Theory and Experimental Program for p-B11 Fusion with the Dense Plasma Focus
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- Lerner, E.J., Krupakar Murali, S. & Haboub, A. J Fusion Energ (2011) 30: 367. doi:10.1007/s10894-011-9385-4
Lawrenceville Plasma Physics Inc. has initiated a 2-year-long experimental project to test the scientific feasibility of achieving controlled fusion using the dense plasma focus (DPF) device with hydrogen-boron (p-B11) fuel. The goals of the experiment are: first, to confirm the achievement of high ion and electron energies observed in previous experiments from 2001; second, to greatly increase the efficiency of energy transfer into the plasmoid where the fusion reactions take place; third, to achieve the high magnetic fields (>1 GG) needed for the quantum magnetic field effect, which will reduce cooling of the plasma by X-ray emission; and finally, to use p-B11 fuel to demonstrate net energy gain. The experiments are being conducted with a newly constructed dense plasma focus in Middlesex, NJ which is expected to generate peak currents in excess of 2 MA. Some preliminary results are reported.
KeywordsDense plasma focusQuantum magnetic field effectNuclear fusionAneutronic Fusion
Controlled nuclear fusion using hydrogen-Boron-11 (p-B11) fuel would constitute a transformative source of electricity with major advantages over any other known source of energy. No neutrons are produced in this reaction, p + B11 → 3 He4, and the released energy is carried only by charged particles. This makes possible the direct conversion of the kinetic energy of these charged particles into electricity without going through the inherently expensive process of using heat to produce steam to run a turbine and generator. It thus opens up the possibly of drastically reducing the cost of electricity generation [1–3].
While a secondary reaction, He4 + B11 → N14 + n, does produce neutrons, they carry only 0.2% of the fusion energy and are low-energy neutrons, which are easily shielded. Thus this fuel makes conceivable the design of a generator that produces insignificant amounts of induced radioactivity, and no radioactive waste. These characteristics give p-B11 very significant operational advantages over deuterium–tritium (DT) fuel.
However, p-B11 presents two major technical challenges that have discouraged funding and research. First, the reaction requires average ion energies above 100 keV, considerably higher than the 40 keV envisioned for DT fuel, and the requirement for plasma density-confinement time product (nτ) is also a factor of 45 times higher for net energy production. Second, the higher atomic charge of boron ions leads to the production of far greater amounts of X-ray energy than DT, and the emission of such X-ray energy cools the plasma, making plasma heating more difficult. We have taken steps to show how both of these technical challenges can be overcome using the DPF.
Dense Plasma Focus
The DPF is a compact and simple device first developed in the 1960s by N. V. Filippov in the USSR and by J. W. Mather in the USA and has been studied by dozens of groups over the last 45 years, resulting a large and rich literature. It consists of two concentric cylindrical electrodes enclosed in a vacuum chamber. The chamber is evacuated to low pressure and then backfilled to several torr with the fuel gas. A pulse of electricity with a rise time of 0.2–10 μs from a capacitor bank is discharged across the electrodes during operation . In operation, the capacitors discharge in a several-microsecond pulse, the gas is ionized and a current sheath, consisting of pinched current filaments, forms and runs down the electrodes. When the sheath reaches the end of the inner electrode (the anode), the filaments pinch together, forming dense, magnetically-confined, hot spots or plasmoids [5, 6]. The plasmoids emit X-rays with energy from several keV to over 100 keV. X-ray pinhole images have demonstrated that the plasmoids are tiny, with radii of tens of microns or less [7–11]. The plasmoids have densities in the range of 1020–1021/cm3. These densities have been measured by a number of independent methods including heavy ion and secondary product fusion [12, 13], CO2 laser scattering , and X-ray line intensities . These plasmoids emit intense beams of accelerated ions and electrons [16–19]. Fusion neutrons are emitted from the device in large quantities (up to 1013 per shot).
The role of the plasmoids in producing the fusion neutrons and the physical processes involved in their formation and maintenance has been hotly debated among DPF researchers for decades. The model that best fits all the existing data makes the role of the plasmoids central to neutron production. This model, initially developed by Bostick and Nardi , and confirmed by observations of several groups over three decades, was elaborated into a more quantitative theory by the present author [20–24]. In this model, the electron beam transfers part of its energy to the plasmoid electrons, which generate X-rays through collisions with nuclei. Through a plasma instability (probably ion-acoustic), the electrons then transfer part of their energy to the ions, with a typical delay (in our experiments) of ~10 ns. Ion collisions, generating fusion reactions and neutrons, then occur . When the ion and electron beams have exhausted the magnetic energy that confines the plasmoid, and partially or wholly evacuated the particles in the plasmoid, the fusion reactions end.
The DPF routinely produces hard X-rays and gamma rays indicating the presence of bremsstrahlung radiation from high-energy electrons colliding with nuclei . Together with independent evidence, this indicated that the hot spots contained ions and electrons at very high energies in the range of interest for advanced fuel fusion [6, 14, 15, 22–24].
The Bostick-Nardi model detailed in  describes the DPF as operating by exploiting a series of natural instabilities in the plasma, with each instability further concentrating the plasma and the magnetic field produced by the currents running through the plasma. In the past few decades, substantial advances have occurred in understanding the basic physics of such instabilities through experiments and observations of space plasma.
In the first instability, the current sheath moving through the plasma between electrodes breaks up into an array of filaments, increasing the density of the plasma and magnetic field strength by a factor of 10–20. The filamentary current sheath, driven by the interaction of its own currents and magnetic field, travels down to the end of the inner hollow electrode, where the filaments converge into a single central pinch region, further concentrating both plasma and magnetic fields. A third instability then kinks the single central filament like an over-twisted phone cord, forming a plasmoid, an extremely dense, magnetically self-confined ball of plasma only tens of microns across. By this time, the density and magnetic fields of the plasma in this small region are tens of thousands of times larger than those present at the start of the process, and a substantial fraction of the energy fed into the device is contained in the plasmoid. A fourth instability causes the magnetic fields at the center of plasmoid to decrease, and these changing magnetic fields induce an electric field, which generates a beam of electrons in one direction and a beam of ions in the other. The electron beam heats the plasmoid electrons which in turn heat the ions, thus igniting fusion reactions. The energy is released in the ion and electron beams and in a burst of X-ray energy from the heated electrons in the plasmoid.
In addition to its very small size, simplicity, and ability to utilize the inherent plasma instabilities (rather than suppressing them), the DPF also has the advantage that the plasmoid is extremely dense. Such a dense plasmoid requires that the ions be confined for only a few thousand orbits, in contrast to the millions of orbits required in tokomaks or most other fusion devices. Thus the high stability of such devices is not required in the DPF, only meta-stability.
Quantum Magnetic Field (QMF) Effect
Lerner theoretically showed  that the problem of high X-ray emission with p-B11could be mitigated through the use of the QMF effect. This effect, first pointed out in the 1970s , and studied in the case of neutron stars , involves the reduction of energy transfer from ions to electrons in the presence of a strong magnetic field. To apply the magnetic effect to the DPF plasmoids, which are force-free configurations, we first note that small-angle momentum transfer parallel to the field can be neglected in these plasmoids, since the ion velocity lies very close to the local magnetic field direction, and Δp||/Δp⊥ ~ sin2 θ, where θ is the angle between the ion velocity and the B field direction .
For Ei = 300 keV, this implies B > 14 GG for p, B > 3.5 GG for α, and B > 1.3 GG for 11B. As will be shown below, such field strengths should be attainable with the DPF.
As calculated , for T = Ti/Eb(M/m) < 1, the coulomb logarithm can drop as low as 0.5 for the heating of electrons by ions, which can only heat electrons that are moving slower than the ions. For the heating of the ions by the much faster thermal electrons, with Te ≫ 1, quantum effects can be ignored and the coulomb logarithm is simply ln(2Te). As a result, the ratio of these two coulomb logarithm terms can be as high as 20, which results in a similar value for Ti/Te. This results in a reduction of X-ray emission by as much as a factor of four.
We have performed 0-D simulations of the plasmoid which include this QMF effect  which show that in this case fusion power can potentially exceed Bremsstrahlung emission by as much as a factor of 2, allowing ignition of the fuel and an 80% burn-up of the fuel in the plasmoid. While not fully realistic, the simulation is adequate to show the impact of the magnetic effect and the possibility for high fusion yields in which the energy emitted in the form of X-rays and ion beams exceeds the total energy input to the plasma by about a factor of two. Of course, this by no means guarantees that direct energy conversion efficiency will be high enough for a practical generator, but it does indicate that this is at least conceivable, and worth investigating experimentally.
The simulation, by its zero-dimensional character, assumes that the plasma in the plasmoid is homogenous. In addition the simulation assumes Maxwellian distributions for the electrons, and hydrogen and boron ions. Helium ions, produced by the fusion reaction, are assumed to cool to a Maxwellian distribution, but the fusion alpha particles are treated separately, as they are slowed by the plasma. In accordance with observation, it is assumed that the ions are all fully ionized.
This simulation also assumes that the plasma radius contracts during compression and then remains stable. Thus it is assumed that as fusion energy is released, countervailing forces prevent the rapid expansion of the plasmoid.
There are good experimental reasons for believing that this is as at least roughly the case, as we explain here. Half the fusion energy released in DD reactions is deposited in the plasmoids by the d + d → p + t reaction. In the plasmoids measured in, for example, Lerner , the additional pressure from this deposited energy would disassemble the plasmoids is a few ps, thousands of times shorter than their observed lifetime of tens of ns, unless the pressure was balanced by an increase in confining forces.
We intend to discuss the role played by centrifugal stabilization in the long lifetime of the plasmoids in a future paper.
Conditions in DPF Plasmoids
Here, τ is plasmoid decay time, RB is the effective resistance of the beam, nc is plasmoid density. However, a modification must be imposed here. For low I and thus low accelerating potentials, all the particles in the plasmoid are evacuated through the beam without carrying all the energy away. In this case the simple model will break down near the end of the plasmoid decay. However, for present purposes a suitable approximation simply reduces the plasma lifetime by the ratio of the accelerating potential to that needed to carry the entire plasmoid energy. To a good approximation this factor turns out to be I/1.4 MA. For I > 1.4 MA, this factor is unity.
These theoretical predictions are in good agreement with results that were obtained experimentally in 2001 with a 1.2 MA DPF . If we use these equations to predict Bc we obtain 0.43 GG, in excellent agreement with the observed value of 0.4 GG. Similarly, the formulae yield nτ = 4.6 × 1013 s/cm3 as compared with the best observed value in  of 9 × 1013 and the average of 0.9 × 1013.
For decaborane with Z = 2.66 and μ = 5.166, with r = 5 cm, I = 2.8 MA these formulae yield Bc = 11 GG and nτ = 6 × 1015.
This is of course a considerable extrapolation—a factor of 60 above the observed values in both B and nτ. However, these conditions can be reached with relatively small plasma focus devices.
The question of heating is more complex. If the electrons simply collided with the plasmoid electrons through Coulomb interaction, heating would be quite inefficient. However, H. Hora  has shown that, for a variety of plasma beam interactions, the electrons behave as if they have a cross section equal to a circle with radius equal to their Compton wavelength, hc/E rather the classical value of e2/E. This of course increases the effective cross section by 1/α2 or 2 × 104. The reason for this relationship is not entirely clear. It occurs only in plasma, since the cross sections of electrons in gas are well known and correspond to the classical result.
Here, dc is collision distance, w is the ratio of electron beam energy to heating energy, Temax is the temperature for w = 1. We take half the peak temperature as Tef, the average electron temperature of the plasmoid, although this ignores the nonlinearity of reaction rate with T. We thus find that the temperature increases nearly as the cube of atomic number of the gas involved. Since nτ increases as z2.4, the nτT product increases as approximately z5. We thus see the model’s strong prediction of fusion yield improvement with increasing μ and z.
It should be noted that the thermonuclear yield, which should be observed as isotropic, scales as I4 while the beam-plasma yield observed as anisotropic scales as I3, so the degree of anisotropy will decrease with increasing current and total yield.
Control of Angular Momentum and Efficiency of Energy Transfer to the Plasmoid
The appropriate choice of cathode radius, peak current and fill pressure is not enough to ensure the efficient formation of a plasmoid that contains the full magnetic energy input to the device. In fact, in few of the existing DPF devices is the efficiency of energy transfer into the plasmoid very high, limiting yield to well below that predicted by the model described in the section “Conditions in DPF Plasmoids”. For high efficiency, control of angular momentum is required, as first pointed out by Lerner and Blake .
The process of plasmoid formation involves the development of a kink instability in the current flow at the pinch and as such requires a certain amount of angular momentum. We here refer to total angular momentum, both of the magnetic field and of the particles. Another way of looking at this is that the axial field/azimuthal field ratio has to be sufficient for the kinking to occur. During the compression phase, angular momentum per unit mass is conserved, so this angular momentum can be derived from angular momentum present in the filament array at the time the compression begins. An approximate measure of the angular momentum per unit mass required can be obtained by the formula 0.5 VAr, here VA is the Alfven’s velocity in the plasmoid and r is the plasmoid radius.
Angular momentum can be imparted to the plasma sheath during the rundown by the interaction of the inward flowing electron flows and any small initial axial magnetic field (e.g., the small axial component of the earth’s magnetic field). The JXB force accelerates the electrons slightly in the azimuthal direction, creating an azimuthal component to the current. This in turn increases the axial magnetic field and thus the azimuthal acceleration of the electrons. In this way, a very small initial magnetic field (or small, random initial azimuthal component in the current created by irregularities in the electrodes) can be rapidly magnified. For example, given a ratio of axial to total magnetic field Bθ/BT = sin θ then any initial axial field will be amplified so that at the end of the run down θ = θieVτ/R, where τ is the run down time and R is the anode radius. Thus final angular momentum per unit mass is VR θieVτ/R where V is the Alfven velocity at the anode radius at peak current. This is a simplified analysis, as in the real case BT is rising rapidly during the early stages of the pulse. However, a numerical analysis using a realistic function for BT shows a very similar result, as at later times, the magnitude of the initial axial field is very small compared with BT, so the amplified field dominates, as in the simplified formula.
Since Vτ/r is proportional to L/R, the angular momentum is sensitively dependent on this ratio. If there is insufficient angular momentum, the plasmoid radius will be reduced in proportion to angular momentum and the total plasmoid energy and mass will be reduced as the cube of angular momentum. This sensitivity to initial very small angular momentum can in part explain the well-known shot-to-shot variability of plasma focus devices. Calculations show that if this natural amplification mechanism is relied on to provide angular momentum and the initial magnetic field is the earth’s ambient field, L/R must be more than about 7 for high efficiency of energy transfer into the plasmoid. Indeed, in the best-performing DPF devices, this ratio exceeds 7 and can be as high as 17, implying that high VA and longer τ are desirable.
The disadvantage of such long electrodes is their high inductance, around 20 nH. Since external inductance must exceed load inductance, total inductance in the system must be around about 45 nH. As Lee  has shown, these considerations lead to limitations on the total amount of current that can be fed into the DPF from a capacitor bank, as the pulse length must increase as capacitance does, unless the charging voltage becomes very high. The high inductance, by forcing up total bank energy, reduces the proportion of that energy that can be converted into the DPF magnetic field. So even if the efficiency of energy transfer from the magnetic field to the plasmoid increases, the total efficiency from capacitor bank to plasmoid does not necessarily increase.
The alternative to relying on amplification of the ambient magnetic field, is to inject angular momentum with a small artificial axial magnetic field, produced by a helical coil. While there have been previous efforts to stabilize DPF pinches with axial fields, these fields have been much greater than those contemplated here, generally thousands of G. If the model described here is valid, too much angular momentum will prevent the plasmoid from being formed and thus drastically reduce fusion yield. Only the optimal amount of field, of the order of a few G, will provide enough angular momentum to just balance the compressional pinch forces and form the largest possible plasmoid.
Viewed in another way, for a given electrode radius and length, the injection of angular momentum will greatly increase the angular momentum and thus the size of the plasmoid, and thus the energy yield from fusion reactions in the plasmoid. Approximately, fusion yield will increase as the third power of the amount of injected angular momentum.
Experiments with Focus Fusion-1
To test the above mentioned, and other theoretical predictions, we have constructed a new DPF facility (called Focus Fusion-1, or FF-1 for short) in Middlesex, NJ, with a 113 μF, 45 kV capacitor bank, which we expect will be able to achieve peak currents above 2 MA.
The design of FF-1 is based on the accumulated experience of the DPF field but represents an advance in two major ways. First, we expect that it will achieve a very high current with a very compact and relatively economical design by achieving a low external inductance of about 15 nH. With a capacitor bank total capacitance of 113 microfarads and a maximum charge potential of 45 kV, we expect the device to achieve a peak current of 2.2 MA using deuterium with 14-cm long anodes and as high as 2.8 MA using heavier gases and shorter anodes. Exclusive of the support structure, the entire device fits in a 2.5 × 2.5 × 1 m volume. Second, for these high peak currents, FF-1 has very compact electrodes, thus generating a high initial magnetic field. The cathode consists of 16 copper rods set at a radius of 5 cm and the anode is a hollow copper cylinder 2.8 cm in radius. At the highest peak current, the field at the anode surface will be around 200 kG, which we estimate is close to the highest field allowed by the mechanical strength of copper. These high fields will make possible high fill gas densities, which in turn will lead to high final densities in the plasmoids. Six different anodes ranging in length from 7 to 14 cm will be used, depending on the fill gas.
The experiment currently uses 13 diagnostic instruments to measure the various parameters of the plasma and the tiny plasmoid. We have listed the instruments by the parameter they are designed to measure. To measure the size and shape of the plasmoids we use:
A 4PICOS Stanford Computer Optics ICCD camera with 0.2 ns minimum exposure time. The ICCD is sensitive to UV radiation down to 180 nm and thus should be able to observe radiation above the plasma frequency of the plasmoids for all deuterium experiments, although not for the heavier gases. In addition to its high temporal resolution, the ICCD currently has a spatial resolution of about 30 microns. We also use an X-ray pinhole camera.
To measure the electron energy distribution we use:
A set of three scintillators and PMT X-ray detectors each located behind a different thickness of copper filter—300 microns, 3, 6 mm. The ratios of the signals from the PMTs measure the average X-ray emission and can be fitted to a model of bremmstralung radiation to obtain the average electron energy with high time resolution. The detectors are shielded by a 5 cm lead brick from the emissions due to the collision of the electron beam and the anode, so that we are measuring X-rays from the plasmoids alone. The detectors are also shielded by copper sheaths from the RF pulse which arrives at the same time as the X-ray pulse.
The energy of the ions is measured by two neutron time of flight (TOF) detectors that consist of two scintillation detectors with PMTs placed at different distances—11 m for the near detector and 17 m for the far detector. The difference in the detection times of the two detectors measures the energies of the neutrons, thus allowing us to differentiate between the 14.7 and 2.5 meV neutrons from the D-T and D-D reactions. In addition, the spread in energies, indicated by the widening of the pulse with distance, gives us a measure of the ion velocities within the plasma. Since X-rays arrive first at the detectors, they produce a distinct signature and can be easily isolated from the neutron signal.
Neutron measurements are made with two BTI bubble detectors, each calibrated by the manufacturer to ±25%. In addition, we have built a silver activation detector exactly following a previously calibrated design . We have cross-calibrated the neutron detector with the bubble detectors and find them in agreement within 10%.
The ion beams are measured with two Rogowski coils built into the drift tube, one at 30 cm from the end of the anode, the other at 100 cm. The coils provide a measure of both the net current of the beams, and by time-of-flight calculations the energy distribution of the ions.
A main Rogowski coil built into the lower (ground) base plate records the dI/dt of the current through the anode and is integrated digitally to provide the current measurement. We have calibrated the current to within 5% by comparing the integrated current with a waveform generated by a detailed run-down simulation developed by Lee  for the measured capacitance, charging voltage and other characteristics of our device. The resulting calibration is also within 5% of that calculated from the inductance of the coil.
Finally, a HV probe measures the voltage on an individual spark plug.
Preliminary Experimental Results
So far, we have fired FF-1 over 500 times since it produced its first pinch on October 15, 2009. We have been somewhat slowed by the poor initial functioning of the switches, supplied by R. E. Beverly and Associates, which required extensive modification and rebuilding. However, we report here on some important preliminary observations which will be elaborated in future papers.
First, the fusion yield, as measured by the silver activation detector, depends sensitively on the time of the pinch. Much other DPF work has shown that the highest yields are obtained when the pinch occurs close to the quarter-cycle time of the device, when the maximum current would occur without the pinch. Here we define the pinch time as the time at which dI/dt has a minimum and thus the rate of energy transfer into the plasmoid is at a maximum. However, most researchers have found that the actual peak in yield occurs with a pinch time somewhat later than the quarter-cycle time, when the current is actually decreasing . This is true for our FF-1 results as well, but only if the axial magnetic field is absent. When the field is present, we find that fusion yield increases significantly—by more than a factor of three—when the pinch time is exactly at the quarter cycle time (within ±3%).
The shots taken at higher current (around 1 MA) and with fill pressures of 18–24 torr show the same pattern, but with an increase in yield of about a factor of 4–5 at the same pinch times. This shot series demonstrated that the short-pinch time regime was not accessible without the additional axial field. Without this imposed field, when the fill pressure was reduced sufficiently for rapid run-downs and thus short pinch times, no pinch occurred.
Comparison with Theory and Tentative Explanation of Results
Comparison of observed and predicted yield, SPTs
Fill pressure (torr)
Peak I (kA)
Our results pose a number of questions that we can provide only tentative answers to at this time. Why are the yields much higher for the STPs than for the LTPs? Why are the STPs obtainable only with the imposed axial field? What is the significance of the bumps in the dI/dt?
It is reasonable to hypothesize that the two precursor bumps are caused by shocks which heat the plasma before the plasmoid is fully compressed, decreasing the amount of energy available for the plasmoid and possibly as well the amount of angular momentum. One likely time for such shocks to occur is in the final stages of the kinking process, when the coils of the kinking filament are approaching each other at high velocity. It is possible that with adequate injected angular momentum, due to the increased axial field, these shocks will be greatly decreased, as the coils will slide past each other in a spiral fashion, rather than colliding head-on. This would allow greater energy transfer other plasmoid and thus the higher yields. Of course, such a conclusion requires a good deal more evidence to be validated.
We note that, as described in the section “Control of Angular Momentum and Efficiency of Energy Transfer to the Plasmoid”, the angular momentum is increasing exponentially with time. If a minimum amount is essential for the formation of the plasmoid, and thus for the pinch, the time needed to reach this minimum will be reduced if the initial axial field is greater. So an alternative explanation of the role of the axial field is that it allows this minimum to be reached before the pinch time, thus allowing the SPTs to take place.
The role of the precise timing of the pinch is less clear. Since the shocks occur before the pinch and presumably determine the energy transferred, the critical periods would logically be before the shocks, when the rapid fall in dI/dt just begins (in our hypothesis, with the start of the kinking process). In that case, the start of this fall in dI/dt occurs when the current is rising for all of the STPs and when it is falling or near zero for all of the LTPs. To be precise, all the STPs have dI/dt > 1011 A/s at the time the pinch begins—a sudden shift to high negative dI/dt—while all the LTPs have dI/dt < 3 × 1010A/s at this time and the vast majority have dI/dt < 0. So the sign of dI/dt seems to be highly significant as well as its magnitude. The question then arises: why does the sign of dI/dt affect the shock process and how does it interact with the imposed axial field?
Tsybenko and Miklaszewkis  looked at the problem of pinch timing based on the observation that, in most DPFs, the maximum yields are slightly after the quarter cycle time. They hypothesized that a tangential discontinuity, which occurs when two adjacent parts of a cylindrical plasma are rotating at different frequencies, could set off instabilities that disrupt the pinch. They show that such an instability is suppressed if the current in the column is varying at a frequency comparable to the instability growth time—around 10 ns. However, in our observations the current varies over a time scale of 6–7 μs, so it is hard to see how this mechanism could be significant.
It is possible that the azimuthal potential induced by the dI/dt can add to or subtract from the angular momentum provided by the axial field coil to produce the optimal amount of angular momentum both for minimizing the shocks and forming the plasmoid, but at the moment a complete explanation of the data is not yet in hand. We expect that experiments now underway will greatly clarify the outstanding theoretical questions.
The authors wish to thank John Thompson for his major contributions to the design and construction of FF-1.