ANULUM / SCPN Reactor Systems / Electrostatic, beam-target and hybrid systems / Physics

The physics of electrostatic, beam-target and hybrid systems

The last family gathers the devices that confine or breed with something other than a closed torus: an electrostatic well that accelerates and recirculates ions toward a dense core, a beam whose kinematics set the reaction rate directly, and a subcritical fission blanket that multiplies the neutrons a fusion source makes. This page collects the closed-form relations the three 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 on a declared operating point at computational_prototype maturity: the printed geometry of a grid, a printed cross-section fit, printed figures of merit. No potential well, sheath, space charge, beam stopping, target density, blanket neutronics or criticality is computed anywhere in these cores; no fusion rate, yield, gain or breakeven statement is made, and the beam-target core states that no energy-gain claim of any kind is made or implied for its family; no value describes or validates any real machine.

1 · The spherical cathode grid of an IEC device

A gridded fusor holds a spherical wire cathode at tens of kilovolts below a concentric anode; ions fall through the transparent grid, converge at the centre, and recirculate until a grid wire or a charge exchange ends their life. The polywell replaces the wire grid with a virtual cathode of electrons trapped in a magnetic cusp, so nothing intercepts the ions. What the IEC core carries is the printed geometry of the grid — its combinatorics, its angles, the fraction of the sphere it leaves open — and the bound that fraction places on recirculation:

$$\alpha_{\mathrm{bridge}} = \arctan\frac{t_{\mathrm{bridge}}}{D_{\mathrm{grid}}}, \qquad N_{\mathrm{globe}} = 2\,n_{\mathrm{long}}(n_{\mathrm{lat}} + 1), \qquad \theta_{\mathrm{sym}} = \frac{360^\circ}{2n}, \qquad \eta = \frac{\sum A_{\mathrm{aperture}}}{4\pi R_{\mathrm{grid}}^2}, \qquad \eta_{\mathrm{circ}} = \frac{\sum 2\pi R^2(1-\cos\theta_{\min})}{4\pi R^2}$$ $$N_{\mathrm{passes}} \le \frac{\eta}{1 - \eta^2}$$Wulfkühler et al., Scientific Reports (2024), open access: eq. 5 bridge half-angle, eq. 12 globe aperture count, the four permissible symmetric grids with the crossing rule $360^\circ/2n$ (3, 6, 9, 15 rings → 8, 24, 48, 120 apertures at 90°, 60°, 45°, 36°), the geometric and circular transparencies and their ratio, and eq. 1 for the pass-count bound, refused outside $(0, 1)$: a grid of zero transparency passes nothing, and a transparency of exactly one is the polywell's virtual cathode, which bounds no number of passes at all. The bound sees no pressure, charge exchange, scattering or ion energy, and the same source states that most devices operate where an ion makes only a few passes. (SCPN-IEC-CORE.)

Anchors: all four bridge angles of the source's table to the three decimals it prints (1.146°, 0.573°, 0.382°, 0.229°); the 50 apertures in the caption of its globe-grid figure; both endpoints of its 8-to-220 globe family; every row of its symmetric-grid table, two of whose counts the same paper states again from two other laboratories; a nine-ring symmetric cathode, which Radel's UWFDM-1325 (2007) reports as built and operated at Wisconsin (20 cm cathode, 0.75 mm wire, 40 cm anode, in a 0.91 × 0.65 m chamber), reports the 48 apertures the other source tabulates — the number crosses two independent documents. Measured: the bridge-angle table cannot settle the form of its own equation, since dropping the arctangent reproduces all four angles to the digit; the pass-count denominator is evaluated factored, $(1-\eta)(1+\eta)$, because over 20029 transparencies the printed difference of squares disagrees at 7994 of them.

2 · One beam, one target: the cross section with the frame stated

A beam-target device makes fusion by kinematics alone — an energetic beam on a fixed or flowing target, or two beams brought to collision — without imploding or magnetically confining a thermal plasma. Its level-0 physics is the total cross section of the six principal light-ion reactions in the Duane fit, evaluated at the energy the fit was made for:

$$\sigma(E) = \frac{A_5 + A_2\big/\!\left[(A_4 - A_3E)^2 + 1\right]}{E\left[e^{A_1/\sqrt{E}} - 1\right]}\ \text{barn},\quad E\ \text{the incident-ion energy with the target at rest}; \qquad E_{\mathrm{equiv}} = 4\,E_{\mathrm{beam}}\ \text{for equal-mass colliding beams}$$ $$\langle\sigma v\rangle = \sqrt{\frac{8}{\pi\mu}}\,(kT)^{-3/2}\int\sigma(E)\,E\,e^{-E/kT}\,\mathrm{d}E, \qquad \dot N = \frac{I}{Ze}, \qquad P = \frac{E_{\mathrm{keV}}\,I_{\mathrm{mA}}}{Z}\ \mathrm{W}$$2019 NRL Plasma Formulary, p. 44: coefficients $A_1\ldots A_5$ for D–D (both branches), D–T, D–³He, T–T and T–³He; ion mass ratios $m_e/m_D = 2.72\times10^{-4} = 1/3670$, $m_e/m_T = 1.82\times10^{-4} = 1/5496$. The frame is the content of this family's physics: under the equal-mass kinematics the configuration uses, a colliding-beam machine reaches the centre-of-mass energy of a stationary-target machine at four times its own beam energy, exact and asserted as an equality; which of the two cross sections is larger is not fixed — below the resonance the colliding machine moves toward it and gains, at it is thrown past and loses. A singly charged beam of one milliampere at one kiloelectronvolt carries exactly one watt. (SCPN-BEAM-TARGET-CORE.)

Anchor, a cross-check inside one document: the formulary prints the Duane coefficients and, further down the same page, ten Maxwellian-averaged D–T reaction rates from 1 to 1000 keV; averaging the fit recovers every one to the printed two significant figures, the residual of at most 1.4 % being the table's own rounding (fifty times as many quadrature intervals move the answer by less than a part in ten thousand). The Gamow boundary at 0.0043 keV is tested from both sides. The cited cross-section works, Bosch & Hale and Wangler, are paywalled and not on file.

3 · A source, a blanket, no chain: the fusion–fission hybrid

A hybrid couples a fusion neutron source to a subcritical fission blanket — $k_{\mathrm{eff}}$ strictly below one as a declared boundary — that multiplies the energy and breeds fissile fuel for ordinary reactors. The hybrid core owns only the coupling; its source's physics belongs to whichever device core is declared. Its level-0 physics is the four figures of merit Saltmarsh, Grimes and Santoro publish:

$$\frac{P_T}{P_F} = \frac{1}{Q'} + 1 + f_n(M - 1), \qquad \eta_{\mathrm{hybrid}} = \eta_H - \frac{1/Q'}{P_T/P_F}, \qquad R_o = \frac{E_{\mathrm{fiss}}}{E_{\mathrm{fus}}}\,\frac{F}{(1-C)(1+\alpha)}, \qquad N_{\mathrm{reactors}} = \frac{R_o}{P_T/P_F}$$ $$R = \frac{1}{Q'}\left(1 - \frac{1}{\eta_H}\right) + 1 + f_n(M-1) + R_o \;=\; \frac{\eta_{HB}}{\eta_H}\frac{P_T}{P_F} + R_o$$ORNL/PPA-79/3 (1979), eqs. 1, 2, 4, 6, 7 and 5. $Q'$ is the engineering Q of the driver, $f_n$ the fraction of fusion power in neutrons (0.8 for D–T, 0.66 for the semicatalysed D–D), $M$ the blanket's energy multiplication, $F$ the fissile breeding rate per source neutron, $C$ the conversion ratio and $\alpha$ the capture-to-fission ratio of the reactors that burn what the blanket breeds; $E_{\mathrm{fiss}} = 200$ MeV. The energy multiplication $M$ and the neutron multiplication $1/(1-k_{\mathrm{eff}})$ the configuration's blanket computes are different quantities; no filed source relates them, so nothing here does either. The number of reactors rises with $Q'$ toward a ceiling set by the blanket, $Q'B/(1+Q'B)$. (SCPN-FUSION-FISSION-HYBRID-CORE.)

Anchors, six numbers the report prints, each recovered from a built record or relation: the 1.33 in the denominator of eq. 17 (exactly, as the same IEEE double); $R_o = 68$ of eq. 19 (68.15); $Q' \approx 1.4$ for electrical self-sufficiency of the molten-salt thorium hybrid (1.396); thorium blankets supporting 3–5× the uranium ones (4.70 fresh, 4.03 exposed); a larger blanket multiplication reaching its ceiling at a lower $Q'$; about 3 % fissile buildup roughly halving the reactor number (2.08 uranium, 2.42 thorium). The filed copy is a scan whose OCR mangles digits, so every value was read off the rendered page. The eq. 7 against eq. 5 agreement is asserted within a tolerance because 317 of 6372 parameter points disagree in the last places: floating-point multiplication is not associative.

4 · Where the family ends

Purely magnetic cusp confinement is the open-field family's; the polywell is here because its energy and reaction workflow is an electrostatic well, whatever its cusp topology.
Beam-driven implosion is the inertial family's, and the dense plasma focus keeps its internal beam-target contribution as its own device truth.
The hybrid's fusion source is never owned here: the core couples to a declared source device core — tokamak, mirror, MIF or other — and makes no nuclear-safety, criticality-safety or licensing claim of any kind.
Solver mathematics stays in SCPN-Fusion-Core.

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Explorer

Grid geometry and the pass bound, the Duane fit in both frames, the hybrid's four figures — live.

The three configurations

What each core owns, excludes and anchors on.

Sources

Every printed value and where it comes from.