Black Body Radiation

Newton’s physics along with the theories of Coulomb, Faraday and Maxwell reigned supreme for a few centuries, but eventually limitations were finally exposed.  In the beginning there was continuous flow, and then Max Planck a German theoretical physicist came along and proposed quantization.  Quantization basically just means, that instead of being continuous, things such as EM radiation, can only exist in multiples of certain values.  Planck came to this conclusion when he solved something called the ‘Ultraviolet Catastrophe’.

Certain objects are called blackbodies because they emit electromagnetic radiation of all wavelengths.  In Physics, an object which radiates perfectly is called a blackbody.  The name is a little misleading, as there is nothing about a blackbody which is black in color.  If you heat a metal object, such as a canon ball, it will start to glow, giving off its own visible light rather than just reflecting light.  A blackbody is something that soaks up every scrap of energy that falls upon it and reflects nothing.  A blackbody isn’t just a perfect absorber, as it is also a perfect emitter.  In one form or another, a blackbody gives back out every bit of energy that it takes in.  If it’s hot enough to give off visible light then it won’t be black at all.  It might glow red, orange, or even white.

We now know that, even at room temperature, all objects are giving off their own light, but it is infrared light, a type of light to which are eyes are not sensitive and is thus invisible unless a thermal imaging camera is used.  Stars despite the obvious fact that they are not black (unless they’re black holes!), act very nearly as blackbodies.  The Sun is an example of a blackbody and most of its light is in the visible spectrum, which is what we see, but the Sun also has light that appears in the UV portion, as well as the infrared range.

According to classical electromagnetism, the number of ways an electromagnetic wave can vibrate a in a 3-dimensional cavity, per unit frequency, is proportional to the square of the frequency.  This means that the power you would get out per unit frequency should follow the Rayleigh-Jeans law, meaning that the power would be proportional to the frequency squared.  So if you put the frequency up higher and higher the power would be unlimited.  Planck said that electromagnetic energy did not follow the classical description.  He said that it could only be emitted in discrete packets of energy proportional to the frequency.  He developed equations where the radiation eventually goes to zero at infinite frequencies, and the total power is finite.  Planck called these packets of energy ‘Quanta’.

During the 19th century much attention was given to the study of heat properties of various objects. An idealized model that was considered was the Black Body, an object which absorbs all incident radiation and then re-emits all this energy again.  We can expect most of the energy at higher frequency, and this energy diverges with frequency.  If we try and sum the energies at each frequency we find that there is an infinite energy in this system!  This paradox was called the Ultraviolet Catastrophe.

The greatest crisis physics has ever known came to a head over afternoon tea on Sunday, October 7th, 1900, at the home of Max Planck in Berlin.  The main theories were in place, all the great discoveries had been made, and only a few minor details needed filling in here and there by generations to come.  It was a view, disastrously wrong but widely held at the time, fueled by technological triumphs and the seemingly all-pervasive power of Newton’s mechanics and Maxwell’s electromagnetic theory.  Planck had been taught by a German physicist named Gustav Kirchhoff, who among other things, laid down some rules about how electrical circuits work (now known as Kirchhoff’s laws) and studied the spectra of light given off by hot substances.  In 1859, Kirchhoff proved an important theorem about ideal objects that he called blackbodies.  Kirchhoff proved that the amount of energy a blackbody radiates from each square centimeter of its surface hinges on just two factors, being the frequency of the radiation and the temperature of the blackbody.  Kirchhoff challenged theorists and experimentalists to figure out and measure (respectively) the energy/frequency curve for this cavity radiation.

Stefan’s Law in 1879, named after Austrian Physicist, Josef Stefan became the first quantitative conjecture based on experimental observation of hole radiation yielding the total power P radiated from one square meter of black surface at temperature T goes as the fourth power of the absolute temperature.  Stefan showed empirically that the radiation of such a body was proportional to the fourth power of its absolute temperatures.  This relationship became known as the ‘Stefan-Boltzmann Law’ after it had been deduced by another Austrian physicist and philosopher Ludwig Eduard Boltzmann in 1884 from thermodynamic considerations.  In 1884, Boltzmann derived this T 4 behavior from theory, where he applied classical thermodynamic reasoning to a box filled with electromagnetic radiation, using Maxwell’s equations to relate pressure to energy density.

By the 1890s, various experimental and theoretical attempts had been made to determine its spectral energy distribution, the curve displaying how much radiant energy is emitted at different frequencies for a given temperature of the black body.  Planck was convinced thermodynamics was the key to understanding nature at the deepest level.  He spent years clarifying the subtleties of the Second Law (that entropy always increases).  He believed the Second Law was rigorously correct, and would eventually be proved so in a more fundamental theory.  And now thermodynamics had made a good start in analyzing black body radiation, with proofs of Stefan’s Law and Wilhelm Wien’s Displacement Law, which stated that the black body radiation curve for different temperatures peaks at a wavelength inversely proportional to the temperature.  Planck thought it very likely that thermodynamics would yield the whole black body radiation curve.  He felt this curve was the key to understanding just how electromagnetic radiation and matter exchanged energy.  This was one of the basic problems in physics, and of obvious technological importance.  In fact, just at that time experimentalists at his university were measuring the black body radiation curve to new levels of precision.

When one thinks of the pioneers of quantum physics, names such as Dirac, Einstein, Bohr, Heisenberg and Schrödinger invariably spring to mind.  However, it was Max Planck’s profound insight into thermodynamics which came from the work he did on black body radiation that set the stage for the revolution to come.  While Planck’s radiation law was readily accepted, the importance of its conceptual novelty, its basis in energy quantization, took several more years to gain notice.  Once it did, physics would never be the same.  Planck was especially intrigued by the formula found in 1896 by his colleague Wilhelm Wien, and he made a series of attempts to derive “Wien’s law” on the basis of the second law of thermodynamics.  By October 1900, however, other colleagues had conducted additional experiments and found definite indications that Wien’s law, while valid at high frequencies, broke down completely at low frequencies.  So Planck went back to work.  He knew that the entropy of the radiation had to depend mathematically upon its energy in the high-frequency region if Wien’s law held there.  He also saw what this dependence had to be in the low-frequency region in order to reproduce the experimental results there.  He guessed, therefore, that he should recombine these two expressions in the simplest possible way, and thus transform the result into a formula relating the energy of the radiation to its frequency.

In 1900, actually some months before Planck’s breakthrough work, John William Strutt, 3rd Baron Rayleigh an English physicist known as Lord Rayleigh was taking a more direct approach to the radiation inside the oven, not even thinking about oscillators in the walls, as he just took the radiation to be a collection of standing waves in a cubical enclosure termed electromagnetic oscillators.  Then in 1905, Sir James Hopwood Jeans an English physicist, astronomer and mathematician derived the spectral distribution of thermal radiation on the basis of the assumption that the classical idea on the uniform distribution of energy is valid.  However, the temperature and frequency dependencies obtained basically differed from Wien’ s relationships.  According to the results of fairly accurate measurements, carried out before that time, and to some theoretical investigations.  Wien’s expression for spectral energy distribution was invalid at high temperatures and long wavelengths.  This circumstance forced Planck to turn to consideration of harmonic oscillators, which have been taken as the sources and absorbers of radiation energy.  Using some further assumptions on the mean energy of oscillators, Planck derived Wien’s and the Rayleigh-Jeans’ laws of radiation.

In the 19th century a major problem for physicists was to predict the intensity of radiation emitted by a black body at a specific wavelength.  Wilhelm Wien made a theory that predicted the overall form of the curve by treating the radiation as gas molecules.  However, at long wavelengths his theory disagreed with experimental data.  Rayleigh and Jeans then produced a formula by considering the radiation within the black body cavity to be made up of a series of standing waves.  They thought that electromagnetic radiation was emitted by oscillating atoms in the walls of the black body and this radiation set up a standing wave between the walls.

Useful as the blackbody curve is, there was a serious problem that the energy distribution from a blackbody did not match scientists’ predictions, which were based on the electromagnetic (EM) wave properties of light.  The predictions suggested that an ideal black body at thermal equilibrium would emit energy proportional to the frequency squared.  So the amount of energy radiated simply gets greater and greater at higher and higher frequencies.  The name Ultraviolet Catastrophe was used to describe this failure of theory because UV light has a higher frequency than visible light.  The total energy output at any frequency is not infinite, because this would conflict with the law of conservation of energy.  Planck, proposed that the electromagnetic energy emitted by a black body could only exist in discrete packets he called quanta, and these were later called photons.

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