# Amateur Topologist

#### Everything but topology.

Tag: particle physics

# Electrons are not like planets

One of my first posts on this blog was about the stability of the atomic nucleus; given that it consists of a bunch of positive/neutral charges clumped together, why doesn’t it fly apart? The answer involves the strong force, which is strong at atomic distances but miniscule at inter-nuclear distances; on distances comparable to that of an atomic nucleus, it’s strong enough to overcome the electromagnetic repulsion. But there’s another question, and it involves the model of orbiting electrons.

Classically (meaning without quantum mechanics), the electron is pictured as a pointlike particle orbiting the nucleus. For simplicitly, we’ll look at the hydrogen atom. The electron orbits at the Bohr radius , $a_0=5.3 \times 10^{-13}$ cm, which can’t really be derived in an easy way from theory; we’ll just take it as a given. So the acceleration the electron undergoes is $q^2 / m_e r^2$ (using cgs units to avoid factors of $1/4\pi \epsilon_0$ and such everywhere), using good old $F=ma$.

However, there is a problem: any accelerating charge radiates energy. The reason for this is, roughly speaking, an accelerating charge has more energy in its electromagnetic field, so you therefore have to expend more energy to accelerate it; since an atom is a closed system, there can be no energy source, so it ‘extracts’ it by spiraling inwards. The derivation of formula is complicated, but it turns out that a particle of charge $q$ accelerating at a rate $a$ will radiate energy at a rate

$P=\frac{2 q^2 a^2}{3 c^3}=\frac{2q^6}{3m_e^2r^4c^3}$

(the second equation what we get when we plug in our value for acceleration).

So what’s the energy of an electron orbiting its atom? The kinetic energy is equal to $\frac{1}{2}mv^2=\frac{1}{2}rF=q^2/2r$, and the potential energy is equal to $-q^2/r$, so the total energy is just

$E=-q^2/2r$

Considering both energy and radiated power as functions of time, we get $dE/dt=-P$; but since the only variable that can change is the radius, we then get the following differential equation for the radius:

$r'=-\frac{4q^4}{3m^2c^3r^2}$

The method of solving this equation isn’t important; I used Mathematica. What is important is the solution:

$r=\left(a_0^3-\frac{4q^4t}{c^3m^2}\right)^{1/3}$

where $a_0$ is the Bohr radius I mentioned earlier. This will obviously become zero at $t_0=\frac{a_0^3m_e^2c^3}{4q^4}$; plugging in the cgs values for these constants, we get that $t_0=3.11 \times 10^{-11}$ seconds. So according to the Bohr model, a hydrogen atom decays in less than the time it takes light to move a centimeter.

So what’s the answer? One is to use the Bohr model, which requires a minimum energy; the electron cannot fall farther into an energy well. This model works to a certain extent, but fails with more complicated atoms; not only that, but it predicts that the hydrogen atom has a minimum nonzero angular momentum, which is not the case. But a full treatment of the failings of the Bohr model goes beyond what I know.

# Why Everything Doesn't Collapse

The strong force, as you may or may not already know, is the force that holds protons and neutrons together. The unusual thing about it is that its magnitude, after a certain point, doesn’t decrease with distance, and is on the order of 100,000 tons. For a subatomic particle, which usually has a mass on the order of 10^-27 grams, this is obviously a huge force, and this is the reason why we can’t isolate a quark: whenever we try to pull one away from other quarks, the energy we have to put into it to move it away eventually gets so large that a new quark-antiquark pair spontaneously forms and combines with the quarks and gluons to form hadrons, in a process aptly called hadronization. So why don’t the quarks in my body attract me to the quarks in the keyboard that I’m typing on?

It’s because protons and neutrons don’t have ‘color’. Color is a property of particles similar to electric charge in that it determines their interaction with a force (in this case, the strong force). It has nothing to do with actual color; it’s just an unfortunate naming coincidence. A quark is either red, green, or blue, and antiquarks are either antired, antigreen, or antiblue. A particle such as a proton or neutron must have zero net color, and since a proton is three quarks, it is therefore made of a red quark, a green quark, and a blue quark. However, since the strong force constantly shifts colors, there is no one quark that is red; rather, there is always a red, green, and blue quark. Now, red + green + blue = ‘white’, just like with normal color, and colorless particles don’t interact via the strong force. So protons and neutrons aren’t attracted to each other (at least not via the strong force).

However, there is a bit of residual force; this is the nuclear force, or what some people call the ‘strong force’, using ‘color force’ for quark-quark forces. This residual force, however, dies off very quickly due to the mathematics of the strong force. You can’t just add up the individual forces between quarks like you do with electromagnetism.

# Categorizing the Particle Zoo

Hadron, baryon, meson, fermion, boson, lepton, quark. All of these are types of fundamental or nearly-fundamental particles, and there’s a good deal of overlap between them. Keeping them straight is challenging; I have a hard time remembering which is which myself (although I’m not a particle physicist, I have read up on it and know more than most.) So I’m going to explain, for both my edification and yours, what all the -ons (and quarks) are.

Fermions are any particle that obey the Pauli Exclusion Principle, which states that no two fermions can have the same state (which includes position, velocity, etc.) Basically, everything that we typically consider to be ‘matter’, such as protons, neutrons, and electrons, are fermions. One defining property of fermions is that they have non-integer (specifically, a half-integer such as 1/2 or 3/2) spin; spin is a property of fundamental particles and particle-like objects that has nothing to do with actually spinning. A lot of stuff in particle physics is meaningless like that, as we’ll see. So, for example, an electron has spin 1/2, and the composite particles known as delta baryons have spin 3/2. Fermions are named after the famous physicist Enrico Fermi, who first formulated the statistical laws that they obey. Because of the way that spin influences statistics, any particle with an odd number of fermions inside it, such as a proton, which has three fermions inside it, is itself a fermion; any particle with an even number of fermions inside it is not. There are 12 fundamental fermions, and they’re split into 6 leptons and 6 quarks.
Bosons have integer spin and do not obey the Pauli Exclusion Principle, so you can have as many bosons in a given state as you want. Everything is either a fermion or a boson, since its spin is either an integer or not an integer. There are five fundamental bosons: the photon, which carries the electromagnetic force, the Z and W bosons, which carry the weak force, the graviton, which carries gravity, and the Higgs boson, which is not yet confirmed to exist (hence the LHC) but is believed to cause particles to have mass in a way similar to how moving objects through water is difficult.

Quarks are one type of fundamental fermion. The name doesn’t have some deep meaning attached to it; it’s from a poem by James Joyce.  Ignoring antimatter, there are six ‘types’ of quarks: up, down, charm, strange, top, and bottom. The names have nothing to do with directions or anything. Each quark also has a color of either red, green, or blue, which has nothing to do with actual color (see what I mean about names not really meaning much?). One interesting point about quarks is that they (and gluons, which I’ll introduce later) are the only particles that have ‘color’ and so participate in the strong interaction. I’ll cover that in a later post, as it’s worthy of its own discussion. For now, all you need to know is that quarks make up protons (two ups and a down), neutrons (two downs and an up), and essentially everything you think of as ‘matter’ except for leptons. One interesting fact about quarks is that they all have charge either 2/3 or -1/3 times the electron charge, which is odd, as the electron charge is fundamental. However, it is impossible to isolate a quark, for reasons I will explain in that later post I mentioned, so it all works out.

Leptons are the other type of fundamental fermion; the six leptons are the electron, the muon, the tau, and the electron, muon, and tau neutrinos. The muon and the tau are just like the electron, only heavier; they are therefore unstable. Muons decay into muon neutrinos, electrons, and electron antineutrinos, and taus typically decay into a tau neutrino and either an electron and an electron antineutrino or a muon and a muon antineutrino, although there are other decay modes such as a negative pion and a tau neutrino. All tau decay modes do contain a tau neutrino, as they take place via the weak force and so the ‘tau lepton number’, which is the sum of the number of taus and tau neutrinos (antiparticles count for -1), is conserved.

Hadrons are simply particles made up of multiple quarks; they’re split into two groups, baryons and mesons. Baryons are hadrons that contain three quarks; they are therefore also fermions. Protons and neutrons are both baryons, being made of two up quarks and a down and two down quarks and an up, respectively; there are too many baryons for me to list here, but there is a good list on Wikipedia. Mesons are hadrons made up of a quark and an antiquark; they are therefore bosons. Again, there are far too many mesons for me to list here, but Wikipedia has a list of mesons.

A list of all known fundamental particles (sans Higgs boson), from Wikipedia