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The energy per proton at the LHC is much larger than what is needed for fusion, protons break up into their constituents easily at this energy and fly away after they interact. In a fusion reactor, one wants the particles to stay within the reactor volume such that the released energy can be transferred to other deuterium/tritium nuclei which then can ...

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Oh, but we do! I'm assuming you mean using the fields to simply collide particles with each other, right? Then that's already being done. For example, take this neat little machine: http://en.wikipedia.org/wiki/Fusor This one runs on the exact same principle you described (though I'm not quite familiar with the inner workings of the LHC). For energy ...

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There isn't exactly a mathematical relationship, but there is a physical one. It is the gravitational compression that causes the increase in temperature in the core of the gas cloud that becomes a star. When the temperature reaches a critical value (in the millions of Kelvin range), hydrogen fusion can occur. This is because the temperature is great enough ...

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The sun gets its energy from the pp-chain. The first step is the two protons forming the diproton (Helium-2): $$\,^1_1H+\,^1_1H\to\,^2_2He+\gamma$$ where the $\gamma$ is the photon (of energy about half an MeV). This quickly $\beta^+$-decays into a deuterium by converting a proton into a neutron: $$\,^2_2He\to\,^2_1D+e^++\nu_e$$ where $e^+$ is the ...

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Not a full answer, but too long for a comment. Maybe this can lead you the right way. I'm going to ignore the changing mass for a moment and assume a ship with a fuel fraction of $0.5$. If we could get enough thrust from the engines to give a $1g$ thrust to the fuel, we could give a $0.5g$ thrust to the ship. If you have an engine with $I_{sp} = x$, and ...

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