Question

Question #861b3

Answer 1

OK, to understand this you must realise that the flux (flow rate) of neutrons is generated according to the volume of material and lost across the surface area of that volume.

Explanation:

Each nucleus has an equivalent chance of decay, so the rate of decay increases with the number of nuclei. This is directly related to the volume of material present, since each nucleus occupies the same volume. The fact that often a fission event will release multiple neutrons (an average of 2.43 in the case of U-235) need not trouble us, it alters the arithmetic, but not the principle.

Thus we see that the surface area to volume ratio is the key factor here. This varies with the geometry of the shape, so to keep things simple we’ll assume we are dealing with a sphere.d

The volume increases in proportion to `r^3`, but the surface area only rises in proportion to the `r^2` so as the number of fissile particles increases, this important ratio governing the neutron balance will fall. This in turn means that more and more neutron flux is being generated i.e. the neutron population rises increasing the chance that a nucleus will absorb a neutron (it’s actually U-236 that is fissile, U-235 is just the naturally occurring form that can take part in the reaction.)

This means as the radius rises, we will reach (and then exceed) a point where the neutron economy is such that there is a reasonable chance of any random fission event releasing sufficient neutrons (that are retained in the volume long enough) to cause a self-sustaining reaction.

This “critical volume” can easily be converted to a critical mass if you know the density. It is surprisingly low in the case of U-235, just a couple of kilos, but would likely be a very inefficient bomb as one of the first effects would be to separate the fissile material.