Small particle accelerators can sit on a desktop, large circular ones can be miles across
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Particle accelerators were first developed by John D. Cockcroft and Ernest T. S. Walton in 1932
Scientists use particle accelerators to collide atoms.
What essentially happens is two nuclei of two different (or the same) elements are smashed together; sometimes (although rarely) parts of the nuclei will stick together instead of getting obliterated or shot out. It is through this process that particle accelerators are used to create elements; however, it is worth noting that these elements are unstable due to the nucleus's atomic radius being to large; it will be greater than the range of the residual strong force.
Yes, they can. In fact, they're absolutely necessary these days due to the energies needed to create them. There are two main labs in the world that focus on doing this. The first is the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt, Germany. The second is Lawrence Berkeley National Laboratory in Berkeley, California.
Yes, a particle used in a particle accelerator must have a charge to be useful in the device. Particle accelerators we use in high energy physics to investigate things all work by applying a moving or shifting magnetic field to accelerate charged particles. We speed these particles up by repeatedly "hitting" them with a magnetic field. Uncharged particles will not respond to this, and canot be used in the devices.