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The Hubbard model combines hopping with a local repulsion,Hopping favours a broad itinerant band; the interaction penalizes opposite-spin double occupancy. For uniform Stoner decoupling of the Hubbard interaction, let $s=n_{\uparrow }-n_{\downarrow }$ be the spin imbalance per site. The interaction is $U(n^2-s^2)/4$, so polarization lowers it by $U{s}^2/4$, at the expense of kinetic energy.

If $D(E_F)$ is the noninteracting density of states per spin per site, small polarization costs $s^2/[4D(E_F)]$. Thereforeis the mean-field ferromagnetic instability criterion. The exchange splitting is $\Delta =Us$, and the paramagnetic response is enhanced as $\chi ={\chi }_0/[1-UD(E_F)]$. A density of states including both spins changes the written factor in the criterion.

The model explains how itinerant bands can polarize without imposing pre-existing localized atomic spins. It is not a claim that every large-$U$ metal becomes ferromagnetic: near half filling the Hubbard model can instead form a Mott insulator with antiferromagnetic superexchange of scale $4t^2/U$. Band filling, bandwidth and fluctuations matter.^[ref] The interaction-energy and kinetic-cost expansion above is the stated mean-field derivation of the enhancement.^[ref]

Assume three-dimensional parabolic band edges, with density-of-states masses $m_c,m_h$ and one valley per edge, each with spin degeneracy two. Their effective densities of states areFor the semiconductor mass-action law, integrating the square-root density of states against the Boltzmann tail gives $n=N_c{e}^{(\mu -E_c)/(k_{B}T)}$ and $p=N_v{e}^{(E_v-\mu )/(k_{B}T)}$. Multiplication cancels the chemical potential, givingIf there are additional valley degeneracies, multiply by their product; anisotropic bands use the appropriate density-of-states masses. In the later equal-mass model, $A=4m^3/(2\pi {\hslash }^2{)}^3$. No numerical mass or valley multiplicity has been invented.

When $\mu =E_c=0$, the conduction-edge occupation is exactly $1/2$, so the nondegenerate approximation for electrons fails at every positive temperature. Define the Normalized complete Fermi-Dirac integralFor the carrier densities with chemical potential pinned to a band edge, with the two specified equal masses, put $N(T)=2[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}$. Direct Fermi-Dirac integration then givesThus $n\propto T^{3/2}$ exactly within the parabolic model. At low temperature, $p\simeq N(T)e^{-E_g/(k_{B}T)}$ is thermally activated; at $k_{B}T\gg E_g$, $p/n\to 1$ from below. Both vanish at zero temperature and increase at fixed chemical potential. The other states that pin $\mu$ can supply the compensating charge; one must not additionally impose $n=p$.

For the Pauli reduction of the pinned-edge mass-action product, the exact product relative to the classical law isIt approaches $f_0$ at low temperature and $f_0^2\simeq 0.58545$ in the formal high-temperature limit. The low-temperature activation factor is unchanged but its prefactor is reduced; the classical mass-action formula overestimates the product. The high-temperature limit here refers to the given unbounded parabolic bands, not a claim that real semiconductor band structures remain unchanged at arbitrarily high temperature.^[ref]

Finally, for a fixed positive $\mu$, the zero-temperature Fermi sphere has radius $k_{\mu }=\sqrt{2m\mu }/\hslash$, soThe low-temperature expansion at fixed $\mu$ gives the fixed-chemical-potential Sommerfeld density increase:The monotonic density at fixed chemical potential in a square-root density of states initially rises quadratically from $n(0)$. For $k_{B}T\gg \mu$, its leading behaviour crosses to $f_{0}N(T)\propto T^{3/2}$. The fixed chemical potential is what allows the particle number to change.

For a complete single-particle state, the Fermi-Dirac distribution isHere ${\mu }_{\mathrm{ch}}$ denotes the chemical potential, distinct from the magnetic-moment symbol in (b). At zero temperature it approaches the Fermi energy $E_F$ and the occupation has a sharp edge. At $T\ll T_F=E_F/k_B$, thermal changes occur within an energy interval of order $k_{B}T$ about that edge.

The Fermi-edge heat-capacity counting argument uses the fact that only order $g(E_F)k_{B}T$ electrons can be rearranged because states well below the edge are blocked by the Pauli exclusion principle. Each active excitation carries energy of order $k_{B}T$, so $U(T)-U(0)\sim g(E_F)(k_{B}T{)}^2$ and $C_{V,e}\propto T$. For a smooth density of states, the Sommerfeld expansion givesHere $g$ is the total single-particle density of states, including spin. In a three-dimensional free-electron gas, $g(E_F)=3N/(2E_F)$, giving $C_{V,e}=({\pi }^2/2)N{k}_B(T/T_F)$. This is the low-temperature normal-state electronic contribution, not the lattice heat capacity or a description across a superconducting transition.^[ref]

Measure energies from the single-particle ground state, and assume the three-dimensional ideal Bose gas in the large-particle-number limit. The harmonic-trap chemical-potential pinning discussed here uses a chemical potential satisfying ${\mu }_{\mathrm{ch}}\le 0$. On cooling, it approaches zero; below the condensation temperature it remains at zero to leading thermodynamic order. Above that temperature it is negative. The detailed negative-chemical-potential branch shape is determined by the density of states; the figure uses the harmonic trap analysed below. The finite-size condensate chemical potential stays slightly below zero rather than being exactly pinned there.

A Bose-Einstein condensate has macroscopic occupation of a single quantum state. A dilute atomic gas needs low temperature and high phase-space density while remaining metastable against ordinary liquid or solid formation. Laser cooling followed by evaporative cooling of atoms in a weak trap can achieve the required regime. For the trapped ideal gas the quantitative threshold is derived below; a uniform ideal gas has $n{\lambda }_T^3=\zeta (3/2)$ at onset. An experimental signature is a narrow low-momentum component appearing within the broader thermal distribution after time-of-flight expansion, as observed in magnetically trapped rubidium-87.^[ref]

Spinless cubic-box wave-vector counting. Impose periodic boundary conditions on a cube of side $L$. The allowed vectors are $\mathbf{k}=(2\pi /L)(n_x,n_y,n_z)$ with integer components. Each state occupies $(2\pi /L{)}^3$ in wave-vector space. For a spinless particle, counting the spherical shell givesThere is no spin multiplicity. This is continuum shell counting; a finite box has discrete levels. Using $ϵ={\hslash }^2{k}^2/(2m)$ gives the free-particle density of states per unit energy and volume

Semiclassical density of states in a central potential. At position $\mathbf{r}$, the available kinetic energy is $ϵ-U(r)$. The local-density approximation integrates the preceding free-particle result over the classically allowed region:This form assumes the allowed radial region is the connected interval $0\le r\le r_{ϵ}$. A nonmonotonic potential would require integrating all allowed shells.

For the isotropic harmonic trap, $r_{ϵ}=\sqrt{2ϵ/(m{\omega }^2)}$. Substituting $r=r_{ϵ}\sin \vartheta$ givesConsequentlyThe exact oscillator ground energy is $3\hslash \omega /2$ above the potential minimum. The harmonic-trap energy-zero convention subtracts that constant, while the displayed density of states is the leading semiclassical continuum expression, neglecting the finite-level corrections produced by that shift. The discrete ground state must be counted separately.

Condensate population. Below the transition, use ${\mu }_{\mathrm{ch}}\simeq 0$ for the excited states. The saturation of harmonic-trap excited states follows from the supplied integral:where $\zeta$ is the Riemann zeta function and $\zeta (3)\simeq 1.202$. Setting $N_{\mathrm{ex}}(T_0)=N$ gives the harmonic-trap critical-temperature prefactor:Above $T_0$ there is no macroscopic ground-state fraction in this approximation. For completeness, above the transition the quantum gas fugacity $z=e^{{\mu }_{\mathrm{ch}}/(k_{B}T)}<1$ satisfies $N=(k_{B}T/\hslash \omega {)}^3{\mathrm{Li}}_3(z)$; the polylogarithm ${\mathrm{Li}}_3$ therefore defines the plotted negative branch. A finite trapped gas has rounded behaviour, weak interactions shift the threshold, and the continuum thermal population is inaccurate when $k_{B}T$ is comparable to or below $\hslash \omega$. The power law is intended for the large-$N$ condensed regime, not an exact discrete-spectrum formula at arbitrarily small temperature.^[ref]