What does the phonon dispersion, that is, the vibrational frequency as

Question

What does the phonon dispersion, that is, the vibrational frequency as

a function of wave vector 𝜔(k), look like for an infinite chain of atoms with
one atom per unit cell?

Answer 1

Not my thing but this might help:

http://users.aber.ac.uk/ruw/teach/334/disprel.php

Answer 2

Thanks anyway

Answer 3

For an infinite chain of atoms with one atom per unit cell, the phonon dispersion relationship can be described as a sinusoidal function.

Assuming a linear chain, the Hamiltonian governing the dynamics of the atoms can be written as:

H = Σ[1/2m (pₙ² + ω₀²xₙ²) + 1/2K(xₙ₊₁ - xₙ)²]

Here, m is the mass of the atoms, pₙ is the momentum of the nth atom, xₙ represents the displacement of the nth atom from its equilibrium position, ω₀ is the natural frequency of each individual atom, K is the force constant, and the term (xₙ₊₁ - xₙ) represents the displacement between adjacent atoms.

To find the phonon dispersion relationship, we can use the equation of motion for this system. Assuming a harmonic solution of the form xₙ(t) = X exp(i(ωt - kRₙ)), where X is the amplitude, ω is the angular frequency, and Rₙ is the position of the nth atom, we can substitute this back into the equation of motion to obtain:

(ω² - ω₀²) X = -2K(X/R)(1 - cos(kR))

Dividing both sides by X and rearranging the terms, we get:

ω² - ω₀² = -2K/R Σ(1 - cos(kR)) exp(-ikR)

Using the identity Σ(1 - cos(kR)) exp(-ikR) = N, where N is the number of atoms in the chain, and summing over all R values, we get:

ω² - ω₀² = -2KN(1 - cos(kR₀))

Here, R₀ is the equilibrium distance between adjacent atoms.

Finally, simplifying the equation, we arrive at:

ω² = ω₀² + 4K/m sin²(kR₀/2)

This is the expression for the phonon dispersion relationship for an infinite chain of atoms with one atom per unit cell. It shows that the square of the vibrational frequency (ω²) depends on the square of the natural frequency (ω₀²) and the force constants (K), as well as the wavevector (k) of the phonons. The dispersion relationship takes the form of a sinusoidal function, with deviations from the natural frequency determined by the force constants and the wavevector.

Answer 4

To understand the phonon dispersion of an infinite chain of atoms with one atom per unit cell, we need to consider the vibrational modes of the system. Each atom in the chain can oscillate around its equilibrium position, leading to the collective vibrations known as phonons.

In this specific case, since there's only one atom per unit cell, the unit cell itself coincides with the atom. So, the concept of wave vector (k) is equivalent to the reciprocal lattice vector (𝛼) in this 1D system.

To visualize the phonon dispersion, we can use a simple model known as the linear chain model. In this model, we consider that atoms are connected by springs of identical force constants. The equilibrium positions of atoms form a regular lattice.

Now, let's discuss the dispersion relation 𝜔(k), which represents the vibrational frequency as a function of wave vector k. In the linear chain model, for an infinite chain of atoms with one atom per unit cell, the dispersion relation can be written as:

𝜔 = 2√(𝑁𝐾/𝑀) |sin(𝛼a/2)|

Where:
- 𝜔 represents the vibrational frequency,
- 𝑁 is the number of unit cells in the chain,
- 𝐾 is the force constant of the spring connecting adjacent atoms,
- 𝑀 is the mass of each atom, and
- 𝛼 is the reciprocal lattice vector associated with the chosen wave vector k,
- 𝑎 is the lattice constant.

From this equation, we observe that the dispersion relation 𝜔(k) is symmetric with respect to 𝛼 = 0. It means the phonon frequencies are the same for k and -k (reciprocal lattice vectors). This is because the chain is infinite and has translational symmetry.

The plot of the phonon dispersion shows the relationship between the vibrational frequency 𝜔 and wave vector k. It typically appears as a band structure graph, with multiple dispersion curves known as branches. In this case, for an infinite chain of atoms with one atom per unit cell, we'll have only one dispersion branch.

Note: The dispersion relation I presented assumes simple harmonic motion, and neglects any anharmonicity effects or other intricacies that may exist in realistic systems.