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.

The four different candidate fuels

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.

The four main processes important for fusion. Two heat the fuel, the external heating and the self-heating by the fusion process itself, and two take energy away from the fuel, the conduction and radiation losses

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.

The ideal ignition temperature is the temperature where the fusion power first exceeds the energy lost to radiation, specifically bremsstrahlung radiation. The different fuels, compared to DT, have lower reactivity and lower fusion power density, and DHe3 and pB also have a higher average atomic number and higher radiation loss.

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.

Fusion power and radiation loss plotted vs temperature for each of the four fuels, allowing the ideal ignition temperature in each case to easily be seen. The required temperatures for DD and DHe3 are higher than DT, and with pB 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.

The ratio of the fusion power to the radiation loss, showing the margin that each fuel has to cope with deleterious additional processes not included this simple analysis.

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.

Fuels for fusion – what are the options?

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

What are the options?

Let’s start at the beginning. The sun is our solar systems resident fusion reactor, why are we talking about options for fuel? Isn’t the objective to build a mini-sun, don’t we just follow the example? Not exactly. The sun fuses hydrogen, normal hydrogen with one proton as the nucleus and nothing else. And eventually the hydrogen turns into helium, natures second element, but the path to get there is quite complicated. The overall process is called the proton-proton chain.

The rate-limiting step in this process is the first one, the fusion of two protons together to form a deuterium nucleus, an isotope of hydrogen with a proton and a neutron, and it is a spectacularly difficult fusion reaction. Protons in the sun wait around for 9 billion years, on average, before eventually fusing. Because of this very slow reaction, the heat output of the sun per unit volume is actually very low, less than the metabolic heat output of a person and about the same as a compost heap. The sun makes up for its poor fusion performance by being truly enormous.

To make fusion work on Earth we need a fuel with much higher energy density, it can’t be hydrogen. There are four candidates commonly discussed. The first is deuterium and tritium (DT), which fuse to produce a helium-4 nucleus, also called an alpha particle, and a neutron. Next there is deuterium and helium-3 (DHe3), which also produces an alpha particle but now with a proton and no neutron. Then there is the pure deuterium case (DD), where deuterium fuses with itself. This reaction has two branches, both of which occur with equal probability. The first produces tritium and a proton, and the second helium-3 and a neutron. And lastly there is normal hydrogen again, but in this case fusing with boron-11, producing three alpha particles.

The four different candidate fuels for fusion power.

These four are the reactions with the highest fusion cross section. The cross section describes the likelihood of fusion occurring as a function of the centre of mass energy of the two reactants. Imagine an experiment where you fire two atomic nuclei, two ions, at each other at a specified velocity. The cross section is the chance that they fuse, which depends on the velocity of the two ions. In fact, this is exactly how experiments to measure the fusion cross section are done, using a particle accelerator. Don’t get it confused with CERN though, the ion energies needed are far less than those needed for particle physics.

For application to fusion power the cross section gets turned into something called the reactivity. Fusion power needs a plasma in thermal equilibrium, a requirement that I will write about in future but not as part of this series, and in a plasma in thermal equilibrium, the ion energies follow a Maxwellian distribution. The ions are in a dynamic equilibrium, they are constantly flying around and constantly bumping into other ions. Sometimes they fuse but most of the time they “scatter”, they bounce off. If we follow an individual ion as it goes on its journey inside the plasma, it would be constantly undergoing collisions. Sometimes it would be going slowly and get whacked by a fast-moving ion, speeding up as a result. Sometimes it will be the fast ion, losing energy to its slower compatriots. It’s a bit like being stuck inside a big crowd.

Overall, looking at all the ions, there is a distribution of different velocities present. Any individual ion will be bouncing around all over, sometimes fast, sometimes slow. But the net result is a population with a clearly defined average energy and a fixed set of velocities present. If the temperature is higher, the average energy is higher, and the more ions we have at the highest velocities. Actually, formally, this should be said the other way around. This is the definition of temperature. When we say a plasma has a particular temperature, what this actually means is we have ions with a particular Maxwellian distribution of velocities; having this distribution defines the temperature.

And this dynamic equilibrium isn’t just a plasma thing, nor this definition of temperature. The molecules in the air around you are constantly bashing into each other in exactly the same way, and the velocities will follow the Maxwellian distribution for room temperature. The average nitrogen molecule bumping into your skin right now is moving at about 940 mph.

Coming back to fusion, the cross section tells us the probability of fusion for a given velocity, and the temperature tells us the velocities we have in the fuel. To find out how much fusion we get, we integrate and count up the contribution of each specific velocity to the total reaction rate. The quantity that results from this integration is called the reactivity.

The reactivities of the four candidate fuels, showing that DT has the highest values at the lowest temperatures.

Plotting the reactivities allows the different candidate fuels to be compared. The first thing that we can see is that DT has the highest reactivity at nearly every temperature, and that the DT reactivity ramps up at the lowest temperature. The second-best reactivity is initially DD, until a temperature of about 24 keV, after which DHe3 takes over. This itself is then surpassed by pB at a temperature of 167 keV, which eventually surpasses DT at 335 keV, after DT has gone past its peak. The DT peak is the biggest number overall, however.

(For reference, 1 keV is about 10,000,000 K)

The different reactivities normalised by the DT reactivity

We can also plot these reactivities a bit differently. The first plot has a logarithmic scale on both axis, which can make even major differences look small. Instead, I’ve plotted them as a ratio of the DT reactivity. This shows the difference very clearly. At lower temperatures, the easiest part of parameter space, the part we want to be in, the DT reactivity is more than 50 times higher than the alternatives. It isn’t until a temperature of 64 keV that DHe3 comes within even 10% of the DT value.

The reactivity tells us how many reactions are happening per unit volume per unit time, and this is not exactly what we want. What we really want is energy, and that means we need to look at the energy released by each reaction. For DT, a single reaction liberates 17.6 MeV of energy. For DD, the value is lower, 3.65 MeV. DHe3 releases the largest amount at 18.3 MeV, and then pB gives 8.7 MeV. To account for the differences in energy released, we can scale the reactivities by the appropriate value. Looking at it this way, DT is the best fuel at all temperatures.

Power density, instead of reactivity. When viewed in these terms, DT is the best fuel over the whole range. The units here are ion number density specific power density per unit volume per unit time, in MeV. I can but apologise…

Having established that DT is best in terms of power density we now need to discuss the potential benefits of other fuels, and there are potential benefits. First, two of these candidate fuels require only naturally occurring inputs. Both tritium and helium 3 don’t exist naturally on Earth in any useful quantity, they will have to be manufactured to be used as fusion fuel. And second, two of the candidate fuels produce neutrons and two do not. This is important because neutron damage is one of the major engineering challenges of fusion. Both DHe3 and pB produce no neutrons, not directly anyway.

The benefits of the different fuel choices. Some occur naturally, others need to be manufactured. And some produce neutrons as a product, and some don’t.

What we have seen here is that DT has the highest power density, but we have also seen that the other fuels do have their advantages. We have not established is what is actually needed for useful power production. It might be that all four options can work great, DT would produce the most power, but the others are fine too. Spoiler alert, this is not the answer, and this is what we will explore in the next few posts.