ap-3-11.dvi

Acta Polytechnica Vol. 51 No. 3/2011

Optimization of a Water Window Capillary Discharge Radiation Source

M. Stefanovič, M. Vrbová

Abstract

Computer modeling of a fast electrical discharge in a nitrogen-filled alumina capillary was performed in order to discover
discharge system parameters that lead to high radiation intensity in the so-called water window range of wavelengths
(2–4 nm). The modeling was performed by means of the two-dimensional RMHD code Z*. The time and spatial
distribution of plasma quantities were used for calculating the ion level populations and for estimating the absorption
of the 2.88 nm radiation line in the capillary plasma, using the FLYCHK code. Optimum discharge parameters for the
capillary discharge water window source are suggested. The heating of the electrodes and the role of capillary channel
shielding were analyzed according to the Z* code.

Keywords: water window radiation, capillary discharge, soft X rays, nitrogen plasma.

1 Introduction
Water window radiation sources are interesting for
application in the biomedical sciences [1]. High ab-
sorption of this radiation by proteins, but very little
absorption by oxygen, makes imaging of living cells
possible.
Wehave looked for optimal parameters of the fast

capillarydischarge radiation source, namely capillary
dimensions, electrodes’ shapes and gas filling pres-
sure, in order to achieve maximum radiation power
in the water window range of wavelengths.

2 RMHD computer modeling

Calculations of plasma quantities were executed us-
ing the two-dimensional RMHD (radiative-magneto-
hydrodynamic) Z*-code [2,3]. The physical model
established in Z* is based on the quasi-neutral mul-
ticharged ion plasma magneto-hydrodynamic equa-
tions with self-consistent electromagnetic fields and
the radiation transport in a 2D axially symmetric
geometry. This enables detailed engineering of the
pinching discharge processes in the capillary. The Z*
computer code engine provides the radial and time
evolutions of the plasma quantities, and also an es-
timation of the radiation generated in the capillary.
These results are based on the input parameters in-
serted into the calculations by the user. These input

Fig. 1: Electrical scheme of the capillary discharge circuit

parametersassume: external electrical circuit param-
eters, discharge geometry andproperties of the filling
gas (pressure, temperature, degree of pre-ionization,
etc.).

2.1 Code input

2.1.1 Electrical scheme of the discharge
system

The capillary discharge is a part of anunder dumped
RLC circuit (Figure 1). In accordance with the ex-
perimental setup [4], the following values are taken
intoaccount for the simulations: capacitor C =15nF
charged to initial voltage U0 =80 kV, external resis-
tivity and inductance of the circuit R = 0.73 Ω and
L = 50 nH, respectively. The capillary itself has its
own impedance, which is incorporated into the code
via the capillary geometry input.

2.1.2 Capillary discharge geometry and
filling gas

Alumina (Al2O3) capillary (magnetic permeabil-
ity μ/μ0 = 8.5, atomic weight A = 102) filled with
nitrogen gas (Z = 7, A = 14) was investigated. Iron
(Z =26, A =55.8) electrodes were presumed.
Figure 2 shows the geometric details of the cap-

illary grid used for the calculations. The capillary is
presented in 2D cylindrical geometry. The left bor-
der of the drawing represents the axis of symmetry.
The outer radius of the capillary is 21mm, while the
inner radius r is varied (0.8, 1.65 and 2.5 mm). The
length of the capillary l represents the distance be-
tween the two electrodes. The simulationswere done
in a range of capillary lengths 40–180mm. The cap-
illary is closed at one end and opened at the other
end. The open end electrode is indented ∼ 0.6 mm

Acta Polytechnica Vol. 51 No. 3/2011

into the channel. The voltage supply electrodes are
connected to the bottom and top right corner of the
grid. Positive voltage is virtually applied to the elec-
trode at the bottom.

Fig. 2: Capillary grid in 2D cylindrical geometry

Thenodesof the grid (Figure2) correspondto the
space coordinatesatwhich thephysical quantities are
calculated. The grid is denser inside the plasma and
around the plasma-electrodes and plasma-capillary
boundaries. The thicknesses of the cathode and an-
ode were chosen to be 9 and 6 mm, respectively.
Nitrogen gas was chosen due to strong spectral

lines in the wavelength region between 2 to 3 nm.
Also, a recent paper [5] confirmed nitrogen as a high
water window emitter. The initial temperatures of
the gas and the capillarywall are presumed ∼ 300K.
A pre-ionized plasma channel with “effective pre-
ionization degree” Z ∼ 10−3 and electron temper-
ature ∼ 0.05 eV is presumed to be created in the gas
cylinder near the alumina wall. Calculations were
made for various initial molecular gas pressures in
the range 13–667Pa.

2.2 Results of modeling

2.2.1 Spatial and time distributions of
plasma quantities

When a charged capacitor is connected to the capil-
lary channel, the electric currentheats the plasma in-
side the channel, and itsmagnetic field causesplasma
pinching. High temperature and dense plasma are
created during the pinch. The plasma quantities

depend substantially on the parameters of the dis-
charge system, namely on the voltage applied, the
discharge geometry and the filling gas. Figure 3 il-
lustrates the distribution of the plasma electron den-
sity at time t = 85 ns, shortly after the pinch.
The electron density is uniform along the z direc-
tion, and decreasesmonotonously in the radial direc-
tion. On the axis, the value of the electron density
is around 1017 cm−3. The electron temperature is
found to be homogeneous in a broad capillary vol-
ume (0 cm < r < 1 mm) having a value of around
Te ∼ 50 eV (Figure 4).

Fig. 3: Radial and longitudinal distribution of electron
density Ne at time t = 85 ns (the units in the figure
are given in Av; density expressed in cm−3 may be ob-
tainedbymultiplying the valueby theAvogadro constant
N a =6.02 × 1023)

Fig. 4: Radial and longitudinal distribution of electron
temperature T e at time t =85 ns

Acta Polytechnica Vol. 51 No. 3/2011

2.2.2 Output radiation estimated by the Z*
engine

The code provides information on the output power
of the radiation generated in the capillary, divided
into 20 different spectral groups. As we are inter-
ested in radiation in the water window region, we
observed the output emission power in group 13 of
the Z* code, corresponding to wavelength interval
2.1 nm < λ < 3.1 nm. We observed the evolution
of the capillary plasma for the filling molecular gas
pressure pm = 80 Pa. Figure 5 shows that the gen-
erated overall radiation (in thewhole spectral range)
Prad has two local maxima. The first radiation peak
is reached at ∼ 15 ns, and the second peak, which
is much higher, appears at t ∼ 85 ns. The second
peak is related to the pinch. The radiation in spec-
tral group 13 (Peuv) has only one maximum at the
pinch time. Its peak value is only 3 times lower than
the peak value of total radiation Prad. The radiation
peaks come later than the current maximum.

Fig. 5: Time dependences of evaluated electric current
(dashed line), total radiation Prad and radiation Peuv in
the selected group 13

Fig. 6: Time dependencies of radiation power Peuv in
group 13 for various initial gas pressures

Also, simulations for various gas initial pres-
sures in the range of 13–667 Pa were executed for
a 1.65mm capillary radius, in order to find the opti-
mum filling gas pressure. The radiation outputs are

very sensitive to changes in the initial gas pressure,
as shown in Figure 6. The highest output radiation
value is achieved for initial gas pressure 80 Pa at
time 85 ns. This is the optimum filling pressure of
the gas for capillary radius r = 1.65 mm. A capil-
lary l = 95 mm in length is considered, but similar
results in terms of optimumpressure are obtained for
all capillary lengths 40–180 mm.
The same procedure for finding the optimum gas

pressure was repeated for two other capillary radii
(0.8 and 2.5mm). The results are shown in Figure 7.
The line in Figure 7 shows the optimum pressure
for the corresponding capillary radius. The optimum
pressure is amonotonouslydecreasing function of the
radius of the capillary.
If we now compare the optimized pressure output

radiation powers for these three capillary radii, we
can see that the highest emission is achieved for cap-
illary radius 0.8 mm, for filling gas pressure 400 Pa
(Figure 8).

Fig. 7: Dependence of optimal pressure on the radius of
the capillary

Fig. 8: Pressure optimized time dependences of radiation
power Peuv in group 13 for different capillary radii

3 Spectrum according to
FLYCHK

The one-dimensional FLYCHK code [6] was used to
estimate the spectral properties of the output radia-
tion from the capillary. We considered the evolution

Acta Polytechnica Vol. 51 No. 3/2011

of the plasma field quantities at the capillary centre
(r = 0 cm, z = l/2, where r and z are radial and
longitudinal coordinates, respectively, and l is cap-
illary length). Similar spectral properties of plasma
are also expected in the cylinder along the capillary
axis 0.3 mm in radius, because of the uniformity of
the plasmaquantities in this region (seeFigures 3, 4).

Fig. 9: Time evolutions of electron density and tempera-
ture calculated by Z*, for gas pressure 80 Pa

Fig. 10: Time dependences of relative nitrogen ion abun-
dances for initial gas pressure 80 Pa (logarithmic scale)

The time evolutions of electron density Ne and
electron temperatureTewere taken from the Z* code
calculations (Figure 9), andwere introducedas an in-
put file into the FLYCHK code.
Non LTE plasma was presumed, and the simu-

lations were executed in the time dynamic regime,
because the electron density is not high enough to
fulfill theMcWhirter criterion [7] for the LTEplasma
model. Optically thick plasma 5 cm in length is pre-
sumed. The resulting evolutions of the relative abun-
dances of nitrogen ions for initial nitrogen pressure
80 Pa are shown in Figure 10 (only 4+, 5+ and 6+

ion fractions are shown, since other fractions have
minor abundances). It is evident that He–like N5+

ions prevail along the current period only after time
t ∼ 50 ns.
Instantaneous spectra at time t = 85 ns for

λ = 2.1.–3.1 nm are shown in Figure 11. We see
that the strongest spectral line lying at the wave-
length λ = 2.8785 nm, corresponding to the quan-

tum transition 1s2–1s2p in helium-like ions, prevails
over the other lines. The optical depth (log(Io/I))
of a 5 cm plasma column for different transitions is
depicted in Figure 12. We see that there is huge ab-
sorption of the 2.88 nm spectral line in the plasma,
i.e. plasmawith such characteristics is almost totally
nontransparent for this radiation. This conclusion is
confirmedby applying the Elton formula to calculate
the absorption coefficient [8].

k = Nl
gu
gl

·
λ3

8πc
(
Δλ
λ

) Aul
Where k is the absorption coefficient, gu and gl are
statistical weights for the upper and lower atomic
level, respectively, Nl is the number of ions on the
ground level, λ is the radiation wavelength, Δλ is
the line width due to Doppler broadening, and c is
speed of light. Presuming Δλ/λ = 1.45 × 10−4 [8],
we calculated the value of the absorption coefficient
k = 110 cm−1, which after multiplying by 5 cm
length gives almost the same result as FLYCHK.

Fig. 11: Spectra for filling gas pressure 80 Pa at the time
of maximum output radiation t =85 ns

Fig. 12: Optical depth of the 5 cmplasma channel for gas
pressure 80 Pa at the time of maximum output radiation
t =85 ns

4 Electrode heating

AZ*engineprovides informationon the temperature
rise of electrodes dTel, from the beginning of the sim-
ulation. We investigated theheating of the electrodes

Acta Polytechnica Vol. 51 No. 3/2011

during the discharge. Heating of the electrodes could
be important when the source is working in a high
repetition regime.
Weconsidered the capillarydischargewith the tip

of one electrode protruded into the capillary chan-
nel (as in the experimental setup from reference [4]
(Figure 13)), and compared it with the system with
electrodes lying in the extension of the capillarywall
and open at both ends (schematically presented in
Figure 14). The protruded top is overheated accord-

Fig. 13: Temperature increase of the electrodes for the
discharge system with the tip drawn into the channel,
after 300 ns from the beginning of the discharge; most
heated parts are magnified

Fig. 14: Temperature increase of the electrodes for the
discharge systemwith electrodes in the linewith the cap-
illary wall 300 ns after the beginning of the discharge

ing to the Z* code. On the other hand, the heating
is two times lower for the discharge geometry in Fi-
gure 14. This leads to the conclusion that the tip of
the electrode in the first scheme has an adverse effect
on the heating of the electrodes.

5 The role of capillary
shielding

In capillarydischarge systems, thedielectric capillary
is often surrounded by another metal in order to re-
duce the overall inductance in the discharge circuit.
Namely, the smaller the inductance, the faster and
higher the current pulse through the capillary will
be. Weperformed simulations to investigatehowdis-
charge current and output radiation depend on the
shielding of the capillary. For this purpose, special
geometry of the capillary was inserted as an input
into the Z* code (see Figure 15). The shield is 7 mm
from the radial center of the capillary.

Fig. 15: Schematic diagram of a shielded capillary, rep-
resented in a cylindrically symmetrical geometry

The time dependences of the output radiation
from the shielded and unshielded capillary, and also
the corresponding discharge currents, are shown in
Figure 16. Themaximumoutput radiation in the Z*
13 radiation group is approximately 15 % higher for
the shielded capillary. At the same time, the current
slope is steeper and the maximum is higher.
We also calculated the distribution of the z-

component of the electric field in the discharge sys-

Acta Polytechnica Vol. 51 No. 3/2011

Fig. 16: Time dependences of the output radiation power
in group 13 and the current for a shielded capillary (full
line) and for an unshielded capillary (dashed line)

Fig. 17: Distribution of the z-component of the electric
field in the discharge for the proposed geometry of the
discharge (the axes are not proportional)

Fig. 18: Distribution of the z-component of the electric
field in the discharge for the proposed geometry of the
discharge (the axes are not proportional)

tem at time 15 ns (the highest electric fields occur
at this time, according to the Z*-code) for a shielded
and unshielded capillary 4 cm in length (the radial
component of the electric field is negligible in com-
parison with the z component). The electrical field
reaches a value of around 45 kV/cm in a shielded
capillary (Figure 17), and only 6 kV/cm in an un-
shielded capillary (Figure 18). Since electrical break-
down in air occurs for an electric field of 30 kV/cm,
we conclude that electrodes must be isolated (either
by sinking into dielectric oil or by separation using
an insulator (i.e. alumina)) in a configurationwith a
shielded capillary.

6 Suggested parameters of
the new source

We propose the parameters of a capillary discharge
water window radiation source on the basis of the
computer modeling described above. The electrical
parameters of the discharge may be as follows: ca-
pacitor C = 15, initial voltage U0 = 80 kV, exter-
nal resistivity R = 0.73 Ω, and external inductance
L =50 nH.
Capillary radius 0.8 mm is most suitable for a

source with gas pressure of 400 Pa. The capillary
channel length should be only 4 cm, due to the
high absorption of thewaterwindow radiation in the
plasma column. Due to the high electric fields in a
system with a shielded capillary, we suggest that an
unshielded capillary be used as the source. The sug-
gested discharge geometry is shown in Figure 18.
The time dependences of the current and radia-

tion power in the Z* 13 radiation group for the pro-
posed system are shown in Figure 19. The highest
XUV emission occurs approximately at the time of
the current maximum, which is around 50 ns.
The electron density and temperature distribu-

tions at the time of the emission peak are shown
in Figures 20 and 21. The electron density is
uniform along the z direction, while it decreases
monotonously in the radial direction. The electron
temperature is homogeneous in abroad capillaryvol-
ume, with a value around 70 eV, suitable for creating
N5+ ions.

Fig. 19: Time dependences of the output radiation power
in group 13 and the current for the proposed geometry

Acta Polytechnica Vol. 51 No. 3/2011

Fig. 20: Radial and longitudinal distribution of electron
density Ne at the time of maximum current (the units in
the figure are given in Av; the density expressed in cm−3

canbeobtainedbymultiplying thevalueby theAvogadro
constant N a =6.02 × 1023)

Fig. 21: Radial and longitudinal distribution of electron
temperature Te at the time of a current maximum

7 Conclusions
The RHMD Z* code was used to model a capillary
pinchingdischarge toget an incoherentandpolychro-
matic “water window” radiation source. The radia-
tion outputs in group 13 (2.1–3.1 nm) were evalu-
ated for different initial gas pressure, different cap-
illary radii and different capillary lengths. The op-
timal radius of the system was proposed (0.8 mm),
with filling gas pressure around 400 Pa and capillary
length 40 mm. Using the FLYCHK code as a post-
processor, the detailed kinetics of nitrogen ions were
computed and the relative abundances of nitrogen
ions were evaluated. After 50 ns, N5+ ions prevail
over the other ions throughout the current period.
FLYCHKsimulations showedveryhighabsorptionof

the 2.88 nm radiation line in the plasma. Electrode
heating was investigated for two different discharge
configurations. The role of capillary shielding was
analyzed by Z*.

Acknowledgement

The authors gratefuly acknowledge professor Sergey
Zakharov for his useful advice on the Z* code and on
pinching discharges. This researchwas supported by
MEYSResearchProjectC42andbyMEYSResearch
Project INGO LA 08024.

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Ing. Miloš Stefanovič
Prof. Miroslava Vrbová, CSc.
E-mail: stefanovic@fbmi.cvut.cz
Department of Natural Sciences
Faculty of Biomedical Engineering
Czech Technical University in Prague
nám. Sítná 3105, Kladno 2, Czech Republic