# Lawson criterion

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 ne and the "energy confinement time" $\tau_E$. 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 refers to this inequality.

## Derivation

Lawson begins by assuming that any fusion reactor contains a hot plasma cloud which has a Gaussian curve of energy. The energy released from fusion in a hot cloud is given by the volumetric fusion equation [2]

$P_E = N_1 \cdot N_2 \cdot v \cdot \sigma(T) \cdot E \, \mathrm{,}$

where PE is the energy from the hot cloud, N1 and N2 are the number densities of the light atoms being fused, $\sigma(T)$ is the nuclear cross section of the fusion reaction at that temperature, v is the average velocity of the light atoms when they collide and E is the energy output per fusion reaction.

Lawson argues next that all hot plasma clouds lose energy through two mechanisms: light radiation and conduction losses.[1] These would be on top of normal energy capture losses (see power plant efficiency). Light is generated in plasma every time a particle accelerates or decelerates (see Larmor formula). This occurs for various reasons, such as electrostatic interactions (Bremsstrahlung) or cyclotron radiation. When electrons in a plasma are above a critical density, this light can be reabsorbed.[3]

For the analysis, Lawson ignores conduction losses and uses a simple expression [2] to estimate light radiation from a hot cloud.

$P_B = 1.4 \cdot 10^{-34} \cdot N^2 \cdot T^{1/2} \frac{\mathrm{Watts}}{\mathrm{cm}^3}$

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 deuteriumtritium reaction

$^2_1\mathrm{D} +\, ^3_1\mathrm{T} \rightarrow\, ^4_2\mathrm{He} \left(3.5\, \mathrm{MeV}\right)+\, ^1_0\mathrm{n} \left(14.1\, \mathrm{MeV}\right)$

to be 30 million degrees (2.6 keV) and for the deuteriumdeuterium reaction

$^2_1\mathrm{D} +\, ^2_1\mathrm{D} \rightarrow\, ^3_1\mathrm{T} \left(1.0\, \mathrm{MeV}\right)+\, ^1_1\mathrm{p} \left(3.0\, \mathrm{MeV}\right)$

to be 150 million degrees (12.9 keV).[1][4]

## Application to fusors and polywells

When applied to the fusor Lawson's analysis shows that conduction and radiation losses are the key impediment to reaching net power. Fusors uses a voltage drop to accelerate and collide ions, resulting in fusion.[5] 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.[6] Regardless, it is argued that radiation is still a major impediment.[7]

## The product nτE

The confinement time $\tau_E$ measures the rate at which a system loses energy to its environment. It is the energy density W (energy content per unit volume) divided by the power loss density $P_{\rm loss}$ (rate of energy loss per unit volume):

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

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).

For illustration, the Lawson criterion for the deuteriumtritium 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 50-50 mixture.[8] Ion density then equals electron density and the energy density of both electrons and ions together is given by

$W = 3n k_{\rm B}T$

where $k_{\rm B}$ is the Boltzmann constant and $n$ is the particle density.

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

$f = n_{\rm d} n_{\rm t} \langle\sigma v\rangle = \frac{1}{4}n^2 \langle\sigma v\rangle$

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.

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

The Lawson criterion, or minimum value of (electron density * energy confinement time) required for self-heating, for three fusion reactions. For DT, nτE minimizes near the temperature 25 keV (300 million kelvins).

The Lawson criterion requires that fusion heating exceeds the losses:

$f E_{\rm ch} \ge P_{\rm loss}$

Substituting in known quantities yields:

$\frac{1}{4}n^2 \langle\sigma v\rangle E_{\rm ch} \ge \frac{3nk_{\rm B}T}{\tau_E}$

Rearranging the equation produces:

$n\tau_{\rm E} \ge L \equiv \frac{12}{E_{\rm ch}}\,\frac{k_{\rm B}T}{\langle\sigma v\rangle}$

(1)

The quantity $\frac{T}{\langle\sigma v\rangle}$ is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product $n\tau_E$. This is the Lawson criterion.

For the deuteriumtritium reaction, the physical value is at least

$n \tau_E \ge 1.5\cdot 10^{20} {\rm s}/\mbox{m}^3$

The minimum of the product occurs near T = 25 keV.

## The "triple product" nTτE

A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, nTτE. 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 p2v>/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:

$n T \tau_{\rm E} \ge \frac{12k_{\rm B}}{E_{\rm ch}}\,\frac{T^2}{\langle\sigma v\rangle}$
The fusion triple product condition for three fusion reactions.

The quantity $\frac{T^2}{\langle\sigma v\rangle}$ is also a function of temperature with an absolute minimum at a slightly lower temperature than $\frac{T}{\langle\sigma v\rangle}$.

For the deuteriumtritium reaction, the minimum of the triple product occurs at T = 14 keV. The average <σv> in this temperature region can be approximated as[9]

$\left \langle \sigma v \right \rangle = 1.1 \cdot 10^{-24} \frac{{\rm m}^3}{\rm s} \left[T {\rm \, in \, keV}\right]^2 \, {\rm ,}$

so the minimum value of the triple product value at T = 14 keV is about

$\begin{matrix} n T \tau_E & \ge & \frac{12\cdot 14^2 \cdot {\rm keV}^2}{1.1\cdot 10^{-24} \frac{{\rm m}^3}{\rm s} 14^2 \cdot 3500 \cdot{\rm keV}} \approx 3 \cdot 10^{21} \mbox{keV s}/\mbox{m}^3 \\ \end{matrix}$

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.

As for tokamaks there is an special motivation for using the triple product. Empirically, the energy confinement time τE is found to be nearly proportional to n1/3/P 2/3. In an ignited plasma near the optimum temperature, the heating power P equals fusion power and therefore is proportional to n2T 2. The triple product scales as

$\begin{matrix}n T \tau_E & \propto & n T \left(n^{1/3}/P^{2/3}\right) \\ & \propto & n T \left(n^{1/3}/\left(n^2 T^2\right)^{2/3}\right) \\ & \propto & T^{-1/3} \\ \end{matrix}$

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.

## Inertial confinement

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 $\tau_E$ is the time that it takes an ion to travel over a distance R at its thermal speed

$v_{th} = \sqrt{\frac{k_{\rm B} T}{m_i}}$

where mi denotes mean ionic mass. The inertial confinement time $\tau_E$ can thus be approximated as

$\begin{matrix} \tau_E & \approx & \frac{R}{v_{th}} \\ \\ & = & \frac{R}{\sqrt{\frac{k_{\rm B} T}{m_i}}} \\ \\ & = & R \cdot \sqrt{\frac{m_i}{k_{\rm B} T}} \mbox{ .} \\ \end{matrix}$

By substitution of the above expression into relationship (1), we obtain

$\begin{matrix} n \tau_E & \approx & n \cdot R \cdot \sqrt{\frac{m_i}{k_B T}} \geq \frac{12}{E_{\rm ch}}\,\frac{k_{\rm B}T}{\langle\sigma v\rangle} \\ \\ n \cdot R & \gtrapprox & \frac{12}{E_{\rm ch}}\,\frac{\left(k_{\rm B}T\right)^{3/2}}{\langle\sigma v\rangle\cdot m_i^{1/2}}\\ \\ n \cdot R & \gtrapprox & \frac{\left(k_{\rm B}T\right)^{3/2}}{\langle\sigma v\rangle}\mbox{ .} \\ \end{matrix}$

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 = <nmi>:

$\rho \cdot R \geq 1 \mbox{ g/cm²}$

Satisfaction of this criterion at the density of solid deuteriumtritium (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 (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.

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:

$\begin{matrix} \mbox{burn-up fraction } & \propto & n^2\langle\sigma v\rangle T^{-1/2}/n \\ & \propto & \left( n T\right)\langle\sigma v\rangle /T^{3/2}\\ \end{matrix}$

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