The **key difference between free electron model and nearly free electron model** is that the free electron model does not take into account the electron interactions and the potential, whereas the nearly free electron model takes into account the potential.

Free electron model is a quantum mechanical model that is useful in describing the behaviour of the charge carriers in a metallic solid. Nearly free electron model is a quantum mechanical model that describes the physical properties of electrons that are moving nearly freely through a crystal lattice of a solid.

### CONTENTS

1. Overview and Key Difference

2. What is Free Electron Model

3. What is Nearly Free Electron Model

4. Similarities – Free Electron Model and Nearly Free Electron Model

5. Free Electron Model vs Nearly Free Electron Model in Tabular Form

6. Summary – Free Electron Model vs Nearly Free Electron Model

## What is Free Electron Model?

Free electron model is a quantum mechanical model that is useful in describing the behaviour of the charge carriers in a metallic solid. This model was developed in 1927 by Arnold Sommerfeld. He combined the classical Drude model with quantum mechanical Fermi-Dirac statistics; therefore, it was named as Drude-Sommerfeld model as well.

It is a very simple model, but it is useful in explaining many experimental findings, including the Wiedemann-Franz law, which relates the electrical conductivity and thermal conductivity, the temperature dependence of electron heat capacity, the shape of the electronic density of states, electrical conductivities, the range of binding energy values, etc. Moreover, this model solves the inconsistencies related to the Drude model, giving an insight into several properties of metals. In addition, this model is predictive when applying it for alkali and noble metals.

When considering the ideas and assumptions of the free electron model, there are four main assumptions; (1) free-electron approximation, which describes the neglection of the interaction between the ions and valence electrons except for the boundary conditions, (2) independent electron approximation, which describes the neglection of the interactions between electrons, (3) relaxation-time approximation, which describes that there is some unknown scattering mechanism in a way that the electron probability of collision is inversely proportional to the relaxation time, and (4) Pauli exclusion principle, which describes that each state of the system can occupy only a single electron.

The name “free electron model” originates from the first two assumptions given above, which describe that electrons act as free particles with a respective quadratic relation between the energy and the momentum.

## What is Nearly Free Electron Model?

Nearly free electron model is a quantum mechanical model that describes the physical properties of electrons that are moving nearly freely through a crystal lattice of a solid. This model is very related to the empty lattice approximation. We can name this model as NFE model or quasi-free electron model. It enables the understanding and calculating of the electronic band structure of metals. Moreover, we can identify this model as an improvement of the free electron model.

We can recognize the nearly free electron model as a modification of the free electron model. Similar to the free electron model, this model also does not take into account the electron-electron interactions (independent electron approximation).

## What are the Similarities Between Free Electron Model and Nearly Free Electron Model?

- Both models are important in quantum mechanics.
- These models describe the behaviour of electrons in systems.
- They use the independent electron approximation.

## What is the Difference Between Free Electron Model and Nearly Free Electron Model?

The free electron model is a quantum mechanical model that is useful in describing the behavior of the charge carriers in a metallic solid. Meanwhile, the nearly free electron model is a quantum mechanical model that describes the physical properties of electrons that are moving nearly freely through a crystal lattice of a solid. The key difference between free electron model and nearly free electron model is that the free electron model does not take into account the electron interactions and the potential, whereas the nearly free electron model takes into account the potential.

The following table summarizes the difference between free electron model and nearly free electron model.

## Summary – Free Electron Model vs Nearly Free Electron Model

Free electron model is a quantum mechanical model that is useful in describing the behavior of the charge carriers in a metallic solid. The nearly free electron model is a quantum mechanical model which describes the physical properties of electrons that are moving nearly freely through a crystal lattice of a solid. The key difference between free electron model and nearly free electron model is that free electron model does not take into account the electron interactions and the potential, whereas the nearly free electron model takes into account the potential.

##### Reference:

1. “Tight Binding and Nearly Free Electrons.” *Open Solid State Notes*.

##### Image Courtesy:

1. “Free-electron DOS” By Pieter Kuiper – Own work (Public Domain) via Commons Wikimedia

2. “Empty-Lattice-Approximation-FCC-bands” By Eg-T2g – Own work (CC BY-SA 3.0) via Commons Wikimedia

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