# What is Fermi-dirac: Definition and 48 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. ### I Has the Fermi-Dirac Integral been solved?

hi guys I have a question about whether or not the Fermi-Dirac Integral has Been solved, because i found a formula on Wikipedia that relates the Fermi-Dirac integral with the polylogarithm function: $$F_{j}(x) = -Li_{j+1}(-e^{x})$$ and in some recent papers they claim that no analytical...
6. ### I The deduction of Fermi-Dirac and Bose-Einstein distrbiutions

I am studyng the deduction of Fermi-Dirac and Bose-Einstein distribution, but I'm not understanding one part. If we have a system of ##N## identical non-interaction particles, with energies levels ##\varepsilon _{l}## and occupation number ##n_{l}## (this is the number of particles with the same...
7. ### A Fermi-Dirac statistics, finding all electron configurations

Hello everyone. I'm having trouble understanding this example: https://ecee.colorado.edu/~bart/book/book/chapter2/ch2_5.htm#2_5_2 In this system of 20 electrons with equidistant energy levels, how is it known that there are only 24 possible configurations, and how are those configurations found?
8. ### I Charge carrier density: Hall experiment vs Fermi-Dirac statistics

Many times, the charge carrier density of a material is determined from a Hall effect experiment, from ##R_H=1/(ne)## (SI units). Where ##R_H## is determined from a measured voltage and other controllable parameters. As far as I know, this simple formula comes from the obsolete Drude's model...
9. ### 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...
10. ### I Is there a simple explanation for Fermi-Dirac statistics?

Is there a simple explanation for fermi-dirac statistics? I can't understand how a particle can "follow" a statistic.
11. ### 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...
12. ### 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...
13. ### 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...
14. ### 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...
15. ### Derivation of FD/BE-distribution using single-particle state

Homework Statement I'm trying to understand a derivation of the Fermi-Dirac and Bose-Einstein distributions. In my textbook Thermal Physics by D. V. Schroeder it says: "The idea is to first consider a "system" consisting of one single-particle state, rather than a particle itself. Thus the...
16. ### 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...
17. ### I Solve Fermi-Dirac Integral: Get Help Now!

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 . on expansion by using the method of taylor's series the result should be where u_f is such that function of u is zero for u greater...
18. ### 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...
19. ### How can I solve special Fermi-Dirac integral at Physics?

Homework Statement I need to solve this integral, $$\int _{-\infty }^{\infty }x\left( \dfrac {1} {1-e^{-x}}+\dfrac {1} {1+qe^{-x}}\right) dx$$ My advisor said its solution will be zero. But i haven't improved it yet. There is important case. This integral is divergent at x=0. So, i should...
20. ### How to find the normalization constant of Fermi-Dirac distribution function?

How to find the normalization constant of Fermi-Dirac distribution function.
21. ### 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)
22. ### 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...
23. ### Why does corrected Maxwell Boltzmann have decimals?

So I have just been reading up on statistical thermodynamics and have no idea why the bose-einstein, fermi dirac and maxwell boltzman are all integers, that makes sense, but then when you make the degenerate correction to the maxwell Boltzmann by dividing by N! we get decimal answers. Why is...
24. ### 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
25. ### Differences between Boltzmann and Fermi-Dirac distributions

Hi All, In relation to the Boltzmann distribution vs the FD/BE distributions in different applications, I have 2 basic questions: 1. The Boltzmann distribution comes most easily from the Canonical Ensemble (constant N, V,T) while the FD/BE come from the Grand Canonical ensemble (constant .mu...
26. ### Fermi-Dirac: Density of electrons

Homework Statement Monovalent copper (one conduction electron per atom) has a density of 9000 kg m-3 and atomic mass of 64 amu (ie. 1 kmole = 64kg). Find: a)The density of conduction electrons per unit volume b)the Fermi energy in electron volts Homework Equations f(E) =...
27. ### 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?
28. ### Is there an analytical way to get average energy of a Fermi-Dirac gas?

The average particle energy of a Fermi-Dirac gas, with zero chemical potential, is about 3.15T, where T is the temperture of this gas. To get the average energy, one needs to do an integration. The integrand is something like \frac{x^3}{e^{x/k_BT}+1}. I could get the result numerically. But...
29. ### Entropy and Internal Energy in Fermi-Dirac statistics

What are the formulas for entropy and internal energy in Fermi-Dirac statistics, and how do I derive them?
30. ### 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...
31. ### 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...
32. ### 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??
33. ### Does the inertia of either a Bose-Einstein or Fermi-Dirac condensate

Does the inertia of either a Bose-Einstein or Fermi-Dirac condensate increase linearly with the number of particles?
34. ### Are there any distributions different from Fermi-Dirac and Bose-Einstein distribution

Please teach me whether it is possible there are any distributions different from Fermi-Dirac and Bose-Einstein distributions.Because the Statistic Theorem only demontrates that integer spin particles can't obey Fermi-Dirac distribution and spin-haft particles can't obey Bose-Einstein distribution.
35. ### Fermi-Dirac statistics at the Fermi level

Hi all, I've search for my question and found no answer. I think it should be pretty simple... Fermi energy corresponds to the last occupied energy, as I understand it. So, energy levels in the Fermi gas are all filled with two electron of opposite spins, up to the Fermi energy. Saying it...
36. ### How to reduce fermi-dirac to maxwell-boltzmann in a solid?

For indistinguishable particles we use fermi-dirac(FD) or bose-einstein(BE), and for distinguishable we use maxwell-boltzmann(MB).For the distinguishable case our prof gave us the example of atoms in solid, because the positions of the atoms are fixed, so they are distinguishable, thus satisfy...
37. ### 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) =...
38. ### 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...
39. ### 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...
40. ### Fermi-Dirac Statistics: What Is It?

What exactly is Fermi-Dirac statistics? Because I think I can't understand fermions without knowing that.Thanks! Sriram
41. ### 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...
42. ### Fermi-Dirac statistics valid for electron gas in metals?

Hello! In my course of solid states physics we use the fermi-dirac statistics for a free electron gas in metals. The fermi wave length of the electrons is about 1 Angström. Now, the wavelength may be intepreted as something as a coherence range - the electron should forget about the state of...
43. ### 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...
44. ### 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...
45. ### Fermi-dirac statistics, Griffiths 5.28

Homework Statement Evaluate the integrals (eqns 5.108 and 5.109) for the case of identical fermions at absolute zero. Homework Equations 5.108 N=\frac{V}{2\pi^{2}}\int_{0}^{\infty}\frac{k^2}{e^{[(\hbar^{2}k^{2}/2m)-\mu]/kT}+1}dk 5.109...
46. ### Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein.

If we have indistinguishable particles, we must use Fermi-Dirac statistics. To Identical and indistinguishable particles, we use Bose-Einstein statistics. And, to distinguishable classical particles we use Maxwell-Boltzmann statistics. I have a system of identical but distinguishable...
47. ### Nice derivations of Maxwell, Fermi-Dirac and Bose-Einstein distributions

Hi all, Does anybody know some reference (even internet one) that explains in detail the derivation of Maxwell´s velocity and/or energy distribution on an ensemble of atoms/molecules ? References to Fermi-Dirac distributions and Bose-Eisntein´s are also welcome. Best Regards, DaTario...
48. ### Exploring Fermi-Dirac Statistics at T=0K

Hey kids, The question I'm having trouble with (this time) is as follows: Show that the Fermi-Dirac distribution function, f_{FD}(E)=\frac{1}{e^{(\frac{E-E_f}{kT})}+1} Has the following functional form at T= 0K (see attachment) Now, the first thing that screamed at me was...