A continuum model of a solid with a free surface: I. Normal mode analysis of the model

A complete analysis is made of the normal modes of an isotropic continuum model of a solid which occupies the region (0, 0, 0) < (x, y, z) < (X, Y, Z), the limit (X, Y, Z) → (∞,∞,∞) being taken. The surface z = 0 is a free surface and that z = Z is fixed; cyclic boundary conditions are used in the x and y directions. The modal displacements (vectors), modal frequencies (as a function of the wave-vectors), and wave-vector-densities are obtained for all types of normal modes; the work is a modification and extension of that of earlier workers.     

Some properties of the spectral flow in semiriemannian geometry

Let be the action integral on a semiriemannian manifold ( , g) defined on the space of the curves z : [0, 1] → joining two given points z₀ and z₁. The critical points of ƒ are the geodesics joining z₀ and z₁. Let s ε [0, 1]. We study the behavior, in dependence of s, of the eigenvalues of the Hessian form of ƒ evaluated at z, restricted to the interval [0, s]. A formula for the derivative of the eigenvalues is given and some applications are shown.     

Oxygen nonstoichiometry and phase transitions of the neodymium-rich Nd1+xBa2−xCu3Oz solid solution

On the basis of chemical, thermal analysis and Cu K-edge X-ray absorption measurements, oxygen content in the Nd_1+xBa_2−xCu₃O_z solid solution was determined between 1000°C in air and 400°C in oxygen for x=0.05–0.9 compositions. It has been observed that the oxygen nonstoichiometry Δz of the Nd_1+xBa_2−xCu₃O_7+x/2−Δz solid solution decreases 2–2.5 times for a large substitution (Δz≈0.3–0.33 for x=0.9), despite of the acclaimed higher total oxygen content. The difference in nonstoichiometry is explained by a higher average value of the copper oxidation state (ACV), which is vital for the solid solution with large x even at elevated temperatures (ACV≈2–2.05 for x>0.3 at 1000°C, PO₂=0.21 atm). On the contrary, the ACV after complete oxygenation is almost constant (about 2.25–2.3) for the whole series. The x-dependence of the oxygen content is not monotonous and structural phase transitions can be observed at x=0.3 and x=0.6, as confirmed by the X-ray diffraction and the Raman scattering spectroscopy. The first well-known transition is connected with the oxygen disorder due to the Nd substitution for Ba at random Ba-sites. In the present work, it is proved by the apical oxygen mode broadening in Raman spectra. Ordering of the Nd and Ba atoms with a subsequent orthorhombic distortion of the lattice may occur even at 1000°C in air due to the second transformation at x≈0.6. The invariable orthorhombicity of the Nd-rich solid solution with x>0.6 is not caused by the oxygen absorption as in the x=0.05 case. Existence of high- and low-temperature orthorhombic modifications of this solid solution has been observed for the first time. Finally, a tentative 3D (z–x–T) diagram is suggested for the Nd_1+xBa_2−xCu₃O_z solid solution up to 1000°C in air, including the new x=0.6–0.9 region.     

利用传递熵对Lorenz系统和Walker环流信息传输方向的分析

基于传递熵方法, 分析Lorenz系统x, y, z三个分量之间的信息传输方向, 并应用温差与垂直速度的再分析资料对Walker环流进行分析. 研究结果表明: 1) 对于Lorenz系统而言, x与y分量之间, y是信息源, x是信息汇; y与z之间, y是信息源, z是信息汇; x与z分量之间的信息传递方向依赖于控制参数r; 且净信息传输的方向不随初值不同而改变; 2)在西太平洋地区, 温差对垂直速度的净信息输送占主导地位, 而赤道东太平洋地区则为垂直速度对温差的净信息输送占主导地位, 这与Walker环流物理机制是一致的, 且海陆热力差异对温差与垂直速度之间的信息输送影响较大; 3)冬季是温差与垂直速度之间的信息输送最强季节, 夏季和秋季次之, 春季最弱, 这可能是存在春季预报障碍的原因之一. 以上结果表明, 传递熵在气象领域有广阔的应用前景, 是测量动力学系统信息传递方向的一种有效方法和工具. 关键词： 传递熵 信息传输 Lorenz系统 Walker环流     

Observing quantum tunneling in perturbation series

We apply Borel resummation to the conventional perturbation series of ground state energy in a metastable potential, V(x) = x²/2 − gx⁴/4. We observe numerically that the discontinuity of the Borel transform reproduces the imaginary part of the energy eigenvalue, i.e., the total decay width due to quantum tunneling. The agreement with the exact numerical value is remarkable in the whole tunneling regime 0 < g 0.7.     

Finite-size scaling properties of the damage distance and dynamical critical exponent for the Ising model

With the damage spreading method, scaling properties of the damage distance on the Ising model with heat bath dynamics are studied numerically. With the parallel flipping scheme, the scaling curves fall on two curves, which depend on the odd or even lattice sizes. The both scaling curves give the consistent dynamical exponent as z = 2.16±0.04 for d = 2 and z = 2.09±0.05 for d = 3, respectively. By shifting one of them, two curves overlap each other perfectly. Meanwhile, all the scaling curves obtained by single-spin flipping processes (with different odd or even lattice sizes) fall on a single curve, from which the consistent dynamical critical exponent with the parallel scheme is obtained z = 2.18±0.02 for d = 2 and z = 2.08±0.04 for d = 3.     

Superconductivity in Ba2−xSrxYCu3Oz prepared at GPa

The compounds Ba_1.4Sr_0.6YCu₃O_z (7.0≤z<7.4) were synthesized at 2 GPa to clarify the effect of high temperature and pressure on the superconducting properties. The structure as observed by powder X-ray diffraction was tetragonal for z≥7.2, and orthorhombic or tetragonal for z<7.1. It has been revealed that high-pressure syntheses suppress the superconductivity in the compounds with z<7.1. Recovery of the superconducting properties occurs above z≥7.2. The effect of Sr substitution on the superconducting properties was also studied. The magnitude of diamagnetism at 5–30 K decreases monotonically from −0.003 to −0.0002 emu/g Oe with increasing x in Ba_2−xSr_xYCu₃O_z (x≥1.2).     

多点边值问题的插值矩阵法及误差分析

插值矩阵法是求解多点边值问题的数值法。本文给出的该法的误差分析,论证了插值矩阵法解得的y(x),y'(x),…,y^(m)(x)有相同的精度,并对二阶方程,给出该法的稳定性证明和收敛阶。     

Globally conformal invariant gauge field theory with rational correlation functions

Operator product expansions (OPE) for the product of a scalar field with its conjugate are presented as infinite sums of bilocal fields V_κ(x₁,x₂) of dimension (κ,κ). For a globally conformal invariant (GCI) theory we write down the OPE of V_κ into a series of twist (dimension minus rank) 2κ symmetric traceless tensor fields with coefficients computed from the (rational) 4-point function of the scalar field.

We argue that the theory of a GCI hermitian scalar field of dimension 4 in D=4 Minkowski space such that the 3-point functions of a pair of 's and a scalar field of dimension 2 or 4 vanish can be interpreted as the theory of local observables of a conformally invariant fixed point in a gauge theory with Lagrangian density .     

Localized canting of spins structure in the spinel oxide system: ZnzTizFe2−x−zCrx−zCoO4

The spinel oxide system Zn_zTi_zFe_2−x−zCr_x−zCoO₄; z=x²; x=0.60, 0.65, 0.70 and 0.80, was studied using neutron diffraction technique, low field DC magnetization measurements (ZFC–FC measurements), magnetic hysterisis, Mössbauer spectroscopy and low field AC susceptibility measurements. All the compositions show significantly less B-site magnetic moments at 10 K temperature derived from neutron diffraction data than the free ions site moments deduced assuming collinear arrangement of spins. This combined with some other features seen in the low temperature neutron diffraction patterns suggest localized canting of spins (LCS) type of magnetic ordering in the present system where a long range order of longitudinal component of moments co-exists along with totally disordered transverse component of moments. The conclusion is also supported by the features seen in the other measurements. The magnetic moments derived from 10 K neutron diffraction data are explained using the LCS approach for different exchange integrals ratios.     

Plastic zone size and crack tip opening displacement of a Dugdale crack interacting with a coated circular inclusion

An analytical investigation on the plastic zone size of a crack near a coated circular inclusion under three different loading conditions of uniaxial tension, uniform tension and pure shear was carried out. Both the crack and coated circular inclusion are embedded in an infinite matrix, with the crack oriented along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small-scale yielding [J. Mech. Phys. Solids 8 (1960) p. 100], two thin strips of yielded plastic zones are introduced at both crack tips. Using the solution for a coated circular inclusion interacting with a single dislocation as the Green's function, the physical problem is formulated into a set of singular integral equations. Using the method of Erdogan and Gupta [Q. J. Appl. Math. 29 (1972) p. 525] and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement.     

Characterization and cathode performance of Li1 − xNi1 + xO2 prepared with the excess lithium method

Samples of Li_{1 − z}Ni_{1 + x}O₂ with various x values were synthesized and their electrochemical properties, phase transitions, and ordering phenomena were investigated comparatively. In order to synthesize samples with a small x value, an excess lithium was used as a starting material to compensate for lithium loss during the calcination process. A stoichiometric sample with a large reversible capacity of more than 200 mAh g⁻¹ is also described.     

Asymmetric bound states via the quadrupled Schrödinger equation

An efficient iterative method of construction of bound states in an asymmetric potential well is suggested and tested on V(x) = ax² + bx³ + cx⁴.     

Scaling limit of deeply virtual compton scattering

I outline a perturbative QCD approach to the analysis of the deeply virtual Compton scattering process γ^*p → γp′ in the limit of vanishing momentum transfer t = (p′ − p)². The DVCS amplitude in this limit exhibits a scaling behavior described by two-argument distributions F(x,y) which specify the fractions of the initial momentum p and the momentum transfer r ≡ p′ − p carried by the constituents of the nucleon. The kernel R(x,y;ξ,η) governing the evolution of the non-forward distributions F(x,y) has a remarkable property: it produces the GLAPD evolution kernel P(x/ξ) when integrated over y and reduces to the Brodsky-Lepage evolution kernel V(y,η) after the x-integration. This property is used to construct the solution of the one-loop evolution equation for the flavor non-singlet part of the non-forward quark distribution.     

Thermodynamics of binary liquid mixtures: application of the Prigogine–Flory–Patterson theory to excess molar volumes of acetonitrile+1-alkanol systems

Excess molar volumes of {(1−x_A)CH₃(CH₂)_n−1OH+x_ACH₃CN} for n=1, 2, 3 or 4 as a function of composition under atmospheric pressure at 288.15, 293.15, 298.15 and 303.15 K have been used to test the applicability of the Prigogine–Flory–Patterson theory. According to the model, interactional contribution is the most important one to explain the V_m^E behavior. Good agreement is only achieved for the mixtures containing methanol (C₁) or 1-butanol (C₄). For the mixtures containing ethanol (C₂) and 1-propanol (C₃), which show an S-shaped V_m^E behavior, the correlation fails.     

Hole density and Tc behaviour in YBa2Cu3O7 − x

The behaviour of the hole number in YBCO systems as a function of the oxygen deficiency x is studied theoretically by calculating the density of electron states for several different superstructures formed by periodic arrangements of the oxygen vacancies. The characteristic features of the electronic properties for chain copper sites with different coordination are pointed out and their relevance in determining the number of carriers for oxygen planes is illustrated. It is shown that the curve of the hole number as a function of x shows plateaus and jumps similar to those found in the T_c versus x curves if the de-oxygenation process takes place by forming fragmented chains. Structures formed by filled and empty chains only give a smooth hole behaviour as a function of the composition. The results of the numerical calculations can be understood by a simple model giving the Fermi energy shift in terms of the concentrations of Cu(1) sites with different coordination.     

Magnetic behaviour of some Mn.III2.VI4 compounds and their alloys

Measurement of the magnetic susceptibility χ as a function of temperature were made on polycrystalline samples from the alloy systems Cd_1−zMn_zGa₂Se₄, Zn_1−zMn_zGa₂Se₄ and Cd_1−zMn_zIn₂Te₄ which had been subjected to various heat treatments. The 1/χ versus T curves indicated that for the Zn---Se alloys, for all values of z, samples slowly cooled to room temperature were antiferromagnetic showing ideal Curie-Weiss behaviour, but for samples quenched from 700°C the behaviour was a mixture of antiferromagnetic and spin-glass. For the Cd---Se alloys, samples from the range (0.6 < z < 1.0) showed very similar behaviour, but in the range (0 < z < 0.6) even the very slowly cooled samples showed a mixture of antiferromagnetic and spin-glass behaviour. For the Cd---Te alloys, all samples, however heat-treated, showed spin-glass form. Values of the Curie-Weiss constant Ф were determined from all of the 1/χ versus T curves, and by comparison with the T(z) phase diagrams for the different alloy systems, the values of Ф were correlated with the ordering of the Mn atoms on the cation lattice. It is shown that the experimental values of Ф can give a very convenient way of determining the type of ordering and the degree of order in such alloys.     

First-principles calculations of thermodynamic properties of TiB2 at high pressure

The equations of state (EOS) and other thermodynamic properties of TiB₂ are investigated using ab initio plane-wave pseudopotential density functional theory method. The obtained results are in good agreement with the experimental measured data and other theoretical calculated ones. Through the quasi-harmonic Debye model, in which the phononic effects are considered, the dependences of relative volume V/V₀ on pressure P, cell volume V on temperature T, and Debye temperature Θ and specific heat C_V on pressure P are successfully obtained.     

Radiation of ultrasound into an anisotropic solid

A half-space, x₃ 0, of a transversely isotropic solid whose axis of symmetry makes an angle with the x₃-axis, is subjected to a spatially uniform time-harmonic distribution of normal surface tractions over a circular area of the plane x₃ = 0. The wave motion radiated into the half-space is investigated. Using an integral representation for the displacement components the problem is first reduced to a system of singular integral equations for the displacements on the surface x₃ = 0. This system is solved by the boundary element method over a truncated area, where use is made of recently derived simplified forms of the Green's functions. The results show the skewing of the beam as the angle between the axis of symmetry of the transversely isotropic solid and the normal to the surface of the solid is increased.     

Two-dimensional shear waves scattering by a large rigidity strip with interface crack

A two-dimensional problem of shear horizontal (SH) waves scattering by a finite width planar elastic (piezoelectric) inclusion partially debonded from its surrounding elastic matrix is investigated using the effective boundary conditions and singular integral equations technique. The case of large rigidity inclusions with blunted tips is considered, in which the upper face of the inclusion is perfectly bonded to the matrix. The debonding region is modeled as interface crack with non-contacting faces. Using the Green theorem the mixed boundary value problem is reduced to a system of the hypersingular integral equations. Numerical results of the scattering fields characteristics are presented. The effects of incidence direction, various material parameters of the strip on the scattering field are discussed and phenomenon of the non-specular reflection of SH waves is considered. The accuracy of the numerical results is confirmed by the use of analytical approximate problem solution of high-frequency SH waves scattering on a finite hard/soft inclusion.