lawson criterion

In nuclear fusion research, the Lawson criterion, first derived by J. D. Lawson 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 n_e and the "energy confinement time" τ_E. 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 product $n_\mbox{e}\tau_\mbox{E}$

The confinement time is simply the energy content W divided by the power lost:

τ_E = W/P_loss

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. In that case, the ion density is equal to the electron density and the energy density of both together is given by

W = 3n_ek_BT

The volume rate f of fusion reactions is

f = n_D n_T <σv> = (1/4)n_e² <σv>

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 E_ch, the energy of the charged fusion products (the neutrons cannot help to keep the plasma hot). In the case of the D-T reaction, E_ch = 3.5 MeV. The Lawson criterion is the requirement that the fusion heating exceed the losses:

f E_ch

\ge

P_loss

(1/4)n_e² <σv> E_ch

\ge

3n_ek_BT/τ_E

n_\mbox{e} \tau_\mbox{E} \ge \frac{12k_\mbox{B}}{E_\mbox{ch}}\,\frac{T}{\langle\sigma v\rangle}

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_eτ_e. This is the Lawson criterion. For the D-T reaction, the physical value is about

n_\mbox{e} \tau_\mbox{E} \ge 10^{20} \mbox{sec}/\mbox{m}^3

This is the product required near T = 10 keV.

The "triple product" $n_\mbox{e}T\tau_\mbox{E}$

A still more useful figure of merit is the "triple product" of density, temperature, and confinement time,

n_\mbox{e}T\tau_\mbox{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 pressure attainable is a constant. When that is the case, the fusion power density is proportional to

p^2\langle\sigma v\rangle/T^2

. Therefore the maximum fusion power available from a given machine is obatined at the temperature where

\langle\sigma v\rangle/T^2

is a maximum. Following the derivation above, it is easy to show the inequality

n_\mbox{e} T \tau_\mbox{E} \ge \frac{12k_\mbox{B}}{E_\mbox{ch}}\,\frac{T^2}{\langle\sigma v\rangle}

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 n/P. In an ignited plasma near the optimum temperature, the heating power P is equal to the fusion power and therefore proportional to n/T. The triple product scales as

nTτ

\propto

nT (n/P)

\propto

nT (n/(n/T))

\propto

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}{\langle\sigma v\rangle}

is also a function of temperature with an absolute minimum at a slightly higher temperature than

\frac{T}{\langle\sigma v\rangle}

. For the D-T reaction, the physical value is about

n_\mbox{e} T \tau_\mbox{E} \ge 10^{21} \mbox{keV-sec}/\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 achieved Lawson at the temperatures it can create, but it cannot create the temperatures at the same time. ITER aims to do both.

Lawson Criterion

The product n_\mbox{e}\tau_\mbox{E}

The "triple product" n_\mbox{e}T\tau_\mbox{E}

See also

The product $n_\mbox{e}\tau_\mbox{E}$

The "triple product" $n_\mbox{e}T\tau_\mbox{E}$