Talk:PlanetPhysics/Wien Displacement Law

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The Wien Displacement Law can be used to find the peak wavelength of a blackbody at a given \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html}. \htmladdnormallink{Planck's radiation law}{http://planetphysics.us/encyclopedia/PlancksRadiationLaw.html} gives us a \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} of $\lambda$ and temperature so we can find the maximum of this function and hence the peak wavelength emitted [1].

So for a given T we have

\begin{equation} f(\lambda) = \frac{2 \pi c^2 h}{\lambda^5} \, \frac{1}{e^{hc/ \lambda kT} - 1} \end{equation}

To find the peak of this function differentiate with respect to $\lambda$ and set it equal to 0

\begin{equation} \frac{df(\lambda)}{d\lambda} = 0 \end{equation}

Use the product rule to carry out this differentiation

\begin{equation} 0 = \frac{-10 \pi c^2 h}{\lambda^6} \, \frac{1}{e^{hc/\lambda kT} - 1} + (\frac{2 \pi c^2 h}{\lambda^5})\frac{d}{d\lambda}(e^{hc/\lambda kT} - 1)^{-1} \end{equation}

Next use the chain rule to get

\begin{equation} 0 = \frac{1}{\lambda^6} \, \frac{-10 \pi c^2 h}{e^{hc/\lambda kT} - 1} + (\frac{2 \pi c^2 h}{\lambda^5}) \, (-(e^{hc/\lambda kT} - 1)^{-2}) \, \frac{d}{d\lambda}(e^{hc/\lambda kT} - 1) \end{equation}

Apply the chain rule again

\begin{equation} 0 = \frac{1}{\lambda^6} \, \frac{-10 \pi c^2 h}{e^{hc/\lambda kT} - 1} + (\frac{2 \pi c^2 h}{\lambda^5}) \, (-(e^{hc/\lambda kT} - 1)^{-2}) \, (-\frac{hc}{\lambda^2 kT}e^{hc/\lambda kT}) \end{equation}

Multiply both sides by $\lambda^6 (e^{hc/\lambda kT} - 1)$

\begin{equation} 0 = -10 \pi c^2 h + (\frac{2 \pi c^3 h^2}{\lambda kT}) \, \frac{e^{hc/\lambda kT}}{(e^{hc/\lambda kT} - 1)} \end{equation}

Pull the e term into the denominator and divide out $2 \pi c^2h$ to get

\begin{equation} \frac{ch}{\lambda kT(1 - e^{-hc/\lambda kT})} - 5 = 0 \end{equation}

This leaves us with a transendental function, which must be solved numerically

Set $\alpha = \frac{ch}{\lambda kT}$ and substitute into above

\begin{equation} \frac{\alpha}{(1 - e^{-\alpha})} - 5 = 0 \end{equation}

After solving this equation for $\alpha$, the result yields Wien's Law

\begin{equation} \alpha = \frac{ch}{\lambda kT} \end{equation}

rearranging

\begin{equation} \lambda = \frac{hc}{\alpha k} \, \frac{1}{T} \end{equation}

A simple way to find $\alpha$ is to use Newton's Method. This can be done by hand or with your favorite numerical \htmladdnormallink{program}{http://planetphysics.us/encyclopedia/Program3.html}. Some matlab routines have been attached to see how to get $\alpha$.

To use Newton's Method we need we rewrite and arrange (8) to get

\begin{equation} F(\alpha) = \alpha - 5 + 5e^{-\alpha} \end{equation}

We also need the first derivative of this so

\begin{equation} \frac{dF(\alpha)}{d\alpha} = 1 - 5e^{-\alpha} \end{equation}

Then through iteration we can converge on the solution

\begin{equation} \alpha_{i+1} = \alpha_i - \frac{F(\alpha_i)}{dF(\alpha_i)} \end{equation}

For our accuracy needs we choose $1\mathsf{x}10^{-8}$ so we stop iterating when

\begin{equation} |\alpha_{i+1} - \alpha_i| < 1\mathsf{x}10^{-8} \end{equation}

In matlab you can run \htmladdnormallink{WienConstant.m}{http://aux.planetphysics.us/files/objects/20/WienConstant.m} which depends on \htmladdnormallink{fWien.m}{http://aux.planetphysics.us/files/objects/20/fWien.m} and \htmladdnormallink{dfWien.m}{http://aux.planetphysics.us/files/objects/20/dfWien.m} and will get a value for $\alpha$. So we see

\begin{equation} \alpha = 4.9651142 \end{equation}

Plugging this value into (10) and evaluating the other constants yields the Wien Displacement Law, which gives the peak wavelength for a given temperature of a blackbody.

\begin{equation} \lambda = \frac{2.897 \mathsf{x} 10^{-3} \, [Km]}{T} \end{equation}

Note that the temperature must be in Kelvin [K] and then $\lambda$ will have units of meters [m]. At different temperatures a blackbody's peak wavelength is displaced, hence the name Wien's Displacement Law.

[1] Krane, K., "Modern Physics." Second Edition. New York, John Wiley \& Sons, 1996.

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