Near-Infrared Spectrum (3000-10 000 cm) of 4-Chlorophenol in Solution

The near-infrared spectra (3000-10 000 cm⁻¹) of 4-chlorophenol-OH and 4-chlorophenol-OD are studied at room temperature in carbon tetrachloride. Some data are also reported for 4-bromophenol-OH and 4-bromophenol-OD. The observed absorptions are assigned to overtones or combinations involving the fundamental transitions which have been recently calculated at the B3LYP/6-311++G(df,pd) level (J. Phys. Chem. A104, 11 685 (2000)). The anharmonicities and the harmonic frequencies of the local ν(OH) and ν(CH) modes and the coupling constants for the binary or ternary combinations are obtained. The extinction coefficients of these transitions are discussed. The half-band width of the ν(OH) and ν(CH) vibrations increases with the absorption frequencies. The experimental harmonic frequencies are obtained from the anharmonicity constants. Comparison with the theoretical frequencies indicates that the low scaling factors ranging from 0.950 to 0.960 characterizing the ν(OH) and ν(CH) vibrations can be accounted for by the large anharmonicity of these vibrations. For these vibrations, the ratio of the experimental harmonic frequencies to the computed frequencies is 0.993. This value does not significantly differ from the average scaling factors of 0.987-0.989 obtained for all the other vibrational modes.     

基于组分Ca2Pr2Cu5O10三元固溶区的范围和晶体结构

 利用X射线衍射分析和Rietveld结构精修方法研究了基于组分Ca₂Pr₂Cu₅O₁₀三元固溶区的范围和晶体结构。根据X射线分析结果和相消失法，为了保持Ca_2+xPr_2-xCu₅O₁₀结构单位晶胞中恒定的氧含量和电价平衡，基于组分Ca₂Pr₂Cu₅O₁₀的三元固溶区的范围可确定为：在Ca_2+xPr_2-xCu₅O₁₀固溶区的富Ca区（Ca/Pr>1），其表达式仍可以表示为一般的化学计量式Ca_2+xPr_2-xCu₅O₁₀（x=0.4, 0.2, 0）；在富Pr区（Ca/Pr<1），其表达式为Ca_1.7Pr_2.2·Cu₅O₁₀、Ca_1.4Pr_2.4Cu₅O₁₀，即：Ca_2+xPr_2-xCu₅O₁₀固溶区范围从Ca_2.4Pr_1.6Cu₅O₁₀至Ca_1.4Pr_2.4·Cu₅O₁₀。研究表明：基于组分Ca₂Pr₂Cu₅O₁₀的三元固溶体晶体结构可以看作是正交晶系NaCuO₂型亚晶胞的无公度相；Ca_2.4Pr_1.6Cu₅O₁₀亚晶胞的点阵常数为a₀=0.282 46(7) nm，b₀=0.636 93(1) nm，c₀=1.067 94(1) nm；其正交超结构的点阵常数为a=5a₀，b=b₀，c=5c₀。Ca_2.4Pr_1.6Cu₅O₁₀的结构也可以用单斜超晶胞来表述，该单斜超晶胞的空间群为P2₁/c，Z=4，a=5a₀，b=b₀，c=c₀/sin β，β=104.79°或136.60°，V=5a₀b₀c₀。在基于组分Ca₂Pr₂Cu₅O₁₀的三元固溶体晶体结构中，Ca和Pr之间在一定范围内是可以相互替代的，而且它们在结构中的占位是无序的。     

From the Solution of the Tsarev System to the Solution of the Whitham Equations

We study the Cauchy problem for the Whitham modulation equations for increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is enumerated by the genus g=0,1,2, ... of the corresponding hyperelliptic Riemann surface. Each of these systems can be integrated by the so-called hodograph transformation introduced by Tsarev. A key step in the integration process is the solution of the Tsarev linear overdetermined system. For each g>0, we construct the unique solution of the Tsarev system, which matches the genus g+1 and g–1 solutions on the transition boundaries.     

Solution for Energies and Mixing of Two 0~+ States in ~(10)He

Using results from various reactions that populate ~(10)He,I conclude that the ground state has E_(2n) = 1.07(7) MeV and the excited 0~+ state is in the region of 2.1-3.1 MeV.The amount of the(sd)~2 component in the ground state is less than about 0.075.     

Parametrization of the Kerr Solution

From the Kerr solution of Ernst equation under Ehlers and unitary transformations, we build a parametrized Kerr solution depending on three parameters, namely the mass, the angular momentum of the source and an adimensional parameter m ₁. Varying m ₁ produces a topological deformation of the ergosphere.     

Solution of the Hartree-Fock equations

We suggest to use the Newton iteration method for constructing a (locally unique) solution of the atomic and nuclear Hartree-Fock equations for an arbitrary number of particles. Our proposal is based on a theorem by Kantorovi and rests on the following points: 1) the two-body potential must satisfy a boundedness condition; 2) the zero-order approximation, used to start the iteration sequence, must satisfy certain conditions, to be proved numerically. Condition 1) holds, for instance, for all local potentials, defined by a bounded function and for a class of nonlocal potentials; it does not hold for local potentials, behaving as 1/r near the origin.This work has been supported in part by Istituto Nazionale di Fisica Nucleare (Sezione di Catania) and by Centro Siciliano di Fisica Nucleare e di Struttura della Materia (Catania).     

Solution of the gauge hierarchy problem

We propose a novel solution to the gauge hierarchy problem in theories with softly broken supersymmetry. Quantum effects can resuscitate classically sick theories, producing the large scale from the small supersymmetry breaking scale. We use this mechanism to construct realistic SU(6) and SU(5) GUTs which do not suffer from gauge hierarchy or fine tuning problems.     

Solution to the problem of Nicolai

We consider the problem of Nicolai on dynamic stability of an elastic cantilever rod loaded by an axial compressive force P and a twisting tangential torque L in continuous formulation. The problem is to find the stability region for non-equal principal moments of inertia of the rod in the space of three parameters: P, L and the parameter α for the ratio of principal moments of inertia. New governing equations and boundary conditions, which form the basis for analytical and numerical studies, are derived. An important detail of this formulation is that the pre-twisting of the rod due to the torque L is taken into account. The singular point on the stability boundary at the critical Euler force P_E is recognized and investigated in detail. For an elliptic cross-section of a uniform rod the stability region is found numerically with the use of the Galerkin method and the exact numerical approach. The obtained numerical results are compared with the analytical formulas of the asymptotic analysis.     

Solution of the relativistic string problem

The quantum dynamics of the free Nambu-Nielson-Susskind string is known to be consistent only in s + 1 dimensional Minkowski space when s = 25 and the intercept α₀ = 1. Ad hoc modifications are required to allow s < 25. The source of this difficulty is pointed out and the system is solved covariantly and ghost free for any value of s > 1 without ad hoc modifications. The intercept value α₀ = 1 is excluded, but α₀ can be fitted to the experimental value. There are only transverse modes, and linearly rising Regge trajectories result as before.     

Rigorous Solution of the Gardner Problem

 We prove rigorously the well-known result of Gardner about the typical fractional volume of interactions between N spins which solve the problem of storing a given set of p random patterns. The Gardner formula for this volume in the limit N, p→∞, p/N→α is proven for all values of α. Besides, we prove a useful criterion for the factorisation of all correlation functions for a class of models of classical statistical mechanics. Received: 2 December 2001 / Accepted: 12 June 2002 Published online: 31 January 2003 Communicated by A. Kupiainen     

Numerical Solution of the Four-Anyon Problem

By making use of the Talmi-Moshinsky transformation bracket, -the Schrödinger equation for four anyons in configuration space has been solved variationally to obtain the eigenenergies and eigenfunctions. The low-lying states have been presented as functions of the statistical parameter. We illustrated how the bosonic states evolve continuously into the fermionic states.     

Solution to the Rarita-Schwinger equations

A systematical method is presented for solving the relativistic wave equations for particles of arbitrary spin. Explicit helicity relativistic wave functions for particles with arbitrary spin are derived rigorously. Received: 22 August 2001 / Revised version: 5 July 2002 / Published online: 9 December 2002 RID="a" ID="a" Corresponding author.     

Schwarzschild Solution on the Brane

In this paper it is shown that Schwarzschild solution is possible in brane world for some specific choices of brane matter and the non-local effects from the bulk. A conformally flat bulk space time with fine-tuned vacuum energy (brane tension) shows that Schwarzschild solution may also be the vacuum solution for brane world scenario.     

Solution of the equation of radiation transport

Solution of a Boltzmann-type radiation-transport equation is considered. The final form of the source function is influenced by the geometry of the medium, the nature of the line broadening, and the accuracy of intermediate steps such as choice of trial functions and determination of integral matrix elements and the eigenvalue of the variational method.     

Solution of the System of Electrodiffusion Equations

The solution of the electrodiffusion equations together with the Poisson equation is examined. It is demonstrated that in the main approximation in a small parameter, the solution is reduced to that of two first-order differential equations. The obtained equations allow the transition processes in systems of charged particles to be examined for an arbitrary time dependence of the external potential.     

Solution of the path integral for the H-atom

The Green's function of the H-atom is calculated by a simple reduction of Feynman's path integral to gaussian form.     

Solution to the equations of the moment expansions

We develop a formula for matching a Taylor series about the origin and an asymptotic exponential expansion for large values of the coordinate. We test it on the expansion of the generating functions for the moments and connected moments of the Hamiltonian operator. In the former case the formula produces the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We choose the harmonic oscillator and a strongly anharmonic oscillator as illustrative examples for numerical test. Our results reveal some features of the connected-moments expansion that were overlooked in earlier studies and applications of the approach.