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Wien displacement law (Definition)

The Wien Displacement Law can be used to find the peak wavelength of a blackbody at a given temperature. Planck's radiation law gives us a function 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

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

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

$\displaystyle \frac{df(\lambda)}{d\lambda} = 0$ (2)

Use the product rule to carry out this differentiation

$\displaystyle 0 = \frac{-10 \pi c^2 h}{\lambda^6} \, \frac{1}{e^{hc/\lambda kT}... ...+ (\frac{2 \pi c^2 h}{\lambda^5})\frac{d}{d\lambda}(e^{hc/\lambda kT} - 1)^{-1}$ (3)

Next use the chain rule to get

$\displaystyle 0 = \frac{1}{\lambda^6} \, \frac{-10 \pi c^2 h}{e^{hc/\lambda kT}... ...\, (-(e^{hc/\lambda kT} - 1)^{-2}) \, \frac{d}{d\lambda}(e^{hc/\lambda kT} - 1)$ (4)

Apply the chain rule again

$\displaystyle 0 = \frac{1}{\lambda^6} \, \frac{-10 \pi c^2 h}{e^{hc/\lambda kT}... ... (-(e^{hc/\lambda kT} - 1)^{-2}) \, (-\frac{hc}{\lambda^2 kT}e^{hc/\lambda kT})$ (5)

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

$\displaystyle 0 = -10 \pi c^2 h + (\frac{2 \pi c^3 h^2}{\lambda kT}) \, \frac{e^{hc/\lambda kT}}{(e^{hc/\lambda kT} - 1)}$ (6)

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

$\displaystyle \frac{ch}{\lambda kT(1 - e^{-hc/\lambda kT})} - 5 = 0$ (7)

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

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

$\displaystyle \frac{\alpha}{(1 - e^{-\alpha})} - 5 = 0$ (8)

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

$\displaystyle \alpha = \frac{ch}{\lambda kT}$ (9)

rearranging

$\displaystyle \lambda = \frac{hc}{\alpha k} \, \frac{1}{T}$ (10)

A simple way to find $\alpha$ is to use Newton's Method. This can be done by hand or with your favorite numerical program. 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

$\displaystyle F(\alpha) = \alpha - 5 + 5e^{-\alpha}$ (11)

We also need the first derivative of this so

$\displaystyle \frac{dF(\alpha)}{d\alpha} = 1 - 5e^{-\alpha}$ (12)

Then through iteration we can converge on the solution

$\displaystyle \alpha_{i+1} = \alpha_i - \frac{F(\alpha_i)}{dF(\alpha_i)}$ (13)

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

$\displaystyle \vert\alpha_{i+1} - \alpha_i\vert < 1\mathsf{x}10^{-8}$ (14)

In matlab you can run WienConstant.m which depends on fWien.m and dfWien.m and will get a value for $\alpha$. So we see

$\displaystyle \alpha = 4.9651142$ (15)

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.

$\displaystyle \lambda = \frac{2.897 \mathsf{x} 10^{-3} \, [Km]}{T}$ (16)

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.



"Wien displacement law" is owned by bloftin.

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Cross-references: program, function, Planck's radiation law, temperature

This is version 10 of Wien displacement law, born on 2004-11-20, modified 2005-08-14.
Object id is 20, canonical name is WienDisplacementLaw.
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Classification:
Physics Classification: 05.70.Ce (Thermodynamic functions and equations of state )
Pending Errata and Addenda
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Discussion
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It's good but need re-arranging by IVE on 2006-10-02 06:46:43
Dear Ben,
Undoubtfully this is a nice example of how one can contribute to PP. But as an encyclopedic entry we don't have to explicitly write down every step in deriving Wien's Law from Planck's Law and numerically computing the Wein constant, in the *main entry*. I suggest in the main article you just tell what the law says and what's its implications in Physics. The detailed steps of calculation and numerical manipulation could be moved to an "attachment" of the defining article.

Thanks

IVE
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