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# Transport Phenomena in Plasma Expansions Containing Hydrogen: A Laser Spectroscopic Study

Stéphane Mazouffre

► To cite this version:

Stéphane Mazouffre. Transport Phenomena in Plasma Expansions Containing Hydrogen: A Laser Spectroscopic Study. Physics [physics]. University of Technology of Eindhoven, Eindhoven, The Netherlands, 2001. English. NNT: tel-04171038

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Transport Phenomena in
Plasma Expansions Containing Hydrogen

A Laser Spectroscopic Study

PROEFSCHRIFT

ter verkrijging van de graad van doctor
aan de Technische Universiteit Eindhoven,
op gezag van de Rector Magnificus, prof.dr. R.A. van Santen,
voor een commissie aangewezen door het College
voor Promoties in het openbaar te verdedigen op
maandag 15 oktober 2001 om 16.00 uur

door

Stéphane Mazouffre

geboren te Limoges, Frankrijk
Dit proefschrift is goedgekeurd door de promotoren:

**prof.dr.ir. D.C. Schram**

en

**prof.dr. H.F. Döbele**

Copromotor:

**dr. R.A.H Engeln**

The work described in this thesis is part of the research program of the Dutch Foundation for Fundamental Research on Matter (FOM) via project 95RG09: "*Thin layer deposition and surface modification. Plasmas in interaction with solid matter*".

It is financially supported by the Dutch Organization for Scientific Research (NWO) as well as the Euratom Foundation.

Printed and bounded by Universiteitsdrukkerij Technische Universiteit Eindhoven

Cover design by Lies Schopmeijer

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Mazouffre, Stéphane

Transport Phenomena in Plasma Expansions Containing Hydrogen: A Laser Spectroscopic Study / by Stéphane Mazouffre.-

Eindhoven: Eindhoven University of Technology, 2001.-

Thesis.-

ISBN 90-386-1839-5

NUGI 812

Subject headings: plasma expansion / laser spectroscopy / hydrogen atom / gas dynamics / surface chemistry
À mon Père,
À ma Mère,
À Aimée,
À Christian.
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# Contents

<table><tbody><tr><td>1. General introduction</td><td>1</td></tr><tr><td>1.1 Plasma expansion: a general physical phenomenon</td><td>2</td></tr><tr><td>1.2 Hydrogen containing plasma jets</td><td>4</td></tr><tr><td>1.3 Transport of particles</td><td>6</td></tr><tr><td>1.4 Experimental facility</td><td>8</td></tr><tr><td>1.5 Aim of this work and outline</td><td>11</td></tr><tr><td>References</td><td>14</td></tr><tr><td>2. Dynamics of a plasma expansion</td><td>17</td></tr><tr><td>2.1 Introduction</td><td>18</td></tr><tr><td>2.2 Governing equations</td><td>18</td></tr><tr><td>2.2.1 The Boltzmann transport equation</td><td>18</td></tr><tr><td>2.2.2 Conservation laws and EOS</td><td>19</td></tr><tr><td>2.2.3 Steady isentropic flow</td><td>21</td></tr><tr><td>2.3 Supersonic expansion: the continuum free jet structure</td><td>22</td></tr><tr><td>2.3.1 Speed of sound and Mach number</td><td>22</td></tr><tr><td>2.3.2 Free jet flow</td><td>22</td></tr><tr><td>2.3.3 Rarefied regime: evanescent shock structure</td><td>26</td></tr><tr><td>2.4 Shock wave</td><td>28</td></tr><tr><td>2.4.1 Properties of a shock wave</td><td>28</td></tr><tr><td>2.4.2 Rankine-Hugoniot relations</td><td>28</td></tr><tr><td>2.4.3 Shock wave thickness</td><td>29</td></tr><tr><td>2.4.4 Shock wave profile</td><td>29</td></tr><tr><td>2.5 Conclusions</td><td>30</td></tr><tr><td>References</td><td>31</td></tr><tr><td>3. Flow characteristics of a supersonically expanding thermal argon plasma</td><td>33</td></tr><tr><td>3.1 Introduction</td><td>34</td></tr><tr><td>3.2 Experimental arrangement</td><td>35</td></tr><tr><td>3.2.1 Cascaded arc</td><td>35</td></tr><tr><td>3.2.2 LIF setup</td><td>35</td></tr><tr><td>3.2.3 Rayleigh scattering</td><td>38</td></tr><tr><td>3.3 Interpretation of the LIF spectra</td><td>39</td></tr><tr><td>3.3.1 Recombination of Ar ions</td><td>39</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td>3.3.2 Coupling between ground-state Ar and Ar[4s]</td>
      <td>41</td>
    </tr>
    <tr>
      <td>3.3.3 Ar velocity distribution function</td>
      <td>42</td>
    </tr>
    <tr>
      <td>3.4 Ar atom temperature, velocity, and density</td>
      <td>44</td>
    </tr>
    <tr>
      <td>3.5 Ar atom forward flux</td>
      <td>48</td>
    </tr>
    <tr>
      <td>3.6 Radial profiles</td>
      <td>49</td>
    </tr>
    <tr>
      <td>3.6.1 Perpendicular temperature</td>
      <td>49</td>
    </tr>
    <tr>
      <td>3.6.2 Radial velocity component</td>
      <td>49</td>
    </tr>
    <tr>
      <td>3.7 Conclusions</td>
      <td>50</td>
    </tr>
    <tr>
      <td>References</td>
      <td>51</td>
    </tr>
    <tr>
      <td><b>4. Two-photon laser induced fluorescence spectroscopy of atomic hydrogen</b></td>
      <td><b>53</b></td>
    </tr>
    <tr>
      <td>4.1 Introduction</td>
      <td>54</td>
    </tr>
    <tr>
      <td>4.2 Two-photon excitation process</td>
      <td>54</td>
    </tr>
    <tr>
      <td>4.3 Excitation and detection of ground-state atomic hydrogen</td>
      <td>56</td>
    </tr>
    <tr>
      <td>4.4 Experimental arrangement</td>
      <td>57</td>
    </tr>
    <tr>
      <td>4.4.1 Generation of narrowband coherent tunable UV radiation</td>
      <td>57</td>
    </tr>
    <tr>
      <td>4.4.2 Fluorescence light monitoring</td>
      <td>58</td>
    </tr>
    <tr>
      <td>4.4.3 Parasitic effects</td>
      <td>59</td>
    </tr>
    <tr>
      <td>4.5 Density measurements</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.5.1 Integrated spectral profile</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.5.2 Absolute densities: calibration methods</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.6 Temperature measurements</td>
      <td>69</td>
    </tr>
    <tr>
      <td>4.7 Velocity measurements</td>
      <td>70</td>
    </tr>
    <tr>
      <td>4.8 Conclusions</td>
      <td>72</td>
    </tr>
    <tr>
      <td>References</td>
      <td>74</td>
    </tr>
    <tr>
      <td><b>5. Anomalous atomic hydrogen shock pattern in a supersonic plasma jet</b></td>
      <td><b>77</b></td>
    </tr>
    <tr>
      <td>5.1 Introduction</td>
      <td>78</td>
    </tr>
    <tr>
      <td>5.2 Experimental arrangement</td>
      <td>79</td>
    </tr>
    <tr>
      <td>5.3 On-axis velocity profile</td>
      <td>79</td>
    </tr>
    <tr>
      <td>5.4 Perpendicular and parallel temperature</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.5 Anomalous H atom transport</td>
      <td>82</td>
    </tr>
    <tr>
      <td>5.5.1 Density profile along the jet axis</td>
      <td>82</td>
    </tr>
    <tr>
      <td>5.5.2 H atom flux in the shock region</td>
      <td>83</td>
    </tr>
    <tr>
      <td>5.5.3 Confinement in the jet core</td>
      <td>85</td>
    </tr>
    <tr>
      <td>5.5.4 Subsonic domain</td>
      <td>85</td>
    </tr>
    <tr>
      <td>5.6 Conclusions</td>
      <td>86</td>
    </tr>
    <tr>
      <td>References</td>
      <td>87</td>
    </tr>
    <tr>
      <td><b>6. Transport of ground-state hydrogen atoms in an Ar-H<sub>2</sub> plasma expansion</b></td>
      <td><b>89</b></td>
    </tr>
    <tr>
      <td>6.1 Introduction</td>
      <td>90</td>
    </tr>
    <tr>
      <td>6.2 Plasma jet and experimental arrangement</td>
      <td>91</td>
    </tr>
    <tr>
      <td>6.3 Transport of atoms along the jet centerline</td>
      <td>93</td>
    </tr>
  </tbody>
</table>
<table><tbody><tr><td>6.3.1</td><td>Argon atoms</td><td>93</td></tr><tr><td>6.3.2</td><td>Hydrogen atoms</td><td>94</td></tr><tr><td>6.4</td><td>Influence of the mass on the transport of radicals</td><td>97</td></tr><tr><td>6.4.1</td><td>Use of a light carrier gas</td><td>97</td></tr><tr><td>6.4.2</td><td>Isotopic effect</td><td>99</td></tr><tr><td>6.5</td><td>Confinement of radicals inside the jet core</td><td>100</td></tr><tr><td>6.5.1</td><td>Radial profiles of the H density</td><td>100</td></tr><tr><td>6.5.2</td><td>Temperature and radial velocity</td><td>101</td></tr><tr><td>6.5.3</td><td>Permeable Ar barrel shock wave</td><td>104</td></tr><tr><td>6.6</td><td>Atomic hydrogen diffusion flux</td><td>105</td></tr><tr><td>6.7</td><td>Conclusions</td><td>107</td></tr><tr><td></td><td>References</td><td>108</td></tr><tr><td></td><td>Appendix 6A: Dissociation degree</td><td>110</td></tr><tr><td></td><td>Appendix 6B: Influence of the nozzle geometry</td><td>111</td></tr><tr><td>7.</td><td>H atom fluid properties in a hydrogen plasma expansion</td><td>113</td></tr><tr><td>7.1</td><td>Introduction</td><td>114</td></tr><tr><td>7.2</td><td>Hydrogen plasma jet and optical system</td><td>115</td></tr><tr><td>7.3</td><td>H<sub>2</sub> density behavior along the jet axis</td><td>116</td></tr><tr><td>7.4</td><td>Axial H atom density and temperature profiles</td><td>118</td></tr><tr><td>7.4.1</td><td>Loss of H atoms by diffusion</td><td>118</td></tr><tr><td>7.4.2</td><td>Perpendicular temperature development</td><td>120</td></tr><tr><td>7.4.3</td><td>Injection of molecular deuterium in the arc</td><td>121</td></tr><tr><td>7.5</td><td>Recombination in the arc nozzle</td><td>122</td></tr><tr><td>7.6</td><td>Axial velocity component and sound speed</td><td>124</td></tr><tr><td>7.7</td><td>Plasma jet vicinity</td><td>126</td></tr><tr><td>7.8</td><td>Conclusions</td><td>127</td></tr><tr><td></td><td>References</td><td>129</td></tr><tr><td>8.</td><td>Behavior of the H atom velocity distribution function within the shock wave of a hydrogen plasma jet</td><td>131</td></tr><tr><td>8.1</td><td>Introduction</td><td>132</td></tr><tr><td>8.2</td><td>Experimental determination of the H atom VDF</td><td>133</td></tr><tr><td>8.3</td><td>Non-Gaussian VDF within the stationary shock wave</td><td>134</td></tr><tr><td>8.4</td><td>Velocity and temperature profiles along the jet axis</td><td>139</td></tr><tr><td>8.5</td><td>Theoretical velocity profile in the shock region: Mott-Smith approach</td><td>141</td></tr><tr><td>8.6</td><td>Conclusions</td><td>143</td></tr><tr><td></td><td>References</td><td>145</td></tr><tr><td>9.</td><td>General Conclusion</td><td>147</td></tr><tr><td>A.</td><td>Invasion of a supersonic jet by the background gas particles</td><td>155</td></tr></tbody></table>
<table><tr><td>B. Recirculation and downstream chemistry</td><td>159</td></tr><tr><td>Summary</td><td>161</td></tr><tr><td>Acknowledgments</td><td>162</td></tr><tr><td>List of publications</td><td>164</td></tr></table>
# Chapter 1

## General introduction

The subject of this thesis is the experimental study of the transport of ground-state hydrogen atoms in thermal plasma expansions. The plasma is generated by a cascaded arc from different gas mixtures: Ar-H₂, He-H₂ and H₂. In all cases, the plasma jet is unmagnetized and weakly ionized. The local properties of H atoms, such as the density, the perpendicular and parallel temperature, and the radial and axial velocity components, are monitored in the course of the plasma expansion by means of Two-photon Absorption Laser Induced Fluorescence (TALIF) spectroscopy. In order to obtain information about the flow of inert neutral particles (Ar, He, H₂), other laser-based diagnostic techniques are available, like cw-Laser Induced Fluorescence and UV-Rayleigh scattering. This work follows the study of the flow of neutrals and charged particles carried out by M.C.M. van de Sanden and R.F.G. Meulenbroeks using Rayleigh-Thomson scattering and Coherent Anti-Stokes Raman Scattering. Here, we focus on the study of the influence of surfaces on the H atom transport mechanisms and on the H atom fluid properties, i.e. coupling between the H fluid and the inert neutral fluid, during the expansion process. We also performed a detailed investigation of the behavior of the atomic velocity distribution function throughout the stationary shock wave of the supersonic plasma jet, to study both the shock wave formation and the invasion of the supersonic domain by the background gas in a rarefied flow regime.

In this introductory chapter, the physical phenomenon of plasma expansion is presented and compared with the expansion of a neutral gas. Furthermore, the experimental facility, including all the different laser-aided diagnostic techniques available, is described, and an outline of this thesis is given.
1.1 Plasma expansion: a general physical phenomenon

The expansion of a plasma from a high temperature and high density region (referred to
as the source) where energy is dissipated<fnref n="1" />, to a low pressure or vacuum environment, is
a very general physical phenomenon. It concerns a large variety of objects and covers a
vast range of dimensions. Furthermore, the physical issues addressed are relevant to many
subjects in science like astrophysics, fluid dynamics, and surface physics.

At large scale plasma expansions encompass objects like supernovae, solar flares, solar outbursts, and phenomena like solar wind and galactic jets [1, 2, 3, 4, 5]. Those systems represent examples of gigantic jets<fnref n="2" /> where the matter ejected out of the source flows into a nearly perfect vacuum until it encounters planets or dense interstellar clouds of gas and dust. Apart from their inconceivable size, such plasma expansions are also characterized by the fact that their structure and properties are also governed by the existence of magnetic fields.

In contrast to huge plasma jets and beams found in the cosmos, one finds at the
opposite side of the dimension scale small and even tiny plasma expansions such as laser
spots [6, 7, 8, 9] and cathode spots [10, 11]. In the case of a laser spot, the plasma is
created by the interaction of a strong electromagnetic field with a target, whose state can
be solid, liquid or gas, and subsequently the plasma expands supersonically from a very
localized region [12, 13]. Laser plasmas allow for a precise control of the jet parameters
like the Mach number and the kinetic energy, and of the jet chemical properties. Such jets
are used for instance for thin layer deposition [14], for the fundamental study of collision
processes as well as for the generation of VUV and X-ray radiation [15]. Laser produced
plasmas are also projected to create modulated plasmas for plasma based accelerators in
order to create bunches of relativistic electrons [16, 17]. Tiny plasma expansion can be
generated using low current vacuum arcs from a very small cathode spot by evaporation
and ionization of a solid target in a low pressure environment. Such plasma flows may be
used for thin film deposition by cathode sputtering, as an ion source for ion implantation
and for particle accelerator injection for fundamental nuclear physics research [18]. In arc
switches and circuit breakers, the conductive properties of the created high density arc
plasma are used [19].

Finally, in between macro and microjets, the physical phenomenon of plasma expan-
sion also includes laboratory scale expansions such as jets produced from thermal plasma
sources like DC, RF, and microwave torches [20], as well as the divertor<fnref n="3" /> region of Toka-

<footnote n="1">¹ Depending on the kind of source the expansion originates from, the energy that drives the flow process is of different form: electromagnetic (arc, laser spot), nuclear (star), or gravitational (black hole).</footnote>

<footnote n="2">² The most extreme jets are galactic jets which can have extensions over distances of several thousands of light-years and for which the flow speed can approach the speed of light. Such plasma expansions are likely to be powered by massive black holes.</footnote>

<footnote n="3">³ In a Tokamak reactor a limiter is a solid surface which serves primarily to protect the reactor wall from the plasma (during disruption for instance). A divertor is a special kind of limiter which includes a magnetic field to guide impurities out of the plasma volume.</footnote>
<table><caption>**Tab. 1.1:** Physical parameters characterizing a variety of plasma expansions. The given values of the parameters represent standard values. Deviations corresponding to specific cases are possible. Even if all plasma expansions mentioned in this table are very similar from the point of view of the underlying physical processes, as can be seen, they cover a broad range of conditions.</caption><thead><tr><th>expansion</th><th>source properties</th><th>flow speed</th><th>jet radius</th><th>ambient density</th></tr></thead><tbody><tr><td>solar flare</td><td>T<sub>e</sub> ≈ 10<sup>4</sup> eV<br>n<sub>0</sub> ≈ 10<sup>20</sup> m<sup>-3</sup><br>n<sub>e</sub> ≈ 10<sup>20</sup> m<sup>-3</sup></td><td>10<sup>6</sup> m s<sup>-1</sup></td><td>r ≈ 10<sup>3</sup> km</td><td>10<sup>6</sup> m<sup>-3</sup></td></tr><tr><td>cascaded arc jet</td><td>T<sub>e</sub> ≈ 1 eV<br>n<sub>0</sub> ≈ 10<sup>23</sup> m<sup>-3</sup><br>n<sub>e</sub> ≈ 10<sup>22</sup> m<sup>-3</sup></td><td>5000 m s<sup>-1</sup></td><td>r ≈ 10 cm</td><td>10<sup>21</sup> m<sup>-3</sup></td></tr><tr><td>laser plasma<br>(X-ray generation)</td><td>T<sub>e</sub> ≈ 100 eV<br>n<sub>0</sub> ≈ 10<sup>24</sup> m<sup>-3</sup><br>n<sub>e</sub> ≈ 10<sup>26</sup> m<sup>-3</sup></td><td>10<sup>5</sup> m s<sup>-1</sup></td><td>sphere<br>r ≈ 5 mm</td><td>10<sup>18</sup> m<sup>-3</sup></td></tr><tr><td>cathode spot</td><td>T<sub>e</sub> ≈ 10 eV<br>n<sub>0</sub> ≈ 10<sup>24</sup> m<sup>-3</sup><br>n<sub>e</sub> ≈ 10<sup>24</sup> m<sup>-3</sup></td><td>10<sup>4</sup> m s<sup>-1</sup></td><td>r ≈ 10 µm</td><td>10<sup>15</sup> m<sup>-3</sup></td></tr></tbody></table>

mak plasmas [3, 21]. Atmospheric thermal plasma jets are used for instance for plasma cutting, welding, heating (chemical waste treatment), annealing, and for plasma spraying, i.e. deposition of thick coatings [22, 23]. Low pressure thermal plasma jets find applications in surface modifications like fast deposition of thin films [24, 25] and micro-circuit etching [26]. They may serve as ion beam and neutral beam sources for research in the field of controlled nuclear fusion.

All previously mentioned kinds of plasma expansions exhibit similarities in the underlying physical mechanisms. They cover a broad range of conditions, as can be seen in the list of physical parameters given in Tab. 1.1 for several sorts of plasma expansions.

To a large extent, the expansion of a thermal plasma resembles a neutral gas expansion [27], a well-known subject in the field of gas dynamics, meaning that in first order the particles density, temperature, and velocity, behave similarly to a free jet expansion [28, 29, 30].

However, a plasma expansion exhibits specific characteristics which make it a richer and more complex process. First, a plasma is not only composed of particles such as stable atoms and molecules, it also contains ions, electrons, photons and radicals<fnref n="4" />. Under certain circumstances, different kind of particles may have a different temperature in the

<footnote n="4">⁴ A radical is an atom or a molecule with a free bond. As examples one can cite atoms like H, N, or C and molecular fragments like CH, NH, SiH₃. Due to their high reactivity radicals play an important role in chemistry, both in volume and at a surface.</footnote>
source and/or in the course of the flow. Moreover, large temperature, density and velocity
gradients are often present already in the source. Second, a plasma expansion is a non-
equilibrium process [31, 32] and, due to recombination effects or due to the action of an
electric or magnetic field, a frozen regime in which there is no energy exchange or energy
transformation is never achieved. Third, the enlarged heat conductivity and viscosity [22]
lead to a non-isentropic supersonic expansion [33]. Three-particle recombination can also
disturb the energy transfer mechanisms between charged particles, and can, therefore,
create non-adiabatic conditions during the expansion process [34]. Fourth, the plasma
nature gives rise to the generation of fields and currents [34, 35], which in turn influence,
or even determine, the currents and thus the power dissipation in the hot region. At
high power densities, the generated currents and fields may be so strong that the plasma
becomes confined by the self-generated magnetic field. Finally, due to plasma-surface
interactions, i.e. interactions with dust particles (e.g. in interstellar medium) or with walls
(nozzle, reactor), radicals generated from molecules in the plasma source, may decouple
from the neutral particles expansion (a two-fluid expansion is then created) and, as a
consequence, radicals may exhibit anomalous transport properties [36]. All these specific
physical processes and mechanisms make the phenomenon of plasma expansion very rich
in view of the possible issues to be studied.

We here restrict ourselves to laboratory scale thermal plasma jets and to the regime
where the plasma remains unmagnetized and singly ionized, in which case the main mass is
contained in the neutrals and thus the expansion resembles the usual gas expansion. At the
same time the study of the intermediate size expansions of thermal plasmas (mm to cm)
may serve as a scale model for both the tens of micrometer-sized and the astrophysical size
plasma jets as to some extent they are driven by similar physical mechanisms. Furthermore,
an important reason to study plasma flow at an intermediate scale is the possibility of
application of a variety of diagnostic techniques, both passive and active, and therefore
opening the way to acquire extensive sets of data.

1.2 Hydrogen containing plasma jets

The fundamental study of hydrogen containing plasma jets<fnref n="5" /> is of particular interest, since it concerns various systems that cover the entire range of plasma expansions.

Atomic hydrogen being the main constituent of both stellar and interstellar matter,
all astrophysical scale plasma expansions represent examples of hydrogen plasma jets. In
the field of controlled nuclear fusion, hydrogen jets plays a specific role [21]. First, one
wants to use H atom sources for neutral beam heating of fusion devices (mainly via the
production of negative hydrogen ions H⁻). Second, the interaction of neutral H atoms
with the reactor wall (in a divertor region for instance) and the resulting formation of new

<footnote n="5">⁵ In order to avoid confusion, we should here emphasize on the difference between a jet and a beam since they are very different media. In a jet, the flow regime is continuum (hydrodynamic) whereas in a beam the flow regime is rarefied (molecular) [28]. If the ambient pressure is low enough, an expansion exhibits a gradual transition from a jet like flow (close to the source) to a beam like flow (far from the source).</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.1:** 3D plot of the density (log scale, in m⁻³) of Ar atoms during the expansion of a cascaded arc generated argon plasma (see section 1.4 for more details) for two different background pressures: 20 Pa (left) and 100 Pa (right). The source conditions are: 3 standard liters per minutes (slm) Ar, pressure = 0.6 atm, temperature = 5000 K, nozzle diameter = 3 mm. These density maps are constructed using simplified 1D hydrodynamic equations (see chapter 2) combined with arbitrary functions that fit the measured radial density profiles. In the subsonic domain, the static pressure is assumed to be constant. A formal calculation using the Navier-Stokes equation would be necessary in order to obtain the exact density behavior. Yet these simple models help the reader to visualize the so-called rarefaction effect (dilution of the fluid into the environment) and the effect of the local pressure on the jet structure (smooth and abrupt jet-background transition at low and high pressure respectively).</figcaption></figure>

products, e.g. H₂ molecules, and the release of impurities due to proton bombardment, have drastic consequences on the stability of the fusion plasma and therefore on device performances in term of energy extraction. Numerous works are nowadays devoted to this so-called hydrogen recycling process [37, 38] and they all require the development of plasma sources capable of generating a high H atom or H ion flux.

From a technological perspective, due to their high reactivity, hydrogen radicals play a key role in many chemical reactions encountered in plasma processes and therefore are of relevance for industrial applications like deposition of thin films and surface modification (etching, passivation). For instance, hydrogen atoms are involved in the plasma-aided fast deposition of thin films of hydrogenated amorphous silicon (a-Si:H) needed for the next solar cell generation. It has been shown recently that hydrogen radicals control the growth process and thus the final film properties [24].

All aforementioned processes that concern intermediate scale expansions, are to a large extent governed by the transport mechanisms of ground-state hydrogen atoms, i.e. H radicals, in an expanding hydrogen containing thermal plasma. In view of the applicability it is therefore of importance to study the H atom flow pattern as well as the influence of plasma-surface interactions on the radical transport properties.
<figure><img src="image_4.png" /><figcaption>**Fig. 1.2:** Axial density profiles of the heavy particle density (left) and of the electron density (right) during the expansion of an argon plasma produced by a cascaded arc (see section 1.4) for 3 different background pressures. The data are taken from the work of M.C.M. van de Sanden [34, 39, 40]. The heavy particle density is measured by Rayleigh scattering and the electron density is obtained from Thomson scattering measurements. In the supersonic domain the density decrease is independent of the ambient pressure. On the contrary, the position and the width of the stationary shock wave as well as the compression strength are influenced by the background pressure. Beyond the shock front, the heavy particle density increases as a consequence of the cooling of the plasma flow by heat transfer to the vessel walls, whereas the electron density decreases as the result of 3-particle recombination. Ignoring the e-ion interactions, those profiles are mainly in agreement with the free jet flow theory.</figcaption></figure>

## 1.3 Transport of particles

### Inert neutrals: the classical transport picture

The flow of inert particles, such as Ar or He atoms and $H_2$ molecules, in a thermal plasma expansion can to a large extent be described with the free jet flow theory developed in the field of gas dynamics [27, 28]. In short, leaving the high pressure region, the gas is accelerated to supersonic velocity due to the conversion of thermal energy gained in the source into kinetic energy and the local particle density decreases due to an increase in the jet cross-section (the so-called rarefaction effect). At some distance from the source the expanding gas collides with the residual ambient gas and a stationary shock wave structure is formed. Behind the shock region, the flow is subsonic and obeys the local pressure conditions. In Fig 1.1, a schematic representation of the inert particle density in an expansion is given. In the case of a plasma, the high temperature leads usually to a quasi-adiabatic expansion. In the work of M.C.M. van de Sanden et al [39, 40], it is demonstrated that the flow of Ar atoms in an argon plasma expansion follows the prediction of the free jet flow theory (see Fig. 1.2).
<figure><img src="image_5.png" /><figcaption>**Fig. 1.3:** Ground-state hydrogen atom density profile measured along the jet centerline of a Ar-H₂ plasma jet for 2 different background pressures (3 slm Ar, 0.5 slm H₂) [36]. The profile is in disagreement with the free jet flow theory. The density decrease in the supersonic domain is pressure dependent (value at the arc exit and slope), there is no density jump across the normal shock wave and the density drops in the subsonic domain. The departure from the classical expansion picture is a direct consequence of plasma-wall interactions, i.e. recombination of H atom at the reactor wall (stainless steel in that case) to form molecular hydrogen.</figcaption></figure>

Reactive neutrals: anomalous transport

Until recently, physicists thought that the transport of radicals in a plasma expansion would not differ from the transport of neutrals. Therefore, the reader can easily imagine the surprise and the excitation when it was discovered that the measured H atom density development along the jet axis of an Ar-H₂ plasma jet was in contradiction with the expectation, as can be seen in Fig 1.3.

The anomalous transport of H radicals finds its origin in the plasma-surface interactions
as it will be shown in this thesis. To please impatient readers, we here give a rapid overview
of what is going on. When a H atom reaches a surface it may recombine to form a H₂
molecule (obviously the efficiency of this process depends strongly on the surface properties
and on the residence time). If the so-called recombination probability is high, the H atom
partial abundance in the background gas is low, that results in the formation of large H
density gradients between the plasma jet and its vicinity. Those gradients are responsible
for two effects: a decoupling between the H atom fluid and the carrier gas fluid during the
expansion and the existence of large H outflow [36]. As a result, only a small fraction of
the hydrogen atoms leaving the source is carried towards the downstream region.
The case of charged particles

Even if the charged particles, i.e. ions and electrons, are not directly of interest in this thesis, it is certainly worthwhile to shortly discuss their influence on the plasma flow. In the case of a fully ionized plasma, the expansion process differs from a gas expansion. Several new effects must be considered: the appearance of a charged particles density gradient in the conservation laws, a higher heat conductivity, the existence of fields and currents. In this work we however restrict ourselves to partially ionized plasmas. As pointed out by M.C.M. van de Sanden et al [39, 34] a partly ionized plasma flow can be explained as a classic supersonic expansion with slight modifications to account for the presence of charged particles. As an example, the electron axial density profile measured in an expanding argon plasma for several pressures is shown in Fig. 1.2.

Another aspect of the presence of electrons and ions is the emergence of a new chemistry which of course has an impact on the flow properties. This is particularly true when molecular gases like hydrogen are added to an atomic plasma (e.g. argon plasma). Charge exchange between an ion and a molecule, followed by the dissociative recombination of the created molecular ion, is a very effective process to dissociate a molecule at low temperature

$$
\begin{array}{l@{\qquad}l@{\qquad}l}
\mathrm{Ar}^{+} + \mathrm{H}_{2}(v=0) \rightarrow \mathrm{ArH}^{+} + \mathrm{H} & k \approx 10^{-15} \mathrm{~m}^{3} \mathrm{~s}^{-1} & \text{at 1 eV,} \\
\mathrm{e} + \mathrm{H}_{2}(v=0) \rightarrow \mathrm{e} + \mathrm{H} + \mathrm{H} & k \approx 10^{-18} \mathrm{~m}^{3} \mathrm{~s}^{-1} & \text{at 1 eV.}
\end{array}
$$

Note that the dissociative recombination of ArH⁺, ArH⁺ + e → Ar + H* is known to be very fast (rate constant around 10⁻¹³ m³ s⁻¹). Thus the charge exchange process leads to the formation of new molecules and radicals (atomic as well as molecular fragments) which play an important role in the plasma chemistry. Furthermore, this process leads to a loss of ionization (i.e. low ionization degree) which in turn influences the expansion behavior.

1.4 Experimental facility

Plasma source: the cascaded arc

The plasma source used in these experiments is a cascaded arc [42, 43] which is an example
of a wall stabilized thermal plasma. It is schematically depicted in Fig. 1.4. The cascaded
arc has been first introduced in the 1950s by Maecker [44]. The arc can be operated in
a wide range of pressure (10⁴ to 10⁸ Pa) and currents (5 to 2000 A). In the standard arc
conditions (gas flow ≈ 3 standard liters per minute (slm), parc ≈ 0.5 atm, Iarc = 40 A
to 60 A), the generated plasma is characterized by a relatively high electron and heavy
particle temperature (Te ≈ Th ≈ 1 eV) and a high electron density (ne ≈ 10²² m⁻³). In a
cascaded arc, a DC discharge is sustained in a channel between 3 cathodes and a grounded
anode to which a nozzle is plugged. The gas is fed through a mass flow controller into the
cathode region. The electrically insulated cascaded plates that compose the channel lead to
a straight arc. All parts are made of copper, except for the cathode tips (tungsten with 2%
lanthanum), the PVC and boron nitride spacers between the plates and the boronnitride
<figure><img src="image_6.png" /><figcaption>**Fig. 1.4:** Plasmas under investigation in this work are generated from a flowing gas using a wall-stabilized cascaded arc. Inside the arc channel ($d = 4$ mm) the pressure is sub-atmospheric (≈ 0.5 atm) and the temperature is around 1 eV. The plasma expands freely from the source nozzle, of which geometry can be varied, into a low pressure medium. The vacuum chamber which is made of stainless steel is 4 m long and has a diameter of 0.36 m. The vessel is divided into 4 observation sections surrounded by 8 Helmholtz coils to provide an axially directed magnetic field (0 to 0.5 T). The background pressure can be varied almost independently from the injected gas flow from 10 Pa to 1 atm. Both the cascaded arc and the temperature controlled substrate are mounted on a movable arm that allows for 3D displacement. In this way spatially resolved measurements can be performed while the diagnostic tools remain fixed.</figcaption></figure>

cover of the cathode feedthrough. The arc is vacuum sealed with O-rings; the cathode, the anode and the cascaded plates are water-cooled. The arc that is used in this work is composed of 4 plates; the channel is 30 mm long and has a diameter of 4 mm. The arc nozzle is made of copper and it has a length of 17 mm. Its geometry can be varied. The cascaded arc is mounted on a moveable arm and can therefore be moved in the 3 directions of space.

In the source the plasma is created from a flowing gas by dissipation of (electric) energy [20]. The supplied energy has to be sufficient enough to ionize and heat the gas/plasma. This will lead to a hot and high pressure source and subsequently to expansion of the generated plasma from that source. Moreover, the production of new electrons and ions has to be large enough to compensate for the losses of charged particles by diffusion and convection. The thermal plasma produced in the cascaded arc is in non-equilibrium and it is ionizing [41, 46]. The expanding plasma is also not in equilibrium, however in contrast with the plasma in the source it is recombining and the temperature is relatively low.
The thermal plasma produced inside the arc channel expands freely from the arc nozzle
into a low pressure vessel made of stainless steel (see Fig. 1.4). The vessel is 4 m long and
has a diameter of 0.36 m; it is divided into 4 segments surrounded by 8 Helmholtz coils
to provide an axially directed magnetic field (0 to 0.5 T). The pumping system consists
of a large roots blower pump (720 1s⁻¹) pumped by a small roots blower pump (140 1s⁻¹)
which in turn is evacuated by a roughing pump (20 1s⁻¹). The pumping capacity of the
large roots blower pump can be varied by means of a valve, and in this way the residual
pressure can be changed almost independently from the gas flow from 10 Pa to 1 atm.
The residence time inside the vacuum chamber is relatively long (≈ 1 s at 20 Pa) and thus
recirculation cells, into which the gas is recycled, are formed around the plasma jet [45].
Those cells play an important role in the downstream chemistry.

Laser-aided diagnostics

Several laser-aided diagnostics are used in the experiments mentioned in this thesis to col-
lect valuable information about the particle transport in an expanding thermal plasma<fnref n="6" />.
We here briefly introduce them; more information can be found in the rest of the manuscript.

* *Two-Photon Laser Induced Fluorescence*

In order to avoid the experimentally demanding generation of high energy (VUV) photons, ground-state hydrogen atoms are excited to the *n*=3 level using two 205 nm (UV) photons [47]. The resulting fluorescence radiation is detected at the Balmer-α line (656 nm). Using this spectroscopic technique one obtains spatially resolved information about the H atom density, temperature and velocity. In order to obtain absolute number density, the fluorescence yield has to be calibrated either via a titration reaction or using a two-photon scheme in krypton.

* cw-Laser Induced Fluorescence*

A continuous wave (cw) laser beam in the near IR is used to probe excited Ar atoms
in the 4s multiplet (both resonant and metastable state) via laser induced fluorescence
spectroscopy. In that way, the local velocity and temperature of the excited Ar atoms are
measured. It can be shown that the properties of excited Ar (but the density) image the
properties of ground-state Ar atoms.

* UV Rayleigh scattering*

The UV laser beam used to excite ground-state H atoms is also used to perform Rayleigh
scattering experiments in order to measure the local heavy particle density (i.e. the neu-
tral density, since the ionization degree is below 1 % in our plasmas). The relatively low
amount of UV photons available (approximately 10¹⁵ and 10¹⁸ photons per pulse at 205 nm

<footnote n="6">⁶ Very often plasma jets properties are monitored using probes such as Langmuir probes, enthalpy probes, and Pitot tubes, which are directly inserted in the jet. However, contrary to laser diagnostics, probes disturb the flow, especially in a supersonic regime. As a consequence, the interpretation of the experimental data is in most cases difficult.</footnote>
and 532 nm, respectively) is partly compensated by the gain in the scattering cross-section $\sigma^R$, since $\sigma^R$ varies in first approximation like $\lambda^{-4}$ [48]. The Rayleigh scattering and the two-photon LIF experiments can be performed simultaneously.

Other laser aided diagnostics have been extensively used in the past, like Thomson scattering [49] and Coherent Anti-Stokes Raman Spectroscopy (CARS) [50]. At the present time, the chemistry in the subsonic domain of the expansion is studied by means of Cavity Ring Down Spectroscopy (CRDS) [51]. In the near future, a VUV-LIF setup is projected to be built in order to probe $H_2$ molecules in a hydrogen expansion [47, 52].

## 1.5 Aim of this work and outline

### Aim: an overall plasma expansion picture

As already suggested in sections 1.2 and 1.3, this thesis deals with the transport phenomena in intermediate scale thermal plasma expansions that contains hydrogen atoms and molecules. Such research is of relevance on a fundamental point of view: detailed dynamic-kinetic study on an intermediate scale plasma expansion with possible extrapolation to expansions at smaller and larger scales. But it also has a large impact on a technological perspective. The field of plasma assisted chemistry, in which radicals and molecules play a key role, is gaining importance. Moreover, researches in the field of controlled nuclear fusion require at present a better understanding in transport and wall-recycling of hydrogen atoms. The thermal plasma jets investigated in this work are generated using a cascaded arc. Such jets are currently used for deposition of thin films: C containing layers [53] and a-Si:H coatings [54] for example. They are also used as particle source for treatment of archeological artefacts [55], and as a light source for spectroscopic applications [56]. Furthermore, a cascaded-arc-produced hydrogen expansion appears to be a good candidate to simulate the $H^+$ flux impinging onto a divertor, one of the critical components of a Tokamak reactor [21].

The aim of this work is therefore to obtain an overall picture of an unmagnetized and weakly ionized plasma expansion in terms of transport properties. To gain a more profound understanding of the basic processes that govern this kind of plasma expansion with in mind not only the gain in knowledge, but also the optimization of aforementioned applications. This present work is the follow up of the work of M.C.M. van de Sanden [34, 40] and R.F.G. Meulenbroeks [50, 57]. Nevertheless, after several years of research on expanding plasmas using active and passive diagnostics, numerous aspects of the transport phenomena in such a medium still have to be investigated or clarified. In order to make the objective of this work clear to the reader, we present a list of questions we will try to answer in the rest of this thesis.

- Which amount of atomic radicals can the plasma source supply? Which fraction of the H atoms leaving the source reaches the downstream region of the expansion?
- In the case of a pure hydrogen plasma jet, can the low ionization degree and the absence of rovibrationally excited H₂ molecules be explained in view of the H atom content?

- What is the structure of a stationary compression shock wave in a rarefied regime? To which extent does the expanding plasma depart from thermodynamic equilibrium?

- Is the invasion of the supersonic domain of the expansion by the residual background gas significant in a rarefied flow regime? How does the mixing of the ambient gas with the jet affect the downstream chemistry?

- Can the interaction between the background gas and the reactor walls influence the transport properties of radicals in a plasma jet?

Trying to answer those questions, more questions will undoubtedly emerge. In like manner, some conclusions drawn in this work will probably have to be modified later on, as new experimental and computational facts arise. In the general conclusion, we present the current view of the transport phenomena in a plasma expansion and we propose near future investigations which, as we hope, will show that we are on the right route.

## Outline of this thesis

The main content of this work consists of seven chapters written in such way that they can be read independently from each other. Apart from the first part in which the main characteristics of an expansion are described, all other parts are organized in such way that this thesis offers a logical trend.

In chapter 2, a detailed description of the continuum free jet flow structure that results from the expansion of a fluid from a high pressure region into a vacuum is given. Moreover, the characteristics of a compression shock wave are presented.

In chapter 3, the transport of argon atoms in an expanding thermal argon plasma seeded with 4% of hydrogen is studied by means of laser induced fluorescence spectroscopy, and by UV Rayleigh scattering. After demonstrating that the properties of metastable Ar atoms image the properties of ground-state Ar atoms, it is experimentally shown that the Ar atom expansion is understandable in term of a Ar free jet flow with a low isentropic exponent. Non-Maxwellian Ar distribution functions are observed, both at the arc exit (influence of the self-generated E field) and across the stationary shock wave (interaction jet-background).

The Two-photon Absorption Laser Induced Fluorescence (TALIF) spectroscopy of atomic hydrogen is explained in chapter 4. A detailed description of the experimental arrangement is presented. It is shown that from a spectral scan over the two-photon transition several quantities can be extracted: the local H atom velocity distribution function, the mean velocity, the temperature (only at thermodynamic equilibrium) and the relative
density. In order to obtain absolute H atom densities, the LIF setup has to be calibrated.
Two calibration methods are presented and compared: titration with NO₂ and comparison
with a two-photon excitation of krypton.

In chapter 5 and chapter 6, the transport of ground state atomic hydrogen in the
expansion of a thermal plasma generated from an Ar-H₂ mixture (14 % H₂) is studied.
The flow of hydrogen atoms is monitored by TALIF, whereas Ar atoms are probed by LIF
as well as by UV Rayleigh scattering. Contrary to Ar atoms, H atoms exhibit anomalous
transport properties. In the course of the plasma expansion, hydrogen atoms decouple from
the argon fluid by a diffusion process as a direct consequence of recombination of H atoms
at the vessel walls. It is shown, on the basis of experimental results, how plasma-surface
interactions can strongly influence the flow pattern of an atomic radical fluid.

The flow properties of the ground state hydrogen atom fluid in the course of the expan-
sion of a weakly ionized hydrogen plasma are examined and compared with the ones of H₂
molecules in chapter 7. The low dissociation degree measured at the source exit implies
that H atoms flow in a H₂ environment. It is clearly proved that as soon as the surface
loss probability is high, the losses of radicals by diffusion, and the subsequent decoupling
between the inert gas fluid and the reactive gas fluid, cannot be avoided even when the
mass of the carrier gas is close to the mass of the radical.

In chapter 8, the evolution of the ground-state hydrogen atom velocity distribution
function (VDF) throughout the stationary normal shock wave of a supersonic hydrogen
plasma jet is studied. A clear departure from thermodynamic equilibrium is observed.
Nevertheless, the H atom VDF can be decomposed into two Maxwellian distributions. One
distribution corresponds to the zone of silence conditions and the other one corresponds to
the conditions in the shock region. The development of the mean axial velocity is modeled
using the Mott-Smith approach. Deviation from the theoretical shock profile is interpreted
in terms of the non-conservation of both the H atom forward flux and momentum across
the shock wave.

Finally in chapter 9, the general conclusions of this thesis work are presented and some
near future perspectives are given.
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# Chapter 2

## Dynamics of a plasma expansion

### Abstract

After presenting the basic hydrodynamic equations that govern the flow of a gas (plasma) in the continuum regime, a set of simplified conservation laws applying to a steady isentropic flow is derived. In first order approximation the expansion of a weakly ionized plasma (ionization degree < 1 %) is similar to the expansion of a hot neutral gas, and can be described using the well established supersonic expansion theory. Therefore a detailed description of the continuum free jet flow structure that results from the free expansion of a fluid from a high pressure region into a vacuum is given. In short, the gas first expands supersonically. After adapting to the local pressure conditions imposed by the residual background gas via the formation of a stationary shock wave system, the gas flows subsonically into the low pressure environment. Departures from the perfect free jet flow picture, when approaching the rarefied (molecular) regime, are also discussed. Finally, the characteristics of a gas shock wave are presented and several theoretical expressions are given for the on-axis density and velocity profile within a shock region.
## 2.1 Introduction

In this chapter we propose to convey to the non specialist reader the basic features of a plasma expansion. All plasmas studied in this thesis (Ar-H₂, He-H₂, D₂-H₂, H₂) are characterized by a low ionization degree at the source exit (< 1 %). The effects of charged particles on the neutral particle transport properties are therefore neglected. As a consequence, the expansion of a weakly ionized plasma can be treated as the expansion of a hot neutral gas [1]. The transport of charged particles in a partially ionized plasma expansion (ionization degree ≈ 10 %) has been studied in the past [2, 3, 4]. It was shown that the electron (ion) flow can to a large extent be described using the neutral gas expansion theory taking into account recombination processes in the energy balance.

A gas expansion corresponds to a free flow from a high pressure ($p_o$) region, often referred to as the source region or reservoir, into a low pressure ($p_{back}$) environment, the so-called background, through an orifice. When the pressure ratio $p_o/p_{back}$ exceeds a critical value (e.g. 2.1 for a monoatomic gas), the gas velocity at the source exit is equal to the local speed of sound (sonic orifice). Then the gas accelerates to supersonic velocity and the flow becomes adiabatic. At some distance from the source, depending on $p_{back}$, the flowing gas collides with the residual background gas that results in the formation of a stationary shock wave through which the supersonic flow adapts to the local subsonic conditions.

First, the general conservation laws which govern the behavior of a fluid on a macroscopic scale are presented and the specific case of a steady adiabatic flow is discussed as it applies well to the description of a neutral gas expansion. The validity of a macroscopic approach for the treatment of an expansion is discussed. Second, on the basis of the equations of hydrodynamics, we describe in detail the continuum free jet flow structure that results from the expansion of a fluid. Finally, a section is devoted to the shock wave properties as it is an important feature of a gas expansion.

## 2.2 Governing equations

### 2.2.1 The Boltzmann transport equation

The Boltzmann transport equation describes the time evolution of the distribution function $f(\vec{r}, \vec{v}, t)$ of a given species in the six dimension phase space [5]. It is a nonlinear equation which describes the irreversible evolution of the fluid towards Local Thermodynamic Equilibrium (LTE). The Boltzmann equation reads

$$ \frac{\partial f}{\partial t} + \vec{v} \cdot \vec{\nabla} f + \frac{\vec{F}}{m} \cdot \vec{\nabla}_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll-rad}} \quad (2.1) $$

where $\vec{F}$ represents forces which act on the system (in a plasma the forces are mainly of electromagnetic origin), $m$ is the mass, and $\vec{\nabla}_v$ is the gradient in the velocity space. The right hand side of Eq. 2.1 describes the changes in the distribution function as a result of elastic and inelastic collisions and as a result of radiative processes.
On a microscopic scale the state of the system, i.e. departure from equilibrium, is entirely defined when the distribution function $f$ is known. If the system is in thermodynamic equilibrium, then the Maxwell distribution function is solution of the Boltzmann transport equation [5]. However it is most of the time difficult to directly access the distribution function and therefore assumptions have to be made about the thermodynamic state of the system. For that reason it is often easier to consider the system on a macroscopic scale and to follow the evolution of quantities like particle density, average velocity, and temperature. The equations describing the evolution in time and space of those thermodynamic quantities, the so-called conservation laws, are deduced from the calculation of the velocity moments of the Boltzmann transport equation. This calculation is based on a decomposition of the velocity $\vec{v}$ in a random component $\vec{v}^r$ and a drift component $\vec{w}$ using the formalism of Braginskii [6].

### 2.2.2 Conservation laws and EOS

The conservation laws represent the conservation of mass, momentum and energy in space and time and they form the gas dynamics set of equation [7]. Those equations can only be used in the hydrodynamic regime, i.e. when the fluid can be considered as a continuous medium. The validity of the hydrodynamic approach is discussed in section 2.3.3.

* Continuity equation

The conservation of mass is insured by the continuity equation

$$ \frac{\partial n}{\partial t} + \vec{\nabla} \cdot (n\vec{w}) = S, \qquad (2.2) $$

where $n$ is the density and $\vec{w}$ the drift velocity. The right hand side represents the source term, i.e. the particles production and destruction due to radiative and collisional processes.

* Momentum conservation

The principle of conservation of momentum is in fact an application of Newton's second law of motion to an element of fluid. The intrinsic momentum equation is given by

$$ nm \frac{\partial \vec{w}}{\partial t} + nm(\vec{w} \cdot \vec{\nabla}) \cdot \vec{w} + \vec{\nabla} p + \vec{\nabla} \cdot \vec{\Pi} = qn(\vec{E} + \vec{w} \times \vec{B}) + \vec{R}, \qquad (2.3) $$

where $m$ is the mass of the species, $p$ is the pressure, $\vec{\Pi}$ is the viscosity tensor, $q$ is the charge of the particle, and $\vec{R}$ represents the exchange of momentum between different species by means of collisions. Both electric field $\vec{E}$ and magnetic field $\vec{B}$ are related to each other via the Maxwell equations [8].
• Energy conservation

The principle of conservation of energy amounts to an application of the first law of ther-
modynamics to a fluid element as it flows. Note that an element of gas has two forms
of energy: kinetic energy and internal or thermal energy (molecular gases have additional
degrees of freedom of vibration and rotation). The intrinsic energy equation reads

$$
\frac{\partial}{\partial t}(n\varepsilon) + \vec{\nabla} \cdot (n\varepsilon \vec{w}) + p\vec{\nabla} \cdot \vec{w} + \vec{\Pi} : \vec{\nabla} \vec{w} + \vec{\nabla} \cdot \vec{q} = Q, \quad (2.4)
$$

where $\varepsilon$ is the internal energy, $\varepsilon = \frac{3}{2}k_B T$ with $k_B$ the Boltzmann constant and $T$ the temperature, $\vec{q}$ the heat flux, and $Q$ the heat transfer term. The latter denotes the energy gain or loss through elastic/inelastic collisions and chemical reactions. For an isotropic fluid the thermal heat flux $\vec{q}$ is given by

$$
\vec{q} = -\kappa \vec{\nabla} \cdot T. \tag{2.5}
$$

This is the Fourier heat conduction law [7], also called general equation of heat transfer.
The coefficient $\kappa$ is the thermal heat conductivity which is usually a function of the tem-
perature [9].

• Equation of state (EOS)

The perfect gas law is used as an equation of state. It reads

$$
p = nk_B T. \tag{2.6}
$$

Dalton’s law relates the partial pressures $p_i$ of a species $i$ to the total pressure of a gas
mixture

$$
p = \sum_{i} p_{i}. \tag{2.7}
$$

• Conducting fluid: additional equations

A plasma can be considered as a conducting fluid since it contains in addition to neu-
tral particles charged particles, i.e. electrons and ions. The transport of charged particles
is strongly influenced by electric and magnetic fields, therefore Maxwell equations have to
be included in the set of governing equations [8]. Furthermore, quasi-neutrality is usually
assumed for the studied plasmas [10], i.e. the electron density approximately equals the
ion density.

Seeing that a plasma contains neutral as well as charged particles, several sets of con-
servation laws are needed, one per kind of particles [11, 3]. If the velocity and temperature
of ions and neutral particles are not too different, then they can be treated as one group:
the heavy particles. The different fluids are coupled together by means of friction and
energy exchange. This coupling is complex, and a detailed analysis of the time scale of
the interactions between the different kinds of particles is needed. The ionization degree of the expanding plasma studied in this thesis is usually low (< 1 %), i.e. the electron(ions) density is low compared to the neutral density, therefore the influence of charged particles on the transport of neutral particles is small and has been ignored.

The knowledge of the conservation laws, the equation of state, and the specific equa-
tions applying to a conducting fluid is not enough to wholly describe a plasma dynamical
flow. One also needs appropriate boundary conditions to find the appropriate solutions of
the equation system.

2.2.3 Steady isentropic flow

The expansion into a vacuum of a neutral gas or a plasma can be considered as a steady
expansion contrary for instance to the expansion of a pulsed laser created plasma for which
time dependent phenomena are of importance [13, 14].

To a good approximation, the flow which is initially in thermodynamic equilibrium can
be assumed to be adiabatic [15], and therefore all dissipative effects, i.e. losses through
line and continuum radiation, heat transfer, are absent. If besides the flow is laminar (high
Reynolds number), the expansion is isentropic (dS = 0 where S is the entropy). Further-
more, in order to go even further in the simplification of the conservation laws, several
assumptions are made: the Lorentz force is neglected (we only consider the transport of
neutral particle in this work) and viscosity (internal dissipation of energy) is also neglected.
The previous remarks lead to a simplification of the conservation laws. We then obtain
the set of equation governing a steady isentropic flow.

• The mass balance reads

$$
\vec{\nabla} \cdot (n\vec{\mathbf{w}}) = 0. \tag{2.8}
$$

• The momentum balance is written as

$$
nm(\vec{\mathbf{w}} \cdot \vec{\nabla}) \cdot \vec{\mathbf{w}} = -\vec{\nabla}p. \quad (2.9)
$$

If the flow is not rotating (curl $\vec{\mathbf{w}}$ = $\vec{0}$) then we obtain

$$
\frac{1}{2} n m \vec{\nabla} w^2 = -\vec{\nabla} p. \qquad (2.10)
$$

• The energy conservation equation is replaced by the Poisson adiabatic law [7]

$$
\left(\frac{n}{n_o}\right)^{\gamma-1} = \frac{T}{T_o}, \qquad (2.11)
$$

where $\gamma = \frac{c_p}{c_V}$ is the specific heat ratio, also called adiabatic exponent or isentropic exponent, which is equal to $\frac{5}{3}$ for monoatomic gases, $\frac{7}{5}$ for rigid diatomic molecules, and $\frac{9}{7}$ for non rigid diatomic molecules.
## 2.3 Supersonic expansion: the continuum free jet structure

### 2.3.1 Speed of sound and Mach number

From the hydrodynamic equations one shows that small adiabatic disturbances like small changes in pressure propagate inside a compressible fluid with speed $c_s$, the so-called speed of sound. The latter is derived from the wave equation which describes the propagation of a density change [12, 16]

$$c_s^2 = \left. \frac{\partial p}{m \partial n} \right|_{dS=0} . \qquad (2.12)$$

Note that the speed of sound is a thermodynamic variable. A sound wave is thus an oscillatory motion with small amplitude in a compressible fluid. The propagation of a sound wave is an adiabatic process in which a change in internal energy is caused by expansion or compression of a fluid element. In an ideal gas the local speed of sound reads

$$c_s = \sqrt{\frac{\gamma k_B T}{m}}. \qquad (2.13)$$

The sound speed is in the order of the mean thermal speed of the gas. The flow velocity of a fluid can be compared to the speed of sound. For this purpose, a dimensionless number, the Mach number, is introduced. The Mach number $M$ is defined as follows

$$M = \frac{w}{c_s}, \qquad (2.14)$$

where $w$ is the drift velocity of the fluid. According to the Mach number, three different flow regimes can be distinguished

- $M > 1$: supersonic flow,

- $M = 1$: sonic flow,

- $M < 1$: subsonic flow.

As we will see, in the case of a gas(plasma) expansion, the flow undergoes a transition from a supersonic regime to a subsonic regime. The transition, which corresponds to the adaptation to the local ambient conditions, is realized via the formation of a shock wave structure.

### 2.3.2 Free jet flow

The basic phenomena of the expansion of a weakly ionized plasma are to some extent similar to those of the expansion of a hot neutral gas, and can thus be described using the adiabatic supersonic expansion theory [1, 17, 18]. In the expansion of a weakly ionized plasma, because of several heating mechanisms, the temperature remains higher than in
neutral gas adiabatic expansions. The expansion of a plasma can thus be described as a
common density rarefaction because of the supersonic expansion, modified to account for
disturbances to an isentropic flow.

When a fluid expands freely through an orifice from a high pressure region (source) into a region with a finite pressure rather than perfect vacuum, a well defined free jet shock wave structure is produced [1, 19, 20], as depicted in Fig. 2.1, due to the interaction of the supersonic flow with the residual background gas. In the hydrodynamic flow regime, the gas(plasma) starts to expand from a sonic orifice where the Mach number *M* equals to 1 and the supersonic flow ends in the formation of a normal stationary shock wave. In the supersonic domain, the density decreases rapidly because of a rarefaction effect (increase in the jet diameter). The gas is accelerated to supersonic velocity over a few source orifice diameter, because of conversion of the thermal energy gained in the source into kinetic energy, i.e. directed motion. As a consequence, the temperature decreases. One has to distinguish between the perpendicular temperature *T*<sub>⊥</sub> and the parallel temperature *T*<sub>||</sub> associated respectively with the velocity distribution perpendicular and parallel to a stream line. If the background pressure is low enough, the collision frequency becomes at one point so low that the flow reaches a steady state [21], in which both the velocity and the parallel temperature are frozen. Behind this limit, *T*<sub>⊥</sub> continues to drop (geometric cooling), and the flow is in non-equilibrium. At the source outlet, the decay in temperature is adiabatic in the case of a neutral gas expansion and the adiabatic exponent γ stays constant. In the case of a plasma, dissipative processes like friction (viscosity) and heat transfer can disturb the cooling mechanism leading to a quasi-adiabatic expansion with a low γ. The isentropic exponent γ depends also on the ionization degree [3, 11, 22]. However, this effect is negligible in view of the very low ion density of all plasmas studied in this thesis.

In radial direction the free jet is limited by the jet boundary and a preceding barrel
shock wave. The internal core, the so-called zone of silence, corresponds to the region
where the flow is not disturbed by the background gas. In this zone, the particles propagate
with a speed higher than the sound speed, and no information can be received from the
surroundings.

At the source exit, the flow is underexpanded (jet static pressure > background pres-
sure), but after some distance, it becomes overexpanded (jet static pressure < background
pressure) and has to adapt the local ambient gas conditions. As a result, a stationary
compression shock wave is formed due to the collision of the moving fluid with the residual
background gas. Over the shock wave the flow experiences a transition from a supersonic
regime to a subsonic regime. Both the position and the thickness of the normal shock wave
depend on the background pressure. Across the shock wave, the flow adapts its pressure
to the local pressure, the gas is compressed and the density increases. Since the forward
flux is conserved, the velocity drops accordingly. Because of collisions, the kinetic energy
is converted back into thermal energy, i.e. random motion, and the temperature increases.
Within the shock wave the entropy increases. Moreover, at low pressure, the flow is lami-
nar in the shock region and turbulent mixing does not occur. The dimensionless Reynolds
<figure><img src="image_7.png" /><figcaption>Fig. 2.1: Schematic view of the continuum free jet flow structure created when a fluid expands freely from a high pressure region into a low pressure region.</figcaption></figure>

number Re [7] is defined as

$$Re = \frac{nmwl}{\mu}, \qquad (2.15)$$

with $l$ a characteristic length scale of the system and $\mu$ the dynamic viscosity. For a free jet flow, if $Re < 10^3$ the flow is laminar, else the flow is turbulent. Inside the shock front of an argon jet ($\mu = 5 \times 10^{-5} \text{ kg m}^{-1} \text{ s}^{-1}$), one finds $Re \approx 30$ at 20 Pa ($l = 10$ cm) and $Re \approx 150$ at 100 Pa ($l = 5$ cm). The features of a shock wave are described in section 2.4.

At the Mach disk location $z_M$, often referred as the end of the shock wave, $M = 1$ and the flow enters the subsonic regime. The position $z_M$ is given by

$$z_M = Cd\sqrt{\frac{p_{\text{stag}}}{p_{\text{back}}}}, \qquad (2.16)$$

where $d$ is the source outlet diameter, $p_{\text{stag}}$ is the stagnation pressure in the source and $p_{\text{back}}$ is the background pressure. The constant $C$ depends slightly on $\gamma$; it equals 0.7 for an atomic gas. The preceding expression can be rewritten in terms of flow and atomic mass number [23]. It has been experimentally discovered by Ashkenas et al [17] and later demonstrated theoretically by Young [24].

Behind the Mach disk, the gas expands subsonically and mixes with the background gas. The evolution of the flow is then mainly geometrically determined (general circulation pattern, residence time). The transport of any particle is controlled by diffusion. The flow velocity decreases because of momentum transfer to the ambient gas particles. The temperature decreases because of heat transfer (to the vessel walls for instance). The pressure is constant, therefore the density increases.
<figure><img src="image_8.png" /><figcaption>Fig. 2.2: Schematic representation of the fluid particle density and velocity development (left) and temperature development (right) along the jet centerline in the course of the fluid expansion.</figcaption></figure>

A schematic representation of the development of the macroscopic quantities along the jet centerline of a free jet is depicted in Fig. 2.2 in the case of a continuum flow regime. In the supersonic domain, a theoretical expression for the on-axis density, temperature and velocity profile of a neutral gas flow can be derived from the hydrodynamic equations. Such expression are useful to compare the a neutral inert gas expansion to a plasma expansion and will be often used in this thesis.

• Density development

Using the conservation laws for a steady isentropic flow (see section 2.2.3), and under the assumption that the particles originate from a virtual point source and flow along straight stream lines, one can obtain the theoretical expression for the density development along the jet centerline

$$n(z) = n_o \frac{z_{\text{ref}}^2}{(z + z_o)^2}. \qquad (2.17)$$

In Eq. 2.17, $n_o$ is the particle density in the source, $z_{\text{ref}}$ is a scaling length (determined by the early expansion and by the source orifice geometry), and $z_o$ is the position of the virtual point source, i.e. the expansion origin. It means that the free jet flow can be described for $z \gg d$ as a source expansion [17, 1, 2]. The expression is valid for an expansion with cylindrical symmetry as well as for an expansion with spherical symmetry [25, 26]. Another expression can be used [23]

$$n(z) = n_o \frac{1}{1 + z^2/z_o^2}. \qquad (2.18)$$

This non-standard expression is connected with the plasma character of the source region with its high temperature, sonic exit velocity, and finite temperature gradients. It allows
for a zero first derivative of the density at the source point in order to accommodate for the convective processes in the source, as well as for field generation.

• Temperature development

For an adiabatic process, the energy equation can be replaced by the Poisson adiabatic formula (Eq. 2.11). Using Eq. 2.18 to express the density, one obtains for the temperature profile in the supersonic domain

$$T(z) = \frac{T_o}{(1 + z^2/z_o^2)^{\gamma-1}}. \qquad (2.19)$$

In the case of a plasma expansion, the cooling mechanism is generally not adiabatic and the value of $\gamma$ is lower than the adiabatic one.

• Velocity development

The velocity along the jet axis is obtained from the simplified momentum balance (Eq. 2.10) using the Poisson adiabatic law (Eq. 2.11). Two situations have to be distinguished. When the system is at rest in the source (expansion from a reservoir, $w(0) = 0$) the velocity reads

$$w(z) = \frac{c_{s,o}}{\sqrt{\gamma - 1}} \sqrt{2 - 2 \left( \frac{n(z)}{n_o} \right)^{\gamma - 1}}. \qquad (2.20)$$

The maximum achievable velocity, that corresponds to the total conversion of the thermal energy into kinetic energy, is

$$w_{\max} = c_{s,o} \sqrt{\frac{2}{\gamma - 1}}. \qquad (2.21)$$

When the fluid is already flowing in the source before it starts expanding (case of thermal arc plasmas under investigation in this thesis) the velocity is given by

$$w(z) = \frac{c_{s,o}}{\sqrt{\gamma - 1}} \sqrt{\gamma + 1 - 2 \left( \frac{n(z)}{n_o} \right)^{\gamma - 1}}, \qquad (2.22)$$

when the fluid is assumed to flow in the source with the speed of sound ($w(0) = c_{s,o}$ or $M(0) = 1$). The maximum velocity is

$$w_{\max} = c_{s,o} \sqrt{\frac{\gamma + 1}{\gamma - 1}}. \qquad (2.23)$$

2.3.3 Rarefied regime: evanescent shock structure

A measure of the validity of the continuum assumption is given by the calculation of the dimensionless Knudsen number $Kn$ [12]
<figure><img src="image_9.png" /><figcaption>Fig. 2.3: Free jet flow in rarefied (molecular) regime. The shock wave structure becomes evanescent and background gas particles invade the final stage of the supersonic domain of the jet.</figcaption></figure>

$$Kn = \frac{\lambda}{l}, \qquad (2.24)$$

where $\lambda$ is the mean free path for momentum exchange and $l$ is a characteristic length scale of the system. If $Kn < 0.2$ then the flow is said to be in hydrodynamic regime and the continuum model is applicable [27]. As $Kn$ approaches unity, the flow enters the free molecular regime, or rarefied regime, and one has to use the molecular dynamics theory.

The use of hydrodynamics to describe a fluid expansion is only strictly valid in the first few millimeters of the flow. At a relatively high background pressure the flow can always satisfactorily be described using hydrodynamic equations. However at low pressure the flow undergoes a transition between hydrodynamic and free molecular regime and gas kinetic theory is needed.

When the Knudsen number starts to become high, $Kn > 0.2$, the fluid can not be approximated as a continuous medium, and departure from the perfect free jet flow picture occurs [28]. In a transition regime between continuum and rarefied, the shock wave structure is diffuse enough to allow penetration, and the zone of silence is invaded by the residual peripheral gas particles, as depicted in Fig. 2.3. This occurs mainly in the final stage of the free jet expansion, where the barrel shock wave becomes evanescent. The invasion process is strongly mass selective (diffusion process) in favor of the light species. It has been studied as a possible means for aerodynamic separation of mixtures and isotopes [29, 30]. Invasion by the ambient gas particles modifies the shock wave formation mechanisms and influence a priori the distribution function within the normal shock wave. Those effects are discussed in chapter 9 and 10. Deep into the rarefied flow regime, the velocity remains supersonic because of scarce collision events and the shock structure vanishes.
## 2.4 Shock wave

### 2.4.1 Properties of a shock wave

A shock wave or shock front is an example of a surface of discontinuity in the field of fluid dynamics [7, 12, 16]. Over a shock wave, the distribution of thermodynamic quantities such as pressure and temperature as well as the distribution of the normal velocity is discontinuous. Certain boundary conditions must be satisfied on a surface of discontinuity: the mass flux, the energy flux as well as the momentum flux must be conserved. This set of conditions forms the so-called shock condition. In the case of an infinitely narrow shock wave it leads to the jump relations for the density, velocity and temperature (see next section). As every thermodynamic variable, entropy is also discontinuous at a shock wave. The entropy increases due to dissipative processes in the shock wave meaning that the gas motion is irreversible. Thus shock waves are a means by which energy can be dissipated in the motion of an ideal fluid.

In general shock waves are classified according to the sign of the pressure jump across the surface: $P = p_2 - p_1$ where the subscript 1 and 2 refer to the quantities ahead of and behind the shock wave respectively. If $P > 0$, respectively $P < 0$, the shock wave is said to be a compression shock, respectively a rarefaction shock [31]. From the entropy balance, and referring to Hugoniot adiabatic, it follows that only compression shocks are possible for the case of a perfect gas with constant specific heats.

For a compression shock wave, the conditions are the following [7]

$$ \begin{gathered} p_1 < p_2, V_1 > V_2, w_1 > w_2 \\ \text{and } w_1 > c_{s,1}, w_2 < c_{s,2}, \end{gathered} $$

where $V$ is the specific volume. Therefore the upstream flow ahead of the shock wave is supersonic whereas the downstream flow behind the shock wave is subsonic.

### 2.4.2 Rankine-Hugoniot relations

The so-called Rankine-Hugoniot relations [7, 12] connect the jump in density, velocity and temperature across the shock wave to the Mach number $M_1$ ahead of the shock front

$$ \frac{n_2}{n_1} = \frac{w_1}{w_2} = \frac{(\gamma + 1)M_1^2}{(\gamma - 1)M_1^2 + 2}, \qquad (2.25) $$

$$ \frac{T_2}{T_1} = \frac{[2\gamma M_1^2 - (\gamma - 1)][(\gamma - 1)M_1^2 + 2]}{(\gamma + 1)^2 M_1^2}. \qquad (2.26) $$

The Rankine-Hugoniot relations represent the conservation of flux and energy across a shock front when the latter is infinitely narrow (proper surface of discontinuity). In the case of very strong shock waves, i.e. $M_1$ tends to infinity, the temperature ratio increases to infinity, whereas the density ratio tends to a constant limit, i.e. $(\gamma + 1)/(\gamma - 1)$. Therefore for a monoatomic gas ($\gamma = \frac{5}{3}$), the maximum achievable compression ratio is 4.
### 2.4.3 Shock wave thickness

The treatment of shock waves as discontinuities of zero thickness is an idealization of inviscid gas dynamics. Shock waves in which the discontinuities are small are not geometrical surfaces with zero thickness, but are in reality transition layers with finite thickness. The thickness of a shock front decreases as the magnitude of the discontinuities increases. To determine the structure and the thickness of the transition layer, viscosity and thermal conductivity of the gas, i.e. dissipative effects, have to be taken into account.

The shock thickness $L$ in the $z$ direction is defined as

$$L = \frac{w(-\infty) - w(+\infty)}{|dw/dz|_{\text{mean}}} = (w_1 - w_2) \frac{dz}{dw}\Big|_{\text{mean}}, \qquad (2.27)$$

according to the Prandtl formalism [32]. In the same manner, the thickness can be defined according to the density jump. Using the continuum theory the shock thickness can also be expressed as a function of the Mach number ahead of the shock front

$$L \approx \frac{8\lambda}{3(M_1 - 1)}, \qquad (2.28)$$

where $\lambda$ is the momentum exchange mean free path of the fluid particles. Thus the thickness of a shock wave is of the same order of magnitude as the mean free path. It follows that the method of gas dynamics, where the gas is treated as a continuous medium, cannot strictly be used to investigate the internal structure of a shock wave (see chapter 9).

When the $Kn < 0.2$ ahead of the stationary shock wave of a free jet (continuum flow regime), the diameter of the jet is constant within the shock region and the Rankine-Hugoniot (RH) relations hold. In the rarefied regime $0.5 < Kn < 0.2$, a departure from the RH relations is possible caused by an increase in the jet diameter but also by the invasion of the jet by the background gas particles.

### 2.4.4 Shock wave profile

#### Hydrodynamic approximation

When the continuum approach is valid, and in the low intensity shock wave approximation (low $M$), the velocity profile across the shock front of a free jet flow can be calculated from the Navier-Stokes equation [12, 33, 34]. This leads to the Taylor shock structure solution for the development of the drift velocity $w$ along the direction $z$ normal to the shock front

$$\frac{\gamma + 1}{2a}(w_1 - w_2)z = \ln \left( \frac{w_1 - w(z)}{w(z) - w_2} \right), \qquad (2.29)$$

where $z = 0$ refers to the center of the shock front and $a$ is a coefficient related to acoustic attenuation [12]. In first order approximation, this coefficient can be estimated using the gas kinetic theory to express the viscosity and the thermal conductivity of the gas. One
obtains $a = \lambda c_s$. From the flux conservation, one can derive the density distribution over the shock wave

$$ \frac{\gamma + 1}{2a} w_1 \left(1 - \frac{n_1}{n_2}\right) z = \ln \left[ \frac{n_2(n(z) - n_1)}{n_1(n_2 - n(z))} \right]. \qquad (2.30) $$

A more complex approach consists of using the Burnett equations of hydrodynamics as they appear from the Chapman-Enskog solution to the Boltzmann equation [35]. More information about hydrodynamic models can be found in reference [34] and [35].

## Solution of the Boltzmann equation

In the case of a strong shock wave or when the continuum assumption is no longer legitimate, the shock structure can only be investigated using the kinetic theory, i.e. by looking directly at the solution of the Boltzmann transport equation. It is a complex problem mainly caused by the nonlinearity of the Boltzmann equation. A possible way of solving the problem is to calculate the numerical solution of the Boltzmann equation by direct Monte-Carlo simulation method [38, 39].

In this thesis we only propose to look at a simple approach that consists of solving the Boltzmann equation assuming that the distribution function within the shock wave can be decomposed into two Maxwellian terms (see chapter 3 and 8). This approach has been first proposed by Mott-Smith [36] about fifty years ago and later extended by several authors [32, 37]. Under the assumption of a bimodal distribution, one obtains an approximate solution to the Boltzmann equation; then using the conservation laws a theoretical density and velocity profile throughout the shock region is determined

$$ \frac{n(z)}{n_1} = \frac{w_1}{w(z)} = \frac{1 + \left(\frac{n_2}{n_1}\right) \exp\left(\frac{4z}{L}\right)}{1 + \exp\left(\frac{4z}{L}\right)}. \qquad (2.31) $$

Eq. 2.31 is often used to model shock wave profiles at the limit of the continuum flow regime. In the case of inert neutral particle expansion, the theoretical profiles are in good agreement with the measured data [40, 41].

## 2.5 Conclusions

The set of equations governing the transport of neutral particles in the expansion of a weakly ionized plasma as well as the main characteristics of the free expansion process have been briefly presented. As we will see in the rest of this thesis, apart from the fact that due to the high plasma temperature the flow is not adiabatic, the inert neutral (e.g. Ar atoms or H₂ molecules) flow is in good agreement with the free jet flow theory. On the contrary, the reactive particle (e.g. H or D atoms) flow deviates significantly from the classical picture. It will be demonstrated step by step that surface chemistry, surface recombination of radical to form stable molecules in that case, strongly influences the transport mechanisms of any kind of radicals in plasma expansions.
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# Chapter 3

## Flow characteristics of a supersonically expanding thermal argon plasma<fnref n="1" />

### Abstract

The transport of neutral argon atoms in an expanding thermal argon plasma seeded with 4 % of hydrogen is studied by means of laser induced fluorescence spectroscopy, on the long lived Ar[4s] atoms, and by UV Rayleigh scattering. The results of the latter confirm the inverse quadratic dependence of the argon atom density with distance from the nozzle in the supersonic part of the expansion. The measured atom density jump through the stationary shock wave can be described with the Rankine-Hugoniot relations. From the Doppler-shifted laser induced fluorescence measurements performed on Ar atoms in the metastable $Ar^*(^3P_2)$ and resonant $Ar^*(^3P_1)$ states, it is inferred that the velocity behavior of the supersonically expanding argon gas can be predicted from the momentum balance, and the temperature from the Poisson adiabatic relation. However, the adiabatic constant is found to be smaller than the adiabatic constant of a neutral argon gas expansion. Non-Maxwellian distribution functions are measured along the jet centerline, both at the arc exit and across the stationary shock wave. The results on temperature and velocity in the subsonic region show that the radius of the plasma jet hardly increases after the stationary shock front, indicating that the flow pattern is geometrically determined.

<footnote n="1">¹ Adapted from:
R. Engeln, S. Mazouffre, P. Vankan, D.C. Schram, N. Sadeghi, accepted for publication in *Plasma Sources Sci. Technol.* (2001).</footnote>
**3.1 Introduction**

Expanding thermal plasmas are interesting both from a fundamental point of view as well as from a view of the applicability. From a fundamental point of view, the plasma expansion is the basis for describing phenomena ranging from astrophysical objects and solar flares, the flow into the divertor region of a Tokamak, to laser spots and vacuum arc spots [1, 2, 3, 4]. In view of the applicability, expanding thermal plasmas are used for deposition of thin films, etching and passivation of surfaces, and treatment of archiological artifacts [5]. Optimal energy conversion from the plasma source, via the expanding plasma, to the gases used for surface treatment or modification, is therefore of eminent importance. The study of the expansion process can substantiate the understanding of the conversion process.

Over the past years, extensive studies have been performed on expanding plasmas. Examples are Thomson-Rayleigh scattering measurements on the expansion properties of pure argon plasmas and argon/hydrogen plasmas, which supplied electron and neutral densities and electron temperatures [6, 7, 8, 9]. Coherent Anti-Stokes Raman scattering experiments on hydrogen plasmas deliver absolute densities of ro-vibrationally excited $H_2$ showing the importance of wall association and re-circulation of background gas [10]. In a study on the influence of wall recombination processes on the transport of radicals, two-photon laser induced fluorescence measurements on hydrogen atoms in an argon/hydrogen plasma are performed and atomic radical densities, temperatures and velocities are obtained [11, 12]. These studies give better insight in the physics of the expanding plasma, like recombination, excitation, wall association and re-circulation.

Starting from a remote ionizing plasma source, the plasma expands supersonically and becomes recombining. The supersonic flow domain ends in the formation of a stationary shock wave. Behind this shock wave the plasma flows subsonically into the background. In the downstream region of the plasma expansion molecules, e.g. $SiH_4$, $CH_4$ or $C_2H_2$, can be injected into the plasma jet [13]. Here they are fragmented via charge-exchange and dissociative recombination reactions into radicals. These molecular radicals are transported by the plasma flow towards a substrate. At the substrate they can stick to the surface and contribute to film growth, reflect from the surface or recombine at the surface forming 'new' species. Depending on the residence time of the gases in the vessel (about 0.5 s at 20 Pa) the newly formed species can re-circulate and subsequently influence the plasma flow pattern. The understanding of the mixing of the injected monomers and newly formed molecules with the plasma jet is important for optimal use of these species in the processes at hand.

In most of the studies and applications of the expanding plasma jet, argon is used as a carrier gas. Due to its relatively high mass, as compared to the one of injected gases like hydrogen, nitrogen or methane, the velocity with which the plasma expands will be to a large extent determined by the Ar velocity. We here report on laser induced fluorescence measurements on metastable argon atoms in an expanding thermal plasma generated from a mixture of argon with a few percent of hydrogen. From these measurements both the axial and radial velocity components and temperatures of the argon atoms are extracted.
The non-Gaussian velocity distributions in the shock region indicate a departure from thermodynamic equilibrium. It will be shown that the velocity and temperature in the supersonic region of the plasma expansion can be described as a neutral gas expansion with a modified value of the adiabatic exponent $\gamma$. Rayleigh scattering measurements are used to determine the total argon atom density and flux in the plasma jet.

## 3.2 Experimental arrangement

### 3.2.1 Cascaded arc

The cascaded arc in which a thermal plasma is created ($\hat{T} \approx 1$ eV) has been described in detail elsewhere [14]. The arc operates at subatmospheric pressure (0.6 bar) It consists of 4 insulated plates with a 3 mm diameter channel. Three cathodes are positioned concentrically around the channel. The nozzle, through which the plasma expands into a low pressure chamber, is composed of a straight channel (6 mm diameter, 10 mm length) followed by a diverging opening (45°, 5 mm length). During the measurements presented here, a current of 40 A is drawn through the arc and an argon flow rate of 3.0 slm (standard liter per minute) is used. A flow of 0.12 slm H₂ is added to the argon flow to decrease the ion density, and thus the metastable density, in order to avoid any significant diminution of the laser intensity due to absorption by metastable atoms present on the laser beam trajectory before reaching the excitation volume. Moreover, H₂ quenches the Ar[4s] atoms present outside of the expanding jet and therefore avoids the reabsorption of the LIF photons by these atoms, before reaching the detector. The influence of molecular hydrogen on the very fast ionization loss in an expanding argon/hydrogen plasma has been reported by Meulenbroeks et al [15]. The electron and argon ion densities are reduced due to charge-exchange of Ar⁺ with molecular hydrogen, followed by dissociative recombination of the ArH⁺ ion forming Ar and excited atomic hydrogen. This leads to an ionization degree of less than 1 %.

### 3.2.2 LIF setup

A schematic view of the Laser Induced Fluorescence setup is depicted in Fig. 3.1. A single-mode cw external cavity diode laser (EOSI LCU 2010M), delivering about 10 mW of radiation, is used to excite Ar* atoms in the expanding thermal plasma. The laser is tuned to the $1s_5 \rightarrow 2p_9$ transition at 811.53 nm or to the $1s_4 \rightarrow 2p_7$ transition at 810.37 nm [16]. The velocity along the laser beam, $w_z$, of the group of atoms in the metastable $1s_5$ and resonant $1s_4$ state absorbing the laser photons at frequency $\nu$ is deduced from the relation

$$w_z = c \times \frac{\nu - \nu_0}{\nu_0}, \qquad (3.1)$$

where $\nu_0$ is the transition frequency for atoms at rest and c the speed of light. In one configuration the laser beam propagates opposite to the flow direction of the plasma jet and
<figure><img src="image_10.png" /><figcaption>**Fig. 3.1:** Schematic view of the experimental LIF setup using a cw laser source and synchronous detection. Also shown is the cascaded arc and the expanding thermal plasma. Spatial scans through the plasma jet can be made by moving the arc relative to the intersection of laser beam and detection volume.</figcaption></figure>

excites the atoms on the center z-axis of the expansion. In another configuration the beam is directed perpendicular to the flow direction. In this case the radial velocity of the atoms excited at different radial positions in the jet is monitored. The laser light is chopped at a frequency of about 800 Hz and the laser induced fluorescence signal is detected under 90° with a photo multiplier tube (PMT). A high pass filter (λ cut-off 780 nm) placed between the plasma and the detector avoids saturation of the detector due to the emission of the plasma. A slit in front of the detector determines the spatial resolution of the measurements to 2 mm in both z and y-direction. The signal from the PMT is analyzed with a lock-in amplifier with an integration time of 500 ms. The signal of the chopper is used as a reference signal for the lock-in amplifier. Part of the laser light is used to measure simultaneously the absorption profile of argon atoms in a low pressure argon lamp (T ≈ 600 K) and the transmission of a Fabry-Perot etalon. The three signals are recorded simultaneously with a 12 bit 200 kHz analog-to-digital converter (National Instruments PCI-6023E). The tuning of the laser is controlled by the lock-in amplifier via a voltage supplied to the piezo connected to the end mirror of the external cavity of the diode laser. In this way mode hop free scans of a few wavenumbers can be recorded. A typical recording of three traces is shown in Fig. 3.2. The LIF signal in Fig. 3.2 is measured at z = 10 mm in the expanding plasma jet after excitation of the Ar*(1s₅) → Ar*(2p₉) transition at 811.53 nm. The linewidth of the laser is in the order of 1 MHz, much narrower than the line width of the recorded Ar transition in the expanding plasma and low pressure argon lamp. The frequency shift between the absorption in the lamp and the LIF signal from the plasma is directly related via Eq. (3.1) to the velocity of the Ar atoms in the expanding plasma. The simultaneously recorded Airy-type transmission of the Fabry-Perot etalon is used to
<figure><img src="image_11.png" /><figcaption>**Fig. 3.2:** A measurement showing three traces: the LIF signal from the excited Ar metastables in the plasma expansion, the absorption spectrum in a low pressure argon lamp, and the transmission of the Fabry-Perot interferometer.</figcaption></figure>

linearize the frequency axis.

## LIF measurements

LIF measurements at different axial positions ($z$) have been performed in the expanding argon plasma jet for two different background pressures, i.e. 20 and 100 Pa, and on two different Ar[4s] → Ar[4p] transitions, i.e. $1s_5 \rightarrow 2p_9$ and $1s_4 \rightarrow 2p_7$. The recorded spectral profile of the transition is a direct measurement of the Ar atom velocity distribution function (VDF) along the laser beam axis. The local mean Ar atom velocity is deduced from the line profile. If the system is in thermodynamic equilibrium (Gaussian VDF), an Ar atom temperature can be defined (related to the width of the line). The area of the peak provides a direct measure for the local density.

Mainly the results on the $1s_5$ state — the lowest metastable state — will be discussed, since the velocity distributions of the $1s_4$ state — the lowest resonant state — showed no significant difference with the former one. In Fig. 3.3 two velocity distributions are shown, measured at $z = 60$ mm. One is recorded after excitation of the $1s_5 \rightarrow 2p_9$ transition, and one after excitation of the $1s_4 \rightarrow 2p_7$ transition. In both cases the same experimental plasma conditions were applied. The small differences could be attributed to a difference in position at which the two spectra are recorded. The day to day reproducibility of the position of the plasma source in the $z$ direction is in the order of 1 millimeter.
<figure><img src="image_12.png" /><figcaption>**Fig. 3.3:** Axial velocity distribution of atoms in the Ar*(³P₂) metastable state (open square) and the Ar*(³P₁) resonant state (filled square) recorded at *z* = 60 mm. The arc is operated at standard conditions; the background pressure is 20 Pa.</figcaption></figure>

### 3.2.3 Rayleigh scattering

The Rayleigh scattering [17, 18] measurements are performed with the UV laser beam used to detect atomic hydrogen by means of two photon laser induced fluorescence. The laser setup is described elsewhere [19, 20] (see also chapters 4 and 6). In short, a 20 Hz frequency doubled Nd:YAG laser beam is used to pump a tunable dye laser, which delivers light around 615 nm. This light is then frequency tripled (KDP + BBO crystals). In this way about 2 mJ of tunable radiation around 205 nm can be obtained (5 ns pulse duration, horizontally polarized). The Rayleigh scattering cross-section increases when the radiation wavelength decreases (λ⁻⁴ dependence) which allows for using a relatively low energy per pulse (compared to the Nd:YAG laser beam for instance). Moreover, the ratio of the scattered light to the stray light increases at low wavelength. The UV radiation is directed into the expansion vessel and the scattered light is detected under 90° with respect to both the laser beam and the plasma jet with a solar blind photomultiplier tube. The current detection limit for the total density in an Ar-H₂ plasma is 10²⁰ m⁻³. The latter does not permit to measure the Ar density accurately at a background pressure below 40 Pa. The Ar density profiles reported here have been measured under slightly different experimental conditions [11]: 3 slm Ar and 0.5 slm H₂. The small difference in the hydrogen flow is taken into account when the density profiles are used in the data analysis.
## 3.3 Interpretation of the LIF spectra

In Fig. 3.4 measured axial velocity distribution functions are shown, recorded at 6 different $z$ positions at a pressure of 20 Pa after excitation of the Ar*(1s$_5$) → Ar*(2p$_9$) transition. The laser frequency is converted to the velocity along the laser propagation direction by using Eq. (3.1). Although the measured spectra originate from metastable Ar atoms, it will be demonstrated that the velocity distributions image the velocity distributions of ground-state Ar atoms.

### 3.3.1 Recombination of Ar ions

Argon atoms in the longlived 4s states that are used for the determination of velocity profiles are formed in a three particle recombination process

$$Ar^+ + e^- + e^- \rightarrow Ar^* + e^- \qquad (3.2)$$

It is well established that all Ar* atoms produced in this three particle recombination cascade down to the 4s states [21, 22]. We will first discuss conditions under which the velocity distribution profile of these longlived atoms is identical to that of the ground state argon atoms. We consider a cross section of $2 \times 10^{-19}$ m$^2$ for Ar-Ar collisions and a cross section of $1 \times 10^{-18}$ m$^2$ for both Ar+-Ar and Ar[4s]-Ar collisions. We also suppose that at any $z$, the electron density will be about 100 times smaller than the argon density [23]. The argon density is determined from the Rayleigh measurements at 42 and 100 Pa (see Fig. 3.9). The gas temperature $T_g$ and the axial velocity $w_z$ are assumed to be equal to the temperature and mean velocity of the metastable Ar atoms. Shown in Fig. 3.5a for a 20 Pa background pressure and in Fig. 3.5b for 100 Pa, are the mean distances, $\lambda_{00}$, $\lambda_{i0}$ and $\lambda_{m0}$, in the laboratory frame, that an Ar, Ar$^+$ or Ar[4s] can travel along the $z$-axis before colliding with an argon atom. All distances are calculated according to the following relation

$$\lambda = \frac{w_z}{n\sigma v_{th}} \qquad (3.3)$$

The thermal velocity $v_{th}$ is deduced from $T_g$ according to

$$v_{th} = \sqrt{\frac{2kT_g}{m}} \qquad (3.4)$$

To consider the kinetics of the four Ar[4s] states, we must first remember that the three particle recombination process is instantaneous, and thus the argon atoms in the metastable and resonant states formed in this process image in first instance the velocity distribution of the ions. The cross section for collisions of Ar$^+$ ions with argon atoms, including both charge transfer and elastic collisions, is about $10^{-18}$ m$^2$ [26]. Consequently, as one can see in Fig. 3.5, for $z < 20$ mm, the path length of the ions, $\lambda_{i0}$, is smaller than 2 mm. This means that ion-neutral collisions are fast enough to equilibrate the velocity distribution of ions and argon atoms. We therefore assume that in view of the smallness of $\lambda_{i0}$ compared to the local gradient lengths, the Ar$^+$ ions and Ar atoms will be in equilibrium at these positions and further downstream.
<figure><img src="image_13.png" /><figcaption>**Fig. 3.4:** Axial velocity distributions of metastable Ar recorded at 6 different z positions along the jet axis at 20 Pa background pressure. The metastable Ar VDF images the ground-state Ar VDF. The full lines show the result of a fit using a bimodal approximation. The dashed lines show the individual Gaussian VDF. Departure from thermodynamic equilibrium occurs both at the source exit and within the stationary shock wave.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 3.5: The mean distance an Ar, Ar⁺ and Ar* can travel along the z-axis before colliding with an argon atom at 20 Pa (a) and 100 Pa (b) background pressure. λₘₑ denotes the mean distance an Ar[4s] atom can travel along the z-axis before being transferred to an adjacent Ar[4s] state.</figcaption></figure>

Surprisingly, at short distances from the exit of the nozzle ($z < 16$ mm at 20 Pa and $z < 8$ mm at 100 Pa) we observe a forward peaking of the velocity distribution (see Fig. 3.4a).

### Influence of the electric field

The forward peaking of the VDF, i.e. the existence of very fast particles, has already been reported in e.g. time-of-flight measurements in a plasma created from a hollow cathode arc [27]. The fast component of the measured VDF may be attributed to metastable atoms born from recombination of Ar⁺ ions accelerated by an electric field generated from the gradient of the charge density in the expanding jet. The amplitude of this field is given by the Boltzmann relation [28]

$$E(z) = \hat{T}_e \frac{\nabla n}{n}. \qquad (3.5)$$

Since the decay of $n_e$ is similar to the decay of the neutral density [6], for $z > 4$ mm, $\nabla n/n$ is approximately 2/z and assuming $\hat{T}_e = 0.3$ eV, identical to the gas temperature, the amplitude of the electric field would be about 120 V/m at $z=5$ mm and 60 V/m at $z=10$ mm. In the valley and after the shock front the gradient and thus the fields are much smaller and reverse. Consequently the fast component should disappear as it is observed.

### 3.3.2 Coupling between ground-state Ar and Ar[4s]

Concerning the Ar[4s] atoms, momentum transfer and metastability-exchange collisions with a cross-section of about $10^{-18}$ m² will mix the Ar*(³P₂) metastable and ground state
atoms. The collision cross-section $\sigma$ is determined from a measured diffusion coefficient $D = 1.7 \times 10^{20}$ molecule m$^{-1}$ s$^{-1}$ at 300 K [29]. The relation between $D$ and $\sigma$ is [34]

$$D = \frac{3\pi}{16\sqrt{2}} \frac{\langle w \rangle}{\sigma} \qquad (3.6)$$

Here $\langle w \rangle$ is the mean velocity of argon atoms at 300 K. We can see in Figs. 3.5a and 3.5b that except for $30 < z < 100$ mm at 20 Pa and $z$ around 20 mm at 100 Pa, the mean path length of metastable atoms, $\lambda_{m0}$, is comparable to, or smaller than the spatial resolution in our experiments. Similarly, the radiative lifetime of the resonant $Ar^*(^3P_1)$ and $Ar^*(^1P_1)$ states is in the order of 10 ns. However, under our experimental conditions, due to the radiation trapping [25], the effective lifetimes of these states become a few microseconds. Consequently, due to this effect, resonant and ground-state argon atoms are very efficiently coupled and can therefore be considered as being fully mixed. As an example, at an argon density of $10^{20}$ m$^{-3}$, the mean free path of the resonant VUV photons is less than 0.1 mm. This means that the velocity distribution of the resonant $Ar^*(^3P_1)$ and ground state atoms are identical. Also, collisions with electrons will transfer $Ar[4s]$ atoms between the resonant $Ar^*(^3P_1)$ and metastable $Ar^*(^3P_2)$ states, via $Ar^*(^3P_2) + e \leftrightarrow Ar^*(^3P_1) + e$. The mean distance $\lambda_{me}$ that $Ar[4s]$ atoms can travel along the $z$-axis before being transferred to the adjacent state is determined by the relation $\lambda_{me} = w_z/(k_e \times n_e)$. For this reaction the rate coefficient $k_e$ is about $2 \times 10^{-13}$ m$^3$s$^{-1}$ [24]. Assuming an electron density with the same $z$-dependence as Ar atoms but 100 times smaller, we calculated the mean path length of a metastable atom before suffering a collision with an electron for 20 Pa (see Fig. 3.5a) and 100 Pa (see Fig.3.5b) background pressure. At 20 Pa, for most positions $\lambda_{me}$ is smaller than the typical gradient length $L_n = n/\nabla n$ (about $z/2$ at these positions); for $30 < z < 100$ mm, the condition is marginally satisfied. At 100 Pa, the condition is fulfilled everywhere. Also, considering the velocity of the expanding jet $w_z \approx 3$ km/s, from the microsecond range lifetime of the resonant states we can conclude that for most positions the argon atoms in the metastable and resonant levels are produced and lost locally [21]. We conclude that for $z > 20(8)$ mm at 20(100) Pa, the velocity distribution function of argon atoms in the $4s$ states is perfectly similar to that of the ground state atoms. They are created from $Ar^+$ ions already in equilibrium with argon atoms and they are strongly coupled to the ground state atoms by collisional and radiative processes. Close to the shock wave, located around $z = 70$ mm at 20 Pa and $z = 30$ mm at 100 Pa, the mean free path for the neutral-neutral collisions becomes larger than the gradient length $L_n$, which means that the velocity distribution of argon atoms can be non-Gaussian in this region.

### 3.3.3 Ar velocity distribution function

#### Departure from thermodynamic equilibrium

The velocity distributions shown in Fig. 3.4 can thus be seen as velocity distributions of neutral Ar atoms. As discussed before, the distributions for $z \le 20$ mm can be distorted due to the existence of an accelerating E-field acting on $Ar^+$ ions. At 20 Pa, for
<figure><img src="image_15.png" /><figcaption>**Fig. 3.6:** Behavior of the content of the metastable Ar VDF at 20 Pa background pressure along the jet axis. (solid triangle). Also shown is the content of both the fast (square) and the slow (circle) component across the shock wave.</figcaption></figure>

$20 < z < 30$ mm the measured distribution can be described to a good approximation with a Gaussian profile, i.e. the flow is in thermodynamic equilibrium. As can be seen in Fig. 3.4c, behind $z = 30$ mm, i.e. in the shock region, the VDF measured along the jet centerline starts to deviate from the equilibrium form. Across the stationary shock wave, the Ar VDF can be decomposed into two Gaussian VDF: one corresponds to the conditions upstream of the shock ("fast" component characterized by a high mean velocity and a low temperature), and the other one corresponds to the conditions in the shock wave ("slow" component characterized by a low mean velocity and a high temperature). The bimodal approximation for the VDF within a shock wave has first been proposed by Mott-Smith [30] and latter developed by Glansdorff [31]. However, in the so-called Mott-Smith model, the slow distribution corresponds to the condition downstream of the shock front, i.e. its mean velocity and its temperature are constant. Across the shock wave, the population of the fast Gaussian component is gradually transferred to the slow component by means of collisions [32, 33]. More information about shock wave properties can be found in chapter 8. Behind the shock region, $z \ge 150$ mm, i.e. in the subsonic region of the expansion, the velocity profile can be described with a single Gaussian again as shown in Fig. 3.4. In this region the flow is in equilibrium and can be described along the jet axis with one mean velocity and one temperature.
Contents of the VDF

In Fig. 3.6 the contents of the VDF measured along the jet axis at 20 Pa background pressure is shown as well as the contents of the two Gaussian distributions that compose the VDF in the shock region. At the beginning of the expansion, for $z < 20$ mm, the Ar[4s] density remains high and changes only by a factor of two. The local metastable production compensates for convective losses due to the rarefaction effect. The production rate of Ar[4s] atoms, through electron-ion recombination, is proportional to $n_e^3 \times T_e^{-4.5}$ [35]. The rarefaction of $n_e$ due to the expansion can be compensated by the diminution of $T_e$ in this region of the plasma jet. From $z = 20$ mm up to the shock wave, the density rarefaction effect (see chapter 2) is the main contributor to the decrease of the Ar[4s] density. However, the density drops slightly faster than $z^{-2}$ due to the efficient loss of Ar[4s] atoms by collisions with electrons and by radiation. The production via three-particle recombination can not compensate for the loss mechanism. Across and behind the shock wave, the Ar[4s] density decreases since Ar[4s] slowly deexcite (mostly by collision with electrons) to form ground-state Ar atom.

Within the shock region, the population of the fast component of the Ar atom VDF gradually diminishes to finally disappear whereas the population of the slow component gradually increases, due to the population transfer, until it remains the only component.

3.4 Ar atom temperature, velocity, and density

The mean velocity of a distribution is determined from the calculation of the 1st moment

$$ \bar{v} = \frac{\int v f(v) dv}{\int f(v) dv}. \qquad (3.7) $$

At thermodynamic equilibrium, i.e. Gaussian VDF, a particle temperature $T$ can be defined that corresponds to the spread in velocity. The 2nd moment of the distribution, i.e.

$$ \bar{v}^2 = \frac{\int (v - \bar{v})^2 f(v) dv}{\int f(v) d(v)}, \qquad (3.8) $$

is used to calculate $T$ via the relation $k_B T = \frac{1}{2} m \bar{v}^2$, where $k_B$ is the Boltzmann constant and $m$ the mass of the particle. The temperature is in fact directly connected to the width of the Gaussian profile. In a non-equilibrium situation, a temperature can not strictly be defined. However, we proposed to define a mean temperature by the density-weighted temperature of each Gaussian component of the VDF

$$ T = \frac{n_1 T_1 + n_2 T_2}{n_1 + n_2}. \qquad (3.9) $$
<figure><img src="image_16.png" /><figcaption>**Fig. 3.7:** Axial profile of the parallel temperature (circle) for two different background pressures. In the shock region the temperature of the fast (square) and the slow (triangle) component of the VDF is plotted. The solid line is the result of a calculation using the Poisson adiabatic law with $\gamma = 1.4$.</figcaption></figure>

Parallel temperature and mean axial velocity

The development of the parallel temperature $T_‖$ (associated with the velocity distribution parallel to a stream line) along the jet centerline is shown in Fig. 3.7 for two background pressures. Also shown in Fig. 3.7 is the temperature of the fast (cold) and slow (warm) component of the VDF within the shock region. From a fit to the measured temperature profile in the supersonic domain using the Poisson adiabatic law (Eq. 2.11) one obtains: $\gamma = 1.4 \pm 0.1$, and $T_o = 5600$ K. The speed of sound in the source can be calculated using Eq. 2.13: $c_{s,o} = 1270 \text{ m s}^{-1}$. The adiabatic exponent $\gamma$ proves to be lower than the expected value for a monoatomic gas (5/3). It may arise from the influence of the ionization degree [36]. Another possibility is that a plasma expansion deviates from an adiabatic expansion because of energy transfer from both the source and the hot background gas. The temperature $T_o$ in the nozzle is lower than the source temperature ($\approx 1$ eV) for two reasons. First, due to the cascaded arc configuration, no energy is supplied to the plasma in the nozzle and therefore the plasma starts to recombine inside the nozzle. Second, due to the nozzle geometry (45° opening angle) the plasma starts to expand, and therefore to cool down, inside the nozzle. In the subsonic domain, the temperature decreases because of heat conduction to the vessel walls.

The measured axial profile of the Ar atom mean velocity is shown in Fig. 3.8 for two different background pressures. Also shown in the graph is the mean velocity of the two components of the VDF across the shock wave. The axial profile of the speed of sound,
<figure><img src="image_17.png" /><figcaption>**Fig. 3.8:** Ar mean axial velocity profile (circle) along the jet axis for two different background pressures. In the shock region the mean velocity of the fast (square) and the slow (triangle) component of the VDF is plotted. Also shown is the speed of sound calculated from the measured parallel temperature (diamond) using Eq. 2.13. The solid line is the result of a calculation using Eq. 2.22.</figcaption></figure>

calculated from the measured $T_\|$, is shown in Fig. 3.8. In the supersonic domain, the
measured velocity profile is modeled using Eq. 2.22. For the two pressures, the plasma
is accelerated up to about six times the sound speed ($M = 6.4$ in front of the shock).
Both the position and the width of the shock wave depend on the background pressure as
expected. The shock wave being formed in a rarefied flow regime, its thickness is large.
At 20 Pa, respectively 100 Pa, the shock wave stretches from $z = 30$ mm to $z = 250$ mm,
respectively from $z = 20$ mm to $z = 200$ mm. The end of the shock is defined as the
position where $M = 1$. The plateau visible inside the shock region at 100 Pa arises from
a second expansion phase due to the step by step adaptation of the Ar static pressure
throughout the shock wave. An interesting feature is observed in the 20 Pa case. The mean
velocity of the fast component of the VDF drops across the shock wave. In other words,
the measured fast component does not exhibit the properties of the supersonic component
defined in the Mott-Smith approximation [30, 31]. The MS model is certainly too simple
to perfectly describe shock wave properties, however, the deviation from the predicted
behavior may be due to the existence at low background pressure of a third equilibrium
velocity component, the so-called scattered component [33, 37]. The good agreement of our
experimental results on average velocity and temperature with the calculations in which
we describe the thermal plasma expansion as if it were a hot neutral gas expansion with a
slightly different adiabatic constant, shows the validity of this approach.
<figure><img src="image_18.png" /><figcaption>**Fig. 3.9:** Ar density profile along the jet centerline of an expanding Ar-H₂ plasma (3 slm Ar and 0.5 slm H₂) measured by means of UV Rayleigh scattering at 42 Pa (triangle) and 100 Pa (circle). Also shown is a calculated density profile in the supersonic domain using equation 2.17. The shock wave profiles (first compression stage) are modeled using the Navier-Stokes equation in the two pressure cases.</figcaption></figure>

Density

As shown in Fig. 3.9, the Ar density has been measured by means of UV Rayleigh scattering for a background pressure of 100 Pa and 42 Pa in a plasma generated from an Ar-H₂ mixture ($\sigma_{Ar}^R = 5 \sigma_{H_2}^R$). In the supersonic domain, the decrease in density arises from a rarefaction effect (decay in $z^{-2}$) and it is pressure independent. The data are fitted using Eq. 2.17 with the following parameters: $n_o = 5 \times 10^{22} m^{-3}$, $z_{ref} = 3$ mm, and $z_o = 3.0$ mm meaning that the expansion starts already inside the arc nozzle as expected with our geometry. In the subsonic domain, the density increases because the plasma flows at constant static pressure. The shock wave is formed in a rarefied flow regime since $Kn = 0.2$ at 100 Pa and $Kn = 0.4$ at 42 Pa. The compression ratio equals 4 at 100 Pa, i.e. the maximum compression ratio for an atomic gas, and 3 at 42 Pa. Seemingly the density jump through the shock wave can well be modeled. The Navier-Stokes equation is used in Fig. 3.9 (the Mott-Smith approach would provide the same outcome). However, only the first compression stage is modeled and the good agreement between the calculation and the experimental data is due to the fact that the shock wave thickness $L$ is a free parameter ($L$ is taken to be the dimension of the first stage in that case, i.e. 25 mm). Using the real shock wave thickness, see Fig 3.8, would lead to a very poor agreement. It means that the shock profile models described in chapter 2 do not account for a step by step change in the static pressure of the flowing gas.
<figure><img src="image_19.png" /><figcaption>**Fig. 3.10:** Ar atom on-axis forward flux at 100 Pa determined from Rayleigh scattering measurements.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 3.11:** Axial mean velocity as function of the parallel temperature in the subsonic domain at 100 Pa background pressure.</figcaption></figure>

## 3.5 Ar atom forward flux

Combining the axial velocity measurement with the Ar density measurement, one can obtain the profile of the Ar atom forward flux along the jet axis. The result is shown in Fig. 3.10 at 100 Pa background pressure. In the supersonic flow domain, the flux decreases as $z^{-2}$ because of the density rarefaction effect. The Ar flux entering the shock wave ($\phi \approx 3.2 \times 10^{25} \text{ m}^{-2}\text{s}^{-1}$ at $z = 20$ mm) matches to a good approximation to the Ar flux behind the shock wave ($\phi \approx 3.8 \times 10^{25} \text{ m}^{-2}\text{s}^{-1}$ at $z = 200$ mm). As expected the flux across the shock region is conserved. The trend observed through the shock results from the adaptation of the Ar static pressure to the background static pressure via several successive compression phases. In the subsonic domain the Ar flux decreases since the velocity decreases because of momentum exchange with the residual background gas.

Analysis of the flow pattern in the expansion chamber using a hydrodynamic code (PHOENICS) [38], reveals that in the continuum flow regime behind the stationary shock front the diameter of the jet stays approximately constant. Since the plasma flows at constant static pressure in the subsonic domain, the drift velocity should be proportional to the temperature. This proves to be indeed the case as shown in Fig. 3.11 at a background pressure of 100 Pa. At low background pressure, i.e. in the rarefied regime, the jet opens and the $w \propto T$ relation is not valid.
<figure><img src="image_21.png" /><figcaption>**Fig. 3.12:** Perpendicular temperature $T_⊥$ versus radial position measured at 20 Pa for two different $z$ positions.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 3.13:** Mean radial velocity component as a function of $r$ measured at 20 Pa for two $z$ positions.</figcaption></figure>

## 3.6 Radial profiles

### 3.6.1 Perpendicular temperature

The radial profiles of the perpendicular temperature $T_⊥$ (associated with the velocity distribution perpendicular to a stream line) measured at 20 Pa in the supersonic domain, see Fig. 3.12, show evidence of the quasi adiabatic cooling effect. They also reveal the structure of the barrel shock wave. The structure visible in the core of the jet at $z = 8$ mm results from a geometric effect. The Ar atoms detected on the side have travelled a longer distance in comparison with those moving along the axis, and therefore they have converted a larger part of their thermal energy into kinetic energy. This leads to a lower temperature at the edges of the jet.

At $z = 20$ mm, this geometric effect is not noticeable. The particles can not convert anymore their thermal energy, and the maximum velocity is reached. Furthermore at this location the flow deviates from thermodynamic equilibrium: $T_⊥ < T_‖$. The pressure is low enough to enable the flow to reach the so-called frozen regime [39].

The rise in $T_⊥$ when going off-axis is due to collisions with Ar atoms in the barrel shock wave. This temperature jump defines the boundary of the plasma jet. Behind the barrel shock wave, $T_⊥$ decreases towards the vessel walls because of heat transfer to the background gas. The gas temperature in the vicinity of the jet is rather low ($T_⊥ ≈ 600$ K).

### 3.6.2 Radial velocity component

As can be seen in Fig. 3.13, the radial profiles of the radial velocity reveals the Ar shock wave structure. Because of the expansion process, the radial velocity first increases when
moving away from the jet axis. When flowing Ar atoms collide with Ar atoms in the
barrel shock wave, the velocity decreases since it is converted into thermal energy, and the
temperature increases accordingly. As expected, the position of the Ar barrel shock as
deduced from the radial velocity profile does coincide with the position deduced from the
corresponding $T_{\perp}$ profile.

In the surrounding of the plasma jet, the measured radial velocity is around zero, whereas it is established that in this zone Ar atoms are moving towards the jet due to the existence of vortices (recirculation cells) [38]. In our conditions, their velocity is however too low to be accurately determined ($w_r \approx 20 \text{ m s}^{-1}$).

## 3.7 Conclusions

Velocity and temperature measurements using Laser Induced Fluorescence spectroscopy on metastable argon atoms in an expanding thermal argon plasma seeded with 4 % of hydrogen have been presented. We demonstrated that the velocity distribution of metastable or resonant atoms reflects the velocity distribution of ground state argon atoms. At low background pressure it is found that within the stationary shock wave the supersonic component of the VDF is decelerated, in disagreement with the Mott-Smith bimodal approximation. This feature may reveal the existence of a third velocity component linked with the invasion of the jet by the background gas.

The LIF measurements combined with the Rayleigh scattering measurements indicate that the expansion of a thermal plasma can to a large extent be described as a hot neutral gas expansion with a smaller adiabatic constant in the supersonic domain due to energy transfer from both the source and the background.

## Acknowledgments

The authors would like to thank Prof. M.C.M. van de Sanden for stimulating discussions. The authors greatly appreciate the skillful technical assistance of M.J.F. van de Sande, A.B.M. Hüsken, and H.M.M. de Jong. This work is part of the research program of the Netherlands Foundation for Fundamental Research on Matter (FOM) and is partially founded by Euratom. It is also financially supported by the Netherlands Organization for Scientific Research (NWO).
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# Chapter 4

## Two-photon laser induced fluorescence spectroscopy of atomic hydrogen

### Abstract

Two-photon Absorption Laser Induced Fluorescence (TALIF) spectroscopy is used to monitor ground-state hydrogen atoms (H) properties in an supersonically expanding plasma. Hydrogen atoms are excited with two 205 nm photons from the $1s^2S_{1/2}$ ground-state to the $n=3$ level, and the fluorescence radiation is detected at the Balmer-$\alpha$ line at 656 nm. A detailed description of the experimental arrangement is presented. It includes: the generation of coherent tunable UV radiation, the detection of the fluorescence light, the signal acquisition and treatment, and a list of all parasitic effects that can disturb the fluorescence yield. It is shown that from a spectral scan over the two-photon transition several quantities can be extracted. The spectral profile directly reflects the local H atom velocity distribution function. At thermodynamic equilibrium, the mean velocity is obtained from the Doppler shift of the peak and temperature from the width of the peak (both in the direction defined by the laser beam). The area of the fluorescence line gives the relative local H atom density. In order to obtain absolute H atom densities, the LIF setup has to be calibrated. Two calibration methods are presented and compared: titration with $NO_2$ in a flow tube reactor and comparison with two-photon excitation of krypton.
## 4.1 Introduction

The experimental study of the transport mechanisms of ground-state (most populated state in a recombining low temperature plasma) hydrogen atoms in the expansion of a thermal plasma created by a cascaded arc requires the application of a diagnostic technique to locally monitor H atom properties such as velocity, temperature and density. The very sensitive Laser Induced Fluorescence (LIF) technique appears to be a good candidate [1]. It is non-intrusive, making it attractive for in-situ measurements, and state selective with high spectral resolution, and measurements can be spatially as well as temporally resolved. This well established method is often used to measure atomic or molecular densities in chemistry [2], in low temperature plasma physics [3, 4], and in the field of controlled nuclear fusion [5]. However due to the large energy spacing involved the detection of ground-state H requires the use of highly energetic photons (VUV domain of the electromagnetic spectrum) of which generation is experimentally demanding [6]. The problems connected with single-photon excitation can be avoided with the application of a two-photon excitation scheme. In this chapter, a detailed analysis of the Two-Photon Absorption Laser Induced Fluorescence (TALIF) spectroscopy of ground state atomic hydrogen is presented that includes among others a description of the experimental arrangement and accurate explanations about the extraction of information from the measured two-photon spectral profile. This technique is currently used for detection of light atomic radicals in plasmas. Several examples of the use of the TALIF diagnostic tool in H containing plasmas can be found in literature: e.g. concentration measurements [7, 8] and surface loss coefficient determination [9].

The excitation and detection scheme for ground-state atomic hydrogen applied in this work is the one that was first demonstrated by Bokor *et al.* [10]. In this scheme H is excited with two 205.14 nm photons from the 1s $^2$S$_{1/2}$ ground-state to the *n*=3 level. The laser induced fluorescence radiation is detected at the Balmer-$\alpha$ line (2↔3 transition), thereby largely avoiding self-absorption of the emitted photons. This has direct consequences for the density range in which the detection technique is applicable. Several schemes for 2-photon excitation of H to *n*=3 and *n*=4 with detection of non-resonant fluorescence, that differ in the number of laser beams and the wavelength combinations used for excitation, the intensity, polarization, and bandwidth of the laser beams, and the wavelength of the fluorescence light, have been directly compared to each other [11]. This detailed comparison shows that the above described scheme for 2-photon excitation of H with a pulsed, narrowband 205 nm laser beam gives the largest average number of fluorescence photons per incident laser photon.

## 4.2 Two-photon excitation process

The theoretical calculation of the fluorescence yields that results from a two-photon excitation has already been performed in detail elsewhere [11, 12, 13, 14]. We here only propose a rapid and simplified description of the calculation in the case of an unsaturated two-
photon excitation (low laser intensity), i.e. the ground-state density remains practically unchanged: $n_1(t) \approx n_1(0)$. In that particular case a rate equation description is justified.

The effect of the laser intensity on the ground-state population $n_1$ can be described by the following differential equation

$$ \frac{d}{dt}n_1(t) = -W(t)n_1(t), \qquad (4.1) $$

where $W(t)$ is the two-photon excitation rate which is expressed with the aid of the generalized two-photon excitation cross-section $\sigma^{(2)}$

$$ W(t) = G^{(2)}\sigma^{(2)}g(\Delta\nu)\left(\frac{\langle\hat{I}(t)\rangle}{h\nu}\right)^2, \qquad (4.2) $$

where $\nu$ is the photon frequency, $h$ is the Planck constant, $\langle\hat{I}(t)\rangle$ is the mean laser pulse power per unit area (in W m$^{-2}$), $g(\Delta\nu = \nu_{12} - 2\nu)$ describes the normalized profile of the two-photon transition (in s) which is composed of the convolution of the laser line profile and atomic absorption line profile, and $G^{(2)}$ is the two-photon statistical factor [15] ($G^{(2)} = 2$ for a stochastic multimode field, e.g. a dye laser beam).

The population of the excited state $n_2$ is given by

$$ \frac{d}{dt}n_2(t) = W(t)n_1(t) - (A_2 + Q + \Gamma(t))n_2(t), \qquad (4.3) $$

where $A_2 = \sum_i A_{2i}$ is the total transition probability of the excited state, $Q$ is the quenching rate, and $\Gamma(t)$ is the photo-ionization rate. The number of generated fluorescence photons per unit volume $n_\gamma$ is given by

$$ n_\gamma = A_{2k} \int_0^\infty n_2(t) dt \qquad (4.4) $$

where $A_{2k}$ is the transition probability of the observed fluorescence spectral channel.

If the loss of population due to photo-ionization is neglected ($\Gamma(t) = 0$), the total amount of fluorescence photons $N_\gamma$ detected at the laser frequency $\nu$ is

$$ N_\gamma = n_1(0) \frac{A_{2k}}{A_2 + Q} G^{(2)} g(\Delta\nu) \sigma^{(2)} \frac{\langle\hat{I}(t)\rangle^2}{(h\nu)^2} F. \qquad (4.5) $$

The factor $F$ is an experimental factor

$$ F = V \frac{\Omega}{4\pi} T \eta, \qquad (4.6) $$

where $V$ is the detection volume, $\Omega$ the detection branch solid angle, $T$ the total transmission of the collection optics, and $\eta$ is the detector quantum efficiency. The time integrated fluorescence signal is therefore proportional to the initial ground-state population and to the square of the laser pulse energy.
<figure><img src="image_23.png" /><figcaption>Fig. 4.1: Energy diagram of the hydrogen atom and two-photon excitation scheme. Two 205 nm photons are used to excite H from the ground-state to the n = 3 level. The fluorescence radiation is detected at the Balmer-α line. The absorption of a third UV photon from the n = 3 level leads to the formation of H⁺.</figcaption></figure>

## 4.3 Excitation and detection of ground-state atomic hydrogen

The excitation and detection scheme for ground-state atomic hydrogen applied in this work is the one that was first demonstrated by Bokor *et al.* in 1981 [10]. In this scheme, depicted in Fig. 4.1, H is excited with two 205.14 nm photons from the 1s $^2S_{1/2}$ ground-state to the n=3 level. The n=3 fine-structure sublevels can not be resolved within the bandwidth of the 205 nm laser radiation (≈ 0.18 cm⁻¹). As a result, in a spectral scan over the 2-photon transition both the 3s $^2S_{1/2}$ and the 3d $^2D_j$ states will be excited, according to the two-photon electric-dipole selection rules: Δℓ = 0, ±2 for a single electron atom where ℓ is the angular momentum. The case of multi-electron atom is extensively discussed in Ref. [16]. The excitation is monitored by detection of the non-resonant fluorescence on the Balmer-α transition at 656.3 nm from the 3d and 3s states to the 2p $^2P_j$ states. Usually it is mentioned that the 3d↔1s transition dominates over the 3s↔1s transition because of the 2-photon absorption cross section ratio [17]

$$ \sigma^{(2)}(3d \leftarrow 1s) = 17.7 \ 10^{-44} \ \mathrm{m}^4, $$

$$ \sigma^{(2)}(3s \leftarrow 1s) = 2.3 \ 10^{-44} \ \mathrm{m}^4. $$
Actually it depends on the degree of saturation of the transition whether the relative
population of 3d and 3s is determined by the relative transition strength or rather by
the ratio of the statistical weights of the respective quantum states. Moreover, we found
evidence for a pure statistical distribution of the population over all the *n*=3 sublevels from
an accurate measurement of the lifetime of the excited state [13]. This can be explained
by a complete *l*-mixing of the quantum states in the electric field associated with the
propagating laser beam [18].

A detailed comparison between various 2-photon excitation schemes of H [11] shows
that the above described scheme with a pulsed, narrowband 205 nm laser beam gives the
largest average number of fluorescence photons per incident laser photon.

4.4 Experimental arrangement

4.4.1 Generation of narrowband coherent tunable UV radiation

A scheme of the experimental laser system used to produce narrowband coherent tunable
UV radiation around 205 nm is depicted in Fig. 4.2. The laser system is based on a
pulsed Nd:YAG/dye laser combination. The second harmonic output beam (500 mJ pulse
energy at 532 nm, 6 ns duration) of an injection-seeded Nd:YAG-laser operating at 20 Hz
(Spectra-Physics/Quanta-Ray, GCR-230) is used to pump a tunable dye laser (Spectra-
Physics/Quanta-Ray, PDL-3). The dye laser has a bandwidth of 0.07 cm⁻<fnref n="1" /> and delivers
radiation around 615 nm (sulforhodamine 640) with a pulse energy of typically 100 mJ. The
dye laser output is frequency doubled in a KD*P crystal (type I). Upon exiting the crystal,
the polarization of the generated blue beam is rotated to coincide with the polarization
of the fundamental red beam using a low-order waveplate (λ/4 for the blue beam). Both
collinear laser beams are input to a BBO crystal (type I) for sum frequency generation.
The two crystals are electronically angle tuned by means of a feed back system to provide
a constant output energy while scanning the dye laser wavelength.

The tripled frequency component is separated from the others and directed towards
the vacuum vessel using high-power, high-reflective dielectric mirrors for 205 nm (45°)
that are at the same time transparent for the other wavelengths. This procedure results
in typically 2 mJ of tunable UV light around 205 nm for excitation of ground-state H.
The UV laser beam is horizontally polarized and its frequency profile is Gaussian<fnref n="1" /> with a
measured bandwidth of 0.18 cm⁻<fnref n="1" /> [19].

The laser wavelength is calibrated by recording simultaneously the molecular iodine
(I₂) absorption spectrum, of which the line positions are tabulated to high accuracy [20].
For this, part of the red dye-laser output is directed through a I₂ absorption cell. This
glass cell has a 2.5 cm diameter and a 60 cm length and contains only iodine (solid and
vapour). The transmitted light intensity is measured behind the cell with a photodiode.
The absorption can be optimized by adjusting the I₂ vapour pressure with a heating wire
around the cell.

<footnote n="1">¹ The laser beam profile has been determined using a two-photon excitation of N atom at 207 nm.</footnote>
<figure><img src="image_24.png" /><figcaption>**Fig. 4.2:** Schematic view of the laser system used to generate narrowband coherent tunable radiation around 205 nm. The output beam of a tunable dye laser is converted into a UV laser beam after indirect third harmonic generation in crystals with nonlinear optical properties. The frequency of the laser beam is calibrated using the absorption spectrum of molecular iodine.</figcaption></figure>

### 4.4.2 Fluorescence light monitoring

The frequency-tripled laser light is directed into the vessel either perpendicular to the plasma expansion axis or counterpropagating with the expansion, as depicted in Fig. 4.3. In the default perpendicular configuration, where the radial velocity component $w_r$ and the perpendicular temperature $T_\perp$ (associated with the velocity distribution perpendicular to a stream line) can be measured, the UV laser light is focused into the plasma with a 60 cm MgF$_2$ lens. When the axial velocity component $w_z$ and the parallel temperature $T_\|$ are to be measured, a 1 m focusing MgF$_2$ lens is used in combination with a quartz right-angle prism that directs the light anti-parallel to the plasma beam. In this way only the prism is exposed to the plasma and the combination of its relatively large distance from the plasma source with its mounting in a watercooled protection cylinder ensures a few hours of measurement time before cleaning of the prism is necessary.

The laser induced fluorescence light originating from the focus is imaged onto an adjustable slit mask, in a direction perpendicular to the plane formed by the laser beam and the symmetry axis of the expansion. This is done by two 10 cm diameter plano-convex lenses with focal lengths of 40 cm. The slit mask defines the spatial resolution (typically 1×1 mm$^2$). The laser induced fluorescence light that passes the slit mask is detected by a gated photo-multiplier (PMT Hamamatsu, R928). The continuous background light emitted by the plasma is strongly reduced by a narrow bandwidth interference filter located in front of the detector (10 nm FWHM, centered at 656 nm).
<figure><img src="image_25.png" /><figcaption>**Fig. 4.3:** Schematic view of the TALIF detection branch. The UV laser beam can be directed either perpendicular or parallel to the flow direction. Spatial scans through the expanding plasma are performed by moving the cascaded arc relative to the intersection of the laser beam and the detection volume.</figcaption></figure>

However, the suppression of the plasma emission by the optical filter is not always sufficient to allow the PMT to be used with maximum gain. This is a well-known problem in the application of the LIF technique in light emitting media like those encountered in plasma and combustion research, which is usually solved by gating the detection. The Hamamatsu PMT R928 is equipped with a gate unit that enables triggered operation of the PMT during a short time interval (typically 2 µs). This reduces the continuous background further by a factor of $2 \times 10^4$, in principle enabling operation of the PMT with maximum gain in all circumstances.

The PMT signal is recorded by a digital sampling oscilloscope (500 Ms s⁻¹, 100 MHz bandwidth). The oscilloscope can be used both for time-integrated and for time-resolved measurements. The signal trace is transferred to the PC where it is analyzed. This enables also baseline corrections and a precise time integration of the signal. The PC also controls the wavelength scanning of the dye laser.

The cascaded arc plasma source is mounted on a translation stage. Spatial scans through the expanding plasma are performed by moving the cascaded arc relative to the intersection of the laser beam and the detection volume.

### 4.4.3 Parasitic effects

Before focusing on the extraction of information from the H atom spectral profile, it is necessary to review the numerous parasitic effects which can disturb the laser induced
fluorescence signal, in order to be able to either avoid them or, when not possible, to quantify and subsequently account for them.

## Collisional quenching

The fluorescence quantum yield $q_f$ is given by the Stern-Vollmer factor

$$q_f = \frac{A_{kl}}{A_k + Q} = A_{kl}\tau_k. \qquad (4.7)$$

This factor is the ratio between the spontaneous emission rate $A_{kl}$ of level $k$ on the transition to level $l$ where the fluorescence is monitored, and the total depopulation rate of the excited state, which is the sum of the total spontaneous emission rate $A_k$ and the non-radiative depopulation (quenching) rate $Q$ of level $k$. The total depopulation rate determines the life-time $\tau_k$ of this state.

The most common uncontrolled depopulation mechanism is the collisional quenching of the upper state, which occurs when numerous collisions take place during the radiative lifetime of the excited state. This is a well-known problem in the application of LIF in experiments at high density, as for instance encountered in combustion research. However, the importance of fluorescence quenching can be easily underestimated in the field of (low-temperature) plasma research, where the quenching by electrons can be very efficient [13].

The occurrence of quenching can be readily checked in a time-resolved LIF experiment, where the effective life-time $\tau_k$ of the upper state $k$ is measured. This lifetime is directly related to the quenching as is indicated by equation 4.7. In fact, a measurement of the lifetime will establish the value of the Stern-Vollmer factor, which potentially allows to correct the density values accordingly. Note that in case the quenching is strong, the fluorescence profile follows in shape the laser pulse and the lifetime can not be extracted from the measured time profile. In a plasma jet generated from an Ar-H₂ mixture or from pure H₂, a decay time of about 10.0 ns has been observed over the entire range of plasma parameters encountered in this plasma, as shown in Fig. 4.4. The longer decay time close to the source outlet is due to the reabsorption of Lyman-β photons (3d←1s transition) by H atoms within the detection volume ($n_H \approx 10^{21} m^{-3}$).

Shortly repeating the discussion in reference [13], this is exactly the decay time expected in the case of a statistical distribution of the population over the $n=3$ sublevels arising from $\ell$-mixing induced by the laser E field. Arguments that the 10 ns decay time is the result of quenching of the 15.5 ns $3d \rightarrow 2p$ natural decay time can be discarded, since the same decay time has been found for a large range of ground state H, electron, and Ar(He, H₂) densities. The value of the collisional quenching rate coefficient can be found in Ref. [21, 22] for several collision partners.

## 3-photon ionization

A characteristic of the applied H excitation scheme is that only one additional laser photon is needed for ionization of the excited atom, as is indicated in figure 4.1. When the laser
<figure><img src="image_26.png" /><figcaption>**Fig. 4.4:** Axial profile of the decay time of the deconvoluted time-resolved fluorescence signal measured in an Ar/H expanding plasma at 20 Pa.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>**Fig. 4.5:** Log-log plot of the fluorescence yield as a function of the laser pulse energy ($n_H = 10^{19} \text{ m}^{-3}$ and $T_H = 800$ K). The slope is equal to 2.05±0.07.</figcaption></figure>

intensity is increased, this ionization pathway will eventually limit the fluorescence yield, as the rate of depopulation via this process starts to compete with the radiative depopulation rate [11].

Depending on the type of plasma and the ionization degree, the laser-induced excited state, ion, and electron densities can be several orders of magnitude higher than the respective stationary densities. A complete model is then needed to quantitatively assess the effect of the pulsed laser excitation on the local plasma parameters. In a concurrent study presently performed in our group, a time-resolved collisional radiative model (CRM) for H has been developed, that models the response of a system of levels to a temporary disturbance of the level population distribution by for instance pulsed laser excitation of one of the levels [13]. This study shows, among others, that for conditions typically encountered in the expanding cascaded arc plasma, the ion and electron densities can temporarily increase easily by a factor of five due to the applied laser pulse. In such a case, the laser diagnostic method can hardly be qualified as ‘non-intrusive’ as far as charged particle density and temperature are concerned.

## Amplified stimulated emission generation

Another uncontrolled depopulation pathway that might occur is amplified spontaneous emission (ASE) of the laser-excited state [23]. The ASE will propagate within a small solid angle around the laser beam axis, and, consequently, the upper state population lost in this way will not be observed by the LIF detection branch. ASE of Balmer-α radiation at 656 nm in a two-photon LIF experiment for detection of ground-state H has been reported
both for measurements in a flame [24], as well as for measurements in glow discharge
plasmas [25, 26]. In these studies, the possibility of explicitly using the ASE signal for the
detection of H is investigated.

Dissociation of molecules

A further phenomenon that can disturb the proportionality between ground-state density and observed LIF signal, is the production of the atoms under investigation by the UV laser itself via photo-dissociation of molecules. This is commonly recognized and described in a number of papers [27, 28, 29]. The effect can even be turned into an advantage by using it to identify an impurity and measure its density via an analysis of the spectral profile of the LIF signal [27, 29]. In the expanding cascaded arc plasma generated from a mixture of Ar and H₂, the only species that could produce H atoms from photo-dissociation is molecular hydrogen. We checked the possibility of 205 nm (≈ 6 eV) photo-dissociation of H₂ by filling the vessel with pure molecular hydrogen at room-temperature. For the highest laser-intensity used in our experiments (2 mJ), we did not observe any H LIF signal for pressures up to 200 Pa. This does not exclude that photo-dissociation of H₂ occurs in the plasma. While the effect is negligible for room temperature H₂, it could still be noticeable for hot molecules when the photo-dissociation cross-section is sufficiently enhanced for vibrationally excited molecules. Photo-dissociation of molecules by the UV laser can also seriously disturb the temperature determination. Depending on the excess energy, the atomic fragments end up with more than their thermal share of kinetic energy. This gives rise to a large Doppler width and an apparent high atom temperature. Observations of high atomic hydrogen temperatures in RF, microwave, and DC discharges have been associated with photolysis of NH₃, H₂S, C₂H₂, C₂H₄, and Si₂H₆ [27, 28, 29]. In fact, the photolytically produced atoms exhibit a spectral profile characteristic of their precursor molecule, and the Doppler width can be used to determine the excess translational energy enabling an identification of the precursor molecule [27].

Saturation broadening

A last effect that can significantly disturb the two-photon spectral profile is saturation broadening. The degree of saturation of a transition is given by the saturation parameter, which is basically the ratio between the induced and the spontaneous emission rate of the excited level [30]. Since the induced emission rate is directly proportional to the excitation cross section, the saturation parameter for a two-photon transition will generally be very small, and thus the required energy to saturate will be high.

Apart from collisional quenching that can be determined from time-resolved measure-
ments, probably the best way to check for the existence of parasitic effect is to measure
the laser intensity dependence of the H atom two-photon LIF signal from the plasma. For
ground-state H atoms a quadratic dependence should be observed for low laser intensities,
as shown in Eq. 4.5, saturating to a lower order dependence for higher intensities. When
photo-dissociation by the UV-laser occurs, a higher order dependence will be observed [27]. Fig. 4.5 shows the LIF signal of atomic hydrogen from the expanding Ar-H₂ plasma as a function of the laser intensity. The slope of the data points indeed equals 2 within the experimental accuracy. From this it is concluded that in this plasma there is no contribution to the H atom LIF signal from photo-dissociation by the UV-laser. Moreover, from the figure it can be seen that the H two-photon excitation does not reach saturation in our experiment and that the H two-photon LIF signal remains in the quadratic regime over the entire range of laser pulse energies used in our measurements.

## 4.5 Density measurements

### 4.5.1 Integrated spectral profile

The density information gained in a LIF experiment, although in principal absolute, is in practice relative in nature. The relative local H atom density is obtained from the integration of the two-photon spectral profile, see Fig. 4.6. Every measured profile is obtained using 256 shots per data point because of the important pulse to pulse energy fluctuation (sd ≈ 25 %) of the UV laser beam.

The usual approach to get quantitative results from a LIF experiment is to calibrate the set-up using an independent source of a well-known amount of the probed species. In the case of stable neutral atoms or molecules this is easily accomplished with e.g. a gas cell containing a fixed pressure of this species. In the case of short-lived species like radicals or ions, calibration is less trivial. For the calibration of the set-up for the atomic hydrogen measurements, two different approaches are used.

### 4.5.2 Absolute densities: calibration methods

#### Titration reaction

The first approach uses a 'flow-tube reactor' as a source of ground-state H atoms, in which the exact amount of atomic hydrogen is determined by a titration. This method is based on well-established flow reactor and titration techniques in chemistry [31], and has already been applied more often in LIF measurements of ground-state H radical densities in plasmas [32, 27]. A scheme of the flow-tube reactor, designed to allow implementation in the expansion chamber, is depicted in Fig. 4.7. A small amount of molecular hydrogen that is diluted in helium flows through a quartz tube that is mounted through the centre of a Beenakker-type microwave cavity [33]. Atomic hydrogen is produced in the flow-tube by dissociation of H₂ in the microwave discharge (typical power of 80 W) that is sustained in the cavity. The main flow continues down the quartz tube and a 0.5 m long teflon tube, transporting H atoms to a position 60 cm downstream from the discharge and only 1 or 2 mm away from the scattering volume of the laser, where they are detected with identical excitation and detection geometry as in the expanding cascaded arc plasma. The tube material has been chosen to minimize the H atom loss between the microwave discharge
<figure><img src="image_28.png" /><figcaption>**Fig. 4.6:** Measured spectral profile of the 3 ← 1 two-photon transition in atomic hydrogen and Gaussian fit. The fluorescence is detected at the Balmer-α line. The area of the peak is a direct measure of the H density (after calibration). The local temperature is deduced from the Doppler broadening of the line after deconvolution with the laser spectral profile. The mean H velocity (in the direction of the laser beam) is determined from the absolute Doppler shift of the line.</figcaption></figure>

and the point where H is detected. The length of the tube ensures that no excited H atoms arrive at the scattering volume of the laser.

The amount of H present in the scattering volume and responsible for the observed signal is determined via a titration with NO₂. The NO₂ comes from a calibrated mixture of 2.11 % NO₂ in He that can be added to the main flow via a small insert that enters the main transport tube near to the end of it. The titration actually determines the H density in the mixing region. To ensure that this density is as close as possible to the H density responsible for the LIF signal in the laser scattering volume, the mixing region should be as close to the exit of the main tube as possible. This distance is small compared to the length of the main tube, and it can therefore be safely assumed that the ground-state H density is constant over the last 3.5 cm of the flow-tube. Upon addition of NO₂, H atoms are lost in the fast reaction

$$H + NO_2 \rightarrow OH + NO, \qquad (4.8)$$

with a rate constant k of 1.3 × 10⁻¹⁰ cm³ s⁻¹ [31]. Solving the symmetric set of rate equations

$$\frac{d}{dt}n_H = -kn_H n_{NO_2} = \frac{d}{dt}n_{NO_2} \qquad (4.9)$$

for the densities n_H and n_{NO₂} of H and NO₂, respectively, shows that when t → ∞, i.e. when the reaction is complete, the remaining H density depends linearly on the initial NO₂ density. Fig. 4.8 shows a typical titration curve, i.e. a measurement of the H fluorescence
<figure><img src="image_29.png" /><figcaption>**Fig. 4.7:** Schematic overview of the flow tube reactor.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>**Fig. 4.8:** Titration curve (800 sccm He, 8 sccm H₂, 85 W, I₂₀₅ = 0.2 mJ).</figcaption></figure>

signal from the flow-tube reactor as a function of the admixed NO₂ flow. The dependence is recorded both in a series with increasing flow and in one with decreasing flow. The H LIF signal decreases linearly with increasing NO₂ flow until it becomes indistinguishable from the background (end point of titration). Given the 1:1 stoichiometric ratio of the titration reaction, the NO₂ density n<sub>NO₂</sub><sup>(T)</sup> at the x intercept in Fig. 4.8, which is necessary to remove all H atoms, equals the original H density n<sub>H</sub> at the beginning of the titration that gives rise to the LIF signal at the y intercept

$$n_H = n_{NO_2}^{(T)} = n_{tot} \frac{\Phi_{NO_2}^{(T)}}{\Phi_{tot}} = \frac{p}{kT} \frac{\Phi_{NO_2}^{(T)}}{\Phi_{tot}}, \quad (4.10)$$

with p and T the pressure and temperature in the vessel, respectively, k the Boltzmann constant, Φ<sub>NO₂</sub><sup>(T)</sup> the NO₂ flow rate. The subscript tot denotes the total density or flow. During the titration, the total flow Φ<sub>tot</sub> is kept constant in order to keep the pressure p in the vessel constant. The increasing flow of the calibrated NO₂/He mixture is balanced by a decreasing additional pure He flow in the small titration tube.

The first equality in equation 4.10 only holds under certain conditions. First of all, the titration reaction should be complete. This condition can be quantified by evaluating the solution of the set of equations 4.9 for the case that the initial NO₂ density n<sub>NO₂,0</sub> approaches the initial H density n<sub>H,0</sub>, i.e. near the end point of titration. This yields

$$n_H(t) = \frac{n_{H,0}}{1 + k n_{H,0} t}. \quad (4.11)$$
The end point of titration ($n_{NO_2,0} = n_{H,0}$) is only sharply determined if $kn_{H,0}t \gg 1$. This is the condition for linearity and complete titration. In addition, the [NO₂]/[H] stoichiometry should be 1:1. Other reactions that could change the stoichiometry should only play a minor role. The secondary reactions OH + OH → H₂O + O and O + OH → O₂ + H are significantly reduced at the low H concentrations in the flow-tube reactor [34, 35], and other interfering reactions such as O + NO₂ → O₂ + NO and OH + H₂ → H₂O + H are made negligible by diluting both NO₂ and H₂ in He [34]. The second equality in equation 4.10 implies the assumption of efficient mixing between the NO₂/He flow from the titration tube and the main flow. A proper density calibration furthermore avoids a ‘self-generated signal’. The occurrence of UV-photolysis of H containing impurities from the flow-tube reactor can be easily checked by examining the Doppler-width of the H LIF signal. This should yield a temperature close to room temperature. The Doppler width of the H LIF signal from our flow-tube reactor corresponds to a temperature close to 400 K.

In the titration shown in Fig. 4.8 the end point of titration is not very sharply determined, and is therefore derived from a linear fit, yielding $\Phi_{NO_2}^{(T)} = 0.71 \pm 0.04$ sccm. The corresponding initial H density can now be derived from equation 4.10 using the settings $p = 240$ Pa, $T = 300$ K, and $\Phi_{tot} = 908$ sccm, resulting in $n_H = (8.2 \pm 0.4) \times 10^{19}$ m⁻³. The real inaccuracy in $n_H$ is however larger (around 25 %) since it is determined by the uncertainty in the value of all parameters (pressure, temperature, flows) and by the uncertainty in the exact state of the chemical reactions involved. The condition for linearity and complete titration can now be evaluated. The reaction time $t$ is determined by the 3.5 cm distance between the end of the titration tube and the exit of the main tube and by the flow velocity. Using the above values yields $t = 2.1$ ms and $kn_{H,0}t = 22$. This shows that the condition for linearity and complete titration is only just fulfilled.

It is of upmost importance to change as little as possible when switching between the recording of the H LIF signal from the flow-tube reactor and that from the expanding thermal plasma. In practice this means that the two measurements should be performed directly after each other. Finally, when converting H LIF signal values from the plasma jet to H densities one should take into account that the H atom temperature in the two recordings will generally be different. This demands to compare only spectrally integrated signal values.

## Two-photon excitation of Kr

The second approach avoids the cumbersome generation of a well-known amount of atomic hydrogen radicals from an independent source. Instead, a LIF signal is generated from a fixed pressure of a different stable species by a comparable two-photon excitation scheme. The H density in the plasma beam can then be determined from the ratio of the LIF signals when the ratio of the two-photon cross sections is known. This method has recently been demonstrated for the calibration of ground-state atomic hydrogen densities [22] as well as of ground-state atomic nitrogen densities [19, 22] by using a two-photon transition in krypton. The method has also been applied to calibrate the fluorescence yield of oxygen atoms via a two-photon transition in xenon [12]. This method is more elegant and experimentally
<figure><img src="image_31.png" /><figcaption>Fig. 4.9: Simplified energy diagram of krypton. Also depicted is the two-photon excitation scheme used to calibrate H densities (to be compared with Fig. 4.1).</figcaption></figure>

easier than the first one, but it relies on critical cross section data.

The basic idea behind a radical density calibration using a known density of a stable species, is to find and use an excitation and detection scheme in a stable species as similar as possible to that of the radical. This warrants a comparable response to the laser excitation, especially when the excitation wavelengths are close to each other, which allows the excitation conditions (laser spectral, temporal, and beam profile, intensity, and focusing) to be the same in the two LIF measurements. The excitation and detection scheme for krypton that is used in the atomic hydrogen density calibration is depicted in figure 4.9. In this scheme Kr is excited with two 204.2 nm photons from the 4p⁶ ¹S₀ ground-state to the 5p'[3/2]₂ state. The resulting non-resonant fluorescence from this state to the 5s'[1/2]₁ state is detected at 826 nm. In both schemes the excitation concerns a two-photon transition, which gives a quadratic response to the laser intensity in the small signal limit (low laser intensity). The wavelengths used for excitation are in addition nearly equal and can actually be generated from the same dye solution in the tunable dye laser. This means in practice that we can switch between H and Kr measurements within an instant; the laser is tuned to the other wavelength and only the fluorescence filter in front of the PMT has to be replaced.

In the quadratic regime and under equal excitation conditions, the ratio $S_{Kr}/S_H$ of the spectrally integrated fluorescence signals of krypton and hydrogen according to equation 4.5, reduces to [12, 19, 22]

$$ \frac{S_{Kr}}{S_H} = \frac{n_{Kr}}{n_H} \frac{\sigma_{Kr}^{(2)}}{\sigma_H^{(2)}} \left( \frac{\nu_H I_{Kr}}{\nu_{Kr} I_H} \right)^2 \frac{A_{Kr} \tau_{Kr}}{A_H \tau_H} \frac{T_{Kr}}{T_H} \frac{\eta_{Kr}}{\eta_H}. \quad (4.12) $$

Here, n refers to the ground-state density, I is the corresponding laser beam energy, ν is the photon frequency, A is the Einstein transition probability for the detected spontaneous emission, τ is the lifetime of the excited state, T is the transmission of the optical
<table><caption>**Tab. 4.1:** Values of the parameters involved in the H density calibration via krypton, using equation 4.12. For the meaning of the symbols, see the text. The transmission *T* of the optical filters in the detection branch for the respective fluorescence wavelengths has been measured with a FT spectrometer. The PMT quantum efficiency *η* for the Kr fluorescence wavelength (826 nm) has been measured relative to the one for the H Balmer-*α* fluorescence (656.3 nm) with a ribbon lamp.</caption><thead><tr><th></th><th>Kr</th><th>H</th></tr></thead><tbody><tr><td>I<sub>laser</sub> (mJ)</td><td>0.12</td><td>0.39</td></tr><tr><td>ν (cm<sup>-1</sup>)</td><td>48972.985</td><td>48746.15</td></tr><tr><td>A (s<sup>-1</sup>)</td><td>2.7 × 10<sup>7</sup></td><td>4.4 × 10<sup>7</sup></td></tr><tr><td>τ (ns)</td><td>35.4</td><td>10.0</td></tr><tr><td>T</td><td>0.95</td><td>0.447</td></tr><tr><td>η (relative)</td><td>0.088</td><td>1.0</td></tr></tbody></table>

detection branch for the detected fluorescence, and *η* is the detector quantum efficiency at the corresponding wavelength. The solid angle that is captured by the detection optics, the detection volume, and the gain of the photon detector are the same in the two LIF measurements. The ratio of the two-photon excitation cross sections has recently been measured to be $\sigma_{Kr}^{(2)}/\sigma_{H}^{(2)} = 0.62$ [22].

For the Kr LIF measurements the vacuum chamber is filled with 10 Pa Kr at room temperature which corresponds to a Kr density of $2.4 \times 10^{21} \text{ m}^{-3}$. The two-photon transition in Kr, that is induced here for the H density calibration, is easily saturated. The transition is for instance known to readily generate ASE, enhanced by its small Doppler width [36]. Since equation 4.12 only holds in the quadratic regime, special care is taken to avoid saturation of the transition. A longpass filter, $T = 0.95$ above 700 nm, is used to reduce the laser stray light impinging onto the PMT. The fluorescence decay times $\tau$ of Kr and H have been measured with a fast PMT (Hamamatsu R5600P-01; 1 ns FWHM single photon response time [13]). As already mentioned, the lifetime of the H $n=3$ excited state has been measured to be 10.0 ns. The lifetime $\tau_{Kr}$ of the Kr 5p'[3/2]$_2$ state is found to be $35.4 \pm 2.7$ ns, in agreement with a value recently measured by Niemi et al [22]. The measured radiative lifetime of krypton is longer than the one usually found in literature ($\tau_{Kr} = 26.9$ ns) [37]. In the calibration procedure reported here, a branching ratio of 0.95 is used for krypton. Table 4.1 summarizes the values of all parameters involved in the H density calibration via krypton, using equation 4.12.

Figure 4.10 shows two axial profiles of the absolute atomic hydrogen density in the supersonic domain of a thermal Ar-H$_2$ plasma expansion. One density profile is calibrated via a NO$_2$ titration in a flow-tube reactor, the other via a two-photon LIF measurement of krypton. The H density values determined by the two different methods agree well with each other. This comparison therefore validates the H density calibration via a two-photon LIF measurement of Kr, which is experimentally easier than the calibration with
<figure><img src="image_32.png" /><figcaption>**Fig. 4.10:** Comparison between absolute atomic hydrogen density profile along the axis of an Ar/H plasma jet (20 Pa) that are calibrated via 2-photon excitation in Kr (triangle) and via a titration in a flow tube reactor with NO₂ (square). Using the latter, a 2-photon cross-section ratio σ<sub>Kr</sub><sup>(2)</sup>/σ<sub>H</sub><sup>(2)</sup> of 0.56 is obtained.</figcaption></figure>

the flow-tube reactor, once the values of the parameters in equation 4.12 are accurately
determined.

Using the H densities determined with the titration method, a two-photon cross-section
ratio σ<sub>Kr</sub><sup>(2)</sup>/σ<sub>H</sub><sup>(2)</sup> equals 0.56 is obtained, in good agreement with the value 0.62 measured by
Niemi *et al* [22].

To conclude this section, the H atom density detection limit that can be achieved un-
der our experimental conditions, has to be given. In an expanding Ar(H₂, He)/H plasma it
is found to be 10¹⁵ m⁻³, which is certainly sufficient to obtain valuable information about
the amount of H atoms that are carried away by a plasma jet.

**4.6 Temperature measurements**

In case of thermodynamic equilibrium (TE), the local atom temperature is derived from
the width of the spectral profile of the two-photon transition<fnref n="2" />. The experimentally ob-
served linewidth is the result of the convolution of all contributions to the width. Under
our experimental conditions, the main contribution comes from the Doppler broadening

<footnote n="2">² In TE, the temperature is related to the second moment of the velocity distribution function (VDF) via the relation kT = (1/2)mv²; a detailed calculation shows however that the temperature can simply be determined from the width of the spectral profile. Note that when the system departs from TE, the temperature is not defined anymore and one has to refer directly to the VDF.</footnote>
that results from the velocity distribution of the ground-state atoms. Collisions (pressure broadening) do not contribute significantly to the broadening of the line within our pressure range ($p < 1$ torr). Another effect that should be seriously considered when determining temperatures from LIF measurements in plasmas, is Stark broadening [38]. In the plasmas investigated in this work, it will not significantly contribute to the experimentally observed linewidth, since the Stark broadening of the two-photon transition in atomic hydrogen, which is mainly determined by the Stark coefficients of the $n=3$ level, is negligible for the range of electron densities and temperatures encountered in the expanding plasma ($n_e \le 10^{19} \text{ m}^{-3}$, $T_e \approx 0.3 \text{ eV}$ [39]).

The general expression for the Doppler width $\Delta\nu_D$ (FWHM) of a two-photon transition, induced by the absorption of one photon with frequency $\nu_1$ and one photon with frequency $\nu_2$ with an angle $\theta$ between their k-vectors, is [15]

$$ \Delta\nu_D = 2(\nu_1^2 + \nu_2^2 + 2\nu_1\nu_2 \cos\theta)^{1/2} \bar{v}/c, \quad (4.13) $$

where $\bar{v}$ is the average velocity of the atom or molecule, and $c$ is the speed of light in vacuum. In the case of absorption of two photons from one single laser beam with a frequency of half the transition frequency $\nu_0$ ($\nu_1 = \nu_2 = \nu_0/2$, $\theta = 0$), and with a Maxwellian distribution function ($\bar{v} = \sqrt{2 \ln 2 \cdot kT/m}$), this expression reduces to the well-known relation between the Doppler width and the temperature $T$

$$ \Delta\nu_D = \nu_0/c\sqrt{8 \ln 2 \cdot kT/m} = 7.16 \times 10^{-7} \nu_0 \sqrt{T/M} \quad (\text{in cm}^{-1}), \quad (4.14) $$

with $k$ the Boltzmann constant, $m$ the mass of the atom or molecule, and $M$ its molecular weight in amu. The Doppler width $\Delta\nu_D$ is convoluted twice with the linewidth $\Delta\nu_L$ of the UV laser radiation ($= 0.18 \text{ cm}^{-1}$ [19]). Assuming a Gaussian spectral profile for the latter, the experimentally observed linewidth $\Delta\nu$ is simply given by: $\Delta\nu^2 = \Delta\nu_D^2 + 2\Delta\nu_L^2$. In the case of atomic hydrogen, the error in $\Delta\nu_D$ — and thus in the temperature — introduced by this squared summation when the laser spectral profile is non-Gaussian, is expected to be small since the observed linewidths are typically one order of magnitude larger than the laser linewidth.

## 4.7 Velocity measurements

Besides determining the velocity spread — and from that the temperature — of the particles in the plasma jet from the Doppler width, in thermodynamic equilibrium the Doppler shift of the center frequency $\nu_0$ of the transition can be used to determine the mean velocity of these particles<fnref n="3" />. The velocity $w$ in the direction of the laser beam of a group of atoms absorbing the laser photons at frequency $\nu$ is given by [40]

$$ w = c \frac{\nu - \nu_0}{\nu_0}. \quad (4.15) $$

<footnote n="3">³ The mean particle velocity is obtained directly from the first moment of the velocity distribution function (VDF). In TE the mean velocity can simply be determined from the shift of the spectral profile.</footnote>
<figure><img src="image_33.png" /><figcaption>**Fig. 4.11:** Example of a spectral scan with simultaneous recording of the H atom 2-photon LIF signal (solid square) and the $I_2$ absorption signal (solid line). The scan is recorded on the jet centerline of an Ar-H$_2$ plasma jet. The horizontal axis is two times the energy of the UV photon used to excite H atoms. This axis is calibrated by the position of the $I_2$ absorption lines.</figcaption></figure>

Previous equation is used to convert the measured spectral profile into the local velocity distribution function (VDF) in the direction defined by the laser beam. The energy distribution function (EDF) can be obtained from the VDF using the kinetic energy formula $E = \frac{1}{2}mw^2$. At thermodynamic equilibrium, the VDF is Gaussian and the EDF is Maxwellian.

To determine the Doppler shift, a Doppler shifted signal must be compared with a non shifted signal, which usually requires two laser beams. An absolute determination of the Doppler shift ‘only’ requires an absolute frequency scale — and then the shift is obtained from a comparison of the center frequency with the literature value — but obviously in that case the laser frequency needs to be accurately calibrated. This is accomplished by using a small fraction of the red dye laser output beam for the simultaneous recording of the absorption spectrum of molecular iodine, as is described in the experimental section 4.4. An example of such a simultaneous recording is given in Fig. 4.11. In the region of the fundamental dye-laser frequency, the $I_2$ spectrum shows many absorption lines that are tabulated in frequency to an accuracy of 0.015 cm$^{-1}$ [20]. The width of the $I_2$ absorption lines, which is at room temperature mainly determined by the underlying hyperfine structure, amounts to ≈ 0.03 cm$^{-1}$. The medium is therefore very well suited for the frequency calibration of pulsed dye-lasers of which the bandwidth is usually in the same order of magnitude (in our case 0.07 cm$^{-1}$), and the method is widely used in spectroscopy.

Since the Doppler shift method determines velocities relative to the velocity of light,
accurate results are not to be expected *a priori*, and it is worthwhile to address the topic of accuracy into somewhat more detail. In the applied method, the error in the velocity has several sources. The error in the calibrated frequency scale depends on the signal to noise (S/N) ratio in the $I_2$ absorption spectrum, and is ultimately limited by the bandwidth of the laser. The attainable minimum error in the (fundamental) frequency scale is $\approx 0.01 \text{ cm}^{-1}$. A further error comes from the determination of the center frequency of the two-photon spectral line from the Gaussian fit. The error in the observed center frequency depends again on the stepsize in the spectral scan and on the S/N ratio of the H LIF signal. In practice, the overall error in the velocity is mainly determined by this contribution.

A last uncertainty concerns the literature value for the transition frequency. Accurate values for the energies of the individual $n=3$ sublevels can for instance be found in [41], but since the maximum energy separation between these levels is smaller than the bandwidth of the 205 nm laser radiation, these levels are not resolved and the resulting center frequency will be at an averaged value. This leaves some room for the exact reference value to take for the velocity determination. The one used in this work is obtained from the experiment itself. In a measurement of the radial velocity in a radial cross section through the expanding plasma jet, symmetry arguments are used to put the radial velocity on the symmetry axis of the expansion to zero. This leads to the value $v_0 = 97492.315 \text{ cm}^{-1}$. This value<fnref n="4" /> is very close to the averaged literature value that accounts for a pure statistical distribution of the population over all the $n=3$ sublevels, which is consistent with the lifetime of the excited state that is observed in this work.

The resulting maximum accuracy that can be attained in the determination of the velocity of atomic hydrogen from the Doppler shift of the transition frequency in the two-photon LIF measurements in the expanding plasma is $\approx 200 \text{ m s}^{-1}$, which is a fraction of less than $10^{-6}$ of the speed of light. This value is consistent with the S/N ratio that we experimentally observe in our velocity measurements. This accuracy is certainly sufficient to gain valuable information about the flow pattern in the expanding plasma, where H atom velocities in the range $0 – 8000 \text{ m s}^{-1}$ are observed.

## 4.8 Conclusions

The TALIF spectroscopic method at 205 nm is perfectly suited for the study of the transport mechanisms of ground-state H(D) atoms in a plasma expansion. From the spectral scan over the two-photon transition, several parameters can be extracted that are of crucial relevance for the understanding of the plasma flow properties. First, the measured spectral profile can be easily transformed into the local H atom velocity distribution function that allows for discussion about the thermodynamic state of the plasma. Second, a mean H atom velocity as well as a temperature when the system is at thermodynamic equilibrium (both are defined in the direction of the laser beam) are obtained. Third, the local absolute

<footnote n="4">⁴ The transport of deuterium atoms (D) in a plasma expansion is also investigated in this thesis. The energy diagram of D of H are linked via the isotopic mass shift relation [42]: $E_D = (\mu_D/\mu_H)E_H$, where $\mu$ is the reduced mass. One obtains $\mu_D/\mu_H = 1.000272099$ and $v_0 = 97518.843 \text{ cm}^{-1}$.</footnote>
H atom density is determined from the area of the fluorescence line after calibration of the LIF setup.

Therefore the TALIF diagnostic technique enable to accurately measure the plasma jet density, velocity and temperature map. From the large dataset that is obtained, the expansion of H atoms can be compared with the classical inert neutral gas expansion picture. This led among others to the discovery of anomalies in the transport of atomic radicals in a plasma expansion. But more than that the application of the TALIF tool to H containing plasma allowed to identify the reason of those anomalies: plasma-wall interactions.

## Acknowledgments

The author would like to thank Dr M.G.H. Boogaarts for his valuable contribution in the fulfillment of this chapter. Professor H.F. Döbele (institute for Laser and Plasma Physics, Essen University, Germany) is gratefully acknowledged for his vital contribution in the initial phase of the TALIF experiment. The author greatly appreciates the skillful technical assistance of M.J.F. van de Sande, A.B.M. Hüsken, and H.M.M. de Jong.
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<figure><img src="image_34.png" /></figure>
Chapter 5

Anomalous atomic hydrogen shock pattern in
a supersonic plasma jet <fnref n="1" />

Abstract

A two-photon LIF study on the transport of ground-state atomic hydrogen in a supersonic
plasma jet, generated from an Ar-H₂ mixture, reveals an unexpected shock pattern. Spatially
resolved hydrogen atom densities, temperatures, and velocities are measured in the expansion of
a thermal plasma source. Whereas both the axial velocity profile and the temperature profile of
hydrogen atoms along the jet centerline can be interpreted in terms of a supersonic expansion of
an Ar-H gas mixture, the H atom density profiles do not satisfy the well established Rankine-
Hugoniot relation leading to a non-conservation of the forward flux. The experimental results
show that H atoms escape from the supersonic expansion by a diffusion process due to strong
density gradients between the core of the jet and its vicinity.

<footnote n="1">¹ Adapted from:
S. Mazouffre, M.G.H. Boogaarts, J.A.M. van der Mullen, D.C. Schram, *Phys. Rev. Lett.* **84**, 2622 (2000).</footnote>
## 5.1 Introduction

Plasma expansion from a high pressure source region into a low pressure region is a general physical phenomenon [1, 2, 3] that covers a broad range of physical systems such as solar flares, remote plasma systems, and vacuum arc spots. Apart from the large differences in scale, those phenomena exhibit numerous similarities. For instance, they are all governed by the same set of equations, and after being accelerated the plasma flow undergoes a transition from a supersonic to a subsonic regime resulting in the formation of a shock wave structure.

From a practical view point it is advantageous to study the physics of expanding plasmas on the intermediate scale as then the system is well suited for diagnostics. In this letter we report on the study of the dynamics of a supersonic plasma jet generated by a cascaded arc [4] from a mixture of Ar and $H_2$, where Ar is used as a carrier gas. We aim at an understanding of the transport of ground-state atomic hydrogen (H) from a reservoir, i.e. the plasma source, to a processed surface. From a fundamental perspective, such an investigation is unique because, contrary to commonly studied supersonic expansions, one of the components of the jet is almost absent in the background gas beyond the barrel shock. Indeed, hydrogen atoms can recombine at the wall to form molecular hydrogen, and, due to the large available surface, the vessel walls act as a huge sink for such a reactive light component.

The study of the transport of atomic hydrogen has also a strong technological relevance because H plays a key role as a chemical agent in many plasma processing applications like plasma enhanced chemical vapor deposition [5, 6]. In addition, remote plasma sources can be used to produce beams of neutral atomic hydrogen, and they may serve for volume production of hydrogen negative ions, both species being of importance in the field of nuclear fusion.

In spite of specific properties such as the presence of charged particles, high viscosity, high thermal conductivity, and large temperature gradients over the radius, so far plasma expansion has been described using the adiabatic supersonic expansion theory [7] developed for the expansion of a neutral gas. This theory describes the continuum free jet shock wave structure [8, 9] that results from the interaction of a supersonic flow with the ambient gas when a gas or a gas mixture expands from a reservoir into a region with a finite pressure rather than a perfect vacuum.

In the case of an expanding argon plasma, it has been demonstrated experimentally [10, 11] that the expansion of the neutrals can be well described in terms of the adiabatic supersonic expansion of an ideal gas. Likewise, supersonic plasma flow created by a cascaded arc from a Ar-$H_2$ mixture should be understandable in terms of the adiabatic supersonic expansion of a Ar-H gas mixture leading to the formation of a well defined free jet shock structure in the continuum regime. Only a segregation effect, the so-called mass focusing effect [9], where the heavier species is relatively more concentrated in the center of the jet, is expected in view of the mass difference between H (1 amu) and Ar (40 amu) atoms. Indeed, both the axial and radial profiles of the hydrogen atom velocity as well as the axial temperature profile are characteristic for a free jet shock structure as we will see.
## 5.2 Experimental arrangement

### Two-photon LIF

In the experiments reported here, ground-state hydrogen atoms are spatially probed using a two-photon excitation laser induced fluorescence (LIF) technique [12, 13]. A 20 Hz Nd:YAG-pumped tunable dye laser is operated around 615 nm. The output of the dye laser is frequency-tripled resulting in 5 ns pulses of tunable UV light around 205 nm with a pulse energy of 2 mJ and a bandwidth of 0.18 cm⁻¹. The frequency-tripled laser light is directed into the vessel either perpendicular to the plasma expansion axis or counterpropagating with the expansion. It should be noticed that the dimensions of the detection volume that can be achieved (<1 mm³) are smaller than the gradient sizes. Hydrogen atoms are excited with two 205 nm photons from the 1s²S ground state to the 3d²D and 3s²S states. The excitation is then monitored by detection of the resulting fluorescence yield on the Balmer-α line at 656 nm. From a spectral scan with the frequency calibrated laser over the two-photon transition the local H atom density, temperature, and velocity along the laser beam are obtained. By applying this method for different positions in the expansion, the H distribution in the plasma beam in terms of density, temperature, and velocity has been completely mapped.

### Cascaded arc operating conditions

The cascaded arc is operated at a 40 A DC current and with a cathode-anode voltage of 100 V. A gas flow of 3.0 standard liters per minute (slm) Ar and 0.5 slm H₂ is used. The diameter of the arc channel is 3 mm, the stagnation pressure is 0.6 bar, and the opening angle of the arc nozzle is 45°. Using a 6:1 Ar-H₂ mixture, at the arc outlet the plasma jet turns into almost a gas jet composed of Ar and H atoms and of residual H₂ molecules, the electrons being consumed immediately at the outlet of the arc channel [14]. Both the amounts of H and H₂ depend on the dissociation degree of the source which is around 30% at the arc outlet with these operating conditions.

## 5.3 On-axis velocity profile

The development of the H drift velocity along the jet centerline, shown in Fig. 5.1, gives proof of an incomplete momentum coupling between H and Ar atoms inside the jet. First the gas is accelerated over a few arc nozzle diameters to a maximum speed of about 4000 ms⁻¹, due to conversion of thermal energy gained in the arc into kinetic energy by means of collisions. This value is not in agreement with a theoretical value of 3500 ms⁻¹ found for the terminal velocity of a 3:1 Ar-H mixture, at a temperature in the nozzle of 5000 K, that expands into vacuum [9]. This indicates that H atoms are not perfectly coupled with Ar atoms. From a comparison with the sound velocity profile (Fig. 5.1), determined from the measured parallel temperature $T_‖$ associated with the velocity distribution parallel to a stream line (see Fig. 5.2), the Mach number M can be estimated. At