# 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.

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.

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”.

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.

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

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

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.

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.