Progress toward fusion energy breakeven and gain as measured againstthe Lawson criterion

Samuel E. Wurzel and Scott C. HsuAdvanced Research Projects Agency-Energy, U.S. Department of Energy, Washington, DC 20585

(*sam.wurzel@hq.doe.gov)

The Lawson criterion is a key concept in the pursuit of fusion energy, relating the fuel density n, pulse duration τ orenergy confinement time τE , and fuel temperature T to the energy gain Q of a fusion plasma. The purpose of thispaper is to explain and review the Lawson criterion and to provide a compilation of achieved parameters for a broadrange of historical and contemporary fusion experiments. Although this paper focuses on the Lawson criterion, it isonly one of many equally important factors in assessing the progress and ultimate likelihood of any fusion conceptbecoming a commercially viable fusion-energy system. Only experimentally measured or inferred values of n, τ orτE , and T that have been published in the peer-reviewed literature are included in this paper, unless noted otherwise.For extracting these parameters, we discuss methodologies that are necessarily specific to different fusion approaches(including magnetic, inertial, and magneto-inertial fusion). This paper is intended to serve as a reference for fusionresearchers and a tutorial for all others interested in fusion energy.

Keywords: fusion energy, nuclear fusion, Lawson criterion, triple product

I. INTRODUCTION

In 1955, J. D. Lawson identified a set of necessary phys-ical conditions for a “useful” fusion system.1 By evaluatingthe energy gain Q, the ratio of energy released by fusion re-actions to the delivered energy for heating and sustaining thefusion fuel, Lawson concluded that for a pulsed system, en-ergy gain is a function of temperature T and the product offuel density n and pulse duration τ (Lawson used t). Whenthermal-conduction losses are included in a steady-state sys-tem (extending Lawson’s analysis), the power gain is a func-tion of T and the product of n and energy confinement timeτE . We call both these products, nτ and nτE , the Lawson pa-rameter. The required temperature and Lawson parameter forself heating from charged fusion products to exceed all lossesis known as the Lawson criterion. A fusion plasma that hasreached these conditions is said to have achieved ignition. Al-though ignition is not required for a commercial fusion-energysystem, higher values of energy gain will generally yield moreattractive economics, all other things being equal. If the en-ergy applied to heat and sustain the plasma can be recoveredin a useful form, the requirements on energy gain for a usefulsystem are relaxed.

Lawson’s analysis was declassified and published in 19572

and has formed the scientific basis for evaluating the physicsprogress of fusion research toward the key milestones ofplasma energy breakeven and gain. Over time, the Lawsoncriterion has been cast into other formulations, e.g., the fusiontriple product3,4 (nT τE ) and “p-tau” (pressure p times τE ),which have the same dimensions (with units of m−3 keV s oratm s) and combine all the relevant parameters convenientlyinto a single value. However, these single-value parametersdo not map to a unique value of Q, whereas unique combina-tions of T and nτ (or nτE ) do. Various plots of the Lawsonparameter, triple product, and “p-tau” versus year achievedor versus T have been published for subsets of experimentalresults,5–8 but to our knowledge there did not exist a compre-hensive compilation of such data in the peer-reviewed litera-

ture that spans the major thermonuclear-fusion approaches ofmagnetic confinement fusion (MCF), inertial confinement fu-sion (ICF), and magneto-inertial fusion (MIF). This paper fillsthat gap.

The motivation to catalog, define our methodologiesfor inferring, and establish credibility for a compilationof these parameters stems from the prior developmentof the Fusion Energy Base (FEB) website (http://www.fusionenergybase.com) by the first author. FEB is a freeresource with a primary mission of providing objective infor-mation to those, especially private investors, interested in fu-sion energy. This paper provides access to the many includedplots, tables, and codes, while also providing context for un-derstanding the history of fusion research9–11 and the tremen-dous scientific progress that has been made in the 65+ yearssince Lawson’s report.

The combination of T and nτ (or nτE ) is a scientific in-dicator of how far or near a fusion experiment is from en-ergy breakeven and gain. Achieving high values of these pa-rameters is tied predominantly to plasma physics and relatedengineering challenges of producing stable plasmas, heatingthem to fusion temperatures, and exerting sufficient control.Since the 1950s, these challenges have driven the develop-ment of the entire scientific discipline of plasma physics,which has dominated fusion-energy research to this day. How-ever, we emphasize that there are many additional consider-ations, entirely independent of but equally important as theLawson criterion, in evaluating the remaining technical andsocio-economic risks of any fusion approach and the likeli-hood of any approach ultimately becoming a commerciallyviable fusion-energy system. These include the feasibility,safety, and complexity of the engineering and materials sub-systems and fuel cycle that impact a commercial fusion sys-tem’s economics12 and social acceptance,13 as illustrated con-ceptually in Fig. 1. The issues of RAMI (reliability, acces-sibility, maintainability, and inspectability)14 and governmentregulation15,16 impact both the economics and social accep-tance. This paper discusses only the progress of fusion energy

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along the axis of energy gain, and we caution the reader not toover-emphasize nor under-emphasize any one axis.

Although we do not further emphasize it in this paper, a dif-ferent scientific metric called the Sheffield parameter8,17 aimsto embody both the required physics performance (like theLawson parameter) and the “efficiency” of achieving that per-formance for MCF concepts. The Sheffield parameter can bethought of as a normalized triple product by explicitly includ-ing the parameter β , which is a measure of how much plasmapressure (related to the triple product) can be confined for agiven magnetic field (which affects cost and engineering dif-ficulty).

Because of these additional considerations, fusion ap-proaches that have achieved the highest values of T and nτ

(or nτE ), i.e., tokamak-based MCF6 and laser-driven ICF,18,19

may not necessarily become the first widely deployed com-mercial fusion-energy systems. In fact, most private fusioncompanies focusing on developing commercial fusion sys-tems have opted for fusion approaches with lower demon-strated values to-date of temperature and Lawson parame-ter because of the expectation that the required economicsand social acceptance may be more readily achievable. Fur-ther discussion of these other considerations are beyond thescope of this paper but are discussed elsewhere in the fusionliterature.8,14,20,21

This paper is organized as follows. Section II defines thekey variables used in the paper and provides plots of the com-piled parameters. Section III provides a review and mathe-matical derivations of the Lawson criterion and the multipledefinitions of fusion energy gain used by fusion researchers.Section IV provides a physics-based justification for the ap-proximations required to compare fusion energy gain acrossa wide range of fusion experiments and approaches. Read-ers primarily interested in seeing and using the data withoutgetting entangled in the details can largely ignore Secs. IIIand IV. Section V provides a summary and conclusions. Theappendices provide supporting information, including data ta-bles of the compiled parameters, additional plots, and consid-eration of advanced fusion fuels (D-D, D-3He, p-11B).

FIG. 1. Progress towards commercially viable fusion energy requiresprogress along three equally important axes. This paper focuses onlyon the axis of energy gain.

II. VARIABLE DEFINITIONS AND PLOTS

This section provides variable definitions (Table I), andplots of compiled Lawson parameters, fuel temperatures, andtriple products. In many places (especially Secs. I, III, and V),we use the generic variables n, T , τ , Q for economy. However,in most of the paper and as indicated in Table I, all these vari-ables have more precise and differentiated versions with var-ious subscripts. The energy unit keV is used for temperaturevariables throughout this paper, and therefore the Boltzmannconstant k is not explicitly shown.

TABLE I: Definitions of variables used in this paper.

Variable Definition

Ti Ion temperatureTe Electron temperatureTi0 Central ion temperatureTe0 Central electron temperature〈Ti〉n Neutron-averaged ion temperatureT Generic temperature, used to refer to either ion

or electron temperature when Ti = Teni Ion densityne Electron densityni0 Central ion densityne0 Central electron densityn Generic density, used to refer to either ion or

electron density when ni = ne in a pure hydro-genic plasma

τ Pulse durationτE Energy confinement timeτeff Effective characteristic time combining pulse

duration and energy confinement time, seeSec. III C

τ∗E Modified energy confinement time, which ac-counts for for transient heating, see Sec. IV A 6

p Plasma thermal pressureV Plasma volume〈σv〉i j Temperature-dependent fusion reactivity be-

tween species i and j (cross section σ timesrelative velocity v of ions averaged over aMaxwellian velocity distribution)

εF Total energy released per fusion reactionεα Energy released in α-particle per D-T fusion

reactionfc Energy fraction of fusion products in charged

particlesPF Fusion powerSF Fusion power densityPc Fusion power emitted as charged particlesSc Fusion power density in charged particlesPn Fusion power emitted as neutronsPB Bremsstrahlung powerSB Bremsstrahlung power densityPext Externally applied heating powerPabs Externally applied power absorbed by fuelEabs Externally applied energy absorbed by fuelPout Sum of all power exiting the plasmaZ Charge state of an ion

Continued on next page

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TABLE I: Definitions of variables used in this paper.

Variable Definition

Z Mean charge state, i.e., ratio of electron to iondensity in a quasi-neutral plasma

Zeff Effective value of charge state. Factor by whichbremsstrahlung is increased as compared to ahydrogenic plasma, see Eq. (41).

η Efficiency of recapturing thermal energy at theconclusion of the confinement duration in Law-son’s second scenario

ηE Efficiency of converting electrical recirculatingpower to externally applied heating power

ηabs Efficiency of coupling externally applied powerto the fuel

ηhs Efficiency of coupling shell kinetic energy tohotspot thermal energy in laser ICF implosions

ηth Efficiency of converting total output power toelectricity

Qfuel Fuel gain. Ratio of fusion power to power ab-sorbed by the fuel

〈Qfuel〉 Volume-averaged fuel gain in the case of non-uniform profiles

Qsci Scientific gain. Ratio of fusion power to exter-nally applied heating power

〈Qsci〉 Volume-averaged scientific gain in the case ofnon-uniform profiles

Qeng Engineering gain. Ratio of electrical power tothe grid to recirculating power

Qwp Wall-plug gain. Ratio of fusion power to inputelectrical power from the grid

Q Generic energy gain. For MCF, this can refer toQfuel or Qsci. For ICF, this refers to Qsci.

Figure 2 plots achieved Lawson parameters versus Ti forMCF, MIF, and ICF experiments, overlaid with contours ofscientific energy gain Qsci, which is the fusion energy releaseddivided by the energy delivered to the plasma fuel (in the caseof MCF) or the target (in the case of ICF). See the remainderof the paper for details on how the relevant data are extractedfrom the primary literature, the mathematical definition ofQsci, and how the effects of non-uniform spatial profiles, im-purities, heating efficiency, and other experimental details aretreated. Figure 3 shows record triple products achieved by dif-ferent fusion concepts versus year achieved (or anticipated tobe achieved) relative to horizontal lines representing variousvalues of Qsci.

Typically, MCF uses τE and ICF uses τ in their respectiveLawson-parameter and triple-product definitions. AlthoughτE and τ have different physical meanings (see Secs. III E andIII F, respectively), they lead to analogous measures of en-ergy breakeven and gain, allowing for MCF and ICF to beplotted together in Figs. 2, 3, and 25. We caution the readerthat sometimes Lawson parameters and triple products maybe overestimated by concept advocates, especially in unpub-lished materials, because τ is used incorrectly in place of τE .

III. LAWSON CRITERION, LAWSON PARAMETER,TRIPLE PRODUCT, AND ENERGY GAIN

In this section, we provide a detailed review of the deriva-tion of the Lawson criterion, following Lawson’s originalpapers.1,2 We then introduce the mathematical definitions ofthe Lawson parameter in the context of idealized MCF andICF scenarios, derive the fusion triple product, and definethree forms of fusion energy gain used by fusion researchers.

Lawson considered the deuterium-tritium (D-T) anddeuterium-deuterium (D-D) fusion reactions:

D+T→ α (3.5 MeV)+n (14.1 MeV) (1)D+D →

50%T (1.01 MeV)+p (3.02 MeV) (2)

D+D →50%

3He (0.82 MeV)+n (2.45 MeV), (3)

where α denotes a charged helium ion (4He2+), p denotes aproton, n denotes a neutron, and 1 MeV = 1.6×10−13 J. Thefusion reactivities 〈σv〉 for thermal ion distributions for thesereactions, as well as the additional reactions,

D+3He→ α (3.6 MeV)+p (14.7 MeV) (4)

p+11B→ 3α (8.7 MeV), (5)

are shown in Fig. 4.As did Lawson, this paper assumes thermal populations

of ions and electrons, i.e., Maxwellian velocity distribu-tions characterized by a temperatures Ti and Te, respectively.Throughout this paper, we assume that ions and electrons arein thermal equilibrium with each other such that T = Ti = Te.Non-equilibrium fusion approaches, where Ti > Te, must ac-count for the energy loss channel and timescale of energytransfer from ions to electrons.24 Analysis of such systemsis not included in this paper. Furthermore, this paper doesnot consider non-thermal ion or electron populations such asthose with beam-like distributions. The latter typically mustcontend with reactant slowing at a much faster rate than thefusion rate. The inherent difficulty (though not necessarilyimpossibility) for non-thermal fusion approaches to achieveQsci > 1 is discussed in Ref. 25.

Lawson’s original papers considered two distinct fusion op-erating conditions. The first is a steady-state scenario in whichthe charged fusion products are confined and contribute to selfheating. The second is a pulsed scenario in which the chargedfusion products escape and energy is supplied over the dura-tion of the pulse. Lawson’s analysis did not address how thefusion plasma is confined and assumed an ideal scenario with-out thermal-conduction losses in both cases.

A. Lawson’s first insight: ideal ignition temperature

Lawson’s first insight was that a self-sustaining, steady-state fusion system without external heating must, at a min-imum, balance radiative power losses with self heating fromthe charged fusion products, as illustrated conceptually inFig. 5. The power released by charged fusion products in a

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FIG. 2. Experimentally inferred Lawson parameters (ni0τ∗E for MCF and nτ for ICF) of fusion experiments vs. Ti0 for MCF and 〈Ti〉n for ICF(see Sec. III for definitions of these quantities), extracted from the published literature (see Tables VI, VII, and VIII). The various contoursin the upper right correspond to the required Lawson parameters and ion temperatures required to achieve the indicated values of scientificgain QMCF

sci for MCF (colored contours) and QICFsci for ICF (solid and dotted black contours), assuming representative density and temperature

profiles, external-heating absorption efficiencies, and D-T fuel. For experiments that do not use D-T, the contours represent a D-T-equivalentvalue of Qsci. The finite widths of the QMCF

sci contours represent a range of assumed impurity levels. See the rest of the paper for details onhow individual data points are extracted and how the QMCF

sci and QICFsci contours are calculated. Note that QMCF

sci & 20 and QICFsci & 100 are likely

needed for practical fusion energy; see Sec. III H and Eq. (27) for discussion and definition, respectively, of Qeng.

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FIG. 3. Triple products (ni0Ti0τ∗E for MCF and n〈Ti〉nτ for ICF; see Sec. III for definitions of these quantities) that set a record for a givenconcept vs. year achieved. Record values for different concepts are shown to illustrate the progress towards energy gain of different conceptsover time. The horizontal lines labeled QMCF

sci represent the minimum required triple product to achieve the indicated values of QMCFsci , assuming

ηabs = 0.9. The horizontal line labeled QICFsci = ∞ represents the required triple product to achieve ignition and propagating burn for ICF,

assuming Ti = 4 keV and ηabs = 0.006. The projected triple-product ranges for SPARC and ITER are bounded above by their projected peaktriple products and below by the stated mission of each experiment (i.e., QMCF

fuel = 2 for SPARC and QMCFfuel = 10 for ITER). Note that QMCF

sci & 20and QICF

sci & 100 are likely needed for practical fusion energy; see Sec. III H and Eq. (27) for discussion and definition, respectively, of Qeng.The NIF shot from August 8, 2021 does not appear in this plot because it did not achieve a record triple product, despite achieving a recordQICF

sci for ICF. This highlights the main limitation of the triple product, i.e., it does not map to a unique value of gain (see Secs. III G and IV Bfor further explanation).

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FIG. 4. Thermal fusion reactivities 〈σv〉 vs. Ti for fusion reactionsshown in the legend. All reactivities are calculated by numericalintegration of velocity-averaged cross sections from Ref. 22 with theexception of p-11B, which is calculated from the parametrization ofRef. 23. Note that the two D-D branches are nearly on top of eachother.

FIG. 5. The steady-state scenario corresponding to Lawson’s firstinsight. Self heating from charged fusion products Pc appears asbremsstrahlung power PB in a steady state plasma of volume V . Fu-sion power emitted as neutrons Pn escapes the plasma and does notcontribute to self-heating. An unspecified, idealized confinementmechanism is assumed, and thermal-conduction is ignored.

plasma of volume V is

Pc = fcPF = fcn1n2

1+δ1,2〈σv〉1,2εFV, (6)

where n1 and n2 are the number densities of the reactants,δ1,2 = 1 in the case of identical reactants (e.g., D-D), andδ1,2 = 0 otherwise (e.g., D-T).

The power emitted by bremsstrahlung radiation is

PB =CBnineZ2T 1/2e V, (7)

FIG. 6. Power produced per unit volume Sc in charged D-T fusionproducts (α particles) and power lost to bremsstrahlung per unit vol-ume SB vs. T in a D-T plasma. When T < 4.3 keV, SB > Sc andignition is not possible (assuming T = Te = Ti).

where CB is a constant and Z = 1 in a hydrogenic plasma.Entering values of density in m−3, temperature in keV, volumein m3, and setting CB = 5.34×10−37 W m3 keV−1/2 gives PBin watts.

If the fusion plasma is to be completely self heated bycharged fusion products (i.e., α , T, p, or He3 in the above reac-tions), then Pc ≥PB is required in order for the plasma to reachignition (ignoring conduction losses for the moment). In thecase of an equimolar D-T fusion plasma, i.e., n/2 = n1 = n2,where n is the total ion number density and Z = 1, and giventhe assumption T = Ti = Te, the condition Pc ≥ PB becomes

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fcn2〈σv〉DT εFV ≥CBn2T 1/2V. (8)

Dividing both sides by V and plotting the resulting fusionpower density Sc = Pc/V (left-hand side) and bremsstrahlungpower density SB = PB/V (right-hand side) versus T in Fig. 6shows that T ≥ 4.3 keV is required for Sc ≥ SB. This tem-perature is known as the ideal ignition temperature because,under the idealized scenario of perfect confinement, ignitionoccurs at this temperature. Note that because n2 cancels onboth sides of Eq. (8), the ideal ignition temperature is inde-pendent of density. In Appendix C, we discuss and show howthe ignition temperature could be modified if bremsstrahlungradiation losses are mitigated.

B. Lawson’s second insight: dependence of fuel energy gainon T and nτ

Lawson’s second insight involves a pulsed scenario where aplasma is heated instantaneously to a temperature T and main-

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FIG. 7. Pulsed scenario corresponding to Lawson’s second insight.At time t = 0, the plasma temperature is instantaneously raised to Tand maintained for a duration τ by externally applied and absorbedpower Pabs. All fusion products escape (no self-heating), and thermalconduction is neglected (ideal confinement). Absorbed power Pabsappears as bremsstrahlung power PB during the pulse duration.

tained at that temperature for time τ , as illustrated conceptu-ally in Fig. 7. In this scenario, bremsstrahlung radiation andall fusion reaction products escape, and heating must comefrom an external source during duration τ . Idealized confine-ment is assumed, and thermal-conduction losses are ignored.

We define the fuel gain Qfuel (Lawson used R) as the ratioof energy released in fusion products to the applied externalenergy that is absorbed by the entire fuel over the duration τ

of the pulse. This absorbed energy is the sum of the instan-taneously deposited energy 3

2 (ne +ni)TV = 3nTV (assumingT = Ti = Te and n = ni = ne) and the energy applied and ab-sorbed over the pulse duration, τPabs. To maintain constantT over duration τ , Pabs = PB is required, and the fuel gain istherefore,

Qfuel =τPF

τPB +3nTV=

PF/(3n2TV )

PB/(3n2TV )+1/(nτ). (9)

Because both PF and PB are proportional to n2V and functionsof T [see Eqs. (6) and (7)], the n2V dependence cancels out,and Qfuel is solely a function of T and nτ ,

Qfuel =〈σv〉εF/12T

CB/3T 1/2 +1/(nτ). (10)

Figure 8 plots Qfuel as a function of T for the indicated valuesof nτ , illustrating that even without self-heating, Qfuel � 1is theoretically possible. Lawson noted that a “useful” sys-tem would require Qfuel > 2, assuming that fusion energy andbremsstrahlung could be converted to useful energy with anefficiency of 1/3, and remarked on the severity of the requiredT and nτ .

In this section, we have assumed that at time t = τ the ex-ternal heating is turned off and none of the applied energyis recaptured. Lawson noticed, however, that if a fraction η

(Lawson used f ) of the thermal energy at the conclusion ofthe pulse duration is recovered and converted into a usefulform of energy (e.g., electrical or mechanical) that could off-set the externally applied energy, the quantity nτ in Eq. (10) isreplaced by nτ/(1−η). The utilization of energy recovery torelax the requirements on nτ for achievement of energy gainis discussed further in Sec. III H.

FIG. 8. Plot of Qfuel vs. T for indicated values of nτ , assuming noself heating and no thermal-conduction losses.

C. Extending Lawson’s second scenario: effect of selfheating and relationship between characteristic times τ and τE

In an effort to capture experimental realities, we ex-tend Lawson’s second scenario to include thermal-conductionlosses and self heating from charged fusion products, as il-lustrated in Fig. 9. The rate of energy leaving the plasma viathermal conduction is characterized by an energy confinementtime τE , which is the time for energy equal to the thermal en-ergy 3nTV to exit the plasma. The power balance over theduration of the constant-temperature pulse is

Pabs +Pc = PB +3nTV/τE . (11)

Applying a similar analysis to that of the previous section,we obtain

Qfuel =〈σv〉εF/12T

CB/3T 1/2− fc〈σv〉εF/12T +1/(nτeff), (12)

where

nτeff = nττE

τ + τE. (13)

FIG. 9. Extension of Lawson’s second scenario. At time t = 0, theplasma temperature is instantaneously raised to T and maintained fora duration τ by absorbed external power Pabs and self-heating powerPc. The sum of absorbed external heating and self-heating appear asbremsstrahlung PB and thermal conduction 3nTV/τE .

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FIG. 10. Plot of Qfuel vs. T for indicated values of effective Law-son parameter nτeff, for a pulsed scenario that includes self heatingfrom charged fusion products and thermal conduction. Self heat-ing reduces the demands on externally applied and absorbed heating.Above a threshold of T and nτeff, Qfuel increases without bound, cor-responding to ignition.

The relationship between the two characteristic times τ and τEis like two resistors in parallel, i.e., it is the smaller of the twothat limits the value of τeff. If τ � τE , the confinement dura-tion τ limits Qfuel because there is limited time to overcomethe initial energy investment of raising the plasma tempera-ture. If τE � τ , the energy confinement time τE limits Qfuelbecause the rate of energy leakage from thermal conductionplaces higher demand on external and self heating. If the twocharacteristic times are of similar magnitude, then both play arole in limiting Qfuel.

Figure 10 plots Qfuel versus T for the indicated values ofnτeff, illustrating that self heating enables ignition (Qfuel→∞)above a threshold of T and nτeff, made possible by the reduc-tion of the denominator of Eq. (12) by amount fc〈σv〉εF/12T .We explore these thresholds in subsequent sections.

D. Scientific energy gain and breakeven

Because external-heating efficiency varies widely acrossfusion concepts, and because the absorption efficiency is in-trinsic to the physics of each concept, we define Pext as theheating power applied at the boundary of the plasma (in thecase of MCF) or the target assembly (in the case of ICF). Thisdefinition of Pext encapsulates all physics elements of the ex-periment. The boundary can typically be regarded as the vac-uum vessel for all concepts, where Pext could be electromag-netic waves for MCF, laser beams for ICF, or electrical cur-rent and voltage for MIF. The previously introduced Pabs isthe fraction ηabs of Pext that is actually absorbed by the fuel,

i.e., Pabs = ηabsPext. The previously defined fuel gain is

Qfuel =PF

Pabs, (14)

and the newly defined scientific gain is

Qsci =PF

Pext= ηabsQfuel < Qfuel. (15)

Whereas Qfuel ignores the plasma-physics losses of the ab-sorption of heating energy into the fuel (e.g., neutral-beamshine-through in MCF or reflection of laser light via laser-plasma instabilities in ICF), Qsci accounts for all plasma-physics-related losses between the vacuum vessel and the fu-sion fuel. Therefore, Qsci is the better metric for assessingremaining physics risk of a fusion concept.

Scientific breakeven is historically defined as Qsci = 1,which is an important milestone in the development of fu-sion energy because it signifies that very significant (but notall) plasma-physics challenges have been retired. Scientificbreakeven has not yet been achieved, although D-T toka-mak experiments such as TFTR and JET from the 1990s andthe NIF experiment of August 8, 202126 have come close(Qsci = 0.27 for TFTR,27 Qsci = 0.64 for JET,28 and Qsci∼ 0.7for NIF29). Because ηabs is much closer to unity in MCF ex-periments, the MCF community often uses Q to refer to Qfuelor Qsci interchangeably.

E. Idealized, steady-state MCF: τE � τ

MCF relies on strong magnetic fields to confine fusion fuel,minimize thermal-conduction losses, and trap the charged fu-sion products for self heating. By the time that Lawson’s re-port was declassified in 1957, the UK, US, and USSR wereall actively developing MCF experiments that included exter-nally applied heating.

Adapting the extension of Lawson’s second insight to thisscenario, we consider the power balance of an externallyheated and self-heated, steady-state plasma. Figure 11 illus-trates this scenario for two different values of energy gain.The power balance and fuel gain of the plasma are describedby Eqs. (11) and (12), respectively, in the limit of steady-stateoperation, i.e., τ → ∞.

To more clearly observe the requirements on nτE and T toachieve certain values of Qfuel, we solve Eq. (12) for nτE inthe steady-state limit (τ � τE ),

nτE =3T

( fc +Q−1fuel)〈σv〉εF/4−CBT 1/2

. (16)

Plotting this expression in Fig. 12 (dashed lines) for D-T fu-sion shows that a threshold value of nτE , which varies withT , is required to achieve a given value of Qfuel. Table II liststhe minimum values of Lawson parameter and correspondingtemperature required to achieve Qfuel = 1 and Qfuel = ∞ forthe indicated reactions. Thus far, spatially uniform profiles ofall quantities are assumed, and geometrical effects and impu-rities are ignored. Later in the paper, we consider the effects of

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FIG. 11. Conceptual illustrations of the steady-state power balancefor two hypothetical steady-state MCF scenarios. The dotted linerepresents the boundary, e.g., vacuum chamber, between the physicsand engineering aspects of the experiment.

TABLE II. Values of minimum niτE and corresponding T for Qfuel =1 and Qfuel = ∞, for different fusion fuels assuming T = Ti = Te,based on Eq. (16) for D-T (see Appendix D for advanced fuels).

Reaction QfuelT

(keV)niτE

(m−3s)

D+T 1 26 2.5×1019

D+T ∞ 26 1.6×1020

Catalyzed D-D 1 107 4.8×1020

Catalyzed D-D ∞ 106 1.5×1021

D+3 He 1 106 2.8×1020

D+3 He ∞ 106 6.2×1020

p+11 B 1 – –p+11 B ∞ – –

nonuniform spatial profiles, different geometries (e.g., cylin-der, torus, etc.), and impurities.

To more clearly observe the requirements on nτE and T toachieve certain values of Qsci, we replace Qfuel with Qsci/ηabsin Eq. (16),

nτE =3T

( fc +ηabsQ−1sci )〈σv〉εF/4−CBT 1/2

. (17)

The ignition contours are identical for Qfuel = ∞ and Qsci = ∞.For MCF experiments, where ηabs is close to unity (ηabs ∼0.9), non-ignition Qsci < ∞ contours are shifted relative totheir respective Qfuel contours only very slightly toward theignition contour (Qfuel,Qsci = ∞), as seen in Fig. 12 (solidlines).

FIG. 12. Lawson parameter nτE vs. T required to achieve indicatedvalues of Qfuel (dashed lines) and Qsci (solid lines), assuming ηabs =0.9 (representative of MCF). Because ηabs is close to unity for MCF,Qfuel and Qsci are nearly coincident (the ignition contours are exactlycoincident) and are often used interchangeably and referred to as Q.

The Lawson criterion, where Pabs → 0 and Qfuel → ∞ inEqs. (11) and (14), respectively, is satisfied for values of nτEand T on or above the Qfuel,Qsci = ∞ curves in Fig. 12. Inthis ignition regime, the plasma is entirely self heated bycharged fusion products, and external heating is zero. Whilethe minimum Lawson parameter required for ignition occursat T ≈ 25 keV, MCF approaches aim for T ≈ 10–20 keV be-cause the pressure required to achieve high gain is minimizedin this lower-temperature range (as discussed in Sec. III G).

F. Idealized ICF: τ � τE

ICF relies on the inertia of highly compressed fusion fuel toprovide a duration to fuse a sufficient amount of fuel to over-come the energy invested in compressing the fuel assembly.In 1971, the concept of using lasers to compress and heat afuel pellet was declassified, first by the USSR and later thatyear by the US.30 In 1972, Nuckolls et al. 18 described thedirect-drive laser ICF concept, where lasers ablate the surfaceof a hollow fuel pellet outward, driving the inner surface to-ward the center. In this scenario the kinetic energy of theinward-moving material is converted to thermal energy of acentral, lower-density “hot-spot” that ignites. The fusion burnpropagates outward through the surrounding denser fuel shell,which finally disassembles. The four-step, “central hot-spotignition” process is illustrated in Fig. 13. Laser indirect-driveICF bathes the fuel pellet in X-rays generated by the interac-tions between lasers and the inside of a “hohlraum” (a metalenclosure surrounding the fuel pellet) to similar effect.

To adapt the extension of Lawson’s second insight, we con-

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sider the energy balance of the hot spot over duration τ , dur-ing which it is inertially confined [Fig. 13(b)]. The sequenceof events that leads to energy delivered to the hot spot are:

1. The laser energy strikes the fuel pellet (or hohlraum);

2. A fraction ηabs of the laser energy is absorbed by thefuel in the form of kinetic energy Eabs of the implodingfuel shell;

3. The imploding shell with energy Eabs does pdV workon the hot spot of volume V , resulting in hot-spot ther-mal energy Ehs = ηhsEabs;

4. If sufficiently high temperature and Lawson parameterare achieved, additional energy τPc is delivered to thehot spot by charged fusion products.

We describe the fuel gain of the hot-spot by applying thefollowing assumptions and modifications to Eq. (12). In thissimplified model, we neglect bremsstrahlung and thermal-conduction losses, i.e., CB → 0 and τE → ∞. While bothprocesses are present in the hot spot, the cold, dense shell islargely opaque to bremsstrahlung and partially insulates thehot spot. In practice (which we also ignore here), both lossmechanisms have the effect of ablating material from the innershell wall into the hot spot, increasing density and decreasingtemperature while maintaining a constant pressure.31 To ac-count for the fraction ηhs of the shell kinetic energy that isdeposited in the hot-spot, the definition of Qfuel becomes,

Qfuel =τPF

Eabs=

τPF

Ehs/ηhs. (18)

We assume that the charged fusion products generated in thehot spot deposit all their energy within the hot spot.

To more clearly observe the requirements on nτ and T toachieve certain values of Qfuel, we solve Eq. (12) for nτ withthe above limits and modifications,

nτ =12T

( fc +ηhsQ−1fuel)〈σv〉εF

, (19)

Plotting this expression in Fig. 14 (dashed lines) for D-T fu-sion shows that a threshold value of nτ , which varies with T ,

FIG. 13. Conceptual schematic of idealized ICF (a) compression, (b)hot-spot ignition, (c) propagating burn of the cold, dense shell, and(d) disassembly.

is required to achieve a given value of Qfuel in an ICF hot spot.We have assumed ηhs = 0.65 based on NIF shot N191007.32

Thus far, reductions in τ due to instabilities, impurities, lossesdue to bremsstrahlung and thermal conduction, and the re-quirements to initiate a propagating burn in the cold, denseshell have been ignored. Later in this paper, we consider someof these effects.

Similarly to the MCF example, the required Lawson pa-rameter and temperature required to reach a certain value ofQsci can be evaluated by replacing ηhsQ−1

fuel with ηabsηhsQ−1sci

in Eq. (19),

nτ =12T

( fc +ηabsηhsQ−1sci )〈σv〉εF

. (20)

For ICF experiments, where ηabsηhs is very low (e.g.,ηabsηhs ∼ 0.006 for indirect-drive ICF), non-ignition (Qsci <∞) contours are shifted relative to their respective Qfuel con-tours strongly toward the ignition contour (Qsci = ∞), as seenin Fig. 14 (solid lines). For this reason, ignition is effec-tively required to achieve scientific breakeven in ICF. Whilethe minimum Lawson parameter required for ignition occursat T ≈ 25 keV, laser-driven ICF approaches aim for hot-spotT ≈ 4 keV (prior to the onset of significant fusion leading tofurther increases in Ti) due to the limits of achievable implo-sion speed, which sets the maximum achievable temperaturedue to pdV heating alone.

Note that our definition of Qfuel for ICF differs slightly fromthe standard definition of ICF fuel gain, Gf, which is the ra-tio of fusion energy to total energy content of the fuel imme-

FIG. 14. Lawson parameter nτ vs. T required to achieve indi-cated values of Qfuel (dashed lines) and Qsci (solid lines), assum-ing ηabsηhs = 0.006 (representative of indirect-drive ICF). Becauseηabsηhs � 1 for laser ICF, contours of Qsci are shifted to be nearlyon top of the the ignition contour relative to their Qfuel counterparts,illustrating that ignition is effectively required to achieve scientificbreakeven for laser ICF.

11

diately before ignition.19 The Lawson parameter of an ICFhot spot is usually framed in terms of the hot-spot ρhsRhs,where ρhs and Rhs are the hot-spot mass density and radius,respectively.19 For the purposes of having a Lawson parame-ter and fuel gain that parallel the MCF case, we proceed withour definition of ICF Qsci, which is the same as the standarddefinition of ICF target gain G.19

The condition for hot-spot ignition for a D-T plasma is,

(nτ)ig =12T〈σv〉εα

, (21)

where εα is the energy of the charged alpha-particle fusionproduct in the D-T fusion reaction. More generally, “igni-tion” has many different meanings in the ICF context.33 The1997 National Academies review of ICF34 addressed the lackof consensus around the definition of ICF ignition by definingignition as fusion energy produced exceeding the laser energy(i.e., Qsci > 1). More recently, the hot-spot conditions neededto initiate propagating burn in the colder, dense fuel shell (an-other definition of ignition) have been quantified.35 These de-tails are discussed further in Sec. IV B.

G. Fusion triple product and “p-tau”

The triple product (nT τE ) and p-tau (pτE ) are commonlyused by the MCF community to quantify fusion performancein a single value. While less common in the ICF community,pτ is sometimes used, and triple product (nT τ) is typicallyused only in the context of comparing ICF to MCF.31 In auniform plasma with n= ni = ne and T = Ti = Te, the relation-ship between triple product and p-tau in both embodiments isnT τ = 1

2 pτ and nT τE = 12 pτE .

An expression for the MCF triple product is obtained bymultiplying both sides of Eq. (16) by T ,

nT τE =3T 2

( fc +Q−1fuel)〈σv〉(T )εF/4−CBT 1/2

. (22)

Figure 15 shows the nT τE required to achieve a specifiedvalue of Qfuel as a function of T (see also Table III). Notethat the minimum triple product needed to achieve ignitionoccurs at a lower T than that of the minimum Lawson param-eter. This lower T is a better approximation of the intendedT of MCF experiments because it corresponds to the mini-mum pressure required to achieve a certain value of Qfuel, andpressure (rather than Lawson parameter) is a more-direct ex-perimental limitation of MCF.

We emphasize the limitation of the triple product (or “p-tau”) as a metric: it does not correspond to a unique valueQfuel or Qsci unless T is specified. While n and τ in the Law-son parameter may be traded off in equal proportions, T mustbe within a fixed range for an appreciable number of fusion re-actions to occur. Appendix A provides a plot of achieved tripleproducts and temperatures analogous to Fig. 2. Appendix Dprovides plots of nT τE vs. T for D-D, D-3He, and p-11B fu-sion.

FIG. 15. Triple product vs. T required to achieve indicated values ofQfuel for MCF [Eq. (22)].

TABLE III. Values of minimum niT τE and corresponding T forQfuel = 1 and Qfuel = ∞ for different fusion fuels assuming T = Ti =Te, based on Eq. (22) for D-T (see Appendix D for advanced fuels).

Reaction QfuelT

(keV)niT τE

(m−3keVs)

D+T 1 14 4.6×1020

D+T ∞ 14 2.9×1021

Catalyzed D-D 1 41 2.9×1022

Catalyzed D-D ∞ 52 1.1×1023

D+3 He 1 63 2.2×1022

D+3 He ∞ 68 5.2×1022

p+11 B 1 – –p+11 B ∞ – –

H. Engineering gain

The previously defined Qsci [Eq. (15)] is the ratio of powerreleased in fusion reactions PF to applied external heatingpower Pext (see Fig. 11), encapsulating the physics of plasmaheating, thermal and radiative losses, and fusion energy pro-duction. Based on conservation of energy in Fig. 11, we canrewrite

Qsci =Pout−Pext

Pext=

PF

Pext, (23)

which is equivalent to Eq. (15).Similarly, the engineering gain,

Qeng =PE

out−PEin

PEin

=PE

grid

PEin

, (24)

is the ratio of electrical power PEgrid (delivered to the grid)

to the input (recirculating) electrical power PEin used to heat,

12

sustain, control, and/or assemble the fusion plasma36 (seeFig. 16). Some fusion designs do not recirculate electri-cal power but rather recirculate mechanical power (see Ap-pendix E). For the case of electrical recirculating power it isstraightforward to show that

Qeng = ηelecηE(ηabsQfuel +1)−1, (25)

where ηE , ηabs, and ηelec are the efficiencies of going fromPE

in → Pext, Pext → Pabs, and Pout → PEout, respectively. Note

that we have included the portion of Pext that is not absorbedby the plasma, i.e., (1− ηabs)Pext, in Pout; this is shown inFig. 11 but not explicitly shown in Fig. 16.

Finally, the “wall-plug” gain,

Qwp =PF

PEin

= ηEPF

Pext= ηEηabs

PF

Pabs, (26)

relates the total fusion power to the power drawn from the grid(i.e., the wall plug) to assemble, heat, confine, and control theplasma. This is a useful energy gain metric for all contempo-rary fusion experiments because they are not yet generatingelectricity. We regard the eventual demonstration of Qwp = 1(not Qfuel or Qsci = 1) as the so-called “Kitty Hawk moment”for fusion energy.

Direct conversion from charged fusion products to electric-ity could be realized with advanced fusion fuels (e.g., D-3Heand p-11B), which produce nearly all of their fusion energy incharged products. This could raise ηelec from approximately40% to > 80% and enable significantly higher Qeng for a givenQfuel or Qsci.

For D-T fusion with a tritium-breeding blanket, the6Li(n,α)T reaction to breed tritium is exothermic (releasing4.8 MeV per reaction), thus amplifying Pout by a factor of ap-proximately 1.15 depending on the blanket design. For thepurposes of this paper, this factor can be considered to be ab-sorbed into ηelec.

Using Qsci = ηabsQfuel, we can rewrite Eq. (25) as

Qeng = ηelecηE(Qsci +1)−1. (27)

Because Qsci encapsulates all the plasma-physics aspects ofboth the absorption efficiency ηabs and fuel gain Qfuel, itis instructive to plot the required combinations of Qsci andηE , assuming ηelec = 0.4 (representative of a standard steamcycle and blanket gain), to achieve certain values of Qeng(see Fig. 17). A convenient rule-of-thumb is that the gain-efficiency product must exceed 10 for practical fusion energy,

FIG. 16. Conceptual schematic of a fusion power plant which recir-culates electrical power. In this system Qeng = PE

grid/PEin .

TABLE IV. Typical efficiency values ηE , ηabs, ηhs, and ηelec fordifferent classes of fusion concepts. Note ηhs is only defined forICF concepts pursuing hot spot ignition. Approximate values of ηabsand ηhs for direct and indirect drive ICF are from Ref. 37 and Ref.32, respectively.

Class ηE ηabs ηhs ηelec

MCF 0.7 0.9 - 0.4MIF 0.9 0.1 - 0.4Laser ICF (direct drive) 0.1 0.06 0.4 0.4Laser ICF (indirect drive) 0.1 0.009 0.7 0.4

FIG. 17. Required combinations of Qsci and ηE in the system shownin Fig. 16 to permit values of Qeng ranging from zero (i.e., PE

grid = 0)to ten (i.e., PE

grid = 10PEin ), where ηelec = 0.4 is assumed.

i.e., QsciηE ≥ 10 (corresponding to Qeng ≈ 3 in Fig. 17), butof course the actual requirement depends on the required eco-nomics of the fusion-energy system.

While the value of ηelec would be around 0.4 for a stan-dard steam cycle for D-T fusion (and higher if an advancedpower cycle is used), the values of ηE and ηabs vary consider-ably depending on the class of fusion concept (see Table IV).For MCF/MIF, ηE > 0.5 is expected (conservatively), mean-ing that Qsci & 20 is required. For laser-driven ICF, ηE ∼ 0.1 isexpected, meaning that Qsci & 100 is required. For an eventualfusion power plant, the required Qsci and Qeng will depend ona number of factors including but not limited to market con-straints (e.g., levelized cost of electricity and desired value ofPE

grid) and the maximum achievable values of ηE , ηelec.In Sec. III B, we noted Lawson’s observation (in the con-

text of his second scenario) that if a fraction η of the plasmaenergy at the conclusion of the pulse is recovered as electri-cal or mechanical energy, the requirement on nτ to achievea given value of Qfuel is reduced by a factor 1/(1−η). Inprinciple, this can be extended to recover Pout with an effi-

13

FIG. 18. Required combinations of Qsci and ηE in the system shownin Fig. 16 to permit values of Qeng ranging from zero (i.e., PE

grid = 0)to ten (i.e., PE

grid = 10PEin ), where ηelec = 0.95 is assumed. Note that

at high ηE and ηelec, net electricity generation (Qeng > 0) is possiblewith Qsci < 1.

ciency ηelec and reinject the recirculating fraction with effi-ciency ηE , thus relaxing the requirements on Qsci to achievea given Qeng. This is shown in Fig. 18, which assumes a highrecovery fraction ηelec = 0.95. If we also assume a high elec-tricity to heating efficiency ηE = 0.9, Qeng = 0.3 (correspond-ing to net electricity) can be achieved with Qsci = 0.5. Whileit may appear counter-intuitive that net electricity can be gen-erated in a system with Qsci < 1, a high ηelec and ηE meanthat most of the recovered heating energy recirculates whilemost of the fusion energy is used for electricity generation.The lower-right quadrant Fig. 18 (corresponding to high re-injection efficiency) illustrates that that net electricity gener-ation (i.e., Qeng > 0) is possible at values of scientific gainbelow break-even (i.e., Qsci < 1).

IV. METHODOLOGIES FOR INFERRING LAWSONPARAMETER AND TEMPERATURE

It is not trivial to infer the component values of the Lawsonparameter and temperature achieved in real experiments. Sim-plifying approximations must be made with certain caveats,both across (e.g., MCF vs. ICF) and within classes (e.g., toka-maks vs. mirrors within MCF) of fusion experiments. In thissection, we describe the methodologies that we use to infer thecomponent values of achieved Lawson parameters and tem-peratures for different fusion classes and concepts, and howthe values can be meaningfully compared against each other.For all values reported here, we require that experimentallyinferred values occur within a single shot or across multiplewell-reproduced shots. An example that we would disqualify

would be to combine the highest Ti achieved in one shot withthe highest ni and τE from a qualitatively different shot.

A. MCF methodology

The analysis presented in Sec. III E assumes that Ti = Teand ni = ne, and that they are spatially uniform and time in-dependent. In real experiments, these assumptions are gener-ally not valid. Because diagnostic capabilities are finite, onlya subset of the complete data (i.e., spatial profiles and timeevolutions) are ever measured and published. Although manyexperiments were not aiming to maximize ni, Ti, and τE asthe goal, we include these experiments because they providehistorical context. Furthermore, the data reported from oneexperiment may not be easily compared to data reported fromanother due to differences in definitions. In the remainder ofthis section, these issues are discussed, and uniform defini-tions are developed.

1. Effect of temporal profiles

Within a particular experiment, the maximum values of ni,Ti, and τE may occur at different times. Where possible wechoose the values of these quantities at a single point in timeduring a “flat-top” time period, the duration of which must ex-ceed τE . Even though the total pulse duration of some MCFexperiments may be of similar magnitude to τE , we only con-sider τE in the Lawson parameter for MCF experiments (asopposed to the expression for τeff in Eq. 13) because we con-sider the progress towards energy gain in MCF to be limitedby thermal-conduction losses and not pulse duration.

In the literature, tables of parameters are commonly pub-lished that report the values of many parameters during sucha flat-top time period. Following this convention, Tables VIand VII list parameters relevant to our analysis. The reportedparameters are Ti0, Te0, ni0, ne0, and τ∗E . Not all experimentshave published temporal evolution of these quantities. In theabsence of such data, we use the values reported with the un-derstanding that it is unknown if they occurred simultaneouslyduring the shot (although, as discussed in the previous para-graph, they must occur in the same shot or in shots intended tobe the same). This deficiency primarily occurs in experimentsprior to 1970 or in small experiments with limited diagnosticcapabilities and niTiτE < 1016 m−3 keV s.

2. Effect of spatial profiles

To quantify the effect of nonuniform temperature and den-sity spatial profiles on the requirements to achieve a certainvalue of Qfuel, which we denote as 〈Qfuel〉 (brackets refer tovolume-averaging over nonuniform profiles), the power bal-ance of Eq. (11) becomes

fc〈SF〉V +Pabs = 〈SB〉V +3〈nT 〉

τEV, (28)

14

where power densities are denoted with variables S, and weassume n = ne = ni (i.e., hydrogenic plasma without impuri-ties) and T = Te = Ti everywhere. Reported/inferred values ofPabs and τE are already global, volume-averaged quantities.

To quantify the profile effect on SF , we introduce

λF = 〈SF〉/SF0, (29)

where SF0 is the fusion power density with spatially uniformTi0 and ni0, and 〈SF〉 is the volume-averaged fusion powerdensity of the nonuniform-profile case with peak values Ti0and ni0. Similarly,

λB = 〈SB〉/SB0 (30)

and

λκ =〈nT 〉n0T0

, (31)

which quantify the nonuniform-profile modifications to thebremsstrahlung power density and thermal energy density, re-spectively.

The result is a modified version of Eq. (11), where profileeffects are captured in the terms λF , λB, and λκ ,

λF fcSF0V +Pabs = λBSB0V +3λκ n0T0

τEV. (32)

From this power balance of the nonuniform-profile case, thepeak value of the Lawson parameter n0τE required to achievea particular value of 〈Qfuel〉 as a function of T0 is

n0τE =

3λκ T0

λF( fc + 〈Qfuel〉−1)〈σv〉εF/4−λBCBT 1/20

,(33)

where

〈Qfuel〉=λF SF0

Pabs/V≡ λF Qfuel. (34)

We adopt the approach of using the same peak (ratherthan average) values of density and temperature when eval-uating Qfuel (uniform spatial profiles) versus 〈Qfuel〉 (nonuni-form spatial profiles), for the practical reasons that peak val-ues are more commonly reported in the literature and that pro-files are often not reported. When using the same peak ratherthan profile-averaged values, spatially nonuniform profiles in-crease rather than decrease the requirements on peak densityand temperature for achieving a given Qfuel.

Next we consider representative profiles in order to quantifythe differences between Qfuel and 〈Qfuel〉 for cylindrical andtoroidal geometries. A wide variety of temperature and den-sity profiles have been observed in fusion experiments. Theseprofiles are typically modeled as functions of normalized ra-dius x = r/a, where a is the device radius for cylindrical sys-tems and the minor radius for toroidal systems with circularcross section. Commonly used and flexible models of densityand temperature profiles are

n(x) = n0(1− x2)νn and T (x) = T0(1− x2)νT , (35)

where n0 and T0 are the central/peak ion or electron densitiesand temperatures, respectively. The values of νn and νT adjustthe sharpness of the peaks of the profiles. In the limit νn→ 0and νT → 0, the peak is infinitely broad and we recover theuniform-profile case. This approach accommodates a widerange of profiles.38,39

From Eqs. (6) and (29),

λF =〈SF〉SF0

=〈n2

i 〈σv〉(Ti(x))〉n2

i0〈σv〉(Ti0), (36)

where the Ti dependence of 〈σv〉 is shown explicitly, resultingin λF being a function of the Ti profile. From Eqs. (7) and(30),

λB =〈SB〉SB0

=〈n2

eT 1/2e 〉

n2e0T 1/2

e0

. (37)

For a cylinder or large-aspect-ratio torus (i.e., R/a � 1,where R and a are the major and minor radii, respectively)with circular cross section and the profiles of Eq. (35), we usethe expressions in Appendix F to obtain

λF =

∫ 10 (1− x2)2νn〈σv〉(Ti(x))2xdx

〈σv〉(Ti0), (38)

which may be evaluated numerically for any tabulated or pa-rameterized values of 〈σv〉(Ti),

λB =∫ 1

0(1− x2)2νn+νT /22xdx =

24νn +νT +2

, (39)

and

λκ =∫ 1

0(1− x2)(νn+νT )2xdx =

11+νn +νT

. (40)

For a torus with circular cross section and arbitrary val-ues of R/a, λF , λB, and λκ must be evaluated numerically(see Appendix F). For profiles with large Shafranov shift, i.e.,magnetic axis shifted toward larger R, the reduction of fusionpower due to profile effects (and hence λF ) is mitigated be-cause the high-temperature region occupies a larger fraction ofthe total volume. Therefore the profiles considered here rep-resent a likely worst-case scenario and provide a lower boundon λF .

To demonstrate the effect of nonuniform profiles on thecontours of 〈Qfuel〉 compared to Qfuel, we consider two setsof profiles. The first is a parabolic profile with νT = 1 andνn = 1, which is a simple approximation of the profiles intokamaks.6 The second is a more strongly peaked tempera-ture profile with νT = 3 and a broader density profile withνn = 0.2, which are representative of profiles in the advancedtokamak or reversed-field pinch.40 For both sets of profiles,we assume T = Ti = Te and n = ni = ne (impurity-free hy-drogenic plasma). Figures 19 and 20 show these two setsof profiles, respectively, along with their corresponding val-ues of λF vs. Ti0 and resulting adjustments to the Qfuel con-tours. For both sets of profiles (Figs. 19 and 20), nonuniform

15

profiles [dashed lines in panel (c)] increase the peak Lawsonparameter needed to achieve a particular value of 〈Qfuel〉 fortemperatures below approximately 50 keV. Additionally, theideal ignition temperature, defined by Eq. (8), is increased. Athigh temperatures approaching 100 keV, where fusion powerexceeds bremsstrahlung by a large factor (see Fig. 6), the ad-justment is equal to the ratio λκ/λF , which is close to unity inthe case of the parabolic profiles, and drops below unity in thecase of the peaked and broad profiles. At intermediate tem-peratures, λF , λB, and λκ all contribute to the modification of〈Qfuel〉 compared to Qfuel.

3. Effect of impurities (and non-hydrogenic plasmas)

Real fusion experiments must contend with the effect ofions with charge state Z > 1. These may be from helium ash,impurities from the first wall, or advanced fuels. These impu-rities increase the bremsstrahlung radiation by a factor

Zeff =ΣiniZ2

ine

, (41)

where i is summed over all ion species in the plasma. Addi-tionally, impurities increase the electron density relative to theion density by a factor of the mean charge state of the entireplasma,

Z = ne/ni, (42)

which reduces ni and therefore PF at fixed pressure.Using these definitions along with the generalized expres-

sion for bremsstrahlung,

PB =CBneT 1/2e Σi(Z2

i ni), (43)

Eq. (33) becomes

ni0τE =

(3/2)λκ(1+ Z)T0

λF( fc +ηabs〈Qsci〉−1)〈σv〉εF/4−λBCBZeffZ2T 1/20

,(44)

and

ni0T0τE =

(3/2)λκ(1+ Z)T 20

λF( fc +ηabs〈Qsci〉−1)〈σv〉εF/4−λBCBZeffZ2T 1/20

,(45)

where λF , λB, and λκ are unchanged because Zeff and Z aretreated as volume-averaged quantities. We have also replacedthe 〈Qfuel〉−1 term with ηabs〈Qsci〉−1, which allows us to in-clude the effect of absorption efficiency.

Each experiment has different values of λF , λB, λκ , Z, Zeff,and ηabs, and therefore each experiment has different 〈Qsci〉contours. It is not feasible to show unique 〈Qsci〉 contours foreach experiment in Figs. 2, 3, and 25. Figure 21 shows finite-width 〈Qsci〉 contours of the peaked and broad profiles whoselower and upper limits correspond to low-impurity (Zeff = 1.5,

(a)

(b)

(c)

FIG. 19. (a) Normalized parabolic profiles (with νT = 1 and νn = 1)of T = Ti = Te and n = ni = ne. (b) Parameter λF vs. Ti0 (λB = 0.286and λκ = 0.333 for these profiles). (c) Peak Lawson parameter vs.T0 for the parabolic profiles (dashed lines) shown in (a) and uniformplasma (solid lines), for Qfuel = 1 (blue) and Qfuel = ∞ (red).

16

(a)

(b)

(c)

FIG. 20. (a) Normalized peaked and broad profiles (with νT = 3 andνn = 0.2) of T = Ti = Te and n = ni = ne. (b) Parameter λF vs. Ti0(λB = 0.345 and λκ = 0.238 for these profiles). (c) Lawson param-eter vs. Ti0 for the profiles (dashed lines) shown in (a) and uniformplasma (solid lines), for Qfuel = 1 (blue) and Qfuel = ∞ (red).

Z = 1.2) and high-impurity (Zeff = 3.4, Z = 1.2) models, re-spectively. These impurity levels correspond to the range ofimpurity levels considered for SPARC41 and ITER.42 For boththe high and low-impurity models, we assume T = Ti = Teand ηabs = 0.9. The finite ranges of 〈Qsci〉 aim to accountfor the main features and uncertainties of a future experimen-tal device that will achieve 〈Qsci〉> 1, and therefore we showfinite-width 〈Qsci〉 contours in Fig. 2 (despite the Qsci labels inthe legend). We emphasize that the finite width of the 〈Qsci〉contours are merely illustrative of the effects of profiles andimpurities and of the approximate values of 〈Qsci〉 that mightbe achieved by SPARC or ITER. To predict 〈Qsci〉with higherprecision would require detailed analysis and simulations.

FIG. 21. Finite-width 〈Qsci〉 contours vs. peak Lawson parameterand T0 bounded by low-impurity (Zeff = 1.5, Z = 1.2) and high-impurity (Zeff = 3.4, Z = 1.2) cases, for peaked and broad spatialprofiles (νT = 3, νn = 0.2). These assumptions are made in plottingthe Qsci contours of Figs. 2, 3, and 25.

4. Inferring peak from volume-averaged values

When only volume-averaged values of density and temper-ature are reported, we infer the peak values from an estimatedvalue of the peaking, T0/〈T 〉 and n0/〈n〉, respectively. De-tailed empirical models of peaking exist for predicting the pro-files of future experiments.43–46 However, for the purposes ofthis paper, we have chosen peaking values on a per-conceptbasis, the values of which are indicated in Table V. Only con-cepts for which peak values must be inferred from reportedvolume-averaged values, along with citations for those val-ues, are listed in Table V. In Tables VI and VII, we append asuperscript asterisk (∗) to peak values inferred from reportedvolume-averaged quantities.

17

TABLE V. Peaking values required to convert reported volume-averaged quantities to peak value quantities.

Concept T0/〈T 〉 n0/〈n〉 Reference

Tokamak 2.0 1.5 46Stellarator 3.0 1.0 47Spherical Tokamak 2.1 1.7 48FRC 1.0 1.3 49 and 50RFP 1.2 1.2 40Spheromak 2.0 1.5 51

5. Inferring ion quantities from electron quantities

When only Te and not Ti is reported, we cannot assumeTi = Te in calculating the triple product without further consid-eration. If the thermal-equilibration time is much shorter thanthe plasma duration, and assuming there are no other effectsthat would give rise to Ti 6= Te, then we can assume Ti = Te. Inthese cases we append a superscript dagger (†) to the inferredvalue of Ti in Tables VI and VII. In cases where both Ti and Teare reported in MCF experiments, we use the reported Ti.

When only ne but not ni is reported, we assume ni = nefor D-T and D-D plasmas. In such cases we append a super-script double dagger (‡) to the inferred value of ni in Tables VIand VII.

6. Accounting for transient heating

All experiments experience a transient start-up phase dur-ing which a portion of the heating power goes into raising theplasma thermal energy Wp = 3nTV (assuming T = Ti = Teand n = ni = ne). There are two self-consistent approaches forderiving an expression for Qfuel that accounts for the effect oftransient heating dWp/dt. In the remainder of this subsection,we closely follow Ref. 52.

The first approach is to group the transient term with Pabs inthe instantaneous power balance which effectively treats thetransient term as a reduction in the externally applied and ab-sorbed heating power,(

Pabs−dWp

dt

)+Pc = PB +

3nTVτE

. (46)

In this approach, the definition of Qfuel is modified, i.e.,

Q∗fuel =PF

Pabs−dWp/dt. (47)

From here, we derive an expression for the Lawson parame-ter following the same steps as Sec. III E, which results in ananalogous expression to Eq. (16) but with Qfuel replaced byQ∗fuel,

nτE =3T

( fc +Q∗−1fuel )〈σv〉εF/4−CBT 1/2

. (48)

From Eq. (46),

τE =Wp

Pheat−dWp/dt, (49)

where

Pheat = Pabs +Pc−PB. (50)

This approach, defined by Eqs. (47)–(50), is the one used byJET and JT-60.

The second approach is to treat the transient heating termas a “loss” term alongside thermal conduction, i.e.,

Pabs +Pc = PB +

(3nTV

τE+

dWp

dt

). (51)

We then define a modified energy confinement time τ∗E whichcharacterizes thermal conduction and transient heating power.

3nTVτ∗E

=3nTV

τE+

dWp

dt. (52)

Combining the latter with Eqs. (50) and (51),

τ∗E =

Wp

Pheat. (53)

From this point, we derive an expression for the Lawson pa-rameter following the same steps as Sec. III E, which resultsin an analogous expression to Eq. (16) but with τE replacedby τ∗E ,

nτ∗E =

3T( fc +Q−1

fuel)〈σv〉εF/4−CBT 1/2. (54)

In this formulation, the definition of instantaneous Qfuel isunchanged from the steady-state value of Eq. (14), and fuelbreakeven occurs at Qfuel = 1, regardless of the value ofdWp/dt. This approach, defined by Eqs. (53), (54), and (15),is the one used by TFTR and consistent with Lawson’s origi-nal formulation.

For the JET/JT-60 approach, fuel breakeven does not nec-essarily occur at Q∗fuel = 1 but rather occurs at a value of Q∗fuelthat depends on the value of dWp/dt. The TFTR/Lawson ap-proach keeps the definition of instantaneous Qfuel the sameas the steady-state Qfuel, and fuel breakeven always occurs atQfuel = 1 regardless of the transient-heating value. Becausea key objective of this paper is to chart the progress of manydifferent experiments toward and beyond Qfuel = 1, we usethe TFTR/Lawson definition for which Qfuel = 1 means thesame thing across different MCF experiments. In practice, thismeans we use τ∗E and Eq. (54) for all MCF experiments. WhenτE and dWp/dt are reported and dWp/dt is nonzero (e.g., JETand JT-60), we calculate and use τ∗E , indicating such caseswith a superscript hash (#) in Tables VI and VII. Some TFTRpublications report τE , requiring the conversion step, and thuswe append a superscript hash for those cases as well.

B. ICF methodology

Direct measurements of plasma parameters are more chal-lenging for ICF. Commonly measured parameters in ICFare fuel areal density ρR (via neutron downscattering), Ti

18

FIG. 22. Representation of an ICF capsule implosion and hot-spotcreation with instability growth: (a) dense fuel shell, with radius Rand thickness ∆, at maximum shell velocity Vi during implosion, (b)fuel assembly at stagnation with the “hot spot” (blue) with effectiveradius Rs, surrounded by the cold, dense fuel (grey). Rayleigh-Taylorinstabilities are shown. If the hot spot reaches high-enough niτ andTi, then it can potentially generate enough fusion energy to initiate apropagating burn into the surrounding cold shell.

and “burn duration” (via neutron time-of-flight), and neutronyield (via various types of neutron detectors). Some exper-iments report an inferred stagnation pressure pstag based onstatistical analysis of other measured quantities and simula-tion databases.

Identifying the requirements for ignition of an ICF capsuleis difficult. The analysis presented in Sec. III F assumes anidealized ICF scenario. Real ICF experiments must contendwith instabilities, impurities, non-zero bremsstrahlung andthermal-conduction losses, and other factors that make it moredifficult to achieve ignition. For the highest-performing ICFexperiments considered here (NIF, OMEGA), a two-stage ap-proach to ignition is pursued, i.e., ignition of a central lower-density “hot spot” followed by propagating burn into the sur-rounding colder, denser fuel, as depicted in Fig. 22. Becauseof the low value of nabs inherent in these experiments, thistwo-stage process is required to achieve Qsci > 1. Therefore,we consider both ignition of the hot spot and a propagatingburn in the dense fuel when we refer to “ignition” in this sec-tion.

Below we describe two methodologies used in this paperfor inferring the Lawson parameter nτ and triple product nT τ

for cases in which pressure is or is not experimentally inferred,respectively.

1. Inferring Lawson parameter and triple product withoutreported inferred pressure

For ICF experiments that do not report experimentally in-ferred values of fuel pressure (i.e., rows with “–” in the pstagcolumn of Table VIII), we employ the methodology of Betti

et al. 31 to infer niτ from other measured ICF experimentalquantities. Here, we state the key logic and equation of thismethodology for the convenience of the reader, but we referthe reader to Ref. 31 for further details, equation derivations,and justifications. It is important to note that Ref. 31 makes asimplifying assumption that thermal-conduction and radiationlosses are negligible (on the timescale of the fusion burn) be-cause of the insulating effects of the dense shell of an ICF tar-get capsule, meaning that Lawson parameters and triple prod-ucts inferred via this method should be considered as upperbounds.

The ICF-capsule shell is modeled as a thin shell with thick-ness ∆ � R, where R is the shell radius, as illustrated inFig. 22. A fraction of the peak kinetic energy of the shell isassumed to be converted to thermal pressure in the hot spot atstagnation. An upper bound on τ is obtained based on the timeit takes for the stagnated shell (at peak compression) to expanda distance of order its inner radius Rs. Significant 3D effectsarising from Rayleigh-Taylor-instability spikes and bubblesat the interface of the shell and hot spot reduce the effectivehot-spot volume by a “yield-over-clean” factor YOCµ , whereµ ∼ 0.4–0.5 is inferred from two simulation databases.53 Withthese and other simplifying assumptions, Betti et al. 31 obtain

nT τ (3D)≈ 4[(ρR)no α

tot(n)Tno α

n ]0.8YOCµ [atms], (55)

with measured total areal density (ρR)no α

tot(n) in g cm−2, andmeasured “burn-averaged” ion temperature T no α

n in keV. Thesuperscript “no α” refers to experimental measurements madewhen α heating is not an appreciable effect (and α heat-ing is turned off in simulations). For ICF experiments with-out reported values of hot-spot pressure, Eq. (55) is used toplot achieved ICF values of Lawson parameters and tripleproducts, where the unit [atm s] is multiplied by 6.333×1020 keV m−3 atm−1 to convert to [m−3 keV s]. Dividing thetriple product by T gives the Lawson parameter nτ .

2. Inferring Lawson parameter from inferred pressure andconfinement dynamics

When the inferred stagnation pressure pstag and the dura-tion of fuel stagnation τstag are reported, the pressure times theconfinement time τ can be calculated directly. However, fol-lowing Christopherson,54 three adjustments are made to τstag,which is defined as the full-width half-maximum (FWHM)of the neutron-emission history (i.e., “burn duration”), to ob-tain an approximation for τ . The first adjustment is that, formarginal ICF ignition, only alphas produced before bang time(time of maximum neutron production) are useful to ignite thehot spot because, afterward, the shell is expanding and the hotspot is cooling, reducing the reaction rate; this introduces afactor of 1/2. The second adjustment is that only a fraction offusion alphas are absorbed by the hot spot; this factor is es-timated to be 0.93. The third adjustment is that, to initiate apropagating burn of the surrounding fuel, an additional factorof 0.71 is applied to account for the dynamics of alpha heatingof the cold shell. Applying these three corrections results in

19

τ ≈ τstag/3 and

nT τ ≈ 12

pstagτ ≈ 16

pstagτstag. (56)

The only exception to this approach is the FIREX experiment,for which we estimate the value of pstagτ directly from thereported values.

3. Adjustments to the required values of Lawson parameterand temperature required for ignition

The ignition requirement derived in Sec. III F ignores anumber of factors that increase the requirements for ignitionof an ICF capsule. We consider these effects to be incorpo-rated in reductions to τ in the previous subsection. Thus, nofurther adjustments are made to the contours of constant Qscidefined by Eq. (20).

4. Differences between ICF and MCF

It is not straightforward to compare the achieved Lawsonparameters and triple-product values between ICF and MCF.While a quantitative approach can be taken via the ignitionparameter χ described in Ref. 31, the approach taken here isqualitative and is reflected in the different Qsci contours forICF and MCF in Figs. 2, 25, and 3.

Firstly, the achieved triple product for ICF is higher thanfor MCF in part because of two assumptions made in theirinference. Following Ref. 31, we assume in ICF that thereare no bremsstrahlung radiation losses due to trapping by thepusher (with a high-enough areal density to be opaque to x-rays) and that the fuel hot-spot pressure is spatially uniform.These assumptions lead to higher values for the inferred Law-son parameter and triple product.

Secondly, whereas Pext and Pabs differ by only a factor of or-der unity in MCF,36 they differ by a factor of & 50 in ICF (seeTable IV). This is due to the low conversion efficiency fromapplied laser energy to absorbed fuel energy. Thus, while bothMCF28 and ICF29 have achieved Qsci ∼ 0.7, ICF has necessar-ily achieved a higher value of Qfuel compared to MCF.

Note further that the horizontal line representing QICFsci =

∞ in Fig. 3 (corresponding to the nT τ value of the contourat 4 keV) is at a higher value than the minimum nT τ valueof the corresponding contour in Fig. 25. This is because Tiin laser ICF experiments (prior to onset of significant fusion)is limited by the maximum implosion velocity at which theshell becomes unstable, corresponding to a maximum Ti ofabout 4 keV. Thus, marginal onset of ignition corresponds tothe required nT τ value at approximately 4 keV. In the caseof NIF N210808, which exceeded the threshold for onset ofignition55, Ti increased due to self heating and τ decreasedbecause of the increased pressure. These effects resulted ina slightly lower triple product compared with previous non-ignition results, which is visible in Fig. 25.

C. MIF/Z-pinch methodology

1. MagLIF

The Magnetized Liner Inertial Fusion (MagLIF)experiment56 compresses a cylindrical liner surroundinga pre-heated and axially pre-magnetized plasma. TheZ-machine at Sandia National Laboratory supplies a largecurrent pulse to the liner along its long axis, compressing it inthe radial direction.

While the solid liner makes diagnosing MagLIF plasmasmore difficult, it is still possible to extract the parametersneeded to estimate the Lawson parameter and triple product.The burn-averaged Ti at stagnation is measured by neutrontime-of-flight diagnostics. The spatial configuration of theplasma column at stagnation is imaged from emitted x-rays.From this spatial configuration and a model of x-ray emis-sion, the effective fuel radius is inferred. The stagnation pres-sure is inferred from a combination of diagnostic signatures.Given the plasma volume, burn duration, and temperature, thepressure was inferred by setting the pressure and mix levelsto simultaneously match the x-ray yield and neutron yield. Inthe emission model used to determine the spatial extent of thestagnated plasma, the pressure in the stagnated fuel is assumedto be spatially constant and the temperature and density pro-files are assumed to be inverse to each other.57 For our pur-poses, we infer an average ni from the stagnation pressure andthe measured burn-averaged Ti.

Finally, the burn time, the duration during which the fuelassembly is inertially confined and hard x-rays (surrogates forfusion neutrons) are emitted, is measured. This duration is anupper bound on τ , and in practice τ is estimated to be equalto it. Data for MagLIF are shown in Table VIII and plotted inFigs. 2, 3, and 25.

2. Z pinch

Z-pinch experiments were one of the earliest approachesto fusion because no external magnetic field is required forconfinement. This simplifies the experimental setup and re-duces costs. Figure 23 shows a representative diagram of a Z-pinch plasma. While fusion neutrons were detected in some ofthe earliest Z-pinch experiments, those fusion reactions werefound to be the result of plasma instabilities generating non-thermal beam-target fusion events (see pp. 91–93 of Ref. 58),which would not scale up to energy breakeven. More recently,however, stabilized Z-pinch experiments have provided evi-dence of sustained thermonuclear neutron production.59,60

Z-pinch plasmas exhibit profile effects perpendicular to thedirection of current flow so the profile considerations dis-cussed in Section IV A apply to Z pinches as well. The ra-dial density profile of Z pinches is typically described by aBennett-type profile61 of the form n(r) = n0/[1 + (r/r0)

2]2

and illustrated in Fig. 24.Assuming T = Ti = Te, n = ni = ne, and a uniform profile

for the plasma temperature, the thermal energy of a Z-pinch

20

FIG. 23. A representation of a Z-pinch plasma of length L, effec-tive radius r0, and electrical current I. Vp is the voltage differencebetween the left and right side of the plasma.

FIG. 24. Bennett-type density profile. In contrast to the parabolicprofiles, the plasma extends beyond the effective radius r0.

plasma can be estimated as

Wp =3T∫

Vn(r)d3r

=3LT∫

∞

0n(r)2πrdr

=6Lπn0T∫

∞

0

r(1+(r/r0)2)2 dr

=3πr20Ln0T.

(57)

The power applied is

Pabs = IVp, (58)

where I is the Z-pinch current and Vp is the voltage across theplasma driving the current along the long axis. Assuming noself heating and that thermal conduction is the primary sourceof energy loss, the τ∗E for the stabilized Z-pinch is

τ∗E =

3πr20Ln0T0

IVp, (59)

and the Lawson parameter for a stabilized Z-pinch is

n0τ∗E =

3πr20Ln2

0T0

IVp. (60)

However, in practice Vp may not be measured directly, andthe voltage across the power supply driving the Z-pinch mayoverestimate Vp. Therefore, evaluations of τ∗E that substitutethe power supply voltage for Vp (as done for FuZE59,60) pro-vide only a lower bound on τ∗E . An upper bound on τ∗E is theflow-through time of the Z-pinch. Our reported value is thelower of the two.

In other Z-pinch approaches like the dense plasma fo-cus (DPF), fusion yields occur from a combination ofnon-Maxwellian ion energy distributions and thermal ionpopulations.62 Because thermal temperatures and τ∗E are typi-cally not well characterized in such approaches, it is not fea-sible to report a reliable, achieved Lawson parameter or tripleproduct. Furthermore, fusion concepts with strong beam-target components may not be scalable to Qfuel > 1.25

3. Other MIF approaches

For other MIF approaches,63 e.g., liner or flux compressionof FRCs or spheromaks, it is difficult to rigorously measureτ∗E due to limited access. A few attempts to quantify τ∗E basedon measurable or calculable parameters, such as particle con-finement time τN , have been proposed.50 In particular, we es-timate τ∗E of FRCs to be τN/3 (for both MIF and MCF).

V. SUMMARY AND CONCLUSIONS

The combination of achieved Lawson parameter nτ or nτEand fuel temperature T of a thermonuclear-fusion conceptare a rigorous scientific indicator of how close it is to en-ergy breakeven and gain. In this work, we have compiled theachieved Lawson parameters and T of a large number of fu-sion experiments (past, present, and projected) from aroundthe world. The data are provided in multiple tables and fig-ures. Following Lawson’s original work, we provided a de-tailed review, re-derivation, and extension of the mathematicalexpressions underlying the Lawson parameter (and the relatedtriple product) and four ways of measuring energy gain (Qfuel,Qsci, Qwp, and Qeng), and explained the physical principlesupon which these quantities are based. Because different fu-sion experiments report different observables, we explainedprecisely how we infer both electron and ion densities andtemperatures and the various definitions of confinement timethat are used in the Lawson-parameter and triple-product val-ues that we report, including accounting for the effects of spa-tial profile shapes (through a peaking factor) and a range inthe level of impurities in the plasma fuel. All data reported inthis paper are based on the published literature or are expectedto be published shortly.

The key results of this paper are encapsulated in Figs. 2,3, and 25, which show that (1) tokamaks and laser-driven ICFhave achieved the highest Lawson parameters, triple products,and Qsci ∼ 0.7; (2) fusion concepts have demonstrated rapidadvances in Lawson parameters and triple products early intheir development but slow down as values approach what isneeded for Qsci = 1; (3) private fusion companies pursuing al-ternate concepts are now exceeding the breakout performanceof early tokamaks; and (4) at least three experiments mayachieve Qsci > 1 within the foreseeable future, i.e., NIF andSPARC in the 2020s and ITER by 2040.

The reason for item (2) in the preceding paragraph is com-monly attributed to the fact that experimental facilities be-came extremely expensive (e.g., $3.5B for NIF according

21

to the U.S. Government Accountability Office, and exceed-ing US$25B for ITER) for making continued and requiredadvances toward energy gain. However, there are two rea-sons that other approaches or experiments might potentiallyachieve commercially relevant energy breakeven and gain on afaster timescale. Firstly, most of the other paths being pursued(i.e., privately funded development paths for tokamaks, stel-larators, alternate concepts, and laser-driven ICF) have lowercost as a key objective, where experiments along the devel-opment path are envisioned to have much lower costs thanNIF and ITER. Secondly, the mature fusion and plasma scien-tific understanding and computational tools, as well as manyfusion-engineering technologies, developed over 65+ years ofcontrolled-fusion research do not need to be reinvented andneed only be leveraged in the development of the alternateand privately funded approaches.

High values of Lawson parameter and triple product, whichare required for energy gain, are a necessary but not sufficientcondition for commercial fusion energy. Additional neces-sary conditions include attractive economics and social accep-tance, including but not limited to considerations of RAMI(reliability, accessibility, maintainability, and inspectability)and the ability to be licensed under an appropriate regulatoryframework. These necessary conditions require additionaltechnological attributes beyond high energy gain, e.g., (1) afusion plasma core that is compatible with both surroundingmaterials and subsystems that survive the extreme fusion par-ticle, heat, and radiation flux, and (2) a sustainable fuel cycle(e.g., tritium breeding, separation, and processing technolo-gies for D-T fusion). Therefore, while this paper’s primaryobjective is to explain and highlight the achieved Lawson pa-rameters (and triple products) of many fusion concepts andexperiments as a measure of fusion’s progress toward energybreakeven and gain, these are not the only criteria for justi-fying continued pursuit of and investment into a given fusionconcept, including concepts using advanced fusion fuels.

Appendix A: Plot of triple products vs. Ti

Figure 25 shows achieved triple products versus Ti, basedon the same data points used in Fig. 2.

Appendix B: Data tables

Table VI provides numerical values of the data for toka-maks, spherical tokamaks, and stellarators. Table VII pro-vides numerical values of the data for “alternate” MCF con-cepts, i.e., not tokamaks or stellarators. Table VIII pro-vides numerical values of the data for ICF and MIF experi-ments. We group lower-density and higher-density MIF ap-proaches with MCF alternate concepts (Table VII) and ICF(Table VIII), respectively.

Appendix C: Effect of mitigating bremsstrahlung losses

If bremsstrahlung radiation losses are mitigated, e.g., inpulsed ICF19 or MIF63,133 approaches with an optically thickpusher,134,135 then the Qfuel and Qsci contours of Figs. 12 and14 can be modified. Figure 26 illustrates the effect of arbi-trarily reducing PB by a factor of 2, i.e., by replacing CB withCB/2 in Eqs. (16) and (22).

Appendix D: Lawson parameters for advanced fusion fuels

The main body of this paper focuses on D-T fusion becauseit has the highest maximum reactivity occurring at the lowesttemperature compared to all known fusion fuels. As a result,the required D-T Lawson parameters and triple products toreach high Qfuel are the lowest and most accessible. However,D-T fusion has two major drawbacks: (i) it produces 14-MeVneutrons that carry 80% of the fusion energy, and (ii) the tri-tium must be bred (because it does not occur abundantly innature due to a 12.3-year half life) and be continuously pro-cessed and handled safely.

Advanced fuels, such as D-3He, D-D, and p-11B, mitigatethese drawbacks to different extents.136 However, becausetheir peak reactivities are all lower and occur at higher tem-peratures compared to D-T, the required Lawson parametersand triple products for these advanced fuels to achieve equiv-alent values of Qfuel are much higher.

Furthermore, at the high temperatures required for ad-vanced fuels, relativistic bremsstrahlung effects become sig-nificant. We utilize the relativistic-correction approximationto Eq. (43) from Ref. 137,

PB =CBneT 1/2e γ(Zeff), (D1)

where

γ(Zeff) = Zeff(1+1.78t1.34)+2.12t(1+1.1t + t2−1.25t2.5)(D2)

and t = Te/mec2.To quantify the Lawson-parameter and triple-product re-

quirements for advanced fuels with non-identical reactantsand reaction products that are immediately removed fromthe plasma (e.g., D-3He and p-11B without ash buildup orsubsequent reactions), we first generalize the expressionfor nτE [Eq. (16)] to account for the effect of relativis-tic bremsstrahlung and the reaction of two ion species withcharge per ion Z1 and Z2, ion number densities n1 and n2, andrelative densities k1 = n1/ne and k2 = n2/ne, respectively.

A more detailed treatment of advanced fuels would need toconsider scenarios in which Te < Ti and account for an ad-ditional term in the power-balance equation for ion energytransfer to electrons. Maintaining Te � Ti has the advantageof reduced bremsstrahlung (especially at high Ti) and lowerplasma pressure for a given Ti. The challenge of such a sce-nario is maintaining Ti > Te for a sufficient duration of timeand with acceptable additional input power. In this section, weonly consider T = Ti = Te, except in the discussion of Fig. 28.

22

FIG. 25. Experimentally inferred, peak triple products of fusion experiments vs. ion temperature, extracted from published literature. See thecaption of Fig. 2 for more details.

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24

TAB

LE

VII

.Dat

afo

rmag

netic

alte

rnat

eco

ncep

ts.

Proj

ect

Con

cept

Yea

rSh

otId

entifi

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ence

T i0

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eV)

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0(m−

3 )τ∗ E (s)

n i0τ∗ E

(m−

3s)

n i0T

i0τ∗ E

(keV

m−

3s)

ZE

TAPi

nch

1957

140k

a-18

0ka

disc

harg

es10

20.

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1020

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1020

0.00

011.

0×

1016

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1014

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TAI

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1977

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mar

y10

30.

01–

1×

1021

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10−

61.

0×

1015

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1013

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TAII

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1984

5961

110

40.

09†

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1020

0.00

013.

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1015

TM

X-U

Mir

ror

1984

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110

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150.

045

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1018

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1018

0.00

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1987

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now

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33†

0.33

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076.

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1990

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flux

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erve

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184.

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1993

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0001

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2001

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60.

792

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0.00

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1016

ZaP

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nch

2003

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25

TAB

LE

VII

I.D

ata

forI

CF

and

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conc

epts

.

Proj

ect

Con

cept

Yea

rSh

otId

entifi

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efer

ence

〈Ti〉 n

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m−

2 )Y

OC

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g(G

bar)

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τ

(atm

s)nτ

(m−

3s)

n〈T〉 n

τ

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m−

3s)

NO

VAL

aser

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AL

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231.

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aser

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131.

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aser

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2009

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aser

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2009

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00.

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2014

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2015

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2015

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1021

26

(a)

(b)

FIG. 26. Contours of Qfuel plotted vs. T and (a) nτE and (b) nT τEfor D-T fusion (assuming T = Ti = Te). Dashed lines represent arbi-trarily reducing bremsstrahlung losses by a factor of 2, i.e., replacingCB by CB/2 in Eqs. (16) and (22).

Accounting for the above,

neτE =(3T/2)(k1 + k2 +1)

( fc +Q−1fuel)k1k2〈σv〉1,2E1,2−CBT 1/2γ(Zeff)

, (D3)

where Zeff = Σ jn jZ2j /ne, and j is summed over the different

reactant species.The relative density for each ion species j that

maximizes138 fusion power for a fixed value of n2e is k j =

1/(2Z j) and Zeff = (Z1 + Z2)/2. Assuming this condition,

Eq. (22) becomes

neT τE =

(3T 2/2)[(2Z1)−1 +(2Z2)

−1 +1]( fc +Q−1

fuel)〈σv〉1,2E1,2/(4Z1Z2)−CBT 1/2γ(Zeff),

(D4)

or equivalently,

niT τE =

(3T 2/2)[(2Z1)−1 +(2Z2)

−1][(2Z1)−1 +(2Z2)

−1 +1]( fc +Q−1

fuel)〈σv〉1,2E1,2/(4Z1Z2)−CBT 1/2γ(Zeff),

(D5)

where we have multiplied both sides of Eq. (D4) by (k1 +k2) = (2Z1)

−1 + (2Z2)−1. This expression ignores syn-

chrotron radiation losses, which may become important at thevery high temperatures required to reach Lawson conditionsfor advanced fuels in magnetically confined systems.

1. D-3He

The D-3He fusion reaction has the advantage that its pri-mary reaction,

D+3He→ α +p (18.3 MeV), (D6)

is aneutronic, where the α is a 4He ion. However, 3Heis not abundant on earth and must be bred via other reac-tions or mined from the moon, both of which involve addi-tional complexity and cost. Also, D-3He will not be com-pletely aneutronic because of D-D reactions. The require-ment for ignition of D-3He ignoring side D-D reactions isniT τ∗E ≥ 5.2× 1022 m−3 keV s at 68 keV (see Fig. 27), 18times higher than for D-T.

2. p-11B

The p-11B fusion reaction has the advantage that its reac-tants are abundant on earth, and the reaction products are threeelectrically charged α particles, potentially allowing for directenergy conversion to electricity. However, this reaction re-quires temperatures around 100 keV, at which bremsstrahlungradiation losses per unit volume exceed fusion power density,and ignition is not possible for a p-11B plasma where Te = Ti,as shown in Fig. 28, which uses the parametrized p-11B fusionreactivity from Ref. 23. The boron and proton concentrationsare set to maximize fusion power for a fixed electron densityas described earlier in this section. Also shown is the effect ofreduced bremsstrahlung if Te is maintained at levels below Ti.We are neglecting the issue of the ion-electron thermal equi-libration time here. Figure 29 shows that only modest valuesof Qfuel are physically possible for Te = Ti, at triple productsthree orders of magnitude higher than that of D-T.

However recent work139 points to a higher reactivity, andgiven certain assumptions, high-Qfuel operation up to and in-cluding ignition may be theoretically possible.

27

(a)

(b)

FIG. 27. Required (a) Lawson parameters and (b) triple productsvs. Ti to achieve the indicated values of Qfuel for D-3He (assumingT = Te = Ti).

3. Fully catalyzed D-D

The D-D fusion reaction has the advantage that its solereactant is abundant on earth. In the fully catalyzed D-Dreaction,140,141 the T and 3He produced as reaction productsundergo subsequent reactions with D, releasing more energy.

FIG. 28. Charged-particle fusion power density Pc (purple line) andbremsstrahlung power density PB for various ratios of Te/Ti vs. Ti forp-11B, showing that PB always exceeds Pc when Te & Ti/3. This plotuses the parameterized p-11B reactivity in Ref. 23. Updated, higherp-11B fusion cross sections139 suggest that ignition may be possiblefor p-11B.137

The reaction paths are

D+D→ T (1.01 MeV)+p (3.02 MeV) (D7)

D+T→ α (3.5 MeV)+n (14.1 MeV) (D8)

D+D→ 3He (0.82 MeV)+n (2.45 MeV) (D9)

D+ 3He→ α (3.6 MeV)+p (14.7 MeV), (D10)

with 62% of the 43.2 MeV released in charged particles (com-pared with only 20% for D-T).

Note that there are other forms of “catalyzed D-D” whichgo by different names in different contexts. For example ex-traction of tritium before the subsequent D-T reaction occursis sometimes called “3He double-catalyzed D-D”.141 Here weonly consider the steady-state reaction path where 3He andT react with D at the same rate as they are created in eachbranch of the D-D reaction. Furthermore, we assume anidealized scenario without synchrotron radiation and that the“ash” α particles and protons immediately exit after deposit-ing their energy and comprise a negligible fraction of ions inthe plasma. Lastly, we assume that D is added at the same rateas it is consumed and that T = Ti = Te.

The ion number density is the sum of the constituent ionnumber densities,

ni = nD +n3He +nT, (D11)

and the electron density is,

ne = nD +2n3He +nT. (D12)

28

(a)

(b)

FIG. 29. Required (a) Lawson parameters and (b) triple products vs.Ti to achieve values of Qfuel assuming T = Ti = Te, for p-11B basedon the p-11B fusion reactivity from Ref. 23.

Requiring that the rate of production of 3He and T are con-sumed at the same rate as they are produced,

12

n2D〈σv〉DD,3He = nDn3He〈σv〉D3He, (D13)

12

n2D〈σv〉DD,T = nDnT〈σv〉DT. (D14)

Rearranging gives the T -dependent, steady-state number den-sity of 3He and T ions, respectively,

n3He =12〈σv〉DD,3He

〈σv〉D3HenD, (D15)

nT =12〈σv〉DD,T

〈σv〉DTnD. (D16)

The total fusion power density is the sum of the power re-leased in its four constituent reactions,

SF =n2

D2〈σv〉DD,3HeEDD,3He +nDn3He〈σv〉D3HeED3He+

n2D2〈σv〉DD,TEDD,T +nDnT〈σv〉DTEDT.

(D17)

The bremsstrahlung power density is

SB =CBn2eT 1/2

e γ(Zeff), (D18)

and from Eq. (41),

Zeff =1ne(nD +4n3He +nT). (D19)

The power lost to thermal conduction per unit volume is

Sκ =(3/2)T (ne +ni)

τE. (D20)

Defining χh and χt as the number density ratios of n3He to nDand nT to nD respectively,

χh ≡n3HenD

=12〈σv〉DD,3He

〈σv〉D3He, (D21)

χt ≡nT

nD=

12〈σv〉DD,T

〈σv〉DT. (D22)

From the steady-state power balance of Eq. (11) and theabove, the Lawson parameter required to achieve fuel gainQfuel at Ti is,

niτE =

T (3+9χh/2+3χt)(1+χh +χt)

( fc +Q−1fuel)〈σv〉DDEtot/4−CB(1+2χh +χt)2T 1/2γ(Zeff)

(D23)

with

Zeff = (1+4χh +χt)/(1+2χh +χt), (D24)

〈σv〉DD = 〈σv〉DD,3He + 〈σv〉DD,T, (D25)

and

Etot = EDD,T +EDT +EDD,3He +ED3He. (D26)

The requirement for ignition of catalyzed D-D is niT τ∗E ≥1.1× 1023 m−3 keV s at T = 52 keV (see Fig. 30), 38 timeshigher than required for D-T.

29

(a)

(b)

FIG. 30. Required (a) Lawson parameters and (b) triple products vs.T to achieve the indicated values of Qfuel for catalyzed D-D (assum-ing T = Te = Ti).

4. Advanced-fuels summary

The extreme requirements for advanced fuels compared toD-T are illustrated in Fig. 31, which shows the required Law-son parameters and triple products vs. Ti required to achieveQfuel = 1 (dashed lines) and Qfuel = ∞ (solid lines) for allof the reactions discussed in this appendix. For all reac-tions except p-11B, Ti = Te is assumed. For p-11B, neither fuelbreakeven nor ignition appears possible when Ti = Te.

(a)

(b)

FIG. 31. Required (a) Lawson parameters and (b) triple productsvs. T to achieve Qfuel = ∞ (solid lines), Qfuel = 1 (dashed lines), andQfuel = 0.5 (dotted line, p-11B only) for the indicated fuels, assumingT = Te = Ti. Neither fuel breakeven (Qfuel = 1) nor ignition (Q = ∞)appears to be possible for p-11B if Te = Ti.

Appendix E: Conceptual power plants with non-electricalrecirculating power

Some fusion designs do not recirculate electrical power butrather capture a portion of the thermal Pout via mechanicalmeans and use it with efficiency ηr as Pext. This is illus-trated in Fig. 32. An example of this approach is the compres-sion of plasma by an imploding liquid-metal vortex driven bycompressed-gas pistons,142 which recapture a fraction of Pout

30

to re-energize the pistons with efficiency ηr for the next pulse.If we define engineering gain in this system as the ratio ofelectrical power to the grid to recirculating mechanical power,then Qeng = PE

grid/Pr, and it is straightforward to show that

Qeng = ηelecηr(Qsci +1)−ηelec. (E1)

This approach has the advantage that net electricity can begenerated (Qeng > 0) with Qsci < 1 if the recirculating effi-ciency ηr is sufficiently high, without advanced fuels or directconversion (i.e., assuming D-T fuel and a standard steam cy-cle ηelec = 0.4). This is due to the fact that the recirculatingpower bypasses the conversion to electricity.

FIG. 32. Conceptual schematic of a fusion power plant that recircu-lates mechanical power with efficiency ηr. In this system, engineer-ing gain is defined as Qeng = PE

grid/Pr.

FIG. 33. Required combinations of Qsci and ηr in the system shownin Fig. 32 to permit values of Qeng ranging from zero (i.e., PE

grid = 0)to ten (i.e., PE

grid = 10Pr), where ηelec = 0.4 is assumed. Note thatat high ηr, net electricity (Qeng > 0) is possible with Qsci < 1 eventhough ηelec is only 0.4, corresponding to D-T fuel and a standardsteam cycle.

Appendix F: Relationships between peak and volume-averagedquantities for MCF

In this appendix, we describe the equations used for vol-ume averaging of plasma parameters for MCF, for the purposeof relating peak values (variables denoted with a subscript of‘0’) to their volume-averaged quantities (denoted with 〈...〉)to, ultimately, relating the peak n0T0τE to an overall Qfuel thataccounts for profile effects in n and T . We denote this as 〈Q〉,even though Qfuel is inherently a volume-averaged quantity.

For any quantity f (x,y), such as n or T , the volume aver-age of f over the plasma cross-sectional surface S (in the x-yplane) is

〈 f 〉=∫∫

S f (x,y)dSA

, (F1)

where A =∫∫

S dS is the area (inside the separatrix or lastclosed flux surface), and axisymmetry is assumed.

1. Cylinder or large-aspect-ratio torus

For a circular cylinder with radius a or a torus with inverseaspect ratio ε = a/R� 1 (where a and R are the minor andmajor radii, respectively), and f (x,y) = f (r) (i.e., circular,concentric flux surfaces with no Shafranov shift), Eq. (F1) be-comes

〈 f 〉=2∫ a

0 r f (r)dra2 . (F2)

For the particular profile

f (x,y) = f (r) = f0

[1−( r

a

)2]S f

, (F3)

where r = (x2 + y2)1/2, Eq. (F2) becomes

〈 f 〉=2 f0

∫ a0 r[1− (r/a)2

]S f dra2 =

f0

1+S f. (F4)

If n = n0[1− (r/a)2]νn and T = T0[1− (r/a)2]νT , then it fol-lows that

〈nT 〉= n0T0

1+νn +νT. (F5)

2. Arbitrary aspect-ratio torus

For an up/down-symmetric torus with arbitrary ε andf (x,y), Eq. (F1) becomes

〈 f 〉=∫ R+a

R−a∫ h(x)

0 x f (x,y)dydx∫ R+aR−a xh(x)dx

, (F6)

where h(x) is the half height of the plasma cross section athorizontal position x as shown in Figure 34. If h(x) and

31

FIG. 34. Cross section of up-down symmetric torus with upperboundary defined by h(x) (shown here as a semi-circle).

f (x,y) = f0 f (x,y) are specified, where f0 is the peak valueof f and max( f ) = 1, then Eq. (F6) can be numerically inte-grated to provide a quantitative relationship between 〈 f 〉 andf0. The function h(x) allows for any plasma cross-sectionalshape, e.g., the highly elongated, D-shaped flux surfaces ofhigh-performance tokamaks.

For the particular case of an up/down-symmetric torus withcircular cross section and f (x,y) as given in Eq. (F3), wherer = [(x−R)2 + y2)1/2, Eq. (F6) becomes

〈 f 〉=f0∫ R+a

R−a∫ h(x)

0 x{1− [(x−R)2 + y2]/a2}S f dydx∫ R+aR−a xh(x)dx

, (F7)

where h(x) = [a2− (x−R)2]1/2. Again, this can be integratednumerically to provide a relationship between 〈 f 〉 and f0.

ACKNOWLEDGMENTS

Most of the first author’s contributions were performedwhile affiliated with Fusion Energy Base prior to joiningARPA-E. We are grateful for feedback on drafts of this pa-per provided by Riccardo Betti, Rob Goldston, Rich Hawry-luk, Omar Hurricane, Harry McLean, Dale Meade, Bob Mum-gaard, Brian Nelson, Kyle Peterson, Uri Shumlak, and GlenWurden. Responsibility for all content in the paper lies withthe authors. Reference herein to any specific non-federal per-son or commercial entity, product, process, or service by tradename, trademark, manufacturer, or otherwise, does not neces-sarily constitute or imply its endorsement, recommendation,or favoring by the U.S. Government or any agency thereof orits contractors or subcontractors.

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n1n2〈σv〉1,2E1,2 = k1k2n2e〈σv〉1,2E1,2 is maximized when k1k2 is maxi-

mized. Quasi-neutrality requires that Z1n1 +Z2n2 = ne, which, after somefurther algebra, results in the expression k1k2 = k2/Z1− k2

2Z2/Z1. This ismaximized when k2 = 1/(2Z2) and k1 = 1/(2Z1). Rather than maximiz-ing fusion power, other optimizations are possible, e.g., minimizing therequired Lawson parameter to achieve a certain value of Qfuel by mini-mizing the entire right-hand side of Eq. (D3) with respect to k1 and k2.However, maximizing fusion power is the simplest choice.

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