Lecture 6

Question 1

Describe the graph of the forces between pairs of atoms

Answer 1

Thick line = potential energy as a function of separation
Thin line = harmonic approximation of the potential

Question 2

What is the name of the graph describing the forces between pairs of atoms?

Answer 2

The Lennard-Jones potential

Question 3

Atoms in a crystal can vibrate about their ___________ positions.

Answer 3

Equilibrium

Question 4

For small amplitude oscillations, the potential energy as a function of separation of neighbouring atoms is _________ (harmonic).

Answer 4

Parabolic

Question 5

What is the restoring force between neighbouring atoms?

Answer 5

The force that brings the two atoms back to their equilibrium separation, proportional to the displacement from equilibrium.

Question 6

What analogy can be used to describe the movement of atoms in a crystal?

Answer 6

They behave like an array of masses coupled by springs.

Question 7

Describe a 1D chain of atoms of mass, m, and a lattice constant, a

Answer 7

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.

Question 8

Give the equation of motion for a 1D chain of atoms of mass, m, with lattice constant, a

Answer 8

C = force constant
u_n = displacement of n-th atom
m = mass

Question 9

What is the equation of motion for a 1D chain of atoms of mass, m, with lattice constant, a, equivalent to?

Answer 9

The force on the n-th atom (that causes motion) due to the atoms either side of it.

Question 10

Give the travelling wave solution for the displacement of the n-th atom in a 1D chain of atoms

Answer 10

u_n = displacement of n-th atom
u0 = amplitude
k = wavenumber = 2π/λ
x = na
ω = angular frequency
t = time
a = lattice constant

Question 11

Give the equation for the dispersion relationship for vibrations of a monatomic 1D chain of atoms with nearest neighbour harmonic forces

Answer 11

ω = angular frequency
k = wavenumber
C = force constant
m = mass
a = lattice constant

Question 12

Describe the shape of the dispersion relationship for vibrations of a monatomic 1D chain of atoms with nearest neighbour harmonic forces

Answer 12

Question 13

Give the equation for the group velocity of the wave from a 1D chain of atoms

Answer 13

v_g = group velocity

Question 14

Give the equation for the phase velocity of a wave from a 1D chain of atoms

Answer 14

v_p = phase velocity

Question 15

Describe the value of the wavenumber (k) of a wave from a 1D chain of atoms in the long wavelength limit

Answer 15

In this limit k is small

Question 16

How is the phase velocity related to the group velocity of a wave from a 1D chain of atoms in the long wavelength limit?

Answer 16

The group velocity and phase velocity are equal.

Question 17

Give the equation equating the group and phase velocities for a 1D chain of atoms in the long wavelength limit

Answer 17

v_g = group velocity
v_p = phase velocity
C = force constant
m = mass
a = lattice constant

Question 18

What is the equation for the group velocity equivalent to in the long wavelength limit for a 1D chain of atoms?

Answer 18

It has a similar form to that of an elastic continuum where B is the bulk modulus and ρ is the density.

Question 19

What can the long wavelength limit for a 1D chain of atoms be used to describe?

Answer 19

It can be used to relate the macroscopic elastic properties of the bulk modulus and density to inter-atomic forces.

Question 20

Describe the value of the wavenumber (k) of a wave from a 1D chain of atoms in the short wavelength limit

Answer 20

k = ± π/a

Question 21

What are the values of ω(k) and dω/dk in the short wavelength limit for a 1D chain of atoms?

Answer 21

ω(k) = non-linear
dω/dk = group velocity = zero

Question 22

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?

Answer 22

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).

Question 23

What is the speed of sound equivalent to in a solid?

Answer 23

The group velocity of the atoms

Question 24

All physical vibration modes correspond to k-vectors within the ______ __________ ____.

Answer 24

First Brillouin zone

Question 25

What is the region of the first Brillouin zone?

Answer 25

• π/a to + π/a

Question 26

Give the equation for any wavevector outside the first Brillouin zone

Answer 26

k’ = wavevector outside the first Brillouin zone
k = wavevector inside the first Brillouin zone
l = integer
a = lattice constant

Question 27

Describe the difference in a wavevector inside the first Brillouin zone compared to outside of it

Answer 27

Red: k-value outside first Brillouin zone (k = 5π/a)
Black: k-value inside the first Brillouin zone (k = π/a)

Question 28

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

Answer 28

u_n = displacement of the n-th atom
k = wavevector
a = lattice constant