The most important idea in science, and why it’s true
If you’re in the mood to read a great book about physics, there is no author I can recommend more highly than Richard Feynman, the great American physicist and teacher. At the beginning of his famous Lectures on Physics delivered at CalTech (which you can find in Six Easy Pieces), Feynman remarked on the most important idea in science:
If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.
There are a few reasons why this is such a powerful statement. First of all, it emphasizes how absolutely crucial the idea of the atom is to our understanding of the universe. If you play the game where you keep asking “why?” to every statement someone makes, at the end of the long sequence of why’s you will come to a statement about atoms (or, if you keep going past that, to the Big Bang). This is especially evident if you play the physicists’ version of the “why” game, wherein you keep asking where did the energy come from? Writ large, the things we do are a result of our psychology, and our psychology is a result of our biology, and our biology — the workings of our mind and body — is a result of or so atoms all pushing and pulling on each other in complicated ways. The atom absolutely reigns supreme in our understanding of Nature.
However, knowing that the world is made of atoms doesn’t do you any good if you don’t know how the atoms interact with one another. That’s why Feynman didn’t end his statement at “all things are made of atoms”; it is imperative to our understanding of the world that we know something about how atoms push and pull on each other. Knowing that atoms attract each other when they’re kind of close but repel when they are really close allows us to predict that atoms can pull together to form large structures without collapsing onto the same point.
But why is it true? What is the origin of the attraction and repulsion between atoms? After all, an atom is a neutral object: it has positive and negative charges that completely cancel each other, so why should it attract or repel another atom? The answer is fairly subtle, and surprisingly poorly-known, given that it is fundamental to the “most important idea in science”.
Not a hypothesis any more
The belief that the universe is made of atoms is at least 2500 years old, but until fairly recently it was just that: a belief. During the last 200 years or so the evidence for “atomism” grew increasingly strong, until it really became impossible to imagine an explanation for things that didn’t involve atoms. But it was still sort of a matter of faith, in the sense that you couldn’t actually see an atom.
That is no longer the case. I am sort of amazed to say that I live in an era when people can actually look at individual atoms through high-powered microscopes. Below is a picture of a metal surface taken with an atomic force microscope. The lighter atoms are tin, and the darker ones are silicon.
Of course an atom is not actually a hard ball; what you’re seeing is the electron cloud that surrounds each atom in a mostly-spherical shape. And these days we can do a lot more than just see atoms; we can actually manipulate them individually. The above image is from this paper, where scientists at Osaka University were able to rearrange silicon atoms one at a time to spell out “Si”.
It’s a little surprising to me that the first real picture of an atom wasn’t met with much fanfare. I guess no one was surprised by it, but still, it would have been nice to have some kind of party to celebrate the culmination of 2,500 years of atomism.
Why atoms attract
Roughly speaking, an atom is made of two parts: the positive nucleus and the negative electron cloud. And, most of the time, these parts exactly cancel each other so that the atom is neutral. Here is a rough picture of one:
When the electron cloud is exactly centered around the nucleus, there is no electric field outside the atom. Thus, the atom should not push or pull on anything else. But it can happen that the electron cloud gets shifted to the side a little bit, so that it is no longer centered around the nucleus. When that happens, electric forces pull on both the nucleus and the electron cloud to bring them back together.
The result is a kind of “atomic springiness”: when the nucleus gets separated a little from the center of the electron cloud, electric forces arise that pull the two back together. And so the electron cloud can oscillate around the nucleus if it is disturbed a little bit.
As it turns out, the electron cloud is always oscillating around the nucleus. This is a result of quantum mechanics, which says that any object in a trap (e.g. the electron cloud stuck in the potential well of the nucleus) will bounce around with a characteristic frequency.
So the electron cloud of an atom is not always centered around the nucleus, but rather is fluctuating back and forth. And when the nucleus and the electron cloud are not centered, there is a resulting dipole moment, which means that there is an electric field surrounding the atom. This field pulls on other nearby atoms.
Imagine, for a moment, two isolated atoms, each of which has an electron cloud oscillating around its nucleus. At first the two atoms may have completely independent oscillations, but after a while the oscillations of one atom start to affect the other atom. The two atoms start to oscillate together (in phase), so that the positive side of one atom is always facing the negative side of the other, and vice-versa. As a result, the atoms begin to pull toward each other. I imagine it something like this:
In physics language, we would say that the dipole field of one atom induces a dipole in the other atom, and we call this the Van der Waals force. It’s pretty weak compared to the attraction between individual charges and it has a short range. Of course, it’s still strong enough to allow geckos to climb on walls.
Why atoms repel
We’ve seen that atoms pull themselves together, but why should they stop? Why not just keep pulling until the two atoms sit right on top of each other?
The answer, once again, comes down to quantum mechanics. This time it involves the Pauli Exclusion principle, which says that two electrons cannot occupy the same space and have the same energy. If we brought our two atoms extremely close together, then their electrons would occupy the same volume. This cannot happen unless some of the electrons move to higher energy levels. So pushing the atoms together requires a lot of energy: enough to move about half the electrons to a higher energy state. And whenever moving in a particular direction costs energy, a force arises that pushes in the opposite direction.
Here is a schematic depiction of the process of pushing two atoms together:
The top row shows the atoms being pushed together and the bottom row (the little horizontal lines) shows the energy levels occupied by the electrons. Once the electron clouds of the two atoms start to overlap, their electrons can no longer have the same energy, but must go to a higher energy state. So the atoms would “prefer” to remain apart, at least enough that their electron clouds don’t overlap strongly.
There are lots of details that I’m omitting here. The strength of the repulsion between atoms depends a lot on the structure of their electron clouds. This is the rich and wonderful playground of chemistry. The hydrogen atom, for example, has essentially no energy cost associated with being pushed into another hydrogen atom because there are two equivalent energy states for each of the electrons to occupy. A helium atom, on the other hand, repels other heliums fairly strongly because there is a big difference between the energy level for the first two electrons and the energy level for the second two. That’s why hydrogen atoms naturally form the molecule but helium atoms remain separated.
Putting it together: Lennard-Jones
The attraction and repulsion of atoms is often put together to form something called the “Lennard-Jones” potential, which looks something like this:
On the attractive side, the force goes to zero like , where r is the distance between atoms. This law is fairly easy to derive by thinking about “one dipole inducing another”. The repulsive side has the force going to infinity like . I don’t know of any derivation of this fairly strange result, and as far as I know it is a purely experimental fact.