Molecular vibration

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Molecular_vibration

A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10¹³ Hz to approximately 10¹⁴ Hz, corresponding to wavenumbers of approximately 300 to 3000 cm⁻¹ and wavelengths of approximately 30 to 3 μm.

For a diatomic molecule A−B, the vibrational frequency in s⁻¹ is given by $\nu =\frac{1}{2\pi }\sqrt{k/\mu }$, where k is the force constant in dyne/cm or erg/cm² and μ is the reduced mass given by $\frac{1}{\mu }=\frac{1}{m_A}+\frac{1}{m_B}$. The vibrational wavenumber in cm⁻¹ is $\tilde{\nu }=\frac{1}{2\pi c}\sqrt{k/\mu },$ where c is the speed of light in cm/s.

Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of different parts of the molecule. In general, a non-linear molecule with N atoms has 3N − 6 normal modes of vibration, but a linear molecule has 3N − 5 modes, because rotation about the molecular axis cannot be observed. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond.

A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = hν, where h is the Planck constant. A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in its ground state. When multiple quanta are absorbed, the first and possibly higher overtones are excited.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, because the potential energy of the molecule is more like a Morse potential or more accurately, a Morse/Long-range potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration–rotation spectra.

Number of vibrational modes

Main article: Degrees of freedom (physics and chemistry)

For a molecule with N atoms, the positions of all N nuclei depend on a total of 3N coordinates, so that the molecule has 3N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates.

A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates.

An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third rotational coordinate.

The number of vibrational modes is therefore 3N minus the number of translational and rotational degrees of freedom, or 3N − 5 for linear and 3N − 6 for nonlinear molecules.

Vibrational coordinates

The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene,

• Stretching: a change in the length of a bond, such as C–H or C–C
• Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group
• Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule
• Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule
• Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups
• Out-of-plane: a change in the angle between any one of the C–H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF₃ when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethylene there are 12 internal coordinates: 4 C–H stretching, 1 C–C stretching, 2 H–C–H bending, 2 CH₂ rocking, 2 CH₂ wagging, 1 twisting. Note that the H–C–C angles cannot be used as internal coordinates as well as the H–C–H angle because the angles at each carbon atom cannot all increase at the same time.

Note that these coordinates do not correspond to normal modes (see § Normal coordinates). In other words, they do not correspond to particular frequencies or vibrational transitions.

Vibrations of a methylene group (−CH₂−) in a molecule for illustration

Within the CH₂ group, commonly found in organic compounds, the two low mass hydrogens can vibrate in six different ways which can be grouped as 3 pairs of modes: 1. symmetric and asymmetric stretching, 2. scissoring and rocking, 3. wagging and twisting. These are shown here:

Symmetrical stretching	Asymmetrical stretching	Scissoring (Bending)
Rocking	Wagging	Twisting

(These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

Symmetry-adapted coordinates

See also: Molecular symmetry

Symmetry–adapted coordinates may be created by applying a projection operator to a set of internal coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four (un-normalized) C–H stretching coordinates of the molecule ethene are given by $${\displaystyle \begin{array}{rl}{Q}_{s1}& =q_1+q_2+q_3+q_4\\ Q_{s2}& =q_1+q_2-q_3-q_4\\ Q_{s3}& =q_1-q_2+q_3-q_4\\ Q_{s4}& =q_1-q_2-q_3+q_4\end{array}}$$ where ${\displaystyle q_1-q_4}$ are the internal coordinates for stretching of each of the four C–H bonds.

Illustrations of symmetry–adapted coordinates for most small molecules can be found in Nakamoto.

Normal coordinates

The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. If the molecule possesses symmetries, the normal modes "transform as" an irreducible representation under its point group. The normal modes are determined by applying group theory, and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO₂, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch:

• symmetric stretching: the sum of the two C–O stretching coordinates; the two C–O bond lengths change by the same amount and the carbon atom is stationary. Q = q₁ + q₂
• asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the other decreases. Q = q₁ − q₂

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

• principally C–H stretching with a little C–N stretching; Q₁ = q₁ + a q₂ (a << 1)
• principally C–N stretching with a little C–H stretching; Q₂ = b q₁ + q₂ (b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.

Newtonian mechanics

The HCl molecule as an anharmonic oscillator vibrating at energy level E₃. D₀ is dissociation energy here, r₀ bond length, U potential energy. Energy is expressed in wavenumbers. The hydrogen chloride molecule is attached to the coordinate system to show bond length changes on the curve.

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere. $${\displaystyle \mathrm{F}=-kQ}$$ By Newton's second law of motion this force is also equal to a reduced mass, μ, times acceleration. $${\displaystyle \mathrm{F}=\mu \frac{d^{2}Q}{d{t}^2}}$$ Since this is one and the same force the ordinary differential equation follows. $${\displaystyle \mu \frac{d^{2}Q}{d{t}^2}+kQ=0}$$ The solution to this equation of simple harmonic motion is $${\displaystyle Q(t)=A\cos (2\pi \nu t);\nu =\frac{1}{2\pi }\sqrt{\frac{k}{\mu }}.}$$ A is the maximum amplitude of the vibration coordinate Q. It remains to define the reduced mass, μ. In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses, m_A and m_B, as $${\displaystyle \frac{1}{\mu }=\frac{1}{m_A}+\frac{1}{m_B}.}$$ The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy. $${\displaystyle k=\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{Q}^2}}$$

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies, ν_i, are obtained from the eigenvalues, λ_i, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule. F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by $${\displaystyle E_n=h\left(n+\frac{1}{2}\right)\nu =h\left(n+\frac{1}{2}\right)\frac{1}{2\pi }\sqrt{\frac{k}{m}},}$$ where n is a quantum number that can take values of 0, 1, 2, ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.

The difference in energy when n (or v) changes by 1 is therefore equal to ${\displaystyle h\nu }$, the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency ${\displaystyle \nu }$ (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one, $${\displaystyle \mathrm{\Delta }n=\pm 1}$$but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n = 2 and n = 1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. To describe vibrational levels of an anharmonic oscillator, Dunham expansion is used.

When it comes to polyatomic molecules, it is common to solve the Schrödinger Equation using Watson's nuclear motion Hamiltonian. Similar as for diatomics, this can be done within the harmonic approximation as stated above. For the anharmonic calculation of vibrational spectra of polyatomic molecules, more sophisticated approaches are used. Prominent examples in computational chemistry are 2nd order vibrational perturbation theory (VPT2) or vibrational configuration interaction theory (VCI).

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate. Likewise, the intensity of Raman bands depends on the derivative of polarizability with respect to the normal coordinate. There is also a dependence on the fourth-power of the wavelength of the laser used.

Black-body radiation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Black-body_radiation

Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spectrum of wavelengths, inversely related to intensity, that depend only on the body's temperature, which is assumed, for the sake of calculations and theory, to be uniform and constant.

As the temperature of a black body decreases, the emitted thermal radiation decreases in intensity and its maximum moves to longer wavelengths. Shown for comparison is the classical Rayleigh–Jeans law and its ultraviolet catastrophe.

A perfectly insulated enclosure which is in thermal equilibrium internally contains blackbody radiation, and will emit it through a hole made in its wall, provided the hole is small enough to have a negligible effect upon the equilibrium. The thermal radiation spontaneously emitted by many ordinary objects can be approximated as blackbody radiation.

Of particular importance, although planets and stars (including the Earth and Sun) are neither in thermal equilibrium with their surroundings nor perfect black bodies, blackbody radiation is still a good first approximation for the energy they emit. 

The term black body was introduced by Gustav Kirchhoff in 1860. Blackbody radiation is also called thermal radiation, cavity radiation, complete radiation or temperature radiation.

Theory

Spectrum

Blacksmiths judge workpiece temperatures by the colour of the glow.

This blacksmith's colourchart stops at the melting temperature of steel

Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature, called the Planck spectrum or Planck's law. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at room temperature most of the emission is in the infrared region of the electromagnetic spectrum. As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. Viewed in the dark by the human eye, the first faint glow appears as a "ghostly" grey (the visible light is actually red, but low intensity light activates only the eye's grey-level sensors). With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises. When the body appears white, it is emitting a substantial fraction of its energy as ultraviolet radiation. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well.

Blackbody radiation provides insight into the thermodynamic equilibrium state of cavity radiation.

Black body

Main article: Black body

All normal (baryonic) matter emits electromagnetic radiation when it has a temperature above absolute zero. The radiation represents a conversion of a body's internal energy into electromagnetic energy, and is therefore called thermal radiation. It is a spontaneous process of radiative distribution of entropy.

Color of a black body from 800 K to 12200 K. This range of colors approximates the range of colors of stars of different temperatures, as seen or photographed in the night sky.

Conversely, all normal matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all wavelengths, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called blackbody radiation.

The concept of the black body is an idealization, as perfect black bodies do not exist in nature. However, graphite and lamp black, with emissivities greater than 0.95, are good approximations to a black material. Experimentally, blackbody radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective. A closed box with walls of graphite at a constant temperature with a small hole on one side produces a good approximation to ideal blackbody radiation emanating from the opening.

Blackbody radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity. In equilibrium, for each frequency, the intensity of radiation which is emitted and reflected from a body relative to other frequencies (that is, the net amount of radiation leaving its surface, called the spectral radiance) is determined solely by the equilibrium temperature and does not depend upon the shape, material or structure of the body. For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is entirely due to emission. In addition, a black body is a diffuse emitter (its emission is independent of direction).

Blackbody radiation becomes a visible glow of light if the temperature of the object is high enough. The Draper point is the temperature at which all solids glow a dim red, about 798 K. At 1000 K, a small opening in the wall of a large uniformly heated opaque-walled cavity (such as an oven), viewed from outside, looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built so that almost all light entering is absorbed by its walls, it will contain a good approximation to blackbody radiation. The spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the spectral radiation intensity plotted versus frequency(or wavelength) is called the blackbody curve. Different curves are obtained by varying the temperature.

The temperature of a Pāhoehoe lava flow can be estimated by observing its color. The result agrees well with other measurements of temperatures of lava flows at about 1,000 to 1,200 °C (1,830 to 2,190 °F).

When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the blackbody curve. This means that the blackbody curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of Kirchhoff's law of thermal radiation: the blackbody curve is characteristic of thermal light, which depends only on the temperature of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in thermodynamic equilibrium. When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium.

In the laboratory, blackbody radiation is approximated by the radiation from a small hole in a large cavity, a hohlraum, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. (This technique leads to the alternative term cavity radiation.) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. Absorption occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole's radiation (that is, the amount of light emitted from the hole at each wavelength) will be continuous, and will depend only on the temperature and the fact that the walls are opaque and at least partly absorptive, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum).

The radiance or observed intensity is not a function of direction. Therefore, a black body is a perfect Lambertian radiator.

Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The emissivity of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength so that the emissivity is a constant. This is known as the gray body assumption.

Nine-year WMAP image (2012) of the cosmic microwave background radiation across the universe.

With non-black surfaces, the deviations from ideal blackbody behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of local thermodynamic equilibrium still follow Kirchhoff's Law: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body; the incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In astronomy, objects such as stars are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect blackbody spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is the hypothetical blackbody radiation emitted by black holes, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles.

A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that, for most practical purposes, such a black body does not emit in the visible range.

The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.

Further explanation

According to the Classical Theory of Radiation, if each Fourier mode of the equilibrium radiation (in an otherwise empty cavity with perfectly reflective walls) is considered as a degree of freedom capable of exchanging energy, then, according to the equipartition theorem of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes, this would imply infinite heat capacity, as well as a nonphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe. In the longer wavelengths this deviation is not so noticeable, as ${\displaystyle h\nu }$ and ${\displaystyle nh\nu }$ are very small. In the shorter wavelengths of the ultraviolet range, however, classical theory predicts the energy emitted tends to infinity, hence the ultraviolet catastrophe. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures (see also Wien's law).

Instead, in the quantum treatment of this problem, the numbers of the energy modes are quantized, attenuating the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The modes that had more energy than the thermal energy of the substance itself were not considered, and because of quantization modes having infinitesimally little energy were excluded.

Thus for shorter wavelengths very few modes (having energy more than ${\displaystyle h\nu }$) were allowed, supporting the data that the energy emitted is reduced for wavelengths less than the wavelength of the observed peak of emission.

Notice that there are two factors responsible for the shape of the graph, which can be seen as working opposite to one another. Firstly, shorter wavelengths have a larger number of modes associated with them. This accounts for the increase in spectral radiance as one moves from the longest wavelengths towards the peak at relatively shorter wavelengths. Secondly, though, at shorter wavelengths more energy is needed to reach the threshold level to occupy each mode: the more energy needed to excite the mode, the lower the probability that this mode will be occupied. As the wavelength decreases, the probability of exciting the mode becomes exceedingly small, leading to fewer of these modes being occupied: this accounts for the decrease in spectral radiance at very short wavelengths, left of the peak. Combined, they give the characteristic graph.

Calculating the blackbody curve was a major challenge in theoretical physics during the late nineteenth century. The problem was solved in 1901 by Max Planck in the formalism now known as Planck's law of blackbody radiation. By making changes to Wien's radiation law (not to be confused with Wien's displacement law) consistent with thermodynamics and electromagnetism, he found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, which is to say that it existed in integer multiples of some quantity. Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect. These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics. These quanta were called photons and the blackbody cavity was thought of as containing a gas of photons. In addition, it led to the development of quantum probability distributions, called Fermi–Dirac statistics and Bose–Einstein statistics, each applicable to a different class of particles, fermions and bosons.

The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases monotonically with temperature. The Stefan–Boltzmann law also says that the total radiant heat energy emitted from a surface is proportional to the fourth power of its absolute temperature. The law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The formula E = σT⁴ is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and σ = 5.670367×10⁻⁸ W·m⁻²⋅K⁻⁴ is the Stefan–Boltzmann constant.

Equations

Planck's law of blackbody radiation

Main article: Planck's law

Planck's law states that $${\displaystyle B_{\nu }(T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{h\nu /kT}-1},}$$ where

• ${\displaystyle B_{\nu }(T)}$ is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ${\displaystyle \nu }$ radiation per unit frequency at thermal equilibrium at temperature ${\displaystyle T}$. Units: power / [area × solid angle × frequency].
• ${\displaystyle h}$ is the Planck constant;
• ${\displaystyle c}$ is the speed of light in vacuum;
• ${\displaystyle k}$ is the Boltzmann constant;
• ${\displaystyle \nu }$ is the frequency of the electromagnetic radiation;
• ${\displaystyle T}$ is the absolute temperature of the body.

For a black body surface, the spectral radiance density (defined per unit of area normal to the propagation) is independent of the angle ${\displaystyle \theta }$ of emission with respect to the normal. However, this means that, following Lambert's cosine law, ${\displaystyle B_{\nu }(T)\cos \theta }$ is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor ${\displaystyle 1/\cos \theta }$ with respect to an area normal to the propagation direction. At oblique angles, the solid angle spans involved do get smaller, resulting in lower aggregate intensities.

The emitted energy flux density or irradiance ${\displaystyle B_{\nu }(T,E)}$, is related to the photon flux density ${\displaystyle b_{\nu }(T,E)}$ through $${\displaystyle B_{\nu }(T,E)=E{b}_{\nu }(T,E)}$$

Wien's displacement law

Main article: Wien's displacement law

Wien's displacement law shows how the spectrum of blackbody radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency.

A consequence of Wien's displacement law is that the wavelength at which the intensity per unit wavelength of the radiation produced by a black body has a local maximum or peak, ${\displaystyle {\lambda }_{\text{peak}}}$, is a function only of the temperature: $${\displaystyle {\lambda }_{\text{peak}}=\frac{b}{T},}$$ where the constant b, known as Wien's displacement constant, is equal to ${\displaystyle \frac{hc}{k}\frac{1}{5+W_0(-5e^{-5})}\approx }$ 2.897771955×10⁻³ m K. ${\displaystyle W_0}$ is the Lambert W function. So ${\displaystyle {\lambda }_{\text{peak}}}$ is approximately 2898 μm/T, with the temperature given in kelvins. At a typical room temperature of 293 K (20 °C), the maximum intensity is at 9.9 μm.

Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by $${\displaystyle {\nu }_{\text{peak}}=T\times \mathrm{5.879...}\times {10}^{10}\mathrm{H}\mathrm{z}/\mathrm{K}.}$$ In unitless form, the maximum occurs when ${\displaystyle e^x(1-x/3)=1}$, where ${\displaystyle x=h\nu /kT}$. The approximate numerical solution is ${\displaystyle x\approx 2.82}$. At a typical room temperature of 293 K (20 °C), the maximum intensity is for ${\displaystyle \nu }$ = 17 THz.

Stefan–Boltzmann law

By integrating ${\displaystyle B_{\nu }(T)\cos (\theta )}$ over the frequency the radiance ${\displaystyle L}$ (units: power / [area × solid angle] ) is $${\displaystyle L=\frac{2{\pi }^5}{15}\frac{k^4{T}^4}{c^2{h}^3}\frac{1}{\pi }=\sigma T^4\frac{\cos (\theta )}{\pi }}$$ by using ${\displaystyle {\int }_0^{\mathrm{\infty }}dx\frac{x^3}{e^x-1}=\frac{{\pi }^4}{15}}$ with ${\displaystyle x\equiv \frac{h\nu }{kT}}$ and with ${\displaystyle \sigma \equiv \frac{2{\pi }^5}{15}\frac{k^4}{c^2{h}^3}=5.670373\times {10}^{-8}\frac{\mathrm{W}}{{\mathrm{m}}^2{\mathrm{K}}^4}}$ being the Stefan–Boltzmann constant.

On a side note, at a distance d, the intensity ${\displaystyle dI}$ per area ${\displaystyle dA}$ of radiating surface is the useful expression $${\displaystyle dI=\sigma T^4\frac{\cos \theta }{\pi d^2}dA}$$ when the receiving surface is perpendicular to the radiation.

By subsequently integrating ${\displaystyle L}$ over the solid angle ${\displaystyle \mathrm{\Omega }}$ for all azimuthal angle (0 to ${\displaystyle 2\pi }$) and polar angle ${\displaystyle \theta }$ from 0 to ${\displaystyle \pi /2}$, we arrive at the Stefan–Boltzmann law: the power j* emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature: $${\displaystyle j^{\star }=\sigma T^4,}$$ We used $${\displaystyle \int \cos \theta d\mathrm{\Omega }={\int }_0^{2\pi }{\int }_0^{\pi /2}\cos \theta \sin \theta d\theta d\varphi =\pi .}$$

Applications

Human-body emission

Much of a person's energy is radiated away in the form of long-wave infrared (LWIR) light. Some materials are transparent in the infrared, but opaque to visible light, as is the plastic bag in this thermal (LWIR) camera image (bottom). Other materials are transparent to visible light, but opaque or reflective in the infrared, noticeable by the darkness of the man's glasses.

The human body radiates energy as infrared light. The net power radiated is the difference between the power emitted and the power absorbed: $${\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}.}$$ Applying the Stefan–Boltzmann law, $${\displaystyle P_{\text{net}}=A\sigma \epsilon \left(T^4-T_0^4\right),}$$ where A and T are the body surface area and temperature, ${\displaystyle \epsilon }$ is the emissivity, and T₀ is the ambient temperature.

The total surface area of an adult is about 2 m², and the mid- and far-infrared emissivity of skin and most clothing is near unity, as it is for most nonmetallic surfaces. Skin temperature is about 33 °C, but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C. Hence, the net radiative heat loss is about $${\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}=100\mathrm{W}.}$$ The total energy radiated in one day is about 8 MJ, or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m²·h), which is equivalent to 1700 kcal per day, assuming the same 2 m² area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.

There are other important thermal loss mechanisms, including convection and evaporation. Conduction is negligible – the Nusselt number is much greater than unity. Evaporation by perspiration is only required if radiation and convection are insufficient to maintain a steady-state temperature (but evaporation from the lungs occurs regardless). Free-convection rates are comparable, albeit somewhat lower, than radiative rates. Thus, radiation accounts for about two-thirds of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal-loss mechanism.

Application of Wien's law to human-body emission results in a peak wavelength of $${\displaystyle {\lambda }_{\text{peak}}=\frac{2.898\times {10}^{-3}\mathrm{K}\cdot \mathrm{m}}{305\mathrm{K}}=9.50\mu \mathrm{m}.}$$ For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micrometer range.

Temperature relation between a planet and its star

Main article: Planetary equilibrium temperature

The blackbody law may be used to estimate the temperature of a planet orbiting the Sun.

Earth's longwave thermal radiation intensity, from clouds, atmosphere and ground

The temperature of a planet depends on several factors:

• Incident radiation from its star
• Emitted radiation of the planet (for example, Earth's infrared glow)
• The albedo effect causing a fraction of light to be reflected by the planet
• The greenhouse effect for planets with an atmosphere
• Energy generated internally by a planet itself due to radioactive decay, tidal heating, and adiabatic contraction due to cooling.

The analysis only considers the Sun's heat for a planet in a Solar System.

The Stefan–Boltzmann law gives the total power (energy/second) that the Sun emits:

The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere.

$${\displaystyle P_{\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{t}}=4\pi R_{\mathrm{S}}^2\sigma T_{\mathrm{S}}^4}$$

1

where

• ${\displaystyle \sigma }$ is the Stefan–Boltzmann constant,
• ${\displaystyle T_{\mathrm{S}}}$ is the effective temperature of the Sun, and
• ${\displaystyle R_{\mathrm{S}}}$ is the radius of the Sun.

The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is:

$${\displaystyle P_{\mathrm{S}\mathrm{E}}=P_{\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{t}}\left(\frac{\pi R_{\mathrm{E}}^2}{4\pi D^2}\right)}$$

2

where

• ${\displaystyle R_{\mathrm{E}}}$ is the radius of the planet, and
• ${\displaystyle D}$ is the distance between the Sun and the planet.

Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction ${\displaystyle \alpha }$ of this energy where ${\displaystyle \alpha }$ is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction ${\displaystyle 1-\alpha }$ of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:

$${\displaystyle P_{\mathrm{a}\mathrm{b}\mathrm{s}}=(1-\alpha )P_{\mathrm{S}\mathrm{E}}}$$

3

Even though the planet only absorbs as a circular area ${\displaystyle \pi R^2}$, it emits in all directions; the spherical surface area being ${\displaystyle 4\pi R^2}$. If the planet were a perfect black body, it would emit according to the Stefan–Boltzmann law

$${\displaystyle P_{\mathrm{e}\mathrm{m}\mathrm{t}\mathrm{b}\mathrm{b}}=4\pi R_{\mathrm{E}}^2\sigma T_{\mathrm{E}}^4}$$

4

where ${\displaystyle T_{\mathrm{E}}}$ is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting ${\displaystyle P_{\mathrm{a}\mathrm{b}\mathrm{s}}=P_{\mathrm{e}\mathrm{m}\mathrm{t}\mathrm{b}\mathrm{b}}}$, is known as the effective temperature. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the Sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits ${\displaystyle \overline{ϵ}}$ of the radiation that a black body would emit where ${\displaystyle \overline{ϵ}}$ is the average emissivity in the IR range. The power emitted by the planet is then:

$${\displaystyle P_{\mathrm{e}\mathrm{m}\mathrm{t}}=\overline{ϵ}{P}_{\mathrm{e}\mathrm{m}\mathrm{t}\mathrm{b}\mathrm{b}}}$$

5

For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:

$${\displaystyle P_{\mathrm{a}\mathrm{b}\mathrm{s}}=P_{\mathrm{e}\mathrm{m}\mathrm{t}}}$$

6

Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, T_P:

$${\displaystyle T_P=T_S\sqrt{\frac{R_S\sqrt{\frac{1-\alpha }{\overline{\epsilon }}}}{2D}}}$$

7

In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet.

Notice that a gray (flat spectrum) ball where ${\displaystyle (1-\alpha )=\overline{\epsilon }}$ comes to the same temperature as a black body no matter how dark or light gray.

Effective temperature of Earth

Substituting the measured values for the Sun and Earth yields:

• ${\displaystyle T_{\mathrm{S}}=5772\mathrm{K},}$
• ${\displaystyle R_{\mathrm{S}}=6.957\times {10}^8\mathrm{m},}$
• ${\displaystyle D=1.496\times {10}^{11}\mathrm{m},}$
• ${\displaystyle \alpha =0.309}$

With the average emissivity ${\displaystyle \overline{\epsilon }}$ set to unity, the effective temperature of the Earth is: $${\displaystyle T_{\mathrm{E}}=254.356\mathrm{K}}$$ or −18.8 °C.

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums). The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the Moon as a good estimate. The albedo and emissivity of the Moon are about 0.1054 and 0.95 respectively, yielding an estimated temperature of about 1.36 °C.

Estimates of the Earth's average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the solar constant (total insolation power density) rather than the temperature, size, and distance of the Sun. For example, using 0.4 for albedo, and an insolation of 1400 W m⁻², one obtains an effective temperature of about 245 K. Similarly using albedo 0.3 and solar constant of 1372 W m⁻², one obtains an effective temperature of 255 K.

Cosmology

The cosmic microwave background radiation observed today is the most perfect blackbody radiation ever observed in nature, with a temperature of about 2.7 K. It is a "snapshot" of the radiation at the time of decoupling between matter and radiation in the early universe. Prior to this time, most matter in the universe was in the form of an ionized plasma in thermal, though not full thermodynamic, equilibrium with radiation.

According to Kondepudi and Prigogine, at very high temperatures (above 10¹⁰ K; such temperatures existed in the very early universe), where the thermal motion separates protons and neutrons in spite of the strong nuclear forces, electron-positron pairs appear and disappear spontaneously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation.

A black body at room temperature (23 °C (296 K; 73 °F)) radiates mostly in the infrared spectrum, which cannot be perceived by the human eye, but can be sensed by some reptiles. As the object increases in temperature to about 500 °C (773 K; 932 °F), the emission spectrum gets stronger and extends into the human visual range, and the object appears dull red. As its temperature increases further, it emits more and more orange, yellow, green, and then blue light (and ultimately beyond violet, ultraviolet).

Light bulb

Tungsten filament lights have a continuous black body spectrum with a cooler colour temperature, around 2,700 K (2,430 °C; 4,400 °F), which also emits considerable energy in the infrared range. Modern-day fluorescent and LED lights, which are more efficient, do not have a continuous black body emission spectrum, rather emitting directly, or using combinations of phosphors that emit multiple narrow spectrums.

The color (chromaticity) of blackbody radiation scales inversely with the temperature of the black body; the locus of such colors, shown here in CIE 1931 x,y space, is known as the Planckian locus.

History

In query 6 of Isaac Newton's Opticks, he states that "Do not black Bodies conceive heat more easily from Light than those of other Colours do, by reason that the Light falling on them is not reflected outwards, but enters into the Bodies, and is often reflected and refracted within them, until it be stifled and lost?", thereby introducing the notion of a black body. In his first memoir, Augustin-Jean Fresnel (1788–1827) responded to a view he extracted from a French translation of Newton's Opticks. He says that Newton imagined particles of light traversing space uninhibited by the caloric medium filling it, and refutes this view (never actually held by Newton) by saying that a black body under illumination would increase indefinitely in heat.

Balfour Stewart

In 1858, Balfour Stewart described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature. Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost and of John Leslie. He wrote, "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." Stewart's statement assumed a general principle: that there exists a body or surface that has the greatest possible absorbing and radiative power for every wavelength and equilibrium temperature.

Stewart was concerned with selective thermal radiation, which he investigated using plates which selectively radiated and absorbed different wavelengths. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-Helmholtz reciprocity principle. His research did not consider that properties of rays are dependent on wavelength, and he did not use tools such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared.

Gustav Kirchhoff

In 1859, Gustav Robert Kirchhoff reported the coincidence of the wavelengths of spectrally resolved lines of absorption and emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber.

Kirchhoff then went on to consider some bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at a temperature T.

Here is used a notation different from Kirchhoff's. Here, the emitting power E(T, i) denotes a dimensioned quantity, the total radiation emitted by a body labeled by index i at temperature T. The total absorption ratio a(T, i) of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature T . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio E(T, i) / a(T, i) of emitting power to absorptivity is a dimensioned quantity, with the dimensions of emitting power, because a(T, i) is dimensionless. Also here the wavelength-specific emitting power of the body at temperature T is denoted by E(λ, T, i) and the wavelength-specific absorption ratio by a(λ, T, i) . Again, the ratio E(λ, T, i) / a(λ, T, i) of emitting power to absorptivity is a dimensioned quantity, with the dimensions of emitting power.

In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers. His theoretical proof was and still is considered by some writers to be invalid. His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorptivity has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) has one and the same value for all bodies. In this report there was no mention of black bodies.

In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio E(T, i) / a(T, i) has one and the same value common to all bodies. Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.

But more importantly, it relied on a new theoretical postulate of "perfectly black bodies," which is the reason why one speaks of Kirchhoff's law. Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewart's reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Planck's black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.

Kirchhoff's proof considered an arbitrary non-ideal body labeled i as well as various perfect black bodies labeled BB. It required that the bodies be kept in a cavity in thermal equilibrium at temperature T. His proof intended to show that the ratio E(λ, T, i) / a(λ, T, i) was independent of the nature i of the non-ideal body, however partly transparent or partly reflective it was.

His proof first argued that for wavelength λ and at temperature T, at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power E(λ, T, BB), with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorptivity a(λ, T, BB) of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorptivity E(λ, T, BB) / a(λ, T, BB) is again just E(λ, T, BB), with the dimensions of power. Kirchhoff considered thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature T. He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio E(λ, T, i) / a(λ, T, i) was equal to E(λ, T, BB), which may now be denoted B_λ (λ, T). B_λ (λ, T) is a continuous function, dependent only on λ at fixed temperature T, and an increasing function of T at fixed wavelength λ. It vanishes at low temperatures for visible wavelengths, which does not depend on the nature i of the arbitrary non-ideal body (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing).

Thus Kirchhoff's law of thermal radiation can be stated: For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature T, for every wavelength λ, the ratio of emissive power to absorptivity has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by B_λ (λ, T). (For our notation B_λ (λ, T), Kirchhoff's original notation was simply e.)

Kirchhoff announced that the determination of the function B_λ (λ, T) was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. Occasionally by historians that function B_λ (λ, T) has been called "Kirchhoff's (emission, universal) function," though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later. Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so. Kirchhoff's 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established. In a more considered account in a book in 1862, Kirchhoff mentioned the connection of his law with Carnot's principle, which is a form of the second law.

According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the "blackbody" radiation that Robert Kirchhoff had first defined in 1859–1860."

Doppler effect

The relativistic Doppler effect causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f': $${\displaystyle f^{\prime }=f\frac{1-\frac{v}{c}\cos \theta }{\sqrt{1-v^2/c^2}},}$$ where v is the velocity of the source in the observer's rest frame, θ is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and c is the speed of light. This can be simplified for the special cases of objects moving directly towards (θ = π) or away (θ = 0) from the observer, and for speeds much less than c.

Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation.

For the case of a source moving directly towards or away from the observer, this reduces to $${\displaystyle T^{\prime }=T\sqrt{\frac{c-v}{c+v}}.}$$ Here v > 0 indicates a receding source, and v < 0 indicates an approaching source.

This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation, which exhibits a dipole anisotropy from the Earth's motion relative to this blackbody radiation field.