Inertial confinement and the state of fusion energy

Take away the magnetic field and nothing holds the plasma but its own mass. A hot sphere of radius R lets go from the outside in, as a rarefaction wave running inward at the sound speed cₛ = √(2kT/mᵢ), and the useful burn lasts about τ ≈ R/(4cₛ) — with cₛ ≈ 1.5 × 10⁸ cm s⁻¹ at 30 keV and R ≈ 120 μm that is 20 ps. For equimolar D-T with n_D = n_T = n/2 the ion density obeys dn/dt = −(n²/2)⟨σv⟩, which integrates to 1/n − 1/n₀ = ⟨σv⟩t/2. The fraction burned by time τ is f = x/(1 + x) with x = n₀⟨σv⟩τ/2. Substitute n₀ = ρ/mᵢ and τ = R/(4cₛ) and x becomes ρR/H_B with H_B = 8mᵢ cₛ/⟨σv⟩: at 30 keV, 8 × 4.15 × 10⁻²⁴ g × 1.5 × 10⁸ cm s⁻¹ / 6.6 × 10⁻¹⁶ cm³ s⁻¹ ≈ 7.5 g cm⁻², and the tabulated H_B(T) sits near 7 g cm⁻² across the 30–50 keV of a burning hot spot, the value this topic uses. The whole burn has collapsed into one areal density, because the confinement time grows as R/cₛ and the reaction rate per ion as ρ⟨σv⟩, so only their product ρR survives. The Maxwell-averaged ⟨σv⟩ is the one from the Gamow-peak topic; it enters only through H_B.

The burn wants ρR ≈ 3 g cm⁻² for f ≈ 30 %. A uniform sphere carries M = (4π/3)ρR³ = (4π/3)(ρR)³/ρ², so at fixed ρR the fuel mass — and with it the yield and the pressure the chamber must swallow — falls as 1/ρ². At cryogenic solid density, 0.25 g cm⁻³, ρR = 3 means R = 12 cm and M = 1.8 kg, whose 30 % burn is 1.8 × 10¹⁴ J, forty kilotonnes: a weapon, not a power plant. Compress a thousandfold to 250 g cm⁻³ and the same ρR needs R = 120 μm and M = 1.8 mg, releasing 180 MJ, a pulse a chamber can take ten times a second. Compression is done by ablation: the laser, or the X-rays it makes inside a gold hohlraum, boils off the outer ablator, and the rocket reaction drives the shell inward at 350–400 km s⁻¹; its kinetic energy stagnates into hundreds of gigabar at the centre. Keeping the cold fuel on a low adiabat, near the Fermi-degenerate minimum energy for that density, is what the shaped multi-shock laser pulse is for.

Heating all 1.8 mg to burn temperature would cost far more drive energy than the yield repays, so the design ignites a central hot spot and lets the alphas carry the burn outward into the cold, dense shell. The hot spot needs (ρR)ₕₛ ≳ 0.3 g cm⁻², because that is the range of a 3.5 MeV alpha in 10 keV D-T (it grows roughly as T⁵⁄⁴) — thinner and the alphas leave before heating — and Tₕₛ ≳ 5 keV, so that alpha heating outruns bremsstrahlung and electron conduction. This is the tokamak's triple product with τ = R/cₛ. Put ρ = 100 g cm⁻³, so n = 2.4 × 10³¹ m⁻³, with R = 30 μm and T = 5 keV: cₛ = 6.2 × 10⁵ m s⁻¹, τ = 48 ps, and nTτ = 5.8 × 10²¹ keV s m⁻³, above the 3 × 10²¹ of the previous topic. The August 2021 NIF shot (1.35 MJ) was the first to pass the ignition form of the criterion, alpha heating outrunning all of the hot spot's losses, after shots of late 2020 and early 2021 had reached a burning plasma, alpha heating exceeding the compression work; the 5 December 2022 shot, 3.15 MJ from 2.05 MJ, was the first with target gain above one, and later shots have raised both yield and gain.

A light fluid pushing a heavy one is unstable, and an implosion does it twice. During acceleration the low-density ablation plasma pushes the dense shell; at stagnation the light hot spot decelerates the dense shell, the inner surface goes unstable, and cold fuel and carbon ablator are pushed into the hot spot, where Z² bremsstrahlung radiates its heat away. A ripple of wavenumber k on a classical interface grows as exp(γt) with γ = √(Akg), A the Atwood number, so short wavelengths grow fastest. Ablation rescues the outer surface: material is stripped from the front at velocity vₐ, and Takabe's fit γ = 0.9√(kg) − 3kvₐ has a cutoff at k_c = 0.09g/vₐ² and its worst mode at k_c/4. With g = 10¹⁴ m s⁻² (400 km s⁻¹ in 4 ns), vₐ = 3 μm ns⁻¹ and R = 1 mm, the cutoff is ℓ_c ≈ 1000, the worst mode ℓ ≈ 250 grows at 2.3 × 10⁹ s⁻¹, and 4 ns of acceleration gives nine e-folds, ×8000, against an in-flight shell 30–40 μm thick. That is why capsule roughness is specified in nanometres, why the fill tube and support tent are studied, and why low-mode hohlraum asymmetry is the other budget line.

Write the ladder down for 5 December 2022. About 300 MJ of electrical energy charged the capacitor banks; 2.05 MJ of ultraviolet light reached the hohlraum, a wall-plug efficiency η_L ≈ 0.7 %; roughly a tenth of that was absorbed by the capsule, and of order 20 kJ ended as kinetic energy of the fuel. The 3.15 MJ of yield gives target gain G = 3.15/2.05 = 1.54, a capsule gain near 15, and an engineering gain of 3.15/300 ≈ 0.01. Full burn of the 0.2 mg of D-T would have been 67 MJ, so the burn fraction was 4.7 %, which f = ρR/(ρR + 7) reads as ρR ≈ 0.35 g cm⁻² for the fuel that burned. For a plant, electrical output Pₑ = ηₜₕ M_b G E_L fᵣₑₚ must repay the laser's draw P_L = E_L fᵣₑₚ/η_L with a recirculating fraction ε = 1/(η_L ηₜₕ M_b G); ε ≤ 0.25 with ηₜₕ = 0.4 and M_b = 1.1 needs η_L G ≳ 10. NIF stands at 0.01. A 10 %-efficient diode-pumped laser would need G ≈ 100, sixty times NIF's, fired at 10 Hz rather than once a day, at targets costing cents rather than the hand-built cryogenic assemblies NIF fires today.

Three rungs remain untouched by any device, magnetic or inertial. Tritium: D-T consumes one triton per 17.6 MeV, so a 1 GW-fusion plant runs 3.5 × 10²⁰ reactions per second and burns 56 kg of tritium a year, against a world civil inventory of a few tens of kilograms. The plant must breed its own from n + ⁶Li → T + ⁴He + 4.8 MeV and the endothermic n + ⁷Li → T + ⁴He + n, with a breeding ratio comfortably above one after neutron leakage, and no blanket has yet closed that cycle. Materials: the 14.1 MeV neutron carries 80 % of the energy through the first wall, and no structural steel has been qualified against decades of that spectrum. Net electricity: JET's 59 MJ in 2021 was 0.33 of its heating power; ITER is built for Q = 10 in the plasma, with D-T operation not scheduled before the late 2030s; NIF's target gains above one are a physics result, not an energy source. Check every headline against the ladder: plasma gain, target gain, engineering gain, and finally electricity on the grid.