Energy Principles of Scientific Breakeven in an Inertial Fusion Experiment

Fusion “scientific breakeven” (i.e., unity target gain $G_{\text{target}}$, total fusion energy out $>$ laser energy input) has been achieved for the first time (here, $G_{\text{target}}\sim 1.5$). This Letter reports on the physics principles of the design changes that led to the first controlled fusion experiment, using laser indirect drive, on the National Ignition Facility to produce target gain greater than unity and exceeded the previously obtained conditions needed for ignition by the Lawson criterion. Key elements of the success came from reducing “coast time” (the time duration between the end of the laser pulse and implosion peak compression) and maximizing the internal energy delivered to the “hot spot” (the yield producing part of the fusion fuel). The link between coast time and maximally efficient conversion of kinetic energy into internal energy is explained. The energetics consequences of asymmetry and hydrodynamic-induced mixing were part of high-yield big radius implosion design experimental and design strategy. Herein, it is shown how asymmetry and mixing consolidate into one key relationship. It is shown that mixing distills into a kinetic energy cost similar to the impact of implosion asymmetry, shifting the threshold for ignition to higher implosion kinetic energy—a factor not normally included in most statements of the generalized Lawson criterion, but the key needed modifications clearly emerge.

Figure 1

A plot of $G_{\mathrm{fuel}}$ estimated from $Y_{\text{total}}/KE$ from theory (solid black line), ensemble simulations database [25] using the hydra code [26] of N210808-like implosions (smaller blue dots), and NIF DT implosion experiments (larger gray dots). Error bars suppressed for clarity but are $\sim 25\%$ on the abscissa and $\sim 15\%$ on the ordinate. Fuel energy gain $G_{\mathrm{fuel}}\sim 160$ resulted from NIF experiment N221204 [13, 14, 15] (exceeded by experiment N220729, not shown), with the next highest, $G_{\mathrm{fuel}}\sim 70$ from N210808 already being in the ignition regime [2, 8, 9, 10]. Unity $G_{\mathrm{fuel}}$ was achieved many years ago [3], when the first indications of alpha-heating feedback were observed. $G_{\mathrm{fuel}}\sim 5$ corresponds to a burning plasma [5, 6, 7], where in the plot above the curve begins to notably curve upward. Thermonuclear instability, causing an explosive increase in $T$ and, therefore, $G_{\mathrm{fuel}}$ is noted on the plot by the vertical dashed line.

Figure 2

Left: calculated yield amplification $Y_{\mathrm{amp}}$ from a numerical model [50] applied to experiment N210808 is plotted against implosion peak kinetic energy for the cases of no mixing into the hot spot ($\overline{Z^2}=1$), moderate mixing ($\overline{Z^2}=1.2$), and higher levels of mixing ($\overline{Z^2}=1.4$). The threshold for significant yield amplification moves to higher levels of KE with increased mixing ($\overline{Z}$). Right: The offset in KE with increased mixing appears to be correspond to the expectation based upon Eq. (9), since the calculated curves overlay if the KE is scaled by ${\overline{Z^2}}^{0.3}$.

Figure 3

Left: calculated yield amplification $Y_{\mathrm{amp}}$ from a numerical model based upon experiment N210808 is plotted against thermal temperature amplification for the cases of no mixing into the hot spot ($\overline{Z^2}=1$), moderate mixing ($\overline{Z^2}=1.2$), and higher levels of mixing ($\overline{Z^2}=1.4$). With more mix, a higher level of yield amplification is required to achieve a fixed level of thermal temperature amplification $T_{\mathrm{amp}}$, thus making ignition harder to achieve. Right: The curves overlay if $Y_{\mathrm{amp}}$ is scaled by $\overline{Z^2}$.

Figure 4

Various published Lawson-like criteria for ICF ignition are plotted with the Table 2 correction for mixing included. Left: The ${\mathrm{GLC}}_L=(\overline{p}/420\mathrm{Gbar})({\overline{R}}_{\mathrm{hs}}/50\mathrm{\mu }\mathrm{m})\ge 1$ criterion [51] (solid curve) becomes $(\overline{p}/420\mathrm{Gbar})({\overline{R}}_{\mathrm{hs}}/50\mathrm{\mu }\mathrm{m})\ge 1+0.24\ln (\overline{Z^2})$ (dashed and dotted curves). In this criterion, $\overline{p}$ and ${\overline{R}}_{\mathrm{hs}}$ are the burn and time-average hot-spot pressure and radius, respectively. Right: The criterion of Ref. [40], which originally included the effect of mixing in the theory, is also shown in the companion Letter [13]. A key feature to note in the right frame is the greatly increased burn-averaged $T_{\mathrm{th}}$ at nearly fixed burn-averaged $p\tau$, as the series of experiments passes the ignition criterion. Gray points are all NIF DT implosion data, while red points are from the Hybrid-E series of experiments.

Figure 5

A subset of fusion yields from DT implosion experiments on the NIF that illustrate the dates of key steps. The different implosion designs listed in the key are explained in detail in Refs. [2, 13], and references therein.