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Absorptance
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In the study of heat transfer, absorptance of the surface of a material is its effectiveness in absorbing radiant energy. It is the ratio of the absorbed to the incident radiant power.[1]

Mathematical definitions

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Hemispherical absorptance

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Hemispherical absorptance of a surface, denoted A is defined as[2]

where

  • ⁠⁠ is the radiant flux absorbed by that surface;
  • ⁠⁠ is the radiant flux received by that surface.

Spectral hemispherical absorptance

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Spectral hemispherical absorptance in frequency and spectral hemispherical absorptance in wavelength of a surface, denoted Aν and Aλ respectively, are defined as[2]

where

  • ⁠⁠ is the spectral radiant flux in frequency absorbed by that surface;
  • ⁠⁠ is the spectral radiant flux in frequency received by that surface;
  • ⁠⁠ is the spectral radiant flux in wavelength absorbed by that surface;
  • ⁠⁠ is the spectral radiant flux in wavelength received by that surface.

Directional absorptance

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Directional absorptance of a surface, denoted AΩ, is defined as[2]

where

  • ⁠⁠ is the radiance absorbed by that surface;
  • ⁠⁠ is the radiance received by that surface.

Spectral directional absorptance

[edit]

Spectral directional absorptance in frequency and spectral directional absorptance in wavelength of a surface, denoted Aν,Ω and Aλ,Ω respectively, are defined as[2]

where

  • ⁠⁠ is the spectral radiance in frequency absorbed by that surface;
  • ⁠⁠ is the spectral radiance received by that surface;
  • ⁠⁠ is the spectral radiance in wavelength absorbed by that surface;
  • ⁠⁠ is the spectral radiance in wavelength received by that surface.

Other radiometric coefficients

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Radiometry coefficients
Quantity SI units Notes
Name Sym.
Hemispherical emissivity ε — Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity εν
ελ 		— Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity εΩ — Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity εΩ,ν
εΩ,λ 		— Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A — Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance Aν
Aλ 		— Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance AΩ — Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance AΩ,ν
AΩ,λ 		— Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance R — Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance Rν
Rλ 		— Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance RΩ — Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance RΩ,ν
RΩ,λ 		— Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T — Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance Tν
Tλ 		— Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance TΩ — Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance TΩ,ν
TΩ,λ 		— Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient μ m−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient μν
μλ 		m−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient μΩ m−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient μΩ,ν
μΩ,λ 		m−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Absorptance

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from Grokipedia
Absorptance is the fraction of incident radiant or that is absorbed by a body or , denoted by the α and ranging from 0 (no absorption) to 1 (complete absorption, as in a blackbody). It represents the efficiency with which a surface or volume retains , such as visible , , or , converting it into or other forms. This property is fundamental in fields like , radiative , and , where it determines how materials interact with and . In , absorptance is interrelated with (ρ, the fraction reflected) and (τ, the fraction transmitted), governed by such that α + ρ + τ = 1 for non-scattering, non-fluorescent materials. Absorptance can be spectral (α_λ, varying with λ due to composition) or total (integrated over a , relevant for sources like solar radiation). It depends on intrinsic properties, including the (n), extinction coefficient (k), and thickness (d), as well as external factors like angle of incidence and polarization. For example, in thin films or coatings, increasing thickness enhances absorptance until saturation, while higher coefficients promote deeper penetration and greater absorption. A key principle linking absorptance to emission is , which states that, for a body in , the spectral absorptance equals the spectral emissivity (ε_λ = α_λ) at the same and , explaining why good absorbers are also efficient emitters. This law underpins applications in solar absorbers (high α in visible, low ε in for selective surfaces) and (tailored α/ε ratios to balance heat gain and loss). Measurements of absorptance often use or calorimetric methods, ensuring accuracy across wavelengths for practical designs.

Basic Concepts

Definition

Absorptance is the fraction of incident absorbed by a surface or medium, denoted by the α, with values ranging from 0 (no absorption) to 1 (total absorption). It quantifies the effectiveness of a in capturing rather than reflecting or transmitting it. Mathematically, absorptance is expressed as the ratio α=PabsorbedPincident,\alpha = \frac{P_\text{absorbed}}{P_\text{incident}}, where PP denotes radiant power. This dimensionless quantity represents the proportion of incoming radiation converted into other forms of energy, typically heat. In the field of radiometry, absorptance is one of the core surface interaction coefficients, alongside reflectance and transmittance, and applies to both opaque materials that fully interact with incident radiation and semi-transparent ones that partially do so. The term absorptance was formalized in mid-20th century literature, building on 19th-century advancements in such as the study of absorption spectra. This general definition extends to specialized forms like hemispherical and directional absorptance, which account for angular variations in incidence.

Physical Principles

Absorptance arises from the of with matter, where incident photons are absorbed through various microscopic processes that convert into internal forms of energy. In materials, primary absorption mechanisms include electronic excitation, where photons promote electrons from valence to conduction bands or higher atomic orbitals, particularly dominant in the and . Vibrational absorption occurs when photons excite molecular or lattice vibrations, often mediated by in solids, which is prevalent in the region. These processes can also involve phonon interactions to conserve in indirect transitions, enabling absorption even when direct electronic jumps are forbidden by . The absorbed energy adheres to the principle of , transforming into through non-radiative relaxation pathways, such as electron-phonon that dissipates excitation as lattice , or into in photochemical reactions. In opaque materials, this leads to localized heating, while in transparent ones, absorption may trigger or other emissions if not fully thermalized. Overall, the absorbed portion contributes to the material's without re-emission of the original , distinguishing it from or reflection. Absorptance depends intrinsically on properties, including atomic and molecular composition that determines spacings for resonant absorption, as well as thickness that influences the probability of photon-material encounters—thinner films may transmit more while thicker ones saturate absorption. Surface conditions, such as roughness, enhance efficiency by increasing the effective through multiple internal reflections and trapping of light. These factors collectively govern how efficiently a captures incident . Ideal absorbers, such as blackbodies, achieve near-unity absorptance across all wavelengths by design, exemplified by cavity structures that prevent escape of radiation through repeated internal absorptions and emissions in . In contrast, selective absorbers exhibit high absorptance in targeted spectral bands, such as the solar spectrum (0.3–2.5 μm), due to engineered compositions like cermets or plasmonic nanostructures that promote resonant excitations while reflecting longer wavelengths to minimize thermal losses.

Types of Absorptance

Directional Absorptance

Directional absorptance, denoted as α(θ,ϕ)\alpha(\theta, \phi), is defined as the fraction of incident radiant power from a specific direction, characterized by the polar θ\theta and azimuthal ϕ\phi, that is absorbed by a surface. This quantity applies to incident along a particular direction onto the surface, making it suitable for analyzing non-diffuse or specular surfaces where absorption varies with the of incidence relative to the surface normal. For an opaque surface, it can be expressed as α(θ,ϕ)=Pabs(θ,ϕ)Pinc(θ,ϕ)\alpha(\theta, \phi) = \frac{P_{\text{abs}}(\theta, \phi)}{P_{\text{inc}}(\theta, \phi)}, where PabsP_{\text{abs}} is the absorbed power and PincP_{\text{inc}} is the incident power from that direction; equivalently, for collimated , α(θ,ϕ)=1−ρ(θ,ϕ)\alpha(\theta, \phi) = 1 - \rho(\theta, \phi), with ρ(θ,ϕ)\rho(\theta, \phi) being the directional . Several factors influence directional absorptance, primarily the orientation of the surface relative to the incident direction, which determines the effective and interaction . For smooth surfaces, Fresnel reflection plays a key role, as the increases with θ\theta for most materials, particularly dielectrics and metals, leading to reduced absorption at oblique angles due to enhanced . On rough or microstructured surfaces, shadowing effects can further modulate α(θ,ϕ)\alpha(\theta, \phi), where incident rays at higher θ\theta may be partially blocked by surface protrusions, reducing the absorbed fraction compared to normal incidence. These angular dependencies are critical in applications involving directed beams, such as interactions or solar on tilted panels. A representative example is observed in many materials, such as polished metals or dielectrics, where directional absorptance is higher at normal incidence (θ=0\theta = 0)—often approaching 0.9 or more—due to minimal Fresnel reflection, but decreases significantly at grazing angles (θ→90∘\theta \to 90^\circ), sometimes dropping below 0.1 as reflection dominates. This behavior underscores the importance of directional effects in designing surfaces for controlled absorption, such as in optical coatings or management systems. Integrating directional absorptance over the hemisphere yields the hemispherical absorptance for diffuse conditions.

Hemispherical Absorptance

Hemispherical absorptance, denoted as α\alpha, is defined as the ratio of the total power absorbed by a surface to the total power incident on that surface from all directions within the incident hemisphere. This quantity accounts for the integrated effect of arriving from diffuse or multidirectional sources, making it essential for scenarios where is not confined to a single . Unlike directional absorptance, which applies to specific incidence angles, hemispherical absorptance provides a comprehensive measure for total energy absorption. The mathematical expression for hemispherical absorptance is α=1Ei∫2πα(θ,ϕ)Ii(θ,ϕ)cos⁡θ dΩ,\alpha = \frac{1}{E_i} \int_{2\pi} \alpha(\theta, \phi) I_i(\theta, \phi) \cos \theta \, d\Omega, where EiE_i represents the total hemispherical incident irradiance, α(θ,ϕ)\alpha(\theta, \phi) is the directional absorptance at polar angle θ\theta and azimuthal angle ϕ\phi, Ii(θ,ϕ)I_i(\theta, \phi) is the incident radiative intensity, cos⁡θ\cos \theta accounts for the projected area of the surface, and dΩ=sin⁡θ dθ dϕd\Omega = \sin \theta \, d\theta \, d\phi is the differential element of solid angle integrated over the 2π\pi steradians of the hemisphere. This formula arises from the radiative transfer principles, where the absorbed power is obtained by integrating the product of directional absorptance and incident intensity over all directions, normalized by the total incident flux. The derivation outlines a weighted average of the directional absorptance, with weights determined by the distribution of the incident intensity and the geometric factor cos⁡θ\cos \theta. Starting from the differential absorbed dQa=α(θ,ϕ)Ii(θ,ϕ)cos⁡θ dΩdQ_a = \alpha(\theta, \phi) I_i(\theta, \phi) \cos \theta \, d\Omega, integration over the hemisphere yields the total absorbed power, which is then divided by the incident Ei=∫2πIi(θ,ϕ)cos⁡θ dΩE_i = \int_{2\pi} I_i(\theta, \phi) \cos \theta \, d\Omega to obtain α\alpha. This approach ensures that surfaces with varying angular response are properly characterized for broadband, multidirectional . Hemispherical absorptance builds on directional absorptance by extending the analysis to the full range of incident directions. In practical contexts, hemispherical absorptance is applied to evaluate overall surface heating under natural conditions, such as or isotropic illumination, where the angular distribution of incident is known or assumed uniform. For instance, in of surfaces exposed to environmental , it facilitates predictions of net and temperature rise, aiding designs in and building applications.

Spectral Absorptance

Spectral Directional Absorptance

Spectral directional absorptance, denoted as α(θ,ϕ,λ)\alpha(\theta, \phi, \lambda), quantifies the fraction of incident from a specific direction, characterized by polar θ\theta and azimuthal ϕ\phi, that is absorbed by a surface at λ\lambda. This property is fundamental in for describing how materials interact with of particular colors or energies from unidirectional sources, such as lasers or collimated . It extends the concept of broadband directional absorptance by resolving the absorption efficiency across the , enabling analysis of wavelength-specific behaviors without angular averaging. The formula for spectral directional absorptance is given by α(θ,ϕ,λ)=Pabs(θ,ϕ,λ)Pinc(θ,ϕ,λ),\alpha(\theta, \phi, \lambda) = \frac{P_{\text{abs}}(\theta, \phi, \lambda)}{P_{\text{inc}}(\theta, \phi, \lambda)}, where Pabs(θ,ϕ,λ)P_{\text{abs}}(\theta, \phi, \lambda) is the absorbed power and Pinc(θ,ϕ,λ)P_{\text{inc}}(\theta, \phi, \lambda) is the incident power at λ\lambda from direction (θ,ϕ)(\theta, \phi). When the incident radiation is characterized by L(λ)L(\lambda), the absorptance relates to the of the surface's absorbed radiance to the incident radiance, often expressed per unit and . This directional specificity arises from the surface's microstructure and orientation relative to the incoming rays, influencing phenomena like polarization-dependent absorption. A key aspect of directional absorptance is the presence of selective absorption bands, where materials exhibit high absorption in certain ranges due to electronic structure. In semiconductors, absorption is sharply cutoff below the material's bandgap , leading to transparent behavior for longer and opaque absorption for shorter ones above the bandgap. In photovoltaic applications, directional absorptance is critical for optimizing conversion from the solar , which peaks in the visible range around 500 nm. Silicon-based solar cells, for example, exhibit α(θ,ϕ,λ)\alpha(\theta, \phi, \lambda) approaching 0.9 for normal incidence in the 400–1000 nm band, aligning with the high irradiance portion of the AM1.5 solar and enabling efficient photon-to-electron conversion, though absorption drops sharply beyond 1100 nm due to the 1.1 eV bandgap. This selectivity underscores the design of antireflection coatings to enhance directional absorptance under varying solar angles.

Spectral Hemispherical Absorptance

Spectral hemispherical absorptance, denoted as α(λ)\alpha(\lambda), represents the of incident at λ\lambda that is absorbed by a surface, averaged over all directions within the incident hemisphere. This property accounts for the directional variation in absorption while resolving the response at specific wavelengths, making it essential for materials exhibiting spectral selectivity. The mathematical definition is given by α(λ)=1Ei(λ)∫2πα(θ,ϕ,λ) Ii(θ,ϕ,λ) cos⁡θ dΩ,\alpha(\lambda) = \frac{1}{E_i(\lambda)} \int_{2\pi} \alpha(\theta, \phi, \lambda) \, I_i(\theta, \phi, \lambda) \, \cos \theta \, d\Omega, where Ei(λ)E_i(\lambda) is the spectral irradiance incident on the surface, α(θ,ϕ,λ)\alpha(\theta, \phi, \lambda) is the directional spectral absorptance, Ii(θ,ϕ,λ)I_i(\theta, \phi, \lambda) is the incident spectral intensity, θ\theta is the polar angle from the surface normal, ϕ\phi is the azimuthal angle, and dΩ=sin⁡θ dθ dϕd\Omega = \sin \theta \, d\theta \, d\phi is the differential solid angle integrated over the hemisphere (solid angle 2π2\pi). This measure derives from the spectral directional absorptance through hemispherical integration at fixed λ\lambda, allowing for accurate of absorbed under arbitrary illumination. The importance of spectral hemispherical absorptance lies in its role for non-blackbody materials, where absorption varies strongly with ; it enables computation of total hemispherical absorptance by integrating α(λ)\alpha(\lambda) weighted by the incident spectral distribution, which is vital for radiative modeling in enclosures, solar devices, and high-temperature systems.

Relation to Reflectance and Transmittance

In , the interaction of incident with a surface or medium is governed by the , which partitions the incident flux into absorbed, reflected, and transmitted components. For non-scattering media, the sum of absorptance (α), (ρ), and (τ) equals unity: α+ρ+τ=1\alpha + \rho + \tau = 1 This equation holds for the total hemispherical quantities, where absorptance represents the fraction of incident absorbed by the , the fraction reflected into the hemisphere, and the fraction passing through to the opposite hemisphere. For opaque surfaces, where no radiation is transmitted (τ = 0), the relation simplifies to absorptance equaling one minus reflectance: α=1−ρ\alpha = 1 - \rho This implies that any radiation not reflected is fully absorbed, a fundamental principle in applications such as solar absorbers and thermal coatings. Similar conservation relations apply to directional and hemispherical variants. For directional properties, considering radiation incident from a specific direction defined by angles θ and φ, the directional absorptance α(θ, φ) plus directional reflectance ρ(θ, φ) equals 1 for opaque surfaces: α(θ,ϕ)+ρ(θ,ϕ)=1\alpha(\theta, \phi) + \rho(\theta, \phi) = 1 Hemispherical quantities, obtained by integrating over the incident hemisphere, follow the same partitioning: α + ρ + τ = 1, with τ = 0 yielding α = 1 - ρ for opaque cases. These directional dependencies are crucial for modeling anisotropic surfaces, such as rough metals or textured coatings, where absorptance varies with incidence angle. In translucent materials, such as certain polymers or biological tissues, τ exceeds zero due to volume absorption and partial transmission through the bulk. Here, the general energy balance α + ρ + τ = 1 persists, but positive τ reduces the effective absorptance α compared to opaque counterparts, as some incident energy exits without being absorbed. This behavior is evident in thin films or leaves, where volume and absorption compete with transmission.

Connection to Emittance via Kirchhoff's Law

establishes a fundamental equivalence between the absorptive and emissive properties of a material body under . Specifically, for a body at TT, the directional absorptance α(λ,θ,ϕ,T)\alpha(\lambda, \theta, \phi, T) equals the directional emittance ε(λ,θ,ϕ,T)\varepsilon(\lambda, \theta, \phi, T), where λ\lambda is the and θ,ϕ\theta, \phi denote the direction of incidence or emission. This relation holds for opaque bodies and extends to the hemispherical or total forms when integrated over appropriate solid angles and wavelengths. The law was first articulated by in papers presented to the Berlin Academy in December 1859 and , with a comprehensive statement appearing in his publication in und Chemie. These works laid the groundwork for the study of by linking absorption to emission, influencing subsequent developments such as Planck's quantum hypothesis. A key implication is that efficient absorbers are necessarily efficient emitters at the same and direction; for instance, a blackbody with α=1\alpha = 1 across all λ,θ,ϕ\lambda, \theta, \phi also has ε=1\varepsilon = 1, achieving perfect balance in equilibrium. This equivalence underpins the reciprocity in but is rigorously valid only in , where the body and its field share the same . In non-equilibrium scenarios, such as solar heating—where short-wavelength solar is absorbed while long-wavelength emission occurs at the cooler body —the serves as an by setting emittance equal to absorptance evaluated at the body's for emission predictions. Such approximations are common in analyses of selective solar absorbers, though they neglect detailed non-equilibrium effects like induced emission or gradients.

Measurement and Applications

Measurement Techniques

Measurement techniques for absorptance encompass laboratory-based methods tailored to specific properties, such as total hemispherical or spectral variants, enabling precise quantification in controlled environments. serves as a direct method for determining total hemispherical absorptance by exposing a sample to known incident radiant power and measuring the resulting temperature rise. The absorbed power is derived from the sample's thermal response, typically using the relation between , temperature change, and time, with absorptance calculated as the ratio of absorbed to incident power. This approach is particularly effective for opaque materials in or controlled atmospheres, providing bulk-averaged values without . Spectrophotometry measures spectral absorptance α(λ) by assessing and across , visible, and near-infrared wavelengths, often employing integrating spheres to capture diffuse hemispherical components. In these systems, a directs monochromatic onto the sample within the sphere, which uniformly redistributes scattered radiation for detection, yielding absorptance via α(λ) = 1 - ρ(λ) - τ(λ) for non-opaque materials. This technique supports broad spectral coverage essential for applications like evaluation. Goniophotometry quantifies directional absorptance α(θ, φ) by rotating the sample relative to a monochromatic source and detector to map angular distributions. For opaque surfaces, absorptance is obtained by subtracting the measured directional from unity, providing insight into angular dependencies under controlled illumination. Standardized procedures, such as ASTM E903, outline spectrophotometric methods for solar absorptance, integrating spectral data weighted by to compute total values. Key challenges in these measurements include mitigating , which can inflate apparent and thus underestimate absorptance, and ensuring meticulous sample preparation to maintain surface uniformity and minimize contamination effects.

Practical Applications

In solar energy systems, high absorptance coatings are essential for maximizing energy capture in photovoltaic panels and collectors. These coatings, often designed with solar absorptance values exceeding 0.9 in the , enable efficient conversion of sunlight into or by minimizing reflection losses. For instance, selective absorbing surfaces in concentrating plants achieve absorptance greater than 0.95 to enhance photothermal efficiency. Thermal management applications leverage low absorptance surfaces to control intake in extreme environments, such as . Coatings with solar absorptance as low as 0.1 on aluminum or silica substrates prevent excessive heating from solar radiation while maintaining controlled emittance for . In stealth technology, materials tuned for specific spectral absorptance reduce infrared signatures by achieving low emissivity in the range, thereby minimizing detectability; this is complemented by radar-absorbing properties for . In , selective absorbers for solar thermal systems are engineered to exhibit high absorptance in the solar spectrum (typically above 0.9) and low emittance in the range (below 0.1), optimizing energy retention as per . These designs, often involving multilayer structures like metal-dielectric composites, are widely used in evacuated tube collectors to boost overall system efficiency. Absorptance plays a key role in through its influence on surface in models, where lower (higher absorptance) over land and ice surfaces amplifies global warming by increasing absorption. For example, reductions in planetary since 2020 have contributed approximately 0.22 K to recent surges by enhancing shortwave uptake. This feedback mechanism is critical in projections of and ice- effects.

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