# Blackbody Radiation

From a historical perspective, blackbody radiation was a key topic of
interest at the end of 19^{th} 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 19^{th} century, there was still
a very fundamental question to resolve at this time:

*Just how do heated bodies radiate? *

At the end of the 19^{th} 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 19^{th} 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]

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 [T^{4}] 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

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]

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]

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 2
^{nd} 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]

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.
*