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When $E-\mu \gg kT$, the unit term in the denominator of the Fermi-Dirac distribution is negligible. The resulting exponentially small occupation is the Boltzmann approximation. It justifies the usual nondegenerate semiconductor carrier formulas; it fails near strongly occupied band states.

For one massless neutrino flavor with one helicity per member and $|{\mu }_{\nu }|\gg T_{\nu }$, the Fermi-Dirac distribution fills momentum states up to the positive chemical potential. If ${\mu }_{\nu }>0$, the neutrino density is ${\mu }_{\nu }^4/(8{\pi }^2)$ and the antineutrino contribution is approximately $3T_{\nu }^4{e}^{-{\mu }_{\nu }/T_{\nu }}/{\pi }^2$. For negative chemical potential the roles reverse. Thus the leading total energy density is even in ${\mu }_{\nu }$ and has no factor of two from the two species. Extra populated internal degrees of freedom do multiply it. This distinction prevents using a positive-degeneracy approximation for an exponentially dilute negative-chemical-potential distribution.

A degenerate relic neutrino has ${\xi }_{\nu }\gg 1$. In the ultrarelativistic limit its Fermi-Dirac distribution approaches a filled momentum sphere, giving for one internal statewhile the antineutrino density is exponentially suppressed.

The density of available one-particle states multiplied by their mean occupation. For equilibrium electrons, $Z(E)=D(E)f(E)$ with the Fermi-Dirac distribution $f$. The integral of $Z$ counts electrons; the integral of $D$ counts available states.

For a vertical optical transition between parabolic energy bands with common carrier temperature, net stimulated emission has occupation factor $f_c(E_c)-f_v(E_v)$. The monotonic Fermi-Dirac distribution makes this positive exactly when the quasi-Fermi-level splitting exceeds the transition photon energy. An allowed transition must also lie above the band gap. Broadening, band degeneracy and a zero-temperature occupation window require their own qualifications; this condition alone does not establish laser threshold.

Symmetry of the excitation spectrum under exchange of conduction band electrons and valence band holes, after choosing a central zero of energy. In a two-band model with equal effective masses and equal degeneracy factors, charge neutrality in an intrinsic semiconductor fixes the chemical potential at that central energy. Indeed the Fermi-Dirac distribution obeys $1-f(-E;\mu =0)=f(E;\mu =0)$, so excitation populations are equal at every positive energy.

The electron single-particle spectral function $A(\mathbf{k},ϵ)=-\mathrm{Im}{G}^R(\mathbf{k},ϵ)/\pi$, in a convention normalized to unit energy integral per orbital and spin, describes addition and removal energies. For the fermionic retarded Green function, $G^R(\mathbf{k},t)=-i\Theta (t)⟨\{c_{\mathbf{k}}(t),c_{\mathbf{k}}^{†}(0)\}⟩$, energy is referenced to the chemical potential. The occupied removal spectrum is $f(ϵ)A(\mathbf{k},ϵ)$, with Fermi-Dirac distribution $f$. In the noninteracting limit, $A$ is a delta peak at the band energy; interactions can shift, broaden, and redistribute its weight.

The Fermi-Dirac distribution follows from Fermi-Dirac statistics and is $n(E)=[e^{\beta (E-\mu )}+1{]}^{-1}$.

The mean occupation of a fermionic one-particle state of energy $E$ isAt zero temperature this becomes the step function $1_{E<E_F}$.

The antisymmetric exchange statistics of identical fermions. Interchanging two one-particle labels reverses the total state, and fermionic creation operators anticommute. Occupation of a one-particle mode is therefore zero or one, giving the Pauli exclusion principle; thermal equilibrium leads to the Fermi-Dirac distribution. Color, flavor, spin and orbital factors must together have the required antisymmetry, even if one factor is symmetric.

The Fermi energy is the highest occupied one-particle energy in the ground state of a noninteracting Fermi gas at zero temperature, measured here relative to the lowest one-particle energy. The Pauli exclusion principle allows only one fermion in each complete one-particle quantum state, including its spin, so adding more particles fills progressively higher energy eigenstates. A cold gas can therefore have substantial kinetic energy even when its temperature is zero. In the thermodynamic limit, with the same energy reference, $E_F=\mu (T=0)$ where $\mu$ is the chemical potential.^[ref]

For a uniform, three-dimensional gas of noninteracting electrons, with spin one-half, equal populations of the two spin states, no applied magnetic field, and nonrelativistic particles of mass $m=m_e$,Here $n$ is the total number density, $V$ the volume, $p_F$ the Fermi momentum, $\hslash =h/(2\pi )$ the reduced Planck constant, and $k_B$ the Boltzmann constant. The electron Fermi temperature $T_F$ expresses the same energy scale in kelvin; it is not the actual gas temperature. Strong degeneracy of a degenerate electron gas means $T\ll T_F$. The momentum formula counts states and survives changing the energy–momentum relation, whereas $E_F=p_F^2/(2m)$ requires $p_F\ll mc$.^[ref]

The reference matters: the displayed $E_F$ excludes rest energy and sets the zero at $p=0$. Including rest energy shifts both the one-particle energy and the zero-temperature chemical potential by $m{c}^2$, leaving occupations unchanged. More generally, adding a constant $C$ to every one-particle energy also adds $C$ to $\mu$; the Fermi-Dirac distribution depends only on their difference. In a finite quantum system the chemical potential at zero temperature can lie between the last occupied and first empty energy eigenstates; its equality with the filling edge is a continuum statement.

At positive temperature, $E_F$ remains the zero-temperature scale associated with the specified number density. The Fermi level $\mu (T)$ must instead satisfy the particle-number conditionwhere $g(E)$ is the extensive density of states, including both spin states. Smearing the Fermi-Dirac distribution while arbitrarily holding $\mu =E_F$ generally changes $N$. This distinction matters when interpreting electron heat capacity and any experiment that infers temperature from an occupation profile.^[ref]

For an ideal emitter able to exchange one fermion with a zero-chemical potential bath, Pauli blocking gives downward rate $A(1-f)$ rather than the bosonic $A(1+f)$. If $u_{\nu }=h\nu {\mathcal{D}}_{\nu }f$ and $C=A/(h\nu {\mathcal{D}}_{\nu })>0$, the rate is $A-C{u}_{\nu }$. Detailed balance with a Fermi-Dirac distribution implies $g_1{B}_{12}=g_{2}C$ and $A/C=h\nu {\mathcal{D}}_{\nu }$. For $d$ massless helicity states, ${\mathcal{D}}_{\nu }=d4\pi {\nu }^2/c^3$. Thus an additive induced-emission coefficient would be negative, expressing blocking rather than a negative total rate. The emitter must change its fermion parity; two ordinary electronic energy levels cannot exchange a single neutrino this way.

Neglecting interactions and thermal pairs, the Fermi-Dirac distribution $f(p)=[\exp ((\epsilon (p)-{\mu }_{\mathrm{kin}})/(k_{B}T))+1{]}^{-1}$ with kinetic energy $\epsilon (p)=\sqrt{m_e^2{c}^4+p^2{c}^2}-m_e{c}^2$ determines the electron contribution to a stellar equation of state. The kinetic chemical potential is fixed by the electron number density, not an independently chosen pressure fraction:The electron kinetic internal energy per volume is $({\pi }^2{\hslash }^3{)}^{-1}{\int }_0^{\infty }{p}^2\epsilon (p)f(p)dp$. In the nondegenerate limit $P_e=n_e{k}_{B}T$; at zero temperature the integrals become the equation of state of a cold electron gas. These are limits of one component and must not be added together as independent electron pressures. When electron-positron thermal pair abundance matters, both charge species must be included, with charge neutrality fixing their net density.

The free electron model approximates conduction electrons by noninteracting particles with the displayed parabolic dispersion relation, filling one-particle states according to Fermi-Dirac distribution. A uniform positive background supplies charge neutrality. Periodic boundary conditions give one spatial state per $k$-space volume $(2\pi {)}^d/V_d$ in $d$ dimensions; the physical spin degeneracy is two. The model captures basic state counting and degeneracy but omits the periodic lattice potential, interaction corrections and detailed electronic band structure.

A Landau quasiparticle is a fermionic excitation connected continuously to a Fermi gas particle when interactions preserve the phase. Near an isotropic Fermi surface, its energy relative to the chemical potential is ${\xi }_p={ϵ}_p-\mu \simeq v_F(p-p_F)$, with Fermi momentum $p_F$ and Fermi velocity $v_F$.^[ref]

The Pauli exclusion principle restricts decay. For a three-dimensional normal Fermi liquid with regular short-range interactions, a representative low-energy lifetime model isHere $E_{\ast }$ is an energy scale and $A$ is dimensionless; sharpness requires $\hslash /{\tau }_p\ll |{\xi }_p|$ at zero temperature.^[ref]

For an original example, stipulate $A=1$ and $T=0$. At $|{\xi }_p|/E_{\ast }=0.02$, the decay energy is $0.0004E_{\ast }$, just $2\%$ of the excitation energy; halving $|{\xi }_p|$ reduces that ratio to $1\%$. Approaching the Fermi surface improves the particle description even though the microscopic interaction has not weakened. This quadratic illustration is three-dimensional; low-dimensional corrections require separate analysis.

With negligible chemical potential, the Fermi-Dirac distribution has $3/4$ of the boson number density per degree of freedom at equal temperature. A species with $g_{\nu }$ populated states therefore has $n_{\nu }/n_{\gamma }=(3g_{\nu }/8)(T_{\nu }/T_{\gamma }{)}^3$, since photons have two states. This remains the number-density ratio after the neutrino becomes massive if its comoving particle number is conserved.

A nearest-neighbour tight-binding model on a bipartite lattice, with identical on-site energy ${ϵ}_0$, has paired single-particle energies ${ϵ}_0\pm x$. At half filling, $f({ϵ}_0+x)+f({ϵ}_0-x)=1$ for the Fermi-Dirac distribution with $\mu ={ϵ}_0$. Thus its fixed-density chemical potential equals ${ϵ}_0$ at all positive temperatures within the isolated-band model. An anisotropic rectangular nearest-neighbour band retains this property; additional same-sublattice hopping or unequal on-site energies can remove it.

The static Lindhard dielectric function describes the real, zero-frequency linear response of a uniform free electron gas. Here ${ϵ}_1=\mathrm{Re}ϵ$, $\mathbf{r}$ is the position operator, and $\mathbf{q}$ is the perturbation wavevector, $q=|\mathbf{q}|$, $V$ the normalization volume, $e>0$ the elementary charge, and ${ϵ}_0$ the vacuum permittivity. The normalized one-electron states are ${\psi }_{\mathbf{k}}=V^{-1/2}{e}^{i\mathbf{k}\cdot \mathbf{r}}$ with $E_{\mathbf{k}}={\hslash }^2{k}^2/(2m_e)$; $f(E)$ is their Fermi-Dirac distribution occupation. Both momentum sums exclude spin; the prefactor two counts the two equal spin populations. The density-operator matrix element is ${\delta }_{{\mathbf{k}}^{\prime },\mathbf{k}+\mathbf{q}}$. Coincident energy denominators are interpreted by the static limit or Cauchy principal value.

A weak total electric potential ${\varphi }_{\mathrm{tot}}$ perturbs an electron by $-e{\varphi }_{\mathrm{tot}}$. First-order state mixing produces $\delta n=e{\Pi }_0(q){\varphi }_{\mathrm{tot}}$, whereThe induced charge is $-e\delta n$. Self-consistent Poisson equation screening then gives ${\varphi }_{\mathrm{ext}}=ϵ{\varphi }_{\mathrm{tot}}$, with $ϵ=1+e^2{\Pi }_0/({ϵ}_0{q}^2)$. This is the random phase approximation: use the independent-electron response to the self-consistent total field.^[ref]

At zero temperature $f(E_{\mathbf{k}})=\Theta (k_F-k)$. Shifting the integration variable in the second occupation term givesThe two angular integrals are equal. Using the supplied integral, with $x=q/(2k_F)$, givesThe limits are $F(0)=1$, $F(1)=1/2$ and $F(x)\sim 1/(3x^2)$ for large $x$. The first recovers Thomas-Fermi screening; the logarithm produces the Kohn singularity of static Lindhard screening at $q=2k_F$.

Use the Fourier transform convention $\rho (\mathbf{q})=\int d^{3}r{e}^{-i\mathbf{q}\cdot \mathbf{r}}\rho (\mathbf{r})$. A point charge $Ze$ has total screened charge $Ze/ϵ(q)$ and induced charge $-Ze[1-ϵ(q{)}^{-1}]$. Angular inversion therefore gives the radial induced-charge screening kernelIn particular, ${\rho }_{\mathrm{ind}}(\mathbf{q}=0)=-Ze$: the total screening charge is negative although its local density can oscillate in sign.

Integrate the sine transform by parts twice; $g(0)=g(\infty )=0$ removes the boundary terms. For the Friedel density tail from a derivative singularity, write $Q=2k_F$ and $g^{\prime \prime }(q)\sim C/(q-Q)$. ThenIndeed, near $Q$ the principal value transform is ${\int }_{-\infty }^{\infty }\sin [(Q+s)r]ds/s=\pi \cos (Qr)$ for $r>0$. Smooth terms do not remove this leading oscillatory long-range contribution. Thus ${\rho }_{\mathrm{ind}}\propto \cos (2k_{F}r)/r^3$.

For a three-dimensional, single-valley gas with two spin states, the Friedel oscillation wavelength from carrier density isLow-temperature scanning tunneling spectroscopy near exposed dopants can map the oscillatory local density of states, with a Fourier feature near $2k_F$ at the Fermi energy. An energy-resolved tunnelling image measures local spectral weight, not directly the bulk energy-integrated charge: comparing occupied-bias maps and accounting for the surface geometry is necessary. In actual GaAs, the background relative permittivity and band effective mass change screening strengths; they do not change this wavelength at the stated density.

A normal Fermi liquid is an interacting fermion phase continuously connected to a Fermi gas without broken symmetry. Its Landau quasiparticles are dressed particle/hole excitations near the Fermi surface, with the same conserved quantum numbers, finite quasiparticle residue and a decay width small compared with their excitation energy. Their reference dispersion already includes zero-temperature interaction renormalization; “bare quasiparticle” here does not mean a free atom.^[ref]

The rate assumes $k_{B}T,|\xi |\ll E_F$, $\xi =ϵ-\mu$, regular scattering amplitudes and density of states, equilibrium Fermi-Dirac distributions, and two-body collisions conserving energy and momentum. Pauli exclusion principle restricts the available phase space. It describes a regular three-dimensional normal liquid, not a one-dimensional interacting electron liquid. On the surface, ${\tau }^{-1}\propto T^2$; at $T=0$ and $\xi >0$, it is proportional to ${\xi }^2$. Hence excitations become increasingly sharp. The quoted asymmetric occupation factor is for particle out-scattering: a hole decay or full spectral width also needs the complementary in-scattering process; it does not imply exponentially stable deep holes.

Symmetry $f_{ab}=f_{ba}$ cancels the quadratic functional's factor one-half, giving the thermal Landau quasiparticle energy shiftHere $a=(\mathbf{k},\lambda )$, and the derivative is with respect to an occupation, not particle density. Replacing the homogeneous momentum sum by an energy integral defines a smooth angularly averaged kernel $H_a(ϵ)$. With $\delta n=n_F(ϵ;\mu ,T)-\Theta ({\mu }_0-ϵ)$,Fixed density gives $\delta \mu =-({\pi }^2/6)(k_{B}T{)}^2{D}^{\prime }({\mu }_0)/D({\mu }_0)+O(T^4)$; fixed chemical potential instead has $\delta \mu =0$. Thus ${\widetilde{ϵ}}_a={ϵ}_a+O(T^2)$ in either case, provided the kernels are smooth and no instability intervenes. This is the Sommerfeld expansion; singular spectra require separate treatment.^[ref]

The leading entropy is therefore that of quasiparticles with their zero-temperature effective mass:$D^{\ast }$ here is the total, spin-summed density of states. The plotted low-temperature slopes require the enhanced masses $5.4m$ at molar volume $26.2\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$ and $2.7m$ at $36.8\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, rather than the bare-mass predictions. At fixed molar particle number, ${\gamma }_{\mathrm{mol}}\propto m^{\ast }{V}_{\mathrm{mol}}^{2/3}$. Agreement is restricted to the low-temperature limit: the measured curves depart markedly from a fixed-mass ideal-gas extrapolation at higher temperature. This supports long-lived quasiparticles with smooth low-energy parameters, but heat capacity alone does not measure their lifetime or prove adiabatic continuity.

To derive fermionic Einstein-coefficient relations, first consider an idealized emitter that can exchange one ultrarelativistic fermion with the bath, with zero chemical potential. Let ${\mathcal{D}}_{\nu }=d4\pi {\nu }^2/c^3$ be its density of states per volume and ordinary frequency, with $d$ populated helicity states; for one neutrino helicity $d=1$, whereas the corresponding photon count is two. The Fermi-Dirac distribution givesPauli blocking changes the final-state factor in downward emission from the bosonic $1+f$ to $1-f$. Define positive coefficients $B_{12},C_{21}$ and $A_{21}=C_{21}h\nu {\mathcal{D}}_{\nu }$. The upward rate per lower emitter is $B_{12}{u}_{\nu }$, and the downward rate per upper emitter isThermal detailed balance, $n_1{B}_{12}{u}_{\nu }=n_2(A_{21}-C_{21}{u}_{\nu })$ with $n_2/n_1=(g_2/g_1)e^{-h\nu /(k_{B}T)}$, yieldsFor example, substituting $e^{h\nu /(k_{B}T)}=x$ reduces the balance equation to $B_{12}=(g_2/g_1)C_{21}$ because $x{f}_{\nu }=1-f_{\nu }$.

If one insists on the original additive notation $A_{21}+B_{21}{u}_{\nu }$, its formal coefficient is negative:A negative correction means inhibited emission into occupied modes, not a negative total decay rate; $0\le f_{\nu }\le 1$ ensures positivity. The dimensions are those of part (a), and the factor of two relative to the photon density is a helicity count, independent of the statistical sign.

There is a physical qualification: ordinary electronic energy levels of the same chemical atom have the same fermion parity, and cannot interconvert by emission or absorption of a single neutrino. A realizable single-fermion emitter must change the relevant particle-number and parity sector, with matching conservation laws. For ordinary atomic levels, neutrino-pair processes would require a different calculation. A bath with nonzero chemical potential would also require the corresponding chemical-equilibrium population ratio. The calculation above is the intended single-mode statistical analogy under explicit assumptions.

For the free electron model, the Fermi energy is the highest occupied one-electron energy at zero temperature, measured relative to the bottom of the free-electron band. The occupied wavevectors fill a disk in two dimensions or a sphere in three dimensions. The radius is the Fermi wavevector $k_F$, withEach allowed wavevector has two spin states. At nonzero temperature, the Fermi-Dirac distribution rounds the occupation boundary; the zero-temperature Fermi energy remains a useful density-defined scale and should not be defined as the thermal energy $k_{\mathrm{B}}T$.