Bare sphere of fissile material is too small to allow the
chain reaction to become self-sustaining as
neutrons generated by
fissions can too easily escape.
Middle: By increasing the mass of the sphere to a critical mass, the reaction can become self-sustaining.
Bottom: Surrounding the original sphere with a
neutron reflector increases the efficiency of the reactions and also allows the reaction to become self-sustaining. The shape with minimal critical mass is a sphere. Bare-sphere critical masses at normal density of some
actinides are listed in the following table. Most information on bare sphere masses is considered classified, since it is critical to nuclear weapons design, but some documents have been declassified. The critical mass for lower-grade uranium depends strongly on the grade: with 45% 235U, the bare-sphere critical mass is around ; with 19.75% 235U it is over ; and with 15% 235U, it is well over . In all of these cases, the use of a neutron reflector like beryllium can substantially drop this amount, however: with a reflector, the critical mass of 19.75%-enriched uranium drops to , and with a reflector it drops to , for example. The critical mass is inversely proportional to the square of the density. If the density is 1% more and the mass 2% less, then the volume is 3% less and the diameter 1% less. The probability for a neutron per cm travelled to hit a nucleus is proportional to the density. It follows that 1% greater density means that the distance travelled before leaving the system is 1% less. This is something that must be taken into consideration when attempting more precise estimates of critical masses of plutonium isotopes than the approximate values given above, because plutonium metal has a large number of different crystal phases which can have widely varying densities. Note that not all neutrons contribute to the chain reaction. Some escape and others undergo
radiative capture. Let
q denote the probability that a given neutron induces fission in a nucleus. Consider only
prompt neutrons, and let
ν denote the number of prompt neutrons generated in a nuclear fission. For example,
ν ≈ 2.5 for uranium-235. Then, criticality occurs when
ν·q = 1. The dependence of this upon geometry, mass, and density appears through the factor
q. Given a total interaction
cross section σ (typically measured in
barns), the
mean free path of a prompt neutron is \ell^{-1} = n \sigma where
n is the nuclear number density. Most interactions are scattering events, so that a given neutron obeys a
random walk until it either escapes from the medium or causes a fission reaction. So long as other loss mechanisms are not significant, then, the radius of a spherical critical mass is rather roughly given by the product of the mean free path \ell and the square root of one plus the number of scattering events per fission event (call this
s), since the net distance travelled in a random walk is proportional to the square root of the number of steps: : R_c \simeq \ell \sqrt{s} \simeq \frac{\sqrt{s}}{n \sigma} Note again, however, that this is only a rough estimate. In terms of the total mass
M, the nuclear mass
m, the density ρ, and a fudge factor
f which takes into account geometrical and other effects, criticality corresponds to : 1 = \frac{f \sigma}{m \sqrt{s}} \rho^{2/3} M^{1/3} which clearly recovers the aforementioned result that critical mass depends inversely on the square of the density. Alternatively, one may restate this more succinctly in terms of the areal density of mass, Σ: : 1 = \frac{f' \sigma}{m \sqrt{s}} \Sigma where the factor
f has been rewritten as
f' to account for the fact that the two values may differ depending upon geometrical effects and how one defines Σ. For example, for a bare solid sphere of 239Pu criticality is at 320 kg/m2, regardless of density, and for 235U at 550 kg/m2. In any case, criticality then depends upon a typical neutron "seeing" an amount of nuclei around it such that the areal density of nuclei exceeds a certain threshold. This is applied in implosion-type nuclear weapons where a spherical mass of fissile material that is substantially less than a critical mass is made supercritical by very rapidly increasing ρ (and thus Σ as well) (see below). Indeed, sophisticated nuclear weapons programs can make a functional device from less material than more primitive weapons programs require. Aside from the math, there is a simple physical analog that helps explain this result. Consider diesel fumes belched from an exhaust pipe. Initially the fumes appear black, then gradually you are able to see through them without any trouble. This is not because the total scattering cross section of all the soot particles has changed, but because the soot has dispersed. If we consider a transparent cube of length
L on a side, filled with soot, then the
optical depth of this medium is inversely proportional to the square of
L, and therefore proportional to the areal density of soot particles: we can make it easier to see through the imaginary cube just by making the cube larger. Several uncertainties contribute to the determination of a precise value for critical masses, including (1) detailed knowledge of fission cross sections, (2) calculation of geometric effects. This latter problem provided significant motivation for the development of the
Monte Carlo method in computational physics by
Nicholas Metropolis and
Stanislaw Ulam. In fact, even for a homogeneous solid sphere, the exact calculation is by no means trivial. Finally, note that the calculation can also be performed by assuming a continuum approximation for the neutron transport. This reduces it to a diffusion problem. However, as the typical linear dimensions are not significantly larger than the mean free path, such an approximation is only marginally applicable. Finally, note that for some idealized geometries, the critical mass might formally be infinite, and other parameters are used to describe criticality. For example, consider an infinite sheet of fissionable material. For any finite thickness, this corresponds to an infinite mass. However, criticality is only achieved once the thickness of this slab exceeds a critical value.
Sphere with tamper The critical mass can be greatly reduced by the use of a
tamper. The following table lists the values for common
neutron reflectors, where λtamp is the mean free path of a neutron within the tamper material before an
elastic scattering. In nuclear weapon design,
natural uranium is primarily used. Tungsten carbide was initially used in the
Little Boy. ==Nuclear weapon design==