Matter & Energy: Planck and Photons

James Richard Fromm

The introduction of the idea of quantization of energy is due to the German physicist Max Planck, who in 1900 postulated that the energy of any oscillator cannot be any desired value, but only the value related to its frequency f (more often symbolized by the Greek lower-case letter nu) by the Planck equation:

E = hf

The constant h appearing in this equation is known as the Planck constant. A precise modern value of this constant is 6.6260755(40) x 10-34 J-s.

The assumption that oscillators can have only the energies which are given by the Planck equation changes the energy of an assembly of oscillators from the value of kT used by Rayleigh and Jeans to the value of hf/(exp(hf/kT) - 1). The equation Planck derived that corresponds to the Rayleigh-Jeans equation is:

E(f) = [8(pi)f 2/c3](hf/(exp(hf/kT) - 1))

This equation gives excellent agreement with observed black-body radiation. By 1919 Ruben and Michel had demonstrated that its predictions were accurate from the temperature of liquid air (113 K) to about 2100 K. This corresponded to a range of total radiation power, or energy, which according to the Stefan-Boltzmann equation is of some 250,000 times. The equation of Planck accurately describes the emission of light at both ends of the electromagnetic spectrum and in addition it quantitatively predicts the shift of the maximum in the emission curve to higher frequencies as the temperature increases.

The interpretation of this equation is not intuitively obvious, but its meaning can be most easily grasped by considering what happens as the ratio hf/kT changes. Both h and k are constants, so the ratio will become smaller as frequency f decreases or as temperature T increases. The exponential function ex or exp(x) can be approximated to any desired accuracy by taking a sufficient number of terms of the series:

exp(x) = 1 + x + x2/2! + x3/3! + x4/4! + ...

It can be seen that the later terms become very small when the value of x is small, approaching the value 1 + x as the value of x decreases, since the factorials in the denominator are continuously increasing. As the ratio hf/kT decreases, then, exp(hf/kT) approaches 1 + (hf/kT). As a consequence, the equation of Planck approaches the equation of Rayleigh and Jeans at lower frequencies.

As the ratio hf/kT increases, which will occur either as frequency increases or as temperature decreases, the effect of the temperature of the body becomes small relative to the effect of the energy quantization restriction. The value of exp(hf/kT) will be much larger than one, so exp(hf/kT) - 1 will approach exp(hf/kT). Since hf/exp(hf/kT) can be rewritten as hfexp(-hf/kT), it is necessary to consider what happens to exp(-x) when the value of x becomes large. The series above will then be:

exp(-x) = 1 - x + x2/2! - x3/3! + x4/4! - ...

The signs of the terms now alternate, and as -x increases the value of exp(-x) will approach one as the alternating terms in x cancel each other out, so hfexp(-hf/kT) approaches hf and the equation simplifies to:

E(f) = [8(pi)f 2/c3]h

This is what would be expected if the effect of the temperature of the black body is small relative to the effects of quantized energy.

The Planck equation is fundamental to all of physics and chemistry, particularly to that part of chemistry concerned with the interaction between light and matter. We shall return to it many times in the course of other sections.

Copyright 1997 James R. Fromm