Why is the triple product nτT used instead of the individual three criteria?

Question

Why is the criteria for fusion represented as a triple product nτT instead of the n,τ,T individually? i.e saying the density has to be above this value, temperature above this value? What are the advantage we gain from representing it this way?

Answer 1

We need to have: $$P_{fusion}>P_{loss}$$
To do so, we begin by defining a new quantity $\tau$ , called the confinement time, which 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 $P_{loss}$ (rate of energy loss per unit volume):
$$\tau=\frac{W}{P_{loss}}$$
A well-known property of gases, which applies equally well to plasmas, is that their energy density W is equal to
$$W=3 k_b n T$$ This formula is a standard result in statistical mechanics and holds whenever there are no impurities in the plasma, an assumption that is justified for fusion reactors because they are typically well insulated. $$P_{fusion}>\frac{W}{\tau}=\frac{3 k_b n T}{\tau}$$

Now we can write: $$n_A n_B \langle {\sigma v}_{A,B} \rangle E_{fusion} >\frac{3 k_b n T}{\tau}$$

Putting $ \frac{3 n^2}{n_A n_B}=12$

$$n T \tau > \frac{12 k_b T^2}{\langle {\sigma v}_{A,B} \rangle E_{fusion} }$$

$\langle {\sigma v}_{A,B} \rangle$ is the averaged cross-section with the Maxwell–Boltzmann distribution.