Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

Free Electron Model of Metals

Reviewed by:
ffImage
hightlight icon
highlight icon
highlight icon
share icon
copy icon
SearchIcon

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:

  1. Free electron approximation 

  2. Independent electron approximation

  3. Relaxation-time approximation

  4. 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:


(Image will be uploaded soon)


Now, we will look at the one-on-one electron theory of metals, involving the following:


  1. Drude free electron theory

  2. Sommerfeld free electron theory

  3. 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.


(Image will be uploaded soon) 


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:

                       

(Image will be uploaded soon)


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. Explain the concept of relaxation and mean free path in the electron theory of metals.

We know that electrons align to a direction when being subjected to the external electric field.


When we switch off the electric field supply, because of the collision, electrons leave their lattice ions, their velocity starts to decrease. The process of the velocity decrease is called relaxation. The relaxation time (t) is the time necessary for reducing the drift velocity to 1/e of its initial value.


The average distance between any two consecutive collisions is called a mean free path (I) of the electron. It is given by:


                  I = vdt

2. Explain quantum free electron theory.

In 1928, to explain the physical properties of free electrons in metals, like photoelectric effect, Compton effect, black body radiation, and superconductivity, Sommerfeld developed a fresh new theory as an application of quantum mechanical concepts and Fermi-Dirac statistics to the free electrons in the metal. This theory was named the quantum free electron theory.


As per the Fermi-Dirac statistics, the probability that a particular energy state with energy “E,” is occupied by an electron is given as;


\[f(E)={\frac{1}{(1+e(E)-EF/KT)}}\]


Where


 EF =  Fermi level (it s the highest energy level at 0 K).

3. What is the Lorentz theory of electric conductivity?

The Lorentz theory of electrical conductivity has the following statements:

  1. The vibrational energy of metallic atoms/ions) about the mean lattice Positions.

  2. The thermal properties of solids depend totally upon energy changes in lattices and free electrons. When an electric field is applied across the metallic solid, the free electrons accelerate in the opposite direction of the electric field.