# Fusion energy gain factor

The explosion of the Ivy Mike hydrogen bomb. The hydrogen bomb is the only known man-made item to achieve fusion energy gain factor larger than 1.

The fusion energy gain factor, usually expressed with the symbol Q, is the ratio of fusion power produced in a nuclear fusion reactor to the power required to maintain the plasma in steady state. The condition of Q = 1, when the power being released by the fusion reactions is equal to the required heating power, is referred to as breakeven.

The power given off by the fusion reactions may be captured within the fuel, leading to self-heating. Most fusion reactions release at least some of their energy in a form that cannot be captured within the plasma, so a system at Q = 1 will cool without external heating. With typical fuels, self-heating does not match the external sources until about Q = 5.

As Q increases past this point, the increasing amount of self-heating eventually removes the need for external heating. At this point the reaction becomes self-sustaining, a condition known as ignition. Ignition corresponds to infinite Q, and is generally regarded as highly desirable for a practical reactor design.

As of 2017, the record for Q is held by the JET reactor in the UK, at 0.67. ITER was originally designed to reach ignition, but is currently designed to reach Q = 10, producing 500 MW of fusion energy with a 50 MW heating system.

## Concept

Q is simply the comparison of the power being released by the fusion reactions in a reactor, Pfus, to the constant heating power being supplied, Pheat. However, there are several definitions of breakeven that consider additional power losses.

### Scientific breakeven

In a successful fusion reactor design, the fusion reactions generate an amount of power designated Pfus. Some amount of this energy, Ploss, is lost through a variety of mechanisms, mostly convection of the fuel to the walls of the reactor chamber and various forms of radiation that cannot be captured to generate power. In order to keep the reaction going, the system has to provide heating to make up for these losses, where Ploss = Pheat to maintain thermal equilibrium.

The most basic definition of breakeven is when Q = 1, that is, Pfus = Pheat. This is sometimes known as scientific breakeven to make this definition clear.[1] Many early machines were not designed to run on actual fusion fuels, notably tritium which presents safety issues, in which case the term extrapolated breakeven was used to define the expected performance running on D-T fuel from the performance using hydrogen or deuterium alone.[2]

### Engineering breakeven

The second definition of breakeven, engineering breakeven, considers the need to extract the energy from the reactor, turn that into electrical energy, and feed that back into the heating system.[2] This closed loop is known as recirculation. In this case, the basic definition changes by adding additional terms to the Pfus side to consider the efficiencies of these processes.

Most fusion reactions release energy in a variety of forms, mostly neutrons and a variety of charged particles like alpha particles. Neutrons are electrically neutral and will travel out of any magnetic confinement fusion (MFE) design, and in spite of the very high densities found in inertial confinement fusion (ICF) designs, they tend to easily escape the fuel mass in these designs as well. This means that only the charged particles from the reactions can be captured within the fuel mass and give rise to self-heating. If the fraction of the energy being released in the charged particles is fch, then the power in these particles is Pch = fchPfus.

If this self-heating process is perfect, that is, all of Pch is captured in the fuel, that means the power available for generating electricity is the power that is not released in that form, or (1 − fch)Pfus. This includes not only the energy in the neutrons but also the losses to the environment through other processes. However, in most designs these other losses are minimized as a design feature, and not easily used for power generation.

In the case of neutrons carrying most of the practical energy, as is the case in the deuterium-tritium (D-T) fuel studied in most designs, this neutron energy is normally captured in a "blanket" of lithium that produces more tritium that is used to fuel the reactor. Due to various exothermic and endothermic reactions, the blanket may have a power gain factor a few percent higher or lower than 100%, but that will be neglected here. The blanket is then cooled and the cooling fluid used in a heat exchanger driving conventional steam turbines. These have an efficiency ηelec which is around 35 to 40%.

Consider a system that uses external heaters to heat the fusion fuel, then extracts the power from those reactions to generate electrical power. Some fraction of that power, frecirc, is needed to recirculate back into the heaters to close the loop. This is not the same as the Pheat because of the self-heating processes. While the system as a whole requires additional power for building heating, lighting, and the confinement system, these are generally much smaller than the heating system requirements.

Considering all of these factors, the heating power can thus be related to the fusion power by the following equation:

${\displaystyle P_{heat}=\eta _{heat}\cdot f_{recirc}\cdot \eta _{elec}\cdot (1-f_{ch})\cdot P_{fus}}$

The fusion energy gain factor is then defined as:

${\displaystyle Q\equiv {\frac {P_{fus}}{P_{heat}}}={\frac {1}{\eta _{heat}\cdot f_{recirc}\cdot \eta _{elec}\cdot (1-f_{ch})}}}$

### Ignition

As the performance of the plasma improves, the amount of self-heating improves as well. Eventually fheat reaches zero, that is, all of the energy needed to keep the plasma at the operational temperature is being supplied by self-heating, and the amount of external energy that needs to be added drops to zero. This point is known as ignition.

Ignition, by definition, corresponds to an infinite Q, but it does not mean that frecirc drops to zero as the other power sinks in the system, like the magnets and cooling systems, still need to be powered. Generally, however, these are much smaller than the energy in the heaters, and require a much smaller frecirc. More importantly, this number is more likely to be near constant, meaning that further improvements in performance will result in more energy that can be directly used for commercial generation, as opposed to recirculation.

### Commercial breakeven

The final definition of breakeven is commercial breakeven, which occurs when the economic value of any net energy left over after recirculation is enough to finance the construction of the reactor.[2] This value depends both on the reactor and the spot price of electrical power.[2]

Commercial breakeven relies on factors outside the technology of the reactor itself, and it is possible that even a reactor with a fully ignited plasma will not generate enough energy to pay for itself. Whether any of the mainline concepts like ITER can reach this goal is being debated in the field.[3]

## Practical example

Most fusion reactor designs being studied as of 2017 are based on the D-T reaction, as this is by far the easiest to ignite, and is energy dense. However, this reaction also gives off most of its energy in the form of a single highly energetic neutron, and only 20% of the energy in the form of an alpha. Thus, for the D-T reaction, fch = 0.2. This means that self-heating does not become equal to the external heating until at least Q = 5.

Efficiency values depend on design details but may be in the range of ηheat = 0.7 (70%) and ηelec = 0.4 (40%). The purpose of a fusion reactor is to produce power, not to recirculate it, so a practical reactor must have frecirc = 0.2 approximately. Lower would be better but will be hard to achieve. Using these values we find for a practical reactor Q = 22.

The one channel of energy loss that is independent of the confinement scheme and practically impossible to avoid is Bremsstrahlung radiation.[dubious ] Like the fusion power density, the Bremsstrahlung power density depends on the square of the plasma density, but it does not increase as rapidly with temperature. By equating the two power densities, one can determine the lowest temperature for which the fusion power can overcome the Bremsstrahlung power. This ignition temperature is about 4 keV for the D-T reaction and about 35 keV for the D-D reaction.

## References

1. ^ "Scientific Breakeven for Fusion Energy" (PDF). Princeton Plasma Physics Laboratory.
2. ^ a b c d Razzak, M. A. "Plasma Dictionary". Nagoya University.
3. ^ Hirsch, Robert (Summer 2015). "Fusion Research: Time to Set a New Path". Issues in Technology. Vol. 31 no. 4.