Open-field devices trade the closed torus for simpler geometry and accept that field lines leave the confinement region.
A magnetic mirror reflects particles from regions of stronger field and loses the ones whose pitch angle is too small; a cusp has no
closed surfaces at all and confines by the sheath-scale apertures at its nulls; a levitated dipole closes its field lines on a
superconducting ring that floats in its own field. This page collects the closed-form relations the family's cores evaluate.
What these relations are. Every relation on this page is one a device core of this family implements as a closed-form evaluation of published scalings and closed forms on a synthetic configuration at computational_prototype maturity. No Fokker–Planck, ambipolar, equilibrium or stability equation is solved; the anchors reproduce numbers and statements the sources print and are not correlations with experimental data; no fusion power, gain, breakeven or reactivity statement is made; no value describes or validates any real machine.
1 · The mirror ratio and the loss cone
A particle moving along a field line toward a region of stronger field converts parallel into perpendicular energy and is reflected when its pitch angle at the midplane exceeds the loss-cone angle. The vacuum mirror ratio is the field at the throat over the field at the midplane; finite plasma pressure deepens the well diamagnetically, and an electrostatic potential drop from midplane to throat widens the cone for ions and narrows it for electrons:
$$R_{\mathrm{vac}} = \frac{B_{\max}}{B_{\min}}, \qquad R_m = \frac{R_{\mathrm{vac}}}{\sqrt{1-\beta}}, \qquad \sin^2\theta_{\mathrm{loss}} = \frac{1}{R_m}\left(1 + \frac{q\,\Delta\phi}{E}\right), \qquad f_{\mathrm{loss}} = 1 - \sqrt{1 - \sin^2\theta_{\mathrm{loss}}}$$Endrizzi et al. 2023 eq. 3.6; Frank et al. 2025 eqs. 2.3–2.5. \(\beta\) is refused outside \([0, 1)\); an electron with \(E \le e\Delta\phi\) is fully confined ("only electrons with energies above 5\,T_e leave"); a cone at or beyond unity is reported as "no trapped region". \(f_{\mathrm{loss}}\) is the isotropic fraction inside the cone and equals the configuration's own loss-cone fraction at zero potential, bit for bit. (SCPN-MIRROR-CORE.)
2 · Collisional time-scales
Three engineering forms from the WHAM physics basis, with the density in \(10^{20}\,\mathrm{m^{-3}}\), the ion mass \(\mu\) in proton masses and the ion charge \(Z\):
$$n_{20}\tau_s = 5\,T_{e,\mathrm{keV}}^{3/2}\,\frac{\mu}{Z^2}\ \mathrm{ms}, \qquad n_{20}\tau_{ii} = \frac{E_{i,\mathrm{keV}}^{3/2}\,\mu}{8 Z^2}\ \mathrm{ms}, \qquad n_{20}\tau_{ee} = 5.8\,T_{e,\mathrm{keV}}^{3/2}\ \mu\mathrm{s}$$Eqs. 3.1–3.3: slowing of fast ions on electrons, ion–ion pitch-angle scattering, electron–electron collisions. The source states the first two are equal when \(T_e = E_i/40^{2/3}\), which the core reproduces to \(10^{-14}\) relative (the source says "about \(E_i/10\)"; the value lies between \(E_i/12\) and \(E_i/10\)).
3 · Confinement scalings and the two regimes
$$n_{20}\tau_p = 250\,E_{b,100\mathrm{keV}}^{3/2}\,\log_{10}R_m\ \mathrm{ms}, \qquad \tau_{\mathrm{GDT}} = \frac{R_m L_p}{c_s} = 5.2\,R_m L_p\,T_{e,\mathrm{keV}}^{-1/2}\ \mu\mathrm{s}, \qquad c_s = \sqrt{T_e/m_i}$$Eqs. 3.4–3.5. The classical scaling equals 250 ms at the reference point (\(n_{20} = 1\), 100 keV, \(R_m = 10\)); the source's statement that beta from 0 to 0.9 gains "only 50 %" is reproduced exactly (factor 1.5 at \(R_{\mathrm{vac}} = 10\)); the gas-dynamic dimensional form reproduces the printed coefficient 5.2 within 3 % at 2.5 proton masses (5.09). Which regime applies — collisionless classical or high-collisionality gas-dynamic — follows the configuration's declaration, never a computed criterion.
4 · What keeps the column stable: finite Larmor radius and adiabaticity
$$m_{\mathrm{crit}} = \frac{2a^2}{L_p\rho_i}, \quad \rho_i = \frac{m_i v}{ZeB_0},\; v = \sqrt{2E_i/m_i}; \qquad \alpha = \frac{L_B}{\rho_\parallel}, \quad \rho_\parallel = \frac{v_\parallel}{\omega_{ci}},\; L_B = \frac{|B|}{|\nabla B|}$$Eq. 3.7 after Ryutov et al. 2011: azimuthal modes above \(m_{\mathrm{crit}}\) are FLR-stabilised; the source's worked case \(a/\rho_i = 4\), \(L_p/a = 10\) gives \(m_{\mathrm{crit}} = 0.8\) ("all \(m \ge 2\) FLR stabilised"). The \(m = 1\) mode is not assessed at level 0. Non-adiabatic effects become significant at \(\alpha \le 10\) (§3.6); \(L_B\) is a declared input and no printed anchor exists.
5 · The tandem mirror: plugging the ends electrostatically
A tandem mirror confines a long central cell between two end cells whose denser plasma raises an ambipolar potential that reflects central-cell ions. The core evaluates the confining potential, Pastukhov's function, the flow-through and radial times and their combination, and the ambipolar-hole loss energy:
$$\phi_i = T_e\ln\frac{n_p}{n_c}, \qquad G(x) = \sqrt{1+\tfrac1x}\,\ln\frac{\sqrt{1+1/x}+1}{\sqrt{1+1/x}-1}, \qquad \tau_f = \sqrt{\pi}\,R_{mc}\frac{l_c}{v_{th,ic}}e^{\phi_i/T_{ic}}, \qquad \tau_\rho = \tfrac14\left(\frac{a_c}{\rho_{ic}}\right)^2\tau_{ii}$$
$$\tau_c = \left(\frac{1}{\tau_{\mathrm{Past}} + \tau_f} + \frac{1}{\tau_\rho}\right)^{-1}, \qquad E_h = \frac{\phi_e}{R_m\sin^2\theta_{\mathrm{NBI}} - 1}$$Frank et al. 2025 eqs. 3.2–3.7 and 4.3 (Rognlien & Cutler 1980 for \(\tau_f\)). \(G(1) = \sqrt2\ln(3 + 2\sqrt2)\) and the monotony of \(G\) are tested; \(n_p \le n_c\) is refused; the hole energy is absent when \(R_m\sin^2\theta \le 1\); the plug electron-confining potential \(\phi_e\) is a declared input, not a result. No printed numerical anchor exists for eq. 3.3; the record's digest is pinned as an immutability fixture.
6 · The magnetic cusp
Opposed coil currents make a field with a central null and point and line cusps; the boundary is everywhere convex toward the plasma, which makes it magnetohydrodynamically stable, and confinement is set by the balance between a high-beta interior and the sheath-scale loss apertures at the cusps. The family's cusp core declares that geometry — spindle and picket-fence classes, coil count, radius and separation, coil current, whether neighbouring currents are opposed — and the diagnostic and clock semantics. It implements no level-0 physics yet, so this portal's explorer has no cusp tab; the cusp page says exactly what is declared. The Polywell is not here: its cusp topology is magnetic, but its energy and reaction workflow is an electrostatic well, and the portfolio standard assigns it to the electrostatic owner.
7 · The levitated dipole: a ring afloat in its own field
A superconducting coil floats in a vacuum vessel, held up against gravity by a levitation coil above it, and the plasma sits on the closed field lines of the ring's own dipole. The core's level-0 physics is the ring as a magnet, its charging by flux conservation, the force that holds it up, and the plasma quantities of one published discharge:
$$NI, \qquad m = \pi R^2\,NI, \qquad I_F = \frac{M_{CF}}{L_F}\,I_C, \qquad B_r^{\text{as printed}} = \frac{Mg}{2\pi a\,NI},\; B_r^{\text{loop}} = \frac{Mg}{2\pi R\,NI}, \qquad \frac{\mathrm{d}B}{\mathrm{d}z} = \frac{Mg}{m}$$
$$B_{\mathrm{res}} = \frac{2\pi m_e f}{e}, \qquad \beta = \frac{2\beta_\perp + \beta_\parallel}{3},\; \beta_\perp = \frac{2\mu_0 p}{B^2}, \qquad \frac{p_0}{p_{\mathrm{sol}}} < \left(\frac{U_{\mathrm{sol}}}{U_0}\right)^\gamma$$Garnier et al., Fusion Eng. Des. 81 (2006) 2371, read off pages rendered at 180 dpi. Two printed numbers come back out of relations they were not fitted to: the resonant field at 6.4 GHz, 0.2286 T against the printed 0.23; and the floating-coil charge at a 300 A charging current, 933 kA-turns against the printed 930 — two inductances and a charging current from two different pages through flux conservation and the printed turn count. "Over 1.5 MA turns" is not reproduced (1.303 at the operational current, 1.322 at maximum charge) and anchors nothing. The printed levitation relation, read literally with \(a\) as the diameter, returns half what the loop force balance requires; both are computed (0.986 and 1.973 mT) and neither is adjusted. (SCPN-LEVITATED-DIPOLE-CORE.)
8 · What the closed forms leave to the solver laboratory
Fokker–Planck loss-cone kinetics, ambipolar potential formation and the \(m = 1\) interchange of the mirror are solver work, not device truth.
Cusp-loss transport and the sheath-scale apertures are declared as configuration facets; nothing evaluates them.
Dipole equilibrium, the compressibility-driven turbulent pinch and the relativistic shift of the electron resonance are outside the levitated-dipole core's closed forms, and its non-claims say so.