Question

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Question : Explain the thermal, acoustic and optical properties of lattice vibrations?
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Answer 1

ANSWER:

Lattice Vibration is the oscillations of atoms in a solid about the equilibrium position.

* Thermal Properties of Lattice Vibrations:-

Heat energy in a solid can be carried by electronic carriers (electrons and holes), lattice waves (phonons), electromagnetic waves, spin waves, or other excitations. In metals, the thermal conductivity is mainly due to the electrons, while in semiconductors and insulators it can be accredited with lattice waves (phonons). The conductivity can simply be given as

K = ∑ Kα (20)

Where, α is an excitation. Thermal conductivities in solids vary dramatically in magnitude and temperature depending on the material and its dimensions. This is due to number of parameters such as lattice defects or imperfections, dislocations, anharmonicity of the lattice forces, carrier concentrations, interactions between the carriers and the lattice waves, interactions between magnetic ions and the Lattice waves, etc.

* Acoustic Properties of Lattice Vibrations:-

Acoustical phonons are phonons whose frequency which goes to zero in the limit of small k. Let’s consider monatomic a linear chain of identical atoms of mass ‘M’ spaced at a distance ‘a’, the lattice constant, connected by invisible Hook's law springs and longitudinal deformations.

• Un =displacement of atom n from its equilibrium position
• Un−1=displacement of atom n-1 from its equilibrium position
• Un+1=displacement of atom n+1 from its equilibrium position

The force on atom ‘n’ will be given by its displacement and the displacement of its nearest neighbors:

Fn = β (Un+1 − 2Un + Un−1)

The equation of motion is:

M ( ∂²Un / ∂t² ) = β (Un+1 − 2Un + Un−1)

where β is a spring constant.

With wave solution

Un = Uno e ^[i(kna±ωt)]

If Uno=Uo and has a definite amplitude.

ω = ± 4βM sin (ka / 2).

* Optical Properties of Lattice Vibrations:-

Compared to the static lattice model that deals with the average positions of atoms in a crystal; lattice dynamics works towards extending the concept of crystal lattice to an array of atoms with finite masses capable of motion. The motion of masses is a superposition of vibrations of atoms around the equilibrium sites induced by the interaction with neighboring atoms. The collective vibration of the atoms within the crystal forms a wave of allowed wavelength and amplitude. For example, as we know that light is said to be a wave motion composed of photons, we can also think of the normal modes of vibration in a solid as being a particle. One major problem with Lattice dynamics is that is hard to find the normal modes of vibration of the crystal. However, lattice dynamic, offers two different ways of finding the dispersion relation within the lattice.

1. Quantum-mechanical approach: Quantum-mechanical approach can be used to obtain phonon's dispersion relation. In order to do so, the solution to the Schrodinger equation for the lattice vibrations must be solved. The quantum-mechanical operators for the normal mode coordinates are substituted in the phonon Hamiltonian by the creation and annihilation operators.
2. Semi-classical treatment of lattice vibrations: The semi-classical treatment gives classical mechanics the use of one additional postulate taken from quantum mechanics, mainly that the energy of lattice vibrations is quantized. The classical motions of any atom are determined by Newton's law of mechanics: force=mass x acceleration. Formally, if r(t)is the position of atom at time t, then

   ∂²r(t) / ∂t² = −1/m ∇ϕ(r,t)

   where m is the atomic mass, and ϕ(r,t)is the instantaneous potential energy of the atom with the other atoms within the crystal.