Applicability Analyses from the Viewpoint of Dynamics of Atoms in Crystal

 

Dynamics of Atoms in Crystal

There are a number of material features, such as chemical properties, material hardness, material symmetry, that can be explained by static atomic structure. There are, however, a large number of technically important properties that can only be understood on the basis of lattice dynamics. These include:

thermal properties,
thermal conductivity, temperature effect, energy dissipation, sound propagation, phase transition,thermal conductivity, piezoelectricity, dielectric and optical properties, thermo-mechanical-electromagnetic coupling properties.

The atomic motion

s,
that are revealed by those features, are not random., Iin fact they are determined by the forces that atoms exert on each other, and most readily described not in terms of the vibrations of individual atoms, but in terms of traveling waves, as illustrated in Fig.1. Those waves are the normal modes of vibration of the system. The quantum of energy in an elastic wave is called a Phonon; a quantum state of a crystal lattice near its ground state can be specified by the phonons present; at very low temperature a solid can be regarded as a volume containing non-interacting phonons. The frequency-wave vector relationship of phonons is called Phonon Dispersion Relation, which is the fundamental ingredient in the theory of lattice dynamics and can be determined through experimental measurements, such as nNeutron scattering, iInfrared spectroscope and Raman scattering, or first principle calculations or phenomenological modeling. Through phonon dispersion relations, the dynamic characteristics of an atomic system can be represented, the validity of a calculation or a phenomenological modeling can be examined, interatomic force constants can be computed, Born effective charge, on which the strain induced polarization depends, can be obtained, various involved material constants can be determined.

.

Fig.1 Typical motions for two atoms in a unit cell,

where 'L' stands for longitudinal, 'T' transverse, 'A' acoustic, 'O' optical

Optical Phonons

Optical phonon branches exist in all crystals that have more than one atom per primitive unit cell. In such crystals, the elastic distortions give rise to wave propagation of two types. In the acoustic type (as LA and TA), all the atoms in the unit cell move in essentially the same phase, resulting in the deformation of lattice, usually referred as homogeneous deformation. In the optical type (as LO and TO), the atoms move within the unit cell, leave the unit cell unchanged, contribute to the discrete feature of an atomic system, and give rise to the internal deformations. In an optical vibration of non-central ionic crystal, the relative displacement between the positive and negative ions gives rise to the piezoelectricity. Optics is a phenomenon that necessitates the presence of an electromagnetic field. In <!--[if !vml]--><!--[endif]--> ferroelectrics the anomalously large Born effective charges produce a giant LO-TO splitting in phonon dispersion relations. This feature is associated

with
to the existence of an anomalously large destabilizing dipole-dipole interaction, sufficient to compensate the stabilizing short-range force and induce the ferroelectric instability. Optical phonons, therefore, appears as the key concept to relate the electronic and structural properties through Born effective charge (Ghosez [1995,1997]). The elastic theory of continuum is the long wave limit of acoustic vibrations of lattice, while optical vibration is the mechanism of a lot of macroscopic phenomena involving thermal, mechanical, electromagnetic and optical coupling effects.

Dynamic Feature of Various Types of Crystals

The dynamic characteristics of crystals depend on crystal structures, as shown in Fig.2, and the binding between the atoms. In metals the atomic cores are surrounded by a more-or-less uniform density of free electrons. This gives metals their electrical conductivity and a nonlocal character of the interatomic potential. Its dynamic feature is represented by the dispersive acoustic vibrations. In ionic crystals, strong Coulomb forces and short-range repulsive forces operate between the ions, and the ions are polarizable. The covalent bond is usually formed from two electrons, one from each atom participating in the bond. These electrons tend to be partially localized in the region between the two atoms and constitute the bond charge. The phonon dispersion relations of ionic and covalent crystals have both acoustic and optical branches, their optical vibrations describe the internal motion of atoms within the primitive basis, as in Fig.1 and Fig.3. In molecular crystal there is usually a large difference between the frequencies of modes in which the molecules move as a united units (the external modes) and the modes that involve the stretch and distortion of the molecules (the internal modes). The framework crystals are similar to molecular crystals in that they are composed of rigid groups. The units are very stiff but linked flexibly to each other at the corner atoms. The phonon dispersion relations, as in Fig.4, of molecular and framework crystals include both acoustic and optical vibrations, and the optical vibrations further include internal modes and external modes.

Phonon Dispersion Relations by Various Microcontinuum Theory

Micromorphic theory (Eringen and Suhubi [1964], Eringen [1999])

Micromophic theory views a material as a continuous collection of deformable particles. Each particle is attached with a microstructure of finite size. The deformation of a micromorphic continuum yields both macro-strains (homogeneous part) and microscopic internal strains (discrete part).

Micropolar theory (Eringen and Suhubi [1964])

When the material particle is considered as rigid, i.e., neglecting the internal motion within the microstructure, micromorphic theory becomes micropolar theory. Therefore, micropolar theory yields only acoustic and external optical modes. They are the translational and rotational modes of rigid units. For molecular crystals or framework crystal, or chopped composite, granular material et al, when the external modes in which the molecules move as rigid units have much lower frequencies and thus dominate the dynamics of atoms, micropolar theory can give a good description to the dynamics of microstructure. It accounts for the dynamic effect of material with rather stiff microstructure.

Assuming a constant microinertia, micropolar theory is identical to Cosserat theory [1902], Compared with micropolar theory, Cosserat theory is limited to problems not involving significant change of the orientation of the microstructure, such as liquid crystal and ferroelctrics.

For isotropic material, the phonon dispersion relations based on a nonlocal theory have been obtained by Eringen [1992] as shown in Fig.7. Remarkable similarity to atomic lattice dynamics solution with Born-von Karman model, and to the experimental results for Aluminum has been reported.

Non local theory takes long-range interatomic interaction into consideration. As a consequence it yields results dependent on the size of a body. Similar to classical continuum theory, the lattice particles are taken without structure and idealized as point masses. Hence, the effect of microstructure does not appear. It is not a theory for material with microstructure, but for material involving long-range interaction. It can be applied to crystal that has only one atom per primitive unit cell at various length scales.

ORLANING COLMENARES

C.I.V.- 18.991.89

Referencias Bibliograficas:

www.seas.gwu.edu/~jdlee/index_files/ijss-applicability.doc