Blackbody Radiation and Bose-Einstein Statistics

Planck's law, Wien's displacement law, the Stefan-Boltzmann law, and h and k from a single spectrum.

Objectives

On completing this experiment you should be able to:

Measure the spectral radiance of a cavity source and fit Planck's law.
Obtain the Planck constant and the Boltzmann constant from the fit.
Show that the Rayleigh-Jeans form fails at short wavelength and the Wien form at long wavelength.
Verify Wien's displacement law and obtain the displacement constant.
Verify the Stefan-Boltzmann law, confirming the fourth power and obtaining sigma.
Relate all of these to the Bose-Einstein occupation of photon modes with zero chemical potential.

Theory

The number of electromagnetic modes per unit volume with frequency between nu and nu + d nu is 8 pi $\nu^2$ d nu / $c^3$ . Photons are bosons whose number is not conserved, so their chemical potential vanishes and the mean occupation of each mode is the Bose-Einstein factor $\frac{1}{\exp(h \nu/kT)}$ - 1). Multiplying the two gives the energy density, and hence the spectral radiance

L_{\lambda} = \frac{2 h c^2}{\lambda^5 (\exp(hc/\lambda k T) - 1)} .

Expanding the exponential for h nu << kT recovers the classical Rayleigh-Jeans result, which diverges when integrated - the ultraviolet catastrophe. Keeping only the leading exponential for h nu >> kT gives Wien's law. Integrating the full expression over all wavelengths gives the Stefan-Boltzmann law with

\sigma = \frac{2 \pi^5 k^4}{15 h^3 c^2} ,

and differentiating gives Wien's displacement law with $\lambda_{max}$ T = hc/(4.965 k).

Key relations

$L_{\lambda}=\frac{2hc^{2}}{\lambda^{5}\left(e^{hc/\lambda kT}-1\right)}$
Planck's law for spectral radiance.
$\bar{n}=\frac{1}{e^{h\nu/kT}-1}$
Bose-Einstein occupation with zero chemical potential.
$\lambda_{\max}T=2.8978\times10^{-3}\,\mathrm{m\,K}$
Wien's displacement law.
$M=\sigma T^{4}$
Stefan-Boltzmann law.

Procedure by part

Part A - The spectrum

Set the cavity temperature and scan the monochromator, recording the detector signal at each wavelength.

Set the cavity temperature and step the wavelength across the whole range, more finely near the peak.
Subtract the dark signal and divide by the detector responsivity given on the apparatus to obtain the spectral radiance.
Fit Planck's law with h and k as free parameters, holding c and the temperature fixed.
Overlay the Rayleigh-Jeans and Wien forms and state where each fails.
Repeat at three temperatures.

Part B - Wien's displacement law

Locate the peak of the spectrum at a series of temperatures.

For each cavity temperature, scan finely around the peak and locate it.
Plot the peak wavelength against $\frac{1}{T}$ and obtain the displacement constant from the gradient.
Compare the value obtained with a wide slit and with a narrow slit and explain the difference.
Deduce the Planck constant from your displacement constant, given k and c.

Part C - The Stefan-Boltzmann law

Measure the total radiant exitance of the cavity with the wide-band pyrometer.

Record the pyrometer reading over the whole temperature range in steps of 50 K.
Plot ln M against ln T and obtain the exponent from the gradient.
With the exponent fixed at four, obtain the Stefan-Boltzmann constant, correcting for the cavity emissivity.
Use your value of sigma together with your value of the displacement constant to deduce h and k independently.

Questions to answer in your report

Derive Planck's law from the Bose-Einstein distribution and the density of electromagnetic modes. Explain why the chemical potential of the photon gas is zero. [10]
Show that your data are inconsistent with the Rayleigh-Jeans law at short wavelength, and explain why the classical result diverges. [8]
The apparent peak of your spectrum shifts when you change the slit width. Explain, and state which measurement you used for the displacement constant and why. [6]