Excerpted from Section 2 of Nuclear Weapons Frequently Asked Qustions by Carey Sublette.
The principle issues that must solved to construct a fission weapon are:
Solving issues 1, 2 and 3 together is greatly complicated by the unavoidable presence of naturally occurring neutrons. Although cosmic rays generate neutrons at a low rate, almost all of these "background" neutrons originate from the fissionable material itself through the process of spontaneous fission. Due to the low stability of the nuclei of fissionable elements, these nuclei will occasionally split without being hit by a neutron. This means that the fissionable material itself periodically emits neutrons.
The process of assembling the supercritical mass must occur in significantly less time than the average interval between spontaneous fissions to have a reasonable chance of succeeding. This problem is difficult to accomplish due to the very large change in reactivity required in going from a subcritical state to a supercritical one. The time required to raise the value of k from 1 to the maximum value of 2 or so is called the reactivity insertion time, or simply insertion time.
It is further complicated by the problem of subcritical neutron multiplication. If a subcritical mass has a k value of 0.9, then a neutron present in the mass will (on average) create a chain reaction that dies out in an average of 10 generations. If the mass is very close to critical, say k=0.99, then each spontaneous fission neutron will create a chain that lasts 100 generations. This persistence of neutrons in subcritical masses further reduces the time window for assembly, and requires that the reactivity of the mass be increased from a value of <0.9 to a value of 2 or so within that window.
Simply splitting a supercritical mass into two identical parts, and bringing the parts together rapidly is unlikely to succeed since neither part will have a sufficiently low k value, nor will the insertion time be rapid enough with achievable assembly speeds.
The key to achieving objectives 1 and 2 is revealed by the fact that the critical mass (or supercritical mass) of a fissionable material is inversely proportional to the square of its density. By contriving a subcritical arrangement of fissionable material whose average density can be rapidly increased, we can bring about the sudden large increase in reactivity needed to create a powerful explosion. As a general guide, a suitable highly supercritical mass needs to be at least three times heavier than a mass of equal density and shape that is merely critical. Thus doubling the density of a pit that is slightly sub- critical (thereby making it into nearly four critical masses) provides sufficient reactivity insertion for a bomb.
Two general approaches have been used for achieving this idea: implosion assembly, and gun assembly. Implosion is capable of very short insertion times, gun assembly is much slower.