Fermi-Dirac Statistics

In Fermi-Dirac statistics the probability of being in a certain energy state is

$$f(E) = \left[1 + \exp\left(\frac{E-E_F}{k T}\right)\right]^{-1}$$

In the area that I'm looking at the texts always assume the population's energy is much greater than the Fermi Energy and so approximate this as the Boltzmann Distribution.

However, I am interested in the probability and am wondering if there is a way I can express the difference $E - E_F$ as a function of the temperature, the work function or some other common parameters?

If anyone has any insight I would appreciate some guidance.

Thank you,

John

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Only a simple thought but Fermi's Energy $E_{F}$ is a parameter, so what you actually want is how does the energy $E$ depends on temperature, $E\equiv E(T)$ –  Jorge May 29 '13 at 21:00
Sorry, but I think this is a non-sense. Here you have a probability $f(E)$ to be in a energy state E. So, if you take global parameters like internal energy $U$, these are mean quantities, for instance, the internal energy $U = <E> = \int dE E f(E)$. So you cannot have any relations between E and U, or with E and any other global parameter. –  Trimok May 30 '13 at 14:30
Did you understood Boltzmann distribution ? Because it's essentially the same thing as the Fermi function: the occupation probability depends on an energy scale and on temperature as well, isn't it ? The Fermi energy $E_{F}$ is a tabulated parameter for materials. Maybe you could find this answer physics.stackexchange.com/a/65624/16689 interesting as well. –  FraSchelle Jun 28 '13 at 22:17
Since, $E - E_F$ is an energy, it can be expressed in terms of energy scale set by temperature. In other words, there always exists a temperature $T$: $k_B T = E - E_F$, where $k_B$ is the Boltzmann constant. Of course I've assumed $E > E_F$. Therefore, $E - E_F$ can be converted to $T$ by dividing it by $k_B$.
So the general idea is to find out the expression of energy (eg. $k_B T$) in terms of the parameter of interest (eg. $T$). Equate the expressions (eg. $k_B T = E - E_F$) and find the energy scale in terms of this parameter of interest.