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Black Body Radiation And The Ultraviolet Catastrophe Essay About Myself

Ultraviolet, not ultraviolence

In some of the past BMC posts, I have blogged about how statistical mechanics, in the 19th century, came perilously close to uncovering quantum mechanics early.  A number of “problems” with statistical mechanics arose due to the classical treatment used.  One of the most serious was the Ultraviolet Catastrophe. This problem was not easily solved with the hand-wavey pseudo-quantum explanations used in some previous cases.  It took a full-on, quantized description of electromagnetism, and helped usher in quantum mechanics to being the foremost theory in subatomic physics.

Black body Radiation

The catastrophe arises from considering a phenomena called black body radiation. A black body is a perfect absorber, and a perfect emitter of radiation.  What that means, is that all the light, x-rays, microwaves, etc. (all forms of electromagnetic radiation) striking the object are completely absorbed (increasing its temperature), and that it is a perfect radiator (it glows because of its temperature).  The glow it gives off increases in frequency as the object heats up (which is why a hot steel will go from red to yellow to white as it heats).  A black body can be used as an approximation for most physical objects, from stars to people, but is one of the many non-physical objects used in physics like massless strings and frictionless planes.  Perhaps Hotblack Desiato could make one, but we haven’t yet.

The Ultraviolet Catastrophe

The problem arose when trying to determine how much power was radiated by the black body.  When physicists in the 19th century attempted to calculate this, they turned to two tools: classical harmonic oscillators (things like vibrations on a string), and the equipartition theorem.  The classical harmonic oscillator theory of radiation told them that the number of oscillation modes in a 3D box is proportional to the frequency of the wave squared, and that the power of a wave was dependent on the frequency of it.  The equipartition theorem told them that each mode of vibration was a degree of freedom, and that it stored a fixed amount of energy, dependent on the temperature.  BIG SCIENCY WARNING BELLS WENT OFF.  These two things give you that as you increase the vibration mode, you get a massive increase in degrees of freedom, and therefore energy, all the way up to infinity.  These calculations told them that any object above absolute zero emits an infinite amount of radiation, most of it in ultra-high energy gamma rays (ultraviolet was the whiz-bang high energy light of the time, hence the name).

Planck Saves the Day

Nice 'stache, Max Planck!

The solution to this catastrophe came from none other than Max Planck.  He modified the idea of using classical harmonic oscillators as a model for radiation by introducing quantized emission.  In other words, the photon: a discrete “packet” of EM radiation.  These packets have energy proportional to their frequency, which means that as you approach infinite frequency, their energy expands to infinity as well.  This dropped the number of available modes in the cavity for higher frequencies, causing the emission to go to zero at very high frequencies (rather than ballooning to infinity).  The result of this modification was a change in how radiation was viewed (which, along with Einstein’s photoelectric effect helped cement the idea of photons) from a classical oscillator to a gas of Bosons (non-excluding, indistinguishable particles).  The result of Planck’s work was Planck’s Law of black body radiation, which accurately describes the spectrum of hot objects, allowing us to easily gauge the temperature of objects by examining their radiation.

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In Physics, we refer to an object which radiates perfectly as a blackbody. The name is a little misleading, as there is nothing about a blackbody which is black in colour. An example of a blackbody is a heated cannon ball, or the surface of stars like the Sun. It has been known since time immemorial that if you heat a metal object, such as a cannon ball, it will start to glow; it actually starts giving off its own visible light rather than just reflecting light. 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 we use a thermal imaging camera.

Wien’s Displacement Law

When an object glows, we can analyse which parts of the spectrum are giving off how much energy. To put it another way, how bright the object is at different wavelengths. When we do this for any object which is radiating like a blackbody we find that the spectra all have the same shape, independent of the temperature of the object or its composition.

The figure below shows the spectra for three different blackbodies. The only difference between the three is their temperature. One is at a temperature of 3,000K, the second one is at 4,000K and the third one is at 5,000K. These three curves explain why an object does not give off visible light when it is at low temperature. 3,000K is pretty hot (nearly 2,700 C), but an object at this temperature is barely giving off any visible light, the peak of the curve is in the infrared. If we came down to room temperature, about 300K, there would be no radiation short-wards of 700nm, so no visible light is to be seen from a room temperature object. These three curves also explain why an object changes colour as we heat it, it goes from red-hot to white-hot to blue-hot. It is why stars have different colours, the cool ones appear red, the hotter ones white and the hottest of all appear blue.

 

Blackbody spectra for three different temperatures, 3000K (red), 4000K (green) and 5000K (blue). Notice the wavelength of the peak of each curves moves as the temperature changes. This is Wien’s displacement law. The x-axis is in nanometres, on this scale the visible part of the spectrum is from about 400 to 700 nm.

 

As you can see from the figure, the size of the curve changes as we change the temperature, and also the wavelength of the peak changes, moving to shorter wavelengths (to the left) as we increase the temperature. The scale on the x-axis is nanometres, so on this scale the kind of wavelengths we can see with our eyes (the visible part of the spectrum) is from about 400 to 700 nm.

In 1893 the German physicist Wilhelm Wien came up with a mathematical formula which allowed one to calculate the wavelength of this peak if one knew the temperature. This equation is known as Wien’s displacement law. It is an empirical law, in the sense that Wien had no physical explanation for why the equation worked, but it did. The physics behind why Wien’s law works did not come until seven years later in 1900 with the work of Max Planck.

We can state Wien’s displacement law as

 

 

where is the temperature in Kelvin. Another law, called the Steffan-Boltzmann law, gives the total amount of power per unit area (Watts per metre squared) that is being emitted at all wavelengths by the blackbody (mathematically this is the same as the area under these curves).

 

 

where is known as the Steffan-Boltzmann constant, and once again is the temperature of the object in Kelvin.

The “ultraviolet catastrophe”

To understand the physics of blackbody spectra, physicists imagined the electromagnetic waves bouncing around inside a closed box, what is often referred to as a cavity. The cavity has a small hole in it, which allows the radiation to escape, this is the radiation we observe. We can see from the curves above that the spectrum (either expressed against wavelength or against frequency (one can go from wavelength to frequency and back by using the wave equation. For electromagnetic radiation, this can be written

 


where is the speed of light ().

The electromagnetic (EM) radiation is produced by an EM field in the cavity. But, there are EM waves with different frequencies. The ensuing EM field is due to the superposition (addition) of all the possible EM waves in the cavity. We can think of the EM waves like pieces of string between two fixed points (the sides of the cavity). So we can draw an analogy to the musical notes that e.g. a guitar string can produce. With a piece of string, only certain standing waves can exist between two fixed points, you cannot get vibrations of any wavelength. Let us suppose the fixed points are a distance apart. As the vibrations need to be zero at the fixed ends, the only wavelengths which can be produced by such a string have to have wavelengths so that where .

The same was believed to be true for EM waves inside a cavity. The EM waves had to have a complete number of half-cycles within the box. Each of these allowed vibrations of the EM field is called a mode.

The number of modes of a particular frequency in the frequency interval to (where is a small increment) is found to be

(this comes about because we can think of the vibrations of the EM field like a harmonic oscillator).
In classical physics, in the 19th century, a theory called the equipartition theorem had been developed, and this stated that each mode would have an energy of where is Boltzmann’s constant. This equipartition theorem had allowed physicists to work out the kinetic energy of molecules of a gas at any given temperature. This was an attempt to apply this equipartition law to EM radiation, which were thought of as waves.

If the energy density (which is the same as the power per unit area) of each mode is , the energy as a function of frequency in a small frequency interval from would be given by

This is the so-called Rayleigh-Jeans law, and notice that is says that the energy density in a frequency interval is proportional to the square of the frequency. If we plot the energy density at a particular frequency as a function of frequency we get a graph which looks like this

 

The so-called “ultraviolet catastrophe”. The black curve shows the observed blackbody curve, the red curve shows the prediction of the Rayleigh-Jeans law.

The black curve in this plot shows the observed spectrum of a blackbody (note, this plot is against frequency not wavelength, so it is backwards from the figure above with the three different spectra. In this plot against frequency, frequency increases to the right, which means we are moving to shorter wavelength).

Now look at the red curve, it is going off to infinity!!! That is to say, the classical theory (the Rayleigh-Jeans law) predicts that there will be an infinite number of modes allowed in the cavity as we go to higher and higher frequency. This means the curve will go off up to infinity as shown, because the power per unit area at each frequency is proportional to , so more and more energy exists in each frequency interval as we go to higher and higher frequencies (shorter and shorter wavelengths). The total area under the curve, the power per unit area, of the blackbody would be infinite and it would be infinitely bright at ultraviolet wavelengths (high frequencies)!!!!! This clearly doesn’t happen, the theory had completely broken down.

This disagreement of the observed behaviour with the prediction of the Rayleigh-Jeans law came to be known as the ultraviolet catastrophe. It showed that there was something deeply flawed in our understanding of physics. Max Planck would solve the problem in 1900, and usher in a complete overthrowing of our understanding of physics, bringing in the age of quantum physics and leading to the demise of classical physics.

 

Addendum

Here is the first part of a series of blogs where I will derive the Rayleigh-Jeans law from first principles, after which I will derive the Planck law.

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Posted in History, Physics, Science | Tagged Classical Physics, Max Planck, Quantum Mechanics, Ultraviolet Catastrophe, Wien's Displacement Law | 12 Comments


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