In nuclear engineering, critical mass is the minimum mass of the fissile material needed for a sustained nuclear chain reaction in a particular setup. The critical mass of a fissile material depends upon its nuclear cross section, density, shape, enrichment, purity, temperature, and surroundings. It is an important parameter of a nuclear reactor core or nuclear weapon and is important in nuclear weapon design.
Critical size is the minimum size of the fissile material needed for a sustained nuclear chain reaction in a particular setup. If the size of the reactor core is less than a certain minimum, too many fission neutrons escape through its surface and the chain reaction is not sustained. A perfect sphere, which has the lowest surface-area-to-volume ratio, gives the minimal critical size.
Criticality
When a nuclear chain reaction in a mass of fissile material is self-sustaining but not growing, the mass is said to be in a critical state, in which there is no increase or decrease in power, temperature, or neutron population.
A numerical measure of a critical mass depends on the effective neutron multiplication factor , the average number of neutrons released per fission event that go on to cause another fission event rather than being absorbed or leaving the material.
A subcritical mass is a mass that does not have the ability to sustain a fission chain reaction. A population of neutrons introduced to a subcritical assembly will exponentially decrease. In this case, known as , .
A critical mass is a mass of fissile material that self-sustains a fission chain reaction. In this case, known as , . A steady rate of spontaneous fission causes a proportionally steady level of neutron activity.
A supercritical mass is a mass which, once fission has started, will proceed at an increasing rate. In this case, known as , . The constant of proportionality increases as increases. The material may settle into equilibrium (i.e. become critical again) at an elevated temperature/power level or destroy itself.
Due to spontaneous fission a supercritical mass will undergo a chain reaction. For example, a spherical critical mass of pure uranium-235 (235U) with a mass of about would experience about 0.3 spontaneous fissions per second. The probability that one such event will cause a chain reaction depends on how much the mass exceeds the critical mass. Fission can also be initiated by neutrons produced by cosmic rays.
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 self-sustaining for only one neutron generation (fuel consumption then makes the assembly subcritical again).
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 A neutron reflector is a layer of material placed around the core that scatters escaping neutrons back into it. Because fewer neutrons leak out, a core of a given fuel and moderator composition reaches criticality at a smaller size, which lowers the amount of fissile material needed.
The gain is usually expressed as the reflector savings, the amount by which the critical dimension shrinks when the reflector is added. For a reflector made of the moderator material, the savings peaks at a thickness of roughly two migration lengths. In thermal reactors a useful approximation is that the savings equals the migration length of the reflector material: roughly 6 cm in a light-water reactor, and more than half a meter in a graphite one.
What matters in a reflector material is that it scatters neutrons well while absorbing few: a large scattering cross section and a small absorption cross section. In thermal reactors the moderator typically doubles as the reflector (ordinary water, heavy water, or graphite), while fast reactors, which have no moderator, use elements of high mass number for reflection, such as steel, uranium, tungsten, or beryllium.
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.Serber, Robert, The Los Alamos Primer: The First Lectures on How to Build an Atomic Bomb, (University of California Press, 1992) Original 1943 "LA-1", declassified in 1965, plus commentary and historical introduction 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 Only a fraction of the neutrons produced by fission are absorbed inside the mass; the rest leak out through the surface. The multiplication factor of a finite system is the infinite-medium value times the nonleakage probability, the fraction of neutrons that are absorbed before escaping.
The nonleakage probability is set by the geometry: it approaches unity as the system is made larger, because a smaller system has more surface area per unit volume and a more strongly curved flux distribution (a larger geometric buckling). This is the mechanism behind the shape effect: for the same volume, a sphere has the smallest surface and therefore the least leakage.
Elastic scattering In elastic scattering a neutron bounces off a nucleus with the total kinetic energy conserved; the neutron is deflected and part of its energy goes into the nucleus. The fraction of energy lost in a single collision depends on the mass of the target nucleus: a neutron can lose all of its energy in a head-on collision with hydrogen, whose nucleus has nearly the same mass, while with carbon it can lose at most about 28% per collision. Each scattering event also lengthens the neutron's path through the material, raising the chance that it is absorbed before it can leak out. 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.Reevaluated Critical Specifications of Some Los Alamos Fast-Neutron Systems
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.