Historical research guide · 0734 · 2012
Modelling of the internal dynamics and density in a tens of joules plasma focus device
Ariel Márquez · José González · Ariel Tarifeño-Saldivia · Cristian Pavez · Leopoldo Soto · Alejandro Clausse
In one minute
A plasma focus is two coaxial electrodes and a capacitor bank. Fire the bank and a sheet of current peels off the insulator, races down the barrel, folds over the end of the anode and collapses onto the axis, squeezing a knot of plasma called the pinch — hot enough and dense enough that deuterium in it fuses. Ariel Márquez, José González, Ariel Tarifeño-Saldivia, Cristian Pavez, Leopoldo Soto and Alejandro Clausse wanted a model of that collapse simple enough to run in seconds and honest enough to trust. They wrote the pinch as a small set of coupled equations, assuming smooth shapes for the velocity and density inside the column rather than solving the full magnetohydrodynamics, and fed it starting conditions from the earlier phases treated as flat shock waves. Then they tested it against a real machine — PF-50J at the Chilean Nuclear Energy Commission, storing only tens of joules — measuring the pinch with a laser interferometer. The model reproduces when the pinch happens and how the electron density is distributed across it.
Connection
Chapter 12 needs fusion-relevant plasma on a bench rather than in a hall, and this is the design tool for that class of machine: a model fast enough to sweep parameters and checked against a device that stores tens of joules. Chapter 9 studies self-organising, self-compressing plasma structures, and the pinch is the cleanest laboratory example of one.
From the source
The pinch compression phase is described by a lumped-parameter model built on the Von Karman method: instead of solving the full compressible magnetohydrodynamic equations, the authors postulate families of radial velocity and density profiles inside the pinch, with exponents of 0.3 for velocity and 2 for density, and integrate the conservation equations over them. The earlier axial and radial phases are modelled as planar shock waves, which supply the pinch equations with their initial conditions.Section II, Model description, Equations 1 to 3; Figure 2
The machine tested is PF-50J, a small fast plasma focus built at the Comisión Chilena de Energía Nuclear: a nominal 160 nanofarad bank charged to between 20 and 30 kilovolts, storing between 32 and 72 joules, delivering 40 to 60 kiloamperes to the plasma in about 150 nanoseconds. The anode is a copper tube of 6 millimetres outer diameter with an effective length of 3.32 millimetres, and the shots reported here ran in deuterium at 6 millibar and 29 kilovolts.Section III, Experiment, first paragraph; Figure 4
The pinch was measured directly rather than inferred from the circuit. A Mach-Zehnder interferometer using an 8 nanosecond pulse from a frequency-doubled Nd:YAG laser at 532 nanometres, read on a CCD at 0.019 millimetres per pixel and inverted through the Abel integral, gave an average on-axis pinch density of about 3.7 × 10²⁴ per cubic metre, a maximum pinch density of about 1.6 × 10²⁵ per cubic metre, and a pinch line density of about 2 × 10¹⁸ per metre. The pinch begins near 129 nanoseconds and lives about 10 nanoseconds.Section III, Experiment, paragraphs 2 to 4; Figures 5, 6 and 8
The same shot that gave those densities also gave neutrons: a yield of 1.6 × 10⁴ neutrons per burst, measured with moderated proportional counters. The model’s own preliminary estimate of the emission is about 10⁴ neutrons per burst, the same order as the measurement.Section III, final paragraph; Section IV, final sentence
The model reproduces the experiment where it counts. Calculated maximum-compression electron densities are 1.6 × 10¹⁹ per cubic centimetre at the axis and 0.25 × 10¹⁹ at the pinch border, and the calculated radial profile matches the interferometric data point for point; the radial sweeping coefficient the model needs, 0.33, is close to the value estimated from the measurements. The maximum densities are also insensitive to a 15 percent change in either profile exponent, so the agreement is not an artefact of the fitted shapes.Section IV, Numerical results; Figures 9, 10, 11 and 12
The strongest control knob is the charging voltage. The sensitivity table gives a coefficient of 6.7 for average pinch density against initial voltage — a one percent change in voltage moves the density by almost seven percent — against minus 1.8 for filling pressure, with minimum pinch radius at minus 2.1 and 1.2 respectively. What to watch is the authors’ own next step: apply the same model to higher-energy devices to look for invariants across energy, electrode geometry and filling pressure, and add validated radiation-production mechanisms inside the pinch.Table I; Section V, Final remarks
The way in
https://doi.org/10.1063/1.3672005How to cite it
Ariel Márquez, José González, Ariel Tarifeño-Saldivia, Cristian Pavez, Leopoldo Soto, Alejandro Clausse (2012) Modelling of the internal dynamics and density in a tens of joules plasma focus device doi:10.1063/1.3672005