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Innuclear engineering,critical mass is the minimummass of thefissile material needed for a sustainednuclear chain reaction in a particular setup. The critical mass of a fissionable material depends upon itsnuclear properties (specifically, itsnuclear fissioncross-section), density, shape,enrichment, purity, temperature, and surroundings. It is an important parameter of anuclear reactor core ornuclear weapon. The concept is important innuclear 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 fissionneutrons escape through its surface and the chain reaction is not sustained. Aperfect sphere, which has the lowestsurface-area-to-volume ratio, gives the minimal critical size.
When a nuclear chain reaction in a mass of fissile material is self-sustaining but not growing, the mass is said to be in acritical state, in which there is no increase or decrease inpower, temperature, orneutron population.
A numerical measure of a critical mass depends on theeffective neutron multiplication factork, 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.
Asubcritical 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 assubcriticality,k < 1.
Acritical mass is a mass of fissile material that self-sustains a fission chain reaction. In this case, known ascriticality,k = 1. A steady rate ofspontaneous fission causes a proportionally steady level of neutron activity.
Asupercritical mass is a mass which, once fission has started, will proceed at an increasing rate.[1] In this case, known assupercriticality,k > 1. The constant of proportionality increases ask increases. The material may settle into equilibrium (i.e. become critical again) at an elevated temperature/power level or destroy itself.
Due tospontaneous fission a supercritical mass will undergo a chain reaction. For example, a spherical critical mass of pureuranium-235 (235U) with a mass of about 52 kilograms (115 lb) would experience around 15 spontaneous fission events per second.[citation needed] 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 bycosmic 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).
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.[2] 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.[3]
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.[4] 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.[5]
Elastic scattering directly relates to escape probability because the bouncing of the particles drastically drops the probability of escaping the mass without collision.[6] 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.[3]
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.[7]
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.
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.
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 toDoppler broadening of the238U 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 inLWR 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.
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 (seeallotropes 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 (enricheduranium-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.
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.[8]
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.[9] 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.[10]
Surrounding a spherical critical mass with aneutron 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.[11]
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.[12] Graphite is used because of its diverse abilities, such as being a moderator and a reflector.[13] 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.[11]
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.[11]
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 atamper. 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.[14] 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-calledboosted configuration.
The shape with minimal critical mass is a sphere. Bare-sphere critical masses at normal density of someactinides 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.[15]
| Nuclide | Half-life (y) | Critical mass (kg) | Diameter (cm) | Ref |
|---|---|---|---|---|
| uranium-233 | 159,200 | 15 | 11 | [16] |
| uranium-235 | 704,000,000 | 52 | 17 | [16] |
| neptunium-236 | 153,000 | 7 | 8.7 | [17] |
| neptunium-237 | 2,144,000 | 60 | 18 | [18][19] |
| plutonium-238 | 87.7 | 9.04–10.07 | 9.5–9.9 | [20] |
| plutonium-239 | 24,110 | 10 | 9.9 | [16][20] |
| plutonium-240 | 6561 | 40 | 15 | [16] |
| plutonium-241 | 14.33 | 12 | 10.5 | [21] |
| plutonium-242 | 375,000 | 75–100 | 19–21 | [21] |
| americium-241 | 432.6 | 55–77 | 20–23 | [22] |
| americium-242m | 141 | 9–14 | 11–13 | [22] |
| americium-243 | 7350 | 180–280 | 30–35 | [22] |
| curium-243 | 29.1 | 7.34–10 | 10–11 | [23] |
| curium-244 | 18.11 | 13.5–30 | 12.4–16 | [23] |
| curium-245 | 8250 | 9.41–12.3 | 11–12 | [23] |
| curium-246 | 4700 | 39–70.1 | 18–21 | [23] |
| curium-247 | 15,600,000 | 6.94–7.06 | 9.9 | [23] |
| berkelium-247 | 1380 | 75.7 | 11.8-12.2 | [24] |
| berkelium-249 | 0.896 | 192 | 16.1-16.6 | [24] |
| californium-249 | 351 | 6 | 9 | [17] |
| californium-251 | 900 | 5.46 | 8.5 | [17] |
| californium-252 | 2.645 | 2.73 | 6.9 | [25] |
| einsteinium-254 | 0.755 | 9.89 | 7.1 | [24] |
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.[26]
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 undergoradiative capture.
Letq denote the probability that a given neutron induces fission in a nucleus. Consider onlyprompt neutrons, and letν denote the number of prompt neutrons generated in a nuclear fission. For example,ν ≈ 2.5 for uranium-235. Then, criticality occurs whenν·q = 1. The dependence of this upon geometry, mass, and density appears through the factorq.
Given a total interactioncross section σ (typically measured inbarns), themean free path of a prompt neutron is wheren is the nuclear number density. Most interactions are scattering events, so that a given neutron obeys arandom 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 thiss), 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 massM, the nuclear massm, the density ρ, and a fudge factorf 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 factorf has been rewritten asf' 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 of239Pu criticality is at 320 kg/m2, regardless of density, and for235U 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 lengthL on a side, filled with soot, then theoptical depth of this medium is inversely proportional to the square ofL, 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 theMonte Carlo method in computational physics byNicholas Metropolis andStanislaw 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.
The critical mass can be greatly reduced by the use of atamper. The following table lists the values for commonneutron reflectors, where λtamp is the mean free path of a neutron within the tamper material before anelastic scattering. In nuclear weapon design,natural uranium is primarily used. Tungsten carbide was initially used in theLittle Boy.
| Minimum critical mass (kg)[27] | |||
|---|---|---|---|
| Tamper | λtamp (cm) | U-235 | Pu-239 |
| Aluminium | 5.595 | 22.0 | 9.7 |
| Beryllium oxide | 2.549 | 8.6 | 3.7 |
| Lead | 5.426 | 21.3 | 9.3 |
| Tungsten carbide | 3.159 | 11.3 | 4.9 |
| Depleted uranium/Uranium-238 | 4.342 | 16.5 | 7.2 |
Until detonation is desired, anuclear 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. InLittle Boy, this was achieved by firing a piece of uranium (a 'doughnut') down agun barrel onto another piece (a 'spike'). This design is referred to as agun-type fission weapon.
A theoretical 100% pure239Pu weapon could also be constructed as a gun-type weapon, like the Manhattan Project's proposedThin Man design. In reality, this is impractical because 100% pure239Pu cannot be achieved. Even "weapons grade"239Pu is contaminated with a small amount of240Pu, 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 ashaped charge surrounding the sphere, increasing the density (and collapsing the cavity, if present) to produce aprompt critical configuration. This is known as animplosion type weapon.
| Assembly name | Location | Year | Fuel | Reactivity timescale | Neutron spectrum | Moderator |
|---|---|---|---|---|---|---|
| Chicago Pile-1 | Metallurgical Laboratory | 1942 | Natural uranium | Delayed | Thermal | Graphite |
| LOPO/Water Boiler | Los Alamos Laboratory | 1944 | Low-enriched uranium | Delayed | Thermal | Light water |
| Prompt | ||||||
| Dragon | Los Alamos Laboratory | 1945 | Highly enriched uranium | Prompt | Fast | None |
| Clementine | Los Alamos Laboratory | 1946 | Weapons-grade plutonium | Delayed | Fast | None |
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