Do Physics, But Learn History
Discovering the Drude model through the lens of a Historian.
Over the past few months, I’ve been reflecting on better ways to learn physics, other than just solving problems. I think one of the best ways to do physics is to learn history. Here’s why.
If you asked me for a hidden gem in physics, I would bring up would be the Drude model. In 1900, Paul Drude wrote a paper on a model of resistance in metals. It was one of the first models that could explain why the ratio of thermal to electrical conductivity was the same in different metals - an observation made fifty years earlier by Wiedemann and Franz. Even now, it’s still taught in introductions to materials physics (0).
It grew on me that it’s somewhat miraculous that physicists in the 1800s actually managed to figure this stuff out in the first place, with very few tools.
And I’ve never really asked how they did it. But to be fair, no one really tells us.
How did Drude figure this out in the first place? What clues did he have to begin to model something as mysterious as metals, in a time when even electricity was not understood? Before writing this essay, I couldn’t find much content on the history and context that came before the Drude model. And despite its importance, a detailed interpretation of his work seem to be missing. Drude’s original paper was written in German, but no English translation is available. When I tried translating Drude’s paper myself, I had more questions than answers.
Despite being able to solve modern problems on this model, this felt like I strong hint that I didn’t understand physics enough. It turned out that I knew nothing about the building blocks that the early materials physicists worked with. I didn’t know the failures in early physics and how they were fixed. I didn’t even know what instruments they made to measure things.
I tried to fill this gap by making a chronology, reasonably suggesting the questions that Drude was trying to answer in his time, and figuring out the tools he had to make his models.
Writing about the model's history helped me understand physics differently. It turns out that it is an elaborate story of frogs, gases, magnetism, and experimentation. Often times, we’re given a perfect version of physics models that we forget how they got there. A physics teacher’s usual advice tends to be to try solve as many problems as possible, but that feels incomplete. Some of the best physicists of our generation have written books on the history of physics. Maybe digging into a little history can’t hurt.
The relationship between thermal and electrical conductivity under different conditions remains an active problem. Perhaps curiosity about its past might lead to questions about its future.
Discovering current
This is a story about conductivity, and I’ll start with a little background on electricity.
It seems that physicists were still confused about how electricity worked, even during the 1900s. The first discovery of electric current actually came only a century earlier, by observing dead frogs. Ten years later, Volta discovered the battery (1). The idea of electric current might feel simple to us now, but no one really knew what it was made of until the late 1800s. I found this fact calming, because it made me feel a less bad about my own confusion when learning simple things.
The key insight that moved things forward were Oersted’s observations (2) that current in wires caused magnetic fields. This allowed scientists to measure, not just with their tongues. In the late 1800s and early 1900s, this ‘current’ was measured with instruments called galvanometers. Nowadays, we call them ammeters.
The early galvanometers came from Johann Salomo, Christof Schweigger and Johann Christian Poggendorff. A rudimentary way of measuring current would be observing how much a compass needle moved when exposed to a wire, and then calibrating it so that measurements could repeatedly be taken. In 1821, Poggendorff realised that coiling this wire amplified its magnetic field, making his instrument more sensitive. He called it a condenser.
The Measurement and Discovery of the Weidemann-Franz Law
This current could now be measured through a metal when an electric field was applied to it. A metric called electric conductivity was the factor that relates the current per unit volume to the strength of the electric field applied to a metal. Similarly, when heat is applied to a metal at a given point, the temperature spreads out in a gradient. The factor that relates the steepness of this gradient to the heat per unit volume applied, is known as the thermal conductivity (3).
Now this is where the physics gets interesting. In 1853, Weidemann and Franz found that electrical conductivity divided by thermal conductivity of a metal at a given temperature, regardless of what type of metal it is, stays the same. In their paper, it was found that the ratios of around 20 different metals were roughly constant.
The Drude Model was one of the first models of conductivity to explain this. I guess this really is every theoretical physicist’s dream, to make some math that explains observations in an experiment.
How did Drude solve it?
In most modern presentations, the Drude model consists of electrons in a metal that bounce off positively charged ions a random way. With some fairly simple calculations, both the electrical and thermal conductivity can be derived, revealing a ratio that stays constant regardless of the type of metal it is. But how did he get there?
After reading and translating his original paper on google translate, I learned that most of his insights were a mix of pretty different things. The first piece was built on the discovery of the electron made by J.J Thompson several years earlier - by the turn of the 1900s, English and German physicists questioned the existence of a more fundamental object than the electric fluid model of current. By noticing that cathode beams tilted slightly under the influence of an electricity, J.J Thompson had developed his theory of 'corpuscles’ (4).
Now suspecting that corpuscles might be in the metal, Drude used the fact that they were charged, along with Lorenz’s electromagnetic force laws to figure out how they accelerated inside it (6). What is surprising though, is how uncertain he was about mass (5), which is something most textbooks don’t mention. He then argues that the
Then, he uses Boltzmann’s gas laws to argue how fast the electrons move on after the collision, saying that that the speed that they leave after the collision is on average dependent on how hot the metal is. Since the average speed is related to the conductivity of a metal, he found that the conductivity must’ve been
𝜎=14𝛼𝑇𝑒2𝑛𝑙𝑢
Here, T is the temperature, alpha is the Boltzmann constant, e is electron charge, n is the density of electrons, u is the average speed of an electron, and l is the average distance an electron travels before knocking into something else.
He also used Boltzmann’s gas laws to figure out the thermal conductivity. To do this, he used a pretty important formula from Boltzmann’s lectures which tells us how much of a quantity (in our case, heat), flows through a portion of a cylinder. Crucially, the formula depends on the temperature gradient, and so Drude was able to calculate thermal conductivity. Physicists usually call this quantity k.
𝑘=13𝛼𝑛𝑙𝑢
Dividing the electrical conductivity by the thermal conductivity gives us, somewhat miraculously, the ratio below. This is only a function of temperature, and doesn’t take into consideration any specific properties about the metal! And so, we have a consistent theory for the Wiedemann-Franz law.
𝑘𝜎=4𝛼23𝑒2𝑇
Closing Thoughts
If anything, I think this proves how influential and successful Boltzmann was at his time, since his ideas on gas theory were used in different ways to try and crack the problem. It also made me realise how really random areas of physics can combine to make discoveries. It’s worth noting that the Wiedemann-Franz law, once thought to only work in normal conditions, is still being observed in more extreme scenarios too.
Appendix
1. The Discovery of Current in Animals
The discovery of electric current began with animals. In 1780, Luigi Galvani found that touching a dead frog with a C-shaped metal contact had, surprisingly, made its legs twitch. It was more than a century before the discovery of the electron, and was one of the first observations of what he thought was an electric fluid contained inside animals. The metal contact was thought to have restored balance between different amounts of electric fluids in the muscles and nerves. Funnily enough, we now know that this is kind of true, depending on how you look at it.
"For, while I with one hand held the prepared frog by the hook fixed in its spinal marrow, so that it stood with its feet on a silver box, and with the other hand touched the lid of the box, or its sides, with any metallic body, I was surprised to see the frog become strongly convulsed every time that I applied this artifice."
Until that point electricity - or current for that matter - wasn't really its own thing. What was known as ‘electricity’ was actually only static electricity. Static electricity was produced from friction, and was known about since the time of the Greeks. At the time, static electricity was thought to be a different thing to the animal electricity in Galvani's frogs.
In 1792, Volta invented the electrostatic pile, but it’s unclear on the motivations he had to lead him there. This pile was a primitive form of battery made of repeated layers of zinc, copper and cardboard that could deliver the first continuous stream of electricity. Upon touch and taste, he felt a continuous shock its electric current, something more than just static electricity.
The pile supported Volta’s idea that frogs didn't twitch from an electric fluid intrinsic to the animal, but rather a more fundamental electricity intrinsic to metals. He thought that electric fluid in his pile that moved through repeated, instantaneous discharges from layer to layer. Whilst natural philosophers debated the workings behind Volta's pile, the French engineer Coulomb had already developed a theory of static electricity in 1780. Coulomb predicted the force between non-moving, electrically charged objects.
Propelled by the novelty of Volta's discovery, and Coulomb's developments in static electricity, the race for a more comprehensive theory behind the electrostatic pile occupied French physicists in the early 1800s, lead by Ampere and Biot. Instead of electricity being a static effect, Ampere proposed that the pile's electricity came from the 'continuous, active impulsion' of an electric fluid - or what we know refer to as current.
2. Oersted’s Discovery
As part of this electrical renaissance, the Danish physicist Oersted noticed that a compass needle moved when it came near a wire connecting the pile’s two ends. Between 1850 and 1900, the physicists Maxwell, Ampere and Faraday had done work to understand that these currents caused magnetic fields, and vice versa. Current at this time was a foggy concept - it was often referred to an as ‘electrical conflict’ - that represented the speed of charge moving through space, and its relationship to electric and magnetic fields was theorised in Maxwell's equations.
In 1832, Faraday realised that a magnetic field could cause a current to be induced in a wire. Electricity and magnetism (and light) were definitively linked by James Clerk Maxwell, in particular in his "On Physical Lines of Force" in 1861 and 1862.
3. Thermal Conductivity
César-Mansuète Despretz's early method of measuring thermal conductivity involved putting evenly spaced thermometers on a rod of the metal that was being measured. He would then heat one end up in molten metal and look at the temperature gradient.
4. J.J Thompson and the Corpuscle
By understanding that electricity applies a force to a charged object, and measuring the deviation of the cathode rays, J.J Thompson he learned that these corpuscles had mass, as well as charge.
The discovery of the corpuscle had paved the way to understanding metals. By 1900, in an attempt to combine Maxwell's laws and the theory of corpuscles, Paul Drude wrote his first model of conduction. It was initially was an attempt to understand metallic reflection.
5. Drude was agnostic about mass, but it didn't matter
Drude's model starts by assuming that these corpuscles live in the metal. In modern textbooks, the electrons are presented as having a mass m, but this doesn't capture the uncertainty Drude had around electron mass at the time.
This uncertainty is captured in his second paragraph. Drude calls corpuscles by an earlier greek word - electron. He avoided calling them corpuscles since the word was associated with mass, and he wasn't sure that corpuscles had mass. He argued that it actually did not matter if they had an intrinsic mass or not, since they would still behave like they did, even if they were intrinsically massless.
By Maxwell's laws, the current from an electron with velocity would induce a magnetic field. Changing its velocity would cause its magnetic field lines to change. For those field lines to change, it would be necessary to apply a force, and so a force would be required to accelerate the corpuscle just like mass would.
This could be interpreted as an interesting insight between the difference between the measured mass and the bare mass of an electron, which is studied in renormalisation today.
6. The Acceleration Of Electrons in a Metal
Drude's Assumptions
As assumptions for his model, Drude only used Boltzmann's gas laws, the size of the charge of the electron, in his thesis. He writes that his theory is
allowed to draw without the use of special hypotheses other than those that have already been preserved in other areas, namely the gas laws, the Loschmidt number for the number of molecules in cm3 of a gas which at OoC exerts the pressure of an atmosphere, and the assumption for the size of the charge of an electron, which is agreed upon in order of magnitude by several people, especially by the works of J. J. Thomson.
He then concludes that there are only two types of such massless electron in metals, a positively charged one and a negatively charged one. From the theory of corpuscles, Drude assumes that the charges of these electrons must be integer multiples of some basic charge, which he calls e.
In his original argument, Drude assumes that electrons could be of different , with the metal consisting of different numbers
Metals by themselves don’t attract or repel anything, which means that they must be neutrally charged. Since Drude knew that metals had zero charge in total, he concluded there must've been areas of positive charge within the metal. Drude didn't know anything about the structure of atoms we know today. In fact, he throws under the rug the issue of corpuscles having any mass at all.
To appreciate the magic of what follows, we’re going to have to do some math. Drude calculated that under the influence of an electric field, the corpuscles would feel a force and start to freely accelerate within the metal. This was what generated current. The force that an electric field gives a charged object is part of what is called the Lorentz force law. This law its strength, multiplied by the charge.
𝐹=𝐸×𝑒
He then used Newton’s laws of motion to get the acceleration. Since acceleration is force divided by mass, the acceleration was given as the number
𝑎=𝑒𝐸𝑚
To then decide what happens next, he made a fairly natural assumption that I think was inspired by gas theory. As they accelerated, the corpuscles would randomly collide with the regions of positive charge. Drude used probabilies to measure this happening, since it was too hard to model the behaviour of the many corpuslces in the system. In the next section, I’ll explain how he figure out the time it took before these electrons got into their first collision.
Over a longer period of time, corpuscles were more likely to hit positive charge. Therefore, Drude's model proposed that the chance of a corpuscle colliding with positive charge in a small period, which we will call dt , was proportional to the time of this small period. Drude assumed that constant of proportionality was some quantity 1/τ (it was inverted so that the math worked out easier later on), where τ was called the relaxation time.
On average, with a derivation that you can try yourself, this meant that a corpuscle would experience an amount of time τ, equal to the relaxation time, before each collision. And since it had accelerated due to an electric field, using our expression for acceleration above, the corpuscles had a speed, on average, of
𝑣=−𝑒𝐸𝜏𝑚
right before colliding.
After a corpuscle collided, Drude assumed that it would continue in a random direction. Its speed after the collision would be proportional to the temperature of the metal, which was inline with gas theory at the time. It wouldn't depend at all on the speed just before it collided.
Current is the amount of charge that travels, multiplied by its velocity. So, if we had N electrons traveling at a velocity v, and the charge for a single electron was, the total charge would be Ne. The total current would be this charge multiplied by their velocity, and so the current I would be given by
𝐼=𝑁×𝑒×𝑣
To simplify the math, and work with numbers that don't rely on the total volume that we're working with, the model used densities, which were the amount of electrons in a given unit volume. If lower case n is the amount of electrons per unit of volume, then our expression for our current density j is
𝑗=𝑛×𝑒×𝑣
But we know that the velocity of each electron is given by the expression above. If we use this value, then the current density j is given by
𝑗=−−𝑛𝑒2𝜏𝑚𝐸
As we mentioned before, electrical conductivity is how the strength of current is related to an electric field field. This gave physicists of the time one of the first models for the electrical conductivity of a metal. By looking at the equation above they could see that the resistivity is given by
𝜎=−−𝑛𝑒2𝜏𝑚
and that the current density relates linearly with the electric field. From this model, physicists were able to calculate the relaxation time tau of a metal, by measuring its electrical conductivity. Using the expression that the average kinetic energy of an electron is propositional to the temperature, one then arrives at the conclusion that the electrical conductivity is given by the expression below, where alpha is the Boltzmann constant, T is the temperature and u is the average velocity that a particle has coming out of a collision.
𝜎=14𝛼𝑇𝑒2𝑛𝑢