*knowledge should be freely accessible to all*

**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 Sessions 1 & 2)**

**Features**

**·**** ****Numerical
Experimental Facility**

**·**** ****Simulates any
Mathers-type plasma focus, computes dynamics**

**·**** ****Design new plasma
focus machines**

**·**** ****Thermodynamics
included; 4 gases: H _{2}, D_{2}, 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. **

**Also includes:**

**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". **

**RADPF5.15dd.xls: 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:**

http://www.intimal.edu.my/school/fas/UFLF/

**Introductory description**

**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 Singapore, Malaysia,
Thailand, India, Pakistan,,
Egypt and **

**The model was written as
a 3 phase (non-radiative) model (in GWBASIC) for an experimental program at the
1991 **

**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:**

**http://www.intimal.edu.my/school/fas/UFLF/**

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 f_{m }and f_{c}) 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 f_{mr} and f_{cr}
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 f_{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 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: f _{m}=0.08, f_{c}=0.7,
f_{mr}=0.15, f_{cr}=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
I_{pinch} from measured I_{total} 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.