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Tl;dr: Can fusion be achieved only by speeding up particles to high enough speeds to smash into each other?

I know that it is necessary for protons, or any oppositely-charged particles for that matter, to have very, very high amounts of energy to be able to get close enough to fuse or "touch". Usually, this involves raising the temperature of a reactor to millions of Kelvin and then accelerating those particles into each other. However, at such a small scale, are temperature and kinetic energy not the same thing? After all, temperature is just average kinetic energy. So, in theory, would it be possible to fuse two protons or particles if they were accelerated fast enough at each other, without raising the temperature very high? Or is there something I am missing here? I've looked all over the internet and can't find anything explaining why exactly this is possible or not possible, other than articles about cold fusion and how it was faked.

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    $\begingroup$ As I have remarked before, you can buy fusion reactors commercial off the shelf. Search term “neutron generator”. They’re typically a simple accelerator based design. $\endgroup$ – dmckee --- ex-moderator kitten Aug 23 '19 at 23:21
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    $\begingroup$ temperature is a classical variable of thermodynamics and can be related to average kinetic energy through statistical mechanics. It has no meaning for individual particles which have a fixed kinetic energy ( except if you decide to call their kinetic energy "temperature" through the average kinetic energy formula) $\endgroup$ – anna v Aug 24 '19 at 10:53
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The "easy" way to get a bunch of particles moving very fast is to make them very hot. If they are hot enough, some of them will fuse when they collide.

While it is possible to speed the particles up in an accelerator/collider instead, and then smack them into each other, this is a hugely inefficient enterprise. The energy release upon fusion is tiny compared to the energy expenditure to rev the particles up to speed in a particle collider.

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One can achieve fusion without high temperatures. However, one cannot achieve net energy production using fusion without high temperatures (based on current knowledge).

You consider fusion using accelerated ions. It is possible, but, as niels nielsen wrote, inefficient: in two colliding beams, most ions will undergo Coulomb scattering, not fusion.

Another low-temperature approach providing fusion is muon-catalyzed fusion, but again, net energy production is not practicable without some future breakthroughs.

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Tl;dr: Can fusion be achieved only by speeding up particles to high enough speeds to smash into each other?

Yes, there is a fundamental limit that needs to be crossed, the coulomb barrier. That requires a certain amount of energy, and the traditional solution is to heat it up until the average velocity is high enough to cross this barrier at a reasonable rate. That has proven difficult...

There is another approach called "colliding beam fusion". In fact, this is the first way fusion was ever generated, by accelerating deuterium ions into a metal foil infused with deuterium. This work also demonstrated the existence of tritium for the first time.

The problem with this approach is that in the vast majority of cases, the ions will reflect off each other at an angle rather than collide. Some very simple math demonstrates that the number of misses times the energy you put into the ions is always many orders of magnitude higher than the energy given off by the fusion events by those ions that do actually collide.

This has led to a cottage industry of methods that aim to "recycle" the ions while keeping their original energy so they get many chances to collide and thus the ratio of energy in to out is improved. One of the more easy-to-understand approaches is the migma concept, which uses a clever magnetic bottle to recirculate the ions through the center of the chamber. Other similar concepts include the fusor and the polywell. A more recent attempt is TAE's design. All of these have fundamental limits that appear to suggest they cannot work, although they are more complex than the "they just miss" of the basic colliding beam concept.

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