A model of hard X-rays emission from free expanding Plasma-Focus discharges

E. Fogliatto, J. González, M. Barbaglia and A. Clausse, A model of hard X-rays emission from free expanding Plasma-Focus discharges, Journal of Physics: Conference Series 511, article 012036, 2014, in the proceedings of ICPP2010 and LAWPP2010. Published at doi.org/10.1088/1742-6596/511/1/012036.

Reproduced in full under the Creative Commons Attribution 3.0 licence stated in the article. Attribution: E. Fogliatto, J. González, M. Barbaglia and A. Clausse, Journal of Physics: Conference Series 511 (2014) 012036, doi:10.1088/1742-6596/511/1/012036. Display equations are restated in words, marked as such; nothing else is altered, and everything outside this article — the summary, the claims and this note — is the site’s own writing.

Abstract

A planar-piston model of the hard x-ray production in Plasma-Focus devices is presented. The model applies Von Karman approximations to represent the inner structure of the pinch. The hard x-ray emission is calculated assuming Bremsstrahlung from the collision on the anode base of an electron current running away from the pinch. The model was applied to analyse the experimental data of a small Plasma Focus without surrounding cathode, finding good agreement.

1. Introduction

Plasma-Focus (PF) devices produce hot (about 0.3 to 1 keV) and dense (about ten to the twenty-fifth per cubic metre) dynamic pinches inducing ionization by means of high voltage discharges in coaxial geometries. Recently, small devices were developed aiming to applications of the pulsed radiation emissions of the pinch. Miniature devices were constructed following a special configuration derived from capillary discharges, where the surrounding cathode is replaced by a circular plate located at the base of the gun. This feature allows the current sheath to expand freely during the run down.

During the compression x-rays are produced by two mechanisms. A soft x-ray source inside the pinch is produced by line radiation, Coulomb interaction between charged particles and recombination. A hard x-ray source of Bremsstrahlung photons are produced by electrons that escape from the pinch and hit the anode base material. This work presents an extension of a lumped parameter code of free expanding PF [1], introducing a model of electron runaway and hard x-ray production. The model is validated with experimental data of a small PF device.

2. Model Description

A PF are basically two coaxial electrodes separated by an insulator material at one end (the anode at the interior). The interelectrode space is filled with low pressure gas and pulsed high voltage is applied between the electrodes. The discharge starts over the insulator from which the plasma sheath takes-off axially accelerated by the magnetic field generated by the current itself. After the current sheath runs over the open end of the central electrode, the plasma becomes rapidly compressed into a small column resulting in very hot and dense plasma (pinch). In cases where the cathode is just a planar plate, the current sheath expands freely radially away from the axis. In the lumped parameter model, the current sheet is represented by planar pistons following a snow-plough approximation. Initially the sheet takes the shape of the insulator and then expands in axial and radial direction, keeping the cylindrical symmetry and capturing gas with certain efficiency [2]. A constant linear mass density is assumed so that gas particles are re-distributed uniformly all over the sheet surface as they are captured, and this approximation is used to estimate the sheet thickness. For details of the evolution equations of the pistons see [1].

The pinch starts when the sheet collapses at the open end of the anode. The pinch is modeled as a plasma cylinder compressed by the Lorentz force and obeying the integral MHD equations. The inner structure of the plasma column is represented by means of Von Karman approximations of velocity and density profiles. The detailed set of equations of the pinch dynamics can be found in [3]. In the present version a current density profile depending on the radial coordinate r is included according to:

(Equation 1, restated in words: the current density at radius r and time t is the boundary current density K at that time, multiplied by the ratio of the radial coordinate to the pinch radius, raised to the power of a shape parameter.)

where R is the pinch radius, K is the current density at the external boundary, and a shape parameter sets how steeply the profile rises. This current profile brings about axial and radial components of electric field inside the pinch, which can be calculated using the Ohm's law:

(Equation 2, restated in words: the electric field plus the cross product of the velocity and the magnetic field equals the Spitzer resistivity multiplied by the current density.)

Where E, V, B, the resistivity and J represent the electric field, velocity, magnetic field, Spitzer's resistivity and current density respectively. Considering a magnetic field with only an azimuthal component

(Equation 3, restated in words: the azimuthal magnetic field at radius r and time t is one half of the vacuum permeability multiplied by the pinch current, divided by two pi times the radius.)

where the permeability is that of vacuum and the current is the pinch current, then the electric field is given by:

(Equations 4 to 6, restated in words: the field has a radial component and an axial component. The axial component combines a resistive term — the resistivity multiplied by the current, by the shape parameter plus two, and by the radial coordinate raised to the shape parameter, over two pi times the pinch radius raised to the shape parameter plus two — with a term driven by the radial velocity of the piston and the permeability. The radial component is set by twice the pinch radius over its length, multiplied by the permeability, the current, the axial velocity and the same radial power law.)

where the axial and radial velocities are those of the cylindrical surfaces respectively.

According to Dreicer theory, a fraction of the pinch electrons escape from the pinch accelerated by the axial field if the magnitude of the axial field is higher than the critical value:

(Equation 7, restated in words: the Dreicer field equals 3.9 times ten to the minus tenth, multiplied by the particle density in particles per cubic centimetre, divided by the temperature in electron volts.)

n being the particle pinch density.

Assuming that the energy of a single runaway electron is proportional to an effective voltage, gives

(Equation 8, restated in words: the electron energy is approximately the electron charge multiplied by the pinch length multiplied by the amount by which the axial field exceeds the Dreicer field.)

where H is the pinch length and q is the electron charge. Hence, using the appropriate conversion factors, the total energy produced by Bremsstrahlung in the anode base can be estimated as proportional to the square of the incident electron energy.

3. Results

In order to validate the model, the numerical results were compared with experimental measurements from a Plasma Focus with open cathode. Table 1 details the geometric and electrical parameters of the device [4]. Fig. 1 shows the measured dependence of the hard X-ray intensity on the charging voltage. The detectors were positioned at 90 degrees from the PF axis, and the Deuterium filling pressure was 16.4 mbar. The curve in Fig. 1 shows the numerical results obtained with the present model. The constant parameters used in the calculations are listed in Table 1. It can be seen that the model is able to follow the experimental trend. In particular, note that below 9 kV very low intensities were observed, which is in agreement with the absence of runaway due to the Dreicer condition. Another remarkable result is that a high exponent of the current density profile was needed to fit the experiments (a shape parameter of 50), indicating that the current is concentrated at the pinch border. The latter is in agreement with previous neutron production models where the current was assumed concentrated at the external boundary [5].

4. Conclusions

A lumped parameter model for estimating hard x-ray production in Plasma Focus discharges has been presented. Numerical calculations show good agreement with experimental data, particularly in predicting the dependence of the radiation intensity with the charging voltage. The model set out here offers a useful tool to calculate and design of open-cathode PF devices applied to production of pulsed beams of charged particles and hard x-rays.

Table 1. Parameters of the experiment and the model

• Bank capacity: 0.8 microfarad
• Charging voltage: 18 to 28 kV
• External inductance: 65 nanohenry
• Anode length: 0.191 cm
• Anode radius: 0.31 cm
• Filling pressure: 16.4 mbar
• Characteristic frequency of the spark gap: 350 per nanosecond
• Characteristic delay of the spark gap: 400 ns
• Efficiency parameter: 0.2
• Current shape parameter: 50

References

1. González J, Barbaglia M, Casanova F and Clausse A 2009 Brazilian Journal of Physics 39 633–637
2. Fogliatto E 2010 Investigación en generación de pulsos de radiación x en equipos plasma focus de baja energía, Instituto Balseiro, Bariloche, Argentina (in Spanish)
3. Márquez A 2010 Caracterización y modelado de fuentes pulsadas de neutrones del tipo plasma focus, Instituto Balseiro, Bariloche, Argentina (in Spanish)
4. Barbaglia M, Bruzzone H, Acuña H, Soto L and Clausse A 2009 Plasma Physics and Controlled Fusion 51 045001 (9pp)
5. González J, Brollo F and Clausse A 2009 IEEE Transactions on Plasma Science 37 2178–2185

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(End of the reproduced article. On this site, the Lee model code — the other widely used lumped-parameter treatment of the plasma focus — is at /library/stm-a9cc428c6d, with its numerical experiments at /library/stm-fbd01b5713; the same Argentine and Chilean groups' modelling and interferometry of tens-of-joules devices are at /library/stm-914ae7ba20 and /library/stm-0291726f47; the neutron side of the same machine architecture is at /library/stm-ef8e9744ca; and the programme aiming a dense plasma focus at aneutronic proton–boron-11 fuel is at /library/stm-8f6027c3eb.)