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Institute for Plasma Focus Studies
Internet Workshop on Numerical Plasma Focus Experiments
Description of Radiative Dense Plasma Focus Computation Package RADPFV5.13.9 - S Lee Model
(Supplementary Notes for Modules 2 & 3)
· Numerical Experimental Facility
· Simulates any Mathers-type plasma focus, computes dynamics
· Design new plasma focus machines
· Thermodynamics included; 4 gases: H2, D2, Ne, Ar, Xe and He
· Model parameters to fit experimental axial, radial phase times
· Radiative phase computes line radiation, recombination and total yield. Computes neutron yield for deuterium operation; based on an improved beam-target model, calibrated at an experimental point.
Plasma Self-absorption based on revised equations presented in File 3; appendix by N A D Khattak.
Time guard feature
Choice of Tapered electrode
Quick choice of specified machines; one click loading of chosen machine; at present 3 machines may be click-loaded: the UNU/ICTP PFF, the NX2 and the PF1000
There are altogether 4 files in this package.
File1: PDF File "Description of Radiative Dense Plasma Focus Computation Package"
File2: PDF File "Theory of Radiative Plasma Focus Model"
File3: PDF file "Appendix by N A D Khattak".
File7: EXCEL file containing the ACTIVE SHEET AND THE PROGRAMME CODE. "Radiative Dense Plasma Focus Computation Code" RADPFV5.14
In addition, there are files for the computation of thermodynamic data needed for this code.
Hint for downloading the EXCEL FILE: Instead of left click to open the file; it is better to right click and select "save target as"; then choose a suitable location e.g. desktop. The saved EXCEL file will be only about 1M. (see last page for more hints on saving/copying )
These files may be downloaded from the following URL:
A simple 2 phase (axial
and radial) model was developed by S.Lee in 1983 as a component of a 3kJ plasma
focus experimental package which became known as the UNU/ICTP PFF. This network
of basically identical 3kJ PF machines, with different experimental and
application emphases, is now operated by groups in countries including
The model was written as
a 3 phase (non-radiative) model (in GWBASIC) for an experimental program at the
The present 5-phase package (axial, radial inward shock, radial reflected shock, slow compression radiative and expanded large column phase) is re-written in Microsoft EXCEL VISUAL BASIC in order to make it available for wider usage.
The model may be adapted to any conventional Mather-type plasma focus by input of machine parameters: inductance, capacitance, electrode radii and length. And operating parameters: charging voltage and fill gas pressure. The thermodynamics (specific heat ratio and charge number as functions of temperature) are included for 6 gases namely hydrogen, deuterium, neon, argon and helium and xenon. The gases may be selected by simply inputting atomic number, molecular weight and dissociation number (2 for deuterium and hydrogen, 1 for the others).
The model has been used in many PhD and Masters Theses. It has also been used for various applications, for example, in the design of a cascading plasma focus (1991); and for estimating soft x-ray yield for the purpose of developing a SXR source for microelectronics lithography (1997). More recently the code has been used to compute pinch current from measured total current waveform (2008). With this technique numerical experiments were run to obtain neutron scaling laws (2008). Use of the code also uncovered a Plasma Focus Pinch Current Limitation Effect (2008).
Five phases of the plasma focus are simulated by the Model code:
1 Axial Phase
2 Radial Inward Shock Phase
3 Radial Reflected Shock Phase
4 Slow Compression (Radiative) Phase
5 Expanded Column Axial Phase
The phases are illustrated by Fig 1 and Fig 2. More details may be obtained from:
Fig 1 (a) Axial Phase Fig 1 (b) Radial Phase
The five phases are summarised as follows (Theory and equations may be obtained from file 2 above):
1. Axial Phase: Described by a snowplow model with an equation of motion (incorporating axial phase model parameters: mass and current factors fm and fc) which is coupled to a circuit equation
2. Radial Inward Shock Phase (See Fig 1): Described by 4 coupled equations using an elongating slug model. The first equation computes the radial inward shock speed from the driving magnetic pressure. The second equation computes the axial elongation speed of the column. The third equation computes the speed of the current sheath, also called the magnetic piston, allowing the current sheath to separate from the shock front by applying an adiabatic approximation. The fourth is the circuit equation. The model parameters, radial phase mass and current factors fmr and fcr are incorporated in the radial phases. Thermodynamic effects due to ionization and excitation are incorporated into these equations, these effects being important for gases other than hydrogen and deuterium. Temperature and number densities are computed during this phase. A communication delay between shock front and current sheath due to the finite small disturbance speed is crucially implemented in this phase.
3. Radial Reflected Shock (RS) Phase: When the shock front hits the axis, because the focus plasma is collisional, a reflected shock develops which moves radially outwards, whilst the radial current sheath piston continues to move inwards. Four coupled equations are also used to describe this phase, these being for the reflected shock moving radially outwards, the piston moving radially inwards, the elongation of the annular column and the circuit. The same model parameters fmr and fcr are used as in the previous radial phase. The plasma temperature behind the reflected shock undergoes a jump by a factor nearly 2.
4. Slow Compression (Quiescent) or Pinch Phase: When the out-going reflected shock hits the in-going piston the compression enters a radiative phase in which for gases such as neon,. radiation emission may actually enhance the compression where we have included energy loss/gain terms from Joule heating and radiation losses into the piston equation of motion. Three coupled equations describe this phase; these being the piston radial motion equation, the pinch column elongation equation and the circuit equation, incorporating the same model parameters as in the previous two phases. Thermodynamic effects are incorporated into this phase. The duration of this slow compression phase is set as the time of transit of small disturbances across the the pinched plasma column. The computation of this phase is terminated at the end of this duration.
5. Expanded Column Phase: To simulate the current trace beyond this point we allow the column to suddenly attain the radius of the anode, and use the expanded column inductance for further integration. In this final phase the snow plow model is used, and two coupled equations are used similar to the axial phase above. This phase is not considered important as it occurs after the focus pinch.
[Note: Transition from Phase 4 to Phase 5 is observed to occur in an extremely short time. This is an important transition which merits efforts to include into the model. It would be an important next step]
Using the Code
Configuring the code
The code may be configured to any Mather-type plasma focus by inputting machine (bank and tube) parameters: inductance, capacitance, electrode radii and length; and operating parameters: charging voltage and fill gas pressure. The thermodynamics (specific heat ratio and charge number as functions of temperature) are included for 6 gases namely hydrogen, deuterium, neon, argon and helium and xenon. The gases may be selected by simply inputting atomic number, molecular weight and dissociation number (2 for deuterium and hydrogen, 1 for the others.
With the bank, tube and operating parameters specified; what remains is to specify the model parameters. As a first trial we may use: fm=0.08, fc=0.7, fmr=0.15, fcr=0.7.
Then we may run the code. The results are the following: waveforms for the total discharge current and tube voltage, axial phase trajectory and speed, radial trajectories for the shock front, current sheath and column length and the corresponding speeds, plasma temperature and radiation yields (Bremsstrahlung, line and recombination) and power; and thermodynamic quantities such as specific heat ratios and charge numbers. These are output in graphical as well as tabular forms. Also computed are plasma pinch current and neutron yield, and energy
distributions, if required.
Note: on the chronology of the development of the Lee model code
1983: 2-phase model developed and presented by S Lee at the Spring College on Radiations in Plasmas, ICTP Trieste published in “Radiations in Plasmas” B McNamara, World Scientific pp 978-87; used in the development of the UNU/ICTP PFF and UNU, ICTP training programs and Colleges (1984, 1986 to 1991); used in PhD theses (T.Y.Tou 1986, K.H.Kwek 1988, J Ali 1990, S Mulyodrono 1993, A Serban 1995)
1991: Extension to
3-phase model (S Lee IEEE Trans Plasma Sci 1991); used for experimental program at the 1991
1995: Implementation of finite small-disturbance speed correction in the radial shock phase first used in PhD thesis (Liu 1997). This is a major feature in the Lee model code. Before this physics was implemented, radial speeds were a factor of nearly 2 too high compared with experiments. Completion of 5-phase model; used in other PhD theses (G Zhang 1999, B Shan 2000).
2000 After discussions with P Lee with a view to wider usage, the code was completely re-written in 2000 in Excel Visual Basics. Used in several recent PhD’s since, notably (A Patran, D Wong, T Zhang) and in many papers. From 2003 onwards, plasma self-absorption and anode taper incorporated. Extension to Xenon.
2007 onwards: Intensive discussions with S H Saw (INTI UC) , P Lee (NTU/NIE), R S Rawat (NTU/NIE) and the AAAPT resulted in push to a new direction of applications of the code. Beam-target mechanism incorporated with realistic simulation of yield resulted in re-examination of neutron scaling laws. Plasma self-absorption and taper features completed. Technique to find Ipinch from measured Itotal waveform published. A new effect of focus pinch current limitation was uncovered. All these activities resulted in the formation of Institute for Plasma Focus Studies to encourage correct usage and innovative applications of the Lee model code. Further development of the code is continuously undertaken. List of papers.