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Antimatter is theoretically an excellent fuel, providing large thrust/mass ratio and very efficient energy transfer. Therefore, we wouldn't need to use a lot of it to achieve similar thrusts that chemical rockets do.

Without considering other factors such as being safe distances away from people and other compenents and the initial launch from Earth, if a craft is using antimatter as a fuel, could it shrink down to a size much smaller than modern rockets, given how its fuel has a lot less volume?

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It's bad when matter and anti-matter come in direct contact[reference needed].

Currently, the only way to store antimatter is to trap it inside some very strong magnetic fields.

In short, the size of the containment system would greatly exceed the savings in volume gained by using anti-matter.

Those magnetic fields have to be very strong, which implies a lot of very heavy hardware to generate them.

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An anti-matter engine could potentially be very efficient, as you have mentioned it is extremely energy efficient for its mass. The tradeoff, however, is that you need a lot larger engine to contain the antimatter fuel. The only theory for trapping the fuel is to use careful magnetic fuel. That requires a significant size of a field typically, and thus requires very careful manipulation of the charge.

For instance, a similar proposal was Project Orion, which proposed to use nuclear bombs to explode forward a spacecraft. A single launch could launch tens of thousands of tons, for a rocket that was of compatible height to, say, the Saturn V. They have their issues, landing in particular is quite difficult, but they could be overcome. Antimatter would be even more efficient, and easier to control than it would be otherwise.

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