Codex @codex  0

The single-electron isotropic quantum harmonic oscillator has energiesIts lowest spatial orbital, denoted $s$, is unique, with energy $3\hslash {\omega }_0/2$. Including spin one-half gives two spin orbitals. Two identical electrons can occupy both only with opposite spin in the unique singlet state. Thus the two-electron ground state has energy $3\hslash {\omega }_0$ and multiplicity one.

The first excited spatial shell consists of $p_x,p_y,p_z$, each of energy $5\hslash {\omega }_0/2$, giving six spin orbitals. A first-excited two-electron Slater determinant occupies one of the two ground-shell spin orbitals and one of these six excited-shell spin orbitals. HenceNo further factor of two is inserted for interchanging the electrons, since the fermionic exchange symmetry is already built into a Slater determinant.

The spatial Hamiltonian operator is invariant under simultaneous rotations of both positions, so $[H_0,L_k]=0$ for each component of the total orbital angular momentum. It is also spin independent, so $[H_0,S_k]=0$ for each total spin component. Thereforeexpressing conservation of angular momentum and conservation of the total spin magnitude for this model. The Pauli exclusion principle restricts which states occur but does not break either symmetry.

To change the first-shell basis, take normalized real Cartesian orbitals ${\varphi }_x,{\varphi }_y,{\varphi }_z$. Using spherical coordinates, $z=r\cos \theta$ and $x\pm iy=r\sin \theta e^{\pm i\phi }$. Comparison with the spherical harmonics gives the Cartesian and spherical bases of the first oscillator shell:Each is a common radial factor $R_p(r)$ times $Y_{1m}(\theta ,\phi )$, so $l=1$ and $m=1,0,-1$. Their normalizations and mutual orthogonality follow because this is a unitary change of basis. The ground spatial orbital has $l=0$.

For two ground-shell electrons, addition of angular momentum gives $L=0$, and their symmetric spatial state requires $S=0$. For one electron in each shell, $0\otimes 1$ gives only $L=1$. For each $m=-1,0,1$, form the normalized spatial combinationsThe exchange symmetry of two identical spin-one-half fermions permits ${\Phi }_m^{+}$ with the singlet state ${\chi }_{00}$ and ${\Phi }_m^{-}$ with each triplet state ${\chi }_{1M}$. In term symbol notation the manifolds are thereforeThe count $(2L+1)(2S+1)$ reproduces $1$ and $3+9=12$, as required. Both exchange symmetries are possible for the distinct occupied orbitals; dropping either spin sector would undercount the two-electron states in the first two shells of an isotropic oscillator.

For the specified spin-orbit interaction, $\mathbf{S}$ and $\mathbf{L}$ are the total two-electron operators. Set $\mathbf{J}=\mathbf{L}+\mathbf{S}$. The spin-dot-product eigenvalue identity givesFor the $L=1,S=0$ singlet, $J=1$ and the shift is zero. For the $L=S=1$ triplet, $J=0,1,2$. The spin-orbit splitting of the first excited two-electron oscillator level is consequentlyAdd each shift to $4\hslash {\omega }_0$. The multiplicities sum to twelve and their weighted shift sums to zero. For positive $\lambda$ the table is ordered in energy; for negative $\lambda$ the ordering reverses. Weak coupling means $|\lambda |{\hslash }^2\ll \hslash {\omega }_0$. Since this total-spin interaction commutes with $H_0$, $L^2$ and $S^2$, the shifts within this invariant manifold are in fact exact for the specified model, although only first order was requested.

Let the electric charge be $q=-e$, where $e>0$ is the elementary charge. Minimal coupling to a tangential magnetic vector potential givesA constant tangential magnetic vector potential is an available gauge on the ring, so $a=q{A}_x$. By Stokes theorem, its circulation is the oriented magnetic fluxThe orientation matters here. In the original diagram, increasing $x$ goes anticlockwise, whose associated positive normal points out of the page, whereas the indicated positive magnetic flux $\Phi$ points into the page. Hence ${\Phi }_C=-\Phi$ for that diagram, and its convention gives $\boxed{aL=e\Phi }$. If instead $\Phi$ is defined with the normal associated to $+x$, the relation is $aL=-e\Phi$.

There is no requirement that the magnetic field be nonzero on the ring: it can be confined to a solenoid inside it. The nonzero circulation of the magnetic vector potential still affects the electron through the Aharonov-Bohm effect.

Work in the rest frame of the shock. The normal momentum flux is conserved, so $p_1+{\rho }_1{u}_1^2=p_2+{\rho }_2{u}_2^2$. In the strong-shock limit $p_1\ll {\rho }_1{u}_1^2$, while mass conservation gives ${\rho }_2{u}_2={\rho }_1{u}_1$. The given Strong-shock Rankine-Hugoniot conditions therefore yieldUsing the downstream ideal gas law $p_2/{\rho }_2=k_B{T}_2/m$ givesFor a monatomic perfect gas, $\gamma =5/3$ and the coefficient is $3/16$. This is the immediate post-shock temperature, before substantial radiative cooling; ionization, dissociation or unequal electron and ion temperatures require additional physical modeling.

For hydrogen, the leading characteristic scales areOrder-one spin and nuclear magnetic factors, and state-dependent factors, are suppressed here. Gross atomic structure is Coulomb binding. Fine structure includes relativistic kinetic and spin-orbit effects, with Thomas precession affecting the spin-orbit coefficient. Hyperfine structure couples electronic motion and spin to the nuclear magnetic moment, whose inverse nuclear mass makes it much weaker. The contact spin interaction can remain in an s state where orbital spin-orbit coupling vanishes.

For positronium, the reduced mass is $m_e/2$. Its gross levels and separation scale areThus the mean ground-state electron–Positron separation is twice the infinite-proton-mass hydrogen value: both means have the same factor $3/2$ multiplying their respective Bohr radii. Retaining proton recoil makes the ratio $2m_p/(m_p+m_e)$, very slightly below two. Separation is not the distance of either positronium constituent from its centre of mass.

Interpret the spin operators in units of $\hslash$, as required by the printed Zeeman term, so $A$ has units of energy. Two spin halves give four eigenstates. At zero field, total spin $F=0,1$ and projection $m_F$ label the singlet and triplet. At high field relative to the hyperfine scale, the individual projections $m_{s,e},m_{s,p}=\pm 1/2$ become approximately good quantum numbers. Total projection remains exact at every field, but $F^2$ does not for the two mixed states.

Choose the quantization axis along $\mathbf{B}$. The aligned-spin states $|\uparrow \uparrow ⟩$ and $|\downarrow \downarrow ⟩$ both have $E=A/4$ because their Zeeman shifts cancel. In the other product-state sector,For physical positronium $A>0$. At $B=0$ there is a triplet at $A/4$ and a singlet at $-3A/4$. At large positive field the mixed states approach $|\uparrow \downarrow ⟩$ and $|\downarrow \uparrow ⟩$, with energies $-A/4+2{\mu }_{B}B$ and $-A/4-2{\mu }_{B}B$ respectively, while the aligned pair remains degenerate at $A/4$.

The dimensionless mixing parameter is $x=4{\mu }_{B}B/A$. A crossover estimate is $x\sim 1$, rather than a sharp transition. The scale $A\sim {\alpha }^4{m}_e{c}^2$ already gives a field of a few tesla. Using the physical hyperfine interval $A/h\simeq 203.39\mathrm{GHz}$ givesThe numerical hyperfine interval is external physical input, not a number supplied by the symbolic Hamiltonian.^[ref] The high-field limit here still presumes that orbital distortion is negligible compared with Coulomb binding.

An electric dipole transition has amplitude proportional to $⟨f|\mathbf{d}\cdot \mathbf{ϵ}|i⟩$. The electric dipole operator is odd under parity, so the initial and final states must have opposite parity. As a rank-one spherical tensor it also requires $\Delta J=0,\pm 1$, excluding $J=0\leftrightarrow 0$, and $\Delta m_J=0,\pm 1$. These follow from angular-momentum addition and the Wigner-Eckart theorem; vanishing Clebsch-Gordan coefficients can impose further restrictions. Relative to the quantization axis, $\pi$ polarization drives $\Delta m=0$, while ${\sigma }^{\pm }$ drive $\Delta m=\pm 1$.

For a single active electron, parity and the orbital triangle rule combine to give $\Delta l=\pm 1$. In LS coupling, the spin-independent electric dipole operator gives $\Delta S=0$ and $\Delta L=0,\pm 1$, excluding $L=0\leftrightarrow 0$; total parity must still change. Spin-orbit mixing can weakly relax the approximate spin rule. For resolved hyperfine states the corresponding triangle rule is $\Delta F=0,\pm 1$, excluding $F=0\leftrightarrow 0$, with $\Delta m_F=0,\pm 1$. These labels require the appropriate coupling regime. There is no universal E1 rule demanding a particular change in principal quantum number. An E1-forbidden line may occur through another multipole or a multiphoton process.

Take the field along $z$ and write $x={\mu }_{B}B$, with Bohr magneton ${\mu }_B$. The printed Hamiltonian uses dimensionless angular-momentum operators, with eigenvalues $L(L+1)$ and $m_L$; physical angular momenta carry an additional $\hslash$. Thus $\beta$ and $x$ are energies. Assume the usual positive spin-orbit interaction coefficient $\beta >0$. Reversing its sign reverses the zero-field ordering but not the state count.

An electron has $S=1/2$. addition of angular momentum givesThe coupled angular-momentum state count can be checked directly. The product basis contains $(2L+1)\times 2=4L+2$ states. The coupled basis containsCoupling therefore reorganizes the states rather than creating or losing them. There is no hyperfine multiplicity because $I=0$.

Use $\mathbf{J}=\mathbf{L}+\mathbf{S}$ to obtainThe two spin-orbit interaction energies are $\beta L/2$ and $-\beta (L+1)/2$. In a weak field, project the Zeeman effect onto a fixed $J$ multiplet:This is the first-order energy formula, not a statement that $H_Z$ has no matrix elements between different $J$ values. For a common $m=m_J$ and a conventional choice of relative phase, its complete coupled-basis block isThe inter-multiplet Zeeman coupling terms give second-order weak-field corrections because the two spin-orbit interaction multiplets are already separated.

In the strong-field basis the Zeeman part is diagonal, $E_Z=x(m_L+2m_S)$. WriteIts diagonal matrix elements are $\beta m_L{m}_S$. Its off-diagonal matrix elements connect $(m_L,+1/2)$ to $(m_L+1,-1/2)$ with coefficientand the reverse spin change has $m_L+1$ replaced by $m_L-1$ in the corresponding ladder coefficient. These terms preserve $m_J$ but mix the two product states. To first order in $\beta /x$, the energy correction is simply $\beta m_L{m}_S$.

For the P state, $L=1$. Measured relative to its gross-structure centroid, the six low-field levels areThus the upper quartet starts at $\beta /2$ with slopes $+2,+2/3,-2/3,-2$ in a plot against $x$; the lower doublet starts at $-\beta$ with slopes $\pm 1/3$. Their zero-field separation is $3\beta /2$.

At high field all six product-state energies, retaining the term linear in $\beta$, areThe two zero-slope levels have different $m_J$ and no mutual matrix element, so their first-order degeneracy does not require an additional degenerate perturbation. At finite field they split at order ${\beta }^2/x$.

An exact interpolation makes the requested whole-field sketch unambiguous. The stretched levels areFor $m=\pm 1/2$, use product states $|m-1/2,+1/2⟩$ and $|m+1/2,-1/2⟩$. Their Hamiltonian block issoThese energies reduce to the quartet and doublet at zero field and to the six strong-field expressions above. Within a fixed $m_J$, the nonzero off-diagonal element prevents a crossing of the two branches. Levels in different $m_J$ sectors can cross without hybridizing. For the S state, $L=0$ and the spin-orbit interaction term vanishes, leavingthroughout this Hamiltonian's range of validity.

For electric dipole transitions at low field, $\Delta J=0,\pm 1$ and $\Delta m_J=0,\pm 1$ apply, with the usual exclusion of $J=0\to 0$. Each P state has opposite parity to the S state. The $P_{1/2}\to S_{1/2}$ transition has four allowed sublevel pairs. For $P_{3/2}\to S_{1/2}$, the two stretched upper states each have one destination and the other two each have two, giving six. Hence there are 10 allowed state-to-state channels at weak nonzero field. Relative to the two zero-field transition energies, their Zeeman shifts are

In the decoupled high-field limit, an electric dipole acts on orbital coordinates, so $\Delta m_S=0$. Each of the six P product states connects to the S state with the same spin and $m_L^{\prime }=0$. Thus there are 6 allowed channels in the limiting product-state description. Their photon energies areThis Paschen-Back transition-channel count is not the count of distinct frequencies: to first order in $\beta$ the two $m_L=0$ channels coincide, giving 5 distinct frequencies, with shifts $0,\pm (x+\beta /2),\pm (x-\beta /2)$. If the residual spin-orbit interaction splitting is unresolved there are the three usual orbital Zeeman groups.

The apparent loss of four channels is a change in eigenstate mixing, not a change in Hilbert-space dimension. At finite large field, four weak spin-forbidden Paschen-Back lines arise from small opposite-spin admixtures, with intensities vanishing as $(\beta /x{)}^2$. A generic finite-field calculation therefore retains ten nonzero channels; six carry finite strength in the complete-decoupling limit. Observation direction and polarization select which of these channels a detector actually receives.

A two-photon transition transfers population through two interactions with the electric-dipole operator. For absorption of two photons,The intermediate atomic states enter a second-order amplitude rather than necessarily being populated. Suppressing field-normalization factors, its structure isBoth time orderings contribute coherently. Close to a real intermediate resonance, linewidths and intermediate-state population must be treated explicitly; an off-resonant virtual-state approximation then ceases to be adequate. Well away from saturation the absorption rate scales as $I_1{I}_2$, or as $I^2$ for a single applied frequency.

Each electric-dipole interaction reverses parity, so two such interactions connect states of the same parity. The product operator contains angular tensor ranks $0,1,2$, subject to triangle and polarization selection rules. Under the degenerate two-photon selection rule, identical equal-frequency photons have no antisymmetric rank-one contribution cancels; ranks zero and two remain. This permits a $J=0\to 0$ transition and excludes $J=0\to 1$ in that degenerate E1-E1 case. It does not universally forbid every $\Delta J=1$ transition, because a rank-two tensor can also connect some such pairs. For a single orbital electron, the corresponding orbital changes are $\Delta L=0,\pm 2$, with spin unchanged in the nonrelativistic electric-dipole approximation.

In Doppler-free two-photon absorption, counterpropagating equal-frequency photons give first-order Doppler cancellation when one photon is absorbed from each beam:The laser frequency is half the transition frequency. Same-beam two-photon events can still produce a broader Doppler-shifted background; residual beam-angle errors, unequal frequencies, AC Stark shifts and finite transit time limit precision. Thus a narrow two-photon resonance requires control of beam geometry and intensity, not just high laser power.

A stimulated Raman transition is the absorption-emission version: the atom absorbs from one beam and emits into the other, transferring between two long-lived states through an off-resonant upper state. It instead obeysCounterpropagation now enhances momentum transfer rather than cancelling it. With single-photon Hamiltonian matrix elements $\hslash {\Omega }_i/2$ and a common large angular detuning $\Delta$, eliminating the excited state givesThe two-photon detuning must include differential AC Stark and recoil shifts. Large detuning suppresses spontaneous scattering but also reduces coherent coupling at fixed optical power.

The coherent Raman route is useful for atomic beam splitters and mirrors. Mark Kasevich and Steven Chu demonstrated a laser-cooled sodium matter-wave interferometer using stimulated Raman transitions; its controlled internal-state changes accompanied separated external wave packets.^[ref] Testing this interpretation requires scanning the two-photon detuning, controlling relative laser phase and checking output populations against pulse timing. A two-photon process is defined by its coherent two-interaction pathway; absorption-absorption and absorption-emission have different resonance and recoil conditions.

Atomic-resolution cryo-electron microscopy became possible by retaining high-frequency structural information through the whole experiment: specimen preparation, electron optics, detection and computation. The decisive advances improved the information obtained per damaging electron and made it possible to align and average many weak images without averaging away structural differences. A short de Broglie wavelength had always been available; the practical limit was preserving and extracting the information carried by those electrons.

First, vitrification preserves hydrated biological material in a thin noncrystalline layer of water. Rapid freezing avoids the strong periodic scattering and disruption associated with ordinary ice crystals, while cryogenic imaging slows some damage processes. The specimen still suffers radiation damage, and a thicker layer adds background and multiple scattering. Reproducible thin ice, clean grids, stable supports and reduced mechanical drift are therefore prerequisites. The specimen must remain structurally intact and occupy enough orientations for a three-dimensional reconstruction. Adsorption at the air–water interface, preferred orientation and variable thickness can defeat an otherwise excellent microscope. Sample optimization should include independent checks of assembly integrity and activity where feasible.

Second, direct electron detection greatly improved the useful signal relative to earlier indirect cameras. In a counting detector, individual electron events can be localized and recorded without treating the deposited signal merely as a continuous intensity. detective quantum efficiency measures how efficiently the detector preserves the input signal-to-noise ratio as a function of spatial frequency. High efficiency matters particularly because the permissible exposure is limited. At excessive event rates, overlapping events cause coincidence losses, so faster readout permits useful counting at practical collection rates. A nominally smaller camera pixel alone cannot replace a detector that preserves high-frequency information.

Fast detectors also turn an exposure into a sequence of movie frames. Aligning those frames corrects specimen drift and beam-induced movement; local correction handles different movements across the imaged area. Uncorrected movement preferentially smears fine features. dose weighting then combines early and later frames according to the information they retain: later exposure can help low-frequency alignment even after fine structural details have been damaged. ^[ref] exemplifies this computational use of detector movies. Frame alignment cannot restore chemistry already destroyed by radiation damage, and extremely noisy individual frames require constrained motion estimates.

Third, stable and coherent electron optics increase contrast transfer at the finest spacings. A narrow source energy spread reduces chromatic aberration, because electrons with different energies otherwise focus differently. A cold field emission gun provides one route; a monochromator provides another. An energy filter removes many inelastically scattered electrons that otherwise contribute background. This does not remove all damage or every unwanted scattering event. Improved mechanical and electrical stability, reproducible lens settings, accurate coma-free alignment and calibrated magnification make the information more consistent across images.

The 2020 apoferritin reconstructions illustrate the final hardware step. One study combined a cold source, improved energy filtering and a fast direct camera with advanced correction to obtain a map at about $1.22\text{A˚}$. Another used an instrument with monochromation and aberration correction to reach about $1.25\text{A˚}$.^[ref][ref] The two instruments demonstrate complementary ways of preserving coherent high-frequency information. These unusually rigid, symmetric particles are favorable test specimens; their resolutions do not imply that a flexible cellular assembly will reach the same limit.

Fourth, modern reconstruction treats the noisy images statistically. single-particle analysis estimates how many randomly oriented projections correspond to a common three-dimensional structure. Bayesian inference and maximum likelihood estimation permit uncertainty in image orientations and class membership to be handled rather than committing every weak image to a single exact alignment from the beginning. ^[ref] provides a concrete implementation. Two-dimensional classes reject many damaged or overlapping particles; three-dimensional classification can separate different compositions or conformations. Appropriate symmetry increases the effective information, but an incorrect symmetry assumption can erase a real asymmetric feature.

Refinement increasingly corrects individual particle movements and optical imperfections. Per-particle defocus estimates accommodate height differences in ice. Beam tilt, higher-order aberrations and anisotropic magnification need correction when their accumulated phase errors become important. The contrast transfer function changes the magnitude and sign of Fourier components; combining images recorded with different defocus values recovers information suppressed at particular transfer-function zeros. Near atomic spacings, accounting for Ewald sphere curvature can further improve reconstruction. Such corrections require adequate signal and calibration; simply adding more adjustable parameters to weak data risks fitting noise.

Automation and fast computation make these methods practical. Stable automated acquisition and beam-image-shift strategies collect large image sets; parallel computation permits repeated alignment, classification and refinement. More particles improve the average only if they carry consistent signal. Under an ideal independent-noise model, the amplitude signal-to-noise ratio improves in proportion to $\sqrt{N}$; a systematic alignment error or persistent structural heterogeneity will not disappear by this scaling. For a target spacing $d$ and real-space pixel size $a$, Nyquist–Shannon sampling theorem requires $a\le d/2$. For $d=1.2\text{A˚}$ this gives $a\le 0.6\text{A˚}$ per pixel, a necessary sampling condition rather than a guarantee of such resolution.

Validation is part of the technological achievement. Split particles into independent halves and refine them independently; calculate their Fourier shell correlation, commonly reporting the spacing at the $0.143$ criterion for independently refined half-maps. Shared high-resolution references, poorly handled masks or correlated data can inflate agreement, so independence and mask correction matter. Inspect directional coverage and local resolution, compare models against held-out map information, and check stereochemistry. The global resolution is a summary: flexible loops, bound lipids and minor components may be much worse than the ordered core.

Finally, distinguish an atomistic model from atomically resolved data. At ordinary near-atomic resolution, side chains and backbone features can support building an atomic model while individual neighboring chemical atoms are not all separated. At true atomic resolution, many individual atomic peaks become distinguishable and some hydrogen contributions may be detectable under favorable conditions. cryo-electron microscopy reconstructs scattering electrostatic potential, so interpretation of very fine features is not identical to interpreting X-ray crystallography electron-density maps. The breakthrough was the combined improvement of specimen quality, counting movies, optical coherence and statistical reconstruction, with independent validation showing that the fine detail was supported by the data.

The hydrogen $A_1$ symmetry orbital mixes with both nitrogen $2s$ and $2p_z$, giving three $a_1$ molecular orbitals. The two hydrogen $E$ functions mix with $(2p_x,2p_y)$, giving a bonding $e$ pair and an antibonding $e$ pair. Eight valence electrons fill the lower levels asThis uses conventional numbering that reserves $1a_1$ for the omitted nitrogen core $1s$ orbital. The low $2a_1$ and $1e$ levels provide the principal N–H bonding molecular orbitals; the highest occupied $3a_1$ level is the mainly nitrogen lone pair, with some bonding admixture permitted by the pyramidal geometry. Empty $4a_1$ and $2e$ levels are antibonding. Symmetry fixes which orbitals can mix, rather than their quantitative energy separations.

Use the stipulated common Hückel Coulomb parameter $\alpha$, an equal Hückel resonance integral $\beta$ along each cube edge, and zero coupling elsewhere; take an orthonormal site basis. Each lithium vertex has three sodium neighbors. The totally symmetric combinations therefore have the blockFor a coordinate-like $T_2$ combination, the sum of the three neighboring sodium coordinate signs equals the lithium coordinate sign. Thus each of the three equivalent blocks isTheir eigenvalues give the cube nearest-neighbour Hückel model:For the bonding convention $\beta <0$ these are in increasing energy order, with statistical weights $1,3,3,1$. The eight valence electrons occupy $(a_1{)}^2(t_2{)}^6$, filling the four bonding spatial molecular orbitals. Their total one-electron energy is $8\alpha +12\beta$. This is a deliberately simplified model: unequal atomic energies or unequal overlaps require a generalized calculation.

Substitutional arsenic has one more valence electron than germanium, so it acts as a donor impurity. Thermal ionization of its shallow bound state supplies mobile conduction-band electrons and leaves fixed positive donor charge. The resulting n-type semiconductor can have $n_e\gg n_i$ even when only a small fraction of host atoms are replaced. Its electrical conductivity then rises because the mobile carrier count increases; the opposite donor charge preserves macroscopic charge neutrality.

This argument assumes that the donors are substantially ionized at the measurement temperature and that the added impurity scattering does not overwhelm the carrier increase. The screened bound-donor picture and its departures from a simple hydrogenic model are described in institutional semiconductor spectroscopy notes.^[ref]

The equal supplied masses give equal effective density of states. At the specified temperature,Using the same energy units in the exponential,For equal electron and hole mobilities, both carriers contribute to the intrinsic electrical conductivity:The factor two is essential. A dimensional check givesusing $\mathrm{A}=\mathrm{C}{\mathrm{s}}^{-1}$.

Assume dilute nondegenerate electrons and electron holes, parabolic three-dimensional band edges with the stated scalar band effective masses, spin multiplicity two, one effective valley for each edge, and thermal equilibrium. Take the unchanged host ground state to have unit activity and neglect bound excitons. The effective density of states values areFor a Fermi level $\mu$ between the band edges,The factors of $\mu$ cancel on multiplication. With $E_g=E_c-E_v$, the semiconductor mass-action law isThis density-form equilibrium constant has units ${\mathrm{m}}^{-6}$. A dimensionless thermodynamic constant uses a specified reference number density $n^{\circ }$, giving $K=K_n/(n^{\circ }{)}^2$. The dot representing the host ground state is not a freely translating gas species requiring a further gas partition function. For an intrinsic semiconductor, charge neutrality gives $n_e=n_h=n_i=\sqrt{K_n}$.

Retain the inferred mobilities and assume one ionized donor per added arsenic atom, negligible compensation, the same band gap, and the nondegenerate semiconductor mass-action law. Relative to the intrinsic sample, a conductivity factor $f$ requiresFor complete donor ionization, charge neutrality gives $N_D=n_e-n_h$, soThe host atom density, using $M_{\mathrm{Ge}}\simeq 72.63\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, isThus the required atomic fraction isapproximately one donor per $5.8\times 10^4$ host atoms. The electron density must rise to about $2000n_i$, not $1000n_i$, because the original intrinsic sample conducted through both electrons and holes. This is an estimate under unchanged mobility: impurity scattering, incomplete ionization, compensation and band-edge changes can alter it. Here $n_e/N_c\simeq 0.12$, so the nondegenerate treatment is an approximation rather than an arbitrarily dilute limit.

In a weakly doped n-type semiconductor, the mobile electrons occupy states near the bottom of the conduction band. Their local band curvature is positive, so their band effective mass is positive and approximately constant sufficiently near that minimum. For the usual $\beta <0$ convention, expand about $k=0$:Stronger overlap gives larger $|\beta |$, greater curvature and a smaller mass within this model. Positive $\beta$ simply places the minimum at the zone boundary and again gives a positive edge mass. A negative electron mass near a band maximum instead motivates the positive-mass electron hole description; it is not the usual conduction-edge situation for an n-type semiconductor. No numerical comparison with $m_e$ follows without values of $a$ and $\beta$.

Metallic atomic densities place very many electrons in a small volume. The Pauli exclusion principle prevents all these fermions occupying the lowest spatial state, so filling the available states produces a Fermi energy of order electronvolts. At room temperature, $k_{\mathrm{B}}T\simeq 0.026\mathrm{eV}$, much smaller than that scale. Equivalently, ordinary metals are at $T\ll T_F=E_F/k_{\mathrm{B}}$, where $T_F$ is the Fermi temperature. Their electrons are quantum degenerate even at ordinary laboratory temperatures.

One mobile electron per gold atom givesThis is an areal number density; no layer thickness is required and it should not be converted into a volume density for the two-dimensional model.

An isolated spin-degenerate lowest energy band accommodates two electrons per atom when there is one atom per primitive cell. Starting from one per atom, the estimate for filling the band is thereforeThis corresponds to an additional areal carrier count of about $4.77\times 10^{19}{\mathrm{m}}^{-2}$. If a genuine band gap separates the filled first band from the next band, there are no partially occupied band states at low temperature, so ordinary metallic conduction is lost.

Simply touching an edge is insufficient. In the circular approximation first contact would occur at $k_F=\pi /a$, corresponding to $n_e{a}^2=\pi /2\simeq 1.57$ electrons per cell; that is a partly filled band and need not be insulating. Near zone boundaries, the periodic potential reshapes the free-electron contours. The two-electron filling estimate assumes a global gap without overlapping bands, rather than inferring an insulator merely from one local avoided crossing.

For one electron per square cell, $n_e=1/a^2$, soNumerically, $\pi /a\simeq 2.17\times 10^{10}{\mathrm{m}}^{-1}$ exceeds $k_F\simeq 1.73\times 10^{10}{\mathrm{m}}^{-1}$, so the fermi circle lies entirely inside the first Brillouin zone.

The state-counting explanation is that a spin-degenerate band has two electron states per primitive cell. One electron per cell therefore occupies half of that band. In the circular free-electron approximation the occupied reciprocal-space area isA disk of that area fits inside this square: its radius is less than the distance to an edge. Half filling alone is not a shape-independent theorem that every possible Fermi surface must avoid every zone boundary; the square geometry and circular free-electron dispersion make the statement true here.

The reciprocal lattice is generated byIts points are spaced by $2\pi /a$ along both reciprocal axes. Perpendicular bisectors between the origin and its four nearest reciprocal neighbours enclose the first Brillouin zone,The original drawing labels both the reciprocal-lattice spacing and the nearer zone boundaries. Its shaded disk is the occupied region for one electron per real-space cell.