# What is Fermi-dirac distribution: Definition and 28 Discussions

In quantum statistics, a branch of physics, the Fermi–Dirac distribution is a probability distribution of particles over energy states in systems consisting of many identical particles that obey the Pauli exclusion principle. It is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently (although Fermi defined the statistics earlier than Dirac).Fermi–Dirac (F–D) statistics apply to identical and non-distinguishable particles with half-integer spin in a system with thermodynamic equilibrium. Additionally, the particles in this system are assumed to have negligible mutual interaction. That allows the multi-particle system to be described in terms of single-particle energy states. The result is the F–D distribution of particles over these states which includes the condition that no two particles can occupy the same state; this has a considerable effect on the properties of the system. F–D statistics apply to particles that are called fermions. It is most commonly applied to electrons, a type of fermion with spin 1/2. Fermi–Dirac statistics are a part of the more general field of statistical mechanics and use the principles of quantum mechanics.
A counterpart to F–D statistics is Bose–Einstein statistics, which apply to identical and non-distinguishable particles with an integer spin (0, 1, 2, etc.). These particles, such as photons (spin 1) and the Higgs bosons (spin 0), are called bosons. Contrary to fermions, bosons do not follow the Pauli exclusion principle, meaning that more than one boson can simultaneously be in the same quantum configuration.
In classical physics, Maxwell–Boltzmann statistics is used to describe particles that are identical and distinguishable.

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1. ### I Fermi energy definition and Fermi-Dirac distribution

1)In my book , there is a definition of fermi energy as topmost filled level in the ground state of an N electron system. This definition holds only for absolute zero,right? If it is not absolute zero,fermi energy is the energy at which the probability of a state being occupied is 50 percent...
2. ### I Is the Fermi-Dirac distribution equal to zero at the state of highest energy?

I`m sorry if this seems too obvious, just trying to clarify something. When Fermi-Dirac distribution is equal to zero , can we assume it is the state of the highest energy? (Because the propability of occupation is zero)
3. ### Fermi-Dirac distribution at T->0 and \mu->\epsilon_0

The limit itself is pretty easy to calculate ##lim_{T->0} \ lim_{\mu->\epsilon_F} \ (e^{\frac{(\epsilon_F - \mu)}{kT}}+1)^{-1} = \frac{1}{2}## But I'm very confused about changing ##\epsilon_\vec{k}## to ##\epsilon_F##. Why do we do this?
4. ### I Another question from Ashcroft and Mermin: Fermi-Dirac Distribution

Good Day : i reached the page 40 of Ashcroft Mermin book and after the equation 2.38 there is this expression of E(a,N) which is equal to Helmoltez Free energy F = U - TS , how this two terms F , E are related ? anyone can provide adequate explanation , and few useful references Best...
5. ### Fraction of occupied states (Fermi-Dirac distribution + DOS)

I just want to clear some confusion I am having with the Fermi-Dirac distribution & density of states (DOS) of a semiconductor, which are given by Say we have a piece of Silicon in equilibrium and its Fermi level lies 0.25 eV below the conduction band edge, i.e. Ec - EF = 0.25 eV. Let us say...
6. ### Average speed of molecules in a Fermi gas

My first most obvious attempt was to use the relation ##<\epsilon> = \frac{3}{5}\epsilon_F## and the formula for kinetic energy, but this doesn't give the right answer and I'm frankly not sure why that's the case. My other idea was to use the Fermi statistic ##f(\epsilon)## which in this case...
7. ### I General Concepts About Fermi-Dirac Distribution

Hello! Thanks for your time reading my questions. When I was studying quantum statistical mechanics, I get so confused about the relations between Pauli's exclusion principle and the Fermi-Dirac distributions. 1. The Pauli's exclusion principle says that: Two fermions can't occupy the same...
8. ### I Fermi-Dirac distribution, when does it break?

Well, the question is if the well known occupation distribution of the energy levels for fermions does break, which means when it is not valid anymore. The Fermi-Dirac distribution reads: ##\displaystyle f_{FD}(E)=\frac{1}{exp\left({\frac{E-\mu}{k_B T}}\right)+1}## And gives the occupation...
9. ### Deriving Fermi-Dirac Distribution misunderstanding

Homework Statement The actual question was deriving Bose-Einstein, but I got confused on the F-D example. I'm basically following the method given here. Homework Equations [All taken directly from the above link] Taylor series: The Attempt at a Solution So after that third equation...
10. ### Stat mech: Fermi-Dirac distribution

Homework Statement Show that the FD distribution can be viewed as giving the probability that a given state ( of the prescribed energy) is occupied. Homework EquationsThe Attempt at a Solution Solution to this problem: I understand the solution,but I took a different approach...
11. ### A quick question about the Fermi-Dirac distribution

Homework Statement An electron has two spin states and a set of energy levels E1,E2,... By the Fermi-dirac distrbution,the mean number of electrons in energy level Ek is https://en.wikipedia.org/wiki/Fermi–Dirac_statistics#Distribution_of_particles_over_energy Does it mean that,for an electron...
12. ### B Neutrinos and Fermi-Dirac Distribution

I'm an A level student currently trying to understand the behaviour and properties of neutrinos, and wanted to check that I've understood the basics of neutrino properties. As neutrinos are half-integer spin particles, can the Fermi-Dirac distribution be used to calculated the probable...
13. ### I Solving Fermi-Dirac Integral: Step-by-Step Guide

i am completely lost. there is an integral in my textbook in fermi dirac statistics whose result is written directly and am not able to understand . it is ∫⌽(u) du /exp.((u-uf)/kt) + 1 from 0 to ∞ expanded by tayor's series to give...
14. ### How to find the normalization constant of Fermi-Dirac distribution function?

How to find the normalization constant of Fermi-Dirac distribution function.
15. ### How to normalize the Fermi-Dirac distribution function?

n=\int^{\infinity}_{infinity}f(E)d^{3}E i am using this formula,,,but i could not normalize it?Any idea or hint
16. ### Derivation of Fermi-Dirac distribution

http://ecee.colorado.edu/~bart/book/book/chapter2/pdf/ch2_5_5.pdfcan you please tell me where f/(f(gi,fi) is from? and also how to get to (2.5.13)
17. ### Why do the energy levels in electronic band structures start at negative values?

In the graphs that I see around the internet I see that the energy axis starts at 0 eV and it goes up. So the electrons have positive energies. But in the electronic band structure, the electrons have negative energies. And if they go to infinity, then their energy becomes 0. So, what is...
18. ### What is the meaning of a fully occupied state in the Fermi-Dirac distribution?

Hi all, The probability that a state is occupied means : 1) Fully Occupied by 2 electrons Spin up and Spin down or 2) Occupied by 1 electron only . Thanks
19. ### Fermi-Dirac distribution for metals

Hello everyone! I'm a little confused. The Fermi-Dirac distribution is about every electron in a metal or only about the valence electrons?
20. ### Quantum Mechanics- statistical physics fermi-dirac distribution.

Homework Statement Consider a free-electron gas at a temperature T such that kT << E_f Write down the expression for the electron number desnity N/V for electrons that have an energy in excess of of E_f. Show by making the change of variables (E-E_f)/kT = x. that the number desnity is...
21. ### Fermi-Dirac distribution normalization

Hi! I have a little question which is puzzling me. Maybe it is a very simple question. It is my understanding that the Fermi-Dirac distribution is a probability density function and, as such, its integral between 0 and infinite should be 1. When T = 0, the integral gives the chemical...
22. ### How Does Temperature Affect Electron Occupation in a Fermi-Dirac Distribution?

An electron state has energy 0.14 eV above the Fermi energy. What is the probability that the electron state will be occupied at T = 200K? do i just use the following formula? the part that throws me off is the "above the Fermi energy" bit. would i just plug that number in for E - Ef??
23. ### What Temperature Gives a 25% Population Probability at 7.00 eV in Copper?

Homework Statement Pleas can you help me figure out what I do wrong? At what temperature is the probability that an energy state at 7.00 eV will be populated equal to 25 percent for copper (EF = 6.95 eV)? Homework Equations The formula for the fermi-Dirac Distribution is f(E) =...
24. ### What Temperature Yields a 25% Population Probability at 7.00 eV in Copper?

Pleas can you help me figure out what I do wrong? At what temperature is the probability that an energy state at 7.00 eV will be populated equal to 25 percent for copper (EF = 6.95 eV)? The formula for the fermi-Dirac Distribution is f(E) = 1/(1+e^((E-EF)/kT)) and looking at the problem I...
25. ### Higgs, Fermi-Dirac distribution, and Pauli exclusion principle

hi, I am studying the Higgs Mechanism these days. And I get two questions. I hope some ones could help me. 1>We know that due to the non-zero VEV, SSB takes place and higgs condensates give masses to bosons and fermions. I wonder that after the SSB and before the universe became as cool...
26. ### Fermi-Dirac Distribution Function

Hey all, I have two questions. 1) The density of electron energy states is given by g(E) = A sqrt E. Evaluate how many quantum states there are with energies between 9.0eV and 9.1eV. Ansewr in terms of the quantity A. 2) Consider an intrinisic semiconductor. Let Nv and Nc be the number...
27. ### General version of fermi-dirac distribution?

general version of fermi-dirac distribution?? merry x-mas everyone! in the Boltzmann distribution every state with energy Ei can be occupied by an arbitrarily large number of molecules. In contrast, if each state can be occupied by only one particle then one needs to use the fermi dirac...
28. ### The Fermi-Dirac distribution function.

Hi. Does anyone know if it is possible to start from the thermal density matrix \hat \rho_T = \frac{e^{-\hat H_0/kT}}{\mathrm{Tr}e^{-\hat H_0/kT}} and from that derive that the single particle density matrix can be written as \rho(p ; p') = \delta_{p,p'} f(\epsilon_p) just by...