Calculating the minimum Lawson parameter for sustained d-d fusion

Question

Calculate the minimum Lawson parameter for sustained deuterium-deuterium fusion in a plasma with an energy of 10 keV.

I have been given the above question as part of a homework assignment and help would be much appreciated.

The Lawson parameter is given as $$ n \tau > \frac{12 k T}{<\sigma v> Q} $$

where $\sigma$ is the fusion cross-section.

I think that Q for d-d reaction is 4 MeV when a proton is produced and 3.3 MeV when a neutron is produced so I assume for the minimum Lawson parameter I should use the larger value of 4 MeV?

Also if the energy of the plasma is 10 MeV should I use this to calculate the temperature of the reaction $T$ using $\frac{3}{2} k T$ or the velocity $v$ using $\frac{1}{2} m v^2$ ?

Answer 1

I hope you have handed in your homework by now. Still, it is a valid question in my opinion, so let my try to answer it.

The Lawson criterion, as you have written it, is the original form. Nowadays, the triple product $nT_i\tau_E$ is more commonly used, but let's stick at the original form since you asked for it.

You are interested in the D-D reaction. As you correctly pointed out, there are two reactions possible (with roughly the same probability):

1. D + D $\rightarrow$ T (1.01 MeV) + p (3.02 MeV)
2. D + D $\rightarrow$ $^3$He (0.82 MeV) + n (2.45 MeV)

The Lawson criterion requires the energy of the fusion products to be used to heat the plasma (and thus sustain the fusion process). If we are considering magnetically confined plasmas, only charged particles are confined and the neutron does therefore not contribute to further heating of the deuterium plasma. Hence, the value of $Q$ evaluates to an average of $$Q=\frac{ 1.01+3.02 + 0.82}{2}\,\mathrm{MeV}=2.425\,\mathrm{MeV}.$$

Using values for the fusion reactivity $\left<\sigma_{fus}v\right>$ from literature, this allows us to plot $n\tau_E$ as a function of the ion temperature:

Looking at 10 keV, a minimum of $n\tau_E\le4.2\cdot10^{22}\,\mathrm{s}/\mathrm{m}^3$ must be achieved.