16.3 Black-body radiation and Planck distribution

Black-Body Radiation and Planck Distribution

Concept of black-body radiation

A black body is an idealized object that absorbs all electromagnetic radiation hitting it, regardless of frequency or angle. Because it's a perfect absorber, it's also a perfect emitter: the radiation it gives off depends only on its temperature, not on what it's made of or its shape.

This concept matters because the observed spectrum of black-body radiation could not be explained by classical physics. The failure to account for that spectrum led directly to the idea of quantized energy levels and, ultimately, to quantum mechanics itself.

Classical physics failure: the ultraviolet catastrophe

Classical physics treated electromagnetic modes in a cavity as continuous oscillators, each carrying an average energy of $kT$ (the equipartition theorem). From this, the Rayleigh-Jeans law predicts spectral radiance that grows without bound as frequency increases:

$B_\nu^{RJ}(T) = \frac{2\nu^2 kT}{c^2}$

At low frequencies this matches experiment reasonably well, but at high frequencies (ultraviolet and beyond) it predicts infinite intensity. Integrating over all frequencies gives infinite total power, which is physically absurd. This dramatic failure became known as the ultraviolet catastrophe and signaled that something fundamental was missing from classical radiation theory.

Derivation of the Planck distribution law

Max Planck resolved the ultraviolet catastrophe in 1900 by proposing that the energy of each electromagnetic oscillator is not continuous but comes in discrete packets (quanta). Each quantum carries energy:

$E = h\nu$

• $h$ is Planck's constant, $6.626 \times 10^{-34}$ J·s
• $\nu$ is the oscillator frequency in Hz

For a single mode at frequency $\nu$, the allowed energies are $0, h\nu, 2h\nu, 3h\nu, \ldots$ Using the canonical partition function for a quantum harmonic oscillator, the mean occupation number of photons in that mode is the Bose-Einstein factor:

$\langle n \rangle = \frac{1}{e^{h\nu / kT} - 1}$

Multiplying the mean energy per mode $\langle n \rangle h\nu$ by the photon density of states (number of modes per unit volume per unit frequency, accounting for two polarizations) gives the Planck spectral radiance:

$B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu / kT} - 1}$

where:

• $B_\nu(T)$ is the spectral radiance (W·sr$^{-1}$·m$^{-2}$·Hz$^{-1}$) at frequency $\nu$ and temperature $T$
• $c$ is the speed of light, $3.00 \times 10^8$ m/s
• $k$ is Boltzmann's constant, $1.381 \times 10^{-23}$ J/K

Limiting behavior worth checking:

• Low frequency ($h\nu \ll kT$): The exponential can be expanded as $e^{h\nu/kT} \approx 1 + h\nu/kT$, so the Planck law reduces to the Rayleigh-Jeans result. Classical physics works fine here.
• High frequency ($h\nu \gg kT$): The $-1$ in the denominator becomes negligible, and the distribution falls off exponentially, avoiding the ultraviolet catastrophe. This limit reproduces Wien's approximation.

Applications of the Planck distribution

Stefan-Boltzmann law from integration. Integrating $B_\nu(T)$ over all frequencies and over the outgoing hemisphere yields the total radiated power per unit area:

$I(T) = \sigma T^4$

where $\sigma = \frac{2\pi^5 k^4}{15 c^2 h^3} = 5.670 \times 10^{-8}$ W·m$^{-2}$·K$^{-4}$ is the Stefan-Boltzmann constant. This tells you that doubling a black body's temperature increases its total emitted power by a factor of 16.

Wien's displacement law. The frequency (or wavelength) at which $B_\nu(T)$ peaks shifts with temperature. In wavelength form:

$\lambda_{\max} T = b$

where $b \approx 2.898 \times 10^{-3}$ m·K. Hotter objects peak at shorter wavelengths.

Stellar spectra. Stars approximate black bodies, so the Planck distribution explains their colors. A cool star like Betelgeuse ($T \approx 3500$ K) peaks in the red/infrared, while a hot star like Rigel ($T \approx 12{,}000$ K) peaks in the blue/ultraviolet. The Hertzsprung-Russell diagram organizes stars by luminosity and surface temperature, both quantities rooted in black-body physics.

Implications for quantum mechanics

The success of Planck's quantization hypothesis had consequences far beyond radiation theory:

• It showed that energy exchange between matter and radiation occurs in discrete amounts, directly contradicting the classical assumption of continuous energy.
• Einstein extended the photon concept in 1905 to explain the photoelectric effect, demonstrating that light itself carries energy in quanta of $h\nu$. This provided independent evidence for quantization and earned him the Nobel Prize.
• The same Bose-Einstein statistics underlying the Planck distribution apply to all bosons, connecting black-body radiation to broader quantum statistical mechanics (superfluidity, Bose-Einstein condensation, laser physics).
• Subsequent developments like Compton scattering and the Bohr model of the atom reinforced the picture of discrete energy levels, solidifying the quantum framework that Planck's work initiated.

In the context of this course, the Planck distribution is your first concrete example of how quantum statistics (specifically, Bose-Einstein statistics for massless, non-conserved bosons with $\mu = 0$) produces physically correct results where classical statistical mechanics fails.