Institute for Plasma Focus Studies

IPFS

knowledge should be freely accessible to all

Internet Workshop on Plasma Focus Numerical Experiments

Module 4; (Follow the instructions in the following notes. You may also wish to refer to the supplementary notes part4supplementary.htm.

Summary:

This module looks at variation of neutron yield with pressure; running PF1000-from short circuit (very high pressure), through optimum pressure to low pressure. The very high pressure of the short circuit shot stops all current sheath motion thus simulating a short circuit. The aim is just to obtain the short-circuit current waveform for comparison with the focusing waveforms. In a second example we also look at variation of SXR with pressure; operating NX2 from short circuit (very high pressure), through optimum pressure to low pressure. In the course of these numerical experiments we take a small detour (during the NX2 experiments) to determine circuit parameters from a short circuit discharge; something very basic, but often overlooked. At the end of the course two additional exercises are given, one comparing computed and measured Y_n vs P_o for the PF400. The other is an outline of an open exercise which gives a glimpse of a frontier area of plasma focus research, that of neutron yield saturation for megajoule devices.

For the PF1000 neutron experiments:

Steps: (a) Configure the code for the PF1000 at 27kV 3.5 Torr Deuterium using model parameters which we had fitted earlier.

(b) Fire the PF1000 at very high pressure, effectively a short circuit.

(d) Place current waveforms at different pressures on the same chart; for comparative study. Completing this chart forms part of Exercise 5.

(e) Tabulate results at different pressures; for comparative study; including speeds, dimensions, duration, temperature and neutron yield. This completes Exercise 5. Discussion.

For the NX2 SXR experiments:

Steps: (a) Configure the NX2 at 11kV 2.6 Torr Neon using fitted model parameters

(b) Fire the NX2 at very high pressure, effectively a short circuit; first introduction to macro code modification.

(c) Detour: Use this short circuit waveform as though it were a measured current waveform, to analyse the lightly damped L-C-R discharge; to fix bank parameters.

(d) Fire NX2 at 5 Torr; as an example of insufficient current drive; over-riding the model’s time-match guard.

(e) Fire NX2 at lower pressures down to 0.5 Torr, looking for optimum SXR yield.

(f) Place current waveforms at different pressures on the same chart, for comparison.

(g) Tabulate results at different pressures; for comparative study; including speeds, dimensions, duration, temperature and neutron yield. Discussion.

The sessions ends with a general consideration of plasma focus yield scaling.

The material:

You need File7RADPFV5.15b (called File7) for the following work. Copy and Paste on your Desktop. You also need the files PF1000pressureblank.xls and NX2pressureblank.xls.These files contain also tabulation blanks for your convenience.

Also provided is file HiRepHiPerformPF.doc from which NX2 Y_sxr vs P_o data is extracted

Three additional files are provided for two additional exercise which you may complete at your leisure later. These are: PF400Yncomparison.xls and an accompanying paper pf400_soto.pdf for the first additional exercise for you to duplicate. The other paper (Saturation….pdf) goes with the final open exercise suggested as an epilogue to this course.

Part 1: Neutron Yield of PF1000 vs Pressure

(a) Preparing Sheet 2 and Configure the code for PF1000

Open File7RADPFV5.13.9b. Copy PF1000pressureblank.xls to Sheet 2 using procedure which we have done before; repeated briefly as follows.

With File7 open; open PF1000pressureblank.xls; click the Edit Tab; scroll down and click 'Move or Copy file'. A window pops out. In the 'To book’: choose ‘File7RADPFV5.13.9b.xls’; then choose ‘move to end’; click ‘OK’. Rename ‘Sheet1(2)’ as Sheet2.

The Current waveforms are now displayed in the chart in Sheet2 of File7.

[PF1000pressureblank.xls has time-current data for several traces, and scrolling to the right, a table of plasma focus properties at various pressures to be filled in, and below that a normalized table; and there are also two charts; one for the current traces at various pressures and one for Y_n, I_peak, I_pinch vs pressures. Have a close look at the opened

sheet to see the locations of the supplied time-current data, the blank spaces for you to fill in the other computed time-current data, the tables with the blank spaces to be filled in, and the partially filled in charts.]

Use the data in PF1000pressureblank.xls to configure.

Bank: L_o=33.5 nH, C_o=1332 mF, r_o=6.1mOhm

Tube: b=16 cm, a=11.55 cm, zo=60 cm

Operation: V_o=27kV, P_o= Torr, MW=4, A=1, At-Mol=2

Model: f_m=0.13, f_c=0.7, f_mr=0.35, f_cr=0.65

(b) Firing a very high pressure shot.

Key in 100,000 Torr at B9. [In the laboratory it is of course impossible to fire such a shot and a physical short-circuit may need to be used at the insulator end of the plasma focus; or fire at the highest safe pressure in argon. In the lab we have used 50 Torr argon, to obtain very approximate results.]

{In the numerical experiment at this high pressure the current sheath only moves a little down the tube, adding hardly any inductance or dynamic loading to the circuit. So it is equivalent to short-circuiting the plasma focus at its input end. In the code there is a loop during the axial phase, computing step- by- step the variables as time is incremented. The loop is broken only when the end of the anode (non-dimensionalised z=1) is reached. In this case we do not reach the end of the anode. However there is an alternative stop placed in the loop that stops the run when (non-dimensionalised time=6 ie nearly 1 full cycle time, 2p, of the short- circuited discharge) is reached. Moreover at the start of the run, the code computes a quantity ALT= ratio of characteristic capacitor time to sum of characteristic axial & radial times. Numerical tests have shown that when this quantity is less than 0.65, the total transit time is so large (compared to the available current drive time) that the radial phase will not be efficiently completed. Moreover because of the large deviation from normal focus behaviour, the numerical scheme and ‘house keeping’ details incorporated into the code may become subjected to numerical instabilities leading to error messages. To avoid these problems a time-match guard feature has been incorporated to stop the code from being run when ALT<0.65. When this happens one

can over-ride the stop; and continue running unless the run is then terminated by Excel for e.g. ‘over-flow’ problems. In that case one has to abandon the run and reset the code.}

Fire the high pressure shot. The Visual Basics Code appears at Statement 430 Stop; with a warning message that pressure is too high. In this case we know what we are doing, and over-ride as follows: Click on ‘Run’ (above the code sheet), and ‘continue’. Another ‘Stop’ appears just below line 485; with a warning about transit time. Click on ‘Run’ and ‘continue’; another ‘Stop’ appears below Line 488.. Click on ‘Run’ and ‘Continue’.

In a little while the run has proceeded and finally the statement “If T > 6 Then Stop” appears; indicating we have completed nearly one cycle of the capacitor discharge.

Now, locate the ‘x’ at the extreme right hand corner of the screen. Click on this ‘x’; pop-up appears with the message ‘This command will stop the debugger’. Click on OK, which brings you back to the worksheet 1.

Copy the data in columns A & B from A20 and B20 to the end of the computed current data (several thousand cells down); ‘paste’ the copied time-current data onto Columns E & F (in the labeled space provided in Sheet 2. Locate the data table by scrolling to the right. Fill in the value of I_peak [read from Fig 1 or from the relevant cell] onto the table against 100,000 Torr; all the other quantities (I_pinch, peak v_a, S, peak v_s …. T and Y_n….being zero. )

(c) Fire at different pressures Place

Fire the next shot at 19 Torr+. As the ALT value is over 0.65, the run proceeds as normal. Copy the time-current data from Columns A & B (from rows 20 down) to Sheet 2 columns G & H. Fill in the table [I_peak_, I_pinch, peak v_a, S, peak v_s … T… Y_n…n_i^** & EINP )

for the data from shot 19 Torr. [+Note: The waveform and data for this point, is already copied for you, to save you some time.]

** The data for n_iis output in column AK in Sheet 1 (you need to scroll way down as these are outputted only for the pinch phase).

Repeat for pressures 14,10, 9, 8, 7.5 , 7, 6, 3.5 , 2 and 1; tabulating the data for all these shots; but copy and paste the time-current data for only selected shots e.g 14, 10, 6 and 2 [some of the shots are pre-filled for you to save you some time]. The list of pressures had been chosen as above, as I carried out the numerical experiments. It was clear that Y_n was increasing rapidly from 14 Torr to 10 Torr. More points were chosen between 10 Torr & 6 Torr until it was obvious that the optimum pressure (for Y_n) was between 8 and 7 Torr.

(d) Place the current waveforms at different pressures on the same chart.

Suggested procedure: To save you time, the comparison chart has already been created for you, and pre-filled with several waveforms namely 19, 7.5, 3.5 and 1 Torr. You only have to fill in the ones for 1,000,000 and 14, 10,6 and 2 Torr in the correct columns indicated by the column headings already placed on Sheet 2. You will note that the computed current waveform for 3.5 Torr falls neatly on the measured current waveform (as you have seen during an earlier exercise precisely with this PF1000 27kV 3.5 Torr current waveform.

(e) Tabulate results at different pressures ; for comparative study; including speeds, dimensions, duration, temperature and neutron yield. Discussion.

This tabulation has already been done as step (c) proceeded above.

In order to chart some of the computed data on one comparative chart, below the table you have filled in, there is another table with each data column normalized to the data at 7.5 Torr, which was found to be the pressure with the highest Y_n. Thus the values of all the data in the normalized table is in the region of 1.

Plot normalized Y_n, I_peak, I_pinch, and radial EINP against P_o.

[As you fill in the table, the normalized quantities are automatically computed, and the chart begins to take the correct shape. At the start the chart is in a jumble because many points have not been filled in, and thus there are erratic zero points all over the place.]

Discussion

Note 1: Look at the change of current waveforms from very high pressures to low pressures. At very high pressures the waveform is a damped sinusoid. At 19 torr the characteristic flattening of the current waveform due to dynamics is already clearly evident. The current peak comes earlier than the unloaded (high pressure) case, the current then droops until the rollover into the dip (due to the increased radial phase loading) at around 15 us. At lower pressures these characteristics remain the same except that the current trace is depressed more and more as speed increases. The peaking (reaching maximum current) also comes earlier and earlier, as does the radial phase rollover of the current trace. At 2.6 Torr, there is hardly any droop, the current waveform showing a distinct flat top leading to the rollover. At 1 Torr the axial speed is now so high that the axial phase is completed in less than 5 us and the current is still rising when it is forced down by the radial phase dynamics.

Note2 : A very important point to note in neutron scaling is that there exist a great deal of confusion and even misleading information in published literature because of sloppy practice with regards to I_peak and I_pinch. These quantities are sometimes treated as one and the same or when a distinction is attempted there is then a confusion between the total current at the time of pinch and I_pinch. For example in the case of PF1000, there appears to be some disappointment(in their publications) that (at 35kV) with the current at more than 2 MA, Y_n is still at best in the mid 10^11; and not at least an order of magnitude higher that one might expect for currents around 2MA. However if you numerically run PF1000 at 35kV you will find that I_pinch is only 1MA; so we are not surprised that the measured yield is at best an order of magnitude down from what you would expect thinking that your current is around 2MA. (scaling at Y_n~I⁴ , a factor of 2 in current gives a factor of 16 in the yield). So it is important that the thinking of yield should be in terms of I_pinch as the relevant scaling parameter. When using the model, the distinction of I_pinch and I_peak is clear.

Coming to the detailed tabulations: As P_o decreases, I_peak decreases, and continues to decrease, because the increasing axial speeds increases the circuit loading, throughout the whole range of pressures. However it is noticed that I_pinch increases from high pressures, peaking in a flat manner at 6 Torr and then drops more sharply towards 1 torr. One factor contributing to the increase is the shift of the pinch time from very late in the discharge

(when discharge current has dropped greatly) to earlier in the discharge (when current has dropped less). That is the main factor for I_pinch increasing despite a decreasing I_peak. At low pressures (e.g.1 Torr), the radial phase now occurs so early that it is forcing the current down early in the discharge. That lowers both the I_peak as well as the I_pinch. These points are clear when you look at the comparative chart of current traces at various pressures.

The radial EINP follows the same pattern as I_pinch, and for the same reasons. The radial EINP computes the cumulative work done by the current sheath in the radial phases.

Looking at the other quantities, we note that the speeds (axial, radial shock and radial piston) and temperature all continue to rise as pressure lowers; similarly S and maximum induced voltage V also increase as pressure is decreased. Pinch length z_max is almost a constant. Minimum pinch radius and pinch duration continue to decrease; the former due to better compression at higher speeds and the latter due to the increased T. The number density progressively drops, due to the decreasing starting numbers, despite the increasing compression.

From the tabulations of the above numerical experiments, it might be useful to consider the beam-target mechanism which we are using to compute the neutron yield. This is summarized in the following note.

[ note: From part4supplementary.htm

Y_b-t= C_n n_iI_pinch²z_p²((lnb/r_p))s/V_max^1/2

where s is the D-D fusion cross section. In the range we are considering we may take s~V_maxⁿ where n~2-3; say we take n=2.5; then we have

Y_b-t~ n_iI_pinch²z_p²((lnb/r_p)) V_max²

The factor z_p²((lnb/r_p)) is practically constant.

Thus we note that it is the behaviour of n_{i ,}I_pinchand V_max as pressure changes that determines the way Y_n increases to a maximum and then drops as pressure is changed.}

[An additional experiment is suggested at the end of these notes, in which you can see how numerical experiments on Y_n vs operating pressure compare with measured results in the case of PF400]

Part 2: Soft X-ray Yield of NX2 with Operating Pressure

Prepare the worksheets for the experiment. Open File7RADPFV5.14.xls. Insert Sheet 2 & Sheet 3.. Copy NX2pressureblank.xls to Sheet 2.

[NX2pressureblank.xls has 2 worksheets, Sheet 2 and Sheet 3. Sheet 2 has time-current data for several traces, and scrolling to the right, a table of plasma focus properties at various pressures to be filled in, and below that a normalized table; and there are also two charts; one for the current traces at various pressures and one for Y_SXR, I_peak, I_pinch vs pressures. Have a close look at the opened sheet to see the locations of the supplied time-current data, the blank spaces for you to fill in the other computed time-current data, the tables with the blank spaces to be filled in, and the partially filled in charts. Sheet 3 has labeled spaces for the computed high pressure current data, a chart and spaces to be filled in for data to be measured from the current waveform.]

Using the same procedures as suggested for the previous PF1000 experiments, copy and paste NX2pressureblank.xls Sheet 2 into Sheet 2 of File7RADPFV5.14.xls. Then copy the data and chart of NX2pressureblank.xls Sheet 3 into Sheet 3 of File7RADPFV5.14.xls.

Sheet 2 and Sheet 3 are now ready to receive the data of the numerical experiments.

(a) Configure the NX2 at 11kV 2.6 Torr Neon using fitted model parameters

We use an earlier version of the NX2 with a lower inductance of 15nH.

The parameters for that version of NX2 were successfully fitted as:

Bank: L_o=15 nH, C_o=28 mF, r_o=2.2 mOhm

Tube: b=4.1cm, a=1.9 cm, zo=5 cm

Operation: V_o=11kV, P_o= Torr, MW=20, A=10, At-Mol=1

Model: f_m=0.1, f_c=0.7, f_mr=0.12, f_cr=0.68

(b) Fire the NX2 at very high pressure, effectively a short circuit; first introduction to macro code modification.

Key in 1,000,000 Torr at B9. [In the laboratory it is of course impossible to fire a shot at such high pressure] {In the numerical experiment at this high pressure the current sheath only moves a little down the tube, adding hardly any inductance or dynamic loading to the circuit. So it is equivalent to short circuiting the plasma focus at its input end. In the code there is a loop during the axial phase, computing step by step the variables as time is incremented. The loop is broken only when the end of the anode (non-dimensionalised z=1) is reached. In this case we do not reach the end of the anode. However there is an alternative stop placed in the loop that stops the run when (non-dimensionalised time=6 ie nearly 1 full cycle time, 2p, of the short circuited discharge) is reached. Moreover at the start of the run, the code computes a quantity ALT= ratio of characteristic capacitor time to sum of characteristic axial & radial times. Numerical tests have shown that when this quantity is less than 0.65, the total transit time is so large (compared to the available current drive time) that the radial phase will not be efficiently completed. Moreover because of the large deviation from normal focus behaviour, the numerical scheme and ‘house keeping’ details incorporated into the code may become subjected to numerical instabilities leading to error messages. To avoid these problems a time-match guard feature has been incorporated to stop the code from being run when ALT<0.65. When this happens one can over-ride the stop; and continue running unless the run is then terminated by Excel for e.g. ‘over-flow’ problems. In that case one has to abandon the run and reset the code.}

We want to use the NX2 in short-circuit mode to illustrate the basic but often overlooked treatment of a lightly damped L-C-R circuit for determining circuit parameters. The method we use requires determining the reversal ratio of the lightly damped discharge. For this purpose we would like to have say 3 cycles of the lightly damped discharge ie we should continue computing until normalized time reaches 6p~20. Since the code has a stop placed at t=6, we need to make a change in this statement in the code.

We have File7 opened. We will now ‘step into’ the code to edit it.

Above the worksheet, locate the control button ‘Tools’. Click on ‘Tools’ then ‘Macro’, then ‘Macros’. Then highlight ‘radpf005’ and click on the button ‘Step Into’. The program code in Visual Basic appears. We have entered the code.

Scroll down to line 580. Just below this line is the Statement “If T > 6 Then Stop”. Change the number ‘6’ to the number ‘20’. Then Exit the code by clicking the ‘x’ at the extreme top right hand corner above the spreadsheet. When drop-down appears with message “This command will stop the debugger” click on the button ‘OK’; bringing us back to Sheet 1.

The code is now configured to run the discharge short-circuited for 3 cycles before stopping.

In a little while the run has proceeded and finally the statement “If T > 20 Then Stop” appears; indicating we have completed more than 3 cycles of the capacitor discharge.

(c) Use this short circuit waveform as though it were a measured current waveform, to analyse the lightly damped L-C-R discharge; to measure discharge period T and reversal ratio f; hence determine L_o and r_o. Only C_o & V_o are assumed to be known.

Copy the current waveform data from Columns A & B and paste to Sheet 3 into the columns A & B starting from A5 & B5; so that we may carry out our little ‘detour’ experiment. To save you some time the chart has been prepared in advance and the current waveform should appear; once the data is pasted correctly starting at A5 and B5.

From the current waveform: measure 3 T (to 3 decimal places); hence obtain T.

Measure the successive peak currents, recording all as positive values. Thus measure:

f₁=I₂/I₁, f₂=I₃/I₂, f₃=I₄/I₃, f₄=I₅/I₄ and f₅=I₆/I₅; and f=(1/5)(f₁+f₂+f₃+f₄+f₅).

We are given C_oand V_o. With the measured T and f (measured from the current waveform) we calculate L_o and r_o and I_o using the following approximations applicable to slightly damped L-C-R discharges:

Lo=T²/(4p²C_o)

ro=-(2/p)Ln(f)(L_o/C_o)^0.5

I_o=pC_oV_o(1+f)/T

We note from this little ‘detour’ that this method gives highly accurate results for lightly damped discharges. In practice the accuracy is limited by experimental features such as electrical noise and electrostatic shielding of the coil which may result in a tilted zero baseline. We also note that it is important for every plasma focus to establish reliable baseline data. First, the capacitance C_o should be reliably known or determined. Then from the value of C_o, L_o and r_o may be fixed; and further I_o deduced to calibrate the monitoring coil.

Also copy the 1,000,000 Torr time-current data to Sheet2 to into the columns provided for this purpose (E & F)

(d) Fire NX2 at 5 Torr; as an example of insufficient current drive; over-riding the model’s time-match guard.

We now proceed to the NX2 SXR vs pressure experiment.

Key in 5 Torr in B9. Fire a shot.

The Visual Basics Code appears at Statement 430 Stop; with a warning message that pressure is too high. In this case we know what we are doing and over-ride as follows: Click on ‘Run’ (above the code sheet), and ‘continue’. Another ‘Stop’ appears just below

line 485; with a warning about transit time. Click on ‘Run’ and ‘continue’; another ‘Stop’ appears below Line 488. Click on ‘Run’ and ‘Continue’.

In a little while the run has completed successfully. In this manner we force the code to run even though the code warns us that the pressure is too high for a good shot.

Copy the time-current data (A20-B20 to several thousand rows down) for this shot and paste into the reserved and labeled space (already done for you in columns Q &R) in Sheet 2. Add the data ( I_peak, I_pinch, Peak v_a, S, Peak v_s, v_p, a_min, z_max, pinch duration etc ) for this shot to the table prepared for this purpose (scroll a little to the right for this table.).

(e) Fire NX2 at lower pressures down to 0.5 Torr, looking for optimum Y_SXR

In a similar way, force the code to run for 4.5 Torr (with an ALT=0.64; so need to force).

Add data to table.

Continue with the following shots: 4 Torr (ALT=0.68, so code runs without ‘Stop’ breaks) 3.5, 3.2, 3, 2.9, 2.8, 2.7, 2.6, 2.4, 2, 1.5, 1, 0.5; adding the data for each shot to the table; but transferring the time-current data to sheet 2 of only those shots in bold [we want to plot a few current traces to see the way the traces evolve with pressure]

(f) Place current waveforms at different pressures on the same chart, for comparison.

The selected current traces are plotted onto the same chart in Sheet 2. When we plot the curve for 2.6 Torr, note that the computed current trace falls neatly over the measured; as these have already been pre-fitted.

(g) Tabulate results at different pressures; for comparative study; including currents, speeds, dimensions, duration, temperature and neutron yield.

Discussion.

We note the way we are computing the neon SXR line radiation; with power of:

Hence the SXR energy generated within the plasma pinch depends on the properties:

Number density ni

Effective charge number Z

Pinch radius rp

Pinch length zf and

Temperature T

and Pinch duration ; since the power is integrated over the pinch duration.

This generated energy is then reduced by the plasma self-absorption which depends primarily on density and temperature; the reduced quantity of energy is then emitted as the SXR yield.

It was first pointed by Liu Mahe in his PhD thesis “Soft X-rays from Compact Plasma Focus” NTU/NIE 1996, that a temperature around 300 eV is optimum for SXR production. Our subsequent experience through numerical experiments suggest that around 2x10⁶ K (below 200eV) seems to be better.

Important note: Unlike the case of neutron scaling, for SXR scaling there is an optimum small range of temperatures (T window) to operate. This could be the most important point to observe for SXR scaling.

With these complicated coupled effects and the small T window I have doubts about such simplistic scaling laws as put forward from time to time: Y_sxr~I_pinch⁴/r_min²???-doubtful

In this present series of experiments on the NX2 we note that a peak yield of 21J is obtained at 2.9 Torr Neon at a temperature of 1.5x10⁶ K (computed at the middle of the pinch duration). This compares well with experimental data in Zhang Guixin’s 1999 PhD thesis, in his series of yield vs pressure experiments at 11.5 kV using the NX2 (in the configuration of our numerical experiments; our measured current waveform was taken from his series of experiments). In that series He obtained a peak yield of 20J at 3.3 Torr with yield fall-off similar to our numerical experiments, although his curve peaks less sharply as our results.

Zhang’s experimental results are plotted as black points on the chart for comparison with the computed Y_sxr vs pressure. Note that the computed yield at optimum pressure is comparable with the measured optimum yield; that the optimum pressure also compare well as is the falloff of yield to either side of the optimum pressure.

General notes on fitting, Yield Scaling and applications of the code

On fitting: We soon learn that one is not able to get a perfect fit; in the sense that you can defend it as absolutely the perfect fit. The way to treat it is that one has got a working fit; something to work with; which gives comparable results with experiments; rather than perfect agreement. There is no such thing anyway; experiments in Plasma Focus (i e on one PF under consistent conditions) give a range of results; especially in yields (factor of 2-5 range is common). So a working fit should still give results within the range of results of the hardware experiment.

Even though a fit may only be a 'working' fit (as opposed to the hypothetical perfect fit) when one runs a series of well planned numerical experiments one can then see a trend e.g. how properties, including yields, change with pressure or how yields scale with I_pinch, or with L_o etc. And if carefully carried out, the numerical experiments can provide, much more easily, results just like hardware experiments; with the advantage that after proper reference to existing experiments, then very quickly one can extend to future experiments and predict probable results.

On scaling: Data used for scaling should be taken from yield-optimized (or at least from near optimized) situations. If one takes from the worst case situations e.g. way out in the high pressure or low pressure regions, the yield would be zero for a non-zero I_pinch. Such data would completely distort the scaling picture.

Not only should the pressure be changed, but there should be consideration for e.g. suitable (or even optimized) I_pinch/a; as the value of I_pinch/a would affect the pressure at which optimized S is achieved.

On directions of work and applications: Efforts on the model code may be applied in at least two directions. The first direction is in the further development of the code; e.g. trying to improve the way the code models the reflected shock region or the pinch region.

The second direction is to apply the model to provide a solution to a particular problem. An example was when it was applied to look at expected improvements to the neutron yield of the PF1000 when L_o is reduced.

Using the model code it was a relatively easy procedure, firing shots as L_o was reduced in steps; optimizing the various parameters and then looking for the optimized neutron yield at the new value of L_o. When this exercise was carried out in late 2007, for PF1000 at 35 kV, unexpectedly it was found that as L_o was reduced from 100nH in steps, in the region around 35nH I_pinch achieved a limiting value; in the sense that as L_o was reduced further towards 5 nH, whilst I_peak continued to increase to above 4MA, I_pinch dropped slightly from its maximum value of 1.05 MA to just below 1 MA. This Pinch Current Limitation Effect could have considerable impact on the future development of the plasma focus. No longer should one think in terms of I_peak, instead it is always I_pinchthat matters. Increasing I_peak does not necessarily mean increasing I_pinch.

On numerical experiments to enhance experience & intuition: Moreover the relationship between I_peak and I_pinch is implicit in the coupling of the equations of circuit and motion within the code which is then able to handle all the subtle interplay of static and dynamic inductances and dynamic resistances and the rapid changes in distributions of various forms of energies within the system. Whilst the intuitive feel of the experienced focus exponents are stretched to the limit trying to figure out isolated or integrated features of these interplays, the simplicity of the underlying physics is captured by the code which then produces in each shot what the results should be; and over a series of shots then reveal the correct trends; provided of course the series is well planned.

So the code may also be useful to provide the numerical experimenter time-compressed experience in plasma focus behaviour; enhanced experience at much reduced time. At the same time the numerical experimenter can in a day fire a number of different machines, without restrictions by time, geography or expense. The problem then becomes one of too much data; sometimes overwhelming the experience and intuition of the numerical experimenter.

On versatility: of the work you have carried out in these 8 sessions plus the additional experiments given below, one of which is practically completed for you to duplicate; the other is completely open for you to explore. Your numerical experiments have included

examining plasma focus behaviour comparing BIG, medium size and small plasma focus, looking for common and scalable parameters. You studied neutron and SXR yields as functions of pressure, comparing computed with experimental data. In 8 sessions involving some 20-30 hours of hands-on work you have ranged over a good sampling of plasma focus machines and plasma focus behaviour.

This was all done with one code the RADPF05.13.9b, the universal plasma focus laboratory facility. We should have the confidence that if we explore the open experiments suggested in 2. below, that could lead us to new areas and new ideas.

Additional Exercises:

1. As an additional exercise which you can look at later, you are provided with an additional file PF400Y_nComparison.xls. This file records data of measured Y_n (from Leopoldo Soto’s paper, also attached) and comparison with computed data using File7RADPF05.14.xls. You will see that the agreement between our computed data and Soto’s published data of neutron yield vs pressure may be considered to be good; features of comparison include the magnitude of the optimum yield, the optimum pressure and the fall-off on each side of the optimum pressure. You may wish to verify the comparison by running the numerical experiments yourself.

2. An open exercise: V Yu Nukulin & S N Polukhin recently (2007) published a paper (attached) discussing the saturation of neutron yield from megajoule PF facilities. Using an analytical method they surmised that in big plasma focus devices if storage energy is increased by essentially increasing storage capacitance C_o then I_peak reaches a limiting value of around 2MA. This is because as C_o increases, the current risetime increases and of necessity the anode length has to be increased. Thus the increased effective inductance on the circuit balances out the increase in C_o. In other words the effective circuit impedance does not go below a limiting value. Hence I_peak reaches the limiting value of 2MA. This thesis is easily tested using our code. Say, starting with the PF1000, keeping voltage and pressure the same, we could increase C_ostarting at say 600uF, increasing in steps of say 200 uF until 5000uF. We could add in other criteria such as keeping I/a approximately constant at some value such as 160kA/cm; ie we vary ‘a’ as C_o is increased; keeping c=b/a constant and r_o/Z₀ a constant where Z_o is the surge impedance (L_o/C_o)^0.5. Then not only can we keep track of I_peak(which Nukulin calls I_max) but more importantly we can keep track of I_pinch as C_o increases. We can then verify (or not) the saturation effect which they surmise (see postscript below). In a sense that already brings us to one frontier of plasma focus research, especially if we keep our minds open as we proceed.

Reference to this course and the Lee model code should be given as follows:

Lee S. Radiative Dense Plasma Focus Computation Package (2008) : RADPF www.plasmafocus.net

End Part 4-End of course- Internet Workshop on Plasma Focus Numerical Experiments

14^th April to 19^th May 2008

[Comments and interaction on the course work and other matters related to plasma focus are welcome at anytime]

Postscript: Numerical experiments at IPFS in connection with the open exercise has resulted in a paper on extending Plasma Focus scaling to multi-megajoule level. Also see list of papers

S Lee

Institute for Plasma Focus Studies, Melbourne- leesing@optusnet.com.au

This activity is carried out in association with AAAPT and

the Plasmas Groups of INTI-UC, Malaysia and NTU/NIE, Singapore