Download Far infrared scattering on plasma crystals - Ruhr

Far infrared scattering
on plasma crystals
Dissertation
zur Erlangung des Grades
eines Doktors der Naturwissenschaften
an der Fakultät für
Physik und Astronomie
der Ruhr-Universität Bochum
vorgelegt von
Jens Ränsch
Bochum 2009
1. Gutachter: Prof. Dr. J. Winter
2. Gutachter: Prof. Dr. H. Soltwisch
Tag der mündlichen Prüfung:
21.04.2009
Diese Arbeit widme ich meiner Familie. Meiner Mutter, die mich stets unterstützt
und mir Halt gibt. Meiner Frau Liudmila, die mir mit ihrer Liebe Kraft und Ausdauer schenkt. Meiner jüngst geborenen Tochter Anna, die so viel Freude in unser
Leben bringt.
Meinem Bruder Martin und meinem Vater – ich werde sie immer in meinem Herzen
tragen.
I devote this work to my family. To my mother, who always stands by me and keeps
me grounded. To my wife Liudmila, who gives me energy and patience through here
love. To my recently born daughter Anna, who brings so much pleasure to our lives.
To my brother Martin and to my father – I will always carry them in my heart.
Contents
1 Introduction
1
2 Theoretical background
5
2.1
Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.1
General principle of lasers . . . . . . . . . . . . . . . . . . .
5
2.1.2
The pump laser . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.3
The FIR laser . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Gaussian optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.3
Capacitively coupled plasma discharges . . . . . . . . . . . . . . . .
16
2.4
Complex plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4.1
OML theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4.2
Non-isotropic plasmas—streaming ions . . . . . . . . . . . .
24
2.4.3
Barrier in the effective potential . . . . . . . . . . . . . . . .
27
2.4.4
Ion-neutral collisions—angular momentum not conserved . .
28
2.4.5
“Closely packed” dust grains . . . . . . . . . . . . . . . . . .
30
2.4.6
Summary of OML theory and neglected effects . . . . . . . .
30
2.4.7
Forces acting on dust particles in a plasma . . . . . . . . . .
31
2.4.8
Producing plasma crystals . . . . . . . . . . . . . . . . . . .
39
Light scattering by small particles . . . . . . . . . . . . . . . . . . .
43
2.5.1
Some basics of scattering theory . . . . . . . . . . . . . . . .
44
2.5.2
Rayleigh scattering by one particle . . . . . . . . . . . . . .
47
2.5.3
Scattered radiant flux at the detector . . . . . . . . . . . . .
50
2.5
3 Setup
55
3.1
The laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.1.1
The CO2 laser . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.1.2
The FIR resonator . . . . . . . . . . . . . . . . . . . . . . .
60
3.1.3
The mirror system . . . . . . . . . . . . . . . . . . . . . . .
64
vii
viii
Contents
3.2
The scattering arrangement . . . . . . . . . . . . . . . . . . . . . .
67
3.3
The plasma chamber . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.4
Setup for calibration scattering experiments . . . . . . . . . . . . .
76
3.5
CCD camera diagnostics and video analysis . . . . . . . . . . . . .
77
4 Results
4.1
81
Properties of the laser system . . . . . . . . . . . . . . . . . . . . .
81
4.1.1
FIR laser operation and beam characteristics . . . . . . . . .
81
4.1.2
The FIR laser power . . . . . . . . . . . . . . . . . . . . . .
84
4.1.3
Characteristics of the beam splitter . . . . . . . . . . . . . .
86
4.2
Results of calibration scattering experiments . . . . . . . . . . . . .
88
4.3
The crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.3.1
Design optimisations for producing plasma crystals . . . . .
91
4.3.2
Extended plasma crystals . . . . . . . . . . . . . . . . . . .
94
4.3.3
Flat crystals . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Scattering by the crystal . . . . . . . . . . . . . . . . . . . . . . . .
99
4.4
5 Conclusion
105
Bibliography
115
Chapter 1
Introduction
A variety of technological reactive plasmas is used to manufacture or treat different types of products. Some examples of such products are micro processors, flat
screens, solar cells, mechanical tools, and lamps. Those plasmas often suffer from
small particles growing and levitating in the plasma volume and falling down onto
the work piece thereby often damaging or even destroying it. A huge research effort
has therefore been initiated to understand growth and behaviour of such nano or
micro particles, often called dust. During the studies of the micro particle behaviour
several groups discovered a new state of dusty plasmas called plasma crystal in the
year 1994 (e.g. [1, 2]1 ).
The dust particles acquire a negative charge due to their accumulation of plasma
electrons and ions and the higher mobility of the electrons. The inter particle electrostatic potential energy is much higher than the individual kinetic energy in a
plasma crystal. The particles move only around their equilibrium positions and
typical crystalline structures can be observed. These structures are similar to those
observed in solids which are usually studied e.g. with X-ray diffraction techniques.
The aim of this work is therefore to develop and qualify a setup for diffraction
experiments on plasma crystals. Methods applied in solid state physics like the powder diffraction or the rotating crystal method are intended to be applied with this
setup. These kinds of experiments reveal information about the global structure,
defect density, and stability of a crystal. Further, dynamic processes like melting,
structural phase transitions, waves, and fluctuations can be investigated using such
methods. They always provide insight into global properties of the crystal like defect density, density of structure domains, and temperature changes through the
1
[1] Chu: Direct observation of coulomb crystals and liquids in strongly coupled rf dusty plasmas, 1994
[2] Thomas: Plasma crystal: Coulomb crystallization in a dusty plasma, 1994
1
2
Chapter 1 Introduction
recording of the Debye–Waller factor.
A different and worldwide extensively applied approach for analysing plasma
crystals uses visible laser illumination and CCD (Charge Coupled Device) cameras to observe the light scattered by the individual particles. The movements of
individual particles can be tracked and analysed with video analysis techniques.
Illuminating the plasma crystal with a thin laser sheet gives a 2D image of a single
lattice plane of the plasma crystal. Valuable information about particle movements
within this small part of the crystal can be extracted and analysed. Other techniques
illuminate the whole plasma crystal with an extended laser beam and observe the
scattered light with two or three CCD cameras obtaining a 3D image of the whole
crystal. However, all such methods suffer from shadowing effects and therefore can
analyse small parts of the crystal or small crystals only.
A different approach uses holographic images of the plasma crystal [3]2 . The 3D
particle positions of all particles of the crystal are encoded within the holographic
image. A drawback of this method is the very time consuming analysis of such
pictures by which the 3D particle coordinates must be calculated from the image.
The calculation time dramatically increases with the number of particles of the
plasma crystal and only small systems can be analysed.
However, the diffraction method proposed in this work may solve the problem of
being restricted to small parts of a plasma crystal. The video analysis is also used in
this work to characterise the plasma crystals while optimising their structure and
stability. The diffraction methods can be tested and evaluated with the results of
the video analysis of small parts of the crystals and vice versa. For the first time it
would be possible to perform the “classical” diffraction experiments on a crystalline
system simultaneously to the direct observation of the individual particles.
Furthermore the method presented here may open the door for investigations
of the dynamics and fluctuations of colloidal plasmas on a new global scale. Wave
phenomena, phase transition dynamics, or temperature fluctuations may be analysed on a global perspective parallel to the video analysis of the individual particle
movements. This can provide information about hitherto not accessible phenomena.
The mean particle distance within a typical plasma crystal roughly lies between
100 µm and 500 µm. Therefore the wavelength range of the far infrared (FIR) has
been chosen to do the diffraction experiments. A FIR wave guide resonator pumped
by a CO2 laser has been built and characterised. A scattering arrangement has been
2
[3] Block: Structural and dynamical properties of Yukawa balls, 2007
3
developed consisting of a Yolo telescope, several tilted mirrors, and a motorised
positioning system. Most window materials are not transparent in the far infrared
region of the spectrum. A cylindrical polymer (TPX) plasma chamber has therefore
been built because it provides roundabout optical and FIR access to the plasma
crystals. The whole setup has been characterised and tested by recording diffraction peaks from a golden mesh deposited on a GaAs wafer. This demonstrates the
principle of the method.
The first part of this dissertation (Chapter 2) provides background information
about lasers and optics, plasma discharges and complex plasmas, and light scattering by small particles. Chapter 3 describes the setup followed by the results in
Chapter 4. A conclusion is given at the end.
4
Chapter 1 Introduction
Chapter 2
Theoretical background
This chapter briefly describes basic principles of the far infrared (FIR) laser system,
important aspects of capacitively coupled plasma (CCP) discharges, complex and
colloidal plasmas, and the theory of light scattering by an ensemble of particles.
2.1 Lasers
The first section deals with lasers in general. The succeeding sections are about the
CO2 pump laser and the FIR resonator itself.
2.1.1 General principle of lasers
Detailed descriptions can be found e.g. in [4, 5, 6, 7, 8]3 . The acronym “laser”
means “light amplification by stimulated emission of radiation”. To build a laser
one needs a medium (in the case at hand it is a gas), an energy source, and a
resonator. In principle the laser medium can be a solid, a fluid, a gas, or a plasma.
Examples of the different laser types are diode lasers, dye lasers, optically pumped
FIR lasers, and CO2 lasers. Such a FIR and CO2 laser are used in this work.
The energy source excites (pumps) the atoms or molecules4 within the laser
medium into a higher energy state. An excited molecule can relax into a lower energy state spontaneously or it can be stimulated to relax. In a laser this stimulation
is done by the laser photons that travel back and forth within the resonator. They
3
[4] Siegman: Lasers, 1986
[5] Eichler: Laser – Bauformen, Strahlführung, Anwendungen, 1998
[6] Kneubühl: Laser, 1991
[7] Das: Lasers and Optical Engineering, 1991
[8] M. Young: Optics and Lasers, 1992
4 The
term “molecule” is used in the following.
5
6
Chapter 2 Theoretical background
disturb the excited molecules and force them to relax into the lower laser level.
The photons that are emitted via this process have the same wavelength and phase
like the already existing photons. The new photons are emitted into all directions
and only the very few that accidentally travel nearly parallel to the resonator axis
are stored within the resonator. The maximum inclination angle between the laser
axis and the direction of the photons which still leads to the storage of the photons
depends on the resonator design.
But the laser photons not only induce stimulated emission of radiation, they can
be absorbed by the laser medium as well. The Einstein coefficient for absorption
of a photon of a given wavelength describes the probability that this photon is
absorbed by a molecule. This molecule is thereby excited from a lower energy state
into a higher one. The Einstein coefficient for stimulated emission describes the
probability of the exact reverse process. These Einstein coefficients have the same
value.
To achieve laser activity the stimulated emission must dominate otherwise laser
photons will just be absorbed. Therefore the molecular energy distribution of the
laser medium must be inverted by the energy source. That means the distribution
is no longer a Boltzmann distribution but there are more molecules in a higher
energy level (the higher laser level) than in a lower (the lower laser level). This is
called “population inversion”.
Additionally the lifetime of the lower laser level must be shorter than that of
the higher level to ensure that the lower level is not filled by relaxed molecules
which would destroy population inversion. If the molecular energy distribution of
the medium is inverted in this way then a weak beam of laser photons that travels through the medium along the resonator will be amplified. The degree of this
amplification is called “small signal gain”. The amplification is due to the additional photons that are produced via stimulated emission and that have the same
wavelength and phase like the original incoming laser photons of the weak beam.
A small fraction (≈ 1%) of the laser photons is allowed to leave the laser through
one mirror of the resonator. This is the laser beam. The photons of this beam all
have the same wavelength and phase which means they are coherent.
2.1.2 The pump laser
In this work a tunable, electrically excited CO2 laser (model PL5, Edinburgh Instruments Ltd.) is used as energy source to pump the far infrared (FIR) laser
7
Fig. 2.1: The CO2 molecule. A: At rest, B: Symmetrical stretching mode, C: Bending
mode, and D: Asymmetrical stretching mode.
active vapour within the home made FIR resonator. The CO2 laser can produce
80 lines with wavelengths between 9.2 and 10.8 µm and a power of 50 W at the
strongest line. The laser active medium of the CO2 laser is a gas mixture consisting
of 7 % CO2 , 18 % N2 , and 75 % He.
The CO2 molecule is a three-atomic linear molecule with double bonds between
the carbon and the oxygen atoms (O=C=O). The CO2 molecule can perform three
normal vibrational modes as sketched in Fig. 2.1 (taken from [9]5 ). The upper laser
level is the asymmetrical stretching mode (part D of Fig. 2.1, [6]6 ). In this mode
the carbon atom and one oxygen atom first approach each other while the second
oxygen atom departs from the carbon atom and then vice versa.
Assuming no coupling between these modes they are denoted by a triple of vibrational quantum numbers n1 –n3 as shown in Fig. 2.1. The superscript of the
second vibrational quantum number n2 designates the angular momentum quantum number l. This vibrational state—the bending vibrational mode—is twofold
degenerated: The carbon atom can oscillate within the plane of this paper (as
sketched) or perpendicular. When these two modes coexist an angular momentum
appears. This gives rise to the angular momentum quantum number l of this mode.
Additionally all vibrational modes are degenerated into several rotational energy
states with rotational quantum numbers J. The molecule can rotate around the
symmetry axis which is perpendicular to the long axis of the molecule. This leads
5
[9] Schornstein: AUFBAU UND CHARAKTERISIERUNG EINES FERNINFRAROT-RESONATORS, 2006
6
[6] Kneubühl: Laser, 1991
8
Chapter 2 Theoretical background
27
25
23 J
21
19
18
24
10P
10R
6
0
collisional excitation
9P2
collisional excitation
(0 00 1)
9 R2
Energy
~0.1meV
v=1 collision
26
24
J 22
20
18
26
24
22 J
20
18
0
(1 0 0)
(0 20 0)
(0 11 0)
N2
v=0
CO2 ground state
(0 00 0)
Fig. 2.2: Energy level and excitation scheme for the CO2 laser with the splitting of
vibrational levels into rotational levels and examples of laser lines.
to a high number of possible transitions between the different vibrational states.
Fig. 2.2 shows a sketch of the CO2 laser energy levels and the excitation and
relaxation scheme. The CO2 molecule can be excited into the first vibrational state
(0 00 1) in two ways: by collisions with excited nitrogen molecules or by collisions
with electrons of the plasma discharge of the CO2 laser.
The energy difference between the first vibrational energy level of nitrogen and
the first vibrational level of CO2 is only about 0.1 µeV. This is by far smaller
than the thermal energy at room temperature (≈ 0.03 eV). Furthermore the nitrogen molecule has no permanent dipole moment. This is the reason why radiative
transitions between different vibrational levels within the same electronic state are
forbidden for nitrogen. Therefore this first vibrational state of nitrogen has a relatively long lifetime which increases the probability of such a collision that transfers
energy from the nitrogen to the CO2 molecule.
The excited CO2 molecule in the asymmetrical stretching mode can relax into
the symmetrical stretching mode ((0 00 1) → (1 00 0)) emitting a wavelength of about
10.4 µm or into the bending vibrational mode ((0 00 1) → (0 20 0)) emitting a wave-
length of about 9.4 µm. The selection rules for vibrational and rotational transitions
are: ∆n = 1, ∆l = 0, ±1, ∆J = ±1, ∆J = 0 forbidden [6].
9
Fig. 2.3: Molecules within the FIR resonator. A: methyl alcohol, B: formic acid.
The different lines of the CO2 laser are denoted according to the selection rules,
energy levels, and wavelength involved in a transition (Fig. 2.2). An example is
the line 9P36. The “9” stands for the wavelength region of this line of 9.4 µm and
denotes a transition into the (0 20 0) vibrational state of the CO2 molecule. The
“P” describes that the transition goes with ∆J = −1. Transitions with ∆J = +1 are
designated with “R”. The different sets of laser lines are therefore referred to as “P
branch” or “R branch”. The “36” of this example denotes the rotational quantum
number J = 36 of the lower state into which the transition occurs.
The de-excitation of the lower laser levels is very important for laser activity
because the population inversion has to be maintained. The lifetimes of the lower
laser levels is very long regarding radiative transitions (1–10 ms, [6]7 ). Therefore
collisions of the CO2 molecules with other molecules or with the walls are necessary.
This is one reason for using helium within the gas mixture. The lower laser levels of
the CO2 molecules are de-excited through collisions with helium and the upper laser
level is almost not affected. Furthermore helium has a high thermal conductivity
and cools the laser gas.
2.1.3 The FIR laser
The vapours of methyl alcohol (CH3 OH) and formic acid (HCOOH) are used as
laser active media in the FIR laser resonator. They are cylindrical, slightly asymmetrical, and have a permanent dipole moment which is necessary for radiative
excitation. These molecules are vibrationally excited by the CO2 laser radiation.
7
[6] Kneubühl: Laser, 1991
10
Chapter 2 Theoretical background
1. vibrational
excited state
radiation
io
at
K’
CO2 laser
K
e
J’+2
J’+1
J’
J’-1
it
xc
J+2
FIR
J+1
J
radiation
J-1
n
re
a
ax
tio
n
l
vibrational
ground state
Fig. 2.4: Energy levels of the FIR active molecules and excitation and relaxation scheme.
For methyl alcohol the vibration and rotation are indicated in Fig. 2.3 (taken from
[9]8 ).
The energy difference between vibrational states of these molecules is about
0.13 eV. This corresponds to a wavelength of about 10 µm. Therefore the CO2 laser
can be used as pumping source. But the pump wavelength of the CO2 laser and
the absorption wavelength of the FIR molecule have to coincide very accurately
because both have only narrow line width.
The molecules emit FIR radiation during their relaxation from a higher rotational
level within the first vibrational state to the adjacent lower rotational level in that
state (Fig. 2.4).
The FIR laser output power is dependent on resonator length and can be calculated within the Manley–Rowe limit using [10]9 :
PF IR =
1 λP
δPP .
2 λF IR
(2.1)
Abbreviations: PF IR and PP : FIR laser and pump laser power, λP and λF IR : pump
laser and FIR laser wavelengths, δ: fraction of pump energy which is absorbed by
the gas within the FIR resonator:
δ =1−
1
.
exp N α(ν)L
(2.2)
Abbreviations: α(ν): frequency dependent absorption coefficient of the pump wavelength, N : effective number of round trips of the pump radiation before dissipated
8
9
[9] Schornstein: AUFBAU UND CHARAKTERISIERUNG EINES FERNIFRAROT-RESONATORS, 2006
[10] Hodges: High-Power Operation and Scaling Behavior of CW Optically Pumped FIR Waveguide Lasers,
1977
11
by losses which are different from absorption by gas, L: resonator length. Equation
(2.2) shows that a longer FIR resonator can produce higher output powers.
The losses within the FIR resonator are significant (≥ 2%, [11]10 ) for:
λ2F IR
L
> 0.05.
a3
(2.3)
Increasing the resonator radius thus leads to a decrease of the losses. Nevertheless,
a resonator with a very large radius is disadvantageous because the molecules are
de-excited through collisions with the wall. The de-excitation rate has to be large
to maintain population inversion which is necessary for laser activity. Furthermore:
In resonators with very large radii the FIR radiation is absorbed by the gas again.
Within a resonator of a small radius the rate of de-excitation of molecules at the
wall is higher and the working pressure is higher. This leads to a higher output
power. But a smaller resonator radius leads to higher propagation losses. Usually
FIR resonators therefore have diameters between 20 and 50 mm.
The pumping and relaxation rates determine the value of the working pressure
for the FIR resonator. The relaxation from a higher rotational level into a lower
one is very fast (time constant τR ∝ 10 nsT orr−1 increases with pressure [12]11 ).
In contrast the relaxation between the vibrational levels is very slow. Therefore the
lower laser level has to be de-excited by collisions either with the resonator wall
or with other laser molecules in the gas phase. The rate of diffusion to the wall is
inversely proportional to pressure and square of resonator diameter: rν,dif f usion ∝
1/(pd2 ). The collision rate within the gas is proportional to gas density and thus
to pressure: rν,collision ∝ n ∝ p.
The pressure can be decreased in order to increase the collision rate with the
wall. But this decreases the collision rate for collisions between molecules in the
gas phase to the same amount. A smaller resonator radius is therefore more suitable
to increase the collision rate with the wall. In this work two resonator diameters
have been used: 48 mm and 32 mm and the smaller diameter resulted in higher FIR
output powers [9]12 .
The optimum working pressure for the FIR laser is also determined by the absorption of pump power which is proportional to pressure. All these effects lead to
a working pressure between 10 and 30 Pa in the present case.
10
[11] Degnan: The Waveguide Laser: A Review, 1976
11
[12] Jacobsson: REVIEW: OPTICALLY PUMPED FAR INFRARED LASERS, 1989
12
[9] Schornstein: AUFBAU UND CHARAKTERISIERUNG EINES FERNINFRAROT-RESONATORS, 2006
12
Chapter 2 Theoretical background
The low pressure of 10 to 30 Pa inside the FIR resonator leads to Dopplerbroadening of the FIR line [12]:
√
∆νF IR = 7.162 × 10−7 ν0 T M −1 .
(2.4)
Abbreviations: ν0 : centre frequency of the FIR line, T : temperature (≈ 300 K), M :
molecular mass in amu. The Doppler line widths of the lines of methyl alcohol
(CH3 OH, M = 32, λ1 = 118.834 µm and λ2 = 170.567 µm) are thus ∆ν1 = 5.5 MHz
and ∆ν2 = 3.9 MHz. The longitudinal mode spacing of the resonator ∆νres =
c/2L = 100 MHz is much broader than these Doppler line widths (L = 1.5 m, fixed
resonator length). The FIR line can be adjusted by tuning the resonator length.
The CO2 laser pump line 9P36 is used to produce a FIR line of λF IR = 170.567 µm.
This pump line has a wavelength of λCO2 = 9.695 µm, a frequency of νCO2 =
30.9 THz, and a Doppler line width of ∆νCO2 ≈ 60 MHz (eq. (2.4), with T = 300 K
and M = 44 amu). Using eq. (2.4) again for methyl alcohol (T = 300 K and
M = 32 amu) gives a line width of ∆νF IR ≈ 70 MHz for absorption of the pump
line. Because of these small line widths the pump and absorption wavelengths have
to coincide very precisely to achieve laser activity as mentioned above. Therefore
only a few FIR lines can be stimulated using one type of gas and different FIR laser
gases have to be used to increase the number of available FIR lines.
13
2.2 Gaussian optics
The diffraction of FIR laser beams is very strong in free space (in contrast to
optical laser beams) and Gaussian beam theory has to be applied to describe their
propagation [4, 5]13 . The intensity of an ideal Gaussian beam is given by [5]:
!
2r2
I(r,z) = Imax exp − 2
.
(2.5)
ω(z)
Abbreviations: Imax : peak intensity at z, r = 0, r: radial coordinate, z: distance from
beam waist in beam direction, ω(z) : beam radius. The beam radius at distance z
from the beam waist writes:
ω(z) = ω0
s
1+
z2
.
zR2
(2.6)
Abbreviations: ω0 : radius of beam waist, zR : Rayleigh length. The Rayleigh length
Fig. 2.5: Propagation of a Gaussian beam. The change of the beam radius from ω0 at
the beam waist to ω(z) at a distant location is shown. The curved dashed lines denote
the wave fronts of the beam. The radius of curvature R(z) of these wave fronts increases
towards the beam waist up to infinity. θ is the asymptotic angle of divergence.
zR is the distance from z = 0 at which the Gaussian beam radius is ω(zR ) =
√
2ω0 .
The Rayleigh length thus marks a range where the beam is only slightly divergent
(Fig. 2.5, taken from [5]). The value b = 2zR is called focal length or confocal
parameter. With zR = πω02 /λ eq. (2.6) writes for the beam diameter:
s
16λ2 z 2
d(z) = d0 1 + 2 4 .
π d0
13
[4] Siegman: Lasers, 1986
[5] Eichler: Laser – Bauformen, Strahlführung, Anwendungen, 1998
(2.7)
14
Chapter 2 Theoretical background
Fig. 2.6: Focussing of a Gaussian beam by a lens. The small beam waist after the focussing leads to a more divergent beam. The case of a spherical mirror is similar with
focal length f = R/2, R denoting the radius of curvature of the mirror. a and a′ denote
the distances of the beam waists to the lens/mirror.
Far away from the beam waist z ≫ zR and z 2 /zR2 ≫ 1 hold and the beam expansion
becomes linear: ω(z) = ω(0) z/zR . The angle of divergence θ thus writes:
ω(z)
ω0
λ
2λ
=
=
=
.
θ∼
=
z
zR
πω0
πd0
(2.8)
The FIR resonator has two plane mirrors and produces a divergent FIR beam.
To focus the FIR beam to a diameter of about 2 cm at the centre of the plasma
chamber a Yolo telescope has been designed and built. This telescope consists of
two spherical mirrors and the desired focal lengths of these mirrors have been
calculated using the following formulae (Fig. 2.6, taken from [5]). The distance of
a beam waist after a focussing element (here: spherical mirror) is given by:
a′ = −f +
f 2 (f − a)
.
(f − a)2 + zR2
(2.9)
Abbreviations: a′ : distance beam waist to spherical mirror after focussing, f : focal
length of mirror (f = R/2, R: radius of curvature), a: distance beam waist to mirror
before focussing, zR : Rayleigh length of beam before focussing (zR = πd20 /4λ).
The beam diameter at the waist after focussing writes:
d′0 = p
d0 f
(a − f )2 + zR2
.
(2.10)
Abbreviations: d′0 : beam diameter at the waist after focussing, d0 : beam diameter
at the waist before focussing.
A computer program has been developed to calculate beam diameters, beam spot
position, and spot size using equations (2.7), (2.9), and (2.10). This program has
15
a graphical interface and has been used to design the mirror arrangement of the
setup.
To ensure that a Gaussian beam passes almost completely through an aperture
the diameter of the aperture should be at least two times the diameter of the beam
[5]. Real laser beams are not ideally Gaussian and have larger diameters. Therefore
the mirrors designed and manufactured for this work are as large as possible.
16
Chapter 2 Theoretical background
2.3 Capacitively coupled plasma discharges
This section briefly describes some aspects of capacitively coupled plasma (CCP)
discharges. There are several books, monographs, and papers about CCP discharges—
see e.g. [13, 14, 15, 16]14 . Therefore only the most important and relevant details
of CCP discharges are discussed here.
Plasma production and maintenance
The plasma is ignited between two parallel plates within a vacuum chamber. A sinusoidal voltage is applied to one of the plates (electrodes) and the other electrode
is grounded. The first free electrons of the discharge are produced through impact
ionisation by incoming cosmic radiation for example. Such electrons are then accelerated in the electric field of the powered electrode. They gain kinetic energy
and ionise other background gas atoms through impact ionisation. The thus newly
created electrons are accelerated and ionise further atoms—an electron avalanche
sets in, the plasma is ignited.
The plasma is maintained through further impact ionisation of background gas
atoms by electrons. The bulk plasma density n0 is obtained from the energy balance
of the system. Equating the absorbed to the lost power gives [14]:
Pabs = n0 uB Aef f ET .
(2.11)
Here Pabs is the total power absorbed in the plasma, uB is the Bohm velocity:
p
uB = kB Te /mi ,
(2.12)
Aef f is the effective wall area of the chamber where plasma particles are absorbed,
and ET is the total energy lost from the system when an electron–ion pair is lost.
ET depends on the electron temperature. Equation (2.11) shows that the plasma
density is proportional to the absorbed power and thus to the input power.
The plasma density also depends on the pressure of the background gas. The
electron–neutral collision frequency ν increases with pressure and so does the ionisation rate. A higher pressure thus results in a higher electron density and thus in
a smaller Debye length.
14
[13] Raizer: Radio-Frequency Capacitive Discharges, 1995
[14] Lieberman: Principles of Plasma Discharges and Materials Processing, Second Edition, 2005
[15] Hargis: The Gaseous Electronics Conference radio-frequency reference cell – A defined parallel-plate radiofrequency system for experimental and theoretical studies of plasma-processsing discharges, 1994
[16] Olthoff: The Gasous Electronics Conference RF Reference Cell – An Introduction, 1995
17
The electron temperature is obtained from the particle balance. Equating particle
loss at surfaces to volume ionisation gives [14]:
n0 uB Aef f = Ki ng n0 V,
(2.13)
where Ki is the rate constant for electron–neutral ionisation (in m3 /s), ng is the
background gas density, and V is the plasma volume. The bulk plasma density n0
cancels out. Inserting the Bohm velocity and Ki → Ki(Te ) [14] one obtains:
mi
Te
=
2
Ki(Te )
kB
ng V
Aef f
2
.
(2.14)
Equation (2.14) shows that the electron temperature depends on the plasma volumeto-surface ratio. Therefore the electron temperature is mainly determined by the
plasma chamber geometry.
Electrical properties of a capacitively coupled plasma
The radio-frequency (rf) power generator of this setup produces a sinusoidal voltage
at 13.56 MHz. The power generator is connected to a matching network consisting
of a high pass filter and a blocking capacitor (chapter 3, p. 74). This blocking
capacitor disconnects the powered electrode from the power generator for direct
current (dc). Therefore (and due to the high mobility of the electrons) the powered
electrode charges up negatively with respect to ground. The corresponding voltage
(relative to ground) is called self-bias [13]15 .
But the grounded electrode also acquires a negative charge—at least with respect
to the plasma. There is a potential drop from the positive plasma potential to the
zero potential of the grounded electrode. The voltage ratio of powered and grounded
electrode (electrode voltages measured with respect to the bulk plasma) depends
on the area ratio of powered and grounded electrode [13]:
q
Vpowered
Agrounded
=
Vgrounded
Apowered
(2.15)
with q ≤ 2.5. The reason for this relation lies in the fact that the same current
has to flow through both electrodes. The smaller electrode therefore draws a higher
current density and has to have a higher potential difference to the bulk plasma
than the larger electrode.
15
[13] Raizer: Radio-Frequency Capacitive Discharges, 1995
18
Chapter 2 Theoretical background
Fig. 2.7:
presheath
Sheath
near
a
and
wall.
ne , ni , n0 : electron, ion, and
gas density, λi : ion mean
free path.
When the powered electrode is connected to ground via a low pass filter the
self-bias drops to zero but the plasma potential increases [13]. Therefore there is a
voltage drop between bulk plasma and electrode as well.
Small dust particles (diameter about 10 µm) which are introduced into the plasma
charge up negatively due to the high mobility of electrons compared to that of
ions. They are repelled by the electric field resulting from the self-bias or from
the grounded electrode. They can be stored, levitating within the plasma some
millimetres to centimetres above the lower electrode. At this position there is a
force balance mainly between gravity and the electric force (sec. 2.4.7, p. 31).
The sheath
As described in the previous subsection the plasma chamber walls charge up negatively and the plasma potential is positive. Therefore there exists a transition layer
in which the potential drops from the bulk plasma value to the wall potential. This
transition layer is called plasma–wall sheath near the wall.
Figure 2.7 schematically shows electron and ion density profiles and the potential
profile in bulk plasma and sheath (taken from [17]16 ).
As can be seen in Fig. 2.7 the ion density is less reduced compared to the electron
density in the sheath close to the wall. This is because the electrons are more mobile
16
[17] Lieberman: PRINCIPLES OF PLASMA DISCHARGES AND MATERIALS PROCESSING, 1994
19
and are reflected by the negative wall potential whereas the ions are attracted by
it. The sheath edge is defined as the point where the ions reach the Bohm velocity
(eq. (2.12), p. 16). At this point the densities of electrons and ions start to diverge
and a positive space charge develops that shields the electrons from the wall. The
Bohm velocity arises when writing down energy and flux conservation of the ions
within a collisionless sheath [13, 14, 18]17 . The thickness of the sheath is a few
electron Debye lengths λDe (eq. (2.25), p. 23).
The ions must enter the sheath with the Bohm velocity to ensure ion flux conservation. Therefore there must exist a presheath in which the ions are accelerated to
that velocity by ambipolar electric fields. In this presheath ion and electron density
are equal. The presheath thickness is of the order of the ion mean free path which
is several times larger than the electron Debye length.
Due to the variation of ion and electron density in presheath and sheath the
ion and electron Debye lengths vary as well in these regions. Therefore the Debye
sheath around a dust particle is not spherically symmetric but deformed or polarised. This polarisation of the Debye sheath leads to a ‘∇λD ’ force acting on dust
particles which is described in section 2.4.7 (p. 33).
The lower part of Fig. 2.7 shows the behaviour of the potential in presheath
and sheath. The potential decreases weakly in the presheath and strongly in the
sheath. Nevertheless, Tomme et al. show that the sheath potential can be well
approximated by a parabola [19]18 . This leads to an electric field that increases
linearly toward the electrode. This electric field gives rise to the electric force that
supports the negatively charged dust particles.
The ions enter the sheath with Bohm velocity and are further accelerated by the
electric field. They stream with supersonic velocity and exert an ion drag force on
the dust particles which levitate in presheath and sheath (sec. 2.4.7, p. 35).
17
[13] Raizer: Radio-Frequency Capacitive Discharges, 1995
[14] Lieberman: Principles of Plasma Discharges and Materials Processing – Second Edition, 2005
[18] Chen: PLASMA PHYSICS AND CONTROLLED FUSION, 1984
18
[19] Tomme: Parabolic plasma sheath potentials and their implications for the charge on levitated dust particles,
2000
20
Chapter 2 Theoretical background
2.4 Complex plasmas
Complex or dusty plasmas consist of neutrals, electrons, ions, and nanometre or micrometre sized particles (so called dust). Under certain experimental conditions the
dust particles can arrange themselves in ordered and stable (crystalline) structures
[1, 2]19 . This state is then called “plasma crystal” which is the subject matter.
A plasma crystal can form when the inter particle potential energy Epot is much
higher than the individual dust kinetic energy Ekin (Epot /Ekin > 170) [20, 21]20 .
Epot depends on the charge of the grains. This steady state charge is predominantly
determined by the balance of ion and electron currents onto the grains in lowtemperature plasmas [22]21 . Other charging currents might be e.g. photo electron
emission due to UV radiation, thermionic and secondary electron emission, field
emission, radioactivity, and impact ionisation, some of which are relevant e.g. in
astrophysical environments [23, 24]22 .
The dust particles become negatively charged in most technological plasmas due
to the high mobility of electrons compared to that of ions. In the following some
theoretical considerations are presented about the charging mechanisms relevant
to the case at hand. The traditional orbital motion limited (OML) theory is developed in some detail. Afterwards different effects are briefly described that can
change currents, grain charge, floating potential of a dust grain, and the potential
distribution around the grain.
2.4.1 OML theory
The simplest way to describe charging currents, acquired charge, and floating potential of a dust grain in a plasma is the orbital motion limited (OML) approach
[25]23 . The OML theory does not include any information or assumption about the
potential distribution around the dust particle. It just uses conservation laws of
energy and angular momentum for ions and electrons to calculate cross sections,
currents, and grain charges. In doing this some simplyfying assumptions are made
19
[1] Chu: Direct Observation of Coulomb Crystals and Liquids in Strongly Coupled rf Dusty Plasmas, 1994
20
[20] Fortov: Complex (dusty) plasmas: Current status, open issues, perspectives, 2005
21
[22] Bouchoule (editor): Dusty Plasmas, 1999
22
[23] Melzer: Introduction to Colloidal (Dusty) Plasmas (Lecture Notes), 2005
[2] Thomas: Plasma Crystal: Coulomb Crystallization in a Dusty Plasma, 1994
[21] Ikezi: Coulomb solid of small particles in plasmas, 1986
[24] Shukla: Introduction to Dusty Plasma Physics, 2002
23
[25] Mott-Smith: THE THEORY OF COLLECTORS IN GASEOUS DISCHARGES, 1926
21
to obtain the different charging currents and cross sections. These are as follows.
Assumptions in OML theory:
1. Isotropic plasma.
2. Maxwellian velocity distribution of electrons and ions within the plasma.
3. No effective barrier in the potential.
4. No collisions between ions and neutrals which implies conservation of angular
momentum of the ions.
5. Isolated dust grains—no “closely packed” grains.
6. Pure electrostatic interaction between electrons, ions, and the dust particles.
7. Shielding is described by the Debye length λD which includes a linearisation
(|eφf | ≪ kB Te ).
Currents within OML theory
Following [23] in writing down the energy balance for an ion coming from the
2
far distant plasma 12 mi vi,0
= 21 mi vi2 + eφf and using the conservation of angular
momentum one obtains the critical impact parameter and thus the cross section
for ion collection for mono-energetic ions:
2eφf
2
σc = πa 1 −
,
2
mi vi,0
φf < 0.
(2.16)
Here a is the dust grain radius, φf is the dust grain floating potential, mi and vi,0
are ion mass and velocity far away from the dust particle. This cross section is
larger than the particle geometric cross section (σ = πa2 ) due to the attraction of
the positively charged ions by the dust (φf < 0).
The ion charging current is given by the cross section for ion collection and the
ion current density: dIiOM L = σc (vi )dji = σc (vi )ni evi f (vi )dvi , where ji = ni evi is
the ion current density, ni and vi are ion density and velocity, and f (vi ) is the ion
velocity distribution which is assumed to be Maxwellian:
f (vi ) =
4πvi2
mi
2πkB Ti
3/2
mi vi2
exp −
2kB Ti
(2.17)
22
with
Chapter 2 Theoretical background
R∞
0
f (vi )dvi = 1. Ti and kB are the ion temperature and Boltzmann’s con-
stant, respectively. The ion charging current is obtained by integration over the
Maxwellian velocity distribution from zero to infinity with the result:
eφf
OM L
2
,
(2.18)
Ii
= πa ni evth,i 1 −
kB Ti
p
where vth,i = 8kB Ti /πmi is the ion thermal velocity (mean velocity in a Maxwell
distribution).
The electron current can be
obtaining the cross section for
derived analogously,
2eφ
f
electron collection: σce = πa2 1 + me v2 which is smaller than the geometric cross
e,0
section because of the repulsion of the electrons by the negatively charged dust grain
(φf < 0). Integration of the electron current density over the Maxwell distribution
gives:
eφf
.
(2.19)
= −πa ne evth,e exp
kB Te
Here the integration starts at a minimum electron velocity given by the floating
p
potential (vmin = −2eφf /me ) because the electrons have to overcome the elecIeOM L
2
trostatic barrier due to the negatively charged dust grains.
Interpreting this equation it is the thermal electron current to a neutral particle
but reduced by the Boltzmann factor because the electrons are repelled by the
negatively charged dust grain.
Potential distribution within OML theory
The steady state floating potential of a grain is obtained by equating electron and
ion current which yields:
eφf
1−
=
kB Ti
r
ne vth,e
exp
ni vth,i
eφf
kB Te
.
(2.20)
This equation can be solved numerically for φf .
The potential distribution in the vicinity of the dust particle is obtained by
solving Poisson’s equation:
e
(ni − ne )
(2.21)
ǫ0
using a Boltzmann distribution for electrons and ions for the simplest case:
eφ(r)
,
(2.22)
ne,i (r) = n0 exp ±
kB Te,i
∆φ =
where the plus sign is for electrons and the minus sign for positive ions. By inserting this into the Poisson equation and linearising (assuming |eφ(r) | ≪ kB Te,i ) one
23
obtains:
φ
,
(2.23)
λ2D
with the linearised Debye length λD defined by the electron and ion Debye lengths:
∆φ =
1
1
1
= 2 + 2 .
2
λD
λDe λDi
These write in particular:
(2.24)
r
ǫ0 kB Te,i
,
(2.25)
e2 n0
where n0 is the electron density in the far distant (quasi-neutral) plasma. In the
λDe,i =
case of spherical symmetry the solution of the linearised Poisson equation (2.23) is
the Debye–Hückel potential or screened Coulomb potential [22]:
r−a
a
.
φ(r) = φf exp −
r
λD
(2.26)
The linearised Debye length is closer to the ion Debye length for plasmas with
Te ≫ Ti as it is the case in this work.
The Debye–Hückel potential has been derived here with the certainly not fulfilled
assumption of ions with a Boltzmann distribution. Nevertheless, a more detailed
discussion of the ion density distribution under OML conditions leads to the same
result for the potential [22]24 .
Grain charge within OML theory
The grain charge Qd is now approximated by assuming the grain capacitance to
be C = 4πǫ0 a(1 + a/λD ) that reduces to the vacuum value for λD ≫ a and ap-
plying Qd = Zd e = Cφf . Here Zd denotes the number of charges on the grain and
φf is the grain floating potential calculated from equation (2.20) (p. 22). This approximation is problematic because it uses the Debye length and therefore includes
the linearisation |eφf | ≪ kB Te which “is not satisfied for a floating sphere” [26]25 .
However, for the case λD ≫ a it yields quite good results at least for conducting
spheres compared to the results of a self-consistent numerical particle-in-cell (PIC)
simulation by Hutchinson [26].
OML theory is the standard theory for the description of particle charging phenomena in an isotropic plasma. However, some effects and phenomena are neglected
but lead to big changes in grain charge, current, and potential distribution. Some
of these effects are described in the next sections with a summary and comparison
in sec. 2.4.6 (p. 30).
24
[22] Bouchoule (editor): Dusty Plasmas, 1999
25
[26] Hutchinson: Ion collection by a sphere in a flowing plasma: 3. Floating potential and drag force, 2005
24
Chapter 2 Theoretical background
2.4.2 Non-isotropic plasmas—streaming ions
The dust grains in the experiments presented here levitate near the lower electrode
in the plasma–wall sheath or presheath region. The plasma is certainly not isotropic
in these regions and the ions are streaming toward the electrode with very high and
even supersonic velocity. Therefore the first two assumptions underlying the OML
theory (isotropy and Maxwell distribution) are not valid in the presented situation.
In the following only the ion currents to dust particles are treated because the ions
have a major effect on structure and dynamics of plasma crystals. Furthermore,
electron streaming has no effect since streaming velocities of electrons are far below the electron thermal velocity [27]26 .
Currents with streaming ions
Drifting ions can be included in the OML theory through a drifting Maxwellian
velocity distribution:
f (vi ) =
4πvi2
mi
2πkB Ti
3/2
mi (vi − ui )2
exp −
2kB Ti
.
(2.27)
Here ui denotes the ion drift velocity. A different ion current to the dust particle
is then obtained if the OML cross section for ion collection is used [20, 28]27 .
When the drift velocity ui is very large (ui ≫ vth,i ) one can replace vth,i with ui
and kB Ti with 21 mu2i in equation (2.18) (p. 22) for the OML ion current yielding:
Ii = πa2 ni eui (1 − 2eφ/mi u2i )
(2.28)
for the ion current to a dust particle in the case of very fast streaming ions and
conducting spheres [23].
Grain charge with streaming ions
The dust grain charge computed by the OML theory is quite accurate for conducting spheres even with ion flow when the Debye length is much larger than the
sphere radius [26]. Hutchinson describes a fully self-consistent numerical approach
which uses the particle-in-cell (PIC) method [26]. Especially insulating spheres are
difficult to describe analytically because they charge up asymmetrically. The results
show that for insulating spheres the OML theory predicts a grain charge that is
26
[27] Lampe: Interactions between dust grains in a dusty plasma, 2000
27
[20] Fortov: Complex (dusty) plasmas: Current status, open issues, perspectives , 2005
[28] Whipple: Potentials Of Surfaces In Space, 1981
25
Fig. 2.8: Dimensionless dust particle charge
z = |Z|e2 /(4πǫ0 akB Te ) of an isolated spher-
ical particle as a function of ion drift velocity ui in Mach numbers or normalised to
vth,i . The calculations are for three different
electron-to-ion density ratios and correspond
to an Ar plasma with Te /Ti = 100. An increasing charge due to a decreasing ion collection cross section (eq. (2.16), p. 21) is followed by a charge decrease due to the positive
space charge near the electrode.
by far too low, meaning that the real charge is more negative. The reason for this
lies in the development of a strong negative potential on the side of the insulating
grain that is downstream to the ion flow. The ion flux reaching the dust surface is
much smaller on this downstream side.
Fortov et al. discuss the behaviour of the particle charge in the sheath region
near the electrode as a function of the ion drift velocity ui [20] (Fig. 2.8, taken
from [20]). The ion velocity increases towards the electrode due to the linearly
increasing (averaged) electric field in the sheath [19]28 . Therefore this discussion is
also a discussion of particle charge as a function of height above the electrode.
At low ui the charge is constant and equal to the OML charge. The charge firstly
increases when ui > vth,i . This is due to the decreasing collection cross section for
ions with increasing ion velocity; see equation (2.16) on page 21. The dust particle
charge decreases again when the ion drift velocity becomes several 10 times the ion
thermal velocity. This is because a positive space charge develops near the electrode
where ui is high. This leads to a higher ion flux to the dust particles compared to
the electron flux. Therefore the particle charge gets less negative. A dust particle
can thus even become positively charged when it comes very close to the electrode.
Potential distribution with streaming ions
The surface potential varies with position on the surface of insulating spheres because of the asymmetrical charging. The local current density is zero in this case.
28
[19] Tomme: Parabolic plasma sheath potentials and their implications for the charge on levitated dust particles,
2000
26
Chapter 2 Theoretical background
Fig. 2.9: Contour plot of the ion density, showing ion focusing, for three different velocities of ion flow. The plot
is presented in the grey-scale topography
style; regions A correspond to ion densities ni < ni0 , and regions B correspond
to ion densities ni > ni0 . The ions are
focusing behind the grain, thus forming
a region with highly enhanced ion density. The distances are given in units of
the electron Debye length.
A conducting sphere acquires a constant surface potential which is not positionally
dependent. The total current density is zero in this case.
An ion cloud is formed downstream the dust particle due to the ion flow toward
the electrode. Ions coming from the distant plasma are attracted by a dust particle.
Therefore the ion trajectories are bended toward the dust particle and eventually
end at the particle surface. But not all ions fall onto the dust grain. So, the dust
grain acts as a focusing lens for the streaming ions. A dust particle located below
will be attracted by the positively charged ion cloud [29, 30, 31]29 . This has important consequences for the particles of a plasma crystal. They can arrange in vertical
strings with one dust grain located directly below another.
A perturbed region of plasma density—a wake—can be formed downstream a
dust particle at low pressure (low collisional damping), high electron-to-ion temperature ratios (low Landau damping), and Mach numbers of the ion velocity around
M = 1 [27]30 . The ion density can show several maxima and minima over several
10 Debye lengths downstream the dust particle [32]31 (Fig. 2.9, taken from [32]).
Lampe et al. show the weakening of the wake effect with increasing pressure
up to ≈ 10 Pa [27]. A pronounced wake is thus not expected in the presented
experiments of this work since the working pressures are much higher (up to 100 Pa).
Nevertheless, the first pronounced maximum in ion density certainly develops in
29
[29] Melzer: Structure and stability of the plasma crystal, 1996
[30] Melzer: Transition from Attractive to Repulsive Forces between Dust Molecules in a Plasma Sheath, 1999
[31] Melzer: Laser manipulation of particles in dusty plasmas, 2001
30
[27] Lampe: Interactions between dust grains in a dusty plasma, 2000
31
[32] Maiorov: Plasma kinetics around a dust grain in an ion flow, 2001
27
the present case and leads to the observed particle string formation. In such a
particle string several dust grains are aligned vertically—one grain directly below
the other.
2.4.3 Barrier in the effective potential
The trajectory of an ion approaching a negative dust grain will be bended and
the ion will eventually fall onto the grain provided the impact parameter is below
the critical one. But if the ion has a high angular momentum (within the central
potential φ(r) of the grain) it can miss the grain even if the impact parameter is
low enough. The ion velocity is simply not strictly enough directed toward the dust
grain in this case. From the view point of the potential this means that there is a
barrier in the potential [20, 27]32 and one has to use an effective potential which is
derived from the energy balance:
1
2
E = Ekin + Erot + Epot = m vr2 + vΘ
+ eφ(r) ,
2
2
with mvΘ
=
2 r2
m2 vΘ
mr 2
=
L2
mr 2
(2.29)
(L is the angular momentum of the ion and r the
distance to the grain). The effective potential thus writes:
Uef f = eφ(r) +
L2
,
2mi r2
(2.30)
with vr and vΘ being radial and angular velocity. Thus some low energy ions can
be reflected from this potential barrier reducing the ion current to the dust grain
and leading to a lower grain charge (more negative). Fortov et al. [20] estimate the
applicability of the OML theory regarding the neglect of this potential barrier to
be good in cases with λD > 5a. Since the potential barrier is very small in most
of the cases [33, 34, 35]33 only ions with low kinetic energy are reflected from this
barrier. Therefore the decrease of the ion current to the dust particle due to the
potential barrier is only a weak effect.
32
[20] Fortov: Complex (dusty) plasmas: Current status, open issues, perspectives, 2005
33
[33] Lampe: Trapped ion effect on shielding, current flow, and charging of a small object in a plasma, 2003
[34] Sternovsky: Ion collection by cylindrical probes in weakly collisional plasmas: Theory and experiment, 2003
[35] Lampe: Effect of Trapped Ions on Shielding of a Charged Spherical Object in a Plasma, 2001
28
Chapter 2 Theoretical background
2.4.4 Ion-neutral collisions—angular momentum not
conserved
The OML theory neglects ion–neutral collisions in the calculation of the grain
charge even though there must be such collisions to maintain the Maxwell distribution in the ambient plasma. It is simply assumed that an ion is coming from a
Maxwellian plasma and eventually hits the grain but certainly without a further
collision. But collisions in the vicinity of a dust grain can be quite important even if
the ion mean free path li is much larger than the screening length λD [20, 33, 34, 35].
Since the working pressures in the case at hand are fairly high (1 to 100 Pa) a discussion of the influence of ion–neutral collisions on dust grain charge and shielding
seems most appropriate.
In the presented case the argon density lies roughly between 2 × 1020 m−3 for
a pressure of 1 Pa and 2 × 1022 m−3 for 100 Pa. An estimation for the total ion–
neutral collision cross section is given in [14]: σitotal ≈ 10−14 cm2 . This cross section
includes elastic scattering and charge transfer collisions. The ion mean free path
−1
li = ng σitotal
then lies between about 4 mm for a pressure of 1 Pa and 0.04 mm
for 100 Pa.
Grain charge with collisions
Especially ion–neutral charge-exchange collisions have a great impact on ion current, grain charge, and shielding. Every such an event creates a new ion with only
the thermal velocity of the background gas. This new ion can then easily be absorbed by the grain which leads to a higher ion current onto the grain and thus
to a lower charge (less negative). The cross section for charge-exchange collisions
between ions and neutrals is higher than that for specular reflection for the relevant
ion energies. Furthermore, some newly created ions may perform orbits around the
dust particle depending on their initial energy and angular momentum. If the kinetic energy is too large, the ion can escape from the particle—if the energy is too
low, it will fall onto the dust grain.
Potential distribution with collisions
Lampe et al. derive a condition for ions to be trapped in the potential well near
the dust grain [33]: r2 φ(r) < a2 φ(a) (r: distance between an ion and the grain, a:
grain radius). Thus a positive shielding cloud is created which is “trapped” in the
29
particle potential very near the grain and moves together with the dust grain. This
is essentially different to the usual Debye shielding: The trapped ion cloud provides
additional shielding against external electric forces thus changing the interaction
with other dust grains. Such an ion cloud can even be polarised leading to vander-Waales interactions between dust particles. In the usual Debye case shielding
is provided by ions and electrons that move through the entire plasma and are not
bounded to a dust grain. These moving ions and electrons cannot screen external
electric forces.
The creation rate of trapped ions is proportional to the ion–neutral collision frequency ν and so is the loss rate of the trapped ions. They can be scattered out of
the potential trap or—more probably—fall onto the grain after a further collision.
Therefore the number of trapped ions within the Debye shield of a dust particle is
independent of the neutral gas pressure in the “weakly” collisional regime [36]34 .
But the creation rate is proportional to the plasma density and so is the density of
the trapped ions.
Currents with collisions
Lampe et al. derive an expression for the ion flux to a dust grain including collisions
(but not streaming ions) which gives the following ion current [33]:
Iicoll
2
≈ πa ni evth,i
eφf
r3
1−
+ 2T
kB Ti a li
.
(2.31)
Here rT is a shielding radius determined by the equality of potential and thermal
energy of an ion: Epot = Eth : eφ(rT ) = − 23 kB Ti . It means that an ion created in a
charge-exchange collision at this distance is likely to be trapped.
In this “weakly” collisional regime the ion current increases with pressure due to
the higher plasma density and accordingly more frequent charge-exchange collisions
within the sheath surrounding the grain. But if the pressure increases to much
higher values eventually the “strongly” collisional regime is reached. In this regime
the ions make many collisions and the motion is mobility controlled [20]. Since
the mobility decreases with increasing pressure the ion flow onto the dust particle
decreases as well. Therefore there exists a maximum in the ion current onto a grain
when the neutral gas pressure is increased.
34
[36] Goree: Ion Trapping By A Charged Dust Grain In A Plasma, 1992
30
Chapter 2 Theoretical background
2.4.5 “Closely packed” dust grains
The quasi-neutrality condition of the plasma has to include the charge on the dust
grains—especially when the dust density is very high: ni = ne + Znd . The electron
density is depleted due to the absorption of electrons by the dust grains making
the ion density larger than the electron density. Therefore the individual dust grain
charge is reduced (less negative) because there are not enough electrons in the
vicinity of a single particle to provide the negative charge that would arise in the
case of an isolated particle in a plasma without other dust particles. The Havnes
parameter describes the strength of this effect: P = |Z|nd /ne . If P ≪ 1 the grain
charges tend to the value of an isolated particle while in the case of P ≫ 1 the
grain charges are reduced significantly [20, 23]35 .
Furthermore, closely packed dust grains can distort trajectories of electrons and
ions when the inter grain distance is smaller than the typical interaction length between electrons/ions and the dust particles. This can also lead to a charge reduction
[20].
2.4.6 Summary of OML theory and neglected effects
The orbital motion limited (OML) theory describes the charging of a dust grain
in a plasma in the most simple way. It assumes an isotropic, Maxwellian plasma
without ion–neutral collisions and potential barrier. The interaction between dust
particles is assumed to be purely electrostatic. Using the conservation of energy and
angular momentum for an incoming ion a cross section for ion collection is derived.
Integration of the product of this cross section with the ion current density over
a Maxwell distribution leads to the ion charging current. The electron charging
current is derived analogously. By equating these currents the floating potential
can be calculated numerically.
Solving Poisson’s equation by assuming a Boltzmann distribution for electrons and
ions one obtains the Debye–Hückel potential with the linearised Debye length.
Finally, the grain charge is calculated inserting a certain expression for the grain
capacitance which is multiplied by the floating potential.
Table 2.1 summarises the effects which are not governed by the OML theory but
lead to a different charge on dust grains.
35
[20] Fortov: Complex (dusty) plasmas: Current status, open issues, perspectives, 2005
[23] Melzer: Introduction to Colloidal (Dusty) Plasmas (Lecture Notes), 2005
31
Mechanism
Grain charge compared to OML
Streaming ions
more negative (strong effect)
Insulating grains
more negative (strong effect with streaming ions)
Barrier in effective potential
more negative (weak effect)
Ion–neutral collisions
less negative (strong effect, trapped ions)
Closely packed grains
less negative (strong effect)
Table 2.1: Effects on grain charge for different mechanisms not included in OML theory.
Streaming ions lead to a higher negative grain charge because less ions reach the
grain on the downstream side and the ion collection cross section decreases with
increasing ion velocity. PIC simulations reveal a higher negative dust grain charge
for insulating particles compared to conducting particles. This is even enhanced by
streaming ions. These effects are important for the presented case since the dust
particles levitate near the lower electrode in the sheath or presheath region of the
plasma where ions are streaming with even supersonic velocity.
A barrier in the potential distribution around a grain arises because of the angular
momentum of incoming ions. But this barrier in the effective potential is only small,
typically. Thus only low energy ions are reflected from it and this effect is of minor
importance.
Ion–neutral collisions in the vicinity of a dust grain increase the ion current to
the grain and thus reduce the grain charge strongly. Especially charge-exchange
collisions are very important. Through such a collision a slow ion is created that
is effectively attracted by the dust grain. The pressures used in this experiment lie
between 1 and 100 Pa and are thus fairly high. Ion–neutral collisions thus play a
significant role.
When the dust number density is very high the electron density is depleted compared to a dust free plasma. Therefore the grain charge is less negative compared
to a single particle within a plasma without any further dust grains. This effect is
of minor importance because dust densities are not that high in the present case.
Different ion currents and potential distributions are listed in table 2.2.
2.4.7 Forces acting on dust particles in a plasma
Dust particles in a plasma are subject to various forces. The relevant forces are
briefly described in this chapter and the appropriate formulae are given. Detailed
32
Chapter 2 Theoretical background
Theory
OML
Streaming ions
Collisions
Ion current Ii
πa2 ni evth,i 1 −
eφf
kB Ti
≈ πa2 ni eui (1 − 2eφf /mi u2i ), ui ≫ vth,i
3
rT
eφf
2
≈ πa ni evth,i 1 − kB Ti + a2 li
Potential φ(r)
φf ar exp − r−a
λD
ion wake, numerics
trapped ions, numerics
Table 2.2: Ion currents and grain potentials for different effects included.
derivations and discussions can be found in several textbooks and articles e.g.
[20, 22, 23, 24]36 . The main forces which determine the levitation height of the dust
particles above the lower electrode are gravity and the electric force. Gravity pulls
the particles out of the plasma volume whereas the electric force confines them.
Some of the forces depend on the ratio of grain radius to Debye length (a/λD ).
This ratio is not generally small as far as plasma crystals are concerned. Debye
spheres can partly overlap making strong coupling between grains possible.
Gravity
Gravity is the most intuitive force of all:
4
F~G = m~g = πa3 ρ~g ,
3
(2.32)
where m is the dust particle mass, ~g the gravitational acceleration, and ρ the particle density. Gravity is thus proportional to the cube of the dust particle radius.
Electric force
The vacuum electrostatic force Qd E is a good approximation of the electric force
provided the dust grain is conducting and the radius is small compared to the
Debye length [22]. The Debye sheath around the dust particle does not screen the
particle from an external electric field E (e.g. originating from the electrode or
from other grains). This is so because the Debye sheath is not attached to the
dust grain but just a local variation of the plasma density. The ions and electrons
that constitute the Debye shielding are moving through the entire plasma. Their
respective densities and velocities are changed in the vicinity of a dust grain, but
they are not attached to the grain.
36
[20] Fortov: Complex (dusty) plasmas: Current status, open issues, perspectives, 2005
[22] Bouchoule: Dusty Plasmas, 1999
[23] Melzer: Introduction to Colloidal (Dusty) Plasmas (Lecture Notes), 2005
[24] Shukla: Introduction to Dusty Plasma Physics, 2002
33
The situation becomes different when collisions are taken into account. Then a
number of ions can be trapped in the potential well near the dust grain (sec. 2.4.4,
p. 28). These trapped ions can move together with the particle and effectively screen
an external electric field.
Applying the capacitor model of section 2.4.1 (p. 20, C = 4πǫ0 a(1 + a/λD )) to
the grain one obtains:
a
~
~
~
FE = Qd E = 4πǫ0 a 1 +
φf E.
λD
Here, a is the particle radius, λD the Debye length, and φf the floating potential
of the particle. This formula reduces to
~
F~E = 4πǫ0 aφf E
(2.33)
in cases with λD ≫ a. The electric force is then proportional to the dust particle
radius.
Effects that can change this expression of the electric force are e.g. polarisation
of the conducting grain and polarisation of the shielding cloud in the sheath of
a discharge. But the force induced by the polarisation of a conducting grain is
exactly cancelled out by an ion drag force that is induced by that polarisation
[37]37 . The same authors derived a force present in sheath and presheath regions
where the Debye length changes spatially. The sheath electric field polarises the
Debye sheath around the particle. This ‘∇λD ’ force writes:
Q2
∇λD
F~∇λD = − d
8πǫ0 (λD + a)2
(2.34)
and points in the direction of decreasing Debye length and thus toward higher
plasma densities (eq. (2.25), p. 23). It supports the electrostatic force in levitating
the dust particles above the lower electrode. Bouchoule et al. calculated the ratio
of the ‘∇λD ’ force to the electrostatic force and found that it is of the order of
a/λD [22]. The ‘∇λD ’ force can therefore only be neglected in cases with λD ≫ a
which is not necessarily true in the case of plasma crystals.
Ion focus
A different electric and attractive force between dust grains can arise in the case of
streaming ions when two or more dust particles are located below each other. As
37
[37] Hamaguchi: Polarization force on a charged particulate in a nonuniform plasma, 1994
34
Chapter 2 Theoretical background
Fig. 2.10: Interparticle forces in a dust molecule.
described in section 2.4.2 (p. 24) an ion cloud can develop beneath a dust particle
when ions are streaming by [27, 29, 30, 31, 32]38 . This means a positive space charge
is generated which attracts the grain directly below. Melzer gives the equation of
motion for the lower particle of a two particle system [30]:
Qupper
Qlower
d
x.
md ẍ = −md β ẋ − (ǫ − 1) d
4πǫ0 d3
(2.35)
Here x is a small deviation from the vertically aligned situation, md is the dust
particle mass, β is the friction coefficient for the dust particles within the neutral
background, and Qupper
and Qlower
are the charges of the upper and lower dust
d
d
grain. ǫ denotes the ratio of attractive (due to ion focus) to repulsive force (due to
negative dust charge). d gives the vertical distance between the upper and lower
particle (Fig. 2.10, taken from [30]).
This electric force caused by the ion focus is of particular importance for the
observed vertical dust particle rows of up to several 10 particles in the presented
experiment.
But in a plasma crystal the situation is more complex. The ion cloud of a neighbour dust molecule within a plasma crystal also attracts the lower particle. Therefore the situation sketched in Fig. 2.10 should be unstable: If the lower particle
moves towards a neighbour dust molecule it is more and more attracted to it and
finally would reach an equilibrium position in the middle between the two dust
38
[27] Lampe: Interactions between dust grains in a dusty plasma, 2000
[29] Melzer: Structure and stability of the plasma crystal, 1996
[30] Melzer: Transition from Attractive to Repulsive Forces between Dust Molecules in a Plasma Sheath, 1999
[31] Melzer: Laser manipulation of particles in dusty plasmas, 2001
[32] Maiorov: Plasma kinetics around a dust grain in an ion flow, 2001
35
molecules. The fact that this does not happen (at low pressures) is due to the ion
stream and the resulting space charge which is not shown in Fig. 2.10 but indicated
in Fig. 2.9 (p. 26). The downstream dust particle of a dust molecule stays relatively
straight downstream the upper particle due to the higher ion density there (region
B in Fig. 2.9, dark regions) compared to regions well aside (region A in Fig. 2.9).
Ion drag
A dust grain in a plasma is continuously bombarded by ions and electrons. The
plasma particles (electrons and ions) thus exert a force (a drag) to the grain. The
force caused by electrons can be neglected because of their low mass. The ion drag
force consists of two parts: (i) the collection force and (ii) the orbit force.
(i) The collection force arises due to collisions of ions with the dust particle. The
ions immediately stick to a dust grain in a collision and transfer their momentum
to the dust particle. Using the cross section for ion collection (equation (2.16), p.
21) the collection force is obtained:
F~coll
2
where vs = u2i + vth,i
1/2
2eφf
= mi vs ni πa 1 −
~ui ,
mi vs2
2
(2.36)
is the mean ion velocity [23]39 . The collection force is
thus proportional to the square of the dust particle radius.
(ii) The orbit force is exerted by ions which are not collected by the grain but
deflected in its electric field. The cross section for Coulomb collisions between an
ion and a dust particle reads after Barnes et al. [22, 38]40 :
!
2
2
λ
+
b
D
π/2
σcCoul = 2πb2π/2 ln
,
b2c + b2π/2
(2.37)
where bπ/2 = eφf a/mi vs2 is the impact parameter (using Qd = 4πǫ0 aφf ) with the
asymptotic scattering angle of π/2 and bc is the critical impact parameter at which
an ion is collected. The logarithm is the so called Coulomb logarithm. This expression only includes ions approaching the dust particle with a critical impact
parameter bc < λD . The orbit force is then given by F~Orbit = ni mi vs σcCoul u~i which
39
[23] Melzer: Introduction to Colloidal (Dusty) Plasmas (Lecture Notes), 2005
40
[22] Bouchoule: Dusty Plasmas, 1999
[38] Barnes: Transport Of Dust Particles In Glow-Discharge Plasmas, 1992
36
Chapter 2 Theoretical background
reads in total:
Barnes
F~Orbit
(eφf a)2
ln
= 2πni
mi vs3
λ2D + b2π/2
b2c + b2π/2
!
u~i .
(2.38)
The orbit force is thus proportional to the square of the dust particle radius as
well.
Khrapak et al. have included ions that approach the dust particle closer than
λD regardless of their initial impact parameter [39]41 . Every ion is included whose
minimal distance to the dust grain is λD . This leads to a considerably higher cross
section for Coulomb collisions in the case of sub-thermal ion flow (ui < vth,i ) and
thus to a higher orbit force. This Khrapak expression for the orbit force is thus
applicable in the presheath region of the discharge. Using again Qd = 4πǫ0 aφf the
orbit force writes [39]:
Khrapak
F~Orbit

√
8 2π
eφf
=
+
ni a2 mi vth,i 1 +
2
3
2mi vth,i
eφf
2
2mi vth,i
!2 
Λ u~i ,
(2.39)
where Λ is the Coulomb logarithm integrated over the shifted Maxwellian distribution:
Λ=2
Z
0
∞
2λD x + ρ0
exp (−x) ln
dx.
2ax + ρ0
(2.40)
2
ρ0 = Qd e/(4πǫ0 mi vth,i
) is the Coulomb radius. The scattering angle is large for ions
with an impact parameter less than ρ0 and small otherwise. The thermal ion velocity
vth,i is used here because ui < vth,i in the presheath region where this formula holds.
In comparison of these both ion drag forces (equations (2.36)—collection force,
(2.38)—orbit force for super-thermal, and (2.39)—orbit force for sub-thermal ion
flow) the following can be concluded: Both ion drag forces are proportional to the
square of the dust particle radius (∝ a2 ) and to the ion drift velocity (∝ ui ).
However, the orbit force is proportional to the inverse cube of the mean velocity
1/2
2
vs = u2i + vth,i
(or to the inverse cube of the thermal ion velocity in the sub-
thermal regime) whereas the collection force has contributions proportional to vs−1
and to vs itself. Thus the orbit force is large at small vs (and small drift velocities
ui ). It is the dominant contribution to the ion drag in the range ui ≤ 4vth,i (see
Fig. 2.11 taken from [23]). The collection force dominates the ion drag at higher
drift velocities.
41
[39] Khrapak: Ion drag force in complex plasmas, 2002
37
Fig. 2.11: Comparison of
ion drag forces. Total force,
collection force, and orbit
orbit
force are shown. The orbit
force is dominant for small
ion drift velocities. This diagram is of general validity
(normalisation to πa2 mi ni ).
The graph is thus independent of dust size, ion mass,
and ion density.
Neutral drag
Ions bombard a dust grain in a plasma and so do neutrals. The neutral drag force
can be calculated using the Epstein formulae for pressures below i.e. 100 Pa:
4
F~N = − πa2 mN nN vth,N (~uD − ~uN ) ,
3
(2.41)
where mN , nN , and vth,N are mass, density, and thermal velocity of the neutrals,
respectively. (~uD − ~uN ) is the relative velocity between dust particles and neutrals
[22]42 . Thus the neutral drag is proportional to the square of the dust particle radius
and to the relative velocity of dust grain and neutrals.
The effects of the neutral drag force are clearly seen in the experiments and
strongly influence plasma crystal formation. Therefore only low and constant flow
rates of Argon gas have been used in this work to minimise neutral drag.
Thermophoresis
A further force can act on the dust particles when the neutral gas temperature
is not uniform. Then the thermal velocity of neutrals coming from the hot side is
higher than the thermal velocity of neutrals coming from the cold side. Thus a net
force is exerted on the dust particle. This force is proportional to the temperature
gradient and reads [22]:
32a2
F~th = −
15vth,N
5π
1+
(1 − α) κT ∇TN ,
32
(2.42)
where κT and TN are the thermal conductivity of the gas and gas temperature,
42
[22] Bouchoule: Dusty Plasmas, 1999
38
Chapter 2 Theoretical background
respectively. Thus the thermophoretic force is proportional to the square of the
dust grain radius and to the temperature gradient.
α is the so called accommodation coefficient which describes the type of neutral
particle reflection considered: α = 0 for specular reflection and α = 1 for perfect
diffuse reflection. Perfect diffuse reflection means that the neutral particle is first
adsorbed at the dust particle surface, reaches the dust particle temperature, and is
then desorbed. The value of α can be adjusted to account for intermediate cases.
Influences of the thermophoretic force on the dust particles have been observed in
this experiment with values of ∇TN as small as 1 Kcm−1 . Experiments are therefore
performed under stable temperature conditions in the laboratory without draught.
The forces discussed so far are present in plasma crystals as well as in the case
of only a few particles in a plasma. The next two forces are found only in plasmas
with a high dust particle density.
Shadowing force
Ions hit a dust particle and exert a force—the ion drag. In a uniform plasma with
no directed ion flow and no density variations the total ion drag to a dust particle
is zero. This is so because equal numbers of ions hit the dust grain from each side
with equal mean velocities. If there are two nearby dust particles—particle A and
B—particle A can collect some ions which would hit particle B if particle A was
not there. The ion drag coming from the inter particle region is smaller than the
ion drag originating from outside. Therefore there exists a net force which pushes
the particles closer to each other. This force is proportional to a4 /r2 where r is the
inter particle distance [40]43 .
Long range repulsion
Far distant dust grains can influence each other via a long range force. This force
is repulsive and originates from the influence of dust particles on ion trajectories.
The trajectory of an ion may be bended due to the presence of other dust particles
in a way that the ion may therefore hit a specific dust particle which it would
miss without the other dust grains. This force is therefore a kind of “dust-particlemediated-ion-drag”. It is proportional to Z 2 a/r3 [40].
43
[40] Tsytovich: Dust plasma crystals, drops, and clouds, 1997
39
Force
Dependencies; equation
Important in. . .
Gravity
a3 , ρparticle ; (2.32)
whole plasma
Electric force
a, φf ; (2.33)
sheath
Ion focus
x, d−3 , Qd ; (2.35)
sheath, presheath
Ion drag—collection
a2 , ni , φf , vs + vs−1 ; (2.36)
sheath
Ion drag—orbit, ui > vth,i
Ion drag—orbit, ui < vth,i
Neutral drag
Thermophoresis
Shadowing force
Long range repulsion
2
a , ni , φ2f , vs−1 (Barnes); (2.38)
−3
a2 , ni , φ2f , vth,i
(Khrapak); (2.39)
a2 , nN , vth,N , ~uD − ~uN ; (2.41)
sheath
a4 , r−2 , r=
b particle separation
dense dust cloud
−1
a2 , vth,N
, −∇TN ; (2.42)
a, r
−3
presheath
whole plasma
whole plasma
dense dust cloud
Table 2.3: Different forces acting on dust particles in complex plasmas, their dependencies on particle radius and plasma properties, and the region where they are important.
Table 2.3 lists the forces discussed and their dependencies.
2.4.8 Producing plasma crystals
The particles in a complex plasma levitate in regions of the discharge where the
different forces are balanced. The force balance is reached near the lower electrode in
the sheath or presheath region of the plasma. A relatively strong electric field exists
in these regions. It is therefore advantageous to have a plasma with an extended
sheath to produce extended plasma crystals.
Sinceqthe sheath thickness lies in the range of several electron Debye lengths with
λDe = ǫ0ek2Bn0Te , a high electron temperature Te and a low electron density n0 are
favourable. The latter is proportional to the input power of a capacitively coupled
discharge (eq. (2.11), p. 16 and [14, 41]44 ). Therefore the experiments were carried
out with the lowest input power possible (5 W or below).
The electron density increases with the discharge pressure (sec. 2.3, p. 16). A
low discharge pressure seems therefore suitable to produce big plasma crystals.
But a high friction force (neutral drag) is needed to damp the dust particle motion
as much as possible. This makes higher pressures more appropriate. Therefore an
44
[14] Lieberman: Principles of Plasma Discharges and Materials Processing – Second Edition, 2008
[41] Boeke: Lithium-Atomstrahl-Spektroskopie als Diagnostik zur Bestimmung von Dichte und Temperatur der
Elektronen in Niedertemperaturplasmen, 2003
40
Chapter 2 Theoretical background
optimum has to be found regarding the discharge pressure.
The electron temperature is not as easily tunable as the electron density. It
is predominantly determined by the production and loss mechanisms of the electrons which are necessary for the maintenance of the discharge. Since the main loss
mechanism are collisions with the wall the discharge geometry sets the electron
temperature (eq. (2.14), p. 17 and [14]).
String formation
The principles of the formation of dust particle strings and crystals are described
e.g. in [42]45 . The ion flow within the sheath where the dust particles are situated
leads to the formation of dust particle strings. A wake develops downstream a
dust particle with a positive space charge directly below the particle. This positive
space charge attracts the nearest downstream neighbour particle (sec. 2.4.7, p. 34).
The dust particle strings are stabilised at higher pressures due to ion–neutral and
dust–neutral collisions. The grain spacing is in the range of λD /3 to λD [42].
At strong vertical confinement the particles within a string are pushed closer
together and a single particle can pop out. The string breaks into smaller strings.
Thus long strings are formed in situations with only weak vertical confinement.
Weak vertical confinement is achieved by lowering the input power (thus increasing
the Debye length and sheath thickness) and increasing discharge pressure. The
latter leads to a smoothing of the wake potential (through ion–neutral collisions)
and thus to a decrease of the attractive force between vertically aligned dust grains.
Furthermore the strings are susceptible to the so called “hose” instability. A
wave-like motion of the string is seen when the string is unstable (Fig. 2.12, left,
taken from [42]). This is essentially a two stream instability caused by the flowing
ions. The grain spacing is not constant within a string but is bigger downstream
a particle. This is due to the increasing ion velocity—ions are accelerated to the
electrode. Therefore the attraction between the dust grains of a string decreases
downstream. This leads to the hose instability with increasing amplitude downstream. Again the string is stabilised by increasing the discharge pressure.
Crystal organisation
When there are several strings the horizontally repulsive forces between the individual grains lead to horizontally hexagonal arrays. This can be seen through the
45
[42] Lampe: Structure and dynamics of dust in streaming plasma: Dust molecules, strings, and crystals, 2005
41
Fig. 2.12: Left: Simulation of the hose instability of a single string [42]. Right: Sketch
of the change in crystal structure due to increased horizontal confinement.
top window of the chamber.
When the horizontal confinement is weak and the vertical confinement strong the
strings arrange in a way that the individual particles are on an equal height in every
string. When the horizontal confinement is strong and the vertical confinement
weaker every second string moves to a higher or lower height so that the particles of
one string are in the middle of the spacing of the particles of a neighbouring string
(Fig. 2.12, right). The horizontal confinement can be adjusted in the presented
experiment by applying a dc voltage to a stainless steel ring on the electrode (sec.
3.3, p. 74).
The possible crystal structures are body-centred-cubic (bcc), face-centred-cubic
(fcc), and hexagonal close-packed (hcp) [43]46 .
Condensation
At low pressures the dust cloud is in a gas-like or fluid-like state with a high kinetic dust temperature (10 to 100 eV). Such high dust kinetic temperatures result
from an ion-dust two-stream instability [42, 44]47 . When the pressure is increased
over a critical pressure pcond the dust cloud condensates abruptly. Ion–neutral and
dust–neutral collisions then effectively damp the instability. The dust kinetic temperature is then approx. room temperature.
Melting
When the pressure is decreased again under a critical pressure pmelt the crystal melts
46
[43] Pieper: Experimental studies of two-dimensional and three-dimensional structure in a crystallized dusty
plasma, 1996
47
[44] Joyce: Instability-triggered phase transition to a dusty-plasma condensate, 2002
42
Chapter 2 Theoretical background
Fig. 2.13: Hysteresis loop traced out by the
dust temperature as the pressure is varied up
and then down (simulation) [42].
and the dust particles acquire a high kinetic temperature. The dust particle kinetic
temperature describes a hysteresis loop when increasing the pressure over pcond and
then decreasing the pressure below pmelt . That means pmelt < pcond as shown in Fig.
2.13 (taken from [42]). This figure shows a simulation of the dust particle kinetic
temperature. But this behaviour is also seen in the presented experiment [45]48 .
48
[45] Aschinger: Struktur und Dynamik von Plasmakristallen, 2008
43
2.5 Light scattering by small particles
The investigation of plasma crystal structures using scattering methods resembles
X-ray diffraction techniques of solid state physics. Similar lattice types occur like
body-centred-cubic (bcc) and face-centred-cubic (fcc).
In order to calculate scattering intensities from a given structure the so called
“structure factor” F is needed. This structure factor is the product of the so called
“atomic form factor” f and a factor which is exclusively determined by the lattice
type:
F =
N
X
j
~
fj exp (i~rj ◦ G)
(2.43)
Abbreviations: F : structure factor, N : number of atoms (for plasma crystals: particles) within a unit cell, f : atomic form factor, ~rj : distance vector from the point
~ reciprocal lattice vector. “ ◦ ” denotes the
of origin to the atoms of a unit cell, G:
scalar product of two vectors.
In the case of identical atoms the atomic form factor can be written in front of the
P
~ is
sum. The latter factor (the geometrical structure factor Fg = N exp (i~rj ◦ G))
j
the same for solid state physics and for the present case whereas the dimensionless
atomic form factor has to be replaced by the so called “efficiency factor” Qsca .
This efficiency factor is for spherical dust particles the scattering cross section Csca
divided by the geometrical cross section:
Qsca =
Csca
.
πa2
(2.44)
To understand this scattering cross section a description of the scattering of light
by small particles seems appropriate. Here the word “light” is synonymic for electromagnetic radiation. The main focus of this section lies on the short discussion
of scattering by particles which are smaller than the wavelength because this is
the case of the experiment under consideration. This type of scattering is called
Rayleigh scattering. For a complete and rigorous treatment see e.g. [46, 47, 48, 49]49 .
Elastic and single scattering are assumed throughout this section. Elastic scattering means there is no change in wavelength through scattering. That implies
49
[46] van de Hulst: Light Scattering by Small Particles, 1981
[47] Bohren: Absorption and scattering of light by small particles, 1983
[48] Born: Principles of Optics, 1975
[49] Jackson: Klassische Elektrodynamik, 1985
44
Chapter 2 Theoretical background
Optical Depth
Scattering Type
τ < 0.1
single scattering
0.1 < τ < 0.3
double scattering
τ > 0.3
multiple scattering
Table 2.4: Definition of single, double, and multiple scattering.
that effects like Raman shifts or electronic excitations are excluded here. Single
scattering means that a scattered wave directly leaves the particle cloud and is not
scattered again. So multiple scattering is exluded as well and it is not necessary to
find a radiation transfer function.
There are two tests to decide if single scattering is dominant in a particle cloud:
(i) Doubling of the particle concentration. Only single scattering is important if the
scattered intensity is doubled as well.
(ii) Measurement of the extinction. The extinction of a beam of light travelling
through a particle cloud is described by exp(−τ ): I = I0 exp(−τ ). τ is the optical
depth of the cloud. Three cases can be distinguished [46] as shown in table 2.4.
2.5.1 Some basics of scattering theory
The 2 × 2 amplitude matrix relates the incoming fields to the scattered fields:
Esca,||
Esca,⊥
!
exp(ik(r − z))
=
−ikr
S2 S3
S4 S1
!
E0,||
E0,⊥
!
(2.45)
Fig. 2.14 shows the scattering geometry and defines the incoming (index 0) and
scattered (index sca) electric fields parallel and perpendicular to the scattering
plane. The scattering plane is defined by the direction of the incident beam and
the direction of the scattered beam.
In a real experiment only intensities can be measured. Therefore the scattering
process needs to be described via incoming and scattered intensities. There are
many possibilities to do this but using the Stokes vectors and the 4 × 4 scattering
45
z
Esca,
particle
h
y
E0,
Fig. 2.14: Scattering geometry. The electric
Esca,||
E0,||
fields of the incident and scattered light are
perpendicular (E⊥ ) to the scattering plane.
scattering plane
v
divided into components parallel (E|| ) and
x
incident beam
θ is the scattering angle.
or Mueller matrix has shown to be very convenient:

Isca

 Qsca

 U
 sca
Vsca



= 1
 k2 r2


S11 S12 S13 S14

 S21 S22 S23 S24

 S
 31 S32 S33 S34
S41 S42 S43 S44

I0



  Q0 


 U .
 0 
V0
(2.46)
The Stokes parameters are:
Isca
=
Qsca =
Usca =
Vsca =
k
2ωµ
k
2ωµ
k
2ωµ
k
i 2ωµ
∗
∗
< Esca,|| Esca,||
+ Esca,⊥ Esca,⊥
>
∗
∗
< Esca,|| Esca,||
− Esca,⊥ Esca,⊥
>
∗
∗
< Esca,|| Esca,⊥ + Esca,⊥ Esca,|| >
∗
∗
< Esca,|| Esca,⊥
− Esca,⊥ Esca,||
>
=
b
=
b
=
b
Intensity
I|| − I⊥
I+45◦ − I−45◦
=
b Iright
circ.
− Ilef t
(2.47)
circ. .
The symbol < · · · > represents the time average and µ the permeability of the
surrounding medium.
The Stokes parameters can be measured using polarisers and λ/4 plates. Isca
is the intensity of the scattered wave which is the sum of scattered parallel and
perpendicular intensities. Qsca is the difference between scattered parallel and perpendicular intensity. (Not to confuse with the efficiency factor of eq. (2.44)!) Usca
denotes the difference of scattered intensities with electric field components rotated
by ±45◦ with respect to the scattering plane. Vsca is the difference between right
circular and left circular polarised scattered intensities.
46
Chapter 2 Theoretical background
Incident Polarisation
Isca k2 r2
Qsca k2 r2
parallel
|S2 |2 I0
|S2 |2 I0
perpendicular
unpolarised
|S1 |2 I0
−|S1 |2 I0
S11 I0
S12 I0
Table 2.5: Stokes parameters for spheres and different incident polarisations. Usca and
Vsca are zero in these cases.
Scattering by spheres
The scattering amplitudes S3 and S4 are zero in the case of spheres:
!
!
!
Esca,||
E0,||
exp(ik(r − z)) S2 0
=
−ikr
Esca,⊥
0 S1
E0,⊥
(2.48)
The scattering functions S1 and S2 write for spheres:
X 2n + 1
(an πn + bn τn )
n(n + 1)
n
X 2n + 1
(an τn + bn πn )
=
n(n + 1)
n
S1 =
S2
(2.49)
The scattering coefficients an and bn and the functions τn and πn are given in [47,
chap. 4]50 .
The stokes parameters follow from equations (2.48) and (2.47):

Isca

 Qsca

 U
 sca
Vsca
S11 =
S33 =



= 1
 k2 r2

1
(|S2 |2
2
1
(S2∗ S1
2

S11 S12
0
0

 S12 S11
0
0

 0
0
S33 S34

0
0 −S34 S33
+ |S1 |2 ) , S12 =
+ S2 S1∗ ) , S34 =
1
2
i
2

I0



  Q0 


 U .
 0 
V0
(|S2 |2 − |S1 |2 )
(S1 S2∗ − S2 S1∗ ) .
(2.50)
(2.51)
The Mueller matrix elements for spheres fulfil the relation:
2
2
2
2
S11
= S12
+ S33
+ S34
.
(2.52)
Table 2.5 shows Stokes parameters of scattered light for spheres and different incident polarisations resulting from the equations above.
50
[47] Bohren: Absorption and scattering of light by small particles, 1983
47
Incident Polarisation
parallel
perpendicular
unpolarised
Isca
2 6
2
16π 4
a
m −1
2
cos
θ
I0
r2
m2 + 2
λ4
2 6
16π 4 m2 − 1
a
I0
2
2
r
m +2
λ4
2
2 6
m −1
a
8π 4
2
(cos θ + 1)
I0
2
2
r
m +2
λ4
Table 2.6: Scattered intensities for different incident polarisations and spheres small
compared with wavelength (Rayleigh scattering). The well known a6 /λ4 dependence is
seen in the formulae.
2.5.2 Rayleigh scattering by one particle
Assumptions underlying the theory of Rayleigh scattering as adopted here:
1. Spherical particles.
2. The external electric field of the light wave is considered to be seen as homogeneous by the particle. Therefore the particles must be small compared to
the wavelength: 2πa ≪ λ.
3. The particles should build up a static polarisation in a short time compared
to the period of the light wave. To meet this condition the size of the particle
should be smaller than the wavelength inside the particle: 2πa ≪ λ/m. Here
m is the refractive index of the dust particle.
4. Homogeneous particles with isotropic polarisability.
The scattering functions S1 and S2 can be obtained using eq. (2.49) [47, chap.
4]51 :
S1 = (3/2)a1
S2 = (3/2)a1 cos(θ)
2 m2 − 1
,
a1 ∼
= −i x3 2
3 m +2
2πa
x=
λ
(2.53)
Inserting these equations into the formulae given in table 2.5 results in scattered
intensities shown in table 2.6.
51
[47] Bohren: Absorption and scattering of light by small particles, 1983
48
Chapter 2 Theoretical background
point P
E
r
q
p
incoming wave
Fig. 2.15: Dipole scattering.
The corresponding Mueller matrix writes:

1
(cos2 θ + 1) 21 (cos2 θ − 1)
0
0
2

1
1
2
2
0
0
9|a1 |2 
 2 (cos θ − 1) 2 (cos θ + 1)

4k 2 r2 
0
0
cos θ
0
0
0
0
cos θ






(2.54)
In the following the term “scattering cross section” is described in more detail.
Energy can be removed out of a beam of light travelling through a sample in two
different ways: scattering and absorption. The total energy scattered by a particle
in all directions can be considered to be equal to the energy of the incident beam
falling on an area Csca . This area is the scattering cross section. The absorption
cross section is defined analogously: The total energy absorbed by a particle is by
definition set equal to the energy of the incident beam falling on an area Cabs . This
area is the absorption cross section.
The sum of these effects—scattering and absorption—is called extinction and is
described by the extinction cross section Cext :
Cext = Csca + Cabs .
(2.55)
The efficiency factors are defined as described above by dividing the cross sections
by the geometrical cross section of the particle.
The formulae given above for scattered intensities can be derived differently with
more physical insight. This is shown in the following.
49
The electric field of a scattered wave in an observation point P is in Gaussian
units [46]:
k2p
~
cos(θ) exp(−ikr)~e.
(2.56)
E=
r
~ electric field, k = 2π/λ: wave number, p: dipole moment, r: disAbbreviations: E:
tance between particle and view point, θ: scattering angle, ~e: unit vector perpendicular to the radius vector as shown in Fig. 2.15 (after [46]).
Following assumptions 2, 3, and 4 at the beginning of this section (p. 47) the
~ 0 is proportional to the incoming electric field E0 where α is
dipole moment p~ = αE
the polarisability of the particle. Incoming and scattered intensities are given by the
absolute value of the according Poyting vectors: I0,sca = c|E0,s |2 /8π (0: incoming,
sca: scattered). Integration of Isca over the surface of a big sphere gives the total
scattered power W = k 4 c|p|2 /3. Dividing this by the incoming intensity I0 results
in the total scattering cross section:
8
Csca = πk 4 |α|2 .
3
(2.57)
For a small sphere the polarisability is given by α = (m2 − 1)a3 /(m2 + 2) where
m = n−iκ is the complex refractive index and a the sphere radius. Using k = 2π/λ
in eq. (2.57) the total scattering cross section then writes:
2 6
128 5 m2 − 1
a
Csca =
π
.
3
m2 + 2
λ4
(2.58)
The a6 /λ4 dependence is seen again.
Fig. 2.16 shows the well known polar diagram of Rayleigh scattered intensity
for unpolarised incident radiation (taken from [46]52 ). From this figure it is clear
that the parallel polarised component of the scattered intensity is not isotropically
distributed over all angles. Under a scattering angle of 90◦ the scattered wave is
linearly polarised perpendicular to the scattering plane (plane of drawing). The
formula for the total scattered intensity Isca with unpolarised incident light reads
[46]:
Isca
8π 4
= 2 1 + cos2 θ
r
m2 − 1
m2 + 2
2
a6
I0 ,
λ4
(2.59)
where r is the distance to the observation point (Fig. 2.15). This formula is identical
to the one given in table 2.6 (p. 47). The cos2 θ contribution corresponds to the
parallel component of the scattered light whereas the other term comes from the
perpendicular component.
52
[46] van de Hulst: Light Scattering by Small Particles, 1981
50
Chapter 2 Theoretical background
Fig. 2.16: Polar diagram for Rayleigh scattering. The scattered intensity is shown for
the case of unpolarised incident radiation. 1 = scattered intensity polarised with electric
field vector perpendicular to plane of drawing (perpendicular polarisation), 2 = scattered
intensity polarised with electric field vector within plane of drawing (parallel polarisation),
1 + 2 = total scattered intensity.
2.5.3 Scattered radiant flux at the detector: particle clouds
Section 2.5.2 gives the formulae necessary to calculate scattered intensities Isca for
single spheres small compared with wavelength. Plasma crystals are considered in
this section. Let N be the number of unit cells of a given plasma crystal and Fg
the geometrical structure factor corresponding to the crystal structure. Now the
scattered radiant flux by such a crystal is estimated. A bcc structure is assumed in
the following.
The radiant flux—which has the dimension of power—at the detector is given
by:
P det = Adet N Fg2 Isca
(2.60)
Abbreviations: P det : radiation power (radiant flux) reaching the detector, Adet :
detector area, N : number of unit cells of the crystal under investigation, Fg : geometrical structure factor (eq. (2.43), p. 43), Isca : scattered intensity from a single
particle.
The input window diameter of the Golay detector is 6 mm and the geometrical
structure factor for the bcc structure is zero or two [50]53 . The number of unit cells
within the plasma crystal can be estimated with the crystal volume and the particle
distances:
53
(20 mm)3 ∼
N=
= 3 × 105 .
3
(0.3 mm)
[50] Kittel: Einführung in die Festkörperphysik, 1993
(2.61)
51
The radiation power which reaches the detector can be estimated using table 2.6 (p.
47). To do this the following values are assumed: Particle radius a = 3.6 µm, real
part of particle refractive index (MF particles) |m| = 1.68, distance particle cloud
to detector: r = 220 mm, wavelength of FIR radiation λ = 118.83 µm, detector area:
Adet = 28 mm2 , incoming intensity of FIR beam I0 ∼
= 0.05 mW/mm−2 , number of
unit cells N ∼
= 3 × 105 , geometrical structure factor for bcc structure Fg = 2. For
incident perpendicular polarisation this results in a radiation power of:
P⊥det ∼
= 8 × 10−11 W ,
a = 3.6 µm.
(2.62)
a = 6 µm.
(2.63)
For particles with 6 µm radius this gives:
P⊥det ∼
= 2 × 10−9 W ,
For an incident parallel polarised beam this value has to be multiplied by cos2 θ
which reduces the radiation power at the detector. The incoming intensity of I0 ∼
=
0.05 mW/mm−2 for the expanded beam of diameter 1 cm at the plasma crystal
corresponds to a FIR laser power density of 0.4 mW/mm2 at the FIR laser output
(product of intensity and beam cross section is constant). This is a typical FIR
laser beam intensity in the diffraction experiments as calibrated with Golay cell
and Pyro detector (sec. 4.1, p. 84).
This calculation does not include losses due to absorption and scattering by the
plasma chamber walls. These effects reduce the radiation power at the detector by
approximately 30% as measured with the Golay detector (see sec. 4.2, p. 90).
Influence of the crystal temperature on the diffraction intensity
The estimations of equations (2.62) and (2.63) do not include the temperature of
the plasma crystal. This will be treated now.
The so called Debye-Waller factor describes the dependence of the total diffraction intensity on the crystal temperature. It is described in [50]54 , appendix A. The
total intensity J scattered by a crystal is:
J=
P det
= N Fg2 Isca
Adet
(2.64)
If the crystal has got a certain temperature T then J is reduced according to:
1
2
2
(2.65)
J(T ) = J0 exp − < u(T ) > G .
3
54
[50] Kittel: Einführung in die Festkörperphysik, 1993
52
Chapter 2 Theoretical background
Debye-Waller factor
1.0
J / J
0.9
2
0
= exp(-1/3 <u > 4
2
2
/ d )
Fig. 2.17: The Debye-Waller factor. The
relative diffracted intensity J/J0 is shown in
J / J
0
0.8
dependence of the root-mean-square displace
√
ment rms(u) = < u2 > in units of the
0.7
0.6
0.5
lattice plane distance d. J/J0 becomes ap-
melting point after
0.4
prox. 0.88 at the melting point after Linde-
Lindemann
0.3
0.2
0.00
0.05
0.10
0.15
0.20
rms(u) / d
0.25
0.30
0.35
mann (rms(u)/d ≥ 0.1). It decreases to 0.5
at rms(u)/d = 0.23.
The exponential factor is the Debye-Waller factor. Abbreviations: J: total diffracted
intensity, < u2(T ) >: mean square displacement of the particles from their lattice
points, G2 : square of the reciprocal lattice vector. Now the Debye-Waller factor is
estimated to clarify the temperature influence on the diffracted signals.
Lindemann gave a criterion for the melting point of a 3D crystal: The root-meansquare displacement of the particles must be greater than 10 % of the inter particle
distance:
q
< u2(T ) > = cb,
(2.66)
with b a typical inter particle distance and c ≥ 0.1. Approximating the typical
particle distance b with the lattice plane distance d, substituting G = 2π/d, and
applying then eq. (2.66) to eq. (2.65) gives:
J(T )
1 2 2
= exp − c 4π .
J0
3
Fig. 2.17 shows this approximation of J/J0 in units of c =
(2.67)
√
< u2 >/d. The
diffracted intensity decreases to ≈ 88 % at the melting point. This shows that even
fluids can be investigated with scattering methods.
Influence of crystal defects and domains on the diffraction intensity
Defects within the crystal structure certainly reduce the diffraction peak intensity.
Formally, a defect reduces the number of unit cells N of the crystal (see eq. (2.64)).
The diffraction intensity therefore decreases linearly with an increasing number of
crystal defects.
The appearance of different crystal domains can lead to a significant reduction
of the intensity of a certain diffraction peak. The Bragg condition is then met for
certain domains only and the rest of the crystal does not contribute to the intensity
53
of the particular diffraction peak. Like defects the appearance of different domains
is represented through the number of unit cells N that contribute to a particular
diffraction peak.
For example: If three different domains are present within the crystal, all with
the same structure and lattice plane distances but with different spacial orientations then the number of unit cells that contribute to a particular diffraction peak
is reduced by a factor of 1/3.
The radiation power reaching the detector is now calculated including the extinction by the chamber walls, the temperature of the plasma crystal, crystal defects,
and crystal domains:
P
det
det N
= TT P X A
− NDef 2
1
2
2
Fg Isca exp − < u(T ) > G
NDom
3
(2.68)
Abbreviations: P det : radiation power reaching the detector, TT P X : transmission of
the TPX chamber, Adet : detector area, N : total number of unit cells calculated with
the crystal volume, NDef : number of defects, NDom : number of domains, Fg : geometrical structure factor, Isca : scattered intensity from a single particle, < u(T ) >:
mean square displacement of the particles from their lattice points, G2 : square of
the reciprocal lattice vector.
Using eq. (2.68) and table 2.6 (p. 47) the total radiation power reaching the
detector can be calculated. The parameter values inserted are listed in table 2.7.
For incident perpendicular polarisation this results in a radiation power of:
P det ∼
= 9.3 × 10−12 W ≈ 10−11 W.
(2.69)
This is too small to be detectable with the Golay cell detector, especially when allowing for a lower FIR power of e.g. only 0.01 mW instead of 0.1 mW at the plasma
crystal. Such a small incident radiation power is possible when blocking parts of
the FIR beam with an orifice to minimise background radiation. Thus the radiation power within a diffraction peak can easily be 10−12 W. Therefore a germanium
detector cooled with liquid helium is used in some experiments. This germanium
detector has a much higher sensitivity than the Golay cell (about four magnitudes
better) and is occasionally provided as a loan by the work group of Prof. Dr. M.
Havenith-Newen, Physikalische Chemie II, Ruhr-Universität Bochum.
The estimations of this chapter imply the assumption that the incoming beam is
a plain wave. This is certainly fulfilled for a single particle but not necessarily for
54
Chapter 2 Theoretical background
Parameter
Notation
Transmission TPX
TT P X
Detector area
Adet
Number of unit cells
N
Number of defects
NDef
Number of domains
NDom
Geometrical structure factor
Fg (bcc)
Particle radius
a
Particle refractive index
m
Distance particles–detector
r
∼
= 220 mm
FIR wavelength
λ
118.83 µm
Intensity FIR beam
I0
q
< u2(T ) >
Root-mean-square displacement
Reciprocal lattice vector
Value
∼
= 0.7
28 mm2
∼
= 3 × 105
0.1 N ∼
= 3 × 104
∼
= 10
2
3.6 µm
1.68
∼
= 0.1 mW/mm−2
G
0.1 d
2π/d
Table 2.7: Values for the calculation of the diffraction power. The transmission of the
TPX chamber includes two walls to account for extinction of the incoming beam and
the diffracted signal. The number of unit cells is calculated invoking crystal volume and
volume of a unit cell. The numbers of defects and domains are roughly estimated from
visual observation using a CCD camera. The value of the root-mean-square displacement
of the particles is chosen to be like at the melting point after Lindemann.
a hole plasma crystal. The FIR beam of the real experiment has a diameter of the
order of one centimetre to assure the incoming wave to be plain.
Chapter 3
Setup
The optimum wavelength of the incident radiation lies in the range of the lattice
plane distances for diffraction experiments on crystals. Therefore crystallography
on solids uses X-rays. For plasma crystals these distances lie in the submm range
(some hundred micrometre). Therefore a radiation source with submm wavelength
is needed. Optically pumped far infrared (FIR) lasers are a well known source for
this radiation [51]55 . They can produce monochromatic FIR radiation with power
high enough for this application. Furthermore the design demands of FIR lasers
are relatively easy to meet because the wavelength is so long. Mirror roughness
for example is not a problem and high quality optical surfaces are not needed.
Therefore it has been decided to build a FIR laser system.
There are several diffraction methods to determine the structure of crystals including Debye–Scherrer for powders and the rotating crystal method for single
crystals. The Debye–Scherrer method uses a monochromatic beam which is scattered by different crystalline domains of a powder. Beam and powder are not moved
in this method and the diffraction peaks are usually recorded by a photographic
film or plate. An equivalent two-dimensional detector is not available for the FIR.
Special two-dimensional detectors are currently developed for specific wavelengths
observed in astronomical investigations but are not available yet.
A low cost and easy-to-handle FIR detector is the Golay cell [52]56 which works
at room temperature and has a theoretical detection limit of 10−9 W. The Golay
detector window has a diameter of 6 mm. Therefore the Golay cell has to be moved
around the crystal very precisely to obtain angle resolved diffraction measurements.
In the rotating crystal method the crystal is rotatable around all three axes with
55
[51] Douglas: Millimetre and submillimetre wavelength lasers, 1989
56
[52] Kimmit: Far-Infrared Techniques, 1970
55
56
Chapter 3 Setup
the incident (white) beam fixed. However, 3D plasma crystals are hardly rotated
without influencing structure and stability. Therefore a mirror system has been
designed which allows to rotate the FIR beam and the detector around the chamber
simultaneously. This mimics the rotating crystal method.
The plasma chamber has to be transparent roundabout for the visible and the
FIR. A material that meets this demand is the polymer TPX (poly-methylpentene).
The chamber is made out of a TPX cylinder with stainless steel top and bottom
parts.
The setup of the laser system, mirror system, and plasma chamber is described
in this chapter. The whole setup is built on two tables. The first one is for the laser
system whereas the second one carries the mirror system and the plasma chamber.
An overview of the experiment is shown in Fig. 3.1.
3.1 The laser system
Setup and characterisation of the laser system is also described by S. Schornstein
[9]57 in detail.
The laser system is situated on an optical table58 . A sketch is shown in Fig.
3.1. The CO2 laser beam has a diameter of 7.5 mm (at the 1/e2 points) and a
divergence of 8.5 mrad at the output coupler of the laser. The beam is deflected
by a mirror (copper coated with gold), focussed by a ZnSe lens (Ø 25.1 mm, focal
length: 600 mm), chopped with a frequency of 10.3 Hz, and again deflected by a
second mirror into the FIR resonator.
3.1.1 The CO2 laser
The CO2 laser PL5 by Edinburgh–Instruments Ltd. is used in this work. It produces
80 lines with wavelengths between 9 and 11 µm and a power of ≈ 50 W on the
strongest line59 . Fig. 3.2 shows a sketch of the CO2 laser. The pyrex tube is 1.3 m
long, double-walled for water cooling, and has a rippled inner surface to ensure
power and stability. A gas flow mix of 7% CO2 , 18% N2 , and 75% He is established
and a dc discharge is ignited between the electrodes and sustained using a voltage
57
[9] Schornstein: AUFBAU UND CHARAKTERISIERUNG EINES FERNIFRAROT-RESONATORS, 2006
58 Newport
RP RelianceT M Sealed Hole Table Top, 2.4 × 1.2 m
59 measured
with CO2 laser power head by Laser2000: PM30, 50W and FieldMaxII -TO, Coherent
Waggon, Golay cell
Laser diode
Yolo telescope
Mirror
Pyro detector
FIR resonator, 1900mm
Chopper
Lens
CO2 laser, 2200mm
Dump
Beam splitter
Tilted mirrors
Laser and plasma chamber tables
Guide rail
Chamber,
electrode,
particles
CCD camera,
objective
Fig. 3.1: The whole setup. The CO2 laser and the FIR laser are situated on the laser table (left). The CO2 laser beam is
collimated by a zinc selenite(ZnSe) lens of focal length 600 mm, chopped with 10.3 Hz, and guided into the FIR resonator via
two gold coated copper mirrors. The FIR laser beam is collimated via a Yolo telescope consisting of two spherical mirrors
with diameters of 100 mm (1st mirror) and 120 mm (2nd mirror) and curvature radii of 850 and 1000 mm. A freezer bag foil
serves as beam splitter and reflects ≈ 10% onto a pyro electric detector (Pyro). Two tilted mirrors guide the FIR beam to the
plasma chamber table (right) to the plasma crystal. The Golay cell is placed on a waggon on a 360◦ guide rail. This waggon is
motorised and can be positioned with an accuracy of 0.1◦ . The mirror system is shown in Fig. 3.11 (p. 67).
57
58
Chapter 3 Setup
ZnSe mirror on piezo
ZnSe window
Golden grating
10 kV
Pyrex pipe
Fig. 3.2: Sketch of the CO2 laser. A gas flow mix of 7 % CO2 , 18 % N2 , and 75 % He is
established within the pyrex tube (gas connections not shown) resulting in a pressure of
3000 Pa. A dc plasma is ignited between the electrodes with a voltage of about 10 kV and
a current of 20 mA. A gold coated grating with 90 lines/mm and a plano-concave ZnSe
window compose the resonator with a length of 1.83 m.
of about 10 kV. A typical dc current of 20 mA is flowing through the discharge. The
pyrex tube is sealed with two zinc selenite (ZnSe) Brewster windows.
The output coupler is a plano-concave ZnSe lens with a radius of curvature of
6 m. A thermally compensated dual cylinder piezo ceramic is attached to the output
coupler to allow fine tuning of the resonator length. A water cooled grating with 90
lines/mm which is gold coated and blazed for 10 µm serves as wavelength selective
element. The cavity length is 1.83 m.
The beam mode is specified by Edinburgh Instruments Ltd. to be TEM00
(> 90%) and M 2 < 1.25. This number is obtained computing:
M2 =
θL ω0,L
,
θG ω0,G
(3.1)
where θL and ω0,L are divergence and beam waist of the laser beam and θG and ω0,G
are the corresponding values of a perfect Gaussian beam with the same wavelength.
A real Gaussian beam therefore has M 2 = 1. Beam diameter and divergence are
specified to 7.5 mm (at output coupler) and 3.5 mRad, respectively.
A self made heat exchanger assures cooling of the pyrex tube. It consists of two
copper pipes with different diameters which are fitted into each other. The inner
one belongs to the closed cycle cooling system of the CO2 laser using water mixed
with an anti-freezer for lower viscosity. The coolant is pumped through the system
by a small commercial aquarium pump. The outer copper pipe is flushed by mains
water which typically has a temperature of 16 ◦ C.
The ZnSe Brewster windows got dirty after about one year of frequent CO2 laser
operation (Fig. 3.3). The laser power then oscillated with an amplitude of about
59
Fig. 3.3: Dirty Brewster window of the
CO2 laser. Dark orange areas are clearly visible. The circles mark areas that intercept
the CO2 laser beam. These areas lower laser
power and disturb the laser mode. The edge
also shows such colouration.
5 W and a period of about 15 s. The FIR power got very low which hints to a mode
change of the CO2 laser beam.
Such colouration effects are known to be caused by oil vapour diffusing out of
the rotary vane pump into the laser discharge tube [53]60 . The oil molecules can
be cracked within the discharge and react with the ZnSe surface. The colouration
occurred in spite of the use of an oil filter that should avoid oil vapour diffusion.
Maybe the oil vapour got into the discharge tube during venting.
The ZnSe Brewster windows were cleaned using lap foils. The contamination
could be removed but some scratches remained on the windows. This is probably
the reason for the lowered CO2 laser power after re-adjustment.
A membrane pump has been tested but resulted in lower laser power due to the
lower pumping speed compared with that of the rotary vane pump (only 2.2 m3 h−1
compared with 3.7 m3 h−1 ). Therefore the rotary vane pump was used again but with
a different venting procedure which avoids oil vapour diffusion into the discharge
tube.
The lapping of the Brewster windows made a re-adjustment of the CO2 laser
necessary. To do this the CO2 laser output coupler has to be replaced by a plate
with a central alignment orifice to define the laser axis (part of the CO2 laser
toolkit). A second orifice is mounted on the grating holder. A HeNe laser beam is
adjusted to travel backwards through FIR resonator and CO2 laser as sketched in
Fig. 3.4. This HeNe laser beam then travels the same path the CO2 laser beam goes
after adjustment. The exact alignment procedure is described in the PL5 CO2 laser
manual.
The laser mode can be checked by placing the second mirror (near the FIR
60
[53] Bründermann: private communication
60
Chapter 3 Setup
A:holder
Mirrors, holders
FIR resonator
B:ground plate
Dump
Beam splitter
CO2 laser
HeNe laser and orifice
Fig. 3.4: CO2 laser adjustment. A HeNe laser beam travels backwards through FIR resonator and CO2 laser. The photo shows a gold coated copper mirror on a special mirror
holder and the third holder.
resonator) onto a third holder (Fig. 3.4). The CO2 laser beam is then deflected
parallel to but behind the FIR resonator onto a chamotte slab. This move of the
mirror does not de-adjust the mirror.
The power achieved is 20 W at the line 9P36 (λ = 9.695 µm). The CO2 laser is
very stable after a warming up time of approximately one hour.
3.1.2 The FIR resonator
A main part of this work was the design, fabrication, and characterisation of the
FIR resonator. It is a wave guide resonator with plane mirrors which is described
in [9] in detail. Fig. 3.5 shows an explicit drawing. The main item is a 1.5 m long
Duran glass pipe of an inner diameter of 48 mm which is vacuum-sealed and fixed
on each side in a stainless steel end piece. A smaller second pipe with inner diameter
of 32 mm is placed into the bigger one using two ring spacers. The spacers are made
of polyethylene and have holes to assure good pumping (Fig. 3.6).
The end pieces have ports for gas inlet and vacuum pump and hold the entrance
and output mirrors. The entrance side is water cooled to avoid refractive index
variations of the ZnSe Brewster window (EF in Fig. 3.5). The stainless steel resonator mirrors (ES, AS) have concave drillings to minimise laser beam distortions.
Different mirror configurations with different surface roughnesses and hole diame-
water cooled quartz pipe
Q
A
A
D
D
C
L
B
C
spacer
AF
K
duran glass pipes
AS
ES
E
water cooling
EF
H
H
D
D
A
R
A : main holder
B : vacuum sealed resonator pipe
C: mounting
D: mirror holder adjustment
A
water cooled quartz pipe
G : resonator length adjustm.
H : entrance window mounting
EF: entrance window
AF: output window
E
F
K
L
: bellows
: support
: output window mounting
: gas inlet
F
G
Q : port for pressure gauge
R : pump port
ES : entrance mirror
AS : output mirror
Fig. 3.5: FIR resonator.
61
62
Chapter 3 Setup
outer pipe
spacer with
holes
inner pipe
Fig. 3.6: Cross section of resonator pipes
and spacer. The spacers fix the inner pipe.
Holes ensure good pumping.
ters have been tested [9]. In the present work both mirrors are polished and have
hole diameters of 2.5 mm (input) and 4 mm (output). The entrance mirror hole is
drilled 3 mm off the centre to avoid back reflections of the CO2 laser beam. The
CO2 laser beam is reflected at the output mirror of the FIR laser and can travel
back into the CO2 laser if the entrance mirror hole is drilled centrally. This would
lower the CO2 laser performance [54]61 .
Two concave stainless steel mirrors have been manufactured with a radius of
curvature of 1000 mm each. The entrance mirror hole diameter is again 2.5 mm and
drilled 3 mm off the centre. The output mirror hole diameter is 4 mm. Both mirrors
are polished as well. This concave mirror cavity is more stable (see e.g. [6]62 ) and
provides better wavelength sensitivity than the plane mirror cavity (see sec. 4.1.1,
p. 81).
Three quartz glass pipes serve as spacers to assure a fixed distance between the
two resonator mirrors. They are flushed by mains water at a constant temperature
of 16 ◦ C. The whole resonator is supported by two triangular holders (A). One
of these holders is not fixed at the table but allowed to slide lengthwise. This is
done to fix the resonator length even when the length of the table changes due to
temperature variations.
The entrance end piece is mounted via two lengthwise ball bearings on two supports (F). This allows adjustment of the resonator length without changing the
mirror tipping. Bellows on entrance and output side ensure mirror adjustments
without vacuum breaks (E). Tipping of the mirrors is accomplished by three mi61
[54] Kuellmann: ENTWICKLUNG EINES OPTISCH GEPUMPTEN FERNINFRAROTLASERS FÜR DEN
EINSATZ IN DER PLASMADIAGNOSTIK UND IN DER MOLEKÜLSPEKTROSKOPIE, 1989
62
[6] Kneubühl: Laser, 1991
63
Fig. 3.7:
Bushings
for
screw
screw nut
quartz tube support.
Left: Bushing with screw.
quartz pipe
quartz pipe
Right: Bushing with screw
nuts.
main
holder
brass
bushing
main
holder
brass
bushing
crometer screws on each side (D).
The output window (AF) is a 3 mm thick quartz plate glued into a polyethylene
holder (K). This holder is also adjustable via two screws and a spring to avoid
possible etalon effects inside the window which could lower the laser power.
The FIR resonator is aligned using a HeNe laser beam as sketched in Fig. 3.8
(p. 65). Yolo telescope and tilted mirrors are not used for resonator adjustment
but for FIR laser beam alignment. The resonator entrance mirror is replaced by
a resonator output mirror with a central hole. The HeNe laser beam then travels
straight through the resonator and is reflected at the output mirror. Observing the
corresponding reflection and back-reflection spots on entrance and output mirror
the mirrors can be adjusted with the micrometre screws (D in Fig. 3.5).
The resonator Duran glass pipes have to be removed for the alignment procedure
and the mirrors have to be of the polished type. If a rough output mirror is intended
to be used during operation the alignment has to be done with a polished mirror
first which is then replaced by the rough one. Alignment apertures can be placed
on a straight rail which is centrally positioned along the resonator axis and fixed
to the table.
The fluids methyl alcohol (CH3 OH) and formic acid (HCOOH) are stored in
polyethylene bottles. The vapours of these fluids are introduced into the resonator
via a flexible polyethylene tube and a port on the entrance side (L in Fig. 3.5).
The pumping is accomplished using a membrane pump63 and a turbo pump64 . A
base pressure well below 5 × 10−3 Pa is achieved measured by a Leybold pressure
gauge65 . The FIR resonator is operated with a small vapour flow to achieve better
stability at pressures between 10 and 30 Pa.
At the beginning of the experiments the water cooled quartz pipes broke due
to disadvantageously designed holders. The tubes were glued into brass bushings
which were attached to the main triangle holders using screws (Fig. 3.7, left). The
63 vacuubrand,
64 Pfeiffer,
model MZ 2C
model TMU 071 P
65 THERMOVAC
TM22
64
Chapter 3 Setup
screw exerted a pressure on brass bushing and quartz pipe which led to the failure
of the pipe. The modified design (Fig. 3.7, right) with threads on the bushings and
screw nuts exerts no pressure and functions very well.
3.1.3 The mirror system
The CO2 laser beam pumps the FIR laser as described in sec. 2.1.3 (p. 9). The
resulting FIR beam is divergent because of the resonator geometry (plane mirrors)
and needs to be focussed onto the plasma crystal. A computer program has been
developed to calculate beam diameters at the plasma crystal using the formulae of
Gaussian beam optics (sec. 2.2, p. 13).
The focussing is done by a Yolo telescope consisting of two home made spherical
mirrors (Fig. 3.1, p. 57). The necessary mirror radii and positions were calculated
using the computer program developed. First and second mirror have radii of curvature of 100 mm and 120 mm, respectively. The mirrors are made of aluminium and
polished to have a low surface roughness. Therefore the mirrors reflect even optical
light very well and a HeNe laser can be used for adjustment. To do this a plate
with a central hole (e.g. a FIR resonator output mirror) is mounted in place of the
FIR resonator entrance mirror and the HeNe laser beam is guided straightly into
the resonator. The HeNe laser beam straightly leaves the FIR resonator through
the output window and follows the beam path of the FIR beam. A sketch of this
adjustment arrangement is shown in Fig. 3.8.
A beam splitter is placed after the Yolo telescope to allow for the observation of
the FIR beam during scattering experiments (Fig. 3.1, p. 57). This beam splitter
is a freezer bag foil which transmits 90% of the FIR beam. About 5% are absorbed
and 5% are reflected as measured with the Pyro detector. This reflected part of the
FIR beam is sufficient to continuously observe the beam with the Pyro detector.
The Pyro detector contains a pyro-electric (ionic) crystal which has a permanent
electric polarisation. When heated the crystal axis parallel to the electric polarisation expands and the distances between the ions of the crystal change which leads
to a charging according to the piezo effect. Furthermore the permanent electric
polarisation changes with temperature. Both effects lead to the charging of the opposite crystal surfaces. The surface charges are not permanent but compensated by
e.g. free electrons. Therefore only temperature changes induce (non-permanent) potential differences between the two opposite surfaces of the crystal. These voltages
can be measured by contacting these surfaces.
65
A
Mirrors
FIR
B
Yolo telescope
resonator
E
C
Beam splitter
CO2 laser
D
E
HeNe laser and orifice
(a) FIR resonator and laser beam adjustment. A HeNe laser (b) Photo showing FIR resonator output window
beam travels through the FIR resonator. The entrance mirror (A), yolo telescope (B + C), beam splitter (D), and
is replaced by an output mirror with central hole.
tilted mirrors (E). The beam path is sketched.
Fig. 3.8: Sketch of FIR laser beam adjustment and photo of the mirror system.
Because of these Pyro detector characteristics only temperature changes can be
detected and the FIR beam has to be chopped. This is accomplished by chopping
the CO2 laser beam. The Pyro signal is then measured by an oscilloscope using the
trigger signal of the chopper.
Two tilted aluminium mirrors deflect the FIR beam to the plasma crystal. On
the plasma chamber table a mirror system can be used to rotate the FIR laser
beam around the plasma crystal. All the mirrors are home made and have a special
design to maximise the mirror surface and to ease the production (Fig. 3.9).
The mirror plate is directly mounted onto the holder and can be adjusted via
two screws. The mirror system on the plasma chamber table is described in section
3.2.
Fig. 3.10 shows reflection coefficients of aluminium for parallel and for perpendicular polarised light of a wavelength of 600 nm. These coefficients don’t differ much
(only 0.06) at an incidence angle of 45◦ . The mirrors of the scattering arrangement
are adjusted to angles around 45◦ . A beam deformation accompanied by a change
Fig. 3.9: Aluminium mirrors.
Left: Mirror with anodised
surfaces.
Right: Mirror components.
66
Chapter 3 Setup
Fig. 3.10: Reflection coefficients of alu-
Reflection coefficients of aluminium
Reflection coefficient
1.00
minium for a wavelength of 600 nm. The re-
= 600 nm, n = 1.2, k = 7.26
0.95
flection coefficients of parallel and perpendic-
0.90
ular polarised light differ by only 0.06 at an
0.85
angle of incidence of 45◦ which is the maximum incidence angle of the scattering ar-
0.80
parallel polarisation
0.75
rangement. Beam distortions due to angle
perpendicular polarisation
dependent reflection coefficients are thus not
0.70
0
10
20
30
40
50
60
70
Angle of incidence in degree
80
90
expected.
of polarisation due to the reflection at the mirrors is thus not expected.
67
3.2 The scattering arrangement
Fig. 3.11 shows a sketch of scattering arrangement and plasma chamber. The FIR
beam is focussed as described in sec. 3.1.3 and guided to the plasma chamber table.
Four additional mirrors can be used to rotate the FIR beam around the plasma
chamber. The first one is fixed whereas the three remaining are mounted on a holder
on a 360◦ guide rail.
The Golay detector is placed on an additional waggon (not shown in the figure
but sketched in Fig. 3.1). Both mirror system and Golay waggon can be motorised.
This allows to rotate FIR beam and Golay cell quasi simultaneously around the
chamber without changing the scattering angle. The rotating crystal diffraction
method is mimicked with this configuration. The mirrors are adjusted using a HeNe
laser beam travelling backwards along the FIR beam path. The HeNe laser beam
spot is guided into the FIR resonator and observed at the FIR resonator entrance
mirror while rotating the mirror system.
The FIR beam can be guided to the plasma crystal straightly without using
the mirrors and therefore with fixed direction. The powder diffraction method is
8
9
10
1
12 2
6
11
3 4
3
FIR laser
7
5
pump
Fig. 3.11: Chamber and mirror system. 1: Dust dispenser, 2: Electrodes, 3: 360◦ guide
rail, 4: Table, 5: Pressure gauge, 6: Mirror for camera observation, 7: Vacuum ports, 8:
Tilted mirrors for FIR beam, 9: Mirror mount on guide rail, 10: fixed mirror mount, 11:
TPX cylinder, 12: HeNe laser holder for adjustment.
68
Chapter 3 Setup
Fig. 3.12: Waggon and guide rail
with gear ring.
Fig. 3.13: Scattering control scheme. Blue
cables are data cables, black cables are for
voltage supply. The dc voltage is modulated
Golay cell
Motor
chopper signal
by a pulse width control which generates rectangular pulses. The pulse width and thereby
the motor speed is controlled with the com-
Lock-In
amplifier
(2x)
puter program “Cockpit”. The voltage polarity can be reversed by a computer controlled
self-made switch. The Golay position is de-
PC
Pulse width
control
9V
termined via an incremental encoder conSwitch
nected to the motor axle. The Golay signal
voltage is recorded with two lock-in amplifiers
(high and low sensitivity).
applied when rotating only the Golay cell around the chamber. A 50 cm long glass
tube with 15 mm inner diameter has been used in some experiments to guide the
FIR beam to the chamber. The beam has been focused into the glass tube with the
Yolo telescope. The glass tube serves as wave guide which results in a beam which
is narrower at the plasma crystal than the beam without the glass tube.
A waggon together with the 360◦ guide rail and its gear ring is shown in Fig.
3.12.
The motorised angle resolved rotation, the Golay cell data storage, and the
plasma control is accomplished via the computer program “Cockpit” developed
in this work. Fig. 3.13 shows motorised rotation and data storage scheme. A dc
voltage with an amplitude of 9 V or 16 V is modulated by a pulse width control.
It generates rectangular dc pulses between 0 V and the maximum amplitude. The
pulse width (the “ON” time) can be set by the computer. This determines the
motor speed. The voltage polarity can also be controlled by a switch actuated by
the computer. This polarity determines the motor speed direction.
Pulse width control and switch are connected to the computer using the National
Instruments Card66 and the connection panel NI BNC 2110. The incremental en66 National
Instruments PCI-MIO-16E-1
69
Fig. 3.14: Angular velocity of the detector
carriage against control voltage—with and
without weight. The two upper curves are
recorded with a supply voltage of 16 V and
the lower curve is for a lower supply voltage
of 9 V. The waggon has been loaded with a
weight of 7.5 kg to obtain the red and blue
Angular velocity in degree per second
5
Angular velocity vs. control voltage
4
3
2
1
no weight, 16 V
0
curves.
with 7.5 kg, 16 V
with 7.5 kg, 9 V
0
1
2
3
4
5
Control voltage in V
Window
Gas
Reflective
membrane
Lenses
Light source
FIR
Photo diode
Absorber
Mesh
(a) Principle of the Golay cell.
(b) Golay cell with aluminium optic
adapter and orifice.
Fig. 3.15: (a) Sketch of Golay principle. The FIR beam heats up absorber and gas.
Heat induced changes of reflective membrane curvature change the photo diode signal. (b)
Photo of Golay cell with optic mount and orifice.
coder signal is recorded by an ADDI DATA card67 .
Fig. 3.14 shows the dependence of the angular velocity of the detector waggon
on the supply voltage and on the weight load of the waggon. The two upper curves
show the angular velocity using a supply voltage of 16 V. This supply voltage is
modulated by the pulse width control. This modulation is controlled by the control
voltage which is the x-axis of the diagram. The angular velocity is independent of
the weight load. The lower curve shows the angular velocity with a lower supply
voltage of 9 V only.
The Golay cell Fig. 3.15 shows the principle of a Golay cell detector (a) and a
photo (b). The FIR radiation passes a HDPE (high density poly-ethylene) window
and heats up an absorber plate. A gas (xenon) is thus heated and expands which
67 APCI
1710, 32 bit
70
Chapter 3 Setup
Chopper-frequency response
300
of the Golay cell
Golay signal in mV
250
200
Fig. 3.16: Dependence of the Golay sensi-
150
tivity on chopper frequency. The error bars
100
are smaller than the point size except of the
50
0
lowest frequency. The chopper frequency is
0
10
20
30
40
50
60
70
Chopper frequency in Hz
80
90
set to 10.3 Hz for the experiments with the
Golay cell.
leads to the bending of a flexible membrane. The membrane back side is reflective
and illuminated by a light source. The light is focussed by a lens system and partly
blocked by a mesh. When the reflected image of the mesh exactly matches the mesh
itself then the photo diode signal is maximum. This signal changes because of heat
induced changes of the curvature of the reflective membrane. Then the reflected
mesh image does not exactly match the mesh itself. The changes of the photo diode
signal are easily measured.
The measurement signal has to be chopped to induce such changes of the photo
diode signal. Fig. 3.16 shows the dependence of the Golay signal on the chopper
frequency. A relatively big error occurs at the very low chopper frequency due to
the unbalance of the chopper blade. The chopper frequency is set to 10.3 Hz for the
experiments with the Golay cell.
The Golay signal is recorded by means of two Lock-In amplifiers68 to increase
the signal-to-noise ratio. One Lock-In amplifier is operated with high sensitivity
to measure diffraction peaks. The 2nd is operated with low sensitivity to precisely
determine the FIR beam forward direction.
Noise reduction The Golay cell has a field of view of FWHM > 30◦ (FWHM:
Full Width at Half Maximum). Therefore an orifice is needed to increase directional
sensitivity and thereby to minimise background noise. An aluminium optic adapter
has been manufactured and attached to the Golay cell. It has four holes to attach
standard optic holders and components. Fig. 3.15 (b) shows a photo of the Golay
cell with mounted optic adapter and orifice.
A black low density polyethylene (LDPE) foil is placed directly in front of the Golay entrance window. It effectively suppresses IR and visible background radiation
68 Stanford
Research Systems, Model SR830 DSP Lock-In Amplifier
71
and stems from the packaging of a laser printer toner.
Three sides of the table and the mirror mounts of the mirror system are provided
with board panels laminated with velours. Velours turned out to be a very good
and cheap absorber material for FIR radiation because of its rough surface. This
avoids background noise.
The Golay cell was originally elastically supported but this turned out to be
disadvantageous. The small shocks and vibrations during the movement of the
Golay waggon and during stops of the waggon induced high signal peaks. Removing
the elastic support eliminated these noise signals.
The germanium detector Some experiments were carried out using a germanium detector in place of the Golay cell. This germanium detector is cooled with
liquid helium and has a four orders of magnitude higher sensitivity than the Golay
cell. Furthermore it can be operated at a frequency of some MHz which is a huge
advantage regarding noise reduction and time resolution. This detector is occasionally borrowed from the work group of Prof. Dr. M. Havenith-Newen, Physikalische
Chemie II, Ruhr-Universität Bochum.
72
Chapter 3 Setup
3.3 The plasma chamber
Fig. 3.17 reproduces the plasma chamber already shown in Fig. 3.11 (p. 67). An
argon plasma is ignited between two parallel aluminium electrodes (2). Dust particles are introduced into the plasma by a dust dispenser (1). This is a small stainless
steel container with holes at the top (for good pumping) and a metal mesh at the
bottom. It is mounted using a spring and can be shaken by knocking at the mount.
The dust particles fall through a 8 mm hole within the upper electrode.
Amplifier
Match
box
6
Power
meter
1
2
13.56 MHz
5
3
PC
4
Valve
control
7
dispenser, 2: Electrodes, 3: Plasma
Pump
Butterfly
valve
Fig. 3.17: Plasma chamber. 1: Dust
crystal, 4: Vacuum ports, 5: TPX
cylinder, 6: Mirror for CCD camera
Gas flow
observation, 7: Pressure gauge.
Different dust particles can be used including melamine resin (melamine formaldehyde, MF) and polystyrene particles of diameters between 3 µm and 20 µm. They
form a plasma crystal under certain plasma parameters (3).
Vacuum ports (4) allow attachment of pressure gauge (7) and gas flow pipe. The
gas flow is introduced near the butter fly valve far away from the chamber. This
minimises gas streams inside the plasma volume which would disturb the plasma
crystals. Butter fly valve and pressure gauge are controlled by a MKS controller69
via the computer program “Cockpit”. The system can be operated at constant
pressure or fixed valve position.
69 MKS
600 Series Pressure Controller
73
A rotary vane pump70 with oil filter or a scroll pump71 pumps the chamber
volume down to a base pressure of about 10 Pa. A turbo molecular pump72 can be
used to obtain lower pressures. The argon gas flow is measured by a MKS mass
flow meter73 and controlled by the MKS mass flow controller74 . Typically gas flows
between 0.5 sccm and 2.0 sccm are used (sccm =
b standard cubic centimetre per
minute).
A TPX cylinder (poly-methylpentene) is used as chamber wall. This polymer
is transparent in the visible and in the FIR region of the spectrum. The cylinder
allows roundabout optical access which is necessary both for camera observations
in the visible and diffraction experiments in the FIR.
Such a TPX cylinder is not easily bought from a supplier because usually suppliers want to sell big numbers of pieces. Therefore a 5 litre TPX measuring cup
is used as plasma chamber which can be obtained from a chemical equipment supplier. The cup was cut at top and bottom to obtain a cylinder with precise edges.
These cylinder edges were then glued into 1 cm deep grooves of aluminium flanges
using silicone.
The top of the plasma chamber has a central window and the upper electrode
has a central hole for top view camera observations. The central hole within the
electrode is closed by a metal mesh to minimise plasma sheath potential distortions.
The top view observations are done using a tilted mirror to be able to place the
CCD camera outside the 360◦ guide rail (Fig. 3.1, p. 57 and Fig. 3.17).
Both electrodes can be powered but in most of the experiments the upper electrode is used. A frequency generator75 provides an rf signal (rf =
b radio frequency)
with frequency of 13.56 MHz and ≈ 2 V peak-to-peak amplitude. A rf amplifier76
amplifies this signal. Forward power and standing wave ratio (SWR) are measured
with a NAP power meter77 directly before a matching network. In the experiments
the forward power typically is 10 W with a standing wave ratio of typical 10 to 20.
The matching network consists of a high pass filter followed by a blocking capacitor (Fig. 3.18). This means the electrical power is capacitively coupled to the
70 Pfeiffer
71 Varian
Vacuum DUO 5
Model IDP3
72 Pfeiffer
Vacuum TMU 261
73 MKS
Instruments, 10 sccm N2 , Viton
74 MKS
Type 247, 4 Channel Readout
75 Agilent
33120A 15MHz Function/Arbitrary Waveform Generator
76 Kalmus
Wideband RF Power Amplifier, 0.001 – 100 MHz, 15 W, 43 dB gain
77 Power
Reflection Meter – NAP 392.4017.02
74
Chapter 3 Setup
blocking
capacitor
rf input
rf output
Fig. 3.18:
low pass
filter
high pass
filter
self bias UDC
Matching network
consisting of high pass filter,
blocking capacitor, and low pass
filter for measuring self-bias or
applying a dc voltage.
electrode of the plasma chamber. The match box has a port to measure the self-bias
(sec. 2.3, p. 17) through a low pass filter. The self-bias has a typical value of -100 V
with an input power of 20 W and -10 V with an input power of 1 W (10 Vpp at a
pressure of 30 Pa, measured with a voltage–current–probe near the chamber).
The port for self-bias measurements also allows to apply an external dc voltage
to the powered electrode. A positive dc voltage of about 1 to 10 V is applied to
the powered electrode to stabilise the plasma crystals in some experiments. This
voltage changes the sheath potential of the powered (upper) electrode. As a result
the plasma crystals can be stretched vertically.
Upper and lower aluminium electrodes are sketched in Fig. 3.19. The upper
electrode has a central hole which is closed by a metal mesh to minimise sheath
potential distortions. This hole allows top view observations as described earlier. A
second hole allows to fill in dust particles.
The lower electrode also has a central and outer mesh to allow dust particles to
fall through. A trash can below the lower electrode stores such dust particles which
were not levitated in the plasma or which fell through the meshes after switching
off the plasma. Two integrated rings can be set to different potentials of up to
Fig. 3.19:
Electrodes of the plasma
chamber. Both have a central and a lateral hole. In the upper electrode they
serve as window and dust input hole. The
holes in the lower electrode are for dust
particle recycling. The lower electrode has
two rings (red and blue in this figure)
+
+
which can be biased separately. Both electrodes and the rings can be powered.
75
400 V (relative to ground). Such voltages deform the sheath potential to provide
an effective particle trap. The body of the lower electrode is always grounded.
The lower electrode surface profile is a crucial parameter in producing plasma
crystals. Therefore the electrode was manufactured with a surface as flat as possible.
It is made of aluminium (like the upper electrode) and the meshes are composed
of several 1.5 mm holes drilled into the material.
Nevertheless, this design turned out to be disadvantageous: The dust particles
were pushed away from the centre. Therefore some modifications were applied including the usage of an stainless steel plate on top of the inner ring. The modifications are described in sec. 4.3.1 in more detail.
A dressler rf generator78 can also be used to power the upper electrode. It is
operated at about 40 W real power (forward minus reflected) and controlled via
the computer program “Cockpit”. The power is damped down to about 5 W using
three T-fittings and dummy loads. This allows to vary the power in 0.25 W steps
and the generator operates at a favourable working point.
78 dressler
CEASAR RF Power Generator, 13.56 MHz
76
Chapter 3 Setup
3.4 Setup for calibration scattering experiments
Experiments have been conducted to calibrate the scattering arrangement. A golden
mesh consisting of 40 µm × 40 µm gold squares with distances of 200 µm has been
deposited onto a small GaAs wafer (≈ 2 cm ×1 cm, Fig. 3.20). The mesh properties
were chosen to assure high diffraction signals and to cover the entire wavelength
range. Therefore the gold squares are made relatively large: The geometrical cross
section is about 15 to 40 times larger than that of the dust particles. The lattice distance was chosen to be 200 µm to separate the diffraction peaks of the
118.83 µm and of the 170.58 µm line for ease of detection.
The GaAs wafer transmits about 80% of the FIR radiation. A standard lens
holder by Linos has served as wafer holder. The holder has been placed onto the
lower electrode near the position of the plasma crytals. Special care has been taken
to minimise background noise and possible reflections of the incoming FIR beam
at electrode and mirror holders. This has been achieved by masking electrode and
holders with velours boards (Fig. 3.20).
The FIR beam has been guided straightly to the gold mesh and diffraction peaks
have been recorded with the Golay cell.
Fig. 3.20: Gold mesh and holder. Left: Microscope image of the gold mesh. The golden
squares (bright spots) are 40 µm × 40 µm and 200 µm apart. Right: Mesh holder on
lower electrode of the plasma chamber. Electrode and chamber are covered with velours
boards to minimise reflections of the FIR radiation.
77
3.5 CCD camera diagnostics and video analysis
The dust particles are illuminated using a diode laser with a wavelength of 682 nm
and a power of about 40 mW79 . An integrated optic produces a laser sheet with
8◦ apex angle and about 117 µm width (lateral Rayleigh length). The scattered light
of the dust particles is observed using a Basler progressive scan CCD camera80 and
a zoom objective81 .
The videos are stored in an 8 bit format (Y8/Y800, 256 levels of brightness)
using the free available program “Virtual VCR”82 . Video conversion from the Y8
to the RGB format is done with the program “Virtual Dub” and video analysis is
accomplished using the software IDL83 .
Several procedures for particle tracking and particle motion analysis have been
provided by Uwe Konopka from the group of Prof. G. E. Morfill84 . They have been
developed further in this project by Andreas Aschinger partly using and complementing the work of other programmers85 . The analysis software and first results
of plasma crystal video analysis are extensively described in [45]86 . Therefore only
a short summary and description is given here. Table 3.1 lists the working scheme
of the analysis software.
1. Video processing and saving
Several filters are applied to optimise the video quality for the particle tracking
procedure. The “Avisynth Frame Server” is used for this task [55]87 . It provides
several different filtering techniques and allows own filter development.
Access to the “Video for Windows” application programming interface is needed
for reading and writing AVI-files with the Avisynth Frame Server. A program package by Oleg Kornilov is used for this task [56]88 .
A graphical preview interface developed in this project by Andreas Aschinger
79 Schäfter
+ Kirchhoff diode laser, P ≤ 40 mW, λ = 682 nm, cw, model: 5LM-8-S325-L + 25CM-660-40-M02-A8-2
80 BASLER
81 Nikon
82 Virtual
83 IDL:
A622F, progressive scan, 1280 × 1024 pixel, 24 fps, up to 500 fps with 200 × 160 pixel field of view
ED AF MIKRO NIKKOR 200 mm 1:4D
VCR: Virtual Video Cassette Recorder
Interactive Data Language
84 Max–Planck–Institut
85 David
für extraterrestrische Physik – Theorie und komplexe Plasmen, Garching, Germany
W. Fanning: “Coyote’s Guide to IDL”, Oleg Kornilov, Matthew W. Craig, NASA–Astronomy–Lib
86
[45] Aschinger: Struktur und Dynamik von Plasmakristallen, 2008
87
[55] Avisynth Frame Server, http://avisynth.org
88
[56] Kornilov: AVI, MPEG, QT reading, writing, and preprocessing,
www.kilvarock.com/freesoftware/dlms/avi.htm
78
Chapter 3 Setup
Task
Methods
1. Video pro-
Filtering using the “Avisynth Frame Server”: Gamma correc-
cessing and
tion, salt’n’pepper, cropping, spacial Gauss filter, saving in
saving
HDF file
2. Find particle
Find optimum intensity threshold, eliminate pixel defects, de-
coordinates
termine intensity-weighted particle centre
3. Determine
Track particles from frame to frame, find optimum search area
particle trajec-
with graphical interface developed, eventually use particle po-
tories
sition prediction
4. Data analysis
Velocity distribution in x and y direction, distribution of |v| ⇒
and illustration
crystal temperature, mean particle velocity against time, particle velocity against height above lower electrode, area density, pair correlation function ⇒ plane distances and crys-
tallinity, Wigner–Seitz cells ⇒ defects
Table 3.1: Working scheme of the analysis software.
allows the application of different filters to selectable frame sequences of a video
and the examination of the filtered video parts. The brightness and contrast filter
is the main filter. It enhances high intensities via the so called Gamma correction.
The bright dust particles are then clearly distinguishable from the background. The
salt’n’pepper filter eliminates pixel defects and background noise. The cropping
filter allows to cut out parts of the video image e.g. such parts in which no particles
are visible.
The Gauss filter changes the intensity distribution of a particle image to obtain
a Gaussian brightness profile. The original intensity distribution is not smooth and
exhibits a strong pixel effect: Neighbouring pixels of the CCD chip can measure very
different intensities which leads to steps in the intensity distribution of a particle
image. The Gauss filter smooths them out which improves the accuracy of the later
determined particle coordinates and velocities.
The data are stored in an HDF file [57]89 . This data format allows the successive
processing of very large data amounts. It is possible e.g. to open a single frame
out of a whole video without opening the video itself. It is also possible to save
attributes together with the data. The applied filtering order or the frame rate of
89
[57] Hirachical Data Format, http://www.hdfgroup.org
79
the video can be saved for example.
2. Find particle coordinates
The dust particles have to be identified within each frame of a video. They must
be found within the 2D intensity distribution of each image. In order to separate
the particles from the background noise an intensity threshold is determined with
the graphical preview interface. Intensity values below the threshold are ignored in
the following analysis.
Care must be taken to obtain the right threshold value. The threshold value must
be chosen so that background noise is suppressed but particles of the crystal plane
of interest are still visible. The centre of intensity of each dust particle image is
calculated analogously to the centre of mass of a given body in basic mechanics.
This centre of intensity is the particle coordinate in the subsequent analysis.
3. Determine particle trajectories
The dust particles must be followed and assigned from video frame to video frame
to obtain particle trajectories. Different methods of particle tracking can be applied:
“Simple–Search”, “Area–Search”, and “Educated Guess”.
Simple–Search The particle search within the actual frame starts at the very
same coordinate of the particle in the last frame. The first particle that is found is
assigned to that particle of the last frame. Wrong assignments are very likely with
this Simple–Search if the particle velocities are too high. Then the particle is too
far away from the last position and a different particle is probably assigned.
Area–Search A search area around the last position is defined. The particle is
looked for only within this area in the next frame. This search area is actually a
velocity limit because fast particles can move out of the search area and cannot
be assigned. This leads to a cut-off at a certain velocity in the calculated velocity
distribution. Furthermore the number of particle trajectories increases because a
particle which cannot be assigned counts as new and additional particle.
On the other hand particle trajectories can mix up if the search area is too large.
Then particles are assigned wrongly and trajectories contain large jumps which are
physically not meaningful. The search area size can be optimised using the already
mentioned graphical interface with preview function.
Educated Guess Using the velocity calculated from the last two frames a prediction for the particle position in the next frame is made. This is a big enhancement
80
Chapter 3 Setup
in many cases. But if the particles perform only small oscillations around a fixed
position this method can lead to a prediction which is too far away from the actual
position. Then the particle cannot be assigned as well. This effect strongly depends
on the ratio of the particle oscillation frequency to the frame rate of the CCD
camera. A higher frame rate reduces this error.
4. Data analysis and illustration
The velocity distribution in x and y direction and |v| can be plotted and analysed.
The particles drift if the maximum of one or both of the 1D velocity distributions is
not at velocity zero. A fit of the |v| distribution gives the crystal temperature if it is
a Maxwell distribution. A diagram of the mean velocity against time shows the long
term stability of the crystal against velocity (temperature) changes. Differences of
the particle velocities in different heights above the lower electrode can also be seen
in a separate diagram. The area density (number of particles divided by field of
view) of the particles can be plotted as well.
The pair correlation function can be calculated and fitted. The positions of the
different peaks are unique to the possible crystal structures (e.g. hcp, bcc) and
give the lattice plane distances. Height and width of the peaks give information
about the crystallinity of the crystal. Smaller widths mean lower temperatures and
smaller heights mean a worse structure. The ratio of peak height to peak width is
the crystallinity.
Defects in the crystal structure can be visualised with the Wigner–Seitz cells.
E.g. each particle has six neighbours in a perfect hcp structure. The fraction of
lattice points with less or more neighbours describes the quality of the crystal.
Chapter 4
Results
4.1 Properties of the laser system
Some properties of laser system and beam characteristics are already described in
[9]90 . Here only the most important results are summarised and new results are
presented.
4.1.1 FIR laser operation and beam characteristics
Plane mirrors
Fig. 4.1 shows the Pyro signal of the λ = 170.58 µm FIR line versus the methanol
vapour pressure within the resonator for different mirror roughnesses and mirror
hole diameters. The combination with polished mirrors and output mirror hole of
4 mm diameter shows the highest FIR signal. The input mirror hole diameter is
2.5 mm in any case. The highest FIR power with this configuration is achieved for
a small pressure range around 15 Pa.
Fig. 4.2 shows FIR laser beam profiles at 55 mm (a) and 1080 mm (b) distance to
the output mirror. The profiles are obtained by precisely moving the Pyro detector
laterally and reading the signal amplitude from the oscilloscope. Apertures of 1 mm
diameter (at 55 mm distance) and 5 mm diameter (at 1080 mm distance) increased
directional sensitivity of the Pyro and minimised background noise. Gauss fits reveal
beam radii of w = 2.2 mm at 55 mm distance and w = 50.3 mm at 1080 mm distance.
The beam diameter at 1080 mm distance gives a divergence of the FIR laser beam
of about 5.3◦ .
90
[9] Schornstein: AUFBAU UND CHARAKTERISIERUNG EINES FERNIFRAROT-RESONATORS, 2006
81
82
Chapter 4 Results
FIR power vs. pressure
6
Pyro signal in V
5
output
output
input
mirror
mirror
mirror
hole
4
3
2
1
roughness
roughness
2mm
rough
2mm
polished
rough
rough
3mm
polished
rough
3mm
rough
4mm
polished
polished
5mm
polished
polished
7mm
polished
rough
rough
0
0.0
0.1
0.2
0.3
0.4
0.5
Pressure in mbar
Fig. 4.1: FIR signal versus pressure for several mirror configurations. Waveguide diameter: 48 mm. The combination of both mirrors polished and a 4 mm output hole gives the
highest FIR laser signal within a small pressure range around 15 Pa. (Input mirror hole
diameter: 2.5 mm in any case.)
120
Beam profile at 55mm distance
= 170.58
m
Mirror hole: 3 mm
80
Mirror type: polished
Pressure: 24 Pa
60
Beam profile at 1080 mm distance
Aperture: 1 mm
Beam radius w = 2.2 mm
40
Measurement
Gauss fit
Pyro signal in mV
Pyro signal in mV
100
8
6
= 170.58
m
Aperture: 5 mm
4
Mirror hole: 3 mm
Mirror type: polished
2
Beam radius w = 50.27 mm
20
Measurement
Gauss fit
0
90
92
94
96
98 100 102 104 106 108
x in mm
(a) FIR beam profile at 55 mm distance.
40
60
80
100
120
140
x in mm
(b) FIR beam profile at 1080 mm distance.
Fig. 4.2: FIR laser beam profiles at 55 mm and 1080 mm distance to the output mirror
of the FIR resonator. x denotes the lateral displacement of the Pyro. 1 mm and 5 mm
apertures have been used to increase directional sensitivity of the Pyro. The output mirror
has been of the polished type with a 3 mm hole. Gauss fits reveal FIR beam radii of 2.2 mm
(a) and 50.3 mm (b).
83
B
A
A
B
Fig. 4.3: 3D visualisation of the FIR beam. Left: IR camera VarioCAM with HDPE lens
and LDPE foil in front of the output side of the FIR resonator. The CO2 laser cover is
visible in the background. Right: IR camera image without LDPE foil. The background
(arrow A) results from the hot output window of the FIR resonator. This is heated by
the CO2 laser beam leaking through the output mirror hole. Arrow B marks the FIR laser
beam. The inset shows an image with LDPE foil.
Setup and result of a 3D visualisation of the FIR laser beam are shown in Fig. 4.3.
An infrared (IR) camera91 has been placed directly into the FIR laser beam near
the output end of the FIR resonator. The IR optics has been replaced by a black
low density polyethylene (LDPE) foil and a high density polyethylene (HDPE)
lens with focal length of 50 mm. The LDPE foil stems from the packaging of a laser
printer toner and effectively suppresses IR and visible background radiation. The
HDPE lens has been glued onto the LDPE foil. It focusses the FIR beam onto the
sensor chip of the IR camera. The experiments with the IR camera are published
in [58]92 .
Concave mirrors
Two concave stainless steel mirrors have been manufactured and installed to further
improve the stability of the FIR laser resonator. These mirrors are polished as
well and they both have a radius of curvature of 1 m. This gives a stable concave
resonator since the resonator length is 1.5 m. The entrance mirror has an offset hole
of 2.5 mm diameter and the output mirror has a central 4 mm diameter hole like the
plane mirrors. The FIR laser with concave mirrors shows a more stable operation
91 VarioCAM
92
by Jenoptik, exclusive distribution by InfraTec
[58] Bründermann: Erste THz-Videos mit einer Silizium-basierten IR-Kamera, 2006
84
Chapter 4 Results
Resonator scan
3
Pyro signal in mW
Pyro signal in mW
3
2
1
118.83
m
170.58
m
2
1
118.83
m
170.58
m
both lines
both lines
0
0
200
400
600
Resonator position in
800
1000
0
600
700
m
(a) Resonator scan with concave mirrors.
800
Resonator position in
900
1000
m
(b) Right part of the left diagram.
Fig. 4.4: Resonator scan with concave mirrors. A clear separation of the lines is not always possible. The resonator is more stable in the right part of the diagram. The entrance
mirror is closer to the resonator pipe there which reduces losses.
and a certain wavelength can be selected more easily. Fig. 4.4 shows the observed
FIR laser lines during a resonator scan. Such resonator scans are shown in [9]93 for
the plane mirror setup. The FIR resonator length is tuned to a value of 4.172 mm
on the micrometer screw reading to obtain the 118.83 µm line. This corresponds to
the resonator position of 883 µm in Fig. 4.4. The entrance mirror is very close to
the resonator pipe at this position. This minimises radiation losses when the FIR
radiation couples out of the resonator pipe to be reflected at the entrance mirror.
This mirror position provides a stable laser signal.
Fig. 4.5 shows pressure dependencies of both lines for plane and concave mirrors.
The optimum working pressure for concave mirrors is higher than for plane mirrors
and lies between 0.25 mbar and 0.35 mbar. The pressure profile of the 118.83 µm line
is broader than that of the 170.58 µm line for concave mirrors.
4.1.2 The FIR laser power
The Golay cell detector has been used to calibrate the FIR laser power. The sensitivity of the Golay cell is 32 kVW−1 . (The sensitivity of the Pyro detector is
not known in the wavelength region used in this experiment.) Since the maximum
radiation power permanently tolerable by the Golay cell is 10 µW the FIR laser
radiation has to be damped. Therefore the procedure of the experiment was as
93
[9] Schornstein: AUFBAU UND CHARAKTERISIERUNG EINES FERNIFRAROT-RESONATORS, 2006
85
Fig. 4.5: Pressure dependence of the FIR
lower signal amplitude with concave mir-
2.5
rors is due to the polishing of the CO2 laser
Brewster windows (sec. 3.1.1, p. 59). For
concave mirrors the optimum pressure is
8
2.0
6
1.5
4
1.0
2
higher and the laser can be operated be-
0.5
tween 0.25 mbar and 0.35 mbar for both
0.0
0.0
lines.
10
Pressure dependence
Pyro signal in V
3.0
Pyro signal in mW
signal with plane and concave mirrors. The
0.1
0.2
118.83
m, concave
170.58
m, concave
170.58
m, plane
0.3
0.4
0
0.5
0.6
Pressure in mbar
follows:
1. Measurement of the FIR laser signal without damping using the Pyro detector.
2. Measurement of the FIR laser signal with damping (by a piece of paperboard)
using the Pyro detector. The damping factor can be calculated from these
measurements.
3. Measurement of the damped laser power using the Golay cell at the very same
position.
The damped laser power can be calculated in absolute numbers invoking the Golay
cell sensitivity. Multiplying this damped laser power by the damping factor gives
the undamped laser power.
The undamped and damped FIR laser signals have been 643 mV ± 23 mV and
1.82 mV ± 0.02 mV measured with the Pyro and using a Lock-In amplifier and the
PC program “Cockpit”. The statistical errors (one sigma) are due to laser fluctuations. These values give a damping factor of about 353.3 ± 13.2. The damped FIR
laser signal has been 126.2 mV ± 2.6 mV measured with the Golay cell. Multiplying
with the damping factor and invoking Golay cell sensitivity and error propagation this gives an undamped FIR laser power of about 1.39 mW ± 0.06 mW in this
experiment.
The Pyro signal was simultaneously recorded by the oscilloscope with a value
of about 1.5 V. The sensitivity of the Pyro is therefore about 1 V(mW)−1 in this
wavelength region. Typical Pyro signals are 4 to 5 V during diffraction experiments
which gives a typical FIR laser power of 4 to 5 mW. This radiation power falls
onto the Pyro detector placed directly at the FIR laser output. The FIR beam
86
Chapter 4 Results
diameter is approximately 4 mm there. The laser power of 5 mW thus corresponds
to an intensity of about 0.4 mW/mm2 .
4.1.3 Characteristics of the beam splitter
A beam splitter can be used to record the FIR laser power during diffraction experiments (see Fig. 3.8, p. 65). The beam splitter consists of a freezer bag foil clamped
onto an aluminium holder. It has a transmission of about 90 % measured with the
Pyro detector.
The FIR laser can produce the two wavelengths 118.83 µm and 170.58 µm simultaneously with certain resonator lengths set (see Fig. 4.4). If there is a difference
in wavelength sensitivity between Golay and Pyro detector or if there is a strong
wavelength sensitivity of the beam splitter then the recording of the FIR beam
with the beam splitter can lead to wrong results.
FIR laser beam signals deflected by the beam splitter have been measured with
the Pyro and the Golay detector to exclude any wavelength sensitivity. Special care
has been taken to operate the laser at only one single wavelength using the concave
mirrors as resonator mirrors in this experiment. Both FIR lines (118.83 µm and
170.58 µm) are partly deflected by the beam splitter. The measured signals have
the same values after correction for the different window diameters and window
materials of Golay and Pyro (Fig. 4.6).
Fig. 4.6: Calibration of the beam splitter.
Signal in
W
10
Calibration of the beam splitter
The Pyro signal is corrected for window di-
9
ameter and window material. Golay win-
8
dow: HDPE, 6 mm diameter, Pyro win-
7
dow: Ge, 5 mm diameter. The beam split-
6
ter partly reflects both lines and the detec-
5
Golay, 118 m
Golay, 170 m
4
Error bars: statistical error due to noise
3
+ 4.3% error of the power calibration
1
2
3
Measurement
Pyro,
118 m
Pyro,
170 m
4
tors measure the reflected signals equally.
The discrepancy between Golay and Pyro in
measurement two results most probably from
a slight misalignment of the Golay cell.
The Golay window has a diameter of 6 mm and is made out of high density
polyethylene (HDPE). One millimetre of HDPE has a transmission of about 90%
in the wavelength region of interest [59]94 . The Pyro window has a diameter of 5 mm
94
[59] Fischbach: Eigenschaften optischer Materialien, 2004
87
and is made out of germanium (Ge). 1.5 millimetre of Ge have a transmission of
about 42% [59]. The ratio of Golay window area to Pyro window area is 1.44. This
is a first correction factor for the Pyro signal. The transmission of 42% of 1.5 mm
of Ge corresponds to 56% transmission for 1 mm Ge roughly assuming a linear
dependence (transmission of 1.0 mm = 4/3 × 0.42). The ratio of the transmissions
of HDPE to Ge is then 90/56 = 1.61. The correction factor for the Pyro signal is
therefore 1.44 × 1.61 ∼
= 2.3.
The thickness of both windows is not known and a difference in this parameter
could rise the need for a third correction. But even without this correction the
deflected signals are equal within the error bars no matter if measured with the
Golay cell or Pyro detector. Only measurement two in Fig. 4.6 shows a difference
between the detectors which is likely due to a misalignment of the Golay cell. This
shows that beam splitter and detectors have no significant wavelength sensitivity
in the wavelength region of interest.
88
Chapter 4 Results
4.2 Results of calibration scattering experiments
The setup described in sec. 3.4 (p. 76) has been used to perform calibration experiments to demonstrate the experimental procedure. The FIR beam has been guided
straightly to a mesh consisting of golden squares (40 × 40 µm2 , 200 µm apart)
deposited on a GaAs wafer. This wafer has been placed on the lower electrode
of the plasma chamber and diffraction peaks have been recorded using the scattering arrangement described in sec. 3.2 (p. 67). The TPX plasma chamber has
been removed in a first experiment. Fig. 4.7 shows the diffraction peaks obtained
with this setup. The black curve shows the diffraction peaks and the main FIR
7
Scattering by golden mesh
6
7
Fig. 4.7: Diffraction peaks of the golden
6
mesh (black curve). Each peak is labelled
5
with diffraction angle, order, and corre-
Signal in nW
5
dsin
FIR beam
= n
35°
4
4
n=1
3
2
-35°
-57°
n=1
1
118 m
n=1
56°
3
n=1
170 m
118 m
170 m
0
-80
-40
-20
0
20
40
Scattering angle in degree
60
80
sponding wavelength. The 118.83 µm and
the 170.58 µm lines are simultaneously
2
present within the FIR beam. The red
1
curve shows the main FIR beam recorded
0
-60
Signal in µW
direct
Maxima:
with the Lock-In amplifier with lower sensitivity.
beam (in forward direction) recorded with the Lock-In amplifier with high sensitivity (max. signal: 200 µV =
b 6.25 nW). The diffraction peaks are distributed
symmetrically around the forward direction. The diffraction angle of each peak can
be calculated using the formula d sin θ = nλ for the diffraction of a planar wave
incident on a mesh (d: lattice constant (200 µm); θ: diffraction angle; n: diffraction order; λ: wavelength). Two lines of 118.83 µm and 170.58 µm wavelength are
simultaneously present within the FIR beam.
Different amplitudes of the peaks of one wavelength result from a slight misalignment of the golden mesh which is not exactly perpendicularly aligned relative
to the FIR beam. The shoulder of the main FIR beam at about -9◦ results from
a reflection of the FIR beam by the mesh holder. It is also seen in the following
diagrams.
The red curve shows the FIR beam recorded with the Lock-In amplifier with low
sensitivity (max. signal: 200 mV =
b 6.25 µW). This curve has been used to determine the forward direction of the FIR beam and the zero point of the x-axis. A
very small peak at -9◦ can be seen in this curve as well.
89
This result shows the principle of the experiment: A FIR diffraction pattern of
a 2D lattice is recorded with a Golay cell. Here, the lattice constant of the golden
mesh is comparable to that of plasma crystals.
The size of the gold squares is larger than that of the dust particles of a plasma
crystal. This is to enhance the diffraction signal of this 2D lattice with one plane
only. Real plasma crystals consist of smaller dust particles (factor roughly 1/3 to 1/2
regarding radius a) but many planes. Invoking the a6 -dependence of the Rayleigh
scattering intensity these smaller particles have scattering intensities 1/729 to 1/64
times the scattering intensity of the gold squares of the test wafer (neglecting the
difference in refractive index). This smaller scattering intensity should be balanced
by the higher number of particles contributing to a diffraction peak for the case of
plasma crystals.
The number of the gold squares constituting the mesh is roughly 2090 determined by simply counting them under a light microscope. A number of about
729 × 2090 = 1.5 × 106 dust particles with radii of 7 µm thus has to contribute to
a diffraction peak in a real diffraction experiment to compensate for the smaller
radius. Then a diffraction peak could be observed even with the Golay cell detector.
In case of the germanium detector a much lower number of dust particles should
be sufficient even invoking domains and temperature of the crystal.
Scattering patterns of the golden mesh have been recorded with and without the
TPX plasma chamber to study the influence of the TPX vessel on the diffraction
signals. The diffraction patterns obtained are shown in Fig. 4.8. The most striking
points in these diagrams are the absence of the 170.58 µm diffraction peak on the
right side of the diagrams (only a small hump is observed at about 56◦ in (a)) and
the appearance of a peak at -23◦ .
The latter again stems from a reflection of the FIR beam by the mesh holder due
to a slight misalignment and should be neglected. It arose after the re-adjustment of
the mesh holder which was necessary to fit the holder into the TPX chamber. The
absence of the 170.58 µm peak on the right is probably due to the misalignment as
well: All peak intensities are lower on the right (Fig. 4.9).
A further reason for the absence of the 170.58 µm peak on the right and for the
angle dependent influence of the TPX chamber can be a change in the FIR laser
90
Chapter 4 Results
8
Maxima:
dsin
n=1
18
118 m
16
14
= n
12
10
6
8
-56°
n=1
4
0
-60
-40
-20
0
20
40
60
18
n=1
Maxima:
dsin
Extinction by
118 m
TPX cylinder
= n
not symmetric
8
36°
6
14
12
8
118 m
n=1
6
170 m
4
2
0
80
16
10
n=1
2
0
-80
12
4
2
20
-35°
4
2
22
FIR beam
-56°
6
170 m
direct
14
10
24
with TPX
16
20
26
Scattering by golden mesh
W
22
36°
118 m
12
FIR beam
18
Signal in
n=1
24
W
Signal in nW
14
direct
Signal in
without TPX
-35°
10
26
Scattering by golden mesh
16
Signal in nW
18
0
-80
-60
-40
-20
0
20
40
60
80
Scattering angle in degree
Scattering angle in degree
(a) Diffraction by golden mesh without TPX chamber.
(b) Diffraction by golden mesh with TPX chamber.
Fig. 4.8: Influence of the TPX chamber on the diffraction pattern. The extinction caused
by the TPX vessel is stronger on the right side. A slight misalignment of the mesh and
FIR laser power variations cannot be excluded.
Fig. 4.9:
out/with TPX. Peak intensities measured
Peak intensity ratio (without/with TPX)
Ratio of peak intensities
3.0
Peak intensity ratios with-
without and with TPX
without TPX related to that measured with
TPX (from Fig. 4.8). The 170.58 µm peak
2.5
at −56◦ is almost not attenuated whereas
2.0
the 118.83 µm peak at 36◦ is attenuated
by a factor of 3. Possible reasons: Mis-
1.5
alignment of the mesh and/or variation of
1.0
laser power during the measurement time
-60
-40
-20
0
20
Scattering angle in degree
40
of about 100 s.
power during the measurement. The angular velocity of the Golay cell has been
about 1.5 degree per second which corresponds to a measurement time of about
100 seconds. Variations of the FIR laser power on this time scale cannot be excluded
because the beam splitter was not yet installed for these experiments.
91
4.3 The crystal
Plasma crystals are produced by dropping dust particles of a defined size into the
argon plasma in this experiment. The argon plasma is operated with a constant flow
of about 1 sccm argon, a pressure between 5 Pa and 120 Pa and a power between
1 W and 20 W. A dust dispenser above the upper electrode drops the dust particles
through a hole in the upper electrode (sec. 3.3, p. 72, 74). Thus the particles fall
directly into the plasma and they are stored within the plasma due to their negative
charge and the positive plasma potential. Particles of diameters between 3 µm and
20 µm are used in the experiments.
The dust particles have high kinetic energies and they are in a gas-like state at
low pressures with the upper electrode powered. Increasing the pressure leads to a
sudden slowdown of the dust particle movement due to increased gas friction and
the crystalline state is eventually reached. The exact pressure for this transition depends on input power, dust particle number density, dust particle size, and voltages
applied to the rings of the lower electrode.
An initially bad dust particle distribution made some changes in the electrode
design necessary. These are described in the next section together with other minor
modifications of the setup which led to more stable plasma crystals. The following
sections describe the two major types of plasma crystals which can be created with
this setup.
4.3.1 Design optimisations for producing plasma crystals
A problem with the material TPX is its sensitivity to UV radiation. UV radiation
causes material defects and makes the TPX more opaque. Because UV radiation is
produced within the argon plasma the TPX chamber got more and more opaque
and had to be changed. This is a principal problem which can only be solved by
using a different chamber material. However, the only suitable material known is
crystalline quartz glass (z–cut) which is also transparent in the visible and in the far
infrared. A polygon chamber could be built out of several crystalline quartz glass
windows. This has the major disadvantage that it would not provide roundabout
optical access to the plasma. Furthermore the FIR transparency of such windows
strongly decreases with increasing angle of incidence (apart from normal incidence).
Therefore the material TPX has been chosen to built the plasma chamber.
92
Chapter 4 Results
Fig. 4.10: Lower electrode configuration
particles
A and dust particle distribution. The regions above the meshes remain dust par-
+
+
ticle free due to plasma–wall sheath deformations.
Crystal
Ring
+
Plate
+
Ring
(a) Lower electrode configuration B with stainless steel (b) Corresponding photograph of the electrode with
ring and plate.
dust cloud.
Fig. 4.11: Lower electrode configuration B and dust cloud—side view. The crystal
from the right photo is inserted into the left sketch. Parameters: MF particle, diameter: 7.32 µm, power: 1.0 W, pressure: 83 Pa, outer ring voltage: −50 V, self bias adjusted
to +8 V.
The original lower electrode design turned out to be disadvantageous. The dust
particles did not arrange themselves in the middle above the lower electrode but
where pushed away from the middle to only one side of the chamber. Fig. 4.10 shows
a sketch of this situation. Stopping the argon gas flow into the plasma chamber did
not change the particle distribution. Thus it seems that potential maxima arose
that pushed the particles away from the centre and away from the outer mesh.
Several modifications of the lower electrode have been tested to optimise the dust
particle distribution. Finally a stainless steel plate with a ring-shaped bottom side
and an additional outer stainless steel ring were placed onto the lower electrode
to cover the central mesh and to increase the radial confinement. This increased
radial confinement reduces the influence of the outer mesh. Furthermore a negative
dc voltage can be applied to the plate which lifts the particles and reduces the
influence of the electrode roughness. A sketch of this new configuration is shown in
Fig. 4.11 (a) and a photograph of the plasma crystal in Fig. 4.11 (b). The crystal
arranges centrally and there are no observable distortions in the global dust particle
distribution.
Different outer ring sizes can be used to vary the lower electrode configuration:
93
The upper edge of the outer ring can be of the same height as the plate or higher.
This varies the radial confinement of the plasma crystals. The radial confinement
is weaker and higher confinement voltages are necessary when using a flat outer
ring. The advantage of this configuration is that the outer ring does not disturb
FIR beam and side view optical observations.
A small funnel has been placed into the small hole of the upper electrode to
improve the dust particle filling. The dust particles thus cannot disperse onto the
upper electrode but fall straightly through the hole.
A lens system consisting of two collecting lenses has been placed onto the upper
electrode right above the central observation hole. The distance between the lenses
is adjustable. A tenfold larger observation area is possible using this lens system.
The electrical matching network is usually detuned to relatively high standing
wave ratios (SWR) of about 20 to reduce dust particle oscillations. This detuning
basically reduces the effective input power but a pure reduction of input power at
the power generator does not have the same effect. The detuning changes the rf
frequency characteristics of the complete electrical system as well. This leads to a
more quiescent plasma which in turn results in more stable crystals in this setup.
The home made match box has been replaced by a commercial one95 . The fine
adjustment can be done more easily and the plasma runs more quiescent with
this matching network. The electrical layout is basically the same as sketched in
Fig. 3.18 but low pass filter and port for measuring or setting the self bias is not
included. These are realised within a separate box which is connected in parallel
to the output of the match box.
Degeneration of gold coated particles
Polystyrene (PS) particles coated with a 100 nm thick gold film have initially been
used to produce plasma crystals for the diffraction experiments. The gold coating
should enhance the diffraction signal expected because of its high refractive index.
Fig. 4.12 displays a scanning electron microscope (SEM) image of such particles
which have been used in a plasma crystal experiment. The particle surface is not
smooth any more but exhibits a craggy structure. An argon ion etching process
is most probably responsible for this surface change. These particles cannot be
used for plasma crystal experiments because they lead to unstable crystals. Lower
95 PALSTAR,
AT-2K Antennentuner, 1.8 MHz – 30 MHz, max. 2000 W power, distributed by Communication
Systems Rosenberg e.K.
94
Chapter 4 Results
Fig. 4.12: Degenerated gold coated micro particles. The particle surfaces are not
smooth after being in the argon plasma but
show a craggy structure. This is most probably the result of an etching process accomplished by the argon ions. The use of gold
coated particles led to unstable plasma crystals.
plasma crystal temperatures and thus more stable crystals can be reached using
uncoated melamine resin (MF) particles.
4.3.2 Extended plasma crystals
Two types of plasma crystals can be produced: extended and flat (21/2D) crystals.
The plasma parameters differ in argon gas pressure and input power. First the extended crystals are described.
The extended crystals are produced with the upper electrode powered and typically have dimensions of 3 cm × 3 cm × 2 cm. This size strongly depends on the
ring voltages applied to the lower electrode. The crystals can be lifted in two ways:
(i) By applying a constant positive dc voltage to the upper powered electrode or
(ii) by applying a negative dc voltage to the inner ring of the lower electrode. In
case (i) the average (positive) plasma potential is increased which pulls the dust
particles into the centre of the discharge. In case (ii) the dust particles are pushed
away from the lower electrode more strongly. This lifting by about 2 cm stabilises
the crystals and the crystal temperature decreases.
In case (ii) (applying a negative voltage to the ring of the lower electrode) the
negative self bias of the upper powered electrode changes as well. This is shown
in Fig. 4.13: The dependence of the self bias of the upper (powered) electrode on
the ring voltages applied to the lower electrode. The self bias increases (gets more
negative) with increasing negative ring voltages.
The particles arrange in vertical strings one particle below another due to the
wake field as already described in sec. 2.4.8 (p. 39). Fig. 4.14 shows an inverted
photo of such a situation and the corresponding 2D pair correlation function of the
95
-5
Self bias against outer ring voltage
-6
-6
-7
-7
Self bias in V
Self bias in V
-5
-8
-8
-9
-9
-10
-10
0
-20
-40
-60
-80
Self bias against inner ring voltage
0
-100
-20
-40
-60
-80
-100
Inner ring voltage in V
Outer ring voltage in V
Fig. 4.13: Self bias against ring voltages. The negative self bias increases with increasing
negative ring voltages. Discharge power: 0.9 W
video (side view).
With increasing pressure the strings are stabilised but can break up at higher
pressures due to the smoothing out of the wake potential. Depending on the outer
ring voltage a structural phase transition can occur. Then the side view shows a
transition from a quadratic to a hexagonal structure.
Pair correlation function
2.5
Pair correlation function
2.0
Power:
~1 W
Pressure:
~80 Pa
Particle diameter: 7
1.5
Mean velocity:
m
0.36 mm/s
Particle distance: 0.47 mm
1.0
0.5
0.5
1.0
1.5
2.0
Particle distance in mm
(a) Photo of strings in a plasma crystal.
(b) Corresponding pair correlation function.
Fig. 4.14: String formation in a plasma crystal. The particles are primarily ordered in
strings one particle below another.
96
Chapter 4 Results
These correspond to the hcp (hexagonal closed packed) and bcc (body centred
cubic) structures in 3D [43]96 . Thus a transition from hcp to bcc can be observed
during pressure increase in this experiment [45]97 . This transition is accompanied
by a change of the plasma itself which resembles the so called α−γ transition [13]98 .
Further investigations are necessary to clarify the exact nature of this transition in
the plasma crystal structure.
Fig. 4.15 shows particle positions and pair correlation functions of extended
(4.15 A – D) and flat (4.15 E – H) crystals. The positions are shown for a time period of one second to illustrate the stability of the crystals. Only small parts (about
3 mm × 3 mm) of the observation area are shown to avoid overloading the pictures.
The corresponding pair correlation functions are calculated including all visible
particles within the observation area. The experimental parameters are given with
the pair correlation functions.
It becomes clear from Fig. 4.15 that the top view stability is better than the side
view stability. This is also seen in the pair correlation functions: The top view gives
higher peak intensities and higher long range correlations (more peaks). This can be
seen by comparing Fig. 4.15 (B) with (D) for extended crystals and by comparing
Fig. 4.15 (F) with (H) for flat crystals.
Reviewing the pair correlation functions the extended crystals are in a fluid like
state. Diffraction peaks from such crystals are expected to be very small and broad.
A reduction of background noise of any kind is thus crucial.
4.3.3 Flat crystals
Flat crystals, sometimes called 21/2D crystals, consist of a small number of horizontal planes only. Three to four planes are possible in this experiment. These crystals
are produced using high pressure and power with the upper electrode powered.
Only two planes are possible when the lower electrode is powered but in a wide
pressure and power range. Then it is no longer possible to lift the crystals which
is unfavourable for scattering experiments because reflections of the FIR beam at
the electrode surface can occur.
Fig. 4.15 (E) – (H) show particle positions and corresponding pair correlation
96
[43] Pieper: Experimental studies of two-dimensional and three-dimensional structure in a crystallized dusty
plasma, 1996
97
[45] Aschinger: Struktur und Dynamik von Plasmakristallen, 2008
98
[13] Raizer: Radio-Frequency Capacitive Discharges, 1995
97
3.0
10.0
extended crystal, top view
Pair correlation function
9.5
9.0
y-position in mm
(B) Pair correlation function
(A) Particle positions over 1 second
8.5
8.0
7.5
extended crystal, top view
2.5
2.0
1.5
1.0
2.9 W
Pressure:
66.6 Pa
Particle diameter: 7
0.5
7.0
Power:
Mean velocity:
m
0.4 mm/s
Particle distance: 0.33 mm
0.0
6.5
6.5
7.0
7.5
8.0
8.5
9.0
9.5
0.2
10.0
0.4
0.6
7.5
1.8
(C) Particle positions over 1 second
7.0
extended crystal, side view
1.0
1.2
1.4
1.6
1.8
2.0
(D) Pair correlation function
Pair correlation function
1.6
6.5
y-position in mm
0.8
Particle distances in mm
x-position in mm
6.0
5.5
5.0
4.5
4.0
extended crystal, side view
1.4
1.2
1.0
0.8
0.6
0.4
Power:
2.9 W
Pressure:
67.4 Pa
Particle diameter: 7
Mean velocity:
0.2
m
0.36 mm/s
Particle distance: 0.34 mm
3.5
3.5
0.0
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
0.0
0.2
0.4
0.6
x-position in mm
13.5
(E) Particle positions over 1 second
1.4
1.6
1.8
12.5
12.0
11.5
11.0
6
5
4
Power:
23.3 W
Pressure:
99 Pa
Particle diameter: 7
Mean velocity:
3
m
0.13 mm/s
Particle distance: 0.29 mm
2
1
10.5
0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
0.0
x-position in mm
0.5
1.0
1.5
2.0
Particle distance in mm
1.8
(G) Particle positions over 1 second
3.5
flat crystal, side view
1.6
(H) Pair correlation function
flat crystal, side view
1.4
y-position in mm
3.0
2.5
2.0
1.5
1.0
1.2
1.0
0.8
0.6
0.4
Power:
23.3 W
Pressure:
99 Pa
Particle diameter: 7
0.5
0.2
Mean velocity:
m
0.13 mm/s
Particle distance: 0.44 mm
0.0
5.0
2.0
flat crystal, top view
Pair correlation function
y-position in mm
1.2
(F) Pair correlation function
7
13.0
y-position in mm
1.0
Particle distance in mm
flat crystal, top view
4.0
0.8
0.0
5.5
6.0
6.5
7.0
7.5
x-position in mm
8.0
8.5
9.0
0.0
0.5
1.0
1.5
2.0
x-position in mm
Fig. 4.15: Particle positions and pair correlation functions of crystals—extended and
flat, side and top view.
98
Chapter 4 Results
functions of a flat crystal. The particles don’t move very far within one second
(E) compared to extended crystals (A). The peaks of the top view pair correlation
function (F) are much higher and more peaks are found than in the case of extended
crystals (B). The crystalline state is clearly reached in this situation. But the side
view of this crystal exhibits more particle movement and thus a much worse pair
correlation function. The side view of flat crystals also exhibits a worse statistics
since only a few particles are seen. This also degrades the pair correlation function.
Fig. 4.16 shows the top view trajectories of the dust particles of the whole observation area of a flat crystal over five seconds. The crystal has been illuminated by
two sheets of laser light to increase the observation area. This has the disadvantage
that the crystal is not homogeneously illuminated. This explains the parts within
the figure without particle trajectories.
It is clearly seen that the crystal is not isotropic. The crystal has different regions with different degrees of stability. E.g. the right part of the picture exhibits
longer trajectories which means more particle movements than the left part. The
movement directions differ as well which can be seen from the orientation of the
trajectories. A more or less stable region has been chosen to obtain Fig. 4.15 (E)
but the corresponding pair correlation function (F) includes the whole observation
area.
Fig. 4.16: Particle trajectories of a flat crystal over 5 seconds—top view.
99
4.4 Scattering by the crystal
Diffraction experiments have been conducted using extended crystals and the Golay
cell as well as the germanium detector. The germanium detector is much more
sensitive and has a higher time resolution than the Golay cell (best: µs instead of
0.1 s). This is necessary as can be inferred from Fig. 4.16: Several crystal domains
are visible and the crystal exhibits regions of higher temperature. These effects
reduce the scattering intensity as described in section 2.5.3 (p. 52).
Fig. 4.17 shows particle positions over one second and the corresponding pair
correlation function of the crystal of the actual diffraction experiment. The crystal
was comparably bad regarding structure and stability. Unfortunately no better
crystal could be established that very day when the germanium detector could be
lend.
Fig. 4.18 shows the result of the plasma crystal diffraction experiment on the
crystal shown in Fig. 4.17 using the germanium detector. An orifice within a piece
of paper board has been used to obtain a narrow beam at the plasma crystal and
to reduce unwanted background radiation. The black curve (upper curve) is the
measurement signal obtained by a motorised scan with the germanium detector.
The scan direction has been from positive angles to negative angles as indicated by
the black (upper) arrow in (a). The large signal between −5◦ and +5◦ is the direct
FIR beam. Several small peaks are visible at negative angles.
The red curve (middle curve) shows the detector signal without plasma crys0.5
2.0
B
Particle positions over
1 second
Pair correlation function
extended crystal, side view
Pair correlation function
y-position in mm
extended crystal, side view
0.4
1.5
1.0
0.5
Power:
~2 W
Pressure:
~50 Pa
Particle diameter: 7
Mean velocity:
m
0.25 mm/s
Particle distance: 0.35 mm
0.3
0.7
0.0
0.8
0.9
1.0
x-position in mm
(a) Particle positions over one second.
1.1
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Particle distance in mm
(b) Corresponding pair correlation function.
Fig. 4.17: Particle positions and pair correlation function of the diffraction experiment
crystal. Structure and stability have been very bad in this experiment.
Chapter 4 Results
3.0
10
insensitive lock-in
8
Scan direction
6
insensitive lock-in
with crystal, sensitive
4
without crystal, sensitive
Diffraction peak?
2
2.5
2.0
with crystal, sensitive
without crystal, sensitive
Diffraction peak?
1.5
1.0
0.5
0.0
0
-20
Germanium detector signal in a.u.
Germanium detector signal in a.u.
100
-15
-10
-5
0
5
10
15
20
Angle in degree
(a) The scan direction of the detector is indicated by
-15 -14 -13 -12 -11 -10
-9
-8
-7
-6
-5
-4
-3
Angle in degree
(b) Detail of the left figure.
the black and red arrows.
Fig. 4.18: Diffraction experiment with the germanium detector. The blue curves show
the signal of the insensitive lock-in amplifier. The black curve is the measured signal
with plasma crystal and the red curve is measured without crystal. There seems to be a
diffraction peak at −12.7 degree.
tal. Some peaks are visible here as well but with lower amplitudes. The peak at
−12.7◦ seems to be missing. This is a hint that this peak could be a real diffrac-
tion peak originating from the plasma crystal. Fig. 4.18 (b) shows a detail of part
(a) which magnifies the peaks. The differences between the diffraction signals with
crystal (black, upper curve) and without crystal (red, lower curve) are clearly seen.
The right peak structure between −3◦ and −10◦ is quite comparable between the
two curves and thus cannot originate from the plasma crystal. However, the peak
at −12.7◦ could be a real plasma crystal diffraction signal.
The scan direction has been reversed for the measurement without plasma crystal
as indicated by the red (lower) horizontal arrow in (a). This led to a small error in
the angle measurement of about 0.4◦ due to a small mechanical tolerance of motor
and gear box. For this reason scans were performed in one and the same direction
only in later experiments.
The blue curves (lowest curves) show the signal recorded with the insensitive
lock-in amplifier. They serve for fixing the zero point of the angle scale. The angle
error of about 0.4◦ is visible comparing the positions of the maxima of these two
blue curves. Both curves exhibit side shoulders which result from diffraction by the
orifice.
The amplitude difference between the two measurements (with and without crystal) can result from a drift of the FIR laser beam intensity to lower values. The
signals therefore have been normalised to a certain peak to eliminate such an effect.
101
Fig. 4.19: Normalised diffraction signals.
The signals have been normalised to the
peak marked with the arrow because the
corresponding peak structure clearly originates from the scattering arrangement and
should not depend on the plasma crystal.
Germanium detector signal in a.u.
Analysis of the measurement
1.5
with crystal
without crystal
normalised to
this peak
1.0
0.5
0.0
-15 -14 -13 -12 -11 -10
-9
-8
-7
-6
-5
-4
-3
Angle in degree
The result of the normalisation is shown in Fig. 4.19.
The peak at −5.8◦ has been chosen to normalise to because the corresponding
peak structure of three close peaks seems to clearly originate from the scattering
arrangement (e.g. from the orifice). It should not change when dropping down the
plasma crystal. This peak structure (between −4◦ and −7◦ ) appears almost identical for both cases after the normalisation. The difference in the curves between
−7◦ and −10◦ are mainly due to the angle measurement error after changing the
scan direction as already mentioned. The amplitude difference at −7.8◦ seems to
be of minor importance and is not much larger than that at −6.5◦ .
The amplitude difference at −12.7◦ is now negligible. This shows that this peak
is not a diffraction peak originating from the plasma crystal.
A possible but not observed diffraction peak of this experiment could have had a
maximum radiation power below 10−14 W since no diffraction peak could be measured with the germanium detector. The negative outcome of this very experiment
reveals some important points:
(1) It is very difficult but essential to get the different experimental parts running
at their optimum at the very same time: FIR laser beam, scattering arrangement,
plasma crystal, and data storage.
(2) Wavelength stability and intensity of the FIR laser have to be further improved. This could probably be done by replacing the stainless steel, hole output
mirror by a mesh output mirror (gold squares on a multi layer film system on a
quartz substrate). This would make a different input mirror necessary with a radius
of curvature larger than the resonator length. Such a mesh output mirror and concave input mirror could not be manufactured and have to be bought. Additionally a
102
Chapter 4 Results
different stabilising system for the whole FIR laser system including the CO2 laser
should be installed. Such a system is available from Edinburgh Instruments Ltd.
The beam splitter concept should be improved to reliably correct the measured
data points for FIR intensity drifts. Different time resolutions of the Pyro, germanium, and Golay cell detectors complicate this and prevented the development of
an automatic data correction program.
(3) The focussing of the FIR beam has to be improved e.g. using an additional
focussing mirror near the plasma chamber. This would make an orifice obsolete.
(4) An adjustment of the FIR beam into the plasma chamber during the experiment is difficult and should be avoided. This is so because additional background
radiation can occur due to e.g. reflections at the electrodes. These then will most
probably be more intense than any diffraction peak. A later calibration is thus
useless.
(5) It is necessary to have an in situ evaluation of the plasma crystals to decide
if structure and stability are good or not. This could not be realised so far: The
video analysis takes too much time to meet experimental needs.
(6) Flat crystals could be an alternative for the scattering experiments because
they are more stable and exhibit a better structure (Fig. 4.15 (E,F)). But then the
adjustment of the FIR beam is more difficult because of the closeness to the lower
electrode. This favours reflections which would cover real diffraction peaks.
The FIR laser beam of 118.83 µm wavelength cannot exactly be focussed to
the plasma crystal because the mirrors of the Yolo telescope have limited radii
of curvature. Larger radii would be preferable but could not be manufactured by
the in-house workshop. Therefore the FIR beam has been relatively broad at the
plasma crystal and an orifice has been used to obtain a more narrow beam. But
this reduces the FIR radiation power reaching the plasma crystal.
To eliminate such losses due to an orifice a 50 cm long pyrex glass tube of inner
diameter of 15 mm has been installed to guide the FIR beam to the TPX plasma
chamber. The FIR beam has been focussed into the glass tube which serves as
wave guide and avoids a broadening of the FIR beam. This narrows the beam and
increases the radiation power at the plasma crystal. But the glass tube has to be
adjusted very carefully to avoid a diffraction pattern originating from the tube
itself. Furthermore only the powder diffraction method can be applied with this
setup and not the pendant of the rotating crystal method because the direction of
103
40
FIR beam profile at the plasma chamber
35
Pyro signal in
W
30
Fig. 4.20: FIR beam profile at the lower
electrode. The beam width is about 7.1 mm
25
20
width = 7.1 mm
15
10
Measurement
Gauss fit
5
with a peak intensity of about 36 µW mea-
0
-12 -10
sured with the Pyro detector.
-8
-6
-4
-2
0
2
4
6
8
10
12
Horizontal position in mm
the incoming FIR beam cannot be changed.
Fig. 4.20 shows the FIR beam profile 2 cm above the lower electrode when using
the glass tube. This beam profile has been recorded with the Pyro detector without
any orifice to avoid lowering the signal. Thus the directional sensitivity has not been
very good. A Gaussian fit of the measurement curve gives a width of about 7.1 mm
and a peak power of 36 µW is reached in this experiment. The measured curve is
not perfectly symmetric which hints to a slight misalignment of the glass tube.
1.0
10
0.9
sensitive
insensitiv (a.u.)
reference signal
0.8
5
0.7
0
0.6
-15
-10
-5
0
5
10
Angle in degree
(a) Scan with de-adjusted glass tube.
1.0
Adjusted glass tube
30
Golay signal in nW
Golay signal in nW
1.1
15
35
0.9
25
0.8
20
0.7
15
sensitive
insensitive (a.u.)
10
0.6
reference signal
0.5
5
0
Pyro reference signal in a.u.
1.2
De-adjusted glass tube
Pyro reference signal in a.u.
20
0.4
-15
-10
-5
0
5
10
Angle in degree
(b) Scan with adjusted glass tube.
Fig. 4.21: Calibration scans for glass tube adjustment. Precise adjustment of FIR beam
and glass tube are necessary to avoid perturbing parasitic peaks. The blue curve (with
circles) shows the Pyro reference signal recorded using the beam splitter right after the
FIR laser output. The FIR laser intensity fluctuations of about 10 to 20 % cannot explain
the parasitic peaks.
Fig. 4.21 shows calibration scans with the Golay cell detector for the glass tube
adjustment. The difference between (a) and (b) is a different focussing of the FIR
beam into the glass tube. In (a) there are several peaks visible besides the main FIR
beam. A careful adjustment of glass tube and FIR beam eliminates such parasitic
104
Chapter 4 Results
peaks as shown in (b). This experiment shows the necessity of a careful adjustment of FIR beam and glass tube to avoid perturbing reflections. Thus beam and
glass tube cannot be adjusted during an actual diffraction experiment but beforehand. This is a major disadvantage when using the glass tube and it is doubtful
if the enhancement of the FIR beam intensity due to the glass tube is such a big
improvement.
However, using only flat crystals for the scattering experiments should ease the
situation because they are more stable. Thus FIR beam adjustments during the
scattering experiments should not be necessary when using flat crystals.
Chapter 5
Conclusion
The aim of this work was the development and qualification of a setup for the
analysis of 3D plasma crystals using FIR (far infrared) diffraction signals. During
the progress of this project the realisation of several elements of the undertaking
turned out to be much more challenging than initially expected. Significant design
and development effort had to be invested in each part of the system starting
with the FIR laser system and ending with the plasma crystals and their control
procedures.
Large 3D plasma crystals had initially been produced in a stainless steel plasma
chamber with four glass windows. This chamber resembles the well known GEC
reference cell (GEC: Gaseous Electronic Conference) but it is smaller and has a
segmented electrode which allows the application of different electric fields. These
crystals had been very stable with volumes of about 3 × 3 × 3 cm3 containing
millions of dust particles. The mono disperse micro particles used have been gold
coated because of the higher refractive index of gold compared to polymer particles.
This gold coating increases scattering intensities expected from these particles.
The power of the incoming radiation necessary to measure diffraction peaks from
those crystals has been calculated using the scattering theory described in this
work. The use of gold coated micro particles has been assumed in this calculation.
A CO2 laser that pumps the FIR resonator has been purchased based on this
estimation and a FIR resonator has been developed. A Golay cell detector has
been purchased which is operated at room temperature and has a minimum signal
limit well above the diffraction intensities estimated.
However, it turned out that the gold coated dust particles could not be used
in actual scattering experiments because they degenerate most probably due to an
argon ion etching process. The surface of the dust particles becomes craggy and long
105
106
Chapter 5 Conclusion
term stable plasma crystals could not be produced using these particles. Therefore,
uncoated mono disperse melamine resin (MF) and polystyrene (PS) particles could
be used only, decreasing the diffraction signal intensities below those expected.
The FIR laser system built in this work consists of a commercial CO2 pump laser
and a home made FIR wave guide resonator. The FIR resonator design allows an
easy change of the wave guide radius, input and output stainless steel mirrors, and
of the laser gas. The FIR laser provides a Gaussian beam of up to 5 mW power
and a wavelength of 118.83 µm and 170.58 µm when using methyl alcohol as laser
medium. The FIR laser system has carefully been characterised and tested. This
includes e.g. the variation of input and output plane mirror roughness and their
hole diameters, the variation of the resonator pipe radius, and the change from
plane mirrors to concave mirrors. The wavelength selectivity and the stability of
the resonator has been improved using the concave instead of plane mirrors.
The CO2 laser Brewster windows got dirty after about one year of frequent operation due to oil vapour originating from the rotary vane pump and diffusing back
into the CO2 laser plasma. The zinc selenite (ZnSe) Brewster windows therefore
had to be removed, cleaned, polished using lap foils, and reinstalled again. This procedure worked quite well but some scratches remained on the windows which could
not be removed by polishing. These are probably responsible for the somewhat
reduced CO2 laser power after the readjustment of the laser.
The CO2 laser power is stabilised by an opto–galvanic stabiliser. It measures
changes of the laser current which are induced by changes of the laser power to
readjust the laser resonator length. However, it turned out that the long term
stability of the whole FIR laser system has to be further improved by enhancing
the frequency stability of the CO2 laser with a feedback system that measures
directly the FIR beam intensity through a beam splitter.
The development of a new plasma vacuum chamber has been a challenging task
of this project. The chamber should provide optical and FIR access to the plasma
crystals—if possible roundabout. Only a few materials suitable for vacuum vessels
meet the requirement to be transparent in the visible and in the far infrared region
of the spectrum. Crystalline quartz windows of the z-cut type fulfil the transparency
demand. But a polygon chamber does not provide roundabout access to the crystals
which is necessary to assure the detection of every possible diffraction peak from
plasma crystals.
The polymer poly-methylpentene (TPX) is a mechanically stable material which
107
meets the demand of roundabout optical and FIR transparency. Thus a plasma
vacuum chamber has been built using a TPX cylinder which is glued into grooves
of aluminium flanges using silicone rubber. This chamber is smaller than the one
previously mentioned and it has a larger window area which means more floating
walls. This changes plasma characteristics and thereby plasma crystal properties
compared to the other chamber.
The material TPX turned out to be sensitive to UV radiation which is produced
by the plasma. It becomes more and more opaque due to microscopically small
imperfections of the material. Furthermore, TPX is elastic to a certain extent but
at the same time prudish. Forces due to repetitive venting and pumping stressed
the material enormously and led to failures of the chamber. Although TPX is not
the perfect material for the plasma chamber it is still used because of the lack of
alternatives.
Several electrode geometries and electrical wirings have been examined to produce plasma crystals. These include the powering of upper, lower, and parts of the
lower electrode as well as the application of different DC and AC electric fields to
the powered as well as to parts of the grounded electrode. Two different plasma
crystal types have thereby been found: extended and flat crystals. The extended
crystals have volumes up to 3 × 3 × 2 cm3 and are not fully stationary. They still
show relatively high mean particle velocities of e.g. 0.36 mms−1 . The flat crystals
are produced using higher power or by powering the lower electrode. They consist
of three to four layers only but are extended over almost the whole electrode area of
10 cm in diameter. They are more stable (mean particle velocity e.g. 0.13 mms−1 )
and exhibit a better defined crystal with less structural domains.
Various control procedures have been developed to stabilise the crystals and to
optimise their structure by reducing their defects and the structural domain density.
The crystal temperature could be reduced through lifting the plasma crystal to
about two centimetres above the lower electrode by applying DC voltages to the
electrodes. This lifted crystal position is not only advantageous for the crystal
quality but also for the diffraction experiment. It eases the FIR beam adjustment
and disturbing reflections from the lower electrode can easily be minimised.
A scattering arrangement has been developed to guide the FIR laser beam into
the plasma chamber and to record diffraction peaks. It consists of a Yolo telescope,
several tilted mirrors, and a motorised circular positioning system to move mirrors
and detector around the chamber. All mirrors and mirror holders have been made
108
Chapter 5 Conclusion
of aluminium and they have been manufactured and polished by the in house
workshop.
The Yolo telescope is composed of two spherical mirrors and focusses the FIR
beam. Two tilted mirrors deflect the beam from the laser table to the plasma
chamber table where it can straightly reach the plasma chamber. Incoming beam
and plasma crystal are not moved and diffraction signals are expected from single
structure domains of the crystals.
A second diffraction method is an analogy to the rotating crystal recording.
But since 3D plasma crystals can not easily be rotated in a well defined way,
the incoming laser beam has to be rotated around the crystal together with the
detector. A mirror system consisting of four mirrors mounted on a circular rail way
accomplishes this beam rotation around the chamber and the detector is placed on
a separate waggon.
Several computer programs have been written e.g. the program “Cockpit” by
which it is possible to control the plasma parameters like pressure and power, to
move the detector waggon and monitor its position, to calculate diffraction peak
positions for different crystal structures and lattice plane distances, and to do
the complete data storage. The program “Beam calculator” has been developed
to calculate beam diameters of the FIR beam on its way along the scattering
arrangement. Position and focal distances of the Yolo telescope mirrors have been
determined using this program which applies the formulae of Gaussian beam optics.
A significant effort has been put into the design, the improvement, and the testing
of the video analysis software. Existing procedures have been adapted, refined, and
customised to ensure the correct analysis of videos taken by laser sheet illumination
and a CCD (Charge Coupled Device) camera. The production of plasma crystals
has been evaluated and refined by judging structure and stability with the video
analysis software.
A 2D mesh that consists of about 2090 golden squares with edge lengths of
40 µm and distances of 200 µm has been deposited on a GaAs wafer (courtesy of
Nadine Vitteriti, Chair for Applied Solid State Physics, Prof. Dr. A. D. Wieck,
Ruhr–University Bochum). Diffraction signals from this golden mesh have successfully been recorded and the influence of the TPX wall and chamber structure has
been analysed. Detailed estimations have been given for the diffraction peak intensities expected from real plasma crystals on the basis of these results. They suggest
that a more sensitive and fast germanium detector should be used in diffraction
109
experiments and that the number of different domains within a plasma crystal has
to be minimised to achieve enough intensity in one diffraction peak.
The presence of several structural domains within the extended 3D plasma crystals remains an unsolved problem. It prevented the recording of a diffraction peak
of an actual plasma crystal. Flat crystals consisting of only three or four planes
may be an alternative to large 3D crystals. They are located nearer to the lower
electrode, however. The adjustment of the FIR beam becomes more critical when
using flat crystals because beam reflections from the near electrode may increase
the incoherent background.
The basis for FIR diffraction experiments on plasma crystals has been developed
in this work. Further refinements of the FIR laser system and the plasma crystal
production and control are necessary to obtain diffraction signals.
110
Chapter 5 Conclusion
List of Figures
2.1
The CO2 molecule and its vibration modes . . . . . . . . . . . . . .
7
2.2
Energy level and excitation scheme for the CO2 laser . . . . . . . .
8
2.3
Molecules within the FIR resonator . . . . . . . . . . . . . . . . . .
9
2.4
Energy levels of the FIR active molecules . . . . . . . . . . . . . . .
10
2.5
Propagation of a Gaussian beam . . . . . . . . . . . . . . . . . . . .
13
2.6
Focussing of a Gaussian beam by a lens . . . . . . . . . . . . . . . .
14
2.7
Sheath and presheath near a wall . . . . . . . . . . . . . . . . . . .
18
2.8
Dust particle charge vs. ion drift velocity . . . . . . . . . . . . . . .
25
2.9
Contour plot of the ion density . . . . . . . . . . . . . . . . . . . .
26
2.10 Interparticle forces in a dust molecule . . . . . . . . . . . . . . . . .
34
2.11 Comparison of ion drag forces . . . . . . . . . . . . . . . . . . . . .
37
2.12 Dust particle strings and crystal organisation . . . . . . . . . . . . .
41
2.13 Hysteresis loop of dust temperature during pressure variation . . . .
42
2.14 Scattering geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.15 Dipole scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.16 Polar diagram for Rayleigh scattering . . . . . . . . . . . . . . . . .
50
2.17 The Debye-Waller factor . . . . . . . . . . . . . . . . . . . . . . . .
52
3.1
The whole setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.2
Sketch of the CO2 laser . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.3
Dirty Brewster window . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.4
CO2 laser adjustment . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.5
FIR resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.6
Cross section of resonator pipes and spacer . . . . . . . . . . . . . .
62
3.7
Bushings for quartz tube support . . . . . . . . . . . . . . . . . . .
63
3.8
Sketch of FIR laser beam adjustment and photo of the mirror system 65
3.9
Aluminium mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.10 Reflection coefficients of aluminium . . . . . . . . . . . . . . . . . .
66
111
112
List of Figures
3.11 Chamber and mirror system . . . . . . . . . . . . . . . . . . . . . .
67
3.12 Waggon and guide rail with gear ring. . . . . . . . . . . . . . . . . .
68
3.13 Scattering control scheme . . . . . . . . . . . . . . . . . . . . . . .
68
3.14 Angular velocity of the detector carriage against control voltage . .
69
3.15 Sketch of Golay principle and photo . . . . . . . . . . . . . . . . . .
69
3.16 Dependence of the Golay sensitivity on chopper frequency . . . . .
70
3.17 Plasma chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.18 Matching network . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.19 Electrodes of the plasma chamber . . . . . . . . . . . . . . . . . . .
74
3.20 Gold mesh and holder . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.1
FIR signal versus pressure . . . . . . . . . . . . . . . . . . . . . . .
82
4.2
FIR laser beam profiles at 55 mm and 1080 mm distance . . . . . .
82
4.3
3D visualisation of the FIR beam . . . . . . . . . . . . . . . . . . .
83
4.4
Resonator scan with concave mirrors . . . . . . . . . . . . . . . . .
84
4.5
Pressure dependence of the FIR signal . . . . . . . . . . . . . . . .
85
4.6
Calibration of the beam splitter . . . . . . . . . . . . . . . . . . . .
86
4.7
Diffraction peaks of the golden mesh . . . . . . . . . . . . . . . . .
88
4.8
Influence of the TPX chamber on the diffraction pattern . . . . . .
90
4.9
Peak intensity ratios without/with TPX . . . . . . . . . . . . . . .
90
4.10 Lower electrode configuration A and dust particle distribution . . .
92
4.11 Lower electrode configuration B and dust cloud—side view . . . . .
92
4.12 Degenerated gold coated micro particles . . . . . . . . . . . . . . .
94
4.13 Self bias against ring voltages . . . . . . . . . . . . . . . . . . . . .
95
4.14 String formation in a plasma crystal . . . . . . . . . . . . . . . . . .
95
4.15 Particle positions and pair correlation functions of crystals . . . . .
97
4.16 Particle trajectories of a flat crystal over 5 seconds—top view. . . .
98
4.17 Particle positions and pair correlation function of the diffraction experiment crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.18 Diffraction experiment with the germanium detector . . . . . . . . . 100
4.19 Normalised diffraction signals . . . . . . . . . . . . . . . . . . . . . 101
4.20 FIR beam profile at the lower electrode . . . . . . . . . . . . . . . . 103
4.21 Calibration scans for glass tube adjustment . . . . . . . . . . . . . . 103
List of Tables
2.1
Grain charge for different mechanisms not included in OML theory
31
2.2
Ion currents and grain potentials for different effects included . . . .
32
2.3
Different forces acting on dust particles in complex plasmas . . . . .
39
2.4
Definition of single, double, and multiple scattering. . . . . . . . . .
44
2.5
Stokes parameters for spheres and different incident polarisations .
46
2.6
Rayleigh scattered intensities . . . . . . . . . . . . . . . . . . . . .
47
2.7
Values for the calculation of the diffraction power . . . . . . . . . .
54
3.1
Working scheme of the analysis software . . . . . . . . . . . . . . .
78
113
114
List of Tables
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Danksagung
Mein herzlicher Dank geht an Herrn Prof. Dr. Jörg Winter, der mir die Durchführung
dieser Arbeit ermöglichte. Seinen Ideen und seiner enthusiastischen Überzeugungskraft
ist das Zustandekommen weiter Teile des Aufbaus zu verdanken. Seit ich an diesem
Lehrstuhl für Experimentalphysik II arbeite, unterstützt mich Herr Winter, wo er
kann. Er gab mir besonders in den persönlich schwierigen Zeiten der letzten Jahre,
in denen mein Bruder und mein Vater starben, den für mich notwendigen Freiraum
und uneingeschränkte Rückendeckung. Dafür danke ich Herrn Winter sehr.
Bei dem gesamten Team des Lehrstuhls für Experimentalphysik II möchte ich
mich bedanken für die freundliche und fröhliche Atmosphäre. Der lockere Umgang
miteinander erleichtert das Arbeiten ungemein.
Besonders bedanke ich mich bei Andreas Aschinger für die Zusammenarbeit in
allen Bereichen der Arbeit und insbesondere für die Umsetzung der Konzepte zur
Videoanalyse. Stefanie Schornstein danke ich für die Mitarbeit bei der Charakterisierung des FIR Resonators. Beide haben das Projekt entscheidend vorangetrieben.
Die Techniker des Lehrstuhls haben wesentlich dazu beigetragen, den Aufbau zu
realisieren. Ich danke deshalb Herrn Karl Brinkhoff für die Hilfe bei der Entwicklung des FIR Resonators und der Plasmakammer. In vielen Diskussionen haben
wir diese entworfen und immer wieder verbessert. Herrn Kai Fiegler danke ich für
die Hilfe bei der Entwicklung des Spiegelsystems zur Lenkung des FIR Strahls in
die Plasmakammer und für die weitere Betreuung des Aufbaus. Er hat auch einen
erheblichen Teil seiner Freizeit in diese Arbeit investiert. Herrn Axel Lang danke
ich für die gesamte und umfassende Betreuung des Labors und die Hilfe bei allen
möglichen Problemen. Herrn Wilhelm Winterhalder und Herrn Michael Konkowski
danke ich für die Anfertigung verschiedenster elektrischer und elektronischer Komponenten, die zur Steuerung des Experiments und zur Datenaufnahme unerlässlich
sind. Eine solche Arbeit wäre ohne die Anstrengungen der Techniker nicht möglich
– vielen herzlichen Dank!
Frau Margot Ocklenburg möchte ich danken für vielfache Hilfen in jeglichen organisatorischen und verwaltungstechnischen Dingen. Es ist oft schwer, den Überblick
zu behalten und ich war stets froh, jemanden zu haben, der sich auskennt.
Herrn Dr. Erik Bründermann gilt mein Dank für zahlreiche Tipps bezüglich der
Detektion der FIR Strahlung und für das Ausleihen und Bedienen des Germanium
Detektors. Durch die Zusammenarbeit mit ihm habe ich viel gelernt.
Ebenso danke ich Herrn Prof. Dr. Henning Soltwisch und Herrn Dr. Carsten
Pargmann für die Beratung und die Tipps beim Aufbau des FIR Resonators.
Ihre Erfahrungen halfen mir beim meiner Arbeit sehr. Weiterhin danke ich Herrn
Soltwisch für die Übernahme des Koreferates.
Mein herzlicher Dank geht auch an Herrn Dr. Uwe Konopka für die Einführung
in das Feld der komplexen Plasmen während meines Besuches in Garching und
die Bereitstellung einiger Routinen zur Auswertung der Partikelbewegungen. Darauf aufbauend konnte die Videoanalyse weiterentwickelt werden. Ebenso danke ich
Herrn Dr. habil. Dietmar Block für die zahlreichen Diskussionen während meines
Besuchs in Kiel und während der vielen Tagungen, auf denen wir uns begegneten.
Seine Kommentare und Ratschläge waren immer sehr erhellend.
Frau Nadine Viteritti danke ich für die Anfertigung des Goldgitters, welches sie
nach mehreren Versuchen auf einen Wafer gebracht hat. Mit diesem Goldgitter
konnte das Arbeitsprinzip des Aufbaus demonstriert werden.
Bei Frau Dr. Janine–Christina Schauer und Herrn Dr. Suk-Ho Hong bedanke
ich mich herzlich für ihre Freundschaft und die unendlichen physikalischen und
teils philosophischen Diskussionen. Unsere gegenseitige Unterstützung hat mir sehr
geholfen und mir viel Rückhalt gegeben.
Zu guter Letzt bedanke ich mich herzlich bei meiner Familie. Bei meiner Frau
Liudmila, die mir mit Ihrer Liebe jederzeit ihre volle Unterstützung gibt. Bei meiner
Mutter, die mir trotz schwieriger Zeiten immer zur Seite Stand und auch bei meinem
Vater, der nie seinen Humor verlor. Er hat mir gezeigt, wie man mit Mut und
Lebensfreude Wunder vollbringen kann.
Lebenslauf
Persönliche Daten
Name
Jens Ränsch
Anschrift
An der Maarbrücke 41
44973 Bochum
Geburtstag
31.01.1978
Geburtsort
Bernburg (Sachsen–Anhalt)
Familienstand
verheiratet, 1 Kind
Hochschulausbildung
10.2003 – heute
Wissenschaftlicher Mitarbeiter am Lehrstuhl für Experimentalphysik II an der Ruhr–Universität Bochm, Promotion in experimenteller Plasmaphysik
10.1998 – 10.2003
Studium der Physik an der Ruhr–Universität Bochum,
Diplomarbeit Untersuchungen zu Wechselwirkungen
zwischen flüssigen Galliumoberflächen und kapazitiv
gekoppelten HF–Plasmen, Abschlussnote: “mit Auszeichnung”
Wehrdienst
07.1997 – 05.1998
Wehrdienst in Düsseldorf als Richtfunker und Kraftfahrer
Schulausbildung
08.1992 – 07.1997
Ingeborg-Drewitz-Gesamtschule in Gladbeck, Abiturnote: 1.0
08.1991 – 08.1992
Gymnasium Süd-Ost in Bernburg
08.1989 – 08.1991
7. Oberschule “Wilhelm Pieck” in Bernburg
09.1984 – 08.1989
Allgemeinbildende polytechnische Oberschule “Karl
Liebknecht” in Bernburg
Preise
05.2003
Studienabschlussstipendium der Ruth und Gert Massenberg-Stiftung für die Studienleistung
02.2004
“ROTARY-UNIVERSITÄTSPREIS 2003 für herausragende Studienleistungen” für die Diplomarbeit, verliehen vom Rotary-Club Bochum-Hellweg
Tätigkeiten neben Studium und Promotion
01.2005 – heute
Mitarbeit im Vorstand des Sportvereins “Biriba Brasil
de Bochum e.V.” als Schriftführer, Organisation diverser
Sportveranstaltungen und Vereinsfahrten, Abwicklung
sämtlicher schriftlicher Korrespondenz
10.2000 – 02.2008
Leitung von Seminaren, Übungsgruppen und Praktika
an der Ruhr–Universität Bochum
10.2003 – 08.2006
Leitung des Physikunterrichts für die MTA-Ausbildung
am Bergmannsheil-Krankenhaus in Bochum
Ränsch, Jens
.........................................................
Name, Vorname
Versicherung gemäß § 7 Abs. 2 Nr. 5 PromO 1987
Hiermit versichere ich, dass ich meine Dissertation selbstständig angefertigt und
verfasst und keine anderen als die angegebenen Hilfsmittel und Hilfen benutzt habe.
Meine Dissertation habe ich in dieser oder ähnlicher Form noch bei keiner anderen
Fakultät der Ruhr-Universität Bochum oder bei einer anderen Hochschule
eingereicht.
Bochum, den ............................................
..........................................................
Unterschrift