Max Planck's Derivation
Blackbody Radiation Law
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Most of us first got the story in its oversimplified form. In December 1900 Max Karl Ernst Ludwig Planck (1858 Apr 23 - 1947 Oct 04) revealed to the scientific community what he had worked out in October and presented to the German Physical Society in a lecture - the formula accurately describing the distribution of energy over the frequencies comprising blackbody radiation. We remember the story because of a strange little proposition that Planck had to assume into his premises in order to obtain a formula that matched the results of experiments.
Planck used an imaginary experiment to guide him as he worked out his derivation. To avoid obtaining an absurd result from his analysis of that experiment, Planck had to postulate that the passive electromagnetic resonators that he imagined in his apparatus could only emit energy in amounts characteristic of the resonators themselves. For an oscillator that has a characteristic frequency of , Planck stated, energy can only come in integer multiples of
in which h represents what Planck himself called the "quantum of action". Thus Planck begat the quantum theory. Well,...sort of.
Reality is rarely as good as the stories we tell about it. Sometimes it's better.
Some of the most profound discoveries in physics originated in observations of the most trivial phenomena. The twitch of a compass needle in a thunderstorm set physicists onto the path that led them to Maxwell's Equations. A spoiled photograph led physicists ultimately to the Standard Model of matter. And a spot of light emerging from a hot stove led them to the quantum theory.
We can best map out the beginning of that latter path by following the plan of Planck's own paper, "On the Law of Distribution of Energy in the Normal Spectrum", which Planck had published in Volume 4 of Annalen der Physik in 1901. But before we do that, we need to see what Planck was actually seeking to accomplish.
By the middle of the Nineteenth Century many people in the social class whence scientists came knew that the light coming from a tiny hole or crack in the side of a stove differs from the light coming directly from the fire inside the stove. They knew that with experience a person with good color discrimination could judge relatively accurately the temperature of the stove from the color of the light emanating from such a hole, just as musicians can, with experience, tell the frequency of any note they hear. Physicists eventually came to call that special manifestation of light cavity (Hohlraum) radiation and then they found that they could give it another name when they discovered that it represents a fundamental problem in the physics of heat and light.
Ever since Isaac Newton (1642 Dec 24 - 1727 Mar 20) used a glass prism to spread a ray of sunlight out into all the colors of the rainbow and to show that those colors comprise the irreducible components of light scientists had wondered what those facts mean. Did those facts offer clues to the knowledge of how Nature makes light and why it does so? In 1800 Friedrich William Herschel (1738 Nov 15 - 1822 Aug 25), the man who discovered Uranus, added another fact to Humanity's knowledge of light, another clue to the answer to the question. As Newton had done, Herschel used a prism to spread sunlight into the multi-colored spectrum and then he put a thermometer into that rainbow to measure how much of the sun's heat came in each color. Then he discovered infrared radiation when he moved the thermometer into the dark zone next to the red light, expecting to see the heating drop to zero and seeing instead that the thermometer gained more heat from that "darkness" than it did from the red light.
Throughout the Nineteenth Century physicists improved that experiment and explored the spectra that they made of light emanating from various sources. In 1814 Joseph von Fraunhofer (1787 mar 06 - 1826 Jun 07) set up an improved version of Newton's experiment. Fraunhofer passed a collimated beam of light through a narrow slit and then sent the light from the slit into a glass prism whose axis of symmetry ran parallel to the slit. With that apparatus Fraunhofer eliminated much of the blurring inherent in Newton and Herschel's version of the experiment. When he brought sunlight into that apparatus he discovered thin bright and dark lines crossing the spectrum. Fraunhofer got a clearer picture, but of what?
In 1859 Gustav Robert Kirchhoff (1824 Mar 12 - 1887 Oct 17) and Robert Wilhelm Bunsen (1811 Mar 31 - 1899 Aug 16) built a version of Fraunhofer's apparatus that used two prisms instead of one. With that improved spectroscope they spread their spectra out wider and thus obtained more accurate measurements of the locations of Fraunhofer's lines on them. Using their expertise as chemists (at a time before chemistry and physics fissioned into two separate disciplines), the two men observed the spectra in light emanating from incandescent samples of the chemical elements known at the time and then discovered that they could match the bright lines in some of those spectra with some of the dark lines in the spectrum obtained from sunlight. From those facts they inferred the existence of the corresponding elements in the sun's atmosphere and even deduced the existence of two new elements, Rubidium and Cesium, from lines that they could not match up with known patterns.
In order to make that inference, Kirchhoff and Bunsen had to assume into their premises a statement to the effect that a chemical element emits radiation preferentially at any wavelength at which it absorbs radiation preferentially. We can prove and verify that statement, much as Kirchhoff did, by performing a simple imaginary experiment in the laboratories of our minds. Imagine a small body suspended (or floating weightless) inside a closed cavity inside a larger body that we keep at a certain temperature. Radiation inside the cavity transfers heat between the bodies until they both have the same temperature and thus come into a state of thermal equilibrium with each other. Because of the work of Rudolf Julius Emanuel Clausius (1822 Jan 02 - 1888 Aug 24) and William Thomson, First Baron Kelvin (1824 Jun 26 - 1907 Dec 17) in 1850, on what we now call the second law of thermodynamics, Kirchhoff knew that a system in thermodynamic equilibrium won't spontaneously go out of it.
Now imagine that we have coated the small body with a thin filter that reflects all radiation perfectly except for a narrow band of wavelengths that it allows to pass. The body will absorb radiation at those wavelengths and no others, so it must emit radiation at those wavelengths and no others. If Reality did not so structure matter that it conforms to that statement, then we would have the ability to contrive a violation of the second law of thermodynamics. We might devise a situation in which the small body reaches the same temperature as the larger body has and yet absorbs heat faster than it radiates heat (or vice versa). But we know that we cannot do that, so we know that Reality conforms to what physicists call the principle of detailed balance.
Stated more broadly as Kirchhoff's law, that principle tells us that any body emits radiation as efficiently as it absorbs radiation. Physicists represent that efficiency as a fraction and for any given body they want to know that fraction as a function of the wavelength or the frequency of the absorbed radiation. But before they could get into that study the physicists of the Nineteenth Century needed to understand the simplest possible case of the interaction between matter and heat radiation. In 1860 Kirchhoff described that simplest case, a body that absorbs radiation with 100% efficiency at all wavelengths. At room temperature such a body appears dead black, so we call it a blackbody, even when it glows white with heat.
Having defined a blackbody by specifying its absorptivity, Kirchhoff wanted to know how the blackbody emits radiation. Wanting to ignore absorption altogether, he imagined a blackbody in total darkness and heated to and kept at some temperature. In that circumstance heat comes out of the body as pure blackbody radiation (a term Kirchhoff coined in 1862). Because a blackbody has perfect absorptivity over all wavelengths, the spectral energy density of the radiation emanating from it must conform to a formula that depends only upon the wavelength of the emitted radiation and the absolute temperature of the blackbody. Such a universal function, Kirchhoff asserted, describing the distribution of energy among the wavelengths or the frequencies of blackbody radiation, would offer profound insight into the relationship between matter and electromagnetic radiation. Thus, determining the proper mathematical description of blackbody radiation became the highly important problem that Planck solved in the autumn of 1900 and described in his 1901 paper.
Spectral measurements gave physicists the real-world touchstones against which to test their ideas on blackbody radiation. Thus Planck began his paper by acknowledging the work of several researchers at the Physikalisch-Technische-Reichsanstalt (Imperial Physico-Technical Institute, the forerunner of modern Germany's National Bureau of Standards). Indeed, because those men were conducting and improving the most advanced experiments in infrared bolometry of the time, Planck could not have made his discovery prior to October 1900, even though he had been working on the blackbody radiation problem for several years.
In July of 1900 Otto Richard Lummer (1860 Jul 17 - 1925 Jul 05) and Ernst Georg Pringsheim (1859 Jul 11 - 1917 Jun 28) revealed the results that they had obtained from an apparatus that Lummer and Ferdinand Kurlbaum (1857 Oct 04 - 1927 Jul 29) had built in 1898. Lummer and Kurlbaum had made a tube of blackened platinum and mounted it inside a closed ceramic tube that had a hole in its side. The cavity radiation coming from that hole precisely mimicked the radiation that would come from a blackbody at the same temperature as that of the platinum tube. Using a bolometer whose design was inspired by that of the bolometer that Samuel P. Langley had used for astronomical measurements since 1880, Lummer and Pringsheim measured the power distribution in the infrared radiation emanating from the hole in their cavity radiator. They measured the heat in wavelengths as long as 8.3 microns (red light has a wavelength of a little less than one micron) as they varied the temperature of the platinum tube from -188 Celsius (-306 Fahrenheit) to +1650 Celsius (+3002 Fahrenheit).
At 1650 Celsius, Wien's displacement law tells us, a blackbody radiates most intensely at a wavelength of 1.486 microns. By making measurements at wavelengths longer than that, Lummer and Pringsheim had extended the realm of their measurements into that part of the infrared spectrum where blackbody radiation conforms less to the law that Wilhelm Wien had proposed in 1896 and more to the Rayleigh-Jeans law. But Lummer and Pringsheim did not obtain a discrepancy between their observations and Wien's law large enough to give them certainty. Later in the year, though, Heinrich Leopold Rubens (1865 Mar 30 - 1922 Jul 17) and Ferdinand Kurlbaum extended the range of measurable wavelengths to 50 microns and obtained results that removed all doubt. On October 07 Rubens revealed his results to Planck over afternoon tea and that evening Planck solved the problem of blackbody radiation.
Planck assumed that Wien's energy distribution law gives a reasonably good account of blackbody radiation, as indeed it does for the shorter wavelengths of the radiation, and that he needed only to re-examine the conditions that Wien had assumed into his derivation, find the term susceptible to change, and replace it with something more suitable. Two paths toward that goal lay open to him and he chose to follow the more difficult of them.
The easier path begins with Wien's distribution law as it was known in 1900,
in which u(ν,T) represents the power density in radiation of frequency ν emanating from a unit area of a blackbody at absolute temperature T, with a and b representing constants that experimenters would have to determine from their observations of blackbody radiation. Following that path, Planck would have sought to meld Wien's law with the radiation law devised by John William Strutt, Third Baron Rayleigh (1842 Nov 12 - 1919 Jun 30) and corrected by Sir James Hopwood Jeans (1877 Sep 11 - 1946 Sep 16),
To that end he would recall that
so that in the limit of frequencies approaching zero (wavelengths becoming extremely long) Wien's law becomes an excellent approximation to
If Planck were then to subtract one from the denominator in Wien's law to produce
then that limit would give it the form of the Rayleigh-Jeans law; that is,
If we give the constants the correct form (a=8πh/c3 and b=h/k), then Equation 6 becomes Planck's version of the radiation law.
But rather than look at the energy distribution directly, Planck focused his attention on the entropy of the cavity radiator that mimics a blackbody. By working out the relation between the entropy S of an irradiated, monochromatic, vibrating resonator and its vibrational energy U, Planck figured that he could exploit the standard thermodynamic relation
to work out the normal energy distribution of heat radiation.
In his Nobel lecture (1920 Jun 02) Planck sketched out a step that he did not mention in his 1901 paper, but which gives a good indication of his thinking in October 1900. Instead of looking at the entropy directly, he chose to focus his attention on the second derivative of the entropy with respect to the resonator energy; that is,
He explained that he chose that particular function "since this has a direct physical meaning for the irreversibility of the energy exchange between resonator and radiation."
Prior to the summer of 1900 physicists had measured cavity radiation that conformed to the statement
in which the minus sign denotes the facts that a) adding energy to a resonator increases its temperature and thus decreases the inverse of its temperature and b) we want to represent our proportionality factors with positive numbers. If we integrate that equation with respect to an increase of resonator energy in light of Equation 9, we get
which we solve for
We recognize that equation as expressing the temperature-dependent factor in Wien's distribution law.
The work of Lummer and Pringsheim and, later, of Rubens and Kurlbaum yielded results that conformed to the statement
If we integrate that equation with respect to an increase in resonator energy in light of Equation 9, we get
which we solve for
We recognize that equation as expressing the temperature-dependent factor in the Rayleigh-Jeans law.
Facing those two statements, Planck conceived a compromise, one that could become either statement under the right conditions. He postulated that
If we integrate that statement with respect to an increase in resonator energy in light of Equation 9, we get
which we solve for
And we recognize that equation as expressing the temperature-dependent factor in Planck's blackbody radiation law.
But Planck had to go much farther than that to obtain the complete function describing the spectral energy density of blackbody radiation. So he devoted the first section of his paper to applying the work of Ludwig Eduard Boltzmann (1844 Feb 20 - 1906 Sep 05) to his problem.
"I. Calculations of the Entropy of a Resonator
as a Function of its Energy"
At the end of the Nineteenth Century any physicist who sought a theoretical understanding of blackbody radiation imagined heating a hollow body that had a small hole drilled in its side. That physicist then imagined that the cavity inside that body contained a large number of electromagnetic dipole resonators of undetermined composition: absorbing and re-emitting radiation more or less at random, those resonators mixed the radiation to ensure that it filled all of the modes of electromagnetic vibration available inside the cavity.
Classical electromagnetic theory, completed by James Clerk Maxwell (1831 Jun 13 - 1879 Nov 05) in the 1860's, provides a straightforward means of calculating the number of vibrational modes inside the cavity. Imagine that we have made the cavity in the shape of a cube whose sides have length L. We heat the cavity's walls to some temperature T and fill the cavity with radiation until the energy in the radiation comes to thermal equilibrium with the energy in the walls. Within the cavity the radiation must take the form of standing waves whose electric fields conform to solutions of the wave equation,
And we have as a boundary condition the statement that the electric field must equal zero on the walls, because otherwise the electric currents that the field would induce in the walls would move a net amount of energy between the walls and the radiation field, thereby destroying our assumed equilibrium. Under that condition we have as a solution of Equation 19
in which λ represents the length of the standing wave.
If we feed that solution back into Equation 19 and then carry out the differentiations, divide out the factors common to all four terms, and then multiply the result through by L2, we obtain
That equation resembles the algebraic expression of the Pythagorean theorem in three dimensions. Indeed, if we require that the sum n1+n2+n3 equal a very large number, then we can interpret that equation as the description of a spherical surface in an abstract n-space. That surface has a radius
and encloses a volume
If we take only the positive values of the integers n1, n2, and n3 that satisfy Equation 21 as representing the modes in the cavity, then the volume of the plus-plus-plus octant of the n-sphere tells us the number of potential modes in the cavity. Multiplying that number by two to acknowledge that each solution of the wave equation comes with two mutually perpendicular polarizations thus gives us the number of modes as
We divide that number by the volume of the cavity to obtain the density of modes and then differentiate with respect to wavelength to get the density of modes per unit wavelength; that is,
But we actually want to know the number of modes per unit frequency, so we exploit the chain rule and the fact that λ=c/ν to get
Once physicists had worked out that equation, they had only to multiply the expression on the far right side of the equality sign by the average energy per mode in the radiation to gain a mathematical description of the spectral density of blackbody radiation. Baron Rayleigh got that particular ball rolling by assuming a direct analogy between the modes in the cavity and the particles in an ideal monatomic gas. He invoked the equipartition theorem and multiplied Equation 26 by kT. Comparison of the result with observations drawn from experiments made clear the fact that electromagnetic vibrations in a cavity resonator do not mimic a classical gas.
Planck took a different approach, not looking at the radiation directly. Asserting equilibrium between the set of resonators in the cavity and the radiation field, he sought to calculate the average entropy of one resonator and then parlay that description into a description of the average energy in the vibrational modes of the radiation field. Classical thermodynamics, of the kind developed largely by Rudolf Clausius, doesn't allow such a derivation, but the statistical thermodynamics developed largely, at that time, by Ludwig Boltzmann does, though it also obliged Planck to assume a strange little proposition into his premises.
Planck wanted to use Boltzmann's famous equation
so he had to devise a way to count the number W of microstates that correspond to a given macrostate of one of his resonators. To that end he postulated M identical resonators inside his hypothetical cavity and defined their macrostate by assuming a total energy E distributed among them such that
in which U represents the average energy of one resonator. That macrostate has entropy
in which S represents the average entropy of one resonator.
In order to calculate a value for W, "it is necessary to interpret E not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energy element E; consequently we must set
where P represents a large integer generally, while the value of E is yet uncertain." In taking that step Planck laid the cornerstone of the quantum theory. In other venues he called it an act of desperation, even though his choice to use Boltzmann's techniques necessitated his use of finite numbers in his calculation of the entropy and he could only get finite numbers by assuming into his premises the quantization of energy. But note his proviso that the value of the energy element remains undetermined.
Now we want to calculate, as Planck did, the number of ways in which we can distribute P energy elements over M resonators. Basic combinatorics tells us that
Because M and P represent very big numbers, we can ignore the minus ones in that formula and we can use Stirling's approximation,
Thus we can rewrite Equation 31 as
That equation gives our macrostate an entropy of
To extract a description of the average entropy of one resonator from that equation we must factor out M, so we note that P=MU/E and write
in which we have exploited the fact that lnAB = lnA + lnB in going from line 2 to line 3. Referring to Equation 29, we divide the above equation by M and obtain
expressing the average entropy of a single resonator. From that equation Planck could have finished his derivation directly, but he chose to take some additional steps first.
"II. Introduction of Wien's Displacement Law"
This famous law, devised by Wilhelm Carl Werner Otto Fritz Franz Wien (1864 Jan 13 - 1928 Aug 30) in 1893, gave Planck an alternative way to derive a formula describing the entropy of a single resonator. But he used it in a form, Wien's distribution law, that most of us find unfamiliar, a form that looks more like a version of the Stefan-Boltzmann law. In that form, though, it gave Planck the means to calculate the entropy through almost purely algebraic manipulation, with very little physics involved.
Wien's distribution law tells us that
in which equation u(λ)dλ represents the volume density of the blackbody radiation manifested in electromagnetic waves with wavelengths between and λ+dλ. We also have T representing the absolute temperature of the blackbody under consideration and 0f(λT) representing some function of the product λT.
Planck tells us that he wants to consider the general case, in which we immerse the blackbody in an arbitrary diathermic medium, a medium whose index of refraction does not necessarily equal 1.000.... But the length of electromagnetic waves depends upon the index of refraction of the medium in which they propagate. Because he wanted to work on describing monochromatic radiation and wanted to minimize the variables in his algebra, Planck re-expressed Equation 37 in terms of the radiation's frequency, which does not depend at all upon the index of refraction of the medium through which the radiation propagates. Taking udν to represent the volume density of the radiation comprising waves with frequencies between ν and ν+dν, he replaced λ with c/ν and dλ with cdν/ν2 and obtained
in which 1f represents a function that may differ from 0f.
Next Planck wanted to remove the speed of light from his unknown function. To that end he invoked what he called "the well-known Kirchhoff-Clausius law" (not so well known now, though: modern sources don't even mention it). According to that rule a body radiates power at a rate inversely proportional to the square of the speed at which the radiation propagates through the medium in which the body is immersed. That means that the irradiance of the heat radiation emanating from a blackbody is also inversely proportional to the square of the speed of light. But the irradiance equals the product of the radiation's energy density and its speed of propagation, so the spectral density must be proportional to the inverse cube of the speed of light. Thus Planck transformed Equation 38 directly into
Now we can absorb the fifth power of T/ν into the undetermined function, so that equation becomes
We already know that the relationship between u and U, the average energy of a single resonator, reflects the number of modes in the cavity resonator; that is,
Equating that to Equation 40, we get
Algebraic theory tells us that we can solve that equation to obtain
or, more relevant to our need,
Now Equation 8 lets us relate that equation to the average energy of one of Planck's resonators, giving us
which we integrate to obtain
Thus we infer that we must describe the average resonator entropy with a function of only the ratio U/ν.
But Equation 36 tells us that we must describe the resonator's average entropy with a function of only the ratio U/E. Planck reconciled those two requirements by writing
in which h represents a factor that converts units of frequency into units of energy (only later did physicists recognize that conversion factor as describing the quantum of action). Thus Planck's description of the average resonator entropy became
Differentiating that equation with respect to U takes us back to Equation 8 in the form
When we solve that equation for U we get
Substituting that result into Equation 41 then gives us Planck's law of blackbody radiation,
"III. Numerical Values"
In a final flourish Planck calculated the values of Boltzmann's constant and of his own new conversion factor. He began by reporting that Ferdinand Kurlbaum had measured the total heat radiating into air from one square centimeter of blackbody at zero Celsius (273 Kelvin) and at one hundred degrees Celsius (373 Kelvin) and had found that the difference came to Δφ=0.0731 watt/cm2 or 7.31x105 erg/sec/cm2. From that datum Planck calculated, by way of the Stefan-Boltzmann law, the energy density of blackbody radiation emanating into air at the absolute temperature T=1:
Then Planck used his own formula (Equation 50) for T=1, integrated over all frequencies, to calculate the same number:
Carrying out the term-by-term integration, he obtained an infinite series
Setting that equal to Equation 52 enabled Planck to calculate
Next Planck reported that Otto Lummer and Ernst Pringsheim had determined the value of λmT, the constant that comprises the content of Wien's displacement law, as
(The current value is 2897.756 micron-degree Kelvin). Then he set up the theoretical calculation of the same constant by differentiating Equation 51 with respect to and set the derivative equal to zero in order to determine the wavelength that corresponds to maximum power output in the radiation. He thus obtained a transcendental equation
which he had to solve numerically to obtain the desired result,
(The current value of the coefficient in the denominator in that equation is 4.9651142). Merging Equations 56 and 58, he got
Calculating the fourth power of that equation and then multiplying the result by Equation 55 gave Planck his proportionality constant directly,
(The current value is h=6.62608x10-34 Joule-sec). Substituting that value into Equation 59 then gave him Boltzmann's constant,
(The current value is k=1.38066x10-23 Joule/degK.).
Thus Planck laid the cornerstone upon which he
and other physicists of the early Twentieth Century built the grand edifice of
the Quantum Theory.
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