In a pinch the confining field is made by the current that flows through the plasma. The Z-pinch drives an axial current whose
azimuthal field squeezes the column; the theta-pinch drives an azimuthal current from a coil so the axial field does the squeezing; the dense
plasma focus runs a current sheath down coaxial electrodes and rolls it into a short, hot pinch at the end. This page collects the
closed-form relations the three device cores evaluate, in the order the family's own literature states them.
What these relations are. Every relation on this page is one a device core of this family implements as a closed-form evaluation of a cited published model on a synthetic configuration at computational_prototype maturity. No equilibrium, stability, compression or transport equation is solved and no eigenvalue problem exists in the cores; the anchors reproduce numbers the sources print, which for the plasma focus are themselves outputs of the source's fitted code, and are not correlations with experimental data. No reactivity, yield, gain, breakeven or confinement-time statement is made; no value describes or validates any real machine.
1 · The Z-pinch: Bennett equilibrium
A column carrying an axial current \(I\) with an ion line density \(N\) balances its kinetic pressure against the magnetic pressure of its own azimuthal field. Integrated over the cross-section this is Bennett's relation; the Bennett profile is the particular equilibrium the core evaluates, with its field, enclosed current and Alfvén time:
$$\frac{\mu_0 I^2}{8\pi} = N k_B (T_e + T_i), \qquad n(r) = \frac{n_0}{(1 + r^2/a^2)^2},\; n_0 = \frac{N}{\pi a^2}, \qquad B_\theta(r) = \frac{\mu_0 I r}{2\pi (a^2 + r^2)}, \qquad I(r) = I\,\frac{r^2}{a^2 + r^2}$$
$$v_A = \frac{B_\theta(a)}{\sqrt{\mu_0 n_0 m_i}}, \qquad \tau_A = \frac{a}{v_A}$$Bennett, Phys. Rev. 45 (1934) 890; Haines, Plasma Phys. Control. Fusion 53 (2011) 093001, §2. Tests close the pressure balance to machine precision, integrate the profile back to the line density, verify Ampère's law at sampled radii and check the \(I^2\), \(1/N\) and \(m_i^{-1/2}\) scalings. (SCPN-Z-PINCH-CORE.)
Sausage and kink, and what stabilises them
The Bennett column is unstable: the \(m = 0\) sausage and \(m = 1\) kink modes grow on the Alfvén time. The core carries the order-of-magnitude growth rate, Kadomtsev's \(m = 0\) criterion evaluated on the Bennett profile in closed form, and the sheared-axial-flow threshold that names the family's second configuration:
$$\gamma \sim k v_A, \qquad -\frac{\mathrm{d}\ln p}{\mathrm{d}\ln r} < \frac{4\gamma_{\mathrm{ad}}}{2 + \gamma_{\mathrm{ad}}\beta}\ \text{with}\ -\frac{\mathrm{d}\ln p}{\mathrm{d}\ln r} = \frac{4x^2}{1+x^2},\; \beta = \frac{1}{x^2},\; x = \frac{r}{a}, \qquad \frac{\mathrm{d}v_z}{\mathrm{d}r} > 0.1\,k v_A$$Haines 2011 §5; Kadomtsev, Reviews of Plasma Physics 2 (1966) 153; Shumlak & Hartman, PRL 75 (1995) 3285. Tests reproduce the published conclusion that the Bennett profile is sausage-unstable at every radius for \(\gamma_{\mathrm{ad}} = 5/3\) and that the reduced criterion flips exactly at \(\gamma_{\mathrm{ad}} = 2\). The wavenumber and the shear are declared inputs; no spectrum is computed.
The Pease–Braginskii current
Above a critical current the bremsstrahlung of a Z-pinch exceeds its ohmic heating and the column contracts radiatively. The core evaluates the published closed form with the NRL Formulary Spitzer and bremsstrahlung coefficients converted to SI:
$$I_{\mathrm{PB}} = \frac{\pi}{\mu_0}\sqrt{\frac{48\,\ln\Lambda}{\sigma_0 A}}\;\frac{1+z}{z}, \qquad \sigma = \sigma_0\frac{(k_BT_e)^{3/2}}{z\ln\Lambda}, \qquad P_{\mathrm{br}} = A\,z^3 n_i^2 (k_BT_e)^{1/2}$$Pease 1957, Braginskii 1958, in the closed form of Klíř, The Study of a Fibre Z-Pinch, CTU Prague (2005), arXiv:physics/0703207, eq. (2.20). For a hydrogenic plasma at \(\ln\Lambda = 10\) the core reproduces 1.37 MA against the literature's ≈ 1.4 MA, and the \(\sqrt{\ln\Lambda}\) scaling. The regime label is an energy-balance disposition of the cited model, not a prediction.
2 · The theta-pinch: sharp-boundary balance and the Scyllac review
A fast-rising axial field from a single-turn coil induces an azimuthal current in the column; the interaction implodes and compresses the plasma. The core's sharp-boundary state is the pressure balance of the Scyllac review, and every further relation needs \(0 < \beta < 1\):
$$\beta = \frac{p}{B^2/2\mu_0}, \qquad T = \frac{p}{2 n e}, \qquad v_A = \frac{B}{\sqrt{\mu_0 n m_i}}, \qquad \tau_A = \frac{L/2}{v_A}$$Quinn et al., Review of Scyllac theta-pinch experiments, LA-UR-73-1053 (1973). \(\beta = 1\) is allowed by the configuration and refused by every sharp-boundary model, never clamped. (SCPN-THETA-PINCH-CORE.)
Toroidal equilibrium with \(l = 1, 0\) fields
To close a theta-pinch into a torus, Scyllac added helical (\(l = 1\)) and bumpy (\(l = 0\)) fields whose plasma excursions produce a force that balances the toroidal drift:
$$\delta_1 = \frac{B_1/B_0}{h a (1 - \beta/2)}, \qquad \delta_0 = -\frac{B_{0l}/B_0}{2(1-\beta)}, \qquad \delta_1\delta_0 = -\frac{2}{(3 - 2\beta)\,h^2 a R}\ \ (\text{eq. 7}), \qquad \frac{B_v}{B_0} = \frac{B_{1,2}}{B_0} = \frac{B_1 B_{0l}}{4 B_0^2}\ \ (\text{eq. 3})$$Anchor: the 5-m sector point of Fig. 2 (\(\beta = 0.85\), \(a = 0.7\) cm, \(R = 2.375\) m, \(h = 0.19\) cm⁻¹) yields a required product of −0.0059 against the measured −0.0064 and the plotted sharp-boundary value ≈ −0.0065, within a declared 10 % tolerance; the quotient forms of the excursions are the ones that reproduce it, which resolves the scan's typographical ambiguity.
\(m = 1\) growth and wall stabilisation
$$\gamma = h v_A\sqrt{-\beta^2\left(\tfrac{a}{b}\right)^4\delta_1^2 + \frac{\beta(4-3\beta)(2-\beta)}{8(1-\beta)}h^2a^2\delta_1^2 + \frac{\beta(3-2\beta)(1-\beta)}{2-\beta}\delta_0^2}, \qquad \left(\frac{a}{b}\right)^4 \ge \frac{(4-3\beta)(2-\beta)(ha)^2}{8\beta(1-\beta)}$$Eqs. 4 and 6 of the review; the wall condition derived from eq. 6 (the source prints eq. 8 without the \(8\beta\) factor, but its worked example \(a = 3\) cm, \(\beta = 0.8\), \(ha = 0.13 \to a/b = 0.4\) is reproduced to 0.399). With the source's stated orders of magnitude for the 5-m sector (\(B \sim 3.6\) T, \(n \sim 2.5\times10^{22}\) m⁻³) the growth rate lands within a factor of two of the source's 1.0 MHz; the source does not print every input of its own calculation, so no tighter statement is made.
End loss of the open-ended column
$$\tau = \tau_{\mathrm{ref}}\,\frac{L}{L_{\mathrm{ref}}}\sqrt{\frac{T_{\mathrm{ref}}}{T_i}}, \qquad L_{\mathrm{ref}} = 5\ \mathrm{m},\; T_{\mathrm{ref}} = 2.7\ \mathrm{keV},\; \tau_{\mathrm{ref}} = 11.5\ \mu\mathrm{s}$$An empirical three-device fit (review p. 16 and Table I) that the source itself contrasts with two disagreeing theoretical models; the Scylla IV-1 and IV-3 rows (2.13 and 9.67 µs) are reproduced within 1 %.
3 · The dense plasma focus: the Lee model in closed forms
A capacitor bank discharges across an insulator between coaxial electrodes; the current sheath lifts, runs down the annulus as a snowplow, rolls over the anode tip and collapses radially into a short dense pinch where instability-driven ion beams and hot spots produce the neutron yield. The core evaluates the closed forms of the Lee model without integrating any of its phases:
$$E_0 = \tfrac12 C_0 V_0^2, \qquad t_0 = \sqrt{L_0 C_0}, \qquad Z_0 = \sqrt{L_0/C_0}, \qquad I_0 = \frac{V_0}{Z_0}, \qquad t_{\mathrm{rise}} = \frac{\pi}{2}t_0, \qquad L_a = \frac{\mu_0}{2\pi}\ln c\; z_0,\; c = \frac{b}{a}, \qquad \beta = \frac{L_0}{L_a}, \qquad \delta = \frac{r_0}{Z_0}$$S. Lee, Plasma Focus Radiative Model: Review of the Lee Model Code, J. Fusion Energ. 33 (2014), eqs. 4–6, 9, 43. The \(E_0\) and \(t_{\mathrm{rise}}\) columns of the twelve-machine table (Saw & Lee, IAEA-TECDOC-1829, Table 1) for PF1000, NX3, INTI and PF400J are reproduced within 2.5 % and 2 %, the table's own rounding. (SCPN-DENSE-PLASMA-FOCUS-CORE.)
Axial and radial phases
$$\alpha = \frac{t_0}{t_a}, \qquad v_a = \frac{z_0}{t_a}, \qquad v_\infty(I) = \left[\frac{f_c^2}{f_m}\,\frac{\mu_0 \ln c}{4\pi^2\rho_0 (c^2-1)}\right]^{1/2}\frac{I}{a}, \qquad \frac{v_r}{v_a} = \left[\frac{(c^2-1)(\gamma+1)}{4\ln c}\right]^{1/2}$$Eqs. 5–7 and eq. 1 at zero acceleration; eqs. 24–28. The terminal snowplow speed at the peak current overestimates the table's \(v_a\) column by 7–14 % on every row, and the tests assert that sign: the deviation is evidence, not noise. At \(c = 3.4\), \(\gamma = 5/3\) the speed ratio is 2.40 against the printed "typically 2.5". Rule-of-thumb geometry (ICTP Table 3): \(r_{\min} = 0.15a\), \(z_{\max} = 1.5a\), radial-shock transit \(5\times10^{-6}a\) s, pinch lifetime \(10^{-6}a\) s for deuterium.
Pinch, beam and neutron closed forms
The pinch-phase forms (eqs. 39–48: density, Bennett temperature, Spitzer resistance, Joule, bremsstrahlung, line and surface emission with both self-absorption branches) and the fast-ion-beam chain (TECDOC eqs. 5–6, items (a)–(k): flux, beam speed, energy flux, power flow, ion current, fluence, ions, beam energy, damage factor) are evaluated against their definitions; the beam chain reproduces the PF1000, NX3 and INTI columns within 3 % and PF400J within 12 %. The tabulated pinch temperature and density are outputs of the integrated code and are not reproduced by the closed forms at the tabulated inputs, which the record states. Two neutron estimates close the chain:
$$Y_{\mathrm{bt}} = C_n\, n_i I_{\mathrm{pinch}}^2 z_p^2 \ln\frac{b}{r_p}\,\frac{\sigma}{U^{1/2}},\; C_n = 8.54\times10^8, \qquad Y_n = 9\times10^{10}\, I_{\mathrm{pinch}}^{3.8}\ \ (I\ \mathrm{in\ MA},\ 0.1\text{–}1\ \mathrm{MA})$$Lee 2014 eq. 50 with a declared cross-section; the empirical scaling law reproduces its own calibration point (\(7\times10^9\) at 0.5 MA) within 10 % and is refused outside its range. Both are consistency instruments at declared inputs; the thermonuclear term is not implemented.
4 · Where the family ends
MagLIF is a pinch by drive — an axial current through a liner — and a magneto-inertial device by physics; it belongs to the magneto-inertial family.
The reversed-field pinch is a toroidal relaxed state, not a linear column; it belongs to the closed magnetic family.
Beam-target fusion with externally accelerated beams is the electrostatic and hybrid family's; the plasma focus's internal beam-target contribution is device truth here.
Solver mathematics stays in SCPN-Fusion-Core; no core of this family copies or dispatches a solver.