INTI International University, Nilai, Malaysia

1. Development of projects based on the Lee Model code- A discussion leading from the simplest to the most profoundS Lee INTI International University, Nilai, Malaysia Institute for Plasma Focus Studies, 32 Oakpark Drive, Chadstone, VIC 3148, Australia

2. The Approach A tutorial approach in the first part to benefit the newcomer; then in the final part integrating the basic aspects to the most profound problems challenging the thinking of the plasma focus community. The simplest and advanced projects (below) can stand alone as numerical experiments or be best synergised with laboratory measurements; the profound requires to integrate all the intuitive and research resources we can muster:

3. Simple Projects • (i) Variation of current waveforms as a function of pressure in various gases- (ii) Collating dynamics, pinch dimensions, plasma conditions and yields (b) (i) Variation of neutron yields with pressure (ii) Variation of dynamics and pinch properties with pressure (iii) Correlation of (ii) with (i) above.

4. Example 1: Variation of current waveforms as a function of pressure • The Universal PF code: RADPFV6.1b • Configure: for PF1000: 27 kV 3.5 Torr D2 (published)

5. Fire the PF1000 • RADPFV6.1b • Look at results: Sheet 1 figures Sheet 1 dataline Sheet 2 figures

6. PF1000c fitted results

7. Collect current waveform 3.5 Torr

8. Collect data PF1000 3.5 Torr

9. SPF1000: Current Waveforms as functions of Pressure

10. PF1000: A Selection of data

11. PF1000: Ynvs Pressure

12. PF1000: Neutron Yield vs Pressure; and physics involved

13. Notes: • 1) Pressure increases, Ipeak increases • 2) Ipinch increases, peaks just before 5 Torr, then drops • EINP follows roughly trend of Ipinch • ni, not plotted, seen from table to increase continuously with presssure • Yn peaks not where Ipinch peaks, but at higher P due to increase in ni • All Competing effects need to be considered • The effects, all regulated by the physics, are automatically included in the model

14. Variations of Project • Different machines- including your own and others • Different gases- D-T mixture for neutrons Neon for neon SXR Ar, N2, O2 for SXR Compare with experimental results- see examples below

15. PF-400J: AEC Chile 1.Fit computed to measured current waveforms to get model parameters 2.Use these fitted model parameters for PF400J to get Yn at various pressures 3. Compare computed with measured Yn (agreement is state-of-the-art)

16. FN-II: U of Mexico 1.fit computed to measured current waveforms to get model parameters 2.Use these fitted model parameters for FN-II to get Yn at various pressures 3. Compare computed with measured Yn (agreement is state-of-the-art)

17. NX2: Neon Ysxrvs P 1. Fit computed to measured current waveforms to get model parameters 2. Use these fitted model parameters for NX2 to get neon Ysxr at various P 3. Compare computed with measured neon Ysxr; and with other relevant data; all as functions of P- present on many graphs; or on 1 normalised graph.

18. Advanced: (c)Scaling properties derived from (a) and (b) above (d) Scaling laws may be developed with comprehensive series of numerical experiments based on suitably designed matrix

19. Scaling Properties Questions:(basis for projects) • What does Ipeak scale with & how? • What does Ipinch scale with & how? • What does axial speed scale with & how? • What does radial speed scale with & how? • What do pinch dimensions, radius & length scale with and how? • What does pinch duration scale with & how? • What do energy distributions (define) scale with & how?

20. Numerical experiments • Study 1 machine, 1 gas, various pressures • Study several machines, several gases, various pressures • Do comparative tabulations and graphs for 1 machine 1 gas; 1 machine several gases, many machines, many gases.

21. Collect measured current waveforms • Each machine to be numerically experimented- we need to fit to get the model parameters, fm,fc,fmr,fcr for the relevant gas. • Run numerical experiments at various P for that gas • Collect data using dataline in row format or in following format

22. Comparative tabulation -link

23. Scaling Law for Yields Questions: (Basis for research projects) • How does Yn scale with stored energy Eo • How does Yn scale with Ipeak • How does Yn scale with Ipinch • Same questions above: for SXR’s for neon, argon, N2, O2, Kr, Xe etc • Same questions above for Ion beams, Fast plasma streams, anode sputtered materials

24. Scaling Laws from Numerical Experiments Most important: • Basis of scaling must be defined: e.g. Optimum yield for each case • Matrix of experiment needs to be properly defined e.g. Fix voltage, fix static inductance, fix b/a then vary: E0, P, z0, a through all combinations • Data to be collected fixed early • Typically needs thousands of shots to go through all combination for each set of E0, P, Z0 and a.

25. Matrix design Example: Scaling law of yield vs E0; fixed L0, fixed c=b/a Additional condition: For Yn: Fix end axial speed at 10 cm/us. Matrix: Fix Energy, fix P, fix z0; vary ‘a’ until optimum ‘a’ Vary z0; vary ‘a’ each zo until obtain optimum ‘z0’ and ‘a’ combination Vary P, vary z0 for each P; vary ‘a’ for each z0 until obtain optimum P, z0 and ‘a’ combination Repeat for each E0

26. Profound: Discussion to optimise plasma focus devices: Tendency to invoke incorrectly ‘matching’ in the Maximum Power Transfer Theorem sense. There are at least 3 separate effects/ mechanisms which are best differentiated, from basic considerations. These are: • Maximum power (energy) transfer • Current & yield limitation as bank inductance L0 is reduced • Neutron & yield deterioration with increase of bank energy E0

27. (a) Maximum power (energy transfer) • There appears to be a systemic misunderstanding about the Maximum Power Transfer Theorem when applied to the plasma focus. • This theorem is applicable to a generator with a fixed resistance Rgen; hence at a given voltage has a maximum power capacity (delivered within itself) when load resistance Rload is zero. As Rload is increased, the total power capacity drops but power is transferred to the load with increasing proportion as Rload is increased; until maximum power transfer occurs at Rload = Rgen; which is said to be the matched condition for Max Power Transfer.

28. The situation of the plasma focus is the converse. • The question here is: With a fixed (though time varying, averaged if you like) load, how do you arrange the generator to give maximum power transfer MPT? • Here we are not at liberty to simply reduce the load resistance or impedance. At an axial speed of 10 cm/us the typical PF has a ‘dynamic resistance’ of some 5 mOhm with little variation among PF’s large and small. A small inductive bank like the UNU ICTP PFF has a surge impedance some 10 times that whilst a large bank like the PF1000 has a surge impedance about the dynamic resistance. A capacitor bank 10 times larger than the PF1000 will have its dynamic resistance overwhelming the surge impedance. • So here because we have little control over the ‘fixed’ resistance and impedance of the load, the question about maximum power transfer should be about variation in the generator impedance. • In such a case MPT theorem does not apply. (although the physics basis on still applies) • One will NOT have best transfer of power when one selects the generator impedance to ‘match’ the load’s ‘averaged’ impedance. • The energy transfer to the load (taking the plasma focus as a whole) will keep increasing towards 100% when the generator impedance is reduced towards zero. Design numerical experiments to test the above conclusion in different gases; D, D-T, neon, Ar etc

29. The Lee code test • We did a trial run with a 28 uF capacitor reducing the Lo in steps from 20 nH down to 0.1nH keeping ‘a’ constant to ensure an approximately constant pinch length hence pinch inductance, also adjusting pressure so that the axial speed is around 10 cm/us [we put fc=fcr=1 to allow full effect of the circuit current]. With reducing L0 there is a progressive increase in total energy dissipated in the plasma focus system until at L0=0.1 nH, (La=1.0 nH, Lp=15.0 nH), the energy dissipated is 36% into the axial phase and 57% into the radial phase and pinch; total energy transferred being 89% of initial stored energy. • The numerical experiments also show that, under the max transfer condition specified by Krishnan (L0=Lp) the energy dissipated in the pinch is 46% (compared to their hypothesized max transfer figure of 25%) ; whilst under max transfer conditions specified by Bernard et al Z0=0.7dL/dt, the energy dissipated in the pinch is 51% (compared to their hypothesized max transfer figure of 43%).

30. Across the literature of Plasma Focus erroneous concept of matching is systemic • These results show that the conclusions of Bernard et al [1974] and Krishnan et al [2009 ] regarding maximum energy transfer are not borne out by numerical experiments based on a charge, energy, mass, momentum consistent model. • On the contrary, the physics require that the lower the generator impedance, the better the % energy transferred • The PF geometry can always be optimised in such a way that more and more % energy is transferred into the pinch as the generator impedance is reduced towards zero. There is no matching impedance or inductance for best transfer. The best % energy transfer occurs by putting the generator impedance/inductance to values much less than the pinch inductance, indeed towards zero. This situation is governed by the same fundamental electrical circuit requirements as the MPT Theorem.

31. (b) Pinch current (& Yn) limitation as L0 goes towards zero • This effect may seem to contradict (a) above but it does not. • More power transferred does not necessarily mean more pinch current or radiation yield. • That is because as L0 is progressively reduced to very small values, the time scale shortens (although not by as much as would be indicated by (L0C0)^0.5 since the dynamics would effectively stretch out this time by loading and distorting the discharge waveform); • zo needs to be increasingly shortened in order to leave drive time for the radial phase which increasingly dominates because of the need to increase anode radius ‘a’ due to the increasing current and the need to keep the drive parameter within limits.

32. At the same time the decrease of L0 increases the coupling of the pinch system with the capacitor. • Under these conditions Ipeak indeed continues to increase as L0 is reduced but the ratio of Ipinch/ Ipeak drops to very low values of even below 0.2 even as the plasma focus is optimised the best one can under the interplay (or conspiracy) of the above complex interactions.

33. So whilst (a) shows that the lowest bank inductance is conducive to the highest energy transfer, the discussion in (b) above indicates the layers of complexity that need to be added when the various time interactions are considered. • This makes it very difficult to develop analytical or conceptual insights. • Yet all it takes is essentially to couple two equations (an electric circuit equation for charge and energy conservation and a Newtonian equation to conserve momentum) [2 additional motion equations for the radial phase] and all these subtle interplay and conspiracy of nature are automatically incorporated! • This demonstrates the encompassing advantages of a simple, flexibly reactive model for numerical experiments Develop matrix of experiments to demonstrate current limitation and effect of Yn in D and D-T; on neon SXR radiation, argon SXR radiation etc

34. C. Neutron Saturation • Used in PF lore to indicate the observation that Yn does not increase further above several hundred kJ. • A study of the data indicate that too much emphasis had been placed on the Frascati experiments which show 3 points of data that since then appeared to have an undue impact on the PF community. • A combination of experimental data, the trends of which have been verified by numerical experiments, the latter also filling in the gaps as well as extending to higher energies beyond experimental data; has resulted in a global scaling law which shows that the experimental results from 100 kJ up towards 1 MJ should actually be interpreted not as neutron saturation but rather as a deterioration of scaling as the index of yield vs Eo goes below 2. • Unfortunately this index goes to a low value of 0.8 at some 25 MJ and even smaller at higher energies. • Indeed at very high energies as this index goes to small enough values the situation may be considered as ‘saturation’. But this saturation is not the same as the historical description of neutron saturation which is suggested as a misnomer/misinterpretation for a deterioration of scaling.

35. Global Scaling Law for Ynvs E0

36. The Physics behind the mis-named ‘neutron saturation’ The reason for the neutron scaling deterioration (with storage energy) is because of the scaling deterioration of Ipeak (leading to scaling deterioration of Ipinch) due to a relatively constant ‘dynamic resistance of 0.5dL/dt’ interacting with a decreasing bank surge impedance which decreases to insignificant values as bank C0 is increased (to increase E0). Based on this reasoning others yields will also experience similar scaling deterioration. Develop matrix of expts to demonstrate scaling deterioration of neutrons, neon SXR, argon SXR etc

37. Scaling further- possibilities • 1. Increase E0, however note: scaling deteriorated already below Yn~E0 • 2. Increase voltage, at 50 kV beam energy ~150kV already past fusion x-section peak; further increase in voltage, x-section decreases, so gain is marginal • Need technological advancement to increase current per unit E0 and per unit V0. • We next extrapolate from point of view of Ipinch

38. Scaling Plasma Focus from Ipinch using present predominantly beam-target in Lee Model code

39. 1. Using above Fig compute Pout at 1 Hz assume efficiency 0.32. Then compute E0 budget to generate the required Ipinch at each point; so as to get Q=2

40. What we need for Focus fusion energy based on D-T (note: 1 D-T neutron has 14.1 MeV of KE) Choose 24 MA point from above graph: Ipinch : 24 MA D-T n from scaling: 3x1019 Kinetic energy: 64 MJ Rep rate: 1 shot per second Then Pfusion (0.3 efficiency): 20 MW If E0=10MJ; input power at 1 Hz 10 MW Net Power 10 MW Technical Requirement: Ipinch= 24MA using E0=10MJ; Rep rate required: 1 Hz

41. Thermonuclear Plasma Focus • Reason why PF fusion is beam-target is PF temp not high enough. • If use additional external heating from present 1 keV to 10 0r 20 keV, then Yth is dominant

42. Thermonuclear Plasma Focus

43. Thermonuclear fusion in PF with additional heating: Technological Targets • Select a point from Fig 3 for discussion • 10MA point at 20keV gives 3x10^19 D-T n /shot • This is equivalent to (Fig 1) b-t at 24 MA • At 1 Hz eff 0.3 (Fig 2) gives 20 MW • If require Q=2 (ie net power of 10 MW) • TechnologicalTargets: 4 MJ to generate 10MA 6 MJ to provide additional heating to 20keV

44. Plasma Focus Reactors • Beam-target regime improvement in technology is required: to generate 24 MA pinch current from 10MJ at 30kV • Thermonuclear regime: plasma focus operation; with 10 MA from 3.5 MJ, no High voltage limit use additional heating (6 MJ budget) to reach 20 keV • Enhancement techniques: Radiative collapse induced by Kr or Xe doping Current injection using current-steps or beam injection

45. Conclusions In this paper we looked at research projects which may be developed in numerical experiments using our code. The experiments ranged from: • Simplest, such as current waveforms and yields as functions of pressure (1 machine, all machines, all gases) • Advanced, such as deriving scaling properties and scaling laws from numerical experiments • Profound- Integration of concepts of maximizing energy transfer with the effects of interaction of times and dimensions on the ratio of pinch current to peak current and yield mechanisms; the ultimate role of the dynamic resistance on yield scaling- for pushing forward the boundaries of Plasma Focus research towards fusion energy. • Numerical experiments indicate: critical technological requirement- development of 2.5A of pinch current per J of stored energy at a level of some 25 MA of pinch current and/ or associated technology of heating the pinch to 20 keV with an energy budget of 6 MJ.

46. THANK YOU Profound Simple Research Projects developed from Plasma Focus Numerical Experiments