Black body spectrum

In physics, Planck's law of black body radiation predicts the spectral intensity of electromagnetic radiation at all wavelengths from a black body at temperature  : Image File history File links Wiens_law. ... Image File history File links Wiens_law. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ...

where the following table provides the definition and SI units of measure for each symbol: Cover of brochure The International System of Units. ...

Symbol	Meaning	SI units of measure
	spectral radiance, energy per unit time per unit surface area per unit solid angle per unit frequency	J•s-1•m-2•sr-1•Hz-1
	frequency	hertz
	temperature of the black body	kelvin
	Planck's constant	joule per hertz
	speed of light	meter per second
	base of the natural logarithm, 2.718281...	dimensionless
	Boltzmann's constant	joule per kelvin

The wavelength is related to the frequency by Radiance and spectral radiance are radiometric measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. ... A pocket watch, a device used to measure time Two distinct views exist on the meaning of time. ... An open surface with X-, Y-, and Z-contours shown. ... A solid angle is the three dimensional analog of the ordinary angle. ... Sine waves of various frequencies; the bottom waves have higher frequencies than those above. ... Sine waves of various frequencies; the bottom waves have higher frequencies than those above. ... The hertz (symbol: Hz) is the SI unit of frequency. ... Fig. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... The joule (symbol: J) is the SI (metric) unit of energy, which is defined as the potential to do work. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation in a vacuum, not just visible light. ... Metre per second (U.S. spelling: meter per second) is an SI derived unit of both speed (scalar) and velocity (vector), defined by distance in metres divided by time in seconds. ... e is the unique number such that the value of the derivative (slope of a tangent line) of f (x)=ex for any value of x is equal to the value of f (x). ... In the physical sciences, a dimensionless number (or more precisely, a number with the dimensions of 1) is a quantity which describes a certain physical system and which is a pure number without any physical units; it does not change if one alters ones system of units of measurement... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... The wavelength is the distance between repeating units of a wave pattern. ...

The law is sometimes written in terms of the spectral energy density Energy density is the amount of energy stored in a given system or region of space per unit volume or per unit mass, depending on the context. ...

which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz). Integrated over frequency, this expression yields the total energy density. The radiation field of a black body may be thought of as a photon gas, in which case this energy density would be one of the thermodynamic parameters of that gas. Volume is how much space a thing has. ... Sine waves of various frequencies; the bottom waves have higher frequencies than those above. ... In physics, a photon gas is a gas-like collection of photons, which come together to form something that has the properties of a conventional gas like hydrogen or neon - including pressure, and temperature. ...

The spectral energy density can also be expressed as a function of wavelength:

as shown in the derivation below.

Max Planck originally produced this law in 1900 (published in 1901) in an attempt to improve upon an expression proposed by Wilhelm Wien which fit the experimental data at short wavelengths but deviated from it at long wavelengths. He found that the above function, Planck's function, fit the data for all wavelengths remarkably well. In constructing a derivation of this law, he considered the possible ways of distributing electromagnetic energy over the different modes of charged oscillators in matter. Planck's law emerged when he assumed that the energy of these oscillators was limited to a set of discrete, integer multiples of a fundamental unit of energy, E, proportional to the oscillation frequency ν: Max Karl Ernst Ludwig Planck (April 23, 1858 – October 4, 1947) was a German physicist. ... 1900 (MCM) was an exceptional common year starting on Monday of the Gregorian calendar, but a leap year starting on Saturday of the Julian calendar. ... 1901 (MCMI) was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 13-day-slower Julian calendar). ... Wilhelm Carl Werner Otto Fritz Franz Wien (January 13, 1864 – August 30, 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to compose Wiens displacement law, which relates the maximum emission of a blackbody to its temperature. ...

.

Planck made this quantization assumption five years before Albert Einstein hypothesized the existence of photons as a means of explaining the photoelectric effect. At the time, Planck believed that the quantization applied only to the tiny oscillators that were thought to exist in the walls of the cavity (what we now know to be atoms), and made no assumption that light itself propagates in discrete bundles or packets of energy. Moreover, Planck did not attribute any physical significance to this assumption, but rather believed that it was merely a mathematical device that enabled him to derive a single expression for the black body spectrum that matched the empirical data at all wavelengths. Einstein redirects here. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... The photoelectric effect is the emission of electrons from matter upon the absorption of electromagnetic radiation, such as ultraviolet radiation or x-rays. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle of a chemical element that retains its chemical properties. ...

Ultimately, Planck's assumption of energy quantization and Einstein's photon hypothesis became the fundamental basis for the later development of Quantum Mechanics. Both scientists would eventually receive (separate) Nobel prizes in recognition of these major contributions to the advancement of physics. Fig. ... Hannes Alfvén (1908–1995) accepting the Nobel Prize for his work on magnetohydrodynamics [1]. List of Nobel Prize laureates in Physics from 1901 to the present day. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ...

Derivation

(See also the gas in a box article for a general derivation.) The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

Consider a cube of side L with conducting walls filled with electromagnetic radiation. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The wavelength λ_i in the three directions orthogonal to the walls can be: In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with...

where the n_i are integers. For each set of integers n_i there are two linear independent solutions (modes). According to quantum theory, the energy levels of a mode are given by:

The quantum number r can be interpreted as the number of photons in the mode. The two modes for each set of n_i correspond to the two polarization states of the photon which has a spin of 1. Note that for r = 0 the energy of the mode is not zero. This vacuum energy of the electromagnetic field is responsible for the Casimir effect. In the following we will calculate the internal energy of the box at temperature T relative to the vacuum energy. In physics, the Casimir effect is a physical force exerted between separate objects, which is due to neither charge, gravity, nor the exchange of particles, but instead is due to resonance of all-pervasive energy fields in the intervening space between the objects. ...

According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by: Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

Here

.

The denominator , is the partition function of a single mode and makes P_r properly normalized: In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

Here we have defined

which is the energy of a single photon. As explained here, the average energy in a mode can be expressed in terms of the partition function: In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

This formula is a special case of the general formula for particles obeying Bose-Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero. In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist (Willard Gibbs and his partner Lauren Berkley), which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given...

The total energy in the box now follows by summing over all allowed single photon states. This can be done exactly in the thermodynamic limit . In this limit, becomes continuous and we can then integrate over this parameter. To calculate the energy in the box in this way, we need to evaluate how many photon states there are in a given energy range. If we write the total number of single photon states with energies between and as , where is the density of states which we'll evaluate in a moment, then we can write: Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. ...

To calculate the density of states we rewrite equation (1) as follows:

where n is the norm of the vector :

For every vector n with integer components larger or equal than zero there are two photon states. This means that the number of photon states in a certain region of n-space is twice the volume of that region. An energy range of corresponds to shell of thickness in n-space. Because the components of have to be positive, this shell spans an octant of a sphere. The number of photon states in an energy range is thus given by:

Inserting this in Eq. (2) gives:

From this equation one easily derives the spectral energy density as a function of frequency u(ν,T) and as a function of wavelength u(λ,T):

where:

u(ν,T) is known as the black body spectrum. It is a spectral energy density function with units of energy per unit frequency per unit volume.

And:

where

This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithms. In this case, however, it is possible to calculate the integral in closed form. Let's first make the integration variable in Eq. (3) dimensionless by substituting : The polylogarithm (also known as de Jonquières function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ...

where J is given by:

We prove this result in the Appendix below. The total electromagnetic energy inside the box is thus given by:

where V = L³ is the volume of the box. (Note - This is not the Stefan-Boltzmann law, which is the total energy radiated by a black body. See that article for an explanation.) Since the radiation is the same in all directions, and propagates at the speed of light (c), the spectral intensity (energy/time/area/solid angle/frequency) is The Stefan-Boltzmann law, also known as Stefans law, states that the total energy radiated per unit surface area of a black body in unit time (known variously as the black-body irradiance, energy flux density, radiant flux, or the emissive power), j*, is directly proportional to the fourth...

which yields

The use of Stirling's formula in the theory of black body radiation

Interestingly, Planck, followed by some other early twentieth-century writers on the black-body radiation such as Debye and de Broglie, used the "first order" Stirling's approximation in a rather unusual way. Instead of using the expression the writers, who all were excellent physicists, preferred either to write it as n! = nⁿ or leave some components out immediately below. Two other eminent physicists, Einstein and Bose each took a different approach in their dealings with the problem and used other approximations. Although the left side of the equation n! = nⁿ is obviously not equal to its right side, which would lead to a gross error in simple calculation (for n=1, 1=1; for n=2, 2=4; for n=3, 6=27 etc., which does not seem reasonable), the substitution n! = nⁿ should be considered in terms of the mathematical operations that followed. Peter Joseph William Debye (March 24, 1884 - November 2, 1966) (born Petrus Josephus Wilhelmus Debije) was a Dutch physical chemist. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... The relative difference between (ln x!) and (x ln x - x) approaches zero as x increases. ... Einstein redirects here. ... Satyendra Nath Bose on an Indian stamp Satyendra Nath Bose /sɐθ.jin. ...

For Planck, they were: (1) n=n+1, (2) substitution to the fraction, (3) neglecting certain terms, (4) differentiation, and (5) integration. Planck's approximation resulted in a kind of "approximation of shape". The relationship between the idea of "energy elements" and the curve that was verified by measurement was demonstrated, but the elements were not separated. In fact the total of the approximations provided a far greater number of "complexions", as Planck called the system states, than the number calculated from the idea of completely separated system elements. This implied some extra microstates or interactions between the elements. The approximation was strongly nonlinear for a small number of elements. The same applies to the derivation of the photoelectric equation by Albert Einstein (the 'light quanta' were not separated). It has been suggested that this article or section be merged with estimation. ... The relative difference between (ln x!) and (x ln x - x) approaches zero as x increases. ...

Actually, none of the papers in question was devoted to the radiation of microsystems and that was perhaps why the writers chose not to give the reason for the approximation. One can only speculate that it could have been equivalent either to a kind of distant interaction being a joint effect of some variables or phenomena not yet known at the time of their writing and discovered later in quantum mechanics, such as spin or the uncertainty principle, or to some still unknown hidden variables . However, a relationship between the lack of separability implied by the early 20th-century thermodynamical analyses of the black body and the contemporary quantum entanglement theories is still missing. In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. ... Interaction is a kind of action which occurs as two or more objects have an effect upon one another. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In quantum physics, the Heisenberg uncertainty principle or the Heisenberg indeterminacy principle — the latter name given to it by Niels Bohr — states that when measuring conjugate quantities, which are pairs of observables of a single elementary particle, increasing the accuracy of the measurement of one quantity increases the uncertainty of... In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ... In science, a mathematical relationship describes how one quantity is related to another. ... Separable quantum states are those without Quantum entanglement. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... It has been suggested that Quantum coherence be merged into this article or section. ...

History

Many popular science accounts of quantum theory, as well as some physics textbooks, contain some serious errors in their discussions of the history of Planck's Law. Although these errors were pointed out over forty years ago by historians of physics, they have proved to be difficult to eradicate. An article by Helge Kragh^[1] gives a lucid account of what actually happened.

Contrary to popular opinion, Planck did not quantize light. This is evident from his original 1901 paper ^[2] and the references therein to his earlier work. It is also plainly explained in his book "Theory of Heat Radiation," where he explains that his constant refers to Hertzian oscillators. The idea of quantization was developed by others into what we now know as quantum mechanics. The next step along this road was made by Albert Einstein, who, by studying the photoelectric effect, proposed a model and equation whereby light was not only emitted but also absorbed in packets or photons. Then, in 1924, Satyendra Nath Bose developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck's law. Fig. ... Einstein redirects here. ... The photoelectric effect is the emission of electrons from matter upon the absorption of electromagnetic radiation, such as ultraviolet radiation or x-rays. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... Satyendra Nath Bose on an Indian stamp Satyendra Nath Bose /sɐθ.jin. ... The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

Contrary to another myth, Planck did not derive his law in an attempt to resolve the "ultraviolet catastrophe", the name given to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is applied to black body radiation. Planck did not consider the equipartion theorem to be universally valid, so he never noticed any sort of "catastrophe" — it was only discovered some five years later by Einstein, Lord Rayleigh, and Sir James Jeans. The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ... The equipartition theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degrees of freedom allowed to the particles of the system. ... Einstein redirects here. ... See also Rayleigh fading Rayleigh scattering Rayleigh number Rayleigh waves Rayleigh-Jeans law External links Nobel website bio of Rayleigh About John William Strutt MacTutor biography of Lord Rayleigh Categories: People stubs | 1842 births | 1919 deaths | Nobel Prize in Physics winners | Peers | British physicists | Discoverer of a chemical element ... Sir James Hopwood Jeans (born Ormskirk, September 11, 1877, died Dorking, September 16, 1946) was a British physicist, astronomer and mathematician who was the first to propose the theory of continuous creation of matter in the universe. ...

Appendix

A simple way to calculate the integral

is as follows. After multiplying the numerator and denominator of the integrand by exp( − x) we can expand the integrand in powers of exp( − x).

Here we have used that is the Riemann zeta function evaluated for the argument 4, which is given by . This fact can be proven by considering the contour integral In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In complex analysis, the evaluation of integrals of real-valued functions along intervals on the real line, is not readily found with certain integrands and methods involving only real variables. ...

Where C_R is a contour of radius R around the origin. In the limit the integral approaches zero. Using the residue theorem the integral can also be written as a sum of residues at the poles of the integrand. The poles are at zero, the positive and negative integers. The sum of the residues yields precisely twice the desired summation plus the residue at zero. Because the integral approaches zero, the sum of all the residues must be zero. The summation must therefore equal minus one half times the residue at zero. From the series expansion of the cotangent function The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ...

we see that the residue at zero is which yields the desired result. In the appendix of the article Stefan-Boltzmann law we give a different derivation of this integral. (See also the polylogarithm article.) The Stefan-Boltzmann law, also known as Stefans law, states that the total energy radiated per unit surface area of a black body in unit time (known variously as the black-body irradiance, energy flux density, radiant flux, or the emissive power), j*, is directly proportional to the fourth... The polylogarithm (also known as de Jonquières function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ...

References

1. ^ Kragh, Helge Max Planck: The reluctant revolutionary Physics World, December 2000
2. ^ Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901) Planck's original 1901 paper.

Annalen der Physik is one of the best-known and oldest (it was founded in 1799) physics journals worldwide. ...

External links

• Radiation of a Blackbody - interactive simulation to play with Planck's law
• Scienceworld entry on the Planck Law
• Interactive Blackbody Radiance Calculator - Includes all common units and an in-depth description of formulas