Critical mass

From Wikipedia, the free encyclopedia

A re-creation of the 1945 criticality accident using the Demon core: a plutonium pit is surrounded by blocks of neutron-reflective tungsten 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.

 

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, the nuclear fission cross-section), its density, its shape, its enrichment, its purity, its temperature, and its surroundings. The concept is important in nuclear weapon design.

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 population.

A numerical measure of a critical mass is dependent on the effective neutron multiplication factor k, 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 k increases. 

A supercritical mass is one in which, once fission has started, it 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 a supercritical mass will undergo a chain reaction. For example, a spherical critical mass of pure uranium-235 will have a mass of 52 kg and will 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 present, the rate of spontaneous fission will be much higher. 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). 

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 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. 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. 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), 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

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.

Nuclide	Half life (y)	Critical mass (kg)	Diameter (cm)
uranium-233	159,200	15	11
uranium-235	703,800,000	52	17
neptunium-236	154,000	7	8.7
neptunium-237	2,144,000	60	18
plutonium-238	87.7	9.04–10.07	9.5–9.9
plutonium-239	24,110	10	9.9
plutonium-240	6561	40	15
plutonium-241	14.3	12	10.5
plutonium-242	375,000	75–100	19–21
americium-241	432.2	55–77	20–23
americium-242m	141	9–14	11–13
americium-243	7370	180–280	30–35
curium-243	29.1	7.34–10	10–11
curium-244	18.1	13.5–30	12.4–16
curium-245	8500	9.41–12.3	11–12
curium-246	4760	39–70.1	18–21
curium-247	15,600,000	6.94–7.06	9.9
berkelium-247	1380	75.7	11.8-12.2
berkelium-249	330 days	192	16.1-16.6
californium-249	351	6	9
californium-251	900	5.46	8.5
californium-252	2.6	2.73	6.9
einsteinium-254	275.7 days	9.89	7.1

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 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 ${\displaystyle \ell ^{-1}=n\sigma }$ 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 ${\displaystyle \ell }$ 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: 

${\displaystyle R_{c}\simeq \ell {\sqrt {s}}\simeq {\frac {\sqrt {s}}{n\sigma }}}$

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 

${\displaystyle 1={\frac {f\sigma }{m{\sqrt {s}}}}\rho ^{2/3}M^{1/3}}$

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: 

${\displaystyle 1={\frac {f'\sigma }{m{\sqrt {s}}}}\Sigma }$

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 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 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.

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 always 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 man, or even 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, 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.

A convenient unit for the measurement of the reactivity is that suggested by Louis Slotin: that of the dollar and cents.

Nuclear chain reaction

From Wikipedia, the free encyclopedia

A possible nuclear fission chain reaction. 1. A uranium-235 atom absorbs a neutron, and fissions into two new atoms (fission fragments), releasing three new neutrons and a large amount of binding energy. 2. One of those neutrons is absorbed by an atom of uranium-238, and does not continue the reaction. Another neutron leaves the system without being absorbed. However, one neutron does collide with an atom of uranium-235, which then fissions and releases two neutrons and more binding energy. 3. Both of those neutrons collide with uranium-235 atoms, each of which fissions and releases a few neutrons, which can then continue the reaction.

 

A nuclear chain reaction occurs when one single nuclear reaction causes an average of one or more subsequent nuclear reactions, this leading to the possibility of a self-propagating series of these reactions. The specific nuclear reaction may be the fission of heavy isotopes (e.g., uranium-235, ²³⁵U). The nuclear chain reaction releases several million times more energy per reaction than any chemical reaction.

History

Chemical chain reactions were first proposed by German chemist Max Bodenstein in 1913, and were reasonably well understood before nuclear chain reactions were proposed. It was understood that chemical chain reactions were responsible for exponentially increasing rates in reactions, such as produced in chemical explosions. 

The concept of a nuclear chain reaction was reportedly first hypothesized by Hungarian scientist Leó Szilárd on September 12, 1933. Szilárd that morning had been reading in a London paper of an experiment in which protons from an accelerator had been used to split lithium-7 into alpha particles, and the fact that much greater amounts of energy were produced by the reaction than the proton supplied. Ernest Rutherford commented in the article that inefficiencies in the process precluded use of it for power generation. However, the neutron had been discovered in 1932, shortly before, as the product of a nuclear reaction. Szilárd, who had been trained as an engineer and physicist, put the two nuclear experimental results together in his mind and realized that if a nuclear reaction produced neutrons, which then caused further similar nuclear reactions, the process might be a self-perpetuating nuclear chain-reaction, spontaneously producing new isotopes and power without the need for protons or an accelerator. Szilárd, however, did not propose fission as the mechanism for his chain reaction, since the fission reaction was not yet discovered, or even suspected. Instead, Szilárd proposed using mixtures of lighter known isotopes which produced neutrons in copious amounts. He filed a patent for his idea of a simple nuclear reactor the following year.

In 1936, Szilárd attempted to create a chain reaction using beryllium and indium, but was unsuccessful. Nuclear fission was discovered and proved by Otto Hahn and Fritz Strassmann in December 1938. A few months later, Frédéric Joliot, H. Von Halban and L. Kowarski in Paris searched for, and discovered, neutron multiplication in uranium, proving that a nuclear chain reaction by this mechanism was indeed possible. 

On May 4, 1939 Joliot, Halban et Kowarski filed three patents. The first two described power production from a nuclear chain reaction, the last one called "Perfectionnement aux charges explosives" was the first patent for the atomic bomb and is filed as patent n°445686 by the Caisse nationale de Recherche Scientifique.

In parallel, Szilárd and Enrico Fermi in New York made the same analysis. This discovery prompted the letter from Szilárd and signed by Albert Einstein to President Franklin D. Roosevelt, warning of the possibility that Nazi Germany might be attempting to build an atomic bomb.

On December 2, 1942, a team led by Enrico Fermi (and including Szilárd) produced the first artificial self-sustaining nuclear chain reaction with the Chicago Pile-1 (CP-1) experimental reactor in a racquets court below the bleachers of Stagg Field at the University of Chicago. Fermi's experiments at the University of Chicago were part of Arthur H. Compton's Metallurgical Laboratory of the Manhattan Project; the lab was later renamed Argonne National Laboratory, and tasked with conducting research in harnessing fission for nuclear energy.

In 1956, Paul Kuroda of the University of Arkansas postulated that a natural fission reactor may have once existed. Since nuclear chain reactions may only require natural materials (such as water and uranium, if the uranium has sufficient amounts of U-235), it was possible to have these chain reactions occur in the distant past when uranium-235 concentrations were higher than today, and where there was the right combination of materials within the Earth's crust. Kuroda's prediction was verified with the discovery of evidence of natural self-sustaining nuclear chain reactions in the past at Oklo in Gabon, Africa, in September 1972.

Fission chain reaction

Fission chain reactions occur because of interactions between neutrons and fissile isotopes (such as ²³⁵U). The chain reaction requires both the release of neutrons from fissile isotopes undergoing nuclear fission and the subsequent absorption of some of these neutrons in fissile isotopes. When an atom undergoes nuclear fission, a few neutrons (the exact number depends on uncontrollable and unmeasurable factors; the expected number depends on several factors, usually between 2.5 and 3.0) are ejected from the reaction. These free neutrons will then interact with the surrounding medium, and if more fissile fuel is present, some may be absorbed and cause more fissions. Thus, the cycle repeats to give a reaction that is self-sustaining. 

Nuclear power plants operate by precisely controlling the rate at which nuclear reactions occur, and that control is maintained through the use of several redundant layers of safety measures. Moreover, the materials in a nuclear reactor core and the uranium enrichment level make a nuclear explosion impossible, even if all safety measures failed. On the other hand, nuclear weapons are specifically engineered to produce a reaction that is so fast and intense it cannot be controlled after it has started. When properly designed, this uncontrolled reaction can lead to an explosive energy release.

Nuclear fission fuel

Nuclear weapons employ high quality, highly enriched fuel exceeding the critical size and geometry (critical mass) necessary in order to obtain an explosive chain reaction. The fuel for energy purposes, such as in a nuclear fission reactor, is very different, usually consisting of a low-enriched oxide material (e.g. UO₂).

Fission reaction products

When a fissile atom undergoes nuclear fission, it breaks into two or more fission fragments. Also, several free neutrons, gamma rays, and neutrinos are emitted, and a large amount of energy is released. The sum of the rest masses of the fission fragments and ejected neutrons is less than the sum of the rest masses of the original atom and incident neutron (of course the fission fragments are not at rest). The mass difference is accounted for in the release of energy according to the equation E=Δmc²: 

 

mass of released energy = ${\displaystyle {\frac {E}{c^{2}}}=m_{\text{original}}-m_{\text{final}}}$

Due to the extremely large value of the speed of light, c, a small decrease in mass is associated with a tremendous release of active energy (for example, the kinetic energy of the fission fragments). This energy (in the form of radiation and heat) carries the missing mass, when it leaves the reaction system (total mass, like total energy, is always conserved). While typical chemical reactions release energies on the order of a few eVs (e.g. the binding energy of the electron to hydrogen is 13.6 eV), nuclear fission reactions typically release energies on the order of hundreds of millions of eVs.

Two typical fission reactions are shown below with average values of energy released and number of neutrons ejected:

${\displaystyle {\begin{aligned}{\ce {^{235}U + neutron ->}}&\ {\text{fission fragments}}+2.4{\text{ neutrons}}+192.9{\text{ MeV}}\\{\ce {^{239}Pu + neutron ->}}&\ {\text{fission fragments}}+2.9{\text{ neutrons}}+198.5{\text{ MeV}}\end{aligned}}}$

Note that these equations are for fissions caused by slow-moving (thermal) neutrons. The average energy released and number of neutrons ejected is a function of the incident neutron speed. Also, note that these equations exclude energy from neutrinos since these subatomic particles are extremely non-reactive and, therefore, rarely deposit their energy in the system.

Timescales of nuclear chain reactions

Prompt neutron lifetime

The prompt neutron lifetime, l, is the average time between the emission of neutrons and either their absorption in the system or their escape from the system. The neutrons that occur directly from fission are called "prompt neutrons," and the ones that are a result of radioactive decay of fission fragments are called "delayed neutrons". The term lifetime is used because the emission of a neutron is often considered its "birth," and the subsequent absorption is considered its "death". For thermal (slow-neutron) fission reactors, the typical prompt neutron lifetime is on the order of 10⁻⁴ seconds, and for fast fission reactors, the prompt neutron lifetime is on the order of 10⁻⁷ seconds. These extremely short lifetimes mean that in 1 second, 10,000 to 10,000,000 neutron lifetimes can pass. The average (also referred to as the adjoint unweighted) prompt neutron lifetime takes into account all prompt neutrons regardless of their importance in the reactor core; the effective prompt neutron lifetime (referred to as the adjoint weighted over space, energy, and angle) refers to a neutron with average importance.

Mean generation time

The mean generation time, Λ, is the average time from a neutron emission to a capture that results in fission. The mean generation time is different from the prompt neutron lifetime because the mean generation time only includes neutron absorptions that lead to fission reactions (not other absorption reactions). The two times are related by the following formula:

${\displaystyle \Lambda ={\frac {l}{k}}}$

In this formula, k is the effective neutron multiplication factor, described below.

Effective neutron multiplication factor

The six factor formula effective neutron multiplication factor, k, is the average number of neutrons from one fission that cause another fission. The remaining neutrons either are absorbed in non-fission reactions or leave the system without being absorbed. The value of k determines how a nuclear chain reaction proceeds:

• k < 1 (subcriticality): The system cannot sustain a chain reaction, and any beginning of a chain reaction dies out over time. For every fission that is induced in the system, an average total of 1/(1 − k) fissions occur.
• k = 1 (criticality): Every fission causes an average of one more fission, leading to a fission (and power) level that is constant. Nuclear power plants operate with k = 1 unless the power level is being increased or decreased.
• k > 1 (supercriticality): For every fission in the material, it is likely that there will be "k" fissions after the next mean generation time (Λ). The result is that the number of fission reactions increases exponentially, according to the equation ${\displaystyle e^{(k-1)t/\Lambda }}$, where t is the elapsed time. Nuclear weapons are designed to operate under this state. There are two subdivisions of supercriticality: prompt and delayed.

When describing kinetics and dynamics of nuclear reactors, and also in the practice of reactor operation, the concept of reactivity is used, which characterizes the deflection of reactor from the critical state. ρ=(k-1)/k. InHour is a unit of reactivity of a nuclear reactor. 

In a nuclear reactor, k will actually oscillate from slightly less than 1 to slightly more than 1, due primarily to thermal effects (as more power is produced, the fuel rods warm and thus expand, lowering their capture ratio, and thus driving k lower). This leaves the average value of k at exactly 1. Delayed neutrons play an important role in the timing of these oscillations. 

In an infinite medium, the multiplication factor may be described by the four factor formula; in a non-infinite medium, the multiplication factor may be described by the six factor formula.

Prompt and delayed supercriticality

Not all neutrons are emitted as a direct product of fission; some are instead due to the radioactive decay of some of the fission fragments. The neutrons that occur directly from fission are called "prompt neutrons," and the ones that are a result of radioactive decay of fission fragments are called "delayed neutrons". The fraction of neutrons that are delayed is called β, and this fraction is typically less than 1% of all the neutrons in the chain reaction.

The delayed neutrons allow a nuclear reactor to respond several orders of magnitude more slowly than just prompt neutrons would alone. Without delayed neutrons, changes in reaction rates in nuclear reactors would occur at speeds that are too fast for humans to control.

The region of supercriticality between k = 1 and k = 1/(1-β) is known as delayed supercriticality (or delayed criticality). It is in this region that all nuclear power reactors operate. The region of supercriticality for k > 1/(1-β) is known as prompt supercriticality (or prompt criticality), which is the region in which nuclear weapons operate.

The change in k needed to go from critical to prompt critical is defined as a dollar.

Nuclear weapons application of neutron multiplication

Nuclear fission weapons require a mass of fissile fuel that is prompt supercritical. 

For a given mass of fissile material the value of k can be increased by increasing the density. Since the probability per distance traveled for a neutron to collide with a nucleus is proportional to the material density, increasing the density of a fissile material can increase k. This concept is utilized in the implosion method for nuclear weapons. In these devices, the nuclear chain reaction begins after increasing the density of the fissile material with a conventional explosive. 

In the gun-type fission weapon two subcritical pieces of fuel are rapidly brought together. The value of k for a combination of two masses is always greater than that of its components. The magnitude of the difference depends on distance, as well as the physical orientation.

The value of k can also be increased by using a neutron reflector surrounding the fissile material.

Once the mass of fuel is prompt supercritical, the power increases exponentially. However, the exponential power increase cannot continue for long since k decreases when the amount of fission material that is left decreases (i.e. it is consumed by fissions). Also, the geometry and density are expected to change during detonation since the remaining fission material is torn apart from the explosion.

Predetonation

If two pieces of subcritical material are not brought together fast enough, nuclear predetonation can occur, whereby a smaller explosion than expected will blow the bulk of the material apart.

 

Detonation of a nuclear weapon involves bringing fissile material into its optimal supercritical state very rapidly. During part of this process, the assembly is supercritical, but not yet in an optimal state for a chain reaction. Free neutrons, in particular from spontaneous fissions, can cause the device to undergo a preliminary chain reaction that destroys the fissile material before it is ready to produce a large explosion, which is known as predetonation.

To keep the probability of predetonation low, the duration of the non-optimal assembly period is minimized and fissile and other materials are used which have low spontaneous fission rates. In fact, the combination of materials has to be such that it is unlikely that there is even a single spontaneous fission during the period of supercritical assembly. In particular, the gun method cannot be used with plutonium.

Nuclear power plants and control of chain reactions

Chain reactions naturally give rise to reaction rates that grow (or shrink) exponentially, whereas a nuclear power reactor needs to be able to hold the reaction rate reasonably constant. To maintain this control, the chain reaction criticality must have a slow enough time-scale to permit intervention by additional effects (e.g., mechanical control rods or thermal expansion). Consequently, all nuclear power reactors (even fast-neutron reactors) rely on delayed neutrons for their criticality. An operating nuclear power reactor fluctuates between being slightly subcritical and slightly delayed-supercritical, but must always remain below prompt-critical. 

It is impossible for a nuclear power plant to undergo a nuclear chain reaction that results in an explosion of power comparable with a nuclear weapon, but even low-powered explosions due to uncontrolled chain reactions, that would be considered "fizzles" in a bomb, may still cause considerable damage and meltdown in a reactor. For example, the Chernobyl disaster involved a runaway chain reaction but the result was a low-powered steam explosion from the relatively small release of heat, as compared with a bomb. However, the reactor complex was destroyed by the heat, as well as by ordinary burning of the graphite exposed to air. Such steam explosions would be typical of the very diffuse assembly of materials in a nuclear reactor, even under the worst conditions.

In addition, other steps can be taken for safety. For example, power plants licensed in the United States require a negative void coefficient of reactivity (this means that if water is removed from the reactor core, the nuclear reaction will tend to shut down, not increase). This eliminates the possibility of the type of accident that occurred at Chernobyl (which was due to a positive void coefficient). However, nuclear reactors are still capable of causing smaller explosions even after complete shutdown, such as was the case of the Fukushima Daiichi nuclear disaster. In such cases, residual decay heat from the core may cause high temperatures if there is loss of coolant flow, even a day after the chain reaction has been shut down (see SCRAM). This may cause a chemical reaction between water and fuel that produces hydrogen gas which can explode after mixing with air, with severe contamination consequences, since fuel rod material may still be exposed to the atmosphere from this process. However, such explosions do not happen during a chain reaction, but rather as a result of energy from radioactive beta decay, after the fission chain reaction has been stopped.

Emission spectrum

From Wikipedia, the free encyclopedia

Emission spectrum of a metal halide lamp.

 

A demonstration of the 589 nm D₂ (left) and 590 nm D₁ (right) emission sodium D lines using a wick with salt water in a flame

 

The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted due to an atom or molecule making a transition from a high energy state to a lower energy state. The photon energy of the emitted photon is equal to the energy difference between the two states. There are many possible electron transitions for each atom, and each transition has a specific energy difference. This collection of different transitions, leading to different radiated wavelengths, make up an emission spectrum. Each element's emission spectrum is unique. Therefore, spectroscopy can be used to identify the elements in matter of unknown composition. Similarly, the emission spectra of molecules can be used in chemical analysis of substances.

Emission

In physics, emission is the process by which a higher energy quantum mechanical state of a particle becomes converted to a lower one through the emission of a photon, resulting in the production of light. The frequency of light emitted is a function of the energy of the transition. Since energy must be conserved, the energy difference between the two states equals the energy carried off by the photon. The energy states of the transitions can lead to emissions over a very large range of frequencies. For example, visible light is emitted by the coupling of electronic states in atoms and molecules (then the phenomenon is called fluorescence or phosphorescence). On the other hand, nuclear shell transitions can emit high energy gamma rays, while nuclear spin transitions emit low energy radio waves. 

The emittance of an object quantifies how much light is emitted by it. This may be related to other properties of the object through the Stefan–Boltzmann law. For most substances, the amount of emission varies with the temperature and the spectroscopic composition of the object, leading to the appearance of color temperature and emission lines. Precise measurements at many wavelengths allow the identification of a substance via emission spectroscopy. 

Emission of radiation is typically described using semi-classical quantum mechanics: the particle's energy levels and spacings are determined from quantum mechanics, and light is treated as an oscillating electric field that can drive a transition if it is in resonance with the system's natural frequency. The quantum mechanics problem is treated using time-dependent perturbation theory and leads to the general result known as Fermi's golden rule. The description has been superseded by quantum electrodynamics, although the semi-classical version continues to be more useful in most practical computations.

Origins

When the electrons in the atom are excited, for example by being heated, the additional energy pushes the electrons to higher energy orbitals. When the electrons fall back down and leave the excited state, energy is re-emitted in the form of a photon. The wavelength (or equivalently, frequency) of the photon is determined by the difference in energy between the two states. These emitted photons form the element's spectrum. 

The fact that only certain colors appear in an element's atomic emission spectrum means that only certain frequencies of light are emitted. Each of these frequencies are related to energy by the formula:

${\displaystyle E_{\text{photon}}=h\nu }$,

where ${\displaystyle E_{\text{photon}}}$ is the energy of the photon, ${\displaystyle \nu }$ is its frequency, and ${\displaystyle h}$ is Planck's constant. This concludes that only photons with specific energies are emitted by the atom. The principle of the atomic emission spectrum explains the varied colors in neon signs, as well as chemical flame test results (described below). 

The frequencies of light that an atom can emit are dependent on states the electrons can be in. When excited, an electron moves to a higher energy level or orbital. When the electron falls back to its ground level the light is emitted. 

Emission spectrum of hydrogen

 

The above picture shows the visible light emission spectrum for hydrogen. If only a single atom of hydrogen were present, then only a single wavelength would be observed at a given instant. Several of the possible emissions are observed because the sample contains many hydrogen atoms that are in different initial energy states and reach different final energy states. These different combinations lead to simultaneous emissions at different wavelengths. 

Emission spectrum of iron

Radiation from molecules

As well as the electronic transitions discussed above, the energy of a molecule can also change via rotational, vibrational, and vibronic (combined vibrational and electronic) transitions. These energy transitions often lead to closely spaced groups of many different spectral lines, known as spectral bands. Unresolved band spectra may appear as a spectral continuum.

Emission spectroscopy

Light consists of electromagnetic radiation of different wavelengths. Therefore, when the elements or their compounds are heated either on a flame or by an electric arc they emit energy in the form of light. Analysis of this light, with the help of a spectroscope gives us a discontinuous spectrum. A spectroscope or a spectrometer is an instrument which is used for separating the components of light, which have different wavelengths. The spectrum appears in a series of lines called the line spectrum. This line spectrum is called an atomic spectrum when it originates from an atom in elemental form. Each element has a different atomic spectrum. The production of line spectra by the atoms of an element indicate that an atom can radiate only a certain amount of energy. This leads to the conclusion that bound electrons cannot have just any amount of energy but only a certain amount of energy. 

The emission spectrum can be used to determine the composition of a material, since it is different for each element of the periodic table. One example is astronomical spectroscopy: identifying the composition of stars by analysing the received light. The emission spectrum characteristics of some elements are plainly visible to the naked eye when these elements are heated. For example, when platinum wire is dipped into a strontium nitrate solution and then inserted into a flame, the strontium atoms emit a red color. Similarly, when copper is inserted into a flame, the flame becomes green. These definite characteristics allow elements to be identified by their atomic emission spectrum. Not all emitted lights are perceptible to the naked eye, as the spectrum also includes ultraviolet rays and infrared lighting. An emission is formed when an excited gas is viewed directly through a spectroscope. 

Schematic diagram of spontaneous emission

 

Emission spectroscopy is a spectroscopic technique which examines the wavelengths of photons emitted by atoms or molecules during their transition from an excited state to a lower energy state. Each element emits a characteristic set of discrete wavelengths according to its electronic structure, and by observing these wavelengths the elemental composition of the sample can be determined. Emission spectroscopy developed in the late 19th century and efforts in theoretical explanation of atomic emission spectra eventually led to quantum mechanics.

There are many ways in which atoms can be brought to an excited state. Interaction with electromagnetic radiation is used in fluorescence spectroscopy, protons or other heavier particles in Particle-Induced X-ray Emission and electrons or X-ray photons in Energy-dispersive X-ray spectroscopy or X-ray fluorescence. The simplest method is to heat the sample to a high temperature, after which the excitations are produced by collisions between the sample atoms. This method is used in flame emission spectroscopy, and it was also the method used by Anders Jonas Ångström when he discovered the phenomenon of discrete emission lines in the 1850s.

Although the emission lines are caused by a transition between quantized energy states and may at first look very sharp, they do have a finite width, i.e. they are composed of more than one wavelength of light. This spectral line broadening has many different causes.

Emission spectroscopy is often referred to as optical emission spectroscopy because of the light nature of what is being emitted.

History

Emission lines from hot gases were first discovered by Ångström, and the technique was further developed by David Alter, Gustav Kirchhoff and Robert Bunsen.

Experimental technique in flame emission spectroscopy

The solution containing the relevant substance to be analysed is drawn into the burner and dispersed into the flame as a fine spray. The solvent evaporates first, leaving finely divided solid particles which move to the hottest region of the flame where gaseous atoms and ions are produced. Here electrons are excited as described above. It is common for a monochromator to be used to allow for easy detection.

On a simple level, flame emission spectroscopy can be observed using just a flame and samples of metal salts. This method of qualitative analysis is called a flame test. For example, sodium salts placed in the flame will glow yellow from sodium ions, while strontium (used in road flares) ions color it red. Copper wire will create a blue colored flame, however in the presence of chloride gives green (molecular contribution by CuCl).

Emission coefficient

Emission coefficient is a coefficient in the power output per unit time of an electromagnetic source, a calculated value in physics. The emission coefficient of a gas varies with the wavelength of the light. It has units of ms⁻³sr⁻¹. It is also used as a measure of environmental emissions (by mass) per MWh of electricity generated.

Scattering of light

In Thomson scattering a charged particle emits radiation under incident light. The particle may be an ordinary atomic electron, so emission coefficients have practical applications.

If X dV dΩ dλ is the energy scattered by a volume element dV into solid angle dΩ between wavelengths λ and λ+dλ per unit time then the Emission coefficient is X. 

The values of X in Thomson scattering can be predicted from incident flux, the density of the charged particles and their Thomson differential cross section (area/solid angle).

Spontaneous emission

A warm body emitting photons has a monochromatic emission coefficient relating to its temperature and total power radiation. This is sometimes called the second Einstein coefficient, and can be deduced from quantum mechanical theory.