The Standing Invitation

Archive for April 2012

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

No Nonsense

with 3 comments

Recently a copy of A J Ayer’s Language, Truth and Logic passed through S I Towers, and it caused quite a stir. It’s a short book and very readable – and, I was amazed to learn, was written when the author was younger than I am. It is a beautifully argued manifesto of logical positivism.

Philosophy, for most people, is the asking of Big Questions. Is there a god? What happens after we die? Does the world disappear when we close our eyes? What is ‘truth’? What is ‘good’? And these questions are called Big Questions precisely because thousands of years of arguing have got us no closer to answering them.

Logical positivism was an attempt to tackle these issues from a different angle. Rather than attempting to answer these questions, the project of the positivists was to decide whether or not the questions could be answered. Here, briefly, is how they set about it.

Forget about what you can see. Think instead about what you can say.

The human vocal apparatus make it possible for you to generate all sorts of noises. Most noises are just that – noises – but some are words. Most combinations of words are nonsense: “Mill food only here bushes pardon speak and.” However, some combinations are full sentences, like “I am wearing shoes” or “The sky is green”.

The important point is that almost everything you could possibly say is actually nonsense. The things that actually mean anything – sentences – are a tiny minority. What is it about these particular utterances that makes them important? Well, sentences have a structure. They obey rules. They are not self-contradictory, like the sentence “X is and is not Y”, which is meaningless and indistinguishable from noise.

In fact, there are only two kinds of sentences that are worth talking about: sentences describing the world, and sentences describing other sentences. Any other kind of sentence is uninteresting, because hearing them does not increase one’s knowledge of the world. It’s just noise.

Now, how do we know which sentences describe the world? That’s easy: these are the sentences that can be checked against what we observe around us. “The sky is green” is an attempt to describe the world, and it is well-phrased, logical, and verifiable. It just happens to be false, because it does not match observations that show the sky is blue. The sentence “I am wearing shoes” is true (at the moment).

If you know all the meaningful, true sentences about the world, and all the meaningful, true sentences about other sentences, you will know everything that it is possible to know about the universe. Obviously, in our lifetimes we will never have this perfect knowledge. There are some things that we will never know. However, adopting this stance gives us a tool for cutting away the layers of nonsense that surround us and prevent us from understanding the world.

Does god exist? If you mean, does he exist in the world, does he have an actual location and mass and velocity we could check, then the answer is – maybe. We don’t know, but we could in principle find out. But if you mean, does he exist somehow outside the world, in a place we can never experience, then there is no question here to answer, because in that case sentences containing the word “god” are meaningless. It is impossible for an atheist to disprove the existence of god, but at the same time, anybody religious who talks about god is just making noises. What happens after we die? Again, things that happen outside the “real world” are not subject to verification, anyone who talks about it is taking nonsense. Likewise the question about the world disappearing when we close our eyes: it’s not a question that can be meaningfully answered. What is truth? Good correspondence between a sentence and observation. What is good? Whatever people say is good; people argue about it, but they argue by appeal to emotion, not to logic, unless it is to show that one’s values are inconsistent.

A lot of this is not new. Hume, much earlier, said that a book that didn’t talk about things observed or calculated should be cast onto the flames because there was nothing in it worth reading. But what the logical positivists added was the system of formal logic developed by Russell and Wittgenstein. For lovers of clarity and precision of writing, the appeal is still strong.

 

REFERENCES

As always, I am not a philosopher, and could easily be getting aspects of this wrong. If so, I would be delighted to be set right by someone who knows more about it than me.

A J Ayer’s book was Language, Truth and Logic. The reference to Hume comes from his Enquiry.

Written by The S I

April 1, 2012 at 2:34 pm