The fuel held together only by its inertia

The product a fusion plasma has to beat found that deuterium and tritium burn when their density times their confinement time passes a value of about $2 \times 10^{20}$ s/m³, and that nothing in the requirement says how the plasma should be held. One end of that line is occupied by magnets. The other end needs no holding at all. Crush a small amount of fuel so hard that it reacts faster than it can expand, and it will burn while flying apart, with the time set by nothing except how long the fuel’s own inertia takes to let go.

Light has a pressure introduced the push of light, $I/c$ on an absorbing surface, and spent most of its length showing how small that is: a tiny fraction of a pascal from sunlight. Laser fusion looks like the place where this pressure finally matters, with the largest lasers in the world pointed at a capsule. It does matter, in a way that seems perverse. Light’s pressure is far too weak to do the job, and the machine works by converting the light into something a thousand times worse at carrying energy per unit of momentum.

The numbers that follow from these ideas determine a capsule’s size, a laser’s energy and the result of the first shot to pass break-even.

A burn that depends on one number #

Take a ball of fuel at some density $\rho$ and radius $R$ and heat it to burning temperature. It will fly apart, starting at its surface, with a rarefaction moving in at the speed of sound $c$. The fuel is effectively gone in a time of about $R/4c$, the disassembly time, with the factor of a quarter coming from the volume being mostly near the surface.

The rate of burning goes as the density of fuel times the reactivity. So the fraction burned in the disassembly time is proportional to $\rho R$, with a coefficient set by temperature alone. Allowing for the fuel depleting as it burns gives a neat form,

$$f = \frac{\rho R}{\rho R + H},$$

with $H \approx 8 m c / \langle\sigma v\rangle$, where $m$ is the average mass per ion. At 30 keV that expression gives about 7.6 g/cm², and detailed calculations give a value of about 7.

The density and the radius do not appear separately, only their product, the areal density, which is the mass of fuel per unit area along a line from the centre to the outside. The product is also how much material an escaping particle must cross. That is a further reason it matters: an alpha particle at 3.5 MeV deposits its energy within about 0.3 g/cm², so a ball with more than that heats itself.

A useful burn asks for about a third of the fuel, so $\rho R$ near 3 g/cm². Solid deuterium–tritium ice has a density of 0.25 g/cm³, and at that density the radius would have to be twelve centimetres. The same calculation says exactly how much fuel that is.

Why the fuel has to be crushed #

The mass of a sphere at a fixed $\rho R$ follows from a few lines of algebra:

$$M = \frac{4\pi}{3}\rho R^3 = \frac{4\pi}{3}\frac{(\rho R)^3}{\rho^2}.$$

The cube of the requirement is fixed, so mass falls as the inverse square of the density. This is the whole argument for compression.

Uncompressed, the fuel weighs 1.8 kg and would release about 44 kilotonnes when a third of it burned, about three times the Hiroshima bomb. That is a weapon. Compress it a thousand times, and the same areal density takes 1.8 milligrams, releasing 185 MJ: a quantity that could plausibly be contained in a chamber several metres across, and a tiny fraction of which could usefully run a power plant several times a second. John Nuckolls, Lowell Wood, Albert Thiessen and George Zimmerman published this argument in Nature in 1972, in a paper that started the field in public, after a decade of classified work.

It is also the reason why a laser could work at all, given that lasers deliver megajoules and not gigajoules. All of the unlikely features of inertial fusion come from the desire to reach that factor of a thousand. Getting the fuel there is a hard problem in several different ways, starting with what compressing solid fuel costs.

Cold is cheap #

Cold matter resists compression through the exclusion principle, by the Fermi pressure that the pressure that is not a temperature introduced and the mass no cold matter can hold up followed to a white dwarf’s limit. The same pressure resists a capsule, and the same formula is used, since a capsule’s fuel at a thousand times solid density is more than a hundred times denser than lead, with its electrons degenerate. For deuterium–tritium, the pressure is about $2.2\,\rho^{5/3}$ megabar with $\rho$ in g/cm³, and it is a very stiff pressure. Yet the energy it costs is modest, since it rises only slowly with density.

Heating to fusion temperatures is more expensive by more than a factor of forty. It follows that a capsule should not be heated overall. Instead, nearly all the fuel is compressed as cold as possible, staying close to its Fermi pressure, and only a small central part is made hot. This is the central hot spot: a few per cent of the mass, compressed last and hardest by the converging shell, reaching 5–10 keV. Its alpha particles then deposit their energy in the surrounding cold, dense fuel, heating that in turn, and a burn wave propagates outward through it in tens of picoseconds.

The design has one more demand. Any heating of the fuel before compression spoils the whole thing, since a warmer fuel is harder to compress. The driving pressure must therefore rise gradually, as a tailored pulse of several shock waves that catch up with one another at the right moment, so that the fuel stays on what is called a low adiabat. A single strong shock at the outset, which would be the simplest way to hit a target hard, would ruin the capsule at the start. This is why the laser pulse at the National Ignition Facility lasts about twenty nanoseconds and rises through four carefully timed steps.

The light does not do the pushing #

The direct way to supply the pressure would be to shine light on the capsule and let radiation pressure do it. The pressure needed at the end of an implosion is hundreds of billions of atmospheres, reached by convergence from a driving pressure of about a hundred million atmospheres. At a radiation temperature of 300 eV, the energy density of the X-rays is enormous by any ordinary standard. It nevertheless falls far short of that.

The reason is a simple one, and connects directly to the fuel a starship needs. Light carries the worst possible momentum for its energy: $p = E/c$. Any other exhaust has more. Matter thrown off at speed $v$ has a momentum $2E/v$ per unit energy, which for $v \ll c$ is much larger than light’s. So absorb the light in the outer layer of the capsule, boil that layer away, and let the resulting plasma stream off at a few hundred kilometres per second. The reaction drives the capsule’s remaining mass inward. At 300 eV the pressure from this ablation is about 86 million atmospheres against the 0.37 million from light alone, a ratio of more than two hundred.

Radiation pressure as such, then, is nearly irrelevant here. Its one role is at the other end of the scale, in the size the light cannot blow away: the same physics of light and plasma is what limits the mass of a star.

The light itself is not usually allowed to touch the capsule at all. In the design used at the National Ignition Facility, 192 ultraviolet beams enter a small gold can, the hohlraum, through holes in its ends, and are absorbed on its inner wall. The wall reradiates as X-rays with a temperature near 300 eV, and that nearly blackbody radiation bathes the capsule at its centre uniformly. It is a way of getting a very even pressure on a sphere, at a cost in energy. Compressing to a thousand times requires the pressure to be symmetric to within about one per cent, since a shell that converges thirtyfold in radius multiplies any lopsidedness by the same factor.

A rocket that fires inward #

Treat the imploding shell as a rocket, and every part of the rocket equation familiar from launch pads comes straight across.

With an exhaust speed near 160 km/s, a capsule that burns off nine-tenths of its mass reaches about 370 km/s, which is about the speed required. The rocket that leaves its fuel at home covers a similar trade from a different angle, since the energy that goes into exhaust is mostly wasted. Here too the kinetic efficiency is poor: it peaks at about a fifth of the mass left and at the implosion’s working point is well below that peak.

Unlike a launch rocket, this one has its payload ahead of its exhaust in a dangerous configuration. A dense shell is being pushed by a light plasma, which is a heavy fluid accelerated by a light one, and any ripple on the interface grows exponentially by the Rayleigh–Taylor instability, as the salt that sinks through a stable sea shows in a far gentler setting. A bump on the capsule’s surface of a few tens of nanometres, the size of a dust speck, can grow by a factor of many thousands during the implosion. If it penetrates the shell, it brings cold material into the hot spot and quenches it. Capsule manufacture is therefore held to a surface smoothness better than a thousandth of the width of a human hair, and the tent that supports the capsule, and the fill tube that delivers its fuel, have been observed to spoil symmetrical implosions by themselves.

Where the energy went #

The result in December 2022 was the first time a laboratory fusion experiment delivered more energy than the laser that drove it. It came about sixty years after the first calculations and a decade after the National Ignition Facility’s original goal date. The ledger shows how far it still stands from a power plant. Of the 2.05 MJ in the light, about a tenth was absorbed by the capsule. Of that, perhaps 15 kJ ended up in the inward motion of the fuel. Most of the rest was lost to heating the can’s walls and to the exhaust. The fuel then gave back two hundred times what it had received, which is why the whole thing works. Fuel gain, the ratio of fusion yield to energy delivered to the fuel, is high and is the figure closest to what a star-builder would mean by ignition.

The upstream number is the sobering one. The lasers at the facility date from the 1990s and draw about 300 MJ to produce those 2.05, a figure that has no bearing on what a modern diode-pumped system could do but does mean the shot returned about one per cent of what it drew from the grid. A power plant would need a target gain above a hundred, with a laser ten times more efficient, firing ten times a second, with a fresh target fired into the chamber and hit with precision each time. Each target at the present level is worth far more than the electricity it could make.

The same inequality, at thirty picoseconds #

It would be easy to think of this as a separate science from the magnetic one, and in most ways it is. But the hot spot at the centre of the imploded fuel has to satisfy the same condition. A pressure $P$ held for a time $\tau$ is what the triple product measures, since a plasma’s pressure is $2nT$ counting ions and electrons, and at the 14 keV minimum

$$P\tau = 2 \times \left(3 \times 10^{21}\ \text{keV s/m}^3\right) \approx 10^{6}\ \text{Pa s},$$

which is about ten atmosphere-seconds. A tokamak reaches this with a few atmospheres held for several seconds. The hot spot in a capsule is at about three hundred thousand million atmospheres, and so it needs only about thirty picoseconds. The shell’s inertia, in stopping against the hot spot, holds it together for roughly that long. Both are the same requirement read at the two ends of the line, and that holding pressure comes from the shell’s kinetic energy being converted to pressure in a shrinking volume.

This also shows why velocity is the critical quantity. The shell’s kinetic energy per gram is $\tfrac12 v^2$, and 370 km/s gives about 68 kJ per milligram; converted to thermal energy with no other loss, that puts the fuel near 2 or 3 keV. The hot spot gets further by a factor of a few, because only the last few per cent of the mass is stopped there, and because alpha heating begins to add its share. A shell moving at 300 km/s falls short of ignition, and one moving at 400 km/s gets there with much greater margin, so the yield is enormously sensitive to speed. In the years before 2021, shots at the National Ignition Facility fell short by what looked like a small factor in pressure, and most of the record gain came from small improvements in symmetry and shell quality that crossed this threshold. A burning plasma, once reached, heats itself much faster than it cools; the same runaway that a tokamak has to control is, here, the very thing being sought, with no time to get out of hand.

What the estimates assume #

The burn fraction formula applies a single temperature and a uniform density to what is really a hot spot surrounded by a cold shell. The value of $H$ depends on temperature, and at the lower temperatures of a real burn it is larger. So the fraction is an estimate that gets the scaling right and not a calculation of any particular capsule. The real December shot burned only about 2 or 3 per cent of its fuel and not a third. Its yield fell short of the 185 MJ that a perfect 1.8 mg ball would give, because it held only about 0.2 mg of fuel at much less than the ideal areal density.

The ablation-pressure line is an empirical fit, and it applies only for the temperatures it was drawn from. The rocket’s constant exhaust speed is a simplification; an actual shell’s exhaust conditions change as it burns. The figures for the fuel’s kinetic energy are quoted as orders of magnitude, since they are inferred through simulation and not measured. What the picture also cannot show is the three-dimensional shape of an implosion, since a hot spot’s symmetry and the mixing of shell into fuel are where most of the differences between one shot and another lie. Shots with nominally the same conditions at NIF have differed in yield by large factors, and the results of repeated experiments are themselves a subject of analysis.

Still open: whether this can ever be cheap #

No capsule built so far has approached the gain that an inertial fusion power plant would need. The direct route to better gain is larger, more robust targets and higher energies, and the record at NIF has continued to climb since the first shot, passing 5 MJ and more. Fast ignition, which would compress first with one pulse and ignite a side region with a separate very short pulse, would remove the need for a central hot spot altogether and in principle allow much higher gain at less compression; it has not been demonstrated at scale. Direct drive, omitting the can altogether, would absorb far more of the laser’s energy at the risk of worse symmetry.

Whether any of these can be manufactured at a million targets a day at a few cents each, as the economics require, has not been tested at all. A capsule and its can now take weeks of skilled work to make. The physics needed to burn a piece of fuel by inertia alone is now in hand. A way to do so a few times a second, reliably and at a profit, is a separate question.

A fuel ball with nothing to hold it burns according to one number, its density times its radius, and reaching $\rho R = 3$ g/cm² with a milligram of fuel forces a thousandfold compression. Light’s own pressure falls more than two hundred times short of the push required, so it is spent boiling off the capsule’s skin, which makes the implosion a rocket: most of the energy goes in the exhaust, a few per cent of it moves the fuel, and the fuel’s burn pays it all back.