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The Cost of Perfection

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Take two particles.*

Particles are fussy things. They don’t like being too close to each other because atomic nuclei repel each other very strongly; at the same time, the shifting clouds of electrons surrounding the nuclei can be mildly attractive to one another, so they don’t like being too far apart either. This balance of attractive and repulsive forces is familiar from all social gatherings: when you’re talking to someone at a party you want to be close enough to hear them, but if they’re standing right in your face it’s uncomfortable.

What this means for particles is that there’s a sweet spot, an optimum separation between the two particles that makes them both happy. It’s the lowest-energy arrangement of particles, in that once they’re at this distance, it would require an input of energy to push them closer together or pull them further apart.

So this idea of the optimum distance between two particles is straightforward. And the same thing applies when you have billions of particles at once. They will move around at random until they find the lowest-energy arrangement, where the average distance between particles is as close as possible to the ideal separation.

When billions of particles try to reach their lowest-energy arrangement, they will try to form a lattice.

Lattices are three-dimensional patterns of points in space. They are infinitely large, infinitely repeating, purely mathematical constructs that can only be approached, never exactly attained. A lattice is a map of where particles should sit in space in order to be at the right distance from each other.

When particles arrange themselves on a lattice, we call this a crystal. Crystals, like diamond, are simply regular arrangements of particles – and they really are very regular, repeat themselves almost perfectly for millions and millions of layers.

But remember, lattices are mathematical ideals, perfect and Platonic, while crystals are real-world lumps of matter. The particles in a crystal may try to reach the perfect state of a lattice, but they will never reach it. There will always be defects – points at which an atom is not sitting where the lattice says it should be. These are the microscopic imperfections that mean the ideal of a lattice will never be attained. Even though all particles in a crystal would benefit from being in a perfect lattice (achieving the optimum separation from other particles), the defects are nevertheless unavoidable.

There are two kinds of defects: extrinsic and intrinsic. Extrinsic defects are easy to understand. They are simply impurities. A diamond crystal is supposedly a regular arrangement of carbon atoms, but since no source of carbon is perfectly pure, no diamond will be perfectly pure. The most common impurity in diamond is a nitrogen taking the place of a carbon. Diamonds, supposedly pure carbon, are typically 1% nitrogen. As well as being impurities in themselves, the presence of a nitrogen atom causes local distortions in the crystal surrounding it, as the adjacent carbons move slightly from their ideal lattice positions in order to compensate for it.

But even if some perfectly pure source of carbon could be found and a diamond crystal grown from it, would that crystal approach a perfect lattice? Not quite, because of the second kind of defect – intrinsic defects. An intrinsic defect occurs when a particle isn’t where it should be.

These intrinsic defects are interesting because they are unavoidable. They are the inevitable consequence of the great trade-off between enthalpy and entropy. Because although there is an energetic benefit to having particles sitting an ideal distance from each other, there is also an energetic cost to having them perfectly ordered. This again makes sense to anybody who has ever tried to organise anything. Of course you want things organised; things are more efficient when they are organised, and so the more organised, the better – but organising things takes time, effort, and energy. Ultimately a compromise has to be reached: you accept the amount of organisation you can achieve for the amount of energy you’re willing to expend on putting things in order.

Although particles like to be separated by an ideal distance, they also like moving around, particularly at high temperatures. For this reason, truly perfect crystals are impossible to grow. Imperfections will always sneak in. In fact, imperfections are necessary. Crystals exist because it is energetically favourable for particles to be organised; but because of the inevitable cost of organising anything, it is also energetically favourable for there to be defects. And the interesting thing is that because the imperfections are the result of particle movement, and movement depends on temperature, it is possible to predict how many imperfections there will be in a crystal at a given temperature. You can’t say where they will be, but you can say how many there will be per cubic centimetre. The defects obey exact laws that can be understood and exploited. They are perfect imperfections.

 

* In this post, by particles I mean atoms, ions, molecules or some colloids; things smaller than atoms behave rather differently.

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Written by The S I

April 9, 2012 at 10:18 am