Thermal Radiation

[12.1] Thermal Radiation

Thermal radiation corresponds to the conversion of thermal energy into electromagnetic wave energy. All matter above the absolute zero temperature undergo thermal movements, causing the charge particles in atoms to undergo acceleration. Accelerated charge particles emit electromagnetic radiation. All physical substances in solid, liquid, or gaseous states can emit energy through this process.

[12.1.1] Spectral Distribution of Radiation Energy

The intensity of energy emitted or absorbed from a warm object is seen to distribute over all possible values of wavelengths (λ) resulting a continuous graph for the spectral distribution of energy as a function of wavelength. The curve at a specific temperature corresponds to a specific peak wavelength whose value decreases when the temperature of the body increases.

[12.2] Black body: A Model

The concept of a black body introduced in 1860 by the German physicist Gustav Kirchhoff to model the spectral distribution of energy of a warm object. The name black body was used to signify that around room temperature most emitted energy occurs within the infrared (IR) region of the electromagnetic spectrum (see the graph), and therefore, inside a dark room the object will appear black (or invisible to human eye). A perfect or ideal black body in thermodynamic equilibrium absorbs all electromagnetic radiation that strikes it. This property is valid for radiation corresponding to all wavelengths and to all angles of incident. Kirchhoff used the perfect black body as a standard to compare the absorption of real objects, and to dene basic properties pertaining to objects undergoing thermal radiation.

Absorptive Power (Absorptivity)

The absorptive power of a body (or a surface) is defined as the ratio of the energy absorbed in a given time (or in a certain time) to the radiant energy incident on it at the same instant of time. It is a measure of heat absorbed by an object.

When heat is incident on the surface an object, a part of it is absorbed and the remaining part is reflected. The absorptive power of a black body is 1 because it absorbs the radiant energy of all wavelength incidents on it. A good absorber is a good radiator. Usually, the absorbed radiation is converted to thermal energy, increasing the object’s temperature. A Black body, for instance, absorbs all incident radiation, and its absorptive power is 1. So, the body which has higher absorptive power has a higher emissive power also. The amount of heat absorbed by the body depends upon the nature of the body.

Absorptive power of a body is defined at a given temperature and wavelength is defined as the ratio of the amount of heat energy absorbed to the amount of heat energy incident on it in a wavelength range.

Therefore absorptive power, a(λ) = Amount of energy absorbed / Amount of energy incident

Emissive Power (Emittance)

Emissivity, ε

[12.2.1] Kirchhoff 's Radiation Law, 1860

Kirchhoff recognized that at thermal equilibrium an object absorbs and emits equal amounts of radiation at any given time. This condition implies that any object must be equally good in their ability to emit and absorb radiation. Kirchhoff's radiation law states that for an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity, or that, α(λ) = ε(λ). (true for any given wavelength at thermal equilibrium)

[12.3] Stephan-Boltzmann Law

In 1879 Austrian physicist Josef Stefan established that the total radiation power P per unit surface area of a black body is proportional to the fourth power of the black body's absolute temperature. Stefan based his derivation on the experimental measurements recorded in 1864 by the Irish physicist John Tyndall on infrared emission by a platinum lament and the corresponding color of the lament. The result was theoretically derived in 1884 by Ludwig Boltzmann.

[12.4] Constructing a Black body: Kirchhoff 's Challenge

Theoretical model of the thermal radiation developed in relation to an ideal black body. Therefore, the concept of an ideal black body plays an important role in understanding thermal radiation. As it was not possible to nd a naturally occurring perfect black body such an entity had to be constructed for the purpose of studying. In 1959 Kirchhoff had implied that an enclosed volume maintained at temperature T without allowing external rays to enter the volume is equivalent to a black body in its behavior.

The practical realization of this idea was a large cavity or an enclosed volume with a small hole. The walls of the enclosure is maintained at a particular temperature T. Radiation can enter and leave the enclosed volume only through the hole. Since the size of the hole is much small compared to the volume of the cavity, radiation entering or leaving through the cavity is a rare event. Therefore, a certain amount of radiation entering or leaving the enclosure does not disturb the thermal equilibrium occurs inside the enclosure and its walls. Once entered the volume, radiation will remain within the volume undergoing multiple reflections at the walls of the cavity. The hole would absorb any radiation that falls upon it. The radiation that leaves the cavity through the hole is as much as the radiation that enters through the hole. Therefore the holes acts as a perfect black body at a particular temperature T.

[12.5] Wien's Displacement Law, 1893

Wien's displacement law describes one of the relations between the emission spectrum of a black body and its temperature. It states that the higher the temperature, the lower the wavelength λ max for which the radiation curve reaches its maximum. The shift to shorter wavelengths corresponds to photons of higher energies.

[12.6] Raleigh-Jean Formula, 1900: Ultraviolet Catastrophe

The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation in all frequency ranges, emitting more energy as the frequency increases. By calculating the total amount of radiated energy (i.e., the sum of emissions in all frequency ranges), it can be shown that a black body is likely to release an arbitrarily high amount of energy. This would cause all matter to instantaneously radiate all of its energy until it is near absolute zero – indicating that a new model for the behaviour of black bodies was needed.

The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 statistical derivation of the Rayleigh–Jeans law. The phrase refers to the fact that the Rayleigh–Jeans law accurately predicts experimental results at radiative frequencies below 10⁵ GHz, but begins to diverge with empirical observations as these frequencies reach the ultraviolet region of the electromagnetic spectrum. Since the first appearance of the term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.

According to the Rayleigh-Jeans' formula for the energy density within a unit wavelength range around the wavelength λ is given by,

In terms of frequency f, the Rayleigh-Jean formula for spectral energy density of black body radiation takes the form given by,

Rayleigh-Jeans' formula agrees with the black body radiation data for high wavelength (low frequency) values. However it deviated badly at short wavelengths. This problem for small wavelengths became known as the ultraviolet catastrophe and was one of the outstanding exceptions that classical physics could not explain.

[12.7] Plank's Radiation Law, 1900

In later 1900, the German physicist Max Planck used the idea that atoms and molecules in a body act like oscillators to absorb and emit radiation. Plank realized that the energies of the oscillating atoms and molecules had to be quantized to correctly describe the shape of the blackbody spectrum. Therefore, Planck hypothesized that the energy of an oscillator having a frequency f is given by:

Using this hypothesis Plank was able to derive an alternative formula for the energy density of black body radiation .This result became known as Plank's radiation law. Plank radiation law could accurately predict the intensity values of the black body radiation that agreed with the experimentally measured values.

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