Supplement 2.2: Planck's Radiation Law (2/2)

Energy of waves

According to the Law of Equipartition of Energy in thermodynamics, the mean kinetic temperature of an atom or another microparticle is determined solely by the absolute temperature $T$ and given by $k T$ , where $k = 1,38 \cdot 10^{- 23} J/K$ is the Boltzmann constant (it is of no relevance right now, that this is a simplified exposition).

This expression meets the mean energy of a wave in a cavity as well. The mean value of the total energy of all waves in a frequency interval $d f$ can be obtained by multiplying the mean energy of the single waves by $d N$ , the total number of waves in that frequency interval:

k T d N = 8 π \frac{f^{2}}{c^{3}} V k T d f

Relating the total energy to the volume, we obtain the energy density of the waves within the frequency interval $d f$ :

d U = u_{f} d f = \frac{k T}{V} d N = 8 π \frac{f^{2}}{c^{3}} k T d f

The equation of spectral energy density

u_{f} = 8 π \frac{f^{2}}{c^{3}} k T

is the Rayleigh-Jeans radiation law. It has been derived by Lord Rayleigh and Sir James Hopwood Jeans in 1900, within a narrow time frame to Planck's radiation law. It describes the radiation of black bodies at low frequencies, shorter than the radiation spectrum's maximum (or: at long wavelengths, longer than the maximum) quite reliably. However, the equation does not have a maximum. Instead, its values increase quadratically towards higher frequencies (or: lower wavelengths), which has been named the 'ultraviolet catastrophe' of this law.

Max Planck by contrast chose a different formulation than $k T$ to approach the waves' energy. He considered the oscillating atoms at the surface of the cavity and set their kinetic energy equal to

E_{n} = n \cdot h f

with $n = 0, 1, 2, 3, ...$ . The quantum number $n$ of the atomic oscillators corresponds to the mode number of the electromagnetic waves. In effect, Planck set the energy of an oscillator equal to the energy of the wave linked to it. This represents a quantization of the radiation, what caused the photon model to evolve.

Classical oscillators ↓

A classical linear oscillator, for example a mass attached to a coil spring, is described by Hooke's law: The restoring force $F$ excerted by the spring is proportional to the displacement $x$ . It equals

F = - D x

with the spring constant $D$ . Linear means, the force behaves proportionally to the mechanical displacement and does not obey any other relation (which would be the case if the spring is over-expanded).

Displacement of a coil spring.
Source: Wikimedia Commons.

The energy stored in the spring is calculated by integrating the force over the displacement:

E = \int F d x = - \frac{1}{2} D x^{2}

For the energy stored in the spring, continuous values are possible.

Quantum mechanical oscillators ↓

A quantum mechanical linear oscillator obeys the same law of forces. Calculating the energy by the help of the Schrödinger equation, one finds that it is quantized through the following equation:

E_{n} = (n + \frac{1}{2}) \cdot h f

by $n = 0, 1, 2, ...$ . The energy of this oscillator can therefore not equal zero but must be at least as high as the zero-point energy

E_{n = 0} = \frac{1}{2} h f

Max Planck could not know the correct quantum mechanical calculation because quantum theory evolved subsequently. However, the difference to the energy equation $E_{n} = n \cdot h f$ used by him does not play a role in calculating his radiation law.

The probability $p_{n}$ of an oscillator being in the state $n$ with the energy $E_{n} = n \cdot h f$ is named occupation probability of the state and it is described by the Boltzmann distribution:

p_{n} = \frac{1}{Z} exp (- \frac{n \cdot h f}{k T})

The occupation probability decreases exponentially to an increase in energy. Temperature plays an important role in this case as well: high energy states are more likely occupied at higher temperatures.

The quantity $Z$ in the denominator of the occupation probability is the so-called state sum:

Z = \sum_{n = 0}^{\infty} exp (- \frac{n \cdot h f}{k T})

The state sum is describable as a geometric series. The result is:

Z = \frac{1}{1 - exp (- h f / k T)}

Prove ↓

With the abbreviation

q = exp (- h f / k T)

becomes $Z = \sum_{n = 0}^{\infty} q^{n}$ . Since $| q | < 1$ , the series converges, therefore:

\sum_{n = 0}^{\infty} q^{n} = lim_{k \to \infty} \sum_{n = 0}^{k} q^{n} = lim_{k \to \infty} \frac{1 - q^{k + 1}}{1 - q} = \frac{1 - lim_{k \to \infty} q^{k + 1}}{1 - q} = \frac{1}{1 - q}

$Z$ limits the occupation probabilities in a manner, so that its sum over all energy values equals 1 (or: 100%):

\sum_{n = 0}^{\infty} p_{n} = 1

The thermic mean value of the quantum number $n$ of the cavity radiation is:

⟨ n ⟩ = \sum_{n = 0}^{\infty} n p_{n} = \frac{1}{Z} \sum_{n = 0}^{\infty} n exp (- n h f / k T)

For easier legibility, we set $h f / k T = α$ , so that the sum expression becomes:

\sum_{n = 0}^{\infty} n exp (- n α) = - \frac{d}{d α} \sum_{n = 0}^{\infty} exp (- n α)

A conversion into the derivation with respect to $α$ simplifies the calculation notably, since the sum has already been calculated on the right-hand side at the beginning. It follows:

- \frac{d}{d α} \sum_{n = 0}^{\infty} exp (- n α) = - \frac{d}{d α} (\frac{1}{1 - exp (- α)}) = \frac{exp (- α)}{{(1 - exp (- α))}^{2}}

Application to the equation for $⟨ n ⟩$ provides the result for $Z$ :

⟨ n ⟩ = \frac{exp (- h f / k T)}{1 - exp (- h f / k T)} = \frac{1}{exp (- h f / k T) - 1}

The mean thermal energy of the cavity radiation is then:

⟨ E ⟩ = ⟨ n ⟩ h f = \frac{h f}{exp (h f / k T) - 1}

This expression replaces the mean kinetic energy $k T$ mentioned at the beginning of this page. Considering this for the equation of spectral energy density, one obtains Planck's radiation equation:

u_{f} = \frac{8 π f^{2}}{c^{3}} \frac{h f}{exp (h f / k T) - 1}

Transition to classical physics ↓

For very high temperatures, namely for $k T ≫ h f$ :

exp (- h f / k T) \approx exp (0) = 1

Which causes $Z \approx 1$ and $p_{n} \approx 1$ , the quantum number $n$ looses relevance.

The series development of the exponential function with negative exponent in Planck's equation can be cut after the second term:

\begin{array}{l} exp (h f / k T) = 1 + h f / k T + \frac{{(h f / k T)}^{2}}{2!} + ... \\ \approx 1 + h f / k T \end{array}

Application to the equation for $⟨ E ⟩$ provides the value $k T$ . Application to the energy density after Planck provides the Rayleigh-Jeans radiation law. This is called the classical limit case of the quantum mechanical relations.

Understanding Spectra from the Earth

Supplement 2.2: Planck's Radiation Law (2/2)

Energy of waves