Equations, data and domains

How a run works

Every laboratory follows the same cycle. Input panels open on entry with a complete, valid parameter set already loaded, so a first calculation is always available.

1. Enter parameters. Controls are grouped by physical role and labelled with their units. Ranges are enforced at the input, and a value outside a model's fit domain is accepted but flagged on the result.
2. Calculate. Enable live recalculation, or press Calculate after editing. Changing an input marks the previous result and its exports as stale rather than leaving an outdated file looking current.
3. Inspect. Result panels, diagnostic tables, charts and the three-dimensional scene all describe the same run. Timeline playback animates a completed calculation; it does not advance a self-consistent plasma state.
4. Export. Save the inputs as an experiment JSON to reload later, and export the results separately with their units, model version and source information attached.

In the reactor and Torus workspaces, the Inputs, Results, Particles and Energy shortcuts jump to a section; closing panels expands the scene, and All panels reopens them. Appearance controls, cameras and layer toggles change the display only.

01 · FusionSim — reactivity and zero-dimensional balance

Reaction-rate density follows from distinct species populations and a Maxwellian relative-energy distribution at the reduced-mass relative temperature.

T_rel = μ(T_p/m_p + T_B/m_B)  ·  R = n_p n_B ⟨σv⟩

⟨σv⟩ = √[8/(πμ)] (kT)^−3/2 ∫ σ(E) E exp[−E/(kT)] dE

The integral is evaluated over 41 published Sikora–Weller evaluation points with log–log interpolation and segment boundaries at every tabulated energy, giving a centre-of-mass domain of approximately 137.4–3480.9 keV after exact invariant-s transformation of the published laboratory axis. No additional endpoint extrapolation is applied.

A separate species-sum radiation comparison (Xie 2024, Eqs. 61–68) evaluates each actual ion charge and splits electron–ion from electron–electron emission over 1–511 keV. It is exported alongside the baseline rather than replacing it, so both figures stay identifiable in the record.

Not included in the balance: synchrotron radiation, impurities, ash accumulation, alpha slowing-down, spatial and kinetic transport, MHD and kinetic stability, driver and magnet power, structures, fuel cycle and electricity conversion.

02 · Isotope Fusion Explorer — nuclear data and kinematics

Atomic masses from the AME2020 slice determine reaction energetics; centre-of-mass energy, closed-channel thresholds and two-body partitions use relativistic invariant-s kinematics with bare-nucleus masses.

Q = (Σm_reactants − Σm_products)c²  ·  s = m_a² + m_b² + 2m_b(m_a + T_lab)

Sequential structural spectrum

The sampler models p+¹¹B→α+⁸Be* followed by ⁸Be*→2α. It draws a user-supplied ground-state fraction or a truncated broad level near 3.03 MeV, samples isotropic decay, performs exact sequential Lorentz boosts, and enforces ΣE_α = Q + E_cm, Σp⃗_α = 0 and the alpha mass shell on every event. Fixed seeds reproduce the same event set exactly.

Product transport is available in the reaction centre-of-mass frame or in a stationary-target laboratory frame, with uniform-field helical motion for charged products and straight-line flight for neutrals. The frame in use is named on every reported quantity.

03–04 · Virtual Reactor Lab and Torus Lab — magnetic configuration screens

The Virtual Reactor and original Torus notebook include analytic configuration screens. The current Torus Observatory additionally implements the explicitly scoped field and equilibrium models listed below. A prescribed field, a manufactured elliptic test and a force-balance problem carry different evidence.

Torus Observatory model inventory

Fixed-boundary results distinguish iterative residual, exact manufactured flux error when available, and boundary error. Imported reconstructions have no local residual history unless supplied by their source. Numerical verification does not establish experimental accuracy; qualification and missing measurements remain explicit.

Mirror field and single-particle invariants

B(z) = B₀[1 + (R_m − 1)(2z/L)²]  ·  θ_LC = asin(1/√R_m)  ·  f_trapped = √(1 − 1/R_m)

Magnetic-moment conservation determines trapping and turning point under a non-relativistic energy partition. Reported diagnostics include the relativistic alpha gyroradius and gyrofrequency, the non-relativistic turning point and harmonic bounce period, beta, an electron-only Debye length under an immobile-ion convention, the stored thermal inventory, and the maximum sampled ρ|∇B|/B over the accessible half-orbit as an adiabaticity proxy.

Toroidal geometry and diagnostic screens

B_φ(R) = B₀R₀/R  ·  B_θ(a) = μ₀I_p/(2πa)  ·  q(a) = aB₀/[R₀B_θ(a)]

Volume uses an elongated torus; pressure and electron density follow the p:B mixture you enter. The cylindrical edge safety factor, beta, a normalised-beta proxy, the Greenwald density comparison, the relativistic 90°-pitch maximum gyroradius and a banana-width proxy are reported as diagnostic screens. Safety-factor and orbit badges are suppressed outside a/R₀ ≤ 0.35, where the cylindrical approximation stops being informative.

The analytic configuration screens do not solve equilibrium. The current Torus solver is restricted to a fixed rectangular Dirichlet boundary; its manufactured mode verifies the elliptic equation and is not a physical pressure/current equilibrium. No model here establishes free-boundary reconstruction, coil optimization, bootstrap current, MHD stability, collisional transport, energy confinement, or reactor performance.

05 · AlphaTrack — image segmentation and morphometry

The pipeline converts the image to luminance, selects an Otsu or manual threshold, labels dark connected components under 4- or 8-connectivity, then computes area, centroid, bounding box, equivalent diameter and pixel-edge compactness for each object before applying morphology bounds and drawing the accepted set.

compactness = 4πA/P²  (four-edge raster perimeter)

Raster sampling and orientation affect compactness, so a sampled disk does not generally score 1. The legacy circularity export key is retained and carries method metadata.

06 · DirectConvert — electrostatic recovery and the adiabatic limit

Two independent models run side by side. The first is a discrete electrostatic stage ledger.

E_rec = qΔφ → E_rec(MeV) = 2Δφ(MV), credited only when 2Δφ(MV) < E_α(MeV)

E_source = E_recovered + E_residual  ·  Ṅ = P_α/Ē_α  ·  I = 2eṄ

Each simulated alpha selects the highest strictly admissible stage, and the energy sum is checked exactly. At the Q/3 mean, 1 MW of alpha power corresponds to approximately 0.691 A of charged-particle current.

First-order transport screens

J_CL = (4/9) ε₀ √(2q/m) V^3⁄2 / d²

Each planar Child–Langmuir row uses the stage-to-stage voltage increment and the current incident on that gap divided by beam area. Every ion enters the first gap, including ions reflected before their collector; later gaps see ions assigned to that or a higher potential in the ideal routing ledger.

Adiabatic large-device limit

C = ε_particle,0B₀²/(mE_field²); solve exp(4x) − C sin²θ exp(x) − 2C cos²θ = 0, then η* = 1 − cos²θ exp(−2x) − sin²θ exp(−x)

The positive root is found by bisection and verified against the published equation and its θ = 0° and 90° analytic limits. This is a single-particle free-energy limit for optimally large geometry, reported for comparison against the stage ledger.

Plant balance

P_direct = P_α·η_stage·η_transport·η_conditioner

P_thermal,e = η_captureη_thermalP_{capturable heat}  ·  P_net = η_site(P_direct + P_thermal,e) − P_aux

A thermal efficiency above the Carnot bound 1 − T_cold/T_hot is rejected as a hard failure. The thermal path is an explicitly shared-reservoir sensitivity: unrecovered particles, conditioning loss and any user-supplied heat share one capture fraction, hot temperature and efficiency.

07 · Reactor 3D — composed system viewer

The composed viewer couples the existing reduced calculations and records each coupling explicitly rather than presenting a single opaque system model.

Numerical checks applied to every run

• Kinetic energy stays constant to round-off in a frozen uniform-field pure-B test.
• Centred finite-difference divergence checks exercise the mirror and torus fields away from coordinate singularities.
• Fixed scientific and visual seeds reproduce the same states; camera, layers and playback are display-only.
• Particle, frame and substep budgets reject oversized runs, and all exported coordinates must remain finite.
• The first sampled analytic-volume crossing freezes the displayed path and is labelled non-terminal.

Release 7.5.2 research models

These models run in separate research panels. Each has a versioned identifier, an independent reference calculation and a result record that states physicalValidationEstablished: false. Full contracts: docs/v752-specs/.

Time-dependent burn and ash — hbf-pb11-burn-0d@1 (FusionSim)

dn_p/dt = S_p − n_p/τ_p − R  ·  dn_B/dt = S_B − n_B/τ_p − R  ·  dn_α/dt = 3f_depR − n_α/τ_α

dW_i/dt = G f_depQR + P_aux,i + P_ei − W_i/τ_E − convective loss  ·  dW_e/dt = (1−G) f_depQR + P_aux,e − P_ei − P_brem − W_e/τ_E − convective loss

R = n_pn_B⟨σv⟩(T_i) from the pinned table (cubic log–log cache, 1–2000 keV, never extrapolated); G is the Stix ion share with E_c = 14.7688 A_αT_e[Σ n_jZ_j²/(n_eA_j)]^2/3; P_ei uses the NRL Plasma Formulary equilibration rate; electrons carry the Maxwell–Jüttner mean energy. Integration: Dormand–Prince 5(4) with an energy and particle ledger. An independent SciPy Radau solution agrees to 1.24×10⁻⁹ over 20 s for the default case and to 3.03×10⁻⁹ over the two recorded scenarios (docs/v752-verification/burn-dynamics-oracle.json). Not modelled: transport profiles, synchrotron and line radiation, fast-alpha slowing-down dynamics.

Collector gap — hbf-collector-gap-1d@1 (DirectConvert)

J_CL = (4ε₀/9)√(2q/m) (E/q)^3/2/d²  ·  η = qV_c/E  ·  J_crit/J_CL = (1 + √(1 − η))³

A planar electrostatic particle-in-cell model (cloud-in-cell weighting, direct Poisson solve, leapfrog push) follows a cold beam into a retarding gap, records which particles are collected or returned, detects a virtual anode and closes an energy ledger. The steady theory lists every non-reflecting state below J_crit and the fully reflecting state for η > 1. An independent shooting solver and a separate NumPy particle-in-cell code agree with it. Not modelled: collisions, secondary electrons, grids, magnetic fields and multi-dimensional geometry.

Exact coil fields (Virtual Reactor Lab, Reactor 3D)

B_z, B_ρ of a circular loop from complete elliptic integrals K(m), E(m), m = 4aρ/[(a+ρ)² + (z−z₀)²]

K and E are computed by the arithmetic–geometric mean with a stable near-axis form; solenoids and thick coils are integrated with adaptive Gauss–Legendre quadrature. A matched two-loop coil set shares the mirror run's B₀, throat position and mirror ratio; for B₀ = 1.5 T, R = 4 and L = 2 m the analytic parabolic mirror differs from it by up to 26 %. A held-out comparison fits only a current scale (and optional offset) on training rows of a field map.

Measured nuclear data comparisons (FusionSim, Isotope Explorer)

χ² = rᵀV⁻¹r,  V = diag(u²_point + u²_baseline) + (n²_set + 0.034²) s sᵀ

EXFOR sets are converted to centre-of-mass energy with the invariant-s relation used for the baseline and compared without changing it; a profiled normalisation factor separates scale from shape. Both correlated terms are built from the baseline values s, not from the measured values: using measured values there produces Peelle's pertinent puzzle, which biases the fit low (for Taskaev 2024 it moves χ² from 49.0 to 144.3). Sampler checks compare branch fractions and Legendre moments (2L+1)⟨P_L(cos θ)⟩ with published coefficients at matching proton energies (tolerance max(1 keV, 0.5 %)).

Shared numerical foundations

• Mass constantsNine mass constants are taken from the unrounded AME2020 table, with eleven masses carrying uncertainties and nine IAEA LiveChart ground-state records bundled with the release. Q-value uncertainty uses signed net mass coefficients before squaring; inter-nuclide covariance, constant uncertainty and electron binding are excluded at this precision.
• Threshold stabilityA common factored Källén momentum prevents negative or non-finite kinetic energy at reaction threshold, and small kinetic energies use factored momentum with rationalised kinetic-energy formulas.
• Thermal reactivity anchorsD+T, D+³He and both D+D branches use Bosch–Hale Table VII coefficients, checked against UKAEA PROCESS with cm³/s converted to m³/s. Identical DD reactants share one deuterium population with a half-pair factor per branch.
• Convergence checkingThe h, h/2, h/4 orbit check doubles substeps as the step is halved, preserving the physical horizon and the seed. Positions at or after the first boundary crossing are excluded, and differing crossing outcomes withhold the order estimate rather than reporting a misleading one.
• RenderingThe GPU path uses procedural physically-based materials with a locally bundled Three.js. The CPU fallback projects the same geometry and instanced transforms with simplified materials, clipping polygons and lines to the camera frustum. Neither path participates in a calculation.

Bundled data sources

Public availability of a source does not imply endorsement by the issuing body, and a bundled slice is not a complete nuclear-data evaluation.