# Numerical Experiments Providing New Insights into Plasma Focus Fusion Devices

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}using a beam–target mechanism [21,22,23,24,25], incorporated in recent versions [26,27] of the code (versions later than RADPFV5.13), resulting in realistic Y

_{n}scaling with I

_{pinch}[21,22]. The versatility and utility of the model are demonstrated in its clear distinction of I

_{pinch}from I

_{peak}[28] and the recent uncovering of a plasma focus pinch current limitation effect [23,24,29] as static inductance is reduced towards zero. Extensive numerical experiments had been carried out systematically resulting in the uncovering of neutron [21,22,30,31,32,33] and SXR [30,31,32,33,34,35,36,37] scaling laws over a wider range of energies and currents than attempted before. The numerical experiments also gave insight into the nature and cause of ‘neutron saturation [31,33,38]. The description, theory, code, and a broad range of results of this “Universal Plasma Focus Laboratory Facility” are available for download from [26].

## 2. The 5-Phase Lee Model Code

_{m}and f

_{c}. The mass swept-up factor f

_{m}accounts for not only the porosity of the current sheet but also for the inclination of the moving current sheet shock front structure, boundary layer effects, and all other unspecified effects which have effects equivalent to increasing or reducing the amount of mass in the moving structure, during the axial phase. The current factor f

_{c}accounts for the fraction of current effectively flowing in the moving structure (due to all effects such as current shedding at or near the back-wall, and current sheet inclination). This defines the fraction of current effectively driving the structure, during the axial phase.

_{mr}and f

_{cr}are incorporated in all three radial phases. The mass swept-up factor f

_{mr}accounts for all mechanisms which have effects equivalent to increasing or reducing the amount of mass in the moving slug, during the radial phase. The current factor f

_{cr}accounts for the fraction of current effectively flowing in the moving piston forming the back of the slug (due to all effects). This defines the fraction of current effectively driving the radial slug.

**Figure 1.**Schematic of the axial and radial phases. The left section depicts the axial phase, the right section the radial phase. In the left section, z is the effective position of the current sheath-shock front structure. In the right section r

_{s}is the position of the inward moving shock front driven by the piston at position r

_{p}. Between r

_{s}and r

_{p}is the radially imploding slug, elongating with a length z

_{f}. The capacitor, static inductance and switch powering the plasma focus is shown for the axial phase schematic only.

**Figure 2.**Schematic of radius versus time trajectories to illustrate the radial inward shock phase when r

_{s}moves radially inwards, the reflected shock (RS) phase when the reflected shock moves radially outwards, until it hits the incoming piston r

_{p}leading to the start of the pinch phase (t

_{f}) and finally the expanded column phase.

_{mr}and f

_{cr}are used as in the previous radial phase. The plasma temperature behind the reflected shock undergoes a jump by a factor close to 2. Number densities are also computed using the reflected shock jump equations.

#### 2.1. From Measured Current Waveform to Modeling for Diagnostics

- Bank parameters, L
_{0}, C_{0}and stray circuit resistance r_{0}; - Tube parameters b, a and z
_{0}and - Operational parameters V
_{0}and P_{0}and the fill gas.

_{m}, f

_{c}, f

_{mr}and f

_{cr}one by one, until the computed waveform agrees with the measured waveform.

_{m}, f

_{c}are adjusted (fitted) until the features (1) computed rising slope of the total current trace and (2) the rounding off of the peak current as well as (3) the peak current itself are in reasonable (typically very good) fit with the measured total current trace (see Figure 3, measured trace fitted with computed trace).

_{mr}and f

_{cr}until features (4) the computed slope and (5) the depth of the dip agree with the measured. Note that the fitting of the computed trace with the measured current trace is done up to the end of the radial phase which is typically at the bottom of the current dip. Fitting of the computed and measured current traces beyond this point is not done. If there is significant divergence of the computed with the measured trace beyond the end of the radial phase, this divergence is not considered important.

_{m}= 0.1, f

_{c}= 0.7, f

_{m}r = 0.12, f

_{cr}= 0.68.

**Figure 3.**The 5-point fitting of computed current trace to the measured (or the reference) current trace. Point 1 is the current rise slope. Point 2 is the topping profile. Point 3 is the peak value of the current. Point 4 is the slope of the current dip. Point 5 is the bottom of the current dip. Fitting is done up to point 5 only. Further agreement or divergence of the computed trace with/from the measured trace is only incidental and not considered to be important.

#### 2.2. Diagnostics-Time Histories of Dynamics, Energies and Plasma Properties Computed from the Measured Total Current Waveform by the Code

#### 2.3. Comments on Computed Quantities

^{2}) peaks at 70% of initial stored energy, and then drops to 30% during the radial phase, as the sharp drop of current more than offsets the effect of sharply increased inductance (Figure 4i). In Figure 4j is shown the work done by the magnetic piston, computed using force integrated over distance method. Also shown is the work dissipated by the dynamic resistance, computed using dynamic resistance power integrated over time. We see that the two quantities and profiles agree exactly. This validates the concept of half Ldot as a dynamic resistance, DR (see Section 6.1). The piston work deposited in the plasma increases steadily to some 12% at the end of the axial phase and then rises sharply to just below 30% in the radial phase. Dynamic resistance is shown in Figure 4k. The values of the DR in the axial phase, together with the bank surge impedance, are the quantities that determine I

_{peak}. The ion number density has a maximum value derived from shock-jump considerations, and an averaged uniform value derived from overall energy and mass balance considerations. The time profiles of these are shown in the Figure 4l. The electron number density (Figure 4m) has similar profiles to the ion density profile, but is modified by the effective charge numbers due to ionization stages reached by the ions. Plasma temperature too has a maximum value and an averaged uniform value derived in the same manner; are shown in Figure 4n. Computed neon soft x-ray power profile is shown in Figure 4o. The area of the curve is the soft x-ray yield in Joule. Pinch dimensions and lifetime may be estimated from Figure 4e,f. The model also computes the neutron yield, for operation in deuterium, using a phenomenological beam-target mechanism [25,26,27]. The model does not compute a time history of the neutron emission, only a yield number Y

_{n}.

## 3. Insight 1-Pinch Current Limitation Effect as Static Inductance Is Reduced towards Zero

_{0}. To investigate this point, experiments were carried out using the Lee Model code. Unexpectedly, the results indicated that whilst I

_{peak}indeed progressively increased with reduction in L

_{0}, no improvement may be achieved due to a pinch current limitation effect [23,24]. Given a fixed C

_{0}powering a plasma focus, there exists an optimum L

_{0}for maximum I

_{pinch}. Reducing L

_{0}further will increase neither I

_{pinch}nor Y

_{n}. The numerical experiments leading to this unexpected result is described below.

_{0}= 1332 μF, operated at 27 kV, 3.5 torr deuterium, has been published [25], with cathode/anode radii b = 16 cm, a = 11.55 cm and anode length z

_{0}= 60 cm. In the numerical experiments we fitted external (or static) inductance L

_{0}= 33.5 nH and stray resistance r

_{0}= 6.1 mΩ (damping factor RESF = r

_{0}/(L

_{0}/C

_{0})

^{0.5}= 1.22). The fitted model parameters are: f

_{m}= 0.13, f

_{c}= 0.7, f

_{mr}= 0.35 and f

_{cr}= 0.65. The computed current trace [21,24,26] agrees very well with the measured trace through all the phases, axial and radial, right down to the bottom of the current dip indicating the end of the pinch phase as shown in Figure 5.

**Figure 5.**Fitting computed current to measured current traces to obtain fitted parameters f

_{m}= 0.13, f

_{c}= 0.7, f

_{mr}= 0.35 and f

_{cr}= 0.65. The measured current trace was for the PF1000 at 27 kV, storage capacity of 1332 μF and fitted static inductance of 33.5 μH.

_{0}was also adjusted to maximize I

_{pinch}as L

_{0}was decreased from 100 nH progressively to 5 nH.

_{peak}increased progressively from 1.66 to 4.4 MA. As L

_{0}was reduced from 100 to 35 nH, I

_{pinch}also increased, from 0.96 to 1.05 MA. However, then unexpectedly, on further reduction from 35 to 5 nH, I

_{pinch}stopped increasing, instead decreasing slightly to 1.03 MA at 20 nH, to 1.0 MA at 10 nH, and to 0.97 MA at 5 nH. Y

_{n}also had a maximum value of 3.2 × 10

^{11}at 35 nH.

_{peak}and then again near the bottom of the almost linear drop to the pinch phase indicated by the arrow pointing to ‘end of radial phase’. The energy equation describing this current drop is written as follows:

_{peak}

^{2}(L

_{0}+ L

_{a}f

_{c}

^{2}) = 0.5I

_{pinch}

^{2}(L

_{0}/ f

_{c}

^{2}+ L

_{a}+ L

_{p}) + δ

_{cap}+ δ

_{plasma}

_{a}is the inductance of the tube at full axial length z

_{0}, δ

_{plasma}is the energy imparted to the plasma as the current sheet moves to the pinch position and is the integral of 0.5(dL/dt)I

^{2}. We approximate this as 0.5L

_{p}I

_{pinch}

^{2}which is an underestimate for this case. δ

_{cap}is the energy flow into or out of the capacitor during this period of current drop. If the duration of the radial phase is short compared to the capacitor time constant, the capacitor is effectively decoupled and δ

_{cap}may be put as zero. From this consideration we obtain

_{pinch}

^{2}= I

_{peak}

^{2}(L

_{0}+ 0.5L

_{a})/(2L

_{0}+ L

_{a}+ 2L

_{p})

_{c}= 0.7 and approximated f

_{c}

^{2}as 0.5.

_{0}is reduced, I

_{peak}increases; a is necessarily increased leading [7] to a longer pinch length z

_{p}, hence a bigger L

_{p}. Lowering L

_{o}also results in a shorter rise time, hence a necessary decrease in z

_{0}, reducing L

_{a}. Thus, from Equation (2), lowering L

_{0}decreases the fraction I

_{pinch}/I

_{peak}. Secondly, this situation is compounded by another mechanism. As L

_{0}is reduced, the L-C interaction time of the capacitor bank reduces while the duration of the current drop increases (see Figure 6, discussed in the next section) due to an increasing a. This means that as L

_{0}is reduced, the capacitor bank is more and more coupled to the inductive energy transfer processes with the accompanying induced large voltages that arise from the radial compression. Looking again at the derivation of Equation (2) from Equation (1) a nonzero δ

_{cap}, in this case, of positive value, will act to decrease I

_{pinch}further. The lower the L

_{0}the more pronounced is this effect.

_{o}to the inductive energetic processes, as L

_{0}is reduced.

#### 3.1. Optimum L_{0} for Maximum Pinch Current and Neutron Yield

_{0}powering a plasma focus, there exists an optimum L

_{0}for maximum I

_{pinch}. Reducing L

_{0}further will increase neither I

_{pinch}nor Y

_{n}. The results of the numerical experiments carried out are presented in Figure 6 and Table 1.

_{0}= 100 nH it is seen (Figure 6) that the rising current profile is flattened from what its waveform would be if unloaded; and peaks at around 12 μs (before its unloaded rise time, not shown, of 18 μs) as the current sheet goes into the radial phase. The current drop, less than 25% of peak value, is sharp compared with the current rise profile. At L

_{0}= 30 nH the rising current profile is less flattened, reaching a flat top at around 5 μs, staying practically flat for some 2 μs before the radial phase current drop to 50% of its peak value in a time which is still short compared with the rise time. With L

_{0}of 5 nH, the rise time is now very short, there is hardly any flat top; as soon as the peak is reached, the current waveform droops significantly. There is a small kink on the current waveform of both the L

_{0}= 5 nH, z

_{0}= 20 cm and the L

_{0}= 5 nH, z

_{0}= 40 cm. This kink corresponds to the start of the radial phase which, because of the large anode radius, starts with a relatively low radial speed, causing a momentary reduction in dynamic loading. Looking at the three types of traces it is seen that for L

_{0}= 100 nH to 30 nH, there is a wide range of z

_{0}that may be chosen so that the radial phase may start at peak or near peak current, although the longer values of z

_{0}tend to give better energy transfers into the radial phase.

_{0}is shown in Table 1. The table shows that as L

_{0}is reduced, I

_{peak}rises with each reduction in L

_{0}with no sign of any limitation. However, I

_{pinch}reaches a broad maximum of 1.05 MA around 40–30 nH. Neutron yield Y

_{n}also shows a similar broad maximum peaking at 3.2 × 10

^{11}neutrons. Figure 7 shows a graphical representation of this I

_{pinch}limitation effect. The curve going up to 4 MA at low L

_{0}is the I

_{peak}curve. Thus I

_{peak}shows no sign of limitation as L

_{0}is progressively reduced. However I

_{pinch}reaches a broad maximum. From Figure 7 there is a stark and important message: one must distinguish clearly between I

_{peak}and I

_{pinch}. In general one cannot take I

_{peak}to be representative of I

_{pinch}.

**Figure 6.**PF1000 current waveforms computed at 35 kV, 3.5 Torr D

_{2}for a range of L

_{0}showing the changes in waveforms as L

_{0}varies.

**Table 1.**Effect on currents and ratio of currents as L

_{0}is reduced-PF1000 at 35kV, 3.5 Torr Deuterium.

L_{0}(nH) | b(cm) | a(cm) | z_{0}(cm) | I_{peak}(MA) | I_{pinch}(M) | Y_{n}(10^{11}) | I_{pinch}/ I_{peak} |
---|---|---|---|---|---|---|---|

100 | 15.0 | 10.8 | 80 | 1.66 | 0.96 | 2.44 | 0.58 |

80 | 16.0 | 11.6 | 80 | 1.81 | 1.00 | 2.71 | 0.55 |

60 | 18.0 | 13.0 | 70 | 2.02 | 1.03 | 3.01 | 0.51 |

40 | 21.5 | 15.5 | 55 | 2.36 | 1.05 | 3.20 | 0.44 |

35 | 22.5 | 16.3 | 53 | 2.47 | 1.05 | 3.20 | 0.43 |

30 | 23.8 | 17.2 | 50 | 2.61 | 1.05 | 3.10 | 0.40 |

20 | 28.0 | 21.1 | 32 | 3.13 | 1.03 | 3.00 | 0.33 |

10 | 33.0 | 23.8 | 28 | 3.65 | 1.00 | 2.45 | 0.27 |

5 | 40.0 | 28.8 | 20 | 4.37 | 0.97 | 2.00 | 0.22 |

**Figure 7.**Effect on currents and current ratio (computed) as L

_{0}is reduced PF1000, 35 kV, 3.5 torr D

_{2}.

_{0}, each set with a different damping factor. In every case, an optimum inductance was found around 30–60 nH with I

_{pinch}decreasing as L

_{0}was reduced below the optimum value. The results showed that for PF1000, reducing L

_{0}from its present 20–30 nH will increase neither the observed I

_{pinch}nor the neutron yield, because of the pinch limitation effect. Indeed, the I

_{pinch}decreases very slightly on further reduction to very small values. We would add that we have used a set of model parameters which in our experience is the most reasonable to be used in these numerical experiments. Variations of the model parameters could occur but we are confident that these variations are not likely to occur with such a pattern as to negate the pinch current limitation effect. Nevertheless these variations should be actively monitored and any patterns in the variations should be investigated.

## 4. Insight 2-Scaling Laws for Neutron

#### 4.1. Computation of Neutron Yield-describing the Beam-target Mechanism

_{b-t}= C

_{n}n

_{i}I

_{pinch}

^{2}z

_{p}

^{2}(ln (b/r

_{p}))σ /U

^{0.5}

_{i}is the ion density, b is the cathode radius, r

_{p}is the radius of the plasma pinch with length z

_{p}, σ the cross-section of the D-D fusion reaction, n- branch [40] and U, the beam energy. C

_{n}is treated as a calibration constant combining various constants in the derivation process.

_{max}of the order of only 15−50 kV. However it is known, from experiments that the ion energy responsible for the beam-target neutrons is in the range 50−150 keV [25], and for smaller lower-voltage machines the relevant energy could be lower at 30-60 keV [16]. Thus in line with experimental observations the D-D cross section σ is reasonably obtained by using U = 3V

_{max}. This fit was tested by using U equal to various multiples of V

_{max}. A reasonably good fit of the computed neutron yields to the measured published neutron yields at energy levels from sub-kJ to near MJ was obtained when the multiple of 3 was used; with poor agreement for most of the data points when for example a multiple of 1 or 2 or 4 or 5 was used. The model uses a value of C

_{n}= 2.7 × 10

^{7}obtained by calibrating the yield [21,22,23,26], at an experimental point of 0.5 MA.

_{max}. However, the usage of the multiple to V

_{max}has some experimental basis due to ion energy measurements. Moreover the value of Vmax in each numerical experiment is calculated from the slug model leading to the slow compression phase whilst it is known experimentally that after the slow compression phase, instability effects set in which will increase the electric fields operating within the pinch. These are the basic arguments supporting the view that the operational beam energy has a value above Vmax. For the thermonuclear component a feasible model to adjust the yield upwards has yet to be suggested.

#### 4.2. Scaling Laws for Neutrons from Numerical Experiments over a Range of Energies from 10 kJ to 25 MJ

_{0}to study the neutrons emitted by PF1000-like bank energies from 10 kJ to 25 MJ.

_{0}, anode length z

_{0}is varied to find the optimum. For each z

_{0}, anode radius a

_{0}is varied so that the end axial speed is 10 cm/µs. The numerical experiments were carried out for C

_{0}ranging from 14 µF to 39,960 µF corresponding to energies from 8.5 kJ to 24.5 MJ [22].

_{n}scaling changes from Y

_{n}~ E

_{0}

^{2.0}at tens of kJ to Y

_{n}~ E

_{0}

^{0.84}at the highest energies (up to 25 MJ) investigated in this series. This is shown in Figure 8.

**Figure 8.**Y

_{n}plotted as a function of E

_{0}in log-log scale, showing Y

_{n}scaling changes from Y

_{n}~ E

_{0}

^{2.0}at tens of kJ to Y

_{n}~ E

_{0}

^{0.84}at the highest energies (up to 25 MJ).

_{n}with I

_{peak}and I

_{pinch}over the whole range of energies investigated up to 25 MJ (shown in Figure 9) are as follows:

_{n}= 3.2 × 10

^{11}I

_{pinch}

^{4.5}and

_{n}= 1.8 × 10

^{10}I

_{peak}

^{3.8}

_{peak}ranges from 0.3 MA to 5.7 MA and I

_{pinch}ranges from 0.2 MA to 2.4 MA.

**Figure 9.**Log(Y

_{n}) scaling with Log(I

_{peak}) and Log(I

_{pinch}), for the range of energies investigated, up to 25 MJ.

_{n}yield measurements, operating conditions and machine parameters including the Chilean PF400J, the UNU/ICTP PFF, the NX2 and Poseidon providing a slightly higher scaling laws: Y

_{n}~ I

_{pinch}

^{4.7}and Y

_{n}~ I

_{peak}

^{3.9}. The slightly higher value of the scaling is because those machines fitted are of mixed 'c' mixed bank parameters, mixed model parameters and currents generally below 1 MA and voltages generally below the 35 kV [21].

_{n}[15] are listed here:

_{n}= 3.2 × 10

^{11}I

_{pinch}

^{4.5}; Y

_{n}= 1.8 × 10

^{10}I

_{peak}

^{3.8}; I

_{peak}(0.3 to 5.7), I

_{pinch}(0.2 to 2.4) in MA.

_{n}~ E

_{0}

^{2.0}at tens of kJ to Y

_{n}~ E

_{0}

^{0.84}at MJ level (up to 25MJ).

## 5. Insight 3-Scaling Laws for Soft X-ray Yield

#### 5.1. Computation of Neon SXR Yield- the Equations Used in the Computation

_{L}is calculated as follows:

_{sxr}= Q

_{L}

**∙**Z

_{n}is the atomic number.

_{i}, effective charge number Z, pinch radius r

_{p}, pinch length z

_{f}and temperature T. It also depends on the pinch duration since in our code Q

_{L}is obtained by integrating over the pinch duration.

_{n}, n

_{i}, Z and T. However, in our range of operation, the numerical experiments show that the self absorption is not significant. It was first pointed out by Liu Mahe [8,11,18] that a temperature around 300 eV is optimum for SXR production. Shan Bing’s subsequent work [9] and our experience through numerical experiments suggest that around 2 × 10

^{6}K (below 200 eV) or even a little lower could be better. Hence unlike the case of neutron scaling, for SXR scaling there is an optimum small range of temperatures (T windows) to operate.

#### 5.2. Scaling Laws for Neon Sxr from Numerical Experiments over A Range of Energies from 0.2 kJ to 1 MJ

_{0}obtained from our numerical experiments.

_{0}(kept at 20 kV); (iii) static inductance L

_{0}(kept at 30 nH, which is already low enough to reach the I

_{pinch}limitation regime [23,24] over most of the range of E

_{0}we are covering) and; (iv) the ratio of stray resistance to surge impedance RESF (kept at 0.1, representing a higher performance modern capacitor bank). The model parameters [26,27] f

_{m}, f

_{c}, f

_{mr}, f

_{cr}are also kept at fixed values 0.06, 0.7, 0.16 and 0.7. We choose the model parameters so they represent the average values from the range of machines that we have studied. A typical example of a current trace for these parameters is shown in Figure 10.

_{0}is varied by changing the capacitance C

_{0}. Parameters that are varied are operating pressure P

_{0}, anode length z

_{0}and anode radius a. Parametric variation at each E

_{0}follows the order; P

_{0}, z

_{0}and a until all realistic combinations of P

_{0}, z

_{0}and a are investigated. At each E

_{0}, the optimum combination of P

_{0}, z

_{0}and a is found that produces the biggest Y

_{sxr}. In other words at each E

_{0}, a P

_{0}is fixed, a z

_{0}is chosen and a is varied until the largest Y

_{sxr}is found. Then keeping the same values of E

_{0}and P

_{0}, another z

_{0}is chosen and a is varied until the largest Y

_{sxr}is found. This procedure is repeated until for that E

_{0}and P

_{0}, the optimum combination of z

_{0}and a is found. Then keeping the same value of E

_{0}, another P

_{0}is selected. The procedure for parametric variation of z

_{0}and a as described above is then carried out for this E

_{0}and new P

_{0}until the optimum combination of z

_{0}and a is found. This procedure is repeated until for a fixed value of E

_{0}, the optimum combination of P

_{0}, z

_{0}and a is found.

_{0}. In this manner after systematically carrying out some 2000 runs, the optimized runs for various energies are tabulated in Table 2.

**Figure 10.**Computed total curent versus time for L

_{0}= 30 nH and V

_{0}= 20 kV, C

_{0}= 30 μF, RESF = 0.1, c = 1.5 and model parameters f

_{m}, f

_{c}, f

_{mr}, f

_{cr}are fixed at 0.06, 0.7, 0.16 and 0.7 for optimized a = 2.285 cm and z

_{0}= 5.2 cm.

**Table 2.**Optimized configuration found for each E

_{0}. Optimization carried out with RESF = 0.1, c = 1.5, L

_{0}= 30 nH and V

_{0}= 20 kV and model parameters f

_{m}, f

_{c}, f

_{mr}, f

_{cr}are fixed at 0.06, 0.7, 0.16 and 0.7, respectively. The v

_{a}, v

_{s}and v

_{p}are the peak axial, radial shock and radial piston speeds, respectively.

E_{0}(kJ) | C_{0}(μF) | a (cm) | z_{0}(cm) | P_{0}(Torr) | I_{peak}(kA) | I_{pinch}(kA) | v_{a}(cm/μs) | v_{s}(cm/μs) | v_{p}(cm/μs) | Y_{sxr}(J) |
---|---|---|---|---|---|---|---|---|---|---|

0.2 | 1 | 0.58 | 0.5 | 4.0 | 100 | 68 | 5.6 | 22.5 | 14.9 | 0.44 |

1 | 5 | 1.18 | 1.5 | 4.0 | 224 | 143 | 6.6 | 23.3 | 15.1 | 7.5 |

2 | 10 | 1.52 | 2.1 | 4.0 | 300 | 186 | 6.8 | 23.6 | 15.2 | 20 |

6 | 30 | 2.29 | 5.2 | 4.2 | 512 | 294 | 8.1 | 24.5 | 15.6 | 98 |

10 | 50 | 2.79 | 7.5 | 4.0 | 642 | 356 | 8.7 | 24.6 | 15.7 | 190 |

20 | 100 | 3.50 | 13 | 4.0 | 861 | 456 | 9.6 | 24.6 | 16.0 | 470 |

40 | 200 | 4.55 | 20 | 3.5 | 1109 | 565 | 10.3 | 24.7 | 16.2 | 1000 |

100 | 500 | 6.21 | 42 | 3.0 | 1477 | 727 | 11.2 | 24.8 | 16.4 | 2700 |

200 | 1000 | 7.42 | 63 | 3.0 | 1778 | 876 | 11.4 | 24.8 | 16.5 | 5300 |

400 | 2000 | 8.70 | 98 | 3.0 | 2079 | 1036 | 11.4 | 24.9 | 16.5 | 9400 |

500 | 2500 | 9.10 | 105 | 2.9 | 2157 | 1086 | 11.5 | 25.1 | 16.7 | 11,000 |

1000 | 5000 | 10.2 | 160 | 3.0 | 2428 | 1261 | 11.4 | 25.2 | 16.7 | 18,000 |

_{sxr}against I

_{peak}and I

_{pinch}and obtain SXR yield scales as Y

_{sxr}~ I

_{pinch}

^{3.6}and Y

_{sxr}~ I

_{peak}

^{3.2}. The I

_{pinch}scaling has less scatter than the I

_{peak}scaling. We next subject the scaling to further test when the fixed parameters RESF, c, L

_{0}and V

_{0}and model parameters f

_{m}, f

_{c}, f

_{mr}, f

_{cr}are varied. We add in the results of some numerical experiments using the parameters of several existing plasma focus devices including the UNU/ICTP PFF (RESF = 0.2, c = 3.4, L

_{0}= 110 nH and V

_{0}= 14 kV with fitted model parameters f

_{m}= 0.05, f

_{c}= 0.7, f

_{mr}= 0.2, f

_{cr}= 0.8) [8,26,27,36], the NX2 (RESF = 0.1, c = 2.2, L

_{0}= 20 nH and V

_{0}= 11 kV with fitted model parameters f

_{m}= 0.10, f

_{c}= 0.7, f

_{mr}= 0.12, f

_{cr}= 0.68) [9,26,27,34,37] and PF1000 (RESF = 0.1, c = 1.39, L

_{0}= 33 nH and V

_{0}= 27 kV with fitted model parameters f

_{m}= 0.1, f

_{c}= 0.7, f

_{mr}= 0.15, f

_{cr}= 0.7) [26,27,23]. These new data points (white data points in Figure 11) contain wide ranges of c, V

_{0}, L

_{0}and model parameters. The resulting Y

_{sxr}versus I

_{pinch}log-log curve remains a straight line, with the scaling index 3.6 unchanged and with no more scatter than before. However the resulting Y

_{sxr}versus I

_{peak}curve now exhibits considerably larger scatter and the scaling index has changed.

**Figure 11.**Y

_{sxr}is plotted as a function of I

_{pinch}and I

_{peak}. The parameters kept constant for the black data points are: RESF = 0.1, c = 1.5, L

_{0}= 30nH and V

_{0}= 20 kV and model parameters f

_{m}, f

_{c}, f

_{mr}, f

_{cr}at 0.06, 0.7, 0.16 and 0.7 respectively. The white data points are for specific machines which have different values for the parameters c, L

_{0}and V

_{0}.

_{pinch}in maintaining the scaling of Y

_{sxr}~ I

_{pinch}

^{3.6}with less scatter than the Y

_{sxr}~ I

_{peak}

^{3.2}scaling particularly when mixed-parameters cases are included, strongly support the conclusion that I

_{pinch}scaling is the more universal and robust one. Similarly conclusions on the importance of I

_{pinch}in plasma focus performance and scaling laws have been reported [21,22,23,24,26,27,28].

_{pinch}scaling rule for Y

_{sxr}not compatible with Gates’ rule [41]. However it is remarkable that our I

_{pinch}scaling index of 3.6, obtained through a set of comprehensive numerical experiments over a range of 0.2 kJ to 1 MJ, on Mather-type devices, is within the range of 3.5 to 4 postulated on the basis of sparse experimental data, (basically just two machines one at 5 kJ and the other at 0.9 MJ), by Filippov [42], for Filippov configurations in the range of energies 5 kJ to 1 MJ.

_{pinch}works well even when there are some variations in the actual device from L

_{0}= 30 nH, V

_{0}= 20 kV and c = 1.5.

_{sxr}= 8.3 × 10

^{3}× I

_{pinch}

^{3.6}

_{sxr}= 600 × I

_{peak}

^{3.2}

_{peak}(0.1 to 2.4), I

_{pinch}(0.07 to1.3) in MA, Y

_{sxr}~ E

_{0}

^{1.6}(kJ range) to Y

_{sxr}~ E

_{0}

^{0.8}(towards MJ).

## 6. Insight 4-Neutron Saturation

_{n}~ E

_{0}

^{2}where E

_{0}is the capacitor storage energy. Such scaling gave hopes of possible development as a fusion energy source. Devices were scaled up to higher E

_{0}. It was then observed that the scaling deteriorated, with Y

_{n}not increasing as much as suggested by the E

_{0}

^{2}scaling. In fact some experiments were interpreted as evidence of a neutron saturation effect [44,45] as E

_{0}approached several hundreds of kJ. As recently as 2006 Krauz [46] and November 2007, Scholz [47] have questioned whether the neutron saturation was due to a fundamental cause or to avoidable machine effects such as incorrect formation of plasma current sheath arising from impurities or sheath instabilities [45]. We should note here that the region of discussion (several hundreds of kJ approaching the MJ region) is in contrast to the much higher energy region discussed by Schmidt at which there might be expected to be a decrease in the role of beam target fusion processes [45,48].

#### 6.1. The Global Neutron Scaling Law

_{n}~ E

_{0}

^{2}scaling held, deterioration of this scaling became apparent above the low hundreds of kJ. This deteriorating trend worsened and tended towards Yn ~ E

_{0}

^{0.8}at tens of MJ. The results of these numerical experiments are summarized in Figure 1 with the solid line representing results from numerical experiments. Experimental results from 0.4 kJ to MJ, compiled from several available published sources [43,45,46,47,49,50,51,52], are also included as squares in the same figure. The combined experimental and numerical experimental results [31,32,33,38] (Figure 12) appear to have general agreement particularly with regards to the Y

_{n}~ E

_{0}

^{2}at energies up to 100 kJ, and the deterioration of the scaling from low hundreds of kJ to the 1 MJ level. The global data of Figure 12 suggests that the apparently observed neutron saturation effect is overall not in significant variance with the deterioration of the scaling shown by the numerical experiments.

**Figure 12.**The global scaling law, combining experimental and numerical data. The global data illustrates Y

_{n}scaling deterioration observed in numerical experiments from 0.4 kJ to 25 MJ (solid line) using the Lee model code, compared to measurements compiled from publications (squares) of various machines from 0.4 kJ to 1 MJ.

#### 6.2. The Cause of Neutron ‘Saturation’ is the Dynamic Resistance

_{m}and f

_{c}. We now represent the plasma focus circuit as shown in Figure 13.

**Figure 13.**Plasma focus circuit schematic. The capacitor bank with static inductance L

_{0}and stray resistance r

_{0}is switched into the plasma focus tube where a fraction f

_{c}of the circuit current I(t) effectively drives the plasma creating a time-varying inductance L(t) in the focus tube.

^{−7}ln(c) dz/dt in SI units. Typically on switching, as the capacitor discharges, the current rises towards its peak value, the current sheet is accelerated, quickly reaching nearly its peak speed and continues accelerating slightly towards its peak speed at the end of the axial phase. Thus for most of its axial distance the current sheet is travelling at a speed close to the end-axial speed. In deuterium the end-axial speed is observed to be about 10 cm/μs over the whole range of devices. This fixes the rate of change of inductance dL/dt as 1.4 × 10

^{−2}H/s for all the devices, if we take the radius ratio c = b/a = 2. This value of dL/dt changes by at most a factor of 2, taking into account the variation of c from low values of 1.4 (generally for larger machines) to 4 (generally for smaller machines). This typical dL/dt may also be expressed as 14 mΩ.

^{2}(dL/dt) + LI(dI/dt)

^{2}):

_{L}= d(½ LI

^{2})/dt = ½I

^{2}(dL/dt) + LI(dI/dt)

_{L}of Equation (7) is not the same as P of Equation (6).

_{L}= (½)(dL/dt)I

^{2}is not associated with the inductive energy stored in L. We conclude that whenever L(t) changes with time, the instantaneous power delivered to L(t) has a component that is not inductive. Hence this component of power (½)(dL/dt)I

^{2}must be resistive in nature; and the quantity (½)(dL/dt) also denoted as half Ldot is identified as a resistance, due to the motion associated with dL/dt ; which we call the dynamic resistance DR [27,31,33,38]. Note that this is a general result and is independent of the actual processes involved. In the case of the plasma focus axial phase, the motion of the current sheet imparts power to the shock wave structure with consequential shock heating, Joule heating, ionization, radiation etc. The total power imparted at any instant is just the amount (½)(dL/dt)I

^{2}, with this amount powering all consequential processes. We denote the dynamic resistance of the axial phase as DR

_{0}.

_{0}= (L

_{0}/C

_{0})

^{0.5}driving a load with a near constant resistance of 7 mΩ. We also assign a value for stray resistance of 0.1Z

_{0}. This situation may be shown in Table 3 where L

_{0}is given a typical value of 30 nH. We also include in the last column the results from a circuit (L-C-R) computation, discharging the capacitor with initial voltage 30 kV into a fixed resistance load of 7 mΩ, simulating the effect of the DR

_{0}and a stray resistance of value 0.1Z

_{0}.

**Table 3.**Discharge characteristics of equivalent PF circuit, illustrating the ‘saturation’ of I

_{peak}with increase of E

_{0}to very large values. The last column presents results using circuit (L-C-R) computation, with a fixed resistance load of 7 mΩ, simulating the effect of the DR

_{0}and a stray resistance of value 0.1Z

_{0}.

E_{0}(kJ) | C_{0}(μF) | Z_{0}(mΩ) | DR_{0}(mΩ) | Z_{total} (mΩ) | I_{peak} = V_{0}/Z_{total}(kA) | I_{peak}, L-C-R(kA) |
---|---|---|---|---|---|---|

0.45 | 1 | 173 | 7 | 197 | 152 | 156 |

4.5 | 10 | 55 | 7 | 67 | 447 | 464 |

45 | 100 | 17 | 7 | 26 | 1156 | 1234 |

135 | 300 | 10 | 7 | 18 | 1676 | 1819 |

450 | 1000 | 5.5 | 7 | 12.9 | 2321 | 2554 |

1080 | 2400 | 3.5 | 7 | 10.8 | 2781 | 3070 |

4500 | 10,000 | 1.7 | 7 | 8.8 | 3407 | 3722 |

45,000 | 100,000 | 0.55 | 7 | 7.6 | 4209 | 4250 |

_{0}we obtain Figure 14, which shows the tendency of the peak current towards saturation as E

_{0}reaches large values; the deterioration of the curve becoming apparent at the several hundred kJ level. This is the case for I

_{peak}= V

_{0}/Z

_{total}and also for the L-C-R discharge with simulated value of the DR

_{0}. In both cases it is seen clearly that a capacitor bank of voltage V

_{0}discharging into a constant resistance such as DR

_{0}will have a peak current I

_{peak}approaching an asymptotic value of I

_{peak}= V

_{0}/DR

_{0}when the bank capacitance C

_{0}is increased to such large values that the value of Z

_{0}= (L

_{0}/C

_{0})

^{0.5}<< DR

_{0}. Thus DR

_{0}causes current ‘saturation’.

**Figure 14.**I

_{peak}versus E

_{0}on log-log scale, illustrating I

_{peak}‘saturation’ at large E

_{0}.

_{n}and I

_{peak}and I

_{pinch}as follows:

_{n}~ I

_{pinch}

^{4.5}

_{n}~ I

_{peak}

^{3.8}

_{peak}will lead to saturation of Y

_{n}.

_{0}is increased to very large values by an increase in C

_{0}, simply due to the dominance of the axial phase dynamic resistance. This makes the total circuit impedance tend towards an asymptotic value which approaches the dynamic resistance at infinite values of E

_{0}. The ‘saturation’ of current inevitably leads to a ‘saturation’ of neutron yield. Thus the apparently observed neutron ‘saturation’ which is more accurately represented as a neutron scaling deterioration is inevitable because of the dynamic resistance. In line with current plasma focus terminology we will continue to refer to this scaling deterioration as ‘saturation’. The above analysis applies to the Mather-type plasma focus. The Filippov-type plasma focus does not have a clearly defined axial phase. Instead it has a lift-off phase and an extended pre-pinch radial phase which determine the value of I

_{peak}. During these phases the inductance of the Filippov discharge is changing, and the changing L(t) will develop a dynamic resistance which will also have the same current ‘saturation’ effect as the Filippov bank capacitance becomes big enough.

#### 6.3. Beyond Presently Observed Neutron Saturation Regimes

_{0}until the surge impedance becomes negligible due to the very large value of C

_{0}. then the ‘saturation’ effect would still be there, but the level of ‘saturation’ would be proportional to the voltage. In this way we can go far above presently observed levels of neutron ‘saturation’; moving the research, as it were into presently beyond-saturation regimes.

_{peak}to beyond 15 MA and I

_{pinch}to over 6 MA. Also multiple Blumleins at 1 MV, in parallel, could provide driver impedance of 100 mΩ, matching the radial phase dynamic resistance and provide fast rise currents peaking at 10 MA with I

_{pinch}value of perhaps 5 MA. Bank energy would be several MJ. The push to higher currents may be combined with proven neutron yield enhancing methods such as doping deuterium with low % of krypton [55]. Further increase in pinch current might be by fast current injection near the start of the radial phase. This could be achieved with charged particle beams or by circuit manipulation such as current-stepping [31,33,56,57]. The Lee model is ideally suited for testing circuit manipulation schemes.

## 7. Conclusions

## References

- Lee, S. Plasma focus model yielding trajectory and structure. In Radiations in Plasmas; McNamara, B., Ed.; World Scientific: Singapore, 1984; Volume II, pp. 978–987. [Google Scholar]
- Lee, S.; Tou, T.Y.; Moo, S.P.; Eissa, M.A.; Gholap, A.V.; Kwek, K.H.; Mulyodrono, S.; Smith, A.J.; Suryadi, S.; Usada, W.; Zakaullah, M. A simple facility for the teaching of plasma dynamics and plasma nuclear fusion. Amer. J. Phys.
**1988**, 56, 62–68. [Google Scholar] [CrossRef] - Tou, T.Y.; Lee, S.; Kwek, K.H. Non perturbing plasma focus measurements in the run-down phase. IEEE Trans. Plasma Sci.
**1989**, 17, 311–315. [Google Scholar] [CrossRef] - Lee, S. A sequential plasma focus. IEEE Trans. Plasma Sci.
**1991**, 19, 912–919. [Google Scholar] [CrossRef] - Jalil, A. Development and studies of a small plasma focus. Ph.D. Dissertation, Universiti Teknologi Malaysia, Kuala Lumpur, Malaysia, 1990. [Google Scholar]
- Potter, D.E. The formation of high-density z-pinches. Nucl. Fusion.
**1978**, 18, 813–823. [Google Scholar] [CrossRef] - Lee, S.; Serban, A. Dimensions and lifetime of the plasma focus pinch. IEEE Trans. Plasma Sci.
**1996**, 24, 1101–1105. [Google Scholar] [CrossRef] - Liu, M. Soft X-rays from compact plasma focus. Ph.D. dissertation, NIE, Nanyang Technological Univ., Singapore, 2006. Available online: http://eprints.ictp.it/327/ (accessed on 25 March 2010). [Google Scholar]
- Bing, S. Plasma dynamics and X-ray emission of the plasma focus. Ph.D. Dissertation, NIE, Nanyang Technological Univ., Singapore, 2000. Available online: http://eprints.ictp.it/99/ (accessed on 25 March 2010). [Google Scholar]
- Serban, A.; Lee, S. Experiments on speed-enhanced neutron yield from a small plasma focus. J. Plasma Phys.
**1998**, 60, 3–15. [Google Scholar] [CrossRef] - Liu, M.H.; Feng, X.P.; Springham, S.V.; Lee, S. Soft X-ray measurement in a small plasma focus operated in neon. IEEE Trans. Plasma Sci.
**1998**, 26, 135–140. [Google Scholar] [CrossRef] - Wong, D.; Lee, P.; Zhang, T.; Patran, A.; Tan, T.L.; Rawat, R.S.; Lee, S. An improved radiative plasma focus model calibrated for neon filled NX2 using a tapered anode. Plasma Sources Sci. Technol.
**2007**, 16, 116–123. [Google Scholar] [CrossRef] - Lee, S. Plasma radiation source lab-computer package 2000–2007. Available online: http://ckplee.myplace.nie.edu.sg/plasmaphysics/ (accessed on 12 March 2010).
- Lee, S. Radiative dense plasma focus model computation package RADPFV5.008 (November 2005). ICTP Open Access Archive, 2005. Available online: http://eprints.ictp.it/85/ (accessed on 12 March 2010).
- Lee, S. Twelve Years of UNU/ICTP PFF—A Review; Abdus Salam ICTP: Trieste, Italy, 1998; pp. 5–34. Available online: http://eprints.ictp.it/31/ (accessed on 12 March 2010).
- Springham, S.V.; Lee, S.; Rafique, M.S. Correlated deuteron energy spectra and neutron yield for a 3 kJ plasma focus. Plasma Phys. Control. Fusion
**2000**, 42, 1023–1032. [Google Scholar] [CrossRef] - Mohammadi, M.A.; Sobhanian, S.; Wong, C.S.; Lee, S.; Lee, P.; Rawat, R.S. The effect of anode shape on neon soft X-ray emissions and current sheath configuration in plasma focus device. J. Phys. D, Appl. Phys.
**2009**, 42, 045 203 (10pp). [Google Scholar] [CrossRef] - Lee, S.; Lee, P.; Zhang, G.; Feng, X.; Gribkov, V.A.; Liu, M.; Serban, A.; Wong, T. High rep rate high performance plasma focus as a powerful radiation source. IEEE Trans. Plasma Sci.
**1998**, 26, 1119–1126. [Google Scholar] [CrossRef] - Bogolyubov, E.P.; Bochkov, V.D.; Veretennikov, V.A.; Vekhoreva, L.T.; Gribkov, V.A.; Dubrovskii, A.V.; Ivanov, Y.P.; Isakov, A.I.; Krokhin, O.N.; Lee, P.; Lee, S.; Nikulin, V.Y.; Serban, A.; Silin, P.V.; Feng, X.; Zhang, G.X. A powerful soft X-ray source for X-ray lithography based on plasma focusing. Phys. Scr.
**1998**, 57, 488–494. [Google Scholar] [CrossRef] - Siahpoush, V.; Tafreshi, M.A.; Sobhanian, S.; Khorram, S. Adaptation of Sing Lee’s model to the Filippov type plasma focus geometry. Plasma Phys. Control. Fusion
**2005**, 47, 1065–1075. [Google Scholar] [CrossRef] - Lee, S.; Saw, S.H. Neutron scaling laws from numerical experiments. J. Fusion Energy
**2008**, 27, 292–295. [Google Scholar] [CrossRef] - Lee, S. Current and neutron scaling for megajoule plasma focus machines. Plasma Phys. Control. Fusion
**2008**, 50, 105 005 (14pp). [Google Scholar] - Lee, S.; Saw, S.H. Pinch current limitation effect in plasma focus. Appl. Phys. Lett.
**2008**, 92, 021 503. [Google Scholar] - Lee, S.; Lee, P.; Saw, S.H.; Rawat, R.S. Numerical experiments on plasma focus pinch current limitation. Plasma Phys. Control. Fusion
**2008**, 50, 065 012 (8pp). [Google Scholar] - Gribkov, V.A.; Banaszak, A.; Bienkowska, B.; Dubrovsky, A.V.; Ivanova-Stanik, I.; Jakubowski, L.; Karpinski, L.; Miklaszewski, R.A.; Paduch, M.; Sadowski, M.J.; Scholz, M.; Szydlowski, A.; Tomaszewski, K. Plasma dynamics in the PF-1000 device under fullscale energy storage: II. Fast electron and ion characteristics versus neutron emission parameters and gun optimization perspectives. J. Phys. D, Appl. Phys.
**2007**, 40, 3592–3607. [Google Scholar] [CrossRef] - Lee, S. Institute for Plasma Focus Studies-Radiative Dense Plasma Focus Computation Package: RADPF. Available online: http://www.plasmafocus.net/IPFS/modelpackage/File1RADPF.htm (accessed on 12 March 2010).
- Internet Workshop on Plasma Focus Numerical Experiments (IPFS-IBC1). Available online: http://www.plasmafocus.net/IPFS/Papers/IWPCAkeynote2ResultsofInternet- basedWorkshop.doc (accessed on 12 March 2010).
- Lee, S.; Saw, S.H.; Lee, P.C.K.; Rawat, R.S.; Schmidt, H. Computing plasma focus pinch current from total current measurement. Appl. Phys. Lett.
**2008**, 92, 111 501. [Google Scholar] - Akel, M.; Al-Hawat, Sh.; Lee, S. Pinch current and soft X-ray yield limitation by numerical experiments on nitrogen plasma focus. J. Fusion Energy. [CrossRef]
- Saw, S.H.; Lee, S. Scaling laws for plasma focus machines from numerical experiments. In Proceedings of IWPDA 2009, Singapore, July 2009.
- Lee, S. Diagnostics and insights from current waveform and modelling of plasma focus. In Proceedings of IWPDA 2009, Singapore, July 2009.
- Saw, S.H.; Lee, S. Scaling the plasma focus for fusion energy considerations. In Tubav Conferences: Nuclear & Renewable Energy Sources, Ankara, Turkey, September 2009; pp. 61–70.
- Lee, S. Nuclear fusion and the Plasma Focus. In Invited paper Tubav Conferences: Nuclear & Renewable Energy Sources, Ankara, Turkey, 28 & 29 September 2009; pp. 9–18.
- Lee, S.; Saw, S.H.; Lee, P.; Rawat, R.S. Numerical experiments on neon plasma focus soft X-rays scaling. Plasma Phys. Contr. Fusion
**2009**, 51, 105013 (8pp). [Google Scholar] [CrossRef] - Akel, M.; Al-Hawat, Sh.; Lee, S. Numerical experiments on soft X-ray emission optimization of nitrogen plasma in 3 kJ plasma focus SY-1 using modified lee model. J Fusion Energy. [CrossRef]
- Saw, S.H.; Lee, P.C.K.; Rawat, R.S.; Lee, S. Optimizing UNU/ICTP PFF plasma focus for neon soft X-ray operation. IEEE Trans. Plasma Sci.
**2009**, 37, 1276–1282. [Google Scholar] [CrossRef] - Lee, S.; Rawat, R.S.; Lee, P.; Saw, S.H. Soft X-ray yield from NX2 plasma focus. J App Phys.
**2009**, 106, 023309. [Google Scholar] - Lee, S. Neutron yield saturation in plasma focus-A fundamental cause. App. Phys Letts.
**2009**, 95, 151503. [Google Scholar] [CrossRef] - Lee, S.; Saw, S.H.; Soto, L.; Moo, S.P.; Springham, S.V. Numerical experiments on plasma focus neutron yield versus pressure compared with laboratory experiments. Plasma Phys. Control. Fusion
**2009**, 51, 075006 (11pp). [Google Scholar] - Huba, J.D. Plasma Formulary. Available online: http://wwwppd.nrl.navy.mil/nrlformulary/NRL_FORMULARY_07.pdf (accessed on 12 March 2010).
- Gates, D.C. X-ray Yield Scaling. In 1978 Proceedings of the IInd Int Conference on Energy Storage, Compression and Switching; Venice, 2, 3239. Plenum Press: New York, 1983. [Google Scholar]
- Filippov, N.V.; Filippova, T.I.; Khutoretskaia, I.V.; Mialton, V.V.; Vinogradov, V.P. Megajoule scale plasma focus as efficient X-ray source. Phys. Lett. A
**1996**, 211, 168–171. [Google Scholar] [CrossRef] - Rapp, H. Measurements referring to plasma focus scaling laws. Phys Lett A
**1973**, 43A, 420–422. [Google Scholar] [CrossRef] - Herold, H.; Jerzykiewicz, A.; Sadowski, M.H. Comparative analysis of large plasma focus experiments performed at IPF, Stuttgart, and at IPJ, Swierk. Nucl Fusion
**1989**, 29, 1255–1260. [Google Scholar] [CrossRef] - Bernard, A.; Bruzzone, H.; Choi, P.; Chuaqui, H.; Gribkov, V.; Herrera, J.; Hirano, K.; Krejci, A.; Lee, S.; Luo, C. Scientific status of plasma focus research. Moscow J Physical Society
**1998**, 8, 93–170. [Google Scholar] - Kraus, V.I. Progress in plasma focus research and applications. In Proceedings of the 33rd EPS Conference on Plasma Physics, Rome, Italy, June, 2006. Plasma Phys. Control. Fusion
**2006**, 48, B221-B229. - Scholz, M. Report at the ICDMP Meeting, ICDMP, Warsaw, Poland, November 2007.
- Schmidt, H. The role of beam target processes in extrapolating the plasma focus to reactor conditions. In Proceedings of the Fifth International Workshop on Plasma Focus and Z-pinch Research, Toledo, OH, USA, June 1987; p. 65.
- Kies, W. Laser and Plasma Technology, Proceedings of Second Tropical College; Lee, S., Tan, B.C., Wong, C.S., Chew, A.C., Low, K.S., Harith, A., Chen, Y.H., Eds.; World Scientific: Singapore, 1988; pp. 86–137. [Google Scholar]
- Herold, H. Laser and Plasma Technology, Proceedings of Third Tropical College; Wong, C.S., Lee, S., Tan, B.C., Chew, A.C., Low, K.S., Moo, S.P., Eds.; World Scientific: Singapore, 1990; pp. 21–45. [Google Scholar]
- Soto, L.; Silva, P.; Moreno, J.; Silvester, G.; Zambra, M.; Pavez, C.; Altamirano, L.; Bruzzone, H.; Barbaglia, M.; Sidelnikov, Y.; Kies, W. Research on pinch plasma focus devices of hundred of kilojoules to tens of joules. Brazil. J. Phys.
**2004**, 34, 1814. [Google Scholar] [CrossRef] - Patran, A.; Rawat, R.S.; Koh, J.M.; Springham, S.V.; Tan, T.L.; Lee, P.; Lee, S. A High Efficiency Soft X-ray Source For Scientific and Industrial Applications. 30th EPS Conference on Plasma Phys. London
**2004**, ECA Vol.28G, 4.213. [Google Scholar] - Chow, S.P.; Lee, S.; Tan, B.C. Current sheath studies in co-axial plasma focus gun. J. Plasma Phys
**1972**, 8, 21. [Google Scholar] [CrossRef] - Decker, G.; Kies, W.; Nadolny, R.; Röwekamp, P.; Schmitz, F.; Ziethen, G.; Koshelev, K.N.; Sidelnikov, V. Micropinch actuation in the SPEED 2 plasma focus. Plasma Sources Sci. Technol.
**1996**, 5, 112. [Google Scholar] [CrossRef] - Rishi, V.; Lee, P.; Lee, S.; Springham, S.V.; Tan, T.L.; Rawat, R.S.; Krishnan, M. order of magnitude enhancement in neutron emission with deuterium-krypton admixture operation in miniature plasma focus device. Appl. Phys. Lett.
**2008**, 93, 101501. [Google Scholar] [CrossRef] - Saw, S.H. Experimental studies of a current-stepped pinch. PhD Thesis, Universiti Malaya, Kuala Lumpur, Malaysia, 1991. [Google Scholar]
- Lee, S. A current-stepping technique to enhance pinch compression. J. Phys. D: Appl. Phys.
**1984**, 17, 733–741. [Google Scholar] [CrossRef]

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, S.; Saw, S.H.
Numerical Experiments Providing New Insights into Plasma Focus Fusion Devices. *Energies* **2010**, *3*, 711-737.
https://doi.org/10.3390/en3040711

**AMA Style**

Lee S, Saw SH.
Numerical Experiments Providing New Insights into Plasma Focus Fusion Devices. *Energies*. 2010; 3(4):711-737.
https://doi.org/10.3390/en3040711

**Chicago/Turabian Style**

Lee, Sing, and Sor Heoh Saw.
2010. "Numerical Experiments Providing New Insights into Plasma Focus Fusion Devices" *Energies* 3, no. 4: 711-737.
https://doi.org/10.3390/en3040711