Why does fusion need to be so hot?

The topic of fuels for fusion is much talked about in the fusion start-up community. What fuel are you using is a common question, particularly from US VC investors.  Yet it is barely discussed in the publicly funded fusion programmes, neither magnetic nor inertial. This post is part of a series where I will outline why it is not discussed in the mainstream of fusion. To me, the answer is simple; it is because deuterium-tritium is the only viable fuel. I will present the analysis and you can make up your own mind.

1. What are the options?
2. Why does fusion need to be so hot? – you are here!
3. The fusion gain limit

The ideal ignition temperature

In the first post in this series, we established the four candidate fuels for fusion power. These fuels have the highest reactivities at the lowest temperatures. The higher the reactivity the more fusion reactions per second, with all else being equal. And we showed that the deuterium-tritium (DT) reaction offers the highest power density across the whole temperature range.

But how did I decide what range to plot? It was a pretty serious range, from 1 keV (or 10 million degrees) all the way up to 1000 keV (a scorching 10 billion degrees). The centre of the Sun is only 15 million degrees, and the surface a measly 5800 K. Although it’s not the surface that does any fusing, 99% of the Sun’s energy comes from its middle 25%.

The reactivities definitely imply the need for high temperatures. The peak of the DT reactivity is at 65 keV. At this temperature we would get the highest fusion power. If it is colder, we get less fusion, if it is hotter, we also get less fusion. We need to establish, however, what is needed for energy gain. We have to invest energy heating the fuel up to the required temperature, we need to get a return on that investment. And we’re not looking for a 5% interest rate here, that isn’t going to deliver economic power production. We need high gain, many times the amount of energy coming out as going in.

A prerequisite for high gain is “ignition” or “self-heating”. There are many processes taking place inside the fuel. Some of these add energy, like the external heating. In magnetic fusion the external heating is supplied by intense radiowaves and microwaves, and so-called neutral beam heaters, which fire high energy particles into the plasma, normally deuterium atoms. In inertial fusion, it is the implosion itself that provides the heating. The implosion does thermodynamic work, pdV work, on the fuel.

There is a second heating mechanism, which is the fusion process itself. Some of the energy released escapes the fuel, some remains. That which remains contributes to the energy balance, helping to heat the plasma.

These two heating mechanisms are balanced by two main processes that take energy away from the fuel. The first of these is conduction of heat to the surroundings, the nature of which differs substantially between approaches. In inertial fusion, the fuel might be surrounded by a shell of much colder material, which means conduction loss will be very important. Or it might be that the fuel is assembled, at the final moment, in vacuum by itself. Conduction needs a material for the heat to flow into, if there is nothing there, there is no conduction. In magnetic fusion, the magnetic fields themselves inhibit conduction, this is almost the point of the approach. Energy is still lost though, the processes involved are complicated, but the end result is that the magnetic bottle is leaky.

The second major energy loss mechanism is universal, present in all approaches, and thankfully it is easier to quantify. This is the energy lost by radiation, which is to say, electromagnetic radiation, photons, light, although not visible. Think about the feeling of warmth you get sitting in front of a fire or stove. You are feeling infrared radiation, light of a particular frequency, and energy is conserved, the light gives energy to you, warming you up, and takes that energy away from the stove. The hotter the fire, the warmer it feels. Or looked at the other way around, the hotter the fire, the more energy it loses by radiation.

In a fusion plasma there is one dominant mechanism producing radiation, which is bremsstrahlung radiation. This word is actually German portmanteau meaning “braking radiation” and in a fusion plasma it occurs whenever an electron is deflected by the presence of an ion. As discussed in the first post, the microphysical picture in the plasma is one of dynamic equilibrium. The ions are flying about crashing into each other, sometimes fusing, sometimes bouncing off. But the electrons are still there too, they can’t disappear, you can’t get rid of them.

When a negatively charged electron encounters an ion, it is deflected on its flight path by the positive charge. The electron can change velocity and direction substantially whilst the ion is basically unaffected. The ion is 3600 times heavier; the electron is like a ping pong ball bouncing off a bowling ball. The energy lost by the electron is converted to photons, which escape the plasma, carrying the energy with them. And at the temperatures for fusion, these are x-ray photons.

We can now put together a simple criterion which will allow us to find the temperature needed for ignition. At low temperatures, the fusion rate has not ramped up yet, and the energy lost by radiation is dominant, meaning that without external heating the fuel would cool down. The point where the fusion energy released first exceeds the radiation loss is called the ideal ignition temperature. It is ideal in the sense that all we are doing here is balancing the fusion power with one specific individual mechanism that produces radiation. A fuller analysis including more losses is going to give a more pessimistic assessment, a higher required temperature.

The ideal ignition temperature also makes assumptions about the fusion energy released. Charged particles produced by fusion, like the alpha particle from the DT reaction, are assumed to stop within the fuel, transferring 100% of their energy into self-heating. The neutrons, on the other hand, are assumed to escape, contributing nothing.

Our criterion is that the energy released by fusion is greater than the energy lost to bremsstrahlung radiation. We can express both processes as power densities, energy released per unit volume per unit time. The answer we will get then doesn’t depend on the plasma geometry in any way, nor on the confinement time. And helpfully both processes depend on density squared, which means that the density cancels out and the result doesn’t depend on whether we at the tenuous densities of magnetic fusion, or the extremes of inertial fusion.

For the different fuels, two parts of the equation change. First, the reactivity is different for the different fuels, and highest for DT. And second, the bremsstrahlung loss depends on the average atomic number of the elements involved. The higher the atomic number, the higher the charge of the ion, the stronger the deflection of a passing electron, and the more energy lost to radiation.

And the answer is that DT needs a temperature of 4.3 keV to ignite, DHe3 comes in second needing 31 keV, and then DD needs 45 keV. The last “option” on the other hand, pB, has no solution, the fusion power never exceeds the radiation loss at any temperature.

It is also interesting to look at the ratio of the fusion power to the radiation loss. This gives a sense of the margin that we have in hand. There is conduction to account for, and other mechanisms of radiation emission that add on top of the bremsstrahlung number.

One example of an additional process is radiation from impurities; impurities make the bremsstrahlung loss worse. Impurities will have a higher atomic number. Tungsten, for example, which is used on the inside of tokamaks, has an atomic number of 74, meaning that the Z squared factor in the equation is about 22,000, implying that with only 0.02% admixture the radiation loss would be doubled.

Depending on the margin that might not be fatal. For DT the maximum ratio of fusion power to bremsstrahlung loss is 33, fusion is 33 times more powerful, and this occurs at a temperature of 39 keV, not much more than the table stakes for the other two fuels. For DHe3 the maximum is 6.5 and for DD it is 2.9, both occurring at extreme temperatures. I really don’t like the idea of a margin less than 10, and less than 3, it isn’t going to work. And for what it’s worth, the maximum ratio for pB is 0.43.

To close, in the first post we identified the four fuels with the highest reactivities. DT had the highest power density, but we didn’t know what else we needed, any of the fuels might have worked even if DT produced the most power. In this post we’ve seen that the need to exceed the radiation loss has boxed us in hugely. DT is still looking good, but the other fuels require higher temperatures and are significantly more marginal. The advantages of DT are stacking up. In the third and next post, we will finally talk about energy gain.

And I will return to pB. Whilst the nail put in the coffin here is pretty big, we’ve not yet considered everything.