Lawson criterion
The Lawson criterion is a figure of merit used in nuclear fusion research.
The concept was first developed by John D. Lawson in a classified 1955 paper[1] that was declassified and published in 1957.
[2] 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"
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" may refer to this value.
On August 8, 2021, researchers at Lawrence Livermore National Laboratory's National Ignition Facility in California confirmed to have produced the first-ever successful ignition of a nuclear fusion reaction surpassing the Lawson's criteria in the experiment.
[3][4] The central concept of the Lawson criterion is an examination of 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 calculated the fusion rate by assuming that the fusion reactor contains a hot plasma cloud which has a Gaussian curve of individual particle energies, a Maxwell–Boltzmann distribution characterized by the plasma's temperature.
[5] 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.
The result is the amount of energy being created by the plasma at any instant in time.
By equating radiation losses and the volumetric fusion rates, Lawson estimated the minimum temperature for the fusion for the deuterium–tritium (D-T) reaction to be 30 million degrees (2.6 keV), and for the deuterium–deuterium (D-D) reaction to be 150 million degrees (12.9 keV).
(energy content per unit volume) divided by the power loss density
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.
, the energy of the charged fusion products (the neutrons cannot help to heat the plasma).
Replacing the function with its minimum value provides an absolute lower limit for the product
A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, nTτE.
The average <σv> in this temperature region can be approximated as[7] 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.
Empirically, the energy confinement time τE is found to be nearly proportional to n1/3/P 2/3[citation needed].
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 in the inertial case it is more usefully expressed in a different form.
is the time that it takes an ion to travel over a distance R at its thermal speed where mi denotes mean ionic mass.
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 =
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.
The fusion power times 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: Thus the optimum temperature for inertial confinement fusion maximises <σv>/T3/2, which is slightly higher than the optimum temperature for magnetic confinement.
Lawson's analysis is based on the rate of fusion and loss of energy in a thermalized plasma.
There is a class of fusion machines that do not use thermalized plasmas but instead directly accelerate individual ions to the required energies.
When applied to the fusor, Lawson's analysis is used as an argument that conduction and radiation losses are the key impediments to reaching net power.
Fusors use a voltage drop to accelerate and collide ions, resulting in fusion.
, it is easy to show that the fusion power is maximized by a fuel mix given by
