Radiation from a hot "black body" and Planck's quantum theory

A spectroscope is used to observe a broad and continuous spectrum of colors of visible light emitted from a hot "black body."

Ingredients: tungsten filament bulb, lamp, variac (variable transformer), spectroscope

Procedure: A complete recipe follows.

1. Use the varaic to control the voltage applied to the light bulb.

2. Observe the emission through the spectrometer.

3. Characterize the range of colors of the visible spectrum. Is the spectrum composed of discreate lines of color, or broad and continuous bands of color?

4. Repeat the experiment at increasing voltage and compare results.

Understanding: The phenomenon of black body radiation is known to anyone that has tended a fire. When a solid object such as a metal poker becomes very hot, it begins to glow red. As it becomes hotter still, the poker will glow a brighter orange. When it is very hot, it will glow white. That is where the term "white hot" comes from. White hot is hotter than red hot.

When the black body radiation is observed through a spectroscope, the "white" light is observed to be composed of the full range of colors of the visible spectrum of electromagnetic radiation. Increasing the voltage of the light increases the temperature of the tungsten filament. The increase in temperature causes the intensity of the light to increase, with more radiation being emitted at higher frequencies.

The solid black body is composed of atoms, and the atoms vibrate in a number of ways. Each vibration has a frequency and a wavelength. The wavelength tells us something about the nature of the motion of atoms in the solid. If the wavelength is on the order of a few Ångströms, the vibration involves an atom or two. If the wavelength is on the order of nanometers, the vibration might involve the collective motion of hundreds of atoms.

There are low frequency vibrations where the atoms move together in a "breathing" pattern of expansion and contraction of the body. Those are known as acoustic modes as the frequencies of vibration correspond to the frequencies of sound. There are high frequency vibrations where atoms tend to vibrate locally against the surrounding atoms. Those are known as optical modes as the frequencies of vibration correspond to frequencies of light. We call each vibration a mode.

How can we understand the relative intensity of the black body radiation as a function of the frequency of the emitted light? One of the vibrational modes is excited in energy. The mode can reduce its energy by relaxing and emitting light energy. The emission process satisfies the conservation of energy as

Eexcited mode → Erelaxed mode + Elight

Planck's revolutionary theory of black body radiation

Max Planck studied this process at the turn of the century. Experimental studies consisted of heating the black body to a well-defined temperature. The body would then be observed using a spectrometer, and the intensity of the light would be recorded as a function of the frequency of the light. Those studies indicated that the intensity of black body radiation would initially increase with increasing frequency of light and then decrease as the frequency continued to increase.

The problem with that experimental result is that the classical laws of physics said that the intensity of the black body radiation should increase with increasing frequency, without ever decreasing!

Planck was able to develop a theory that fit the experimental data. He assumed that when the excited mode of the black body would relax, light energy was given off, and the light energy was proportional to the frequency of the light emitted

Elight = nhν

where n is a counting number, n=1,2,3... The constant h has units of energy per unit time and is known as Planck's constant. Boltzmann's constant can be used to convert temperature to a corresponding energy. Planck's constant can be used to convert frequency of motion to a corresponding energy.

Planck assumed that the intensity of light emitted at a given frequency would be determined by

Intensity(ν) = N(ν) p(ν, T)

where N(ν) is the number of modes in the solid of a given frequency ν, and p(ν, T) is the probability of emitting light at a frequency ν at a temperature T. For a solid black body the number of vibrational modes increases as

N(ν) = constant ν2

And by Boltzmann's probability we find that the probability of emitting light energy at a given frequency would be

p(ν, T) = hν /(exp(hν/kT)-1)

With these assumptions, Planck was able to show that (1) at low frequencies the intensity of the radiation increased as the frequency of the emitted light increased

Intensity(ν) = constant ν3

and (2) at high frequencies the intensity decreased exponentially with increasing frequency

Intensity(ν) = constant ν3 exp(-hν/kT)

At intermediate frequencies, the intensity would be a maximum.

The most important result of Planck's theory was his conjecture that the energy of the light was a multiple of a "quantum" unit of energy proportional to the frequency of the light

E = hν

It was not clear why his theory made sense, but mathematically it was able to predict the result of the experiments. It would take a number of years before Planck's inspired conjecture would be understood conceptually.

The Boltzmann probability and black body radiation

Question: The visible range of the electromagnetic spectrum extends from 400 nm to 700 nm. If we divide it into six ranges of 50 nm each, we can step from red (700-650 nm) to orange (650-600 nm) to yellow (600-550 nm) to green (550-500 nm) to blue (500-450 nm) to violet (450-400 nm). That is only approximate, but it is easily remembered and useful to recall.

Compute the energy of red (650 nm), yellow (570 nm) and violet (400 nm) light using Planck's constant. After computing the energies, compute the corresponding temperatures of each of the colors of light.

You can check your answers here.