Lawson criterion

Extensions into nτE

The confinement time ${\displaystyle {\tau }_E}$ measures the rate at which a system loses energy to its environment. The faster the rate of loss of energy, ${\displaystyle P_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}}$, the shorter the energy confinement time. It is the energy density ${\displaystyle W}$ (energy content per unit volume) divided by the power loss density ${\displaystyle P_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}}$ (rate of energy loss per unit volume):

${\displaystyle {\tau }_E=\frac{W}{P_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}}}$

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 at the same rate the plasma loses energy in order to maintain the fusion conditions. This energy can be supplied by the fusion reactions themselves, depending on the reaction type, or by supplying additional heating through a variety of methods.

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.^a Ion density then equals electron density and the energy density of both electrons and ions together is given, according to the ideal gas law, by

${\displaystyle W=3nT}$

where ${\displaystyle T}$ is the temperature in electronvolt (eV) and ${\displaystyle n}$ is the particle density.

The volume rate ${\displaystyle f}$ (reactions per volume per time) of fusion reactions is

${\displaystyle f=n_{\mathrm{d}}{n}_{\mathrm{t}}⟨\sigma v⟩=\frac{1}{4}{n}^2⟨\sigma v⟩}$

where ${\displaystyle \sigma }$ is the fusion cross section, ${\displaystyle v}$ is the relative velocity, and ${\displaystyle ⟨⟩}$ denotes an average over the Maxwellian velocity distribution at the temperature ${\displaystyle T}$.

The volume rate of heating by fusion is ${\displaystyle f}$ times ${\displaystyle E_{\mathrm{c}\mathrm{h}}}$, the energy of the charged fusion products (the neutrons cannot help to heat the plasma). In the case of the D-T reaction, ${\displaystyle E_{\mathrm{c}\mathrm{h}}=3.5\mathrm{M}\mathrm{e}\mathrm{V}}$.

The Lawson criterion requires that fusion heating exceeds the losses:

${\displaystyle f{E}_{\mathrm{c}\mathrm{h}}\ge P_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}}$

Substituting in known quantities yields:

${\displaystyle \frac{1}{4}{n}^2⟨\sigma v⟩E_{\mathrm{c}\mathrm{h}}\ge \frac{3nT}{{\tau }_E}}$

Rearranging the equation produces:

The quantity ${\displaystyle T/⟨\sigma v⟩}$ is a function of temperature with an absolute minimum. Replacing the function with its minimum value provides an absolute lower limit for the product ${\displaystyle n{\tau }_E}$. This is the Lawson criterion.

For the deuterium–tritium reaction, the physical value is at least

${\displaystyle n{\tau }_E\ge 1.5\cdot {10}^{20}\frac{\mathrm{s}}{{\mathrm{m}}^3}}$

The minimum of the product occurs near ${\displaystyle T=26\mathrm{k}\mathrm{e}\mathrm{V}}$.

Extension into the "triple product"

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 p²<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:

${\displaystyle nT{\tau }_{\mathrm{E}}\ge \frac{12}{E_{\mathrm{c}\mathrm{h}}}\frac{T^2}{⟨\sigma v⟩}}$

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

For the D-T reaction, the minimum occurs at T = 14 keV. The average <σv> in this temperature region can be approximated as

${\displaystyle ⟨\sigma v⟩=1.1\cdot {10}^{-24}{T}^2\frac{{\mathrm{m}}^3}{\mathrm{s}},\mathrm{T}\mathrm{i}\mathrm{n}\mathrm{k}\mathrm{e}\mathrm{V},}$

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

${\displaystyle \begin{array}{ccc}nT{\tau }_E& \ge & \frac{12\cdot {14}^2\cdot {\mathrm{k}\mathrm{e}\mathrm{V}}^2}{1.1\cdot {10}^{-24}\frac{{\mathrm{m}}^3}{\mathrm{s}}{14}^2\cdot 3500\cdot \mathrm{k}\mathrm{e}\mathrm{V}}\approx 3\cdot {10}^{21}\text{keV s}/{\text{m}}^3\end{array}(3.5\cdot {10}^{28}\text{K s}/{\text{m}}^3)}$

This number has not yet been achieved in any reactor, although the latest generations of machines have come close. JT-60 reported 1.53×10²¹ keV.s.m⁻³. 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 n²T ². The triple product scales as

${\displaystyle \begin{array}{ccc}nT{\tau }_E& \propto & nT\left(n^{1/3}/P^{2/3}\right)\\ & \propto & nT\left(n^{1/3}/{\left(n^2{T}^2\right)}^{2/3}\right)\\ & \propto & T^{-1/3}\end{array}}$

The triple product is only weakly dependent 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 in the inertial case it is more usefully expressed in a different form. A good approximation for the inertial confinement time ${\displaystyle {\tau }_E}$ is the time that it takes an ion to travel over a distance R at its thermal speed

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

where m_i denotes mean ionic mass. The inertial confinement time ${\displaystyle {\tau }_E}$ can thus be approximated as

${\displaystyle \begin{array}{ccc}{\tau }_E& \approx & \frac{R}{v_{th}}\\ \\ & =& \frac{R}{\sqrt{\frac{k_{\mathrm{B}}T}{m_i}}}\\ \\ & =& R\cdot \sqrt{\frac{m_i}{k_{\mathrm{B}}T}}\text{ .}\end{array}}$

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

${\displaystyle \begin{array}{ccc}n{\tau }_E& \approx & n\cdot R\cdot \sqrt{\frac{m_i}{k_{B}T}}\ge \frac{12}{E_{\mathrm{c}\mathrm{h}}}\frac{k_{\mathrm{B}}T}{⟨\sigma v⟩}\\ \\ n\cdot R& ⪆& \frac{12}{E_{\mathrm{c}\mathrm{h}}}\frac{{\left(k_{\mathrm{B}}T\right)}^{3/2}}{⟨\sigma v⟩\cdot m_i^{1/2}}\\ \\ n\cdot R& ⪆& \frac{{\left(k_{\mathrm{B}}T\right)}^{3/2}}{⟨\sigma v⟩}\text{ .}\end{array}}$

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>:

${\displaystyle \rho \cdot R\ge 1\mathrm{g}/{\mathrm{c}\mathrm{m}}^2}$

Satisfaction of this criterion at the density of solid D-T (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 ~ 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 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:

${\displaystyle \begin{array}{ccc}\text{burn-up fraction }& \propto & n^2⟨\sigma v⟩T^{-1/2}/n\\ & \propto & \left(nT\right)⟨\sigma v⟩/T^{3/2}\end{array}}$

Thus the optimum temperature for inertial confinement fusion maximises <σv>/T^3/2, which is slightly higher than the optimum temperature for magnetic confinement.

Non-thermal systems

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. The best-known examples are the migma, fusor and polywell.

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. The voltage drop is generated by wire cages, and these cages conduct away particles.

Polywells are improvements on this design, designed to reduce conduction losses by removing the wire cages which cause them. Regardless, it is argued that radiation is still a major impediment.