Free Electron In Metals (Success & Failure) - Coggle Diagram
Free Electron In Metals (Success & Failure)
Success of classical free electron theory: Ohm's law
Consider a metal pure of length, ℓ and area cross section, A. An electric field is applied to the metal piece. And the electron are accelerated with velocity, v.
The success of Classical free electron theory: Relationship between σ and 𝜿
Consider a uniform metallic rod containing free electron
Let A and B be the two cross section of temperature T and (T - dT) sepearted by a distance of mean free path (λ). Heat flows from hot end ‘A’ to cold end ‘B’.
During the collisions, the electron
The success of classical free electron theory: Wiedemann-Franz Law
The ratio between thermal conductivity and electrical conductivity of a material is directly proportional to the absolute T of the material
Failure of classical free electron theory: Lorentz number
The ratio between thermal conductivity and electrical conductivity of a material is constant
where L is the Lorentz number (2.44 ×
L = 1.11 × 10−8 WΩK−2
Hence it is found that the classical value of Lorentz number is only half of the experimental value,
L = 2.44 × 10−8 WΩK−2
Failure of classical free electron theory: Heat capacity
According to Drude model:
The average kinetic energy of each electron = 3/2 kBT
However when we measure, we find the electronic contribution is only 1% of this. (only small fraction of electron can be excited to higher level).
C contributed by electron is much smaller than the calculated results.
as such, Drude model fails to predict C for metal.
Failure to correctly account for the T dependence of resistivity in metals
It is not a relationship as shown in the experiment. But for most metals as shown in figure.
For some metal such as copper, resistivity is nearly proportional to temperature as shown in figure
A non-linear region always exists at very low T, and the resistivity usually reaches some finite value as the T approaches absolute zero
This residual resistivity near absolute zero is caused primarily by the collision of electrons with impurities and imperfections in the metal.
In contrast, high T resistivity (the linear region) is predominantly characterized by collisions between electrons and metal atoms.
Failure to correctly account for the T dependence of thermal conductivity
Substitute eq. (4) into eq. (3), we obtained k is directly proportional with T ½ whereas experimentally, k behaves in a more complex manner as seen in the graph below: