Near-Field Radiative Heat Transfer¶

Thermally driven electromagnetic exchange across subwavelength gaps, where evanescent channels supplement propagating radiation and can carry heat far above the corresponding far-field blackbody benchmark.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2356
Origin domain: physics
Subdomain: nanoscale heat transfer
Aliases: NFRHT

Core Idea¶

Near-field radiative heat transfer (NFRHT) is thermally driven electromagnetic energy exchange between material bodies close enough that the fields which decay away from one surface can reach and be absorbed by another. In the familiar far field, only propagating electromagnetic waves carry energy from body to body. At separations comparable to or smaller than the thermally important wavelengths, that propagating sector remains present, but evanescent modes with parallel wave vector larger than the vacuum wave number open additional channels. Polder and Van Hove's foundational calculation treated two half-spaces at different temperatures across a vacuum gap and predicted a strong increase in radiative transfer as the gap narrowed.^[1] The modern field extends that mechanism to particles, finite bodies, structured materials, and many-body systems.^[2]

Fluctuational electrodynamics supplies the constitutive account. Lossy matter contains random thermally excited currents. The fluctuation-dissipation theorem relates their correlations to temperature and the imaginary, dissipative part of the material response. Maxwell's equations propagate the fields from those sources through the geometry; absorption in another body converts the field energy into heat. For two bodies at uniform temperatures \(T_1\) and \(T_2\), a Landauer form exposes the balance:

\[ P_{\mathrm{net}} =\int_0^\infty \frac{d\omega}{2\pi} \bigl[\Theta(\omega,T_1)-\Theta(\omega,T_2)\bigr] \,\mathcal{T}(\omega), \]

where \(\Theta(\omega,T)=\hbar\omega/[\exp(\hbar\omega/k_BT)-1]\) is the mean thermal oscillator energy used in the net heat current, and \(\mathcal{T}(\omega)=\sum_n\tau_n(\omega)\) is the total transmission summed over electromagnetic channels. In the passive two-terminal formulation, each channel probability satisfies \(0\leq\tau_n\leq1\). Equal temperatures make the net power zero because the two directional contributions balance, not because microscopic emission stops.^[2]

For two parallel half-spaces separated by a gap \(d\), the channels may be indexed by frequency, polarization, and lateral wave vector \(\kappa\). Propagating channels have \(\kappa<k_0=\omega/c\); evanescent channels have \(\kappa>k_0\), an imaginary normal wave-vector component, and a gap factor that decays approximately as \(e^{-2|q_v|d}\). Because the latter channels are absent from the usual far-field Stefan-Boltzmann count, sufficiently small gaps can yield a flux above the corresponding far-field blackbody benchmark. This is not unlimited energy and not a violation of thermodynamics. It is a change in the accessible channel set, subject to material loss, geometry, separation, nonlocality, atomic structure, and detailed balance.^[3]^[2]

The locked identity is bodies at specified temperatures + thermally fluctuating electromagnetic sources + a subwavelength coupling geometry + propagating and evanescent transmission channels + absorption and directional balance -> net radiative heat flow. Evanescent transfer may dominate, resonant surface polaritons may enhance it, and the result may be super-Planckian relative to a far-field reference; none of those three outcomes alone is the abstraction.

Structural Signature¶

the thermal bodies — two or more material objects with defined temperatures or temperature fields;
the temperature imbalance — the nonequilibrium occupation difference that drives net heat current, while allowing bidirectional exchange at all temperatures;
the separating geometry — a vacuum or dielectric gap, or another configuration with separations or feature sizes comparable to thermally populated wavelengths;
the fluctuating sources — random currents or polarizations inside dissipative matter, with correlations fixed by fluctuation-dissipation relations;
the material response — frequency-dependent permittivity, permeability, conductivity, or a more general local or nonlocal susceptibility that fixes emission, propagation, reflection, and absorption;
the electromagnetic propagator — Green functions, scattering operators, reflection matrices, or an equivalent Maxwell solver connecting source fields to receiving bodies;
the propagating sector — channels capable of carrying energy to the far field;
the evanescent sector — spatially decaying channels that do not transport energy to infinity in the gap direction but can mediate body-to-body transfer when the receiver lies inside their decay length;
the transmission spectrum — \(\mathcal{T}(\omega)\), or its wave-vector-, polarization-, and body-resolved form, which weights each thermally occupied channel;
the absorption sink — dissipative degrees of freedom in the receiving body that turn electromagnetic field energy into heat;
the directional balance — power from body 1 to body 2 minus power from body 2 to body 1, generalized in many-body settings to all emitters and the environment;
the regime test — comparison of gap, feature sizes, wavelength, and material length scales to determine when ray optics, local response, dipole, proximity, or continuum approximations are valid;
the benchmark — an explicitly declared far-field blackbody or propagating-channel reference, not an assumed universal ceiling.

Recognition test. A case is NFRHT when a thermal population difference drives electromagnetic exchange in a geometry where near-field channels materially alter the transfer relative to a propagating-wave model, and when the calculation or measurement separates that radiative contribution from contact conduction, convection, and other carriers. A nanometre gap alone is insufficient; a large heat flux alone is insufficient; and an evanescent optical field generated by a laser is insufficient because it is not thermally driven radiative exchange.

What It Is Not¶

Not heat conduction. Conduction moves energy through matter by phonons, electrons, ions, or molecular interactions. A vacuum gap is a clean NFRHT geometry precisely because it can suppress ordinary contact conduction. At subnanometre separations, electron and phonon tunneling can compete with photons, so a pure fluctuational-electrodynamics model may no longer identify the whole measured current.^[2]
Not convection. No moving bulk fluid is required. Gas conduction or convection caused by imperfect vacuum is an experimental background, not part of the electromagnetic mechanism.
Not far-field thermal radiation. Far-field exchange retains propagating modes and admits view-factor or ray-optical descriptions. NFRHT adds wave interference and evanescent coupling that those descriptions omit.
Not the electromagnetic near field in general. An antenna's reactive field, a laser-excited plasmon, or an evanescent wave in total internal reflection is not NFRHT unless thermal fluctuations and a temperature-dependent energy balance are the source of the heat current.
Not particle tunneling. “Photon tunneling” is shorthand for coupling through evanescent electromagnetic modes. It is not electron transmission through a potential barrier, although both use exponentially decaying solutions.
Not synonymous with resonance. Surface phonon or plasmon polaritons can create narrow, strong channels when spectra match, but broadband frustrated total-internal-reflection modes and nonresonant channels can also transfer heat.
Not necessarily super-Planckian. Near-field channels make flux above a chosen far-field blackbody comparison possible, not automatic. Poor spectral overlap, large gaps, weak absorption, geometry, or mismatch can keep the actual transfer small.
Not a universal hard distance cutoff. “Near” depends on frequency, temperature, material dispersion, and geometry. The thermal wavelength is a diagnostic scale, not a phase boundary at which one formula abruptly replaces another.
Not a violation of the second law. Passive two-body transfer has zero net value at equal temperatures and points from hotter to colder bodies away from equilibrium. Exceeding a far-field channel benchmark does not reverse detailed balance.

Scope of Application¶

The canonical planar problem uses two parallel half-spaces at temperatures \(T_1\) and \(T_2\), separated by a vacuum gap \(d\). Translation symmetry permits decomposition by \(\kappa\) and polarization, making explicit the propagating and evanescent sectors. It is the workhorse for identifying surface-wave resonances, frustrated modes, material trends, and idealized bounds.^[1]^[2]

Finite geometry is also inside scope. Sphere-sphere, sphere-plane, tip-plane, nanoparticle, grating, thin-film, and arbitrary-shape configurations require multipole, scattering-matrix, boundary-element, volume-integral, or discrete-dipole treatments. Narayanaswamy and Chen's two-sphere analysis, for example, tested where dipole and proximity approximations succeed or fail rather than assuming that one planar formula covers every shape.^[4] Many-body NFRHT adds multiple scattering, nonadditive exchange, environmental channels, relaxation toward equilibrium, and reciprocal or nonreciprocal transport networks.^[2]

Applications include near-field thermophotovoltaics, where a hot emitter couples a tailored spectrum to a nearby photovoltaic cell; nanoscale thermal management and proposed rectifiers, transistors, or switches; scanning thermal microscopy and local thermometry; thermal lithography; and localized heating. These applications instantiate the same mechanism only when the transported energy is thermal radiation. A near-field optical transducer driven coherently for heat-assisted magnetic recording may be technologically adjacent but is not automatically an NFRHT device, because its source can be externally driven light rather than spontaneous thermal fluctuations.

The ordinary local-continuum theory has a validity envelope. At extremely small gaps, spatial dispersion, atomic granularity, surface chemistry, roughness, charge transfer, phonon coupling, and electron tunneling can matter. Kittel and colleagues measured tip-surface transfer below \(10^{-8}\) m and reported a marked departure from a macroscopic fluctuational-electrodynamics prediction, illustrating why “smaller gap” cannot be extrapolated without checking material length scales.^[5] The abstraction includes that diagnostic boundary; it does not promise that one constitutive model remains exact down to contact.

Clarity¶

Use four questions to decide whether NFRHT is the right model.

What is the source? Identify thermal current fluctuations and the body temperatures. If a coherent external source fixes the field, the problem is near-field optics or photothermal absorption, not necessarily NFRHT.
What crosses the separation? Verify that the reported component is electromagnetic. Estimate or measure gas, solid-contact, electron, and phonon contributions rather than labeling all nanoscale heat flow “radiative.”
Which channel sector matters? Compare a propagating-only result with the full wave calculation. A substantial \(\kappa>\omega/c\) contribution is the cleanest modal diagnostic of near-field radiative transfer in planar geometry.
Which approximations survive? Declare gap and object sizes, temperature range, material optical data, local versus nonlocal response, surface roughness, and geometry approximation. “Near field” does not validate a dipole, proximity, or local-response model by itself.

The blackbody comparison must also be named. For parallel plates, the conventional far-field benchmark is the Stefan-Boltzmann difference \(\sigma(T_1^4-T_2^4)\) under ideal propagating-channel transmission. A measured or calculated flux above that reference is super-Planckian in the specified comparison. It does not establish an unbounded flux, and it should not be silently compared with the emission of a different area, geometry, bandwidth, or angular acceptance.

Manages Complexity¶

NFRHT replaces an intractable narrative about countless microscopic emitters with a modular calculation. Fluctuation-dissipation theory specifies the source statistics; Maxwell response specifies propagation and multiple scattering; a channel or operator trace specifies transmission; Bose occupation specifies temperature weighting; and the directional difference specifies net heat. Each part can be varied without redefining the phenomenon.

The channel view is especially compressive. Instead of treating “super-Planckian” transfer as anomalous, it asks how many channels are available, how well each transmits, which are propagating or evanescent, and which frequencies lie in the thermal window. For passive two-terminal transfer, the individual \(\tau_n\leq1\) bound separates improvement by opening more channels from improvement by raising transmission in existing channels.^[2] This framing also makes numerical-method choice tractable: planar symmetry suggests Fresnel coefficients; small particles suggest dipoles or multipoles; arbitrary bodies suggest scattering or integral operators; many bodies require a terminal-resolved balance.

The abstraction further disciplines experiments. A gap-dependent increase is evidence only after conductive backgrounds, alignment, roughness, temperature calibration, and view factors are controlled. Ottens and colleagues measured two macroscopic sapphire plates over millimetre-to-micrometre separation and found an increase consistent with near-field theory, directly demonstrating evanescent heat exchange without contact.^[6] A tip experiment probes smaller scales and lateral localization but has different geometry and may expose continuum breakdown. The framework explains why those measurements are complementary rather than interchangeable.

Abstract Reasoning¶

Several inferences follow from the structural signature.

Gap filtering. Because an evanescent component decays across the gap, decreasing \(d\) admits larger lateral wave vectors. In a local planar quasistatic regime, the number of materially contributing channels per area often scales roughly as \(1/d^2\), but atomic and nonlocal cutoffs prevent treating that trend as universal to \(d=0\).^[2]
Thermal-window selection. Increasing temperature changes the Bose occupation and shifts the frequencies carrying most heat. A material resonance outside that window contributes little even if it is optically sharp.
Spectral matching. Two surfaces with compatible surface-polariton frequencies can couple strongly through the gap. Detuning, loss, or mismatch lowers the relevant transmission, connecting the node to Resonance and Impedance Mismatch without reducing it to either prime.
Loss is generative and limiting. Dissipation supplies fluctuating sources and absorption, yet excessive loss can broaden or damp resonant channels. A lossless idealization can support fields while failing to represent thermal emission and absorption correctly.
Equilibrium cancellation. At \(T_1=T_2\), each body still emits, but reciprocal directional powers cancel. A nonzero claimed two-body net heat flux at equal temperature signals an omitted reservoir, drive, non-equilibrium degree of freedom, or accounting error.
Many-body nonadditivity. A third object can scatter or open channels between two others, so total exchange need not equal a sum of isolated pairwise coefficients. Terminal temperatures and environmental radiation must be included consistently.^[2]
Approximation reversal. A condition such as “sphere radius much larger than gap” does not by itself guarantee a proximity approximation for heat transfer. The field's spectral and multipolar content must be checked against an exact or converged solution.^[4]

These are conditional deductions, not slogans. Each depends on passivity, reciprocity, locality, geometry, or scale assumptions that must be stated when used.

Knowledge Transfer¶

The abstraction transfers within thermal photonics because the same roles recur even when the solver changes. A planar-film designer maps modes to \((\omega,\kappa,\mathrm{polarization})\); a nanoparticle researcher maps them to multipoles; an arbitrary-body calculation maps them to singular or scattering channels; a many-body model maps them to terminal-to-terminal coefficients. In every case, the analyst identifies thermal sources, a propagator, dissipative receivers, transmission channels, and a net balance.

It also supports disciplined comparison with adjacent transport theories. The Landauer form resembles coherent electron and phonon transport: reservoir occupations differ, channels transmit energy, and a sum over transmission probabilities yields current. That structural analogy can transfer numerical methods and bounding ideas. It does not make photons, electrons, and phonons the same carrier or authorize adding their currents without an interface model. At subnanometre gaps, the useful transfer is precisely the question of how several carrier theories meet without double counting.

Outside physics, words such as “tunneling,” “resonance,” or “near-field influence” are analogies only. The full NFRHT identity does not recur in organizations, markets, or social networks because the fluctuation-dissipation relation, Maxwell fields, thermal wavelengths, and electromagnetic absorption are indispensable. The portable skeleton belongs to primes such as Coupling, Environmental Coupling Strength, Wave, Resonance, and Non-Locality; the present node remains a domain-specific realization.

Examples¶

Parallel polar dielectrics. Two silicon-carbide or silica half-spaces at different temperatures are brought within a subwavelength vacuum gap. Thermally generated fields excite surface phonon-polariton channels. Compatible resonances on the facing surfaces and the evanescent gap factor produce a strong, spectrally concentrated flux. The bodies, gap, fluctuating sources, high-\(\kappa\) channels, absorption, and temperature difference instantiate every mandatory role.^[3]^[2]

Metallic plates. Metals may show a different modal composition: in standard local calculations, transverse-electric evanescent modes associated with fluctuating eddy currents can dominate over the surface-plasmon picture at thermally relevant frequencies. The case is still NFRHT because the identity is modal thermal exchange, not a requirement that one named polariton supply it.^[2]

Macroscopic sapphire plates. Ottens and colleagues measured heat flow between separated sapphire plates near room temperature as a function of gap and temperature difference. The observed rise toward small separation agreed with theoretical expectations and demonstrated a no-contact evanescent contribution.^[6] The example is valuable because the objects were macroscopic while the gap placed the relevant radiation in its near-field regime; object size alone does not decide the classification.

Scanning thermal microscope. A heated probe tip and a planar sample exchange thermal radiation across a nanoscale vacuum gap. The finite curvature and small active region require more than a parallel-plate view factor. Kittel and colleagues' very-small-gap discrepancy also exposes the local-continuum boundary, making this both an application and a failure-mode example.^[5]

Two spheres. Two nonoverlapping spheres at different temperatures exchange radiative energy. A dipole model works only when the spheres are sufficiently small relative to relevant wavelengths, and a proximity approximation must be validated rather than inferred solely from a small gap. A multipole solution preserves the same source-channel-sink structure while changing the basis.^[4]

Near-field thermophotovoltaics. A hot selective emitter is separated from a photovoltaic receiver by a subwavelength gap. Evanescent coupling can increase above-bandgap photon delivery and alter spectral selectivity. The useful output depends not merely on total heat flux but on matching the emitter spectrum, cell bandgap, gap, parasitic channels, and electrical conversion. It is an NFRHT application when the source radiation is thermally generated.^[2]

Many-body control. A third nanoparticle or structured body placed between two terminals can scatter, relay, suppress, or redirect thermally populated electromagnetic channels. The total current may be nonadditive because each body's field is multiply scattered. The example extends rather than abandons the identity: there are more sources, sinks, and pair-resolved transmission coefficients, but the same fluctuation, propagation, absorption, and balance roles remain.^[2]

Structural Tensions¶

Enhancement versus validity. Smaller gaps open higher-\(\kappa\) channels, but the same reduction eventually exposes nonlocal response, atomic structure, roughness, and competing carriers. The regime promising the largest continuum enhancement is also where continuum extrapolation is least secure.
Resonance versus bandwidth. Sharp surface modes can produce high spectral transmission, while useful devices may require broad heat throughput or bandwidth matched to a receiver. Strong peak enhancement need not maximize integrated or converted power.
Loss as source versus loss as damping. Dissipation is required by the fluctuation-dissipation mechanism for emission and absorption, yet loss also damps coherence and can reduce resonant build-up.
Flux versus efficiency. Increasing total radiative heat transfer can increase unwanted sub-bandgap or parasitic power. Thermophotovoltaic performance requires spectral and electrical accounting, not the largest heat current alone.
Ideal geometry versus experimental control. Infinite parallel plates make modes transparent but require extreme parallelism and gap control in practice. Sphere-plane and tip-plane geometries ease alignment while complicating exact inversion and area normalization.
Local simplicity versus microscopic fidelity. Frequency-only optical constants make fluctuational electrodynamics tractable. Spatial dispersion and atomic models add fidelity at the cost of extra material information and computation.
Two-body clarity versus many-body realism. A two-terminal Landauer balance is readily interpreted. Environments and additional bodies introduce multiple scattering, nonadditivity, and possibly nonreciprocal terminal relations, demanding a complete reservoir account.
Far-field benchmark versus universal bound. Stefan-Boltzmann exchange is the correct propagating-channel benchmark for the stated planar comparison, but presenting it as a universal ceiling creates a false paradox. Near-field bounds require separation-, geometry-, and material-aware constraints.

Structural–Framed Character¶

This node is strongly structural. Temperatures, source correlations, electromagnetic response, channel transmission, absorption, and net power can be specified independently of a particular institution or human judgment. “Near,” “super-Planckian,” and even the choice of a blackbody benchmark introduce conventions, but those conventions are operationally declared and do not create the underlying mechanism. The structural-framed aggregate is therefore \(0.10\): the vocabulary requires some contextual scale choices, while the causal and mathematical commitments are physical.

Structural Core vs. Domain Accent¶

The portable core is unequal reservoirs + a coupling channel + transmission spectrum + directional balance -> net flow. Coupling, Resonance, Wave, Environmental Coupling Strength, and Non-Locality can each recover part of that skeleton across substrates.

The domain accent is indispensable: Bose thermal occupation, random electric currents linked to dissipative susceptibility by fluctuation-dissipation theory, Maxwell propagation, propagating versus evanescent electromagnetic sectors, thermal wavelength, material absorption, and radiative heat flux. Remove those commitments and the result is generic transport or coupling, not Near-Field Radiative Heat Transfer. This is why the candidate has autonomous domain-specific identity but fails the prime transfer bar.

Coupling is the minimal constitutive parent. NFRHT instantiates coupling by specifying two or more material subsystems, a thermal electromagnetic interaction channel, its frequency- and geometry-dependent strength, and the timescale or conductance of exchange. The relation is not exact coverage: Coupling does not supply fluctuation-dissipation sources, evanescent modes, thermal occupation, material response, or the radiative balance equation.

Environmental Coupling Strength is a close quantitative view because NFRHT computes the rate or conductance of energy exchange across a separation. It is related rather than a second parent: the candidate is the domain mechanism and theory of the transfer, not merely the scalar strength assigned to an already defined boundary link. Wave supplies propagation and interference; Resonance explains spectral enhancement; Non-Locality becomes relevant when response depends on spatial wave vector; and Impedance Mismatch and Coupling Efficiency clarifies reflection and spectral matching. None covers the complete source-channel-sink balance, and adding all of them still leaves the thermal electromagnetic constitutive law unstated.

Relationships to Other Abstractions¶

Current abstraction Near-Field Radiative Heat Transfer Domain-specific

Parents (1) — more general patterns this builds on

Near-Field Radiative Heat Transfer is a kind of Coupling Prime

Coupling is the minimal constitutive parent.

Hierarchy path (1) — routes to 1 parentless root

Near-Field Radiative Heat Transfer → Coupling

Neighborhood in Abstraction Space¶

Near-Field Radiative Heat Transfer sits in a sparse region of the domain-specific corpus (93^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Greenhouse Effect. Atmospheric absorption and re-emission alter planetary energy balance over macroscopic paths. It is not evanescent exchange across a subwavelength gap.
Albedo. Albedo is a reflected fraction of incident radiation, not thermally generated near-field energy exchange.
Urban Heat Island or Marine Heatwave. These are climate-temperature patterns, not electromagnetic transport mechanisms. Their semantic proximity comes from the word “heat,” not shared structure.
Thermal radiation. Thermal radiation is the broader electromagnetic emission and exchange category. NFRHT is the wave-optical, close-separation regime in which normally nonpropagating fields participate materially.
Near-field thermal radiation. This phrase sometimes names the broader field, including local density of states, coherence, forces, and emission without a second receiving body. It should not be treated as an automatic exact alias without contextual review.
Radiative heat transfer. The broader process includes far-field exchange describable with propagating modes and view factors. NFRHT is not a spelling variant of that genus.
Casimir force. Both may be analyzed with fluctuating electromagnetic fields, but Casimir interactions concern force or free energy rather than thermally driven net heat current.
Heat-assisted magnetic recording. A near-field transducer can use coherently driven optical power to heat a recording medium. That apparatus is not itself proof of NFRHT unless the transferred energy is sourced by thermal fluctuations.
Quantum or electron tunneling. These transfer particles or charge through forbidden regions; NFRHT's “photon tunneling” denotes evanescent electromagnetic coupling.
Contact heat transfer. Once bodies touch—or electron and phonon channels become comparable at an ultrashort gap—the measured current cannot be assigned to NFRHT without carrier-resolved modeling.

References¶

[1] D. Polder and M. Van Hove, “Theory of Radiative Heat Transfer between Closely Spaced Bodies,” Physical Review B 4, 3303–3314 (1971). doi:10.1103/PhysRevB.4.3303. registry ↩a ↩b

[2] S.-A. Biehs, R. Messina, P. S. Venkataram, A. W. Rodriguez, J. C. Cuevas, and P. Ben-Abdallah, “Near-field radiative heat transfer in many-body systems,” Reviews of Modern Physics 93, 025009 (2021). doi:10.1103/RevModPhys.93.025009; author preprint. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m

[3] K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surface Science Reports 57, 59–112 (2005). doi:10.1016/j.surfrep.2004.12.002. registry ↩a ↩b

[4] A. Narayanaswamy and G. Chen, “Thermal near-field radiative transfer between two spheres,” Physical Review B 77, 075125 (2008). doi:10.1103/PhysRevB.77.075125. registry ↩a ↩b ↩c

[5] A. Kittel, W. Müller-Hirsch, J. Parisi, S.-A. Biehs, D. Reddig, and M. Holthaus, “Near-Field Heat Transfer in a Scanning Thermal Microscope,” Physical Review Letters 95, 224301 (2005). doi:10.1103/PhysRevLett.95.224301. registry ↩a ↩b

[6] R. S. Ottens et al., “Near-Field Radiative Heat Transfer between Macroscopic Planar Surfaces,” Physical Review Letters 107, 014301 (2011). doi:10.1103/PhysRevLett.107.014301. registry ↩a ↩b