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nuclear engineering Nuclear engineering is the branch of engineering concerned with the application of breaking down atomic nuclei ( fission) or of combining atomic nuclei ( fusion), or with the application of other sub-atomic processes based on the principles of n ...
, a critical mass is the smallest amount of
fissile In nuclear engineering, fissile material is material capable of sustaining a nuclear fission chain reaction. By definition, fissile material can sustain a chain reaction with neutrons of thermal energy. The predominant neutron energy may be t ...
material needed for a sustained
nuclear chain reaction In nuclear physics, a nuclear chain reaction occurs when one single nuclear reaction causes an average of one or more subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The specific nu ...
. The critical mass of a fissionable material depends upon its
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space *Nuclear ...
properties (specifically, its
nuclear fission Nuclear fission is a reaction in which the nucleus of an atom splits into two or more smaller nuclei. The fission process often produces gamma photons, and releases a very large amount of energy even by the energetic standards of radio ...
cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...
), density, shape,
enrichment Enrichment may refer to: * Behavioral enrichment, the practice of providing animals under managed care with stimuli such as natural and artificial objects * Data enrichment, appending or enhancing data with relevant context from other sources, se ...
, purity, temperature, and surroundings. The concept is important in
nuclear weapon design Nuclear weapon designs are physical, chemical, and engineering arrangements that cause the physics package of a nuclear weapon to detonate. There are three existing basic design types: * pure fission weapons, the simplest and least technically ...
.


Explanation of criticality

When a nuclear chain reaction in a mass of fissile material is self-sustaining, the mass is said to be in a ''critical'' state in which there is no increase or decrease in power, temperature, or
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
population. A numerical measure of a critical mass is dependent on the
effective neutron multiplication factor In nuclear physics, a nuclear chain reaction occurs when one single nuclear reaction causes an average of one or more subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The specific nu ...
, 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. When ''k'' = 1, the mass is critical, and the chain reaction is self-sustaining. A ''subcritical'' mass is a mass of fissile material 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, ''k'' < 1. A steady rate of spontaneous fissions causes a proportionally steady level of neutron activity. The constant of proportionality increases as increases. A ''supercritical'' mass is one which, once fission has started, will proceed at an increasing rate. The material may settle into equilibrium (''i.e.'' become critical again) at an elevated temperature/power level or destroy itself. In the case of supercriticality, ''k'' > 1. Due to
spontaneous fission Spontaneous fission (SF) is a form of radioactive decay that is found only in very heavy chemical elements. The nuclear binding energy of the elements reaches its maximum at an atomic mass number of about 56 (e.g., iron-56); spontaneous breakd ...
a supercritical mass will undergo a chain reaction. For example, a spherical critical mass of pure
uranium-235 Uranium-235 (235U or U-235) is an isotope of uranium making up about 0.72% of natural uranium. Unlike the predominant isotope uranium-238, it is fissile, i.e., it can sustain a nuclear chain reaction. It is the only fissile isotope that exi ...
(235U) with a mass of about would experience around 15 spontaneous fission events per second. The probability that one such event will cause a chain reaction depends on how much the mass exceeds the critical mass. If there is
uranium-238 Uranium-238 (238U or U-238) is the most common isotope of uranium found in nature, with a relative abundance of 99%. Unlike uranium-235, it is non-fissile, which means it cannot sustain a chain reaction in a thermal-neutron reactor. However ...
(238U) present, the rate of spontaneous fission will be much higher. Fission can also be initiated by neutrons produced by
cosmic rays Cosmic rays are high-energy particles or clusters of particles (primarily represented by protons or atomic nuclei) that move through space at nearly the speed of light. They originate from the Sun, from outside of the Solar System in our ...
.


Changing the point of criticality

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. These examples only outline the simplest ideal cases:


Varying the amount of fuel

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). Similarly, 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.


Changing the 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.


Changing the 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 In atomic physics, Doppler broadening is broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting (or absorbing) particles result in different Do ...
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 The light-water reactor (LWR) is a type of thermal-neutron reactor that uses normal water, as opposed to heavy water, as both its coolant and neutron moderator; furthermore a solid form of fissile elements is used as fuel. Thermal-neutron reacto ...
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.


Varying the density of the mass

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 Uranium-235 (235U or U-235) is an isotope of uranium making up about 0.72% of natural uranium. Unlike the predominant isotope uranium-238, it is fissile, i.e., it can sustain a nuclear chain reaction. It is the only fissile isotope that exi ...
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.


Use of a neutron reflector

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 Beryllium is a chemical element with the symbol Be and atomic number 4. It is a steel-gray, strong, lightweight and brittle alkaline earth metal. It is a divalent element that occurs naturally only in combination with other elements to for ...
metal. This reduces the number of neutrons which escape the fissile material, resulting in increased reactivity.


Use 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.


Critical size

The critical size is the minimum size of a nuclear reactor core or nuclear weapon that can be made for a specific geometrical arrangement and material composition. The critical size must at least include enough fissionable material to reach critical mass. 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.


Critical mass of a bare sphere

The shape with minimal critical mass and the smallest physical dimensions is a sphere. Bare-sphere critical masses at normal density of some
actinide The actinide () or actinoid () series encompasses the 15 metallic chemical elements with atomic numbers from 89 to 103, actinium through lawrencium. The actinide series derives its name from the first element in the series, actinium. The info ...
s 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 20% 235U it is over 400 kg; with 15% 235U, 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. 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 neutron In nuclear engineering, a prompt neutron is a neutron immediately emitted ( neutron emission) by a nuclear fission event, as opposed to a delayed neutron decay which can occur within the same context, emitted after beta decay of one of the fis ...
s, 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 A barn is an agricultural building usually on farms and used for various purposes. In North America, a barn refers to structures that house livestock, including cattle and horses, as well as equipment and fodder, and often grain.Allen G ...
), the
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
of a prompt neutron is \ell^ = n \sigma where ''n'' is the nuclear number density. Most interactions are scattering events, so that a given neutron obeys a
random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
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 \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 \simeq \ell \sqrt \simeq \frac 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 1 = \frac \rho^ M^ 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: 1 = \frac \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 In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to ''transmitted'' radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power throug ...
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 Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...
'' in computational physics by Nicholas Metropolis and
Stanislaw Ulam Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...
. 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.


Criticality in nuclear weapon design

Until detonation is desired, a
nuclear weapon A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions ( thermonuclear bomb), producing a nuclear explosion. Both bomb ...
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 In nuclear engineering, 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 (specifically, its nuclear fissi ...
either because they are too small or unfavorably shaped. To produce detonation, the pieces of uranium are brought together rapidly. In
Little Boy "Little Boy" was the type of atomic bomb dropped on the Japanese city of Hiroshima on 6 August 1945 during World War II, making it the first nuclear weapon used in warfare. The bomb was dropped by the Boeing B-29 Superfortress ''Enola Gay'' p ...
, this was achieved by firing a piece of uranium (a 'doughnut') down a
gun barrel A gun barrel is a crucial part of gun-type weapons such as small firearms, artillery pieces, and air guns. It is the straight shooting tube, usually made of rigid high-strength metal, through which a contained rapid expansion of high-pres ...
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 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 In nuclear physics, a nuclear chain reaction occurs when one single nuclear reaction causes an average of one or more subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The specific nu ...
) 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 A shaped charge is an explosive charge shaped to form an explosively formed penetrator (EFP) to focus the effect of the explosive's energy. Different types of shaped charges are used for various purposes such as cutting and forming metal, ini ...
surrounding the sphere, increasing the density (and collapsing the cavity, if present) to produce a
prompt critical In nuclear engineering, prompt criticality describes a nuclear fission event in which criticality (the threshold for an exponentially growing nuclear fission chain reaction) is achieved with prompt neutrons alone (neutrons that are released immed ...
configuration. This is known as an '' implosion type weapon''.


Prompt criticality

The event of fission must release, on the average, more than one free neutron of the desired energy level in order to sustain a chain reaction, and each must find other nuclei and cause them to fission. Most of the neutrons released from a fission event come immediately from that event, but a fraction of them come later, when the fission products decay, which may be on the average from microseconds to minutes later. This is fortunate for atomic power generation, for without this delay "going critical" would be an immediately catastrophic event, as it is in a nuclear bomb where upwards of 80 generations of chain reaction occur in less than a microsecond, far too fast for a human, or even a machine, to react. Physicists recognize two points in the gradual increase of neutron flux which are significant: critical, where the chain reaction becomes self-sustaining thanks to the contributions of both kinds of neutron generation,In the description of the Soviet equivalent of the CP1 startup at the University of Chicago in 1942, the long waits for those tardy neutrons is described in detail and
prompt critical In nuclear engineering, prompt criticality describes a nuclear fission event in which criticality (the threshold for an exponentially growing nuclear fission chain reaction) is achieved with prompt neutrons alone (neutrons that are released immed ...
, where the immediate "prompt" neutrons alone will sustain the reaction without need for the decay neutrons. Nuclear power plants operate between these two points of reactivity, while above the prompt critical point is the domain of nuclear weapons and some nuclear power accidents, such as the
Chernobyl disaster The Chernobyl disaster was a nuclear accident that occurred on 26 April 1986 at the No. 4 reactor in the Chernobyl Nuclear Power Plant, near the city of Pripyat in the north of the Ukrainian SSR in the Soviet Union. It is one of only two n ...
. A convenient unit for the measurement of the reactivity is that suggested by Louis Slotin: that of the
dollar Dollar is the name of more than 20 currencies. They include the Australian dollar, Brunei dollar, Canadian dollar, Hong Kong dollar, Jamaican dollar, Liberian dollar, Namibian dollar, New Taiwan dollar, New Zealand dollar, Singapore dollar, ...
and cents.


See also

*
Criticality (status) In the operation of a nuclear reactor, criticality is the state in which a nuclear chain reaction is self-sustaining—that is, when reactivity is zero. In supercritical states, reactivity is greater than zero. Applications Criticality is the no ...
* Criticality accident *
Nuclear criticality safety Nuclear criticality safety is a field of nuclear engineering dedicated to the prevention of nuclear and radiation accidents resulting from an inadvertent, self-sustaining nuclear chain reaction. Nuclear criticality safety is concerned with mitig ...
* Geometric and material buckling


References

{{DEFAULTSORT:Critical Mass Mass Nuclear technology Radioactivity Nuclear weapon design Nuclear fission