The fusion gain limit

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?
  3. The fusion gain limit – you are here!

The gain limit

To get fusion reactions to happen the fuel must be hot. In the last post we looked at how hot, discussing the criterion for the fusion power to exceed the radiation loss, and how this criterion defines the “ideal ignition temperature”. Heating the fuel is an energy investment that we must make to get fusion conditions. To turn fusion into a power source, the energy we get back must be more than the energy we put in, and the ratio of the two is called the gain.

Roughly, that is. The exact nature of the definition is something that people get a bit argumentative about. Obviously, the ultimate balance is the total energy input requirement of the entire plant, balanced against the total output. The output energy is not the same as the fusion energy, there is always an efficiency associated with power conversion. Any thermal cycle, for example boiling water to make steam and turn a turbine, has a thermodynamic efficiency. And direct energy capture, as proposed by some, will not be perfectly 100% efficient. Then there are all the plant auxiliaries to run, things like pumps. Plus other losses, for example in transformers or just plain old electrical resistance. They all eat into the gross energy output.

A high-level view of energy flowing around a fusion power plant

Having to think about all of this is not very helpful when working on the physics of the problem. We need a definition that can guide us in the right way without getting bogged down in engineering that is better addressed later. If the yardstick we are using to understand progress requires a detailed electrical layout of the power plant to compute, it’s not going to be too useful. The fusion power out is pretty clear, the question is what to use as the denominator. There are at least three different choices which, when used in the right way, are all helpful.

The first is the total energy in the fuel. That is to say, the total internal energy of the fuel. This is the very direct energy cost of heating up the fuel to the required temperature. Using this definition is helpful because there are some very simple and fundamental calculations that can be done, which I will come back to shortly. The definition does get a bit complicated because the fusion process itself heats up the fuel, adding internal energy, so the formal definition is the maximum total internal of the fuel if the fusion process was magically switched off. If we are talking about deuterium-tritium (DT) fuel, it is like the total internal energy if it was just deuterium with no tritium (DD), which has the effect of basically turning off the reactions due to the much lower reactivity of pure deuterium. When defined this way, we speak about “fuel gain”.

The next definition is “target gain”, or in magnetic fusion simply “Q”, and to define the energy input in this case we draw an imaginary box around the reaction chamber. We look at the energy that goes into the reaction chamber, passing through the wall of the imaginary box. For laser inertial fusion, this is the energy in the laser light. For First Light, it is the projectile kinetic energy. For a tokamak we look at the total heating power in all forms. This definition is useful because it cleanly separates physics from engineering. If you have the same energy going into the reaction chamber in the same form, the physics of the fusion process itself will be the same. Improvements in the efficiency of the driver or the heating systems improve the overall energy balance, but the target gain or plasma Q is the same.

The third definition I will advance then takes into account the energy input to the heating system, rather than the output. This is clearly getting much closer to the needs of the power plant, but we’re just looking at the heating systems. We are still not considering all the plant auxiliaries. When using the energy input to the driver, the resulting ratio is called the “engineering gain”.

How the different definitions of fusion gain correspond to the different energy inputs

The most common term is target gain, or Q. If you are reading something from the fusion field and there is no additional qualification about the type of gain being discussed, it is probably a fair assumption that it is target gain. This does lead to a lot of “gotcha!” moments where people feel the need to point out that the driver efficiency hasn’t been considered. It’s a bit frustrating to be honest, it’s kind of like pointing out to a solar developer that the sun doesn’t shine at night. We know.

Moving on, there is a very simple calculation that we can do in terms of fuel gain, one of the reasons this is a helpful definition. For a plasma in thermal equilibrium we can write down an equation for the average energy per particle.

Now we get the benefit of working with keV as our unit of temperature, where k, Boltzmann’s constant, is equal to one. This allows us to very easily convert the ideal ignition temperatures we found previously into internal energy of the fuel, the energy we must invest to get it hot. And this is the minimum possible energy required, this is literally just the energy of the particles themselves, we are not accounting for any inefficiencies here. Nor are we accounting for energy lost to radiation or any other energy sink, which will matter if we are right at the ideal ignition temperature.

This expression is the average energy per particle. To calculate the possible gain we also need to understand the total number of particles and the energy release. We can do this by looking on a pairwise basis. All four candidate fuels are binary reactions with two inputs. If we look at how many particles we have as inputs to the reaction, we can define the internal energy, and then we can look at the energy released from fusion in each case.

The only trick is to remember that the electrons involved can’t just disappear. They are still there, and they have the same average energy as the ions, they count towards the total. For DT fuel, we have two hydrogen isotopes, which means each ion has one electron, and overall we have four particles. For the boron in the pB reaction, we have to pay to heat up five electrons, plus the boron ion itself, and then the hydrogen constitutes a further ion-electron pair, a total of eight particles.

The different atomic structures of the different fuels for fusion

Going through and crunching the numbers, these are the values.

FuelNumber of particlesEnergy released (MeV)Ideal ignition temperature (keV)Max fuel gain
Maximum fuel gain based on the ideal ignition temperature for each fuel. There is no temperature where the fusion power exceeds the radiation loss for proton-boron (pB) fuel, so this entry is blank

It shouldn’t be a surprise that DT has the highest possible gain. After that we have DHe3, and then DD. I didn’t include pB in here because the fusion power cannot exceed the radiation losses*, and so the ideal ignition temperature is not defined. We can also look at the temperature with the maximum ratio of the fusion power to the radiation loss, which at least gives a well-posed temperature for pB. In this case the higher temperatures across the board mean more invested energy and lower maximum fuel gain.

* Again, I will come back to pB in a later point

FuelTemperature of max power (keV)Max fuel gain
Maximum fuel gain based on the temperature at which the fusion power most greatly exceeds the radiation loss

As the fuel gain is a function of temperature, we can plot it. This really makes the advantage of DT very clear. The maximum gains are a lot higher.

Maximum fuel gain vs temperature for the four fuels, showing the extent of the advantage of DT

And to finish, I am characterising this as the maximum possible fuel gain. The arguments above about the internal energy being the absolute minimum energy investment are part of that, but there is also another simple reason. This calculation assumes that the entire fuel mass is fused, a “burn fraction” of 100%. This is not possible in any real system. The burn fractions in inertial fusion can reach more than 50%, but not 100%. And in magnetic fusion, even 2% is a high number. We need to account for this, and the different efficiencies in the rest of the system, to know if power production is possible.

That will be the subject of the next post, which will show that power production is likely not possible; the gain limit is too low, even for DT. But fortunately for fusion power, there are ways to work around the limit, to change the nature of the process and tip the scales of the energy balance. Although whether you can do this depends on your approach.

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.

The nuclear physics of why tritium is a challenge for fusion engineering

The production, or breeding, of tritium is one of the major engineering challenges of fusion. The easiest fusion reaction uses deuterium and tritium; it requires the lowest temperature and can return the most energy. The core reaction fuses one deuterium and one tritium, producing one alpha particle, which is to say a helium 4 nucleus, and one neutron. The alpha particle has one fifth of the energy (3.5 MeV) and the neutron takes the remaining four fifths (14.1 MeV).

The basic DT fusion reaction

This introduces a challenge because tritium is not a stable isotope. The half-life is 12 years; medium length on a human timescale, but nothing on geological or astrophysical timescales. This means that there is no natural abundance, it decays too quickly, and therefore any fusion power plant must produce its own tritium in a closed cycle.

There are many reasons that tritium is challenging. It is not a stable isotope, as in, it is radioactive. It is also hydrogen, which makes it “leaky”, able to percolate through containment structures. And it is also chemically identical to hydrogen, as in, it is flammable. This all complicates the plant required to extract, purify, store, and recirculate produced tritium into the machine. For these reasons tritium is difficult, but these things are solvable. There is, however, a deep-seated nuclear physics constraint that makes closing the fuel cycle, producing enough tritium, very challenging for some fusion power plant designs.

Tritium is produced using a reaction between the neutron from the fusion reaction itself and lithium. The output of this reaction is another helium 4 and one tritium. The problem is that we have used exactly one tritium to make exactly one neutron, to make exactly one tritium. To make this work we would need to capture every single neutron. Every single one must react with lithium and produce tritium. And we’d have to usefully extract every single atom of tritium produced. Neither of these things are possible.

The fuller “fuel chain”, including the production of tritium using lithium

Fortunately, this is still a simplification, one commonly used and it is normally where the story stops. In fact, there are two stable isotopes of lithium, lithium 6 and lithium 7, and the reaction with the neutron is different between the two. With lithium 7, we get a neutron back out again. This neutron can then go on to interact with another lithium and produce a second tritium. For one neutron in, we can get more than one tritium out.

The even fuller fuel chain, now accounting for the two stable isotopes of lithium. The lithium 7 reaction produces a neutron as an output, which allows further tritium production, essential to closing the loop and being tritium self-sufficent

But again, this is STILL a simplification! There are many things that can happen when a neutron collides with a lithium nucleus. The TENDL-2019 database of neutron cross sections has 88 entries for lithium. All of these entries describe a different possible outcome from the event. Only two produce tritium, and only five produce further neutrons, of which the reaction with lithium 7 mentioned above is the most likely. The rest are mainly different types of “scattering”, different ways that the neutron just bounces off the lithium nucleus.

Each of these processes has a different likelihood of occurring, and that likelihood or “cross-section”, depends on the energy of the neutron coming in. The lithium 7 reaction that gives both a neutron and a tritium works well at the “birth” energy of the neutron, 14.1 MeV, the amount of energy it has just after fusing. However, the lithium 7 reaction is endothermic, meaning that the energy of the output neutron is less than the input neutron, which means that the probability of another lithium 7 reaction is lower. You can’t have an endless process going round and round producing more and more tritium each time. And anyway, scattering is overall the more likely outcome, which also lowers the neutrons energy and makes the lithium 7 reaction less likely. This reaction falls off a cliff as the neutron energy goes down, and that is the direction that nature is taking it.

Eventually, at low energies, when we have “thermal neutrons”, the tritium producing reaction with lithium 6 becomes the most likely process, so thermal neutrons produce tritium very well, but this reaction consumes the neutron.

Neutron cross-section data for both lithium 6 and lithium 7, highlighting the tritium and neutron producing reactions. Nuclear data taken from Jon Shimwell’s super helpful plotter.

So, thinking of the process as one reaction with lithium 6 and another with lithium 7 is a pretty good approximation. But the outcome of all the detail about energies, cross-sections, and scattering is that there is a limit to the maximum number of tritium atoms one can produce per neutron. In the field this is called the “tritium breeding ratio” and the nuclear physics says it can never be higher than 2. It can be changed by changing the ratio of the two isotopes of lithium, a common proposal for fusion power plant designs struggling to close the loop and produce enough tritium, a very challenging proposal as there is no supply chain for enriched lithium, but it can’t get above this limit of 2.

There are also other ways to introduce neutron multiplication. There are spallation-like reactions with both beryllium and lead that are often discussed. Beryllium is often introduced through the use of FLiBe, a blend of the salts lithium fluoride and beryllium fluoride, acting as a molten salt coolant. And lead is often introduced through the use of lead-lithium eutectic mixture as the coolant. Both of these options lead to the liberation of a neutron from the target nucleus, i.e. the beryllium or the lead, and produce a lighter isotope of the same as the other product. Regardless of all the options, and even combining these with the option of enrichment of the lithium, it is not possible to produce more than two tritium atoms per neutron.

And these additions that can improve the tritium production do not come for free. Beryllium is highly toxic, as is lead. FLiBe introduces fluorine chemistry, and yes, tritiated hydrofluoric acid will be produced. Lead also has a highish atomic number and the range of different isotopes produced rather challenges the “no radioactive waste” statement, polonium being one of the eventual products.

Some suggest using fissionable material, e.g. uranium. The neutron multiplication possible by this route is higher, and the tritium breeding ratio can be greater than two. But why bother? My view is that if you are going to have uranium in your plant, you may as well build a fission plant. You will have all the same difficulties to deal with and fission already works.

This fundamental limit on the amount of tritium that can be produced per neutron is the deep-seated reason that closing the fuel cycle is a challenge. If you cannot capture more than 50% of the neutrons for tritium production, you fundamentally cannot close the loop and be tritium self-sufficient. Space used for other things eats into the total amount of neutrons being captured. In magnetic fusion, neutral beam heaters, RF heaters, the magnets themselves, the central column in a tokamak, the first wall and a host of other things all eat into the space available. In laser inertial fusion, the entrance ports for the many laser beams have the same impact. And in designs that are cylindrical in nature, the ends of the cylinder cannot typically be used for tritium production.

Fortunately for First Light, we can capture 99% of the neutrons in the flowing lithium coolant inside the reaction vessel. We can achieve a tritium breeding ratio of up to ~1.5 simply with normal lithium, no additives, and no enrichment. Any when I say fortunately, what I of course mean is, we deliberately designed it that way.

Learn more here…

The NIF gain result, how they did it and why power plant capsules will be easier to design

We have heard a lot about what NIF did a few weeks ago, but how did they do it? This is my understanding, at least, based on the technical briefing from the announcement.

Indirect Drive

First things first, NIF uses indirect drive, a detail often skipped for a lay audience. The laser does not shine directly onto the fuel capsule. Instead, it illuminates the inside of a gold cylinder. When laser light is absorbed in a high-Z material like gold, it mainly heats it up. The inside of the gold cylinder gets so hot that it emits x-rays.

Gold is very effective at both absorbing and emitting x-ray radiation. Inside the cylinder x-ray photons are constantly being emitted, travelling across the cylinder, and being absorbed. If you paused time, so that all the photons stopped for a moment, you could look at where they are, where they are going, and the energy they have. This sea of photons is the “radiation field” inside the cavity. The constant emission and absorption bring this radiation field into thermal equilibrium with the temperature of the gold wall. What you get is a blackbody source of x-rays, with a Planckian spectrum. The gold cylinder has become a “hohlraum”.

The NIF lasers shine on the inside of a gold cylinder, heating it up and forming a bath of x-rays. It is the x-rays that hit the capsule, not the lasers, causing it to implode.

It is the x-rays that then impinge on the capsule. The capsule is made of a low-Z material, carbon, and when this kind of material absorbs radiation, it heats up, like the gold, but also expands outwards very rapidly. Every action has an equal and opposite reaction; the capsule implodes.

The trouble with this is that what you want is perfectly uniform x-ray radiation everywhere on the capsule surface. What you get is not uniform. There are ends in the gold cylinder, which have to be there to let the lasers in, but they inevitably let some x-rays out. Then there are a finite number of laser beams. This means that the inside wall of the cylinder is not uniform in temperature, it has hot spots where the lasers hit and cooler spots elsewhere. And unfortunately, the capsule itself is in the middle, right in the way of any x-ray trying to cross directly through the centre of the cylinder. Drive asymmetry is inevitable.

The pattern of laser absorption on the inside of a hohlraum from NOVA, a predecessor to NIF. The walls are not uniform in temperature, there are hot spots where the lasers hit.

The Available Energy is Fixed

There is a decade-long story of design evolution at NIF. The first megajoule shot, from Aug 2021, was the HYBRID-E design, and it is almost certain that the Dec 2022 gain shot was also HYBRID-E. The HYBRID approach (“high yield big radius implosion design”, apparently) uses bigger capsules. But the whole design was not scaled up. Proportionally, the shell of the capsule was thinner to maintain the required implosion velocity.

This brings us to one of the key trade-offs for inertial fusion – you have a fixed budget of energy; the facility is the constant. A bigger capsule will have more fuel and higher potential yield. It will also lead to longer confinement time, one of the three parameters that make up the fusion triple product. But the mass of the capsule will be larger; at fixed energy, the implosion velocity must be lower. It is the velocity which sets, more than anything else, the temperature, another part of the triple product. Fusion performance is a function of both mass and velocity.

Fusion performance is a function of both mass and velocity, with more of both best. But the facility energy is fixed, which puts a cap on kinetic energy and leads to complex trade-offs.

Unless you thin the capsule down, make it bigger but with proportionally thinner walls, exactly the HYBRID strategy. Unfortunately, that walks straight into a second key aspect of fusion, which is instability. The thinner the shell the less robust it will be to instabilities. Any particular instability will progress at a given speed, and all else being equal, with a thinner shell the perturbations will be proportionally larger. And like drive asymmetry, seeds for instability are inevitable. They can come from defects in the shell, engineering structures such as the “tent” which holds the capsule in the centre of the cylinder, or the fill tube used to introduce the fuel.

The decade-long design challenge has been to make best use of the total laser energy, whilst keeping the drive uniform, and without falling foul of instabilities.

The HYBRID-E Design

The HYBRID designs use bigger capsules with thinner walls. The HYBRID-E design, specifically, uses a capsule that is also proportionally larger when compared to the hohlraum. This leads to drive asymmetry. The “equator” of the capsule, the bit halfway along the length of the gold cylinder, gets less x-rays. If you imagine standing on that point of the capsule and looking out at the hohlraum wall, as the capsule gets larger, you see less of the wall “head on”. You see the surface at a higher angle. Like the sun hitting the Earth at high latitudes, the energy per unit area is less.

To fix this, the HYBRID-E design uses “cross-beam energy transfer” or CBET. This is a seriously complex quantum and kinetic phenomenon, the essence of which is that two laser beams overlapping with each other in a plasma environment can exchange energy. It is sort of like a gravitational slingshot but swapping energy between laser beams instead a satellite and a planet. On NIF there are beams coming in at four different angles. Some energy is borrowed from the ends of the cylinder and redirected towards the middle through control of CBET, compensating for the lack of drive at the equator.

Typical energy transfers between beam angles in the HYBRID-E design. A significant proportion of the beam energy at 44 degrees is reapportioned.

CBET is controlled by separating the wavelengths of the different lasers by a small amount. It is a very empirical science but with a database of previous shots to study, and with a shoot-then-correct approach, implosions can be tuned from oval final states back to round.

Controlling the wavelength separation of the lasers controls the cross-beam energy transfer (CBET), allowing implosions to be tuned from oblate to round to prolate.

Another important feature of HYBRID-E seems to be smaller laser entrance holes. This reduces the x-ray losses and makes the overall hohlraum more efficient. A more efficient hohlraum gives more flexibility in using that fixed total energy, allowing the pulse to be extended slightly, also a key feature of HYBRID-E.

Dec 2022 Gain Shot

So what did they do to get to gain? They said they made the shell thicker. That is going to increase the mass and therefore drop the implosion velocity. Past work shows that going to DT ice thickness of 65 um, compared to 55 um, dropped the implosion velocity from 400 to 385 km/s; small changes have large effects. The key is that the Dec 2022 shot was “matched with higher laser energy”. The laser had 8% more energy, which implies 4% more velocity, and which in principle takes you back up to 400 km/s.

They also described the thicker shell as having “more margin”, another way of saying more robustness to instabilities. They said the Aug 2021 shot was their “most pristine shell ever”, whereas the Dec 2022 shot was described as having “tungsten inclusions in large number”, i.e. lots of little specks of tungsten in the shell. Tungsten is very dense so each of these little specks locally increases the mass. Each one becomes a tiny anchor on the implosion, locally slowing the velocity and distorting the shape. A thicker shell means more of this can be tolerated.

They also said that the thicker shell “burns more fuel”. This is to do with the amount of shell left at the end of the implosion. Most of the diamond is ablated away, typically less than 10% of the total mass remains. Whatever does remain holds the fuel in place. If there is more left, it holds it a little longer, the confinement time is increased, and more fuel can be burnt.

And the last piece is the symmetry. They said that they had a shot in Sept 2022 with a 1.22 MJ yield, and that the only change between that shot the Dec one was tuning the symmetry, i.e. controlling the CBET, i.e. tuning the wavelength separation. Sept will have been their best guess at an optimised design, which they then will have observed to be slightly oval. Reaching back into the data archive and figuring out the just-so tweak to get it back to round has raised the yield from 1.22 to 3.05 MJ.

More Energy Makes Everything Easier

Bigger is easier for fusion, and now it is hopefully clear why. Overall, an 8% increase in laser energy led to 230% increase in fusion yield. With more laser energy, NIF could have just gone straight for the bigger, thicker capsule and still maintained the implosion velocity. The delicate balance played here between mass, velocity and instabilities wouldn’t have been so delicate.

And tweaking of the implosion symmetry through the very complex cross-beam energy transfer process could also have been avoided. We know, now, that that particular blob of round plasma they created in the Dec 2022 shot ignites. Now imagine an oval blob of plasma, but one where the round version fits entirely inside the oval. If the round one ignites, the bigger oval one will ignite too. But, of course, the bigger oval requires more energy.

Power plants will use higher energy drivers than NIF. More energy gives you more margin; capsules for power plants will be easier to design. And, of course, the energy of the driver itself is not really the key parameter, not for a plant, it is how much it costs that matters, and how much energy it produces. This is one of the deepest reasons why I believe First Light Fusion can succeed. We have a much cheaper driver, we can afford more energy, and we will have a more robust approach because of it.


How NIF got to gain, explainer by @FLF_Nick! #fusion

Our New Design Point for Inertial Fusion Explained

Two weeks ago, we published a paper in the Philosophical Transactions of the Royal Society. The paper is an economic model for inertial fusion and demonstrates a new design point for a power plant. Objectively the most important conclusion is probably that fusion can generate power for the same price as renewables, which are currently our cheapest source of energy. Fusion will never be “too cheap to meter”, that was always a pipedream, but it can be the cheapest source of baseload power.

Objectively, this probably is the most important message, but it’s what our new design point means for the engineering and physics challenges that I’m most excited about. It is natural to expect a direction with lower cost to be more difficult but, in this case, it is both lower cost and lower risk, for reasons I will try to explain.

All inertial fusion, in the most general sense, works in the same way and involves a big machine called a “driver”. At First Light this is a pulsed power machine that electromagnetically launches a projectile, like a railgun. At the National Ignition Facility, the driver is a laser, at Sandia National Lab, the driver is a different type of pulsed power machine.

The driver puts a huge amount of energy into a small “target” in a very short space of time. In First Light terms, the projectile hits the target. The target focusses the energy into fusion fuel, which burns very quickly and releases a pulse of energy.

It is less than one millionth of a second from the projectile hitting the target to the fusion event, and the pulse of energy released is shorter than a billionth of a second. The way you make continuous power with inertial fusion is to repeat the process again and again. Inertial fusion is a pulsed process; it’s like an internal combustion engine: inject fuel, spark, burn, reset.

Inertial fusion is a pulsed process, like an internal combustion engine. To make continuous power, you repeat again and again: inject fuel, spark, burn, reset.

When designing a power plant this means we have an extra free parameter. The same amount of power can be generated with a big energy release per shot and a slow frequency, or a smaller energy release and higher frequency; power is energy times frequency.

P = Ef

Given that we are free to pick, the crucial question is, which is the cheapest? A larger energy release might require a bigger driver, which will cost more, but the slower frequency means less targets are used, so they will cost less. What the paper does is build the simplest possible fully formed levelised cost model, accounting for all the main costs.

The levelised cost is the most common way to compare different energy technologies. Specifically, it is the price for which the energy would need to be sold to make the net present value of the investment zero. That is, on average you must sell the energy for more than this price to make a profit. The normal unit of levelised cost is dollars per megawatt-hour ($/MWh). To give some indicative numbers, solar and onshore wind are $35 / MWh, offshore wind $50 / MWh, gas $80 / MWh, and nuclear $100 / MWh.

The basic trade-off given above doesn’t seem too difficult, but it is more complex than just that. An example of a more complex issue is the lifetime of the driver, which is one of the difficult engineering challenges. Our current driver, Machine Three, uses spark gap switches with a lifetime of perhaps 1000 shots, nowhere near long enough for a power plant.

Even with a longer lifetime, one of the major costs in the model is the replacement of driver components as they wear out. This means that if the frequency is slower, the driver lasts proportionally longer, and this affects the final cost.

The model has fourteen parameters and the paper uses a Monte Carlo approach to explore how these parameters interact and affect the levelised cost. The key result is how the cost varies with the frequency.

The essential feature of The Monte Carlo method is randomness, in fact it is named after the Monte Carlo casino in Monaco for precisely this reason. The method is often used to assess integrals, but here is it used to explore the properties of the model. If the model had two parameters, we could pick a range of 20 values for each, and that would give 400 combinations. But it has fourteen, which would mean more than a billion billion combinations. This is the problem the Monte Carlo approach solves. The input parameters for the model are picked randomly from given ranges and the resulting cost is found. With a big enough sample size, the characteristics of the system can be understood whilst running the model far fewer times.

Levelised cost of energy (LCOE) in dollars per megawatt-hour ($/MWh) vs frequency . The results show an optimum frequency around 0.2 Hz, or once every 5 seconds.

If you’re new to Monte Carlo analysis, this plot probably needs some explanation. The model has been run repeatedly, varying all of the parameters all at the same time. Even though everything is changing from one point to the next, we can still just plot the cost against any individual parameter such as frequency. When we do this, we don’t get a nice simple line, we get a cloud of points. There are many points with the same frequency but different costs, that is the influence of the other parameters. What we are interested in is the lower edge of the cloud. This is the minimum cost at that frequency regardless of everything else. What the lower edge of the cloud shows is that there is an optimum around 0.2 Hz.

Previous studies are, more often than not, discussing much higher frequencies, around 5 Hz. This seems surprising given the optimum we have just found. To see what is going on we need another piece of information, which is the energy per shot. What I’ve done is colour the points on the plot, putting them into three buckets, low, medium and high energy.

Levelised cost of energy (LCOE) in dollars per megawatt-hour ($/MWh) vs frequency, where the points have been split into three groups depending on the energy released per shot. This shows that the optimum frequency gets slower for larger energy per shot.

What this shows is that the optimum depends on the energy per shot. For low energy, the yellow points, the optimum is around 3 Hz. This energy per shot matches that assumed in previous studies. This is important because it gives us confidence in our model; the previous authors were at the optimum, but within their design space. The crucial thing that changes the conclusion is a larger energy per shot, something the First Light plant design can handle.

Digging in further also revealed something else very interesting. For low energy per shot, the plants near the optimum would be very big. I had a limit at 2 GW electrical (GWe) and these plants are right up against that limit. Higher energy can, counter-intuitively, lead to a smaller plant overall.

Levelised cost of energy (LCOE) in dollars per megawatt-hour ($/MWh) vs frequency, where the points have been split into three groups depending on the plant power. This shows that the lowest cost slower frequency designs are generally also smaller plants.

These findings, taken together, are huge. Not only is it simply cheaper, but everything about this new design point is also easier. The engineering risk is reduced on two major counts. First, the slower the better. Many of the engineering challenges specific to inertial fusion, including the driver lifetime, lessen as the frequency falls.

Second, the smaller the better. At a plant size of 150 MWe, the balance of plant is very low risk. Instead of being something bespoke, it is more likely to be something off the shelf, a design borrowed, which reduces risk hugely.

The smaller plant size also translates to a much lower capital cost, which makes financing a plant much easier. The upfront price tag is the other cost that matters, besides the levelised cost. Again, this couples back to the engineering risk because you can afford to iterate

A large driver and a high energy per shot also reduces the physics risk. I don’t have the space to substantiate that here, but it really does.

This is what excites me about the paper. The model points the way to something that could be very cost competitive, but this is about the destination. Everyone in fusion gets told the “20 years away and always will be” joke. The criticism that sits behind this line is not that fusion power isn’t worth it, but that it’s too difficult. What this paper has found is a road to the destination that is a lot easier to traverse, that is what is important here.

And the key thing, in the end, that makes all this work is a cheaper driver technology, which is exactly what First Light is trying to prove.

First Light’s fusion driver, Machine 3. A pulsed power driver of this kind is relatively low cost. This allows economical construction of high energy drivers, which is one method to increase the energy released per shot, unlocking the new plant design point. Machine Three isn’t designed for rep rated operation, but it is part of a roadmap in the right direction.

Check out this blog post by @FLF_Nick about @FLFusion power plant design. #fusion