FREE ELECTRON THEORY. ARC. TOPICS TO BE COVERED. Chief Characteristics of Metals. Metal possesses high electrical and thermal conductivity Metals obey Ohm’s law Conductivity of metals decreases with rise of temperature Metals obey Wiedemann -Franz law
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+ mechanics, so he began with a classical model.
+Free Electron Model
Schematic model of metallic crystal,
such as Na, Li, K, etc.
Equilibrium positions of the atomic cores are positioned on the crystal lattice
and surrounded by a sea of conduction electrons.
In presence of applied electric field, electrons move in a specific direction. This directional motion of the free electrons is called DRIFT.
Average velocity gained during this drift motion is called DRIFT VELOCITY.
Steady state drift velocity produced for unit electric field is called MOBILITY (μ)
Fig. Ref. Google
Relaxation Time ( mechanics, so he began with a classical model. 𝜏)
When the applied electric field is switched off, the electrons again undergo collision. The electron gas resumes its equilibrium condition. Such a process which leads to the establishment of equilibrium in a system from which it was previously disturbed is called the relaxation process. The time taken for this process is RELAXATION TIME.
Mean free path ( mechanics, so he began with a classical model. λ)
It is the average distance travelled by the conduction electron between successive collisions with the lattice ions.
Mean collision time ()
The average time taken by an electron between two successive collisions of an electron with lattice points during its motion. (averaging is done over a large number of collisions)
An mechanics, so he began with a classical model. electric field is applied. The equation of motion of free electron of mass is
Integrating, we get
If is the average time between collisions then the average drift velocity isDrift Velocity Expression
E mechanics, so he began with a classical model.
If J = current density for electric field E, then
, where σ= conductivity
Amount of charge passing per unit time = - mechanics, so he began with a classical model.
So, current density
J = = = ……… (1)
Distance covered =
From (1) and (2),
This is the form of Ohm’s Law in terms of free electrons.
Wiedemann and Franz law states that the ratio of thermal and electrical conductivity of all metals is constant at a given temperature
(for room temperature and above).
Later it was found by L. Lorenz that this constant is proportional to the absolute temperature
L = Lorentz Number
Drawbacks of Classical Free Electron Theory mechanics, so he began with a classical model.
Drawbacks (continued..) mechanics, so he began with a classical model.
FERMI LEVEL mechanics, so he began with a classical model.
: Highest energy level occupied by electrons at Absolute zero. All the energy states upto Fermi level are OCCUPIED and all energy levels above Fermi level are VACANT.
FERMI ENERGY: Energy corresponding to Fermi Level.
Constant for a particular system
Probability of an electron occupying a particular energy level ‘E’ is given by
At T = 0 K and for E < E , f(E) = 1
for E > E , f(E) = 0
For lower energies,
ftends to 1.
For higher energies,
ftends to 0.
On increasing the temperature, electrons get excited to higher energy level.
Distribution of electrons in different energy levels gets determined by Fermi-Dirac Distribution function.
Fermi Distribution Function at Different Temperatures mechanics, so he began with a classical model.
For temperatures greater than zero, Fermi function plot begins to fall close to E
and at E = E , f(E) =
FERMI mechanics, so he began with a classical model. VELOCITY= velocity associated with Fermi Energy
i.e. velocity of electrons occupying Fermi Level
= 3.2 X 1.6 X J
FERMI TEMPERATURE = Temperature associated with Fermi energy
Ref: Eisberg, R. and Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. New York: Wiley, 1985.
DENSITY OF ENERGY STATES mechanics, so he began with a classical model.
CONCEPT OF DENSITY OF ENERGY STATES simplifies OUR CALCULATION. It helps in finding
the number of energy states with a specific energy.
Density of states (DOS) of a system describes the number of available states in a unit volume per unit energy range.
In a system, if N(E) = number of electrons with energy E,
g(E) = number of energy states with energy E,
f(E) = probability of an electron to occupy energy state E,
N(E) dE = g(E)dEf(E)
We consider a free electron of mass ‘m’ trapped inside a cubical metal block of side length, ‘a’.
According to quantum mechanics, energy of the free electron,
----- (1) where h = Planck’s constant
Let us consider a space of points represented by coordinate system along the three mutually perpendicular directions.
Let each point with integer values of the coordinates represent an energy state.
Let n be the radius vector from origin (0,0,0) to a point represented by ().
So, + -------- (2)
All points on the surface of the sphere of radius ‘n’ will have the same energy.
As per quantum condition, values of are restricted to be positive.
Only in one octant of the sphere, each point corresponds to only positive values of .
SPACE OF POINTS cubical metal block of side length, ‘a’.
If n = radius of cubical metal block of side length, ‘a’.sphere whose octant encloses all the points upto an energy ‘E’, then
Number of allowed energy values upto an energy E
= number of points in the octant of sphere of radius ‘n’
We consider another sphere of radius n+dn whose octant encloses all points upto an energy ‘E+dE’, then
Number of allowed energy values upto an energy E + dE
= number of points in the octant of sphere of radius ‘n + dn’
So, number of allowed energy states in energy range dE
= number of points in the space between two octant shells of radii
n and n+dn
= (volume of space between two octant shells of radii n and n+dn)
X (number of points / unit volume)
Since are all integers,
A unit volume of plot consists of
Just one point
If g(E) = number of energy states per unit energy range, then number of energy states in the energy interval dE
So, g(E) dE =
From (1), = ---------- (5)
Differentiating (5) , we get
So, ----------- (7)
Using (4), (6) and (7), we get then number of energy states in the energy interval
Each energy value is applicable to two energy states, one for an electron with spin-up, and the other for an electron with spin down (Pauli’s exclusion principle).
So, the number of allowed energy states in the energy interval dE
Hence, the number of energy states present in unit volume having energy values lying between E and E + dE (DOS) is given by
Density of energy states for a free electron gas then number of energy states in the energy interval
General Expression for =
TASK: Find out the number of electrons present per unit volume of a cubical metal block at absolute zero temperature
Thermionic Emission then number of energy states in the energy interval
The emission of electrons from a metal under the effect of thermal energy is called
Emitted electrons are called THERMIONS.
Electrons are free to move inside the metal
Electrons cannot come out of the metal surface on its own as high potential barrier is present at
but when the temperature of the metal is sufficiently high, electrons gain sufficient energy to
overcome the barrier and ESCAPE from the metal surface
Free electron theory assumes that the potential within the metal is constant.
The minimum energy to be supplied to the electron for its emission from the metal is termed as
WORK FUNCTION (Ф) of the metal
Richardson’s Equation then number of energy states in the energy interval
If W = minimum energy of the electron for its emission from the surface, E = Fermi energy of the metal,
then, Ф = W – E = work function of the metal.
No. of energy states / unit volume in energy range E to E + dE,
So, density of energy states per unit volume in momentum range p to p+dp,
We construct a plot in ‘momentum space’ such that each point represents a particular combination of
momenta components of an electron along x-, y- and z-directions. So,
Volume element in momentum space,
From (1) and (2), the density of states in momentum space, then number of energy states in the energy interval
= --------- (3)
Hence, no. of electrons/unit volume having momenta in the range and + and
Now, we consider the metal plate to be in Y-Z plane. Electrons will be emitted in a direction perpendicular to Y-Z plane
i.e. along x-axis. Only those electrons will be emitted whose energy, E > W.
. So, 1 in denominator can be neglected.
Standard Integral Form then number of energy states in the energy interval
So, current density,
J then number of energy states in the energy interval