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## Derivation of Planck’s radiation law – part 2

In the first part of this blog (here), I described how experimenters at the Physikalisch-Technische Reichsanstalt (PTR) determined the true spectrum of blackbody radiation during the 1890s, By the year 1900, primarily by the work of Heinrich Rubens, Ferdinand Kurlkbaum, Ernst Pringsheim and Otto Lummer, the complete spectrum, from the ultraviolet through the visible and into the infrared, was known for the very first time. As the true shape of the blackbody spectrum started to emerge from this experimental work, theoreticians tried to find a theory to explain it.

The first to meet with any success was Wilhem Wien. As I mentioned in the first part of this blog, in 1893 he came up with his displacement law, which gave a very simple relationship between the wavelength of the peak of the spectrum and its temperature.

$\lambda_{peak} = \frac{ 0.0029 }{ T }$

where $\lambda_{peak}$ is the wavelength of the peak in metres, and $T$ is expressed in Kelvin.

By 1896 Wien had come up with a theory to explain the shape of the spectrum (even though the shape in the infrared was not fully known at that time). In what we now call ‘Wien’s distribution law’ or ‘Wien’s approximation’, he tried to explain the blackbody spectrum using thermodynamic arguments, and assuming that the gas molecules obeyed the Maxwell-Boltzmann speed distribution for molecules (or atoms) in a gas. I will not derive that explanation here, but if any readers wish me to derive it I can do so at a later date.

Wilhelm Wiens, who in 1893 came up with Wiens displacement law, and in 1896 with the Wien distribution.

## Wien’s distribution law (1896)

What Wien suggested was that the energy of a black body in the wavelength interval $d \lambda$ was given by

$E_{ \lambda } d \lambda = \frac{ A }{ \lambda ^{5} } f( \lambda T) d \lambda$

Wien found, using the Maxwell-Boltzmann distribution law for the speed of atoms (or molecules) in a gas, that the form of the function $f( \lambda T)$ was

$f( \lambda T ) = e^{ -a / \lambda T }$

and so

$\boxed{ E_{ \lambda } d \lambda = \frac{ A }{ \lambda ^{5} } e^{ -a / \lambda T } d \lambda }$

where $A \text{ and } a$ were constants to be determined.

If we wish to express this in terms of frequency $\nu$ instead of wavelength $\lambda$ then we need to remember that, from the wave equation, $c = \nu \lambda$ and so $\lambda = c/\nu$. But, we also need to rewrite $d\lambda$ in terms of $d\nu$ and to do this we write

$\nu = \frac{ c }{ \lambda } \rightarrow d \lambda = \frac{ -c }{ \nu^{2} }\; d \nu$

We can ignore the minus sign as it is just telling us that as the frequency increases the wavelength decreases, and so substituting for $\lambda \text{ and } d\lambda$ we can write
that the energy in the frequency interval $d \nu$ is given by

$E_{\nu} d \nu = \frac { A \nu^{5} } { c^{5} } e^{ -a \nu / cT } \frac{ c }{ \nu^{2} } d \nu$

$\boxed{ E_{\nu} d \nu = A^{\prime} \nu^{3} e^{ -a^{\prime} \nu / T } d \nu }$

where $A^{\prime} \text{ and } a^{\prime}$ are also just constants to be determined.

## Wien’s ‘law’ breaks down

As I will show next week, Wien’s distribution law gave good (but not perfect) agreement with the blackbody curve on the short-wavelength side of the peak (what we now call the ‘Wien-side’ of the peak). But, as experimental results on the long-wavelength side started to emerge from the PTR, it became clear that his ‘law’ did not work on that side; it broke down on the long-wavelength side and showed very poor agreement with the actual observed curve.

Next week, in part 3 of this blogpost, I will also describe how and why Planck got involved in the problem, and what the solution he concocted was; the law which would correctly describe the blackbody spectrum and usher in the quantum age.

## Derivation of Planck’s radiation law – part 1

One of my most popular blogposts is the series I did on the derivation of the Rayleigh-Jeans law, which I posted in three parts (part 1 here, part 2 here and part 3 here). I have had many thousands of hits on this series, but several people have asked me if I can do a similar derivation of the Planck radiation law, which after all is the correct formula/law for blackbody radiation. And so, never one to turn down a reasonable request, here is my go at doing that. I am going to split this up into 2 or 3 parts (we shall see how it goes!), but today in part 1 I am going to give a little bit of historical background to the whole question of deriving a formula/law to explain the shape of the blackbody radiation curve.

## ‘Blackbody’ does not mean black!

When I first came across the term blackbody I assumed that it meant the object had to be black. In fact, nothing could be further from the truth. As Kirchhoff’s radiation laws state

A hot opaque solid, liquid, or gas will produce a continuum spectrum

(which is the spectrum of a blackbody). The key word in this sentence is opaque. The opaqueness of an object is due to the interaction of the photons (particles of light) with the matter in the object, and it is only if they are interacting a great deal (actually in thermal equilibrium) that you will get blackbody radiation. So, examples of objects which radiate like blackbodies are stars, the Cosmic Microwave Background, (which is two reasons why astronomers are so interested in blackbody radiation), a heated canon ball, or even a canon ball at room temperature. Or you and me.

Kirchhoff’s 3 radiation laws, which he derived in the mid-1800s

Stars are hot, and so radiate in the visible part of the spectrum, as would a heated canon ball if it gets up to a few thousand degrees. But, a canon ball at room temperature or you and me (at body temperature) do not emit visible light. But, we are radiating like blackbodies, but in the infrared part of the spectrum. If you’ve ever seen what people look like through a thermal imaging camera you will know that we are aglow with infrared radiation, and it is this which is used by Police for example to find criminals in the dark as the run across fields thinking that they cannot be seen.

The thermal radiation (near infrared) from a person. The differences in temperature are due to the surface of the body having different temperatures in different parts (e.g. the nose is usually the coldest part).

Kirchhoff came up with his radiation laws in the mid-1800s, he began his investigations of continuum radiation in 1859, long before we fully knew the shape (spectrum) of a blackbody.

## Germans derive the complete blackbody spectrum

We actually did not know the complete shape of a blackbody spectrum until the 1890s. And the motivation for experimentally determining it is quite surprising. In the 1880s German industry decided they wanted to develop more efficient lighting than their British and American rivals. And so they set about deriving the complete spectrum of heated objects. In 1887 the German government established a research centre, the Physikalisch-Technische Reichsandstalt (PTR) – the Imperial Institute of Physics and Technology, one of whose aims was to fully determine the spectrum of a blackbody.

PTR was set up on the outskirts of Berlin, on land donated by Werner von Siemens, and it took over a decade to build the entire facility. Its research into the spectrum of blackbodies began in the 1890s, and in 1893 Wilhelm Wien found a simple relationship between the wavelength of the peak of a blackbody and its temperature – a relationship which we now call Wien’s displacement law.

Wien’s displacement law states that the wavelength of the peak, which we will call $\lambda_{peak}$ is simply given by

$\lambda_{peak} = \frac{ 0.0029 }{ T }$

if the temperature $T$ is expressed in Kelvin. This will give the wavelength in metres of the peak of the curve. That is why, in the diagram below, the peak of the blackbody shifts to shorter wavelengths as we go to higher temperatures. Wien’s displacement law explains why, for example, an iron poker changes colour as it gets hotter. When it first starts glowing it is a dull red, but as the temperature increases it becomes more yellow, then white. If we could make it hot enough it would look blue.

The blackbody spectra for three different temperatures, and the Rayleigh-Jeans law, which was behind the term “the UV catastrophe”

By 1898, after a decade of experimental development, the PTR had developed a blackbody which reached temperatures of 1500 Celsius, and two experimentalists working there Enrst Pringsheim and Otto Lummer (an appropriate name for someone working on luminosity!!) were able to show that the blackbody curve reached a peak and then dropped back down again in intensity, as shown in the curves above. However, this pair and others working at the PTR were pushing the limits of technology of the time, particularly in trying to measure the intensity of the radiation in the infrared part of the spectrum. By 1900 Lummer and Pringsheim had shown beyond reasonable doubt that Wien’s ad-hoc law for blackbody radiation did not work in the infrared. Heinrich Rubens and Ferdinand Kurlbaum built a blackbody that could range in temperature from 200 to 1500 Celsius, and were able to accurately measure for the first time the intensity of the radiation into the infrared. This showed that the spectrum was as shown above, so now Max Planck knew what shape curve he had to find a formula (and hopefully a theory) to fit.

In part 2 next week, I will explain how he went about doing that.