An assembly in which a chain reaction is just possible is called critical, and is said to have obtained criticality. In such an assembly, without new input of free neutrons, e.g. from spontaneous fissions, the reaction will on average be just sustained, and in the case of a steady input of new free neutrons, the reaction will increase linearly. A more than critical assembly is termed supercritical. An assembly that is capable of sustaining a chain reaction without needing the contribution of delayed neutrons is called prompt critical (and is therefore also supercritical). Even larger masses are called superprompt critical.

If an assembly is less than critical, then with a steady input of new free neutrons the fission reaction will reach a steady state, and the assembly is said to be subcritical.

The realisation that a supercritical assembly is not necessarily prompt critical is attributed to Enrico Fermi, and made the construction of a nuclear reactor using a fission chain reaction possible. Any prompt critical assembly will explode if not rapidly brought below prompt criticality.

Critical mass of a bare sphere

In the case of a sphere surrounded by a neutron reflector the critical mass is about 15 kg for uranium-235 (20 to 25 kg for a gun-type assembly) 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.

The critical mass for lower-grade uranium depends strongly on the grade: with 20 % U-235, and surrounded by a 4 cm thick beryllium neutron reflector, it is over 400 kg; with 15 % U-235, it is well over 1000 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 $\backslash nu$ denote the number of prompt neutrons generated in a nuclear fission. For example, $\backslash nu\; \backslash simeq\; 2.5$ for uranium-235. Then, criticality occurs when $\backslash nu\; q\; =\; 1$. The dependence of this upon geometry, mass, and density appears through the factor $q$.

Given a total interaction cross section $\backslash sigma$ (typically measured in barns), the mean free path of a prompt neutron is $\backslash ell^\{-1\}\; =\; n\; \backslash sigma$ 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 $\backslash ell$ 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:

$R\_c\; \backslash simeq\; \backslash ell\; \backslash sqrt\{s\}\; \backslash simeq\; \backslash frac\{\backslash sqrt\{s\}\}\{n\; \backslash sigma\}$

Note again, however, that this is only a rough estimate.

In terms of the total mass $M$, the nuclear mass $m$, the density $\backslash rho$, and a fudge factor $f$ which takes into account geometrical and other effects, criticality corresponds to

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

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, $\backslash Sigma$:

$1\; =\; \backslash frac\{f\text{'}\; \backslash sigma\}\{m\; \backslash sqrt\{s\}\}\; \backslash Sigma$

where the factor $f$ has been rewritten as $f\text{'}$ to account for the fact that the two values may differ depending upon geometrical effects and how one defines $\backslash Sigma$. 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 $\backslash rho$ (and thus $\backslash Sigma$ 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.

Weapon design

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 blow itself apart 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.

Criticality pathways tutorial

• Criticality via additional mass

"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 equillibrium state).

1. 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).

2. 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-degree above the surroundings) and then decrease back to room temperature after a period of time (fuel consumption eventually makes the assembly sub-critical).

3. An exactly critical, room temperature mass will be 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.

• Criticality via geometry

Start with a fixed but sufficient amount U-235 marbles with holes drilled though in the x, y and z directions, slightly off-center so that they do not intersect at the center of the marble. Thread slender strong rods through beads and build up a 3-D simple cubic lattice, like those models of salt (NaCl) in your chemistry class. The difference is that you can slide the beads along the rods. Grab opposite corners of the cubic lattice and play with them a little, growing and shrinking the cubic lattice. Suddenly, push as hard and fast as you can, to bring all the marbles together, all at once. No human can push hard and fast enough for a detonation, but in principle that could be the result. The high explosives in an implosion-type nuclear weapon provide trivial thermal and no chemical factors to the nuclear detonation; their important contribution is sudden compression and, therefore, geometry alone, just like bringing two sub-critical objects of U-235 together to form a critical mass in a gun type weapon, as mentioned above.

It is often assumed that the implosion-type nuclear weapon compresses metallic fuel beyond its natural density; normally solids and liquids are considered incompressible. A better mechanism (more reactivity insertion) is to explosively collapse a partially hollow sub critical mass into a more solid, super critical mass. In a macroscopic sense, collapsing a hollow fuel assembly is equivalent to compressing a low density material into a higher density material.

Global parameters:

The practical engineering calculations involve two parameters: mass and radius of sphere. Start with sufficient mass, compress it into a smaller sphere via high explosives. Boom. One other useful relationship: The more mass you start with, provided it is not already critical, the less you have to compress it to get critical mass. Most other calculations are related to maximizing the conversion of mass to energy before the pieces fly apart.

If we do not use a trigger, but assemble a sphere of supercritical mass of U-235, a chain reaction may or may not be caused by a single spontaneous decay, with one of the neutrons produced causing a new decay. See nuclear chain reaction. Since there are typically at least tens of such free neutrons per second, it takes only a fraction of a second until the chain reaction starts.

See also

• Nuclear chain reaction
• Nuclear weapon design
• Criticality accident

Mass | nuclear technology | Radioactivity

Massa crítica | Kritische Masse | Masa crítica | Masse critique (réaction nucléaire) | מסה קריטית | Massa critica | 臨界量 (原子力) | Kritische massa | Masa krytyczna | Massa crítica | Kritisk massa | 臨界質量