Inertial confinement relies on the target's own inertia: a driver compresses and heats a small fuel capsule faster than it can
disassemble. Lasers do it with light, directly on the capsule or through the X-rays of a hohlraum; ion and electron beams do it with mass;
a hypervelocity projectile does it with a collision. This page collects the closed-form relations the family's three cores evaluate,
and it keeps the family's own habit of saying which printed number comes back and which does not.
What these relations are. Every relation on this page is one a device core of this family implements as a closed-form evaluation on a declared operating point at computational_prototype maturity: algebra a filed source prints, never a simulation. No radiation hydrodynamics, transport, laser–plasma interaction, beam deposition, shock or burn calculation is performed anywhere in these cores; the ignition floors are necessary algebraic conditions on a design, never a prediction that a design ignites; no value describes, approximates or validates any real machine or shot.
1 · The hot-spot ignition condition, in the four forms the laser review prints
A compressed capsule ignites when its central hot spot is dense enough along a radius to stop the alpha particles and hot enough for them to be made. Craxton et al. write that condition four equivalent ways, and the laser core carries all four with the energy relation that turns one into the next:
$$\rho R_{hs} \ge 0.3\ \mathrm{g\,cm^{-2}},\; T_i \ge 5\ \mathrm{keV}; \qquad P_{hs} > 100\ \mathrm{Gbar}\,\frac{100\ \mu\mathrm{m}}{R_{hs}}; \qquad P_{hs} > 250\ \mathrm{Gbar}\left(\frac{f_k E_k}{10\ \mathrm{kJ}}\right)^{-1/2}; \qquad R_{hs} < 40\ \mu\mathrm{m}\sqrt{\frac{f_k E_k}{10\ \mathrm{kJ}}}$$
$$f_k E_k = 2\pi P_{hs} R_{hs}^3, \qquad P = \frac{2 T_i\rho}{2.5\,m_p}$$Craxton et al., Phys. Plasmas 22 (2015) 110501, eqs. 3–5. \(f_k E_k\) is the fraction of the shell's kinetic energy coupled into the hot spot. The 40-µm coefficient is recovered exactly, as the same IEEE double, from the two printed pressure coefficients and the printed reference radius. Measured rather than assumed: the three printed coefficients close on the energy relation 0.53 % high at every energy from 1 kJ to 1 MJ (the rounding of the printed coefficients), and the pressure coefficient computed from the review's own floors is 115 Gbar where it prints 100. (SCPN-ICF-LASER-CORE.)
The figures a one-dimensional design is quoted by
$$\alpha_{DT} = \frac{P_{\mathrm{shell}}}{2.2\,\rho^{5/3}}, \qquad \mathrm{IFAR} = \frac{R_{\mathrm{abl}}}{\Delta R}\ \text{at}\ R = \tfrac23 R_{\mathrm{inner},0}, \qquad C_r = \frac{R_{\mathrm{inner},0}}{R_{\mathrm{inner,stag}}}, \qquad \eta_{\mathrm{hydro}} = \frac{E_k}{E_{\mathrm{abs}}},\; E_{\mathrm{abs}} = f_{\mathrm{abs}} E_{\mathrm{inc}}$$The DT shell adiabat against the Fermi-degenerate pressure, the in-flight aspect ratio at the evaluation point the review fixes, the convergence ratio (whose definition carries a condition no code here can enforce: alpha deposition switched off), and the hydrodynamic efficiency against absorbed rather than incident energy. Anchor: the review's 1.5 MJ direct-drive design, 1700 µm target radius with 37 µm ablator and 160 µm ice, absorbed fraction 0.95 (exact at 1425 kJ), \(\eta = 0.067\), IFAR 24.3, \(C_r = 23\), adiabat 1.6, hot-spot pressure 215 Gbar. The initial inner radius 1503 µm and the evaluation point 1002 µm are exact from three printed lengths.
Fuel, yield and gain
$$m_{\mathrm{fuel}} = \tfrac43\pi\left(R_o^3 - R_i^3\right)\rho, \qquad Y = m_{\mathrm{fuel}}\, f_b\, \varepsilon_{DT}, \qquad G = \frac{Y}{E_{\mathrm{inc}}}, \qquad \varepsilon_{DT} = \frac{17.6\ \mathrm{MeV}}{m_D + m_T}$$The specific energy is built from the two nuclear masses, not carried as a rounded constant. Not reproduced, and recorded as tests so it stays visible: the review's one-dimensional gain of 48 is not recovered from its printed geometry at its printed 20 % burnup with solid fuel at standard density (0.25 g/cm³, which the review does not print); that reconstruction gives about 57. The gap is physical — a quoted burnup applies to the fuel that assembles and burns — and neither figure is used as an anchor. The review's worked "120 to 180 Gbar" from its own energy-form floor comes out 112 to 125 at the stated inputs.
2 · Mass instead of light: the beam-driven capsule
An ion beam deposits its energy in a converter or directly in the ablator over a range measured in grams per square centimetre; the driver-side physics is accelerator transport and final focus. The beam core evaluates the illumination geometry of a multibeam arrangement, the mass inventory of a three-layer capsule, and the coupling chain from driver to capsule:
$$A_{\mathrm{spot}} = \pi a b, \qquad r_{\mathrm{eff}} = \sqrt{ab}, \qquad N_{\mathrm{beams}} = 2\,N_{\mathrm{side}}, \qquad F = \frac{E_{\mathrm{driver}}/N_{\mathrm{beams}}}{\pi a b}, \qquad n = \frac{\ln(R_2/R_1)}{\ln(E_2/E_1)}$$
$$G_{\mathrm{target}} = \eta_c\,\eta_e\,G_{\mathrm{capsule}}, \qquad E_{\mathrm{abs}} = \eta_c\eta_e E_{\mathrm{driver}}, \qquad A_{\mathrm{enclosure}} = \frac{4\pi r^2}{\text{ratio}}$$Ho, Harte & Tabak, UCRL-JC-118161 (1994): a capsule of 2.34 mm pellet radius, fuel between 1.8 and 2.12 mm, ablator 1.85 g/cm³, solid fuel 0.25, vapour 0.3 mg/cm³ — 25.45 mg of ablator against 3.87 mg of fuel and 0.0073 mg of vapour; 430 MJ from 1 MJ absorbed, capsule gain exactly 430; coupling efficiency 0.21, system gain 80, capsule-to-enclosure area ratio 0.075. Callahan-Miller & Tabak, UCRL-JC-131974 (1998): elliptical spots 4.15 × 1.8 mm (\(r_{\mathrm{eff}}\) 2.7331 for a printed 2.7) and 2.78 × 1.0 mm (1.6673 for a printed 1.67), eight and sixteen beams per side; three gain cases the review truncates rather than rounds (370/6.35 = 58.27 printed 58; 413/7.4 = 55.81 printed 55; 436/3.3 = 132.12 printed 132); three range–energy pairs no single power law joins (exponents 1.12, 1.25, 1.19). The cited driver-window review is behind a subscription and not on file; both filed preprints describe heavy-ion drivers, and nothing here is evidence about the electron-beam configuration the core also owns. (SCPN-ICF-BEAM-CORE.)
3 · Kinetic energy as the driver: the impacted target
A macroscopic projectile at hypervelocity converts its kinetic energy on impact into compression and heating. The impact core evaluates what a flying plate carries, the fuel in a plane slab and in a solid sphere, and where the fuel ends up — by conservation of mass in one axis, and as the cube of a radial factor in convergence:
$$\rho t = \frac{m}{A}, \qquad t = \frac{\rho t}{\rho_{\mathrm{mat}}}, \qquad \frac{E}{A} = \frac{m v^2}{2A}, \qquad \rho R_{\mathrm{slab}} = \rho\, t_{\mathrm{fuel}}, \qquad t_c = \frac{t_0}{C}, \qquad \frac{\rho_c}{\rho_0} = C_r^3, \qquad \rho R_{\mathrm{sphere}} = \rho\, r$$Proceedings of the Impact Fusion Workshop, LA-8000-C (Los Alamos, 1979), read off pages rendered at 170 dpi. Christiansen's worked plane case: a uranium plate (18.8 g/cm³) at 200 km/s presenting a 1 cm² face, areal density 0.052 g/cm² exactly, 1.04 MJ/cm² to one unit in the last place, thickness 2.7659×10⁻³ cm printed as the truncation 2.7×10⁻³; the fuel slab from 0.01 of the cryogenic density 0.213 g/cm³ to 4.2 of it, a compression ratio of 420, compressed thickness 2.3810×10⁻³ printed as 2.3×10⁻³. Marshall's convergent alternative: a 1 mm solid sphere compressed tenfold in radius reaches 2.13 g/cm² against a printed 2. A driven slab gains no areal density — its density rises exactly as its thickness falls — while a converging sphere gains the square of its radial factor; at these dimensions the two differ by a factor of 1000, which is the convergent paper's objection to the plane scheme stated as arithmetic. No filed source pairs the two schemes. (SCPN-ICF-IMPACT-CORE.)
Not reproduced, recorded rather than absorbed into a tolerance: the convergent target's printed mass of 0.84 mg (0.8922 mg at the cryogenic density the volume prints elsewhere), and "nearly 400 GJ" for a one-gram burn against the 337.5 GJ the nuclear masses give. No burn-up fraction exists anywhere in the impact core: no filed source prints one and none is invented.
4 · Where the family ends
Magnetised liner inertial fusion is a pulsed-power implosion of a premagnetised, preheated liner; it belongs to the magneto-inertial family, as does the slower material liner the impact core excludes.
Non-imploding beam-target fusion — fixed targets and colliding beams without implosion — is the electrostatic and hybrid family's.
Radiation hydrodynamics, transport, laser–plasma interaction and burn physics stay in SCPN-Fusion-Core; no core of this family copies or dispatches a solver.