Sommerfeld Expansion & Chemical Potential

In summary, the speaker is trying to understand the relationship between chemical potential and temperature using the Sommerfeld expansion. They are using it on the density of charge and have encountered difficulty in inserting a term with μ. They have searched for information on this topic and found an excerpt that explains the concept using the Taylor series.
  • #1
RicardoMP
49
2
Hi!
I'm trying to show how the chemical potential depends on the temperature and I'm advised to use the Sommerfeld expansion. I'm using it on the density of charge [tex] n=\int^{+\infty}_{-\infty} \rho(\epsilon)n_Fd\epsilon [/tex], which gives [tex] n=\int^{\mu}_{0} \rho(\epsilon)d\epsilon +\frac{\pi^2}{6}(k_BT)^2\frac{d\rho(\epsilon)}{d\epsilon}|_{E=\mu}[/tex].
After having a hard time trying to insert a term with a [tex] \mu [/tex] in it, I searched on google and found the following excerpt that I uploaded. I don't understand the second paragraph and how it gives the [tex] (\mu-E_F)\rho(\epsilon) [/tex] term.
Thank you for your time.
 

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  • #2
RicardoMP said:
Hi!
I'm trying to show how the chemical potential depends on the temperature and I'm advised to use the Sommerfeld expansion. I'm using it on the density of charge [tex] n=\int^{+\infty}_{-\infty} \rho(\epsilon)n_Fd\epsilon [/tex], which gives [tex] n=\int^{\mu}_{0} \rho(\epsilon)d\epsilon +\frac{\pi^2}{6}(k_BT)^2\frac{d\rho(\epsilon)}{d\epsilon}|_{E=\mu}[/tex].
After having a hard time trying to insert a term with a [tex] \mu [/tex] in it, I searched on google and found the following excerpt that I uploaded. I don't understand the second paragraph and how it gives the [tex] (\mu-E_F)\rho(\epsilon) [/tex] term.
Thank you for your time.
Nevermind! Taylor Series, duh! xD
 

FAQ: Sommerfeld Expansion & Chemical Potential

What is the Sommerfeld Expansion?

The Sommerfeld Expansion is a mathematical technique used in statistical mechanics to calculate the properties of a system at a non-zero temperature. It is based on expanding the Boltzmann distribution in a power series and is commonly used to study the thermodynamic properties of systems with a large number of particles.

What is the chemical potential?

The chemical potential is a thermodynamic quantity that represents the amount of work needed to add or remove a particle from a system. In other words, it is a measure of the energy required to change the number of particles in a system at constant temperature and pressure. It is also related to the tendency of a system to undergo a chemical reaction.

How is the Sommerfeld Expansion used to calculate the chemical potential?

The Sommerfeld Expansion involves expanding the thermodynamic potential, such as the Helmholtz free energy or Gibbs free energy, in a power series. The coefficient of the first term in the expansion is related to the chemical potential. Therefore, by finding the coefficient, the chemical potential can be calculated.

What are the limitations of the Sommerfeld Expansion?

The Sommerfeld Expansion is only accurate for systems with a large number of particles. It also assumes that the particles are non-interacting and the potential energy is negligible compared to the kinetic energy. In addition, it is only valid for systems at low to moderate temperatures where quantum effects are significant.

What are the practical applications of the Sommerfeld Expansion and chemical potential?

The Sommerfeld Expansion and chemical potential are widely used in the fields of statistical mechanics, thermodynamics, and quantum mechanics. They have practical applications in various areas such as material science, solid state physics, and chemistry. They are also used to study the thermodynamic properties of gases, liquids, and solids, and to understand the behavior of complex systems at non-zero temperatures.

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