Lawson Criterion

Consider a thermonuclear plasma that consists of an optimal 50%-50% mixture of deuterium and tritium ions, as well as electrons. Suppose that there are no impurity ions or helium ash particles (i.e., thermalized alpha particles) present in the plasma. Quasi-neutrality [21] demands that

$$n_D = n_T = \frac{n_e}{2},$$ (1.10)

where $n_e$ is the number density of electrons. Suppose that the two ions species have the same temperature, $T_e$ (measured in energy units), as the electrons. The total thermal energy density of the plasma is thus [21]

$$W \equiv \frac{3}{2}\,n_D\,T_e + \frac{3}{2}\,n_T\,T_e + \frac{3}{2}\,n_e\,T = 3\, n_e\,T_e.$$ (1.11)

The rate of nuclear fusion reactions per unit volume is [see Equation (1.6)]

$$f \equiv n_D\,n_T\,\langle \sigma\,v\rangle_{DT}(T_e) = \frac{n_e^{\,2}}{4}\,\langle \sigma\,v\rangle_{DT}(T_e).$$ (1.12)

In order to achieve a self-sustaining nuclear fusion reaction in a thermonuclear plasma, the fusion heating power per unit volume, $f\,{\cal E}_\alpha$, must exceed the energy loss rate per unit volume, $P_{\rm loss}$. (Recall that the alpha particles produced by nuclear fusion reactions heat the plasma, whereas the neutrons exit the plasma without heating it.) Thus, we require

$$f\,{\cal E}_\alpha \geq P_{\rm loss}.$$ (1.13)

Let us write

$$P_{\rm loss} = \frac{W}{\tau_E}.$$ (1.14)

Here, the energy confinement time, $\tau _E$, is a measure of how long the plasma's thermal energy is confined within the plasma before escaping. Note that, at this stage, we are making no statement about the nature of the energy loss mechanism. In fact, Equation (1.14) can be thought of as the definition of $\tau _E$. The previous four equations can be combined to give

$$n_e\,\tau_E \geq F_{\rm lawson}(T_e),$$ (1.15)

where

$$F_{\rm lawson}(T_e)= \frac{12\,T_e}{\langle \sigma\,v\rangle_{DT}(T_e) \,{\cal E}_\alpha}.$$ (1.16)

Figure 1.2: The Lawson function, $F_{\rm lawson}$, versus the electron temperature, $T_e$.

Figure 1.2 plots $F_{\rm lawson}$ as a function of the electron temperature, $T_e$. It can be be seen that $F_{\rm lawson}(T_e)$ attains a minimum value of $1.49\times 10^{20}\,{\rm s}/{\rm m}^3$ when $T_e= 25.67$ keV. Thus, we conclude that a self-sustaining nuclear fusion reaction is only possible in a thermonuclear plasma if

$$n_e\,\tau_E \geq 1.49\times 10^{20}\,{\rm s}/{\rm m}^3.$$ (1.17)

This criterion is known as the Lawson criterion [41].

Figure 1.3: The triple product function, $F_{\rm triple}$, versus the electron temperature, $T_e$.

In conventional magnetic confinement devices, $n_e$ and $T_e$ can be varied over a wide range of values. However, the maximum value of the plasma pressure, which is proportional to $n_e\,T_e$, is fixed by plasma stability considerations [26,64]. (See Section 1.10.) Now, according to Equations (1.15) and (1.16), the criterion for a self-sustaining nuclear fusion reaction can be recast in the form

$$n_e\,T_e\,\tau_E \geq F_{\rm triple}(T_e),$$ (1.18)

where

$$F_{\rm triple}(T_e)= \frac{12\,T_e^{\,2}}{\langle \sigma\,v\rangle_{DT}(T_e) \,{\cal E}_\alpha}.$$ (1.19)

Given that the maximum value of the product $n_e\,T_e$ is fixed, it follows that fusion reactivity is maximized at the temperature that minimizes the function $F_{\rm triple}(T_e)$. As illustrated in Figure 1.3, $F_{\rm triple}(T_e)$ attains a minimum value of $2.76\times 10^{21}\,{\rm keV\,s}/{\rm m}^3$ when $T_e= 13.54$ keV. Thus, a more useful form of the Lawson criterion is

$$n_e\,T_e\,\tau_E \geq 2.76\times 10^{21}\,{\rm keV\,s}/{\rm m}^3.$$ (1.20)

Here, $n_e\,T_e\,\tau_E$ is known as the fusion triple product, and is the conventional figure of merit for thermonuclear fusion reactions in magnetic confinement devices [64].