A re-creation of the 1945 criticality accident using the Demon core: a plutonium pit is surrounded by blocks of neutron-reflectivetungsten carbide. The original experiment was designed to measure the radiation produced when an extra block was added. The mass went supercritical when the block was placed improperly by being dropped.
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 population.
A numerical measure of a critical mass depends on the effective 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.
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, called subcriticality, k < 1.
A critical mass is a mass of fissile material that self-sustains a fission chain reaction. In this case, called criticality, k = 1. 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.[1] In this case, called supercriticality, k > 1. The constant of proportionality increases as k 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 52 kilograms (115lb) would experience around 15spontaneous 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. If there is uranium-238 (238U) present, the rate of spontaneous fission will be much higher.[citation needed] Fission can also be initiated by neutrons produced by cosmic rays.
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: 1K 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 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.
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 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 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.[2] 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
Top: A sphere of fissile material is too small to allow the chain reaction to become self-sustaining as neutrons generated by fissions can too easily escape.
Middle: By increasing the mass of the sphere to a critical mass, the reaction can become self-sustaining.
Bottom: Surrounding the original sphere with a neutron reflector increases the efficiency of the reactions and also allows the reaction to become self-sustaining.
The shape with minimal critical mass and the smallest physical dimensions 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.[3]
The critical mass for lower-grade uranium depends strongly on the grade: with 20% 235U it is over 400kg; with 15% 235U, it is well over 600kg.
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 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 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:
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
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 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 320kg/m2, regardless of density, and for 235U at 550kg/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.
Criticality in nuclear weapon design
If two pieces of subcritical material are not brought together fast enough, nuclear predetonation (fizzle) can occur, whereby a very small explosion will blow the bulk of the material apart.
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 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.
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,[14] and prompt critical, 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.
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 radioactive decay.
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 or "positive feedback loop" of these reactions. The specific nuclear reaction may be the fission of heavy isotopes. A nuclear chain reaction releases several million times more energy per reaction than any chemical reaction.
A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat from nuclear fission is passed to a working fluid, which in turn runs through steam turbines. These either drive a ship's propellers or turn electrical generators' shafts. Nuclear generated steam in principle can be used for industrial process heat or for district heating. Some reactors are used to produce isotopes for medical and industrial use, or for production of weapons-grade plutonium. As of 2022, the International Atomic Energy Agency reports there are 422 nuclear power reactors and 223 nuclear research reactors in operation around the world.
In nuclear engineering, fissile material is material that can undergo nuclear fission when struck by a neutron of low energy. A self-sustaining thermal chain reaction can only be achieved with fissile material. The predominant neutron energy in a system may be typified by either slow neutrons or fast neutrons. Fissile material can be used to fuel thermal-neutron reactors, fast-neutron reactors and nuclear explosives.
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:
In nuclear engineering, a neutron moderator is a medium that reduces the speed of fast neutrons, ideally without capturing any, leaving them as thermal neutrons with only minimal (thermal) kinetic energy. These thermal neutrons are immensely more susceptible than fast neutrons to propagate a nuclear chain reaction of uranium-235 or other fissile isotope by colliding with their atomic nucleus.
The nuclear fuel cycle, also called nuclear fuel chain, is the progression of nuclear fuel through a series of differing stages. It consists of steps in the front end, which are the preparation of the fuel, steps in the service period in which the fuel is used during reactor operation, and steps in the back end, which are necessary to safely manage, contain, and either reprocess or dispose of spent nuclear fuel. If spent fuel is not reprocessed, the fuel cycle is referred to as an open fuel cycle ; if the spent fuel is reprocessed, it is referred to as a closed fuel cycle.
Uranium-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 exists in nature as a primordial nuclide.
A neutron reflector is any material that reflects neutrons. This refers to elastic scattering rather than to a specular reflection. The material may be graphite, beryllium, steel, tungsten carbide, gold, or other materials. A neutron reflector can make an otherwise subcritical mass of fissile material critical, or increase the amount of nuclear fission that a critical or supercritical mass will undergo. Such an effect was exhibited twice in accidents involving the Demon Core, a subcritical plutonium pit that went critical in two separate fatal incidents when the pit's surface was momentarily surrounded by too much neutron reflective material.
A criticality accident is an accidental uncontrolled nuclear fission chain reaction. It is sometimes referred to as a critical excursion, critical power excursion, divergent chain reaction, or simply critical. Any such event involves the unintended accumulation or arrangement of a critical mass of fissile material, for example enriched uranium or plutonium. Criticality accidents can release potentially fatal radiation doses if they occur in an unprotected environment.
A subcritical reactor is a nuclear fission reactor concept that produces fission without achieving criticality. Instead of sustaining a chain reaction, a subcritical reactor uses additional neutrons from an outside source. There are two general classes of such devices. One uses neutrons provided by a nuclear fusion machine, a concept known as a fusion–fission hybrid. The other uses neutrons created through spallation of heavy nuclei by charged particles such as protons accelerated by a particle accelerator, a concept known as an accelerator-driven system (ADS) or accelerator-driven sub-critical reactor.
In nuclear engineering, a prompt neutron is a neutron immediately emitted 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 fission products anytime from a few milliseconds to a few minutes later.
A boosted fission weapon usually refers to a type of nuclear bomb that uses a small amount of fusion fuel to increase the rate, and thus yield, of a fission reaction. The neutrons released by the fusion reactions add to the neutrons released due to fission, allowing for more neutron-induced fission reactions to take place. The rate of fission is thereby greatly increased such that much more of the fissile material is able to undergo fission before the core explosively disassembles. The fusion process itself adds only a small amount of energy to the process, perhaps 1%.
Plutonium-239 is an isotope of plutonium. Plutonium-239 is the primary fissile isotope used for the production of nuclear weapons, although uranium-235 is also used for that purpose. Plutonium-239 is also one of the three main isotopes demonstrated usable as fuel in thermal spectrum nuclear reactors, along with uranium-235 and uranium-233. Plutonium-239 has a half-life of 24,110 years.
Gun-type fission weapons are fission-based nuclear weapons whose design assembles their fissile material into a supercritical mass by the use of the "gun" method: shooting one piece of sub-critical material into another. Although this is sometimes pictured as two sub-critical hemispheres driven together to make a supercritical sphere, typically a hollow projectile is shot onto a spike, which fills the hole in its center. Its name is a reference to the fact that it is shooting the material through an artillery barrel as if it were a projectile.
Nuclear reactor physics is the field of physics that studies and deals with the applied study and engineering applications of chain reaction to induce a controlled rate of fission in a nuclear reactor for the production of energy. Most nuclear reactors use a chain reaction to induce a controlled rate of nuclear fission in fissile material, releasing both energy and free neutrons. A reactor consists of an assembly of nuclear fuel, usually surrounded by a neutron moderator such as regular water, heavy water, graphite, or zirconium hydride, and fitted with mechanisms such as control rods which control the rate of the reaction.
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.
In nuclear power technology, burnup is a measure of how much energy is extracted from a primary nuclear fuel source. It is measured as the fraction of fuel atoms that underwent fission in %FIMA or %FIFA as well as, preferably, the actual energy released per mass of initial fuel in gigawatt-days/metric ton of heavy metal (GWd/tHM), or similar units.
Hybrid nuclear fusion–fission is a proposed means of generating power by use of a combination of nuclear fusion and fission processes.
A pressurized heavy-water reactor (PHWR) is a nuclear reactor that uses heavy water (deuterium oxide D2O) as its coolant and neutron moderator. PHWRs frequently use natural uranium as fuel, but sometimes also use very low enriched uranium. The heavy water coolant is kept under pressure to avoid boiling, allowing it to reach higher temperature (mostly) without forming steam bubbles, exactly as for a pressurized water reactor. While heavy water is very expensive to isolate from ordinary water (often referred to as light water in contrast to heavy water), its low absorption of neutrons greatly increases the neutron economy of the reactor, avoiding the need for enriched fuel. The high cost of the heavy water is offset by the lowered cost of using natural uranium and/or alternative fuel cycles. As of the beginning of 2001, 31 PHWRs were in operation, having a total capacity of 16.5 GW(e), representing roughly 7.76% by number and 4.7% by generating capacity of all current operating reactors.
References
↑ Hewitt, Paul G. (2015). Conceptual Physics (12thed.). 300 Beach Drive NE, 1103, St. Petersburg: Pearson. p.666. ISBN978-1-292-05713-2.{{cite book}}: CS1 maint: location (link)
↑ Serber, Robert, The Los Alamos Primer: The First Lectures on How to Build an Atomic Bomb, (University of California Press, 1992) ISBN0-520-07576-5 Original 1943 "LA-1", declassified in 1965, plus commentary and historical introduction
1 2 3 Dias, Hemanth; Tancock, Nigel; Clayton, Angela (2003). "Critical Mass Calculations for 241Am, 242mAm and 243Am"(PDF). Challenges in the Pursuit of Global Nuclear Criticality Safety. Proceedings of the Seventh International Conference on Nuclear Criticality Safety. Vol.II. Tokai, Ibaraki, Japan: Japan Atomic Energy Research Institute. pp.618–623.
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