Paul Drude: A Life in Five Chapters

Chapter 1 · Göttingen, and the Letter c

From mathematics to optics

1863 – 1894 · Brunswick · Göttingen · Freiburg · Berlin

Paul Karl Ludwig Drude was born at Brunswick in July 1863, the son of a physician.

He began at Göttingen in *mathematics*, and switched to *physics* under *Woldemar Voigt* — a decision he attributed to Voigt's teaching and to finding that mathematics without a physical object to attach it to did not hold him.

His doctorate in 1887 was on *optics*: the reflection and refraction of light at the surfaces of *absorbing* crystals, which is difficult mathematics and required him to work out the optical constants of a range of substances.

He stayed in optics for over a decade and became a leading authority. His *Lehrbuch der Optik* of 1900 was a standard text across Europe for a generation. He developed *ellipsometry* — determining the properties of a thin surface film from the change in polarisation of light reflected off it — which is now a routine technique in semiconductor manufacture.

And he is the reason the speed of light is written *c*.

The symbol had been used occasionally before; various authors used *V* or other letters. Drude adopted *c* consistently in his optics textbook, which was widely read, and it became standard. It probably derives from the Latin *celeritas*, speed.

So one of the most recognisable symbols in physics — the one in *E = mc²* — is there because of a textbook choice by a man almost nobody has heard of.

He held posts at Göttingen and Leipzig, and became professor at Giessen in 1900.

And in 1897 *J. J. Thomson* announced the discovery of the *electron*.

“The electrons in a metal may be treated as a gas.”

— Attributed to Paul Drude, on the free-electron model (1900)

Why this matters

The c in E = mc² is written that way because Drude used it consistently in a widely read optics textbook.

You have the optics and the letter. What would you ask him?

Ask Drude

• “Why is the speed of light written c?”
• “Why did you leave mathematics for physics?”
• “What is ellipsometry good for?”
• “How much does a textbook shape a subject?”
• “What did Thomson's electron open up for you?”

Chapter 2 · A Metal Is a Box of Gas

The free-electron model, 1900

1900 · Giessen

Drude's model, published in *Annalen der Physik* in 1900, rests on one bold simplification.

Suppose that in a metal, each atom gives up one or more of its outer electrons, which become *free* — not attached to any particular atom. What remains behind is a rigid framework of heavy *positive ions*, fixed in position.

And suppose the free electrons move through the metal *exactly as the molecules of a gas move through a box*: travelling in straight lines at high speed, colliding, moving randomly in all directions with no net motion.

The kinetic theory of gases was thoroughly developed by 1900, and Drude simply *borrowed it wholesale* and applied it to electrons in a metal. The electrons collide with the fixed ions as gas molecules collide with obstacles.

Now apply an *electric field*. The random motion continues, but superimposed on it is a slow *drift* in the direction of the field. That drift is the *electric current*.

And the *collisions* — electrons scattering off the ion framework — are what oppose the drift. That is *resistance*.

So electrical conduction and resistance both come out of one picture.

Then *heat*. Heat one end of a metal bar and the electrons at that end move faster. They carry that extra energy along as they travel, and deposit it where they collide at the cooler end.

So the *same* free electrons that carry the current also carry the *heat*.

That explains, immediately, why *metals are good conductors of both electricity and heat*, and why the two properties go together — which is an obvious fact about metals that nobody had explained.

It is a crude model. Drude knew it was crude. He treated the electrons as an ordinary classical gas obeying ordinary statistics, and assumed the collisions randomise the electron's velocity completely.

Why this matters

The same free electrons carry both the current and the heat, which is why good electrical conductors are also good thermal conductors.

You have the electron gas and the fixed ions. What is your question?

Ask Drude

• “What exactly are you supposing about a metal?”
• “What is an electric current, on your account?”
• “What causes resistance?”
• “Why are metals good at conducting heat too?”
• “How much did you borrow from the kinetic theory of gases?”

Chapter 3 · The Ratio

Reproducing a number measured forty-seven years earlier

1853 – 1900 · Giessen

Here is why the model was taken seriously, and it is a beautiful piece of scientific reasoning.

In *1853*, *Rudolph Franz* and *Georg Wiedemann* had made a purely empirical observation about metals. Divide a metal's *thermal* conductivity by its *electrical* conductivity, and the answer is *roughly the same for all metals* — and it is proportional to the *absolute temperature*.

They had no explanation whatever. It was a regularity in a table of measurements, sitting there unexplained for forty-seven years.

Now apply Drude's model. Calculate the electrical conductivity from the electron gas: it depends on the number of electrons, their charge, their mass and the time between collisions. Calculate the thermal conductivity: it depends on the number of electrons, their heat capacity, their speed and the time between collisions.

And then *divide one by the other*.

Most of the properties of the electron *cancel*. The number of electrons cancels. The collision time cancels. The mass largely cancels.

What is left is a quantity depending only on the *temperature* and on *fundamental constants* — Boltzmann's constant and the electron's charge.

So the ratio should be the same for all metals and proportional to temperature. Which is exactly what Wiedemann and Franz had measured.

And the *numerical value* Drude calculated came out close to the measured one.

That is the strongest kind of evidence a model can have. Drude was not fitting a parameter to data. He was reproducing a number somebody else had measured, for another purpose, half a century earlier, with no theory in mind at all.

There is a wrinkle. Drude's agreement was partly a *fortunate cancellation* of two errors: he used a classical heat capacity for the electrons, which is far too large, and a classical speed, which is far too small. The two mistakes largely cancel in the ratio. The quantum theory later got the right answer for the right reasons.

He did not know that. The agreement was real, and it convinced people, and it was partly luck.

Why this matters

Drude reproduced a ratio measured forty-seven years earlier for another purpose with no theory attached — which is the strongest evidence a model can have.

You have the cancelling properties and the matching number. What would you ask?

Ask Drude

• “What did Wiedemann and Franz actually measure?”
• “Why do the electron's properties cancel in the ratio?”
• “Why is reproducing an old measurement such strong evidence?”
• “Was the agreement partly luck?”
• “Would you have believed the model without the ratio?”

Chapter 4 · What It Cannot Do

A heat capacity metals do not have

1900 – 1928 · Giessen · Leiden · Munich

The model's failures are as instructive as its success, and Drude was clear about most of them.

It says *nothing about what holds a metal together*. Why does a lump of copper cohere? Why can a metal be *hammered flat* and *drawn into wire* when a salt crystal shatters? The mobile electron sea is in fact the right answer — the ions can slide past one another without breaking bonds because the electron sea holds them regardless — but Drude did not draw that out.

It says nothing about *why one metal differs from another*: why copper conducts better than iron, why mercury is liquid, why tungsten melts at three and a half thousand degrees.

And it makes a *definite wrong prediction*.

If a metal contains a gas of free electrons, that gas should contribute to the metal's *heat capacity*. Heating a metal should require energy to heat the electron gas as well as to make the ions vibrate. Classical kinetic theory says each electron should absorb an amount of energy proportional to temperature, and there is roughly one free electron per atom.

So a metal should have a heat capacity roughly *half as large again* as a non-metal with the same number of atoms.

Metals do not. They obey the same *Dulong–Petit* value as everything else. The electron gas contributes essentially *nothing* to the heat capacity at ordinary temperatures.

This was recognised immediately and it was a serious problem. *Hendrik Lorentz* reworked the model more rigorously in 1905 and made the difficulty *worse*, not better.

The resolution came in 1927 and 1928, from *Arnold Sommerfeld*, using the *Fermi–Dirac statistics* developed by Fermi and Dirac in 1926.

Electrons are not an ordinary gas. They obey the *Pauli exclusion principle*: no two can occupy the same state. So the electrons in a metal are stacked into energy levels right up to a high level called the Fermi energy, and *only the few electrons near the top* can absorb a small amount of thermal energy — the rest have no empty state to move into.

So only a tiny fraction of the electrons participate, and the heat capacity contribution is small.

Sommerfeld kept Drude's picture entirely and replaced the *statistics*. The Drude–Sommerfeld model is what an undergraduate learns.

Why this matters

Drude's electron gas predicts a heat capacity metals do not have, and the fix was to keep his picture and replace classical statistics with quantum ones.

You have the missing heat capacity and the Pauli principle. What is your question?

Ask Drude

• “Why should an electron gas add to a metal's heat capacity?”
• “Why does it not?”
• “Why can a metal be hammered flat?”
• “Did Lorentz make things better or worse?”
• “How much of your model survived?”

Chapter 5 · Forty-Two

The Annalen, a chair at Berlin, and a death nobody explained

1900 – 1906 · Giessen · Berlin

In 1900 Drude became *editor of the Annalen der Physik*, the most important physics journal in Germany and one of the most important in the world.

That is a position of very considerable influence: the editor decides what German physics publishes.

And it produced one decision that matters enormously. In 1905 the *Annalen* received four papers from an unknown patent clerk in Bern named *Albert Einstein* — on the photoelectric effect, on Brownian motion, on special relativity, and on mass–energy equivalence.

Drude published all four.

Einstein had no academic position, no reputation and no institutional backing, and the special relativity paper contained no references to any other work. It would have been entirely ordinary to reject them or to demand revisions. Drude, on the recommendation of Max Planck as an advisor to the journal, published them.

That is 1905 — the *annus mirabilis* — and it happened in Drude's journal on his watch.

In 1905 he was appointed to the chair of physics at the *University of Berlin*, brought in by *Max Planck*. It is one of the most senior positions in German science and he was forty-two.

He was also elected to the Prussian Academy of Sciences and delivered his inaugural address in June 1906.

*Five days later*, on 5 July 1906, he took his own life.

No explanation was ever established. Contemporaries speculated about the pressure of the new position, about depression, about his health. His family said nothing publicly. Planck, who had brought him to Berlin, was reportedly shattered.

He was forty-two. He left a wife and four children.

His model was superseded and it is still taught, because the picture of a metal as a lattice of positive ions in a sea of mobile electrons is *his*, and the quantum theory of solids was built by *repairing* it rather than by replacing it.

Why this matters

Drude edited the journal that published Einstein's four 1905 papers from an unknown patent clerk, and he died by his own hand the following year at forty-two.

You have the four papers, the chair and the five days. What would you ask?

Ask Drude

• “What did you make of Einstein's four papers?”
• “Why publish an unknown clerk with no references?”
• “What did editing the Annalen involve?”
• “What did the Berlin chair mean to you?”
• “Why is a crude model still worth teaching?”

What Drude changed

Every school and undergraduate description of a metal — positive ions in fixed positions with a sea of mobile electrons around them — is Drude's picture, and the quantum theory of solids was built by repairing it rather than by replacing it. The Wiedemann–Franz agreement remains one of the most quoted successes of a deliberately crude model, and the c in E = mc² is written that way because of his textbook.

A debate that continues

Drude's agreement with the Wiedemann–Franz ratio depended partly on two large errors cancelling — a classical heat capacity too high and a classical speed too low — and his model predicts an electronic heat capacity that metals do not have.

Keep exploring — ask Drude

• “How crude may a useful model be?”
• “What else could be borrowed from the theory of gases?”
• “Which paper are you proudest to have published?”

Related lives

• J. J. Thomson — Who Weighed What Nobody Could See
• Albert Einstein — Nobel Laureate · Author of Relativity
• Georg Ohm — Whose Proportion Was Called A Fancy
• Niels Bohr — Architect of the Quantum Atom

Related themes

Metals and conduction · Electrons in solids · Models and approximations