Monthly Archives: March 2012

The Universe Is Made Of…

Since we have been digging pretty deep lately I thought a step back might be in order. One of my favorite inspirational series is the Symphony of Science videos, musical versions of popular science content like the Cosmos series with Carl Sagan. Many of the videos are beautiful and thought-provoking, and describe the amazing things you can see by looking at the world through the lens of science, but most pertinent for what we have been talking about is the one about the quantum world. Enjoy!

Spin, Rotation, and a Plate Trick

In my introduction to the quantum number spin, I mentioned that particles can have half-integer or integer spin, and that which they have deeply affects their behavior. This is not an easy statement to understand, especially without seeing the math. The allowed values for spin come from solutions to quantum mechanical energy equations. But what do differences in these values mean? How does a spin-1/2 particle behave differently than a spin-0 particle?

One major difference is in the behavior under rotation. When we try to calculate how rotation affects a particle with spin-0, we find that it doesn’t matter: the particle is indistinguishable before and after any rotation. However, a spin-1 particle requires a 360° rotation to return to its initial state, and a spin-2 particle requires a 180° rotation to return to its initial state. This may seem strange, but what it means is that the spin value describes the symmetry of the particle. If you imagine a deck of cards, the spin-2 particles are like face cards that look the same when rotated 180°. Spin-1 particles are like number cards which must be rotated 360° to look the same as they did when they started. Particles with integer spin are called bosons, after the Indian physicist Satyendra Bose.

There are no playing cards which must be rotated 720° in order to look the same, and yet this is the case with spin-1/2 particles. There are few macroscopic objects that can demonstrate this property, but one of them is your hand! Place any object on your hand, palm up, and rotate it without dropping your palm. After 360° you will find your arm to be pretty contorted, but after 720° of rotation your arm has regained its initial position! Another way to think of it is that, instead of a 360° rotation bringing the object back to its initial state, which would be like multiplying by 1, the 360° brings the object to another state like multiplying by -1, and then an additional 360° rotation multiplies by (-1)*(-1) which equals 1. Every spin-1/2 particle shares this behavior, such as quarks (the constituents of protons and neutrons) and electrons. We call these particles fermions, after the physicist Enrico Fermi.

That factor of -1 becomes important because of the idea in quantum mechanics that particles are interchangeable or identical. That is, we cannot tell one specific electron from another. Mathematically, you can state this by writing a function that describes the positions of two particles, and seeing what happens to that function when you exchange the particles. If you do this, what you find is that bosons are symmetric under particle interchange and the function stays the same, but fermions are antisymmetric under particle exchange, and the function is multiplied by -1.

This idea, that bosons are symmetric and fermions are antisymmetric under exchange of identical particles, is called the spin-statistics theorem. A thorough proof requires relativity and quantum field theory, but the fundamental cause is the differing rotational behavior due to spin as a measure of symmetry. One very important consequence of all of this is that if you have two fermions occupying the same state, and you exchange them, you find that the function describing their position cancels out to zero. This is a mathematical statement of the Pauli exclusion principle forbidding two fermions from being in the same quantum mechanical state!

On the other hand, we find that bosons are perfectly happy to all pile into the same quantum mechanical state, at least at low temperatures. This is the concept behind the Bose-Einstein condensate, the state of matter experimentally realized only 30 years ago in which bosons can be cooled into occupying the same state.

I hope this makes the connections between spin, the Pauli exclusion principle, and particle types clearer. But if nothing else, the rotational exercise with an object on your hand, better known as Feynman’s plate trick, is fun at parties.

The Rainbow of Bonds

Now that we have looked at the broader picture of what a bond is, we can go a little deeper. Bonds can be easy or hard to break, they can involve particle exchange between atoms, they can be the result of transient forces, and they can react in a variety of ways. There is a rainbow of bond types to explore, but we can focus on a few primary examples.

We’ll start with the stronger sort of bonds: those that involve direct transfer of electrons between atoms. For example, say we have two neighboring atoms, one with an empty low-energy state and one with an outer electron that’s all alone at a high-energy state. If the states are similarly shaped, both atoms can lower their overall energy when the extra electron moves to the low-energy state. The atom that gave up the electron is now positively charged, and the atom that accepted the electron is negatively charged, so there is an electrostatic force attracting them. Charged atoms are also called ions, so we say that these two atoms have an ionic bond. And it’s possible to have ionic bonds involving more than one electron, if an atom has two or three electrons to donate which another atom can accept. A common example of ionic bonding is table salt, which has a sodium atom donate an electron to a chlorine atom.

It’s also possible for two atoms to share a pair of electrons, so that the electron cloud overlaps with both atomic nuclei. If the electrons in question have oppositely aligned spins, they can have the same energy without being in the same quantum mechanical state. This is called covalent bonding. It happens most often when the two atoms in question are comparably attractive to electrons, for example if they are the same type of atom. Graphite, or pencil lead, is one form of carbon that has covalent bonds. So is graphene, the atomically thin version of graphite whose discovery (and extraordinary properties) recently garnered a Nobel prize in physics.

Ionic and covalent bonds tie atoms together very tightly, and can be linked together to form complexes with many bonded atoms. These complexes are known as molecules. But large numbers of atoms can also share electrons diffusely, so that the electrons aren’t localized to a single atom or a pair of atoms. This is called metallic bonding, so-called because delocalized electrons are found in metals. The free electrons move around the atomic nuclei like a sea moving around rocks, only weakly bound to them. The mobility that electrons have in metals is why we say that metals have high ‘electrical conductivity’: it is easy to pass an electrical current, which just consists of individual electrons, through a metal. As a special case of metallic bonding, it’s also possible to have partially delocalized electrons in small molecules, which is the basis of organic chemistry.

Another way to weakly bind atoms comes from the fact that charge is separated in an atom, between the positively charged nucleus and the negatively charged electron cloud. Imagine that the cloud is slightly distorted, by a passing electrical field or by a random fluctuation. If the electron cloud is not symmetric around the nucleus at that moment, there will be a distance between the center of the positive charge and the center of the negative charge, and a force because of the opposite charges. This is called a dipole in electromagnetism, because of the two oppositely charged poles. And if you have two next to each other, they will try to align so that the negative side of one dipole is near the positive side of the other. What starts as a small fluctuation can cause a slight reordering over a large material, because of the dipoles attempting to align. This dipole-dipole interaction is another weak form of bonding. It can happen with induced dipoles, as I’ve described, or between permanent dipoles which are common in molecules.

There is also a lone form of chemical bonding which doesn’t rely solely on electrons. The hydrogen atom, with its single proton and single electron, is pretty small and pretty reactive. So it’s actually possible for two atoms to share a third atom, hydrogen, which means that both the electron and the proton are in energy states that minimize the total system energy. The hydrogen bond is partly covalent, since the hydrogen electron is usually paired with a second electron. But the separation of the proton and electron also induces a dipole, making hydrogen bonding a dipole-dipole interaction. Hydrogen bonding may sound like a strange beast, and it is, but it is an important factor in the chemical behavior of water which is essential to life as we know it.

What is spin?

First the basics: spin is an intrinsic property of matter, like charge or mass. It is measurable in the real world by observing interactions with magnetism, and is the basis of technologies like MRI and hard disk drives!

We of course recognize the verb ‘to spin’, which means to rotate around a fixed axis the way that wheels, figure skaters, and the Earth do. But the word spin is also used to describe a fundamental property of particles. We have already talked a little about a fundamental property, charge, which was useful because a lot of the important forces at the atomic scale are electromagnetic and thus related to charge. And we remember that mass, another fundamental property, determines how matter interacts via the gravitational force. Spin is a bit different.

The idea of particles having an intrinsic spin first arose during the development of quantum mechanics, when Wolfgang Pauli and others noticed that part of the mathematical solution for particle states resembled angular motion, as if the particles were physically spinning around an axis. But unlike spinning at the macroscopic scale, quantum spin can only occur at a few discrete values: integer and half-integer multiples of ħ, the reduced Planck constant. The allowed values of spin are clustered around zero, and the ħ factor is dropped by convention because particle physicists like to make things look simple. So a photon, the quantum of light, has spin 0, whereas electrons and quarks, which make up protons and neutrons, have spin 1/2. There are also particles with spin 1, 3/2, and 2. As with charge, spin is reminiscent of a behavior we see in the macroscopic world, but its values are quantized into a few allowed values.

Spin can have one of two polarities, meaning we can have an electron with spin +1/2 and one with spin -1/2. And charged particles like the electron actually respond to magnetic fields differently if they have positive or negative spin! This is because the motion of a charged particle creates a small magnetic moment, which will be aligned in one direction for positive spin and the opposite direction for negative spin. This is the basis of the famous Stern-Gerlach experiment, in which atoms with one free electron are sorted by their spin under the influence of a magnetic field. But it’s also the basis of nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI), two related techniques for determining the composition and structure of either chemical substances or human patients! Strong magnetic fields can be used to align spins within any object, and how quickly the spins decay back to their original orientation gives information about what is inside the object. Currently, researchers are trying to build circuits that use spin instead of charge to carry information, which is called ‘spintronics’.

But at a more basic level, when we talked about chemical bonds we skipped over the importance of spin. The reason spin matters for bonding is due to the Pauli exclusion principle, the idea that no two electrons can share the same quantum state. In the development of quantum mechanics, it became clear from the data that even if all the available energy states were mathematically accounted for, there still seemed to be a degeneracy in which two electrons shared what was thought to be the same quantum state. This can be explained with a new quantum number, which we call spin. So spin is another factor of the electron cloud shape and is critical in the understanding of chemical bonding.

But there are actually even more strange things about spin than I can fit in this post, including the fact that the Pauli exclusion principle only applies to particles with half-integer spin! Half-integer and whole-integer spin particles are fundamentally different from each other, in some pretty interesting ways, but why is a story for another time!