Pressure dependence of argon cluster size for different nozzle geometries

Abstract

We experimentally study Rayleigh scattering from a cluster jet produced by high pressure argon gas expanding into vacuum through four different nozzles (a supersonic slit nozzle, a slit nozzle, a conical nozzle, and a sonic nozzle). The scattering signal intensity and the scattering image are recorded by photomultiplier tube and charge-coupled device camera, respectively. Based on the scattering image, the atom density in the gas flow is estimated. This allows for the comparison of the dependence of average cluster size on argon gas backing pressure between the nozzles. The experimental results show that the planar expansion developed from the supersonic slit and the slit nozzles exhibits the higher atom density than the axisymmetric expansion from the conical and the sonic nozzles. The slit nozzle is shown to have the highest pressure dependence of average cluster size. It is found that the supersonic slit nozzle is more favorable to the large clusters than the slit nozzle under the backing pressure of up to 50 bars, though it has the lower pressure dependence of average cluster size.

Full text

Page 1

Pressure dependence of argon cluster size for different nozzle geometries
Guanglong Chen,1Byunghoon Kim,1Byungnam Ahn,1,2and Dong Eon Kim1,a?
1Department of Physics, Pohang University of Science and Technology (POTECH), Pohang 790-784,
Republic of Korea
2Vacuum Measurement Technology, Pohang 790-320, Republic of Korea
?Received 4 April 2009; accepted 22 July 2009; published online 2 September 2009?
We experimentally study Rayleigh scattering from a cluster jet produced by high pressure argon gas
expanding into vacuum through four different nozzles ?a supersonic slit nozzle, a slit nozzle, a
conical nozzle, and a sonic nozzle?. The scattering signal intensity and the scattering image are
recorded by photomultiplier tube and charge-coupled device camera, respectively. Based on the
scattering image, the atom density in the gas flow is estimated. This allows for the comparison of
the dependence of average cluster size on argon gas backing pressure between the nozzles. The
experimental results show that the planar expansion developed from the supersonic slit and the slit
nozzles exhibits the higher atom density than the axisymmetric expansion from the conical and the
sonic nozzles. The slit nozzle is shown to have the highest pressure dependence of average cluster
size. It is found that the supersonic slit nozzle is more favorable to the large clusters than the slit
nozzle under the backing pressure of up to 50 bars, though it has the lower pressure dependence of
average cluster size. © 2009 American Institute of Physics. ?doi:10.1063/1.3204974?
I. INTRODUCTION
Atomic clusters, a state of matter intermediate between
molecules and solids, have been studied for many years since
Becker et al.1first reported the cluster formation. For more
than a decade, the interaction of intense femtosecond laser
pulses with clusters has been an active area of research, and
important findings have been reported: the generation of x
rays, energetic ions and electrons,2–4and the intense femto-
second laser-driven nuclear fusion.5To understand the laser-
cluster interaction, the size information of these weakly
bound van der Waals clusters is required. The atomic clusters
are usually formed in the adiabatic expansion of high pres-
sure gas into vacuum through a nozzle. Hence the gas prop-
erties, source conditions, and the nozzle geometry affect the
cluster formation and thus the properties of the cluster jet.
However, the difficulty to describe the cluster formation has
prevented a quantitative theory from being developed.6Ex-
perimentally, many methods have been developed to estimate
the cluster size.6–12As a feasible approach, Rayleigh scatter-
ing measurement has been widely used.10–15From Rayleigh
scattering measurement, however, the determination of abso-
lute cluster size is not practical.11Traditionally, for the axi-
symmetric expansion of the rare gas, Hagena’s scaling law
Nc?????2.35between the average atomic cluster size Nc?the
number of atoms per cluster? and the Hagena’s empirical
parameter ???=Kdeq
guideline for the average cluster size, where K is a constant
related to the gas property, P0the gas backing pressure in
mbar, T0the gas temperature in kelvin before expansion, and
deq
theequivalentdiameter
=0.74d/tan??? for a conical nozzle, d is the throat diameter
of nozzle, and ? is the expansion half angle?.6,16–18There
have been many works that, using Rayleigh scattering from
0.85P0/T0
2.29? has served primarily as a
ofnozzlein
?m
?deq
the cluster jets, studied the pressure dependence of the scat-
tering signal intensity SRSfor the axisymmetric expansion
?SRS?P0
the pressure dependence of the average cluster size is Nc
?P0
condensed into clusters. The results were roughly in agree-
ment with the pressure dependence obtained from Hagena
scaling law and confirmed that the scaling law could predict
the pressure dependence of cluster size for the axisymmetric
expansion.
However, so far, the experimental study on the scaling of
SRSor Ncwith P0for the planar expansion is scarce. Very
recently, DeArmond et al.15reported their measurement of
Rayleigh scattering from argon clusters formed in a planar
expansion through a 15 cm long slit nozzle and found that
the scattering signal exhibits the higher pressure dependence
for the planar expansion than for the axisymmetric expan-
sion.
In this paper, Rayleigh scattering method is employed to
investigate the argon cluster formation from four nozzles
with different geometries, i.e., the supersonic slit nozzle, the
slit nozzle, the conical nozzle, and the sonic nozzle ?a pho-
tomultiplier tube ?PMT? and a charge-coupled device ?CCD?
camera are used to detect the scattering signal and image the
scattered light, respectively?. Due to difficulties in the deter-
mination of absolute cluster size only from Rayleigh scatter-
ing measurement, this work assesses the pressure depen-
dence of the relative cluster size for these nozzles. The
spatial profiles of Rayleigh scattering signal from CCD im-
age are used to estimate the cross section of gas flow and
thus the atom density. It is shown that although the slit
nozzle, corresponding to the planar expansion, exhibits the
highest dependence of average cluster size on the gas back-
ing pressure, the supersonic slit nozzle is more helpful to
form large clusters under usual cluster experimental condi-
tions.
2.6–3.6?.8–13These experimental results revealed that
1.6–2.6under the assumption that all of the atoms are
a?Electronic mail: kimd@postech.ac.kr.
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II. DEPENDENCE OF CLUSTER SIZE
The Rayleigh scattering cross section for a spherical
cluster is given by
?4?i2− 1
? ?r6
i2+ 2?,
?1?
where r is the cluster radius, ? the wavelength of the laser,
and i the refractive index of the cluster medium.10Since the
average cluster size Ncis proportional to r3for the spherical
cluster, ??Nc
SRS? nc? ? ncNc
2, Rayleigh scattering signal is then given by
2,
?2?
where ncis the number of clusters in the scattering region.
Under the assumption that all the atoms are condensed into
clusters in the gas expansion, the number of clusters in the
scattering region can be expressed as
nc? nl/Nc,
?3?
for a given laser beam, where n is the atom number density
in the scattering region and l is the dimension of the scatter-
ing region along the laser beam. Thus, Rayleigh scattering
signal can be rewritten as
SRS? nlNc.
Using Eq. ?4? and the experimental results about the pressure
dependence of Rayleigh scattering signal SRS??P0??, the re-
lation between the average cluster size Ncand the gas back-
ing pressure P0can be expressed as
Nc? SRS/nl ? ?P0??/nl.
?4?
?5?
For a stable gas flow, at a distance of a few nozzle-orifice
diameters downstream, the flow velocity v reaches its final
value ??/??−1??1/2?2kT0/m?1/2, where ? is the specific heat
ratio, k is the Boltzmann constant, and m is the atomic
mass.18,19From the continuity equation,
n0v0A0= nvA,
where n0, v0=?2kT0/m?1/2, and A0are the atom number den-
sity, the flow velocity, and the area of the gas flow at the
nozzle throat, respectively, and n, v, and A are the respective
values in the scattering region;18the average cluster size for
a given backing pressure P0can be expressed as
Nc? SRSA/?n0A0l?.
Because n0is proportional to P0and A0is the same ?A0
=?d2/4? for the different nozzles, i.e., the area of the 0.5
mm diameter orifice in the pulsed valve in our work, the
average cluster size can also be given by
?6?
?7?
Nc? SRSA/?P0l?.
?8?
III. EXPERIMENTS
The experimental setup is shown in Fig. 1?a?. A pulsed
valve ?Parker series 99? with a 0.5 mm diameter orifice is
used. The 0.5 mm diameter orifice shown in Fig. 1?b? is itself
used as a sonic nozzle. The other nozzles shown in Fig. 1?b?
are directly placed on the top of the 0.5 mm diameter orifice.
The slit width of the slit nozzle and the supersonic slit nozzle
is 0.5 mm. The supersonic slit, the slit, and the conical
nozzles have the same dimension for nozzle exit, D. The half
opening angle for the supersonic slit and the conical nozzles
is 24.2°. In our experiment, the repetition rate of the pulsed
valve is set to be 1/12 Hz, and the pressure of 3
?10−5Torr is maintained before the operation of the pulsed
valve using turbo molecular pumps ?Balzers TPH330?. Ar-
gon clusters are produced in the adiabatic expansion of high
pressure argon gas through one of these nozzles into vacuum.
The He–Ne laser ?UNIPHASE 1125, 632.8 nm and 10 mW?
beam is focused by a lens with a 20 cm focal length and
passes through the argon gas flow perpendicularly at 2.5 mm
above from the nozzle exit, as shown in Fig. 1?a?. The laser
beam diameter in the gas jet is estimated to be less than
1 mm. Care is taken to reduce background lights from the
vacuum chamber ?59?53?30 cm3? walls as much as pos-
sible. For the slit nozzle and the supersonic slit nozzle, the
laser beam propagates along the slit direction. A 2 in. diam-
eter lens with a 7.5 cm focal length is mounted about 19 cm
away from the laser beam and is used to collect and image
the 90° Rayleigh scattering light onto a head-on type PMT
??1 kV is applied?. The axis of the lens is mutually perpen-
dicular to the laser beam and the gas jet. The output signal
from the PMT is recorded by a 1 GHz bandwidth digital
oscilloscope ?Teketronix TDS5104?. Meanwhile, a CCD
camera located at the opposite side of the PMT is used to
image the 90° Rayleigh scattering light from the cluster jet.
Note that the collection solid angle subtended by the optical
system described above is large enough to cover all the scat-
tering regions for different nozzles, and one optical setup is
used for different nozzles.
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FIG. 1. ?Color online? Schematic diagram of ?a? experimental setup and ?b?
nozzle geometries ?d=0.5 mm, D=5.0 mm, L=5.0 mm, and the sonic
nozzle is made of stainless steel and the others are made of brass?. The
upper and the lower figures in ?b? are the side cross-sectional view and the
top view of the nozzles, respectively.
053507-2 Chen et al.J. Appl. Phys. 106, 053507 ?2009?
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Page 3

IV. RESULTS AND DISCUSSION
First, to adjust the valve opening time and guarantee the
steady state gas flow, which is critical to the cluster forma-
tion, the time-resolved Rayleigh scattering measurement at a
gas backing pressure of 50 bars was performed for the su-
personic slit nozzle. Figure 2 is the scattering signal recorded
by the PMT for different valve opening times, showing that
Rayleigh scattering signal reaches the steady state and has
the flat top profiles when the pulsed valve opening time is
longer than 3 ms. These data imply that both gas flow and
cluster formation thus reach the steady state when the valve
opening time is not less than 3 ms under our experimental
conditions. It was also true for other nozzles. All the experi-
ment data were obtained at the valve opening time of 3 ms,
unless otherwise mentioned, and were quite reproducible.
The dependence of the Rayleigh scattering signal SRS
and the average cluster size Ncon the gas backing pressure
for different nozzles is shown in Fig. 3. Every data point was
obtained by averaging 40 gas pulses. Figure 3?a? indicates
that the scattering signal SRSfor the supersonic slit and the
slit nozzles is much higher than that for the conical and the
sonic nozzles. Furthermore, for the planar gas expansion, the
supersonic slit nozzle demonstrates the higher scattering sig-
nal than the slit nozzle, and for the axisymmetric gas expan-
sion, the scattering signal for the supersonic nozzle ?conical
nozzle? is higher than that for the sonic nozzle. By fitting
SRS??P0??to the scattering signal, we obtain ? of
3.27?0.17, 3.76?0.16, 3.24?0.19, and 3.43?0.12 for the
supersonic slit, the slit, the conical, and the sonic nozzles,
respectively. The power ? for the conical and the sonic
nozzles ?corresponding to the axisymmetric expansion? are
close to the previous measurements.10–15Furthermore, it is
found that the pressure dependence of 3.76 for the slit nozzle
is higher than that for other nozzles. This indicates that for
the planar expansion developed through the slit nozzle, the
pressure dependence of scattering signal is different from
that for the axisymmetric expansion. This is in agreement
with the result by DeArmond,15where the pressure depen-
dence between a 15 cm long slit nozzle and a sonic nozzle
was compared. However, it is interesting to note that for the
supersonic slit nozzle, the pressure dependence of scattering
signal is lower than that for the slit nozzle and close to that
for the conical nozzle. We have also investigated the pressure
dependence of Rayleigh scattering signal at the different dis-
tances ?from 1.5 to 3.5 mm? downstream from the nozzle
exit and found that the change in ? at the different distances
is within ?4.5%.
To compare the average size of clusters produced from
the different nozzles, the information about the dimension l
and the area A of the scattering region is required by Eq. ?8?.
The CCD image from the scattered light was used to esti-
mate the scattering dimension l and the scattering area A of
the gas flow where the laser beam passed through. Figure 4
shows the typical CCD images at 2.5 mm away from the
nozzle exit for different nozzles at a backing pressure of 50
bars. For the supersonic slit and the slit nozzles, to estimate
the scattering area, the transverse image of the scattered light
was also recorded after the slit direction was set to be per-
pendicular to the laser beam. Figures 4?b? and 4?d? are the
scattering images of the gas flow along the slit direction,
while Figs. 4?e? and 4?f? are the transverse images perpen-
-202468 1012
0.0
0.1
0.2
0.3
0.4
Trigger
Scattering signal (arb.u.)
Time (ms)
5 ms
4 ms
3 ms
2 ms
1 ms
FIG. 2. ?Color online? Time-resolved Rayleigh scattering signal from PMT
at a backing pressure of 50 bars ?argon gas? for the supersonic slit nozzle for
different valve opening times.
10
0.01
0.1
1
10
100
Rayleigh scattering signal (arb.u.)
Gas backing pressure (bars)
sonic
conical
slit
supersonic slit
010 2030 4050
0
2
4
6
8
10
12
14
16
18
20
22
Relative average cluster size
_atom number per cluster(arb.u)
Gas backing pressure (bars)
sonic
conical
slit
supersonic slit
(b)
(a)
FIG. 3. ?Color online? Pressure dependence of ?a? Rayleigh scattering signal
and ?b? the average cluster size for different nozzles. The solid lines repre-
sent the fit to the data.
053507-3 Chen et al. J. Appl. Phys. 106, 053507 ?2009?
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Page 4

dicular to the slit direction for the supersonic slit and the slit
nozzles, respectively. It can be seen that for the slit or the
supersonic slit nozzle, the dimension of gas flow perpendicu-
lar to the slit is much narrower than that along the slit. Thus
we can think that the gas flow from the slit or the supersonic
slit nozzle behaves like a planar expansion. The white curves
in Fig. 4 show the profiles along the laser beam at the center
of the image, i.e., the spatial distribution of the scattered
light. It is found that these nozzles can produce the roughly
uniform profile except for the sonic nozzle, and the gas ex-
pansion through the conical or the supersonic slit nozzle
forms the wider gas flow, i.e., the gas flow with the larger
dimension.
We define the dimension of the scattering region in the
gas flow as the length of scattering region where the scat-
tered light intensity is higher than 3% of the maximum in-
tensity. This definition is based on the fact that the dimension
of the scattering region estimated from the CCD image
should be close to the slit length D ?5.0 mm? due to the zero
expansion angle for the slit nozzle. Considering the differ-
ence in the scattered light intensity for different nozzles, we
first normalized the spatial intensity profile of the scattered
light. The dimension l was then taken at the 3% level in the
normalized profile and estimated to be 7.3, 5.0, 7.0, and 5.2
mm for the supersonic slit, the slit, the conical, and the sonic
nozzles, respectively. For the axisymmetric expansion ?the
sonic and the conical nozzles?, the area of the scattering re-
gion A can be obtained immediately using ?l2/4. For the
planar expansion ?the supersonic slit and the slit nozzles?, the
required transverse dimensions of the scattering region are
estimated to be about 2.4 mm from the transverse images.
Then we estimate the scattering area A at a backing pressure
of 50 bars to be 17.1, 12.0, 38.0, and 20.8 mm2for the
supersonic slit, the slit, the conical, and the sonic nozzles,
respectively. Based on Eq. ?6? and the estimated areas A, the
ratios of the corresponding atom number density in the scat-
teringregionamongthese
2.2:3.2:1.0:1.8 at a backing pressure of 50 bars for the super-
sonic slit, the slit, the conical, and the sonic nozzles, respec-
tively. Clearly, the gas flows from the supersonic slit and the
slit nozzles have higher atom density due to the slit geom-
etry. It is interesting to note that for the sonic nozzle, atom
density is higher than for the conical nozzle in our work. The
reason is discussed below. For the sonic nozzle, the scatter-
ing region is at 2.5 mm above from the nozzle exit, i.e., 2.5
mm above the orifice in the pulsed valve. While for the coni-
cal nozzle, although the scattering region is also at 2.5 mm
above from the conical nozzle exit, it is actually 7.5 mm
above the orifice in the pulsed valve due to the nozzle length
L ?Fig. 1?b??. Thus it is not surprising that the conical nozzle
corresponds to lower atom density than the sonic nozzle.
Note that the atom number density ratio between the slit and
the sonic nozzles is much lower than that calculated by the
equation in Table I in Ref. 18. This is understood because the
slit configuration ?the area of the gas flow at the nozzle throat
A0=dD? in Ref. 18 is different from our slit nozzle configu-
ration ?A0is ?d2/4?.
After estimating the area of the scattering region A and
the dimension of the scattering region l, we use Rayleigh
scattering signal SRSto calculate the relative average cluster
sizes for different nozzles at a backing pressure of 50 bars
using Eq. ?8?. If we use the A and l at the 50 bars pressure for
the lower backing pressures, the relative cluster sizes at the
lower backing pressures can be obtained, as shown in Fig.
3?b?. By fitting these relative cluster sizes to the pressure
power dependence of Nc??P0?B, B was estimated to be
2.16?0.11, 2.73?0.10, 2.23?0.13, and 2.45?0.10 ?only
including the uncertainty in Rayleigh scattering signal? for
the supersonic slit, the slit, the conical, and the sonic nozzles,
respectively. Note that for the axisymmetric gas expansion,
the conical nozzle corresponds to lower pressure dependence
than the sonic nozzle. As in the pressure dependence of Ray-
leigh scattering signal, the pressure dependence of average
cluster size is the highest for the slit nozzle. Moreover the
pressure dependence of average cluster size for the super-
sonic slit nozzle is close to that for the conical nozzle. Al-
though at present it is not easy to explain this result due to
the lack of theory, we think that the reason for the fact that
nozzlesare givenby
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FIG. 4. CCD images of the scattered light for ?a? the sonic, ?b? the slit
?along the slit direction?, ?c? the conical, and ?d? the supersonic slit ?along
the slit direction? nozzles at an argon gas backing pressure of 50 bars. ?e?
and ?f? are the transverse images ?perpendicular to the slit direction? for the
slit and the supersonic slit nozzles, respectively. The laser beam direction is
shown as the arrow. Due to the weaker scattering signal for sonic and coni-
cal nozzles, images ?a? and ?c? were obtained at the valve opening time of 5
ms. ?Note that the scales of signal intensity for the different nozzles are
different.?
053507-4 Chen et al. J. Appl. Phys. 106, 053507 ?2009?
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Page 5

the pressure dependence for the supersonic slit nozzle is
lower than that for the slit nozzle could result from different
expansion angle. Unlike the slit nozzle, which has a zero
opening expansion angle ??=0 in Fig. 1?b??, the opening
expansion angle of supersonic slit nozzle can make the gas
expansion along the y direction. This can be seen from Fig.
4, where the length of scattering region ?y direction? is
longer than the length of nozzle exit for the supersonic slit
nozzle, while the length of scattering region is close to the
length of nozzle exit for the slit nozzle. Thus the gas expan-
sion from the supersonic slit nozzle is different from that
from the slit nozzle ?mainly along x direction? but is similar
to a 2 dimensional expansion ?not only along x but also along
the y direction?.
Note that the high pressure dependence of cluster size
?large B? only means a fast increasing trend in the cluster
size with the backing pressure. As shown in Fig. 3?b?, al-
though the conical nozzle corresponds to the lower pressure
dependence, the relative average cluster size for the conical
nozzle is estimated to be 1.8 times larger than that for the
sonic nozzle at a backing pressure of 50 bars. This is in
agreement with the fact that the supersonic geometry of coni-
cal nozzle is helpful for the large cluster formation for the
axisymmetric gas expansion.6To further understand the re-
sult, we quantitatively compare the relative average cluster
size for the conical nozzle with that for the sonic nozzle
using Hagena scaling law.6From the equivalent diameter
model,18the equivalent diameter of the conical nozzle ?deq
=0.74d/tan???? is about 1.64 times larger than that of the
sonic nozzle. Thus the average cluster size is about 2.6 times
larger than that for the sonic nozzle based on Hagena scaling
law Nc?????B?B=2.23 for our experimental results?. This
result is roughly in agreement with our estimated value ?1.8?
from the experiment data. The deviation could result from
the assumption in the equivalent diameter model that the
streamline of the flow field is straight. Based on this model,
the dimension of the scattering region at 2.5 mm downstream
from the nozzle exit for the conical nozzle is about 7.3 mm,
while the estimated dimension from the CCD image is 7.0
mm. The other possible reason is that, as discussed above,
the distances between the scattering region and the orifice in
the pulsed valve are different for the sonic and the conical
nozzles, while the average cluster size could be related to the
distance.12
Figure 3?b? shows that for given gas source conditions,
the average cluster size for the supersonic slit nozzle is the
largest in our work. The average cluster size is about 1.5
times larger than that produced from the slit nozzle at back-
ing pressure of 50 bars. Specifically, if, as practiced by
others,13–15we take the cluster size Ncto be 100 atoms per
cluster when the scattering signal is first detectable ?the sig-
nal was first detected at 5 bars for the supersonic slit nozzle
in our case?, the average cluster size Ncat backing pressure
of 50 bars is estimated to be about 14 000 ?r=5.4 nm? at-
oms per cluster using the pressure dependence of ?P0?2.16.
However for the slit nozzle, the average cluster size Ncat a
backing pressure of 50 bars is estimated to be 9600 atoms
?r=4.7 nm? based on the relative cluster size shown in Fig.
3?b?. These results could indicate that similarly to the axi-
symmetric gas expansion, the supersonic geometry of the
supersonic slit nozzle is also helpful for the larger cluster
formation.
It is interesting to find from Fig. 3?b? that the average
cluster size produced from the slit nozzle is larger than that
from the conical nozzle at a higher gas backing pressure. For
example, the relative cluster size for the slit nozzle is larger
than that for the conical nozzle by a factor of 2.2 at a backing
pressure of 50 bars. The reason for this could result from the
higher atom density in the gas flow for the slit nozzle as
shown above. In our work, the conical nozzle has an exit
whose diameter ?5 mm? is equal to the slit length of the slit
nozzle and thus has a large half opening angle of 24.2°. If a
conical nozzle with a smaller opening angle is used, which
can produce the gas jet with the same atom density as that
from the present slit nozzle under a given backing pressure,
the average cluster size from the conical nozzle could be
expected to be larger than that from the slit nozzle due to
supersonic geometry of the conical nozzle as discussed
above.
V. CONCLUSIONS
Using the scattered light intensity measurement by a
PMT together with the scattered light image by a CCD, the
pressure dependence of relative average cluster size has been
investigated for an argon cluster jet developed from super-
sonic slit, slit, conical, and sonic nozzles. It is shown that the
planar expansion developed from the supersonic slit and the
slit nozzles exhibits the higher atom density than the axisym-
metric expansion, and the slit nozzle corresponds to the high-
est pressure dependence of average cluster size. For the su-
personic slit nozzle, unlike the slit nozzle, the pressure
dependence is close to that for a conical nozzle. As known in
axisymmetric gas expansion for the conical nozzle, it is
found that although the supersonic slit nozzle has the lower
pressure dependence than the slit nozzle, its supersonic ge-
ometry is still helpful in the case of a slit geometry for the
formation of large clusters under the usual experimental con-
ditions. The detailed comparison of average cluster size ?not
the pressure dependence of cluster size? between a slit nozzle
and a conical nozzle with different half opening angles
would be the interesting subjects for further work.
ACKNOWLEDGMENTS
This work has been supported by BK21 project, Basic
Research Program ?Grant No. K12F-2008-313-C00356?
funded by Korean Research Foundation, and Global Re-
search Laboratory Program of the National Research Foun-
dation.
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