定积分第一中值定理的改进与应用

本文重新表述了定积分第一中值定理的证明,并改进了该定理,对于改进了的定积分第一中值定理还给出了证明及一些应用实例.

积分中值定理的证明与应用

本文首先证明了定积分第一中值定理,接着利用定积分第一中值定理给出了积分中值定理的证明。

中国古代文学教学改革尝试(上)

古代文学的教学问题是执教者十分关注和经常思考的重要问题。上课伊始,当堂测验,了解学生的真实水准,可以因材施教。课堂教学中重传道、授业、解惑,做到教书育人。先测验、明源流、定第一、评得失,教学效果会好于以前的教学。

First principles calculation of intermetallic compounds in FeTiCoNiVCrMnCuAl system high entropy alloy

The structural, electronic and elastic properties of common intermetallic compounds in FeTiCoNiVCrMnCuAI system high entropy alloy were investigated by the first principles calculation. The calculation results of formation enthalpy and cohesive energy show that FeTi, Fe2Ti, AlCrFe2, Co2Ti, AlMn2V and Mn2Ti phases may form in the formation process of the alloy. Further studies show that FeTi, FezTi, AlCrFe2, Co2Ti and AlMn2V phases with higher shear modulus and elastic modulus would be excellent strengthening phases in high entropy alloy and would improve the hardness of the alloy. In addition, the partial density of states was investigated for revealing the bonding mode, and the analyses on the strength of p-d hybridization also reveal the underlying mechanism for the elastic properties of these compounds.

A first-principles study on elastic properties and stability of Ti_xV_(1-x)C multiple carbide

The structure,stability and elastic properties of di-transition-metal carbides TixV1-xC were investigated by using the first-principles with a pseudopotential plane-waves method.The results show that the equilibrium lattice constants of TixV1-xC show a nearly linear reduction with increasing addition of V.The elastic properties of TixV1-xC are varied by doping with V.The bulk modulus of Ti0.5V0.5C is larger than that of pure TiC,as well as Ti0.5V0.5C has the largest C44 among TixV1-xC（0≤x≤1）,indicating that Ti0.5V0.5C has higher hardness than pure TiC.However,Ti0.5V0.5C presents brittleness based on the analysis of ductile/brittle behavior.The Ti0.5V0.5C carbide has the lowest formation energy,indicating that Ti0.5V0.5C is more stable than all other alloys.

First-principles study of phase stability and electronic properties of RhZr

First-principles calculations were carried out to investigate the structural stabilities and electronic properties of RhZr.The plane wave based pseudopotential method was used,in which both the local density approximation（LDA） and the generalized gradient approximation（GGA） implanted in the CASTEP code were employed.The internal positions of atoms in the unit cell were optimized and the ground state properties such as lattice parameter,density of state,cohesive energies and enthalpies of formation of ortho-RhZr and cubic-RhZr were calculated.The calculation results indicate that ortho-RhZr can form more easily than cubic-RhZr and the ortho-RhZr is more stable than cubic-RhZr.The density of states（DOS） reveals that the strong bonding in the Rh-Zr and Rh-Rh or Zr-Zr interaction chains accounts for the structural stability of ortho-RhZr and the hybridization between Rh-4d states and Zr-4d states is strong.

Study on structural stability,elastic and electronic properties for β-Ti under pressure based on first principles

The structural stability, elastic and electronic properties under pressure at 0 K for β-Ti have been investigated by per-forming first-principles calculations. With the increase of pressure, the structure of β-Ti becomes stabler, which is further con-firmed by the calculation for density of state （DOS）. The phase transition pressure of is about 64. 3 GPa, which is consist-ent with other theoretical predictions （63. 7 GPa） and the experimental result （50 GPa）. The pressure dependence of elastic constants shows that the low-pressure limit for a mechanically stable β-Ti is about 50 GPa with low Young？s modulus value of about 30. 01 GPa, which approaches the value of a human bone （30 GPa）. In addition, the pressure dependence of bulk modu-lus B, shear modulus G, Young’s modulus E,Poisson’s ratio σ,aggregate sound velocities,and ductility/brittleness under different pressures were also discussed. B, G and E ascend monotonously with increasing pressure, while a descends. β-Ti re-mains ductile by analysis of B/G under considered pressures.

First-principles lattice stability of Fe, Ru and Os

Lattice constants, total energies and densities of states of transition metals Fe, Ru and Os with BCC, FCC and HCP structures were calculated by the GGA+PBE functional and the ultrasoft pseudo-potential plane wave method, and compared with those of the first-principles projector augmented wave (PAW) method, CALPHAD method and experimental data. The results show that the lattice stability of this work is △GBCC-HCP>△GFCC-HCP>0, agreeing well with those of PAW method in the first-principles and CALPHAD method except for BCC-Fe. And the densities of state of HCP-Ru and Os have an obvious character of stable phase, agreeing completely with the results of the total energy calculations. Further analyses of atomic population show that the transition rate of electrons from s to p state for HCP, FCC and BCC crystals increases from Fe to Os, and a stronger cohesion, a higher cohesive energy or a more stable lattice between atoms of heavier metals are formed.

First-principle investigation on stability of Co-doped spinel λ-Mn_(4-x)Co_xO_8

The mechanism of stability of Co-doped spinel λ-MnO_2 that is referred to as spinel Li_xMn_2O_4 (x=0) was studied by using the first-principle calculation method. The total energy and formation enthalpy can be decreased remarkably due to the Co substation, resulting in a more stable structure of λ-Mn_xCr_(2-x)O_4. The bond order and DOS analysis were given in detail to explain the nature of stability improvement. The calculated results show that as the content of Co dopant increases, the bond order of Mn—O becomes larger and the peak of density of states around Fermi level shifts toward lower energy. The charge density distribution illustrates that the Mn—O bonding is ionic and partially covalent, and the covalent Mn-O bonding becomes stronger with the increase of Co dopant content. The results confirm that the Codoping will enhance the stability of λ-MnO_2 and hence improve the electrochemistry performance of Li_xMn_2O_4.

Stable High-Energy Density Super-Atom Clusters of Aluminum Hydride

With the concept of super-atom, first principles calculations propose a new type of super stable cage clusters AlnH3n that are much more energetic stable than the well established clusters, AlnHn＋2. In the new clusters, the aluminum core-frame acts as a super-atom with n vertexes and 2n A1-A1 edges, which allow to adsorb n hydrogen atoms at the top-site and 2n at the bridge-site. Using Al12H36 as the basic unit, stable chain structures, （Al12H36）m, have been constructed following the same connection mechanism as for （A1H3）n linear polymeric structures. Apart from high hydrogen percentage per molecule, calculations have shown that these new clusters possess large heat of formation values and their combustion heat is about 4.8 times of the methane, making them a promising high energy density material.

First-Principle Calculations for Thermodynamic Properties of LiBC Under High Temperature and High Pressure

The thermodynamic properties of LiBC are investigated by using the full-potential linearlzed muffin-tin orbital method （FP-LMTO） within the frame of density functional theory （DFT） and using the quasi-harmonic Debye model. The dependencies of the normalized lattice parameters a/a0 and c/c0, the ratio （c/a）/2, the normalized primitive volume V/V0 on pressure and temperature are successfully obtained. It is found that the interlayer covalent interactions （Li-B bonds or Li-C bonds） are more sensitive to temperature and pressure than intralayer ones （B-C bonds）, as gives rise to the extreme lattice anisotropy in the bulk hop LiBC.

Thermal Relativity

The group G of general coordinate transformations on the thermodynamic configuration space ε spanned by all the extensive variables keeps the first law of thermodynamics invariant. One can introduce a metric with Lorentzian signature on the space ε, with the corresponding line element also being invariant under the action of G. This line element is identi6ed as the square of the proper entropy. Thus the second law of thermodynamics is also formulated invariantly and this lays down the foundation for the principle of thermal relativity.

Thermal Entangled Quantum Refrigerator Working with Three-Qubit Heisenberg XX Model

An entangled quantum refrigerator working with a three-qubit one-dimensional isotropic Heisenberg XX model in a constant external magnetic field is constructed in this paper. Based on the quantum first law of thermodynamics, the expressions for several basic thermodynamic quantities such as the heat transferred, the net work and the coefficient of performance are derived. Moreover, the influence of the thermal entanglement on the basic thermodynamic quantities is investigated. Several interesting features of the variation of the basic thermodynamic quantities with the thermal entanglement in zero and nonzero magnetic field are obtained.

Entropy Corrections for a Charged Black Hole of String Theory

We study the entropy of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger （GMGHS） charged black hole, originated from the effective action that emerges in the low-energy of string theory, beyond semiclassical approxi- mations. Applying the properties of exact differentials for three variables to the first law thermodynamics we derive the quantum corrections to the entropy of the black hole. The leading （logarithmic） and non leading corrections to the area law are obtained.

Simplification and Application of Boyd Membrane-diffusion Equation

The having-been-used-for-50-year Boyd membrane diffusion Equation-In(1 - F) = R t can be deduced into F = kt through using Maclanrin expansion equation and the Lagerange remainders. The latter is a simple membrane diffusion equation, which is available to judge if the exchanging course of the resin obeys the rules of membrane-diffusion mechanism more conveniently.

Solution of Schrdinger Equation for Two-Dimensional Complex Quartic Potentials

We investigate the quasi-exact solutions of the Schrodinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x1 ＋ ip3, y = x2 ＋ ip4, px= p1＋ ix3, py= p2 ＋ ix4. Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetrie one, are also worked out.

First Principles Study of AI-Li Intermetallic Compounds

The structural properties, heats of formation, elastic properties, and electronic structures of four compositions of binary A1-Li intermetallics, A13Li, A1Li, A12Li3, and A14Li9, are ana- lyzed in detail by using density functional theory. The calculated formation heats indicate a strong chemical interaction between A1 and Li for all the A1-Li intermetallics. In partic- ular, in the Li-rich A1-Li compounds, the thermodynamic stability of intermetallics linearly decreases with increasing concentration of.Li. According to the computational single crystal elastic constants, all the four A1-Li intermetallic compounds considered here are mechani- cally stable. The polycrystalline elastic modulus and Poisson＇s ratio have been deduced by using Voigt, Reuss, and Hill approximations, and the calculated ratios of bulk modulus to shear modulus indicate that the four compositions of binary A1-Li intermetallics are brittle materials. With the increase of Li concentration, the bulk modulus of A1-Li intermetallics decreases in a linear manner.

Topological quantum catalyst: Dirac nodal line states and a potential electrocatalyst of hydrogen evolution in the TiSi family

Topological nodal line（DNL） semimetals, a closed loop of the inverted bands in its bulk phases, result in the almost flat drumhead-like non-trivial surface states（DNSSs） with an unusually high electronic density near the Fermi level. High catalytic active sites generally associated with high electronic densities around the Fermi level, high carrier mobility and a close-to-zero free energy of the adsorbed state of hydrogen（？G_（H＊）≈0） are prerequisite to design alternative of precious platinum for catalyzing electrochemical hydrogen production from water. By combining these two aspects, it is natural to consider if the DNLs are a good candidate for the hydrogen evolution reaction（HER） or not because its DNSSs provide a robust platform to activate chemical reactions. Here, through first-principles calculations we reported a new DNL TiSi-type family, exhibiting a closed Dirac nodal line due to the linear band crossings in k_y=0 plane.The hydrogen adsorbed state on the surface yields ？G_（H＊） to be almost zero and the topological charge carries participate in HER. The results highlight a new routine to design topological quantum catalyst utilizing the topological DNL-induced surface bands as active sites, rather than edge sites-, vacancy-,dopant-, strain-, or heterostructure-created active sites.

Quantum heat engine cycle working with a strongly correlated electron system

A new model of a quantum heat engine （QHE） cycle is established, in which the working substance consists of an interacting electrons system. One of our purposes is to test the validity of the second law of thermodynamics by this model, which is more general than the spin-1/2 antiferromagnetic Heisenberg model since it would recover the spin model when the on-site Coulomb interaction U is strong enough. On the basis of quantum mechanics and the first law of thermodynamics, we show no violation of the second law of thermodynamics during the cycle. We further study the performance characteristics of the cycle by investigating in detail the optimal relations of efficiency and dimensionless power output. We find that the efficiency of our engine can be expressed as η = t22/t21 in the large-U limit, which is valid even for a four sites QHE.

The Generalized Thermodynamic Temperature and the New Expressions of the First and the Second Law of Thermodynamics

The classical thermodynamics reflects the significant relationship between the heat and the temperature. On the basis of the relationships, according to the mathematical derivation, this paper structures the conceptions of generalized heat, generalized thermodynamic temperature, generalized entropy and so on. The series of conceptions in the classical thermodynamics is merely a special case of the generalized thermodynamics. Based on these conceptions of generalized thermodynamics, this paper presents the new expressions of the first law and the second law of thermodynamics. In other words, these expressions are endued with new explanations. The Eq. LZ = kTS given by this paper provides theoretical basis for these new expressions.