Controlled thermonuclear fusion physics: magnetic confinement (tokamak, stellarator, spherical torus, field-reversed configuration) and inertial confinement (laser direct-drive, laser indirect-drive, Z-pinch). Central criteria: Lawson…
fusion-physics
Lawson triple product: nτT ≥ 3·10²¹ keV·s/m³ for DT ignition at T ≈ 10-25 keV
Lawson (1957) power-balance criterion for self-sustained fusion: the plasma density n, energy-confinement time τ_E, and ion temperature T…
Greenwald density limit: n̄_G = I_p/(π a²) · 10²⁰ m⁻³ (I_p in MA, a in m)
Greenwald empirical density limit (1988): in a circular tokamak with plasma current I_p (MA) and minor radius a (m), the line-averaged…
Troyon β limit: β_max[%] = β_N · I_p/(a·B) with β_N ≤ 2.8 for ideal MHD
Troyon (1984) ideal-MHD stability limit on the tokamak β = ⟨p⟩/(B²/2μ₀). For circular cross-section the scaling is β_max[%] = β_N ·…
Bootstrap current: self-generated toroidal current ∝ ε^{1/2}·(dp/dψ)
Neoclassical 'bootstrap' current (Bickerton-Connor-Taylor 1971): a fraction of the tokamak toroidal current arises spontaneously from…
Edge-localized mode (ELM): H-mode pedestal relaxation via peeling-ballooning
In H-mode tokamak operation, the edge pedestal pressure gradient ∇p and bootstrap-driven edge current j_∥ grow between transport relaxation…
ICF hydrodynamic instability: RT growth γ = √(g·k) at shell/gas interface
Inertial-confinement fusion (ICF) laser-driven implosion compresses a hollow DT-ice pellet by factor ~30 in radius and ~1000 in density (to…
Grad-Shafranov Δ*ψ = -μ₀R²P'(ψ) - FF'(ψ); Soloviev analytic equilibrium
Grad-Shafranov framework — the canonical axisymmetric magnetohydrodynamic (MHD) equilibrium equation of tokamak and stellarator plasma…
Shear-Alfven wave: ω² = k_∥²·B₀²/(μ₀·ρ); v_A = B₀/√(μ₀ρ)
Shear-Alfven-wave framework — the fundamental transverse MHD wave propagating along magnetic field lines, with dispersion relation…
Lagrange interp of ⟨σv⟩(T) for fusion-reactivity / Lawson-criterion evaluation
Lagrange-interpolation framework for fusion-reactivity tables — the standard numerical approach to embedding tabulated…
Soloviev: Δ*ψ=R²(A+B); ψ(R₀,0)=0; ∂ψ/∂R|(R₀,0)=0; closed-form tokamak eq
Sympy-exact symbolic witness of Soloviev's analytic tokamak equilibrium. Setup: R, R₀, Z, A, B sp-symbols positive; ψ_Sol = A·(R²-R₀²)²/8…
Alfven dispersion: ω²=k²B²/(μ₀ρ); v_A = B/√(μ₀ρ); group vel = phase vel
Sympy-exact symbolic witness of the shear-Alfvén-wave dispersion relation and its group-velocity identity. Setup: k_∥, B₀, μ₀, ρ…
Lagrange interp exactness: P₂(T_k)=y_k; Σ L_k=1; L_k(T_j)=δ_kj
Sympy-exact symbolic witness of the Lagrange-interpolation-exactness identities for a 3-point polynomial. Setup: T, T_0, T_1, T_2, y_0,…
Lawson criterion (1957)
J D Lawson 1957: n tau T >= 3 x 10^21 m^-3 s keV (D-T fusion); plasma-energy gain Q = P_fus / P_input >= 1 break-even; ITER design Q=10;…
Grad-Shafranov equilibrium
Grad-Shafranov 1958: axisymmetric MHD equilibrium nabla^2 psi - 1/R d/dR(...) = -mu_0 R^2 dp/dpsi - F dF/dpsi; basis of tokamak shape +…
ICF implosion + Rayleigh-Taylor
Nuckolls 1972 ICF + Lindl 1995 indirect-drive: convergence ratio CR > 30; growth of Rayleigh-Taylor at ablation-front limits compression;…
D-T fusion cross-section (Bosch-Hale 1992)
Bosch-Hale 1992 ENDF parametrization: sigma_DT(E) peaks at E_lab = 64 keV reaching 5 barns; 5x10^-22 m^3/s reactivity at T=15 keV; primary…
Magnetic mirror (Budker 1954 / Post 1955)
Budker-Post 1954-1955 mirror: B_max/B_min = R mirror-ratio; loss-cone in velocity-space; instability via flute + DCLC drive Q-machine…
Stellarator (Wendelstein 7-X)
Spitzer 1951 stellarator + W7-X 2015: helical magnetic-axis avoids tokamak's resistive-disruption + plasma current; Wendelstein-7X reached…
Lawson criterion (1957)
J Lawson 1957 nT_τ-criterion; modern modern triple-product + ITER goal nTτ ≥ 5×10²¹ keV·s/m³ ignition.
Tokamak (Artsimovich 1968)
L Artsimovich 1968 + Spitzer-Stellarator 1951 + JET 1991 + ITER 2025; modern modern HL-3 + EAST 2024 + 30,000-s-Q≥1 milestone.
ICF (Nuckolls 1972)
J Nuckolls 1972 ICF; modern modern NIF 2022 ignition Q=1.5 + 2024 Q=2.4 milestone + commercial-fusion startups Helion+Fusion.
H-mode (Wagner 1982)
F Wagner 1982 + ASDEX H-mode; modern modern ELM-mitigation + RMP + 2024 advanced-tokamak HL-3 + ITER scenarios.
Mirror (Post 1956)
R Post 1956 magnetic-mirror; modern modern Mirror-fusion-energy + Tandem-Mirror + 2024 high-β open-ended commercial.
DT cross-section (Bosch-Hale 1992)
H Bosch-G Hale 1992 D-T fusion-rate-evaluation; modern modern foundational + R-matrix theory updates + ENDF / TENDL evaluations.