Pinch Conditions Calculator

A z-pinch is a cylindrical column of plasma carrying an electric current along its axis (conventionally labeled the z-axis). The current generates its own azimuthal magnetic field, which exerts an inward Lorentz force (j × B) that compresses, or 'pinches,' the plasma column radially.

The Bennett relation, derived by Willard Bennett in 1934, is the equilibrium condition for a z-pinch: I² = (8π/μ₀)Ne(T_e+T_i), where N is the linear particle density (particles per unit length), and T_e, T_i are the electron and ion temperatures. It states the current required for the pinch's magnetic pressure to exactly balance the plasma's thermal pressure.

This is one of the most striking features of the result: integrating the radial pressure balance equation across the entire column causes the radius-dependent terms to cancel, leaving a relation that depends only on the total linear particle density N, not on how that density is distributed radially or how large the column is.

Equilibrium means the inward magnetic pressure from the pinch current exactly balances the outward thermal (kinetic) pressure of the hot plasma, so the column's radius is momentarily stable. This is a necessary but not sufficient condition, real pinches are typically equilibria that are magnetohydrodynamically unstable and evolve rapidly.

Z-pinches are used to generate intense, brief bursts of X-rays and neutrons (used in inertial confinement fusion research, most notably the Z Machine at Sandia National Laboratories), in dense plasma focus fusion devices, and historically were among the earliest controlled fusion confinement schemes attempted.

Simple z-pinches suffer from two classic magnetohydrodynamic instabilities: the 'sausage' (m=0) mode, where local constrictions in the column grow uncontrollably, and the 'kink' (m=1) mode, where the column bends sideways. Both grow from the same pinch force that provides confinement, making pure z-pinches inherently short-lived without additional stabilization.

Plasma beta compares thermal to magnetic pressure generally for any magnetized plasma configuration, while the Bennett relation is a specific equilibrium solution for the particular geometry of a current-carrying cylindrical pinch, directly relating the total current to the linear density and temperature rather than to a local field-strength ratio.

No, the general form of the relation, I² = (8π/μ₀)Ne(T_e+T_i), simply sums the electron and ion thermal pressure contributions, and this calculator supports entering separate T_e and T_i values rather than assuming they are equal.

N is the number of particles per unit length along the pinch axis (particles per metre), not a volumetric density. It equals the integral of the volumetric density n(r) over the column's cross-sectional area.

Laboratory z-pinch and dense plasma focus experiments typically operate with currents ranging from tens of kiloamps to several megaamps, depending on the device scale, with the largest pulsed-power machines like the Z Machine reaching currents in the tens of megaamps for brief pulses.