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Encyclopedia > Lawson criterion

In nuclear fusion research, the Lawson criterion, first derived[1] by John D. Lawson in 1955 and published[2] in 1957, is an important general measure of a system that defines the conditions needed for a fusion reactor to reach ignition, that is, that the heating of the plasma by the products of the fusion reactions is sufficient to maintain the temperature of the plasma against all losses without external power input. As originally formulated the Lawson criterion gives a minimum required value for the product of the plasma (electron) density ne and the "energy confinement time" τE. Later analyses suggested that a more useful figure of merit is the "triple product" of density, confinement time, and plasma temperature T. The triple product also has a minimum required value, and the name "Lawson criterion" often refers to this inequality. The deuterium-tritium (D-T) fusion reaction is considered the most promising for producing fusion power. ... 1955 (MCMLV) was a common year starting on Saturday of the Gregorian calendar. ...

Contents

The product neτE

The confinement time τE measures the rate at which a system loses energy to its environment. It is the energy content W divided by the power loss Ploss (rate of energy loss):


tau_E = frac{W}{P_{rm loss}}


For a fusion reactor to operate in steady state, as magnetic fusion energy schemes usually entail, the fusion plasma must be maintained at a constant temperature. Thermal energy must therefore be added to it (either directly by the fusion products or by recirculating some of the electricity generated by the reactor) at the same rate the plasma loses energy (for instance by heat conduction to the device walls or radiation losses like bremsstrahlung). (help· info), (from the German bremsen, to brake and Strahlung, radiation, thus, braking radiation), is electromagnetic radiation produced by the acceleration of a charged particle, such as an electron, when deflected by another charged particle, such as an atomic nucleus. ...


For illustration, the Lawson criterion for the D-T reaction will be derived here, but the same principle can be applied to other fusion fuels. It will also be assumed that all species have the same temperature, that there are no ions present other than fuel ions (no impurities and no helium ash), and that D and T are present in the optimal 50-50 mixture.[3] In that case, the ion density is equal to the electron density and the energy density of both together is given by


W = 3nekBT


where kB is the Boltzmann constant. Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...


The volume rate f (reactions per volume per time) of fusion reactions is


f = n_D n_T langlesigma vrangle = frac{1}{4}n_e^2 langlesigma vrangle


where σ is the fusion cross section, v is the relative velocity, and < > denotes an average over the Maxwellian velocity distribution at the temperature T. To meet Wikipedias quality standards, this article or section may require cleanup. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...


The volume rate of heating by fusion is f times Ech, the energy of the charged fusion products (the neutrons cannot help to keep the plasma hot). In the case of the D-T reaction, Ech = 3.5 MeV.

The deuterium-tritium L function (minimum neτE needed to satisfy the Lawson criterion) minimizes near the temperature 25 keV (300 million kelvins).
The deuterium-tritium L function (minimum neτE needed to satisfy the Lawson criterion) minimizes near the temperature 25 keV (300 million kelvins).

The Lawson criterion is the requirement that the fusion heating exceed the losses: Image File history File links DT_ntauE.svg Summary Logarithmic plot of the Lawson criterion for deuterium-tritium fusion, or the minimum product of electron number density times energy confinement time needed to maintain the fusion plasma at a constant temperature. ... Image File history File links DT_ntauE.svg Summary Logarithmic plot of the Lawson criterion for deuterium-tritium fusion, or the minimum product of electron number density times energy confinement time needed to maintain the fusion plasma at a constant temperature. ... Deuterium, also called heavy hydrogen, is a stable isotope of hydrogen with a natural abundance in the oceans of one atom in 6400 of hydrogen (see VSMOW; the abundance changes slightly from one kind of natural water to another). ... Tritium (symbol T or 3H) is a radioactive isotope of hydrogen. ... The electronvolt (symbol eV, or, rarely and incorrectly, ev) is a unit of energy. ...


f E_{ch} ge P_{loss}


leftrightarrow


frac{1}{4}n_e^2 langlesigma vrangle E_{ch} ge frac{3n_ek_BT}{tau_E}


leftrightarrow


n_{rm e} tau_{rm E} ge L equiv frac{12}{E_{rm ch}},frac{k_{rm B}T}{langlesigma vrangle}


The quantity frac{T}{langlesigma vrangle} is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product neτe. This is the Lawson criterion.


For the D-T reaction, the physical value is at least


n_{rm e} tau_{rm E} ge 1.5times10^{20} {rm s}/mbox{m}^3


The minimum of the product occurs near T = 25 keV. The electronvolt (symbol eV, or, rarely and incorrectly, ev) is a unit of energy. ...


The "triple product" neTτE

A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, neTτE. For most confinement concepts, whether inertial, mirror, or toroidal confinement, the density and temperature can be varied over a fairly wide range, but the maximum pressure attainable is a constant. When that is the case, the fusion power density is proportional to p^2langlesigma vrangle/T^2. Therefore the maximum fusion power available from a given machine is obtained at the temperature where langlesigma vrangle/T^2 is a maximum. Following the derivation above, it is easy to show the inequality Inertial confinement fusion using lasers rapidly progressed in the late 1970s and early 1980s from being able to deliver only a few joules of laser energy (per pulse) to a fusion target to being able to deliver tens of kilojoules to a target. ... A magnetic mirror is a magnetic field configuration where the field strength changes when moving along a field line. ...


n_{rm e} T tau_{rm E} ge frac{12k_{rm B}}{E_{rm ch}},frac{T^2}{langlesigma vrangle}


For the special case of tokamaks there is an additional motivation for using the triple product. Empirically, the energy confinement time is found to be nearly proportional to n1/3/P2/3. In an ignited plasma near the optimum temperature, the heating power P is equal to the fusion power and therefore proportional to n2/T2. The triple product scales as A split image of the largest tokamak in the world, the JET, showing hot plasma in the right image during a shot. ...

nTτ propto nT (n1/3/P2/3) propto nT (n1/3/(n2/T2)2/3) propto T -1/3

Thus the triple product is only a weak function of density and temperature and therefore a good measure of the efficiency of the confinement scheme.


The quantity frac{T^2}{langlesigma vrangle} is also a function of temperature with an absolute minimum at a slightly higher temperature than frac{T}{langlesigma vrangle}.


For the D-T reaction, the physical value is about

n_{rm e} T tau_E ge 10^{21} mbox{keV s}/mbox{m}^3

This number has not yet been achieved in any reactor, although the latest generations of machines have come close. For instance, the TFTR has achieved the densities and energy lifetimes needed to achieve Lawson at the temperatures it can create, but it cannot create those temperatures at the same time. ITER aims to do both. The Tokamak Fusion Test Reactor (TFTR) was an experimental fusion test reactor built at Princeton Plasma Physics Laboratory (in Princeton, New Jersey) circa 1980. ... Cutaway of the ITER Tokamak Torus in casing. ...


Inertial confinement

The Lawson criterion applies to inertial confinement fusion as well as to magnetic confinement fusion but is more usefully expressed in a different form. Whereas the energy confinement time in a magnetic system is very difficult to predict or even to establish empirically, in an inertial system it must be on the order of the time it takes sound waves to travel across the plasma: Inertial confinement fusion using lasers rapidly progressed in the late 1970s and early 1980s from being able to deliver only a few joules of laser energy (per pulse) to a fusion target to being able to deliver tens of kilojoules to a target. ... Magnetic confinement fusion is an approach to fusion energy that uses magnetic fields to confine the fusion fuel in the form of a plasma. ...

tau_E approx R/sqrt{k_BT/m_i}

Following the above derivation of the limit on neτE, we see that the product of the density and the radius must be greater than a value related to the minimum of T3/2/<σv>. This condition is traditionally expressed in terms of the mass density ρ:

ρR > 1 g/cm²

To satisfy this criterion at the density of solid D-T (0.2 g/cm³) would require an implausibly large laser pulse energy. Assuming the energy required scales with the mass of the fusion plasma (Elaser ~ ρR3 ~ ρ-2), compressing the fuel to 103 or 104 times solid density would reduce the energy required by a factor of 106 or 108, bringing it into a realistic range. With a compression by 103, the compressed density will be 200 g/cm³, and the compressed radius can be as small as 0.05 mm. The radius of the fuel before compression would be 0.5 mm. The initial pellet will be perhaps twice as large since most of the mass will be ablated during the compression. Ablation is defined as the removal of material from the surface of an object by vaporization, chipping, or other erosive processes. ...


The fusion power density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burn-up of the fuel is probably more useful. The burn-up should be proportional to the specific reaction rate (n²<σv>) times the confinement time (which scales as T-1/2) divided by the particle density n:

burn-up fraction ~ n²<σv> T-1/2 / n ~ (nT) (<σv>/T3/2)

Thus the optimum temperature for inertial confinement fusion is that which maximizes <σv>/T3/2, which is slightly higher than the optimum temperature for magnetic confinement.


External links

  • 50 years of the Lawson criteria
  • The mathematical derivation is reproduced here.

See also

The fusion energy gain factor, usually expressed with the symbol Q, is the ratio of fusion power produced in a nuclear fusion reactor to the power required to maintain the plasma in steady state. ...

Notes

  1. ^ J.D. Lawson's original report on fusion reactors (initially classified)
  2. ^ J. D. Lawson, Proceedings of the Physical Society B, Vol. 70, p. 6 (1957), available here.
  3. ^ It is straightforward to relax these assumptions. The most difficult question is how to define n when the ion and electrons differ in density and temperature. Considering that this is a calculation of energy production and loss by ions, and that any plasma confinement concept must contain the pressure forces of the plasma, it seems appropriate to define the effective (electron) density n through the (total) pressure p as n = p/2Ti. The factor of 2 is included because n usually refers to the density of the electrons alone, but p here refers to the total pressure. Given two species with ion densities n1,2, atomic numbers Z1,2, ion temperature Ti, and electron temperature Te, it is easy to show that the fusion power is maximized by a fuel mix given by n1/n2 = (1+Z2Te/Ti)/(1+Z1Te/Ti). The values for nτ, nTτ, and the power density must be multiplied by the factor (1+Z1Te/Ti)×(1+Z2Te/Ti)/4. For example, with protons and boron (Z=5) as fuel, another factor of 3 must be included in the formulas. On the other hand, for cold electrons, the formulas must all be divided by 4 (with no additional factor for Z>1).


  Fusion power  v  d  e 
Atomic nucleus | Nuclear fusion | Nuclear power | Nuclear reactor | Timeline of nuclear fusion
Plasma physics | Magnetohydrodynamics | Neutron flux | Fusion energy gain factor | Lawson criterion
Methods of fusing nuclei

Magnetic confinement: Tokamak - Spheromak - Stellarator - Reversed field pinch - Field-Reversed Configuration - Levitated Dipole
Inertial confinement: Laser driven - Z-pinch - Bubble fusion - Farnsworth–Hirsch Fusor
Other forms of fusion: Muon-catalyzed fusion - Pyroelectric fusion - Cold fusion The Sun is a natural fusion reactor. ... A semi-accurate depiction of the helium atom. ... The deuterium-tritium (D-T) fusion reaction is considered the most promising for producing fusion power. ... A nuclear power station. ... Core of a small nuclear reactor used for research. ... Timeline of significant events in the study and use of nuclear fusion: 1929 - Atkinson and Houtermans used the measured masses of light elements and applied Einsteins discovery that E=mc² to predict that large amounts of energy could be released by fusing small nuclei together. ... A Plasma lamp In physics and chemistry, a plasma is an ionized gas, and is usually considered to be a distinct phase of matter. ... Magnetohydrodynamics (MHD) (magnetofluiddynamics or hydromagnetics), is the academic discipline which studies the dynamics of electrically conducting fluids. ... neutron flux n : the rate of flow of neutrons; the number of neutrons passing through a unit area in unit time via dictionary. ... The fusion energy gain factor, usually expressed with the symbol Q, is the ratio of fusion power produced in a nuclear fusion reactor to the power required to maintain the plasma in steady state. ... The deuterium-tritium (D-T) fusion reaction is considered the most promising for producing fusion power. ... Magnetic Fusion Energy (MFE) is a sustained nuclear fusion reaction in a plasma that is contained by magnetic fields. ... A split image of the largest tokamak in the world, the JET, showing hot plasma in the right image during a shot. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Stellarator magnetic field and magnets A stellarator is a device used to confine a hot plasma with magnetic fields in order to sustain a controlled nuclear fusion reaction. ... Reversed-Field Pinch is a toroidal magnetic confinement scheme. ... A Field-Reversed Configuration (FRC) is a device developed for magnetic fusion energy research that confines a plasma on closed magnetic field lines without a central penetration. ... A Levitated Dipole is a unique form of fusion reactor technology using a solid superconducting torus, magnetically levitated in the reactor chamber. ... Inertial confinement fusion using lasers rapidly progressed in the late 1970s and early 1980s from being able to deliver only a few joules of laser energy (per pulse) to a fusion target to being able to deliver tens of kilojoules to a target. ... In inertial confinement fusion (ICF), nuclear fusion reactions are initiated by heating and compressing a target – a pellet that most often contains deuterium and tritium – by the use of intense laser or ion beams. ... The Z machine at Sandia National Laboratories in Albuquerque, New Mexico. ... Bubble fusion or sonofusion is the common name for a nuclear fusion reaction hypothesized to occur during sonoluminescence, an extreme form of acoustic cavitation; officially, this reaction is termed acoustic inertial confinement fusion (AICF) since the inertia of the collapsing bubble wall confines the energy causing a rise in temperature. ... U.S. Patent 3,386,883 - fusor — June 4, 1968 The Farnsworth–Hirsch Fusor, or simply fusor, is an apparatus designed by Philo T. Farnsworth to create nuclear fusion. ... Muon-catalyzed fusion is a process allowing nuclear fusion to take place at room temperature. ... Pyroelectric fusion is a technique for achieving nuclear fusion by using an electric field generated by pyroelectric crystals to accelerate ions of deuterium (tritium might also be used someday) into a metal hydride target also containing detuerium (or tritium) with sufficient kinetic energy to cause these ions to fuse together. ... Charles Bennett examines three cold fusion tests cells at the Oak Ridge National Laboratory, USA Cold fusion cell at the US Navy Space and Naval Warfare Systems Center, San Diego, CA (2005) Cold fusion is a theoretical fusion reaction that occurs near room temperature and pressure using relatively simple devices. ...

List of fusion experiments

Magnetic confinement devices
ITER (International) | JET (European) | JT-60 (Japan) | Large Helical Device (Japan) | KSTAR (Korea) | EAST (China) | T-15 (Russia) | DIII-D (USA) | ASDEX Upgrade (Germany) | TFTR (USA) | NSTX (USA) | NCSX (USA) | Alcator C-Mod (USA) | LDX (USA) | H-1NF (Australia) | MAST (UK) | START (UK) | TCV (Switzerland) | DEMO (Commercial) Experiments directed toward developing fusion power are invariably done with dedicated machines which can be classified according to the principles they use to confine the plasma fuel and keep it hot. ... Cutaway of the ITER Tokamak Torus in casing. ... Split image of JET with right side showing hot plasma during a shot. ... JT-60 (JT stands for Japan Torus) is the flagship of Japans magnetic fusion program, run by the Japan Atomic Energy Research Institute (JAERI), and the Naka Fusion Research Establishment in Ibaraki Prefecture, Japan. ... Categories: Stub | Nuclear technology ... The KSTAR, or Korean Superconducting Tokamak Advanced Reactor is a magnetic fusion device being built at the Korea Basic Science Institute in Daejon, South Korea. ... The Experimental Advanced Superconducting Tokamak (EAST, internally called HT-7U) is a project being undertaken to construct an experimental superconducting tokamak magnetic fusion energy reactor in Hefei, the capital city of Anhui Province, in eastern China. ... The T-15 is a Russian nuclear fusion research reactor, based on the (Russian-invented) tokamak design. ... DIII-D or D3-D is the name of a tokamak machine developed in the 1980s by General Atomics in San Diego, USA, as part of the ongoing effort to achieve magnetically confined fusion. ... The ASDEX Upgrade divertor tokamak (Axially Symmetric Divertor EXperiment) went into operation at the Max-Planck-Institut für Plasmaphysik, Garching in 1991. ... The Tokamak Fusion Test Reactor (TFTR) was an experimental fusion test reactor built at Princeton Plasma Physics Laboratory (in Princeton, New Jersey) circa 1980. ... The National Spherical Torus Experiment (NSTX) is an innovative magnetic fusion device that was constructed by the Princeton Plasma Physics Laboratory (PPPL) in collaboration with the Oak Ridge National Laboratory, Columbia University, and the University of Washington at Seattle. ... The National Compact Stellarator Experiment (NCSX) is a plasma confinement experiment being conducted at the Princeton Plasma Physics Laboratory. ... Alcator C-Mod is a tokamak, a magnetically confined nuclear fusion device, at the MIT Plasma Science and Fusion Center. ... The Levitated Dipole Experiment (LDX) is a project devoted to researching a type of nuclear fusion which utilizes a floating superconducting torus to provide an axisymmetric magnetic field which is used to contain plasma. ... The H-1 flexible Heliac is a three field-period helical axis stellarator located in the Research School of Physical Sciences and Engineering at the Australian National University. ... The Mega Ampere Spherical Tokamak, or MAST experiment is a nuclear fusion experiment in operation at Culham since December 1999. ... The Small Tight Aspect Ratio Tokamak, or START was a nuclear fusion experiment that used magnetic confinement to hold plasma. ... Tokamak à Configuration Variable (TCV): inner view, with the graphite-claded torus. ... The word demo may refer to one of the following. ...


Inertial confinement devices
Laser driven:
ICF lasers in the United States: NIF (USA) | OMEGA laser (USA) | Nova laser (USA) | Novette laser (USA) | Nike laser (USA) | Shiva laser (USA) | Argus laser (USA) | Cyclops laser (USA) | Janus laser (USA) | Long path laser (USA) | 4 pi laser (USA)
ICF lasers in other nations: LMJ (France) | GEKKO XII (Japan) | ISKRA lasers (Russia) | Vulcan laser (UK) | Asterix IV laser (Czech Republic) | HiPER laser (European)
Non-laser driven:
Z machine (USA) | PACER (USA)
A construction worker inside NIFs 10 meter target chamber. ... The Laboratory for Laser Energetics (LLE) is a scientific research facility which is part of the University of Rochesters south campus, located in Rochester, New York. ... The Nova laser was a laser built at the Lawrence Livermore National Laboratory in 1984 and which conducted advanced inertial confinement fusion experiments until its dismantling in 1999. ... The Novette target chamber with two laser chains visible in background. ... Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 Kj by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. ... The Shiva laser was an extremely powerful 20 beam infrared neodymium glass (silica glass) laser built at Lawrence Livermore National Laboratory in 1977 for the study of inertial confinement fusion and long-scale-length laser-plasma interactions. ... Argus laser overhead view. ... The single beam Cyclops laser at LLNL around the time of its completion in 1975. ... The Janus laser as it appeared in 1975. ... The Long Path laser was an early high energy infrared laser at the Lawrence Livermore National Laboratory used to study inertial confinement fusion. ... Physicist Frank Rainer (inset), who was involved in laser research and development at LLNL since 1966, holds the target chamber seen at the center of the larger picture. ... Laser Mégajoule (LMJ) is an experimental inertial confinement fusion (ICF) device being built in France by the French nuclear science directorate, CEA. Laser Mégajoule plans to deliver about 1. ... GEKKO XII is a high-power 12-beam neodymium doped glass laser at the Osaka Universitys Institute for Laser Engineering completed in 1983, which is used for high energy density physics and inertial confinement fusion research. ... The ISKRA-4 and ISKRA-5 lasers are lasers which were built by the Russian federation at RFNC-VNIIEF in Arzamas-16() with the ~2Kj output ISKRA-4 laser being completed in 1979 and the ~30Kj output ISKRA-5 laser which was completed in 1989. ... The Vulcan laser is an 8 beam 2. ... HiPER is an experimental laser-driven inertial confinement fusion (ICF) device currently undergoing preliminary design for possible construction in the European Union starting around 2010. ... Zork universe Zork games Zork Anthology Zork trilogy Zork I   Zork II   Zork III Beyond Zork   Zork Zero   Planetfall Enchanter trilogy Enchanter   Sorcerer   Spellbreaker Other games Wishbringer   Return to Zork Zork: Nemesis   Zork Grand Inquisitor Zork: The Undiscovered Underground Topics in Zork Encyclopedia Frobozzica Characters   Kings   Creatures Timeline   Magic   Calendar... The PACER project, carried out at Los Alamos National Laboratory in the mid-1970s, explored the possibility of a fusion power system that would involve exploding small hydrogen bombs (fusion bombs)—or, as stated in a later proposal, fission bombs—inside an underground cavity. ...


See also: International Fusion Materials Irradiation Facility The International Fusion Material Irradiation Facility, also known as IFMIF, is an international scientific research program designed to test materials for suitability for use in a fusion reactor. ...


  Results from FactBites:
 
Fusion: Introduction and Beyond (771 words)
lso known as the Lawson number or condition, this slightly more advanced concept is a quantitative measure that fusion scientists use to measure their progress towards achieving fusion at a practical level.
When this number is large enough, the fusion reactions release the same amount of energy that was used to start the reactions, also known as breakeven.
is the Lawson criterion for the deuterium-tritium reaction at around 100 million K. What this also tells us is that the plasma can either consist of a lot of particles for a short period of time (inertial confinement) or few particles confined for a long period of time (magnetic confinement).
  More results at FactBites »

 
 

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