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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.

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