Fusion energy gain factor

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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, or in some sources, scientific 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 in fusion reactors is not expected to match the external sources until at least Q = 5. If Q increases past this point, increasing self-heating eventually removes the need for external heating. At this point the reaction becomes self-sustaining, a condition called ignition. Ignition corresponds to infinite Q, and is generally regarded as highly desirable for a practical reactor design.

Over time, several related terms have entered the fusion lexicon. As a reactor does not cover its own heating losses until about Q = 5, the term engineering breakeven is sometimes used to describe a reactor that produces enough electricity to provide that heating. Above engineering breakeven a machine would produce more electricity than it uses, and could sell that excess. A machine that can sell enough electricity to cover its operating costs, estimated to require at least Q = 20, is sometimes known as economic breakeven. Additionally, fusion fuels, especially tritium, are very expensive, so many experiments run on various test gasses like hydrogen or deuterium. A reactor running on these fuels that reaches the conditions for breakeven if tritium was introduced is said to be operating at extrapolated breakeven.

As of 2017, the record for Q is held by the JET tokamak in the UK, at Q = (16 MW)/(24 MW) ≈ 0.67, first attained in 1997. ITER was originally designed to reach ignition, but is currently designed to reach Q = 10, producing 500 MW of fusion power from 50 MW of injected thermal power. The highest record for extrapolated breakeven was posted by the JT-60 device, with Qext = 1.25.


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.


In 1955, John Lawson was the first to explore the energy balance mechanisms in detail, initially in classified works but published openly in a now-famous 1957 paper. In this paper he considered and refined work by earlier researchers, notably Hans Thirring, Peter Thonemann, and a review article by Richard Post. Expanding on all of these, Lawson's paper made detailed predictions for the amount of power that would be lost through various mechanisms, and compared that to the energy needed to sustain the reaction.[1] This balance is today known as the Lawson criterion.

In a successful fusion reactor design, the fusion reactions generate an amount of power designated Pfus.[a] 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.[2]

The most basic definition of breakeven is when Q = 1,[b] that is, Pfus = Pheat.

Some works refer to this definition as scientific breakeven, to contrast it with similar terms.[3][4] However, this usage is rare outside certain areas, specifically the inertial confinement fusion field, where the term is much more widely used.

Extrapolated breakeven[edit]

Since the 1950s, most commercial fusion reactor designs have been based on a mix of deuterium and tritium as their primary fuel; others fuels have been studied for a variety of reasons but are much harder to ignite. As tritium is radioactive, highly bioactive and highly mobile, it represents a significant safety concern and adds to the cost of designing and operating such a reactor.[5]

In order to lower costs, many experimental machines are designed to run on test fuels of hydrogen or deuterium alone, leaving out the tritium. In this case, the term extrapolated breakeven is used to define the expected performance of the machine running on D-T fuel based on the performance when running on hydrogen or deuterium alone.[6]

The records for extrapolated breakeven are slightly higher than the records for scientific breakeven. Both JET and JT-60 have reached values around 1.25 (see below for details) while running on D-D fuel. When running on D-T, only possible in JET, the maximum performance is about half the extrapolated value.[7]

Engineering breakeven[edit]

Another related term, 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.[6] 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.

In the case of neutrons carrying most of the practical energy, as is the case in the 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 the self-heating processes are providing some of the required energy. While the system as a whole requires additional power for building climate control, lighting, and the confinement system, these are generally much smaller than the plasma heating system requirements.

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

where is the efficiency that power supplied to the heating systems is turned into heat in the fuel, as opposed to lost in the equipment itself, and is the efficiency achieved when turning the heat into electrical power, for instance, through the Rankine cycle.

The fusion energy gain factor is then defined as:


As the temperature of the plasma increases, the rate of fusion reactions grows rapidly, and with it, the rate of self heating. In contrast, the non-capturable energy losses like x-rays do not grow at the same rate. Thus, in overall terms, the self-heating process becomes more efficient as the temperature increases, and less energy is needed from external sources to keep it hot.

Eventually Pheat 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 plasma performance will result in more energy that can be directly used for commercial generation, as opposed to recirculation.

Commercial breakeven[edit]

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.[6] This value depends both on the reactor and the spot price of electrical power.[6][8]

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.[9]

Practical example[edit]

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.

Transient vs. continual[edit]

Many early fusion devices operated for microseconds, using some sort of pulsed power source to feed their magnetic confinement system and used the confinement as the heating source. Lawson defined breakeven in this context as the total energy released by the entire reaction cycle compared to the total energy supplied to the machine during the same cycle.[7]

Over time, as performance increased by orders of magnitude, the reaction times have extended from microseconds to seconds, and in ITER, on the order of minutes. In this case definition of "the entire reaction cycle" becomes blurred. In the case of an ignited plasma, for instance, Pheat may be quite high while the system is being set up, and then drop to zero when it is fully developed, so one may be tempted to pick an instant in time when it is operating at its best to determine Q. A better solution in these cases is to use the original Lawson definition averaged over the reaction to produce a similar value as the original definition.[7]

However, there is a complication. During the heating phase when the system is being brought up to operational conditions, some of the energy released by the fusion reactions will be used to heat the surrounding fuel, and thus not be released. This is no longer true when the plasma reaches its operational temperature and enters thermal equilibrium. Thus, if one averages over the entire cycle, this energy will be included as part of the heating term, that is, some of the energy that was captured for heating would otherwise have been released in Pfus and is therefore not indicative of an operational Q.[7]

Operators of the JET reactor argued that this input should be removed from the total:


That is, Ptemp is the amount of energy needed to raise the internal energy of the plasma. It is this definition that was used when reporting JET's record 0.67 value.[7]

Some debate over this definition continues. In 1998, the operators of the JT-60 claimed to have reached Q = 1.25 running on D-D fuel, thus reaching extrapolated breakeven. However, this measurement was based on the JET definition of Q*. Using this definition, JET had also reached extrapolated breakeven some time earlier.[10] If one considers the energy balance in these conditions, and the analysis of previous machines, it is argued the original definition should be used, and thus both machines remain well below break-even of any sort.[7]

Scientific breakeven at NIF[edit]

Although most fusion experiments use some form of magnetic confinement, another major branch is inertial confinement fusion (ICF) that mechanically presses together the fuel mass (the "target") to increase its density. This greatly increases the rate of fusion events and lowers the need to confine the fuel for long periods. This compression is accomplished by heating a lightweight capsule holding the fuel so rapidly that it explodes outwards, driving the fuel mass on the inside inward in accordance with Newton's third law. There are a variety of proposed "drivers" to cause the implosion process, but to date most experiments have used lasers.[11]

Using the traditional definition of Q, Pfus / Pheat, ICF devices have extremely low Q. This is because the laser is extremely inefficient; whereas for the heaters used in magnetic systems might be on the order of 70%, lasers are on the order of 1.5%. For this reason, Lawrence Livermore National Laboratory (LLNL), the leader in ICF research, has proposed another modification of Q that defines Pheat as the energy delivered by the driver, as opposed to the energy put into the driver. This definition produces much higher Q values, and changes the definition of breakeven to be Pfus / Plaser = 1. On occasion, they referred to this definition as "scientific breakeven".[12][13] This term was not universally used, other groups adopted the redefinition of Q but continued to refer to Pfus = Plaser simply as breakeven.[14]

On 7 October 2013, the BBC announced that LLNL had achieved scientific breakeven in the National Ignition Facility (NIF) on 29 September.[15][16][17] In this experiment, Pfus was approximately 14 kJ, while the laser output was 1.8 MJ. By their previous definition, this would be a Q of 0.0077. However, for this press release, they re-defined Q once again, this time equating Pheat to be only the amount energy delivered to "the hottest portion of the fuel", calculating that only 10 kJ of the original laser energy reached the part of the fuel that was undergoing fusion reactions. This release has been heavily criticized in the field.[18][19]


  1. ^ This was denoted PR in Lawson's original paper, but changed here to match modern terminology.
  2. ^ In Lawson's original paper, the term Q was used to denote the total energy released by the individual fusion reactions, in MeV, and R referred to the power balance.[1] Later works used Q to refer to the power balance, as it is used in this article.



  1. ^ a b Lawson 1957, p. 6.
  2. ^ Lawson 1957, pp. 8-9.
  3. ^ Karpenko, V. N. (September 1983). "The Mirror Fusion Test Facility: An Intermediate Device to a Mirror Fusion Reactor". American Nuclear Society. 4 (2P2). doi:10.13182/FST83-A22885.
  4. ^ 17th IAEA Fusion Energy Conference. 19 October 1998.
  5. ^ Jassby, Daniel (19 April 2017). "Fusion reactors: Not what they're cracked up to be". Bulletin of the Atomic Scientists.
  6. ^ a b c d Razzak, M. A. "Plasma Dictionary". Nagoya University.
  7. ^ a b c d e f Meade 1997.
  8. ^ "Glossary". Lawrence Livermore National Laboratory.
  9. ^ Hirsch, Robert (Summer 2015). "Fusion Research: Time to Set a New Path". Issues in Technology. Vol. 31 no. 4.
  10. ^ "JT-60U Reaches 1.25 of Equivalent Fusion Power Gain". 7 August 1998. Archived from the original on 6 January 2013. Retrieved 5 December 2016.
  11. ^ Pfalzner, S. (2006). An Introduction to Inertial Confinement Fusion (PDF). CRC Press. pp. 13–24.
  12. ^ Moses, Edward (4 May 2007). Status of the NIF Project (Technical report). Lawrence Livermore National Laboratory. p. 2.
  13. ^ Ahlstrom, H. G. (June 1981). "Laser fusion experiments, facilities, and diagnostics at Lawrence Livermore National Laboratory". Applied Optics. 20 (11): 1903. doi:10.1364/AO.20.001902.
  14. ^ Assessment of Inertial Confinement Fusion Targets. National Academies Press. July 2013. p. 45, 53.
  15. ^ Rincon, Paul (7 October 2013). "Nuclear fusion milestone passed at US lab". BBC News.
  16. ^ Ball, Philip (12 February 2014). "Laser fusion experiment extracts net energy from fuel". Nature.
  17. ^ "Latest Fusion Results from the National Ignition Facility". HiPER. 13 February 2014.
  18. ^ "Scientific Breakeven for Fusion Energy" (PDF). FIRE.
  19. ^ Clery, Daniel (10 October 2013). "Fusion "Breakthrough" at NIF? Uh, Not Really …". Science.