There is a problem throwing protons or any other nuclei at nuclei - and that's Coulomb barrier.
Most of time, a proton either simply bounces back, and the energy is lost as heat, or else also emits a x-ray, and energy is also lost as heat, until the proton slows down and will cause no reaction.
Whereas neutrons are uncharged. Send a neutron into a matter - it often bounces back, but it's still a neutron. Eventually it will cause a nuclear reaction, with the sole exception of very pure helium 4.
Say, transmute mercury into gold?
Easy.
Mercury consists of 7 stable isotopes. 196, 198-202, and 204.
Since 198-202 are all stable, sending neutrons at mercury 198 to 201 will produce pure energy - which is not waste because that's the heat output of the reactor.
Mercury 202 and 204 are transmuted into thallium, and 196 into gold.
Now, the composition of mercury is:
- 196 - 0,15 %
- 198 - 10 %
- 199 - 17 %
- 200 - 23 %
- 201 - 13 %
- 202 - 30 %
- 204 - 7 %
Mercury 196 is scarce... but fortunately, the neutron cross-sections of mercury isotopes are not equal.
They are:
- 196 - 3200
- 198 - 2
- 199 - 2150
- 200 - 1,4
- 201 - 7,8
- 202 - 5
- 204 - 0,43
Now, what's the fractional cross-section of each?
- 196 - 4,8
- 198 - 0,2
- 199 - 360
- 200 - 0,32
- 201 - 1,0
- 202 - 1,5
- 204 - 0,03
So - an overwhelming majority (360 barns) of neutrons will produce heat in mercury 199. The other 6 isotopes have a combined cross-section of 7,8 barns - slightly over 2 % of total - of which 4,8 barns produce gold, 1,5 barns produce thallium and 1,5 barns produce also heat.
Mercury boils at 360 degrees, gold at 3000 degrees and thallium at 1500 degrees. So in the boiler, mercury boils and powers the turbines, while gold cannot boil and is collected.
Mercury cooled reactor can work, despite the high cross-section - as proven by Clementine.
However, it is not clear how effective mercury cooled reactors are, compared to other reactors which do not produce gold.