The Quantum Theory of Solids

The Puzzle

Nineteenth-century physics predicted that every solid should need the same amount of heat, per atom, to warm by one degree. For many materials near room temperature this rule, named after Dulong and Petit, works well. But it fails badly in the cold. Carbon in the form of diamond falls short even at room temperature, and as any solid is cooled toward absolute zero, the heat it can absorb per degree drops toward nothing. Classical physics had no explanation.

Einstein's Idea

Planck had introduced energy quanta in 1900 to explain the light given off by hot objects, and Einstein had extended the idea to light itself in 1905. In 1907 he asked what would happen if the atoms in a solid, vibrating about their positions, could also only hold energy in discrete steps of size $h\nu$.

He modeled a solid as a collection of identical oscillators, all vibrating at one frequency. At high temperatures there is plenty of thermal energy to go around, the steps do not matter, and the classical rule is recovered. At low temperatures there is not enough energy to excite even one step, the vibrations "freeze out," and the heat capacity collapses. The formula above describes the whole curve with a single adjustable number, the Einstein temperature ${\theta }_E$.

Why It Mattered

This was the first application of quantum ideas to the behavior of matter, as opposed to light. It showed that the quantum was not a peculiarity of radiation but something general, and it helped convince leading physicists to take the idea seriously. The first Solvay Conference of 1911, where the quantum was the central topic, was prompted in large part by Walther Nernst's low-temperature measurements, which confirmed Einstein's prediction.

How It Held Up

The model captures the essential behavior but is too simple in detail. Real atoms in a crystal do not vibrate independently at one frequency; they share a spectrum of collective vibrations. In 1912 Peter Debye refined the theory accordingly, giving the correct behavior at the lowest temperatures. Einstein's version remains the standard first example in textbooks, where it is known as the Einstein solid.

Where It Stands

The quantum theory of solids that began here grew into the physics behind semiconductors, and so behind every transistor and computer chip.