A critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction. The critical mass of a fissionable material depends upon its nuclear properties (e.g. the nuclear fission crosssection), its density, its shape, its enrichment, its purity, its temperature, and its surroundings.
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The term critical refers to an equilibrium fission reaction (steadystate or continuous chain reaction); this is where there is no increase or decrease in power, temperature, or neutron population.
A numerical measure of a critical mass is dependent on the neutron multiplication factor, k, where:
where f is the average number of neutrons released per fission event and l is the average number of neutrons lost by either leaving the system or being captured in a nonfission event. When k = 1 the mass is critical.
A subcritical mass is a mass of fissile material that does not have the ability to sustain a fission reaction. A population of neutrons introduced to a subcritical assembly will exponentially decrease, typically rapidly. In this case, k < 1. A steady rate of spontaneous fissions causes a proportional steady level of neutron activity. The constant of proportionality increases as k increases.
A supercritical mass is one where there is an increasing rate of fission. The material may settle into equilibrium (i. e. become critical again) at an elevated temperature/power level or destroy itself (disassembly is an equilibrium state). In the case of supercriticality, k > 1.
The point, and therefore the mass, where criticality occurs may be changed by modifying certain attributes, such as fuel, shape, temperature, density, and the installation of a neutronreflective substance. These attributes have complex interactions and interdependencies, this section explains only the simplest ideal cases.
It is possible for a fuel assembly to be critical at near zero power. If the perfect quantity of fuel were added to a slightly subcritical mass to create an "exactly critical mass", fission would be selfsustaining for one neutron generation (fuel consumption makes the assembly subcritical).
If the perfect quantity of fuel were added to a slightly subcritical 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 fuel consumed during fission brings the assembly back to subcriticality once again.
A mass may be exactly critical, but not a perfect homogeneous sphere. Changing the shape to be closer to a perfect sphere will make the mass supercritical. Conversely, changing the shape to be further from a sphere will decrease its reactivity, making it subcritical.
A mass may be exactly critical at a particular temperature. Fission and absorption crosssections decrease with the inverse of relative neutron velocity. As fuel temperature increases, neutrons of a given energy appear faster and thus fission/absorption is less likely. This is not unrelated to doppler broadening of the U238 resonances, but is common to all fuels/absorbers/configurations. Neglecting the very important resonances, the total neutron cross section of every material exhibits an inverse relationship with relative neutron velocity. Hot fuel is always less reactive than cold fuel (over/under moderation in LWR is a different topic). Thermal expansion associated with temperature rise also contributes a negative coefficient of reactivity, since fuel atoms are farther apart. A mass that is exactly critical at room temperature would be subcritical in an environment anywhere above room temperature due to thermal expansion alone.
The higher the density, the lower the critical mass. The density of a material at a constant temperature can be changed by varying the pressure or tension or by changing crystal structure (see Allotropes of plutonium). An ideal mass will become subcritical if allowed to expand or conversely the same mass will become supercritical if compressed. Changing the temperature may also change the density, however the effect on critical mass is then complicated by the temperature effects (See Changing the temperature), and by whether the material expands or contracts with increased temperature. Assuming the material expands with temperature (enriched Uranium 235 at room temperature for example), at an exactly critical state, it will become subcritical if warmed, hence lower density, or conversely the same mass will become supercritical if cooled, hence higher density. Such a material is said to have a negative temperature coefficient of reactivity to indicate that its reactivity decreases when its temperature increases. Using such a material as fuel means fission reduces as the fuel temperature increases.
Surrounding a spherical critical mass with a neutron reflector further reduces the mass needed for criticality. A common material for a neutron reflector is beryllium metal. This reduces the number of neutrons which escape the fissile material, resulting in increased reactivity.
In a bomb, a dense shell of material surrounding the fissile core will contain, via inertia, the expanding fissioning material. This increases the efficiency. Because a bomb relies on fast neutrons (not ones moderated by reflection with light elements, as in a reactor) the tamper in a bomb is not functioning as a neutron reflector. Also, if the tamper is (e.g. depleted) uranium, it can fission due to the high energy neutrons generated by the primary explosion. This can greatly increase yield, especially if even more neutrons are generated by fusing hydrogen isotopes, in a socalled boosted configuration.
The shape with minimal critical mass and the smallest physical dimensions is a sphere. Baresphere critical masses at normal density of some actinides are listed in the following table.
Nuclide  Critical Mass (kg) 
Diameter (cm) 
Ref 

protactinium231  750±180?  45±3?  
uranium233  15  11  ^{[1]} 
uranium235  52  17  ^{[1]} 
neptunium236  7  8.7  ^{[2]} 
neptunium237  60  18  ^{[3]}^{[4]} 
plutonium238  9.04–10.07  9.59.9  ^{[5]} 
plutonium239  10  9.9  ^{[1]}^{[5]} 
plutonium240  40  15  ^{[1]} 
plutonium241  12  10.5  ^{[6]} 
plutonium242  75–100  1921  ^{[6]} 
americium241  55–77  2023  ^{[7]} 
americium242m  9–14  1113  ^{[7]} 
americium243  180–280  3035  ^{[7]} 
curium243  7.34–10  1011  ^{[8]} 
curium244  (13.5)–30  (12.4)–16  ^{[8]} 
curium245  9.41–12.3  1112  ^{[8]} 
curium246  39–70.1  1821  ^{[8]} 
curium247  6.94–7.06  9.9  ^{[8]} 
californium249  6  9  ^{[2]} 
californium251  5  8.5  ^{[2]} 
The critical mass for lowergrade uranium depends strongly on the grade: with 20% U235 it is over 400 kg; with 15% U235, 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 uranium235. 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 Pu239 criticality is at 320 kg/m², regardless of density, and for U235 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 implosiontype 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.
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 unfavourably shaped. To produce detonation, the uranium is brought together rapidly. In Little Boy, this was achieved by firing a piece of uranium (a 'doughnut'), down a gun barrel onto another piece, (a 'spike'), a design referred to as a guntype fission weapon.
A theoretical 100% pure Pu239 weapon could also be constructed as a guntype weapon. In reality, this is impractical because even "weapons grade" Pu239 is contaminated with a small amount of Pu240, which has a strong propensity toward spontaneous fission. Because of this, a reasonably sized guntype weapon would suffer nuclear reaction before the masses of plutonium would be in a position for a fullfledged explosion to occur.
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 present) to produce a prompt critical configuration. This is known as an implosion type weapon.
