Phy592 Chapter 3: Lattice Vibrations and Phonons

Questions and Answers

What are lattice vibrations?

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

The vibration of atoms is completely independent of each other.

Lattice vibrations can explain sound velocity, thermal properties, elastic properties, and optical properties of materials. (correct)

In the Einstein model, atoms are assumed to oscillate independently of each other.

True

What is the equation of motion for the nth atom in a one-dimensional lattice wave?

Mu_{n} = K (u_{n+1} - 2u_{n} + u_{n-1})

The Law of Dulong Petit gives the heat capacity for many solids as _______.

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Match the scientist with their contributions: 1. Boltzmann 2. Einstein 3. Debye

Boltzmann = Constructed a model for heat capacity in solids Einstein = Proposed atoms are in identical harmonic wells with oscillation frequency Debye = Discovered a quantum mechanical treatment for atom oscillations Signup and view all the answers

What can be used to describe lattice vibration in a one-dimensional monoatomic solid?

Group velocity Vg and phase velocity Vp Signup and view all the answers

State the expression for the frequency difference 𝜟𝛚 between the optical and acoustic branches at q = 0 for a diatomic one-dimensional lattice in terms of masses M1 and M2 and the interatomic force constant 𝒂.

Frequency difference 𝜟𝛚 = √(𝑎/(M1 + M2)) Signup and view all the answers

What are the two modes of vibrations that exist in a one-dimensional alternating chain of two types of atoms connected by a spring?

Acoustic and optical modes Signup and view all the answers

What can be computed based on the 𝛚 vs q plot in a one-dimensional alternating chain of two types of atoms?

Maximum frequency gap 𝜟𝛚max between the optical branch and the acoustic branch Signup and view all the answers

Study Notes

Lattice Vibrations and Phonons

Lattice vibrations can explain sound velocity, thermal properties, elastic properties, and optical properties of materials.
Lattice vibration is the oscillation of atoms in a solid about their equilibrium position, which forms a regular lattice due to the binding of atoms to neighboring atoms.

Vibration of Monoatomic Lattices (1D)

A monoatomic lattice consists of identical atoms with identical masses and equilibrium spacing 'a' between atoms.
The vibration of neighboring atoms is not independent of each other and can be modeled using harmonic forces and normal modes of vibrations, called lattice waves.

Equation of Motion for Monoatomic Lattice

The equation of motion for the nth atom can be derived by considering nearest-neighbor interaction and Newton's equation.
The equation of motion is Mu'' = K(un+1 - 2un + un-1), where Mu'' is the acceleration of the nth atom, K is the spring constant, and un is the displacement of the nth atom.

Dispersion Relation for Monoatomic Lattice

The dispersion relation for vibrations of the one-dimensional monoatomic harmonic chain is ω(k) = √(2K/m) |sin(ka/2)|, where ω is the angular frequency, k is the wavevector, K is the spring constant, m is the mass of the atom, and a is the equilibrium spacing between atoms.
The dispersion relation is periodic in k-space with periodicity 2π/a, and the Brillouin Zone (BZ) is defined as the unit cell in k-space centered around k = 0.

Vibrations of a One-Dimensional Diatomic Chain

A diatomic lattice consists of two types of atoms with different masses (M1 and M2) and spring constants (κ1 and κ2) that alternate along the one-dimensional chain.
The equation of motion for the diatomic lattice can be derived by considering nearest-neighbor interaction and Newton's equation.
The dispersion relation for the diatomic lattice exhibits two branches: the acoustic branch and the optical branch.

Acoustic and Optical Modes

The acoustic mode has linear dispersion as k → 0 and corresponds to sound waves.
The optical mode has a higher energy vibration and requires a certain amount of energy to excite.
In a diatomic chain, the optical mode corresponds to the vibration of the lighter atom, while the acoustic mode corresponds to the vibration of the heavier atom.

Specific Heat of Solids

The specific heat of solids is due to the vibration of atoms, which requires an input of heat energy.
The Law of Dulong-Petit states that the specific heat of many solids is proportional to the atomic mass.
Einstein's model assumes that each atom is in a harmonic well created by the interaction with its neighbors and has an oscillation frequency ω.
Debye's model treats the lattice vibrations as sound waves and assumes that the frequency of the lattice vibrations covers a wide range of values.

Debye's Calculation

Debye realized that the oscillation of atoms is the same as sound waves and should be quantized the same way as Planck had quantized light waves.
Debye's model provides a better treatment of the quantum mechanics of oscillations of atoms.