具有在位势的一维双原子链中杂质引起的局域振动

对具有在位势且含杂质的一维双原子链的晶格振动方程组进行求解,得到了局域振动的解析解, 给出了在位势对局域振动影响的基本特征,并简要讨论了局域模的存在形式.     

具有在位势的一维双原子链中杂质引起的对称局域模

对具有在位势且含杂质的一维双原子链的晶格振动方程组进行求解,得到了局域振动的解析解,给出了在位势对局域振动影响的基本特征,并简要讨论了对称局域模的存在形式.     

Nonlinear excitations in a diatomic chain

We have set up field equations for a diatomic chain of atoms, including nonlinear single site potentials. Some particular solitary wave solutions are found for the case of equal displacement fields. These solutions propagate with a fixed characteristic velocity. Some preliminary molecular dynamics results are also reported.     

Traveling solitons in modulated elastic media

A diatomic elastic chain with nonlinear interparticle interaction placed in a nonlinear external potential is considered. A detailed classification of possible two-parametric traveling gap-and near-gap-soliton solutions for this system is given for different relative strengths of the nonlinearities. Analytical expressions are derived for traveling near-gap solitons in the absence of an external field.     

PARAMETRIC SIMULTONS IN ONE-DIMENSIONAL NONLINEAR LATTICES

Parametric simultaneous solitary wave (simulton) excitations are shown to be possible in nonlinear lattices. Taking a one-dimensional diatomic lattice with a cubic potential as an example, we consider the nonlinear coupling between the upper cut-off mode of acoustic branch (as a fundamental wave) and the upper cut-off mode of optical branch (as a second harmonic wave). Based on a quasi-discreteness approach the Karamzin－Sukhorukov equations for two slowly varying amplitudes of the fundamental and the second harmonic waves in the lattice are derived when the condition of second harmonic generation is satisfied. The lattice simulton solutions are given explicitly and the results show that these lattice simultons can be nonpropagating when the wave vectors of the fundamental wave and the second harmonic waves are exactly at π/a (where a is the lattice constant) and zero, respectively.     

Parameterized effective potential for molecules and its optimization in the single-determinant approximation

A parameterized effective potential for molecules is developed in which the Coulomb and exchange parts of the Hartree-Fock potential are expressed in terms of the electron-nucleus interaction potential. Within the framework of the single-determinant approximation, equations for determining the parameters of the effective potential and of the finite basis set are obtained. In contrast to the equations of the Hartree-Fock method, the solution procedure for the equations for the parameterized effective potential does not require self-matching of the orbital expansion coefficients in terms of the basis set functions. Using the calculation of the ground state energy of diatomic hydrides as an example, the dependence of the effective potential parameters on the number of particles in the system is analyzed in detail. Based on this analysis, a simple two-parameter model is proposed. The results of calculations in terms of this model deviate from those obtained by the precision Hartree-Fock method by 0.02–0.06%.     

Three-to-one internal resonances in a general cubic non-linear continuous system

A general continuous system with an arbitrary cubic non-linearity is considered. The non-linearity is expressed in terms of an arbitrary cubic operator. Three-to-one internal resonance case is considered. A general approximate solution is presented for the system. Amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed in the general sense. The sufficiency condition for such resonances to occur is derived. Finally the algorithm is applied to a beam resting on a non-linear elastic foundation.     

Normal modes of oscillations of one-dimensional monoatomic and diatomic lattices

A polynomial equation is obtained for the solutions of the vibrational frequencies of one-dimensional monoatomic and diatomic lattices with particles connected by identical springs, but with arbitrary springs connecting the end particles to rigid walls. The exact expressions of the different normal modes of oscillations of the linear chain of particles for monoatomic, diatomic and defective lattices are derived in a straightforward way. As special cases of our problem we have considered the effects of different end springs on the vibrational frequencies. One interesting result is that very high frequencies are allowed when the ends of the diatomic lattice are rigidly fixed with the boundary walls.     

Partial differential equations possessing Frobenius integrable decompositions

Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Bäcklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.     

General behavior of longitudinal optical phonons in cubic diatomic crystals

A general relationship for longitudinal optical (LO) phonons in diatomic cubic crystals is introduced. The analysis of spectroscopic data of diatomic cubic crystals reveals that LO frequencies correlate directly with bulk moduli in a nearest-neighbor fashion. The proportionality constant is found to be dependent only on lattice geometries. Transverse effective charges can be obtained when the Lyddane-Sachs-Teller is incorporated into the model. Our calculations are in excellent agreement with experimental values and theoretical calculations. Some interesting implications of the model are discussed.     

Rotation and vibration of diatomic molecule oscillator with hyperbolic potential function

The explicit expressions of energy eigenvalues and eigenfunctions of bound states for a three-dimensional diatomic molecule oscillator with a hyperbolic potential function are obtained approximately by means of the hypergeometric series method. Then for a one-dimensional system, the rigorous solutions of bound states are solved with a similar method. The eigenfunctions of a one-dimensional diatomic molecule oscillator, expressed in terms of the Jacobi polynomial, are employed as an orthonormal basis set, and the analytic expressions of matrix elements for position and momentum operators are given in a closed form.     

具有在位势的一维双原子链晶格振动的长声学波图像

通过数值求解具有在位势的一维双原子链晶格振动的运动方程组,给出了具有不同大小在位势的长声学波振动模图像,直观展示出在位势使原胞中两种原子的振动位移不同,并且随在位势的增大,振动位移之差显著增大.     

Approximate Analytical Solutions of Dirac Equation with Spin and Pseudo Spin Symmetries for the Diatomic Molecular Potentials Plus a Tensor Term with Any Angular Momentum

Approximate analytical solutions of the Dirac equation are obtained for some diatomic molecular potentials plus a tensor interaction with spin and pseudospin symmetries with any angular momentum. We find the energy eigenvalue equations in the closed form and the spinor wave functions by using an algebraic method. We also perform numerical calculations for the Pöschl-Teller potential to show the effect of the tensor interaction. Our results are consistent with ones obtained before.     

Algebraic approach to thermodynamic properties of diatomic molecules

A simple model extending Lie algebraic techniques is applied to the analysis of thermodynamic vibrational properties of diatomic molecules. Local anharmonic effects are described by means of a Morse-like potential and the corresponding anharmonic bosons are associated with the SU(2) algebra. The total number of anharmonic bosons, fixed by the potential shape, is determined for a large number of diatomic molecules. A vibrational high-temperature partition function and the related thermodynamic functions are derived and studied in terms of the parameters of the model. The idea of a critical temperature is introduced in relation to the specific heat. A physical interpretation in terms of a quantum deformation associated with the model is given.     

Relaxation algorithm to hyperbolic states in Gross–Pitaevskii equation

A new version of the relaxation algorithm is proposed in order to obtain the stationary ground-state solutions of nonlinear Schrödinger-type equations, including the hyperbolic solutions. In a first example, the method is applied to the three-dimensional Gross–Pitaevskii equation, describing a condensed atomic system with attractive two-body interaction in a non-symmetrical trap, to obtain results for the unstable branch. Next, the approach is also shown to be very reliable and easy to be implemented in a non-symmetrical case that we have bifurcation, with nonlinear cubic and quintic terms.     

The analysis of rovibrational motion equations of a diatomic molecule in the linear regime

The motion equations of diatomic molecules are here extended from the absolute vibrational case to a more general and real rotational and vibrational (rovibrational) case. The rovibrational Hamiltonian is heuristically formed by substituting the respective number and angular momentum operators for the vibrational and rotational quantum numbers in the energy eigenvalues of a diatomic molecule which was first introduced by Dunham. The motion equations of observable quantities such as the position and linear momentum are then determined by implementing the well-known Heisenberg relation in quantum mechanics. We face with the second-order imaginary differential equations for describing the temporal variations of the relative position and the linear momentum of two oscillating atoms, which are coupled in the x–y horizontal plane. The possible rovibrational oscillations are distinguished by the three quantum numbers n, l and m associated with the energy and angular momentum quantities. It is finally demonstrated that the simultaneous solutions of rovibrational equations satisfy the energy conservation during all quantised oscillations of a diatomic molecule in space.     

Response of a semi-infinite cubic lattice to uniform electric fields

Self-consistent field equations for the dipole moments of point polarizable atoms in slabs of cubic lattices with a uniform applied electric field are constructed. Results of Toeplitz operator theory are used to characterize the ranges of atomic polarizability for which there are unique solutions to the equations. A normal mode analysis of the frequency spectrum of the coupled dipole lattice is given and is used in the interpretation of the results for the simple cubic lattice. Approximate solutions of the self-consistent field equations for the semi-infinite lattice are constructed which display the exponential approach of the atomic dipoles to their infinite lattice value with increasing penetration into the lattice.