Blackbody radiation

The explanation of this experiment was proposed by the German physicist Max Planck in 1901. Every object emits energy from its surface in the form of electromagnetic radiation.

Thus, in order to discuss this experiment, we need a little background on electromagnetic waves. Some examples of waves are:

1.

Vibrations of a string

2.

Sound waves

3.

Electromagnetic (light) waves

4.

Water waves

In general, the mathematical form of a wave in motion in one dimension is

$$A(x,t) = A_0\cos\left({2\pi x \over \lambda} - 2\pi \nu t\right)$$

where $A_0$ is the wave amplitude, $\lambda$ is its wavelength, and $\nu$ is its frequency. A snapshot of this wave form at $t=0$ is shown in the figure below:

Figure 3:

If we wait a time $t=1/4\nu$ later, the wave looks as shown below:

Figure 4:

Thus, $\nu$ is the number of wave crests (maxima) that pass a given point per unit time (e.g. seconds). Thus, the units of frequency are $crests/second$ or $cycles/second = Hz$ (Hertz), however, it is only the inverse time that is important in dimensional analysis. Given that the time $\tau = 1/\nu$ is the time for a wave crest to move a distance $\lambda$, the speed of the wave is

$$v = {\lambda \over \tau} = \lambda \nu$$

Visible light, microwaves, X-rays, ultraviolet radiation,... all are forms of electromagnetic (EM) radiation. Maxwell's theory (introduced in the late 19th century) predict that such waves are composed of electric and magnetic fields, both wave-like and perpendicular to each other. Thus, a free EM wave appears as follows:

Figure 5:

If the propagation occurs in the $x$-direction, then the electric and magnetic field components are given by

$${\bf E}(x,t) = {\bf E}_0\cos\left({2\pi x \over \lambda} - 2\pi \nu t\right)$$

$${\bf B}(x,t) = {\bf B}_0\cos\left({2\pi x \over \lambda} - 2\pi \nu t\right)$$

where ${\bf E}\perp {\bf B}$. These expressions for the electric and magnetic fields are solutions of the so-called ``free-space Maxwell equations'' (Maxwell's equations are the fundamental equations of the theory of electromagnetism). The speed of the wave $c=\lambda\nu$ is the speed of light $c=2.9979246\times 10^8$ m/s. Visible light spans 400$\times$10$^{-9}$m - 700$\times$10$^{-9}$m or 400-700 nm. The lower end of this range corresponds to red light while the upper end to blue light.

Examples of blackbody radiation include heating elements on electric stoves and incandescent lightbulbs. The intensity of the radiation depends on the wavelength and on temperature. The solid curves in the figure below show the blackbody radiation spectrum (as a function of the wavelength of the radiation) for different temperatures:

Figure 6:

The temperature corresponding to the blue curve is higher than the temperature corresponding to the red curve. Since $\lambda = c/\nu$, we can also use the frequency $\nu$ to characterize the radiation spectrum.

Classical mechanics predicts that the intensity $I$ of emitted radiation at frequency $\nu$ is given by

$$I(\nu) = {8\pi k_{\rm B}T \nu^2 \over c^3}$$

where $k_{\rm B}$ is Boltzmann's constant, $k_{\rm B} = 1.38066\times 10^{-23}$ J/K. The classical curves in the above figure are obtained by setting $\nu = c/\lambda$ in this expression:

$$I(\lambda) = {8\pi k_{\rm B}T \over c\lambda^2}$$

and are shown as the dashed curves in the figure. Note that as $\lambda \rightarrow 0$, $I\rightarrow \infty$, which is known as the ultraviolet catastrophe.

In the classical theory, blackbody radiation is modeled as the radiation emitted from oscillating charged particles at the object's surface. These oscillations are produced by the thermal motions of the charged particles. If we treat each particle as a simple harmonic oscillator, then we can easily understand how Planck was able to explain blackbody radiation.

A simple harmonic oscillator is described by a Hooke's law force

$$F = -k(x-x_0)$$

where the force $F$ arises from the displacement $x$ from the equilibrium position $x_0$ of a charged particle attached to a spring. Here $k$ is the spring constant, which measures the stiffness of the spring.

Figure 7:

The force $F$ is derived from the potential energy curve

$$V(x) = {1 \over 2}k(x-x_0)^2$$

and produces the following motion in $x$:

$$x(t) = x_0 + A\cos(2\pi\nu t + \phi)$$

where

$$\nu = {1 \over 2\pi}\sqrt{k \over m}$$

and $\phi$ is known as the phase angle. Its value is determined by the initial position $x(0)$ and initial momentum $p(0)$ of the oscillator. The energy

$$E = {p^2 \over 2m} + {1 \over 2}k(x-x_0)^2$$

can take on any value, as determined by the initial conditions, but once its value is set, it does not change over time because energy is conserved.

Planck's ingenious idea was to propose the following radical hypothesis: What if the energy $E$ could not take on just any value but only certain discrete values called ``quanta'' of energy. He suggested that higher frequency motion meant higher energy. By the way, in classical mechanics, the energy is only determined by the amplitude of the oscillator, not the frequency. Thus, he proposed that the energy quanta be multiplies of the frequency

$$E \propto n\nu$$

where $n$ is an integer $n=0,1,2,...$. The constant of proportionality needed to give energy units he called $h$, which is now known as Planck's constant. Thus, the discrete energy values were proposed to be determined by

$$E = nh\nu$$

Since there are many oscillating charged particles, we need to consider a very large number of identical oscillating systems. At Planck's time, it was known that the probability that a collection of identical systems at the same temperature $T$ but starting from different classical initial conditions would have an energy $E$ was proportional to the Boltzmann factor

$${\rm probability} \propto e^{-E/k_{\rm B}T}$$

Let us recall the definition of a probability. Generally, given $N$ possible events $\epsilon_1,...,\epsilon_N$, the probability that an event $\epsilon_n$ will occur is defined to be

$${\rm probability\ that\ }\epsilon_n\;{\rm occurs} \equiv P(\... ...} \over {\rm Total\ number\ of\ ways\ any\ event\ can\ occur}}$$

Planck applied this formula to his quantized energy values. Let the event $\epsilon_n$ be a measurement of the energy of the system at temperature $T$ that yields an energy $E = nh\nu$. The probability of such an event, according to the above formula is

$${\rm probability} = {e^{-nh\nu/k_{\rm B}T} \over \sum_{n=0}^{\infty}e^{-nh\nu/k_{\rm B}T} }$$

Note that if we sum both sides over $n$, we must get 1. That is, the sum over $n$ is the probability of finding the system with any allowable energy, which, of course, must be 1. The denominator is of the form

$$\sum_{n=0}^{\infty} \left(e^{-h\nu/k_{\rm B}T}\right)^n$$

and since $0<\exp(-h\nu/k_{\rm B}T)<1$, it follows that the sum is nothing more than a geometric series for which the following summation formula applies

$$\sum_{n=0}^{\infty} r^n = {1 \over 1-r}$$

where $0<r<1$. Applying this to the probability formula, we obtain

$${\rm probability} = e^{-nh\nu/k_{\rm B}T}\left(1-e^{-h\nu/k_{\rm B}T}\right)$$

Similarly, the average energy of such a collection of systems is

$$\bar{E} = {\sum_{n=0}^{\infty} nh\nu e^{-nh\nu/k_{\rm B}T} \over \sum_{n=0}^{\infty} e^{-nh\nu/k_{\rm B}T}}$$

These sums can be worked out (with a lot of algebra not presented here) with the result

$$\bar{E} = {h\nu \over e^{h\nu/k_{\rm B}T}-1}$$

From this average energy formula, Planck was able to show that the intensity of emitted radiation from the blackbody would be given by

$$I(\nu) = {8\pi h\nu^3 \over c^3}{1 \over e^{h\nu/k_{\rm B}T}-1}$$

or using $\nu = c/\lambda$

$$I(\lambda) = {8\pi h \over \lambda^3}{1 \over e^{hc/\lambda k_{\rm B}T} -1}$$

Plotting this curve as a function of $\lambda$ yields the solid lines in the radiation spectra shown above.

By fitting this expression to the experimental data, he was able to determine a value for $h$, which is now accepted to be $h=6.6208\times 10^{-34}$J$\cdot$s. Planck's formula actually reduces to the classical formula in the limit that the difference between successive quantized energy values $\Delta E = h\nu$ is small compared to $k_{\rm B}T$, i.e.

$${h\nu \over k_{\rm B}T} \ll 1$$

When this is true, the exponential can be approximated using the formula $e^x \approx 1+x$, so that

$$e^{h\nu/k_{\rm B}T} \approx 1 + {h\nu \over k_{\rm B}T}$$

In this case

$$I(\nu) \rightarrow {8\pi h\nu^3 \over c^3} {1 \over 1 + h\nu/k_{\rm B}T -1}$$

$$= {8\pi h\nu^3 \over c^3}{k_{\rm B}T \over h\nu}$$

$$= {8\pi k_{\rm B}T \nu^2 \over c^3}$$

Planck's proposed theory about energy quantization was able to explain the blackbody radiation spectra, yet it constituted a radical departure from classical physics.