Blackbody Radiation

Definition:   A blackbody is an object with a low reflectivity and has capability of emitting and absorbing all frequencies of radiation uniformly.  A good model of a blackbody is a closed empty container with a pinhole so that radiation is reflected inside the container many times and comes to thermal equilibrium with the container before emitted.  So radiation leaks out from the pinhole is the characteristics of the radiation within the container.

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Experiment:  Heat up a container having a pinhole then measure the radiant emittance, %$ \rho $% (energy per unit volume per unit wavelength, also known as the energy density distribution, i.e. the energy spectral density) as a function of the temperature of the container and the radiant wavelength.

Observations:

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1. The energy spectral density has a maximum and the peak shifts toward shorter wavelength (or higher frequency) as the temperature increased.   A metal rod when heated, its color changes from red to yellow to blue as the temperature increases.
2. The total energy density (area under the curve), energy per unit volume, is proportional to the fourth power of the temperature, i.e. %$\propto { T}^{ 4}$%.

Classical Mechanics Interpretation (Rayleight-Jeans Law)

Assumption: Blackbody radiation consists of a set of electromagnetic standing waves behaving like oscillators inside the container.  Each frequency of light is represented by one oscillator and the presence of light of a certain frequency is considered as the excitation of the oscillator of that frequency.

Using the classical equipartition theorem to calculate the average energy of the oscillators, Rayleigh-Jeans arrived

%MATHMODE{ \rho = \frac{ 8 \pi kT}{ { \lambda }^{ 4} } }%

where k is the Boltzmann constant.

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Results:  

1. In the long wavelength limit, %$ \lambda \rightarrow \infty $%, Reyleigh-Jeans expression agrees well with observations.
2. However, in the short wavelength limit, it diverges, i.e.%$ \varrho \rightarrow \infty$% when %$ \lambda \rightarrow 0$%. The area under the curve, i.e. the total energy density is infinite when %$T \neq 0$%. In other words, classical theory predicts the blackbody emits an infinite amount of energy at all temperatures above the absolute 0 K.

The Plank's Quantum hypothesis 

Plank's observation:   The problem with the classical theory is in the high frequency range.  Experimental observation shows that more high-frequency radiation is observed when the temperature is increased.  Thus, as the temperature increases, more energy is given to the blackbody, and therefore more high-frequency oscillators can be excited. 

From this reasoning, Plank proposed energy of the radiation is proportional to the frequency, %$E \propto \nu $%.  Note this contradicts with classical theory where energy of an oscillator is proportional to the square of the amplitude and independent of its frequency.  Furthermore, Plank showed that if the energy can only take discrete values, the quantization of energy, namely %$E = nh \nu $%,  where n is a positive integer (n = 0, 1, 2, ...) and h is a proportional constant, then the spectral density has the form:

%MATHMODE{ \rho = \frac{ 8 \pi hc}{ { \lambda }^{ 5} } \left ( \frac{ 1}{ { e}^{ \frac{ hc}{ \lambda kT} } -1} \right )}%

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With a single adjustable parameter h, Plank was able to reproduce the experimental data at all temteratures. 

Interestingly, Plank's hypothesis of energy quantization was not accepted by most scientists at the time since the only justification for it is the agreeement with the blackbody radiation experimental observations.  Such an agreement was regarded by most at the time as pure luck and arbitrary.   The support for Plank's hypothesis only came after Einstein used it to explain the photoelectric effects.

Afterthought:   Radiation light of frequency %$ \nu $% can be thought as consisting of 0, 1, 2, .. particles called photons, each having an energy of %$h \nu $% known as quanta.  In other words, light waves have particle-like properties!

-- ThanhTruong - 16 Jul 2007