Anti matter is a rare exotic substance that could be used as extremely efficient energy storage
Anti matter has the opposite charge
For example an anti-proton has -1 or a "positron" (positive electron) has a +1 charge. This means atoms and everything made out of antimatter behave just like matter. Anti neutrons exist as well, they are just made out of quarks with the opposite charges which add up to 0. A fundamental trait of anti-matter is that when it collides with "regular" matter it release all of its energy.
Anti matter and the big bang
During the big bang an equal amount of matter and antimatter were created. But for some reason as far as we know in the end an unequal amount existed. For every billion antimatter particles a billion and one matter particles were created. We have theories but no solid evidence as to why this happened. This is one of the mysteries of the early universe.
fuel and energy
The main reason why it has been studied so much (by governments) is because it has a 100% conversion rate of mass into energy. This means it is the most efficient energy storage possible. This is E =MC^2 joules of energy. A single gram of antimatter can run an average computer for 1.7 * 10^12 years.Or about 12 times the age of the universe. That however doesn't factor in something. Anti matter also turns the same amount of matter into energy when they collide. The full amount of energy could run your computer for about 24 times the age of the universe. (if it only takes about 500 watts.) For comparison last I heard a nuclear fission explosion was at about 0.5% efficiency. No wonder the military wants to build bombs out of this stuff.
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Proof of anti-matter
We have been "making it" for a while now, just a few atoms at a time. However, some people have said "it doesn't exist" putting it nicely. Here is proof, get over it:
https://home.cern/topics/antimatter
The antimatter section of CERN the top particle physics "group" in the world.
Or how about the first observation of them was in the 1920's.
http://www.actforlibraries.org/proof-of-antimatter/
It was an experiment by Carl Anderson. He basically said, " Why are there negative and positive signs all over in relativity and if the square root of something has two answers why can't the universe?" Well turns out it can. He observed anti electrons or "positrons".
How did he do this? Well he was photographing a cloud chamber. Cloud chambers allow you to "see" particles. A cloud chamber is basically a container filled with water/alcohol vapor that is ionized when a particle goes through. This allows you to see the path of any particle. If you put magnets on each side you can change the path enough that you can tell what it is. Well, turns out he either found protons with like 1999/2000th of their mass missing or positive electrons. They accelerated to the negative side of the electric field as fast as an electron but with the wrong charge. He decided to write about the positive electrons and not small protons.
Antimatter has been proven without a doubt and studies at CERN are going on now. As far as we can figure only the charge is different and everything else is the same. Which means we still have nothing to go on for the big bang inequality.
Creation of anti-matter
Matter-antimatter pairs are created in a lot of situations in nature. For example gamma rays which usually create electron-positron pairs, which annihilate right away and go back into being a gamma ray.
They are also created in something called vacuum fluctuations. Basically pairs of particles are created non-stop and annihilate almost instantly. In many high energy reactions they are created but usually live short lives as well. For example the sun produces it in solar flares.
Humans make them in extremely complex ways. We shoot particles at each other until they collide. We give them as much energy as possible so they can have a chance of creating it. Sadly it only happens one in a million collisions or less. We then use magnetic fields to separate them from normal matter and contain them.
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Antimatter at this point is the best energy storage, but nowhere near the most efficient for us. This means we can use it in ships and moving objects without need of a battery, but unless we find more efficient ways to create it this will be a large drain on our electrical system.
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