Critical mass

Factors

The mass where criticality occurs may be changed by modifying certain attributes such as fuel, shape, temperature, density and the installation of a neutron-reflective substance. These attributes have complex interactions and interdependencies. The factors listed are for the simplest/ideal cases.

Shape

A mass may be exactly critical without being a perfect homogeneous sphere. More closely refining the shape toward a perfect sphere will make the mass supercritical. Conversely changing the shape to a less perfect sphere will decrease its reactivity and make it subcritical.

Temperature

A mass may be exactly critical at a particular temperature. Fission and absorption cross-sections increase as the relative neutron velocity decreases. 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 ²³⁸U 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 increase also contributes a negative coefficient of reactivity since fuel atoms are moving farther apart. A mass that is exactly critical at room temperature would be sub-critical in an environment anywhere above room temperature due to thermal expansion alone.

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 the ambient temperature after a period of time, because fuel consumed during fission brings the assembly back to subcriticality once again.

Density

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 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 to lower density or become supercritical if cooled to 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 decreases as the fuel temperature increases.

Presence of a neutron reflector

Surrounding a spherical critical mass with a neutron reflector further reduces the mass needed for criticality. By using a neutron reflector to increase the likelihood of a fission event, it simultaneously increases the fuel efficiency of the entire reaction. As fuel becomes more efficient, the reaction requires a lower critical mass.

Common reflector materials include beryllium, graphite, and heavy metals like tungsten carbide, with the choice of material depending on the specific reactor design and the type of neutrons being managed. When looking at the specific materials commonly used, beryllium has been identified as an effective material due to its low neutron absorption cross-section. Graphite is used because of its diverse abilities, such as being a moderator and a reflector. Heavy metals such as tungsten carbide and stainless steel are common as they reduce the amount of neutrons that escape during the reaction by a considerable amount. By reflecting neutrons, reflectors can make a subcritical mass of fissile material critical; they can also increase the rate of fission in an already critical mass.

Neutron reflectors are commonly applied in various nuclear technologies, including nuclear reactors, nuclear weapons, and even neutron supermirrors. Nuclear reactors commonly rely on neutron reflectors due to their ability to exponentially speed up and increase the likelihood of fission and fusion reactions occurring. Nuclear weapons have historically utilized neutron reflectors to increase the rate of reaching the critical mass in the use of the weapons. Lastly, neutron reflectors are vital to nuclear physics research regarding the reflection of neutron beams, otherwise known as nuclear supermirrors.

Presence of a tamper

In a bomb, a dense shell of material surrounding the fissile core will contain, via inertia, the expanding fissioning material, which increases the efficiency. This is known as a tamper. A tamper also tends to act as a neutron reflector. Because a bomb relies on fast neutrons (not ones moderated by reflection with light elements, as in a reactor), the neutrons reflected by a tamper are slowed by their collisions with the tamper nuclei, and because it takes time for the reflected neutrons to return to the fissile core, they take rather longer to be absorbed by a fissile nucleus. But they do contribute to the reaction, and can decrease the critical mass by a factor of four. 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 so-called boosted configuration.

Escape probability

During a nuclear episode, isolated neutrons are produced because of the division of particles. As those neutrons break away from their nucleus, they have free rein to disperse throughout space. The probability that a subatomic particle and a target nucleus will not collide in an environment that contains those released neutrons is known as the escape probability. Active neutrons are less likely to escape without interfering with other subatomic particles in environments with a higher critical mass. This is because more room stimulates greater particle movement and, therefore, more collisions.

Elastic scattering

Elastic scattering explains that when particles collide with other particles, they respond entirely elastically and bounce off while retaining the same kinetic energy as before the collision. The duration and length of a neutron’s path are exponentially larger due to elastic scattering; this is because the rebounding of the particle is in a different direction.

Elastic scattering directly relates to escape probability because the bouncing of the particles drastically drops the probability of escaping the mass without collision. In an atomic environment with a large critical mass, there are inherently more opportunities for neutron collision as the retention of energy allows for continuous scattering.

Additionally, materials that have a high elasticity potential and therefore higher elastic scattering allow the use of less fissile material because a reduced critical mass is required. This conservation of fissile material is essential to the future of nuclear science because the cheaper and more convenient it is to retain criticality, the more investment into developing nuclear innovation there will be.

Values

Bare sphere

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 185 kilograms (408 lb); with 19.75% ²³⁵U it is over 780 kilograms (1,720 lb); and with 15% ²³⁵U, it is well over 1,350 kilograms (2,980 lb). In all of these cases, the use of a neutron reflector like beryllium can substantially drop this amount, however: with a 5 centimetres (2.0 in) reflector, the critical mass of 19.75%-enriched uranium drops to 403 kilograms (888 lb), and with a 15 centimetres (5.9 in) reflector it drops to 144 kilograms (317 lb), 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 ${\displaystyle {\ell }^{-1}=n\sigma }$ 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 ${\displaystyle \ell }$ 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:

${\displaystyle R_c\simeq \ell \sqrt{s}\simeq \frac{\sqrt{s}}{n\sigma }}$

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

${\displaystyle 1=\frac{f\sigma }{m\sqrt{s}}{\rho }^{2/3}{M}^{1/3}}$

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:

${\displaystyle 1=\frac{f^{\prime }\sigma }{m\sqrt{s}}\mathrm{\Sigma }}$

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 λ_tamp 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

Until detonation is desired, a nuclear weapon must be kept subcritical. In the case of a uranium gun-type 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 pieces of uranium are 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'). This design is referred to as a gun-type fission weapon.

A theoretical 100% pure ²³⁹Pu weapon could also be constructed as a gun-type weapon, like the Manhattan Project's proposed Thin Man design. In reality, this is impractical because 100% pure ²³⁹Pu cannot be achieved. Even "weapons grade" ²³⁹Pu is contaminated with a small amount of ²⁴⁰Pu, which has a strong propensity toward spontaneous fission. Because of this, a reasonably sized gun-type weapon would suffer nuclear reaction (predetonation) before the masses of plutonium would be in a position for a full-fledged 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.

Quantum field theory

Quantum chromodynamics

Quantum chromodynamics is the theory that explains the formation of atomic nuclei through the analysis of subatomic and atomic particle interactions. Similar to critical mass, quantum chromodynamics theory has a critical point that defines both the minimum temperature and potential necessary for the transition of the first-order phase to take place.

Dynamic chiral symmetry breaking

Dynamic chiral symmetry breaking is the spontaneous formation of a mass through the accumulation of fermions, which are a category of subatomic particles. Its theory describes critical mass as the minimum quark mass required to separate subatomic particles. When a division of particles takes place, it leaves gaps in the fermion mass that allow for increased particle movement. As particles move, they stimulate energy gain, which ultimately increases the chances of jump-starting a nuclear reaction.

Notable firsts