Free Electron Model of Metals Meaning
In solid-state physics, the free electron model of metals portrays the metals composed of quantum electronic gas. This electronic gas is responsible for high electrical and thermal conductivity.
A Free Electron Model of Metals Considers the Following Four Assumptions:
Free electron approximation
Independent electron approximation
Relaxation-time approximation
Pauli exclusion principle
In the nutshell, the name of this model comes from the first two assumptions, as each electron is treated as a free particle with a respective quadratic relation between energy and momentum.
However, there is an electron gas theory of metals, like the Drude Sommerfeld model, Drude Lorentz free electron theory, which we will discuss in detail.
Do You Know?
In 1927, the Free Electron Model of Metals was developed principally by Arnold Sommerfeld, who later combined the classical Drude model with quantum mechanical Fermi - Dirac statistics, and hence we call it the Drude Sommerfeld model.
Electron Gas Theory of Metals
The additional information for the free electron model of metals is that these models can be very predictive when applied to alkali and noble metals. The most common noble metals include Gold (Au), Silver (Ag), Osmium (Os), Rhodium (Rh), Palladium (Pd), Iridium (Ir), etc.
Do You Know the history of the electron theory of solids? Well! It lies hereunder:
The development of the electron theory of solids started early in the 20th century with the declaration of Drude-Lorenze free electron theory, the Sommerfeld model of free electron theory, and zone theory, etc.
The electron theory of solids, in its initial stages, was only a model for metals but later on, with further development, it became applicable to metals and non-metals.
Do you know the properties of Electron Gas? If not, let’s understand the significance of electron gas in metals followed by their properties:
Electron Gas in Metals
The statement for an electron gas as per the Free electron model is:
Electrons in metal are considered to form a uniform Fermi gas. A Fermi gas is an ideal gas, a state of matter which is an assembly of many non-interacting fermions (move freely linearly without deflection by collisions).
Fermions are particles that follow Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in specific, particles with half-integer spin.
A free-electron Fermi gas looks like the following:
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Now, we will look at the one-on-one electron theory of metals, involving the following:
Drude free electron theory
Sommerfeld free electron theory
Drude Lorentz free electron theory
Drude Free Electron Theory
The Drude free electron theory was discovered by Paul Drude in 1900. It is the theory of electrical and thermal conduction in a metal. It is also the application of the kinetic theory of gases to metals, which is considered the electron gas.
Drude free electron model consists of mobile negatively charged particles, electrons that are confined in metal by attraction to immobile positively charged ions.
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In the above diagram, we can see an isolated atom with a nucleus charge of eZa placed in metal.
Z - Valence Electrons/ Conduction Electrons/Electrons
These electrons are weakly bound to the nucleus, and they wander away from their parent atoms, that’s why they participate highly in chemical reactions.
However, Za - Z are core electrons that remain bound to the nucleus to form a metallic ion. Since they’re tightly bound, so they hardly participate in chemical reactions.
Now, let us understand the Sommerfeld free electron theory with the help of the Sommerfeld free electron model to understand the Sommerfeld theory of electrical conductivity:
Sommerfeld Model of Free Electron Theory
The Sommerfeld model considers electrons the free particles that are non-interacting with atomic nuclei. This the reason for the model being called the free electron model or Sommerfeld free electron model. These free particles are placed in a cubic box of size L * L * L with periodic boundary conditions.
The solutions to the Schrödinger equation of a free particle are planes, given as:
ψ ⍺ exp (ik * r)
Here,
k - electron wave vector. K takes a discrete value 2L(nx, ny, nz). The plane waves have eigenvalues illustrated by the following equation:
\[\epsilon (k)=\frac{\hslash^2k^2}{2m}\]
Also,
m = the mass of the electron. The graph for ∊(k) of function “k” in the 1-D system for Sommerfeld theory of metals is:
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Here, each black dot is a probable electron state.
We have another theory that considers a free movement of electrons inside the container (however, different from the one discussed in the Sommerfeld free electron theory and that is Drude Lorentz theory. Now, let’s understand this theory in detail.
Drude Lorentz Free Electron Theory
A classical free electron theory of metals developed by both Drude and Lorentz is called the Drude-Lorenz theory. The statement is given below:
A metal comprises electrons that are free to move about in the crystal-like molecules of gas in a container. Here, the condition of gas molecules is ideal, which means that their mutual repulsion is ignored, i.e, the potential energy is taken zero.
Do You Know?
Any charged particle, when subjected to the applied electric field, shows electrical conductivity, To explain this concept, we have the Sommerfeld Theory of Electrical Conductivity.
Sommerfeld Theory of Electrical Conductivity
Since electrons make Brownian motion, for which, we need an applied electric field that makes electron drift with a velocity (drift velocity) by being aligned with the direction opposite to that of \[\vec{E}\].
So, the electrical conductivity s is given as;
s = neμ
Here,
n = no of electrons
e = charge
= mobility of a charge carrier
Mobility = drift velocity per unit electric field
\[\mu =\frac{v_d}{\vec{E}}\]
The unit of “s” is Ohm-m.
FAQs on Free Electron Model of Metals
1. What is the classical free electron model of metals?
The classical free electron model, proposed by Drude and Lorentz, describes a metal as a lattice of positive ions immersed in a 'gas' of free electrons. It assumes that the valence electrons detach from their atoms and move randomly throughout the metal's volume, much like molecules in a gas. These free electrons are considered solely responsible for the high electrical and thermal conductivity of metals.
2. What are the key assumptions of the classical free electron theory?
The classical free electron theory is built on several fundamental assumptions:
- The valence electrons are completely free to move within the metal's boundaries, with their movement restricted only by the metal's surface.
- The electrostatic force between the free electrons and the fixed positive ions, as well as the repulsion between electrons themselves, is considered negligible.
- The free electrons undergo collisions with the positive ions, and these collisions are considered elastic.
- The motion and velocity distribution of the electrons follow the laws of the classical kinetic theory of gases.
3. How does the free electron model explain the high conductivity of metals?
The model explains the high conductivity of metals through the collective movement of free electrons:
- Electrical Conductivity: When an external electric field is applied, the free electrons experience a force that causes them to accelerate. This results in a net directional movement, or drift velocity, opposite to the field, creating a substantial electric current.
- Thermal Conductivity: If one end of a metal is heated, the free electrons in that region gain kinetic energy. These high-energy electrons then move rapidly towards the colder end, transferring their energy through collisions, which efficiently conducts heat.
4. What are relaxation time and mean free path in the context of this model?
In the free electron model, these terms define the electron's collision dynamics. The relaxation time (τ) is the average time interval between two successive collisions of a free electron with the lattice ions. It represents how long an electron accelerates under an electric field before being scattered. The mean free path (λ) is the average distance an electron travels during this time, i.e., between two consecutive collisions.
5. What were the major failures of the classical free electron model?
While useful, the classical model failed to explain several experimental observations. Its key limitations include:
- Specific Heat: It incorrectly predicted a much higher electronic contribution to the specific heat capacity of metals than what was observed.
- Temperature Dependence of Resistivity: It could not accurately explain why the electrical resistivity of metals is proportional to the absolute temperature.
- Magnetic Properties: It was unable to explain phenomena like paramagnetism and diamagnetism in metals.
- Quantum Effects: It could not account for quantum phenomena such as the photoelectric effect, Compton effect, or superconductivity.
6. How does the quantum free electron theory improve upon the classical model?
The quantum free electron theory, or Sommerfeld model, improves upon the classical model by applying quantum mechanics and Fermi-Dirac statistics. This leads to crucial corrections:
- It correctly states that only electrons with energies near the Fermi level participate in conduction, which resolves the discrepancy in specific heat calculations.
- It provides a more accurate theoretical basis for the temperature dependence of conductivity.
- It introduces the concept of quantised energy levels for electrons, laying the groundwork for the band theory of solids, which explains the difference between conductors, semiconductors, and insulators.
7. What is the significance of Fermi energy in the quantum model?
Fermi energy (EF) is a fundamental concept in the quantum free electron model. It represents the highest energy level occupied by an electron in a metal at absolute zero temperature (0 K). Due to the Pauli Exclusion Principle, electrons must fill progressively higher energy states. The existence of Fermi energy explains why not all electrons contribute to conductivity and why they possess significant kinetic energy even at 0 K.
