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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 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). (helpÂ· 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 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.

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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