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Expand a single-electron wavefunction as $|\psi ⟩={\sum }_n{\psi }_n|n⟩$. Project the Schrödinger equation onto $⟨n|$ to obtainTranslation symmetry suggests the Bloch state ${\psi }_n=N^{-1/2}{e}^{ikna}$ on a periodic chain of $N$ sites. Substitution givesTake the conventional positive hopping $t>0$. The first Brillouin zone is $-\pi /a\le k<\pi /a$; the band has minimum ${ϵ}_0-2t$ at $k=0$ and maximum ${ϵ}_0+2t$ at the zone edges. Its width is $4t$. A negative hopping interchanges the extrema but does not change the density of states after replacing $t$ by $|t|$.

There are $N$ allowed wavevectors and two spin states per wavevector. For the total spin-summed density of states $D(E)$, each interior band energy has two roots, $\pm k$. Using the spacing $2\pi /(Na)$ givesThus the density of states per atom isIt is symmetric about ${ϵ}_0$ and has integrable inverse-square-root Van Hove singularities at both band edges. Its integral over the band is two states per atom. The plot labels the dispersion extrema and both singular edges.

Monovalency gives one electron per atom, so the two-spin-state band is half filled. At zero temperature, occupying $-k_F<k<k_F$ gives $2(Na/2\pi )(2k_F)=N$. HenceRelative to the band minimum, the Fermi energy is $2t$.

The density of states obeys $g({ϵ}_0+x)=g({ϵ}_0-x)$. At $\mu ={ϵ}_0$, the Fermi-Dirac distribution satisfiesPairing the energies above and below the band centre therefore leaves exactly half of the two available states per atom occupied, at every positive temperature. Increasing $\mu$ strictly increases the mean particle number, so this solution at fixed half filling is unique. ConsequentlyThis particle-hole symmetry of a half-filled bipartite band argument uses the symmetric tight binding band and fixed half filling; it is not a general temperature-independence property of chemical potential in metals with additional bands or an asymmetric density of states.

The Boltzmann carrier density of a two-dimensional parabolic band can be derived directly. Use one isotropic parabolic conduction band, $E(\mathbf{k})=E_c+{\hslash }^2{k}^2/(2m_e^{\ast })$, with two spin states per wave vector. In the nondegenerate limit the Fermi-Dirac distribution reduces to its Boltzmann factor. The number per area is thereforeThe integral uses ${\int }_0^{\infty }k{e}^{-a{k}^2}dk=1/(2a)$. ThusEquivalently, the two-dimensional density of states is constant, $D_c=m_e^{\ast }/(\pi {\hslash }^2)$, and $n=N_c{e}^{-\beta (E_c-\mu )}$ with $N_c=D_c{k}_{B}T$. These expressions include twofold spin degeneracy and one valley. With a general total band degeneracy $g_c$, replace the factor $1/\pi$ by $g_c/(2\pi )$; extra valleys or occupied confinement subbands would change the coefficient.

Let the sample area be $A$, set the band touching energy to zero, and write $m^{\ast }=m_e^{\ast }=m_h^{\ast }$. Assume one electronic valley in each energy band with two spin states. The near-edge parabolic dispersions areUnder periodic boundary conditions, each spin occupies reciprocal-space area $(2\pi {)}^2/A$. Counting the quantum states inside a circle givesIts derivative is the constant two-dimensional free-electron density of states. Defining $D=A{m}^{\ast }/(\pi {\hslash }^2)$, the signed-energy density of states isThe total electronic density of states is therefore $D$ on either side of zero, not $2D$ at every signed energy. If instead one counts positive-energy excitations, the electron and hole species each have density of states $D$, giving $2D$ when summed. A common additional valley degeneracy multiplies all these quantities by that factor. The effective mass approximation assumes thermally sampled energies remain within the near-edge regions of both energy bands.

For the degenerate electron gas, let $D_g=Am/(\pi {\hslash }^2)$ include both spin states. The Sommerfeld expansion givesFor the gapless symmetric intrinsic semiconductor, part (d) gave the same coefficient with $D_s=A{m}^{\ast }/(\pi {\hslash }^2)$. Thus for equal sample areas and equal spin and valley degeneracies,The equality often intended in this comparison requires the same density of states. The source sets $m_e^{\ast }=m_h^{\ast }$ but does not explicitly equate them to the electron gas mass $m$. Without that additional assumption the ratio is $m^{\ast }/m$, or more generally $D_s/D_g$ if sample areas or degeneracy factors differ. For example, $m^{\ast }=2m$ with equal areas satisfies the stated equal-effective mass condition in the semiconductor and gives ratio two. The filled Fermi sea does not add a factor $N_e/N_c$: only the thermal shell near its Fermi energy contributes to its heat capacity.

Use the stipulated noninteracting impurity band approximation, with one donated electron per donor and two available spin states per donor orbital. The energy band is half filled. Approximating its spin-summed density of states per volume near the Fermi energy by $D_v(E_F)\simeq 2n/W$ gives the zero-temperature Pauli paramagnetismWith $\alpha =1$,A factor of order one depends on the density of states shape; using the conduction band $g$-factor rather than silently setting $g=2$ matters here.

To locate the comparison with the magnitude of the undoped Larmor-Langevin susceptibility, put $x=n^{-1/3}/a_b>1$. Numerically,For the stated atomic-scale estimate $R=a_0$ in part (b), equality with $|{\chi }_0|=2.599\times 10^{-5}$ requiresThus the dilute-branch crossover iswell below $a_b^{-3}=9.45\times 10^{23}{\mathrm{m}}^{-3}$. Within this model the positive spin response exceeds the negative background magnitude for lower dopant densities on this dilute branch: the exponential narrowing of $W$ wins over the loss of donors. Indeed $d\ln (e^x/x^3)/dx=1-3/x>0$ for $x>3$. This is not a claim of ferromagnetism.

The degenerate Pauli paramagnetism calculation requires $k_{B}T\ll W$, with an unimportant order-one filling factor. At the estimated crossing,The weak-field condition also requires $|g|{\mu }_{B}B\ll W$. For a different assumed atomic $R$, use $|{\chi }_0|=2.599\times 10^{-5}(R/a_0{)}^2$ in the displayed equation for $x$. An order-of-magnitude background $|{\chi }_0|\sim 10^{-5}$ gives the same density order and a crossover scale of order $10^{-5}\mathrm{K}$. At higher temperature than this extremely narrow bandwidth permits, the $n/W$ enhancement cannot be used; the independent-donor Curie law is the appropriate thermal scale if interactions remain negligible. At very small dopant density, disorder, localization and Coulomb interactions can also invalidate the assumed noninteracting impurity band. The formal divergence as $n\to 0$ is therefore a limit of the stated zero-temperature band model, not a prediction for isolated donors at a fixed nonzero temperature.

Let $\beta =1/(k_{B}T)$, with $k_B$ the Boltzmann constant, $T$ the absolute temperature, and $\mu$ the common equilibrium chemical potential. The conduction band minimum and valence band maximum are ${ϵ}_c$ and ${ϵ}_v$; their separation is the band gap $E_g={ϵ}_c-{ϵ}_v$. The positive parameters $m_e,m_h$ in the supplied density of states functions are the band effective masses of electrons and electron holes, not necessarily the bare electron mass.

The electron occupation is the Fermi-Dirac distribution $f(ϵ)=[e^{\beta (ϵ-\mu )}+1{]}^{-1}$. The appropriate statistical factors are $f_c=f$ for electrons and $f_v=1-f$ for electron holes. In a nondegenerate semiconductor, ${ϵ}_c-\mu \gg k_{B}T$ and $\mu -{ϵ}_v\gg k_{B}T$, so the Boltzmann approximation givesFor the conduction band, set $u=ϵ-{ϵ}_c$ and then $\xi =\beta u$. The supplied integral yieldsFor the valence band, set $u={ϵ}_v-ϵ$ to obtainThe effective density of states factors $N_c,N_v$ include the two spin states already present in the supplied density of states functions. Multiplication cancels the chemical potential and gives the semiconductor mass-action lawHere $n_i=\sqrt{N_c{N}_v}{e}^{-E_g/(2k_{B}T)}$ is the intrinsic carrier concentration. Equality $n=p=n_i$ additionally requires an electrically neutral intrinsic semiconductor. The product law also applies to an equilibrium doped semiconductor while both populations remain nondegenerate. The derivation assumes parabolic energy bands, the stated spin degeneracy, and a single common chemical potential.

A magnetic field quantizes closed semiclassical orbits into Landau levels. Their passage through the Fermi energy modulates the density of states and scattering, producing Shubnikov-de Haas oscillations of resistivity. The fundamental frequency for a regular extremal orbit perpendicular to the field isThus two distinct fundamentals ordinarily identify two inequivalent extremal areas, rather than two arbitrary points or two spin directions automatically giving two different geometric areas.

The particle-hole equality of tetragonal extremal orbit areas is a special constraint of the exact half-filled nearest-neighbor band stated here. For $B\parallel z$, its regular closed endpoint sections are the electron contour at $k_{z}c=\pi$ and the hole contour at $k_{z}c=0$ described in b. Translation $(k_{x}a,k_{y}a)\mapsto (k_{x}a+\pi ,k_{y}a+\pi )$ changes the sign of both cosines and maps one enclosed pocket onto the other. Their areas are equal. Direct integration givesConsequently this bare model predictsnot two distinct regular fundamental frequencies. The intermediate $k_{z}c=\pi /2$ contour is a separatrix with a divergent cyclotron period, not another ordinary smooth extremal pocket to insert into the Onsager formula. The larger occupied-area slice at $k_z=0$ is the complement of a hole pocket on the Brillouin torus; counting that complement as a second enclosed planar orbit would give a spurious ratio.

For comparison, an additional small-$k_x,k_y$ parabolic approximation would replace the exact surface by $(k_{x}a{)}^2+(k_{y}a{)}^2\simeq 4+\cos (k_{z}c)$ and suggest a rough belly-to-neck ratio $5/3$. That approximation is uncontrolled at half filling and erases the actual topology; it cannot be asserted as an exact prediction of the supplied cosine band. The observation of genuinely different frequencies in a real material therefore needs further information, such as longer-range hopping, a departure from half filling, extra bands, symmetry breaking or a resolved spin-dependent effect. The universal area-ratio inference is valid, but a nonunit numerical ratio is not determined consistently by the complete stated band assumptions alone.^[ref]

The conduction and valence dispersions are two cones touching at the Dirac point, with slopes $\pm \hslash v_F$. Here $\mathbf{k}$ is measured relative to one valley centre. Each cone has a density of states proportional to $|E|$, which vanishes at the touching energy. Gate-induced carriers satisfy approximately $n=C_g(V_g-V_D)/e$ per unit area. Negative and positive offsets from $V_D$ produce holes and electrons, respectively. Away from neutrality a simple charge carrier mobility model gives $\sigma \simeq e\mu |n|$, so resistance decreases on either side of a maximum near $V_D$. Disorder, electron–hole puddles, contacts and quantum transport round the peak; the cone dispersion alone fixes neither its height nor its width, nor does it require exact electron–hole symmetry in a real device.

A direct derivation of the magnetic spectrum starts from the single-valley Dirac Hamiltonian operator $H=v_F({\sigma }_x{\pi }_x+{\sigma }_y{\pi }_y)$. Introduce oscillator operators using the magnetic length ${\ell }_B=\sqrt{\hslash /(e|B|)}$. Up to exchanging sublattices or valleys,Squaring gives adjacent oscillator number operators, henceThe positive expression printed in the source describes the electron branch only; the negative-energy partners must also be included. The same square-root dependence follows from quantizing circular orbit area, with the Dirac Berry phase cancelling the usual half-integer offset.

Each spin and valley contributes $Ae|B|/h$ states in area $A$. Including two spins and two valleys gives $N_{\mathrm{LL}}=4Ae|B|/h$, or areal degeneracy $4e|B|/h$. The zero-energy level is half-filled at charge neutrality: the carrier density counts departures from that state.

For $n=10^{16}{\mathrm{m}}^{-2}$, the filling factor and fourfold graphene Hall sequence areIn a plateau, ${\rho }_{xx}\simeq 0$ and $|{\rho }_{xy}|=h/({\nu }_N{e}^2)$. The Hall sign reverses between electron and hole doping; the sketch uses its magnitude to avoid tensor-index sign conventions. Representative plateau-centre fields are $20.68$, $6.893$, $4.136$ and $2.954\mathrm{T}$ for $\nu =2,6,10,14$. Between plateaux, crossing extended states in broadened levels gives longitudinal peaks and Hall steps. Their ideal filling-centre positions have $\nu =4j$, namelyThus peaks near $10.34$, $5.17$ and $3.45\mathrm{T}$ separate the first prominent plateaux as field increases. Many weak oscillations accumulate toward zero field and are damped there. Finite-width plateaux require localized states between extended-state energies; an exactly clean Landau spectrum does not by itself prescribe plateau widths. The illustration uses an explicitly schematic broadening, not inferred experimental widths.

At $6.5\mathrm{T}$, $\nu =6.36$, close to the $\nu =6$ plateau. With $v_F=10^6\mathrm{m}{\mathrm{s}}^{-1}$, $E_1=92.5\mathrm{meV}$ and $E_2=130.8\mathrm{meV}$. The relevant adjacent gap near the chemical potential is therefore $\Delta =E_2-E_1\simeq 38.3\mathrm{meV}$, rather than the gap from the zero level. Resolving levels requires quantum-lifetime broadening $\hslash /{\tau }_q\lesssim \Delta$ and thermal broadening appreciably smaller than $\Delta$, giving the order-of-magnitude limitsThe upper temperature is an energy-scale estimate, not a guaranteed sharp-feature temperature. For oscillation amplitudes the thermal damping factor is $X/\sinh X$, $X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c^{\ast })$, with $\hslash {\omega }_c^{\ast }=\hslash eB{v}_F^2/E_F\simeq 36.7\mathrm{meV}$ and $E_F=\hslash v_F\sqrt{\pi n}$. Requiring $X\lesssim 1$ gives the more conservative scale $T\lesssim 22\mathrm{K}$ for little thermal damping. Visible weaker features may persist above that scale; their actual cutoff depends on resolution, disorder and temperature-dependent scattering. The transport lifetime and quantum lifetime also need not coincide.

In BCS theory, an electron distorts the positively charged lattice, and the retarded response can attract another electron. Within a phonon-energy shell around the Fermi surface, the effective attraction can overcome the screened Coulomb repulsion and destabilize the normal Fermi sea toward Cooper pairs. In conventional elemental superconductors the pairs predominantly join time-reversed states of opposite momentum and spin, forming a spin singlet.

In the weak-coupling approximation,Here $N(0)$ is the appropriate single-spin density of states and $V_{\mathrm{eff}}>0$ the effective attractive interaction in a consistent normalization. Both the phonon scale and the exponential coupling matter. A large Fermi-level density of states, often associated with partially filled narrow $d$ bands, can raise $T_c$; weak coupling or a small density of states can suppress it even in a good conductor. With force constants and coupling fixed, ${\omega }_D\propto M^{-1/2}$ gives the conventional isotope trend. This is not a universal monotonic ranking of elements by atomic mass: phonon stiffness, electron–phonon matrix elements, Coulomb effects and magnetism vary between elements. Magnetic pair breaking helps explain why some elements do not follow a naive density-of-states expectation.

In weak-coupling BCS theory, a useful scale isHere the phonon spectrum sets the attractive-interaction cutoff, $N(0)$ is the consistently defined Fermi-level density of states and $V_{\mathrm{eff}}$ the effective pairing attraction after repulsion is accounted for. Light hydrogen supports high vibrational frequencies; strong electron–phonon coupling and favourable metallic density of states can raise $T_c$. Pressure changes bonding, phase stability, electronic structure and phonon frequencies. It is not simply an independent multiplier in this expression. In a real strongly coupled hydride, Eliashberg theory uses the full spectral function and a screened Coulomb pseudopotential; the elementary exponential is a mechanism guide rather than a controlled quantitative fit to $203\mathrm{K}$.

Measure a four-terminal resistance that falls below the instrumental bound, and test that a magnetic field suppresses the transition. Independently detect diamagnetic screening and, where feasible, field-cooled flux expulsion or an AC susceptibility signal with controlled cell background. A resistance drop alone can reflect contact changes or a filamentary path; a consistent magnetic response is important evidence for superconductivity. A reproducible isotope shift upon replacing hydrogen by deuterium tests the involvement of lattice motion, although it is not by itself proof of superconductivity.

Extreme-pressure samples are microscopic and enclosed in a diamond-anvil cell. Reliable separate electrical contacts, homogeneous pressure, pressure calibration during cooling, chemical stability and the tiny magnetic signal relative to the anvils and gasket are practical difficulties. Structural identification matters because a sulfur-rich hydrogen compound formed under pressure need not retain the starting stoichiometry.

Historical correction: the original primary report associates the $203\mathrm{K}$ transition with pressures near $155\mathrm{GPa}$, while roughly $90\mathrm{GPa}$ is the metallization scale, not that same transition point. The printed pairing of $203\mathrm{K}$ with $90\mathrm{GPa}$ should not be treated as an established measured pair. The statement about the highest conventional transition temperature is also a historical statement in this 2018 examination.^[ref]

A perfectly clean two-dimensional system has sharply degenerate Landau levels and integer filling only at isolated fields for fixed density. Real disorder broadens each level into a band, with states localized in the tails and extended states near its centre. When the chemical potential moves through a localized region, changing field can change the occupation of those localized states without changing the number of conducting extended channels. The Hall conductance then remains $p{e}^2/h$ and the Hall voltage at fixed current remains $I_{x}h/(p{e}^2)$ over a finite field interval.

In the same regime, separated edge channels and localized bulk states suppress dissipative backscattering, so ${\rho }_{xx}$ is very small. At a transition the chemical potential reaches the next band of extended states; longitudinal conduction rises and the Hall conductance changes to another integer. This is a mobility-gap explanation: a plateau does not require the density of states itself to vanish, nor does it require the carrier density to grow linearly with field. Low temperature, sufficient mobility and suitable contacts are needed to resolve it.

This physical explanation is determined by the surviving question text. Particular measured plateau heights and widths cannot be quoted because the source graph is absent, as recorded in part (b).

Use a normal-metal--insulator--superconductor junction with a weak tunnelling barrier. For a nearly constant normal-metal density of states, the differential conductance probes the thermally broadened superconducting quasiparticle density of states. At sufficiently low temperature, the ideal BCS theory result isThere is very little single-electron tunnelling for $|eV|<\Delta$, while a sharp onset occurs at $|eV|=\Delta$. The gap can therefore be obtained from the two symmetric onset voltages, $\Delta =e|V_{\mathrm{onset}}|$. For this one-superconductor junction the threshold is $\Delta /e$, not $2\Delta /e$.

Integrating at zero temperature gives the useful current sketchThe curve is odd, nearly flat inside the gap, and approaches the normal-state line $V/R_N$ at large $|V|$. The ideal gap-edge singularity is in the slope, not in the current itself. At finite temperature below the transition the onset rounds and thermally activated subgap current appears. Measuring several temperatures distinguishes this rounding from the closing of $\Delta (T)$.

Ultraviolet photoelectron spectroscopy probes occupied valence states through photoemitted electron energies. Angle-resolved photoemission spectroscopy adds emission direction and therefore parallel crystal momentum, revealing occupied band dispersions and Fermi contours. For a clean periodic surface, $k_{\parallel }=\sqrt{2m_e{E}_{\mathrm{kin}}}\sin \theta /\hslash$ modulo a surface reciprocal-lattice vector. A two-dimensional surface band has no intrinsic perpendicular dispersion; varying photon energy helps distinguish it from bulk bands, while allowing for matrix-element changes. A simple surface-state example is a parabolic band $E({\mathbf{k}}_{\parallel })=E_0+{\hslash }^2{k}_{\parallel }^2/(2m_{\ast })$.

Scanning tunneling spectroscopy measures local differential conductance. With a suitable featureless tip and weak tunneling, $dI/dV$ samples the local density of states, giving occupied and unoccupied features on opposite bias sides. It provides spatial localization, defects and standing-wave patterns, while ordinary spectroscopy at one point does not directly give the complete momentum-resolved band structure. For the parabolic two-dimensional band, the ideal density of states has a step at its onset, while spatial interference can supply information about allowed wavevectors. Inverse photoemission spectroscopy accesses unoccupied states by injecting electrons and measuring emitted photons. Optical absorption and electron energy loss spectroscopy instead probe allowed excitation energies and collective modes, with their own selection rules and momentum ranges. These methods supply complementary information; none automatically returns the full interacting band structure without matrix elements, resolution and surface effects.

For the Pt core-level spectra, electrons suffering inelastic losses emerge with reduced kinetic energy and appear at larger inferred binding energies. Loss tails and secondary-electron contributions therefore generate the background, principally on the high-binding-energy side of a core peak. The two clean-surface components correspond to different environments of the same Pt core-level branch. The stated values imply a surface core-level shiftReduced coordination at a surface narrows the $d$ band. In a simple almost-filled-band picture, maintaining its occupation moves the band center upward relative to the Fermi level; associated local-potential changes can move core levels upward too, reducing their binding energy. This is a plausible initial-state explanation of the negative shift. Final-state screening of the core hole also matters, so the measured shift alone does not uniquely determine a local charge or separate these mechanisms.

After adsorption, Pt atoms directly bonded to CO have a different local potential, hybridization and core-hole screening. In carbon monoxide adsorption on platinum, carbon-end donation and metal-to-molecule back-donation change the Pt–CO bond and hence the Pt core binding energy. The additional weak high-binding-energy component is naturally assigned to these chemically modified surface Pt atoms. Its small weight is compatible with only a minority of surface sites being occupied; a quantitative coverage estimate requires fitted areas and appropriate attenuation assumptions.

The new component is centered at approximately $71.9$–$72.0\mathrm{eV}$ in the supplied plot. Relative to bare surface Pt at $70.5\mathrm{eV}$, the core-level chemical shift is thereforeUsing the bulk peak as the reference would instead give about $+1.0$–$1.1\mathrm{eV}$, which answers a different comparison. The plotted axis runs toward decreasing binding energy to the right, so the additional peak on its left is a positive binding-energy shift. The estimate is a graphical reading, with no claim to sub-tenth-eV precision or a unique charge-transfer value.

Periodic boundary conditions give wavevectors $(2\pi /L)(n_x,n_y)$. In the continuum, large-box limit each state occupies area $(2\pi /L{)}^2$ in wavevector space. A circular annulus therefore contains $[L^2/(2\pi {)}^2]2\pi kdk$ spinless states. Since $ϵ={\hslash }^2{k}^2/(2m)$, the density of states isAt finite $L$ the exact spectrum is discrete; this formula is its continuum density.

With electron spin the two Zeeman branches are ${ϵ}_{\mathrm{kin}}\mp \mu H$. The requested calculation includes this spin coupling and neglects orbital coupling to the field; for a perpendicular field, Landau levels would change the result. It is appropriate for a field parallel to the ideal two-dimensional plane, or for the stated spin-only model. Taking $\mu H\ge 0$, the lower branch starts at $-\mu H$, and the upper one at $\mu H$, soConstruct the ground state by filling the lowest one-electron levels subject to the Pauli exclusion principle. If the Fermi energy ${ϵ}_F$ exceeds $\mu H$, the two populations are $N_{+}=g_0({ϵ}_F+\mu H)$ and $N_{-}=g_0({ϵ}_F-\mu H)$. ConsequentlyThe ground-state energy, including the Zeeman term, follows by integrating the energy over each occupied branch:

In a large volume, one one-particle state occupies momentum-space volume $h^3/V$. A spherical shell of radius $p$ has volume $4\pi p^{2}dp$, so for spin degeneracy $g$ the density of states is $4\pi gV{p}^{2}dp/h^3$. With ultra-relativistic dispersion $E=cp$ and zero chemical potential, the Bose-Einstein distribution givesThe convergent dimensionless integral is ${\pi }^4/15$, obtained by expansion into exponentials and summing $\sum n^{-4}$. Therefore $ϵ\propto T^4$ and $\alpha =4$. Photons are massless bosons with two polarizations; equilibrium emission and absorption do not conserve photon number and force ${\mu }_{\gamma }=0$.

For recombination, chemical equilibrium imposes ${\mu }_p+{\mu }_e={\mu }_H+{\mu }_{\gamma }={\mu }_H$, including rest energies in the chemical potentials. Divide the three nonrelativistic number-density expressions to obtainHere $m_p{c}^2+m_e{c}^2-m_H{c}^2=I$. Neglect the relative proton/hydrogen mass difference and take the ground-state degeneracy ratio $g_p{g}_e/g_H=1$ (electron and proton spins give $2\cdot 2$ and hydrogen gives $4$). Charge neutrality gives $n_p=n_e$. This yields the stated Saha equation

With $n_e=X_e{n}_B$ and $n_H=(1-X_e)n_B$, the ratio asked for isEquivalently the prefactor is $16\pi \zeta (3)\eta (2\pi {)}^{-3/2}[kT/(m_e{c}^2){]}^{3/2}$. Because photons vastly outnumber baryons, even a small high-energy tail can keep hydrogen ionized. Neutralization therefore requires exponential suppression of that tail far below $kT=I$, explaining the much lower recombination temperature. The equilibrium formula is an estimate; the actual expanding-universe recombination kinetics eventually depart from Saha equilibrium.

In the nondegenerate regime, $\mu$ lies well inside the gap, with $\mu \gg kT$ and $E_g-\mu \gg kT$. Then conduction occupations are approximately $e^{(\mu -E)/(kT)}$, while the hole probability in a valence state is approximately $e^{(E-\mu )/(kT)}$. Using the near-edge density of states and the supplied gamma integral givesThereforeThis assumes the edge approximations are valid over the thermally populated ranges. For an intrinsic semiconductor $n=p$, so each carrier density is proportional to $T^{3/2}{e}^{-E_g/(2kT)}$. Both mobile conduction electrons and holes contribute to conductivity, $\sigma =|e|(n{b}_e+p{b}_h)$ with mobilities $b_e,b_h$. The exponentially varying carrier densities explain strong temperature dependence even when the mobilities vary only algebraically.

Donors supply electrons and acceptors remove electrons to create holes. With complete ionization and negligible minority carriers, the majority densities are $n\simeq N_d$ on the n side and $p\simeq N_a$ on the p side. The corresponding isolated chemical potentials areCarrier diffusion at contact leaves positively charged donors and negatively charged acceptors in a depletion layer. At equilibrium the electrochemical potential is constant; the electron energy shift is $-|e|\varphi$. Hence the n-minus-p electric potential isAll $N$ quantities must have a consistent normalization: if $N_c,N_v$ are state densities per volume, $N_d,N_a$ are dopant densities rather than bare atom counts. The formula neglects incomplete ionization, degeneracy and changes in band structure.

At a common temperature the two equilibrium junction/contact effects do not drive a perpetual current. At different junction temperatures, electrons and holes transport entropy differently, giving different Seebeck coefficients on the two sides. The loop has a thermoelectric electromotive force, oriented as ${\int }_{T_1}^{T_2}(S_n-S_p)dT$, and therefore generally carries current when closed. This is the Seebeck effect: energy is supplied by heat flow from hot to cold. The temperature dependence of the junction barrier is useful intuition, but an exact terminal voltage is not obtained simply by subtracting two equilibrium built-in potentials; it also depends on bulk transport and contact conditions. The given equilibrium density-of-states data do not determine the numerical thermopowers.

The chemical potential is $\mu =(\partial E/\partial N{)}_{S,V}$, equivalently $(\partial F/\partial N{)}_{T,V}$: it is the free-energy cost of adding a particle at fixed temperature and volume. With units $k_B=1$, a large reservoir has entropy $S_R(E_{\mathrm{tot}}-E,N_{\mathrm{tot}}-N)=S_R^0-E/T+\mu N/T$ to first order. Since the number of compatible reservoir states is proportional to its exponential, the grand canonical ensemble assignsFor independent fermions, a one-particle state of energy $\epsilon$ has occupation zero or one. Its partition factor is $1+e^{-(\epsilon -\mu )/T}$, so the mean occupation is the Fermi-Dirac distribution $[e^{(\epsilon -\mu )/T}+1{]}^{-1}$. Multiplication by the given density of states yields $dN=C{\epsilon }^{1/2}d\epsilon /[e^{(\epsilon -\mu )/T}+1]$.

At zero temperature the occupation is a step function, filled below $\mu$ and empty above it, so $\mu ={\epsilon }_F$ and $N=(2C/3){\epsilon }_F^{3/2}$. At fixed particle number, the low-temperature shift of $\mu$ is $O(T^2/{\epsilon }_F)$ and does not affect the leading estimate. In the narrow interval of width $T$ above ${\epsilon }_F$, replace the density of states by $C\sqrt{{\epsilon }_F}$ and put $u=(\epsilon -{\epsilon }_F)/T$. ThenEquivalently, the fraction above the Fermi energy is asymptotic to $(3\log 2/2)T/{\epsilon }_F$.

Measure energy from the single-particle ground state and write $z=e^{\mu /(k_{B}T)}\le 1$. The Bose-Einstein distribution gives the number in excited levels asHere ${\mathrm{Li}}_s(z)={\sum }_{j\ge 1}{z}^j/j^s$, obtained directly from the supplied integral identity. At the Bose-Einstein condensation threshold, $z\uparrow 1$ and the excited states are just able to hold all $N$ particles. ThusThe ground-state population is treated separately from the continuum density of states.

Each one-particle momentum state occupies volume $(2\pi \hslash {)}^3/V$ in momentum space. Counting the spherical shell gives the density of statesThe increasing dispersion law permits inversion, and the density is zero below any minimum allowed energy. No additional spin degeneracy is included here.

For noninteracting particles, the classical Boltzmann sum factorizes into $z^N$. Dividing by $N!$ accounts for indistinguishability without quantum exchange corrections, giving the partition function $Z=z^N/N!$, with $z=\int g(ϵ)e^{-ϵ/(kT)}dϵ$. Differentiation with respect to inverse temperature givesFixed mode indices have momenta proportional to $L^{-1}$ because the boundary quantization condition fixes $pL/\hslash$. As $V=L^3$, $\partial p/\partial V=-p/(3V)$ and $\partial ϵ/\partial V=-p{ϵ}^{\prime }(p)/(3V)$. The pressure $P=kT{\partial }_V\log Z$, evaluated by differentiating the mode sum, therefore obeys

For nonrelativistic particles $ϵ=p^2/(2m)$, so $PV=2E/3$ and $E=3NkT/2$. For massless relativistic particles $ϵ=cp$, so $PV=E/3$ and $E=3NkT$. Both obey $PV=NkT$.

The nonrelativistic thermal wavelength is ${\lambda }_{\mathrm{th}}=h/\sqrt{2\pi mkT}$, and $z=V/{\lambda }_{\mathrm{th}}^3$. If ${\lambda }_{\mathrm{th}}\ll (V/N{)}^{1/3}$, then $N{\lambda }_{\mathrm{th}}^3/V\ll 1$: occupation per thermal quantum state is small. Wave packets scarcely overlap and exchange corrections to Maxwell-Boltzmann distribution are negligible, justifying the classical formulas.