Log in

No account? Create an account
Previous Entry Share Next Entry
Nuclear physics, part 1: non-nuclear chemistry
A while a go I had a project of writing 500 words, more or less, in a blog post a day. I wish to start up with that again, as I feel it's nicely creative. Then, I tried to do it in the form of fiction. Now, I won't make it exclusively fiction, but a mix of fiction and non-fiction.

500 words, more, or less, is a minimum. If I feel I'm on a roll, and have time, I won't stop there. If I finish what I want to say in less than 500 words, but it's sufficient, I'll stop. But I need to say something substantial to qualify. "Went to work today, car died, cat died, boo hoo" is a tweet, not 500 words more or less.

The first thing I want to talk about is nuclear physics, aimed at the non-physicist. I am not a physicist myself, although I know I have real physicists who read my LJ. As such, this discourse will have errors in it.

There are two kinds of errors that may be in here: "Lies we tell to students" are simplifications which are mostly accurate, but leave out lots of complicating details. An explanation of sex which begins "when a man and a woman love each other very much" is fundamentally a lie we tell to students. This series of articles will be full of lies we tell to students, either because I'm intentionally simplifying, or because that's what I've been told. I ask that folks recognize this, and understand that this isn't the complete story.

The other kind of error is where I am mistaken about something fundamental and it's not as simple as lies we tell to students. I might say something as definite that I believe or suspect that is completely wrong. Here I would like those who know better to step in. Obviously, I can't give an example of something I incorrectly think is true.

With those caveats,

Since this series is on nuclear physics, not chemistry, the section on chemistry will be brief, and will mainly lay a foundation for comparison when talking about nuclear stuff later. So beware, massive simplifications follow.

Roughly speaking, chemistry is mediated primarily by the electromagnetic interaction between atoms, and, again, roughly speaking an atom is made up of a cloud of light electrons each with a negative electric charge, and a heavy nucleus with a positive electric charge.

There are a few things which govern chemical reactions:

Electromagnetic forces
Minimal energy
Quantized states
Pauli Exclusion
Heisenberg Uncertainty

Electromagnetic force causes like charges to repel, unlike charges to attract, and further says that this force is related to the square of the distance between the charges. It takes energy to push like charges together or to pull unlike charges apart. Conversely, you release energy when you let like charges go apart or unlike charges come together. In chemistry, electromagnetic forces are what bind electrons to a nucleus: each electron is a small negative charge, each nucleus has a positive charge. The electrons are attracted to the nucleus by the EM force.

The "minimal energy" idea probably has a better name, but basically it is the idea that systems favor low-energy states over high-energy states. If there are two configurations of a system that it is possible to transition between, the system will most likely be in the state with the overall lowest energy. An apple on the ground is a lower energy state than an apple in a tree, so if the stem of the apple gets weak enough, the apple will fall from the high energy state of in the tree to the low energy state of on the ground.

There are many things which can get in the way: Often times its not possible to go from high-energy state A to low-energy C directly, but the system must pass through higher-energy state B in between. In which case, the transition from A to B is not favored, and nothing takes place (but when A->B happens, then B->C is favored over B->A and so A->B->C will likely happen). So even though C might be the lowest energy state, things might be stuck in A.

If you've heard (correctly) that energy is conserved, you might be wondering what happens to the energy freed by transitioning to a lower-energy state, and where the energy would come from to allow the A->B transition. The answer is that the energy isn't lost; it instead is transferred to a different form. In chemistry, atoms and molecules don't just sit there: they vibrate, they twist, they move, they spin, etc. Each of these modes (moving, vibrating, spinning, etc) takes energy, and when two things with all these different modes bump into each other, the collision redistributes the energy between the different modes: A fast moving helium atom hitting a slow ethanol molecule might yield a slower helium atom and a spinning ethanol molecule, for instance. Sometimes that transfer of energy goes into the right mode to allow an A->B transition, thus allowing A->B->C. In which case the B->C transition releases energy which can make its way to some other part of the system. So there's always a bit of extra energy in the system sloshing around from mode to mode.

Roughly speaking, "temperature" is a measure of this extra energy. When things get hot, there's a lot of extra energy sloshing about, and rare, low-to-high energy transitions can take place. When things get cooler, there's little extra energy, and things tend to stick to low-energy states. When things get real cold, high-energy states are very rare, and quantum effects start to show up. Which brings us to quantum states.

In many situations, a particle in a system can be described by discrete quantities: an electron bound to an atom can be in shell 1, shell 2, shell 3, but not shell 2.71818 for instance. If something causes a particle to go from one of these discrete states to another discrete state, it can't go partway, it's either all or none. Since each state usually has a different energy level, to change from one state to another requires a discrete quantity or "quantum" of energy. Hit a helium atom in its ground state with another with an energy of collision of 15eV (eV is a measure of energy used at this scale) and it will remain in the ground state and rebound like an ideal billiard ball. If the energy of collision is 25eV then some of the energy of collision could cause one of the electrons to jump to a higher energy state making the collision less "elastic". Going from one discrete quantum state to another is a "quantum leap".

Each of the discrete states a particle in a system could be in is a "quantum state", and is commonly described by a series of numbers or symbols that describe that state. The quantum states come from solutions to equations describing the system as a whole.

For instance, an electron bound to an atom can be described by it's principle quantum number (n = 1, 2, 3, ...), it's angular momentum (l = 0, .., n-1), and it's magnetic quantum number (m = -l, ..., l), and it's "spin" (s = -1/2, 1/2). So one way of describing an electron state would be (n=2,l=1,m=0,s=-1/2). This is cumbersome, so a short-cut was developed (it's possible the shortcut was developed before the long-cut) of referring to l=0 states as "s" (for "sharp"), l=1 states as "p" (for "principal"), l=3 as "d" (for "diffuse") l=4 as "f" (for "fundamental"), and so on. The n values are given, followed by the l letter, and then a subscript describing the m value (the s values are less important chemically). So an electron in the state I gave would also be described as 2pz.

Quantum states are important in an atom because of the Pauli Exclusion Principle, which states that two electrons cannot be in the same quantum state. So if an electron is already in the 1s spin-up state in an atom, a second electron can't also be in the 1s spin-up state in the same atom. This forces additional electrons to be in higher and higher energy states, while simultaneously making them more and more loosely bound -- it takes less energy to pull the 7s electron off of an atom of francium than it takes to pull the 2s electron off of an atom of lithium.

And finally, for this overview, we come to Heisenberg's Uncertainty Principle, which basically explains why atoms have the size they do. Heisenberg noted that some quantities in a quantum system (like the position and momentum of a particle) are fundamentally linked in a way which means that knowing one precisely prevents knowing the other precisely. A particle with a well-defined position has a poorly-defined momentum, and a particle with a well-defined momentum has a poorly-defined position. This is expressed mathematically as "ΔxΔp>h" where h is Planck's constant, and is very small but non-zero. It's exact value is unimportant right now.

The best classical analogy I've heard for this phenomenon is a chirp of noise: If you play a short chirp of noise into a frequency analyzer, it will come out with a diffuse spread of frequencies, no matter how "pure" you can make the noise. The shorter the chirp, the more spread out the frequencies are. This is because in order for the chirp to have a beginning and end, the chirp must be a collection of frequencies which cancel outside of where the chirp, well, chirps. The shorter the chirp, the wider the band of frequencies needed to cancel outside the chirp. Conversely, if you use a synthesizer to play a mix of a band of frequencies, you can get beats or chirps. The wider the band, the shorter you can get the chirps, but narrow bands yield long chirps. A single frequency gives a chirp of unlimited duration. In signal processing, they say that a short duration in the "time domain" means a wide bandwidth in the "frequency domain", and vice versa. Quantum mechanics speak of the "position domain" and "momentum domain" and use much the same maths to convert between them.

Where this is important for this discussion is that mass is part of the momentum -- the Δp part. A larger mass means that the Δp is inherently larger so the Δx can be smaller. It also puts a minimum size limit on an atom: because an electron is bound to an atom, it's energy is limited. Special relativity relates energy, mass, and momentum in such a way that tight bounds can be computed for Δp. Therefore, Δx, the space we can localize a bound electron to, has to be greater than h/Δp. For a hydrogen atom, Δx works out to be around an angstrom (within an order of magnitude). The relatively large mass of the nucleus (about 2000 times more massive for H, 480,000 Plutonium) compared to an electron means that the Δx can be minuscule compared to the electron cloud. This will be important later, as well.

I'm sure that's way more than 500 words, so I think I'll stop here and pick it up again tomorrow.

  • 1
This would appear to be an invitation for someone to say, "Damn. Now we'll never know his momentum."

  • 1