Thermal Radiation, Black Body Theory and the Birth of Quantum Physics

Introduction

The transfer of heat at a distance was known as thermal radiation long before Kirchoff's times, and explained why a fire could transfer heat at a distance even at ambient temperature lower than 0º Celsius. By 1860, it was accepted that heat radiation, as well as visible ligth, travelled across the ether as waves in this fluid that permeated the whole space, in the same form that sound waves did through the air. 

Only after that Maxwell proved in 1865 that electromagnetic waves, propagating at the speed of light c, were the carriers for thermal radiation, visible light and other types (UV, X rays and higher frequencies), scientific theories appeared embracing this new field of physics. Maxwell also proved that moving charges (oscillators) were the responsibles for producing these electromagnetic waves, which verified the Euler's wave equation for three dimensions, and propagated as orthogonal electrical and magnetic oscillations, transversal to the direction of the wave. The EM waves also verified the relationship c = 𝜆f, as in sound waves. 

The following graphic represent the ranges of wavelenghts in the electromagnetic spectrum, and goes from radio up to gamma frequencies, generated in nuclear reactions and in the outer space.

 

Thermal radiation, at Earth's surface and in normal enviroments, is conducted almost exclusively by infrared and visible light. Only in particular places, near radar installations and strong radio transmitters, microwaves and lower frequencies have an aditional impact (which can be dangerous to living beings).

Kirchoff's studied the transfer of heat and the behavior of opaque and black bodies using the old concept of thermal radiation, which didn't invalidated his work and theories due to the affinity with Maxwell's theory.

Besides his work on emissivity and absorptivity of thermal radiation by opaque and black bodies, Kirchoff left a challenge to others: to find the spectral radiation density equation of a perfect black body cavity, with an absorptive power of 100% of the incident thermal radiation. He explicitly said that such a function J(λ,T) had only two variables: the wavelength of the thermal radiation and the absolute temperature of the cavity, in thermal equilibrium.

It took about 40 years until Max Planck found the correct solution to this theorem, what also the led to find that this radiation was composed of discrete "quantum of action", creating the basis for the development of the Quantum Physics. Also, this led to the acceptance that ligth had a dual behavior, as a wave and also as a particle, depending on the practical application about emission, absorption or transmission.

Kirchoff, Maxwell, Boltzmann, Stefan, Hertz, Lummer, Kurlbaum, Wein, Rayleigh, and Planck are just the most known within a number of contributors in thermodynamics and electromagnetism.

Definitions and Properties of a Black Body

A perfect black body has to verify the Stefan-Boltzmann law for the total radiated energy and the Planck's law for spectral intensity radiance per unit of wavelength or frequency. It can be an arbitrary surface or a cavity, at which the total absorbed energy equals to the total emitted energy, so it verifies that

e_𝜆  =  a_𝜆

A perfect black body is defined as an object made by materials that allow them to absorb all of the incident thermal radiation at every wavelenght it contains, and to emit equal amount of it. A perfect black body does not allow transmissivity of energy (passing through), difraction or reflection of incoming energy. For cavities, a small hole is made to allow measurements of the radiation, as it does not disturb the internal equilibrium. 

Under thermal equilibrium of the system, a black body emits radiation with a Planck's spectrum, which has frequency components only function of the absolute temperature of the body (ºKelvin).

The next graphic shows that, at temperatures of 5000 ºK (near the Sun's surface temperature), the emitted radiation peaks in the visible range (visible light amount almost 50% of the energy received by the Earth). The temperature for the generation of X-rays and gamma rays in substantial amounts is above 50000 ºK and up to millions of ºK, and is related to processes in nuclear reactions, high energy particles and in the field of astrophysics, which are not being contemplated here.

The construction of black bodies for research and industrial uses involves two aspects: the coating used and the material used under the coat, to provide density to the body. The problem is that materials that behave like black bodies (graphite) are difficult to manage when building practical devices of arbitrary dimensions. Many materials can have an absorptivity a_𝜆 close to 1 in the visible spectral range, but fails to maintain this value at infrared or ultraviolet regions. 

For instance, using polished metals (before the coating) poses the problem that energy absorption occurs at a small distance from the surface (a few microns), lefting the remaining thickness useless for our purpose. In the case of non metallic materials, the opposite happens, requiring a larger thickness to absorb the energy. At any case, both types of materials have to whitstand high temperatures without deformation or melting.

Kirchoff's postulate of a perfect black body requires a material with an infinitesimal width, not practical for the construction of the body. Instead, using metallic coating of "black" materials like platinum provide a body with a few microns of thickness and a reflectivity lower than 1% of the incident energy, like the Sun's ligth.

In the practice, currently are being used materials like: platinum black, gold black, carbon black (soot coat) and special black paints over absorbing substrates. In many cases, the coating has defects in a given range of energy, for which a patch is used for measurements in the faulty range of wavelenghts.

A black body has the maximum emissive power E(𝜆,T) at any given temperature, when is compared to any other object. 

Ideally, a perfect absorber is also a perfect emitter of thermal radiation, with a thermal equilibrium between the black body and its enclosure. In practice, this property has to be substained at temperatures reaching thousands of ºK. It is required that absorption and emission occur in a homogeneous, isotropic medium, generated by the equilibrium of the radiation that fills the cavity. The orientation and position of the body within the cavity is irrelevant if the energy emissivity is omnidirectional. 

The black body has to verify the second law of thermodynamics, which forbids an increase in radiant energy with a decrease in the temperature of the closed system. By this law, the total amount of energy (integrating the contribution at every wavelength) that a black body emits has to increase with a proportional increase in the equilibrium temperature of the system.

In practice, the reflectance of the inner surface of the cavity can be greater than zero, so the radiation that enters from the small hole is reflected continuously, losing energy by absorption until all of it is absorbed by the cavity. The cavity walls may be made black and rough to help the absorption process. Hollow spheres or cylinders with one side opened can be used as black body emiters, allowing the insertion of small bodies for measuring purposes. For measurements at high temperatures, ceramic cavities are used and also patches of absorbent coatings are used to correct properties within a range of wavelengths to be employed to measure.

Color temperature measurement is derived from the black body theory and is a comparative value used in the visible range of ligth, which follows the maximum values given by Wien's law of displacement on Planck's spectral radiation law. It is known that colors at incandescence bodies follow an empirical rule of relationship with its temperature, being opaque at low or medium temperatures, dark red as temperature increases and then passing through yellow orange colors, to finally reach white colors at very high temperatures. This knowledge, using known laws of radiation provides a way to determine the temperature by color impression of the radiation of a thermal radiator. Currently, many manufacturers provide black body equipment in a wide range of values of wavelengths, temperatures and also provide several calibrators and instruments.

The following figure is a simplified schematic of a small black body cavity radiation generator. These devices are  manufactured today for scientific, industrial and military applications, with a wide range of parameters in different models, like: temperature range (up to 3000 K), wavelengths (covering a wide spectrum in bands) and different sizes and materials for the cavity. They have properties similar to that of Kirchoff's studies in the XIX century, which stimulated a decades long reseach to find the spectrum of its thermal radiation.

The following figure is a simplified drawing of a modern, compact black-body cavity based radiation generator. The Aperture Wheel provides different solid angles for spectral intensity.

Kirchoff Law of Thermal Radiation and the Black Body Theorem

From 1859 to 1862, Kirchoff worked the problem of emission and absorption of thermal radiation by gaseous matter and different objects, either opaque or black bodies. In particular, and using spectroscopy, in 1859 he discovered three basic principles of it, which allowed to understand how spectra of different types are created:

• 1.  A heated solid, liquid or gas, under high pressure, irradiates a continuous spectrum.
• 2.  A hot gas of low density (low pressure) produces a bright line or an emission line spectrum.
• 3. When a source of a continuous spectrum behind a cool gas of low density is viewed, a dark spectral line appears at the spectrum. The wavelength of the dark line  depends on the particular matter under study (absorption phenomena, explained about 50 years later with Bohr's model of atoms).

Kirchoff worked on the thermal behavior of bodies and cavities, finding that real bodies radiate and absorb thermal radiation with lower efficiency than black bodies, and studied properties of emissivity, absorptivity, reflection, difraction and transmissivity of different materials, using spectral analysis at different wavelengths and temperatures. His work had a wider prospect than the theories of the english physicist Stewart, who was working on the same field  by 1858, and whose work Kirchoff acknowledged when he published his theory.

In 1859, Kirchoff presented to the Berlin Academy of Science his theory about the exchange of radiant heat between two bodies, with his  law for thermal radiation, formally proven in 1861. This advance in the theory of heat was published at the Annalen der Physik in 1860, being re-printed in London the same year, as "On the Relation between the Radiating and Absorbing Powers of different Bodies for Light and Heat".

Kirchoff expressed: "For rays of the same wavelength at the same temperature, the ratio of the emissive power to the absorptivity is the same for all bodies." Kirchoff cited tests from de la Provostaye and Desains. 

In his theoretical development, Kirchoff analyzed the exchange of radiant heat between two plates of infinite  size, faced one to each other, which exchanged thermal radiation. Then,  for any material used for the plates: 

ε_𝜆 / α_𝜆 = J(λ, T)

Emissivity ε_𝜆 and absorptivity α_𝜆 are dimensionless, and have values between 0 and 1. The formula is based on thermodynamic equilibrium conditions and, for this, there was no need to use molecular or atomic properties of the materials, which is a probe of the genius of Kirchoff.

Also, the spectral structure of the function J(λ, T) was a challenge that lasted 41 years, until Planck found a definite solution, valid for the entire spectrum, by the end of 1900. The final solution required new theories like Maxwell's electromagnetism (1865) and Boltzmann's statistical thermodynamics (1887) and the use of black body cavities filled with isotropic electromagnetic radiation.

Emissivity ε_𝜆 represent the capability of a body to emit thermal radiation at a given wavelength λ, being at a given temperature T, compared with the emissive power of a perfect black body.

Absorptivity α_𝜆 represent the capability of a body to absorb thermal radiation at a given wavelength λ, being at a given temperature T, compared with the absorptive power of a perfect black body.

As it is treated ahead in this article, Kirchoff proposed a couple of "thought experiments" using black body cavities with an arbitrary body placed inside and a small hole to measure a sample of the thermal radiation escaping from it. When he used a black body instead on an arbitrary object, his law can be expresed as:

ε_𝜆 = α_𝜆

This relationship is verified at any temperature, when thermal equilibrium has been achieved within the cavity, and express that any radiation emitted by the body is absorbed by the cavity walls, and reciprocally. This is a mandatory condition at any wavelength, to preserve the second law of thermodynamics and to have a spectral density of thermal radiation that is continuous on the entire range of frequencies, from 0 to ∞ .

For his work, Kirchoff conceptualized the existance of a perfect black body, which has the property to absorb any incident thermal radiation, of any wavelength. This imply that, for visible light, this body has an absolute darkness, because it absorb every spectral component of the light, without any reflection, refraction or even transmission through the body.

To reach to his final expression, Kirchoff used several "thought experiments" using black body cavities.

Defining black bodies as perfect absorbers and perfect emitters of thermal radiation, Kirchoff expressed that the spectral emissivity ε_𝜆 of the body in the cavity, under thermal equilibrium, is:

ε_𝜆 = E(𝜆) / E_b(𝜆)   , what gives   E(𝜆) = ε_𝜆 . E_b(𝜆)

where E(𝜆)  and  E_b(𝜆) are spectral surface emissive power functions of the body and the cavity, respectively, and have units [Joule.m^-2.𝜇m^-1]. 

To reach a thermodynamic equilibrium, the cavity has to absorb completely the spectral emissive power of the body placed in the cavity, defining absorptivity α_𝜆  as a dimensionless factor that verifies:

E(𝜆) = α_𝜆  . E_b(𝜆)

The result of combining both equations gives another form of the Kirchoff's law for thermal radiation:

ε_𝜆 = α_𝜆

In the special case that the object is a black body, the relationship ε_𝜆 = α_𝜆 apply for the body and the cavity under thermal equilibrium, because the internal radiation density is isotropic, so it applies in both directions. 

From this law also is deduced that if a body, at a given temperature T, doesn't absorb radiation at a given 𝜆, the it also doesn't emit at the same 𝜆. This explain the spectral lines of absorption.

The case of different emissivities is solved by Kirchoff's law, because if the body has low emissivity, it also has low absorptivity. With this law, a body with lower temperature can't radiate more heat than what it receives from the walls at higher temperature, and this preserve the second law of thermodynamics.

In a more advanced "thought experiment", Kirchoff proposed to use two perfect cavities with black walls that had a black body C enclosed in cavity S1 with a small hole 1 on it, and a cavity S2 with a hole 2, which could be closed by a black surface. He required that the whole system be kept at a constant temperature by using a cover with a perfect reflecting surface, used to heat the cavities until a thermal equilibrium was achieved.

In the hypothesis, the black body like C is a perfect absorber of incident rays, and as the temperature of C is constant, the intensity of the absorbed rays must be equal to that of the emitted rays. Replacing the cover 2 with a perfectly reflecting spherical mirror aligned with the hole 1, would allow measurements to be done. 

The use of a dual cavity would prove that thermal radiation from C would fill the cavity S2 and still let go out some thermal radiation through the hole 2. In the cavity S2, under thermal equilibrium as S1, the walls must absorb and emit the same amount of spectral energy to maintain such equilibrium. Due to this condition, the absorbed energy is α_𝜆 J(λ,T), and has  to be equal to the ε_𝜆 J(λ,T)  energy emitted by the walls. 

For grey or opaque bodies, the relationship ε_𝜆 = α_𝜆 is not verified at every wavelength.

This effect of real elements is a problem to manufacture black bodies for science and industry even today, as maintaining this fundamental property over an extremely high bandwidth is very difficult.

Kirchoff's definition of a perfect black body included an infinitesimal thicknesses (discarded decades later), a complete absorbtion of incident thermal radiation, and null reflectivity or transmissivity.

On ideal black bodies, if the relationship ε_𝜆 = α_𝜆 is integrated over the entire range of λ, results in:  

Total Emissivity E(T) = Total Absorptivity A(T).

One common misconception about Kirchoff's blackness of a body is that it remains black notwithstanding the temperature at which it's heated. Most materials remains black up to 600 ºC, after which start to change the color from red to orange-yellow to white, in the range of 600 ºC to 3000 ºC, as more energy from shorter wavelenghts contributes to the total radiation. Only a few selected materials, like graphite, keep black.

Stefan-Boltzmann Law of Thermal Radiation

The experimental studies of Tyndall (1863) translated to german by A. Muller in his 1865 book, and the work of the French physicists Dulong and Petit, inspired in 1879 to Joseph Stefan, who derived his law stating that the heat radiation from a body is proportional to T⁴, with T being its absolute temperature in Kelvin degrees.

The Stefan's law is:

j = 𝜎T⁴

where j is the emitted power per unit area, 𝜎  a the Stefan's constant and T the temperature of the surface of the body, measured in Kelvin degrees (273.15 + ºCelsius). The transformation of ºK in ºC was a critical step of Stefan's theory, when using the experimental data of Tyndell.

Stefan estimated the constant 𝜎 value as 4.5x10^-8 Watt.m^-2.K^-4, using available rates of emission of energy (cal.cm^-2.min^-1) and emissivity (which had a wide dispersion), and used it to calculate the temperature of the Sun's surface, obtaining an average of 5700 K, close to the current value of 5778 K. Modern value of  𝜎 is 5.67x10^-8 Watt.m^-2.K^-4.

Stefan's law has been applied to a wide range of problems, in particular at astrophysics, where it's used to calculate the temperature of stars based on their luminosity and stellar radius, treating them as black bodies.

Even when proved to be very useful, even today, it didn't solve the Kirchoff's theorem about J(λ,T) because his formula only depends on the absolute temperature T.

From 1861 to 1863, Tyndall conducted several experiments with absorption of thermal radiation by gaseous matter, subject that had failed by then as it was believed that gases transmited thermal radiation without obstacles. Some of the Tyndall's findings were that water vapour absorbed eight times more heat than pure air, what led him to theorize that it had an important rol in the Earth's climate, thought that he extended to CO₂.  Wullner's book translated Tyndall's papers, remarking that the quantity of emitted heat increase much more rapidly than the temperature. He included his own estimations about Tyndall's measurements about intensity of radiation vs. observed color and assigned 525 ºC to a measured radiation for dark red color and 1200 ºC to a measured white red color.

Stefan was puzzled by what he read, and transformed Tyndall's ºC in ºK, after which he calculated the ratio between both temperatures in ºK, obtaining a value of 11.6 which was about 1.846⁴. This value was close to the reported increase in radiation between both colors, what drove him to postulate that "this observation caused me to take the heat radiation as proportional to fourth power of the absolute temperature".

In 1884, his former student Ludwig Boltzmann provided a mathematical proof of his law, now known as the Stefan–Boltzmann law, and valid only for perfect black bodies. This law has been extended to astrophysics, assuming that cosmic objects behave like perfect black bodies. This allows calculations of the temperature of the planets of the Solar System, their moons and the stars, based on its luminosity. Also, it has been used for estimations of the average temperature of the Earth.

The Stefan's formula  j = 𝜎T⁴ [units: Watt.m^-2.K^-4] is the emitted power per surface of a black body. Within a black body cavity, the energy density per unit volume u = 4/c.j or u = 4.c^-1.𝜎.T⁴ [units: Watt.m^-3.Hz^-1.K^-4].

This radiation, filling the cavity under thermal equilibrium, exerts a pressure over the walls equal to:

p = u /3 = 4/3.c^-1.𝜎.T⁴

Boltzmann applied this relation to express a change in U (the internal energy of black body of volume V) due to a pressure p applied on the perfectly reflexing surface by perpendicular electromagnetic radiation, and expressed its relationship as 

p= U/ V.

The fundamental thermodynamic relation is generally expressed as a microscopic change in internal energy due to microscopic changes in entropy and volume for a closed system in thermal equilibrium as follows:

dU = T dS – p dV

Here, U is internal energy, T is absolute temperature, S is entropy, p is pressure, and V is volume.

Dividing the expression of dU by dV , fixing T, and using the Maxwell Relations on S, V, P and T, is obtained:

 

From the energy density u, the total energy U is obtained as  U = uV, where the energy density of radiation only depends on the temperature, therefore:

After substituting different values in the former equation (Maxwell's radiation pressure p = u/3),

The partial derivative between u and T can be replaced by general derivative du/dT, as U is only function of T:

which has the general solution u = AT⁴, where A is the Stefan's constant 𝜎.  This expression is known as the Stefan-Boltzmann law, which current  value is  5.67x10^-8 Watt.m^-2.K^-4, and expressed in modern terms:

Contributions from Ludwig Boltzmann

Once James Clerk Maxwell published his Theory of Electromagnetism in 1865, this discovery took a while to be understood by the scientific community. Then, the true nature of the thermal radiation was understood: It had an electromagnetic nature with an undulatory behavior and could be generated by moving or oscillating particles with charges. Maxwell introduced the energy density of a plane, monochromatic and unpolarized electromagnetic wave with a peak electrical value E (in Watts/m³ and contemplating the contribution of the electrical and magnetic fields) as:

Introducing the intensity of an EM wave as I = u.c, and the momentum of an EM wave as p= I/c = u, it was deduced that if a black body is filled with isotropic EM radiation with energy density u, then the radiation pressure on the walls was p = u/3.

Boltzmann worked on the theory of heat radiation and made important contributions to the theory of heat using his statistical mechanics theory (field that he founded, along with Maxwell) which was a bridge between the macro-world and the atomic and molecular world, and  by which he explained the behavior of matter under heat modifuing the basis of classical thermodynamics. 

Boltzmann pursued, to some extent, to find how radiation was emited and absorbed by a black body. This was a goal of many scientists like Hertz, Heaviside, Planck, Wien, Rayleigh, Thiesen and many others. Boltzmann extended the kinetical theory of gases (Maxwell in 1860, Boltzmann in 1872 and 1877). An enhanced theory, known as Maxwell-Boltzmann distribution for molecular speeds in a gas, with a statistical approach to the subject made feel very uncomfortable the community of physicists by then, which were not used to the statistical view of Newton's mechanics neither the view of a world of atoms and molecules.

However, his work inspired Wilhelm Wien to use molecular resonators and exponential distributions of the energy  to find a solution to the spectral radiation intensity of a black body cavity.

Boltzmann used the concept of thermodynamic probability as the number of microstates of gas particles that correspond to the current macrostate of matter, and he showed that its logarithm is proportional to entropy in a simple formula

S = k lnW

In this law, Boltzmann introduced the constant k = R/N is  the amount of energy gained per particle of gas in a mol of matter with temperature increments  (R is the Regnault’s constant and N the Avogadro’s number). The constant k (1.38×10⁻²³ Joules/Kelvin) is fundamental to express the energy factor kT in the theories of black body radiation. With this factor, Boltzmann suggested that the energy levels of a physical system could be discrete, which was a curious anticipation of the concepts of Planck about energy radiation.

Boltmann helped on the view of energy filling the cavity propagating omnidirectionally, being a problem to model tridimensional "resonators", with three degrees of freedom (axis x, y, and z). 

In 1887, Boltzmann reformulated the ideal gas law as P.V = N.k.T (in 1834, Clapeyron derived it as P.V=n.R.T), being P: Pressure, V: Volume, N: Avogrado's number, R: Regnault’s constant and T: absolute temperature. 

Using the statistical mechanics (Maxwell-Boltzmann) with ideal monoatomic gasses it is shown, being v the average translational speed of an atom of mass m, the formula can be expressed as:

P.V = 1/3 N.m.v² = N.k.T,    or that     1/3 m.v² = k.T

Using the average kinetic energy of any given particle, it is obtained for an atom in the ideal monoatomic gas:

KE_a = 1/2 m.v² = 3/2 k.T

As the velocity v has three degrees of freedom (x, y, z), then if the oscillations are restricted to a single axis, the average energy per atom (or ideal particle) is: 

E_a = ½ k.T

 And this expression for average energy for a single axis of freedom was used with linear arrangements of the resonators in the theories used by Wien, Planck and others.

Wien's Displacement Law

In 1893, Wilhelm Wien published his paper on "The upper limit of the wavelengths which can occur in the heat radiation of solid bodies" at  the Annalen der Physik journal, based on his experimental work while working as an assistant at the  Berlin PTR (Imperial Physical Technical Institute). This paper, that expressed a simple relationship between the peak wavelength 𝜆_max at which the radiating energy of a black body reaches his maximum value and the absolute temperature T at which it happens, made him famous at a young age.

His work was based on adiabatic expansion or compression caused by thermal radiation on an enclosure, and he also expressed that a black body radiation spectrum was an unknown function f(ʋ,T) multiplied by ʋ³. He would propose, three years later the exponential nature of this function, when he published his 1896 paper.

The Wien's Law of Displacement covers a wide range of frequencies between infrared and ultraviolet types of radiation in a black body cavity, and is expressed as:

𝜆_max = b/T   , being b = 2900 𝝁m.K^-1

The current value of b is 2.8977729×10⁻³ m⋅K (US National Institute of Standards and Technology. June 2015), and Wien's Law of Displacement has an important place today, at many applications like (Wikipedia):

Relationship between color, temperature and radiating heat of metals by passing from black to red, orange-red and reaching white at high temperatures, as it is heated in an oven or by a torch. This allows to produce color maps relating color of objects, wavelength of radiation and absolute temperature to determine, for instance: peak radiation of Sun's surface, color of stars according to its temperature (Stefan's law), peak wavelength radiation of mammals used to create passive infrared cameras, apparent color of lighting sources (with deviations from black body radiation), and many other practical applications at industry and science.

Even when the formula was an approximation, as it failed at lower frequencies, the simple expression allows calculations for the emission of a black body at any temperature only by using the peak value at a reference temperature. Wien was 29 years old when he published this paper, and kept working as an assistant at the PTR, where he teamed with O. Lummer during 1894 and 1895 to find experimental results of the radiation of black bodies, which Lummer converted to graphics. This work allowed every physicist involved in the black body radiation problem to have a grasp at the form adopted by the spectral density as a function of 𝜆.

Wien's Law of Spectral Intensity of a Black Body Radiation

Prior to Wien's law, the first attempt to find the Kirchoff's J(𝜆,T) function was performed in 1894 by Friedrich Paschen, an experimental physicist working at the same PTR laboratory as Wien. He used glowing bodies for his experiment and the results were limited and poor.

In June 1896, Wilhelm Wien published the first analytical solution for the spectral distribution of thermal radiation as a funcion of its wavelength and absolute temperature within a black body cavity. His paper "On the distribution of energy in the emission-spectrum of a black body" was published at the Annalen der Physik journal, and was based on his experimental work with O. Lummer between 1894 and 1895 at the Berlin PTR (Imperial Physical Technical Institute), the most advanced institution for measurements in the world. 

Wien's theory used exponential functions, with basis on the Maxwell-Boltzmann distribution of velocities of molecules on ideal gases and the Stefan's law of total energy emitted by a black body.

Wien theorized that the electric charges within the "molecular resonators" produced the EM radiation that filled the cavity, and theorized that the spectral energy distribution 𝝋_𝝀 per unit 𝜆 should follow the equation:

𝝋_{𝝀 = F(𝜆) e}^-c/𝜆T

where the c constant and the function F(𝜆) were defined later. F(𝜆)  is  related to the density of modes of resonance per unit of volume. Wien stablished that the final proof for his formula was that the total energy of the spectral distribution 𝝋_𝝀 should respect the Stefan's law. 

He expressed this hypothesis stating that this relationship was mandatory:

To find the expression F(𝜆), Wien developed several calculations by substituing the variable 𝜆 and using a series expansion of the new function  F(c/yT) to count the modes, demonstrating that  F(𝜆) = C 𝜆^-5, and finally expressed his  spectral formula:

𝝋_{𝝀 = C 𝜆}^-5 _e^-c/𝜆T

The constants C and c were deduced empirically, based on his prior experimental work with O. Lummer. 𝝋_𝝀 represents the energy emitted per spectral unit per unit time per unit area of emitting surface, per unit solid angle. In physical units its value is expressed as Joule.sec^-1.m^-2.sr^-1 and a function of 𝜆.

Wien's equation can be presented in terms of frequency, as follows (c represents the speed of ligth):

𝝋_{𝝀 = A ʋ}^-5 c^-4 _e^-bʋ/cT

Wien's paper had a great impact on Planck, who had been working theoretically on the same subject since 1894, using the concept of "hertzian oscillators" that Hertz envisioned as molecular resonators, even when Planck was reluctant to mention molecules. Wien's formula behaved well in the short infrared and visible regions of the spectrum, as initial measurements by Lummer and Warburg at the PTR in 1896 showed. But Wien's formula had discrepances with measurements in lower IR ranges and low temperatures, as Lummer and E. Pringsheim proved, early in 1900, with more advanced equipment using temperatures up to 2000 ºK.

At the same PTR, between 1899 and 1900, the experimental physicists Rubens and Kurlbaum were working on black bodies' radiation measurements in the long infrared range, at wavelengths close to 40 𝝁m by using new composite crystals, and found similar discrepances with Wien's formula. When Rayleigh's paper was published by June 1900, with critics to Wien's formula and a new proposal, the formula worked well up to about a couple of hundred of 𝜇m, divergin rapidly with shorter wavelengths. Rubens informed to Planck on the discrepances of Rayleigh's and Wien's formulae and, with this information Planck could develop a better solution, by interpolating both theories.

In a very short lapse, Planck presented his provisory paper "On an improvement of Wien's spectral equation" to the German Physical Society on October 1900, and his final solution with the paper "On the Law of Energy Distribution in the Normal Spectrum" on December 14, 1900. After each presentation, Planck published both papers at the same Annalen der Physik. The December paper was a fully developed theory with the values of his "h" constant plus the "k" Boltzmann's constant, which he had informed to the GPS in advance.

The October's presentation initiated a discussion between Planck and Wien about the origin of the formula, but Wien dropped the issue after Planck's final theory, and continued with his prolific work on other fields. By example, he discoverered the proton in 1898 when analyzing ionized Hydrogen gases and found it mass almost equal to that of the Hydrogen atom, narrowing the result using the electron, discovered in 1897 by J.J. Thomson. He was also one of the founders of the mass spectrometry science.

Wien won the 1911 Nobel Prize in Physics for his Displacement Law and his work on black bodies.

Wien was inspired by the Maxwell-Boltzmann distribution, so his solution can be explained from it, making the origin explicit. For Wien, the intensity of radiation is proportional to the number of resonators (simple molecules in the gas). Using this distribution the energy per resonator is given by:

E_v ~ e^-1/2mv2/kT

Wien assumed that the frequency of the radiation was proportional to the kinetic energy of molecules, as:

 ½ mv²= aʋ,  so  E_v ~ e^-aʋ/kT

Computing the resonators per unit of volume, the energy per unit of frequency is as follows:

E_{ʋ = B ʋ}³ _e^-bʋ/cT

where c represents the speed of light, while B and b are constants to be determined.

Rayleigh-Jeans Law of Spectral Intensity of a Black Body Radiation

In June 1900, Rayleigh published his radiation formula, based on the amount of standing EM waves that fills a cubic cavity, and arguing that Wien's spectral formula didn't verify the equipartition of energy (every degree of freedom radiates the same kT energy). Basing his work on his experience with waves and using the wave equation theory, applied it to the thermal radiation in a cavity.

Rayleigh wrote a short paper without justifications, and his formula worked very well at wavelengths down to 100 𝜇m, but diverged towards infinity when the wavelength 𝜆 approached to zero. In 1905, Jeans noted that the 1900's Rayleigh formula had a mistake about the count of modes, which increased the formula by 8. Jeans, with Rayleigh's consent, published this finding in 1905, and the formula was named Rayleigh-Jeans.

In 1911, when P. Ehrenfest analyzed the developments in the black body theory, called it "the Rayleigh-Jeans ultraviolet catastrophe".

The Rayleigh-Jeans formula, expressed in units of frequency, is:

One of many ways to recreate the original work made by Rayleigh is as follows:

In a cube with sides of L size filled with electromagnetic radiation in thermal equilibrium, standing EM waves can't take any path, except existing along every axis xyz. Also, the standing waves must have a null value at both ends to avoid energy dissipation and conserve thermal equilibrium.

The standing waves must satisfy the classical wave equation 𝛻²E  for axis xyz, which is:

𝜕²E/𝜕t² = c²(𝜕²E/𝜕x²+ 𝜕²E/𝜕y²+ 𝜕²E/𝜕z²)

The general solution for the wave equation, under these restrictions, is:

E = E₀ sin(n₁𝜋x/L) sin(n₂𝜋y/L) sin(n₃𝜋z/L) sin(2𝜋ct/L)

Using this expression in the wave equation, gives a value:

 (n₁𝜋/L)²+(n₂𝜋/L)²+(n₃𝜋/L)² = (2𝜋/𝜆)²

or

 n₁² + n₂² + n₃² = (2L/𝜆)²

The values n_W adopt a large combination of modes in the tridimensional space of the cube, which have to be counted to calculate the total amount of standing waves within the cavity.

To count the number of modes of resonance, Rayleigh replaced the cube with a sphere of radius R (which has not the same volume). Imaging a tridimensional grid which contains N_W modes, is required to discard negative values originate by the cuadratic expression of each mode n_W. The volume, expressed in number of modes is, for Rayleigh, a good approximation to the real value if the size L is much greater than the standing wave 𝜆.

Also, it has to be accounted 2 standing waves per mode (orthogonal polarization) and, by halving the number of modes per axis (Jeans contribution). it gives a total reduction of 2/8 in the number (2 due to polarization and 1/8 by discarding half the modes per axis). The final solution for N, the total number of modes, is:

N = (𝜋/3) (n₁² + n₂² + n₃²)^3/2 = (8𝜋L³/3𝜆³)

The number of modes per unit of wavelength 𝜆 per unit of volume at the cube is:

(-1/L³) (dN/d𝜆) = (8𝜋/𝜆⁴)

The negative sign in the derivative dN/d𝜆 is originated by the decreasing number of modes with increasing values of 𝜆.  Assuming that every standing wave has the same energy (principle of equipartition of energy), each one contributes with an average energy kT. In this case, the energy density per 𝜆 unit is:

du/d𝜆 = (1/L³) (dE/d𝜆) = (8𝜋kT/𝜆⁴)

which is the Rayleigh-Jeans law of spectral distribution of energy in a black body. The classical result of the law of radiation, expressing the spectral energy as a function of frequency ʋ (the density of modes is the expression within the parentheses) is:

du/dʋ = (8𝜋ʋ²/c³) kT

Planck's Law of Spectral Intensity of a Black Body Radiation

Planck created a turning point in the history of physics, introducing the minimal possible value of an electromagnetic energy as 𝜖 = hʋ, when he developed his theory for spectral radiation in a black body cavity. 

Planck understood that h was a fundamental constant of nature, and introduced the concept of 𝜖 = hʋ as a quantum of action. He stated that no other value of energy of a single electromagnetic wave could be lower than hʋ, what clashed with the classical physics, with infinitesimally small values. This concept created a new field of science, the quantum physics, which developed slowly in the next 25 years, after contributions from Einstein, Bohr, de Broglie, Schrodinger, Heisenberg, Born, Dirac and plenty other scientists.

Also, this new concept about the quantification of EM energy in n.hʋ packets conduced to the forced acceptance about a dual behavior in electromagnetism: waves and particles, depending on the application.

Planck, a theoretical physicist specialized in thermodynamics, focused in the problem of the spectral formula for the black body radiation since 1894, being his main concern the second law of thermodynamics. He used "hertzian oscillators" (model proposed by Hertz), thermodynamics and the theories of Wien and Rayleigh to obtain his final formula in 1900. Also, he used the results of experimental research at the Berlin PTR institute. 

For Planck, hertzian oscillators were minimal units of matter (equivalent to Wien's molecular resonators) which, by emission and absorption of electromagnetic waves, were responsible for the behavior of radiation in black body cavities. Having published several papers on the subject prior to 1900, which were proved to be wrong, in October 1900 and on December 14, 1900, Planck gave two lectures at the German Physical Society, prior publishing his two papers with the right theory, at the Annalen der Physik journal:

• 1900 paper (October lecture): "On an improvement of Wien's spectral equation"
• 1901 paper (December lecture): "On the Law of Energy Distribution in the Normal Spectrum"

The October paper was an outline of his final theory, and was an empirical approach that merged Wien's and Rayleigh's formulae, but whitout specific values nor the presentation of the 𝜖 = hʋ concept. In the second and final paper, presented in December, Planck introduced his famous formula with the values of h and k.

Planck's 1900 paper:  "On an improvement of Wien's spectral equation"

At the end of this paper, a transcript of his lecture at the German Physics Society on October 19, 1900, Planck introduced a draft of his radiation formula for a black body cavity. Planck was polite and cautious when he begun with critics about the Wien's work, published 4 years ago, without open criticism.

Planck stated that Wien's formula for entropy versus energy was wrong, when considered the linear adition of the energy of n identical processes, which lead to:

d²S/dU² = constant/U

For larger wavelengths, Planck proposed to use:  

d²S/dU² = constant/U²

while keeping Wien's entropy formula for shorter wavelengths. Merging both, the final proposal was to use:

when is it's complemented with Wien’s displacement law, gives a radiation formula with two constants:

 Having presented this modification, Planck had less than two months to find the theory around this formula.

Planck's 1901 paper:  "On the Law of Energy Distribution in the Normal Spectrum"

On December 14, 1900, Planck presented his complete theory about the radiation formula for a black body cavity at the German Physics Society (published in 1901 at the Annalen der Physik journal), which included the demonstration of the development of the "h" constant and its value, along with Boltzmann's k constant.

By then, Planck wasn't a believer in atomicism, so he avoided relating his hertzian oscillators to molecules, as Wien did. He proposed a change in the Wien's postulate about the entropy of N identical resonators by stating that the energy U of each single stationary vibrating resonator has to be averaged in time, instead of assuming that the energy of each resonator is the same.

Then, the energy of N resonators is U_N = N.U and its entropy is S_N = N.S, being S the entropy of a single resonator. Then, the total entropy S_N corresponds to the disorder with which the total energy U_N is distributed among N resonators. This difference leads to the discretness of energy changes.

Using Boltzmann's formula for entropy S = k log W, being W the probability of a number of microstates that correspond to a given macrostate of radiation, Planck will find the probability W it confers to N resonators the total energy U_N. Introducing h here, he states that U_N is a discrete quantity with a number of finite equal parts. By calling each part 𝜖 as the finite energy, he follows with U_N = P. 𝜖, with P being a large integer.

Using P and N, and calling any distribution of the P elements of energy along N resonators as a Boltzmann's "complex", Planck introduces the R value as the total number of all possible complexes, each one generated by assigning an integer probability to every arrangement that can be associated with every number N (1, 2, 3, 4, 5, ……, N+P-1) for a given N and P.

Using the Stirling's theorem, it gives a first approximation as N! =N^N, R first approximation is:

Planck assumed that the probability W is proportional to the number R of all possible complexes formed by distribution of the energy U_N among the N resonators. Introducing R instead of W in S_N = k log R, it gives:

S_N = k log R =k [(N+P) log(N+P) – N logN – P logP]

Considering previous formulae, then P/N = U/𝜖 and, by using S_N /N = S, the entropy S of a single resonator as a function of its energy is given by:

S = k [(1+ U/𝜖) log(1+ U/𝜖) – U/𝜖 log U/𝜖]

Using a modified version of the Thiesen's law of radiation applied to monochromatic, the modified Thiesen's formula is:

u = ʋ³.c^-3.f(T/ʋ)

Planck uses the formula K = ʋ².c^-2.U  from his previous paper on the irreversible radiation processes (U is the energy of a single resonator), in order to go from the energy density u to the energy U, along with (his words)  the well-known u = 8.𝜋.K.c^-1  (K is the intensity of a linearly polarized monochromatic ray), Planck obtains the relation u = 8.𝜋. ʋ².c^-3. U.

Using u = ʋ³.c^-3.f(T/ʋ), derives U = ʋ.f(T/ʋ), which Planck transforms in T = ʋ.f(U/ʋ) without explanations.

Introducing the entropy S of a single resonator as  dS/dU = 1/T, Planck obtains:

dS/dU = ʋ^-1.f(U/ʋ)   or, integrated,  S = f(U/ʋ)

Planck concludes that: "the entropy of a resonator vibrating in an arbitrary diathermic medium depends only on the variable U/ʋ, containing besides this only universal constants."

Introducing his constant h in the formula of S, Planck finds that the energy element 𝜖 must be proportional to the frequency ʋ, and then:   

𝜖 = hʋ

and consequently, being h and k universal constants:

S = k [(1+ U/(hʋ)) log(1+ U/(hʋ)) – U/(hʋ)  log U/(hʋ)]

Planck wrote that by substitutions at the equation T = ʋ.f(U/ʋ), he obtains:

1/T = k/(hʋ) log(1+(hʋ/U) 

and that, given the relationship between u and U, then the energy distribution law is (Joule.m^-3.Hz^-1):

Note: u is the density of the energy (in the frequency range between ʋ and ʋ + dʋ in a black body radiation at an absolute temperature T.

or, by introducing substitutions in terms of wavelength 𝜆 instead of the frequency (Joule.m^-3.m^-1):

Planck finishes his 1901 paper, and using experimental findings at the PTR (Kurlbaum, Lummer-Pringswim),  and his own equation 𝜆_mT= chk^-1/4.9651 (from experimental data), he finds relationships between h and k as:

k⁴.h^-3 = 1.1682 . 10¹⁵    and    h.k^-1 = 4.866.10^-11

Finally, Planck finds the values of h and k as:

h = 6.55.10^-27 erg.sec

k = 1.346.10^-16 erg/deg

Steradians and Solid Angles 

The following figure shows relationships between steradians and solid angles in a sphere. From Wikipedia: "Solid angles are often used in astronomy, physics, and in particular astrophysics. The solid angle of an object that is very far away is roughly proportional to the ratio of area to squared distance. Here "area" means the area of the object when projected along the viewing direction".

Solid angles can be defined in steradians (sr), as is shown at the figure. If the dome at the sphere of radius r has an area A equal to r², then the solid angle 𝛺 is 1 steradian, as it is viewed from the center of the sphere.

If the aperture angle 2𝜃  is 180º, then the solid angle 𝛺 has a value of 2𝜋 sr, which is one hemisphere. The whole sphere has 4𝜋 sr solid angle, and this is the reason by which 4𝜋 is often used to divide a given value, in order to obtain its distribution on the spherical volume around in sr values, which are adimensional.

Using steradians, the solid angle is expressed over the plane that cut the sphere in half, containing the origin and the line 2a, which is normal to the radius r at the figure. This method reduces a tridimensional shape of an sphere described in cartesian coordinates to a plane xy, and is commonly used to express the angle 𝛺 . 

Steradians (sr) can be converted to degrees (º or deg) or to arcmin (1/60 degrees), with the relationships:

1 steradian = 1 rad = (180/π) degrees = 57.3 deg = 3437.7 arcmin

In the case that an adimensional surface has to be involved, the following relationships apply:

1 steradian = 1 rad² = (180/π)² square degrees = 3282.8 deg² = 1.18 x 10⁷ arcmin²

The use of steradians to deal with solid angles and spherical sectors has been adopted since the XIX century.

Quote from Wikipedia: "In geometry, a spherical sector is a portion of a sphere defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap."

For instance, radiant intensity can be measured in watts per steradian (W⋅sr⁻¹). The steradian is considered since 1995 as an SI derived unit.

Radiometric Quantities

In the field of Radiometry there are several radiometric quantities (units) that are used to measure radiant energy (Joules) and radiant flux (Watts), depending on how they are measured in a tridimensional space (per unit surface, per unit solid angle, per unit wavelength, per unit frequency or per total hemispheric volume or area). The different names being used may cause confusion if the radiometric quantity is not clearly defined.

This is an excerpt from Wikipedia's article Irradiance:

"In radiometry, irradiance is the radiant flux (power) received by a surface per unit area. The SI MKS unit of irradiance is the watt per square metre (W.m^-2). In astronomy the CGS unit erg per square centimetre per second (erg·cm⁻²·s⁻¹) is often used. Irradiance is usually called intensity because it has the same physical dimensions, but this is avoided in radiometry where such usage leads to confusion with radiant intensity."

"Spectral irradiance is the irradiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The two forms have different dimensions: spectral irradiance of a frequency spectrum is measured in watts per square metre per hertz (W·m^−2·Hz⁻¹), while spectral irradiance of a wavelength spectrum is measured in watts per square metre per metre (W·m⁻³), or more commonly watts per square metre per nanometre (W·m⁻²·nm⁻¹)."

Some of the SI (MKS) radiometry units are:

Q_e: Radiant Energy (Joule): Energy of electromagnetic radiation.

Φ_e: Radiant Flux (Watt): Radiant energy per unit time. Sometimes is called "radiant power".

Spectral Radiant Energy inside a cavity: The original formulae u and E from Planck.

W_u(ʋ,T) = u = 8𝜋hʋ³c^-3(e^hʋ/kT-1)^-1     [units:  Joule.m^-3. Hz ^-1]

W_e(𝜆,T) = E = 8𝜋hc𝜆^-5(e^ch/k𝜆T-1)^-1     [units:  Joule.m^-3. 𝜇m ^-1]

Spectral Radiant Exitance M_e(𝜆,T) at the aperture in the cavity: Radiant exitance of a surface per unit wavelength (or frequency). This is sometimes also confusingly called "spectral intensity".

M_e(𝜆,T) = (c/4).W_e𝜆(𝜆,T)      [units:  Watt.m^-2.m^-1]

Monochromatic Irradiance or Spectral Flux Density at the aperture in the cavity: Radiance of a surface per unit frequency or wavelength per unit solid angle. A directional quantity and differs from Planck's formula by (c/4𝜋). This is sometimes confusingly called "spectral intensity".

L_ʋ(ʋ,T) = (c/4𝜋).W_u(ʋ,T) = 2hʋ³c^-2(e^hʋ/kT-1)^-1    [units:  Watt.m^-2.Hz^-1.sr^-1]

L_e(𝜆,T) = (c/4𝜋).W_e𝜆(𝜆,T) = 2hc²𝜆^-5(e^ch/k𝜆T-1)^-1    [units:  Watt.m^-2.𝜇m ^-1.sr^-1]

E_e: Flux Density or Irradiance (Watt.m^-2): Radiant flux received by a surface per unit area. This is sometimes confusingly called "intensity".

L_e,Ω: Radiance (Watt.m^-2.sr^-1): Radiant flux emitted, reflected, transmitted or received by a surface, per unit projected area per unit solid angle. This is a directional quantity. This is sometimes also confusingly called "intensity".

Some scientists involved with the development of the black body radiation

Balfour Stewart (Scotland, UK, 1828-1887): A physicist, Stewart studied the radiation of heat, the Sun and Solar flares, meteorology, and terrestrial magnetism. He wrote several books and papers, and is one of the first to work with the theory of radiation in opaque bodies. In 1858, Stewart determined that the thermal radiation of a body is not a surface phenomenom but it happens throughout the interior of it, and that the radiative and absorptive powers of a substance must be equal. He generalized the concept to any opaque body under thermal equilibrium and, at his law, stated that the coefficient of emission plus the coefficient of reflexion were equal to one.

 Gustav Kirchhoff (Germany, 1824-1887): A physicist and chemist, Kirchhoff pioneered the fields of electrical circuits, spectroscopy and the study of black-body radiation in cavities (coining the "black body" term). Kirchhoff formulated four universal laws.  He formulated his circuit laws were while he was a student, and stated his law of thermal radiation in 1859. Along with Bunsen, invented the spectroscope, discovered caesium and rubidium in 1861, pioneered the identification of the elements in the Sun and created three laws of spectroscopy. Kirchhoff contributed to optics by applying Maxwell's equations to the Huygens principle and created the Kirchhoff's law of thermochemistry. Also, he published numerous papers on his different fields of investigation and, as a professor, teached to Eötvös, Nichols, Mendeleev, Planck, Schröder, Noether and other notable scientists.

James Clerk Maxwell (Scotland, UK, 1831-1879): Physicist and polymath, Maxwell created an entire new field of physics known as electromagnetism. Also was one of the founding fathers of statistical mechanics with his kinetic theory of gas, along with Boltzmann. He contributed to astronomy; theory of color; theory of human vision; invented the color photograph, electricity and magnetism and, as a experimental physicist, created different instruments for measuring physical units, and re-structured the Cavendish laboratory at Cambridge. Besides his breakthrough theory about electromagnetism and other theories, published his monumental Treatise on Electricity and Magnetism, at which he treated every single aspect of both fields, known by then. His scientific legacy is considered at the same height as Newton's foundations for classical mechanics.

Joseph Stefan (Austria, 1835-1893): Physicist, mathematician and poet, published about 80 scientific articles covering electromagnetism, vector algebra, kinetic theory of heat, calculation of inductivity, skin effect on conductors of high frequency currents,  created the Stefan law for thermodynamics (and measure the Sun's surface temperature), Stefan's problems at mathematics, the Stefan's equation for civil engineering, Stefan Flow, Stefan Number, Stefan Tube and several other co-named fundamental values and formulae. His main contributions, however, are centered on thermal theories.

Heinrich Hertz (Germany, 1857-1894): An experimental physicist, Hertz is famously known for his proof of the existance of electromagnetic waves in 1887, 22 years after Maxwell's theory. Hertz designed the first transmitter and receiver based on resonating dipoles tuned at about 50 MHz, even when he didn't foresee any practical application. The announcement of his discovery caused a worldwide wave of experimenters on the new field, what led to the discovery and practical uses of "Hertzian waves", likes in the cases of Tesla and Marconi. He pioneered the study of photoelectric effects using cathode rays and thin foils of metals like aluminium. Von Lenard, a student of Hertz, extended the study up to X-rays, and adviced Röntgen how to produce them. Herz also actively worked on contact mechanics (deformation on solids under motion) and had early interests on meteorology, contributing with a new kind of hygrometer. Besides publishing several papers at scientific journals, Hertz had profesorship positions at several Universities and the recognitions for his work on radio waves extended worldwide. Hertz died at an early age (36 y.o.), due to a disease.

Ludwig E. Boltzmann (Austria, 1844-1906): Physicist and philosopher, is considered, along with J.C. Maxwell one of the founding fathers of statistical mechanics, prior to Gibbs. Boltzmann was a pioneer in the study of the atomic structure of matter through his many contributions to thermodynamics. His statistical approach to thermodynamics led him to the theoretical discover or enhance of several theories, named after him. Examples are: Boltzmann constant; Maxwell–Boltzmann distribution and statistics; Boltzmann's transport equation; Stefan–Boltzmann constant and law; Boltzmann's factor; Boltzmann's distribution; etc. He was offered proffesorships and lectured at several Universities, covering physics, mathematics and philosophy, being highly appreciated in the last field.  The modern view of the second law of thermodynamics as the law of entropy or disorder is due to his work on this field. Another contribution is the Boltzmann's equation, which describes the dynamics of an ideal gas, by using a statistical approach to the position and momentum of particles under a density distribution of a cloud of points in a phase space. Boltzmann received numerous awards and honors, being member of the Imperial Austrian Academy of Sciences, the Royal Swedish Academy of Sciences, Foreign Member of the Royal Society, and President of the University of Graz in 1887. Boltzmann spent his final years fighting to defend his theories against his peers, who didn't share his view of a universe based in atoms and molecules. Suffering from depression, resigned his positions at the University, and commited suicide while on vacation at Italy in 1906. On his tombstone, there is  the inscription of his famous entropy formula, S = k ln W.

Otto R. Lummer (Germany, 1860-1925): Physicist and researcher, worked in the fields of optics and thermal radiation. Along with E. Gehrcke, developed the Lummer–Gehrcke interferometer, co-designed and built the Arons–Lummer mercury-vapor lamp, and co-designed with E. Brodhum an improved version of a photometer invented by R. Bunsen, totalizing more than a dozen of developments and seven patents. Lummer became professor at the University of Breslau in 1905. Lummer's research on radiating energy, working with W. Wien at the Physical-Technical Reichsanstalt (PTR) in Berlin by 1893, led him to construct an almost perfect black-body radiator, which had been conceived only as a theoretical abstraction. The results provided Wien with the information to develop and publish his theory of radiation of a black body in 1896. Lummer teamed with Kurlbaum and, since 1899, with E. Pringsheim to perform measuremments of the distribution of energy in black-body radiation up to 2000 degrees. Both knew about the wrong behavior of the theory of Wien, and carried out more experiments over a larger temperature intervals and wavelength range.  In 1899, Lummer tried to modify the Wien's law of radiation, working with the mathematician Eugen Jahnke (1863-1921), but he found a particular criticism from W. Wien  and his work was surpassed by the Planck`s solution the same year. Lummer was involved in more than 130 scientific publications, some of fundamental importance, and was twice proposed (1910-1911) by Emil Warburg for the Nobel Prize, by sharing it with Planck and Wienn "because of their success in the experimental and theoretical research of radiation laws".

Wilhelm Wien (Germany, 1864-1928): A physicist, Wien was involved in thermal radiation theory, black body research, electromagnetism, ionized gasses and mass spectrometry. Anticipated the discovery of the proton by 14 years, when in 1898, identified a positive particle equal in mass to the hydrogen atom (the electron had been discovered in 1897). In 1893, using a combination of electromagnetic theory and thermoelectricity, Wien found his law of displacement relating temperature and wavelength in the radiation of a black body, while working at the PTR as assistant. After this success, which made him famous Wien teamed with Lummer at the PTR, in 1894-1895 to experiment on the radiation of a black body, to complement his prior finding. In 1895, after trying different materials and type of surfaces, finished the experiments and concluded that a black body radiation was a natural radiation that can be created in a closed cavity with a small opening for measurements, while keeping the box at a steady temperature. 

Using the recolection of results of the work with Lummer, Wien developed his theory in 1896, and published the paper "About the energy distribution in the emission spectrum of a black body" at the Annalen der Physik. Wien was inspired by Maxwell and Boltzmann's works on electromagnetism and statistical mechanics, and merged both fields. He entered in a public dispute with Planck around 1900, about the origin of both theories and its true validity, due to Planck approach criticizing Wien's work. It was evident the rol of experimental findings when criticism arose around the use of empiricism in Wien and Planck theories. Wien his work with Lummer (1894), and Planck used the findings of Lummer-Pringsheim (1900) and Rubens-Kurlbaum (1899-1900), performed at the same Berlin PTR (Physical-Technical Reichsanstalt) laboratory. Wein lost interest on the discussion when Planck pulled the card "empirical vs. theoretical" approach to a research, as he was a firm believer that a physicist should have skills at experimental and theoretical physics simultaneously. 

Wein excelled in both fields of physics, published numerous papers and held profesorship positions at four prominent Universites, being appointed succesor of W. Röntgen, in 1900 and 1919. Wien received the Nobel Prize in 1911 for "his discoveries regarding the laws governing the radiation of heat". Wien's displacement law is a fundamental basis of modern devices and instrumentation for the measurement of temperature.

Lord John Rayleigh (England, 1842-1919):  An experimental and theoretical physicist and also a polymath, Lord Rayleigh (John William Strutt) spent his entire career at Cambridge University, making an impressive amount of contributions to different fields of physics. He was awarded with the Nobel Prize in 1904 "for his investigations of the densities of the most important gases and for his discovery of argon in connection with these studies". This excerpts from Wikipedia sumarize his work: "Rayleigh provided the first theoretical treatment of the elastic scattering of light by particles much smaller than the light's wavelength (Rayleigh scattering), which notably explains why the sky is blue. He studied and described transverse surface waves in solids (Rayleigh waves). He contributed extensively to fluid dynamics, with concepts such as the Rayleigh number (a dimensionless number associated with natural convection), Rayleigh flow, the Rayleigh–Taylor instability, and Rayleigh's criterion for the stability of Taylor–Couette flow. He also formulated the circulation theory of aerodynamic lift. In optics, Rayleigh proposed a well known criterion for angular resolution".

His law for black-body radiation later played an important role in birth of quantum mechanics. His scientific papers from 1869 to 1919 were recopiled into six volumes by the Cambridge University Press.

James Jeans (England, 1877-1946): A physicist, astronomer and mathematician, Jeans contributed to the fields of kinetic and dynamical theories of gases, astronomy and cosmology, electricity and magnetism and philosophy of science. Author of more than a dozen books covering different fields, he is known by his contribution to the Rayleigh–Jeans law (black body radiation), made in 1905 when he found a numerical mistake at the Rayleigh law (1900). He received numerous awards and prizes and was knighted in 1928.

Max Planck (Germany, 1858-1947): Theoretical physicist, known as one of the founding fathers of Quantum Physics, along with Niels Bohr, he devoted most of his career at thermodynamics and quantum theories. His revolutionary 1900 lecture about the discrete nature of the electromagnetic radiation, previously thought as being analog, with infinitesimal possible values of energy changed the face of modern physics. His scientific life spans for more than 60 years, with unvaluable contributions to the modern world like the concept of "quantum of action", later carried by photons. 

His initial field of work was the theory of heat and he focused on the thermal radiation of a black body by 1894, which he considered a fundamental subject for the validity of the second law of thermodynamics. After the proof of the existance of electromagnetic waves by H. Hertz in 1884, Planck adopted the concept of "tiny hertzian resonators" capable of produce EM waves, proved to be of fundamental importance on his theory 15 years later.

Planck started to study the black body radiation after Wien publications at 1903 (Wien's displacement law) and at 1906 (energy distribution in the emission spectrum of a black body). He focused on the concept of resonators avoiding references to the real constituents of matter (atoms and molecules) as emiters or absorbers of radiation. Initially convinced at the results of Wien law of radiation, discrepances marked by Lummer and Pringsheim early in 1900, made Planck aware of other ways, particularly when he received information from H. Rubbens about the experiments conducted by him and F. Kurlbaum at the same PTR, about radiation being emited by a black body in the infrared range up to wavelengths of 50 microns. The measurements showed great deviation from Wien and Rayleigh's formulae, which moved Planck to modify his theory and to make public a hint, on October 19, 1900, about his improvement of the Wien equation. 

On December 14, 1900, Planck presented his final theory at the German Physics Society, being this date regarded as the birth of the Quantum Theory. The introduction of the "quanta of action" h made possible that Bohr, 13 years after, presented a viable theory about the equilibrium of a Hydrogen atom, isolated or under radiation, by being at discrete quantum levels of energy or performing absorption or emission of energy in quantum units hf. Planck published two papers, in 1900 and 1901, the later cementing his theory: "On an improvement of Wien's spectral equation" and "On the Law of Energy Distribution in the Normal Spectrum". Since 1901, Planck's career took a different course, as he embarked into the new quantum mechanics and Einstein's relativity for the next 30 years, making contributions too extense to sumarize here. 

 

In 1885, Planck started a career as a professor in Theoretical Physics, and finished it at the Berlin University in 1926, after four decades in this field. Planck kept giving lectures and held different prestigious positions like Chief Editor at the Annalen der Physik, President of the German Physical Society (DPG), President of the Kaiser Wilhelm Society (Max Planck Society since 1948), foreign member of the Royal Netherlands Academy of Arts and Sciences, gave his name to the Max Planck medal (the highest medal by the DPG since 1928), and several others. Planck was awarded with the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".