Pushing the limits of existing plasma focus towards 1016 fusion neutrons with Q=0.01

Sing LEE

1 Institute for Plasma Focus Studies,32 Oakpark Drive,Chadstone VIC3148,Australia

2 Physics Department,University of Malaya,Kuala Lumpur 50603,Malaysia

3 INTI International University,Nilai 71800,Malaysia

4 Fuse Energy Technologies,Napierville QC J0J 1L0,Canada

Abstract Existing conventional megajoule plasma focus machines with 2-3 MA are producing fusion neutron yields of several times 1011 in deuterium operation,the fusion yields predominantly being the beam-gas target.Increasing the current to 10 MA and using 50%-50% D-T mixture will scale the neutron yield towards 1016 D-T fusion neutrons.In this work,we derive the Lawson criterion for plasma focus devices with a beam-target fusion neutron mechanism,so that we may glimpse what future technological advancements are needed for a break-even Q=1 plasma focus.We perform numerical experiments with a present-day feasible 0.9 MV,8.1 MJ,11 MA machine operating in 100 Torr in 50%-50% D-T mixture.The Lee Code simulation gives a detailed description of the plasma focus dynamics through each phase,and provides plasma and yield parameters which show that out of 1.1×1019 fast beam ions produced in the plasma focus pinch,only 1.24×1014 ions take part in beam-target fusion reactions within the pinch,producing the same number of D-T neutrons.The remnant beam ions,numbering at least 1019,exit the focus pinch at 1.9 MeV,which is far above the 115 keV ion energy necessary for an optimum beam-target cross-section.We propose to regain the lost fusion rates by using a high-pressure D-T-filled drift-tube to attenuate the energy of the remnant beam ions until they reach the energy for the optimum fusion cross-section.Such a fusion enhancement tube would further harvest beam-target fusion reactions by increasing the interaction path length(1 m)at increased interaction density(6 atm).A gain factor of 300 is conservatively estimated,with a final yield of 3.7×1016 D-T neutrons carrying kinetic energy of 83.6 kJ,demonstrating Q=0.01.

Keywords:plasma focus simulation,neutron enhancement,fusion harvester,plasma focus

1.Introduction

Current plasma focus(PF)machines for research on conventional megaampere(MA)include the PF1000[1,2]and the NIR(neutron imaging radiography)[3],both operating around the 2-2.5 MA level.The PF1000 is a 40 kV,1332 mF machine with 1 MJ at full voltage operated for the International Centre for Dense Magnetized Plasmas(ICDMP)in Poland.The NIR at Lawrence Livermore National Laboratory,USA,is a 1 MJ machine operating at up to 100 kV with four parallel modules of Marx generators,with currents at the level of 2.5 MA.According to plasma focus scaling of D-D neutron yield[4,5],Yn=1.8×1010(Ipeak)3.8(Ipeakin MA).This scaling was derived on the basis of numerical experiments using the Lee Code over a range of peak discharge currentIpeakof 0.3-5.7 MA.Similarly,experimental results from 50 kA to almost 3 MA produce a scaling law ofYn=8×109(Ipeak)4.4[6].The PF1000[1]reports that its good shots have neutron yields around several times 1011,whilst the NIR reports[3]best yields of 3.8×1011.Both these yields are in agreement with both scaling laws,which giveYnin the range of(2.5-6)×1011(numerically derived scaling law given above)and(1.7-4.5)×1011(experimentally derived scaling law given above),respectively.On the other hand,the HAWK plasma focus at the Naval Research Laboratory,USA,uses a 640 kV high inductance(607 nH 150 mW,stiff)current source to power an injected plasma upstream of an on-axis gas puff,producing 5×1010D-D neutrons at 0.67 MA[6,7].This yield is 13 times and 37 times in excess of that produced by a conventional dense plasma focus(DPF)according to the above-mentioned(Ipeak)3.8scaling and(Ipeak)4.4scaling,respectively.A five times enhancement(energy-based comparison)in neutron yield was observed for SPEED 1[8],which also used a highvoltage stiff(160 mW)current source.The energy-based comparison of HAWK with PF1000[6]shows a similar enhancement factor.It may be postulated that the high impedance of the current source reduces the drop in the current(current dip)during the pinch phase.This reduction is responsible for much of the enhancement.The method of mass delivery of HAWK gives better control and is reported to eliminate trailing mass and restrikes,thus enabling 100%current delivery to the pinch.The use of increasingly higher voltages[3,6,7]with designed high bank impedance and new methods of mass delivery[6]represent innovative approaches and new interest from the leading large laboratories.This concerted effort of these and other large laboratories may be expected to overcome the problem of the socalled neutron saturation in plasma focus.The problem of an apparent neutron limit was summarized in 2012 by Struve and Freeman[9],with data showing that the Los Alamos National Laboratory DPF 6.5 reached 2×1012D-D neutrons,a number which seemed unsurpassable and which was claimed by Nikulin and Polukhin[10]in 2007 as due to neutron saturation imposed by the need to increase anode length with increasing capacitance.Lee[11],in 2009,showed that the Nikulin and Polukhin scenario was erroneous and that the current and neutron‘saturation’was a misnomer for current and neutron scaling deterioration.Numerical experiments using the Lee Code further suggest that,to progress experiments to higher currents,it is advantageous[4,5,11]to go to higher voltages,rather than to try to increase the current by simply increasing the bank capacitance.This is because the speed of the plasma focus current sheath constitutes a‘dynamic’resistance of several mW during the rise of the discharge current.This resistance is beginning to dominate the total circuit impedance,so that the reduction of bank impedance by increasing the already very large bank capacitance leads to a situation of diminishing returns,resulting in a deteriorating scaling of the current.Thus,at several MA,a voltage of 100 kV may be efficient but,at 10 MA,1 MV may be advantageous.Nikulin and Polukhin[10]also pointed out the advantages of higher voltages.The experience of SPEED 2[12]suggested that at 300 kV an upper limit to sheath energy density caused sheath destruction,prevention of which required an additional surface irradiation conditioning process or an artificially coppered insulator surface.The gas puff and gas injection methods of HAWK[6]appear to have eliminated not just such insulator problems but also the re-strike problems[3]associated with trailing mass that plague conventional plasma focus machines.Thus,it appears that high voltages approaching MV may be feasible and should be the basis in our quest to develop bigger plasma focus machines.To envisage what the direction could be,we derive the Lawson criterion[13]for the plasma focus.In this derivation it is necessary to start with the fusion mechanism.The first question that needs to be asked is whether the fusion mechanism in a plasma focus is the beam-gas target or thermonuclear.

It is now known with certainty that plasma focus machines,from small sub-kJ to large MJ devices,all operate with the same speed factors,resulting in axial speeds of around 10 cm μs-1and radial speeds around 20-30 cm μs-1;the wider range of radial speeds are due to the range of the radius ratioc=b/a,wherebis the cathode radius andais the anode radius[5,14].Such speeds induce voltages ofIdL/dt(whereLis the tube inductance andIis the tube current)that is easily computed to be in the range of tens to hundreds of kV.The induced voltages are largest when the current sheath is compressed by its own magnetic field to a small radius in the radial phase.These high inductive voltages accelerate the beam ions to tens and hundreds of keV and beyond.A comprehensive review of the experimental evidence,on neutron production mechanisms[2]in experiments performed in plasma focus machines world-wide,clearly shows the properties and the dominance of these fast beam ions on the neutron production processes in the plasma focus pinch.On the other hand,the kinetic energies associated with the highly supersonic plasma sheaths are shown to generate temperatures in the stagnated pinched regions of less than 0.5 keV.To show the stark contrast between observed beam ion energies and plasma temperatures within the context of fusion mechanisms,figures 1 and 2 represent a good visual summary.Figures 1 and 2 relate the observed plasma focus beam ion energies to the beam-gas target fusion cross-sections σ[15,16]and,respectively,the observed focus pinch temperatures to the thermonuclear fusion reactivity〈σv〉[15,16].

Figure 1.The D-D beam-gas target fusion cross-section σ in cm2.

Figure 2.D-D fusion reactivity〈σv〉in cm3 s-1 for thermonuclear fusion.

Figures 1 and 2 contrast that the plasma focus beam ions have energies near the optimum of the D-D beam-gas target fusion cross-sections,whereas the plasma focus plasma temperatures are so low that the fusion reactivity〈σv〉are 8 to 14 orders of magnitude below the optimum thermonuclear cross-section.Moreover,the bigger plasma focus machines,for example PF1000 and NIR,have lower radial speeds due to their smaller values of the cathode to anode radius ratioc,typically 1.5 compared with small PFs,which tend to havecabout 3.Therefore,the bigger PFs have gross pinch temperatures close to 0.1 keV.Experiments have suggested that instabilities in pinches[17]may further inject energy from the magnetic fields into the plasma to the extent of 3 to 4 times that of the kinetic energies.This still does not alter the situation significantly.Thus,from energy considerations,conventional plasma focus machines are cold from a thermonuclear fusion point of view.Whilst these remarks may not apply to composite(hybrid)Z-pinches such as magnetized linear inertial fusion(MAGLIF)it must be noted that those composite systems invariably require a two-step process,which can be designed to lead to superior end-point compressions and temperatures[18].For conventional,even highvoltage,plasma focus machines,there is no escaping the fact that the gross pinch is cold from a thermonuclear point of view.The fusion cross-sections for D-T have the same general features as D-D,but the values of the cross-sections are generally 100 times bigger.Therefore,plasma focus machines operating in D-T also produce fusion neutrons through the beam-gas target mechanism.Hence,to derive the Lawson criterion[13]for plasma focus machines,we use the beamgas target fusion mechanism.

Figure 3.The value of the beam-target cross-section parameter σ/U1/2 versus U for D-T:σ is in cm-2.Note:the optimum value of σ/U1/2 is found to be 5×10-25 cm2 keV-0.5 at beam energy U of 115 keV.

Figure 4.The discharge current and scaled tube voltage(×10)of DPFQ0.01.

2.Lawson criterion for beam-gas target PF

2.1.Developing the Lawson criterion

We ask the question:how many D-T neutrons(from beamtarget)do we get per unit pinch energy?Modelling by the Lee Code[2,5,19,20]provides the number of beam-target neutrons[21,22]as follows:

whereIpinchis the current flowing through the pinch at the start of the slow compression phase;rpandzpare the pinch radius and length at the end of that phase;andniis the pinch ion density.Here,Cnis a constant which,in practice,we had calibrated with an experimental point.Here,all the quantities are in SI units(unless otherwise stated)with the beam ion energyU=3Vmax(Vmaxis the maximum induced tube voltage)in keV and the constantCn=1.4×107(a calibrated value)[21,22].The pinch energy[23]at temperatureTis

wherek=Boltzmann constant,γ=specific heat ratio=5/3,andZeff=1(for fully ionized D-T plasma).

We assume an equilibrium pinch,equate the confining magnetic pressure to the hydrostatic plasma pressure and we use:

We divideYb-tbyEpinch,replacinginYb-twith the RHS of equation(3),and we get the required number of beamtarget neutrons per unit pinch energy.Note:((ln(b/rp))～2 andzp～1.4afor a fully ionized hollow anode DPF[24,25],whereais the anode radius.

Thus,as a first step we obtain the general scaling for the number of D-T DPF beam-target neutrons

This general scaling stipulates that the number of beam-target neutrons depends on the pinch ion density,the anode radiusaand the energy of the D-T beam ions through the fusion crosssection parameter[σ/U1/2].

Table 1.The machine configuration of the 1 m PF at 1.2 MV 6 atm D-T operation for DPFQ1.

Using the optimum value of σ/U1/2(see figure 3),we obtain the optimized scaling as:

A D-T neutron has energy of 14.1 MeV,i.e.2.26×10-12J.By conservatively estimating thatEpinch～10% of the stored energyE0,then we may take the throughput ratio

ForQ＞1(i.e.better than break-even)

This may be taken as the equivalent Lawson criterion for the PF.

2.2.An example

As an example,we take:a=1 m then forQ＞1

ni＞1.7 ×1026m-3(at optimum beam energy of 115 keV)

(i.e.6 atm of fill pressure is sufficient for break-even,allowing for 50% particle loss from pinch).

We ran several series of numerical experiments witha=1 m at 4000 Torr.We concluded with the following configuration,shown in table 1.

The required peak current is 350 MA,and the break-even point(DPFQ1)requires extreme conditions,far beyond what is technologically proven in present-generation DPF machines.For example,the highest pressure that DPFs have been operated at is not much more than 50 Torr[26,27],less than 0.1 atm.Not least among the problems of operating at such high pressures is the localization of currents,leading to sparks and filaments causing asymmetry and breakup of the current sheet.The discharge of currents of the order of 10 MA and more is found to be associated with electrode damage.This indeed threatens the integrity of the whole structure and diagnostic accessories since the explosive powers of the submicrosecond transfer of just multi-megajoule between parts of the system exceed that of exploding dynamite sticks within the confines of the system.The structural engineering of suchlarge plasma focus systems is itself a formidable task,as would be experienced in MAGLIF experiments[28,29].

Table 2.The machine configuration[5,14,19]of DPFQ0.01:a=15 cm,at 0.9 MV,100 Torr D-T.

We note that this example ofQ=1 plasma focus is based on a single-step compression.It was shown from conservation of energy and momentum that in inertial fusion schemes,a two-step compression is much more efficient[18,30]than a single-pulse compression.For example,the MAGLIF concept[28,29]uses a laser to provide the first stage of heating,a liner capsule and high current compression as second stage compression,with simulated break-even at about 60 MA.We have shown that a plasma focus operated with a current step[31]achieves much improved efficiency in terms of neutron yield per unit stored energy.It is expected that an improved two-stage scheme coupled with a method to trap some of the remnant fast beam ions(to be discussed below)would further lower the energy and current requirements of the break-even plasma focus to be competitive with other inertial fusion schemes.More work is planned for further simulation of such a two-step or hybrid plasma focus system.

3.Achievable plasma focus with Q=0.01

3.1.Proposed configuration

As an intermediate step towards a DPFQ1 project,a technologically feasible device is proposed-DPFQ0.01 to reachQ～0.01.Linear Transformer Driver LTD technology is progressing towards efficient and reliable implementation at the MV level[32-34],and can be adapted for this configuration.The bank and tube parameters are given in table 2.These parameters were decided upon after numerous runs using the Lee Code discussed below.It is pertinent to note that even this technologically feasible device would still operate at 8 MJ,with currents of just over 10 MA,and would still present formidable challenges of proper current sheath formation and the engineering of the anode and structural integrity mentioned above.

3.2.Simulation and results

3.2.1.Description of the LEE Code.The Lee Code is a widely used radiation-coupled code[2,5,14,19]for simulating the plasma focus in various gases,including H2,He,N2,Ne,Ar,Kr,Xe,D2and D-T mixtures.It divides the plasma focus dynamics into five sub-phases,namely the axial phase,the radial inward shock phase,the reflected shock phase,the slow compression pinch phase and the expanded column post-pinch phase.The axial phase uses a snow-plow model,simply for time-matching purposes.The radial inward shock phase uses a slug model separating the magnetic piston from the shock front,and incorporates the all-important signal delay between the current sheath and shock front,as does the reflected shock phase.The pinch phase uses a radiationcoupled equation producing soft x-ray(SXR)yields in neon and nitrogen that agree with measured values,and demonstrates realistic effects of radiative cooling and radiative collapse when operated in the noble gases[23].In each phase,the equation(s)of motion are coupled to the circuit equation.Thermodynamics is implemented in the equations of motion.In the formulation,energy,charge,mass and momentum conservation is carefully maintained.Inductive voltages are considered as the driver of the observed fast ion beams.Realistic simulations are achieved for axial and radial dynamics[35],fast ion beams[36]and relativistic electron beams[37],and post-pinch fast plasma flows[38],with all these having been verified against experimental measurements.The simulated neutron yields in D and D-T have been compared extensively with experimental measurements in various machines,most recently by Marciniaket alin the Polish machine PF-24[39],with wider agreement than achievable by any other codes[2,5].The code has been used in planning and designing machines,for diagnostic references and for predicting yields of SXR,fast beam ions,plasma flows and fusion neutrons[2,5].Scaling laws and insights have been developed using this code[2,5,14,20-23].The scaling has been verified from sub-kJ machines to the MJ PF1000,particularly for neutron yield in terms of anode radius and discharge currents[2,5].The general and specific verification of this code has been reviewed recently by the ICDMP[2].Recent use of the code includes developing the concept of thermalization of the plasma focus using a tapered anode which,by increasing the value of(I/a)in the tapered pinch,increases the temperature of the focus pinch,for the pinch to transition from a beam-target fusion source to one of a thermonuclear fusion source[40,41].By adjusting the taper ratio,the pinch temperature is tunable up to 200 keV,sufficient for aneutronic fusion applications[27,42].

3.2.2.Simulation results.The Lee Code was run with the configuration of table 2.The main results for DPFQ0.01 are summarized in table 3.The simulated current and voltage waveforms are shown in figure 4.The current sheath accelerates along the axial phase,reaching a speed of 8 cm μs-1with a current rising to 9 MA,at the end of the axial phase at 1.18 μs.As the current sheath transitions into the radial phase the current continues rising,until a peak current of 11.1 MA is reached at 1.9 μs.The current then starts to drop and reaches 9.8 MA at 2.32 μs,as the radially inwardmoving shock wave hits the axis 30 ns after reaching a peak speed of 19.4 cm μs-1,starting the reflected shock phase.Meanwhile,the current sheath,portrayed as a magnetic piston in the code,reaches a peak speed of 12.7 cm μs-1,200 ns before the inward radial shock hits the axis,and slows rapidly to 3 cm μs-1,as the radially inward-moving shock hits the axis.The reflected shock wave moves radially outwards at a speed just under 7 cm μs-1and reaches the slowly inwardmoving magnetic piston after another 0.5 μs.The hardly compressing pinch phase lasts 314 ns.The axial phase takes 1.18 μs,whilst the radial phase takes a total period of 1.98 μs.The peak inductive voltage of 635 kV is reached at the time the piston reaches its peak speed,just before the inward radial shock front hits the axis.

Table 3.The computed properties of DPFQ0.01.

The axial and radial mass and plasma current factors in table 3 are assumed model parameters based on experience with a range of machines.The radial phase current dip is not as severe as that typically observed in smaller plasma focus machines.This is due to two factors:(a)the large ratioa/z0of value 3 resulting in a relatively long-duration pinch,and(b)the relatively large surge impedance of the capacitor bank.This large surge impedance is made possible by the high voltage and is deliberately designed to minimize the current drop to optimize the energy transfer to the pinch.The voltage spike corresponding to the pinch dynamics peaks at 635 kV.This is an important indicator that is associated with the energy of the fast beam ions accelerated by the plasma focus mechanism.These fast beam ions are responsible for the fusion reactions.The neutrons will be emitted from a plasma pinch with radius of 3.3 cm and length of 22.9 cm.Half the length of the pinch will protrude from the hollow anode and the other half will extend inside the hollow anode.The neutrons are assumed to be emitted nearly isotropically with a small forward bias.

3.2.3.Discussion of the simulation results.The code computes aQof 3×10-5,with a beam ion energy of 1.9 MeV.This excessively high ion energy has dropped the fusion cross-section parameter[σ/U1/2]by about 100 times from the optimum value(see figure 3).The operational pressure is 100 Torr;therefore,the exit path in the DPF chamber does not attenuate the beam ion energy sufficiently towards the optimum for fusion.The code shows that the number of beam ions available for fusion is 1.1×1019,and only 1.24×1014of those ions have been involved in fusion in the pinch.Hence,there is a remnant beam consisting of at least 1019ions at 1.9 MeV exiting the plasma focus pinch.A suitable fusion-enhancing tube with length of 1 m containing 6 atm D-T gas is proposed to be placed downstream of the plasma focus,to harvest a proportion of this remnant beam for fusion reactions.This high-pressure section is separated from the DPF chamber by a sub-millisecond shutter,which opens momentarily to allow the remnant ion beam to enter this fusion harvest tube.The D-T ions exit the pinch in a beam with divergence around 20 degrees[1,2,36].A beam-shaper with a suitable magnetic field could be designed to reduce the divergence so that a large fraction of the remnant ions travels down the fusion harvest tube.

Three factors are expected to contribute to the beam-gas target fusion yield of the harvest tube.First,upon traversing the harvest tube,based on consideration of beam attenuation through the high-pressure tube,the remnant beam ion energy drops from above MeV towards 115 keV,the energy for optimum fusion cross-section;thus,σ increases.In figure 3,it is seen that this recovery of σ contributes up to 100 times to the fusion yield of the harvest tube.Second,the number density of the gas serving as a target to the remnant beam ions is 12 times that of the pinch target densityni.Third,the interaction distance of the remnant beam ion is 5 times that of the interaction distance of the beam ions within the pinch lengthzp.From equation(1)it is seen that beam-target interaction is proportional to target density.It may also be shown from the same equation that the beam-target interaction is also proportional to interaction length.In this situation,the above second and third factors will have a contribution proportional to the ratio of the tube density to pinch density multiplied by the ratio of tube interaction to the pinch length.The product of these two factors is 60.This should be confirmed by Monte Carlo N Particle calculations,which are being planned but are outside the scope of this study.In the meantime,we feel confident that the above discussion allows us to conservatively estimate a minimum number of 300 for the enhancement of the fusion harvesting tube,leading to 3.7×1016D-T neutrons carrying kinetic energy of 83.6 kJ.The result is aQ=0.01 device.

4.Conclusion

Scaling using a beam-gas target mechanism in the plasma focus suggests that a 1.2 MV,5 atm D-T DPF with a peak current of 350 MA would suffice for break-even at stored capacitor energy of 7 GJ in a device designated as DPFQ1.The required current could be significantly reduced to below 100 MA with a two-step compression design.Even so,a device of such a scale is beyond the present-day capabilities when one considers that the largest currents reported from the biggest national laboratories are far less than 100 MA.A present-day technologically feasible point(DPFQ0.01:900 kV(8 MJ),11 MA,100 Torr withQ～0.01)is proposed for initial tests.The Lee Code is used to find a suitable configuration of such an operational point.The numerical experiments confirm that all the parameters of such a machine are technologically feasible.The simulations also suggest the necessity to add a high-pressure fusion harvest tube[43]to address the problem of excessive remnant beam ions exiting the plasma focus pinch region and to further enhance beamtarget fusion reactions by increasing the interaction path length at increased interaction density.A final yield of 3.7×1016D-T neutrons carrying kinetic energy of 83.6 kJ is conservatively estimated,demonstratingQ=0.01.Detailed computation is planned,to verify and quantify the enhancement mechanisms of the proposed harvest tube.It is important to note that the Lee Code has been tested by comparison with experiments only up to the level of 2.5 MA,12 cm anode radius and 50 Torr.All the results beyond these levels discussed in this paper await the next generation of machines for comparison with experimental results.