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FREE ELECTRON IN METALS II - Coggle Diagram
FREE ELECTRON IN METALS II
Fermi–Dirac distribution function
The probability that a particular state having energy E is occupied by one of the electrons in a solid is
where f(E) is called the Fermi–Dirac distribution function and EF is called the Fermi energy.
The Fermi function f(E) specifies how many of the existing states at the energy E will be filled with electrons.
The function f(E) specifies, under equilibrium conditions, the probability that an available state at an energy E will be occupied by an electron. It is a probability distribution function.
Consider T > 0 K
As T increases, the distribution rounds off slightly.
Because of thermal excitation, states near and below EF lose population and states near and above EF gain population.
The Fermi energy EF also depends on temperature, but the dependence is weak in metals.
Consider T = 0 K
At 0 K, all states having energies less than the Fermi energy are occupied and all states having energies greater than the Fermi energy are vacant.
Temperature dependence of Fermi-Dirac distribution
Quantum-based free-electron theory of metals
It remedies the shortcomings of the classical model by taking into account the wave nature of the electrons.
In this model, based on the quantum particle under boundary conditions analysis model, the outer-shell electrons are free to move through the metal but are trapped within a three-dimensional box formed by the metal surfaces.
Therefore, each electron is represented as a particle in a box.
Particles in a box are restricted to quantized energy levels.
One-dimensional region of space
A particle confined to a one-dimensional region of space, called the particle-in-a-box.
In classical viewpoint, if a particle is bouncing elastically back and forth along the x axis between two impenetrable walls separated by a distance L.
Because the walls are impenetrable, there is zero probability of finding the particle outside the box, so the wave function 𝜑(x) must be zero for x < 0 and x > L.
Energy of particle
If a particle of mass m is confined to move in a one-dimensional box of length L, the allowed states have quantized energy levels given by
This expression shows that the energy of the particle is quantized.
The lowest allowed energy corresponds to the ground state, which is the lowest energy state for any system.
Three-dimensional region of space
In this model, we require that 𝜑(x, y, z) = 0 at the boundaries of the metal.
It can be shown that the energy for such an electron is
where me is the mass of the electron and nx, ny, and nz are quantum numbers.
As we expect, the energies are quantized, and each allowed value of the energy is characterized by this set of three quantum numbers (one for each degree of freedom).
Density of states function
Under this assumption, the number of allowed states per unit volume that have energies between E and E + dE is
where g(E) is called the density of states function
Number of electrons per unit volume
If a metal is in thermal equilibrium, the number of electrons per unit volume N(E) dE that have energy between E and E + dE is equal to the product of the number of allowed states per unit volume and the probability that a state is occupied; that is,
