Radiation laws and Planck’s light quanta

The quantum theory of absorption and emission of radiation announced in 1900 by Planck ushered in the era of modern physics. He proposed that all material systems can absorb or give off electromagnetic radiation only in “chunks” of energy, quanta E, and that these are proportional to the frequency of that radiation E = hν. (The constant of proportionality h is, as noted above, called Planck’s constant.)

Planck was led to this radically new insight by trying to explain the puzzling observation of the amount of electromagnetic radiation emitted by a hot body and, in particular, the dependence of the intensity of this incandescent radiation on temperature and on frequency. The quantitative aspects of the incandescent radiation constitute the radiation laws.

The Austrian physicist Josef Stefan found in 1879 that the total radiation energy per unit time emitted by a heated surface per unit area increases as the fourth power of its absolute temperature T (Kelvin scale). This means that the Sun’s surface, which is at T = 6,000 K, radiates per unit area (6,000/300)⁴ = 20⁴ = 160,000 times more electromagnetic energy than does the same area of Earth’s surface, which is taken to be T = 300 K. In 1889 another Austrian physicist, Ludwig Boltzmann, used the second law of thermodynamics to derive this temperature dependence for an ideal substance that emits and absorbs all frequencies. Such an object that absorbs light of all colours looks black, and so was called a blackbody. The Stefan-Boltzmann law is written in quantitative form W = σT⁴, where W is the radiant energy emitted per second and per unit area and the constant of proportionality is σ = 0.136 calories per metre²-second-K⁴.

The wavelength or frequency distribution of blackbody radiation was studied in the 1890s by Wilhelm Wien of Germany. It was his idea to use as a good approximation for the ideal blackbody an oven with a small hole. Any radiation that enters the small hole is scattered and reflected from the inner walls of the oven so often that nearly all incoming radiation is absorbed and the chance of some of it finding its way out of the hole again can be made exceedingly small. The radiation coming out of this hole is then very close to the equilibrium blackbody electromagnetic radiation corresponding to the oven temperature. Wien found that the radiative energy dW per wavelength interval dλ has a maximum at a certain wavelength λ_m and that the maximum shifts to shorter wavelengths as the temperature T is increased, as illustrated in Figure 8. He found that the product λ_mT is an absolute constant: λ_mT = 0.2898 cm-K.

Wien’s law of the shift of the radiative power maximum to higher frequencies as the temperature is raised expresses in a quantitative form commonplace observations. Warm objects emit infrared radiation, which is felt by the skin; near T = 950 K a dull red glow can be observed; and the colour brightens to orange and yellow as the temperature is raised. The tungsten filament of a lightbulb is T = 2,500 K hot and emits bright light, yet the peak of its spectrum is still in the infrared according to Wien’s law. The peak shifts to the visible yellow when the temperature is T = 6,000 K, like that of the Sun’s surface.

It was the shape of Wien’s radiative energy distribution as a function of frequency that Planck tried to understand. The decrease of the radiation output at low frequency had already been explained by Lord Rayleigh in terms of the decrease, with lowering frequency, in the number of modes of electromagnetic radiation per frequency interval. Rayleigh, following the principle of equipartition of energy, assumed that all possible frequency modes could radiate with equal probability. Since the number of frequency modes per frequency interval continues to increase without limit with the square of the frequency, Rayleigh’s formula predicted an ever-increasing amount of radiation of higher frequencies instead of the observed maximum and subsequent fall in radiative power. A possible way out of this dilemma was to deny the high-frequency modes an equal chance to radiate. To achieve this, Planck postulated that the radiators or oscillators can emit electromagnetic radiation only in finite amounts of energy of size E = hν. At a given temperature T, there is then not enough thermal energy available to create and emit many large radiation quanta hν. More large energy quanta hν can be emitted, however, when the temperature is raised. Quantitatively the probability of emitting at temperature T an electromagnetic energy quantum hν is where k is Boltzmann’s constant, well known from thermodynamics. With c = λν, Planck’s radiation law then becomes

This is in superb agreement with Wien’s experimental results when the value of h is properly chosen to fit the results. It should be pointed out that Planck’s quantization refers to the oscillators of the blackbody or of heated substances. These oscillators of frequency ν are incapable of absorbing or emitting electromagnetic radiation except in energy chunks of size hν. To explain quantized absorption and emission of radiation, it seemed sufficient to quantize only the energy levels of mechanical systems. Planck did not mean to say that electromagnetic radiation itself is quantized, or, as Einstein later put it, “The sale of beer in pint bottles does not imply that beer exists only in indivisible pint portions.” The idea that electromagnetic radiation itself is quantized was proposed by Einstein in 1905, as described in the subsequent section.

Photoelectric effect

Hertz discovered the photoelectric effect (1887) quite by accident while generating electromagnetic waves and observing their propagation. His transmitter and receiver were induction coils with spark gaps. He measured the electromagnetic field strength by the maximum length of the spark of his detector. In order to observe this more accurately, he occasionally enclosed the spark gap of the receiver in a dark case. In doing so, he observed that the spark was always smaller with the case than without it. He concluded, correctly, that the light from the transmitter spark affected the electrical arcing of the receiver. He used a quartz prism to disperse the light of the transmitter spark and found that the ultraviolet part of the light spectrum was responsible for enhancing the receiver spark. Hertz took this discovery seriously because the only other effect of light on electrical phenomena known at that time was the increase in electrical conductance of the element selenium with light exposure.

A year after Hertz’s discovery, it became clear that ultraviolet radiation caused the emission of negatively charged particles from solid surfaces. Thomson’s discovery of electrons (1897) and his ensuing measurement of the ratio m/e (the ratio of mass to charge) finally made it possible to identify the negative particles emitted in the photoelectric effect with electrons. This was accomplished in 1899 by J.J. Thomson and independently by Philipp Lenard, one of Hertz’s students. Lenard discovered that for a given frequency of ultraviolet radiation the maximum kinetic energy of the emitted electrons depends on the metal used rather than on the intensity of the ultraviolet light. The light intensity increases the number but not the energy of emitted electrons. Moreover, he found that for each metal there is a minimum light frequency that is needed to induce the emission of electrons. Light of a frequency lower than this minimum frequency has no effect regardless of its intensity.

In 1905 Einstein published an article entitled “On a Heuristic Point of View about the Creation and Conversion of Light.” Here he deduced that electromagnetic radiation itself consists of “particles” of energy hν. He arrived at this conclusion by using a simple theoretical argument comparing the change in entropy of an ideal gas caused by an isothermal change in volume with the change in entropy of an equivalent volume change for electromagnetic radiation in accordance with Wien’s or Planck’s radiation law. This derivation and comparison made no references to substances and oscillators. At the end of this paper, Einstein concluded that if electromagnetic radiation is quantized, the absorption processes are thus quantized too, yielding an elegant explanation of the threshold energies and the intensity dependence of the photoelectric effect. He then predicted that the kinetic energy of the electrons emitted in the photoelectric effect increases with light frequency ν proportional to hν − P, where P is “the amount of work that the electron must produce on leaving the body.” This quantity P, now called work function, depends on the kind of solid used, as discovered by Lenard.

Einstein’s path-breaking idea of light quanta was not widely accepted by his peers. Planck himself stated as late as 1913 in his recommendation for admitting Einstein to the Prussian Academy of Sciences “the fact that he [Einstein] may occasionally have missed the mark in his speculations, as, for example, with his hypothesis of light quanta, ought not to be held too much against him, for it is impossible to introduce new ideas, even in the exact sciences, without taking risks.” In order to explain a quantized absorption and emission of radiation by matter, it seemed sufficient to quantize the possible energy states in matter. The resistance against quantizing the energies of electromagnetic radiation itself is understandable in view of the incredible success of Maxwell’s theory of electromagnetic radiation and the overwhelming evidence of the wave nature of this radiation. Moreover, a formal similarity of two theoretical expressions, in Einstein’s 1905 paper, of the entropy of an ideal gas and the entropy of electromagnetic radiation was deemed insufficient evidence for a real correspondence.

Einstein’s prediction of the linear increase of the kinetic energy of photoemitted electrons with the frequency of light, hν − P, was verified by Arthur Llewelyn Hughes, Owen Williams Richardson, and Karl Taylor Compton in 1912. In 1916 Robert Andrews Millikan measured both the frequency of the light and the kinetic energy of the electron emitted by the photoelectric effect and obtained a value for Planck’s constant h in close agreement with the value that had been arrived at by fitting Planck’s radiation law to the blackbody spectrum obtained by Wien.