Blackbody Radiation

1From a historical perspective, blackbody radiation was a key topic of interest at the end of 19th century, because it was linked to the important development of the electric light bulb, which required an understanding of the underlying physics. While much scientific progress had been made throughout the 19th century, there was still a very fundamental question to resolve at this time:

Just how do heated bodies radiate? 

At the end of the 19th century, the concept of heat was known to be linked to the atoms of a solid vibrating and that atoms appeared to be some sort of configuration of charge particles.  Equally, following the publication of Maxwell’s equations, in 1864, it was believed that an oscillating charge would emit electromagnetic radiation. From Maxwell’s equations, it was also predicted that this radiation would travel at the speed of light and that both light, and infrared heat radiation, could be characterized in terms of an electromagnetic wave.  As such, it was assumed that when a body was heated, the microscopic structures, i.e. atoms, would increase in vibration and that the associated oscillating charges would radiate, leading to radiation of heat and light.

So what is meant by the phrase ‘blackbody’ radiation? 

To some extent, the radiation from any given heated body depends on the specific material being heated. For example, while some transparent materials like glass absorb very little light, opaque materials like reflective metals, also absorb little light. In contrast, a material like black soot will absorb most of the heat and light that falls on it. Although we have not really addressed the question above, it is logical to raise another question at this point:

How is the material behaviour outlined above explained in terms of an electromagnetic wave interacting with charges in the material?

Although it was not initially realized, it is said that the question above can only be fully addressed in terms of quantum theory, which did not exist at the end of the 19th century. However, by experimentation, it was clear that a material like soot could absorb radiation and transform that energy into heat. In 1859, Gustav Kirchhoff had defined the law of thermal radiation, which then led to the idea of a perfect blackbody:

“At thermal equilibrium, the emissivity of a body equals its absorptivity.”

As such, an object at some given, non-zero temperature will radiate electromagnetic energy. If it is a perfect ‘blackbody’, it will absorb all the light falling on it, and will radiate energy according to some given formula. So, in this context, any object at any temperature above absolute zero will radiate to some extent, where the intensity and frequency distribution of the radiation depends on the material of the object. Therefore, in order to be specific in the formulation of any law, the nature of the material had to be compared with a perfect absorber and emitter, i.e. aligning to Kirchhoff’s law above, which is described as a ‘blackbody’. Of course, it was realized that a ‘blackbody’ was essentially an ideal concept, but one that might be approximated by an insulated box with a small hole in it. The basic idea being that any radiation that went through the hole would be trapped inside, where the energy of the radiation would be absorbed inside the box with little chance of escaping back out through the hole. Equally, when this logic is reversed, an oven with a tiny hole in the side would approximate a perfect emitter. In the spirit of empirical science of that age, the challenge was then to use this idea to formulate a law that described the energy/frequency curve for a ‘blackbody’. In 1879, Jožef Stefan forwarded what became known as Stefan's law:

[1]      1

This equation states that the total power [P] radiated from one square metre of a black surface, at temperature [T], follows the fourth power of the absolute temperature. Five years later, in 1884, Lugwig Boltzmann derived the [T4] relationship from theory by applying classical thermodynamic reasoning to a box filled with electromagnetic radiation, using Maxwell’s equations to relate pressure to energy density. In 1895, Wien and Lummer had carried out an experiment, which provided an insight to the relationship of intensity versus wavelength associated with a blackbody radiation

2

Wien also developed an initial displacement law that suggested the wavelength distribution of thermal radiation from a blackbody, at any temperature, has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced, as shown on the graph above. As such, the average thermal energy associated with each frequency-wavelength was expected to depend on the ratio of frequency with temperature [f/T]. Originally, it was thought that Wien’s displacement law would be a description of the complete spectrum of thermal radiation, but it was eventually shown to fail to accurately describe longer wavelength radiation, i.e. low frequency. The first successful theoretical analysis of the data provided in the graph above was presented by Max Planck in 1900.  In his analysis, Planck had modelled a blackbody radiator as a system of oscillating charges, which radiated heat inwards and when in thermodynamic equilibrium, these oscillations were also driven by the radiated energy. However, Planck found that he could only account for the observed curves, if the oscillators in the model did not radiate energy continuously, as expected by a classical theory, but rather could only lose or gain energy in chunks. These ‘chunks’ of energy became known as quanta, defined by the relationship [E=hf], where [f] corresponded to the frequency of the oscillation and [h] is known as Planck’s constant with a value of [6.626*10-34 joule.sec].  The formulation of Planck’s analysis can be presented in a number of forms:

[2]      2

The other constants introduced are the speed of light [c] and Boltzmann’s constant [k]. However, Planck's law is sometimes also written in terms of the spectral energy density per unit volume:

[3]      3

While Planck’s formula agreed with the experimental evidence by introducing the idea of energy quantization, it is probably true to say that nobody really understood the significance of this concept, which included Planck himself. In many ways, Planck’s formula appeared to be based on a number of somewhat implausible and even contradictory assumptions, which simply gave the correct result. The full mathematical derivation of Planck’s equations, based on the energy attributed to conceptual oscillators is quite involved and is not necessarily helpful in gaining an initial perception of quantization, which is the fundamental concept of interest in this discussion. Therefore, the following approach is not intended to be rigorous, but might be more helpful in visualizing the basic concepts at work. If we start with the idea of just a ‘1-dimensional wave, we might visualize the basic relationship between energy [E] and frequency [f]. If you tie a rope at one end and hold the other in your hand, you can set up a ‘standing wave’  by shaking this end of the rope up and down. If you do this slowly, i.e. less energetically, you get a long wavelength, i.e. low frequency. However, if you shake the rope as fast as possible, i.e. more energetically, you get a multiple standing waves of a shorter wavelength, i.e. higher frequency. As such, we are making a simple analogy in which there appears to be direct and proportional relationship between energy [E] and frequency [f].

But what have standing waves got to do with blackbody radiation?

In 1900, some months before Planck’s discovery, Lord Rayleigh was taking a different approach to the problem of blackbody radiation in which he assumed the radiation to be a collection of standing waves confined within a cubical enclosure. If we run with this idea, we might realise that only wavelengths that were small enough to fit within the enclosure would be allowed. Of course, this would still allow an almost infinite number of shorter wavelengths to conceptually exist within an enclosure of some given size. However, in the context of classical physics, the idea of ‘equipartition’ of energy also requires that any system in thermal equilibrium must assign equal energy to each degree of freedom, i.e. 1, 2 or 3 dimensional motion.

Note: In Planck’s approach, the idea of equipartition is not mention. However, he does essentially address this issue via the idea of entropy and the 2nd law of thermodynamics, which suggests that all components of any system will collapse towards some state of thermal equilibrium. However, we might still be left wondering where the quantization of energy fits into this picture?

With respect to the energy distribution shown in the graph above, we know that only a finite amount of energy [E] is being input into heating the blackbody radiator to some given temperature. So we need a mechanism that will distribute the energy, in accordance to the equipartition assumption outlined, while also aligning to the distribution shown in the graph. Without some sort of quantization of energy being introduced, the potentially infinite number of high frequency standing waves would share near-zero energy. However, if we now also put a restriction on the minimum energy, as defined by Planck’s energy equation, we can reach a different conclusion, which then helps explain the energy distribution of the graph.

[4]      4

In essence, the graph above is showing 5 different experiments, where an oven is heated to a different temperature [T], which in-turn represents a different total energy input. This energy still has to be evenly distributed across all allowable wavelengths, in accordance with equipartition, with one additional proviso. Should the distributed energy fall below the threshold defined by [4], this wavelength-frequency cannot be supported. As such, it is this restriction that limits the high frequency distribution to a finite number of higher frequency waves, which ultimately explains the cut-off point on the shorter wavelength scale of the graph. As the temperature [T] is increased, corresponding to an increase in energy, there is a higher proportion of this energy distributed across the shorter wavelengths. As such, this explains why the peak of each curve is biased to the left and moves to the left with increased energy.

Footnote:

Without wishing to undermine Planck’s role in the development of quantum theory, by his own admission, he did not really understand the significance of quantization required to solve the problem of blackbody radiation. In fact, nobody realized the importance of what Planck had done and his work was widely seen as just a clever technical fix, which provided the right answer according to experimental results. However, this situation would start to change, in 1905 when Einstein released his paper of the Photoelectric Effect.