IPFS

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Institute for Plasma Focus Studies

Internet Workshop on Numerical Plasma Focus Experiments

(Supplementary Notes for Sessions 7-8)

[in part extracted from file 2Theory.pdf from: http://www.intimal.edu.my/school/fas/UFLF/ ] and from various papers

Radiation Terms

The Bremsstrahlung loss term may be written as:

Recombination loss term is written as:

The line loss term is written as:

where dQ/dt is the total power gain/loss

of the plasma column

By this coupling, if, for example, the radiation loss is severe, this would lead to a large value of inwards. In the extreme case, this leads to radiation collapse, with r_p going rapidly to zero, or to such small values that the plasma becomes opaque to the outgoing radiation, thus stopping the radiation loss.

This radiation collapse occurs at a critical current of 1.6 MA (the Pease-Braginski current) for deuterium. For gases such as Neon or Argon, because of intense line radiation, the critical current is reduced to even below 100kA, depending on the plasma temperature.

Plasma Self Absorption and transition from volumetric emission to surface emission

Plasma self absorption and volumetric (emission described above) to surface emission of the pinch column have been implemented in the following manner.

The photonic excitation number (see File 3 Appendix by N A D Khattak) is written as follows:

M = 1.66 x 10 ^-15r_p Z_n ^0.5 n_i / (Z T^1.5) with T in eV, rest in SI units

The volumetric plasma self-absorption correction factor A is obtained in the following manner:

A₁ = (1 + 10^-14n_i Z) / (T ^3.5))

A₂ = 1 / AB₁

A = A₂ ^{(1 + M)}

Transition from volumetric to surface emission occurs when the absorption correction factor goes from 1 (no absorption) down to 1/e (e=2.718) when the emission becomes surface-like given by the expression:

where the constant const is taken as 4.62x10^-16 to conform with numerical experimental observations that this value enables the smoothest transition, in general, in terms of power values from volumetric to surface emission.

Where necessary another fine adjustment is made at the transition point adjusting the constant so that the surface emission power becomes the same value as the absorption corrected volumetric emission power at the transition point. Beyond the transition point (with A less than 1/e) radiation emission power is taken to be the surface emission power.

Neutron Yield

http://www.intimal.edu.my/school/fas/UFLF/

Adapted from the following papers (with modifications for erratum)

Pinch current limitation effect in plasma focus (This version includes an Erratum)

S. Lee and S. H. Saw, Appl. Phys. Lett. 92, 021503 (2008), DOI:10.1063/1.2827579

Copyright (2008) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. This article appeared in (citation above) and may be found at

http://link.aip.org/link/?APPLAB/92/021503/1

Neutron Scaling Laws from Numerical Experiments (This version includes an Erratum)

S Lee and S H Saw, J of Fusion Energy, DOI: 10.1007/s10894-008-9132-7

published first online 20 February 2008 at http://dx.doi.org/10.1007/s10894-008-9132-7

"The original publication is available at www.springerlink.com."

Neutron yield is calculated with two components, thermonuclear term and beam-target term.

The thermonuclear term is taken as:

dY_th = 0.5n_i²( 3.142)r_p²z_f<sv>(time interval)

Where <sv> is the thermalised fusion cross section-velocity product corresponding to the plasma temperature, for the time interval under consideration. The yield Y_th is obtained by summing up over all intervals during the focus pinch.

The beam-target term is derived using the following phenomenological beam-target neutron generating mechanism¹⁷, incorporated in the present RADPFV5.13. A beam of fast deuteron ions is produced by diode action in a thin layer close to the anode, with plasma disruptions generating the necessary high voltages. The beam interacts with the hot dense plasma of the focus pinch column to produce the fusion neutrons. In this modeling each factor contributing to the yield is estimated as a proportional quantity and the yield is obtained as an expression with proportionality constant. The yield is then calibrated against a known experimental point.

The beam-target yield is written in the form: Y_b-t ~n_b n_i (r_p²z_p) (s v_b) t

where n_b is the number of beam ions per unit plasma volume, n_i is the ion density, r_p is the radius of the plasma pinch with length z_p, s the cross-section of the D-D fusion reaction, n- branch¹⁸, v_b the beam ion speed and t is the beam-target interaction time assumed proportional to the confinement time of the plasma column.

Total beam energy is estimated¹⁷ as proportional to L_pI_pinch², a measure of the pinch inductance energy, L_p being the focus pinch inductance. Thus the number of beam ions is N_b~L_pI_pinch²/v_b² and n_b is N_b divided by the focus pinch volume. Note that L_p~ln(b/r_p)z_p , that⁴ t~r_p~z_p , and that v_b~U^1/2 where U is the disruption-caused diode voltage¹⁷. Here ‘b’ is the cathode radius. We also assume reasonably that U is proportional to V_max, the maximum voltage induced by the current sheet collapsing radially towards the axis.

Hence we derive: Y_b-t= C_n n_i I_pinch²z_p²((lnb/r_p))s/V_max^1/2 (1)

where I_pinch is the current flowing through the pinch at start of the slow compression phase; r_p and z_p are the pinch dimensions at end of that phase. Here C_n is a constant which in practice we will calibrate with an experimental point.

The D-D cross-section is highly sensitive to the beam energy so it is necessary to use the appropriate range of beam energy to compute s. The code computes V_max of the order of 20-50 kV. However it is known¹⁷, from experiments that the ion energy responsible for the beam-target neutrons is in the range 50-150keV¹⁷, and for smaller lower-voltage machines the relevant energy ¹⁹ could be lower at 30-60keV. Thus to align with experimental observations the D-D cross section s is reasonably obtained by using beam energy equal to 3 times V_max.

A plot of experimentally measured neutron yield Y_n vs I_pinch was made combining all available experimental data^{2,4,12,13,17,19-22}. This gave a fit of Y_n=9x10¹⁰I_pinch^3.8 for I_pinch in the range 0.1-1MA. From this plot a calibration point was chosen at 0.5MA, Y_n=7x10⁹ neutrons. The model code²³ RADPFV5.13 was thus calibrated to compute Y_b-t which in our model is the same as Y_n.

Notes on The total current and I_peak, the plasma current and I_pinch

Extracted From: Computing Plasma Focus Pinch Current from Total Current Measurement

S. Lee, S. H. Saw, P. C. K. Lee, R. S. Rawat and H. Schmidt, Appl Phys Letters 92, 111501 (2008) DOI:10.1063/1.2899632

The total current I_total waveform in a plasma focus discharge is easily measured using a Rogowski coil. The peak value I_peak of this trace is commonly taken as a measure of the drive efficacy and is often used to scale the yield performance of the plasma focus. This is despite the fact that yields should more consistently be scaled to focus pinch current I_pinch, since it is I_pinch which directly powers the emission processes. The reason many researchers use I_peak instead of I_pinch for scaling is simply that while I_peak is easily measured, I_pinch, which is the value of the plasma sheath current Ip at time of pinch, is very difficult to measure even in large devices where it is possible to place magnetic probes near the pinch. This measurement is also inaccurate and perturbs the pinch. In a small device, there is no space for such a measurement.

The relationship between _Ipinch and I_peak is not simple and has only been recently elaborated. It primarily depends on the value of the static inductance L₀ compared to the dynamic inductances of the plasma focus. As L₀ is reduced, the ratio I_pinch / I_peak drops. Thus, yield laws scaled to I_peak will not consistently apply when comparing two devices with all parameters equal but differing significantly in L₀. Better consistency is achieved when yield laws are scaled to I_pinch. In this paper, we propose a numerical method to consistently

Distinguishing the I_total waveform from the I_p waveform

A measured trace of I_total is commonly obtained with a Rogowski coil wrapped around the plasma focus flange through which is fed I_total discharged from the capacitor bank between the coaxial electrodes across the back wall. A part of I_total, being the plasma sheath current I_p, lifts off the back-wall insulator and drives a shock wave axially down the coaxial space. We denote f_c as the current fraction I_p/_Itotal for the axial phase and f_cr for the radial phases. In modeling it is found that a reasonable value for initial trial for f_c is 0.7 with a similar first trial value for f_cr . However in a DPF78 experiment f_c was found to vary from 0 at the start of the axial phase rising rapidly above 0.6 for the rest of the axial phase. In the radial phase f_cr was found to stay above 0.6 before dropping to 0.48 at the start of the pinch and then towards 0.4 as the pinch phase progressed. These Stuttgart results confirm a complex relationship between the waveforms of I_total and I_p.

The performance of a plasma focus is closely linked to the current I_pinch actually participating in the focus pinch phase rather than the total current flowing in the circuit. It is a common practice to take I_peak or some representative fraction of it as I_pinch. Another practice is to take the value of I_total at the time of the pinch as I_pinch Whilst in their special cases this practice could be justifiable, the distinction of I_p from I_total should generally be clearly made. We emphasize that it should be the value of I_p at the time of pinch which is the relevant value for the purpose of yield scaling. The practice of associating yield scaling with the total current waveform (i.e. taking I_peak or I_total at estimated pinch time) would be justifiable if there were a linear relationship between the waveforms of I_total and I_p. However as shown by the Stuttgart experiments the actual relationship is a very complex one which we may ascribe to the interplay of the various electro-dynamical processes including the relative values of static inductance L_o, tube inductance and the dynamic resistances which depend on the tube geometry and plasma sheath speeds. This relationship may change from one machine to the next. Whilst these electro-dynamical processes and other relevant ones such as radiation are amenable to modeling there are other machine effects such as back wall restriking (for example due to high induced voltages during the pinch phase) which can almost unpredictably affect the relationship between I_total and I_p during the crucial radial phases. Hence it is not only simplistic to discuss scaling in terms of the I_total waveform (i.e. taking Ipeak or the value of Itotal at the estimated time of pinch) but also inconsistent. One of the most important features of a plasma focus is its neutron production. The well-known neutron yield scaling, with respect to current, based on various compilations of experimental data, is Y_n ∼ I_pinch^x where x is varied in the range 3–5. In a recent paper , numerical experiments using a code was used to derive a scaling with x = 4.7. Difficulties in the interpretation of experimental data ranging across big and small plasma focus devices include the assignment of the representative neutron yield Y_n for any specific machine and the assignment of the value of I_pinch. In a few larger machines attempts were made to measure I_pinch using magnetic probes placed near the pinch region, with uncertainties of 20%. Moreover the probes would have affected the pinching processes. In most other cases related to yield scaling data compilation or interpretation I_pinch is simply assigned a value based on the measurement of peak total current I_peak or the value of total current at the observed current dip.

The difficulties in distinguishing Ipinch from I_total are obviated in numerical experiments using the Lee Model [In a typical simulation, the I_total trace is computed and fitted to a measured I_total trace from the particular focus. Three model parameters for fitting are used: axial mass swept-up factor f_m, current factor fc and radial mass factor f_mr. A fourth model parameter, radial current factor, f_cr may also be used. When correctly fitted the computed Itotal trace agrees with the measured I trace in peak amplitude, rising slope profile and topping profile which characterize the axial phase electro-dynamics. The radial

phase characteristics are reflected in the roll-over of the current trace from the flattened top region, and the subsequent current drop or dip. Any machine effects, such as restrikes, current sheath leakage and consequential incomplete mass swept up, not included in the simulation physics is taken care of by the final choice of the model parameters, which are fine-tuned in the feature-by-feature comparison of the computed I_total trace with the measured I_total trace. Then there is confidence that the computed gross dynamics, temperature, density, radiation, plasma sheath currents, pinch current and neutron yield may also be realistically compared with experimental values.

A note on scaling:

Scaling of yields to say I_pinch should be carried out using yields which are at optimum, or at least near optimum. If one indiscriminately uses any data one may end up with completely trivial or misleading results. For example if a point is used at too high or low pressure (away from the optimum pressure) then there may be zero yield ascribed to values of I_pinch.

¹ Lee S 1984 Radiations in Plasmas ed B McNamara (World Scientific) pp 978–87

Also: S. Lee in Laser and Plasma Technology, edited by S. Lee, B. C. Tan, C. S.

Wong, & A. C. Chew. World Scientific, Singapore, (1985), pp. 387–420.

²S. Lee, T. Y. Tou, S. P. Moo, M. A. Elissa, A. V. Gholap, K. H. Kwek, S.

Mulyodrono, A. J. Smith, Suryadi, W. Usala, & M. Zakaullah, Am. J.

Phys. 56, 62 (1988).

³T. Y. Tou, S. Lee, & K. H. Kwek, IEEE Trans. Plasma Sci. 17, 311 (1989).

⁴S. Lee and A. Serban, IEEE Trans. Plasma Sci. 24, 1101 (1996).

⁵D. E. Potter, Phys. Fluids 14, 1911 (1971).

⁶M. H. Liu, X. P. Feng, S. V. Springham, and S. Lee, IEEE Trans. Plasma

Sci. 26, 135(1998).

⁷S. Lee, P. Lee, G. Zhang, X. Feng, V. A. Gribkov, M. Liu, A. Serban, and

T. Wong, IEEE Trans. Plasma Sci. 26, 1119 (1998).

⁸S. Bing, “Plasma dynamics and x-ray emission of the plasma focus,” Ph.D. thesis, NIE, (2000) in ICTP Open Access Archive: http://eprints.ictp.it/99/

⁹S. Lee, in http://ckplee.myplace.nie.edu.sg/plasmaphysics/ (2000 & 2007).

¹⁰S. Lee in ICTP Open Access Archive: http://eprints.ictp.it/85/ (2005).

¹¹V. Siahpoush, M. A. Tafreshi, S. Sobhanian, and S. Khorram, Plasma

Phys. Controlled Fusion 47, 1065 (2005).

¹²S. Lee, Twelve Years of UNU/ICTP PFF-A Review (1998) IC, 98 (231);

A.Salam ICTP, Miramare, Trieste ( in ICTP OAA: http://eprints.ictp.it/31/ ).

¹³L. Soto, P. Silva, J. Moreno, G. Silvester, M. Zambra, C. Pavez, L. Altamirano, H. Bruzzone, M. Barbaglia, Y. Sidelnikov, and W. Kies, Braz. J. Phys. 34, 1814 (2004).

¹⁴H. Acuna, F. Castillo, J. Herrera, and A. Postal, International Conference

on Plasma Sci, 3–5 June 1996 (unpublished), p. 127.

¹⁵C. Moreno, V. Raspa, L. Sigaut,& R. Vieytes, Appl. Phys. Lett. 89, 15 (2006).

¹⁶D. Wong, P. Lee, T. Zhang, A. Patran, T. L. Tan, R. S. Rawat, and S. Lee,

Plasma Sources Sci. Technol. 16, 116 (2007).

¹⁷V A Gribkov, A Banaszak, B Bienkowska, A V Dubrovsky, I Ivanova-Stanik,

L Jakubowski, L Karpinski, R A Miklaszewski, M Paduch, M J Sadowski, M Scholz,

A. Szydlowski, and K. Tomaszewski, J. Phys. D 40, 3592 (2007).

¹⁸J. D. Huba, 2006 Plasma Formulary, p. 44. http://wwwppd.nrl.navy.mil/nrlformulary/NRL_FORMULARY_07.pdf

¹⁹S. V. Springham, S. Lee, and M. S. Rafique, Plasma Phys. Controlled

Fusion 42, 1023 (2000).

²⁰W. Kies, in Laser and Plasma Technology, Proceedings of Second Tropical

College, edited by S. Lee, B. C. Tan, C. S. Wong, A. C. Chew, K. S. Low, H. Ahmad, and Y. H. Chen World Scientific, Singapore, (1988), pp. 86–137.

²¹H. Herold, in Laser and Plasma Technology, Proceedings of Third Tropical

College, edited by C. S. Wong, S. Lee, B. C. Tan, A. C. Chew, K. S.

Low, and S. P. Moo World Scientific, Singapore, (1990), pp. 21–45.

²²A. Patran, R. S. Rawat, J. M. Koh, S. V. Springham, T. L. Tan, P. Lee, & S. Lee, 31st EPS Conf on Plasma Phys London, (2004), Vol. 286, p. 4.213.

²³S Lee Radiative Dense Plasma Focus Computation Package (2008) RADPF: http://www.intimal.edu.my/school/fas/UFLF/

Reference to this course and the Lee model code should be given according to the following format:

Lee S. Radiative Dense Plasma Focus Computation Package (2008) : RADPF www.plasmafocus.net www.intimal.edu.my/school/fas/UFLF/