[NF.7] How to Measure Quality of Nuclear Fusion

This article is on the second term I skipped in the temperature NF article, where I wrote about the recent EAST tokamak success. That term is called "fusion product" or popularly "the triple product".

Before I start, I will repeat the guilty sentence: "Let's ignore (for now) what is on the vertical axis and focus on the horizontal one that is showing ion temperature.", where vertical axis showed: "Fusion Product P_i τ_E (atmosphere secs)".

Source: Figure 10.7 from McCracken & Stott, "Fusion - The Energy of Universe"

Physical units

We can see that units are [atmosphere * seconds], which in SI units is simply [Pascal * seconds]. Finally, physically we see that fusion product talks about the product of pressure and time, where the pressure is that of ions ("i" in an index) and time is connected to energy (hence "E" in subscription). The higher the product the better nuclear fusion plasma.

But how did we ended up with this? What is the physics behind? I will give my best to explain it in the following sections.

Reaction probability

When I was explaining nuclear fusion I shared with you following table (which I made some time ago):

Cross-section in area units is the way for the nuclear physicist to express the probability ;)

In the caption of a figure, we can see that [barn] units represent the probability for reaction to happen on the given temperature. For D-T reaction maximum is about 120 keV.

Including particle movement

However, when particles have the temperature, they also have kinetic energy, that means movement. The movement or velocity defines for how long particles can interact. One can see this with the following macroscopic parallel:

• pedestrians can talk a lot while walking, but their "interaction" can be too weak due to their low energy
• car drivers cannot talk (let's ignore windows and the roof!) as they are moving too fast, while their energy would be enough for interaction
• bikers can talk shortly and their energy is enough for "reaction" to happen

Back to the physics. This interaction with the inclusion of velocity we will name "reactivity". This was also given in my blog on nuclear fusion basics in the following graph:

Image Source

For D-T reaction, the maximum of reactivity is around 65 keV. We are going down with the temperature, as predicted with the pedestrian-bike-car example, and that is good!

Ignition

For an energy engineer, ignition means limit when the system starts to sustain itself without putting any input into it. The input is usually heat, and it is also the case for the nuclear fusion power plant.

Remember that products of D-T reaction are an alpha particle (helium nucleus) and neutron. Electrically neutral neutron leaves the systems as there are no physical means of making it stay. On the other hand, the alpha particle is charged and it stays. Henceforth, the nuclear fusion power plant gets ignited when alpha particles created due to the fusion reaction can sustain the temperature of the system. In science, this threshold is known as Lawson criterion.

After doing the math for D-T reaction with the inclusion of temperature, one gets that best performance is around 25 keV. Even smaller temperature than earlier. GOOD!

Optimal machine

Finally, when fixing the machine parameters (mostly related to the magnetic field) one can show that the most optimal temperature for the nuclear fusion power plant is approximately 15 keV!!!

Optimal parameters

The above analysis can be interpreted in the means of the triple product:

n_i * T_i * τ_E > 3 * 10²¹ [keV * s / m³].

Here T_i is ion temperature, n_i is ion density (concentration) and τ_E is ion energy confinement time. The former term represent for how long does particle keeps its energy, i.e. before loosing its energy. Recover from the basic physics that pressure is equal to product of temperature and density, which brings us to the expression from the initial image.

Knowing that optimal ion temperature is around 15 keV, and taking typical tokamak density of 10²⁰m^-3 we need confinement time to be around 2 seconds.

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Note that optimal temperature was already obtained in the last 30 years, as I explained already. The problem is with the combination of density and confinement time. Note that the optimum was never achieved. ITER aims to have the confinement time about 1 second with density of 5*10²⁰m^-3.

I hope I did not overwhelm you with the terms from nuclear physics. If so, please feel free to ask anything in the comments. I will either try to answer you in the comment or even make the new post.