1. Review of Plasma Focus Numerical Experiments
S H Saw1,2, S Lee1,2,3
1Nilai University, Nilai, Malaysia
2Institute for Plasma Focus Studies, Melbourne, Australia
3University of Malaya, Kuala Lumpur, Malaysia

2. Plasma Focus Numerical Experiments- Outline of Lecture
Development, usage and results
Basis and philosophy
Reference for Diagnostics
Insights and frontiers
Continuing development- Ion beam modelling, radiative collapse and HED, throughput scaling to breakeven

3. UNU ICTP PFF- 3 kJ Plasma Focus Designed for
UNU ICTP PFF- 3 kJ Plasma Focus Designed for International Collaboration within AAAPT Background

4. Design of the UNU/ICTP PFF- 3kJ Plasma Focus System Background

5. UNU/ICTP PFF- placed at ICTP, 1988 Background
Network: Malaysia, Singapore, Thailand, Pakistan, India, Egypt,
Similar machines with designs based on or upgraded:
Zimbabwe, Syria, USA, Bulgaria, Iran

6. The Code Intro code
It was realized that the laboratory work should be complemented by computer simulation.
A 2-phase model was developed in 1983
We continually developed the code to its 5-phase form
It now includes thermodynamics data so the code can be operated in H2, D2, D-T, N2, O2, He, Ne, Ar, Kr, Xe.
We have used it to simulate a wide range of plasma focus devices from the sub-kJ PF400 (Chile) , the small 3kJ UNU/ICTP PFF (Network countries), the NX2 3kJ Hi Rep focus (Singapore), medium size tens of kJ DPF78 & Poseidon (Germany) to PF1000, the largest in the world.
An Iranian Group has modified the model, calling it the Lee model, to simulate Filippov type plasma focus .

7. Review of Lee Code Plasma Focus Numerical Experiments Intro code
The code10 couples the electrical circuit with PF dynamics, thermodynamics and radiation.
Using standard circuit equations and equations of motion adapted for the axial and radial phases, the code is consistent in
(a) energy,
(b) charge and
(c) mass.

8. Development of the code Intro code
It was described in and used in the design and interpretation of experiments12-15.
This evolved into a 5-phase code incorporating finite small disturbance speed16, radiation and radiation-coupled dynamics17-19,
It was web-published20 in 2000 and
Plasma self- absorption was included20 in 2007

9. Axial Phase- snow plough type
Axial Phase: Snow- plow type equation of motion coupled to circuit equation

10. Inward radial shock phase - 4 coupled equations namely: inward radial shock, elongation, inward radial piston and circuit, respectively as follows:

11. Reflected Shock RS phase
Reflected shock radial phase- RS moving out, piston moving in and column elongation coupled to circuit equation- total of 4 coupled equations

12. Pinch Phase – Key equation: radiation- coupled dynamics
Piston dynamics depends on dI/dt, dzf/dt and dQ/dt.
dI/dt is computed with coupled circuit equation;
dQ/dt is calculated with the various components being the bremsstrahlung, recombination and line radiation and Joule heating.

13. Usage Intro code
It has been used extensively as a complementary facility in several machines, for example: UNU/ICTP PFF12,14,15,17-19, NX219,22, NX119, DENA23, AECS
It has also been used in other machines for design and interpretation including Chile’s sub-kJ PF and other machines24, Mexico’s FNII25 and the Argentinian UBA hard x-ray source26.
More recently KSU PF (US), NX3 (Singapore), FoFu I (US) and several Iranian machines APF, Tehran U, AZAD U

14. Information derived Intro code
Information computed includes
axial and radial dynamics11,17-23, pinch properties
SXR emission characteristics and yield17-19, 22, 27-33,
design of machines10,12,24,26,
optimization of machines10,22, 24,30 and adaptation to Filippov-type DENA23.
Speed-enhanced PF17 was facilitated.

15. Information Derived Intro code
Scaling Properties;
Constancy of energy density (per unit mass) across range of machines14
Hence same temperature and density14
Constancy of drive current density I/a relating to the speed factor14
(I/a)/r0.5
Scaling of pinch dimensions & lifetime14 with anode radius ‘a’:
pinch radius ratio rp/a =constant
pinch length ratio zp/a=constant
pinch duration ratio tp/a=constant

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

17. Recent development and Insights Intro code
PF neutron yield calculations34
Current & neutron yield limitations35 with reducing L0
Wide-ranging neutron scaling laws
Wide-ranging soft x-ray scaling laws in various gases
Neutron saturation36,37- cause and Global Scaling Law
Radiative collapse 38
Current-stepped PF39
Extraction of diagnostic data33,40-42
Anomalous resistance data43,44 from current signals
Benchmarks for Ion Beams- scaling with E0.

18. Philosophy of our Modelling Philosophy
Experimental based
Utility prioritised
To cover the whole process- from lift-off, to axial, to all the radial sub-phases; and recently to post-focussed phase which is important for advanced materials deposition and damage simulation.

19. Priority of Basis Philosophy
Correct choice of Circuit equations coupled to equations of motion ensures:
Energy consistent for the total process and each part of the process
Charge consistent
Mass consistent
Fitting computed current waveform to measured current waveform ensures:
Connected to the reality of experiments

20. Priority of Results Philosophy
Applicable to all PF machines, existing and hypothetical
Current Waveform accuracy
Dynamics in agreement with experiments
Consistency of Energy distribution
Realistic Yields of neutrons, SXR, other radiations; Ions and Plasma Stream (latest-Benchmarks); in conformity with experiments
Widest Scaling of the yields
Insightful definition of scaling properties
Design of new devices; e.g. Hi V & Current-Step
Design new experiments-Radiative cooling & collapse

21. Philosophy, modelling, results &
Philosophy, modelling, results & applications of the Lee Model code Philosophy

22. Numerical Experiments Philosophy
Range of activities using the code is so wide
Not theoretical
Not simulation
The correct description is:
Numerical Experiments

23. UPFLF-The Code Control Panel- configured for PF1000 Demo
L0 nH C0 mF b cm a cm z0 r0 mW fm fc fmr fcr V0 P0 M.W. A At/Molecular

24. PF1000, ICDMP Poland, the biggest plasma focus in the world
PF1000, ICDMP Poland, the biggest plasma focus in the world Firing the PF Demo

25. Fitting: 1. L0 fitted from current rise profile 2
Fitting: 1. L0 fitted from current rise profile 2. Adjust model parameters (mass and current factors fm, fc, fmr, fcr) until computed current waveform matches measured current waveform (sequential processes shown below) Demo

26. PF1000 fitted results Demo

27. PF1000: Yn Focus & Pinch Properties as functions of Pressure Demo

28. Plasma Focus- Numerical Experiments leading Technology Insights
Numerical Experiments- For any problem, plan matrix, perform experiments, get results- sometimes surprising, leading to new insights
In this way, the Numerical Experiments have pointed the way for technology to follow

29. NE showing the way for experiments and technology Insights
PF1000 (largest PF in world): 1997 was planning to reduce static inductance so as to increase current and neutron yield Yn. They published their L0 as 20 nH
Using their published current waveform and parameters we showed
a. their L0 =33 nH
b. their L0 was already at optimum
c. that lowering their L0 would be a waste of effort and resources

30. As L0 was reduced from 100 to 35 nH - As expected
Results from Numerical Experiments with PF For decreasing L0- from 100 nH to 5 nH Insights 1
As L0 was 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.
Yn also had a maximum value of 3.2x1011 at 35 nH.

31. Pinch Current Limitation Effect - Insights 1
L0 decreases higher Ipeak bigger a longer zp bigger Lp
L0 decreases shorter rise time shorter zo smaller La
L0 decreases, Ipinch/Ipeak decreases

32. Pinch Current Limitation Effect Insights 1
L0 decreases, L-C interaction time of capacitor decreases
L0 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 L0

33. Pinch Current Limitation Effect Insights 1
A combination of two complex effects
Interplay of various inductances
Increasing coupling of C0 to the inductive energetic processes as L0 is reduced
Leads to this Limitation Effect
Two basic circuit rules: lead to such complex interplay of factors which was not foreseen; revealed only by extensive numerical experiments

34. Neutron yield scaling laws and neutron saturation problem Insights 2
One of most exciting properties of plasma focus is
Early experiments show: Yn~E02
Prospect was raised in those early research years that, breakeven could be attained at several tens of MJ .
However quickly shown that as E0 approaches 1 MJ, a neutron saturation effect was observed; Yn does not increase as much as expected, as E0 was progressively raised towards 1 MJ.
Question: Is there a fundamental reason for Yn

35. Global Scaling Law Insights 2 Scaling deterioration observed in numerical experiments (small black crosses) compared to measurements on various machines (larger coloured crosses) Neutron ‘saturation’ is more aptly portrayed as a scaling deterioration-Conclusion of IPFS-INTI UC research
S Lee & S H Saw, J Fusion Energy, (2008)
S Lee, Plasma Phys. Control. Fusion, 50 (2008)
S H Saw & S Lee.. Nuclear & Renewable Energy Sources Ankara, Turkey, 28 & 29 Sepr 2009.
S Lee Appl Phys Lett 95, (2009)
Cause: Due to constant dynamic resistance relative to decreasing generator impedance

36. Scaling for large Plasma Focus Scaling 1
Targets:
IFMIF (International fusion materials irradiation facility)-level fusion wall materials testing
(a major test facility for the international programme to build a fusion reactor)- essentially an ion accelerator

37. IPFS numerical Experiments:
Fusion Wall materials testing at the mid-level of IFMIF: 1015 D-T neutrons per shot, 1 Hz, 1 year for dpa- Gribkov Scaling 1
IPFS numerical Experiments:

38. Operating Parameters: 35kV, 14 Torr D-T E0=8.2 MJ
Possible PF configuration: Fast capacitor bank 10x PF1000-Fully modelled- 1.5x1015 D-T neutrons per shot Scaling 1
Operating Parameters: 35kV, 14 Torr D-T
Bank Parameters: L0=33.5nH, C0=13320uF, r0=0.19mW
E0=8.2 MJ
Tube Parameters: b=35.1 cm, a=25.3 cm z0=220cm
Ipeak=7.3 MA, Ipinch=3.0 MA
Model parameters 0.13, 0.65, 0.35, 0.65

39. Ongoing IPFS numerical experiments of Multi-MJ Plasma Focus Scaling 1

40. 50 kV modelled- 1.2x1015 D-T neutrons per shot Scaling 1
Operating Parameters: 50kV, 40 Torr D-T
Bank Parameters: L0=33.5nH, C0=2000uF, r0=0.45mW
E0=2.5 MJ
Tube Parameters: b=20.9 cm, a=15 cm z0=70cm
Ipeak=6.7 MA, Ipinch=2.8 MA
Model parameters 0.14, 0.7, 0.35, 0.7
Improved performance going from 35 kV to 50 kV

41. IFMIF-scale device Scaling 1
Numerical Experiments suggests the possibility of scaling the PF up to IFMIF mid-scale with a PF1000-like device at 50kV and 2.5 MJ at pinch current of 2.8MA
Such a system would cost only a few % of the planned IFMIF

42. Scaling further- possibilities Scaling 2
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

43. Scaling from Ipinch using present predominantly beam-target : Yn=1
Scaling from Ipinch using present predominantly beam-target : Yn=1.8x1010Ipeak3.8; Yn=3.2x1011Ipinch4.4 (I in MA) Scaling 2

44. SXR Scaling Laws Scaling 3
First systematic studies in the world done in neon as a collaborative effort of IPFS, INTI IU CPR and NIE Plasma Radiation Lab:
Ysxr = 8300× Ipinch3.6
Ysxr = 600 × Ipeak in J (I in MA).
Scaling laws extended to Argon, N and O by M Akel AEC, Syria in collaboration.

45. Special characteristics of SXR-for applications Scaling 3
Not penetrating; for example neon SXR only penetrates microns of most surfaces
Energy carried by the radiation is delivered at surface
Suitable for lithography and micro-machining
At low intensity - applications for surface sterilisation or treatment of food
at high levels of energy intensity, Surface hammering effect;, production of ultra-strong shock waves to punch through backing material; or as high intensity compression drivers in fusion scenarios

46. Compression- and Yield- Enhancement methods Scaling 4
Suitable design optimize compression
Role of high voltage
Role of special circuits e.g current-steps
Role of radiative cooling and collapse

47. A more dramatic example: radiative collapse in Kr, measured in the INTI PF on the basis of a current measurement
Fig 1. Fitting the computed current trace to the measured current trace of INTI PF at 12 kV 0.5 Torr Kr (shot 631). (Note the two curves have a close fit. Without radiation, the current (not shown) has a much smaller dip.

48. Expanding the current trace to show the region of the dip
Fig 1a Expanded view of the fitting

49. The fitting to the measured current waveform gives the radial trajectory revealing strong radiative collapse to very small radius
Having fitted the computed current trace to the measured current trace, the resulting radial trajectory indicates strong radiative collapse to very small radius, as shown in the following Figure. The radial trajectory hypothetically without radiation is also computed and shown for comparison.

50. Comparing radial trajectory with radiation (purple) and (hypothetically) without radiation (black)

51. Expanding the time scale to show details of radiative collapse region

52. From a measured current waveform, the parameters of the radiative collapse are derived.
The peak compression region is magnified and shown in the above figure.
The current values are normalized by 145 kA, the Pline is normalized by 3.7x1012 W and the radius ratio kp=rp/a is multiplied by 20.
The pinch compresses to a radius of cm corresponding to a radius ratio (pinch radius normalized to anode radius) of T
The radiative collapse is ended when plasma self-absorption attenuates the intense line radiation. The rebound of the pinch radius is also evident in Fig 3.
The line radiation leaving the plasma is also plotted (in normalized unit) to show its correlation to the trajectory in order to show the effect of the radiation on the compression.
This intense compression, despite the low mass swept in factor of fmr= 0.11, reaches 3.7 x 1026 ions m-3, which is 15 times atmospheric density (starting from less than 1/1000 of an atmospheric pressure).
The energy pumped into the pinch is 250 J whilst 41 J are radiated away in several ns, most of the radiation occurring in in a tremendous burst of 50 ps at peak compression with a peak radiation power of almost 4 x1012 W.
The energy density at peak compression is 4 x 1013 J m-3 or 40 kJ mm-3.
Thus from a measured current waveform in this Kr discharge, the parameters of this intense HED is measured showing the high density achieved .

53. All this information from just one measured current waveform

54. Recent developments
Modelling of: Ion beam fluence Post focus axial shock waves Plasma streams Anode sputtered material Radiatively enhanced compressions Energy Throughput scaling

55. Plasma Focus Pinch Latest photo taken by Paul Lee on INTI PF

56. Emissions from the PF Pinch region Latest
+Mach500 Plasma stream
+Mach20 anode material jet

57. Highest pre-pinch radial speed>25cm/us M250
Sequence of shadowgraphs of PF Pinch- M Shahid Rafique PhD Thesis NTU/NIE Singapore Latest
Highest post-pinch axial shock waves speed ~50cm/us M500
Highest pre-pinch radial speed>25cm/us M250

58. Slow Copper plasma jet 2cm/us M20
Much later…Sequence of shadowgraphics of post-pinch copper jet S Lee et al J Fiz Mal 6, 33 (1985) Latest
Slow Copper plasma jet 2cm/us M20

60. Extracted from V A Gribkov presentation: IAEA Dec 2012

61. Flux out of Plasma Focus
Charged particle beams
Neutron emission when operating with D
Radiation including Bremsstrahlung, line radiation, SXR and HXR
Plasma stream
Anode sputtered material

62. Modelling the flux Latest
Ion beam number fluence is derived from beam-plasma target considerations as:
Fibt = Cn Ipinch2zp[ln(b/rp)]/ (prp2 U1/2)
ions m-2
All SI units:calibration constant Cn =8.5x108; calibrated against experimental point at 0.5MA
Ipinch=pinch current
zp=pinch length
b=outer electrode, cathode radius
rp=pinch radius
U=beam energy in eV where in this model U=3x Vmax (max dynamic induced voltage)
These values are computed by our code

63. Table 1: Parameters of a range of Plasma Focus and
Table 1: Parameters of a range of Plasma Focus and computed Ion Beam characteristics Latest
Machine
PF1000
DPF78
NX3
INTIPF
NX2
PF-5M
PF400J
E0 (kJ)
486
31.0
14.5
3.4
2.7
2.0
0.4
L0 (nH) 33 55 50
110 20 40
V0 (kV) 27 60 17 15 14 16 28
'a' (cm)
11.50
4.00
2.60
0.95
1.90
1.50
0.60
c=b/a
1.4
1.3 2
2.2
1.7
Ipeak (kA)
1846
961
582
180
382
258
129
Ipinch (kA)
862
444
348
122
220
165 84
zp (cm)
18.8
5.5
3.8
2.8
2.3
0.8
rp (cm)
2.23
0.62
0.13
0.31
0.22
0.09
t (ns)
255
41.0
36.5
7.6
30.0
12.2
5.1
Vmax (kV) 42
68.3 35 25 22
32.3 18

64. 3.9 3.2 5.7 3.6 3.4 2.4 2.6 Machine IB Ion Fluence (x1020m-2) PF1000
Latest
Machine
IB Ion Fluence (x1020m-2)
PF1000
3.9
DPF78
3.2
NX3
5.7
INTI
3.6
NX2
3.4
PF5M
2.4
PF400J
2.6
IB Ion Flux (x1027m-2s-1)
1.5
7.8
15.6
46.7
11.5
19.6
50.4
Mean Ion Energy (keV)
126
205
105 75 66 97 54
IB Energy Fluence (x106 J m-2)
10.6
9.6
4.3
3.7
2.2
IB Energy Flux (x1013 W m-2)
3.1
25.8
26.3
56.4
12.0
30.6
43.2
Ion Number (x1014)
6100
390
280 19
110 37
5.9
IB Energy (J)
12248
1284
479 23
111 58
5.1
(% E0)
(2.5)
(4.1)
(3.3)
(0.7)
(2.8)
(1.3)
IB current (kA)
380.0
152.4
124.8
40.0
56.7
49.1
18.6
IB Damage Ftr (x1010 Wm-2s0.5)
1.6
5.2
5.0
4.9
2.1
Ion Speed (cm/ms)
347
443
317
269
250
305
226
Ion Number per kJ (x1014)
12.6
12.7
19.4
5.6
40.1
18.1
15.1
Plasma Stream Energy (J)
39120
394
1707
249
369 92 17
(8.1)
(12.0)
(7.4)
(13.7)
(4.5)
Plasma Stream Speed (cm/ms)
18.2
23.1
24.2
47.4
20.1
48.6
35.7

65. Plasma Focus Numerical Experiments- Conclusions: We have covered
Development, usage and results
Basis and philosophy
Reference for Diagnostics
Insights and frontiers
Continuing development- Ion beam modelling, radiative collapse for HED and energy throughput scaling.

66. References
10S Lee, Radiative Dense Plasma Focus Computation Package: RADPF websites)
11 S Lee in Radiation in Plasmas Vol II, Ed B McNamara, Procs of Spring College in Plasma Physics (1983) ICTP, Trieste, p , ISBN , Published byWorld Scientific Publishing Co, Singapore (1984)
12S 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. Amer J Phys 56, 62 (1988)
13T.Y.Tou, S.Lee & K.H.Kwek. IEEE Trans Plasma Sci 17, (1989)
14S Lee & A Serban, IEEE Trans Plasma Sci 24, (1996)
15 SP Moo, CK Chakrabarty, S Lee - IEEE Trans Plasma Sci 19, (1991)
16D E Potter, Phys Fluids 14, 1911 (1971)
17A Serban and S Lee, Plasma Sources Sci and Tehnology, 6, 78 (1997)
18M H Liu, X P Feng, SV Springham & S Lee, IEEE Trans Plasma Sci. 26, 135 (1998)
19S Lee, P.Lee, G.Zhang, X.Feng, V.A.Gribkov, M.Liu, A.Serban & T.Wong. IEEE Trans Plasma Sci, 26, 1119 (1998)
20S.Lee in (archival website) (2012)
21S. Lee in ICTP Open Access Archive: (2005)
22D.Wong, P.Lee, T.Zhang, A.Patran, T.L.Tan, R.S.Rawat & S.Lee. Plasma Sources, Sci & Tech 16, 116 (2007)
23V. Siahpoush, M.A.Tafreshi, S. Sobhanian, & S. Khorram. Plasma Phys & Controlled Fusion 47, 1065 (2005)

67. References
24L. Soto, P. Silva, J. Moreno, G. Silvester, M. Zambra, C. Pavez, L. Altamirano, H. Bruzzone, M. Barbaglia, Y. Sidelnikov & W. Kies. Brazilian J Phys 34, 1814 (2004)
25H.Acuna, F.Castillo, J.Herrera & A.Postal. International conf on Plasma Sci, 3-5 June 1996, conf record Pg127
26C.Moreno, V.Raspa, L.Sigaut & R.Vieytes, Applied Phys Letters 89(2006)
27S. Lee, R S Rawat, P Lee and S H Saw, J. Appl. Phys. 106, (2009)
 28S. H. Saw and S. Lee, Energy and Power Engineering, 2 (1), (2010)
 29M. Akel, Sh Al-Hawat, S H Saw and S Lee, J Fusion Energy, 29, 3, (2010)
 30S H Saw, P C K Lee, R S Rawat, S Lee, IEEE Trans Plasma Sci, 37, (2009)
31Sh. Al-Hawat, M. Akel, S H Saw, S Lee, J Fusion Energy, 31, 13 – 20, (2012)
 32Sh Al-Hawat, M. Akel , S. Lee, S. H. Saw, J Fusio Energy 31, (2012)
 33S Lee, S H Saw, R S Rawat, P Lee, A.Talebitaher, A E Abdou, P L Chong, F Roy,
A Singh, D Wong and K Devi, IEEE Trans Plasma Sci 39, (2011)
 34S Lee and S H Saw, J Fusion Energy, 27, (2008)
 35S. Lee and S H Saw, Appl. Phys. Lett., 92, (2008)
 36S Lee. Plasma Physics Controlled Fusion, (2008)
37S Lee. Appl. Phys. Lett (2009) 

68. References
38S Lee, S. H. Saw and Jalil Ali, J Fusion Energy DOI: /s First Online 26 Feb (2012)
 39S Lee and S H Saw, J Fusion Energy DOI: /s First Online 31 January (2012) 
40 S Lee, S H Saw, P C K Lee, R S Rawat and H Schmidt, Appl Phys Lett 92, (2008)
 41S H Saw, S Lee, F Roy, PL Chong, V Vengadeswaran, ASM Sidik, YW Leong & A
Singh, Rev Sci Instruments, 81, (2010)
 42 S Lee, S H Saw, R S Rawat, P Lee, R Verma, A.Talebitaher, S M Hassan, A E Abdou,
Mohamed Ismail, Amgad Mohamed, H Torreblanca, Sh Al Hawat, M Akel, P L Chong, F
Roy, A Singh, D Wong and K Devi, J Fusion Energy 31,198–204 (2012)
 43S Lee, S H Saw, A E Abdou and H Torreblanca, J Fusion Energy 30, (2011)
 44F M Aghamir and R A Behbahani, J. Plasma Physics: doi: /S in press (2012)
 45 S.Lee, S.H.Saw, L..Soto, S V Springham, S P Moo, Plasma Phys and Control. Fusion, (11pp) (2009)
46 S.P. Chow, S. Lee and B.C. Tan, J Plasma Phys, (1972).