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What is it that make nuclear fusion so difficult at low temperatures? There is plasma existing at low temperatures, so what does it make for fusion to be so difficult to reach at ambient temperatures?

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Basically, what needs to happen in a fusion reaction is that two nuclei approach each other very close. They don't want to do that mainly because of their extremely strong electrostatic repulsion - both nuclei are positively charged. The force of repulsion grows quadratically with decreasing separation ($\frac{1}{r^2}$), and the distances required for the strong force to bind nuclei together are extremely short.

Our so far best bet is to just make nuclei go very fast and bash them into each other. They will still slow down as they approach each other, but a small fraction can overcome the repulsion and fuse together. The magnetic fields in most fusion reactors are needed to stabilise the plasma inside the reactor vessel. We don't want that they touch the outside, of course.

The high temperature is required simply because a hot plasma (or gas, for that matter) has a high average particle speed (Maxwell distribution). These fast particles are the ones that are able to fuse.

With a low temperature such as room temperature, only a very very small part of the nuclei move at a sufficiently high speed to fuse together.

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