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Assume the atmosphere is transparent to incoming shortwave radiation, as required for the stated infrared-only radiative-equilibrium model. Global energy balance givesPut $D={\tau }_{\infty }$ and $F=I_{+}-I_{-}$. Radiative equilibrium requires $dF/d\tau =0$, hence the two equations imply $I_{+}+I_{-}=2\sigma T^4$. Substitution gives $d{I}_{+}/d\tau =d{I}_{-}/d\tau =-F/2$. At the top there is no incident infrared flux, $I_{-}(D)=0$, and $I_{+}(D)=F=L_{\infty }$. ThereforeThe black ground supplies $I_{+}(0)=\sigma T_g^4$. ThusThe ground also satisfies $\sigma T_g^4=L_{\infty }+I_{-}(0)$, showing explicitly that its outgoing flux balances absorbed sunlight plus downward atmospheric radiation. In pure radiative equilibrium the air just above the ground need not have the ground's temperature: $\sigma T(0{)}^4=L_{\infty }(1+D)/2$, whereas $\sigma T_g^4=L_{\infty }(2+D)/2$. That discontinuity is relevant to surface-driven convection in part (d).

The global mean absorbed solar flux is $L_{\infty }=0.8\times 584/4=116.8\mathrm{W}{\mathrm{m}}^{-2}$. Using $\sigma =5.6704\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ givesFor the requested graphical parcel estimate, start a dry adiabat at the warm ground, $T_{\mathrm{ad}}(z)=240.5-4.4z$ with $z$ in km, and intersect it with the radiative atmospheric profile $179.1[1+1.25e^{-z/10}{]}^{1/4}$. Solving givesThe original diagram below shows the intersection and the initial ground–air temperature discontinuity. A parcel heated towards the ground temperature remains buoyant until nearly one scale height in this model.

The atmospheric radiative lapse rate itself is only about $3.05\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ at the bottom and decreases with height, smaller than the stated adiabatic $4.4\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$. Thus the interior radiative atmosphere is statically stable; the driver of this particular estimate is the hotter ground and discontinuity at the lower boundary, not a radiative lapse rate everywhere steeper than adiabatic. Convection would smooth that discontinuity and redistribute energy.

This is a diagnostic depth using the unadjusted radiative ground and atmospheric temperatures, not a fully energy-conserving radiative–convective equilibrium after adjustment. A self-consistent adjustment changes the surface temperature, infrared fluxes and boundary height. Mars also has spectral CO${}_2$ absorption, dust, day–night variation, seasonal changes and geographically uneven heating; a gray, horizontally averaged, constant-scale-height column with fixed lapse rate captures none of those details. The calculation indicates the possible importance of a deep surface-driven mixed layer under the stated assumptions, not a universal Martian boundary-layer measurement.

Retaining the integrated energy flux from part (i), slow changes of depth and latitude obeyAt fixed depth and higher absolute latitude in the same hemisphere, $|f|=2\Omega |\sin \phi |$ increases, so $A_0\propto \sqrt{|f|}$ increases. The offshore trapping width $L_R=c/|f|$ narrows, concentrating the same transported energy closer to the coast.

This uses a local Kelvin-mode approximation on a slowly varying background, despite the original constant-$f$ derivation. The scale of variation must be long enough for the wave to adjust without substantial reflection, scattering or dissipation. It is not a rule to extrapolate through the equator, where the coastal trapping scale and hemisphere-dependent propagation rule change.

The Kelvin-wave coastal amplitude scaling follows from the flux. For a wave with peak coastal amplitude $A_0$, $\overline{v\eta }=(c/H)A_0^2{e}^{-2r/L_R}/2$. Integrating the supplied flux over the offshore coordinate gives its magnitudeThe direction is the alongshore propagation direction; the expression above is positive magnitude. For slowly varying depth, negligible dissipation/reflection and no other source or sink, the transported energy per time is conserved. At fixed latitude,The coastal amplitude increases on entering shallower water. This is stronger than the familiar fixed-width $H^{-1/4}$ shoaling law because the Kelvin mode's offshore width also contracts, $L_R\propto \sqrt{H}$. The comparison is a slowly varying trapped-mode result; abrupt topography can scatter or reflect energy and invalidate simple flux conservation for one outgoing mode. The amplitude panel in solution illustrates the scaling.

Put $K=(k^2+m^2{)}^{1/2}$ and $P=p^{\prime }/{\rho }_0$. Acting on the Fourier mode, $D_0=-i(\hat{\omega }-Uk)=-i\omega$. Continuity and horizontal momentum give $u=-mw/k$ and $P=\omega u/k$. The buoyancy equation gives $g{\rho }^{\prime }/{\rho }_0=i{N}^{2}w/\omega$. Substitution into vertical momentum yieldsThe two signs describe the two frequency branches; when the intrinsic frequency is required positive, $\omega =N|k|/K$. For a stationary disturbance $\hat{\omega }=0$, so $\omega =-Uk>0$. With $U>0$ this requires $k<0$ and selects the minus sign, $\hat{\omega }=Uk-Nk/K$. The stationary dispersion condition isDifferentiate the unconstrained dispersion before imposing stationarity. The stationary mountain-wave group velocity is obtained from the ground-frame dispersion; before imposing stationarity it isOn the stationary locus this becomesDifferentiating after substituting $K=N/U$ as though all neighbouring modes were stationary would incorrectly erase the frequency gradients.

For upward energy propagation from the lower boundary, choose $m<0$, since $k<0$ and ${\hat{c}}_{gz}>0$. It requires real nonzero $m$, henceThe downstream ray inclination above horizontal isIt is fixed for a given monochromatic forcing, with the two signs of $m$ giving the upward/downward rays.

Frame qualification for the requested phase angle: the normal intrinsic phase velocity is ${\mathbf{c}}_p=\omega (k,m)/K^2$. Because $\omega =-Uk$, the boxed stationary group velocity is exactly $-{\mathbf{c}}_p$: its angle to intrinsic phase propagation is $\pi$, and it is perpendicular to the stationary phase lines. By contrast the intrinsic group velocity ${\mathbf{c}}_g=\widehat{{\mathbf{c}}_g}-U\hat{\mathbf{x}}$ satisfies ${\mathbf{c}}_g\cdot (k,m)=0$, so its angle to intrinsic phase propagation is $\pi /2$. In the ground frame the phase velocity of a strictly stationary pattern is zero and has no nonzero propagation direction. For example $k=-0.6N/U$, $m=-0.8N/U$ gives $\widehat{{\mathbf{c}}_g}=(0.36U,0.48U)$ and ${\mathbf{c}}_p=(-0.36U,-0.48U)$. These distinctions prevent applying the usual zero-flow perpendicularity rule to two velocities expressed in different frames.

A mountain width is not an exactly specified sinusoidal wavelength. Taking its characteristic horizontal inverse length as $|k|\sim 1/a$ with $a=10\mathrm{km}$ gives $|k|\sim 10^{-4}{\mathrm{m}}^{-1}$. Since $N/U=10^{-3}{\mathrm{m}}^{-1}$, the stationary vertical wavenumber isThis is close to the long-horizontal-scale/hydrostatic estimate $2\pi U/N=6.28\mathrm{km}$. If instead the quoted width is identified with a sinusoidal wavelength ${\lambda }_x=10\mathrm{km}$, then $|k|=2\pi /{\lambda }_x$ and ${\lambda }_z\simeq 8.1\mathrm{km}$. Thus several kilometres, roughly $6$–$8\mathrm{km}$ under these two common width conventions, is the justified estimate; the terrain profile would fix its actual forcing spectrum.

For the inverse-width choice, both $k$ and $m$ are negative for upward energy radiation. The stationary phase lines $kx+mz=\text{constant}$ tilt upstream with increasing height, while energy propagates upward and slightly downstream. Numerically $\widehat{{\mathbf{c}}_g}\simeq (0.10,0.995)\mathrm{m}{\mathrm{s}}^{-1}$, with ray inclination about $84.3^{\circ }$. Intrinsic phase propagation points upstream and downwards; the ground pattern itself remains fixed. The original sketch below uses this explicit convention.

A ground observer may see stationary lenticular or wave clouds where ascent cools sufficiently moist air, with continuing air flow through a cloud whose position remains almost fixed. Descending regions can clear or warm. An aircraft crossing the pattern encounters alternating updraughts and downdraughts; gliders can exploit the lift. Strong amplitudes, shear or rotors may cause turbulence, but a linear gravity wave can also be smooth, and clear-air waves need not produce any visible cloud. Mountain height, humidity and nonlinear stability are not supplied, so neither a vertical-wind amplitude nor a definite atmospheric precipitation forecast follows.

Assuming the mean wind remains $10\mathrm{m}{\mathrm{s}}^{-1}$, the weakly stratified upper layer hasFor the inverse-width estimate $|k|\sim 10^{-4}{\mathrm{m}}^{-1}$, it is near the turning threshold $m=0$, with negligible upward group-velocity component. Components of larger $|k|$ are evanescent aloft and are substantially reflected; only horizontal wavelengths exceeding $2\pi U/N_{\mathrm{upper}}\simeq 62.8\mathrm{km}$ can propagate vertically there. For the alternative $10\mathrm{km}$ sinusoidal wavelength, $m_{\mathrm{upper}}^2<0$ and the evanescent decay length is about $1.6\mathrm{km}$.

The sharp inversion and change of stability can therefore cap the upward-radiating disturbance. Reflection between the ground and cap may trap energy in the lower layer and allow downstream lee-wave trains or resonant guided modes, with permitted wavelengths depending on the interface and lower-layer structure. Reflection, leakage and possible resonance are distinct: a specified $5\mathrm{km}$ height does not by itself fix a unique mode amplitude or wavelength.

Wave clouds may become confined below the cap or organized into persistent lee bands if there is enough moisture. Inversion-limited vertical mixing can also retain low cloud, haze or pollutants; sufficiently strong trapped motions can promote rotors or turbulence. These are possible changes, not inevitable weather outcomes: humidity, wave amplitude, detailed stability and wind shear are unspecified. The new structure suppresses upward energy transmission for the dominant short horizontal scales even though the lower atmosphere can still support waves.

Take the two-dimensional electron gas in the $xy$ plane, the Hall bar along $x$, and $\mathbf{B}=B\hat{\mathbf{z}}$ with $B>0$ perpendicular to it. An electron has charge $-e$. The confining electron potential energy $U(y)$ is approximately flat inside a wide strip and rises at either boundary; smooth quenched disorder adds hills and valleys. For a slowly varying edge, each spin-resolved Landau level approximately followsHere $s=\pm 1$, and $Y$ is the guiding centre. For $\mathbf{A}=(-By,0,0)$, $Y={\ell }_B^2{k}_x$ and the edge group velocity isThus electrons move along $+x$ at the upper edge and $-x$ at the lower edge. The Lorentz force gives counterclockwise bulk cyclotron motion in this convention. At a hard boundary its interrupted circles become directed skipping orbits; a smooth boundary gives the corresponding guiding-centre drift. Smooth bulk quenched disorder generally gives closed drift contours around localized hills or valleys, with extended trajectories appearing near a broadened level's critical energy. Realistic electrostatic screening can produce alternating compressible quantum Hall edge strips and incompressible quantum Hall edge strips, so the unscreened bending curves are an effective sketch rather than an exact self-consistent edge profile.

In the original diagram, contacts are ordered around the electron propagation direction: upper-left $1\to 2\to 3$, then lower-right $4\to 5\to 6$. Let $I_1=I$, $I_4=-I$ and the other leads be ideal equilibrating voltage probes. Define $R_H=(V_2-V_6)/I$ and $R_{xx}=(V_2-V_3)/I$. These choices fix the voltage sign; reversing the field reverses chirality and the Hall polarity. For spin-resolved plateau index $N$,At fixed sheet carrier density and increasing field, $N$ decreases, giving successively higher Hall plateaus; longitudinal peaks occur between them. Unresolved spin branches can suppress odd plateaus. The sketch uses resistance magnitudes, and every integer step is not promised in an arbitrary sample.

For Fermi-energy oscillations in a two-dimensional electron gas, first distinguish fixed density from fixed reservoir chemical potential. At negligible temperature the former is determined by state counting. Each resolved Landau level contains $eB/h$ states per unit area, soIgnoring spin and broadening for the counting illustration, if $N<\nu <N+1$ then $N$ levels are full and level $N$ is partly full: $\mu =E_N$. At fixed $n_s$, increasing $B$ increases both level energy and degeneracy. The partly filled level rises in energy until the lower levels can contain all electrons; $\mu$ then drops to the next lower level. The resulting sawtooth is periodic in $1/B$, with $\Delta (1/B)=e/(n_{s}h)$ per resolved branch. At an exactly integer filling in an ideal gap, any chemical potential within the gap gives the same zero-temperature particle number; the plotted discontinuity denotes its limiting values.

At fixed $B$ while $n_s$ increases, the ideal chemical potential stays at a partly occupied level and steps upward after each additional density $eB/h$. It therefore has a staircase superposed on the rising zero-field trend, not a repeated downward sawtooth. Landau-level broadening and finite temperature smooth both ideal graphs. With an externally pinned chemical potential, density changes instead; the fixed-density sawtooth cannot simultaneously be imposed. In a realistic disordered sample, the Fermi energy sweeps localized states between transport-carrying level centres. The following schematic omits spin for clarity and does not attempt an interaction-corrected density of states.

For $N$ transmitted spin-resolved quantum Hall edge states, the Landauer-Büttiker formula reduces, in that contact order, toZero probe currents imply $V_2=V_3=V_1$ and $V_5=V_6=V_4$. Hence $I=(N{e}^2/h)(V_1-V_4)$ and the boxed resistance results follow. The spin-resolved conductance quantum arises because the one-dimensional density of states times velocity is $1/h$ per unit energy; a reservoir imbalance supplies $e^2\Delta V/h$ per channel. Its value does not depend on edge slope or sample width.

On a plateau, opposite-moving edge channels are separated by a wide bulk with a mobility gap. Ordinary elastic scattering on one edge has no counterpropagating channel into which to reflect; closed bulk orbits do not connect the contacts. Changing $B$ or $n_s$ can alter the occupancy of localized states without changing the number of transmitting edge channels. This accounts for the finite plateau width, which an entirely clean set of sharp levels would not explain. Accuracy requires negligible backscattering between opposite edges, low enough current to avoid breakdown, equilibrating contacts and negligible probe loading. The finite-width plateau and the precision of its quantized value are related but distinct observations.

An antidot is a depleted potential hill, not a potential well. Its closed edge circulates oppositely to the outer boundary. In a constriction, tunnelling through an antidot orbit can transfer an electron from a forward edge into the returning edge: resonant antidot backscattering produces conductance dips. For $N$ otherwise open spin-resolved channels and one active resonance,At symmetric coupling a single resonance can remove $e^2/h$; asymmetry or thermal averaging reduces its depth. If the background constriction is closed and the resonant route instead supplies forward transmission, peaks occur. The lead connectivity must therefore accompany the conductance sketch. In the fixed-area single-particle limit, successive alignments change enclosed flux by ${\Phi }_0=h/e$, giving the Aharonov-Bohm antidot periodThe effective orbit area, not merely the etched island area, enters. Charging, screening and spin can modify this simple sequence; two coupled spin branches can yield a near-$h/(2e)$ pattern without changing the elementary flux quantum.^[ref]

A Landau quasiparticle is an electron-like excitation dressed by the interacting medium. Adiabatic continuity connects low-energy states of an ordinary Fermi liquid to those of a free Fermi gas without changing their particle statistics or conserved quantum numbers. The excitation has a renormalized dispersion, effective mass, magnetic response and finite lifetime. Pauli exclusion principle restricts collisions near the Fermi surface; in a regular three-dimensional Fermi liquid,Thus the width is small compared with excitation energy sufficiently near the surface at low temperature. This permits particle-like counting and transport even when microscopic interactions are strong. It does not make every interaction disappear: the Landau quasiparticle energy functional contains residual interactions,Specific heats and magnetic susceptibilities need not equal their bare-gas values. Logarithmic corrections can modify the lifetime scale in two dimensions, and a Luttinger liquid or a fractional Hall fluid is not an ordinary weakly perturbed electron gas.

An electron-like excitation in a normal metal and a neutral fermionic excitation in normal helium-3 are Landau quasiparticles. More generally, phonons, magnons and excitons are particle-like collective excitations, with their own statistics and quantum numbers; they are not all Landau quasiparticles. composite fermions are another emergent description, reorganizing correlations in a partly occupied Landau level rather than simply replacing the electron mass in an ordinary band.

Attach two statistical flux quanta ${\Phi }_0=h/e$ to each electron. Even attachment preserves fermionic exchange statistics, while the average statistical field opposes the applied field:This is an effective description of correlations, not literal flux tubes produced by small solenoids. An integer number of filled effective levels can give an incompressible electronic fractional Hall state. This is the correspondence between integer Hall quantization of composite fermions and selected fractional Hall states of electrons.^[ref]

Setting the signed effective filling to integer $p$ givesPositive $p$ gives $1/3,2/5,3/7,\dots$ below $1/2$. Negative $p=-q$, $q>0$, gives $q/(2q-1)$ above it, including $2/3,3/5,4/7,\dots$. The sign records reversed $B^{\ast }$. At $\nu =1/2$, the mean effective field vanishes, so the simplest description is a compressible composite-fermion sea rather than another member with finite integer $p$. Fractional charged excitations of gapped Hall states and the charge used in this semiclassical composite-fermion orbit description should not be conflated.

A direct probe away from the incompressible fractions is composite-fermion geometric resonance. One implementation patterned a high-mobility GaAs/AlGaAs Hall bar into periodic and random antidot arrays. electron beam lithography followed by an approximately $130\mathrm{nm}$ reactive-ion etch made arrays with periods $400$–$700\mathrm{nm}$. Four-terminal low-current measurements swept a perpendicular magnetic field at low temperature; red-light illumination varied density and the modulation. Periodic arrays showed commensurability structure near half filling and a Hall anomaly; matched random arrays did not. Unpatterned sections supplied another reference.^[ref]

The semiclassical explanation uses, for a fully spin-polarized sea,The mass cancels from the orbit radius. A selected orbit fits the lattice when $2R_c^{\ast }\simeq Ca$, with a geometry-dependent constant $C$ of order unity. Pinned orbits around antidots then alter longitudinal transport, producing features nearThe simplest single-antidot scale has $C\simeq 1$; detailed peak positions require the actual potential and orbit family. The two field directions bend orbits oppositely. The diagnostic is an orbit scale controlled by the small residual $B^{\ast }$, rather than the much larger applied $B$. At the same density the fully polarized $k_F^{\ast }$ is $\sqrt{2}$ times the zero-field spin-degenerate electron value; this is an independent counting test.

Calibrate density from independent Hall/filling measurements and compare several lattice periods. Use small excitation and temperature sweeps, and check periodic, random and unpatterned regions. Density variation changes both $B_{1/2}$ and the orbit scale, so simply plotting against uncorrected field can mislead. mean free path and density inhomogeneity limit resolution; illumination can soften or substantially change the imposed potential, so it cannot automatically be treated as density adjustment alone. These qualifications are part of the inference from geometry to emergent particles, rather than a claim that every feature near half filling is a new fractional plateau.

For a double quantum dot, write $(n_L,n_R)$ for occupation. Tune a bias window allowing the cycle $(0,1)\to (1,1)\to (0,2)\to (0,1)$. Two electrons in the same lowest orbital must form a singlet state by the Pauli exclusion principle. The $(0,2)$ triplet requires an excited orbital, with an energy cost ${\Delta }_{ST}$ above the accessible window. Approximately spin-conserving tunnelling transfers $S(1,1)$ into $S(0,2)$, but cannot transfer $T(1,1)$ into that singlet. Once a triplet loads, transport stops until spin relaxation or mixing occurs: this is Pauli spin blockade. The reverse bias can avoid trapping in this sequence, producing rectification. Coulomb charging sets the allowed occupancies, but the extra blockade is a spin-selection restriction, not merely an energetically forbidden charge state.

magnetically driven spin-blockade lifting uses electron paramagnetic resonance. A transverse oscillating magnetic field drives spin transitions atFlipping one spin in $|\uparrow \uparrow ⟩$ gives $|\downarrow \uparrow ⟩=(|T_0⟩-|S⟩)/\sqrt{2}$, which has a singlet component and can resume charge transfer. A perfectly uniform simultaneous rotation of two identical spins preserves their triplet character; effective relative rotation needs unequal local resonance conditions or another singlet–triplet mixing mechanism. In a real GaAs double dot, inhomogeneous nuclear fields can supply that difference. Hyperfine mixing, spin-orbit effects, cotunnelling and ordinary relaxation also produce leakage and must be distinguished from driven resonance.

A demonstrated device used surface Ti/Au gates to define two dots in a GaAs/AlGaAs electron gas $90\mathrm{nm}$ below the surface. An insulated on-chip gold coplanar stripline supplied the oscillating magnetic field. With the dots biased in spin blockade, RF excitation produced current peaks at the two field polarities satisfying resonance. Peak positions varied linearly with frequency, giving $|g^{\ast }|=0.35\pm 0.01$. RF applied to a side gate instead gave negligible response, testing magnetic rather than electric excitation. Gate pulses and RF bursts further produced coherent spin-rotation oscillations in the charge readout.^[ref]

An ideal isolated resonant spin has Rabi oscillation probability $P_{\mathrm{flip}}(t)={\sin }^2({\Omega }_{R}t/2)$. For a linearly oscillating field of peak amplitude $B_{\mathrm{ac}}$, the rotating-wave convention is ${\Omega }_R=|g^{\ast }|{\mu }_B{B}_{\mathrm{ac}}/(2\hslash )$. Repeated initialization, isolated manipulation and blockade-sensitive readout can therefore map a damped oscillatory singlet yield into an average current. Nuclear fluctuations, readout rates and dephasing change the envelope and contrast; an ideal sine squared is not a reconstruction of experimental data. Compare RF on/off, resonance tracking with frequency, drive-amplitude dependence and detuning controls. Current enhancement alone does not prove coherent rotations; the pulse-duration oscillations provide the further time-domain test.

In Landau gauge for a uniform magnetic field $\mathbf{A}=(-By,0,0)$, the guiding centre is $Y=\hslash k_x/(eB)$. The group velocity isThus the upper edge in the original trajectory diagram carries electrons to the right and the lower edge to the left. Reversing the magnetic field reverses both arrows. Semiclassically these are skipping trajectories; quantum mechanically the guiding centres label propagating quantum Hall edge states. The larger orbital-plus-spin energy crosses the Fermi energy farther toward the interior, because less confinement energy is needed. Opposite spins in one orbital branch are therefore spatially separated.

In a clean bulk with the Fermi energy between Landau levels, there are no extended bulk states at that energy. Occupied bulk cyclotron orbits lie below it; one must not draw them as extra conducting Fermi-level channels. Smooth disorder can instead produce localized bulk contours without adding a direct source-to-drain channel.

For the spin separation at a smooth quantum Hall edge, linearize the confinement energy at a crossing. Compensating a spin energy difference by a potential-energy change givesHere $V^{\prime }$ is interpreted as the specified slope of electron potential energy. If it denotes electrostatic voltage per length instead, the denominator is $e|V^{\prime }|$. The two edges of a neck and the two sides of an antidot have oppositely directed slopes; the formula is a positive local separation on any one edge. For local filling two, only the $n=0$ spin pair is occupied at the neck minimum: $E_{0,+}+U_{\min }<E_F<E_{1,-}+U_{\min }$. Raising both branches with the confinement produces two crossings on each side, as shown in the original neck sketch. This fixed-slope estimate precedes self-consistent electrostatic screening.

An antidot is a depleted potential hill. guiding centres circulate around its closed equipotentials, with chirality opposite to the outer sample edge. A phase-coherent round trip contributes magnetic phase $2\pi BS/{\Phi }_0$, with ${\Phi }_0=h/e$. A localized orbit resonant with the incident energy can transfer an electron between extended channels. If the outgoing channel returns toward the injecting reservoir, the process is resonant reflection and reduces electrical conductance. If coupling instead connects an otherwise blocked incident channel to the far reservoir, it is resonant transmission and adds electrical conductance. Transmission peaks require that forward connectivity; a closed orbit by itself does not guarantee them. Interchannel scattering can change which orbit participates.^[ref]

For the antidot resonance conductance bounds with one active spin-resolved resonance, writeHere ${\Gamma }_L,{\Gamma }_R$ are energy broadenings; the corresponding escape rates are ${\Gamma }_L/\hslash ,{\Gamma }_R/\hslash$. In a reflection-dominated configuration with two otherwise transmitted channels, $G=(e^2/h)(2-\mathcal{L})$: a $2e^2/h$ background can dip to $e^2/h$. In a transmission-dominated configuration with one spectator channel and the second otherwise blocked, $G=(e^2/h)(1+\mathcal{L})$: peaks rise from $e^2/h$ toward $2e^2/h$. These are the two ideal channel-count examples plotted below. More generally, $G=(e^2/h)(N_{\mathrm{open}}-R)$ or $(e^2/h)(N_{\mathrm{dir}}+T)$; a different spectator count changes the integer offset. The constriction filling alone does not specify every tunnel coupling or resonance height. Equal couplings permit unit resonant transfer in the isolated-level model; asymmetry, temperature and dephasing reduce the contrast. Holding the effective orbit area approximately fixed gives $\Delta B\simeq h/(eS)$ for one branch, not a universal charging-corrected period.

In quantum-dot versus antidot resonance topology, a quantum dot is a potential well holding a finite electron island. Its closed guiding-centre circulation is reversed relative to an antidot at the same field. With two barriers in series, current follows source $\to$ dot $\to$ drain; off resonance there need not be a direct conducting path. A gate shifts a level or an addition electrochemical potential into alignment with the reservoirs and produces a electrical conductance peak. For an isolated coherent spin-resolved level, the zero-temperature peak is at most $e^2/h$ and reaches it only for equal couplings; unresolved noninteracting spin degeneracy doubles this bound. In weak sequential tunnelling, peak height depends on rates and temperature rather than being quantized. Charging produces Coulomb blockade peaks with spacing set by addition energy and gate lever arm. An antidot also supports discrete orbit/charging resonances, but can be side-coupled to already open channels and produce dips. Gate voltage changes both orbit area and confinement, whereas a magnetic sweep directly changes enclosed flux.

Coulomb interactions require a self-consistent confinement potential. A gradual decrease in electron density cannot leave arbitrary uncompensated donor charge at no cost. Partly occupied Landau levels have a large charge response: electrostatic screening flattens their effective energy near the Fermi energy over compressible quantum Hall edge strips. Between them, integer-filled incompressible quantum Hall edge strips have a gap, weak density response and most of the potential variation. In an idealized cross-section the effective branch thus alternates between flat, partially occupied portions and rising portions crossing a gap; its carrier density varies oppositely. The bare levels still have Landau level and spin splittings locally, but the common electrostatic potential is no longer an imposed straight slope. exchange interaction can additionally renormalize spin gaps. Widths must follow electrostatics and the density gradient, so the unscreened $E_Z/|V^{\prime }|$ separation is not a quantitative screened-strip width.^[ref]

A modulation-doped GaAs-AlGaAs electron gas is confined on the GaAs side of the interface. The larger-gap AlGaAs supplies a conduction-band barrier; charge transfer from remote donors and the resulting electric field bend the bands into an approximately triangular well. transverse electronic mode wavefunctions are confined normal to the interface but free in its plane. electrons settle into the well until their electrochemical potential agrees with the donor/contact environment. An undoped spacer separates mobile electrons from ionized donors, reducing impurity scattering. Equilibrium has one Fermi level across the structure; the conduction-band edge is not flat. Surface depletion and the cap also affect the profile. The original drawing illustrates one occupied confinement transverse electronic mode; a single spatial electron gas need not in general imply only one occupied vertical transverse electronic mode.

molecular beam epitaxy can grow a clean GaAs buffer/channel, an undoped AlGaAs spacer, a Si-doped AlGaAs supply region and a protective GaAs cap on a suitable substrate. Control composition, interface quality and background impurities during growth; placing dopants directly in the channel defeats the principal mobility advantage. Define a mesa or isolation region, fabricate low-resistance ohmic contacts to the buried gas, and pattern fine metal split gates by electron beam lithography and the lift-off process. Negative gate bias depletes the underlying gas, leaving a narrow channel between the gates. Check contact resistance, leakage, carrier density and mobility; operation with ballistic electron transport requires channel length shorter than the relevant scattering lengths and smooth confinement on the wavelength scale. The lithographic gap does not equal the electrical width because of lateral depletion.

For a long uniform channel supporting ballistic electron transport, transverse confinement creates thresholds $E_n$ with continuous longitudinal propagation above each threshold. One directed spin branch of a transverse electronic mode has states per length $dk/(2\pi )$ and group velocity $v={\hslash }^{-1}dE/dk$. Hence $vdk/(2\pi )=dE/h$: velocity cancels the one-dimensional density of states. Including two unresolved spin states,For low temperature and small bias, a fully open transverse electronic mode has $T_n(E_F)=1$, givingThe assumptions are ideal reservoirs/contacts, negligible reflection and interactions that would invalidate the independent-subband description, and unresolved spin splitting. A resolved spin branch contributes $e^2/h$. The finite two-terminal electrical resistance is a contact/mode-injection effect, despite the absence of dissipative scattering inside the ideal channel.

For voltage partition in a ballistic constriction, a short saddle has strong local electric-field and electrostatic screening variation near its entrances and barrier, rather than the uniform Ohmic voltage gradient of a diffusive wire. Right-moving incident states inherit the source distribution and left-moving states the drain distribution; there is no unique local equilibrium chemical potential inside an ideal ballistic saddle. The electrostatic potential must be determined self-consistently. In a symmetric rigid-barrier approximation, the barrier follows the mean lead potential and half the electrochemical bias lies on each side. This is a stated approximation, not a universal spatial voltage profile.

For a saddle $U=U_0-m^{\ast }{\omega }_x^2{x}^2/2+m^{\ast }{\omega }_y^2{y}^2/2$, transverse electronic mode thresholds are $E_n=U_0+\hslash {\omega }_y(n+1/2)$. The smooth-barrier transmission is $T_n(E)=[1+\exp (-2\pi (E-E_n)/(\hslash {\omega }_x)){]}^{-1}$, approaching a step for a sharp threshold on the bias/thermal scale.^[ref]

Let ${\mu }_{S,D}=E_F\pm eV/2$ and assume no additional bias dependence of $T_n$. At zero temperature,When the source and drain sample $N+1$ and $N$ thresholds, the differential electrical conductance is $(N+1/2)2e^2/h$. These half plateaus arise from the two ends of the bias window, not fractional carriers. The dc ratio $I/V$ need not share their plateau values. Heating, spin splitting and interaction-induced features must be distinguished experimentally.

In bias spectroscopy of one-dimensional subbands, plot differential electrical conductance or its gate derivative against gate bias and source-drain bias. Threshold branches occur where $E_n(V_g)={\mu }_S$ or ${\mu }_D$. When one bias-window edge meets $E_n$ and the other $E_{n+1}$,Thus the intersections of adjacent threshold branches calibrate the one-dimensional spacing directly in energy. A gate lever arm $\alpha =-e^{-1}\partial E_n/\partial V_g$ can then convert gate intervals to energy; it should be measured rather than guessed. Asymmetric bias partition, a moving barrier or a gate-dependent spacing modifies slopes and plateau weights, requiring the corresponding self-consistent bias model.

A single gate-defined dot can trap one electron whose lower and upper Zeeman states define $|0⟩,|1⟩$. Initialize by energy-selective loading or relaxation, coherently rotate the spin with a resonant magnetic drive or electrically generated spin-dependent drive, and use neighbouring-dot exchange interaction for two-qubit operations. The dot supplies confinement; a full qubit also requires control, coherence and calibrated readout. A nearby quantum point contact or single-electron transistor detects charge capacitively, so spin-to-charge conversion is needed.

In energy-selective spin readout, set the reservoir chemical potential between the two spin-dependent removal energies. The upper state can tunnel out; the lower remains. Reloading can create a temporary empty-dot signal that a time-resolved charge detector records. Resolving this window requires sufficiently small electron-temperature broadening and stable level alignment. The escape should be faster than spin relaxation, but the empty interval must remain long enough for detection. Useful controls include known-state preparation, sensor-noise measurement, varied waiting time and readout pulses, and checks for missed events or thermally activated false escape.^[ref]

tunnel-rate-selective spin readout instead places both candidate removal energies on the allowed side of the reservoir and uses different escape rates ${\Gamma }_f\gg {\Gamma }_s$. After a window $\tau$ with ${\Gamma }_f^{-1}\ll \tau \ll {\Gamma }_s^{-1}$, the fast state has usually changed charge while the slow state has not. It does not require the reservoir Fermi edge to resolve the state splitting, although it still requires controlled charge occupation, substantial rate contrast and relaxation slower than the mapping. The contrast must have a physical origin. A demonstrated single-dot implementation uses two-electron singlet and triplet states: Pauli symmetry puts the triplet's second electron in a higher orbital with stronger lead coupling. Nearly degenerate states can then be distinguished by rate rather than by their energy difference.^[ref]

For a bare one-electron Zeeman pair in the same orbital with an unpolarized lead, different tunnel rates are not automatic. A spin-selective barrier/reservoir or another spin-dependent mechanism is needed. The two-electron singlet/triplet example is a different state encoding/readout arrangement, not evidence that the bare single-electron pair already has useful rate contrast.

For the optimal tunnel-rate readout time, assume no relaxation and equal prior probabilities. The fast-state missed-event probability is $e^{-{\Gamma }_f\tau }$ and the slow-state false-event probability is $1-e^{-{\Gamma }_s\tau }$. Their average is minimized atFinite detector bandwidth, preparation errors, state relaxation and unwanted reloading alter this optimum. energy selection is direct for a resolved single-electron spin but sensitive to the narrow alignment window; rate selection has a wider energy operating window but requires engineered rate contrast and a carefully timed charge measurement. Neither method is an ideal nondestructive spin measurement merely because the charge sensor itself is noninvasive.

A high-mobility modulation-doped GaAs-AlGaAs electron gas is grown by molecular beam epitaxy as a clean layered semiconductor heterostructure. An undoped gallium arsenide region is followed by an undoped aluminium gallium arsenide spacer and a donor-doped barrier region, commonly with a protective cap. The closely matched crystal structures allow a high-quality heterojunction. The spacer supplies a crucial separation: the mobile electrons reside away from the ionized donors, reducing impurity scattering compared with putting the donors directly in the conducting layer.

The larger conduction band energy on the aluminium gallium arsenide side permits donor electrons to transfer into gallium arsenide. The resulting charge separation creates band bending; together with the band offset, it forms a roughly triangular quantum well at the interface. The normal-direction wavefunction is confined, while motion parallel to the interface remains approximately free:When only the lowest confinement subband is occupied, this is a two-dimensional electron gas. It is the quantized normal motion, not a physically zero layer thickness, that makes the low-energy system two-dimensional.

At zero magnetic field, one occupied spin-degenerate parabolic subband has constant density of states per area per energy, $D_0=m^{\ast }/(\pi {\hslash }^2)$ above its edge. A perpendicular field quantizes the in-plane orbital motion into Landau levels:Here $g^{\ast }$ is the material's effective spin factor and $e$ is the positive elementary charge; no free-electron $g=2$ assumption is made for this semiconductor. Each resolved spin branch contains $e|B|/h$ states per unit area. The ideal density of states becomeswith Landau-level broadening in a real sample. The Landau filling factorcounts the carrier density in units of one spin-resolved level's degeneracy. Two unresolved spin branches have twice that degeneracy; resolved branches must not each be counted twice.

Disorder localizes states in the broadened level tails, while extended states occur near level centres. As $B$ varies, occupation of localized states can change without changing the number of current-carrying edge channels. This produces a finite-width quantum Hall plateau, rather than quantization only at isolated fields. In a mobility gap, longitudinal resistivity approaches zero and the Hall magnitude is $R_K/N=h/(N{e}^2)$ for $N$ occupied transmitting branches. When the chemical potential passes through extended states, longitudinal dissipation peaks and the Hall value changes to another plateau. A spectral density of states can remain nonzero where transport is localized. Very strong disorder can destroy the resolved quantum Hall regime; the statement is not that any amount of disorder improves it.

The original plot is schematic, not fitted data. At fixed sheet density, increasing $|B|$ decreases Landau filling factor, so the Hall magnitude climbs through $R_K/N$ plateaus while longitudinal peaks separate the plateaus. The signed Hall voltage depends on carrier and probe conventions.

For the gate calculation, the target sheet density is fixed by the desired Landau filling factor:Use top-gate density tuning in the parallel-plate approximation. The geometric capacitance per unit area isThereforeThe gate must become more negative relative to the electron gas, reducing its density. The voltage is the change from the gate setting associated with the stated initial density; if that setting is defined as zero, it is the required gate voltage. Work-function offsets, trapped charge and quantum capacitance are neglected. An absolute voltage referenced to some other uncalibrated potential cannot be inferred from geometry alone. Numerical values use exact SI $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and $e=1.602176634\times 10^{-19}\mathrm{C}$, and ${ϵ}_0\simeq 8.8541878188\times 10^{-12}\mathrm{F}{\mathrm{m}}^{-1}$.^[ref]

For the subsequently specified five-channel plateau, the chemical potential between the fifth and sixth resolved branches means $N=5$, not ten. The physical contact order in the PDF is $1,2,3,4$ along one side of the mesa. Use ideal equilibrating Ohmic contacts and the Landauer-Büttiker formula, rather than inferring a longitudinal measurement merely from the contacts being drawn on one edge.

Choose one field polarity so the chiral order is $1\to 2\to 3\to 4\to 1$. With $G_H=5e^2/h$,Contacts 2 and 4 are voltage probes, so $I_2=I_4=0$. With $V_3=0$, this gives $V_2=V_1$, $V_4=V_3=0$, and $I_1=G_H{V}_1$. Hencefor this circulation. Reversing the field reverses the contact order and gives $R_{13,24}=-h/(5e^2)$ with the same voltage ordering. The resistance magnitude is $5.16\mathrm{k\Omega }$; it is not zero. Contact 2 samples the outgoing source potential and contact 4 the outgoing drain potential for the chosen chirality.

The antidot is a depleted island supporting closed quantum Hall edge states. Tunnelling between those states and opposite propagating edges can produce resonant backscattering and resistance peaks as $B$ varies. One electronic Aharonov-Bohm phase winds by $2\pi$ when flux through a fixed effective orbit area $S$ changes by $h/e$. Using the supplied area as that effective area,With the two spin-split branches of the lowest orbital level occupied, the common Coulomb-dominated antidot oscillation model has two charge rearrangements per added $h/e$ flux. Both branches contribute to charge buildup; screening and their capacitive coupling can make tunnelling through the outer edge occur twice during the cycle. It then givesThis is two electron-tunnelling resonances per flux cycle, not evidence that the mobile particles have charge $2e$. The primary charging model specifically addresses this two-spin regime.^[ref]

The sketch shows resistance peaks in a weak-backscattering configuration, with arbitrary height and linewidth because tunnel couplings and temperature are unspecified. A two-spin count alone does not force two independent noninteracting resonance series to be exactly half a flux period apart: their relative positions also depend on spin splitting and confinement. Thus $51.7\mathrm{mT}$ describes the stated charging-model interpretation; $103.4\mathrm{mT}$ is the single-electron fixed-area flux period. The actual enclosed edge-orbit area can differ from the geometrically depleted area.

For bulk filling two and total forward transmission $\mathcal{T}$ in spin-resolved units, ideal chiral probes give the longitudinal four-terminal resistanceResonant loss of transmission therefore gives peaks. Their size, background and detailed shape cannot be calculated from the supplied area and bulk filling alone. If the gates instead make a tunnelling path that enhances forward transmission, the resonant contrast can be reversed; the plotted peak convention is explicitly the backscattering one.

In a zone with $dv/dz<0$, a descending ray bends towards the vertical instead of turning back towards the surface. A ray cannot have its ordinary smooth diving-ray turning point within that decreasing-speed interval. There are rays turning above the zone and rays that penetrate it and turn deeper, after speed has increased sufficiently. The latter spend extra time traversing slower material, so their branch is delayed relative to a simple extrapolation of the original uniform-gradient curve.

Turning depths within the low-velocity zone are missing. In a piecewise-gradient example this can leave a range interval without a returning diving ray: a geometrical shadow zone between the shallow and deeper branches. The lower row of the original plot shows that situation, with the interrupted shallow branch and the delayed deeper branch. The boundaries are profile-dependent; reflected, refracted or diffracted energy can still arrive, and smoothing an idealized velocity maximum can change the branch limits. Thus the robust signature is the interruption/reorganization of the ordinary turning-ray family, not a universal claim of zero recorded signal. The supplied description alone does not specify unique numerical travel times or shadow-zone widths.

Earthquake location. The hypocentre is rupture initiation; its surface projection is the epicentre. Record calibrated arrival times at stations of known coordinates, identify P wave and S wave phases, and predict their travel times through a stated velocity model. Earthquake location by arrival times fitsfor the three source coordinates and origin time $t_0$. Linearization supplies residual equations $\delta t_i=\delta t_0+\nabla T_i\cdot \delta \mathbf{x}$. A grid search or iterative weighted least squares inversion can use pick uncertainties, while station geometry controls whether those parameters are well constrained. Four independent arrival constraints are the minimum for four unknowns, not a guarantee of a reliable solution.

In a homogeneous medium, an arrival-time difference between an S wave and a P wave cancels origin time and estimates source-to-station distance,Several distances intersect to locate the event; real layered or three-dimensional Earth structure requires curved-ray travel times. Nearby stations and diverse takeoff angles help resolve depth. Clock errors, phase misidentification, poor azimuthal coverage and incorrect velocities introduce uncertainty or systematic bias. Closely spaced earthquakes can be relocated precisely relative to each other because common-path errors partly cancel, even when their absolute locations remain uncertain. Report a location uncertainty and model assumptions, not merely a best-fit point.

H. F. Reid's earthquake-cycle model. A fault can remain locked while long-term tectonic motion slowly deforms the surrounding rocks. Elastic strain energy and shear stress accumulate. When resistance is overcome, rapid fault slip releases part of that stored energy as seismic waves, frictional heat and other irreversible work. The adjoining rocks partly recover their elastic distortion while retaining a permanent displacement discontinuity across the fault: this is elastic rebound. Surface markers initially crossing the fault become bent during loading and offset after slip.^[ref]

A simple sketch is a slow rise of stress followed by a rapid stress drop, then renewed loading; the plate displacement continues throughout. Geodetic deformation, observed fault offsets and earthquake stress changes can test the picture. Actual cycles need not repeat with identical slips or times: creep, afterslip, stress transfer and heterogeneous friction complicate the idealization. The release need not remove all pre-existing stress. Elastic rebound explains storage and release; it is not a deterministic earthquake clock.

Use the flat two-layer model, with thickness $h=30\mathrm{km}$, crustal shear speed ${\beta }_1=3.5\mathrm{km}{\mathrm{s}}^{-1}$ and mantle speed ${\beta }_2=4.5\mathrm{km}{\mathrm{s}}^{-1}$. For SH-wave travel times in a crustal waveguide, unfolding each reflected path gives all reflection orders at once:with $x$ in kilometres and times in seconds. The direct wave begins at the origin; reflected orders have intercepts $60m/3.5=17.14m\mathrm{s}$ and approach the direct-wave slope at large distance.

The critical angle, measured from the vertical, is $i_c=\arcsin (3.5/4.5)\simeq 51.1^{\circ }$. Associated critical refraction and its multiples giveEach head-wave line is tangent to its reflection curve at the critical distance. In particular the primary head wave exists before it becomes the first arrival; it overtakes the direct wave only around $170\mathrm{km}$.

The slope of a reflected travel-time curve is $p=d{t}_m/dx=\sin i/{\beta }_1$. For $x<x_{c,m}$, $p<1/{\beta }_2$, so the mantle supports outgoing shear radiation and successive reverberations leak energy. For $x>x_{c,m}$, $1/{\beta }_2<p<1/{\beta }_1$: the mantle field is evanescent and ideal interface reflection is total. These leaking and trapped SH-wave reverberations explain the annotations in the original travel-time drawing below. A reflected ray is not itself a separate Love mode: constructive interference of the trapped contributions, including reflection phase, selects the normal modes.

For a horizontally propagating mode with ${\beta }_1<c<{\beta }_2$, define vertical wavenumbersThe free upper surface and welded interface impose the Love-wave dispersion conditionwhere ${\mu }_j={\rho }_j{\beta }_j^2$ are shear moduli. Each phase-velocity branch falls towards $3.5\mathrm{km}{\mathrm{s}}^{-1}$ with increasing frequency. Higher branches join $4.5\mathrm{km}{\mathrm{s}}^{-1}$ at their low-frequency cutoffs. At cutoff $s\to 0$, soThus the first three overtones have maximum periods about $10.77$, $5.39$ and $3.59\mathrm{s}$. The fundamental mode has $n=0$ and no positive-frequency cutoff in this flat layer-over-half-space model: it approaches $c={\beta }_2$ as $f\to 0$, with ever-deeper penetration. The phase-velocity sketch uses an explicitly illustrative equal-density choice, because the question gives speeds but no density or rigidity ratio. That missing ratio changes detailed branch curvature, not the cutoff values or the requested branch topology.

For the actual finite spherical Earth, the very-long-period continuation must instead be described by discrete toroidal normal modes, not a flat infinite waveguide at arbitrary wavelength. The flat-model answer therefore does not assert a continuum of arbitrarily low-frequency global nonrigid oscillations. A global lowest oscillation frequency cannot be computed from the supplied crust-and-half-space data alone.^[ref]

Elastic rebound. Tectonic motion loads a fault whose shallow portion is locked. Rock on either side deforms and stores elastic strain energy while slip is inhibited. When failure permits rapid slip, part of the stored energy becomes seismic radiation and part is dissipated in friction and damage. The adjacent rock partly relaxes, while a permanent displacement remains across the fault. Geodetic strain accumulation and coseismic offsets support this loading–release picture.^[ref]

The model is useful for brittle crustal faulting but is not a prediction of precise recurrence, rupture size or frictional evolution. Aseismic creep can release deformation without a large sudden event. Viscoelastic flow, afterslip, fluid-pressure changes and interacting fault segments complicate a simple locked-fault cycle. Deep-focus earthquakes and non-shear source components require additional physics; not every earthquake source is adequately described by a shallow brittle double couple. Rebound also need not release all the pre-existing stress.

A thin negative-gradient velocity zone. In horizontal layering, use the seismic ray parameter $p=\sin i/v$, the conserved horizontal slowness. A ray turns when $pv=1$. Downward passage through decreasing speed makes $pv$ smaller, so a surface ray cannot turn in the negative-gradient interior. Rays either turn above the zone or penetrate it and turn below where speed has recovered sufficiently. This produces missing turning depths and a seismic shadow range from a low-velocity zone in the ideal ray model.

Retain all branches usingThe travel-time slope is $dT/dx=p$. Just below the critical value that permits passage beyond the upper speed maximum, the ray makes a finite extra excursion through the low-velocity region. Range can jump, and the deep branch can fold back before resuming its normal trend, giving multiple arrivals outside the shadow. A first-arrival curve alone hides some of these branches. Define the seismic intercept time byThe $x$–$p$ representation displays the gap and fold; the $\tau$–$p$ representation has separate shallow and deep branches and a branch jump at the critical ray parameter for the piecewise-linear cusp model. Do not join them by invented turning rays. Diffraction and finite-frequency energy soften a purely geometrical shadow.

The figures use an explicit illustrative profile, not measured Earth parameters: speed rises from 1 to 2 over depths 0 to 1, falls to 1.8 by depth 1.2, then rises with its original gradient. The asthenosphere beneath oceanic lithosphere is an important setting for a low-velocity zone. A major P-wave speed reduction also occurs at the boundary between the mantle and the outer core of the Earth; S waves do not propagate through the liquid outer core of the Earth.^[ref]

Use geometric rays for body-wave arrivals and a traction-free upper surface. A compact isotropic explosion source has moment tensor proportional to ${\delta }_{ij}$ and produces direct P body radiation in a homogeneous isotropic medium. Thus the basic near-offset arrival is the direct upper-layer P wave, with time $X/{\alpha }_1$. A pure explosion is not a shear fault: it does not automatically emit a direct S wave of comparable strength. Interfaces can nevertheless convert its P energy into SV motion, and the free surface can support Rayleigh waves.

Downward P energy reflects from the interface to give a PP reflection returning to the receiver; for a horizontal interface its ray is symmetric and its time is $\sqrt{X^2+4H^2}/{\alpha }_1$. Other transmitted P energy enters the faster half-space. At $\sin i_c={\alpha }_1/{\alpha }_2$, a critical ray travels along the interface and feeds an upgoing seismic head wave, often labeled Pn in this simplified crustal model. Its ideal travel-time branch isIt requires enough offset for the two inclined upper-layer legs, $X\ge 2H\tan i_c$. Once $X$ exceeds the larger first-arrival crossover distance $X_c=2H\sqrt{({\alpha }_2+{\alpha }_1)/({\alpha }_2-{\alpha }_1)}$, this branch arrives before the direct upper-layer wave because it spends most of the path in faster material. A transmitted ray descending into an otherwise uniform infinite half-space does not return as a separate reflection without an additional reflector.