International Workshop on Plasma Science & Applications, 27 & 28 October, Tehran, Iran

1. International Workshop on Plasma Science & Applications, 27 & 28 October, Tehran, Iran Review of Plasma Focus Numerical Experiments Scaling and Compression Enhancement S H Saw and S Lee INTI International University, 71800 Nilai, Malaysia Institute for Plasma Focus Studies, Chadstone VIC 3148 Australia e-mail: leesing@optusnet.com.au sorheoh.saw@newinti.edu.my

2. Outline • Lee Model code- relates to physical reality through measured • current trace; results are reference points for diagnostics • Axial phase scaling properties • axial speed, speed factor, energy density • Radial phase scaling properties • -same speeds, temperatures, densities • -pinched plasma dimensions, lifetimes • Scaling Laws: Neutrons and SXR • Global Scaling Laws for neutrons • New developments: • instability phase modelling, current-stepping, radiative collapse, Plasma Focus- Scaling Properties and Scaling Laws

3. The Plasma Focus Plasma focus: small fusion device, complements international efforts to build fusion reactor Multi-radiation device - x-rays, particle beams and fusion neutrons Neutrons for fusion studies 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. The 5 phases of Lee Model code Includes electrodynamical and 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) Extension to phase 4a- instability-induced resistive phase Crucial technique of the code: Current Fitting

5. 30 mF, 15 kV HV The Plasma Dynamics in Focus Radial Phase Axial Accelaration Phase Inverse Pinch Phase

6. The Lee Model Code Realistic simulation of all gross focus properties Couples the electrical circuit with plasma focus dynamics, thermodynamics and radiation (1984,1990) 5-phase model; axial & radial phases Includes plasma self-absorption for SXR yield (2000) Includes neutron yield, Yn, using a beam–target mechanism(2007)

7. The Plasma Focus Axial Phase Radial Phases

8. Philosophy of Current fittings (1/3) The current trace of the focus is the best indicators of gross performance. The exact time profile of the current trace is governed by the bank parameters, the focus tube geometry and the operational parameters. It depends on the mass swept-up and drive current fractions and their variations. These parameters determine the dynamics, specifically the axial and radial speeds which in turn affect the profile and magnitudes of the current. There are many underlying mechanisms (see following 2 slides) which are not simply modeled. The detailed current profile is influenced by these effects and during the pinch phase also reflects the Joule heating and radiative yields. At the end of the pinch phase the profile reflects the sudden transition from a constricted pinch to a large column flow. Thus the current powers all dynamic, electrodynamic, thermodynamic and radiation processes in the various phases. Conversely all dynamic, electrodynamic, thermodynamic and radiation processes in the various phases affect the current. The current waveform contains information on all the dynamic, electrodynamic, thermodynamic and radiation processes that occur in the various phases. This explains the importance attached to matching the computed total current trace to the measured total current trace in the procedure adopted by the Lee model code. Once matched, the fitted model parameters assure that computation proceeds with all physical mechanisms accounted for, in the gross energy & mass balance sense.

9. Philosophy of Current fittings (2/3) So we relate to reality through a measured current trace computed current waveform is adjusted to fit measured current waveform Adjustment by model parameters fm, fc, fmr, fcr; account for all factors affecting mass flow and force field flows not specifically modelled including all KNOWN and UNKNOWN effects. When adjustments are completed so that the computed waveform fit the measured waveform, the computed system is energetically and mass-wise equivalent to the real system.

10. Philosophy of Current fittings (3/3) All inaccurate model effects are accounted for by the fitting: Known effects that might deviate from our modelling include: Geometrical, including our assumed geometry Our assumed structures and distributions Mass shedding & current sheet CS porosity Current shedding, fragmenting, leakage & inclination Non uniformity & inhomogeneity of CS and plasma; boundary layer effects Radiation & thermodynamics Ejection of mass caused by necking curvatures Once current-fitted, all unspecified even unknown effect are also accounted for in terms of energy and mass

11. From Measured Current Waveform to Modelling for Diagnostics (1/4) 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

12. Step 2: Fitting the computed current waveform to the measured waveform – (connecting with reality) (2/4) 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.

13. Fitting computed Itotalwafeform to measuredItotaltwaveform: the 5-point fit (3/4)

14. Once fitted: model is energy-wise & mass-wise equivalent to the physical situation (4/4) 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

15. 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

16. NumericalDiagnostics- Example of NX2 2/3

17. Numerical Diagnostics- Example of NX2 3/3

18. Scaling Properties 3 kJ machine Small Plasma Focus 1000 kJ machine Big Plasma Focus

19. Comparison (Scaling) - 1/2 Important machine properties: UNU ICTP PFF PF1000 E0 3kJ at 15 kV 600kJ at 30kV I0 170 kA 2MA ‘a’ 1 cm 11.6 cm

20. Comparison (Scaling) - 2/2Important Compressed Plasma Properties Density of plasma Temperature of plasma These two properties determine radiation intensity energy radiated per unit volume per unit lifetime of plasma) Size of plasma Lifetime of plasma These two properties together with the above two determine total yield.

21. Compare Temperatures UNU ICTP PFF PF1000 (from LS lecture) (this estimate) Axial speed 10 12 cm/us Radial speed 25 20 cm/us Temperature 1.5x106 1x106 K Reflected S 3x106 2x106 KAfter RS comes pinch phase which may increase T a little more in each case Comparative T: about same; several million K

22. Compare Number Density – 1/2 During shock propagation phase, density is controlled by the initial density and by the shock-’jump’ density Shock density ratio=4 (for high T deuterium) RS density ratio=3 times On-axis density ratio=12 Initial at 3 torr n=2x1023 atoms m-3 RS density ni=2.4x1024 m-3 or 2.4x1018 per cc Further compression at pinch; raises no. density higher typically to just under 1019 per cc.

23. Compare Number Density – 2/2 Big or small PF initial densitysmall range of several torr Similar shock processes Similar final density

24. Big PF and small PFSame density, same temperature Over a range of PFs smallest 0.1J to largest 1 MJ; over the remarkable range of 7 orders of magnitude- same initial pressure, same speeds This leads to conclusion that all PF’s: Same T, hence same energy (density) per unit mass same n, hence same energy (density) per unit volume Hence same radiation intensity

25. Next question: How does yield vary? Yield is Intensity x Volume x Lifetime Dimensions and lifetime of the focus will determine radiation and particle yields. How do dimensions and lifetime of compressed plasma vary from PF to PF? Look at experimental observations

26. Comparing large and small PF’s- Dimensions and lifetimes- putting shadowgraphs side-by-side, same scale Anode radius 1 cm 11.6 cm Pinch Radius: 1mm 12mm Pinch length: 8mm 90mm Lifetime ~10ns order of ~100 ns

27. Comparing small (sub kJ) and large (thousand kJ) Plasma FocusScaling Properties: size (energy) , current, speed and yield Scaling properties-mainly axial phase Table 1

28. Scaling of anode radius ‘a’ with current (I) and storage energy (E0) Scaling properties-machine properties Peak current Ipeak increases with E0. Anode radius ‘a’ increases with E0. Current per cm of anode radius (ID) Ipeak /a : narrow range 160 to 210 kA/cm for all machines

29. Speeds, Speed Factor SF and Yn Observed Peak axial speed va : 9 to 11 cm/us. SF (speed factor) (Ipeak /a)/r0.5 : narrow range 82 to 100 (kA/cm) per Torr 0.5 D Fusion neutron yield Yn : 106 for PF400-J to 1011 for PF1000 (Yn 5 orders of magnitude for storage energy of 3 orders of magnitude)

30. Comparing small (sub kJ) & large (thousand kJ) Plasma FocusScaling Properties: size (‘a’) , T, pinch dimensions & duration Scaling properties-mainly radial phase

31. Moreover number densities are also the same for big and small focus, in the pinch compressed plasma around 1019/cc

32. Focus Pinch T, dimensions & lifetime with anode radius ‘a’ Scaling properties-mainly radial phase Dimensions and lifetime scales as the anode radius ‘a’. rmin/a (almost constant at 0.14-0.17) zmax/a (almost constant at 1.5) Pinch duration narrow range 8-14 ns/cm of ‘a’ Tpinchis measure of energy per unit mass. Quite remarkable that this energy density varies so little (factor of 5) over such a large range of device energy (factor of 1000).

33. Rule-of-thumb scaling properties, (subject to minor variations caused primarily by the variation in c=b/a) over whole range of device 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

34. Further equivalent Scaling Properties Constant axial phase energy density (Speed Factor (I/a)/r0.5, speed) equivalent to constant dynamic resistance I/a approx constant since r has only a relatively small range for each gas

35. Scaling Conclusions • Energy Storage in Plasma Focus spans 7 orders of magnitudes • We discussed the scaling of properties • 1. Machine properties • - Ipeak, anode radius ‘a’, storage energy; (Ipeak/a)=constant • 2. Axial phase scaling properties • - axial speed, speed factor, energy density; constant • 3. Radial phase scaling properties • -same speeds, temperatures, densities; constant • hence same radiation intensities and • 4. Radiation/particle yields depend only on • -pinched plasma dimensions, lifetimes: depend on ‘a’, on I • The current dominates Plasma Focus Yields

36. Insights Ground-breaking Insights published 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

37. 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)

38. 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) Summary-Scaling Laws (2/2)

39. Global scaling law, combining experimental and numerical data- Yn scaling , numerical experiments from 0.4 kJ to 25 MJ (solid line), compared to measurements compiled from publications (squares) from 0.4 kJ to 1 MJ. What causes the deterioration of Yield scaling?

40. What causes current scaling deterioration and eventual saturation? 1/2 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

41. What causes current scaling deterioration and eventual saturation? 2/2 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

42. Into the Future Anomalous Resistive Modelling Current Stepping Radiative Collapse

43. Current-Step method to enhance compressions

44. Ongoing IPFS numerical experiments of Multi-MJ, High voltage MJ and Current-step Plasma Focus

45. Current step increases compression

46. Comparison Deuterium PF Case Study: 50 kV 777 uF stepped with 200kV 20uF total energy=1.37 MJ produces 1013 D-D neutrons Compared with 90kV 777 uF energy 3 MJ Produces 1013 D-D neutrons Current-step has improved compression and the yield per unit input energy

47. Radiative Cooling and CollapseIn Kr and Xe, intense line radiation reduces pinch plasma pressure allowing magnetic piston to compress further. Numerical experiments show that a complete column collapse to very small radii is possible. Thus radiative cooling/collapse can enhance compression This mechanism may be instrumental in the observed neutron enhancement of Kr-doped deuterium PF Numerical Experiments will be carried out to understand and use this mechanism better

48. Numerical Experiments on 3 kJ PFdemonstrating Kr pinch column undergoing radiative collapse of whole column 0.1 Torr 0.5 Torr

49. Conclusion Numerical experiments have played a significant role in the understanding of scaling properties and scaling laws of the plasma focus We continue to push the boundaries of this understanding upwards by studying yield enhancement methods developed within the numerical experiments.

50. International Workshop on Plasma Science & Applications, 27 & 28 October, Tehran, Iran Review of Plasma Focus Numerical Experiments Scaling and Compression Enhancement S H Saw and S Lee INTI International University, 71800 Nilai, Malaysia Institute for Plasma Focus Studies, Chadstone VIC 3148 Australia e-mail: leesing@optusnet.com.au sorheoh.saw@newinti.edu.my