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| {{About|nuclear fission reactions}}
| | Most people genuinely believe that the first thing you will need is a site, In regards to earning money online. You can definitely generate profits in a large amount ways when you have your own website, but you don't require this to be able to start getting some cash.<br><br>~ Article Writing may bring in a fantastic number of prospects! It's much like blogging as you write records that interest other's. Inturn they'll visit your other websites attached to your Posts.<br><br>Currently what you need are back links, back links are merely a link on the website other than yours, that convey the clicker to your website. The web link should be your key-phrase.<br><br><br><br>A website is very important for network marketers trying to control the internet for their business success. Just how can a blog support my business you ask? Effectively, it serves as a central link in which all the of the internet actions can be connected. If you are an integral part of social network websites it's really a great way to ship the brand new people you relate genuinely to for more information about you and about what content you're expressing. This can be a good way to company YOU.<br><br>A successful [http://www.linkedin.com/pub/jordan-kurland/a/618/581 Jordan Kurland] approach is forum participation. Article answers to questions folks are asking. It is possible to set up a reputation on your own as someone obtaining information and is reliable. By expressing suggestions and views in a website marketing forum you are able to brand yourself being an expert.<br><br>There are many affiliate programs out there such as for example Ebay and Amazon but we'll just speak about Clickbank here. It's absolve to join and it's an excellent stock of electronic products. These are goods that customers get immediate pleasure from simply because they camera immediately get their purchase and start using it.<br><br>Literal Bears I'm Jealous Of is named to be the most effective enjoyable website of all time. This site was started as Bears I Am Jealous Of which was fully dedicated to hairy males. The internet site is later changed into Literal Bears I'm Jealous Of with the aim to remember all good bears in literature.<br><br>Irrespective of where you choose is the greatest place for you, make sure to talk with everybody. Don't keep in touch with them. As mentioned above, they'll be-gone in a heartbeat if they dropped theyAre being sold to. Don't you should be that company. Become A true person. Do not be afraid of showcasing some of your best work, many profitable weddings or content customers - especially with videos. Reply to questions and comments really and communicate in similar communities. When you become a trusted individual locally you're in, you'll be surprised what social-media can-do on your wedding organization. |
| {{Merge from|Critical size|discuss=Talk:Critical size#Merger proposal|date=January 2014}}
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| {{Refimprove|date=May 2012}}
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| [[File:Partially-reflected-plutonium-sphere.jpeg|thumb|right|350px|As part of a re-creation of a 1945 [[criticality accident]], a [[plutonium pit]] is surrounded by blocks of [[neutron reflector|neutron-reflective]] [[tungsten carbide]]. The original experiment was designed to measure the radiation produced when an extra block was added. Instead, the mass went supercritical when the block was placed improperly by being dropped.]]
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| 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 [[atomic nucleus|nuclear]] properties (specifically, the [[nuclear fission]] [[cross section (physics)|cross-section]]), its [[density]], its [[shape]], its [[Isotope separation|enrichment]], its purity, its [[temperature]], and its surroundings. The concept is important in [[nuclear weapon design]].
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| == Explanation of criticality==
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| 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.
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| A numerical measure of a critical mass is dependent on the [[Nuclear chain reaction#Effective neutron multiplication factor|effective neutron multiplication factor]] {{math|<var>k</var>}}, 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 <math>k = 1</math>, the mass is critical, and the chain reaction is barely self-sustaining.
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| 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, <math>k < 1</math>. A steady rate of spontaneous fissions causes a proportionally steady level of neutron activity. The constant of proportionality increases as {{math|<var>k</var>}} increases.
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| A ''supercritical'' mass is one where there is an increasing rate of fission. The material may settle into equilibrium (''i.e.'' become critical again) at an elevated temperature/power level or destroy itself, by which equilibrium is reached. In the case of supercriticality, <math>k > 1</math>. | |
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| ==Changing the point of criticality==
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| 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. This section explains only the simplest ideal cases.
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| *'''Varying the amount of fuel'''
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| 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 one neutron generation (fuel consumption makes the assembly subcritical).
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| 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 [[Kelvin|K]] above the ambient temperature) and then decrease back to room temperature after a period of time, because fuel consumed during fission brings the assembly back to subcriticality once again.
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| *'''Changing the shape'''
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| 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.
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| *'''Changing the temperature'''
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| 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 U238 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.
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| *'''Varying the density of the mass'''
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| 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.
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| *'''Use of a neutron reflector'''
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| Surrounding a spherical critical mass with a [[Neutron reflector#Neutron reflector|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.
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| *'''Use of a tamper'''
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| In a bomb, a dense shell of material surrounding the fissile core will contain, via inertia, the expanding fissioning material. This increases the efficiency. 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), because 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.<ref>Serber, Robert, ''The Los Alamos Primer: The First Lectures on How to Build an Atomic Bomb'', (University of California Press, 1992) ISBN 0-520-07576-5 Original 1943 "LA-1", declassified in 1965, plus commentary and historical introduction</ref> 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.
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| ==Critical mass of a bare sphere==
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| [[File:Critical mass.svg|right|251px|thumb|''Top:'' A [[sphere]] of fissile material is too small to allow the [[chain reaction]] to become self-sustaining as [[neutron]]s generated by [[nuclear fission|fission]]s can too easily escape.<br/><br/> ''Middle:'' By increasing the mass of the sphere to a critical mass, the reaction can become self-sustaining.<br/><br/> ''Bottom:'' Surrounding the original sphere with a [[Neutron reflector#Neutron reflector|neutron reflector]] increases the efficiency of the reactions and also allows the reaction to become self-sustaining.]]
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| The shape with minimal critical mass and the smallest physical dimensions is a sphere. Bare-sphere critical masses at normal density of some [[actinide]]s are listed in the following table.
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| {{Critical mass}}
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| The critical mass for lower-grade uranium depends strongly on the grade: with 20% U-235 it is over 400 kg; with 15% U-235, it is well over 600 kg.
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| 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.
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| Note that not all neutrons contribute to the chain reaction. Some escape and others undergo [[Neutron capture|radiative capture]].
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| Let ''q'' denote the probability that a given neutron induces fission in a nucleus. Let us consider only [[prompt neutron]]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''.
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| Given a total interaction [[cross section (physics)|cross section]] σ (typically measured in [[Barn (unit)|barns]]), the [[mean free path]] of a prompt neutron is <math> \ell^{-1} = n \sigma </math> 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 <math> \ell </math> 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:
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| <math>
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| R_c \simeq \ell \sqrt{s} \simeq \frac{\sqrt{s}}{n \sigma}
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| </math>
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| Note again, however, that this is only a rough estimate.
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| 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
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| <math>
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| 1 = \frac{f \sigma}{m \sqrt{s}} \rho^{2/3} M^{1/3}
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| </math>
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| which clearly recovers the aforementioned result that critical mass depends inversely on the square of the density.
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| Alternatively, one may restate this more succinctly in terms of the areal density of mass, Σ:
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| <math>
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| 1 = \frac{f' \sigma}{m \sqrt{s}} \Sigma
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| </math>
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| where the factor ''f'' has been rewritten as <math> f' </math> 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 Pu-239 criticality is at 320 kg/m<sup>2</sup>, regardless of density, and for U-235 at 550 kg/m<sup>2</sup>.
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| 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.
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| 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. | |
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| 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 <math> L </math> on a side, filled with soot, then the [[optical depth]] of this medium is inversely proportional to the square of <math> L </math>, 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.
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| 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. 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.
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| 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.
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| ==Criticality in nuclear weapon design==
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| [[File:Nuclear predetonation.svg|right|thumb|250px|If two pieces of subcritical material are not brought together fast enough, nuclear predetonation ([[Fizzle (nuclear test)|fizzle]]) can occur, whereby a very small explosion will blow the bulk of the material apart.]]
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| Until detonation is desired, a [[nuclear weapon]] must be kept subcritical. In the case of a uranium 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 uranium is 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'), a design referred to as a ''[[gun-type fission weapon]]''.
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| A theoretical 100% pure Pu-239 weapon could also be constructed as a gun-type weapon, like the Manhattan Project's proposed [[Thin Man (nuclear bomb)|Thin Man]] design. 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 suffer nuclear reaction before the masses of plutonium would be in a position for a full-fledged explosion to occur.
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| 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 ''[[Nuclear weapon design#Implosion-type weapon|implosion type weapon]]''.
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| ==See also==
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| * [[Nuclear chain reaction]]
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| * [[Nuclear weapon design]]
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| * [[Criticality accident]]
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| * [[Nuclear criticality safety]]
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| * [[Geometric and Material Buckling]]
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| ==References==
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| {{reflist|2}}
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| {{DEFAULTSORT:Critical Mass}}
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| [[Category:Mass]]
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| [[Category:Nuclear technology]]
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| [[Category:Radioactivity]]
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| [[Category:Nuclear weapon design]]
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