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. Quasineutrality [21] demands that

(1.10) 
where is the number density of electrons. Suppose that the two ions species have the same temperature, (measured in energy units), as the electrons. The total
thermal energy density of the plasma is thus [21]

(1.11) 
The rate of nuclear fusion reactions per unit volume is [see Equation (1.6)]

(1.12) 
In order to achieve a selfsustaining nuclear fusion reaction in a thermonuclear plasma, the fusion heating power per unit volume,
, must exceed the
energy loss rate per unit volume,
. (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

(1.13) 
Let us write

(1.14) 
Here, the energy confinement time, , 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 . The previous
four equations can be combined to give

(1.15) 
where

(1.16) 
Figure 1.2:
The Lawson function,
, versus the electron temperature, .

Figure 1.2 plots
as a function of the electron temperature, . It can be be seen that
attains a
minimum value of
when
keV. Thus, we conclude that a selfsustaining nuclear fusion reaction is only possible in a
thermonuclear plasma if

(1.17) 
This criterion is known as the Lawson criterion [41].
Figure 1.3:
The triple product function,
, versus the electron temperature, .

In conventional magnetic confinement devices, and can be varied over a wide range of values. However, the maximum value of the plasma pressure, which is proportional to , 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 selfsustaining nuclear fusion reaction can be recast in the form

(1.18) 
where

(1.19) 
Given that the maximum value of the product is fixed, it follows that fusion reactivity is maximized at the temperature that
minimizes the function
. As illustrated in Figure 1.3,
attains a
minimum value of
when
keV. Thus, a more useful form of the Lawson criterion is

(1.20) 
Here,
is known as the fusion triple product, and is the conventional figure of merit for thermonuclear fusion reactions in magnetic confinement devices [64].