5 th International Conference on the Frontiers of Plasma Physics and Technology

1. 5th International Conference on the Frontiers of Plasma Physics and Technology 18-22 April 2011, Singapore The Plasma Focus- Numerical Experiments Leading Technology Sor Heoh Saw 1,2 and Sing Lee 1,2,3 1INTI International University, 71800 Nilai, Malaysia 2Institute for Plasma Focus Studies, 32 Oakpark Drive, Chadstone, VIC 3148, Australia 3Nanyang Technological University, National Institute of Education, Singapore 637616 e-mails: sorheoh.saw@newinti.edu.my; leesing@optusnet.com.au

2. The Plasma Focus- Numerical Experiments Leading Technology Outline of talk: Numerical experiments assist design; provide reference for diagnostics and provide guidance for implementation of technology for Plasma Focus devices. Pointed the way to development of the Nanofocus, extending proven scalable range of DPF devices from 0.1J to MJ over unprecedented 7 orders of magnitude in storage energy Important guidance uncovered by numerical experiments include: (1) Plasma current limitation effect, so show that it is futile to lower static inductance below around 20 nH (2) Scaling laws of neutron yield and soft x-ray yield as functions of E0 & I (3) Deterioration of scaling laws due to dynamic impedance; so need to go to yield enhancement techniques: high-Z seeding, higher voltage operation & circuit manipulation

3. The Plasma Focus 1/2 • Plasma focus: small fusion device, complements international efforts to study nuclear fusion • Multi-radiation device - x-rays, particle beams and fusion neutrons • Soft XR applications include microelectronics lithography and micro-machining • Large range of device-from J to thousands of kJ • Experiments-dynamics, radiation, instabilities and non-linear phenomena

4. Scaling Properties 3 kJ machine Small Plasma Focus 1000 kJ machine Big Plasma Focus These two Images are shown to geometrical scale: Energy scaling : 300 times

5. Radial Compression (Pinch) Phase of the Plasma Focus

6. Our early numerical work [Lee & Serban IEEE Trans Plasma Sci 24 (1996) 1101]already showed the following scaling rules-of-thumb • Axial phase energy density (per unit mass) constant • Radial phase energy density (per unit mass) constant • Pinch radius ratio constant • Pinch length ratio constant • Pinch duration per unit anode radius constant

7. The constant energy density scaling pointed the way to development of so-called nanofocus Nanofocus: an ultra-miniature dense pinch plasma focus device with operating at 0.1 J L Soto et al- Plasma Sources Sci & Tech 18 (2009) 015007 pinch Sub-mm anode

8. Scaling Rules-of-thumb predicted numerically are used to design nanofocus- DPF’s operate over 7 orders of magnitude: 0.1J to 1MJ- • Axial phase energy density (per unit mass) constant • Radial phase energy density (per unit mass) constant • Pinch radius ratio constant • Pinch length ratio constant • Pinch duration per unit anode radius constant

9. Extending DPF’s to operate over an unprecedented scalable 7 orders of magnitude: 0.1J to 1MJ- This is an example of: Numerical Experiments Leading Technology

10. The Plasma Focus Axial Phase Radial Phases

11. The 5-phases of Lee Model code Includes electrodynamical- & radiation- coupled equations to portray the REGULAR mechanisms of the: axial (phase 1) radial inward shock (phase 2) radial RS (phase 3) slow compression radiation phase (phase 4) the expanded axial post-pinch phase (phase 5) Crucial technique of the code: Current Fitting

12. The Lee Model code-Comprehensive Numerical Experiments The approach of the Lee Model code To model the plasma dynamics & plasma conditions Then obtain insights into scaling properties Then uncover scaling laws Critical to the approach: Model is linked to physical reality by the current waveform

13. Insights 1/2 The Lee model code has produced ground-breaking insights no other plasma focus codes has been able to produce These insights have led to other examples of Numerical Experiments leading Technology

14. Ground-breaking Insights published- Numerical Experiments leading Technology Limitation to Pinch Current and Yields-Appl Phys Letts. 92 (2008) S Lee & S H Saw: an unexpected, important result Neutron Yield Scaling-sub kJ to 1 MJ-J Fusion Energy 27 (2008) S Lee & S H Saw- multi-MJ- PPCF 50 (2008) S Lee Neon Soft x-ray Scaling-PPCF 51 (2009) S Lee, S H Saw, P Lee, R S Rawat Neutron Yield Saturation- Appl Phys Letts. 95 (2009) S Lee Simple explanation of major obstruction to progress

15. From Measured Current Waveform to Modelling for Diagnostics1/2 Procedure to operate the code: Step 1: Configure the specific plasma focus Input: Bank parameters, L0, C0 and stray circuit resistance r0; Tube parameters b, a and z0 and Operational parameters V0 and P0 and the fill gas

16. Step 2: Fitting the computed current waveform to the measured waveform-(connecting with reality)2/2 Ameasured discharge current Itotal waveformfor the specific plasma focus is required The code is run successively. At each run the computed Itotal waveform is fitted to the measured Itotal waveform by varying model parameters fm, fc, fmr and fcr one by one, one step for each run, until computed waveform agrees with measured waveform. The 5-Point Fit: First, the axial model factorsfm, fcare adjusted (fitted) until (1) computed rising slope of the Itotal trace and (2) the rounding off of the peak current as well as (3) the peak current itself are in reasonable (typically very good) fit with the measured Itotal trace. Next, adjust (fit) the radial phase model factors fmr and fcr until - (4) the computed slope and - (5) the depth of the dip agree with the measured Itotal waveform.

17. Example : NX2-Plasma SXR Source1/4 NX2 11.5kV, 2 kJ 16 shots /sec; 400 kA 20J SXR/shot (neon) 109 neutrons/shot (D)

18. Example of current fitting: Given any plasma focus : e.g. NX2 16 shots/sec Hi Rep 2/4 Bank parameters: L0=15nH; C0=28uF; r0=2 mW Tube parameters: b=4.1 cm, a=1.9 cm, z0=5cm Operation parameters: V0=11kV, P0=2.6 Torr in Neon The UPFLF (Lee code) is configured (by keying figures into the configuration panel on the EXCEL sheet) as the NX2 INPUT: OUTPUT: NX2 current waveform NX2 dynamics & electrodynamics NX2 plasma pinch dimensions & characteristics NX2 Neon SXR yield

19. Fitting computed Itotal waveform to measured Itotal waveform: the 5-point fit 3/4

20. Once fitted: model is energy-wise & mass-wise equivalent to the physical situation4/4 All dynamics, electrodynamics, radiation, plasma properties and neutron yields are realistically simulated; so that the code output of these quantities may be used as reference points for diagnostics

21. Numerical Diagnostics- Example of NX2Time histories of dynamics, energies and plasma properties computed by the code1/3Last adjustment, when the computed Itotal trace is judged to be reasonably well fitted in all 5 features, computed times histories are presented (NX2 operated at 11 kV, 2.6 Torr neon) Computed Itotal waveform fitted to measured Computed Tube voltage Computed Itotal & Iplasma Computed axial trajectory & speed

22. NumericalDiagnostics- Example of NX2 2/3

23. Numerical Diagnostics- Example of NX23/3

24. More on theThe Lee Model Code 1/3 • Realistic simulation of all gross focus properties • Couples the electrical circuit with plasma focus dynamics, thermodynamics and radiation (Lee 1983, 1984) • 5-phase model; axial & radial phases • Includes plasma self-absorption for SXR yield (Lee 2000) • Includes neutron yield, Yn, using a beam–target mechanism (Lee & Saw 2008, J Fusion energy)

25. Numerical Experiments leading TechnologyPinch current limitation effect in plasma focus(S. Lee and S. H. Saw, Appl. Phys. Lett. 92, 021503 (2008), DOI:10.1063/1.2827579) • Pinch current limitation effect- Ipinch does not increase beyond a certain value however low Lo, the static inductance is reduced to. • Decreasing the present Lo of the PF1000 machine will neither increase the pinch current nor the neutron yield, contrary to expectations.

26. Results from Numerical Experiments with PF1000 - For decreasing Lo - from 100 nH to 5 nH • As Lowas reduced from 100 to 35 nH - As expected • Ipeak increased from 1.66 to 3.5 MA • Ipinch also increased, from 0.96 to 1.05 MA • Further reduction from 35 to 5 nH • Ipeak continue to increase from 3.5 to 4.4 MA • Ipinch decreasing slightly to - Unexpected  1.03 MA at 20 nH,  1.0 MA at10 nH, and  0.97 MA at 5 nH. • Ynalso had a maximum value of 3.2x1011 at 35 nH.

27. Energy distribution in the system at the end of the axial phase and at the end of the pinch-(1/2) • The energy equation describing this current drop is written as follows: • 0.5Ipeak2(Lo + Lafc2) = 0.5Ipinch2(Lo/fc2 + La + Lp ) + Where La= inductance of the tube at full axial length zo. = energy imparted to the plasma as the current sheet moves to the pinch position = integral of 0.5(dL/dt)I2 ~ 0.5LpIpinch2 (an underestimate for this case) =energy flow into or out of the capacitor during this period of current drop. = 0 (capacitor is effectively decoupled-duration of the radial phase is short compared to the capacitor time constant) • Ipinch2 = Ipeak2(Lo + 0.5La)/(2Lo + La + 2Lp) (Note : fc=0.7, fc2~0.5)

28. Pinch Current Limitation Effect - (1/3) • Lo decreases higher Ipeakbigger a zp longer bigger Lp • Lo decreases shorter rise time shorter zo smaller La Lo decreases, Ipinch/Ipeak decreases

29. Pinch Current Limitation Effect - (2/3) • Lo decreases, L-C interaction time of capacitor decreases • Lo decreases, duration of current drop increases due to bigger a Capacitor bank is more and more coupled to the inductive energy transfer  Effect is more pronounced at lower Lo

30. Pinch Current Limitation Effect - (3/3) • A combination of two complex effects • Interplay of various inductances • Increasing coupling of Co to the inductive energetic processes as Lo is reduced

31. Conclusions – (1/2) • Several sets of Numerical results For PF1000 with different damping factors indicate • Optimum inductances are around 30-60 nH with Ipinch decreasing for Lo below optimum value • Reducing Lo from its present 20-30 nH will increase neither Ipinch nor Yn

32. Conclusions – (2/2) • For a fixed Co powering a plasma focus, there exist an optimum Lo for maximum Ipinch • Reducing Lo will increase neither Ipinch nor Yn • Because of the Pinch Current Limitation Effect

33. The numerical experiments and discussions – 4/7 • Figure 2. PF1000 current waveforms (computed) at 35 kV, 3.5 Torr D2 for a range of Lo.

34. The numerical experiments and discussions – 6/7 • Figure 3. Effect on currents and current ratio (computed) as Lo is reduced-PF1000, 35 kV,3.5 Torr D2.

35. Computation of Neutron yield (1/2) • Adapted from Beam-target neutron generating mechanism (ref Gribkov et al) • A beam of fast deuteron ions close to the anode • Interacts with the hot dense plasma of the focus pinch column • Produces the fusion neutrons • Given by: Yb-t= Cn niIpinch2zp2(ln(b/rp))σ /U0.5 where ni = ion density b = cathode radius, rp = radius of the plasma pinch column with length zp, σ = cross-section of the D-D fusion reaction, n- branch, U= beam energy, and Cn = calibration constant

36. Computation of Neutron yield (2/2) Note: • The D-D cross-section is sensitive to the beam energy in the range 15-150 kV; so it is necessary to use the appropriate range of beam energy to compute σ. • The code computes induced voltages (due to current motion inductive effects) Vmax of the order of only 15-50 kV. However it is known, from experiments that the ion energy responsible for the beam-target neutrons is in the range 50-150keV, and for smaller lower-voltage machines the relevant energy could be lower at 30-60keV. • In line with experimental observations the D-D cross section σ is reasonably obtained by using U= 3Vmax. • The model uses a value of Cn =2.7x107 obtained by calibrating the yield at an experimental point of 0.5 MA.

37. Computation of Neon SXR yield (1/2) Neon SXR energy generated YSXR = Neon line radiation QL QL calculated from: • where : • Zn= atomic number, • ni = number density , • Z = effective charge number, • rp = pinch radius, • zf = pinch length and • T = temperature • QL is obtained by integrating over the pinch duration. NOTE

38. Computation of Neon SXR yield (2/2) Note: The SXR yield is the reduced quantity of generated energy after plasma self-absorption which depends primarily on density and temperature The model computes the volumetric plasma self-absorption factor A derived from the photonic excitation number M which is a function of the Zn,ni, Z and T. In our range of operation the numerical experiments show that the self absorption is not significant. Liu Mahe (1999) first pointed out that a temperature around 300 eV is optimum for SXR production. Shan Bing’s (2000) subsequent work and our experience through numerical experiments suggest that around 2x106 K (below 200 eV) or even a little lower could be better. Hence for SXR scaling there is an optimum small range of temperatures (T window) to operate.

39. Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (2/4) • Computed Total Current versus Time • For L0 = 30nH; V0 = 20 kV; C0 = 30 uF; RESF = 0.1; c=1.5 • Model parameters : fm = 0.06, fc = 0.7, fmr =0.16, fcr = 0.7 • Optimised a=2.29cm; b=3.43 cm and z0=5.2 cm.

40. Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (3/4) • Ysxr scales as: • E01.6at low energies in the sub-kJ to several kJ region. • E00.76 at high energies towards 1MJ.

41. Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (4/4) • Scaling with currents • Ysxr~Ipeak3.2 (0.1–2.4 MA) and • Ysxr~Ipinch3.6 (0.07-1.3 MA) • Black data points with fixed parameters RESF=0.1; c=1.5; L0=30nH; V0=20 kV and model parameters fm=0.06, fc=0.7, fmr=0.16, fcr=0.7. • White data points are for specific machines with different values for the parameters :c, L0, V0 etc.

42. Summary-Scaling Laws (1/2) • The scaling laws obtained (at optimized condition) for Neutrons: • Yn~E02.0 at tens of kJ to • Yn~E00.84 at the highest energies (up to 25MJ) • Yn =3.2x1011Ipinch4.5(0.2-2.4 MA) • Yn=1.8x1010Ipeak3.8 (0.3-5.7MA)

43. Summary-Scaling Laws (2/2) • The scaling laws obtained (at optimized condition) for neon SXR: • Ysxr~E01.6at low energies • Ysxr~E00.8 towards 1 MJ • Ysxr~Ipeak3.2 (0.1–2.4 MA) and • Ysxr~Ipinch3.6 (0.07-1.3 MA)

44. Global Scaling Law for NeutronsCombining experimental & numerical Experimental data from 0.4kJ to 1 MJ & beyond What causes the deterioration of Yield scaling?

45. What causes current scaling deterioration and eventual saturation? 1/3 The axial speed loads the discharge circuit with a dynamic resistance The same axial speed over the range of devices means the same dynamic resistance constituting a load impedance DR0 Small PF’s : have larger generator impedance Z0=[L0/C0] 0.5 than DR0 As energy is increased by increasing C0, generator impedance Z0 drops

46. What causes current scaling deterioration and eventual saturation? 2/3 At E0 of kJ and tens of kJ the discharge circuit is dominated by Z0 Hence as E0 increases, I~C0-0.5 At the level typically of 100 kJ, Z0 has dropped to the level ofDR0; circuit is now no longer dominated byZ0; and current scaling deviates fromI~C0-0.5, beginning of current scaling deterioration. At MJ levels and above, the circuit becomes dominated by DR0, current saturates

47. Deterioration and eventual saturation of Ipeakas capacitor energy increases Axial phase dynamic resistance causes current scaling deterioration as E0 increases

48. In numerical experiments we showed: Yn~Ipinch4.5 Yn~Ipeak3.8 Hence deterioration of scaling of Ipeakwill lead to deterioration of scaling of Yn.

49. What causes current scaling deterioration and eventual saturation? 3/3 Analysis using the Lee model code has thus shown that the constancy of the dynamic resistance causes the current scaling deterioration resulting in the deterioration of the neutron yield and eventual saturation. This puts the global scaling law for neutron yield on a firmer footing

50. Into the Future Beyond Saturation Plasma Focus?