Supplement 2.4: Wien's Displacement Law

Wavelength representation

The spectral energy density distribution of the radiation field of a black body is:

The wavelength ${\lambda }_{max}$ at the maximum of Planck's radiation curve over the temperature can be calculated from its derivative by equalising it with zero (extrema problem). For a clear overview we set

With the new variable, the radiation law becomes:

By the help of the quotient rule, the derivate will be:

By equating with zero and reducing, a transcendental equation for the extremum $x\text{'}$ can be found:

As with most transcendental equations and in contrast to algebraic equations, it cannot be solved exactly, but only approximately.

How do we find the solution? ↓ ↑

First we form the equation to

$x\text{'}=5\left(1-e^{-x\text{'}}\right)$

In a graph, the left and right sides are plotted over $x\text{'}$. The intersection of the two curves indicates the value of $x\text{'}$ we are looking for. It is approximately 5.

The left and right sides of the transcendental equation plotted separately, with the common intersection as the solution sought.
Source: Rainer Reuter, University of Oldenburg, Germany

More precisely, $x\text{'}$ can be determined iteratively. For this purpose, one considers numerical sequences of $x{\text{'}}_{n+1}$ that result from previous values of $x{\text{'}}_n$ with the aim of continuously bringing the left and right sides of the relation

$x{\text{'}}_{n+1}\approx 5\left(1-e^{-x{\text{'}}_n}\right)$

closer together. In this way, the true value can be determined with any degree of accuracy: $\underset{n\to \infty }{lim}x{\text{'}}_n\to x{\text{'}}_{true}$.

One finds:

With $h=6.6\cdot {10}^{-34}\text{Js}$, $k=1.38\cdot {10}^{-23}\text{J<mo>/</mo>K}$ and $c=3\cdot {10}^{10}\text{m<mo>/</mo>s}$ it follows Wien's displacement law:

Frequency representation

It this case it is not possible either, to deduce the maximum $f_{max}$ from the maximum of the wavelength representation on the left ${\lambda }_{\text{max}}$ with the relation $f\lambda =c$. The graphics of the Planck curves in the wavelength and frequency representation show, that their maxima of $f\lambda =c$ do not match.

The calculation therefore has to emanate in this case as well from Planck's radiation law in the form of the frequency representation. The process of calculation is comparable with the one shown in the left column. The adequate equations are given below for the sake of completeness.

The spectral energy density of the black body as a function of the frequency is:

The radiation law with the new variable $hf/kT=x$:

Differentation with respect to the frequency:

The transcendent equation for the calculation of the extremum $x\text{'}$

with the solution $x\text{'}=h{f}_{max}/kT\simeq 2.\mathrm{821...}$.

Wien's displacement law hence becomes: