Progress toward fusion energy breakeven and gain as measured against the Lawson criterion

Wurzel, Samuel E.; Hsu, Scott C.

doi:10.1063/5.0083990

The Lawson criterion is a key concept in the pursuit of fusion energy, relating the fuel density n, pulse duration τ or energy confinement time τ_E, and fuel temperature T to the energy gain Q of a fusion plasma. The purpose of this paper is to explain and review the Lawson criterion and to provide a compilation of achieved parameters for a broad range of historical and contemporary fusion experiments. Although this paper focuses on the Lawson criterion, it is only one of many equally important factors in assessing the progress and ultimate likelihood of any fusion concept becoming a commercially viable fusion-energy system. Only experimentally measured or inferred values of n, τ or τ_E, 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 fusion researchers and a tutorial for all others interested in fusion energy.

I. INTRODUCTION

In 1955, J. D. Lawson identified a set of necessary physical conditions for a “useful” fusion system.¹ By evaluating the energy gain Q, the ratio of energy released by fusion reactions to the delivered energy for heating and sustaining the fusion fuel, Lawson concluded that for a pulsed system, energy gain is a function of temperature T and the product of fuel density n and pulse duration τ. When thermal-conduction losses are included in a steady-state system (extending Lawson's analysis), the power gain is a function of T and the product of n and energy confinement time τ_E. We call both these products, $n τ$ and $n τ_{E}$ ⁠, the Lawson parameter. The required temperature and Lawson parameter for self-heating from charged fusion products to exceed all losses is known as the Lawson criterion. A fusion plasma that has reached these conditions is said to have achieved ignition. Although ignition is not required for a commercial fusion-energy system, higher values of energy gain will generally yield more attractive economics, all other things being equal. If the energy applied to heat and sustain the plasma can be recovered in a useful form, the requirements on energy gain for a useful system are relaxed.

Lawson's analysis was declassified and published in 1957² and has formed the scientific basis for evaluating the physics progress of fusion research toward the key milestones of plasma energy breakeven and gain. Over time, the Lawson criterion has been cast into other formulations, e.g., the fusion triple product^3,4 (⁠ $n T τ_{E}$ ⁠) and “p-tau” (pressure p times τ_E), which have the same dimensions (with units of m⁻³keV s or atm s) and combine all the relevant parameters conveniently into a single value. However, these single-value parameters do not map to a unique value of Q, whereas unique combinations of T and $n τ$ (or $n τ_{E}$ ⁠) do. Various plots of the Lawson parameter, triple product, and “p-tau” vs year achieved or vs T have been published for subsets of experimental results,^5–8 but to our knowledge, there did not exist a comprehensive compilation of such data in the peer-reviewed literature that spans the major thermonuclear-fusion approaches of magnetic confinement fusion (MCF), inertial confinement fusion (ICF), and magneto-inertial fusion (MIF). This paper fills that gap.

The motivation to catalog, define our methodologies for inferring, and establish credibility for a compilation of these parameters stems from the prior development of the Fusion Energy Base (FEB) website (http://www.fusionenergybase.com) by the first author. FEB is a free resource with a primary mission of providing objective information to those, especially private investors, interested in fusion energy. This paper provides access to the many included plots, tables, and codes, while also providing context for understanding the history of fusion research^9–11 and the tremendous scientific progress that has been made in the 65+ years since Lawson's report.

The combination of T and $n τ$ (or $n τ_{E}$ ⁠) is a scientific indicator of how far or near a fusion experiment is from energy breakeven and gain. Achieving high values of these parameters is tied predominantly to plasma physics and related engineering challenges of producing stable plasmas, heating them to fusion temperatures, and exerting sufficient control. Since the 1950s, these challenges have driven the development of the entire scientific discipline of plasma physics, which has dominated fusion-energy research to this day. However, we emphasize that there are many additional considerations, entirely independent of but equally important as the Lawson criterion, in evaluating the remaining technical, economic, and societal risks of any fusion approach and the likelihood of any approach ultimately becoming a commercially viable fusion-energy system. These include the feasibility, absolute power levels or yield per pulse, safety, and complexity of the engineering and materials subsystems and fuel cycle that impact a commercial fusion system's economics¹² and social acceptance,^13,14 as illustrated conceptually in Fig. 1. The issues of RAMI (reliability, accessibility, maintainability, and inspectability)¹⁵ and government regulation^16,17 impact both the economics and social acceptance. This paper discusses only the progress of fusion energy along the axis of energy gain, and we caution the reader not to over-emphasize nor under-emphasize any one axis.

FIG. 1.

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Progress toward commercially viable fusion energy requires progress along three equally important axes. This paper focuses only on the axis of energy gain.

Although we do not further emphasize it in this paper, a different scientific metric called the Sheffield parameter^8,18 aims to embody both the required physics performance (like the Lawson parameter) and the “efficiency” of achieving that performance for MCF concepts. The Sheffield parameter can be thought of as a normalized triple product by explicitly including the parameter β, which is a measure of how much plasma pressure (related to the triple product) can be confined for a given magnetic field (which affects the cost and engineering difficulty).

Because of these additional considerations, fusion approaches that have achieved the highest values of T and $n τ$ (or $n τ_{E}$ ⁠), i.e., laser-driven ICF^19,20 and tokamak-based MCF,⁶ may not necessarily become the first widely deployed commercial fusion-energy systems. In fact, most private fusion companies focusing on developing commercial fusion systems have opted for fusion approaches with lower demonstrated values to date of temperature and Lawson parameter because of the expectation that the required economics and social acceptance may be more readily achievable. Further discussion of these other considerations is beyond the scope of this paper but is discussed elsewhere in the fusion literature.^8,15,21,22

This paper is organized as follows. Section II provides plots of the compiled parameters. Section III provides a review and mathematical derivations of the Lawson criterion and the multiple definitions of fusion energy gain used by fusion researchers. Section IV provides a physics-based justification for the approximations required to compare fusion energy gain across a wide range of fusion experiments and approaches. Readers primarily interested in seeing and using the data without getting entangled in the details can largely ignore Secs. III and IV. Section V provides a summary and conclusions. The Appendices provide supplementary material, including data tables of the compiled parameters, additional plots, and consideration of advanced fusion fuels (D-D, D-³He, p-¹¹B).

II. PLOTS

This section provides plots of compiled Lawson parameters, fuel temperatures, and triple products. These plots are generated with a code that is available for download.²³ In many places (especially Secs. I, III, and V), we use the generic variables n, T, τ, and Q for economy. However, in most of the paper and as indicated in the Nomenclature, these variables are more precisely differentiated with various subscripts. The energy unit keV is used for temperature variables throughout this paper, and therefore, the Boltzmann constant k is not explicitly shown.

Figure 2 plots achieved (and projected to be achieved) Lawson parameters vs T_i for MCF, MIF, and ICF experiments. Colored contours indicate the requirements of MCF experiments to achieve indicated values of scientific energy gain $Q_{sci}$ (labeled $Q_{sci}^{MCF}$ on the plot), which is the fusion power divided by the power delivered to the plasma. The black curve labeled ${(n τ)}_{ig, hs}^{ICF}$ indicates the requirements of ICF experiments needed to achieve hot-spot ignition. See the remainder of this paper for details on how the relevant data are extracted from the primary literature, the mathematical definitions of $Q_{sci}$ and ${(n τ)}_{ig, hs}^{ICF}$ ⁠, and how the effects of non-uniform spatial profiles, impurities, heating efficiency, and other experimental details are treated. Figure 3 shows record triple products achieved by different fusion concepts vs year achieved (or projected to be achieved) relative to horizontal lines representing various values of $Q_{sci}^{MCF}$ and ${(n T τ)}_{ig, hs}^{ICF}$ ⁠.

FIG. 2.

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Experimentally inferred Lawson parameters (⁠ $n_{i 0} τ_{E}^{*}$ for MCF and $n τ$ for ICF) of fusion experiments vs $T_{i 0}$ for MCF and ${⟨ T_{i} ⟩}_{n}$ for ICF (see Sec. III for definitions of these quantities), extracted from the published literature (see Tables V–VII). The colored contours correspond to the Lawson parameters and ion temperatures required to achieve the indicated values of scientific gain $Q_{sci}^{MCF}$ for MCF. The black curve labeled ${(n τ)}_{ig, hs}^{ICF}$ corresponds to the Lawson parameters and ion temperatures required to achieve hot-spot ignition for ICF. We assume 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-equivalent value. The finite widths of the $Q_{sci}^{MCF}$ contours represent a range of assumed impurity levels. See the rest of this paper for details on how individual data points are extracted and how the $Q_{sci}^{MCF}$ and ${(n τ)}_{ig, hs}^{ICF}$ contours are calculated.

FIG. 3.

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Triple products (⁠ $n_{i 0} T_{i 0} τ_{E}^{*}$ for MCF and $n {⟨ T_{i} ⟩}_{n} τ$ for ICF; see Sec. III for definitions of these quantities) that set a record for a given concept vs year achieved. Record values for different concepts are shown to illustrate the progress toward energy gain of different concepts over time. The horizontal lines labeled $Q_{sci}^{MCF}$ represent the minimum required triple product to achieve the indicated values of $Q_{sci}^{MCF}$ ⁠, assuming $η_{abs} = 0.9$ ⁠. The thickness of these lines is equal to the thickness of the equivalent contours in Fig. 16 at their minimum values. The horizontal line labeled ${(n T τ)}_{ig, hs}^{ICF}$ represents the required triple product to achieve ignition in an ICF hot spot, assuming T_i = 4 keV. The NIF shot from August 8, 2021, does not appear in this plot because its triple product tied the previous record. It did, however, achieve hot-spot ignition and a record $Q_{sci}$ for ICF,²⁴ see Sec. IV B 4. The projected triple-product ranges for SPARC and ITER are bounded above by their projected peak triple products and below by the stated mission of each experiment (i.e., $Q_{fuel}^{MCF} = 2$ for SPARC and $Q_{fuel}^{MCF} = 10$ for ITER).

Typically, MCF uses τ_E and ICF uses τ in their respective Lawson-parameter and triple-product definitions. Although τ_E and τ have different physical meanings (see Secs. III E and III F, respectively), they lead to analogous measures of energy breakeven and gain, allowing for MCF and ICF to be plotted together in Figs. 2, 3, and 16. We caution the reader that sometimes Lawson parameters and triple products may be overestimated by concept advocates, especially in unpublished materials, because τ is used incorrectly in place of τ_E.

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

In this section, we provide a detailed review of the derivation of the Lawson criterion, following Lawson's original papers.^1,2 We then introduce the mathematical definitions of the Lawson parameter in the context of idealized MCF and ICF scenarios, derive the fusion triple product, and define three forms of fusion energy gain used by fusion researchers.

Lawson considered the deuterium-tritium (D-T) and deuterium-deuterium (D-D) fusion reactions,

D + T \to α (3.5 MeV) + n (14.1 MeV),

(1)

D + D \underset{50 %}{\to} T (1.01 MeV) + p (3.02 MeV),

(2)

D + D \underset{50 %}{\to}^{3} He (0.82 MeV) + n (2.45 MeV),

(3)

where α denotes a charged helium ion (⁴He²⁺), p denotes a proton, n denotes a neutron and 1 MeV $= 1.6 \times 10^{- 13}$ J. The fusion reactivities $⟨ σ v ⟩$ for thermal ion distributions for these reactions as well as the additional reactions,

D +^{3} He \to α (3.6 MeV) + p (14.7 MeV),

(4)

p +^{11} B \to 3 α (8.7 MeV),

(5)

are shown in Fig. 4.

FIG. 4.

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Thermal fusion reactivities $⟨ σ v ⟩$ vs T_i for fusion reactions shown in the legend. All reactivities are calculated by numerical integration of velocity-averaged cross sections from Ref. 25 except p-¹¹B, which is calculated from the parametrization of Ref. 26. Note that the two D-D branches are nearly on top of each other.

As did Lawson, this paper assumes thermal populations of ions and electrons, i.e., Maxwellian velocity distributions characterized by temperatures T_i and T_e, respectively. Throughout this paper, we assume that ions and electrons are in thermal equilibrium with each other such that $T = T_{i} = T_{e}$ ⁠. Non-equilibrium fusion approaches, where T_i > T_e, must account for the energy loss channel and timescale of energy transfer from ions to electrons.²⁷ Analysis of such systems is not included in this paper. Furthermore, this paper does not consider non-thermal ion or electron populations such as those with beam-like distributions. The latter typically must contend with reactant slowing at a much faster rate than the fusion rate. The inherent difficulty (though not necessarily impossibility) for non-thermal fusion approaches to achieve $Q_{sci} > 1$ is discussed in Ref. 28. Unless stated otherwise, this paper assumes that all charged fusion products thermalize in the plasma and contribute to self-heating and that all uncharged fusion products (neutrons) exit the plasma without contributing to self-heating. Finally, in this paper, the small Mach-number limit is taken, i.e., the kinetic pressure is small compared to the thermal pressure.

Lawson's original papers considered two distinct fusion operating conditions. The first is a steady-state scenario in which the charged fusion products are confined and contribute to self-heating. The second is a pulsed scenario in which the charged fusion products escape and energy is supplied over the duration of the pulse. Lawson's analysis did not address how the fusion plasma is confined and assumed an ideal scenario without 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 minimum, balance radiative power losses with self-heating from the charged fusion products, as illustrated conceptually in Fig. 5. The power released by charged fusion products in a plasma of volume V is

P_{c} = f_{c} P_{F} = f_{c} \frac{n_{1} n_{2}}{1 + δ_{1, 2}} {⟨ σ v ⟩}_{1, 2} ε_{F} V,

(6)

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

FIG. 5.

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The steady-state scenario corresponding to Lawson's first insight. Self-heating from charged fusion products P_c appears as bremsstrahlung power P_B in a steady-state plasma of volume V. Fusion power emitted as neutrons P_n escapes the plasma and does not contribute to self-heating. An unspecified, idealized confinement mechanism is assumed, and thermal-conduction is ignored.

The power emitted by bremsstrahlung radiation is

P_{B} = C_{B} n_{i} n_{e} Z^{2} T_{e}^{1 / 2} V,

(7)

where C_B is a constant and Z = 1 in a hydrogenic plasma. Entering values of density in m⁻³, temperature in keV, volume in m³, and setting $C_{B} = 5.34 \times 10^{- 37}$ W m³ keV^−1/2 gives P_B in watts.

If the fusion plasma is to be completely self-heated by charged fusion products (i.e., α, T, p, or He³ in the above reactions), then $P_{c} \geq P_{B}$ is required in order for the plasma to reach ignition (ignoring conduction losses for the moment). In the case of equimolar D-T fusion plasma, i.e., $n / 2 = n_{1} = n_{2}$ ⁠, where n is the total ion number density and Z = 1, and given the assumption $T = T_{i} = T_{e}$ ⁠, the condition $P_{c} \geq P_{B}$ becomes

\frac{1}{4} f_{c} n^{2} {⟨ σ v ⟩}_{D T} ε_{F} V \geq C_{B} n^{2} T^{1 / 2} V .

(8)

Dividing both sides by V and plotting the resulting fusion power density $S_{c} = P_{c} / V$ (left-hand side) and bremsstrahlung power density $S_{B} = P_{B} / V$ (right-hand side) vs T in Fig. 6 shows that $T \geq 4.3$ keV is required for $S_{c} \geq S_{B}$ ⁠. This temperature is known as the ideal ignition temperature because, under the idealized scenario of perfect confinement, ignition occurs at this temperature. Note that because n² cancels on both sides of Eq. (8), the ideal ignition temperature is independent of density. In Appendix B, we discuss and show how the ignition temperature could be modified if bremsstrahlung radiation losses are mitigated.

FIG. 6.

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Power produced per unit volume S_c in charged D-T fusion products (α particles) and power lost to bremsstrahlung per unit volume S_B vs T in a D-T plasma. When T < 4.3 keV, S_B > S_c and ignition is not possible (assuming $T = T_{e} = T_{i}$ ⁠).

B. Lawson's second insight: Dependence of fuel energy gain on T and $n$ τ

Lawson's second insight involves a pulsed scenario where a plasma is heated instantaneously to a temperature T and maintained at that temperature for time τ (Lawson used t), as illustrated conceptually in Fig. 7. In this scenario, bremsstrahlung radiation and all fusion reaction products escape, and therefore, heating must come from an external source during duration τ. Idealized energy confinement is assumed, i.e., thermal-conduction losses are ignored. Immediately following the instantaneous temperature increase and throughout the remainder of the pulse, the absorbed heating power is balanced by bremsstrahlung radiation. The steady-state power balance of the plasma is

\frac{d (3 n T V)}{d t} = P_{abs} - P_{B} = 0.

(9)

FIG. 7.

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Pulsed scenario corresponding to Lawson's second insight. At time t = 0, the plasma temperature is instantaneously raised to T and maintained for a duration τ by externally applied and absorbed power $P_{abs}$ ⁠. All fusion products escape (no self-heating), and thermal conduction is neglected (ideal confinement). Absorbed power $P_{abs}$ appears as bremsstrahlung power P_B during the pulse duration.

We define the fuel gain $Q_{fuel}$ (Lawson used R) as the ratio of energy released in fusion products E_F to the applied external energy $E_{abs}$ that is absorbed by the entire fuel over the duration τ of the pulse. This absorbed energy is the sum of the instantaneously deposited energy $\frac{3}{2} (n_{e} + n_{i}) T V = 3 n T V$ (assuming $T = T_{i} = T_{e}$ and $n = n_{i} = n_{e}$ ⁠) and the energy applied and absorbed over the pulse duration, $τ P_{abs}$ ⁠. To maintain constant T over duration τ, $P_{abs} = P_{B}$ is required, and the fuel gain is, therefore,

Q_{fuel} = \frac{E_{F}}{E_{abs}} = \frac{τ P_{F}}{τ P_{B} + 3 n T V} = \frac{P_{F} / (3 n^{2} T V)}{P_{B} / (3 n^{2} T V) + 1 / (n τ)} .

(10)

Because both P_F and P_B are proportional to $n^{2} V$ and functions of T [see Eqs. (6) and (7)], the $n^{2} V$ dependence cancels out, and $Q_{fuel}$ is solely a function of T and $n τ$ ⁠,

Q_{fuel} = \frac{⟨ σ v ⟩ ε_{F} / 12 T}{C_{B} / 3 T^{1 / 2} + 1 / (n τ)} .

(11)

Figure 8 plots $Q_{fuel}$ as a function of T for the indicated values of $n τ$ ⁠, illustrating that even without self-heating, $Q_{fuel} ≫ 1$ is theoretically possible. Lawson noted that a “useful” system would require $Q_{fuel} > 2$ ⁠, assuming that fusion energy and bremsstrahlung could be converted to useful energy with an efficiency of 1/3, and remarked on the severity of the required T and $n τ$ ⁠.

FIG. 8.

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Plot of $Q_{fuel}$ vs T for indicated values of $n τ$ ⁠, assuming no self-heating and no thermal-conduction losses.

In this section, we have assumed that at time $t = τ$ ⁠, the external heating is turned off and none of the applied energy is recaptured. Lawson noticed, however, that if a fraction η (Lawson used f) of the thermal energy is recovered at the conclusion of the pulse and converted into a useful form of energy (e.g., electrical or mechanical) that could offset the externally applied energy, the quantity $n τ$ in Eq. (11) is replaced by $n τ / (1 - η)$ ⁠. The utilization of energy recovery to relax the requirements on $n τ$ for the achievement of energy gain is discussed further in Sec. III H.

C. Extending Lawson's second scenario: Effect of self-heating and relationship between characteristic times τ and τ_E

In an effort to capture experimental realities, we extend Lawson's second scenario to include thermal-conduction losses and self-heating from charged fusion products, as illustrated in Fig. 9. The rate of energy leaving the plasma via thermal conduction is characterized by an energy confinement time τ_E, which is the time for energy equal to the thermal energy 3nTV to exit the plasma via thermal conduction. The power balance over the duration of the constant-temperature pulse is

\frac{d (3 n T V)}{d t} = P_{abs} + P_{c} - P_{B} - \frac{3 n T V}{τ_{E}} = 0.

(12)

FIG. 9.

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Extension of Lawson's second scenario. At time t = 0, the plasma temperature is instantaneously raised to T and maintained for a duration τ by absorbed external power $P_{abs}$ and self-heating power P_c. The sum of absorbed external heating and self-heating appear as bremsstrahlung P_B and thermal conduction $3 n T V / τ_{E}$ ⁠.

Applying a similar analysis to that of Sec. III B, we obtain

Q_{fuel} = \frac{⟨ σ v ⟩ ε_{F} / 12 T}{C_{B} / 3 T^{1 / 2} - f_{c} ⟨ σ v ⟩ ε_{F} / 12 T + 1 / (n τ_{eff})},

(13)

where

n τ_{eff} = n \frac{τ τ_{E}}{τ + τ_{E}} .

(14)

The relationship between the two characteristic times τ and τ_E is that of two resistors in parallel, i.e., it is the smaller of the two that drives the value of $τ_{eff}$ ⁠. If $τ ≪ τ_{E}$ ⁠, the confinement duration τ limits $Q_{fuel}$ because there is limited time to overcome the initial energy investment of raising the plasma temperature. If $τ_{E} ≪ τ$ ⁠, the energy confinement time τ_E limits $Q_{fuel}$ because the rate of energy leakage from thermal conduction places higher demand on external and self-heating. If the two characteristic times are of similar magnitude, then both play a role in limiting $Q_{fuel}$ ⁠.

Figure 10 plots $Q_{fuel}$ vs T for the indicated values of $n τ_{eff}$ ⁠, illustrating that self-heating enables ignition (⁠ $Q_{fuel} \to \infty$ ⁠) above a threshold of T and $n τ_{eff}$ ⁠, made possible by the reduction of the denominator of Eq. (13) by amount $f_{c} ⟨ σ v ⟩ ε_{F} / 12 T$ ⁠. We explore these thresholds in Secs. III E and III F.

FIG. 10.

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Plot of $Q_{fuel}$ vs T for indicated values of effective Lawson parameter $n τ_{eff}$ ⁠, for a pulsed scenario that includes self-heating from charged fusion products and thermal conduction. Self-heating reduces the demands on externally applied and absorbed heating. Above a threshold of T and $n τ_{eff}, Q_{fuel}$ increases without bound, corresponding to ignition.

D. Scientific energy gain and breakeven

Because external-heating efficiency varies widely across fusion concepts, and because the absorption efficiency is intrinsic to the physics of each concept, we define $P_{ext}$ as the heating power applied at the boundary of the plasma (in the case of MCF) or the target assembly (in the case of ICF). This definition of $P_{ext}$ encapsulates all physics elements of the experiment. The boundary can typically be regarded as the vacuum vessel for all concepts, where $P_{ext}$ could be neutral beams for MCF, laser beams for ICF, or electrical current and voltage for MIF. The previously introduced $P_{abs}$ is the fraction $η_{abs}$ of $P_{ext}$ that is actually absorbed by the fuel, i.e., $P_{abs} = η_{abs} P_{ext}$ ⁠. The previously defined fuel gain is

Q_{fuel} = \frac{P_{F}}{P_{abs}},

(15)

and the newly defined scientific gain is

Q_{sci} = \frac{P_{F}}{P_{ext}} = η_{abs} Q_{fuel} < Q_{fuel} .

(16)

Whereas $Q_{fuel}$ ignores the plasma-physics losses of the absorption of heating energy into the fuel (e.g., neutral-beam shine-through in MCF or reflection of laser light via laser-plasma instabilities in ICF), $Q_{sci}$ accounts for all plasma-physics-related losses between the vacuum vessel and the fusion fuel. Therefore, $Q_{sci}$ is the better metric for assessing the remaining physics risk of a fusion concept.

Scientific breakeven is historically defined as $Q_{sci} = 1$ ⁠, which is an important milestone in the development of fusion energy because it signifies that very significant (but not all) plasma-physics challenges have been retired. Scientific breakeven has not yet been achieved, although D-T tokamak experiments such as TFTR and JET from the 1990s and the NIF experiment of August 8, 2021 have come close (⁠ $Q_{sci} = 0.27$ for TFTR,²⁹ $Q_{sci} = 0.64$ for JET,³⁰ and $Q_{sci} = 0.72$ for NIF).²⁴ Because $η_{abs}$ is much closer to unity in MCF experiments, the MCF community often uses Q to refer to $Q_{fuel}$ or $Q_{sci}$ interchangeably. For example, the SPARC ion-cyclotron heating system may achieve $η_{abs} = 0.9$ (single-pass absorption),³¹ and the neutral-beam heating system for ITER may achieve $η_{abs} \geq 0.95$ ⁠.³²

E. Idealized, steady-state MCF: $τ_{E} ≪ τ$

MCF relies on strong magnetic fields to confine fusion fuel, minimize thermal-conduction losses, and trap the charged fusion products for self-heating. By the time that Lawson's report was declassified in 1957, the UK, US, and USSR were all actively developing MCF experiments that included externally applied heating.

Adapting the extension of Lawson's second insight to this scenario, we consider the power balance of an externally heated and self-heated, steady-state plasma. Figure 11 illustrates this scenario for two different values of energy gain. The power balance and fuel gain of the plasma are described by Eqs. (12) and (13), respectively, in the limit of steady-state operation, i.e., $τ \to \infty$ ⁠.

FIG. 11.

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Conceptual illustrations of the steady-state power balance for two hypothetical steady-state MCF scenarios. The magnitude of external heating in scenarios (a) and (b) is the same. However, the fusion power and, therefore, $Q_{fuel}$ and $Q_{sci}$ in scenario (b) are all 20 times larger than that of scenario (a). The dotted line represents the boundary, e.g., vacuum chamber, between the physics and engineering aspects of the experiment.

To more clearly observe the requirements on $n τ_{E}$ and T to achieve certain values of $Q_{fuel}$ ⁠, we solve Eq. (13) for $n τ_{E}$ in the steady-state limit (⁠ $τ ≫ τ_{E}$ ⁠),

n τ_{E} = \frac{3 T}{(f_{c} + Q_{fuel}^{- 1}) ⟨ σ v ⟩ ε_{F} / 4 - C_{B} T^{1 / 2}} .

(17)

Plotting this expression in Fig. 12 (dashed lines) for D-T fusion shows that a threshold value of $n τ_{E}$ ⁠, which varies with T, is required to achieve a given value of $Q_{fuel}$ ⁠. Table I lists the minimum values of the Lawson parameter and corresponding temperature required to achieve $Q_{fuel} = 1$ and $Q_{fuel} = \infty$ for the indicated reactions. Thus far, spatially uniform profiles of all quantities are assumed, and geometrical effects and impurities are ignored. Later in the paper, we consider the effects of nonuniform spatial profiles, different geometries (e.g., cylinder, torus, etc.), and impurities.

FIG. 12.

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Lawson parameter $n τ_{E}$ vs T required to achieve indicated values of $Q_{fuel}$ (dashed lines) and $Q_{sci}$ (solid lines), assuming $η_{abs} = 0.9$ (representative of MCF). Because $η_{abs}$ is close to unity for MCF, $Q_{fuel}$ and $Q_{sci}$ are nearly coincident (the ignition contours are exactly coincident) and are often used interchangeably and referred to as Q.

TABLE I.

Values of minimum $n_{i} τ_{E}$ and corresponding T for $Q_{fuel} = 1$ and $Q_{fuel} = \infty$ for different fusion fuels assuming $T = T_{i} = T_{e}$ based on Eq. (17) for D-T (see Appendix C for advanced fuels).

Reaction	$Q_{fuel}$	T (keV)	$n_{i} τ_{E}$ (m⁻³ s)
$D + T$	1	26	2.5 × 10¹⁹
$D + T$	$\infty$	26	1.6 × 10²⁰
Catalyzed D-D	1	107	4.8 × 10²⁰
Catalyzed D-D	$\infty$	106	1.5 × 10²¹
${D +}^{3} He$	1	106	2.8 × 10²⁰
${D +}^{3} He$	$\infty$	106	6.2 × 10²⁰
${p +}^{11} B$	1	⋯	⋯
${p +}^{11} B$	$\infty$	⋯	⋯

Reaction	$Q_{fuel}$	T (keV)	$n_{i} τ_{E}$ (m⁻³ s)
$D + T$	1	26	2.5 × 10¹⁹
$D + T$	$\infty$	26	1.6 × 10²⁰
Catalyzed D-D	1	107	4.8 × 10²⁰
Catalyzed D-D	$\infty$	106	1.5 × 10²¹
${D +}^{3} He$	1	106	2.8 × 10²⁰
${D +}^{3} He$	$\infty$	106	6.2 × 10²⁰
${p +}^{11} B$	1	⋯	⋯
${p +}^{11} B$	$\infty$	⋯	⋯

To more clearly observe the requirements on $n τ_{E}$ and T to achieve certain values of $Q_{sci}$ ⁠, we replace $Q_{fuel}$ with $Q_{sci} / η_{abs}$ in Eq. (17),

n τ_{E} = \frac{3 T}{(f_{c} + η_{abs} Q_{sci}^{- 1}) ⟨ σ v ⟩ ε_{F} / 4 - C_{B} T^{1 / 2}} .

(18)

The ignition contours are identical for $Q_{fuel} = \infty$ and $Q_{sci} = \infty$ ⁠. For MCF experiments, where $η_{abs}$ is close to unity (⁠ $η_{abs} \sim 0.9$ ⁠), non-ignition $Q_{sci} < \infty$ contours are shifted relative to their respective $Q_{fuel}$ contours only very slightly toward the ignition contour (⁠ $Q_{fuel}, Q_{sci} = \infty$ ⁠), as seen in Fig. 12 (solid lines).

The Lawson criterion, where $P_{abs} \to 0$ and $Q_{fuel} \to \infty$ in Eqs. (12) and (15), respectively, is satisfied for values of $n τ_{E}$ and T on or above the $Q_{fuel}, Q_{sci} = \infty$ curves in Fig. 12. In this ignition regime, the plasma is entirely self-heated by charged fusion products, and external heating is zero. While the minimum Lawson parameter required for ignition occurs at $T \approx 25$ keV, MCF approaches aim for $T \approx 10$ –20 keV because the pressure required to achieve high gain is minimized in this lower-temperature range (as discussed in Sec. III G).

F. Idealized ICF: $τ ≪ τ_{E}$

ICF relies on the inertia of highly compressed fusion fuel to provide a duration to fuse a sufficient amount of fuel to overcome the energy invested in compressing the fuel assembly. In 1971, the concept of using lasers to compress and heat a fuel pellet was declassified, first by the USSR and later that year by the US.³³ In 1972, Nuckolls et al.¹⁹ described the direct-drive laser ICF concept, where lasers ablate the surface of a hollow fuel pellet outward, driving the inner surface toward the center. In this scenario, the kinetic energy of the inward-moving material is converted to the thermal energy of a central, lower-density “hot spot” that ignites. The fusion burn propagates outward through the surrounding denser fuel shell, which finally disassembles. The four-step, “central hot-spot ignition” process is illustrated in Fig. 13. Laser indirect-drive ICF bathes the fuel pellet in x rays generated by the interactions between lasers and the inside of a “hohlraum” (a metal enclosure surrounding the fuel pellet) to similar effect.

FIG. 13.

View large Download slide

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

To adapt the extension of Lawson's second insight, we consider the energy balance of the hot spot over duration τ, during which it is inertially confined [Fig. 13(b)]. The sequence of events that leads to energy delivered to the hot spot are as follows:

The laser energy strikes the fuel pellet (or hohlraum);
a fraction $η_{abs}$ of the laser energy is absorbed by the fuel in the form of kinetic energy E_abs of the imploding fuel shell;
the imploding shell with energy E_abs does $p d V$ work on the hot spot of volume V, resulting in hot-spot thermal energy $E_{h s} = η_{hs} E_{abs}$ ⁠;
if sufficiently high temperature and Lawson parameter are achieved, additional energy $τ P_{c}$ is delivered to the hot spot by charged fusion products.

We describe the fuel gain of the hot spot by applying the following assumptions and modifications to Eq. (13). In this simplified model, we neglect bremsstrahlung and thermal-conduction losses, i.e., $C_{B} \to 0$ and $τ_{E} \to \infty$ ⁠. While both processes are present in the hot spot, the cold, dense shell is largely opaque to bremsstrahlung and partially insulates the hot spot. In practice (which we also ignore here), both loss mechanisms have the effect of ablating material from the inner shell wall into the hot spot, increasing density, and decreasing temperature while maintaining a constant pressure.³⁴ To account for the fraction $η_{hs}$ of the shell kinetic energy that is deposited in the hot spot, the definition of $Q_{fuel}$ becomes

Q_{fuel} = \frac{τ P_{F}}{E_{abs}} = \frac{τ P_{F}}{E_{hs} / η_{hs}} .

(19)

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

To determine the requirements on $n τ$ and T to achieve certain values of $Q_{fuel}$ ⁠, we solve Eq. (13) for $n τ$ setting C_B = 0 (i.e., no hot-spot bremsstrahlung losses) and $τ_{E} = \infty$ (i.e., no hot-spot thermal-conduction losses),

n τ = \frac{12 T}{(f_{c} + η_{hs} Q_{fuel}^{- 1}) ⟨ σ v ⟩ ε_{F}} .

(20)

Plotting this expression in Fig. 14 (dashed lines) for D-T fusion shows that a threshold value of $n τ$ ⁠, which varies with T, is required to achieve a given value of $Q_{fuel}$ of an ICF hot spot. We have assumed $η_{hs} = 0.65$ based on NIF shot N191007.³⁵ Because this simple model only describes the hot spot and does not include an increase in temperature due to self-heating nor the dynamics of propagating burn into the cold fuel, we only consider it valid for values of $Q_{fuel} \leq 1$ ⁠, where self-heating energy is small compared to the hot-spot energy.

FIG. 14.

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Lawson parameter $n τ$ vs T required to achieve indicated values of hot-spot $Q_{fuel}$ (dashed lines), assuming $η_{hs} = 0.65$ (representative of indirect-drive ICF). The curve labeled ${(n τ)}_{ig, hs}^{ICF}$ is the requirement for hot-spot ignition.

Note that our definition of $Q_{fuel}$ for an ICF hot spot differs slightly from the standard definition of ICF fuel gain, G_f, which is the ratio of fusion energy to the total energy content of the fuel immediately before ignition.²⁰ ICF target gain G is, however, identical to our definition of $Q_{sci}$ ⁠.

Unlike the idealized MCF example, we cannot extend this model to an expression for the required values of $n τ$ and T required to achieve a certain value of $Q_{sci}$ because of the inherently dynamic nature of an ICF implosion. The total fusion energy released depends on the extent to which the fusion burn propagates into the cold fuel. This, in turn, depends on $ρ_{hs} R_{hs}$ of the expanding hot spot, where $ρ_{hs}$ and $R_{hs}$ are the hot-spot mass density and radius, respectively. It also depends on the achieved implosion symmetry. We instead consider the requirements on $n τ$ and T such that the energy of charged fusion products released in the hot spot equals the energy delivered to the hot spot by $p d v$ work,

τ P_{c} = 3 n T V .

(21)

Because we consider other loss mechanisms to be small, this is effectively the ignition condition for an ICF hot spot, i.e., the condition where self-heating exceeds all losses and the energy initially deposited into the hot spot, which itself is lost upon expansion. Solving the above equation for $n τ$ gives the required Lawson parameter for hot-spot ignition,

{(n τ)}_{ig, hs}^{ICF} = \frac{12 T}{⟨ σ v ⟩ ε_{α}},

(22)

where $ε_{α}$ is the energy of the charged alpha-particle fusion product in the D-T fusion reaction. The solid line in Fig. 14 illustrates the required ${(n τ)}_{ig, hs}^{ICF}$ as a function of hot-spot temperature to achieve hot-spot ignition. Thus far, reductions in τ due to instabilities, impurities, losses due to bremsstrahlung and thermal conduction, and the requirements to initiate a propagating burn in the cold, dense shell have been ignored. Later in this paper, we consider some of these effects.

More generally, “ignition” has many different meanings in the ICF context.³⁶ The 1997 National Academies review of ICF³⁷ addressed the lack of consensus around the definition of ICF ignition by defining ignition as fusion energy produced exceeding the laser energy (i.e., $Q_{sci} > 1$ ⁠). More recently, the hot-spot conditions needed to initiate propagating burn in the colder, dense fuel shell (another definition of ignition) have been quantified.³⁸

The achievement of hot-spot Lawson parameters and temperatures that meet the ${(n τ)}_{ig, hs}^{ICF}$ requirement for a given temperature signifies hot-spot ignition. While the minimum Lawson parameter required for ignition occurs at $T \approx 25$ keV, laser-driven ICF approaches aim for hot-spot $T \approx 4$ keV (prior to the onset of significant fusion leading to further increases in T_i) due to the limits of achievable implosion speed, which sets the maximum achievable temperature due to $p d V$ heating alone. These details are discussed further in Sec. IV B.

G. Fusion triple product and “p-tau”

The triple product (⁠ $n T τ_{E}$ ⁠) and p-tau (⁠ $p τ_{E}$ ⁠) are commonly used by the MCF community to quantify fusion performance in a single value. While less common in the ICF community, $p τ$ is sometimes used, and the triple product (⁠ $n T τ$ ⁠) is typically used only in the context of comparing ICF to MCF.³⁴ In a uniform plasma with $n = n_{i} = n_{e}$ and $T = T_{i} = T_{e}$ ⁠, the relationship between triple product and p-tau in both embodiments is $n T τ = \frac{1}{2} p τ$ and $n T τ_{E} = \frac{1}{2} p τ_{E}$ ⁠.

An expression for the MCF triple product is obtained by multiplying both sides of Eq. (17) by T,

n T τ_{E} = \frac{3 T^{2}}{(f_{c} + Q_{fuel}^{- 1}) ⟨ σ v ⟩ (T) ε_{F} / 4 - C_{B} T^{1 / 2}} .

(23)

Figure 15 shows the $n T τ_{E}$ required to achieve a specified value of $Q_{fuel}$ as a function of T (see also Table II). Note that the minimum triple product needed to achieve ignition (Fig. 15) occurs at a lower T than that of the minimum Lawson parameter (Fig. 12). This lower T is a better approximation of the intended T of MCF experiments because it corresponds to the minimum pressure required to achieve a certain value of $Q_{fuel}$ ⁠, and pressure (rather than the Lawson parameter) is a more-direct experimental limitation of MCF.

FIG. 15.

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Triple product vs T required to achieve indicated values of $Q_{fuel}$ for MCF [Eq. (23)].

TABLE II.

Values of minimum $n_{i} T τ_{E}$ and corresponding T for $Q_{fuel} = 1$ and $Q_{fuel} = \infty$ for different fusion fuels assuming $T = T_{i} = T_{e}$ based on Eq. (23) for D-T (see Appendix C for advanced fuels).

Reaction	$Q_{fuel}$	T (keV)	$n_{i} T τ_{E}$ (m⁻³ keV s)
$D + T$	1	14	4.6 × 10²⁰
$D + T$	$\infty$	14	2.9 × 10²¹
Catalyzed D-D	1	41	2.9 × 10²²
Catalyzed D-D	$\infty$	52	1.1 × 10²³
${D +}^{3} He$	1	63	2.2 × 10²²
${D +}^{3} He$	$\infty$	68	5.2 × 10²²
${p +}^{11} B$	1	⋯	⋯
${p +}^{11} B$	$\infty$	⋯	⋯

Reaction	$Q_{fuel}$	T (keV)	$n_{i} T τ_{E}$ (m⁻³ keV s)
$D + T$	1	14	4.6 × 10²⁰
$D + T$	$\infty$	14	2.9 × 10²¹
Catalyzed D-D	1	41	2.9 × 10²²
Catalyzed D-D	$\infty$	52	1.1 × 10²³
${D +}^{3} He$	1	63	2.2 × 10²²
${D +}^{3} He$	$\infty$	68	5.2 × 10²²
${p +}^{11} B$	1	⋯	⋯
${p +}^{11} B$	$\infty$	⋯	⋯

We emphasize the limitation of the triple product (or “p-tau”) as a metric: it does not correspond to a unique value $Q_{fuel}$ or $Q_{sci}$ unless T is specified. While n and τ in the Lawson parameter may be traded off in equal proportions, T must be within a fixed range for an appreciable number of fusion reactions to occur.

Figure 16 provides a plot of achieved triple products and temperatures analogous to Fig. 2. Appendix C provides plots of $n T τ_{E}$ vs T for D-D, D-³He, and p-¹¹B fusion.

FIG. 16.

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

H. Engineering gain

The previously defined $Q_{sci}$ [Eq. (16)] is the ratio of power released in fusion reactions P_F to applied external heating power $P_{ext}$ (see Fig. 11), encapsulating the physics of plasma heating, thermal and radiative losses, and fusion energy production. Based on the conservation of energy in Fig. 11, we can rewrite

Q_{sci} = \frac{P_{out} - P_{ext}}{P_{ext}} = \frac{P_{F}}{P_{ext}},

(24)

which is equivalent to Eq. (16).

Similarly, the engineering gain,

Q_{eng} = \frac{P_{out}^{E} - P_{in}^{E}}{P_{in}^{E}} = \frac{P_{grid}^{E}}{P_{in}^{E}},

(25)

is the ratio of electrical power $P_{grid}^{E}$ (delivered to the grid) to the input (recirculating) electrical power $P_{in}^{E}$ used to heat, sustain, control, and/or assemble the fusion plasma³⁹ (see Fig. 17). Some fusion designs do not recirculate electrical power but rather recirculate mechanical power (see Appendix D). For the case of electrical recirculating power, it is straightforward to show that

Q_{eng} = η_{elec} η_{E} (η_{abs} Q_{fuel} + 1) - 1,

(26)

where η_E, $η_{abs}$ ⁠, and $η_{elec}$ are the efficiencies of going from $P_{in}^{E} \to P_{ext}, P_{ext} \to P_{abs}$ ⁠, and $P_{out} \to P_{out}^{E}$ ⁠, respectively. Note that we have included the portion of $P_{ext}$ that is not absorbed by the plasma, i.e., $(1 - η_{abs}) P_{ext}$ ⁠, in $P_{out}$ ⁠; this is shown in Fig. 11 but not explicitly shown in Fig. 17.

FIG. 17.

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Conceptual schematic of a fusion power plant that recirculates electrical power. In this system, $Q_{eng} = P_{grid}^{E} / P_{in}^{E}$ ⁠.

Finally, the “wall-plug” gain,

Q_{wp} = \frac{P_{F}}{P_{in}^{E}} = η_{E} \frac{P_{F}}{P_{ext}} = η_{E} η_{abs} \frac{P_{F}}{P_{abs}},

(27)

relates the total fusion power to the power drawn from the grid (i.e., the wall plug) to assemble, heat, confine, and control the plasma. This is a useful energy gain metric for all contemporary fusion experiments because they are not yet generating electricity. We regard the eventual demonstration of $Q_{wp} = 1$ (not $Q_{fuel}$ or $Q_{sci} = 1$ ⁠) as the so-called “Kitty Hawk moment” for fusion energy.

Direct conversion from charged fusion products to electricity could be realized with advanced fusion fuels (e.g., D-³He and p-¹¹B), which produce nearly all of their fusion energy in charged products. This could raise $η_{elec}$ from approximately 40% to > 80% and enable significantly higher $Q_{eng}$ for a given $Q_{fuel}$ or $Q_{sci}$ ⁠.

For D-T fusion with a tritium-breeding blanket, the ⁶Li(n,α)T reaction to breed tritium is exothermic (releasing 4.8 MeV per reaction), thus amplifying $P_{out}$ by a factor of approximately 1.15 depending on the blanket design. For the purposes of this paper, this factor can be considered to be absorbed into $η_{elec}$ ⁠.

Using $Q_{sci} = η_{abs} Q_{fuel}$ ⁠, we can rewrite Eq. (26) as

Q_{eng} = η_{elec} η_{E} (Q_{sci} + 1) - 1.

(28)

Because $Q_{sci}$ encapsulates all the plasma-physics aspects of both the absorption efficiency $η_{abs}$ and fuel gain $Q_{fuel}$ ⁠, it is instructive to plot the required combinations of $Q_{sci}$ and η_E, assuming $η_{elec} = 0.4$ (representative of a standard steam cycle and blanket gain), to achieve certain values of $Q_{eng}$ (see Fig. 18). A convenient rule-of-thumb is that the gain-efficiency product must exceed 10 for practical fusion energy, i.e., $Q_{sci} η_{E} \geq 10$ (corresponding to $Q_{eng} \approx 3$ in Fig. 18), but of course, the actual requirement depends on the required economics of the fusion-energy system.

FIG. 18.

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Required combinations of $Q_{sci}$ and η_E in the system shown in Fig. 17 to permit values of $Q_{eng}$ ranging from zero (i.e., $P_{grid}^{E} = 0$ ⁠) to ten (i.e., $P_{grid}^{E} = 10 P_{in}^{E}$ ⁠), where $η_{elec} = 0.4$ is assumed.

While the value of $η_{elec}$ would be around 0.4 for a standard steam cycle for D-T fusion (and higher if an advanced power cycle is used), the values of η_E and $η_{abs}$ vary considerably depending on the class of fusion concept (see Table III). For MCF/MIF, $η_{E} > 0.5$ is expected (conservatively), meaning that $Q_{sci} ≳ 20$ is required. For laser-driven ICF, $η_{E} \sim 0.1$ is expected, meaning that $Q_{sci} ≳ 100$ is required. For an eventual fusion power plant, the required $Q_{sci}$ and $Q_{eng}$ will depend on several factors including but not limited to market constraints (e.g., levelized cost of electricity and desired value of $P_{grid}^{E}$ ⁠) and the maximum achievable values of η_E, $η_{elec}$ ⁠.

TABLE III.

Typical efficiency values η_E, $η_{abs}, η_{hs}$ ⁠, and $η_{elec}$ for different classes of fusion concepts. Note that $η_{hs}$ is only defined for ICF concepts pursuing hot-spot ignition. Approximate values of $η_{abs}$ and $η_{hs}$ for direct and indirect drive ICF are from Refs. 40 and 35, respectively. The approximate value of $η_{E}$ for MIF is from Ref. 41.

Class	η_E	$η_{abs}$	$η_{hs}$	$η_{elec}$
MCF	0.7	0.9	⋯	0.4
MIF	0.5	0.1	⋯	0.4
Laser ICF (direct drive)	0.1	0.06	0.4	0.4
Laser ICF (indirect drive)	0.1	0.009	0.7	0.4

Class	η_E	$η_{abs}$	$η_{hs}$	$η_{elec}$
MCF	0.7	0.9	⋯	0.4
MIF	0.5	0.1	⋯	0.4
Laser ICF (direct drive)	0.1	0.06	0.4	0.4
Laser ICF (indirect drive)	0.1	0.009	0.7	0.4

In Sec. III B, we noted Lawson's observation (in the context of his second scenario) that if a fraction η of the plasma energy after the pulse is recovered as electrical or mechanical energy, the requirement on $n τ$ to achieve a given value of $Q_{fuel}$ is reduced by a factor $1 / (1 - η)$ ⁠. In principle, this can be extended to recover $P_{out}$ with efficiency $η_{elec}$ and reinject the recirculating fraction with efficiency η_E, thus relaxing the requirements on $Q_{sci}$ to achieve a given $Q_{eng}$ ⁠. This is shown in Fig. 19, which assumes a high recovery fraction $η_{elec} = 0.95$ ⁠. If we also assume high electricity to heating efficiency $η_{E} = 0.9, Q_{eng} = 0.3$ (corresponding to net electricity) can be achieved with $Q_{sci} = 0.5$ ⁠. While it may appear counter-intuitive that net electricity can be generated in a system with $Q_{sci} < 1$ ⁠, a high $η_{elec}$ and η_E mean that most of the recovered heating energy recirculates while most of the fusion energy is used for electricity generation. The lower-right quadrant of Fig. 19 (corresponding to high re-injection efficiency) illustrates that net electricity generation (i.e., $Q_{eng} > 0$ ⁠) is possible at values of scientific gain below break-even (i.e., $Q_{sci} < 1$ ⁠).

FIG. 19.

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Required combinations of $Q_{sci}$ and η_E in the system shown in Fig. 17 to permit values of $Q_{eng}$ ranging from zero (i.e., $P_{grid}^{E} = 0$ ⁠) to ten (i.e., $P_{grid}^{E} = 10 P_{in}^{E}$ ⁠), where $η_{elec} = 0.95$ is assumed. Note that at high η_E and $η_{elec}$ ⁠, net electricity generation (⁠ $Q_{eng} > 0$ ⁠) is possible with $Q_{sci} < 1$ ⁠.

IV. METHODOLOGIES FOR INFERRING LAWSON PARAMETER AND TEMPERATURE

It is not trivial to infer the component values of the Lawson parameter and temperature achieved in real experiments. Simplifying approximations must be made with certain caveats, both across (e.g., MCF vs ICF) and within classes (e.g., tokamaks vs mirrors within MCF) of fusion experiments. In this section, we describe the methodologies that we use to infer the component values of achieved Lawson parameters and temperatures for different fusion classes and concepts, and how the values can be meaningfully compared against each other. For all values reported here, we require that experimentally inferred values occur within a single shot or across multiple well-reproduced shots. An example that we would disqualify would be to combine the highest T_i achieved in one shot with the highest n_i and τ_E from a qualitatively different shot. The year listed for each shot is the year in which the shot occurred if the year is reported, or it is an approximation of the year the shot occurred based on publication submission dates.

A. MCF methodology

The analysis presented in Sec. III E assumes that T_i = T_e and n_i = n_e, and that they are spatially uniform and time independent. In real experiments, these assumptions are generally not valid. Because diagnostic capabilities are finite, only a subset of the complete data (i.e., spatial profiles and time evolutions) are ever measured and published. Although many experiments were not aiming to maximize n_i, T_i, and τ_E as the goal, we include these experiments because they provide historical context. Furthermore, the data reported from one experiment may not be easily compared to data reported from another due to differences in definitions. In the remainder of this section, these issues are discussed, and uniform definitions are developed.

1. Effect of temporal profiles

Within a particular experiment, the maximum values of n_i, T_i, and τ_E may occur at different times. Where possible we choose the values of these quantities at a single point in time during a “flat-top” time period, the duration of which must exceed τ_E. Even though the total pulse duration of some MCF experiments may be of a similar magnitude to τ_E, we only consider τ_E in the Lawson parameter for MCF experiments [as opposed to the expression for $τ_{eff}$ in Eq. (14)] because we consider the progress toward energy gain in MCF to be limited by thermal-conduction losses and not pulse duration.

In the literature, tables of parameters have commonly been published that report the values of many parameters during such a flat-top time period. Following this convention, Tables V and VI list parameters relevant to our analysis. The reported parameters are $T_{i 0}, T_{e 0}, n_{i 0}, n_{e 0}$ ⁠, and $τ_{E}^{*}$ ⁠. Not all experiments have published the temporal evolution of these quantities. In the absence of such data, we use the values reported with the understanding that it is unknown if they occurred simultaneously during the shot (although, as discussed in the previous paragraph, they must occur in the same shot or in shots intended to be the same). This deficiency primarily occurs in experiments prior to 1970 or in small experiments with limited diagnostic capabilities and $n_{i} T_{i} τ_{E} < 10^{16}$ m⁻³keV s.

2. Effect of spatial profiles

To quantify the effect of nonuniform temperature and density spatial profiles on the requirements to achieve a certain value of $Q_{fuel}$ ⁠, which we denote as $⟨ Q_{fuel} ⟩$ (brackets refer to volume-averaging over nonuniform profiles), the power balance of Eq. (12) becomes

f_{c} ⟨ S_{F} ⟩ V + P_{abs} = ⟨ S_{B} ⟩ V + \frac{3 ⟨ n T ⟩}{τ_{E}} V,

(29)

where power densities are denoted with variables S, and we assume $n = n_{e} = n_{i}$ (i.e., hydrogenic plasma without impurities) and $T = T_{e} = T_{i}$ everywhere. Reported/inferred values of $P_{abs}$ and τ_E are already global, volume-averaged quantities.

To quantify the profile effect on S_F, we introduce

λ_{F} = \frac{⟨ S_{F} ⟩}{S_{F 0}},

(30)

where $S_{F 0}$ is the fusion power density with spatially uniform $T_{i 0}$ and $n_{i 0}$ ⁠, and $⟨ S_{F} ⟩$ is the volume-averaged fusion power density of the nonuniform-profile case with peak values $T_{i 0}$ and $n_{i 0}$ ⁠. Similarly,

λ_{B} = \frac{⟨ S_{B} ⟩}{S_{B 0}},

(31)

and

λ_{κ} = \frac{⟨ n T ⟩}{n_{0} T_{0}},

(32)

which quantify the nonuniform-profile modifications to the bremsstrahlung power density and thermal energy density, respectively.

The result is a modified version of Eq. (12), where profile effects are captured in the terms λ_F, λ_B, and $λ_{κ}$ ⁠,

λ_{F} f_{c} S_{F 0} V + P_{abs} = λ_{B} S_{B 0} V + \frac{3 λ_{κ} n_{0} T_{0}}{τ_{E}} V .

(33)

From this power balance of the nonuniform-profile case, the peak value of the Lawson parameter $n_{0} τ_{E}$ required to achieve a particular value of $⟨ Q_{fuel} ⟩$ as a function of T₀ is

\begin{matrix} n_{0} τ_{E} = \frac{3 λ_{κ} T_{0}}{λ_{F} (f_{c} + {⟨ Q_{fuel} ⟩}^{- 1}) ⟨ σ v ⟩ ε_{F} / 4 - λ_{B} C_{B} T_{0}^{1 / 2}}, \end{matrix}

(34)

where

⟨ Q_{fuel} ⟩ = \frac{λ_{F} S_{F 0}}{P_{abs} / V} \equiv λ_{F} Q_{fuel} .

(35)

We adopt the approach of using the same peak (rather than average) values of density and temperature when evaluating $Q_{fuel}$ (uniform spatial profiles) vs $⟨ Q_{fuel} ⟩$ (nonuniform spatial profiles), for the practical reasons that peak values are more commonly reported in the literature and that profiles are often not reported. When using the same peak rather than profile-averaged values, spatially nonuniform profiles increase rather than decrease the requirements on peak density and temperature for achieving a given $Q_{fuel}$ ⁠.

Next, we consider representative profiles in order to quantify the differences between $Q_{fuel}$ and $⟨ Q_{fuel} ⟩$ for cylindrical and toroidal geometries. A wide variety of temperature and density profiles have been observed in fusion experiments. These profiles are typically modeled as functions of normalized radius $x = r / a$ ⁠, where a is the device radius for cylindrical systems and the minor radius for toroidal systems with circular cross section. Commonly used and flexible models of density and temperature profiles are

n (x) = n_{0} {(1 - x^{2})}^{ν_{n}} and T (x) = T_{0} {(1 - x^{2})}^{ν_{T}},

(36)

where n₀ and T₀ are the central/peak ion or electron densities and temperatures, respectively. The values of ν_n and ν_T adjust the sharpness of the peaks of the profiles. In the limit $ν_{n} \to 0$ and $ν_{T} \to 0$ ⁠, the peak is infinitely broad and we recover the uniform-profile case. This approach accommodates a wide range of profiles.^42,43

From Eqs. (6) and (30),

λ_{F} = \frac{⟨ S_{F} ⟩}{S_{F 0}} = \frac{⟨ n_{i}^{2} ⟨ σ v ⟩ (T_{i} (x)) ⟩}{n_{i 0}^{2} ⟨ σ v ⟩ (T_{i 0})},

(37)

where the T_i dependence of $⟨ σ v ⟩$ is shown explicitly, resulting in λ_F being a function of the T_i profile. From Eqs. (7) and (31),

λ_{B} = \frac{⟨ S_{B} ⟩}{S_{B 0}} = \frac{⟨ n_{e}^{2} T_{e}^{1 / 2} ⟩}{n_{e 0}^{2} T_{e 0}^{1 / 2}} .

(38)

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. (36), we use the expressions in Appendix E to obtain

λ_{F} = \frac{\int_{0}^{1} {(1 - x^{2})}^{2 ν_{n}} ⟨ σ v ⟩ (T_{i} (x)) 2 x d x}{⟨ σ v ⟩ (T_{i 0})},

(39)

which may be evaluated numerically for any tabulated or parameterized values of $⟨ σ v ⟩ (T_{i})$ ⁠,

λ_{B} = \int_{0}^{1} {(1 - x^{2})}^{2 ν_{n} + ν_{T} / 2} 2 x d x = \frac{2}{4 ν_{n} + ν_{T} + 2},

(40)

and

λ_{κ} = \int_{0}^{1} {(1 - x^{2})}^{(ν_{n} + ν_{T})} 2 x d x = \frac{1}{1 + ν_{n} + ν_{T}} .

(41)

For a torus with circular cross section and arbitrary values of R/a, λ_F, λ_B, and $λ_{κ}$ must be evaluated numerically (see Appendix E). For profiles with a large Shafranov shift, i.e., magnetic axis shifted toward larger R, the reduction of fusion power due to profile effects (and hence λ_F) is mitigated because the high-temperature region occupies a larger fraction of the total volume. Therefore, the profiles considered here represent a likely worst-case scenario and provide a lower bound on λ_F.

To demonstrate the effect of nonuniform profiles on the contours of $⟨ Q_{fuel} ⟩$ compared to $Q_{fuel}$ ⁠, we consider two sets of profiles. The first is a parabolic profile with $ν_{T} = 1$ and $ν_{n} = 1$ ⁠, which is a simple approximation of the profiles in tokamaks.⁶ The second is a more strongly peaked temperature profile with $ν_{T} = 3$ and a broader density profile with $ν_{n} = 0.2$ ⁠, which are representative of profiles in the advanced tokamak or reversed-field pinch.⁴⁴ For both sets of profiles, we assume $T = T_{i} = T_{e}$ and $n = n_{i} = n_{e}$ (impurity-free hydrogenic plasma). Figures 20 and 21 show these two sets of profiles, respectively, along with their corresponding values of λ_F vs $T_{i 0}$ and resulting adjustments to the $Q_{fuel}$ contours. For both sets of profiles (Figs. 20 and 21), nonuniform profiles [dashed lines in panel (c)] increase the peak Lawson parameter needed to achieve a particular value of $⟨ Q_{fuel} ⟩$ for temperatures below approximately 50 keV. Additionally, the ideal ignition temperature, defined by Eq. (8), is increased. At high temperatures approaching 100 keV, where fusion power exceeds bremsstrahlung by a large factor (see Fig. 6), the adjustment is equal to the ratio $λ_{κ} / λ_{F}$ ⁠, which is close to unity in the case of the parabolic profiles, and drops below unity in the case of the peaked and broad profiles. At intermediate temperatures, λ_F, λ_B, and $λ_{κ}$ all contribute to the modification of $⟨ Q_{fuel} ⟩$ compared to $Q_{fuel}$ ⁠.

FIG. 20.

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(a) Normalized parabolic profiles (with $ν_{T} = 1$ and $ν_{n} = 1$ ⁠) of $T = T_{i} = T_{e}$ and $n = n_{i} = n_{e}$ ⁠. (b) Parameter λ_F vs $T_{i 0}$ (⁠ $λ_{B} = 0.286$ and $λ_{κ} = 0.333$ for these profiles). (c) Peak Lawson parameter vs T₀ for the parabolic profiles (dashed lines) shown in (a) and uniform plasma (solid lines) for $Q_{fuel} = 1$ (blue) and $Q_{fuel} = \infty$ (red).

FIG. 21.

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(a) Normalized peaked and broad profiles (with $ν_{T} = 3$ and $ν_{n} = 0.2$ ⁠) of $T = T_{i} = T_{e}$ and $n = n_{i} = n_{e}$ ⁠. (b) Parameter λ_F vs $T_{i 0}$ (⁠ $λ_{B} = 0.345$ and $λ_{κ} = 0.238$ for these profiles). (c) Lawson parameter vs $T_{i 0}$ for the profiles (dashed lines) shown in (a) and uniform plasma (solid lines) for $Q_{fuel} = 1$ (blue) and $Q_{fuel} = \infty$ (red).

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

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

Z_{eff} = \frac{Σ_{i} n_{i} Z_{i}^{2}}{n_{e}},

(42)

where i is summed over all ion species in the plasma. Only free-free bremsstrahlung is considered in this paper; bound-free bremsstrahlung and synchrotron radiation are ignored. Additionally, impurities increase the electron density relative to the ion density by a factor of the mean charge state of the entire plasma,

\bar{Z} = n_{e} / n_{i},

(43)

which reduces n_i and, therefore, P_F at a fixed pressure.

Using these definitions along with the generalized expression for bremsstrahlung,

P_{B} = C_{B} n_{e} T_{e}^{1 / 2} Σ_{i} (Z_{i}^{2} n_{i}),

(44)

Equation (34) becomes

\begin{matrix} n_{i 0} τ_{E} = \frac{(3 / 2) λ_{κ} (1 + \bar{Z}) T_{0}}{λ_{F} (f_{c} + η_{abs} {⟨ Q_{sci} ⟩}^{- 1}) ⟨ σ v ⟩ ε_{F} / 4 - λ_{B} C_{B} Z_{eff} {\bar{Z}}^{2} T_{0}^{1 / 2}}, \end{matrix}

(45)

and

\begin{matrix} n_{i 0} T_{0} τ_{E} = \frac{(3 / 2) λ_{κ} (1 + \bar{Z}) T_{0}^{2}}{λ_{F} (f_{c} + η_{abs} {⟨ Q_{sci} ⟩}^{- 1}) ⟨ σ v ⟩ ε_{F} / 4 - λ_{B} C_{B} Z_{eff} {\bar{Z}}^{2} T_{0}^{1 / 2}}, \end{matrix}

(46)

where λ_F, λ_B, and $λ_{κ}$ are unchanged because $Z_{eff}$ and $\bar{Z}$ are treated as volume-averaged quantities. We have also replaced the ${⟨ Q_{fuel} ⟩}^{- 1}$ term with $η_{abs} {⟨ Q_{sci} ⟩}^{- 1}$ ⁠, which allows us to include the effect of absorption efficiency.

Each experiment has different values of λ_F, λ_B, $λ_{κ}, \bar{Z}, Z_{eff}$ ⁠, and $η_{abs}$ ⁠, and therefore, each experiment has different $⟨ Q_{sci} ⟩$ contours. It is not feasible to show unique $⟨ Q_{sci} ⟩$ contours for each experiment in Figs. 2, 3, and 16. Figure 22 shows finite-width $⟨ Q_{sci} ⟩$ contours of the peaked and broad profiles whose lower and upper limits correspond to low-impurity (⁠ $Z_{eff} = 1.5, \bar{Z} = 1.2$ ⁠) and high-impurity (⁠ $Z_{eff} = 3.4, \bar{Z} = 1.2$ ⁠) models, respectively. These impurity levels correspond to the range of impurity levels considered for SPARC⁴⁵ and ITER.⁴⁶ For both the high and low-impurity models, we assume $T = T_{i} = T_{e}$ and $η_{abs} = 0.9$ ⁠. The finite ranges of $⟨ Q_{sci} ⟩$ aim to account for the main features and uncertainties of a future experimental device that will achieve $⟨ Q_{sci} ⟩ > 1$ ⁠, and therefore, we show finite-width $⟨ Q_{sci} ⟩$ contours in Fig. 2 (despite the $Q_{sci}$ labels in the legend). We emphasize that the finite width of the $⟨ Q_{sci} ⟩$ contours is merely illustrative of the effects of profiles and impurities and of the approximate values of $⟨ Q_{sci} ⟩$ that might be achieved by SPARC or ITER. To predict $⟨ Q_{sci} ⟩$ with higher precision would require detailed analysis and simulations.

FIG. 22.

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Finite-width $⟨ Q_{sci} ⟩$ contours vs peak Lawson parameter and T₀ bounded by low-impurity (⁠ $Z_{eff} = 1.5, \bar{Z} = 1.2$ ⁠) and high-impurity (⁠ $Z_{eff} = 3.4, \bar{Z} = 1.2$ ⁠) cases for peaked and broad spatial profiles (⁠ $ν_{T} = 3, ν_{n} = 0.2$ ⁠). These assumptions are made in plotting the $Q_{sci}$ contours of Figs. 2, 3, and 16.

4. Inferring peak from volume-averaged values

When only volume-averaged values of density and temperature are reported, we infer the peak values from an estimated value of the peaking, $T_{0} / ⟨ T ⟩$ and $n_{0} / ⟨ n ⟩$ ⁠, respectively. Detailed empirical models of peaking exist for predicting the profiles of future experiments.^47–50 However, for the purposes of this paper, we have chosen peaking values on a per-concept basis, the values of which are indicated in Table IV. Only concepts for which peak values must be inferred from reported volume-averaged values, along with citations for those values, are listed in Table IV. In Tables V and VI, we append a superscript asterisk (⁠ $^{*}$ ⁠) to peak values inferred from reported volume-averaged quantities.

TABLE IV.

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

Concept	$T_{0} / ⟨ T ⟩$	$n_{0} / ⟨ n ⟩$	References
Tokamak	2.0	1.5	50
Stellarator	3.0	1.0	51
Spherical Tokamak	2.1	1.7	52
FRC	1.0	1.3	53 and 54
RFP	1.2	1.2	44
Spheromak	2.0	1.5	55

Concept	$T_{0} / ⟨ T ⟩$	$n_{0} / ⟨ n ⟩$	References
Tokamak	2.0	1.5	50
Stellarator	3.0	1.0	51
Spherical Tokamak	2.1	1.7	52
FRC	1.0	1.3	53 and 54
RFP	1.2	1.2	44
Spheromak	2.0	1.5	55

TABLE V.

Data for tokamaks, spherical tokamaks, and stellarators. ^*Peak value of density or temperature has been inferred from volume-averaged value as described in Sec. IV A 4. ^†Ion temperature has been inferred from electron temperature as described in Sec. IV A 5. ^‡Ion density has been inferred from electron density as described in Sec. IV A 5. ^#Energy confinement time $τ_{E}^{*}$ (TFTR/Lawson method) has been inferred from a measurement of the energy confinement time τ_E (JET/JT-60) method as described in Sec. IV A 6.

Project	Concept	Year	Shot identifier	References	$T_{i 0}$ (keV)	$T_{e 0}$ (keV)	$n_{i 0}$ (m⁻³)	$n_{e 0}$ (m⁻³)	$τ_{E}^{*}$ (s)	$n_{i 0} τ_{E}^{*}$ (m⁻³ s)	$n_{i 0} T_{i 0} τ_{E}^{*}$ (keV m⁻³ s)
T-3	Tokamak	1969	H_z = 25 kOe, I_z = 85 kA discharges	67–69	0.3	1.05	2.25 × 10¹⁹ $^{‡}$	2.25 × 10¹⁹	0.003	6.8 × 10¹⁶	2.0 × 10¹⁶
Model C	Stellarator	1969	ICRH heated	70	0.2^†	0.2	5 × 10¹⁸ $^{‡}$	5 × 10¹⁸	0.0001	5.0 × 10¹⁴	1.0 × 10¹⁴
ST	Tokamak	1971	10 cm limiter, 42 kA	71	0.5	1.45	4 × 10¹⁹ $^{‡}$	4 × 10¹⁹	0.0034	1.4 × 10¹⁷	6.8 × 10¹⁶
ST	Tokamak	1972	12 cm limiter	72	0.4	0.8	6 × 10¹⁹ $^{‡}$	6 × 10¹⁹	0.01	6.0 × 10¹⁷	2.4 × 10¹⁷
TFR	Tokamak	1974	Molybdenum limiter	73	0.95	1.8	7.1 × 10¹⁹ $^{‡}$	7.1 × 10¹⁹	0.019	1.3 × 10¹⁸	1.3 × 10¹⁸
PLT	Tokamak	1976	22149-231	74	1.54	1.86	5.2 × 10¹⁹ $^{‡}$	5.2 × 10¹⁹	0.04	2.1 × 10¹⁸	3.2 × 10¹⁸
Alcator A	Tokamak	1978	8.7 T discharge	75	0.8	0.9	1.5 × 10²¹ $^{‡}$	1.5 × 10²¹	0.02	3.0 × 10¹⁹	2.4 × 10¹⁹
W7-A	Stellarator	1980	Zero current	76	0.545	0.316	9.6 × 10¹⁹ $^{‡}$	9.6 × 10¹⁹	0.0165	1.6 × 10¹⁸	8.6 × 10¹⁷
TFR	Tokamak	1981	Iconel limiter	73	0.95	1.2	1.61 × 10²⁰ $^{‡}$	1.61 × 10²⁰	0.034	5.5 × 10¹⁸	5.2 × 10¹⁸
TFR	Tokamak	1982	Carbon limiter	73	0.95	1.5	9 × 10¹⁹ $^{‡}$	9 × 10¹⁹	0.025	2.2 × 10¹⁸	2.1 × 10¹⁸
Alcator C	Tokamak	1984	Multiple pellet injection	77	1.5	1.5	1.5 × 10²¹ $^{‡}$	1.5 × 10²¹	0.052	7.8 × 10¹⁹	1.2 × 10²⁰
ASDEX	Tokamak	1988	23349-57	78	0.8	1	7.50 × 10¹⁹ $^{‡}$ ^*	⋯	0.12	9.0 × 10¹⁸	7.2 × 10¹⁸
JET	Tokamak	1991	26087	79	18.6	10.5	4.1 × 10¹⁹	5.1 × 10¹⁹	0.8^#	3.3 × 10¹⁹	6.1 × 10²⁰
JET	Tokamak	1991	26095	79	22.0	11.9	3.4 × 10¹⁹	4.5 × 10¹⁹	0.8^#	2.7 × 10¹⁹	6.0 × 10²⁰
JET	Tokamak	1991	26148	79	18.8	9.9	2.4 × 10¹⁹	3.6 × 10¹⁹	0.6^#	1.4 × 10¹⁹	2.7 × 10²⁰
JT-60U	Tokamak	1994	17110	80	37	12	4.2 × 10¹⁹	5.5 × 10¹⁹	0.3^#	1.3 × 10¹⁹	4.7 × 10²⁰
TFTR	Tokamak	1994	68522	81	29.0	11.7	6.8 × 10¹⁹	9.6 × 10¹⁹	0.18^#	1.2 × 10¹⁹	3.5 × 10²⁰
TFTR	Tokamak	1994	76778	81	44	11.5	6.3 × 10¹⁹	8.5 × 10¹⁹	0.19^#	1.2 × 10¹⁹	5.3 × 10²⁰
TFTR	Tokamak	1994	80539	81	36	13	6.7 × 10¹⁹	1.02 × 10²⁰	0.17^#	1.1 × 10¹⁹	4.1 × 10²⁰
TFTR	Tokamak	1995	83546	81	43	12.0	6.6 × 10¹⁹	8.5 × 10¹⁹	0.28^#	1.8 × 10¹⁹	7.9 × 10²⁰
JT-60U	Tokamak	1996	E26939	82	45.0	10.6	4.35 × 10¹⁹	6 × 10¹⁹	0.26^#	1.1 × 10¹⁹	5.1 × 10²⁰
JT-60U	Tokamak	1996	E26949	82	35.5	11.0	4.3 × 10¹⁹	5.85 × 10¹⁹	0.28^#	1.2 × 10¹⁹	4.3 × 10²⁰
JET	Tokamak	1997	42976	30	28	14	3.3 × 10¹⁹	4.1 × 10¹⁹	0.51^#	1.7 × 10¹⁹	4.7 × 10²⁰
DIII-D	Tokamak	1997	87977	83	18.1	7.5	8.5 × 10¹⁹	1 × 10²⁰	0.24^#	2.0 × 10¹⁹	3.7 × 10²⁰
START	Spherical Tokamak	1998	35533	84	0.2^†	0.2	1.02 × 10²⁰ $^{‡}$ ^*	⋯	0.003	3.1 × 10¹⁷	6.1 × 10¹⁶
JT-60U	Tokamak	1998	E31872	85	16.8	7.2	4.8 × 10¹⁹	8.5 × 10¹⁹	0.69^#	3.3 × 10¹⁹	5.6 × 10²⁰
W7-AS	Stellarator	2002	H-NBI mode	86	2.28^*	⋯	1.10 × 10²⁰ $^{‡}$ ^*	⋯	0.06	6.6 × 10¹⁸	1.5 × 10¹⁹
HSX	Stellarator	2005	QHS configuration	87	0.45^†	0.45	2.5 × 10¹⁸ $^{‡}$	2.5 × 10¹⁸	0.0006	1.5 × 10¹⁵	6.8 × 10¹⁴
MAST	Spherical Tokamak	2006	14626	88	3	2	3 × 10¹⁹ $^{‡}$	3 × 10¹⁹	0.05	1.5 × 10¹⁸	4.5 × 10¹⁸
LHD	Stellarator	2008	High triple product	89	0.47	0.47	5 × 10²⁰ $^{‡}$	5 × 10²⁰	0.22	1.1 × 10²⁰	5.2 × 10¹⁹
NSTX	Spherical Tokamak	2009	129041	90	1.2	1.2	5 × 10¹⁹ $^{‡}$	5 × 10¹⁹	0.08	4.0 × 10¹⁸	4.8 × 10¹⁸
EAST	Tokamak	2012	41079	91	1.2^†	1.2	2 × 10¹⁹ $^{‡}$	2 × 10¹⁹	0.04	8.0 × 10¹⁷	9.6 × 10¹⁷
EAST	Tokamak	2012	41195	92	0.9	⋯	3 × 10¹⁹ $^{‡}$	3 × 10¹⁹	0.04	1.2 × 10¹⁸	1.1 × 10¹⁸
EAST	Tokamak	2014	48068	92	1.2	⋯	6.1 × 10¹⁹ $^{‡}$	6.1 × 10¹⁹	0.037	2.3 × 10¹⁸	2.7 × 10¹⁸
KSTAR	Tokamak	2014	7081	93	2	⋯	4.80 × 10¹⁹ $^{‡}$ ^*	⋯	0.1	4.8 × 10¹⁸	9.6 × 10¹⁸
EAST	Tokamak	2015	56933	92 and 94	2.1	1.8	8.5 × 10¹⁹ $^{‡}$	8.5 × 10¹⁹	0.054	4.6 × 10¹⁸	9.6 × 10¹⁸
C-Mod	Tokamak	2016	1160930033	95	2.5^†	2.5	5.5 × 10²⁰ $^{‡}$	5.5 × 10²⁰	0.054	3.0 × 10¹⁹	7.4 × 10¹⁹
C-Mod	Tokamak	2016	1160930042	95	6^†	6	2 × 10²⁰ $^{‡}$	2 × 10²⁰	0.054	1.1 × 10¹⁹	6.5 × 10¹⁹
ASDEX-U	Tokamak	2016	32305	96	8	5	5 × 10¹⁹ $^{‡}$	5 × 10¹⁹	0.056	2.8 × 10¹⁸	2.2 × 10¹⁹
EAST	Tokamak	2016	71320	92 and 97	1.8	2.0	5.5 × 10¹⁹ $^{‡}$	5.5 × 10¹⁹	0.036	2.0 × 10¹⁸	3.6 × 10¹⁸
W7-X	Stellarator	2017	W7X 20171207.006	98–100	3.5	3.5	8 × 10¹⁹ $^{‡}$	8 × 10¹⁹	0.22	1.8 × 10¹⁹	6.2 × 10¹⁹
EAST	Tokamak	2018	78723	92	1.94	⋯	5.58 × 10¹⁹ $^{‡}$	5.58 × 10¹⁹	0.045	2.5 × 10¹⁸	4.9 × 10¹⁸
Globus-M2	Spherical Tokamak	2019	37873	101	1.2	⋯	1.19 × 10²⁰ $^{‡}$ ^*	⋯	0.01	1.2 × 10¹⁸	1.4 × 10¹⁸
SPARC	Tokamak	2025	Projected	45 and102	20	22	4 × 10²⁰ $^{‡}$	4 × 10²⁰	0.77	3.1 × 10²⁰	6.2 × 10²¹
ITER	Tokamak	2035	Projected	32, 46, 103–105	20	⋯	1 × 10²⁰ $^{‡}$	1 × 10²⁰	3.7	3.7 × 10²⁰	7.4 × 10²¹

Project	Concept	Year	Shot identifier	References	$T_{i 0}$ (keV)	$T_{e 0}$ (keV)	$n_{i 0}$ (m⁻³)	$n_{e 0}$ (m⁻³)	$τ_{E}^{*}$ (s)	$n_{i 0} τ_{E}^{*}$ (m⁻³ s)	$n_{i 0} T_{i 0} τ_{E}^{*}$ (keV m⁻³ s)
T-3	Tokamak	1969	H_z = 25 kOe, I_z = 85 kA discharges	67–69	0.3	1.05	2.25 × 10¹⁹ $^{‡}$	2.25 × 10¹⁹	0.003	6.8 × 10¹⁶	2.0 × 10¹⁶
Model C	Stellarator	1969	ICRH heated	70	0.2^†	0.2	5 × 10¹⁸ $^{‡}$	5 × 10¹⁸	0.0001	5.0 × 10¹⁴	1.0 × 10¹⁴
ST	Tokamak	1971	10 cm limiter, 42 kA	71	0.5	1.45	4 × 10¹⁹ $^{‡}$	4 × 10¹⁹	0.0034	1.4 × 10¹⁷	6.8 × 10¹⁶
ST	Tokamak	1972	12 cm limiter	72	0.4	0.8	6 × 10¹⁹ $^{‡}$	6 × 10¹⁹	0.01	6.0 × 10¹⁷	2.4 × 10¹⁷
TFR	Tokamak	1974	Molybdenum limiter	73	0.95	1.8	7.1 × 10¹⁹ $^{‡}$	7.1 × 10¹⁹	0.019	1.3 × 10¹⁸	1.3 × 10¹⁸
PLT	Tokamak	1976	22149-231	74	1.54	1.86	5.2 × 10¹⁹ $^{‡}$	5.2 × 10¹⁹	0.04	2.1 × 10¹⁸	3.2 × 10¹⁸
Alcator A	Tokamak	1978	8.7 T discharge	75	0.8	0.9	1.5 × 10²¹ $^{‡}$	1.5 × 10²¹	0.02	3.0 × 10¹⁹	2.4 × 10¹⁹
W7-A	Stellarator	1980	Zero current	76	0.545	0.316	9.6 × 10¹⁹ $^{‡}$	9.6 × 10¹⁹	0.0165	1.6 × 10¹⁸	8.6 × 10¹⁷
TFR	Tokamak	1981	Iconel limiter	73	0.95	1.2	1.61 × 10²⁰ $^{‡}$	1.61 × 10²⁰	0.034	5.5 × 10¹⁸	5.2 × 10¹⁸
TFR	Tokamak	1982	Carbon limiter	73	0.95	1.5	9 × 10¹⁹ $^{‡}$	9 × 10¹⁹	0.025	2.2 × 10¹⁸	2.1 × 10¹⁸
Alcator C	Tokamak	1984	Multiple pellet injection	77	1.5	1.5	1.5 × 10²¹ $^{‡}$	1.5 × 10²¹	0.052	7.8 × 10¹⁹	1.2 × 10²⁰
ASDEX	Tokamak	1988	23349-57	78	0.8	1	7.50 × 10¹⁹ $^{‡}$ ^*	⋯	0.12	9.0 × 10¹⁸	7.2 × 10¹⁸
JET	Tokamak	1991	26087	79	18.6	10.5	4.1 × 10¹⁹	5.1 × 10¹⁹	0.8^#	3.3 × 10¹⁹	6.1 × 10²⁰
JET	Tokamak	1991	26095	79	22.0	11.9	3.4 × 10¹⁹	4.5 × 10¹⁹	0.8^#	2.7 × 10¹⁹	6.0 × 10²⁰
JET	Tokamak	1991	26148	79	18.8	9.9	2.4 × 10¹⁹	3.6 × 10¹⁹	0.6^#	1.4 × 10¹⁹	2.7 × 10²⁰
JT-60U	Tokamak	1994	17110	80	37	12	4.2 × 10¹⁹	5.5 × 10¹⁹	0.3^#	1.3 × 10¹⁹	4.7 × 10²⁰
TFTR	Tokamak	1994	68522	81	29.0	11.7	6.8 × 10¹⁹	9.6 × 10¹⁹	0.18^#	1.2 × 10¹⁹	3.5 × 10²⁰
TFTR	Tokamak	1994	76778	81	44	11.5	6.3 × 10¹⁹	8.5 × 10¹⁹	0.19^#	1.2 × 10¹⁹	5.3 × 10²⁰
TFTR	Tokamak	1994	80539	81	36	13	6.7 × 10¹⁹	1.02 × 10²⁰	0.17^#	1.1 × 10¹⁹	4.1 × 10²⁰
TFTR	Tokamak	1995	83546	81	43	12.0	6.6 × 10¹⁹	8.5 × 10¹⁹	0.28^#	1.8 × 10¹⁹	7.9 × 10²⁰
JT-60U	Tokamak	1996	E26939	82	45.0	10.6	4.35 × 10¹⁹	6 × 10¹⁹	0.26^#	1.1 × 10¹⁹	5.1 × 10²⁰
JT-60U	Tokamak	1996	E26949	82	35.5	11.0	4.3 × 10¹⁹	5.85 × 10¹⁹	0.28^#	1.2 × 10¹⁹	4.3 × 10²⁰
JET	Tokamak	1997	42976	30	28	14	3.3 × 10¹⁹	4.1 × 10¹⁹	0.51^#	1.7 × 10¹⁹	4.7 × 10²⁰
DIII-D	Tokamak	1997	87977	83	18.1	7.5	8.5 × 10¹⁹	1 × 10²⁰	0.24^#	2.0 × 10¹⁹	3.7 × 10²⁰
START	Spherical Tokamak	1998	35533	84	0.2^†	0.2	1.02 × 10²⁰ $^{‡}$ ^*	⋯	0.003	3.1 × 10¹⁷	6.1 × 10¹⁶
JT-60U	Tokamak	1998	E31872	85	16.8	7.2	4.8 × 10¹⁹	8.5 × 10¹⁹	0.69^#	3.3 × 10¹⁹	5.6 × 10²⁰
W7-AS	Stellarator	2002	H-NBI mode	86	2.28^*	⋯	1.10 × 10²⁰ $^{‡}$ ^*	⋯	0.06	6.6 × 10¹⁸	1.5 × 10¹⁹
HSX	Stellarator	2005	QHS configuration	87	0.45^†	0.45	2.5 × 10¹⁸ $^{‡}$	2.5 × 10¹⁸	0.0006	1.5 × 10¹⁵	6.8 × 10¹⁴
MAST	Spherical Tokamak	2006	14626	88	3	2	3 × 10¹⁹ $^{‡}$	3 × 10¹⁹	0.05	1.5 × 10¹⁸	4.5 × 10¹⁸
LHD	Stellarator	2008	High triple product	89	0.47	0.47	5 × 10²⁰ $^{‡}$	5 × 10²⁰	0.22	1.1 × 10²⁰	5.2 × 10¹⁹
NSTX	Spherical Tokamak	2009	129041	90	1.2	1.2	5 × 10¹⁹ $^{‡}$	5 × 10¹⁹	0.08	4.0 × 10¹⁸	4.8 × 10¹⁸
EAST	Tokamak	2012	41079	91	1.2^†	1.2	2 × 10¹⁹ $^{‡}$	2 × 10¹⁹	0.04	8.0 × 10¹⁷	9.6 × 10¹⁷
EAST	Tokamak	2012	41195	92	0.9	⋯	3 × 10¹⁹ $^{‡}$	3 × 10¹⁹	0.04	1.2 × 10¹⁸	1.1 × 10¹⁸
EAST	Tokamak	2014	48068	92	1.2	⋯	6.1 × 10¹⁹ $^{‡}$	6.1 × 10¹⁹	0.037	2.3 × 10¹⁸	2.7 × 10¹⁸
KSTAR	Tokamak	2014	7081	93	2	⋯	4.80 × 10¹⁹ $^{‡}$ ^*	⋯	0.1	4.8 × 10¹⁸	9.6 × 10¹⁸
EAST	Tokamak	2015	56933	92 and 94	2.1	1.8	8.5 × 10¹⁹ $^{‡}$	8.5 × 10¹⁹	0.054	4.6 × 10¹⁸	9.6 × 10¹⁸
C-Mod	Tokamak	2016	1160930033	95	2.5^†	2.5	5.5 × 10²⁰ $^{‡}$	5.5 × 10²⁰	0.054	3.0 × 10¹⁹	7.4 × 10¹⁹
C-Mod	Tokamak	2016	1160930042	95	6^†	6	2 × 10²⁰ $^{‡}$	2 × 10²⁰	0.054	1.1 × 10¹⁹	6.5 × 10¹⁹
ASDEX-U	Tokamak	2016	32305	96	8	5	5 × 10¹⁹ $^{‡}$	5 × 10¹⁹	0.056	2.8 × 10¹⁸	2.2 × 10¹⁹
EAST	Tokamak	2016	71320	92 and 97	1.8	2.0	5.5 × 10¹⁹ $^{‡}$	5.5 × 10¹⁹	0.036	2.0 × 10¹⁸	3.6 × 10¹⁸
W7-X	Stellarator	2017	W7X 20171207.006	98–100	3.5	3.5	8 × 10¹⁹ $^{‡}$	8 × 10¹⁹	0.22	1.8 × 10¹⁹	6.2 × 10¹⁹
EAST	Tokamak	2018	78723	92	1.94	⋯	5.58 × 10¹⁹ $^{‡}$	5.58 × 10¹⁹	0.045	2.5 × 10¹⁸	4.9 × 10¹⁸
Globus-M2	Spherical Tokamak	2019	37873	101	1.2	⋯	1.19 × 10²⁰ $^{‡}$ ^*	⋯	0.01	1.2 × 10¹⁸	1.4 × 10¹⁸
SPARC	Tokamak	2025	Projected	45 and102	20	22	4 × 10²⁰ $^{‡}$	4 × 10²⁰	0.77	3.1 × 10²⁰	6.2 × 10²¹
ITER	Tokamak	2035	Projected	32, 46, 103–105	20	⋯	1 × 10²⁰ $^{‡}$	1 × 10²⁰	3.7	3.7 × 10²⁰	7.4 × 10²¹

TABLE VI.

Data for magnetic alternate concepts. ^*Peak value of density or temperature has been inferred from volume-averaged value as described in Sec. IV A 4. ^†Ion temperature has been inferred from electron temperature as described in Sec. IV A 5. ^‡Ion density has been inferred from electron density as described in Sec. IV A 5. ^#Energy confinement time $τ_{E}^{*}$ (TFTR/Lawson method) has been inferred from a measurement of the energy confinement time τ_E (JET/JT-60) method as described in Sec. IV A 6.

Project	Concept	Year	Shot identifier	Reference	$T_{i 0}$ (keV)	$T_{e 0}$ (keV)	$n_{i 0}$ (m⁻³)	$n_{e 0}$ (m⁻³)	$τ_{E}^{*}$ (s)	$n_{i 0} τ_{E}^{*}$ (m⁻³ s)	$n_{i 0} T_{i 0} τ_{E}^{*}$ (keV m⁻³ s)
ZETA	Pinch	1957	140 kA–180 kA discharges	106	0.09	0.03	1 × 10²⁰ $^{‡}$	1 × 10²⁰	0.0001	1.0 × 10¹⁶	9.0 × 10¹⁴
ETA-BETA I	RFP	1977	Summary	107	0.01	⋯	1 × 10²¹	⋯	1 × 10^–6	1.0 × 10¹⁵	1.0 × 10¹³
TMX-U	Mirror	1984	2/2/84 S21	108	0.15	0.045	2 × 10¹⁸	2 × 10¹⁸	0.001	2.0 × 10¹⁵	3.0 × 10¹⁴
ETA-BETA II	RFP	1984	59611	109	0.09^†	0.09	3.5 × 10²⁰ $^{‡}$	3.5 × 10²⁰	0.0001	3.5 × 10¹⁶	3.2 × 10¹⁵
ZT-40M	RFP	1987	330 kA discharge	110	0.33^†	0.33	9.60 × 10¹⁹ $^{‡}$ ^*	⋯	0.0007	6.7 × 10¹⁶	2.2 × 10¹⁶
CTX	Spheromak	1990	Solid flux conserver	111	0.18	0.18	4.50 × 10¹⁹ $^{‡}$ ^*	⋯	0.0002	9.0 × 10¹⁵	1.6 × 10¹⁵
LSX	FRC	1993	s 2	53	0.547	0.253	1.30 × 10²¹ $^{*}$	⋯	0.0001	1.3 × 10¹⁷	7.1 × 10¹⁶
MST	RFP	2001	390 kA discharge	44	0.396	0.792	1.20 × 10¹⁹ $^{‡}$ ^*	⋯	0.0064^#	7.7 × 10¹⁶	3.0 × 10¹⁶
FRX-L	FRC	2003	2027	112	0.09	0.09	4 × 10²²	4 × 10²²	3.3 × 10^–6	1.3 × 10¹⁷	1.2 × 10¹⁶
ZaP	Z Pinch	2003	Unknown	113	0.1	⋯	9 × 10²² $^{‡}$	9 × 10²²	3.7 × 10^–7	3.3 × 10¹⁶	3.3 × 10¹⁵
FRX-L	FRC	2005	3684	114	0.18^*	⋯	4.81 × 10²² $^{‡}$ ^*	⋯	3.3 × 10^–6	1.6 × 10¹⁷	2.9 × 10¹⁶
TCS	FRC	2005	9018	115	0.025	0.025	6.50 × 10¹⁸ $^{‡}$ ^*	⋯	4 × 10^–5	2.6 × 10¹⁴	6.5 × 10¹²
SSPX	Spheromak	2007	17524	116	0.5^†	0.5	2.25 × 10²⁰ $^{‡}$ ^*	⋯	0.001	2.2 × 10¹⁷	1.1 × 10¹⁷
GOL-3	Mirror	2007	Unknown	117	2	2	7 × 10²⁰	7 × 10²⁰	0.0009	6.3 × 10¹⁷	1.3 × 10¹⁸
TCSU	FRC	2008	21214	115	0.1	0.1	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	7.5 × 10^–5	9.7 × 10¹⁴	9.7 × 10¹³
RFX-mod	RFP	2008	24063	118	1^†	1	3 × 10¹⁹ $^{‡}$	3 × 10¹⁹	0.0025	7.5 × 10¹⁶	7.5 × 10¹⁶
MST	RFP	2009	pellets	119	0.6	0.7	4 × 10¹⁹ $^{‡}$	4 × 10¹⁹	0.007	2.8 × 10¹⁷	1.7 × 10¹⁷
MST	RFP	2009	w/o pellets	119	1.3	1.9	1.2 × 10¹⁹ $^{‡}$	1.2 × 10¹⁹	0.012	1.4 × 10¹⁷	1.9 × 10¹⁷
IPA	FRC	2010	Unknown	120	0.85^*	⋯	5.20 × 10²¹ $^{‡}$ ^*	⋯	1 × 10^–5	5.2 × 10¹⁶	4.4 × 10¹⁶
Yingguang-I	FRC	2015	150910-01	121	0.2^*	⋯	4.81 × 10²² $^{‡}$ ^*	⋯	1 × 10^–6	4.8 × 10¹⁶	9.6 × 10¹⁵
C-2U	FRC	2017	46366	122, 123	0.68^*	⋯	2.47 × 10¹⁹ $^{‡}$ ^*	⋯	0.00024	5.9 × 10¹⁵	4.0 × 10¹⁵
FuZE	Z Pinch	2018	Multiple identical shots	62	1.8	⋯	1.1 × 10²³ $^{‡}$	1.1 × 10²³	1.1 × 10^–6	1.2 × 10¹⁷	2.2 × 10¹⁷
GDT	Mirror	2018	Multiple identical shots	124	0.45^†	0.45	1.1 × 10¹⁹ $^{‡}$	1.1 × 10¹⁹	0.0006	6.6 × 10¹⁵	3.0 × 10¹⁵
C-2W	FRC	2019	104989	125	1.0^*	⋯	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	0.003	3.9 × 10¹⁶	3.9 × 10¹⁶
C-2W	FRC	2019	107322	125	0.6^*	⋯	2.08 × 10¹⁹ $^{‡}$ ^*	⋯	0.0012	2.5 × 10¹⁶	1.5 × 10¹⁶
C-2W	FRC	2020	114534	126	1.8^*	⋯	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	0.0015	2.0 × 10¹⁶	3.5 × 10¹⁶
C-2W	FRC	2021	118340	127	3.5^*	⋯	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	0.005	6.5 × 10¹⁶	2.3 × 10¹⁷

Project	Concept	Year	Shot identifier	Reference	$T_{i 0}$ (keV)	$T_{e 0}$ (keV)	$n_{i 0}$ (m⁻³)	$n_{e 0}$ (m⁻³)	$τ_{E}^{*}$ (s)	$n_{i 0} τ_{E}^{*}$ (m⁻³ s)	$n_{i 0} T_{i 0} τ_{E}^{*}$ (keV m⁻³ s)
ZETA	Pinch	1957	140 kA–180 kA discharges	106	0.09	0.03	1 × 10²⁰ $^{‡}$	1 × 10²⁰	0.0001	1.0 × 10¹⁶	9.0 × 10¹⁴
ETA-BETA I	RFP	1977	Summary	107	0.01	⋯	1 × 10²¹	⋯	1 × 10^–6	1.0 × 10¹⁵	1.0 × 10¹³
TMX-U	Mirror	1984	2/2/84 S21	108	0.15	0.045	2 × 10¹⁸	2 × 10¹⁸	0.001	2.0 × 10¹⁵	3.0 × 10¹⁴
ETA-BETA II	RFP	1984	59611	109	0.09^†	0.09	3.5 × 10²⁰ $^{‡}$	3.5 × 10²⁰	0.0001	3.5 × 10¹⁶	3.2 × 10¹⁵
ZT-40M	RFP	1987	330 kA discharge	110	0.33^†	0.33	9.60 × 10¹⁹ $^{‡}$ ^*	⋯	0.0007	6.7 × 10¹⁶	2.2 × 10¹⁶
CTX	Spheromak	1990	Solid flux conserver	111	0.18	0.18	4.50 × 10¹⁹ $^{‡}$ ^*	⋯	0.0002	9.0 × 10¹⁵	1.6 × 10¹⁵
LSX	FRC	1993	s 2	53	0.547	0.253	1.30 × 10²¹ $^{*}$	⋯	0.0001	1.3 × 10¹⁷	7.1 × 10¹⁶
MST	RFP	2001	390 kA discharge	44	0.396	0.792	1.20 × 10¹⁹ $^{‡}$ ^*	⋯	0.0064^#	7.7 × 10¹⁶	3.0 × 10¹⁶
FRX-L	FRC	2003	2027	112	0.09	0.09	4 × 10²²	4 × 10²²	3.3 × 10^–6	1.3 × 10¹⁷	1.2 × 10¹⁶
ZaP	Z Pinch	2003	Unknown	113	0.1	⋯	9 × 10²² $^{‡}$	9 × 10²²	3.7 × 10^–7	3.3 × 10¹⁶	3.3 × 10¹⁵
FRX-L	FRC	2005	3684	114	0.18^*	⋯	4.81 × 10²² $^{‡}$ ^*	⋯	3.3 × 10^–6	1.6 × 10¹⁷	2.9 × 10¹⁶
TCS	FRC	2005	9018	115	0.025	0.025	6.50 × 10¹⁸ $^{‡}$ ^*	⋯	4 × 10^–5	2.6 × 10¹⁴	6.5 × 10¹²
SSPX	Spheromak	2007	17524	116	0.5^†	0.5	2.25 × 10²⁰ $^{‡}$ ^*	⋯	0.001	2.2 × 10¹⁷	1.1 × 10¹⁷
GOL-3	Mirror	2007	Unknown	117	2	2	7 × 10²⁰	7 × 10²⁰	0.0009	6.3 × 10¹⁷	1.3 × 10¹⁸
TCSU	FRC	2008	21214	115	0.1	0.1	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	7.5 × 10^–5	9.7 × 10¹⁴	9.7 × 10¹³
RFX-mod	RFP	2008	24063	118	1^†	1	3 × 10¹⁹ $^{‡}$	3 × 10¹⁹	0.0025	7.5 × 10¹⁶	7.5 × 10¹⁶
MST	RFP	2009	pellets	119	0.6	0.7	4 × 10¹⁹ $^{‡}$	4 × 10¹⁹	0.007	2.8 × 10¹⁷	1.7 × 10¹⁷
MST	RFP	2009	w/o pellets	119	1.3	1.9	1.2 × 10¹⁹ $^{‡}$	1.2 × 10¹⁹	0.012	1.4 × 10¹⁷	1.9 × 10¹⁷
IPA	FRC	2010	Unknown	120	0.85^*	⋯	5.20 × 10²¹ $^{‡}$ ^*	⋯	1 × 10^–5	5.2 × 10¹⁶	4.4 × 10¹⁶
Yingguang-I	FRC	2015	150910-01	121	0.2^*	⋯	4.81 × 10²² $^{‡}$ ^*	⋯	1 × 10^–6	4.8 × 10¹⁶	9.6 × 10¹⁵
C-2U	FRC	2017	46366	122, 123	0.68^*	⋯	2.47 × 10¹⁹ $^{‡}$ ^*	⋯	0.00024	5.9 × 10¹⁵	4.0 × 10¹⁵
FuZE	Z Pinch	2018	Multiple identical shots	62	1.8	⋯	1.1 × 10²³ $^{‡}$	1.1 × 10²³	1.1 × 10^–6	1.2 × 10¹⁷	2.2 × 10¹⁷
GDT	Mirror	2018	Multiple identical shots	124	0.45^†	0.45	1.1 × 10¹⁹ $^{‡}$	1.1 × 10¹⁹	0.0006	6.6 × 10¹⁵	3.0 × 10¹⁵
C-2W	FRC	2019	104989	125	1.0^*	⋯	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	0.003	3.9 × 10¹⁶	3.9 × 10¹⁶
C-2W	FRC	2019	107322	125	0.6^*	⋯	2.08 × 10¹⁹ $^{‡}$ ^*	⋯	0.0012	2.5 × 10¹⁶	1.5 × 10¹⁶
C-2W	FRC	2020	114534	126	1.8^*	⋯	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	0.0015	2.0 × 10¹⁶	3.5 × 10¹⁶
C-2W	FRC	2021	118340	127	3.5^*	⋯	1.30 × 10¹⁹ $^{‡}$ ^*	⋯	0.005	6.5 × 10¹⁶	2.3 × 10¹⁷

5. Inferring ion quantities from electron quantities

When only T_e and not T_i is reported, we cannot assume T_i = T_e in calculating the triple product without further consideration. If the thermal-equilibration time is much shorter than the plasma duration, and assuming that there are no other effects that would give rise to $T_{i} \neq T_{e}$ ⁠, then we can assume T_i = T_e. In these cases, we append a superscript dagger (⁠ $^{†}$ ⁠) to the inferred value of T_i in Tables V and VI. In cases where both T_i and T_e are reported in MCF experiments, we use the reported T_i.

When only n_e but not n_i is reported, we assume n_i = n_e for D-T and D-D plasmas. In such cases, we append a superscript double dagger (⁠ $^{‡}$ ⁠) to the inferred value of n_i in Tables V and VI.

6. Accounting for transient heating

All experiments experience a transient startup phase during which a portion of the heating power goes into raising the plasma thermal energy $W_{p} = 3 n T V$ (assuming $T = T_{i} = T_{e}$ and $n = n_{i} = n_{e}$ ⁠). There are two self-consistent approaches for deriving an expression for $Q_{fuel}$ that accounts for the effect of transient heating $d W_{p} / d t$ ⁠, where

\frac{d W_{p}}{d t} = P_{abs} + P_{c} - P_{B} - \frac{3 n T V}{τ_{E}} .

(47)

In the remainder of this subsection, we closely follow Ref. 56.

The first approach is to group the transient term with $P_{abs}$ in the instantaneous power balance which effectively treats the transient term as a reduction in the externally applied and absorbed heating power,

(P_{abs} - \frac{d W_{p}}{d t}) + P_{c} = P_{B} + \frac{3 n T V}{τ_{E}} .

(48)

In this approach, the definition of $Q_{fuel}$ is modified, i.e.,

Q_{fuel}^{*} = \frac{P_{F}}{P_{abs} - d W_{p} / d t},

(49)

where $Q_{fuel}^{*}$ is the ratio of fusion power to absorbed heating power minus the portion that is being used to increase the plasma temperature. From here, we derive an expression for the Lawson parameter following the same steps as Sec. III E, which results in an analogous expression to Eq. (17) but with $Q_{fuel}$ replaced by $Q_{fuel}^{*}$ ⁠,

n τ_{E} = \frac{3 T}{(f_{c} + Q_{fuel}^{* - 1}) ⟨ σ v ⟩ ε_{F} / 4 - C_{B} T^{1 / 2}} .

(50)

From Eq. (48),

τ_{E} = \frac{W_{p}}{P_{heat} - d W_{p} / d t},

(51)

where

P_{heat} = P_{abs} + P_{c} - P_{B} .

(52)

This approach, defined by Eqs. (49)–(52), is the one used by JET and JT-60.

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

P_{abs} + P_{c} = P_{B} + (\frac{3 n T V}{τ_{E}} + \frac{d W_{p}}{d t}) .

(53)

We then define a modified energy confinement time $τ_{E}^{*}$ which characterizes thermal conduction and transient heating power,

\frac{3 n T V}{τ_{E}^{*}} = \frac{3 n T V}{τ_{E}} + \frac{d W_{p}}{d t} .

(54)

Combining the latter with Eqs. (52) and (53),

τ_{E}^{*} = \frac{W_{p}}{P_{heat}} .

(55)

From this point, we derive an expression for the Lawson parameter following the same steps as Sec. III E, which results in an analogous expression to Eq. (17) but with τ_E replaced by $τ_{E}^{*}$ ⁠,

n τ_{E}^{*} = \frac{3 T}{(f_{c} + Q_{fuel}^{- 1}) ⟨ σ v ⟩ ε_{F} / 4 - C_{B} T^{1 / 2}} .

(56)

In this formulation, the definition of instantaneous $Q_{fuel}$ is unchanged from the steady-state value of Eq. (15), and fuel breakeven occurs at $Q_{fuel} = 1$ ⁠, regardless of the value of $d W_{p} / d t$ ⁠. This approach, defined by Eqs. (55), (56), and (16), is the one used by TFTR and consistent with Lawson's original formulation.

For the JET/JT-60 approach, fuel breakeven does not necessarily occur at $Q_{fuel}^{*} = 1$ but rather occurs at a value of $Q_{fuel}^{*}$ that depends on the value of $d W_{p} / d t$ ⁠. The TFTR/Lawson approach keeps the definition of instantaneous $Q_{fuel}$ the same as the steady-state $Q_{fuel}$ ⁠, and fuel breakeven always occurs at $Q_{fuel} = 1$ regardless of the transient-heating value. Because a key objective of this paper is to chart the progress of many different experiments toward and beyond $Q_{fuel} = 1$ ⁠, we use the TFTR/Lawson definition for which $Q_{fuel} = 1$ means the same thing across different MCF experiments. In practice, this means we use $τ_{E}^{*}$ and Eq. (56) for all MCF experiments. When τ_E and $d W_{p} / d t$ are reported and $d W_{p} / d t$ is nonzero (e.g., JET and JT-60), we calculate and use $τ_{E}^{*}$ ⁠, indicating such cases with a superscript hash (⁠ $^{#}$ ⁠) in Tables V and VI. Some TFTR publications report τ_E, requiring the conversion step, and thus, we append a superscript hash for those cases as well. In evaluating $τ_{E}^{*}$ from published data, the bremsstrahlung losses are often not reported and are considered to be small compared to $P_{abs}$ and may be neglected.

B. ICF methodology

Direct measurements of plasma parameters are more challenging for ICF. Commonly measured parameters in ICF are fuel areal density ρR (via neutron downscattering), T_i and “burn duration” (via neutron time-of-flight), and neutron yield (via various types of neutron detectors). Some experiments report an inferred stagnation pressure $p_{stag}$ based on a statistical analysis of other measured quantities and simulation databases.

Identifying the requirements for the ignition of an ICF capsule is difficult. The analysis presented in Sec. III F assumes an idealized ICF scenario. Real ICF experiments must contend with instabilities, impurities, non-zero bremsstrahlung and thermal-conduction losses, and other factors that make it more difficult to achieve ignition. For the highest-performing ICF experiments considered here (NIF, OMEGA), a two-stage approach to ignition is pursued, i.e., ignition of a central lower-density “hot spot” followed by propagating burn into the surrounding colder, denser fuel, as depicted in Fig. 23. Because of the low value of $η_{abs}$ inherent in these experiments, this two-stage process is required to achieve $Q_{sci} > 1$ ⁠. Therefore, we consider both ignition of the hot spot and the onset of propagating burn in the dense fuel when we refer to “ignition” in this section.

FIG. 23.

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Representation of an ICF capsule implosion and hot-spot creation with instability growth: (a) dense fuel shell, with radius R and thickness Δ, at maximum shell velocity V_i during the implosion and (b) fuel assembly at stagnation with the “hot spot” (blue) with effective radius R_s, surrounded by the cold, dense fuel (grey). Rayleigh–Taylor instabilities are shown. If the hot spot reaches high-enough $n_{i} τ$ and T_i, then it can potentially generate enough fusion energy to initiate a propagating burn into the surrounding cold shell.

Below we describe two methodologies used in this paper for inferring the Lawson parameter $n τ$ and triple product $n T τ$ for cases in which pressure is or is not experimentally inferred, respectively.

1. Inferring Lawson parameter and triple product without reported inferred pressure

For ICF experiments that do not report experimentally inferred values of fuel pressure (i.e., rows with “⋯” in the $p_{stag}$ column of Table VII), we employ the methodology of Ref. 34 to infer $n_{i} τ$ from other measured ICF experimental quantities. Here, we state the key logic and equation of this methodology for the convenience of the reader, but we refer the reader to Ref. 34 for further details, equation derivations, and justifications. It is important to note that Ref. 34 makes a simplifying assumption that thermal-conduction and radiation losses are negligible (on the timescale of the fusion burn) because of the insulating effects of the dense shell of an ICF target capsule, meaning that Lawson parameters and triple products inferred via this method should be considered as upper bounds.

TABLE VII.

Data for ICF and MIF concepts.

Project	Concept	Year	Shot identifier	References	${⟨ T_{i} ⟩}_{n}$ (keV)	T_e (keV)	$ρ R_{tot (n)}^{n o (α)}$ (g/cm⁻²)	YOC	p_stag (Gbar)	τ_stag (s)	$P τ$ (atm s)	$n τ$ (m⁻³ s)	$n {⟨ T ⟩}_{n} τ$ (keV m⁻³ s)
NOVA	Laser ICF	1994	100 atm. fill	128	0.9	⋯	⋯	⋯	16	5 × 10^–11	0.26	9.2 × 10¹⁹	8.3 × 10¹⁹
OMEGA	Laser ICF	2007	47206	129 and 130	2.0	⋯	0.202	0.1	⋯	⋯	1.23	1.9 × 10²⁰	3.9 × 10²⁰
OMEGA	Laser ICF	2007	47210	129 and 130	2.0	⋯	0.182	0.1	⋯	⋯	1.13	1.8 × 10²⁰	3.6 × 10²⁰
OMEGA	Laser ICF	2009	55468	130	1.8	⋯	0.300	0.1	⋯	⋯	1.55	2.7 × 10²⁰	4.9 × 10²⁰
OMEGA	Laser ICF	2009	Unknown	130	1.8	⋯	0.240	0.1	⋯	⋯	1.29	2.3 × 10²⁰	4.1 × 10²⁰
OMEGA	Laser ICF	2013	69236	131	2.8	⋯	⋯	⋯	18	1.15 × 10^–10	0.68	7.7 × 10¹⁹	2.1 × 10²⁰
NIF	Laser ICF	2014	N140304	132	5.5	⋯	⋯	⋯	222	1.63 × 10^–10	11.86	6.8 × 10²⁰	3.8 × 10²¹
MagLIF	MagLIF	2014	z2613	133	2.0	⋯	⋯	⋯	0.56	1.38 × 10^–9	0.76	1.2 × 10²⁰	2.4 × 10²⁰
OMEGA	Laser ICF	2015	77068	134	3.6	⋯	⋯	⋯	56	6.6 × 10^–11	1.21	1.1 × 10²⁰	3.8 × 10²⁰
MagLIF	MagLIF	2015	z2850	133	2.8	⋯	⋯	⋯	0.6	1.62 × 10^–9	0.96	1.1 × 10²⁰	3.0 × 10²⁰
NIF	Laser ICF	2017	N170601	132	4.5	⋯	⋯	⋯	320	1.6 × 10^–10	16.78	1.2 × 10²¹	5.3 × 10²¹
NIF	Laser ICF	2017	N170827	132	4.5	⋯	⋯	⋯	360	1.54 × 10^–10	18.17	1.3 × 10²¹	5.7 × 10²¹
FIREX	Laser ICF	2019	40558	135	⋯	2.1	⋯	⋯	2	4 × 10^–10	0.79	1.2 × 10²⁰	2.5 × 10²⁰
NIF	Laser ICF	2019	N191007	35	4.52	⋯	⋯	⋯	206	1.51 × 10^–10	10.20	7.1 × 10²⁰	3.2 × 10²¹
NIF	Laser ICF	2020	N201101	136	4.61	⋯	⋯	⋯	319	1.18 × 10^–10	12.34	8.5 × 10²⁰	3.9 × 10²¹
NIF	Laser ICF	2020	N201122	136	4.65	⋯	⋯	⋯	297	1.37 × 10^–10	13.34	9.1 × 10²⁰	4.2 × 10²¹
NIF	Laser ICF	2021	N210207	136	5.23	⋯	⋯	⋯	339	1.07 × 10^–10	11.89	7.2 × 10²⁰	3.8 × 10²¹
NIF	Laser ICF	2021	N210220	136	5.13	⋯	⋯	⋯	371	1.35 × 10^–10	16.42	1.0 × 10²¹	5.2 × 10²¹
NIF	Laser ICF	2021	N210808	24	10.9	⋯	⋯	⋯	569	8.9 × 10^–11	16.60	4.8 × 10²⁰	5.2 × 10²¹

Project	Concept	Year	Shot identifier	References	${⟨ T_{i} ⟩}_{n}$ (keV)	T_e (keV)	$ρ R_{tot (n)}^{n o (α)}$ (g/cm⁻²)	YOC	p_stag (Gbar)	τ_stag (s)	$P τ$ (atm s)	$n τ$ (m⁻³ s)	$n {⟨ T ⟩}_{n} τ$ (keV m⁻³ s)
NOVA	Laser ICF	1994	100 atm. fill	128	0.9	⋯	⋯	⋯	16	5 × 10^–11	0.26	9.2 × 10¹⁹	8.3 × 10¹⁹
OMEGA	Laser ICF	2007	47206	129 and 130	2.0	⋯	0.202	0.1	⋯	⋯	1.23	1.9 × 10²⁰	3.9 × 10²⁰
OMEGA	Laser ICF	2007	47210	129 and 130	2.0	⋯	0.182	0.1	⋯	⋯	1.13	1.8 × 10²⁰	3.6 × 10²⁰
OMEGA	Laser ICF	2009	55468	130	1.8	⋯	0.300	0.1	⋯	⋯	1.55	2.7 × 10²⁰	4.9 × 10²⁰
OMEGA	Laser ICF	2009	Unknown	130	1.8	⋯	0.240	0.1	⋯	⋯	1.29	2.3 × 10²⁰	4.1 × 10²⁰
OMEGA	Laser ICF	2013	69236	131	2.8	⋯	⋯	⋯	18	1.15 × 10^–10	0.68	7.7 × 10¹⁹	2.1 × 10²⁰
NIF	Laser ICF	2014	N140304	132	5.5	⋯	⋯	⋯	222	1.63 × 10^–10	11.86	6.8 × 10²⁰	3.8 × 10²¹
MagLIF	MagLIF	2014	z2613	133	2.0	⋯	⋯	⋯	0.56	1.38 × 10^–9	0.76	1.2 × 10²⁰	2.4 × 10²⁰
OMEGA	Laser ICF	2015	77068	134	3.6	⋯	⋯	⋯	56	6.6 × 10^–11	1.21	1.1 × 10²⁰	3.8 × 10²⁰
MagLIF	MagLIF	2015	z2850	133	2.8	⋯	⋯	⋯	0.6	1.62 × 10^–9	0.96	1.1 × 10²⁰	3.0 × 10²⁰
NIF	Laser ICF	2017	N170601	132	4.5	⋯	⋯	⋯	320	1.6 × 10^–10	16.78	1.2 × 10²¹	5.3 × 10²¹
NIF	Laser ICF	2017	N170827	132	4.5	⋯	⋯	⋯	360	1.54 × 10^–10	18.17	1.3 × 10²¹	5.7 × 10²¹
FIREX	Laser ICF	2019	40558	135	⋯	2.1	⋯	⋯	2	4 × 10^–10	0.79	1.2 × 10²⁰	2.5 × 10²⁰
NIF	Laser ICF	2019	N191007	35	4.52	⋯	⋯	⋯	206	1.51 × 10^–10	10.20	7.1 × 10²⁰	3.2 × 10²¹
NIF	Laser ICF	2020	N201101	136	4.61	⋯	⋯	⋯	319	1.18 × 10^–10	12.34	8.5 × 10²⁰	3.9 × 10²¹
NIF	Laser ICF	2020	N201122	136	4.65	⋯	⋯	⋯	297	1.37 × 10^–10	13.34	9.1 × 10²⁰	4.2 × 10²¹
NIF	Laser ICF	2021	N210207	136	5.23	⋯	⋯	⋯	339	1.07 × 10^–10	11.89	7.2 × 10²⁰	3.8 × 10²¹
NIF	Laser ICF	2021	N210220	136	5.13	⋯	⋯	⋯	371	1.35 × 10^–10	16.42	1.0 × 10²¹	5.2 × 10²¹
NIF	Laser ICF	2021	N210808	24	10.9	⋯	⋯	⋯	569	8.9 × 10^–11	16.60	4.8 × 10²⁰	5.2 × 10²¹

The ICF-capsule shell is modeled as a thin shell with thickness $Δ ≪ R$ ⁠, where R is the shell radius, as illustrated in Fig. 23. A fraction of the peak kinetic energy of the shell is assumed to be converted to thermal pressure in the hot spot at stagnation. An upper bound on τ is obtained based on the time it takes for the stagnated shell (at peak compression) to expand a distance of order its inner radius R_s. Significant 3D effects arising from Rayleigh–Taylor-instability spikes and bubbles at the interface of the shell and hot spot reduce the effective hot-spot volume by a “yield-over-clean” factor $Y O C^{μ}$ ⁠, where $μ = 0.5$ is inferred from two simulation databases.⁵⁷ With these and other simplifying assumptions, Betti et al.³⁴ obtain

n T τ (3 D) \approx 4 {[{(ρ R)}_{tot (n)}^{no α} T_{n}^{no α}]}^{0.8} Y O C^{μ} [atm s],

(57)

with measured total areal density ${(ρ R)}_{tot (n)}^{no α}$ in g cm⁻² and measured “burn-averaged” ion temperature $T_{n}^{no α}$ in keV. The superscript “ $no α$ ” refers to experimental measurements made when α heating is not an appreciable effect (and α heating is turned off in simulations). For ICF experiments without reported values of hot-spot pressure, Eq. (57) is used to plot achieved ICF values of Lawson parameters and triple products, where the unit [atm s] is multiplied by $6.333 \times 10^{20}$ keV m⁻³atm⁻¹ to convert to [m⁻³keV s]. Dividing the triple product by T gives the Lawson parameter $n τ$ ⁠.

2. Inferring Lawson parameter from inferred pressure and confinement dynamics

When the inferred stagnation pressure $p_{stag}$ and the duration of fuel stagnation $τ_{stag}$ are reported, the pressure times the confinement time τ can be calculated directly. However, following Christopherson et al.,⁵⁸ three adjustments are made to $τ_{stag}$ ⁠, which is defined as the full-width half-maximum (FWHM) of the neutron-emission history (i.e., “burn duration”), to obtain an approximation for τ. The first adjustment is that, for marginal ICF ignition, only alphas produced before bang time (time of maximum neutron production) are useful to ignite the hot spot because, afterward, the shell is expanding and the hot spot is cooling, reducing the reaction rate; this introduces a factor of 1/2. The second adjustment is that only a fraction of fusion alphas are absorbed by the hot spot; this factor is estimated to be 0.93. The third adjustment is that, to initiate a propagating burn of the surrounding fuel, an additional factor of 0.71 is applied to account for the dynamics of alpha heating of the cold shell. Applying these three corrections results in $τ \approx τ_{stag} / 3$ and

n T τ \approx \frac{1}{2} p_{stag} τ \approx \frac{1}{6} p_{stag} τ_{stag} .

(58)

The only exception to this approach is the FIREX experiment, for which we estimate the value of $p_{stag} τ$ directly from the reported values.

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

The ignition requirement derived in Sec. III F ignores several factors that increase the requirements for the ignition of an ICF capsule. We consider these effects to be incorporated in reductions to τ in Sec. IV B 2. Thus, no further adjustments are made to ${(n τ)}_{ig, hs}^{ICF}$ as defined in Eq. (22).

4. Differences between ICF and MCF

It is not straightforward to compare the achieved Lawson parameters and triple-product values between ICF and MCF. While a quantitative approach can be taken via the ignition parameter χ described in Ref. 34, the approach taken here is qualitative and is reflected in different $Q_{sci}^{MCF}$ and ${(n τ)}_{ig, hs}^{ICF}$ or ${(n T τ)}_{ig, hs}^{ICF}$ contours in Figs. 2, 3, and 16.

First, the achieved triple product for ICF is higher than for MCF in part because of two assumptions made in its inference. Following Ref. 34, we assume in ICF that there are no bremsstrahlung radiation losses due to trapping by the pusher (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 Lawson parameter and triple product.

Second, whereas $P_{ext}$ and $P_{abs}$ differ by only a factor of order unity in MCF,³⁹ they differ by a factor of $≳ 50$ in ICF (see Table III). This is due to the low conversion efficiency from applied laser energy to absorbed fuel energy. Thus, while both MCF³⁰ and ICF have achieved $Q_{sci} \sim 0.7$ ⁠, ICF has necessarily achieved a higher value of $Q_{fuel}$ compared to MCF.

Note further that the horizontal line representing ${(n T τ)}_{ig, hs}^{ICF}$ in Fig. 3 (corresponding to the $n T τ$ value of the contour at 4 keV) is at a higher value than the minimum $n T τ$ value of the corresponding contour in Fig. 16. This is due to the fact that T_i in laser ICF experiments (prior to the onset of significant fusion) is limited by the maximum implosion velocity at which the shell becomes unstable, corresponding to a maximum T_i of about 4 keV. Thus, the marginal onset of ignition corresponds to the required $n T τ$ value at approximately 4 keV. In the case of NIF N210808, which exceeded the threshold for the onset of ignition,²⁴ T_i increased due to self-heating and τ decreased because of the increased pressure. These effects resulted in a triple product equal to the previous record non-ignition result, yet clearly in the ignition regime, which is visible in Fig. 16. For this reason, NIF N210808 does not appear in Fig. 3, illustrating an additional limitation of the triple-product metric in the context of ICF.

C. MIF/Z-pinch methodology

1. MagLIF

The Magnetized Liner Inertial Fusion (MagLIF) experiment⁵⁹ compresses a cylindrical liner surrounding a pre-heated and axially pre-magnetized plasma. The Z-machine at Sandia National Laboratory supplies a large current pulse to the liner along its long axis, compressing it in the radial direction.

While the solid liner makes diagnosing MagLIF plasmas more difficult, it is still possible to extract the parameters needed to estimate the Lawson parameter and triple product. The burn-averaged T_i at stagnation is measured by neutron time-of-flight diagnostics. The spatial configuration of the plasma column at stagnation is imaged from emitted x-rays. From this spatial configuration and a model of x-ray emission, the effective fuel radius is inferred. The stagnation pressure is inferred from a combination of diagnostic signatures. Given the plasma volume, burn duration, and temperature, the pressure was inferred by setting the pressure and mix levels to simultaneously match the x-ray yield and neutron yield. In the emission model used to determine the spatial extent of the stagnated plasma, the pressure in the stagnated fuel is assumed to be spatially constant and the temperature and density profiles are assumed to be inverse to each other.⁶⁰ For our purposes, we infer an average n_i from the stagnation pressure and the measured burn-averaged T_i.

Finally, the burn time, the duration during which the fuel assembly is inertially confined and hard x rays (surrogates for fusion neutrons) are emitted, is measured. This duration is an upper bound on τ, and in practice, τ is estimated to be equal to it. Data for MagLIF are shown in Table VII and plotted in Figs. 2, 3, and 16.

2. Z pinch

Z-pinch experiments were one of the earliest approaches to fusion because no external magnetic field is required for confinement. This simplifies the experimental setup and reduces costs. Figure 24 shows a representative diagram of a Z-pinch plasma. While fusion neutrons were detected in some of the earliest Z-pinch experiments, those fusion reactions were found to be the result of plasma instabilities generating non-thermal beam-target fusion events (see pp. 91–93 of Ref. 61), which would not scale up to energy breakeven. More recently, however, stabilized Z-pinch experiments have provided evidence of sustained thermonuclear neutron production.^62,63

FIG. 24.

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A representation of a Z-pinch plasma of length L, effective radius r₀, and electrical current I. V_p is the voltage difference between the left and right sides of the plasma.

Z-pinch plasmas exhibit profile effects perpendicular to the direction of current flow so the profile considerations discussed in Sec. IV A apply to Z pinches as well. The radial density profile of Z pinches is typically described by a Bennett-type profile⁶⁴ of the form $n (r) = n_{0} / {[1 + {(r / r_{0})}^{2}]}^{2}$ and illustrated in Fig. 25.

FIG. 25.

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Bennett-type density profile. In contrast to the parabolic profiles, the plasma extends beyond the effective radius r₀.

Assuming $T = T_{i} = T_{e}, n = n_{i} = n_{e}$ ⁠, and a uniform profile for the plasma temperature, the thermal energy of a Z-pinch plasma can be estimated as

\begin{matrix} W_{p} = & 3 T \int_{V} n (r) d^{3} r \\ = & 3 L T \int_{0}^{\infty} n (r) 2 π r d r \\ = & 6 L π n_{0} T \int_{0}^{\infty} \frac{r}{{(1 + {(r / r_{0})}^{2})}^{2}} d r \\ = & 3 π r_{0}^{2} L n_{0} T . \end{matrix}

(59)

The power applied is

P_{abs} = I V_{p},

(60)

where I is the Z-pinch current and V_p is the voltage across the plasma driving the current along the long axis. Assuming no self-heating and that thermal conduction is the primary source of energy loss, $τ_{E}^{*}$ for the stabilized Z-pinch is

τ_{E}^{*} = \frac{3 π r_{0}^{2} L n_{0} T_{0}}{I V_{p}},

(61)

and the Lawson parameter for a stabilized Z-pinch is

n_{0} τ_{E}^{*} = \frac{3 π r_{0}^{2} L n_{0}^{2} T_{0}}{I V_{p}} .

(62)

However, in practice, V_p may not be measured directly, and the voltage across the power supply driving the Z-pinch may overestimate V_p. Therefore, evaluations of $τ_{E}^{*}$ that substitute the power supply voltage for V_p (as done for FuZE^62,63) provide only a lower bound on $τ_{E}^{*}$ ⁠. An upper bound on $τ_{E}^{*}$ is the flow-through time of the Z-pinch. Our reported value is the lower of the two.

In other Z-pinch approaches such as the dense plasma focus (DPF), fusion yields occur from a combination of non-Maxwellian ion energy distributions and thermal ion populations.⁶⁵ Because thermal temperatures and $τ_{E}^{*}$ are typically not well characterized in such approaches, it is not feasible to report a reliable, achieved Lawson parameter or triple product. Furthermore, fusion concepts with strong beam-target components may not be scalable to $Q_{fuel} > 1$ ⁠.²⁸

3. Other MIF approaches

For other MIF approaches,⁶⁶ e.g., liner or flux compression of FRCs or spheromaks, it is difficult to rigorously measure $τ_{E}^{*}$ due to limited access. A few attempts to quantify $τ_{E}^{*}$ based on measurable or calculable parameters, such as particle confinement time τ_N, have been proposed.⁵⁴ In particular, we estimate $τ_{E}^{*}$ of FRCs to be $τ_{N} / 3$ (for both MIF and MCF).

V. SUMMARY AND CONCLUSIONS

The combination of achieved Lawson parameter $n τ$ or $n τ_{E}$ and fuel temperature T of a thermonuclear-fusion concept is a rigorous scientific indicator of how close it is to energy breakeven and gain. In this work, we have compiled the achieved Lawson parameters and T of a large number of fusion experiments (past, present, and projected) from around the world. The data are provided in multiple tables and figures. Following Lawson's original work, we provided a detailed review, re-derivation, and extension of the mathematical expressions underlying the Lawson parameter (and the related triple product) and four ways of measuring energy gain (⁠ $Q_{fuel}, Q_{sci}, Q_{wp}$ ⁠, and $Q_{eng}$ ⁠) and explained the physical principles upon which these quantities are based. Because different fusion experiments report different observables, we explained precisely how we infer both electron and ion densities and temperatures and the various definitions of confinement time that are used in the Lawson-parameter and triple-product values that we report, including accounting for the effects of spatial profile shapes (through a peaking factor) and a range in the level of impurities in the plasma fuel. All data reported in this paper are based on the published literature or are expected to be published shortly.

The key results of this paper are encapsulated in Figs. 2, 3, and 16, which show that (1) tokamaks and laser-driven ICF have achieved the highest Lawson parameters, triple products, and $Q_{sci} \sim 0.7$ ⁠; (2) fusion concepts have demonstrated rapid advances in Lawson parameters and triple products early in their development but slow down as values approach what is needed for $Q_{sci} = 1$ ⁠; (3) private fusion companies pursuing alternate concepts are now exceeding the breakout performance of early tokamaks; and (4) at least three experiments may achieve $Q_{sci} > 1$ within the foreseeable future, i.e., NIF and SPARC in the 2020s and ITER by 2040.

The reason for item (2) in the preceding paragraph is commonly attributed to the fact that experimental facilities became extremely expensive (e.g., US$3.5B for NIF according to the U.S. Government Accountability Office, and exceeding US$25B for ITER) for making continued and required advances toward energy gain. However, there are two reasons that other approaches or experiments might potentially achieve commercially relevant energy breakeven and gain on a faster timescale. First, most of the other paths being pursued (i.e., privately funded development paths for tokamaks, stellarators, alternate concepts, and laser-driven ICF) have lower costs as a key objective, where experiments along the development path are envisioned to have much lower costs than NIF and ITER. Second, the mature fusion and plasma scientific understanding and computational tools, as well as many fusion-engineering technologies, developed over 65+ years of controlled-fusion research do not need to be reinvented and need only be leveraged in the development of the alternate and privately funded approaches.

High values of Lawson parameter and triple product, which are required for energy gain, are a necessary but not sufficient condition for commercial fusion energy. Additional necessary conditions include attractive economics and social acceptance, including but not limited to considerations of RAMI (reliability, accessibility, maintainability, and inspectability) and the ability to be licensed under an appropriate regulatory framework. These necessary conditions require additional technological attributes beyond high energy gain, e.g., (1) a fusion plasma core that is compatible with both surrounding materials and subsystems that survive the extreme fusion particle, heat, and radiation flux, and (2) a sustainable fuel cycle (e.g., tritium breeding, separation, and processing technologies for D-T fusion). Therefore, while this paper's primary objective is to explain and highlight the achieved Lawson parameters (and triple products) of many fusion concepts and experiments as a measure of fusion's progress toward energy breakeven and gain, these are not the only criteria for justifying the continued pursuit of and investment into a given fusion concept, including concepts using advanced fusion fuels.

ACKNOWLEDGMENTS

Most of the contributions of S.E.W. were performed while affiliated with Fusion Energy Base prior to joining ARPA-E. This work was funded by Fusion Energy Base and ARPA-E. We are grateful for feedback on drafts of this paper provided by Riccardo Betti, Rob Goldston, Rich Hawryluk, Omar Hurricane, Harry McLean, Dale Meade, Bob Mumgaard, Brian Nelson, Kyle Peterson, Uri Shumlak, and Glen Wurden. Responsibility for all content in the paper lies with the authors. Reference herein to any specific non-federal person or commercial entity, product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof or its contractors or subcontractors.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study. The extracted data from other publications and codes used to make the plots in this paper are available for download.²³

NOMENCLATURE

$E_{abs}$: Externally applied energy absorbed by the fuel
f_c: Energy fraction of fusion products in charged particles
n: The generic density used to refer to either ion or electron density when n_i = n_e
n_e: Electron density
$n_{e 0}$: Central electron density
n_i: Ion density
$n_{i 0}$: Central ion density
${(n T τ)}_{ig, hs}^{ICF}$: Temperature-dependent triple product required to achieve ICF hot-spot ignition
${(n τ)}_{ig, hs}^{ICF}$: Temperature-dependent Lawson parameter required to achieve ICF hot-spot ignition
p: Plasma thermal pressure
$P_{abs}$: Externally applied power absorbed by the fuel
P_B: Bremsstrahlung power
P_c: Fusion power emitted as charged particles
$P_{ext}$: Externally applied heating power
P_F: Fusion power
P_n: Fusion power emitted as neutrons
$P_{out}$: Sum of all power exiting the plasma
Q: Generic energy gain. For MCF, this can refer to $Q_{fuel}$ or $Q_{sci}$ ⁠. For ICF, this refers to $Q_{sci}$
$Q_{eng}$: Engineering gain. The ratio of electrical power to the grid to recirculating power
$Q_{fuel}$: Fuel gain. The ratio of fusion power to the power absorbed by the fuel
$⟨ Q_{fuel} ⟩$: Volume-averaged fuel gain in the case of non-uniform profiles
$Q_{sci}$: Scientific gain. The ratio of fusion power to externally applied heating power
$⟨ Q_{sci} ⟩$: The volume-averaged scientific gain in the case of non-uniform profiles
$Q_{wp}$: Wall-plug gain. The ratio of fusion power to input electrical power from the grid
S_B: Bremsstrahlung power density
S_c: Fusion power density in charged particles
S_F: Fusion power density
T: Generic temperature, used to refer to either ion or electron temperature when T_i = T_e
T_e: Electron temperature
$T_{e 0}$: Central electron temperature
T_i: Ion temperature
$T_{i 0}$: Central ion temperature
${⟨ T_{i} ⟩}_{n}$: Neutron-averaged ion temperature
V: Plasma volume
Z: Charge state of an ion
$Z_{eff}$: The effective value of the charge state. The factor by which bremsstrahlung is increased as compared to a hydrogenic plasma, see Eq. (42).
$\bar{Z}$: Mean charge state, i.e., the ratio of electron to ion density in a quasi-neutral plasma
ε_F: Total energy released per fusion reaction
$ε_{α}$: Energy released in α-particle per D-T fusion reaction
η: The efficiency of recapturing thermal energy after the confinement duration in Lawson's second scenario
$η_{abs}$: The efficiency of coupling externally applied power to the fuel
η_E: The efficiency of converting electrical recirculating power to externally applied heating power
$η_{elec}$: The efficiency of converting total output power to electricity
$η_{hs}$: The efficiency of coupling shell kinetic energy to hotspot thermal energy in laser ICF implosions
${⟨ σ v ⟩}_{i j}$: Temperature-dependent fusion reactivity between species i and j (cross section σ times the relative velocity v of ions averaged over a Maxwellian velocity distribution)
τ: Pulse duration
τ_E: Energy confinement time
$τ_{E}^{*}$: Modified energy confinement time, which accounts for transient heating, see Sec. IV A 6
$τ_{eff}$: Effective characteristic time combining pulse duration and energy confinement time, see Sec. III C

APPENDIX A: DATA TABLES

Table V provides numerical values of the data for tokamaks, spherical tokamaks, and stellarators. Table VI provides numerical values of the data for “alternate” MCF concepts, i.e., not tokamaks or stellarators. Table VII provides numerical values of the data for ICF and MIF experiments. We group lower-density and higher-density MIF approaches with MCF alternate concepts (Table VI) and ICF (Table VII), respectively.

APPENDIX B: EFFECT OF MITIGATING BREMSSTRAHLUNG LOSSES

If bremsstrahlung radiation losses are mitigated, e.g., in pulsed ICF²⁰ or MIF^66,137 approaches with an optically thick pusher,^138,139 then the $Q_{fuel}$ and $Q_{sci}$ contours of Figs. 12 and 14 can be modified. Figure 26 illustrates the effect of arbitrarily reducing P_B by a factor of 2, i.e., by replacing C_B with $C_{B} / 2$ in Eqs. (17) and (23).

FIG. 26.

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Contours of $Q_{fuel}$ plotted vs T and (a) $n τ_{E}$ and (b) $n T τ_{E}$ for D-T fusion (assuming $T = T_{i} = T_{e}$ ⁠). Dashed lines represent arbitrarily reducing bremsstrahlung losses by a factor of 2, i.e., replacing C_B by $C_{B} / 2$ in Eqs. (17) and (23).

APPENDIX C: LAWSON PARAMETERS FOR ADVANCED FUSION FUELS

The main body of this paper focuses on D-T fusion because it has the highest maximum reactivity occurring at the lowest temperature compared to all known fusion fuels. As a result, the required D-T Lawson parameters and triple products to reach high $Q_{fuel}$ are the lowest and most accessible. However, D-T fusion has two major drawbacks: (i) it produces 14-MeV neutrons that carry 80% of the fusion energy, and (ii) the tritium must be bred (because it does not occur abundantly in nature due to a 12.3-year half life) and be continuously processed and handled safely.

Advanced fuels, such as D-³He, D-D, and p-¹¹B, mitigate these drawbacks to different extents.¹⁴⁰ However, because their peak reactivities are all lower and occur at higher temperatures compared to D-T, the required Lawson parameters and triple products for these advanced fuels to achieve equivalent values of $Q_{fuel}$ are much higher.

Furthermore, at the high temperatures required for advanced fuels, relativistic bremsstrahlung effects become significant. We utilize the relativistic-correction approximation to Eq. (44) from Ref. 142,

P_{B} = C_{B} n_{e} T_{e}^{1 / 2} γ (Z_{eff}),

(C1)

where

γ (Z_{eff}) = Z_{eff} (1 + 1.78 t^{1.34}) + 2.12 t (1 + 1.1 t + t^{2} - 1.25 t^{2.5})

(C2)

and $t = T_{e} / m_{e} c^{2}$ ⁠.

To quantify the Lawson-parameter and triple-product requirements for advanced fuels with non-identical reactants and reaction products that are immediately removed from the plasma (e.g., D-³He and p-¹¹B without ash buildup or subsequent reactions), we first generalize the expression for $n τ_{E}$ [Eq. (17)] to account for the effect of relativistic bremsstrahlung and the reaction of two ion species with charge per ion Z₁ and Z₂, ion number densities n₁ and n₂, and relative densities $k_{1} = n_{1} / n_{e}$ and $k_{2} = n_{2} / n_{e}$ ⁠, respectively.

A more detailed treatment of advanced fuels would need to consider scenarios in which T_e < T_i and account for an additional term in the power-balance equation for ion energy transfer to electrons. Maintaining $T_{e} ≪ T_{i}$ has the advantage of reduced bremsstrahlung (especially at high T_i) and lower plasma pressure for a given T_i. The challenge of such a scenario is maintaining T_i > T_e for a sufficient duration of time and with acceptable additional input power. In this section, we only consider $T = T_{i} = T_{e}$ ⁠, except in the discussion of Fig. 28. Accounting for the above,

n_{e} τ_{E} = \frac{(3 T / 2) (k_{1} + k_{2} + 1)}{(f_{c} + Q_{fuel}^{- 1}) k_{1} k_{2} {⟨ σ v ⟩}_{1, 2} E_{1, 2} - C_{B} T^{1 / 2} γ (Z_{eff})},

(C3)

where $Z_{eff} = Σ_{j} n_{j} Z_{j}^{2} / n_{e}$ ⁠, and j is summed over the different reactant species.

The relative density for each ion species j that maximizes¹⁴² fusion power for a fixed value of $n_{e}^{2}$ is $k_{j} = 1 / (2 Z_{j})$ and $Z_{eff} = (Z_{1} + Z_{2}) / 2$ ⁠. Assuming this condition, Eq. (23) becomes

\begin{matrix} n_{e} T τ_{E} = \frac{(3 T^{2} / 2) [{(2 Z_{1})}^{- 1} + {(2 Z_{2})}^{- 1} + 1]}{(f_{c} + Q_{fuel}^{- 1}) {⟨ σ v ⟩}_{1, 2} E_{1, 2} / (4 Z_{1} Z_{2}) - C_{B} T^{1 / 2} γ (Z_{eff})}, \end{matrix}

(C4)

or equivalently,

\begin{matrix} n_{i} T τ_{E} = \frac{(3 T^{2} / 2) [{(2 Z_{1})}^{- 1} + {(2 Z_{2})}^{- 1}] [{(2 Z_{1})}^{- 1} + {(2 Z_{2})}^{- 1} + 1]}{(f_{c} + Q_{fuel}^{- 1}) {⟨ σ v ⟩}_{1, 2} E_{1, 2} / (4 Z_{1} Z_{2}) - C_{B} T^{1 / 2} γ (Z_{eff})}, \end{matrix}

(C5)

where we have multiplied both sides of Eq. (C4) by $(k_{1} + k_{2}) = {(2 Z_{1})}^{- 1} + {(2 Z_{2})}^{- 1}$ ⁠. This expression ignores synchrotron radiation losses, which may become important at the very high temperatures required to reach Lawson conditions for advanced fuels in magnetically confined systems.

1. D-³He

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

D +^{3} He \to α + p (18.3 MeV),

(C6)

is aneutronic, where the α is a ⁴He ion. However, ³He is not abundant on earth and must be bred via other reactions or mined from the moon, both of which involve additional complexity and cost. Also, D-³He will not be completely aneutronic because of D-D reactions. The requirement for ignition of D-³He ignoring side D-D reactions is $n_{i} T τ_{E}^{*} \geq 5.2 \times 10^{22}$ m⁻³keV s at 68 keV (see Fig. 27), 18 times higher than for D-T.

FIG. 27.

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Required (a) Lawson parameters and (b) triple products vs T_i to achieve the indicated values of $Q_{fuel}$ for D-³He (assuming $T = T_{e} = T_{i}$ ⁠).

2. p-¹¹B

The p-¹¹B fusion reaction has the advantage that its reactants are abundant on earth, and the reaction products are three electrically charged α particles, potentially allowing for direct energy conversion to electricity. However, this reaction requires temperatures around 100 keV at which bremsstrahlung radiation losses per unit volume exceed fusion power density, and ignition is not possible for a p-¹¹B plasma where T_e = T_i, as shown in Fig. 28, which uses the parametrized p-¹¹B fusion reactivity from Ref. 26. The boron and proton concentrations are set to maximize fusion power for a fixed electron density as described earlier in this section. Also shown is the effect of reduced bremsstrahlung if T_e is maintained at levels below T_i. We are neglecting the issue of the ion-electron thermal equilibration time here. Figure 29 shows that only modest values of $Q_{fuel}$ are physically possible for T_e = T_i, at triple products three orders of magnitude higher than that of D-T.

FIG. 28.

View large Download slide

Charged-particle fusion power density P_c (purple line) and bremsstrahlung power density P_B for various ratios of $T_{e} / T_{i}$ vs T_i for p-¹¹B, showing that P_B always exceeds P_c when $T_{e} ≳ T_{i} / 3$ ⁠. This plot uses the parameterized p-¹¹B reactivity in Ref. 26. Updated, higher p-¹¹B fusion cross sections¹⁴³ suggest that ignition may be possible for p-¹¹B.¹⁴¹

FIG. 29.

View large Download slide

Required (a) Lawson parameters and (b) triple products vs T_i to achieve values of $Q_{fuel}$ assuming $T = T_{i} = T_{e}$ for p-¹¹B based on the p-¹¹B fusion reactivity from Ref. 26.

However, recent work¹⁴³ points to a higher reactivity, and given certain assumptions, high- $Q_{fuel}$ operation up to and including ignition may be theoretically possible.

3. Fully catalyzed D-D

The D-D fusion reaction has the advantage that its sole reactant is abundant on earth. In the fully catalyzed D-D reaction,^144,145 the T and ³He produced as reaction products undergo subsequent reactions with D, releasing more energy. The reaction paths are

D + D \to T (1.01 MeV) + p (3.02 MeV),

(C7)

D + T \to α (3.5 MeV) + n (14.1 MeV),

(C8)

D + D \to^{3} He (0.82 MeV) + n (2.45 MeV),

(C9)

D +^{3} He \to α (3.6 MeV) + p (14.7 MeV),

(C10)

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

Note that there are other forms of “catalyzed D-D” which go by different names in different contexts. For example, extraction of tritium before the subsequent D-T reaction occurs is sometimes called “ $^{3} He$ double-catalyzed D-D.”¹⁴⁵ Here, we only consider the steady-state reaction path where $^{3}$ He and T react with D at the same rate as they are created in each branch of the D-D reaction. Furthermore, we assume an idealized scenario without synchrotron radiation and that the “ash” α particles and protons immediately exit after depositing their energy and comprise a negligible fraction of ions in the plasma. Finally, we assume that D is added at the same rate as it is consumed and that $T = T_{i} = T_{e}$ ⁠.

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

n_{i} = n_{D} + n_{^{3} He} + n_{T},

(C11)

and the electron density is

n_{e} = n_{D} + 2 n_{^{3} He} + n_{T} .

(C12)

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

\frac{1}{2} n_{D}^{2} {⟨ σ v ⟩}_{DD,^{3} He} = n_{D} n_{^{3} He} {⟨ σ v ⟩}_{D^{3} He},

(C13)

\frac{1}{2} n_{D}^{2} {⟨ σ v ⟩}_{DD, T} = n_{D} n_{T} {⟨ σ v ⟩}_{DT} .

(C14)

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

n_{^{3} He} = \frac{1}{2} \frac{{⟨ σ v ⟩}_{DD,^{3} He}}{{⟨ σ v ⟩}_{D^{3} He}} n_{D},

(C15)

n_{T} = \frac{1}{2} \frac{{⟨ σ v ⟩}_{DD, T}}{{⟨ σ v ⟩}_{DT}} n_{D} .

(C16)

The total fusion power density is the sum of the power released in its four constituent reactions,

\begin{matrix} S_{F} = \frac{n_{D}^{2}}{2} {⟨ σ v ⟩}_{DD,^{3} He} E_{DD,^{3} He} + n_{D} n_{^{3} He} {⟨ σ v ⟩}_{D^{3} He} E_{D^{3} He} \\ + \frac{n_{D}^{2}}{2} {⟨ σ v ⟩}_{DD, T} E_{DD, T} + n_{D} n_{T} {⟨ σ v ⟩}_{DT} E_{DT} . \end{matrix}

(C17)

The bremsstrahlung power density is

S_{B} = C_{B} n_{e}^{2} T_{e}^{1 / 2} γ (Z_{eff}),

(C18)

and from Eq. (42),

Z_{eff} = \frac{1}{n_{e}} (n_{D} + 4 n_{^{3} He} + n_{T}) .

(C19)

The power lost to thermal conduction per unit volume is

S_{κ} = \frac{(3 / 2) T (n_{e} + n_{i})}{τ_{E}} .

(C20)

Defining χ_h and χ_t as the number density ratios of $n_{^{3} He}$ to $n_{D}$ and $n_{T}$ to $n_{D}$ respectively,

χ_{h} \equiv \frac{n_{^{3} He}}{n_{D}} = \frac{1}{2} \frac{{⟨ σ v ⟩}_{DD,^{3} He}}{{⟨ σ v ⟩}_{D^{3} He}},

(C21)

χ_{t} \equiv \frac{n_{T}}{n_{D}} = \frac{1}{2} \frac{{⟨ σ v ⟩}_{DD, T}}{{⟨ σ v ⟩}_{DT}} .

(C22)

From the steady-state power balance of Eq. (12) and the above, the Lawson parameter required to achieve fuel gain $Q_{fuel}$ at T_i is

\begin{matrix} n_{i} τ_{E} = \frac{T (3 + 9 χ_{h} / 2 + 3 χ_{t}) (1 + χ_{h} + χ_{t})}{(f_{c} + Q_{fuel}^{- 1}) {⟨ σ v ⟩}_{DD} E_{tot} / 4 - C_{B} {(1 + 2 χ_{h} + χ_{t})}^{2} T^{1 / 2} γ (Z_{eff})} \end{matrix},

(C23)

with

Z_{eff} = (1 + 4 χ_{h} + χ_{t}) / (1 + 2 χ_{h} + χ_{t}),

(C24)

{⟨ σ v ⟩}_{DD} = {⟨ σ v ⟩}_{DD,^{3} He} + {⟨ σ v ⟩}_{DD, T},

(C25)

and

E_{tot} = E_{DD, T} + E_{DT} + E_{DD,^{3} He} + E_{D^{3} He} .

(C26)

The requirement for ignition of catalyzed D-D is $n_{i} T τ_{E}^{*} \geq 1.1 \times 10^{23}$ m^{− 3}keV s at T = 52 keV (see Fig. 30), 38 times higher than required for D-T.

FIG. 30.

View large Download slide

Required (a) Lawson parameters and (b) triple products vs T to achieve the indicated values of $Q_{fuel}$ for catalyzed D-D (assuming $T = T_{e} = T_{i}$ ⁠).

4. Advanced-fuels summary

The extreme requirements for advanced fuels compared to D-T are illustrated in Fig. 31, which shows the required Lawson parameters and triple products vs T_i required to achieve $Q_{fuel} = \infty$ (solid lines), $Q_{fuel} = 1$ (dashed lines), and $Q_{fuel} = 0.5$ (dotted line, p-¹¹B only) for the reactions discussed in this appendix. For all reactions, T_i = T_e is assumed. For p-¹¹B, neither fuel breakeven nor ignition appears possible when T_i = T_e.

FIG. 31.

View large Download slide

Required (a) Lawson parameters and (b) triple products vs T to achieve $Q_{fuel} = \infty$ (solid lines), $Q_{fuel} = 1$ (dashed lines), and $Q_{fuel} = 0.5$ (dotted line, p-¹¹B only) for the indicated fuels, assuming $T = T_{e} = T_{i}$ ⁠. Neither fuel breakeven (⁠ $Q_{fuel} = 1$ ⁠) nor ignition (⁠ $Q = \infty$ ⁠) appears to be possible for p-¹¹B if T_e = T_i.

APPENDIX D: CONCEPTUAL POWER PLANTS WITH NON-ELECTRICAL RECIRCULATING POWER

Some fusion designs do not recirculate electrical power but rather capture a portion of the thermal $P_{out}$ via mechanical means and use it with efficiency η_r as $P_{ext}$ ⁠. This is illustrated in Fig. 32. An example of this approach is the compression of plasma by an imploding liquid-metal vortex driven by compressed-gas pistons,¹⁴⁶ which recapture a fraction of $P_{out}$ to re-energize the pistons with efficiency η_r for the next pulse. If we define engineering gain in this system as the ratio of electrical power to the grid to recirculating mechanical power, then $Q_{eng} = P_{grid}^{E} / P_{r}$ ⁠, and it is straightforward to show that

Q_{eng} = η_{elec} η_{r} (Q_{sci} + 1) - η_{elec} .

(D1)

FIG. 32.

View large Download slide

Conceptual schematic of a fusion power plant that recirculates mechanical power with efficiency η_r. In this system, engineering gain is defined as $Q_{eng} = P_{grid}^{E} / P_{r}$ ⁠.

FIG. 33.

View large Download slide

Required combinations of $Q_{sci}$ and η_r in the system shown in Fig. 32 to permit values of $Q_{eng}$ ranging from zero (i.e., $P_{grid}^{E} = 0$ ⁠) to ten (i.e., $P_{grid}^{E} = 10 P_{r}$ ⁠), where $η_{elec} = 0.4$ is assumed. Note that at high η_r, net electricity (⁠ $Q_{eng} > 0$ ⁠) is possible with $Q_{sci} < 1$ even though $η_{elec}$ is only 0.4, corresponding to D-T fuel and a standard steam cycle.

This approach has the advantage that net electricity can be generated (⁠ $Q_{eng} > 0$ ⁠) with $Q_{sci} < 1$ if the recirculating efficiency η_r is sufficiently high, without advanced fuels or direct conversion (i.e., assuming D-T fuel and a standard steam cycle $η_{elec} = 0.4$ ⁠). This is illustrated in Fig. 33 and is due to the fact that the recirculating power bypasses the conversion to electricity.

APPENDIX E: RELATIONSHIPS BETWEEN PEAK AND VOLUME-AVERAGED QUANTITIES FOR MCF

In this appendix, we describe the equations used for volume averaging of plasma parameters for MCF, to relate peak values (variables denoted with a subscript of ‘0’) to their volume-averaged quantities (denoted with $⟨ \dots ⟩$ ⁠) to, ultimately, relating the peak $n_{0} T_{0} τ_{E}$ to an overall $Q_{fuel}$ that accounts for profile effects in n and T. We denote this as $⟨ Q ⟩$ ⁠, even though $Q_{fuel}$ is inherently a volume-averaged quantity.

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

⟨ f ⟩ = \frac{\int \int_{S} f (x, y) d S}{A},

(E1)

where $A = \int \int_{S} d S$ is the area (inside the separatrix or last closed flux surface), and axisymmetry is assumed.

1. Cylinder or large-aspect-ratio torus

For a circular cylinder with radius a or a torus with an inverse aspect ratio $ε = a / R ≪ 1$ (where a and R are the minor and major radii, respectively), and $f (x, y) = f (r)$ (i.e., circular, concentric flux surfaces with no Shafranov shift), Eq. (E1) becomes

⟨ f ⟩ = \frac{2 \int_{0}^{a} r f (r) d r}{a^{2}} .

(E2)

For the particular profile,

f (x, y) = f (r) = f_{0} {[1 - {(\frac{r}{a})}^{2}]}^{S_{f}},

(E3)

where $r = {(x^{2} + y^{2})}^{1 / 2}$ ⁠, Eq. (E2) becomes

⟨ f ⟩ = \frac{2 f_{0} \int_{0}^{a} r {[1 - {(r / a)}^{2}]}^{S_{f}} d r}{a^{2}} = \frac{f_{0}}{1 + S_{f}} .

(E4)

If $n = n_{0} {[1 - {(r / a)}^{2}]}^{ν_{n}}$ and $T = T_{0} {[1 - {(r / a)}^{2}]}^{ν_{T}}$ ⁠, then it follows that

⟨ n T ⟩ = \frac{n_{0} T_{0}}{1 + ν_{n} + ν_{T}} .

(E5)

2. Arbitrary aspect-ratio torus

For an up/down-symmetric torus with arbitrary ε and f(x, y), Eq. (E1) becomes

⟨ f ⟩ = \frac{\int_{R - a}^{R + a} \int_{0}^{h (x)} x f (x, y) d y d x}{\int_{R - a}^{R + a} x h (x) d x},

(E6)

where h(x) is the half-height of the plasma cross section at horizontal position x as shown in Fig. 34. If h(x) and $f (x, y) = f_{0} \bar{f} (x, y)$ are specified, where f₀ is the peak value of f and $max (\bar{f}) = 1$ ⁠, then Eq. (E6) can be numerically integrated to provide a quantitative relationship between $⟨ f ⟩$ and f₀. The function h(x) allows for any plasma cross-sectional shape, e.g., the highly elongated, D-shaped flux surfaces of high-performance tokamaks.

FIG. 34.

View large Download slide

Cross section of an up-down symmetric torus with an upper boundary defined by h(x) (shown here as a semi-circle).

For the particular case of an up/down-symmetric torus with circular cross section and f(x, y) as given in Eq. (E3), where $r = {[{(x - R)}^{2} + y^{2})}^{1 / 2}$ ⁠, Eq. (E6) becomes

⟨ f ⟩ = \frac{f_{0} \int_{R - a}^{R + a} \int_{0}^{h (x)} x {1 - [{(x - R)}^{2} + y^{2}] / a^{2}}^{S_{f}} d y d x}{\int_{R - a}^{R + a} x h (x) d x},

(E7)

where $h (x) = {[a^{2} - {(x - R)}^{2}]}^{1 / 2}$ ⁠. Again, this can be integrated numerically to provide a relationship between $⟨ f ⟩$ and f₀.

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Progress toward fusion energy breakeven and gain as measured against the Lawson criterion

I. INTRODUCTION

II. PLOTS

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

A. Lawson's first insight: Ideal ignition temperature

B. Lawson's second insight: Dependence of fuel energy gain on T and nτ

C. Extending Lawson's second scenario: Effect of self-heating and relationship between characteristic times τ and τE

D. Scientific energy gain and breakeven

E. Idealized, steady-state MCF: τE≪τ

F. Idealized ICF: τ≪τE

G. Fusion triple product and “p-tau”

H. Engineering gain

IV. METHODOLOGIES FOR INFERRING LAWSON PARAMETER AND TEMPERATURE

A. MCF methodology

1. Effect of temporal profiles

2. Effect of spatial profiles

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

4. Inferring peak from volume-averaged values

5. Inferring ion quantities from electron quantities

6. Accounting for transient heating

B. ICF methodology

1. Inferring Lawson parameter and triple product without reported inferred pressure

2. Inferring Lawson parameter from inferred pressure and confinement dynamics

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

4. Differences between ICF and MCF

C. MIF/Z-pinch methodology

1. MagLIF

2. Z pinch

3. Other MIF approaches

V. SUMMARY AND CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

NOMENCLATURE

APPENDIX A: DATA TABLES

APPENDIX B: EFFECT OF MITIGATING BREMSSTRAHLUNG LOSSES

APPENDIX C: LAWSON PARAMETERS FOR ADVANCED FUSION FUELS

1. D-3He

2. p-11B

3. Fully catalyzed D-D

4. Advanced-fuels summary

APPENDIX D: CONCEPTUAL POWER PLANTS WITH NON-ELECTRICAL RECIRCULATING POWER

APPENDIX E: RELATIONSHIPS BETWEEN PEAK AND VOLUME-AVERAGED QUANTITIES FOR MCF

1. Cylinder or large-aspect-ratio torus

2. Arbitrary aspect-ratio torus

References

B. Lawson's second insight: Dependence of fuel energy gain on T and $n$ τ

C. Extending Lawson's second scenario: Effect of self-heating and relationship between characteristic times τ and τ_E

E. Idealized, steady-state MCF: $τ_{E} ≪ τ$

F. Idealized ICF: $τ ≪ τ_{E}$

1. D-³He

2. p-¹¹B