Development of an absorption model
for gas discharge lamp simulation
Utveckling av absorptionsmodell f¨
or simulering av en
gasurladdningslampa
Absorption model for plasma simulation
Master’s thesis, 30 hp, Computational materials science Spring 2021
Malm¨o University
Supervisor: Henrik Hartman, Department of Materials Science and Applied Math-ematics
Tetra Pak®
Supervisor: Anders Floderus, Tetra Pak®Packaging Solutions Master’s Thesis 2021
Department of Materials Science and Applied Mathematics
Copyright © 2021 Oscar Vigstrand Typeset in LATEX
Malm¨o University, Sweden 2021
Abstract
Ultraviolet (UV) light has been used for disinfection purposes for over 100 years. Irradiation by UV light is a method to disinfect surfaces in order to prevent micro-biological growth. At Tetra Pak this is of great importance as they are manufacturer of filling machines. Those filling machines must ensure a certain level of sterility on all packages produced. The irradiation process can be simulated using Geant4 which is a software package that tracks particles through matter. The simulation model used today does not consider the absorption of photons inside of medium-pressure UV lamps. By understanding the absorption that takes place in the lamp, one can quantify how changes in the design would impact the emitter output. In this master’s thesis, the aim is to develop a model that can describe the interaction of photons with a medium-pressure UV lamp. An absorption model was suggested and developed with the assumption of local thermodynamical equilibrium and ex-isting Hg radiative data. A simulation including the collision process in Geant4 was used. In this collision process the non-radiative transition probabilities was assumed to be the same as that of the radiative, this was done in order to demon-strate how it can be done. It resulted in collisions populating other states allowing more transitions to be present in the final output spectrum. The collision process and a method for computing the Einstein’s emission coefficient with the software package General Relativistic Atomic Structure Package is proposed as future work.
Sammanfattning
I ¨over 100 ˚ar har ultraviolet (UV) ljus anv¨ants till desinficering. UV bestr˚alning ¨ar en metod f¨or att desinficera ytor med m˚alet att f¨orhindra mikrobiologisk tillv¨axt. F¨or Tetra Pak som ¨ar ledande inom tillverkning av fyllmaskiner ¨ar det extra vik-tigt. F¨orpackningarna inuti fyllningsmaskinerna m˚aste garantera en viss niv˚a av sterilitet f¨or alla f¨orpackningar. Dagens simuleringar av medeltrycks UV lampa utf¨ors i Geant4 som ¨ar ett mjukvarupaket som m¨ojligg¨or f¨oljandet av partiklar genom olika medium. Detta g¨ors utan att ta h¨ansyn till absorptionen av fotoner. Genom att f¨orst˚a absorptionen som sker i lampans gas kan man kvantifiera hur f¨or¨andringar i design skulle p˚averka emittorns utg˚aende effekt. I detta examen-sarbete ¨ar m˚alet att utveckla en modell som kan beskriva hur fotoner v¨axelverkar med gasen i en medeltrycks UV lampa. En modell utvecklas och f¨oresl˚as med an-tagandet att lokalt termodynamisk j¨amvikt r˚ader och att enbart Hg str˚alnings data anv¨ands. En simulering med en kollisionsprocess i Geant4 inkluderades. I denna kollisionsprocess antas den icke-optiska ¨overg˚angssannolikheten vara densamma som f¨or de optiska ¨overg˚angarna. Detta inkluderades f¨or att demonstrera hur en s˚adan process kan g˚a till. Detta resulterade i att kollisionerna populerade andra tillst˚and vilket gjorde att dessa ¨overg˚angar visade sig i utg˚aende spektrum. Kolli-sionsprocessen och en metod f¨or att ber¨akna Einsteins emissions koefficient med mjukvarupaketet General Relativistic Atomic Structure Package f¨oresl˚as ¨aven som framtida arbete.
Acknowledgements
I would like to thank my supervisors Henrik Hartman and Anders Floderus, for all the help and interesting discussion during this thesis. All your feedback and input has been invaluable. Thank you for all the support and enthusiasm during this thesis.
A special thanks to J¨orgen Ekman and Per J¨onsson for taking the time to help me with some questions during this time. Also, thank you for your valuable input in GRASP.
Finally, I must express my very profound gratitude towards my family and friends. Thank you for the support and encouragement along the way.
Contents
1 Introduction 1
1.1 Aim . . . 2
1.2 Limitations . . . 2
2 Theory 3 2.1 Spectral line shapes . . . 4
2.2 Irradiance . . . 5
2.3 Einstein’s coefficients . . . 6
2.4 Radiative transition probabilities . . . 6
2.5 Line broadening . . . 7 2.5.1 Doppler broadening . . . 7 2.5.2 Pressure broadening . . . 8 2.5.3 Natural broadening . . . 8 2.5.4 Instrumental broadening . . . 8 2.6 Beer’s law . . . 9 3 UV emitters 10 3.1 Medium-pressure UV lamps . . . 11 3.1.1 Self-saturation . . . 12 4 Plasma 13 4.1 Local thermodynamical equilibrium . . . 13
4.2 Partial LTE . . . 14
4.3 Boltzmann relation . . . 14
4.4 Saha equation . . . 14
4.5 Collision-induced transitions . . . 15
5 Plasma simulation model 16 5.1 Model parameters . . . 16
5.2 Initial spectrum . . . 17
5.3 Photon absorption . . . 17
5.4 Photon emission . . . 18
6 Results and discussion 19 6.1 Electron number density . . . 19
6.2 Transition probability . . . 20
6.3 Initial spectrum . . . 22
6.4 Photon absorption . . . 24
CONTENTS CONTENTS
7 Conclusions 27
Nomenclature
c Speed of light (« 2.998 ¨ 108 ms´1)
λ Wavelength
ν Frequency (ν “ c{λ)
τ Radiative lifetime of excited state gj Statistical weight for upper state gi Statistical weight for lower state Ni Number of atoms in lower state Nj Number of atoms in upper state
ne Number density of electrons e Electron charge (« 1.602 ¨ 10´19J) me Electron mass (« 9.109 ¨ 10´31 kg)
m Mass of particle
Zj Partition function for upper state Zi Partition function for lower state Ei Energy of lower state i
Ej Energy of upper state j Te Electron temperature
T Gas temperature
J Total angular momentum
kB Boltzmann constant (« 1.381 ¨ 10´23JK´1)
I Irriadiance
h Planck’s constant (« 6.626 ¨ 10´34 Js) κ Absorption coefficient
∆ν1{2, ∆λ1{2 Spectral linewidth
σ Absorption cross-section / Standard deviation x0 Absorption length
LG
D Doppler profile
fvel Velocity distribution function Aji Einsteins emission coefficient
Bji Einsteins coefficient for stimulated emission Bij Einsteins coefficient for absorption
fij Oscillator strength (f -value) between states 0 Vacuum permittivity (« 8.854 ¨ 10´12Fm´1)
n Particle number density λ0 Wavelength at centre of peak LG Gaussian line profile
LL Lorentzian line profile LV Voigt line profile
Chapter 1
Introduction
Ultraviolet (UV) irradiation is short-wave light that has several applications. There are different ways of producing UV radiation, e.g. by using a LED or a gas dis-charge lamp. A gas disdis-charge lamp contains an ionized gas that exhibits plasma properties. The lamp usually contains a noble gas and for disinfection purposes, it is common to use mercury (Hg) as an additional substance. The Hg is vapor-ized when starting the lamp and then mixes with the noble gas [1]. The resulting spectrum from a Hg UV lamp is due to the Hg’s atomic structure.
Tetra Pak® is a global food packaging and processing company that manu-factures filling machines. The inside of the packages that are filled in the filling machines needs to be sterile in order to prevent microbiological growth and en-sure better shelf life. A typical procedure to obtain sterile packaging is a combi-nation of both hydrogen peroxide and ultraviolet disinfection. The packages are first exposed to hydrogen peroxide, followed by a UV source [2] that irradiates the packaging surface. The UV radiation damages the microorganisms residing on the surface by disrupting their nucleic acids. This prevents them from reproduc-ing [3, 4]. Once the microorganisms are incapable of reproducreproduc-ing, they no longer pose a health risk [5].
The certainty to which the UV irradiation disrupts the nucleic acids that may reside on the packaging surface is limited. It is possible to simulate this irradiation process in order to get a better understanding of the disinfection efficacy. With these simulations, the system can be improved to make the process more effective. Also, if any changes to the lamp or its housing are made, the simulations should be able to estimate the impact on the emitter output due to these changes.
As a leading company within liquid food packaging, it is important that Tetra Pak® can assure safe packaging. Today, simulations of UV irradiation are done in Geant4. It is a toolkit that allows one to simulate and track particles through matter [6]. Geant4 is based on the Monte Carlo method, which is a method that solves numerical problems by drawing random numbers from known probability distributions. The goal of these simulations is to predict the exposure from a UV source. The current model does not account for the absorption of UV light and as photons traversing the gas have a probability of being absorbed, this will affect the final output.
1.1. AIM CHAPTER 1. INTRODUCTION
1.1
Aim
This master’s thesis aims to develop a model that can describe the interaction of photons with the plasma inside a medium-pressure UV lamp. By researching relevant topics and publications, an absorption model is proposed and evaluated. The evaluation of the proposed model is done by implementing the absorption model in the Geant4 simulation.
1.2
Limitations
This thesis does not treat non-local thermodynamical equilibrium plasmas. Param-eters such as Einstein coefficients are taken from available databases and are not calculated. The overall goal is not to develop an ideal absorption model but to de-velop one that works with few assumptions and has the possibility to be improved later on. The implementation of the model in Geant4 is outside of the scope of this thesis and will be done by a specialist at Tetra Pak.
Chapter 2
Theory
Electrons in atoms can occupy certain energy levels. By delivering energy to the system, an electron can be excited to a higher energy level. The atom can then de-excite by emitting a photon whose energy is equal to the energy difference between the levels [7]. This process occurs because quantum mechanical systems strive to be in the lowest possible energy state. Figure 2.1 illustrates the excitation of an electron due to the absorption of a photon and the emission when a photon is emitted due to de-excitation. In figure 2.2 some of the energy levels of Hg are illustrated.
Figure 2.1: The figure shows the excitation and de-excitation of a hydrogen atom. The electron can be excited from a state with lower energy to a state with higher energy by absorbing a photon whose energy hν is equal to the energy difference between the states (a). The electron can then de-excite back to the lower energy state by emitting a photon with energy hν (b).
2.1. SPECTRAL LINE SHAPES CHAPTER 2. THEORY
Figure 2.2: Energy level diagram for Hg showing some of the energy levels and transitions between states. The thick line at the top marks the ionization energy for Hg.
2.1
Spectral line shapes
A spectrum can be analysed by looking at the line shape of each transition. Each transition gives rise to a profile due to broadening effects. One of the important parameters is the full width at half maximum (FWHM), see figure 2.3. The line profiles are usually described by either a Gaussian distribution, Lorentzian distri-bution or a convolution of both (Voigt). The line shapes are given by
LGpλ; λ0, σq “ A σ?2πe pλ´λ0q2 2σ2 , (Gaussian) (2.1) LLpλ; λ0, γq “ A π γ pλ ´ λ0q2` γ2 , (Lorentzian) (2.2) LVpλ; λ0, σ, γq “ LGpλ; λ0, σq b LLpλ; λ0, γq, (Voigt) (2.3) where A is the integral of the peak, λ denotes wavelength and λ0is the wavelength at the centre of the peak. The FWHM for a Gaussian profile is related to its stan-dard deviation σ by ∆λ1{2“ 2
?
2.2. IRRADIANCE CHAPTER 2. THEORY
by ∆λ1{2 “ 2γ. Convolutions are in general not numerically favourable. There-fore, the Voigt function can in addition to convolution, be obtained by computing the Faddeeva function and taking the real part of the complex error function. The different line profiles are illustrated in figure 2.3.
Figure 2.3: The top figure shows how the broadening is usually quantified, by the full width at half maximum (FWHM) of the peak. It shows a Gaussian line profile and the corresponding FWHM. The figure below shows common line profiles for analysing broadening effects. The Lorentzian profile has greater tails than the Gaussian profile but a smaller amplitude at the centre wavelength. The Voigt profile is a convolution of the other two.
2.2
Irradiance
Irradiance I is given by
I “ P
A, (2.4)
where P is the incident power and A is the surface area [8]. The quantity used in this thesis is spectral irradiance. It is the irradiance of a surface per unit frequency or wavelength [9].
2.3. EINSTEIN’S COEFFICIENTS CHAPTER 2. THEORY
2.3
Einstein’s coefficients
Einstein’s coefficients are atomic parameters that describe the probability of ab-sorption and emission. If Einstein’s emission coefficient (Aji) is known, the ab-sorption (Bij) and stimulated emission (Bji) coefficients can be obtained as
Bji “ c3 8πhν3Aji “ gi gj Bij, (2.5)
where ν is the frequency, h is Planck’s constant, c is the speed of light and g are the g-factors (also known as statistical weights). The g-factors are determined based on the quantum structure and are given by 2J ` 1, where J is the total angular momentum of the excited state. Einstein’s coefficients can be related to the oscillator strength f (sometimes referred to as the f-value) as
fij “ ´ gj gi fji “ 40meh e2 νBij, (2.6)
where 0 is the permittivity of vacuum and me is the free electron mass [8]. The transition rate Ajican be related to the monochromatic absorption cross-section
σijpνq “ gj gi
c2
8πν2AjiPpνq, (2.7)
where Ppνq is the normalized line profile [10]. The normalized line profile is defined as
ż
Ppνqdν “ 1. (2.8)
Inserting equation 2.5 and 2.6 in 2.7 yields an expression that relates the absorp-tion cross-secabsorp-tion to the oscillator strength.
σij “ e2 40mec
fijPpνq (2.9)
Equation 2.9 corresponds well with the expression in [11]. The spectral data needed can be found in the literature and various websites, e.g., NIST [12].
2.4
Radiative transition probabilities
Figure 2.4 shows three states that are allowed to transition to a state that is de-noted 4. If there are several possible ways to excite an electron to state 4, there is a finite probability for the electron to decay into any of these other channels. As an example, three transitions are occurring from different lower levels to the same upper level denoted 4. If the electron is excited from state 1 to 4 there is a probability for the decay to move the electron into any state connected to state 4. This probability depends on the Aji value as given by
P pj Ñ iq “ řAji iAji
2.5. LINE BROADENING CHAPTER 2. THEORY
Figure 2.4: The figure shows an energy level diagram. It contains four states denoted 1, 2, 3 and 4. An electron in state 4 can decay into any of the lower levels. Each decay path is called a channel. The probability for the decay to occur via a given channel is called its branching fraction.
2.5
Line broadening
There are several causes for a line broadening. The most important broaden-ing mechanisms in the context of medium-pressure plasma modellbroaden-ing are Doppler broadening, pressure broadening, natural broadening and instrumental broaden-ing. The different broadening mechanisms give rise to different line profiles [8]. Line broadening is measured by its FWHM, as was illustrated at the top of figure 2.3. Depending on the broadening, the FWHM will vary.
2.5.1
Doppler broadening
The frequency ν of the emitted photon is shifted due to the thermal motion of atoms or molecules. The Doppler effect depends on the mass m and temperature T of the emitter. The line profile is given by
LGDpνq “ c 2πνji fvel ´ cν ´ νji νji ¯ , (2.11)
where c is the speed of light, νji is the central frequency of the line profile and fvel is the velocity distribution function of the emitters [13]. If the function fvelfollows a Maxwell velocity distribution of temperature T the following relation is valid:
∆ν1/2 νji “ ∆λ1/2 λji “ 2 c 2 ln 2kBT mc2 (2.12)
where m is the particle mass, and ∆ν1/2 and ∆λ1/2 are the FWHM in Hz and nm, respectively [14]. In local thermodynamical equilibrium, the Doppler temperature equals the plasma’s excitation temperature in Boltzmann’s relation.
2.5. LINE BROADENING CHAPTER 2. THEORY
2.5.2
Pressure broadening
Pressure broadening (or collision broadening) follows a Lorentzian profile and occurs due to particles colliding. Both a broadening and a shift of the spectral line occur due to pressure broadening. The pressure broadening can be calculated as
∆ν1{2 “ bp, (2.13)
where p is the plasma pressure and b is a pressure broadening coefficient [15]. The pressure broadening coefficient has experimentally been measured to be around 10 MHz per Torr of a pressure broadened gas. With this approximation, it is possible to compute a rough estimate of the pressure broadening.
2.5.3
Natural broadening
Excited states in quantum mechanical systems states have natural lifetimes before they decay. This means that they have a finite radiative lifetime. Natural broaden-ing occurs due to the Heisenberg uncertainty relation. It states that 2∆E∆τ „ }, where ∆E is the uncertainty on the energy of the emitted photon and ∆τ is the uncertainty on the lifetime of the excited state [8]. The uncertainty appears be-cause of electrons not having an infinite lifetime τ in their excited states. The short-lived states have large uncertainties in energy. The FWHM is given by
∆ν1{2 “ ∆E
h “ 1
4π∆τ. (2.14)
The natural broadening has a Lorentzian profile. By analysing the FWHM for nat-ural broadening, it is possible to calculate the lifetime of the corresponding excited state. The natural broadening is negligible compared to the Doppler broadening and the pressure broadening. The lifetime of an excited state also yields a relation to Einstein’s emission coefficient [8]
Aji “ 1 τ.
2.5.4
Instrumental broadening
If an ideal spectrometer were to measure a single monochromatic wavelength, it would appear as a straight vertical line (a Dirac delta function). If the spectrom-eter is not ideal, the monochromatic wavelength will give rise to a broad peak in the recorded spectrum. This is called instrumental broadening [16]. Suppose that Doppler broadening is the dominant non-instrumental broadening mecha-nism. Then there are three cases to consider in terms of the instrument and its bandwidth. In the first case, the bandwidth is very narrow, so that the instrumen-tal broadening is much smaller than the Doppler broadening. It is then possible to analyse the profile with any of the previously mentioned methods. Suppose instead that the instrumental broadening is close to the Doppler broadening, re-sulting in a convolution of the two. Then it is more difficult to determine what is causing the broadening of the profile. The last case occurs if the instrumental broadening is much greater than any other broadening mechanism. Then it is not possible to say anything about the system by analysing the broadening.
2.6. BEER’S LAW CHAPTER 2. THEORY
2.6
Beer’s law
In absorption spectroscopy Beer’s law (also known as Beer-Lambert’s law) for a homogeneous medium is
Ipνq “ I0pνqe´nσpνqL“ I0pνqe´κpνqL. (2.15) It describes the attenuation of light when it passes through a medium. The equa-tion relates the attenuated irradiance to the number density n of the medium traversed by the light, the absorption cross-section σ, the absorption coefficient κ and the absorption path length L [17]. The optical depth is given by nσpνqL. If nσpνqL " 1, the medium is considered to be optically thick. Rearranging equa-tion 2.15 yields a relaequa-tion between the absorpequa-tion cross-secequa-tion and the absorpequa-tion coefficient
Chapter 3
UV emitters
UV light can be emitted with, e.g., a gas discharge lamp or a LED. In a Hg vapour lamp, the gas is ionized by an electric discharge. Charged particles then move between atoms and sometimes collide with an atom causing it to either be ionized or for the electron to be excited [18]. When the electrons in the excited states are de-excited they produce electromagnetic radiation. The radiation that is produced depends on the atomic structure of the gaseous medium. The electronic structure of Hg is illustrated with a Bohr model in figure 3.1. It is important to note that the electrons can only move in certain discrete orbits and not in intermediate orbits [18], meaning that the electrons are limited to certain discrete energy levels. The spectrum of the emitted radiation also depends on the pressure of the gas. This thesis is concerned with medium-pressure emitters, which are typically operated at a pressure of a few bars. Figure 3.2 shows selected parts from a UV emitter CAD model. The emitter consists of a UV lamp and a reflector surrounded by some housing. The purpose of the reflector is to increase the output power of the system by reflecting additional photons through the emitter exit windows, which are located at the base of the housing. Understanding the absorption of photons in the lamp gas enables one to quantify how changes in, e.g., the reflector would impact the emitter output.
Figure 3.1: Schematic of Hg atom according to Bohr’s model. The electronic shell structure can also be expressed as [Xe] 4f145d106s2.
3.1. MEDIUM-PRESSURE UV LAMPS CHAPTER 3. UV EMITTERS
Figure 3.2: Illustration of a UV emitter with a UV source and a reflector sur-rounded by a housing. [19]
3.1
Medium-pressure UV lamps
Two important types of gas discharge lamps for disinfection are low-pressure UV lamps and medium-pressure UV lamps. A low-pressure UV lamp only emits a single wavelength at 253.6 nm (monochromatic). When the pressure is increased atoms begin to collide. This results in other excited states being populated and gives rise to a broader spectrum. The number density is also increased and with this, saturation occurs. This means that no more absorption can occur in the 253.6 nm line which results in more photons being able to traverse the gas. With more photons traversing the gas a greater output power is also obtained. Both low-pressure UV lamps and medium-low-pressure UV lamps are what is called germicidally efficient and mean that they operate within the germicidal range (200-280 nm). An example of a spectrum for a medium-pressure UV lamp can be seen in figure 3.3. The spectrum shows the example output of a lamp in the range of 200-400 nm.
3.1. MEDIUM-PRESSURE UV LAMPS CHAPTER 3. UV EMITTERS
Figure 3.3: An example of an emitted spectrum from a medium-pressure UV lamp. Each peak correspond to a transition due to excitation [19]. This spectrum is ob-tained from Heraeus holding GmbH through Tetra Pak and is not the real measured spectrum due to confidentiality.
3.1.1
Self-saturation
Self-saturation occurs when a transition saturates its own emission. If absorp-tion is strong enough in a transiabsorp-tion so that the photons emitted at the resonance wavelength are immediately re-absorbed. This results in that only the surface of the gas emit at this wavelength. The photons inside the bulk of the plasma that vibrate at this particular resonance wavelength are shifted away from the centre, giving rise to a broadening effect before escaping the lamp. In a spectrum, the self-saturation of a transition can appear as the wings of the peak broadens, as illustrated in figure 3.4.
Figure 3.4: Figure illustrates how the self-saturation effect can appear when look-ing at the line profile of a transition. The self-saturation can be seen by looklook-ing at the wings as they starts to grow vertically.
Chapter 4
Plasma
Matter can exist in different states: liquid, solid, gas and plasma. A plasma is formed when any state of matter contains enough free charged particles so that electromagnetic forces dominate the system dynamics. The plasma is governed by Maxwell’s equations and the Lorentz force. If the electric field (Eq and mag-netic field (B) from Maxwell’s equation are known, the Lorentz force provides the trajectory (xj) and velocity (vj) of each particle. If each particle trajectory and ve-locity is known, Maxwell’s equation provides the electric and magnetic field [20]. In a gaseous medium, it is not necessary for a high level of ionization in order for it to exhibit plasma properties [21]. Plasma is not common on earth, even though it is one of the most common states of matter in space.
4.1
Local thermodynamical equilibrium
In plasma, complete equilibrium can never be obtained in a laboratory because its radiation results in radiation loss. Electrons follow the Maxwell energy distri-bution because the electron collisions are much faster than ion collisions. This means that the temperature of the ions can be neglected. In local thermodynam-ical equilibrium (LTE), the temperature T in the Maxwell-Boltzmann distribution, the Boltzmann distribution and the Saha distribution can be assumed to be the same locally. If LTE cannot be assumed, the electromagnetic radiation is described by Planck’s distribution function. The electron density required to obtain complete LTE is neě 1.4 ¨ 1020 ´E j´ Ei eV ¯3´k BTe eV ¯1{2 m´3, (4.1)
where Te is the electron temperature [13] and the unit comes from the constant. In LTE, the distribution of electrons or particles is solely determined by the col-lision processes [22]. By assuming LTE, the absorption and emission of photons in the plasma can be determined given only its temperature, pressure and chem-ical composition. If the number of electrons or particles cannot be determined from the Boltzmann distribution, a non-LTE condition will apply [23]. Non-LTE conditions usually appear in X-ray or extreme UV plasma sources [24].
4.2. PARTIAL LTE CHAPTER 4. PLASMA
4.2
Partial LTE
The majority of laboratory plasma’s do not reach complete LTE. However, they can reach partial LTE (PLTE) if the population of different ionization states still follow the Saha relation. In PLTE, the population in the higher energy levels continues to be governed by the Boltzmann distribution [13]. This means that the energy difference between two excited states is usually small. However, the energy differ-ence between the ground state and an excited state may be large enough so that it is no longer governed by the Boltzmann distribution.
4.3
Boltzmann relation
The plasma density in relation to its potential in thermal equilibrium can be de-scribed by the Boltzmann relation [25]. When modelling plasma, each transition’s number density is distributed according to the Boltzmann relation. The Boltzmann relation is defined as nj ni “ gj gi e´pEj´Eiq{kBT, (4.2)
where T is the gas temperature, ni the number density in lower state and nj the number density in the upper state.
4.4
Saha equation
The Saha equation describes the distribution of atoms in different ionization states. It relates to the change in ionization states when varying the temperature and pres-sure and can be derived from statistical mechanics [26]. For a medium-prespres-sure UV lamp, there may exist singly ionized Hg atoms which are determined by the Saha equation. Looking at the process Hg`
`e´ é Hg, the Saha equation becomes Nj Ni “ 2Zj neZi ´2πm ekBT h2 ¯3{2 e´∆E{kBT, (4.3)
where Ni is the number of atoms in the lower ionization state, Nj is the number of atoms in the upper ionization state and ne is the electron number density. The partition function is given by
Z “ 8 ÿ k“0
gke´pEk`1´Ekq{kBT. (4.4)
The partition function is a cumulative density of the one given by the Boltzmann relation in equation 4.2. The prerequisite for the Saha equation to be valid is to assume LTE or PLTE and obey the Maxwell-Boltzmann velocity distribution. It is also possible to solve the Saha equation using the pressure instead of the free electron number density Pe “ nekBTe.
4.5. COLLISION-INDUCED TRANSITIONS CHAPTER 4. PLASMA
4.5
Collision-induced transitions
The absorption or emission due to inelastic collisions is vital in order to model the medium-pressure UV lamp. They arise because of the intermolecular interactions between two or more bodies. If the plasma is assumed to be in LTE, the electron-collision rates exceed the radiative rates [27]. It is the rates of electron-collisions that controls the relative populations. Plasma modelling depends on collision strength and collision cross-section of various collision processes. The collision strength is a measure of the collision between two bodies [28].
When a collision occurs, the electron may be captured into an excited state. The excited state can then decay by emitting a photon [29]. The collision distorts the electronic structure of the atom and this is what causes the formation of a transient electric dipole moment [30]. The collision process involves an induced transient electric dipole moment that appears due to colliding particles. This can give rise to a forbidden transition. Figure 4.1 illustrates other states becoming populated due to collisions. The probability for each transition due to collisions are needed in order to model a medium-pressure UV lamp.
Figure 4.1: Figure shows an energy level diagram. The radiative and collisional induced decay channels are illustrated. The collision give rise to an induced tran-sient dipole moment that results in other states being populated. All energy levels Eis arbitrary and represents different energy levels such as Ei, Ej, Ekand Ep. The energy of the ground state is given by E0.
Chapter 5
Plasma simulation model
5.1
Model parameters
The key parameters for the plasma model are gas temperature, electron tempera-ture, total pressure, partial pressure of Hg and spectral line data. The total pres-sure of the lamp determines the prespres-sure broadening and the partial prespres-sure of Hg determines the number density in the Boltzmann relation. The gas tempera-ture can be estimated from a lamp temperatempera-ture profile as seen in figure 5.1 and the electron temperature can be computed from the ideal gas law, Ppartial “ nekBTe. The spectral line data is obtained from NIST and some of the Hg data can be seen in table 5.1. If PLTE is assumed, the total gas temperature can be used to calculate the relative number density of lower state i from Boltzmann relation
ni “ n0 gi g0
e´Ei{pkBT q, (5.1)
where n0 is obtained from the ideal gas law.
Figure 5.1: Example of a temperature profile with a colour bar showing the colour scale for a simulated UV lamp [19].
5.2. INITIAL SPECTRUM CHAPTER 5. PLASMA SIMULATION MODEL
Table 5.1: The table shows some of the peaks that are present in the spectrum in figure 3.3. Lines included are some of the ones with existing Aji values from NIST [12]. Each line is presented with its transition energy, g-factors and Einstein’s emission coefficient Aji.
λ
λλ[nm] EEEiii [eV] EEEjjj [eV] Transition gggiii gggjjj AAAjijiji [s´1´1´1] 184.95 0.000 6.704 1S 0 é1P1 1 3 7.46e8a 237.68 4.886 10.101 3P 1 é1S0 3 1 9.11e3b 237.83 4.667 9.879 3P 0 é3D1 1 3 3.6e6a 253.65 0.000 4.887 1S 0 é3P1 1 3 8.40e6a 265.20 4.886 9.560 3P 1 é3D2 3 5 3.9e7a 265.51 4.886 9.555 3P 1 é1D2 3 5 1.1e7a 275.28 4.667 9.170 3P 0 é3S1 1 3 6.1e6a 289.36 4.886 9.170 3P 1 é3S1 3 3 1.57e7a 292.54 5.461 9.698 3P 2 é3S1 5 3 8.0e6a 296.73 4.667 8.845 3P 0 é3D1 1 3 4.6e7a 302.15 5.461 9.563 3P 2 é3D3 5 7 5.1e7a 302.35 5.461 9.560 3P 2 é3D2 5 5 9.4e6a 302.75 5.461 9.555 3P 2 é1D2 5 5 1.98e6a 312.57 4.886 8.852 3P 1 é3D2 3 5 6.6e7a 334.15 5.461 9.170 3P 2 é3S1 5 3 1.68e7a 365.02 5.461 8.856 3P 2 é3D3 5 7 1.29e8a 365.48 5.461 8.852 3P 2 é3D2 5 5 1.84e7a 366.28 5.461 8.845 3P 2 é3D1 5 3 3.6e6a 366.33 5.461 8.845 3P 2 é1D2 5 5 3.1e8c aEinstein coefficient is taken from reference [12].
bEinstein coefficient computed with GRASP, private communication. cEinstein coefficient is taken from reference [31].
5.2
Initial spectrum
The volume emission coefficient of a certain line in a spectrum is proportional to Einstein’s emission coefficient. If the electron is de-excited from state j to i in an isotropic and unpolarized plasma [32] the volume emission coefficient [8, 11] in frequency space is given by
jν “ hνAjinjPpνq (5.2) and has the units Wm´3Hz´1.
5.3
Photon absorption
From Beer’s law the absorption coefficient can be obtained as
κijpνq “ σijni “ e2 40mec
5.4. PHOTON EMISSION CHAPTER 5. PLASMA SIMULATION MODEL
The absorption length is then found by taking the inverse of the absorption coeffi-cient
x0 “ 1 κijpνq
. (5.4)
By utilizing both the partial pressure of Hg and temperature in the lamp, it is possible to compute the number density in a lower state, the absorption coefficient and the absorption length.
5.4
Photon emission
Once the initial spectrum has been created, it is used as an input in the Geant4 simulation. At this stage, the photons will interact with the plasma, where some of the photons are scattered or/and absorbed. The spectrum that is obtained from the Geant4 simulation is not the same as the initial spectrum. The photons are gener-ated by the initial spectrum that then goes through the Geant4 simulation. There it interacts with the gas and is subjected to different effects such as self-saturation. The spectrum that comes out from the Geant4 simulation is the recorded spectrum as illustrated in figure 5.2.
Produce photons according
to the initial spectrum
Record the spectrum
that escapes the lamp
Figure 5.2: Figure illustrates a 3-dimensional model of the lamp containing the gas. The photons that are produced inside the lamp corresponds to the inital spectrum. Once the photons have traversed the gas the recorded spectrum is obtained [19].
Chapter 6
Results and discussion
6.1
Electron number density
The number density is computed with different partial pressures that are suited for the simulation. The partial pressures that are used are 150 and 1500 µPa. The total number density from the ideal gas law as mentioned before can be seen in table 6.1. The values are computed with a temperature of 2000 K and a pressure of 1500 µPa. It is evident that the number density in the ground state is several magnitudes larger than the excited states, as expected.
Table 6.1: Number density for lower transition state i for each line with data from NIST [12]. A temperature of 2000 K and a pressure of 1500 µPa is used.
EEEiii [eV] EEEjjj [eV] ni [cm´3] 0.000000 é 6.703662 5.432228e10 4.886494 é 10.10141 7.941840e-2 4.667383 é 9.878884 9.433125e-2 0.000000 é 4.886495 5.432228e10 4.886495 é 9.560150 7.941840e-2 4.886495 é 9.554715 7.941840e-2 4.667383 é 9.170012 9.433125e-2 4.886495 é 9.224990 7.941840e-2 4.886495 é 9.170012 4.707970e-3 5.460625 é 9.697563 9.433125e-2 4.667383 é 8.844537 4.707970e-3 5.460625 é 9.562823 4.707970e-3 5.460625 é 9.560150 4.707970e-3 5.460625 é 9.554715 7.941840e-2 4.886495 é 8.851985 4.707970e-3 5.460625 é 9.170012 4.707970e-3 5.460625 é 8.856338 4.707970e-3 5.460625 é 8.851985 4.707970e-3 5.460625 é 8.844537 4.707970e-3 4.667383 é 7.730455 9.433125e-2 4.886495 é 7.926077 7.941840e-2
6.2. TRANSITION PROBABILITY CHAPTER 6. RESULTS AND DISCUSSION
Table 6.1 – continued from previous page EEEiii [eV] EEEjjj [eV] P pj Ñ iqP pj Ñ iqP pj Ñ iq [%] 6.703662 é 9.720887 2.083353e-6 6.703662 é 9.560150 2.083353e-6 6.703662 é 9.554715 2.083353e-6 4.886495 é 7.730455 7.941840e-2 6.703662 é 9.224990 2.095476e-6 6.703662 é 9.170012 2.083353e-6 5.460625 é 7.730455 4.735366e-3 6.703662 é 8.851985 2.083353e-6 6.703662 é 8.844537 2.083353e-6 6.703662 é 8.844171 2.083353e-6 7.730455 é 9.771569 5.411985e-9 7.926077 é 9.914254 5.785449e-10 7.926077 é 9.771569 5.785449e-10 7.730455 é 9.524891 5.411985e-9 7.926077 é 9.529814 5.785449e-10 8.618957 é 10.05151 1.037657e-11 8.856338 é 10.22333 1.836277e-11 8.618957 é 9.878884 1.037657e-11 8.636963 é 9.880321 2.804253e-11 8.636963 é 9.876651 2.804253e-11 6.703662 é 7.926077 2.083353e-6 7.730455 é 8.839436 5.411985e-9 7.730455 é 8.828580 5.411985e-9 6.703662 é 7.730455 2.083353e-6 8.540829 é 9.562823 8.157841e-11 8.618957 é 9.557250 1.037657e-11 8.636963 é 9.560150 2.804253e-11 8.636963 é 9.557250 2.804253e-11 8.636963 é 9.554715 2.804253e-11 7.926077 é 8.839436 5.785449e-10 7.730455 é 8.636963 5.411985e-9 7.730455 é 8.618957 5.411985e-9 7.730455 é 8.540829 5.411985e-9
6.2
Transition probability
If only the transitions with Ajivalues in NIST database are considered in the model it results in branching fractions shown in table 6.2. The probabilities are calculated according to equation 2.10. More transitions can occur for each state. Some of the transitions may be in the visible or IR range. These transitions are not taken into account when calculating these probabilities. Including the transitions in the visible and IR range will make a difference and change the result by alternating the decay probabilities. This affects the UV range as well since the Ajiratio in equation
6.2. TRANSITION PROBABILITY CHAPTER 6. RESULTS AND DISCUSSION
2.10 will change. It might result in some weaker peaks within the germicidal range. The difficulties lie in a reliable value of Aji to get more precise results, especially as some of the transitions Aji values does not exist in databases.
Table 6.2: Probability for emission from an upper state to a lower state. The branching fractions are calculated according to equation 2.10 and is only treating the transitions with Aji values found at NIST.
Ei EEii [eV] EEEjjj [eV] P pj Ñ iqP pj Ñ iqP pj Ñ iq [%] 0.0000000 é 6.70366230 100.0 4.6673829 é 9.87888390 99.9 0.0000000 é 4.88649459 100.0 4.8864945 é 9.56014960 5.9 4.8864945 é 9.55471450 33.6 4.6673829 é 9.17001199 23.3 4.8864945 é 9.22499000 56.7 4.8864945 é 9.17001190 3.6 5.4606247 é 9.69756300 100.0 4.6673829 é 8.84453740 10.6 5.4606247 é 9.56282339 56.1 5.4606247 é 9.56014960 83.7 5.4606247 é 9.55471450 37.6 4.8864945 é 8.85198489 1.5 5.4606247 é 9.17001199 39.4 5.4606247 é 8.85633759 100.0 5.4606247 é 8.85198489 98.4 5.4606247 é 8.84453740 24.4 4.6673829 é 7.73045509 22.5 4.8864945 é 7.92607670 99.6 6.7036622 é 9.72088699 100.0 6.7036622 é 9.56014960 6.5 6.7036622 é 9.55471450 8.8 4.8864945 é 7.73045509 18.1 6.7036622 é 9.22499000 43.3 6.7036622 é 9.17001199 33.5 5.4606247 é 7.73045509 36.2 6.7036623 é 8.85198489 0.02 6.7036623 é 8.84453740 64.9 6.7036623 é 8.84417140 100.0 7.7304551 é 9.77156899 90.9 7.9260767 é 9.91425399 100.0 7.9260767 é 9.77156899 9.1 7.7304551 é 9.52489100 100.0 7.9260767 é 9.52981439 100.0 8.6189569 é 10.0515179 100.0 8.8563376 é 10.2233300 100.0 8.6189569 é 9.87888399 0.04
6.3. INITIAL SPECTRUM CHAPTER 6. RESULTS AND DISCUSSION
Table 6.2 – continued from previous page Ei EEii [eV] EEEjjj [eV] P pj Ñ iqP pj Ñ iqP pj Ñ iq [%] 8.6369628 é 9.88032100 100.0 8.6369628 é 9.87665099 100.0 6.7036623 é 7.92607670 0.3 7.7304551 é 8.83943599 11.8 7.7304551 é 8.82857950 100.0 6.7036623 é 7.73045509 23.1 8.5408289 é 9.56282339 43.9 8.6189569 é 9.55724999 5.0 8.6369628 é 9.56014960 3.7 8.6369628 é 9.55724999 94.9 8.6369628 é 9.55471450 20.0 7.9260767 é 8.83943599 88.1 7.7304551 é 8.63696280 100.0 7.7304551 é 8.61895699 100.0 7.7304551 é 8.54082899 100.0
6.3
Initial spectrum
The spectrum obtained from experimental data has, in our case, dominating in-strumental broadening, which means that it is not possible to say anything by fitting a line profile to the data. However, the FWHM is still required as an input when generating the emission and absorption spectrum. Since the FWHM cannot be obtained by analysing the experimental spectrum, a rough estimate of the pres-sure broadening from equation 2.13 is used instead. The natural broadening are not included due to being negligible compared to the pressure broadening. The Doppler broadening should be included with higher temperatures. In figure 6.1 both pressure broadening and Doppler broadening has been included but as the pressure broadening is several magnitudes larger, it dominates.
By using the number density in the upper state (nj) it is possible to calculate the volume emission. The emission obtained from equation 5.2 with temperature 2000 K, partial pressure 1500 µPa and total pressure of 150 kPa can be seen in figure 6.1. It is seen that the 6.074 eV and the 4.886 eV transitions are dominating and are very narrow. These transitions are several magnitudes larger than the rest and results in all other transitions being suppressed. Figure 6.2 shows all transitions individually with a Lorentzian profile. It is evident that most photons are created in the 4.886 eV transition as it is dominating.
6.3. INITIAL SPECTRUM CHAPTER 6. RESULTS AND DISCUSSION
Figure 6.1: Initial spectrum created inside the lamp with both pressure broadening and Doppler broadening. The 6.704 eV and 4.886 eV transitions are dominating. It is seen that the majority of photons are created at the 4.886 eV peak. All three profiles are included to see the difference but as the broadening is mainly follow-ing a Lorentzian profile it is seen that the Voigt profile is close to the Lorentzian. The partial pressure used for the number density is 1500 µPa and the pressure broadening is computed with a total pressure of 150 kPa.
Figure 6.2: Each transition is plotted individually to illustrate all the suppressed peaks with a Lorentzian profile. The partial pressure used for the number density is 1500 µPa and the pressure broadening is computed with a total pressure of 150 kPa.
6.4. PHOTON ABSORPTION CHAPTER 6. RESULTS AND DISCUSSION
6.4
Photon absorption
The absorption length is calculated according to equation 5.4 with the same values as for the initial spectrum. The partial pressure is set to 1500 µPa in order for any photons to be able to escape the gas. If the partial pressure is higher, the absorption length becomes too short in terms of what a CPU can manage. A too short absorption length results in a too short iteration step that is taken within the simulation, meaning that the photons will never exit the gas. Absorption of photons primarily occurs at the 6.704 eV and the 4.886 eV transitions due to the partial pressure and the number density in the lower state. Number density in the lower state is much denser at these transitions and thus, the photons will not be able to travel as far without being absorbed. This results in two peaks that suppresses all other transitions and only showing the one at 4.886 eV and 6.704 eV as seen in figure 6.3. Similar to the emission spectrum, the absorption length in figure 6.4 shows all transition individually with a Lorentzian profile in order to better illustrate the suppressed transitions. It is seen in both figures that all photons at each resonance wavelength have a high probability of being absorbed.
Figure 6.3: Absorption length as a function of photon energy. It is seen that most of the photons corresponding to transitions 6.704 eV and 4.886 eV has a probability of being absorbed quickly. The partial pressure used for the number density is 1500 µPa and the pressure broadening is computed with a total pressure of 150 kPa.
6.5. PHOTON EMISSION CHAPTER 6. RESULTS AND DISCUSSION
Figure 6.4: Absorption length as a function of photon energy. Figure illustrates each transition individually with a Lorentzian profile to illustrate the suppressed transitions. The partial pressure used for the number density is 1500 µPa and the pressure broadening is computed with a total pressure of 150 kPa.
6.5
Photon emission
Figure 6.5 shows the resulting spectra once the photons have traversed the gas. The upper part in figure 6.5 shows the spectrum from a lamp with a partial pres-sure of 150 µPa and the lower part shows the spectrum from a lamp with a partial pressure of 1500 µPa. If there are no collisions occurring, only the 6.704 eV and 4.886 eV transitions will be seen. A photon is created by the 4.886 eV transition that becomes re-absorbed until it eventually gets emitted due to pressure broad-ening. This emitted photon is then absorbed by the 6.704 eV transition, resulting in that this peak appear in the spectrum.
An estimate of the collision process has been tested where the 6.704 eV and the 4.886 eV transitions are allowed to decay both radiatively and non-radiatively. There is a greater probability of non-radiative decay due to the existence of several non-radiative processes and only one radiative process. The non-radiative decay (e.g., due to collisions) is given by the same Aji values as that of the radiative transitions. This is not correct, but it gives an idea of how it can be done. What is missing is the correct probability for each non-radiative transition. It is this non-radiative process that gives rise to all other peaks as the collisions populate all states.
When the partial pressure is increased from 150 to 1500 µPa, the absorption from the 4.886 eV transition increases. The self-saturation is clearly seen in the wings of the 6.704 eV transition. The self-saturation occurs when the absorption becomes strong enough so that the photons at the centre of the peak are immedi-ately re-absorbed. This means that only the surface of the gas emits this energy. The spike at 0.1 eV is due to a cutoff energy in the simulation and is to be over-looked. The actual partial pressure in the lamp is most likely to be close to 150 kPa, but due to not having infinite CPU power, it is not possible to simulate it with
6.5. PHOTON EMISSION CHAPTER 6. RESULTS AND DISCUSSION
this model.
0 1 2 3 4 5 6 7 8 Photon energy [eV] 0 5000 10000 15000 20000 25000 30000
Emitted spectral power [W/eV]
Pa
µ
With transitions due to collisions, 150 Pa
µ
No transitions due to collisions, 150
0 1 2 3 4 5 6 7 8 Photon energy [eV] 0 5000 10000 15000 20000 25000 30000
Emitted spectral power [W/eV]
Pa µ With transitions due to collisions, 1500
Pa µ No transitions due to collisions, 1500
Figure 6.5: Figures showing the resulting spectra once the photons have traversed the gas. If no collisions are present the only transitions that occurs is the ones seen at 4.886 and 6.704 eV. If collisions are introduced this results in that they will populate other states and thus give rise to more peaks in the spectrum.
Chapter 7
Conclusions
This master thesis has developed and suggested an absorption model for the inter-actions within a medium-pressure UV lamp. This model has been used to model the absorption in the plasma.
The model is validated against the observed spectrum emitted by the lamp. The absorption and emission spectrum evaluation cannot be done through Geant4 as the collision process is not properly described. However, the simulated spectrum reproduces selected features of the observed spectrum.
The collision process is essential for the simulation in Geant4 to reproduce the medium-pressure UV lamp spectrum. It seems that the absorption model could work for a medium-pressure UV lamp simulation if the collision-induced transition probabilities are implemented correctly.
However, there is an issue with CPU power as the absorption length is very short. It is not possible to use this model at operational pressure as the absorption length becomes too short. A more accurate model will be obtained if more Einstein emission coefficients are computed.
Both the number density and the radiative decay has been calculated for all transitions with an Aji value in NIST. The Aji values may be computed from dif-ferent software packages. One software package is the General Relativistic Atomic Structure Package (GRASP). The computations of the Einstein emission coefficient are non-trivial and should be done carefully to get a reasonable estimate.
Chapter 8
Future work
The collision process needs to be described and implemented. In order to capture the physics in a medium-pressure UV lamp, the redistribution between different energy levels needs to be accounted for. A method for estimating the Einstein emission coefficient should also be suggested and implemented. One Einstein emission coefficient that does not exist on the NIST database or other references has been calculated. The transition energies obtained from the computation agrees well with the value on NIST. This indicates that the new calculations can be used to calculate more transitions in future work. The calculations were made with the General Relativistic Atomic Structure Package (GRASP) [33]. A method to deal with the short absorption length is also left for future work.
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![Figure 3.1: Schematic of Hg atom according to Bohr’s model. The electronic shell structure can also be expressed as [Xe] 4f 14 5d 10 6s 2 .](https://thumb-eu.123doks.com/thumbv2/5dokorg/3950115.73692/20.892.330.557.806.1064/figure-schematic-according-bohr-model-electronic-structure-expressed.webp)
![Figure 5.1: Example of a temperature profile with a colour bar showing the colour scale for a simulated UV lamp [19].](https://thumb-eu.123doks.com/thumbv2/5dokorg/3950115.73692/26.892.127.770.766.876/figure-example-temperature-profile-colour-showing-colour-simulated.webp)
