4. Free electron model Flashcards
The removal of the _______ ________ leaves a positively charged ion core (not a nucleus as it still has the core electrons).
Valence electrons
What is the free electron model?
The assumption that the electrons in a metal behave like a gas of free particles. It is also assumed that the charge density of ion cores is spread evenly throughout the metal so that electrons move in a constant electrostatic potential.
What is the free electron model also known as?
The particle in a box approximation
Describe the particle in a box approximation
Electrons within a crystal are considered to be moving independently in a square potential well of finite depth. The edges of the well correspond to the boundaries of the metal. The corrugations at the bottom of the real energy well are ignored here.
State the time-independent Schrodinger equation
m = electron mass
ψ = electron wavefunction
ε = electron energy
What assumption is made when calculating the energy of a particle in a box?
The potential inside the box equals 0.
What type of waves are found within a particle in a box?
Standing wave solutions because ψ = 0 outside the box.
How is the energy of an electron found using the free electron model?
- Adopt periodic boundary conditions for the wavefunctions.
- State the equation for the running solutions for the electron wavefunction.
- Find the 3 wavevector components that satisfy the periodic boundary conditions.
- Equate the wave function to momentum.
- Equate to momentum to energy.
Give the equation for the periodic boundary condition of a particle in a box
ψ = wavefunction
x, y, z = position
L = box length
Give the equation for the running solution for an electron wavefunction
ψ = wavefunction
x, y, z = position
V = L³ = volume
k = wave vector
State the wavevector components that satisfy the periodic boundary conditions for a particle in a box
k = wave vector
p, q, r = integers
L = box length
Give the equation that relates wavefunction to momentum for a particle in a box
p = momentum
λ = wavelength
k = wave vector
Give the equation that relates momentum to energy for a particle in a box
ε = energy
m = mass
v = velocity
p = momentum
k = wave vector
Each quantum state can be represented as a point in ______. The energy of that state is proportional to the _____ __ ____ _______ ____ ___ _______.
k-space
square of its distance from the origin
What is a distribution function?
A function that explains where the electrons all are in terms of energy at a given temperature. From this, we can calculate some ‘real’ properties of metals such as electrical conductivity and heat capacity
What is the energy density of states?
The number of electron states per energy range.
What is the occupation function?
A function that explains how the electrons will occupy the quantum states at a particular temperature.
Give the equation for the distribution function
n(ε,T) = distribution function
g(ε) = energy density
f(ε,T) = occupation function
How many quantum states are there between ε and ε + dε?
g(ε)dε
How many quantum states are there with k-vectors of length between k and k + dk?
g(k)dk
g(k) = density of k-states
Give the equation for the number of k-states
g(k) = density of k-states
V = crystal volume
k = k-state
How many electrons can fit in each k-state?
2
Give the equation for the energy density of states
g(ε) = energy density
V = crystal volume
m = mass
ε = energy
The Pauli exclusion principle says each electron state can accommodate only ____ ________.
one electron
What does the Fermi-Dirac function show?
It shows the probability that a state is occupied at a given temperature.
Give the equation for the Fermi-Dirac function (occupation function)
f = occupation function
ε_F = Fermi energy
k_B = Boltzmann’s constant
T = temperature
What is the probability of occupying a state at the Fermi energy?
Exactly 1/2
What order do electrons fill k-space in?
From the lowest energy states to the highest energy (the Fermi energy).
What is the Fermi sphere?
A sphere in k-space whose radius is equal to the Fermi wavevector. At the surface of this sphere, the energy is equal to the Fermi energy.
How are the number of free electrons in a crystal found?
By dividing the volume of the Fermi sphere by the volume of each electron state.
Give the equation for the number of free electrons in a crystal
N = number of electrons
k_F = Fermi wavevector
V = volume of crystal
Give the equation for the Fermi wavevector
k_F = Fermi wave vector
n = N / V = electron density
N = number of electrons
V = crystal volume
Give the equation for the Fermi energy
ε_F = Fermi energy
k_F = Fermi wave vector
m_e = electron mass
Electrons with the maximum wavevector have the _______ wavelength.
Minimum
Give the equation that relates wavelength to wavevector
λ = wavelength
k = wavevector
Describe be shape of a density of states graph
Describe the shape of an occupation function graph
Describe the shape of a distribution function graph
What happens to the distribution of occupied quantum states as a metal is heated above T=0?
Some quantum states above the Fermi energy will become occupied, and some states below it will become empty.
Give the equation for the heat capacity of electrons
C_v = heat capacity
∂ε_tot = change in total energy
∂T = change in temperature
Give the equation for the number of electrons in a particular energy range
N = number of electrons
g(ε) = energy density
f(ε,T) = occupation function
Give the equation for the total energy of a metal
ε_tot = total energy
ε = energy
g(ε) = energy density
f(ε,T) = occupation function
Which electrons can change their energy state?
Electrons within ~kB T of the Fermi energy (as the temperature is raised above 0K).
Give the equation for the number of electrons that can change their energy state
N = number of electrons
g(ε_F) = Fermi energy density
k_B = Boltzmann’s constant
T = temperature
Give the equation for the energy gained by electrons in a metal
ε_gain = energy gain
g(ε_F) = Fermi energy density
k_B = Boltzmann’s constant
T = temperature
Give the equation showing the exact solution of the heat capacity of electrons
C_v = heat capacity
g(ε_F) = Fermi energy density
k_B = Boltzmann’s constant
T = temperature
γ = electronic specific heat constant
When is the electron heat capacity comparable to the lattice heat capacity?
At very low temperatures
Give the equation for the experimental value of heat capacity
C_v = heat capacity
T = temperature
γ = electronic specific heat constant
β = lattice specific heat constant
Why doesn’t the electron gas contribute significantly to the heat capacity of metals?
Because only the few electrons close to the Fermi level can take up thermal energy.
Give the equation for the force on an electron in an applied electric field
F = force
e = electron charge
E = electric field
Give the equation for the momentum of a free electron
m = mass
v = velocity
mv = momentum
k = wavevector
Give the equation for the force on an electron in an electric field AND a magnetic field
F = force
m = electron mass
v = velocity
t = time
k = wavevector
e = electron charge
E = electric field
B = magnetic field
What is the impact of applying an electric field on the Fermi sphere?
It shifts the Fermi sphere. The displacement depends on how drastically the field strength and length of application changes the wavevector.
Give the equation for the change in wavevector that shifts the Fermi sphere
k = wavevector
e = electron charge
E = electric field
t = time
How do electron collisions impact the rate of displacement of the Fermi sphere in an electric field
When collisions occur they cause the displacement to maintain a steady state rather than to increase in acceleration steadily. This is because the change in wavevector depends on the collision time (a constant) rather than time (a variable).
Give the equation for the change in wavevector that shifts the Fermi sphere when there are electron collisions
k = wave vector
e = electron charge
E = electric field
τ = collision time
Give the equation for incremental velocity
v = incremental velocity
e = electron charge
τ = collision time
m = electron mass
E = electric field
What is the electron mobility?
The constant of proportionality that relates |v| to |E|
Give the equation for the electron mobility
µ = electron mobility
e = electron charge
τ = collision time
m = electron mass
Give the equation for the electric current density
j = electric current density
n = number of electrons
q = charge
v = incremental velocity
τ = collision time
e = electron charge
m = electron mass
E = electric field
σ = conductivity
Give the equation for conductivity
σ = conductivity
n = number of electrons
τ = collision time
e = electron charge
m = electron mass
µ = electron mobility
Is electron-photon scattering temperature dependent?
Yes (it has a temperature-dependent collision time that tends to infinity as temperature tends to 0K)
Is electron-defect scattering temperature dependent?
No
Give the equation for the collision time due to scattering in a metal
τ = collision time
τ_ph = temperature-dependent collision time (electron-photon scattering)
τ_0 = temperature-independent collision time (electron-defect scattering)
What is resistivity?
The reciprocal of conductivity
Give the equation for resistivity
ρ = resistivity
m = electron mass
n = number of electrons
e = electron charge
τ = collision time
τ_ph = temperature-dependent collision time (electron-photon scattering)
τ_0 = temperature-independent collision time (electron-defect scattering)
ρ1 = ideal resistivity
ρ0 = residual resistivity
What is Matthiessen’s rule?
The relationship between temperature and relative resistance for a metal.
It implies that two different samples of the same metal, one pure and one not, should exhibit identically shaped resistivity v. temperature curves but with an offset corresponding to the residual resistivity (which is in turn related to the metal purity).
Why do electrons move from hot to cold?
Because electrons travelling from hotter regions of the metal carry more thermal energy than those travelling from cooler regions, resulting in a net flow of heat.
Give the equation for electron kinetic energy at the Fermi energy
ε_F = Fermi energy
m = electron mass
v_F = Fermi velocity
Give the equation for the thermal conductivity of free electron gas (in terms of electronic specific heat)
K = thermal conductivity
C_v = electronic specific heat
v_F = Fermi velocity
l = mean free path between collisions
Give the full equation for the thermal conductivity of free electron gas
K = thermal conductivity
n = free electron density of the crystal
k_B = Boltzmann constant
T = temperature
τ = collision time
m = electron mass
State the Wiedmann-Franz law
The ratio of electrical conductivity to thermal conductivity is predicted to be the same for all metals.
Give the equation for the Weidmann-Franz law
K = thermal conductivity
σ = electrical conductivity
k_B = Boltzmann constant
T = temperature
e = electron charge
What is the Hall effect?
When a conductor is placed in a magnetic field, an electric field will build up perpendicular to both the magnetic field and the current.
Give the equation for the Hall effect
E_H = electric field
R_H = Hall coefficient
B = magnetic field
j = electric current density
Give the equation for the electric field of an electromagnetic wave
E(z,t) = electric field in the z-direction
E0 = field amplitude
k = wave number
z = position
ω = angular frequency
t = time
State the equation for the electric field of an electromagnetic wave in terms of the dielectric function
E(z,t) = electric field in the z-direction
E0 = field amplitude
ω = angular frequency
ε = energy
c = speed of light
z = position
t = time
What is the plasma frequency?
The frequency that an electron gas oscillates at about its equilibrium position when freed from restraint. It occurs because any displacement of the electron gas with respect to the lattice sets up a restoring electric field.
Give the equation for the polarization of a solid
P = polarisation
n = free electron density
e = electron charge
x = position
E = electric field
m = electron mass
ω = angular frequency
State the general relation between the electric field, E, and the dielectric displacement field, D
D = dielectric displacement field
ε = relative permittivity
ε0 = permittivity of free space
E = electric field
P = polarisation
Give the equation for the relative permittivity
ε = relative permittivity
P = polarisation
ε0 = permittivity of free space
E = electric field
Give the equation for plasma frequency
ω_p = plasma frequency
n = free electron density
e = electron charge
m = electron mass
ε0 = permittivity of free space
What determines whether an electron is ‘shiny’?
The plasma frequency. If the wavelength of incident light is smaller than the plasma frequency then there isn’t sufficient energy for transitions so the wave is reflected.
What is the plasma energy equal to?
What information is missing from the free electron model?
The lattice of the crystal
What is the free electron model good at explaining?
- Density of states
- Distribution function
- Heat capacity
- Electrical conductivity
- Thermal conductivity
- Hall effect (to some extent)
- Why metals are shiny
