Lecture 7 Flashcards
How do the modes of a crystal change when the basis has more than one atom type?
There are additional vibrational modes
Describe the model of a diatomic chain of atoms where the masses m1 and m2 (where m1 > m2) alternate
a = lattice parameter
u_n = displacement of n-th atom
Atom n is at a distance na from the origin. Atom n+1/2 is at a distance (n+1/2)a from the origin.
Give the two equations of motion for a diatomic chain of atoms
Give the travelling wave solution for the displacement of the n-th atom in a diatomic chain of atoms
u_n = displacement of the n-th atom
A = amplitude
Give the travelling wave solution for the displacement of the (n+1/2)-th atom in a diatomic chain of atoms
u_n+1/2 = displacement of the (n+1/2)-th atom
B = amplitude
Give the equation for the dispersion relationship for vibrations of a diatomic chain of atoms
ω = angular frequency
C = force constant
m1, m2 = mass
k = wavevector
a = lattice constant
Describe the shape of the dispersion relationship for vibrations of a diatomic chain of atoms with nearest neighbour harmonic forces
What is the optical mode of the dispersion relationship for vibrations of a diatomic chain of atoms?
The branch that occurs when the square root term is positive. m1 and m2 oscillate in anti-phase.
Give the equation for the optical mode of the dispersion relationship for vibrations of a diatomic chain of atoms
Describe the limiting case of the optical mode
m1»_space; m2
The heavy atoms are almost stationary and only the light atoms vibrate.
How can optical modes be driven?
By oscillating the electric fields of photons.
What is the acoustic mode of the dispersion relationship for vibrations of a diatomic chain of atoms?
The branch that occurs when the square root term is negative. m1 and m2 oscillate in phase.
Give the equation for the optical mode of the dispersion relationship for vibrations of a diatomic chain of atoms
What is a longitudinal wave?
Atoms that oscillate along the direction of propagation
What is a transverse wave?
Atoms that oscillate perpendicular to the propagation direction. There are two distinct polarisations.
What are the 3 modes that a 3D crystal has?
1 longitudinal mode and 2 transverse modes for each k-value.
What are the 4 possible types of modes for a vibrating 1D crystal?
LA: longitudinal acoustic mode
TA: transverse acoustic mode
LO: longitudinal optical mode
TO: transverse optical mode
Describe the graph of the dispersion relations for a 1D crystal lattice
What k-states are allowed for a finite crystal?
Only specific quantised k-states
There is only one allowed k-state per length ____ of the k-axis in k-space.
2π/L
Give the equation for the total number of allowed states of a 1D chain of atoms in k-space
2π/L = area of one allowed k-state
a = lattice constant
N = number of allowed states
There is only one allowed k-state per volume ____ of the k-axis in k-space.
(2π)³/V
Give the equation for the total number of allowed states of a 3D crystal in k-space
(2π)³/V = area of one allowed k-state
a = lattice constant
N = number of allowed states
The number of allowed k-states is equal to the number of atoms in the _____ or _______.
Chain
Crystal
How many allowed states are there for a 1D chain of atoms / a 3D crystal
3N
(from 2 transverse and 1 longitudinal mode)
Give the equation for the density of k-states in 1D
Give the equation for the density of angular frequency states in 1D
Give the equation for the density of k-states in 3D
Give the equation for the density of angular frequency states in 3D
