ap-4-10.dvi


Acta Polytechnica Vol. 50 No. 4/2010

Capillary Discharge Parameter Assessment for X-ray Laser Pumping

J. Hübner

Abstract

This paper assigns optimum capillary discharge characteristics with respect to reaching the maximum emission gain on
wavelength l = 18.2 nm and corresponding to Balmer α transition H-like carbon. The computer modelling of the capil-
lary discharge evolution is carried out using the NPINCH programme, using a one-dimensional physical model based on
MHD equations. The information about the capillary discharge evolution is processed in FLY, FLYPAPER, FLYSPEC
programmes, enabling the population to be modelled on specific levels during capillary discharge.

Keywords: capillary discharge, X-ray laser, MHD (magnetohydrodynamic) simulation, population inversion, gain optimal-
ization, carbon.

1 Introduction
Non-stationaryplasmaof a fast capillary electrical dis-
charge was studied as a potential active medium for a
soft X-ray laser. The aim was to achieve an optimum
set of characteristics for lasing at 18.2 nm wavelength
H-like carbon C5+ in an alumina capillary. Optimum
in the sense of reachingamaximumgainvalue for each
set of parameters.
Capillary pinch dynamics is determined by many

selected parameters: capillary geometry (radius and
capillary length), the substance of the capillary, the
initial filling pressure, and the electric current time
dependance. This dependance, in particular, is given
by an electric circuit which is joined to the capillary.
A capillary discharge Z-pinch acting as a medium

for a soft X-ray laser uses ASE, the “Amplified Spon-
taneous Emission” effect, and electron-collisional re-
combination pumping. The main variable for ASE is
the gain, and the most important goal of this paper is
to find specific parameters of a capillary to obtain the
maximum gain.
Electron-collisional recombination, sometimes re-

ferred to as “three body” recombination, is the in-
verse process to electron-collisional ionization fromex-
cited levels. The combined recombination and radia-
tive downward cascading process is illustrated by the
equation

X(i+1)+o +2e → X
i+
n + e → X

i+
u + e (1)

It is assumed that the ions exist plentifully in
the initial state, which is one stage of ionization
higher than that of the lasing ions. Note that this is
not a self-containedpopulation-replenishment scheme.
Hence, replenishment depends upon further reioniza-
tion, which is provided in our case by the release of
the next current pulse.
It was assumed that ablation from inner walls has

only an imperceptible effect, and so it was not in-

cluded in the calculations. The capillary discharge
plasma quantities were calculated by means of the
NPINCH [2] code under the one-dimensional, two-
temperature, one-fluidMHDapproximation. The out-
put data was then processed by means of the FLY
code, which enables the creation of a history file of the
population on the individual levels along the capillary
axis. On the basis of the populations history, the gain
and the gain history can be calculated.

2 Optimization

Optimization itself involves searching the maximum
in the four-dimensional space of the parameters. Two
parameters relate to the current-impulse shape, the
third parameter is initial density (or pressure), and
the fourth is the radius of the capillary.
The right choice in the range of parameters is

of fundamental importance. The results of our
study should be instrumental for experimenters in the
CAPEX-U [4] project. The range of parameters there-
fore has to be related to their experimental equipment,
with the view to verifying the results. Consequently,
the choice of the first parameter was obvious because
an alumina capillary r = 1.6 mm in radius is used
in CAPEX-U. The choice of the current pulse had to
coincide with the source in CAPEX-U. The electric
current in a capillary was fitted from measuring by
function.

If it = I0 sin

(
πt

2t0

)
exp

(
−

t

t1

)
(2)

The chosen parameters are therefore associated
with equation 2: current peak value Imax and
dI/dt|t=0. Two cases were calculated Imax ≈ 30 kA
and Imax ≈ 60 kA, on behalf of dI/dt|t=0 the area
within the range of value 0.5 ·1012–2.5 ·1012 A/s was
chosen.

52



Acta Polytechnica Vol. 50 No. 4/2010

Table 1: Parameters of pulse shape for Imax =30 kA

dI/dt|t=0 Imax tmax I0 t0 t1
(A/s) (kA) (ns) (kA) (ns) (ns)

0.750 · 1012 30.023 67.6 33.900 71 570
0.923 · 1012 30.031 54 33.082 56.3 570
1.000 · 1012 30.003 49.5 32.786 51.5 570
1.250 · 1012 30.049 39.4 32.229 40.5 570
1.500 · 1012 30.015 32.5 31.799 33.3 570

Table 2: Parameters of pulse shape for Imax =60 kA

dI/dt|t=0 Imax tmax I0 t0 t1
A/s (kA) (ns) (kA) (ns) (ns)

0.923 ·1012 60.056 115.15 74.213 126.3 570
1.000 ·1012 60.029 105.05 72.765 114.3 570
1.250 ·1012 60.006 82.020 69.630 87.5 570
1.500 ·1012 60.044 67.878 67.8 71 570
1.750 ·1012 60.027 57.57 66.51 59.7 570
2.000 ·1012 60.002 50 65.27 51.5 570
2.250 ·1012 60.005 43.63 64.88 45.3 570
2.500 ·1012 60.095 39.59 64.45 40.5 570

The last parameter is initial density N, accord-
ing to the theoretical analysis and in order to achieve
the gain maximum, the maximum electron den-
sity for carbon has to be Ne ≈ 3 · 1020. To
accomplish this requirement, we chose the range
N =0.5 · 1017–5 ·1017 cm−3.

3 Gain evaluation
The selected parameters for each case served as input
data for NPINCH. A very useful output of the calcu-
lation was the dens.dat file, which contained a time
history of electron and ion density, and electron tem-
perature. This file futher served as input for the FLY
and the FLYPAPER code, and by means of these we
obtained the time histories of populations on set lev-
els N3 for level n = 3 and N2 for level n = 2 H-like
carbon ions. Inversion function F [3] is represented by
Equation 3, where the statistical weights gu/gl are for
our transition 9/4, and Nl, Nu presents the lower and
upper laser state population densities.

F =1−
(
2.25

N1
Nu

)
(3)

The cross-section with corresponding coefficients
for our transition λul = 13.3818 · 10−7 cm, Aul =
5.72 · 1010 s−1 can be illustrated as

σstim =
λ3

8πcΔλ/λ
Aul =

1.62 · 104 · (18.2178)3 · 5.72 ·1010

8 · π ·2.997925 · 1010 ·
√
2π

= (4)

2.966136 · 10−15cm2

And finally, G after cross-section substitution can
be illustrated as

G =2.966136 · 10−15NuF cm−1 (5)

For each calculation, we listed the parameter data,
information about the maximum value and the time
of reaching gain, the pinch time (the time when the
maximum of electron density Ne is reached).

4 Imax =30 kA

The results for Imax = 30 kA, are in Fig. 1. The
maximum gain value was achieved at dI/dt|t=0 =
1 ·1012 A/s, initial carbon density 1.4 ·1017 cm−3 and
Gmax ≈ 0.05 cm−1. The calculated data resembled
the Gauss function, so they were fitted by this func-
tion, and the deviation between the calculated data
and the fitted function was minimal. The most infor-
mation is shown in Fig. 2. The best gainwas achieved
when the time of reaching currentmaximumwas equal
to the pinch time.

Fig. 1: Maximum Gmax against initial density for various
current derivations at Imax =30 kA. The data is fitted by
the Gauss function

Fig. 2: Comparison of the achieving time ofmaximumgain
TG, and current TI in dependence on dI/dt|t=0

5 Imax = 60 kA

For the casewhere Imax =60 kA is chosen, the results
are shown in thenextfigure. Themaximumgainvalue

53



Acta Polytechnica Vol. 50 No. 4/2010

Gmax was reachedatdI/dt|t=0 =2.25·1012 A/s, initial
carbon density 4.1 · 1017 cm−3 and Gmax ≈ 3.4 cm−1.
This data too was fitted by the Gauss function.

Fig. 3: Maximum Gmax against initial density for various
current derivations at Imax =60 kA. The data is fitted by
the Gauss function

Fig. 4: Comparison of achieving time ofmaximumgain TG
and current TI in dependence on dI/dt|t=0

6 Example
Now we can take a closer look at a particular case
to show the history of certain variables. The param-
eters of a capillary discharge N0 = 4.0 · 1017 cm−3,
dI/dt|t=0 = 2 · 1012 A/s, r0 = 1.6 mm and Imax =
60 kAhave been chosen. We will plot the curve for an
individual quantity, such as electron density, electron
temperature, inversion of population, and gain.

Fig. 5: Behaviour of electron density for parameters N0 =
4.0·1017 cm−3, dI/dt|t=0 =2·1012 A/s, r0 =1.6 mmand,
Imax =60 kA

Fig. 6: Behaviour of electron temperature for parameters
N0 =4.0·1017 cm−3, dI/dt|t=0 =2·1012 A/s, r0 =1.6mm,
and Imax =60 kA

Fig. 7: Behaviour of the population on levels n = 2
a n = 3 for parameters N0 =4.0 · 1017 cm−3, dI/dt|t=0 =
2 ·1012 A/s, r0 =1.6 mm, and Imax =60 kA

Fig. 8: Behaviour of inversion function F and gain (cm−1).
In addition, the figure contains a marked pinch time tp =
44nsand time of reaching currentmaximum tImax =55 for
parameters N0 =4.0 ·1017 cm−3, dI/dt|t=0 =2 ·1012 A/s,
r0 =1.6 mm and Imax =60 kA

Figures 6 and 7 demonstrate the behaviour of elec-
tron density and temperature. They show pinch time
tp =44 ns, Nemax =3.8 ·1020 cm−3, Temax =110 eV.
After that, a rapid drop in these quantities followsdue
to adiabatic expansion. The electron density drops by
nearly two orders in 5 ns and the electron temperature
drops by nearly 90 eV. These are ideal conditions for
creating population inversion at the desired levels.

54



Acta Polytechnica Vol. 50 No. 4/2010

Figure 8 shows that at time 46.5 ns, 2.5 ns after
pinch time, both the inversion function and gain had
an above zero value. At time 47.2 ns, the gain maxi-
mum Gmax =3.2 cm

−1 is reached, and just about one
ns later the inversion function also reaches its max-
imum and starts to fall steadily. However, the gain
begins to drop very rapidly, due to rapid density de-
cay of the whole mass (both electrons and ions, which
are lasing) along to axix. It follows from the gain be-
haviour that the hypothetical laser pulse could be 2 ns
in length.
Howdoes the initial density influence the pinchbe-

haviour? With increasing initial density themaximum
density also increases. At the same time, the pich time
is delayed. The effect is totally contrary to the elec-
tron temperature. As the initial density increases the
maximum electron temperature value decreases.

7 Conclusion
The optimal conditions for achieving maximum gain
for initial density and electric current pulse shape
have been obtained for two different current maxima,
Imax = 30 kA, Imax = 60 kA and capillary radius
r = 1.6 mm. Figures 2 and 4 show clearly that the
time of achieving the currentmaximum and the pinch
time have to be synchronized for optimal current dif-
ferentiation at zero time, so that both times will be
almost identical.

Gmax = 0.05 cm
−1 was achieved at N0 = 1.4 ·

1017 cm−3, dI/dt|t=0 =1 ·1012 A/s, r0 =1.6 mm and
Imax =30 kA.

Gmax = 3.4 cm
−1 was achieved at N0 = 4.1 ·

1017 cm−3, dI/dt|t=0 = 2.25 · 1012 A/s, r0 = 1.6 mm
and Imax =60 kA. The assumed pulse length is 2 ns.

Acknowledgement

The research described in this paper was supervised
by Ing. Pavel Vrba, CSc., IPP AS CR in Prague, and
was supportedby theCzechGrantAgencyundergrant
No. 202/08/057.

References

[1] Rocca, J. J.: Table-top soft x-ray lasers,Review of
Scientific Instruments, 1999, vol. 70, no. 10.

[2] Razinkova, T. L., Sasorov, P. V.: Program
NPINCH na počítání dynamiky z-pinče, Moskva,
1998.

[3] Elton, R. C.: X-Ray Laser, Academic Press, New
York, 1990.

[4] Koláček, K. et al.: 10th ICXRL2006, Berlin, Ger-
many (poster – 14 09 Kolacek.pdf).

Jakub Hübner
E-mail: bukajus@centrum.cz
Dept. of Physical Electronics
Faculty of Nuclear Science and Physical Engineering
Czech Technical University
Břehová 7, 115 19 Prague 1, Czech Republic

55