Blackbody Radiation Exploration

History

In physics, a black body is defined as any object which absorbs all electromagnetic radiation that falls upon it. The black body also emits radiation with wavelengths solely determined by the black body’s temperature. For instance, black bodies appear black when cold. This is why the term “black body” was coined by German physicist Gustav Kirchoff, who studied electrical circuits and spectroscopy along with developing a theory for the nature of black-body radiation emitted by heated objects. There are no “ideal blackbodies” (objects that only absorb and emit EM radiation; they do not reflect it) in nature, but graphite comes fairly close to representing the thermal radiation emission expected from an ideal black body.

Implications for Quantum Mechanics

Figure 1

It was the studying of the laws of the black body that eventually led to the study of quantum mechanics. Figure 1 shows the relationship between the intensity of the incident light and the resulting intensity of radiation. As the wavelength increases, the peak of the radiation intensity curve goes to lower and lower energy. The “classical theory” curve shown on the graph is predicted by the Rayleigh-Jeans law. This law, proposed in the early 20^th century, attempts to describe the intensity of electromagnetic radiation at all wavelengths from a blackbody at a certain temperature by using classical arguments. The Rayleigh-Jeans law was successful in predicting electromagnetic radiation from black bodies at high wavelengths and low energy, but rapidly (and incorrectly) approached infinity as the wavelengths became shorter. This disagreement between theory and experimental results is known as the ultraviolet catastrophe. This model incorrectly assumed that energy in blackbody systems was continuous, rather than quantized. As a result, the Rayleigh-Jeans law proved incorrect for applications of quantum mechanics.

The Ultraviolet Catastrophe

The classical theory states that a body can absorb or emit energy in any quantity and continuously. Given that a blackbody only emits energy as heat (corresponding to long wavelength, low frequency, low energy ER), classical theory implies that no matter how short the wavelength or high the frequency (how high the energy) of the absorbed ER is, the blackbody should have no problem continuously emitting that energy as heat (low energy ER) instead of as higher-energy forms of ER. The higher the incident energy, the more intense thermal transitions are in the molecules of the blackbody, leading to more intense heat energy emitted. The key of this prediction is that as the wavelength of the incident radiation approaches 0 (and its energy therefore approaches infinity) the heat emissions of a blackbody should correspondingly approach infinity. This is known as the Ultraviolet Catastrophe.

Why so catastrophic? Because it catastrophically failed at predicting the experimental results. In the experiment, as wavelength decreased the intensity of emitted heat increased, and then abruptly dropped to zero. Clearly a classical interpretation of energy was not working. How then could quantum mechanics explain the results?

Recall that quantization of energy means that energy can only be absorbed by a given substance at discrete and specific values. If the energy of an incident photon matches one of those values, all of that energy is absorbed at once. If not, no energy is absorbed. At progressively shorter wavelengths, the probability that incident energy will increase the molecular kinetic energy rather than provoke a higher energy transition in a given substance gets smaller. Hence, as wavelength shortens and energy increases within the range that corresponds to thermal transitions, the intensity of the emitted heat increases. As wavelengths get even shorter and energy moves into the range not associated with thermal transitions (ultraviolet, X-ray, and gamma ray wavelengths), there is a rapid drop-off in the intensity of heat emitted by the blackbody.

Planck’s Solution

If the shape of the blackbody radiation curve could be explained by quantum principles, the next logical step would be to incorporate those principles into a mathematical underpinning for the curve. Enter Max Planck. Plank assumed that the electromagnetic modes of a particle were first of all quantized in energy, and that energy could be given by the frequency times some constant h. Planck also knew that a given mode would have a probability of being occupied that was related to its energy hv, given by

(e^(hv/kT)-1)^-1

Therefore, the average energy of a given mode would be given by energy times probability, or

hv(e^(hv/kT)-1)^-1

Note that as frequency v increases from 0, the denominator starts at nearly 0 but grows at an exponential rate compared to the numerator hv, meaning that at low frequencies the average energy of a mode approaches infinity, and at high frequencies approaches 0.

Planck also knew that intensity could be given by energy times the density of a given mode. The density of a given mode is given by

(8πv²/c³)

which approaches infinity at high frequency and 0 at low frequency. Planck’s final equation for Intensity as a function of frequency is

I = (8πv²/c³)hv(e^(hv/kT)-1)^-1

which, since one factor approaches 0 at low frequency and the other approaches 0 at high frequency, keeps the intensity from approaching infinity and averts the ultraviolet catastrophe. Planck’s model, once the constant h was determined, fit experimental data remarkably well.

Significance of Planck’s Constant ‘h’

Planck’s equation derived using statistical mechanics and the quantization of energy is:

This gives the intensity of the radiation at a given frequency and temperature. The other values in this equation are constants: c is the speed of the light, the constants π and e, and k and h are additional constants (whose values were unknown at the time). Using experimental data, Planck managed to calculate them; his value for h was off by 1%, while his value for k differed by only 2.5% from currently accepted values. Both of these are universal constants and appear frequently in chemistry and physics.

In Planck’s original paper, h results strictly from the mathematics. He assumed quantization of energy, but not much more. He considered this constant more a theoretical curiosity than anything else. As dramatic (and accurate) as his proposal was, in 1905 the physics community would be turned upside down by a young physicist by the name of Albert Einstein. Einstein was studying the photoelectric effect. The effect is when light hits the surface of a material (typically a metal) and electrons are released. Despite classical mechanics predictions, the kinetic energy of the electrons was proportional to the frequency of the light, and if the number of electrons emitted was greater than zero, than the number of electrons was proportional to the intensity of the light. When modeling this behavior for a variety of substances, he obtained the following relation:

KEMax=hν-Φ

The maximum kinetic energy of electrons released is proportional to the frequency minus a constant that varies by solid (the work function). This was an experimental result. Perhaps surprisingly, Planck’s constant seems to have appeared out of nowhere. Planck assumed that energy changes are discrete, but Einstein then proposed that light consists of photons with quanta of energy equal to¹:

E=hν

He then went on to explore the particle nature of light, which results from the above assumptions. Although Planck had found a mathematical constant, Einstein had a theoretical proposal into what that constant implied, specifically how it is related to the energy of photons in light. For these results, Planck and Einstein each earned a Nobel Prize (the former in 1918, the latter in 1921)².

This was the beginning of quantum physics. By 1930, de Broglie’s matter waves had been introduced, the Schrodinger equation had been worked out, Born’s probabilistic interpretation had been created, Heisenberg’s matrix mechanics led to the uncertainty principle, and many other results had been theorized and verified. Examples of results include electron microscopes, x-ray crystallography and other spectroscopic methods, and ultimately how we understand atoms and bonding today. Besides changes in chemistry, quantum mechanics and general relativity paved the way for modern physics, which led to things like computers and the atomic bomb. The significance of the above cannot be understated.

Conclusions:

The study of quantum mechanics became increasingly important as theories and concepts of classical physics began to fail. The Rayleigh-Jeans law represents one of the most catastrophic failures of classical mechanics in the late 19^th and early 20^th centuries. Were it not for such revolutionary minds such as Einstein and Planck, we might still be trying to decipher the reason behind the ultraviolet catastrophe and the motions and actions of particles too small for the human eye to see. The study of quantum mechanics continues, with efforts to further understand the way electrons, atoms and molecules act together to provide us with the effects of life that we see today.

References:

Oxtoby, D, W. Introduction to Quantum Chemistry: Chemical Principles, Part 1; Cengage Learning: Mason, Ohio, 2008.

Yale Department of Chemistry. “Planck." http://www.chem.yale.edu/~chem125/125/history99

/7BondTheory/planck.htm (accessed Oct 23, 2010).

http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html