Numerical Experiments on Ion Beams from Plasma Focus S H Saw 1,2 & S Lee 1,2,3 1 INTI International University, Nilai, Malaysia 2 Institute for Plasma Focus Studies, Chadstone, VIC 3148, Australia 3 University of Malaya, Kuala Lumpur, Malaysia Siam Physics Congress SPC2013 Thai Physics Society on the Road to ASEAN Community March 2013

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Summary- Previous work Much work using variety of diagnostics reported on plasma focus ion beams, mainly experimental Confusing picture- even units are confusing un-correlated across devices and experiments No benchmark or scaling patterns appears to have been reported until: Our Previous work: We adapted beam- gas target neutron yield mechanism for D beams from plasma focus Our Previous results: (first plasma focus results on ion beam scaling- D)  Ion number fluence: x ions m -2 ; independent of E 0  Ion Number: x ions per kJ; dependent on E 0

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Summary- New work Our New work: First principle derivation of ion number flux and fluence equations applicable to all gases. New results:  Fluence, flux, ion number and ion current decrease from the lightest to the heaviest gas  Energy fluence, energy flux and damage factors are constant from H 2 to Ne; but increase for the 3 high-Z gases Ar, Kr and Xe due to to radiative collapse.  The FIB energy has a range of 4-9% E 0.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Brief review of existing work Measurements of ion beams from PF’s have produced a wide variety of results using different units; Less correlated than expected No discernable pattern or benchmarks. In summarizing experimental results, Bernard el al 1 in 1996, it was reported that total yields of ions reach sr -1 depending on energetics and experimental conditions. In a single discharge fast ions are emitted from point-like (sub- mm) sources mostly as narrow micro-beams with duration times of 2-8 ns forming intense bunches having total powers reaching to W.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Brief Review: Takao et al kJ (43  F, 30 kV 500 kA peak current) PF published nitrogen ion beam power brightness of 0.23 GW cm -2 sr (maximum ion energy at 0.5 MeV) using a solid anode; and 1.6 GW cm -2 sr (maximum ion energy of 1 MeV) using a hollow anode. Peak ion current densities of A cm -2 (50-60 ns) were recorded. [which would give a beam ion current <1kA!!]

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Brief Review- Bhuyan et al 3, 2011 reported beam ion densities of x10 19 m -3 (ion energy keV) at the aperture of Faraday Cups for 0-25 degree angular positions 6 cm from the anode top in a 40 kV 2.2 kJ neon PF. Track densities (CR-39 film) had a maximum value of 10.9x10 9 tracks m -2 at 30 degrees. In another experiment, Bhuyan 6 operating a 1.8 kJ methane PF quoted a flux of 2x10 22 m -2 s -1 multiple-charged carbon ions ( keV).

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Brief Review- Kelly et al 4 UBA PF II (4.75 kJ 30 kV) in nitrogen; used Faraday Cup and Thomson spectra to measure nitrogen ions ( keV) recording 3.2x10 13 ions/sterad with energy content of 0.74 J/sterad. In another experiment 5 same machine in dueterium, Kelly surmised a total number of deuterons at keV.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Brief Review- Szydlowski et al 7 foil-covered CR-39 track detectors states that the PF-1000 generates 10 5 ions/mm 2 (energy above several dozen keV) with neutron yields of neutrons/shot.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Brief review- Bostick et al kJ PF using TOF and ion filters recorded fluence of (MeV.sr) -1 for the energy spectrum of deuterons ( MeV) with FWHM of ns and (MeV.sr) -1 (at 1-9 MeV) with FWHM of ns.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Brief Review- Summary Many different experiments Many different machines Different gases Many types of diagnostics Many sets of data Data- some total (FC), some sampling (track detectors) ie not all ions recorded Different perspectives, different units: number sr -1 ; bunch power in W; beam power brightness in GW cm -2 sr ; ion current densities in A cm -2 ; beam ion densities in m -3 ; tracks m -2 ; ions/sterad ; J/sterad ; total ion numbers; flux in m -2 s -1 ; ion fluence in (MeV.sr) -1 Correlation among experiments? Benchmarking? Scaling? Obvious errors of orders of magnitude!!

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Our numerical experiments on Ion Beams The ion beam numerical experiments adds on as a branch to the integrated view which our numerical experiments strive to present of the plasma focus

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Philosophy, modelling, results & applications of the Lee Model code Philosophy

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Latest development Latest Modelling : Ion beam fluence Post focus axial shock waves Plasma streams Anode sputtered material

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Summary of basic physical picture

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Plasma Focus Pinch with plasma stream (Paul Lee- INTI PF) Plasma Focus Pinch with plasma stream (Paul Lee- INTI PF)

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Emissions from the PF Pinch region Emissions from the PF Pinch region Mach500 Plasma stream Mach20 anode material jet

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Sequence of shadowgraphs of PF Pinch- ( M Shahid Rafique PhD Thesis NTU/NIE Singapore 2000) Highest post-pinch axial shock waves speed ~50cm/us M500 Highest pre-pinch radial speed>25cm/us M250

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Extracted from V A Gribkov presentation: IAEA Dec V N Pimenov 2008 Nukleonika 53:

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Comparing large and small PF’s- Dimensions and lifetimes- putting shadowgraphs side-by-side, same scale Lifetime ~10ns order of ~100 ns Anode radius 1 cm 11.6 cm Pinch Radius: 1mm 12mm Pinch length: 8mm 90mm

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy 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

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Basic Definition of Ion Beam characteristics Beam number fluence F ib (ions m -2 ) Beam energy fluence (J m -2 ) Flux =fluence x pulse duration Beam number flux F ib  (ions m -2 s -1 ) Beam energy flux (W m -2 )

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Our starting point: the mechanism described by V. A. Gribkov et al in J Phys. D 40, 3592 (2007) Beam of fast deuteron ions produced by diode action in thin layer close to the anode; plasma disruptions generating the high voltages. Beam interacts with the hot dense plasma of focus pinch column to produce the fusion neutrons. In our modeling of the neutron yields, each factor contributing to the yield is estimated as a proportional quantity. The yield is obtained as an expression with proportionality constant. The neutorn yield is then calibrated against a known experimental point. We start with the neutron yield (rather than the ion beam number) because our method requires a calibration point; and for neutron yields by placing published yields on a chart we can obtain a good fitted calibration point. Had we started with the D ion beam number, we would not have been able to get a reliable calibration point

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Ion beam flux and fluence equations Ion beam flux J b =n b v b where n b = number of beam ions N b divided by volume of plasma traversed v b = effective speed of the beam ions. All quantities in SI units, except where otherwise stated. Note that n b v b has units of ions per m -2 s -1.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy We derive n b from pinch inductive energy considerations. Total number of beam ions N b (each ion mass Mm p, speed v b ) has KE= (1/2) N b Mm p v b 2 where m p =1.673x kg is proton mass; M=mass number of ion e.g. neon ion has mass number M=20. Assume this KE is imparted by a fraction f e of the inductive pinch energy (1/2) L p I pinch 2 where L p =(  /2p) (ln[b/r p ])z p ; where  =4p x10 -7 Hm-1, b=outer electrode of PF carrying the return current, rp= pinch radius and zp= length of the pinch. The pinch current Ipinch is the value taken at start of pinch. Thus: (1/2) N b Mm p v b 2 = (1/2) f e (m/2p) (ln[b/r p ]) z p I pinch 2 ; n b = N b /(pr p 2 z p ) n b = (m/[2p 2 m p ]) (f e /M) {(ln[b/r p ])/(r p 2 )} (I pinch 2 / v b 2 ) – (1)

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy We derive v b from the accelerating voltage taken as the diode voltage U Each ion mass Mm p, speed v b, effective charge Z eff is given KE (1/2) Mm p v b 2 by diode voltage U. Therefore: (1/2) Mm p v b 2 = Z eff eU where e is the electronic (or unit) charge 1.6x C; Hence v b = (2e/m p ) 1/2 (Z eff /M) 1/2 U 1/2 – (2)

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy From (1) multiplying both sides of equation by v b, we have Algebraic manipulations: n b v b = (m/[2p 2 m p ]) (f e /M) {(ln[b/r p ])/(r p 2 )} (I pinch 2 / v b ) Eliminate v b on RHS of this equation by using Eqn (2) gives J b =n b v b = (m/[2p 2 m p ])(f e /M){(ln[b/r p ])/(r p 2 )}(I pinch 2 )(m p /2e) 1/2 (M/Z eff ) 1/2 /U 1/2 = (m/[2.83p 2 (em p ) 1/2 ])(f e /[M Z eff ] 1/2 ){(ln[b/r p ])/(r p 2 )}(I pinch 2 )/U 1/2 Noting that: (m/[2.83p 2 (em p ) 1/2 ]) = 2.74x We have: Result: Flux = J b = 2.75x10 15 (f e /[M Z eff ] 1/2 ){(ln[b/r p ])/(r p 2 )}(I pinch 2 )/U 1/2 ions m -2 s -1 (3)

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy The fluence is the flux multiplied by pulse duration t; Thus: Fluence: J b  = 2.75x10 15 t (f e /[M Z eff ] 1/2 ){(ln[b/r p ])/(r p 2 )}(I pinch 2 )/U 1/2 ions m -2 (4)

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Value of f e The parameter f e is the fraction of energy converted into beam energy from the inductive energy of the pinch. By analyzing neutron yield data 1,3,4 and pinch dimensional and temporal relationships 15 we estimate a value of f e =0.14. This condition f e =0.14 is equivalent to ion beam energy of 3%-6% E 0 in the case when the pinch inductive energy holds 20% -40% of E 0. Our extensive study of high performance low inductance plasma focus classified 16 as Type 1 shows that this estimate of f e is consistent with data.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy We summarise the assumptions: 1.Ion beam flux J b is n b v b with units of ions m -2 s Ion beam is produced by diode mechanism (ref). 3.The beam is produced uniformly across the whole cross- section of the pinch 4.The beam speed is characterized by an average value v b. 5.The beam energy is a fraction f e of the pinch inductive energy, taken as 0.14 in the first instance; to be adjusted as numerical experiments indicate. 6.The beam ion energy is derived from the diode voltage U 7.The diode voltage U is proportional to the maximum induced voltage V max ; with U=3V max (ref) taken from data fitting in extensive earlier numerical experiments.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Procedure The value of the ion flux is deduced in each situation (specific machine using specific gas) by computing the values of Z eff, r p, I pinch and U by configuring the Lee Model code with the parameters of the specific machine and specific gas.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Example: Numerical Experiment for NX2 based on following fitted parameters: L 0 =20 nH, C 0 =28 uF, r 0 =2.3 m  b=4.1cm, a= 1.9 cm, z 0 =5 cm f m =0.08, f c =0.7, f mr =0.2, f cr =0.7 V 0 =14 kV, P 0 = within appropriate P range for each gas

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Range of Pressures PF axial run-down time covers a range which encompasses at least from 0.5 to to 1.3 of the short-circuit rise time 1.57*(L 0 /C 0 ) 0.5. The matched condition with the strongest energy transfer into the plasma focus pinch is well covered within the range; also the range covers conditions of high enough pressures that the focus pinch is almost not ocurring as defined by the condition that the reflected shock is barely able to reach the rapidly decelerating magnetic piston.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Collection of data For each shot the dynamics is computed and displayed by the code; which also calculates and displays the ion beam properties. For H 2, D 2, He, N 2 and Ne the procedure is relatively simple even though Ne already exhibits enhanced compression due to radiative cooling.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy RESULTS Fig 2(a) shows a typical PF discharge current computed for NX2 and fitted to the measured discharge current in order to obtain the model parameters f m, f c, f mr and f cr 32,33,41. Fig 2(b) shows the computed radial trajectories of the radially inward shock wave, the reflected radially outward shock wave, the piston trajectory and the pinch length elongation trajectory. Range of pressures: widest for lightest gas H 2 (1 Torr -70 Torr ). For D 2 and He Torr; for Ne we successfully ran numerical experiments Torr; N 2 from Torr; Xe to 1.8 Torr.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Fig 3 illustrates the different compression of the PF pinch. In H 2, D 2 & He radius ratio ~0.15 up to 10 Torr then rises towards 0.2. For N 2 the radius ratio drops from 0.15 to about 0.13 over range of operation. Ne shows signs of enhanced compressions 3- 5 Torr; smaller radius ratio to 0.08 at 4 Torr. Ar shows strong radiative collapse with radius ratio of 0.04 (cut-off value) around 2.0 Torr. Kr strong radiative collapse from Torr; Xe from 0.3 to 1.5 Torr.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Fig. 4a shows the flux in ions m -2 s -1. H 2 : 6x10 27 at 1 Torr, rises to a peak 1.9x10 28 at 25 Torr; pressure of best energy transfer for NX2 in H 2. The D 2 and He curve show same trend but lower peak flux values at 15 Torr. N 2 shows same trend peaking at 3.6x10 27 at 3 Torr. Ne shows an accentuated peak of 6.6x10 27 at 4 Torr due to radiative enhanced compression. Ar flux is even more accentuated with 8x10 27 at 2 Torr. For Kr although the radiative collapse is more severe than Ar, flux is flat at 1.4x10 27 at 1 Torr.; this is due to the much greater energy per ion. Xe shows the same flat flux curve as Kr with a flat central value around 6x Conclusion: Beam ion flux drops as the mass number increases, with accentuating factors provided by radiatively enhanced compression.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Fig 5a shows the fluence in ions m -2. The shape of the curves and the trend with gases are very similar to the flux The peak values of the fluence (ions m -2 ) range from 8x10 20 for H2 decreasing to 6x10 18 for Xe; with clearly radiation enhanced values of 2x10 20 and 1.7x10 20 for Ar and Ne respectively..

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Figure 6 a-c show that the beam ion number per kJ range from for the lightest gases decreasing to 1.5x10 12 for Xe in the radiative enhanced regime.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Although the beam ion number is the lowest (see Fig 6) for the heaviest gases Ar Kr and Xe, yet these beams also carry the largest amounts of energy at 8-9% E 0 compared to around 5-8% for the other gases. This is because the energy per ion more than compensate for the low numbers.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy The damage factor defined as power flow density multiplied by (pulse duration) 0.5. This quantity is considered to be important for assessing the utility of a beam for damage simulation of plasma-facing wall materials in fusion test reactors. The results show that the heaviest ions produce the biggest damage factors.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy

IV Conclusion In this paper we deduce the flux equation of ion beams in plasma focus for any gas using experimental data from the case of deuterons to obtain a calibration constant for energy fraction. We configure the Lee Model code as the NX2 using best estimated average model mass and current factors obtained from fitting the computed current traces of several gases with experimentally measured current traces. The flux equation is incorporated into the code and the number and energy flux and fluence from different gases are computed together with other relevant properties. The results portray the properties of the ion beam at the pinch exit.

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Results: The ion fluence range from 7x10 20 for the lightest gas H2 decreasing through the heavier gases until a value of 1.7x10 20 for Ar and decreases further dramatically to 0.03 x10 20 for Xe. The very small fluence value of Xe is due to the very large energy of the Xe ion, estimated to have average charge state Z eff of 28 and accelerated by exceedingly large electric fields induced in the radiative collapse

School and Training Course on Dense Magnetized Plasma as a Source of Ionizing Radiations, their Diagnostics and Applications 8-12 October 2012, ICTP, Trieste, Italy Ion Beam Benchmarks: Mather PF Latest ion number fluence ions m -2 The number of beam ions: beam ions per kJ for PF’s with typical static inductances L 0 of nH. Total beam energy: for NX2 Beam current: of I peak. Beam energy fluence: J m -2 Beam Energy Flux: W m -2 The independence from E 0 of the ion beam fluence is likely related to the constancy of energy density (energy per unit mass) that is one of the key scaling parameters of the PF throughout its E 0 range of sub kJ to MJ 14,28.