Critical mass
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- Supercritical redirects here; for supercritical fluid, see supercritical fluid. For supercritical bifurcations, see pitchfork bifurcation.
The critical mass of fissile material is the amount needed for a sustained nuclear chain reaction. The critical mass of a fissionable material depends upon its nuclear properties (e.g. the nuclear fission cross-section) and physical properties (in particular the density), its shape, and its enrichment. Surrounding fissionable material by a neutron reflector reduces the needed mass for criticality. See also neutron radiation.
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[edit] Explanation
"Critical" implies an equilibrium (steady-state) fission reaction; there is no increase in power/temperature/neutron population. "Subcritical" implies an inability to sustain a fission reaction; a population of neutrons introduced to a subcritical assembly will decrease in number over time. "Supercritical" implies an increasing rate of fission until natural feedback mechanisms cause the reactor to settle into equilibrium (i. e. be critical) at an elevated temperature/power level or destroy itself (disassembly is an equilibrium state).
It is possible for an assembly to be critical at near zero power. If the perfect quantity of fuel was added to a slightly sub critical mass to create an "exactly critical mass", fission would be self sustaining for one neutron generation (fuel consumption makes the assembly sub-critical).
If the perfect quantity of fuel was added to a slightly sub critical mass to create a "barely supercritical mass", the temperature of the assembly would increase to an initial maximum (for example: 1 K above the ambient temperature) and then decrease back to room temperature after a period of time because the fuel consumption during the process eventually makes the assembly sub-critical again.
An exactly critical, room temperature mass will become sub critical if warmed and supercritical if cooled. Intrinsically, fission becomes less probable as fuel temperature increases (negative coefficient of reactivity). Also contributing a negative coefficient of reactivity is thermal expansion; a decrease in fuel density associated with increase in temperature also makes the fission reaction less probable.
[edit] Critical mass of a bare sphere
The shape with minimum critical mass is a sphere. This can be further reduced by surrounding the sphere with a neutron reflector.
In the case of a bare sphere the critical mass is about 50 kg for uranium-235 and 10 kg for plutonium 239.
Bare-sphere critical masses of some other isotopes whose half-lives exceed 100 years are compiled in the following table.
| Isotope | Critical Mass | Link |
|---|---|---|
| protactinium-231 | 750±180 kg | |
| uranium-233 | 15 kg | [1] |
| uranium-235 | 52 kg | [2] |
| neptunium-236 | 7 kg | [3] |
| neptunium-237 | 60 kg | [4],[5] |
| plutonium-238 | 9.04–10.07 kg | [6] |
| plutonium-239 | 10 kg | [7],[8] |
| plutonium-240 | 40 kg | [9] |
| plutonium-241 | 12 kg | [10] |
| plutonium-242 | 75–100 kg | [11] |
| americium-241 | 55–77 kg | Dias et. al. |
| americium-242m | 9–14 kg | ibid. |
| americium-243 | 180–280 kg | ibid. |
| curium-243 | 7.34–10 kg | [12] |
| curium-244 | (13.5)–30 kg | [13] |
| curium-245 | 9.41–12.3 kg | [14] |
| curium-246 | 39–70.1 kg | [15] |
| curium-247 | 6.94–7.06 kg | [16] |
| californium-249 | 6 kg | [17] |
| californium-251 | 5 kg | [18] |
The critical mass for lower-grade uranium depends strongly on the grade: with 20 % U-235 it is over 400 kg; with 15 % U-235, it is well over 600 kg.
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, so 1% more, which compensates 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 rough 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. Others undergo radiative capture.
Let q denote the probability that a given neutron induces fission in a nucleus. Let us consider only prompt neutrons, and let ν denote the number of prompt neutrons generated in a nuclear fission. For example, 
for uranium-235. Then, criticality occurs when νq = 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 
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 
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:
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
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:
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 Pu-239 criticality is at 320 kg/m², regardless of density, and for U-235 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 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, so that the problem reduces 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.
[edit] Weapon design
Until detonation is desired, a nuclear weapon must be kept subcritical. In the case of a uranium bomb, this can be achieved by keeping the fuel in a number of separate pieces, each below the critical size either because they are too small or unfavorably shaped. To produce detonation, the uranium is brought together rapidly. In Little Boy, this was achieved by firing a smaller piece of uranium down a gun barrel into a corresponding hole in a larger piece, a design referred to as a gun-type fission weapon.
A theoretical 100% pure Pu-239 weapon could also be constructed as a gun-type weapon. In reality, this is impractical because even "weapons grade" Pu-239 is contaminated with a small amount of Pu-240, which has a strong propensity toward spontaneous fission. Because of this, a reasonably sized gun-type weapon would suffer nuclear reaction before the masses of plutonium would be in a position for a full-fledged explosion to occur. Even accounting for Pu-240 impurity, a gun type weapon could still be constructed. It would not be a very practical weapon, however, as it would have to be very long in order to accelerate a mass of plutonium to very high velocities to overcome the effects just mentioned. A better solution exists.
Instead, the plutonium is present as a subcritical sphere (or other shape), which may or may not be hollow. Detonation is produced by exploding a shaped charge surrounding the sphere, increasing the density (and collapsing the cavity, if that was present) to produce a prompt critical configuration. This is known as an implosion type weapon.
