Some of those are categories rather than specific particles. For instance, “fermions” are particles that obey Fermi-Dirac statistics and that includes protons, neutrons and electrons. (The other definition is they’re particles with noninteger spin, like one half, one and a half, two and a half, and so on.) The other category covers stuff like photons and…most of the bosons, maybe all of them. Bosons obey Bose-Einstein statistics instead, and have integer spin. They also split particles into baryons and leptons, which are categories that meant “high mass” and “low mass”*. So while there are certainly a lot of particles, the terminology can make it seem even worse than it is. 🙂
Some of these particles are said to “mediate” a force, which means they appear when a field is acting. Photons appear when you interact with an electromagnetic field, W+, W- and Z^0 bosons are manifestations of the weak force, Higgs bosons go with the Higgs field (please don’t ask me anything about that one), you get gluons with the strong force, and we figure there should be a gravity particle, the graviton, but we have yet to confirm that one.
* *Those categories have had their definitions tweaked since, I believe, and now we have baryon = made of three quarks, meson = made of two quarks, and lepton = not made of quarks. And if you’re made of one quark, you’re called a quark. :)*
Physics doesn’t really do *why* – they just exist. Asking that is the same as asking why *anything* exists.
Their purpose? Some quarks are stable inside of protons and neutrons (each is made of 3 quarks), some carry forces (photons are how the electromagnetic force moves through space), and a lot of them don’t really exist out in nature except for super high energy levels. These ones almost immediately fall apart when we create them. So, it’s really only a fraction of the fundamental particles that are “important” in the functioning of our daily world.
Neutrinos are stable, commonly produced as byproducts of solar fusion, but have almost no mass so they move almost the speed of light, and no electromagnetic charge so they can fly straight through the electron cloud of an atom. About 100 trillion neutrinos shoot harmlessly through your body (and the whole Earth) every single second, just flying in a direction away from the Sun forever.
The way we learn about subatomic particles is kind of like taking two cars, smashing them together in a head on collision, and trying to figure out how the car was built by the path of the debris flying through the air.
As for the bits of debris, we can tell some things about charge and mass by how their paths get bent by electric or magnetic fields.
Then there’s a whole lot of simulations and statistics to see if the debris that our models predict would come out of said collisions actually does.
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