Lecture 6 Flashcards
Describe the graph of the forces between pairs of atoms
Thick line = potential energy as a function of separation
Thin line = harmonic approximation of the potential
What is the name of the graph describing the forces between pairs of atoms?
The Lennard-Jones potential
Atoms in a crystal can vibrate about their ___________ positions.
Equilibrium
For small amplitude oscillations, the potential energy as a function of separation of neighbouring atoms is _________ (harmonic).
Parabolic
What is the restoring force between neighbouring atoms?
The force that brings the two atoms back to their equilibrium separation, proportional to the displacement from equilibrium.
What analogy can be used to describe the movement of atoms in a crystal?
They behave like an array of masses coupled by springs.
Describe a 1D chain of atoms of mass, m, and a lattice constant, a
Blue disks: the equilibrium position of each atom
Yellow disks: the position of each atom in the present of a lattice vibration at some time, t.
The n-th atom is displaced from its equilibrium position by a displacement, u_n.
Give the equation of motion for a 1D chain of atoms of mass, m, with lattice constant, a
C = force constant
u_n = displacement of n-th atom
m = mass
What is the equation of motion for a 1D chain of atoms of mass, m, with lattice constant, a, equivalent to?
The force on the n-th atom (that causes motion) due to the atoms either side of it.
Give the travelling wave solution for the displacement of the n-th atom in a 1D chain of atoms
u_n = displacement of n-th atom
u0 = amplitude
k = wavenumber = 2π/λ
x = na
ω = angular frequency
t = time
a = lattice constant
Give the equation for the dispersion relationship for vibrations of a monatomic 1D chain of atoms with nearest neighbour harmonic forces
ω = angular frequency
k = wavenumber
C = force constant
m = mass
a = lattice constant
Describe the shape of the dispersion relationship for vibrations of a monatomic 1D chain of atoms with nearest neighbour harmonic forces
Give the equation for the group velocity of the wave from a 1D chain of atoms
v_g = group velocity
Give the equation for the phase velocity of a wave from a 1D chain of atoms
v_p = phase velocity
Describe the value of the wavenumber (k) of a wave from a 1D chain of atoms in the long wavelength limit
In this limit k is small
How is the phase velocity related to the group velocity of a wave from a 1D chain of atoms in the long wavelength limit?
The group velocity and phase velocity are equal.
Give the equation equating the group and phase velocities for a 1D chain of atoms in the long wavelength limit
v_g = group velocity
v_p = phase velocity
C = force constant
m = mass
a = lattice constant
What is the equation for the group velocity equivalent to in the long wavelength limit for a 1D chain of atoms?
It has a similar form to that of an elastic continuum where B is the bulk modulus and ρ is the density.
What can the long wavelength limit for a 1D chain of atoms be used to describe?
It can be used to relate the macroscopic elastic properties of the bulk modulus and density to inter-atomic forces.
Describe the value of the wavenumber (k) of a wave from a 1D chain of atoms in the short wavelength limit
k = ± π/a
What are the values of ω(k) and dω/dk in the short wavelength limit for a 1D chain of atoms?
ω(k) = non-linear
dω/dk = group velocity = zero
Why do the vibrations of atoms in a 1D lattice for a 1D chain of atoms form a standing wave solution in the short wavelength limit?
Because k = ± π/a and λ = 2a which forms a solution that can’t propagate with adjacent atoms moving in antiphase (i.e. a standing wave).
What is the speed of sound equivalent to in a solid?
The group velocity of the atoms
All physical vibration modes correspond to k-vectors within the ______ __________ ____.
First Brillouin zone
What is the region of the first Brillouin zone?
- π/a to + π/a
Give the equation for any wavevector outside the first Brillouin zone
k’ = wavevector outside the first Brillouin zone
k = wavevector inside the first Brillouin zone
l = integer
a = lattice constant
Describe the difference in a wavevector inside the first Brillouin zone compared to outside of it
Red: k-value outside first Brillouin zone (k = 5π/a)
Black: k-value inside the first Brillouin zone (k = π/a)
Give the equation for the physical displacement of the n-th atom in a 1D chain of atoms given that the wavevector, k, is outside the first Brillouin zone
u_n = displacement of the n-th atom
k = wavevector
a = lattice constant
