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say you had a large quantity of a heavy element such as uranium... AND you put a massive amount of energy into it, so that it began to undergo nuclear fission and transformed to a significantly smaller element(s). Would it be possible to use this energy to now begin a process of fusion, to output energy? And then once it's fused beyond a certain size, back to fission? Basically the idea is to extract back and forth as much energy as possible from the original sample.

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Starting with a heavy element, the most energy you can extract is by reducing it to the most stable form of normal baryonic matter, which is stuff somewhere around iron on the periodic table. You could conceivably get to iron through running back and forth through cycles of fusion and fission, but in reality I think it would be very inefficient and take more energy than you got out. If you want to be really unrealistic, the ultimate theoretical limit would be to convert your input matter into a microscopic black hole and then extract the energy from the Hawking radiation. – Ben Crowell May 30 '13 at 20:26
Not really. Heavy fissionable elements are products of supernovae, not just simple fusion (the buck stops at iron IIRC). On a much smaller scale, there are fission-fusion-fission nuclear weapon designs but they don't make the full chain that you mention and are just a means to get more bang for the buck... – Deer Hunter May 30 '13 at 20:27
You can't get energy out of a transformation from A to B to A any more than you can get energy out of rolling a ball down a hill and then pushing it back up again. If you get energy going one way you have to supply energy going the other. – dmckee May 30 '13 at 21:22
When we right now produce energy do we perform fission until we hit iron? Or do we stop earlier – frogeyedpeas May 30 '13 at 22:09
up vote 2 down vote accepted

The first iteration -- fission driven fusion -- is exactly what happens in thermonuclear weapons. There the energy released by fission of heavy elements is used to fuse lighter elements (in practice isotopes of hydrogen, maybe with the help of lithium). In staged nuclear weapons design this fusion energy could be used to additionally drive the thermonuclear (that is, fusion) explosion of a larger scale.

One can also foresee the second iteration -- the neutrons released by the fusion of deuterium and tritium used to power the fission reaction of U-238 (or other heavy element). The U-238 by itself is not fissile by means of ~1MeV neutrons released in fission but can be split by energetic 14MeV neutrons from D-T fusion reaction. That single step (fusion driven fission) is also exploited in the suggested hybrid reactor designs.

As to further iterations -- fusion of elements heavier than hydrogen or fission of elements lighter than uranium or thorium -- these, while possible theoretically and achievable in accelerators, are not usable for power generation either controlled (like in a reactor), or explosive (like a bomb).

Let us also note the theoretical limits for this type of energy extraction -- the nuclear binding energy (already noted in other answer) is one such limit for ordinary matter (composed of atoms).

Another limit is imposed by the conservation of baryon number: all nuclear reactions conserve it so the total number of nucleons is constant and maximum energy extracted would be only a fraction of $M c^2$.

In nature several processes provides energy conversion rates exceeding the nuclear binding energy for atoms without violating the baryon number conservation:

There are also theoretical models that allow total energy conversion through physical processes beyond the Standard Model of particle physics. For instance magnetic monopole predicted by some such models would be catalyst of proton decay. None of such processes has been (reliably) observed so far.

Finally black holes formation and their subsequent evaporation through Hawking radiation also totally converts mass to energy -- however the timescales involved are (much) greater than the current age of the Universe.

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just for the sake of theoretical argument (and to fuel my curiosity), could a machine that makes, small black holes, feeds small amounts of matter, and quickly has them radiate, perform total mass energy conversion? – frogeyedpeas Jun 2 '13 at 4:07
Yes. While looking at at wikipedia pages for Hawking radiation I noticed this entry which discusses such energy conversion for propulsion. See original paper for discussion of finer points. – user23660 Jun 3 '13 at 3:22

The nuclear binding energy goes as displayed there. You can see that if you start with a heavy element (right hand side of the curve) and breaking it into 2 smaller elements (center of the curve), you end up with more energy. Say you want to break it up even more. Then the left part of the curve drops. It means that you will loose energy doing so. As a consequence, when you fuse these elements, you will only gain exactly the energy that you lost before.

Note that, any machine you do will have a finite efficiency, so overall you will loose energy. The solution is to find the fuel in the nature around us.

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