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1 Blackbody radiation derivation of Planck‘s radiation low

Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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Page 1: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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Blackbody radiation derivation of

Planck‘s radiation low

Page 2: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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Classical theories of Lorentz and Debye:

• Lorentz (oscillator model):

– Electrons and ions of matter were treated as a simple harmonic oscillators (springs) subject to the driving force of applied E-M fields; matter becomes polarized by induction of electric dipoles

• It models optical properties of materials and provides theory of refraction, reflectance and absorption

• Analogy between the classical and the quantum-mechanical descriptions:– E.g., excitation frequency vs. resonance frequency, probabilities of transition to all other

quantum states vs. damping factors

• Debye(relaxation model):

– The E-M field causes polarization of matter containing permanent electric dipoles leading to the partial alignment of the dipoles along the electric field against the counteracting tendency toward disorientation caused by thermal buffeting. The restoring force tries to return a polarized region to an unpolarized state is thus the statistical tendency toward random orientation of the dipoles; the dipole restoring tendency leads to oscillation of the electric polarization

• It models the optical constants of liquids at certain frequencies

Page 3: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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Laws of radiation

• Kirchoffs law of thermal radiation (1860)

At thermal equilibrium, the emissivity of a body (or surface) equals its absorptivity.

Introduction of black body radiation concept;

A black body is an object that absorbs all light that falls on it ( no light is reflected or transmitted). The object appears black when it is relatively cold.

Page 4: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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Wien's law

- It accurately describes the spectrum of thermal radiation from objects at the short wavelength

-the hotter an object is, the shorter the wavelength at which It will emit most of its radiation,

-the frequency for maximal intensity or peak power radiation

λ is the peak wavelength in metersT is the temperature of the blackbody in Kelvins (K)B is a constant of proportionality, called Wien's displacement

constant and equals 2.897 768 5(51) × 10–3 m K

Rayleigh-Jeans, Wien and Planck law‘s for a body of 8 mK

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• Rayleigh–Jeans Lawit describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at a given temperature:

c is the speed of light, k is Boltzmann's constantT is the temperature in kelvins.

It predicts an energy output that diverges towards infinity as wavelengths grow smaller. This was not supported by experiments and the failure has become known as the ultraviolet catastrophe.

Page 6: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths.

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Derivation of Planck‘s radiation law

• Assumptions:

A cavity in a material that is maintained at constant temperature T

The emission of radiation from the cavity walls is in equilibrium with the radiation that is absorbed by the walls

The radiation field in an empty volume in thermal equilibrium with a container at T can be viewed as a superposition of standing harmonic waves

M. Planck, Ann. Phys. Vol. 4, p.553 (1901)

• Blackbody cavity: schematic

T, ϖi

Page 8: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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Practical cavity: example

Page 9: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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• The radiation field in an empty volume (V=L3) is in thermal equilibrium with container at temperature T.

This can be viewed as a superposition of standing harmonic waves (oscillators modes).

Radiation field in an empty volume

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The mode density

The waves are solutions of the wave equation:

Taking into accountthe boundary conditions:

The solutions of equation have the form:

with sin (ki L) = 0 from which follows ki L = ni for i = 1; 2; 3 and ni = 1; 2; …

**

*

Page 11: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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Inserting equations ** in * one obtains:

From which follows:

Page 12: Blackbody radiation derivation of Planck‘s radiation lowtulej/Spectroscopy... · • Kirchoffs law of thermal radiation (1860) At thermal equilibrium, the emissivity of a body (or

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The number of modes with angular frequency between 0 and ω

is

where represents the number density of oscillators which

corresponds to all combinations of n1 ,n2 and n3 which fulfill the equation:

Equation*** is the equation of a sphere (see Fig. 2) with radius:

***

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Diagram of the possible (n1 , n2 , n3 ) combinations

n1 ,n2 ,n3 are positive and therefore

Since ω

= 2πν, one obtains from Equation ****

****

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The mode density for a given polarization is therefore

Since two polarization directions have to be considered for each mode, the mode density is twice larger than given above and amounts to:

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The energy density

Each mode has an energy kT and the energy of the radiation field in a volume V at temperature T and between frequency ν

and ν+νdν

is:

The energy density (i.e., the energy per unit volume) is therefore given by

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The energy density (i.e., the energy per unit volume) is therefore given by

a relation known as Rayleigh-Jeans' law. The expression is valid for but incorrect for , as it predicts that in this case the energy density should become infinite. This physically incorrect property of the equation would lead to what has been termed \UV catastrophy".

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Planck's radiation law is derived by assuming that each radiation mode can be described by a quantized harmonic oscillator with energy

Referencing the energy of each oscillator to the ground state (v = 0) of the oscillators:

one can determine the average energy of an oscillator using statistical mechanics:

with vhν

the energy of the oscillator and the probability ( >=1) that)the oscillator has the energy vhν

(Boltzmann factor)

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Making the substitution

The energy density is the product of the mode density per unit volume and the average energy of the modes and is therefore given by:

Planck's law for the energy density of the radiation eld (M. Planck, Verh. Deutsch. Phys. Ges. 2, 202 and 237 (1900)

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The total energy UT V per unit volume is

The heat capacity of free space can be determined to be:

At high T, UTV becomes very large and eventually suficiently large that

electron-positron pairs can be formed, at which point vacuum fills with matter.

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Applications

• Temperature measurements of astrophysical objects

• Cosmic microwave background radiation

• Color temperatureis a characteristic of visible light that has important applications in photography, videography, publishing and other fields. The color temperature of a light source is determined by comparing its chromaticity with a theoretical, heated black-body radiator. The temperature (in kelvin) at which the heated black-body radiator matches the color of the light source is that source's color temperature

• Infrared thermometers measure temperature using blackbody radiation (generally infrared) emitted from objects. They are sometimes called laser thermometers if a laser is used to help aim the thermometer, or non-contact thermometers to describe the device’s ability to measure temperature from a distance. By knowing the amount of infrared energy emitted by the object and its emissivity, the object's temperature can be determined.

• Combustion: laser induced incandescence for soot particle measurements, two-color

pirometry