**IPFS**

*knowledge** should
be freely accessible to all*

**Institute for Plasma Focus Studies**

**Internet Workshop on Numerical Plasma Focus Experiments**

**(Supplementary Notes
for Sessions 7-8)**

[in part extracted from file 2Theory.pdf from:** **http://www.intimal.edu.my/school/fas/UFLF/
] and from various papers

**Radiation Terms**

The Bremsstrahlung loss term may be
written as:

_{}

Recombination loss term is written as:

_{}

The line loss term is written as:

_{}

where dQ/dt is the total power
gain/loss

of the plasma column

By this coupling, if, for example, the radiation loss _{} is severe, this would
lead to a large value of _{} inwards. In the extreme case, this leads to radiation
collapse, with r_{p} going rapidly to zero,
or to such small values that the plasma becomes opaque to the outgoing
radiation, thus stopping the radiation loss.

This radiation collapse occurs at a critical current of 1.6 MA (the
Pease-Braginski current) for deuterium. For gases such as Neon or Argon, because of
intense line radiation, the critical current is reduced to even below 100kA,
depending on the plasma temperature.

**Plasma Self
Absorption and transition from volumetric emission to surface emission**

Plasma self absorption and volumetric (emission described above) to
surface emission of the pinch column have been implemented in the following
manner.

The photonic excitation number (see File 3 Appendix by N A D Khattak) is written as
follows:

M = 1.66 x 10 ^{-15}r_{p} Z_{n}^{ 0.5}
n_{i} / (Z T^{1.5}) with T in eV,
rest in SI units

The volumetric plasma self-absorption correction factor A is
obtained in the following manner:

A_{1} = (1 + 10^{-14}n_{i}
Z) / (T^{ 3.5}))

A_{2} = 1 / AB_{1}

A = A_{2}^{ (1 + M)}

^{ }

Transition from volumetric to surface emission occurs when the
absorption correction factor goes from 1 (no absorption) down to 1/e (e=2.718)
when the emission becomes surface-like given by the expression:

_{}

where the constant *const*
is taken as 4.62x10^{-16 }to conform with numerical experimental
observations that this value enables the smoothest transition, in general, in
terms of power values from volumetric to surface emission.

Where necessary another fine adjustment is made at the transition
point adjusting the constant so that the surface emission power becomes the
same value as the absorption corrected volumetric emission power at the
transition point. Beyond the transition point (with A
less than 1/e) radiation emission power is taken to be the surface emission
power.

**Neutron Yield**

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

Adapted from the following papers (with modifications for erratum)

**Pinch
current limitation effect in plasma focus **(This version includes an Erratum)

**S. Lee
and S. H. Saw, Appl. Phys. Lett. 92, 021503 (2008),
DOI:10.1063/1.2827579**

*Copyright
(2008) American Institute of Physics. This article may be downloaded for
personal use only. Any other use requires prior permission of the author and the
American Institute of Physics.* *This
article appeared in (citation above) and may be found at *

http://link.aip.org/link/?APPLAB/92/021503/1

**Neutron
Scaling Laws from Numerical Experiments **(This
version includes an Erratum)

**S Lee and S H Saw, J of Fusion Energy, DOI:
10.1007/s10894-008-9132-7**

published first
online 20 February 2008 at** http://dx.doi.org/10.1007/s10894-008-9132-7 **

**"The original publication is available at
www.springerlink.com."**

Neutron yield is calculated with two components, thermonuclear term
and beam-target term.

The thermonuclear term is taken as:

dY_{th} = 0.5n_{i}^{2}(
3.142)r_{p}^{2}z_{f}<sv>(time interval)

Where <sv> is the thermalised fusion cross section-velocity product
corresponding to the plasma temperature, for the time interval under
consideration. The yield Y_{th} is
obtained by summing up over all intervals during the focus pinch.

The beam-target term is derived using the following phenomenological
beam-target neutron generating mechanism^{17}, incorporated in the
present RADPFV5.13. A beam of fast deuteron ions is produced by diode action in
a thin layer close to the anode, with plasma disruptions generating the
necessary high voltages. The beam interacts with the hot dense plasma of the
focus pinch column to produce the fusion neutrons. In this modeling each factor
contributing to the yield is estimated as a proportional quantity and the yield
is obtained as an expression with proportionality constant. The yield is then
calibrated against a known experimental point.

The beam-target yield is written in
the form: Y_{b-t }~n_{b}
n_{i} (r_{p}^{2}z_{p}) (s v_{b}) t

where n_{b} is the
number of beam ions per unit plasma volume, n_{i }is the ion density, r_{p}
is the radius of the plasma pinch with length z_{p}, s the cross-section of the
D-D fusion reaction, n- branch^{18}, v_{b} the beam ion speed
and t is the beam-target interaction time assumed proportional to the
confinement time of the plasma column.

Total beam energy is estimated^{17}
as proportional to L_{p}I_{pinch}^{2}, a measure of the
pinch inductance energy, L_{p} being the focus pinch inductance. Thus
the number of beam ions is N_{b}~L_{p}I_{pinch}^{2}/v_{b}^{2}
and n_{b} is N_{b} divided by the focus pinch volume. Note that
L_{p}~ln(b/r_{p})z_{p} , that^{4} t~r_{p}~z_{p}
, and that v_{b}~U^{1/2} where U is the disruption-caused diode
voltage^{17}. Here b is the cathode radius. We also assume reasonably
that U is proportional to V_{max}, the maximum voltage induced by the
current sheet collapsing radially towards the axis.

Hence we derive: **Y _{b-t}= C_{n} n_{i }I_{pinch}^{2}z_{p}^{2}((lnb/r_{p}))**

_{ }where I_{pinch} is
the current flowing through the pinch at start of the slow compression phase; r_{p}
and z_{p} are the pinch dimensions at end of that phase. Here C_{n}
is a constant which in practice we will calibrate with an experimental point.^{}

The D-D cross-section is highly
sensitive to the beam energy so it is necessary to use the appropriate range of
beam energy to compute s. The code computes V_{max}
of the order of 20-50 kV. However it is known^{17}, from experiments
that the ion energy responsible for the beam-target neutrons is in the range
50-150keV^{17}, and for smaller lower-voltage machines the relevant
energy ^{19} could be lower at 30-60keV. Thus to align with
experimental observations the D-D cross section s is reasonably obtained by
using beam energy equal to 3 times V_{max}.

A plot of experimentally measured
neutron yield Y_{n} vs I_{pinch} was made combining all
available experimental data^{2,4,12,13,17,19-22}. This gave a fit of Y_{n}=9x10^{10}I_{pinch}^{3.8
}for I_{pinch} in the range 0.1-1MA. From this plot a calibration
point was chosen at 0.5MA, Y_{n}=7x10^{9} neutrons. The model
code^{23} RADPFV5.13 was thus calibrated to compute Y_{b-t}
which in our model is the same as Y_{n}.

Notes
on **The total current and I _{peak}, the plasma current and I_{pinch}**

Extracted From: **Computing Plasma Focus
Pinch Current from Total Current Measurement **

**S. Lee, S. H. Saw, P. C. K. Lee, R. S. Rawat
and H. Schmidt, Appl Phys Letters 92, 111501 (2008)
DOI:10.1063/1.2899632**

The
total current I_{total}
waveform in a plasma focus discharge is easily measured using a Rogowski coil. The peak value I_{peak} of
this trace is commonly taken as a measure of the drive efficacy and is often
used to scale the yield performance of the plasma focus. This
is despite the fact that yields should more consistently be scaled to focus
pinch current I_{pinch},
since it is I_{pinch} which
directly powers the emission processes. The reason many researchers use I_{peak}
instead of I_{pinch} for
scaling is simply that while I_{peak} is
easily measured, I_{pinch},
which is the value of the plasma sheath current Ip* *at
time of pinch, is very difficult to measure even in large devices where it is
possible to place magnetic probes near the pinch. This
measurement is also inaccurate and perturbs the pinch. In a small device, there
is no space for such a measurement.

The
relationship between _{I}_{pinch}_{ }and I_{peak} is
not simple and has only been recently elaborated. It
primarily depends on the value of the static inductance L_{0}
compared to the dynamic inductances of the plasma focus. As L_{0} is
reduced, the ratio I_{pinch}_{ }/ I_{peak}
drops. Thus, yield laws scaled to I_{peak} will
not consistently apply when comparing two devices with all parameters equal but
differing significantly in L_{0}.
Better consistency is achieved when yield laws are scaled to I_{pinch}. In
this paper, we propose a numerical method to
consistently

**Distinguishing
the **I_{total}
waveform from the I_{p}**
waveform**

A
measured trace of I_{total}_{ }is
commonly obtained with a Rogowski coil wrapped around
the plasma focus flange through which is fed I_{total} discharged from the
capacitor bank between the coaxial electrodes across the back wall. A part of I_{total},
being the plasma sheath current I_{p}, lifts
off the back-wall insulator and drives a shock wave axially down the coaxial
space. We denote f_{c} as
the current fraction I_{p}/_{I}_{total} for
the axial phase and f_{cr} for
the radial phases. In modeling it is found that a reasonable value for initial
trial for f_{c} is
0.7 with a similar first trial value for f_{cr}
. However in a DPF78 experiment f_{c}_{ }was
found to vary from 0 at the start of the axial phase rising rapidly above 0.6
for the rest of the axial phase. In the radial phase f_{cr} was found to stay above
0.6 before dropping to 0.48 at the start of the pinch and then towards 0.4 as
the pinch phase progressed. These _{total}_{ }and I_{p}.

The
performance of a plasma focus is closely linked to the current I_{pinch} actually
participating in the focus pinch phase rather than the total current flowing in
the circuit. It is a common practice to take I_{peak}_{ }or
some representative fraction of it as I_{pinch}. Another practice is to
take the value of I_{total}_{ }at the
time of the pinch as I_{pinch}
Whilst in their special cases this practice could be justifiable, the
distinction of I_{p}_{ }from I_{total}_{ }should
generally be clearly made. We emphasize that it should be the value of I_{p} at
the time of pinch which is the relevant value for the purpose of yield scaling.
The practice of associating yield scaling with the total current waveform (i.e.
taking I_{peak} or I_{total}_{ }at
estimated pinch time) would be justifiable if there were a linear relationship
between the waveforms of I_{total} and I_{p}.
However as shown by the Stuttgart experiments the actual relationship is a very
complex one which we may ascribe to the interplay of the various
electro-dynamical processes including the relative values of static inductance L_{o}, tube inductance and the
dynamic resistances which depend on the tube geometry and plasma sheath speeds.
This relationship may change from one machine to the next. Whilst these
electro-dynamical processes and other relevant ones such as radiation are
amenable to modeling there are other machine effects such as back wall restriking (for example due to high induced voltages during
the pinch phase) which can almost unpredictably affect the relationship between
I_{total} and I_{p}
during the crucial radial phases. Hence it is not only simplistic to discuss
scaling in terms of the I_{total}
waveform (i.e. taking Ipeak or
the value of Itotal at
the estimated time of pinch) but also inconsistent. One of the most important
features of a plasma focus is its neutron production. The well-known neutron
yield scaling, with respect to current, based on various compilations of
experimental data, is Y_{n} ∼ I_{pinch}^{x} where
x is varied in the range 35. In a recent
paper , numerical experiments using a code was used to
derive a scaling with x = 4.7. Difficulties in the interpretation of
experimental data ranging across big and small plasma focus devices include the
assignment of the representative neutron yield Y_{n} for
any specific machine and the assignment of the value of I_{pinch}. In a few larger machines attempts
were made to measure I_{pinch} using
magnetic probes placed near the pinch region, with uncertainties of 20%.
Moreover the probes would have affected the pinching processes. In most other
cases related to yield scaling data compilation or interpretation I_{pinch} is simply assigned a value based on
the measurement of peak total current I_{peak}_{ }or the
value of total current at the observed current dip.

The
difficulties in distinguishing Ipinch from I_{total}_{ }are obviated in numerical
experiments using the Lee Model [In a typical simulation, the I_{total} trace
is computed and fitted to a measured I_{total} trace
from the particular focus. Three model parameters for fitting are used: axial
mass swept-up factor f_{m},
current factor fc and radial mass factor f_{mr}. A fourth model parameter, radial current factor, f_{cr} may also be used. When correctly fitted the
computed Itotal trace
agrees with the measured I trace in peak amplitude, rising
slope profile and topping profile which characterize the axial phase
electro-dynamics. The radial

phase characteristics
are reflected in the roll-over of the current trace from the flattened top region,
and the subsequent current drop or dip. Any machine effects, such as restrikes, current sheath leakage and consequential
incomplete mass swept up, not included in the simulation physics is taken care
of by the final choice of the model parameters, which are fine-tuned in the
feature-by-feature comparison of the computed I_{total} trace with the measured I_{total}
trace. Then there is confidence that the computed gross dynamics, temperature,
density, radiation, plasma sheath currents, pinch current and neutron yield may
also be realistically compared with experimental values.

A note on scaling:

Scaling of yields to
say I_{pinch} should be carried out using
yields which are at optimum, or at least near optimum. If one indiscriminately
uses any data one may end up with completely trivial or misleading results. For
example if a point is used at too high or low pressure (away from the optimum
pressure) then there may be zero yield ascribed to values of I_{pinch}.

^{1} Lee S 1984 *Radiations in
Plasmas *ed B McNamara (World
Scientific) pp 97887

Also: S. Lee in *Laser and Plasma
Technology*, edited by S. Lee, B. C. Tan, C. S.

Wong, & A. C. Chew.
World Scientific,

^{ }

^{2}S. Lee, T. Y. Tou, S. P. Moo, M. A. Elissa, A.
V. Gholap, K. H. Kwek, S.

Mulyodrono, A. J. Smith, Suryadi,
W. Usala, & M. Zakaullah,
Am. J.

Phys. **56**, 62 (1988).

^{ }

^{3}T. Y. Tou, S. Lee, & K. H. Kwek,
IEEE Trans. Plasma Sci. **17**, 311 (1989).

^{ }

^{4}S. Lee and A. Serban, IEEE Trans. Plasma Sci. **24**, 1101 (1996).

^{ }

^{5}D. E. Potter, Phys. Fluids **14**, 1911 (1971).

^{ }

^{6}M. H. Liu, X. P. Feng, S. V. Springham, and S.
Lee, IEEE Trans. Plasma

Sci. **26**, 135(1998).

^{ }

^{7}S. Lee, P. Lee, G.
Zhang, X. Feng, V. A. Gribkov,
M. Liu, A. Serban, and

T. Wong, IEEE
Trans. Plasma Sci. **26**, 1119 (1998).

^{ }

^{8}S. Bing, Plasma
dynamics and x-ray emission of the plasma focus, Ph.D. thesis, NIE, (2000) in
ICTP Open Access Archive: http://eprints.ictp.it/99/

^{ }

^{9}S. Lee, in http://ckplee.myplace.nie.edu.sg/plasmaphysics/ (2000 & 2007).

^{ }

^{10}S. Lee in ICTP
Open Access Archive: http://eprints.ictp.it/85/ (2005).

^{ }

^{11}V. Siahpoush, M. A. Tafreshi, S. Sobhanian, and S. Khorram, Plasma

Phys. Controlled
Fusion **47**, 1065 (2005).

^{ }

^{12}S. Lee, Twelve
Years of UNU/ICTP PFF-A Review (1998) IC, 98 (231);

A.Salam ICTP, Miramare,

^{ }

^{13}L. Soto, P. Silva,
J. Moreno, G. Silvester, M. Zambra,
C. Pavez, L. Altamirano, H.
Bruzzone, M. Barbaglia, Y. Sidelnikov, and W. Kies, Braz. J.
Phys. **34**, 1814 (2004).

^{ }

^{14}H. Acuna, F. Castillo, J. Herrera, and A. Postal,
International Conference

on Plasma Sci, 35 June 1996 (unpublished), p. 127.

^{ }

^{15}C. **89**, 15 (2006).

^{ }

^{16}D. Wong, P. Lee,
T. Zhang, A. Patran, T. L. Tan, R. S. Rawat, and S. Lee,

Plasma Sources
Sci. Technol. **16**, 116 (2007).

^{ }

^{17}V A Gribkov, A Banaszak,
B Bienkowska, A V Dubrovsky,
I Ivanova-Stanik,

L Jakubowski, L Karpinski, R A Miklaszewski, M Paduch, M J Sadowski, M Scholz,

A. Szydlowski, and K. Tomaszewski,
J. Phys. D **40**, 3592 (2007).

^{ }

^{18}J. D. Huba, 2006 Plasma Formulary, p. 44. http://wwwppd.nrl.navy.mil/nrlformulary/NRL_FORMULARY_07.pdf

^{ }

^{19}S. V. Springham, S. Lee, and M. S. Rafique,
Plasma Phys. Controlled

Fusion 42,
1023 (2000).

^{ }

^{20}W. Kies, in Laser and Plasma Technology, Proceedings of Second
Tropical

College, edited by
S. Lee, B. C. Tan, C. S. Wong, A. C. Chew, K. S. Low, H. Ahmad, and Y. H. Chen World
Scientific, Singapore, (1988), pp. 86137.

^{ }

^{21}H. Herold, in Laser and Plasma Technology, Proceedings of
Third Tropical

College, edited by C. S.
Wong, S. Lee, B. C. Tan, A. C. Chew, K. S.

Low, and S. P. Moo
World Scientific,

^{ }

^{22}A. Patran, R. S. Rawat, J. M. Koh, S. V. Springham, T. L. Tan,
P. Lee, & S. Lee, 31st EPS Conf on Plasma Phys London, (2004), Vol. 286, p.
4.213.

^{ }

^{23}S Lee **Radiative**** Dense Plasma
Focus Computation Package (2008) **RADPF: http://www.intimal.edu.my/school/fas/UFLF/

**Reference to this course and the Lee model code
should be given according to the following format:**

**Lee S.
Radiative Dense Plasma Focus Computation
Package (2008) : RADPF www.plasmafocus.net www.intimal.edu.my/school/fas/UFLF/ **