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A fusion energy gain factor, usually expressed with the symbol ''Q'', is the ratio of
fusion power Fusion power is a proposed form of power generation that would generate electricity by using heat from nuclear fusion, nuclear fusion reactions. In a fusion process, two lighter atomic nucleus, atomic nuclei combine to form a heavier nucleus, whi ...
produced in a
nuclear fusion Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is manifest ...
reactor to the power required to maintain the
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
in
steady state In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
. 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 energy 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 combustion, and is generally regarded as highly desirable for practical reactor designs. Ignition corresponds to infinite ''Q'', in which case no energy input is required to start self sustaining fusion reactions in the plasma. Over time, several related terms have entered the fusion lexicon. Energy that is not captured within the fuel can be captured externally to produce electricity. That electricity can be used to heat the plasma to operational temperatures. A system that is self-''powered'' in this way is referred to as running at engineering breakeven. Operating above engineering breakeven, a machine would produce more electricity than it uses and could sell that excess. One that sells enough electricity to cover its operating costs is sometimes known as economic breakeven. Additionally, fusion fuels, especially
tritium Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleus o ...
, are very expensive, so many experiments run on various test gasses like
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
or
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
. A reactor running on these fuels that reaches the conditions for breakeven, if tritium was introduced, would be operating at breakeven, and this theoretical threshold is referred to as extrapolated breakeven. In 2021, the record for ''Q'' was held by the
National Ignition Facility The National Ignition Facility (NIF) is a laser-based inertial confinement fusion (ICF) research device, located at Lawrence Livermore National Laboratory in Livermore, California, United States. NIF's mission is to achieve fusion ignition wit ...
in the US, at ''Q'' = (1.35 MW)/(1.9 MW) ≈ 0.70, first attained in August 2021. The highest record for extrapolated breakeven was posted by the
JT-60 JT-60 (short for Japan Torus-60) is a large research tokamak, the flagship of Japan's magnetic fusion program, previously run by the Japan Atomic Energy Research Institute (JAERI) and currently run by the Japan Atomic Energy Agency's (JAEA) Naka ...
device, with ''Q''ext = 1.25, slightly besting JET's earlier 1.14.
ITER ITER (initially the International Thermonuclear Experimental Reactor, ''iter'' meaning "the way" or "the path" in Latin) is an international nuclear fusion research and engineering megaproject aimed at creating energy by replicating, on Earth ...
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. On December 13, 2022, the U.S. Department of Energy announced that a gain factor greater than 1 was achieved by the
National Ignition Facility The National Ignition Facility (NIF) is a laser-based inertial confinement fusion (ICF) research device, located at Lawrence Livermore National Laboratory in Livermore, California, United States. NIF's mission is to achieve fusion ignition wit ...
at
Lawrence Livermore National Laboratory Lawrence Livermore National Laboratory (LLNL) is a federal research facility in Livermore, California, United States. The lab was originally established as the University of California Radiation Laboratory, Livermore Branch in 1952 in response ...
in California using
inertial confinement fusion Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with thermonuclear fuel. In modern machines, the targets are small spherical pellets about the size of ...
, delivering 2.05 MJ to generate 3.15 MJ from the resulting fusion reaction. This is roughly equivalent to a gain factor ''Q'' = 1.54 and is the first time that a gain factor Q ≥ 1 has been achieved in the history of nuclear fusion, besides in thermonuclear weapons.


Concept

''Q'' is simply the comparison of the
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
being released by the fusion reactions in a reactor, ''P''fus, to the constant heating power being supplied, ''P''heat, in normal operating conditions. For those designs that do not run in the steady state, but are instead pulsed, the same calculation can be made by summing all of the fusion energy produced in ''P''fus and all of the energy expended producing the pulse in ''P''heat. However, there are several definitions of breakeven that consider additional power losses.


Breakeven

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 Hans Thirring (March 23, 1888 – March 22, 1976) was an Austrian theoretical physicist, professor, and father of the physicist Walter Thirring. He won the Haitinger Prize of the Austrian Academy of Sciences in 1920. Together with the mathemati ...
, 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. This balance is today known as the
Lawson criterion The Lawson criterion is a figure of merit used in nuclear fusion research. It compares the rate of energy being generated by fusion reactions within the fusion fuel to the rate of energy losses to the environment. When the rate of production is ...
. In a successful fusion reactor design, the fusion reactions generate an amount of power designated ''P''fus. Some amount of this energy, ''P''loss, 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 ''P''loss = ''P''heat to maintain thermal equilibrium. The most basic definition of breakeven is when ''Q'' = 1, that is, ''P''fus = ''P''heat. Some works refer to this definition as scientific breakeven, to contrast it with similar terms. However, this usage is rare outside certain areas, specifically the
inertial confinement fusion Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with thermonuclear fuel. In modern machines, the targets are small spherical pellets about the size of ...
field, where the term is much more widely used. Inertial devices, and many similar concepts, do not attempt to maintain equilibrium but simply capture the energy produced. In this case, ''P''heat considers all of the energy needed to produce the reaction, whether it be direct heating or other systems such as laser or magnetic compression.


Extrapolated breakeven

Since the 1950s, most commercial fusion reactor designs have been based on a mix of
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
and
tritium Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleus o ...
as their primary fuel; other fuels have attractive features 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. 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. 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.


Engineering breakeven

Another related term, engineering breakeven, considers the need to extract the energy from the reactor, turn that into electrical energy, and feed some of that back into the heating system. This closed loop sending electricity from the fusion back into the heating system is known as ''recirculation''. In this case, the basic definition changes by adding additional terms to the ''P''fus side to consider the efficiencies of these processes. D-T reactions release most of their energy as
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s and a smaller amount as charged particles like
alpha particle Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be produce ...
s. Neutrons are electrically neutral and will travel out of any
magnetic confinement fusion Magnetic confinement fusion is an approach to generate thermonuclear fusion power that uses magnetic fields to confine fusion fuel in the form of a plasma. Magnetic confinement is one of two major branches of fusion energy research, along with i ...
(MFE) design, and in spite of the very high densities found in
inertial confinement fusion Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with thermonuclear fuel. In modern machines, the targets are small spherical pellets about the size of ...
(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 ''f''ch, then the power in these particles is ''P''ch = ''f''ch''P''fus. If this self-heating process is perfect, that is, all of ''P''ch 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 − ''f''ch)''P''fus. In the case of neutrons carrying most of the practical energy, as is the case in the D-T fuel, this neutron energy is normally captured in a "
blanket A blanket is a swath of soft cloth large enough either to cover or to enfold most of the user's body and thick enough to keep the body warm by trapping radiant body heat that otherwise would be lost through convection. Etymology The term ...
" of
lithium Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense solid el ...
that produces more tritium that is used to fuel the reactor. Due to various
exothermic In thermodynamics, an exothermic process () is a thermodynamic process or reaction that releases energy from the system to its surroundings, usually in the form of heat, but also in a form of light (e.g. a spark, flame, or flash), electricity (e ...
and
endothermic In thermochemistry, an endothermic process () is any thermodynamic process with an increase in the enthalpy (or internal energy ) of the system.Oxtoby, D. W; Gillis, H.P., Butler, L. J. (2015).''Principle of Modern Chemistry'', Brooks Cole. p. ...
reactions, the blanket may have a power gain factor MR. MR is typically on the order of 1.1 to 1.3, meaning it produces a small amount of energy as well. The net result, the total amount of energy released to the environment and thus available for energy production, is referred to as ''P''R, the net power output of the reactor. The blanket is then cooled and the cooling fluid used in a
heat exchanger A heat exchanger is a system used to transfer heat between a source and a working fluid. Heat exchangers are used in both cooling and heating processes. The fluids may be separated by a solid wall to prevent mixing or they may be in direct contac ...
driving conventional
steam turbine A steam turbine is a machine that extracts thermal energy from pressurized steam and uses it to do mechanical work on a rotating output shaft. Its modern manifestation was invented by Charles Parsons in 1884. Fabrication of a modern steam turbin ...
s and generators. That electricity is then fed back into the heating system. Each of these steps in the generation chain has an efficiency to consider. In the case of the plasma heating systems, \eta_ is on the order of 60 to 70%, while modern generator systems based on the
Rankine cycle The Rankine cycle is an idealized thermodynamic cycle describing the process by which certain heat engines, such as steam turbines or reciprocating steam engines, allow mechanical work to be extracted from a fluid as it moves between a heat sourc ...
have \eta_ around 35 to 40%. Combining these we get a net efficiency of the power conversion loop as a whole, \eta_, of around 0.20 to 0.25. That is, about 20 to 25% of P_ can be recirculated. Thus, the fusion energy gain factor required to reach engineering breakeven is defined as: Q_E \equiv \frac = \frac To understand how Q_E is used, consider a reactor operating at 20 MW and ''Q'' = 2. ''Q'' = 2 at 20 MW implies that ''P''heat is 10 MW. Of that original 20 MW about 20% is alphas, so assuming complete capture, 4 MW of ''P''heat is self-supplied. We need a total of 10 MW of heating and get 4 of that through alphas, so we need another 6 MW of power. Of the original 20 MW of output, 4 MW are left in the fuel, so we have 16 MW of net output. Using ''MR'' of 1.15 for the blanket, we get ''PR'' about 18.4 MW. Assuming a good \eta_ of 0.25, that requires 24 MW ''PR'', so a reactor at ''Q'' = 2 cannot reach engineering breakeven. At ''Q'' = 4 one needs 5 MW of heating, 4 of which come from the fusion, leaving 1 MW of external power required, which can easily be generated by the 18.4 MW net output. Thus for this theoretical design the ''QE'' is between 2 and 4. Considering real-world losses and efficiencies, Q values between 5 and 8 are typically listed for magnetic confinement devices, while inertial devices have dramatically lower values for \eta_\text and thus require much higher ''QE'' values, on the order of 50 to 100.


Ignition

As the temperature of the plasma increases, the rate of fusion reactions grows rapidly, and with it, the rate of self-heating. In contrast, 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 ''P''heat 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. In the case of D-T fuel, where only 20% of the energy is released as alphas that give rise to self-heating, this cannot occur until the plasma is releasing at least five times the power needed to keep it at its working temperature. Ignition, by definition, corresponds to an infinite ''Q'', but it does not mean that ''f''recirc 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 ''f''recirc. 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

The final definition of breakeven is commercial breakeven, which occurs when the economic value of any net electricity left over after recirculation is enough to pay for the reactor. This value depends both on the reactor's
capital cost Capital costs are fixed, one-time expenses incurred on the purchase of land, buildings, construction, and equipment used in the production of goods or in the rendering of services. In other words, it is the total cost needed to bring a project to a ...
and any financing costs related to that, its
operating cost Operating costs or operational costs, are the expenses which are related to the operation of a business, or to the operation of a device, component, piece of equipment or facility. They are the cost of resources used by an organization just to main ...
s including fuel and maintenance, and the
spot price In finance, a spot contract, spot transaction, or simply spot, is a contract of buying or selling a commodity, security or currency for immediate settlement (payment and delivery) on the spot date, which is normally two business days after the ...
of electrical power. 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 operating well beyond engineering breakeven will not generate enough electricity rapidly enough to pay for itself. Whether any of the mainline concepts like
ITER ITER (initially the International Thermonuclear Experimental Reactor, ''iter'' meaning "the way" or "the path" in Latin) is an international nuclear fusion research and engineering megaproject aimed at creating energy by replicating, on Earth ...
can reach this goal is being debated in the field.


Practical example

Most fusion reactor designs being studied 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, ''f''ch = 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 ''η''heat = 0.7 (70%) and ''η''elec = 0.4 (40%). The purpose of a fusion reactor is to produce power, not to recirculate it, so a practical reactor must have ''f''recirc = 0.2 approximately. Lower would be better but will be hard to achieve. Using these values we find for a practical reactor ''Q'' = 22. Considering ITER, we have a design that produces 500 MW of energy for 50 MW of supply. If 20% of the output is self-heating, that means 400 MW escape. Assuming the same ηheat = 0.7 and ηelec = 0.4, ITER (in theory) could produce as much as 112 MW of heating. This means ITER would operate at engineering breakeven. However, ITER is not equipped with power-extraction systems, so this remains theoretical until follow-on machines like
DEMO Demo, usually short for demonstration, may refer to: Music and film *Demo (music), a song typically recorded for reference rather than release * ''Demo'' (Behind Crimson Eyes), a 2004 recording by the band Behind Crimson Eyes * ''Demo'' (Deafhea ...
.


Transient vs. continual

Many early fusion devices operated for microseconds, using some sort of pulsed power source to feed their
magnetic confinement Magnetic confinement fusion is an approach to generate thermonuclear fusion power that uses magnetic fields to confine fusion fuel in the form of a plasma. Magnetic confinement is one of two major branches of fusion energy research, along with i ...
system while using the compression from 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. Over time, as performance increased by orders of magnitude, the reaction times have extended from microseconds to seconds, and in
ITER ITER (initially the International Thermonuclear Experimental Reactor, ''iter'' meaning "the way" or "the path" in Latin) is an international nuclear fusion research and engineering megaproject aimed at creating energy by replicating, on Earth ...
, 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 a high, or infinite, ''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. There is an additional 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 to the environment. 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''. Operators of the JET reactor argued that this input should be removed from the total: Q^* \equiv \frac where: P_\text = \frac That is, ''P''temp is the power applied to raise the internal energy of the plasma. It is this definition that was used when reporting JET's record 0.67 value. Some debate over this definition continues. In 1998, the operators of the
JT-60 JT-60 (short for Japan Torus-60) is a large research tokamak, the flagship of Japan's magnetic fusion program, previously run by the Japan Atomic Energy Research Institute (JAERI) and currently run by the Japan Atomic Energy Agency's (JAEA) Naka ...
claimed to have reached ''Q'' = 1.25 running on D-D fuel, thus reaching extrapolated breakeven. This measurement was based on the JET definition of Q*. Using this definition, JET had also reached extrapolated breakeven some time earlier. 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.


Scientific breakeven at NIF

Although most fusion experiments use some form of magnetic confinement, another major branch is
inertial confinement fusion Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with thermonuclear fuel. In modern machines, the targets are small spherical pellets about the size of ...
(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 using some form of "driver". There are a variety of proposed drivers, but to date, most experiments have used
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s. Using the traditional definition of ''Q'', ''P''fus / ''P''heat, ICF devices have extremely low ''Q''. This is because the laser is extremely inefficient; whereas \eta_ for the heaters used in magnetic systems might be on the order of 70%, lasers are on the order of 1%. For this reason,
Lawrence Livermore National Laboratory Lawrence Livermore National Laboratory (LLNL) is a federal research facility in Livermore, California, United States. The lab was originally established as the University of California Radiation Laboratory, Livermore Branch in 1952 in response ...
(LLNL), the leader in ICF research, has proposed another modification of ''Q'' that defines ''P''heat as the energy delivered by the driver to the capsule, as opposed to the energy put into the driver by an external power source. That is, they propose removing the laser's inefficiency from the consideration of gain. This definition produces much higher ''Q'' values, and changes the definition of breakeven to be ''P''fus / ''P''laser = 1. On occasion, they referred to this definition as "scientific breakeven". This term was not universally used; other groups adopted the redefinition of ''Q'' but continued to refer to ''P''fus = ''P''laser simply as breakeven. On 7 October 2013, LLNL announced that it had achieved scientific breakeven in the
National Ignition Facility The National Ignition Facility (NIF) is a laser-based inertial confinement fusion (ICF) research device, located at Lawrence Livermore National Laboratory in Livermore, California, United States. NIF's mission is to achieve fusion ignition wit ...
(NIF) on 29 September. In this experiment, ''P''fus was approximately 14 kJ, while the laser output was 1.8 MJ. By their previous definition, this would be a ''Q'' of 0.0077. For this press release, they re-defined ''Q'' once again, this time equating ''P''heat 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. On 17 August 2021, the NIF announced that in early August 2021, an experiment had achieved a ''Q'' value of 0.7, producing 1.35 MJ of energy from a fuel capsule by focusing 1.9 MJ of laser energy on the capsule. The result was an eight-fold increase over any prior energy output. On December 13, 2022, the U.S. Department of Energy announced that NIF had exceeded the previously elusive ''Q'' ≥ 1 milestone on December 5, 2022. This was achieved by producing 3.15 MJ after delivering 2.05 MJ to the target, for an equivalent ''Q'' of 1.54.


Notes


References


Citations


Bibliography

* } * * {{fusion power Fusion power Energy Energy economics