Watching a documentary, I heard Michio Kaku telling that when he was young, he built a home-made particle accelerator. What I would like to know is if I, myself can do so? And how?

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Whether you can do so is a question that only you can answer, and you can answer it by trying to build one. A web search for "homemade cyclotron" yields many interesting articles, videos and sites. Whether those techniques are within your level of skill and budget, how should we know? –  Eric Lippert Jul 8 at 16:09
In the general sense all you need is a source of particles and an accelerating field. Both exist in a fluorescent tube or a CRT. With a Van der Graaff generator you can make quite high voltages - 10's of kV - with which to accelerate electrons / make "lightning". But even a BB gun is a "particle" accelerator. You might have to be more specific. –  Floris Jul 8 at 19:25

It is not so hard, but it won't be able to generate enough high energetic particles.

The best example for a particle accelerator is a CRT (cathod ray tube), which you can find in every CRT monitor or TV. It can generate around $40keV$ electrons. (LHC generates $3.5TeV$ protons, thus it is around a million times stronger).

Only a particle accelerator isn't enough, if you want to make experiments with it, you need some analytic/measurements devices also. In the current accelerators, they are nearly so complex and costly as the main accelerating device.

There is also a device capable to be built in home, it is the Farnsworth fusor:

Maybe it is not a particle accelerator in the classical sense, it creates enough strong field to be able to fuse deuterons (although it is doing this with terrible efficiency, around $10^{-8}$). You can see a Farnsworth fusor scematic below:

There is a whole community of home fusors which can be found here.

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Your ratio is off: it's more like 100 million. –  Ryan Reich Jul 8 at 14:25
That's a great photo. –  iamnotmaynard Jul 8 at 15:35
Removed my old comment; didn't see you had already linked to fusor. Great resource! –  TylerH Jul 8 at 16:50
@RyanReich Sorry, I think, I was clear - if not, I considered your change suggestion very helpful. –  Peter Horvath Jul 8 at 20:14

Well, there are a lot of different types of particle accelerators.

Besides the already suggested cathode ray tube, another rather small accelerator would be a cyclotron. The first one was built by Lawrence and Livingston, "a device about 4.5 inches in diameter used a potential of 1,800 volts to accelerate hydrogen ions up to energies of 80,000 electron volts." I guess it would be possible to get a high voltage power source and hydrogen, but really building a working device would probably take a long time.

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And how can i ionize hydrogen atoms? –  George Smyridis Jul 8 at 12:27
The ionization energy of nuclear hydrogen is 13,6eV, which corresponds to electromagnetic radiation of 91nm wavelength (extreme UV). So you could use a "light" source emitting a wavelength smaller than 91nm and just direct it at the hydrogen. Another way would be to heat it up. However, there might be easier solutions, e.g. by applying an electric field, but i'm not sure about this. Interesting question, though. –  Timitry Jul 8 at 15:15
Another idea: Maybe you could use a thin stripper foil (usually made from carbon) and direct your gas with some pressure through it. This is how negative ions are converted into positive ones in e.g. a Tandem Van de Graaff accelerator. However, i don't know if it will work with "slow" particles from a gas bottle. –  Timitry Jul 9 at 7:57