeSe by chemical-vapor-transport (CVT) growth. No signature of the electronic nematicity, but an evident metal-to-nonmetal crossover with increasing temperature, is detected in the normal state of the present hydrothermal samples. Interestingly, a higher superconducting critical temperature $T_{\rm c}$ of 13.2 K is observed compared to a suppressed $T_{\rm c}$ of 9 K in the presence of the nematicity in the CVT FeSe. Moreover, the upper critical field in the zero-temperature limit is found to be isotropic with respect to the field direction and to reach a higher value of $\sim $42 T, which breaks the Pauli limit by a factor of 1.8.

We present the superconducting (SC) property and high-robustness of structural stability of kagome CsV$_{3}$Sb$_{5}$ under in situ high pressures. For the initial SC-I phase, its $T_{\rm c}$ is quickly enhanced from 3.5 K to 7.6 K and then totally suppressed at $P \sim 10$ GPa. With further increasing pressure, an SC-II phase emerges at $P \sim 15$ GPa and persists up to 100 GPa. The $T_{\rm c}$ rapidly increases to the maximal value of 5.2 K at $P=53.6$ GPa and slowly decreases to 4.7 K at $P=100$ GPa. A two-dome-like variation of $T_{\rm c}$ in CsV$_{3}$Sb$_{5}$ is concluded here. The Raman measurements demonstrate that weakening of $E_{\rm 2g}$ mode and strengthening of $E_{\rm 1g}$ mode occur without phase transition in the SC-II phase, which is supported by the results of phonon spectra calculations. Electronic structure calculations reveal that exertion of pressure may bridge the gap of topological surface nontrivial states near $E_{\rm F}$, i.e., disappearance of $Z_{2}$ invariant. Meanwhile, the Fermi surface enlarges significantly, consistent with the increased carrier density. The findings here suggest that the change of electronic structure and strengthened electron-phonon coupling should be responsible for the pressure-induced reentrant SC.

We systematically measure the superconducting (SC) and mixed state properties of high-quality CsV$_{3}$Sb$_{5}$ single crystals with $T_{\rm c} \sim 3.5$ K. We find that the upper critical field $H_{\rm c2}(T)$ exhibits a large anisotropic ratio of $H_{\rm c2}^{ab}/H_{\rm c2}^{c} \sim 9$ at zero temperature and fitting its temperature dependence requires a minimum two-band effective model. Moreover, the ratio of the lower critical field, $H_{\rm c1}^{ab}/H_{\rm c1}^{c}$, is also found to be larger than 1, which indicates that the in-plane energy dispersion is strongly renormalized near Fermi energy. Both $H_{\rm c1}(T)$ and SC diamagnetic signal are found to change little initially below $T_{\rm c} \sim 3.5$ K and then to increase abruptly upon cooling to a characteristic temperature of $\sim $2.8 K. Furthermore, we identify a two-fold anisotropy of in-plane angular-dependent magnetoresistance in the mixed state. Interestingly, we find that, below the same characteristic $T \sim 2.8$ K, the orientation of this two-fold anisotropy displays a peculiar twist by an angle of 60$^{\circ}$ characteristic of the Kagome geometry. Our results suggest an intriguing superconducting state emerging in the complex environment of Kagome lattice, which, at least, is partially driven by electron-electron correlation.

High resolution angle-resolved photoemission spectroscopy (ARPES) measurements are carried out on CaKFe$_4$As$_4$, KCa$_2$Fe$_4$As$_4$F$_2$ and (Ba$_{0.6}$K$_{0.4}$)Fe$_2$As$_2$ superconductors. Clear evidence of band folding between the Brillouin zone center and corners with a ($\pi$,$\pi$) wave vector has been found from the measured Fermi surface and band structures in all the three kinds of superconductors. A dominant $\sqrt{2} \times \sqrt{2}$ surface reconstruction is observed on the cleaved surface of CaKFe$_4$As$_4$ by scanning tunneling microscopy (STM) measurements. We propose that the commonly observed $\sqrt{2} \times \sqrt{2}$ reconstruction in the FeAs-based superconductors provides a general scenario to understand the origin of the ($\pi$,$\pi$) band folding. Our observations provide new insights in understanding the electronic structure and superconductivity mechanism in iron-based superconductors.

We show that a suitable combination of flat-band ferromagnetism, geometry and nontrivial electronic band topology can give rise to itinerant topological magnons. An $SU(2)$ symmetric topological Hubbard model with nearly flat electronic bands, on a Kagome lattice, is considered as the prototype. This model exhibits ferromagnetic order when the lowest electronic band is half-filled. Using the numerical exact diagonalization method with a projection onto this nearly flat band, we can obtain the magnonic spectra. In the flat-band limit, the spectra exhibit distinct dispersions with Dirac points, similar to those of free electrons with isotropic hoppings, or a local spin magnet with pure ferromagnetic Heisenberg exchanges on the same geometry. Significantly, the non-flatness of the electronic band may induce a topological gap at the Dirac points, leading to a magnonic band with a nonzero Chern number. More intriguingly, this magnonic Chern number changes its sign when the topological index of the electronic band is reversed, suggesting that the nontrivial topology of the magnonic band is related to its underlying electronic band. Our work suggests interesting directions for the further exploration of, and searches for, itinerant topological magnons.

Magnon–magnon coupling in synthetic antiferromagnets advances it as hybrid magnonic systems to explore the quantum information technologies. To induce magnon–magnon coupling, the parity symmetry between two magnetization needs to be broken. Here we experimentally demonstrate a convenient method to break the parity symmetry by the asymmetric structure. We successfully introduce a magnon–magnon coupling in Ir-based synthetic antiferromagnets CoFeB(10 nm)/Ir($t_{_{\scriptstyle \rm Ir}}=0.6$ nm, 1.2 nm)/CoFeB(13 nm). Remarkably, we find that the weakly uniaxial anisotropy field ($\sim $20 Oe) makes the magnon–magnon coupling anisotropic. The coupling strength presented by a characteristic anticrossing gap varies in the range between 0.54 GHz and 0.90 GHz for $t_{_{\scriptstyle \rm Ir}} =0.6$ nm, and between 0.09 GHz and 1.4 GHz for $t_{_{\scriptstyle \rm Ir}} = 1.2$ nm. Our results demonstrate a feasible way to induce magnon–magnon coupling by an asymmetric structure and tune the coupling strength by varying the direction of in-plane magnetic field. The magnon–magnon coupling in this highly tunable material system could open exciting perspectives for exploring quantum-mechanical coupling phenomena.

The V$_{2}$C compound, belonging to the group of two-dimensional transition metal carbonitrides, or MXenes, has demonstrated a promising electrochemical performance in capacitor applications in acidic electrolytes; however, there is evidence to suggest that V$_{2}$C is unstable in an acidic environment. On the other hand, the performance of V$_{2}$C in neutral aqueous electrolytes is still moderate, and has not yet been systematically studied. The charge storage mechanism in a V$_{2}$C electrode, employed in neutral aqueous electrolytes, is investigated via cyclic voltammetry testing and in situ x-ray diffraction (XRD). Good specific capacitances are achieved, specifically 208 F/g in 0.5 M Li$_{2}$SO$_{4}$, 225 F/g in 1 M MgSO$_{4}$, 120 F/g in 1M Na$_{2}$SO$_{4}$, and 104 F/g in 0.5 M K$_{2}$SO$_{4}$. Using in situ XRD, we observe that, during the charge and discharge process, the $c$-lattice parameter shrinks or expands by up to 0.25 Å in MgSO$_{4}$, and 0.29 Å in Li$_{2}$SO$_{4}$ which demonstrates the intercalation/de-intercalation of cations into the $d$-V$_{2}$C layer.