Exploring Fermi-Dirac Statistics at T=0K

In summary, the conversation is about understanding and showing the functional form of the Fermi-Dirac distribution function at T=0K. It is shown that the function has the form of a step function, where for E>Ef the probability is 0 and for E<Ef the probability is 1. The conversation also discusses how the Fermi-level is determined and why this leads to the specific form of the distribution function at T=0K.
  • #1
tyco05
161
0
Hey kids,

The question I'm having trouble with (this time) is as follows:

Show that the Fermi-Dirac distribution function,

[tex] f_{FD}(E)=\frac{1}{e^{(\frac{E-E_f}{kT})}+1} [/tex]

Has the following functional form at T= 0K
(see attachment)


Now, the first thing that screamed at me was the division by T in the exponential bit. If T=0, what is going on!?

The obvious things are:

E>Ef then f(E) = 0

and

E<Ef then f(E) = 1.

I'm just really confused at how I can show that the function has that form at T=0K

Any ideas?

Cheers
 

Attachments

  • FD-distribution.bmp
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  • #2
I don't understand your question. Didn't you just show the function has that form?
[tex]\lim_{T \rightarrow 0}f_{FD}(E)=\left\{ \begin{array}{ll}1 & \mbox{if} E<E_f\\ \frac{1}{2} & \mbox{if} E=E_f \\0 & \mbox{if} E>E_f[/tex]
 
  • #3
I'm glad somebody else doesn't understand the question either.

They want me to 'show' that the distribution has the (attached pic) form at T=0.

The real problem I'm having is how do I "show" that it has that form? Via two lines of maths? That's it?
 
  • #4
You HAVE just shown it. By taking the limits.
So yeah, that's it. :)
 
  • #5
Well, just use the definition of the Fermi-level... It is the the maximum energy-level at T = 0 K. Just fill up all the available energy-levels with all available electrons. The last electron is placed at the highest energylevel which is called the Fermi-level. Ofcourse all this is done at zero Kelvin. That is why the distribution function has the drawn form. All levels are filled up (probability one) till the Fermi-level. Above this level there are no filled levels since the Fermi-level is the highest. It is just by QM-definition of the Fermi-level


regards
marlon
 

What is Fermi-Dirac Statistics?

Fermi-Dirac Statistics, also known as Fermi statistics, is a branch of quantum statistics that describes the behavior of particles with half-integer spin, such as electrons, in a system at thermal equilibrium.

How is Fermi-Dirac Statistics different from Bose-Einstein Statistics?

Fermi-Dirac Statistics describes the behavior of particles with half-integer spin, while Bose-Einstein Statistics describes the behavior of particles with integer spin. Additionally, Fermi-Dirac Statistics obey the Pauli exclusion principle, meaning that two particles cannot occupy the same quantum state, while Bose-Einstein Statistics do not have this restriction.

What is the Fermi-Dirac Distribution Function?

The Fermi-Dirac Distribution Function is a mathematical function used to describe the probability of a particle occupying a specific energy state in a system at thermal equilibrium. It takes into account the energy of the particle, the temperature of the system, and the chemical potential.

What is the significance of the Fermi-Dirac Distribution in solid state physics?

In solid state physics, the Fermi-Dirac Distribution is used to describe the distribution of electrons in a material. It helps to explain properties such as electrical conductivity, thermal conductivity, and the behavior of materials under different temperatures and pressures.

How does Fermi-Dirac Statistics influence the behavior of electrons in a material?

Fermi-Dirac Statistics governs the behavior of electrons in a material by determining their energy distribution. This, in turn, affects the material's properties such as electrical conductivity and thermal conductivity. It also plays a role in phenomena such as superconductivity and the Hall effect.

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