# Why Doesn't the Stefan-Bolzmann Law Deter Controlled Fusion Engineers From Seeking Extreme Temperatures To Maximize the $nτT$ Triple Product?

The triple product figure of merit for controlled nuclear fusion, nτT (density * time * temperature) would seem to favor high density approaches over high temperature. Radiative loss goes up as the fourth power of temperature due to the Stefan-Bolzmann law. This means a fusion plasma's time is inversely proportional to the fourth power of temperature.

Yet we see many if not most of the major efforts to achieve controlled nuclear fusion as relatively uninterested in increasing density, inertial confinement being the exception. Indeed, magnetic confinement systems have received the most funding, yet confine their plasmas for extended periods of time in direct communication with a vacuum.

Is there some reason optimizing for density isn't the dominant approach?

• I can have liquid hydrogen sit around in a cryo vessel all day long without anything happening pretty much ever. Jul 22 at 18:44
• Take the density ratio of liquid hydrogen to tokamak plasma. Multiply that ratio times the hydrogen liquefaction temperature (23 K). What temperature do you get? Jul 22 at 21:57

The Stefan-Bolzmann law only applies to opaque objects. The plasma in a tokamak is almost perfectly transparent.

If the Stefan-Bolzmann law applied to the plasma in ITER, it would have the luminosity of a star! 100 million kelvin to the fourth power is a huge number. Radiative loss would be the least of its problems ツ

The energy losses don't scale as $$T^4$$ - that would be appropriate for a blackbody, but the fusion plasma is optically thin, does not absorb incident radiation and therefore cannot be a blackbody.

At temperatures of $$\sim 10^8$$K, the dominant radiation loss mechanism will be optically thin thermal bremsstrahlung and the total power scales only as $$n^2 T^{1/2}$$.

That favours low densities, and when you also factor in that the fusion reaction rate scales as something like $$n^2 T^2$$ (according to the first source you cite), then you can see that this points you towards having a high $$T$$ rather than a high $$n$$, given that the product $$nT$$ is fixed by the pressure containment properties of the reactor.

Yet we see many if not most of the major efforts to achieve controlled nuclear fusion as relatively uninterested in increasing density, inertial confinement being the exception

I'm not sure you could call it "the exception" considering it's basically one of two major approaches to fusion and has received many billions of dollars in funding, second only to ITER.

The early papers on the topic, notably Nuckolls famous Nature article of 1972, suggested that implosion energies on the order of 10 kJ would be needed to reach breakeven conditions. KMS Fusion started getting neutrons the next year at ~1 kJ and so everyone was happy.

But then it simply refused to scale. All attempts to move from the 1 kJ to 10 kJ regime failed to improve the yields, and the 10k Shiva was a failure by any measure. From that point on we've been chasing one problem after another and we're still not there after another 45 years and 2.5 orders of magnitude more power.

The problem is that the plasma simply does not want to be compressed. You are compressing a light gas/liquid/frozen layer onto a near vacuum and that's basically a formula for R-T instabilities and shock-driven problems and all sorts of other issues. The result is that you get bits of the fuel driving into the center before others and the spherical compression you need to reach the required density is extremely difficult to actually achieve.

These issues can be overcome with additional compression, but the testing in Halite/Centurion suggested that 100 MJ of energy would be needed and that is well beyond anything we know how to build.

There are "middle road" approaches like MIF and MAGLIF that attempt to reach moderate density for longer times, but they have been beset by instabilities and have proven rather unimpressive in practice. Nevertheless, a number of private companies like General Fusion continue to predict practical fusion any day now using these methods.