Black-body radiation & Planck

A black-body is a perfect energy absorber: it absorbs all incoming frequencies of electromagnetic radiation. This is converted to heat and then re-emitted as radiation that has a specific spectrum and intensity that depends on the temperature of the body.

Scientists performed this test with many materials, and the results were consistent. The problem was that Maxwell’s equations predicted that at high temperatures, the amount of ultraviolet should increase, not decrease. Maxwell’s equations seemed to only work at low frequencies. Rayleigh and Wien modified the equations to make them work with low and high frequencies, but no single set of equations worked with low, high, and middle-range frequencies.

This is where Max Planck comes in. He hypothesized that electrons can only vibrate with certain allowed frequencies. That is to say, electromagnetic radiation is discrete rather than continuous:

continuousdiscrete
A continuous hill and discrete staircase as an analogy for Planck’s hypothesis

Similar to the elementary charge for electricity, the energy conveyed by radiation can only be a multiple of an elementary amount. We can calculate the energy carried by a single photon with

`E = hf`,

where `h` is Planck’s constant, equal to 6.63 × 10−34 J·s, and `f` is the frequency, measured in hertz (Hz). When Planck’s hypothesis is taken into account, the black-body frequency distribution results agree with the theory. The continuous graph of intensity versus frequency can instead be thought of as a vertical bar graph where the vertical axis represents the number of photons that have the particular energy value on the horizontal.

Planck’s idea of “allowed” values shows up in the Bohr model of the atom. In the old Rutherford model, the electrons orbit the nucleus. There were many problems with this model. For one thing, it couldn’t explain the emission spectra of elements. Bohr’s explanation was that only some electron orbitals are allowed, and each one has a specific energy. Electrons can absorb energy and jump to a higher level. When they fall to a lower level, they emit energy in the form of a photon. The energy, or frequency, of the photon depends on the difference in energy between the two levels.