In nuclear fusion research, the Lawson criterion, first derived on fusion reactors (initially classified) by John D. Lawson in 1955 and published in 1957,^{[1]} 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 n_{e} and the "energy confinement time" .
Later analysis 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^{[citation needed]} refers to this inequality.
The central concept of the Lawson criterion is the energy balance for any fusion power plant, using a hot plasma. This is shown below:
Net Power = Efficiency × (Fusion − Radiation Loss − Conduction Loss)
Lawson calculates the fusion rate by assuming that any fusion reactor contains a hot plasma cloud which has a Gaussian curve of energy. Based on that assumption, he estimates the first term, the fusion energy coming from a hot cloud using the volumetric fusion equation.^{[2]}
Fusion = Number Density of Fuel A × Number Density of Fuel B × Cross Section(Temperature) × Energy Per Reaction
This equation is typically averaged over a population of ions which has a normal distribution. For his analysis, Lawson ignores conduction losses. In reality this is nearly impossible, practically all systems lose energy through mass leaving. Lawson then estimated^{[2]} the radiation losses using the equation below.
where N is the number density of the cloud and T is the temperature.
By equating radiation losses and the volumetric fusion rates Lawson estimates the minimum temperature for the fusion for the deuterium–tritium reaction
to be 30 million degrees (2.6 keV) and for the deuterium–deuterium reaction
to be 150 million degrees (12.9 keV).^{[1]}^{[3]}
When applied to the fusor Lawson's analysis is used as an argument that conduction and radiation losses are the key impediment to reaching net power. Fusors use a voltage drop to accelerate and collide ions, resulting in fusion.^{[4]} The voltage drop is generated by wire cages, and these cages conduct away particles. Polywell are improvements on this design, designed to reduce conduction losses by removing the wire cages which cause them.^{[5]} Regardless, it is argued that radiation is still a major impediment.^{[6]}
The confinement time measures the rate at which a system loses energy to its environment. It is the energy density (energy content per unit volume) divided by the power loss density (rate of energy loss per unit volume):
For a fusion reactor to operate in steady state, 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. The plasma loses energy through mass (conduction loss) or light (radiation loss) leaving the chamber.
For illustration, the Lawson criterion for the deuterium–tritium 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 deuterium and tritium are present in the optimal 5050 mixture.^{[7]} Ion density then equals electron density and the energy density of both electrons and ions together is given by
where is the Boltzmann constant and is the particle density.
The volume rate (reactions per volume per time) of fusion reactions is
where is the fusion cross section, is the relative velocity, and denotes an average over the Maxwellian velocity distribution at the temperature .
The volume rate of heating by fusion is times , the energy of the charged fusion products (the neutrons cannot help to heat the plasma). In the case of the deuterium–tritium reaction, .
The Lawson criterion requires that fusion heating exceeds the losses:
Substituting in known quantities yields:
Rearranging the equation produces:


(1) 
The quantity is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product . This is the Lawson criterion.
For the deuterium–tritium reaction, the physical value is at least
The minimum of the product occurs near .
A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, nTτ_{E}. 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 attainable pressure p is a constant. When such is the case, the fusion power density is proportional to p^{2}<σv>/T^{ 2}. The maximum fusion power available from a given machine is therefore reached at the temperature T where <σv>/T^{ 2} is a maximum. By continuation of the above derivation, the following inequality is readily obtained:
The quantity is also a function of temperature with an absolute minimum at a slightly lower temperature than .
For the deuterium–tritium reaction, the minimum of the triple product occurs at T = 14 keV. The average <σv> in this temperature region can be approximated as^{[8]}
so the minimum value of the triple product value at T = 14 keV is about
This number has not yet been achieved in any reactor, although the latest generations of machines have come close. JT60 reported 1.53x10^{21} keV.s.m^{−3}.^{[9]} 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.
As for tokamaks there is a special motivation for using the triple product. Empirically, the energy confinement time τ_{E} is found to be nearly proportional to n^{1/3}/P^{ 2/3}. In an ignited plasma near the optimum temperature, the heating power P equals fusion power and therefore is proportional to n^{2}T^{ 2}. The triple product scales as
The triple product is only weakly dependant on temperature as T^{ 1/3}. This makes the triple product an adequate measure of the efficiency of the confinement scheme.
The Lawson criterion applies to inertial confinement fusion (ICF) as well as to magnetic confinement fusion (MCF) but is more usefully expressed in a different form. A good approximation for the inertial confinement time is the time that it takes an ion to travel over a distance R at its thermal speed
where m_{i} denotes mean ionic mass. The inertial confinement time can thus be approximated as
By substitution of the above expression into relationship (1), we obtain
This product must be greater than a value related to the minimum of T^{ 3/2}/<σv>. The same requirement is traditionally expressed in terms of mass density ρ = <nm_{i}>:
Satisfaction of this criterion at the density of solid deuterium–tritium (0.2 g/cm³) would require a laser pulse of implausibly large energy. Assuming the energy required scales with the mass of the fusion plasma (E_{laser} ~ ρR^{3} ~ ρ^{−2}), compressing the fuel to 10^{3} or 10^{4} times solid density would reduce the energy required by a factor of 10^{6} or 10^{8}, bringing it into a realistic range. With a compression by 10^{3}, 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.
The fusion power density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burnup of the fuel is probably more useful. The burnup should be proportional to the specific reaction rate (n^{2}<σv>) times the confinement time (which scales as T^{ 1/2}) divided by the particle density n:
Thus the optimum temperature for inertial confinement fusion maximises <σv>/T^{3/2}, which is slightly higher than the optimum temperature for magnetic confinement.
Mathematical derivation: http://wwwfusionmagnetique.cea.fr/gb/fusion/physique/demo_ntt.htm
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