DIPLOMARBEIT. Design of an FTIR spectroscopy system for metal-oxide single crystals

DIPLOMARBEIT

Design of an FTIR spectroscopy system for metal-oxide single crystals

zur Erlangung des akademischen Grades Diplom-Ingenieur/in

unter Anleitung von

Ao.Univ.Prof. Dipl.-Ing. Dr.techn. Michael Schmid

im Rahmen des Studiums Technische Physik

ausgeführt am Institut für Angewandte Physik

der Fakultät für Oberflächenphysik der Technischen Universität Wien

eingereicht von

David Andreas Rath BSc 01225498

Die approbierte Originalversion dieser Diplom-/

Masterarbeit ist in der Hauptbibliothek der Tech- nischen Universität Wien aufgestellt und zugänglich.

http://www.ub.tuwien.ac.at

The approved original version of this diploma or master thesis is available at the main library of the Vienna University of Technology.

http://www.ub.tuwien.ac.at/eng

Abstract

The aim of this thesis was to set the basis for the implementation of the commer- cially available Fourier-transform infrared (FTIR) spectrometer Bruker Vertex 80v to an ultra-high-vacuum-system (TPD chamber). This system will be used to in- vestigate adsorbates on metal oxide surfaces with infrared reflection absorption spectroscopy (IRAS). The signal from such surfaces is lower than the signal ob- tained from metal surfaces, mostly investigated by this technique.

In this work the reflectivity with the influence of the adsorbate on the sample is discussed. Sensitivity and band shape analysis lead to design criteria for the incidence angle Θ and cone angle restriction. Also theoretical considerations are discussed to set design criteria for the design of GRISU (Grazing incidentInfrared absorptionSpectroscopy Unit). Here, the illumination on the sample is estimated and used to approximate the focal length of the illumination mirror. These de- sign criteria were used to design the optical setup for GRISU. GRISU is optimised for high throughput on the signal area and a minimised signal of the background area. Apertures at the positions of the intermediate images of the system enable the shaping of the illumination area on the sample. For the optimisation, the optical path of the spectrometer was modelled and its validity is checked by com- parison of simulated and measured intensity maps.

The optical design of GRISU provides up to 28 times higher intensities on the detector then the optical design with two parabolic mirrors suggested by Bruker Optics.

Kurzzusammenfassung

Ziel dieser Arbeit war die Entwicklung einer theoretischen Basis zur Implemen- tierung des kommerziell erhältlichen Fourier-Transformations-Infrarotspektrometers (FTIR-Spectrometer) Bruker Vertex 80v an einer Ultrahochvakuum-Kammer (TPD- Kammer). Mit diesem System sollen Adsorbate auf Metalloxidoberflächen mithilfe von Infrarot-Reflexions-Absorptions-Spektroskopie (kurz IRAS) untersucht wer- den. Das Signal solcher Oberflächen ist niedriger als das Signal von Metallober- flächen, welche meistens mit IRAS untersucht werden.

In dieser Arbeit wird das Reflexionsvermögen unter dem Einfluss eines Adsorbates auf der Oberfläche untersucht. Die Sensitivitätsanalyse sowie die Untersuchung der Absorptionsbanden liefern Werte für den optimalen Einfallswinkel Θ und für den Öffnungswinkel κ. Auch die Intensität der Infrarotstrahlung auf der Probe wird abgeschätzt. Sie liefert einen Startwert für das optische Design. Diese hergeleit- eten Konstruktionsbedingungen wurden genutzt, um das optische System GRISU (Grazing incident Infrared absorption Spectroscopy Unit) zu entwerfen. GRISU ist so optimiert, dass maximaler Strahlungsdurchsatz für die Reflexion an der Sig- nalfläche und minimalen Durchsatz für die Reflexion an der Hintergrundfläche erre- icht wird. Blenden an den Positionen der Zwischenbilder des Systems ermöglichen ein Anpassen der Beleuchtung an die Adsorbatfläche. Zum Optimieren von GRISU wurde ein optisches Modell des Spektrometers erstellt. Gemessene und simulierte Intensitätsverteilungen werden miteinander verglichen um die Gültigkeit des Mod- ells zu überprüfen.

Im Vergleich zum optischen System mit zwei Parabolspiegeln, welches von Bruker Optics vorgeschlagen wurde, ermöglicht GRISU, eine um bis zu 28-mal höhere Intensität am Detektor.

Acknowledgements

I would first like to thank my supervisor Michael Schmid, who was always avail- able to answer any questions, supported me whenever I had to solve problems and guided me in the right direction if it was necessary.

Beside my supervisor, I would also like to thank Ulrike Diebold and Gareth Parkin- son, who gave me the opportunity to work on this project.

I am also grateful for the assistance given by Jiri Pavelec, who was always available for constructive discussions.

I would also like to thank Saghafi Saiedeh PhD from the Institute of Solid State Electronics, department of Bioelectronics, who helped us with the optical ray trac- ing program Zemax.

Finally, I would like to thank my parents for supporting me over my entire years of study. This achievement would not have been possible without them.

Contents

1 Introduction 1

1.1 Infrared spectroscopy . . . 1

1.2 Basic principle of an of an FTIR . . . 3

1.3 Resolution and J-stop . . . 6

1.4 Liouville’s theorem and optical throughput . . . 6

1.5 The TPD chamber . . . 8

2 Design considerations 11 2.1 Estimation of the sample illumination . . . 11

2.2 Reflectivity and Absorption . . . 15

2.2.1 Theory . . . 15

2.2.2 Analysis of the reflectivity . . . 18

Sensitivity . . . 18

Band shape analysis . . . 28

2.3 Tilting error of the cryostat and the sample mount . . . 33

2.4 Chamber restrictions . . . 36

2.5 Summary of the design considerations . . . 37

3 Optical models 39 3.1 Spectrometer model . . . 39

3.1.1 Measurement of the beam profile . . . 40

3.2 GRISU . . . 43

3.2.1 Simulation . . . 45

3.3 Discussion of the optical models . . . 48

3.4 Summary of the optical models . . . 49

4 Conclusion 51

Appendix Bibliography

1 Introduction

1.1 Infrared spectroscopy

Infrared spectroscopy is a widely used technique employed in different areas of research and industry. Commercially available spectrometers are easy to use and in most cases the measurements can be performed in very short times (< 1 min).

Infrared spectroscopy is used in chemistry to characterise and identify molecules.

A similar application is to use the spectrometer for investigations of molecules ad- sorbed at surfaces. In this work we focus on the implementation of a commercial FTIR spectrometer (Bruker Vertex 80v) to an ultra high vacuum (UHV) system for the investigation of adsorbates on single-crystal metal oxides. In the follow- ing chapters the theoretical basics of infrared spectroscopy and infrared reflection absorption spectroscopy will be explained. Furthermore the theoretical consider- ations were used to design an optical system, which allows to use the commercial infrared spectrometer in combination with a UHV chamber to study adsorbates on metal oxide surfaces.

The basic principle of infrared spectroscopy is the interaction between electro- magnetic radiation and molecules. Depending on the type of molecule, different frequencies are absorbed leading to characteristic absorption bands in the infrared spectra. In infrared spectroscopy the wavenumber ˜ν is used as quantity instead of the frequency. The wavenumber is defined as [1]

˜ ν 1

λ ν

c [cm^✏1] (1.1)

where λ is the wavelength in cm, ν is the frequency and c is the speed of light.

The unit of ˜ν is cm^✏1 and proportional to the frequency and energy.

Most of the vibrational-rotational bands of molecules are located in the mid- infrared region ranging from about 400 cm^✏1 to 4000 cm^✏1[2]. Prominent examples in this wavenumber region are water and carbon dioxide which can be found in air and are usually seen in infrared spectra [3]. One main criterion for molecules to show infrared activity is a change of the dipole moment during vibrational excita- tion. This is the case when the transition dipole moment is not zero [4]. A dipole moment arises due to different centres of positive charge (nucleus) and negative

charge (e^✏) of a molecule. To excite vibrations the incoming electromagnetic field needs a component pointing in the direction of the dipole moment. One can un- derstand the vibrations and the band positions in the spectra easily by looking at the concept of the harmonic oscillator [5] where

ν˜➀

➽k

µ with (1.2)

µ m1 m2

m1✔m2

. (1.3)

Here, a massless spring (representing the chemical bond) with spring constant k connects two masses m1 and m2. µ is the reduced mass. One can see that an increase of the reduced mass and a decrease of the spring constant lower the wavenumber. With a decrease of µ and an increase in bond strength ˜ν rises. The classical model can only be used to understand the concept. In reality the vibra- tional states are quantised. The quantised energy states of a diatomic molecule can be understood by the quantum mechanical harmonic oscillator but a more suitable theoretical description uses the Morse potential [6].

The number of vibrational modes of a molecule depends on the number of atomsN and on its structure. In the gas phase, linear molecules show 3N✏5 and nonlinear molecules show 3N✏6 vibrational modes [2]. The appearance of the bands in the spectra is influenced by the aggregate state. Molecules in gas phase show usually a lot of sharp peaks arising from coupling of vibrations with the rotational degrees of freedom. The bands of liquids and solids show broader bands due to stronger interaction between the molecules. Absorption bands observed from molecules adsorbed on the surface show a shift due to additional bonds to the surface and changed bonds of the molecule.

One method used to investigate the vibrational modes of adsorbates on surfaces is infrared reflection absorption spectroscopy (IRAS). Here, the infrared beam is reflected on a sample. One measurement is performed without an adsorbate and a second one with an adsorbate. The signal observed is a differential reflectivity and can be written according to [7] as

∆R R0

R₀✏R R0

, (1.4)

where R is the reflectivity of the surface with adsorbate and R0 is the reflectivity without adsorbate. A more detailed description of the reflection on a surface with adsorbate based on reference [8] will be given in chapter 2.

2

On metal oxides the expected signal is by about two orders of magnitude lower than on metals [9].

In difference to metal oxides, the surface selection rules apply for IRAS measure- ments on metals [10]. These selection rules can be described by means of image dipoles. A dipole moment oriented parallel to the surface causes an image dipole in opposite direction in the metal surface. This image dipole suppresses the dipole moment originating from the parallel oriented surface dipole. A dipole oriented perpendicular to the surface gets reinforced by the image dipole. Here, the dipole moment is amplified, which leads to higher absorption bands.

The lower signal intensity requires an optimized optical design with high through- put to increase the signal-to-noise ratio and the use of a liquid nitrogen (LN2) cooled mercury cadmium telluride (MCT) detector [9].

To measure infrared spectra, a Fourier-transform infrared (FTIR) spectrometer is used. In this work the spectrometer Bruker Vertex 80v is used. It is equipped with a standard glowbar, a narrow band MCT detector and a KBr beam splitter.

The basic principle of such a spectrometer is described in chapter 1.2.

1.2 Basic principle of an of an FTIR

A Fourier-transform infrared (FTIR) spectrometer features several advantages over grating spectrometers. It measures all wavenumbers of the source spectrum within a short time (typically ❅1 s). This fact is known as Fellgett’s advantage or multi- plex advantage [2, 11]. FTIR spectrometer feature a✂60 times higher throughput, compared to grating spectrometers, which are using slits to separate the resolution elements. This higher throughput is called Jacquinot’s advantage [11]. FTIRs also feature a high wavenumber accuracy due to the use of laser interferometry (Connes Advantage) [12].

The typical interferometer combined with a reflection setup is shown in figure 1.1. The description focuses on the Bruker Vertex 80v. The first important part is the illumination stage. Here, the radiation for the source is collected by a mirror (source mirror) and then focused in the position of the first aperture. This aper- ture is called J-stop. In our configuration a U-shaped glowbar (SiC) is used. In the illumination stage it is very important to gather as much radiation as possible. To increase the intensity transferred to the interferometer one can increase the tem- perature [13] of the source. Due to the higher intensity of the infrared radiation better signal-to-noise ratios (SNR) can be reached. To transfer all of the radiation to the interferometer, the source mirror and the collimation mirror have to fulfil, κ1 κ2 (see figure 1.1). All of these steps can be easily explained by the theorem

of Liouville (see chapter 2.1).

The J-stop is a very crucial part of the spectrometer. It defines the maximum resolution of the spectrometer. In modern FTIR spectrometer this aperture is variable in diameter. A small J-stop goes along with high resolution and a lower throughput. In case of adsorbates on metal oxide surfaces a spectral resolution of 4 cm^✏1 is enough which allows the use of bigger J-stop settings. The connection between J-stop and resolution will be explained in section 1.3.

After the J-stop the beam gets collimated by a parabolic mirror for the Michelson interferometer. In the interferometer the beam splits up in two parts. In the ideal case 50 % of the intensity is reflected at the fixed mirror and the other 50 % are reflected on the moving mirror. After the reflection on the mirrors both beams are recombined behind the beam splitter. The two beams interfere according to the optical path difference L introduced by the moving mirror. L is the opti- cal path difference between the two recombined beams and twice the mechanical movement in case of a perpendicular incidence of the beam on the mirror. In our case the beam is then decoupled to GRISU (Grazing incident Infrared absorption Spectroscopy Unit) (see chapter 3.2). The interfered beam is then guided to the sample, reflected and directed to the detector. Depending on the position of the moving mirror different intensities are detected. The recorded signal is the inten- sity versus optical path difference L, and then gets fast Fourier transformed.

For studies of molecules on dielectric surfaces, a polariser is used to switch be- tween s and p polarisation. This is useful due to the fact that the reflectivity is different for dissimilar polarisations and dipole moment orientations (see chapter 2.2.2). This property can be exploited to determine, for example, orientation of the molecule on the surface [9].

Furthermore, a modern FTIR includes laser interferometry to precisely measure L to set a wavenumber standard for the measurement (Connes advantage). In the Bruker Vertex 80v a helium-neon laser with a wavelength (in air) of λHeN e

632.8 nm [6] equivalent to a wavenumber of ˜νHeN e= 15803 cm^✏1 is used. The HeNe laser light beam is passed through the same interferometer used for the infrared radiation. It interferes and gets detected (see figure 1.1). The laser detector (photo diode) detects a sinusoidal signal if the moving mirror in the interferometer changes position. This signal can be converted to optical path difference L. Additionally, the laser is also used for the auto-alignment system of the spectrometer.

4

h

fixed mirror moving mirror beam splitter laser mirror

L/2 α

Sp ec tro m et er

ø ≈ 40 m m

f

CM

κ2

h

detector source mirror

J-stoplaser beam splitterlaser mirror laser detector linear polariser source κ

exit X1

GR IS U

Figure 1.1: Schematic illustration of the main components of the spectrometer in reflection setup

1.3 Resolution and J-stop

The maximum resolution

∆˜νmax

1 Lmax

(1.5) reached by a Michelson interferometer is defined by the maximum optical path difference Lmax [2, 14]. The higher the obtained resolution the higher the optical path length difference L has to be. The resolution of a spectrometer is not only dependent on the resolution of the interferometer inside the spectrometer. It is also defined by the optical setup. The combination of J-stop diameter and focal distance of the collimation mirror defines the divergence α of the beam entering the interferometer. This divergence also sets a limit for resolution. Therefore the achievable resolution of the spectrometer is a combination of maximum optical path difference Lmax and beam divergence α inside the interferometer. The limit for the divergence angle [2] in the spectrometer is

αmax

➽ ∆˜ν ν˜max

[rad]. (1.6)

The divergenceαin the interferometer of the spectrometer is set by the diameter of the J-stop and the focal length of the collimation mirror. For the parameters ˜νmax

4000 cm^✏1 representing the highest wavenumber one wants to measure, resolution

∆˜ν 4 cm^✏1 and fCM 100 mm describing the focal length of the collimation mirror in figure 1.1, an allowed J-stop diameter

dJ 2 tan➆

➈

➽ ∆˜ν

˜ νmax

➇

➉ fCM (1.7)

of 6.2 mm can be estimated. It is important to mention that this is just a simpli- fied concept to get better understanding of the connection between interferometer resolution and spectrometer resolution. A more detailed description can be found in [14].

1.4 Liouville’s theorem and optical throughput

Liouville’s theorem is a very basic concept in physics stating that the phase space of a particle or light beam cannot be compressed. In optics the phase space consists of the position space represented by the image and the momentum space describing the cone angle [15]. The image produced by an optical system extends over a certain area. In the ideal case all the rays in the optical system contribute to the image. There, the area of the image can be identified as position space.

6

The bigger the image the bigger the position space. In geometrical optics a beam is represented by single rays. Every ray can be assigned to a k-vector following the relations

❙Ñk❙ 2π⑦λ (1.8)

Ñ

p Òh Ñk (1.9)

whereλ is the wave length,hÒ is the reduced Planck constant andpÑis the momen- tum. To connect this parameter to an optical system it is helpful to have a look at a focussing lens.

Let’s assume a monochromatic parallel beam is focused by a convex lens onto the focal plane. The rays are refracted with different angles behind the lens de- pending on their radial position. Rays refracted further outside feature a bigger angle than the central rays. Every different direction of the rays contributes to the momentum space. A simple way of describing all the different ray directions is the solid angle.

f' EP

DEP κmax

Figure 1.2: Schematic drawing illustrating the f-number, EP is the entrance pupil, D_EP is the entrance pupil diameter, f^➐ is the focal length of the lens and κmax the maximum half cone angle

Combining position space (image area) and momentum space (solid angle) leads to the optical throughput criteria [16]

A Ω invariant (1.10)

valid for paraxial optics, with A indicating the image area and

Ω 2π ❼1✏cos❼κ➁➁ (1.11)

describing the solid angle [17] for a rotationally symmetric beam. κ is the half angle of the beam cone. With equation 1.10 one can obtain a decreasing image area A when Ω increases. Increasing Ω has the consequence of increasing κ (see

equation 1.11 and figure 1.2). So bigger angles κ or solid angles Ω lead to smaller focal spots. The capability of small focussing is also described by the f-number defined as

f-number f^➐ DEP

(1.12) with f^➐ as the focal length of the system and D_EP is the entrance pupil diameter.

It is usually written as "f" followed by the reciprocal f number, e.g. f/2.8. A small focal point can be achieved with low f-number optics. A high f-number decreases the half cone angle κ and increases the spot size on the sample.

1.5 The TPD chamber

The TPD (temperature programmed desorption) chamber is a surface chemistry setup with the main focus on performing high-quality TPD spectra. A detailed ex- planation of the UHV chamber can be found in the work of Pavelec et al. [18, 19].

This UHV system, shown in figure 1.3, reaches a base pressure of 5✕10^✏11 mbar.

The chamber has an upper and a lower level. A molecular beam [20] and a mass spectrometer are located in the lower level. The molecular beam is used to adsorb molecules on the sample surface in a very defined way. The adsorbate area created by the molecular beam has a diameter of 3.5 mm with a relative dose of 97.7 % [19]. The mass spectrometer is located 45° rotated to the molecular beam to per- form temperature programmed desorption (TPD) measurements (see figure 2.20).

Furthermore, the low-energy electron diffraction (LEED) analyser is located on the opposite side of the mass spectrometer. The chamber flange, where the LEED is mounted, will be used to place a flange with infrared transparent windows to separate the UHV in the chamber from the medium-vacuum (❆10^✏4mbar) in the spectrometer. The sample is mounted to a sample holder which itself is connected to a rotatable and in xyz direction moveable cryostat.

Further techniques implemented in the chamber are X-ray photoelectron spec- troscopy (XPS), ultraviolet photoelectron spectroscopy (UPS) and low-energy ion scattering (LEIS).

The infrared reflection absorption spectroscopy setup GRISU (Grazing incident Infrared absorption Spectroscopy Unit), will be an additional analysis technique in our already existing TPD chamber. The implementation of GRISU into the existing chamber leads to spacial restrictions influencing the optical design, which will be discussed in the following chapter.

8

Figure 1.3: (a) 3D view of the UHV chamber with the key instruments; (b), (c) schematic drawing of the instruments in the upper and lower level [18]

2 Design considerations

To design an optical layout enabling the use of a commercial FTIR in combination with our TPD chamber (see section 1.5) some theoretical considerations have to be taken into account. These factors concerning the focal spot at grazing incident, reflectivity, and maximum sensitivity to adsorbates on the surface lead to design criteria for the mechanical and optical design. The theoretical investigation will give a basis for optimisation of the system. Also some design criteria set by the current UHV chamber, such as the diameter of adsorbate area on the crystal or angle of the sample mount are considered. The treatment of all these factors is necessary due the low signal gained on oxide single crystals [9]. The goal is to increase the intensity on the sample and thus the signal-to-noise ratio (SNR) in the final spectra. Higher intensity on the sample can be achieved by the use of a short back focal length of the illumination mirror realising a small illumination area. The optimisation of the throughput with suitable apertures for the optical components increases the intensity on the sample and the detector. Also the correct size of the image on the detector is essential for high intensity.

In principle there are two main parameters determining the signal on the detector.

The first one is the intensity on the sample at grazing incident, which is connected to the diameter of the focal spot. The second parameter is connected to the reflectivity on the sample and the small amount of adsorbates at (sub-) monolayer coverage. Also alignment errors in the optical system lead to intensity losses at the detector.

2.1 Estimation of the sample illumination

In this section simple geometrical considerations of how the grazing incidence on the sample influences the intensity on the adsorbate are presented. Furthermore, the theorem of Liouville (discussed in chapter 1.4) is applied to estimate the pos- sible focal spot size on the sample which is then connected to restrictions given by the molecular beam spot.

An important point is to maximise the intensity on the signal area (area with adsorbed molecules). In our case the signal area is a circular spot with a diameter of about 3.5 mm created on a sample by a molecular beam. On the one hand, this

limited area enables the user to adsorb molecules in a very defined way [19]. On the other hand, it sets the requirement to the optical system to focus very well.

Thus, one has to find a compromise between spot size and cone angle.

The Liouville theorem (see chapter 1.4) can be used to estimate maximum cone angleκ of illumination beam. For this, one needs to know some parameters of the internal optical spectrometer components shown in figure 1.1. The focal length fCM of the collimation mirror with an aperture of about dCM 40 mm is 100 mm.

It is located directly after the J-stop in the spectrometer. The J-stop is in the focal plane of the collimation mirror. The diameter of the J-stop is defined by the maximum resolution discussed in chapter 1.3. Here, one gets a J-stop diameter dJ 6 mm.

Equation 1.10 defines the calculation of the optical throughput. The aperture of the focussing element df is assumed to be 40 mm and the desired diameter of the focal spot matches the diameter of the molecular beam spot with 3.5 mm.

Using formula 1.10 and formula 1.11 gives a half cone angle κ of 19.5 °. The re- lated focal length for the focussing element in front of the sample is approximately f_f^➐ 57 mm.

In this estimation the losses in the optical path are not considered and the po- sitions of the intermediate images are assumed to be in the focal planes of the optical elements. Furthermore, it is a paraxial calculation.

Optimum sensitivity to adsorbate vibrations requires grazing incidence on the sam- ple. The calculations in chapter 2.2 show that an incidence angle of about 80 ° to 85 ° is beneficial to maximise sensitivity. Due to grazing incidence of the beam on the sample the spot radius

rx

r0

cos❼Θ➁ (2.1)

along the x axis, increases with increasing angle of incidence Θ. r0 is calculated with

r0

dJ df

4 fCM tan❼κ➁ (2.2)

wheredJ is the diameter of the J-stop, df is the diameter of the focussing element before the sample,fCM is the front focal length of the collimation mirror inside the spectrometer (see figure 1.1) andκis the half cone angle. Here, the assumption was used, that the reproduction scale β^➐ (see chapter 3.2) is defined by the front focal length of collimation mirror and the back focal length of the illumination mirror.

The result of the grazing incidence of the beam is an elliptical illumination of the sample. Figure 2.1 and figure 2.2 illustrate this behaviour. In figure 2.1 one spots two problems. Firstly, an elliptical illumination area Ai appears. Secondly,

12

Illumiation area A_i Signal area A_s

Sample Sample Sample

incidence beam

Sample

x y z y x z

Θ

Θ+κ > 90°

Θ> 0°

κ

Θ

Figure 2.1: Schematic sketch of the sample illumination at different incidence an- gles Θ, illumination by a conical beam with half opening angleκ: Left image: Θ = 0 °, middle image: Θ ❆0 °, right image: Θ✔κ❆90 ° intensity in the signal areaAsis lost if Θ✔κexceeds 90°. To estimate the intensity on the signal area at grazing incidence, the following assumptions were made. A parallel beam with radiusr0 ry hits the sample in the centre of the signal areaAs. Furthermore, the case Θ✔κ ❆ 90 ° was not considered. Also the inhomogeneous intensity distribution of the illumination area is not covered.

Illumiation area A_i Signal area A_s A_intersect

x y z

r_x r_y

Figure 2.2: Sketch of the intersection area used for calculations. Left drawing:

illumination area Ai bigger than signal area As, right drawing: partly illuminated signal area As

At θ 0 ° the illuminated circular area has a radius ry r0. With increasing incidence angle Θ, the radius rx of the illumination area increases in x direction according to equation 2.1. ry is independent from Θ. An increase of the illumina- tion area over the signal area (seen in figure 2.2) leads to a loss of intensity atAs. By calculating the intersection areaAintersectbetween illumination area Ai and the

signal area As one can estimate the loss of intensity at Ai. The intersection area Aintersect depends on the one hand on the incidence angle Θ influencing rx. On the other hand, the spot diameterr0 depends onκ(see equation 2.2). One has to keep in mind that the intensity estimation is still based on a parallel beam with radius r0. κ is only used to calculate the radius of the illumination area.

Figure 2.3: Contour plot of the area factorγA showing the loss of intensity on the signal area As dependent on κ and Θ. White lines indicate κ 19 ° and the sensitivity at Θ 60 ° and Θ 80 °

We describe the illumination of the signal area As by the area factor γA. It is defined by

γA

Aintersect

Ai

Aintersect

As

. (2.3)

The first term in equation 2.3 describes the illumination of the signal area. The second term is a compensation factor for the back focal length of the illumination mirror. This factor keeps the back focal length within a reasonable limit for the mechanical design. γA is equal to one if Ai As Aintersect. If Ai extends over the signal area As and As Aintersect then γA❅1 (figure 2.2, left). An intersection area smaller than the signal area As also causes γA❅1 shown in the right part of figure 2.2. Here, the illumination radius r0 is smaller than the radius of As but the whole signal area is not utilised for the measurement. The area factor γA is

14

proportional to the intensity on the signal area.

Figure 2.3 shows the area factor γA on the signal area. A maximum is visible for κ ☎ 19 ° up to incidence angles of about 15 °. The relevant region for inves- tigation is below κ 19 °. With increasing angle of incidence, γA decreases. An increasing cone angleκleads to an increase ofγA. Above Θ☎30 ° it is not possible any more to concentrate the radiation just on the signal area As. An incidence angle of Θ = 80 ° shows an area factor γA of approximately 0.17. At Θ = 60 ° and κ 19 °, γA ☎0.5. Here, the change of the incidence angle form 60 ° to 80 ° decreases the area factor γA by a factor of 2.9.

It is important to keep in mind that these values are only valid for equal area illumination. In reality more intensity is centred to the middle of the spot (see chapter 3) because of vignetting in the optical system. That means the beam is cut unintentionally or intentionally somewhere before the focus point. Also aberra- tions have an influence on inhomogeneous illumination. The approximation above gives lowerγAfactors than in reality. The area factor refers to the intensity of the beam at the location of the focussing element.

2.2 Reflectivity and Absorption

2.2.1 Theory

A theory for reflection on surfaces covered with adsorbates is very important for the design of an infrared reflection absorption (IRAS) setup especially when metal oxide single crystals are probed. Compared to metals the reflectivity of these samples is low, implicating reduced intensity, on the detector. Furthermore, the amount of adsorbed molecules covering the surface is in the order of one monolayer or less, leading to small absorption. These difficulties required us to take special care about the throughput of the optical system and choosing the correct angle of incidence on the sample. The main principle of this technique is the measurement of infrared radiation reflected on a sample (see chapter 1).

To describe the reflection on a bare surface, the Fresnel equations can be used [21]. These formulas describe the transmission and reflection of electromagnetic radiation on a flat interface between two different media for a given the polarisa- tion. Molecules adsorbed on a surface require a different theory to describe the reflection and transmission. One way to include the influence of the adsorbate on the reflection is the use of a three-layer model described in reference [22, 23]. In this thesis, the equations for the reflectivity of the system developed by Langreth

[8] are used. Here, a macroscopic approach is used to derive generalized Fresnel reflectivity formulas with s and p polarisation including a polarisable surface. In the following part calculations for the reflectivity of the surface with and without adsorbate based on [8] will be shown. This will give a set of design criteria for the optical system mostly considering the incidence angle of the radiation on the surface, to gain maximum sensitivity and signal in the final setup.

The performed calculations take p polarisation and s polarisation of the incidence radiation into account. The p polarisation splits in two different components of the electric field. In the first case the dipole moment is oriented parallel to the surface.

In the second case the dipole moment is oriented orthogonal to the surface. The adsorbate is modelled with the polarisability α. For polarisability calculations a Lorentzian oscillator model also used in reference [24] was utilised. To model the polarisability

α❼ν˜➁ αe✔ αv

1✏ ❼ν˜⑦ν˜0➁ ❻ ❼ν˜⑦ν˜0✔i❻γ⑦ν˜0➁ (2.4) was used for the calculations. αe is the electronic polarisability, αv describes the vibrational polarisability, ˜ν₀ is the resonance wave number, ˜ν is the wave number (equation 1.1) and γ characterises the line width.

As a result, the formulas for the fractional change ∆R⑦R, dependent on the inci- dence angle Θ (measured to the normal of the surface), including s and p polari- sation can be written as

∆Rs

R0s ✏16πkcos❼Θ➁ Im❿Nsα_Õ

ǫ✏1➄ and (2.5)

∆Rp

R0p

16πkcos❼Θ➁ Im➀ǫ²❼Nsα_Ù➁ ✏ ❼ǫζ²✔ǫ✏1➁❼Nsα_Õ➁

❼ǫ✏1➁❼ǫζ²✏1➁ ➅ . (2.6) Ns is the surface density of oscillators per unit area. α_Ù is the polarisability perpendicular to the surface andα_Õ is the polarisability parallel to the surface. In this work, ∆Rσ is defined differently compared to the work of Langreth [8]:

∆Rσ R0σ✏Rσ . (2.7)

Equation 2.7 is a common definition of ∆Rσ used in the IRAS-community, where

∆Rσ ❆0 indicates absorption. R0σ describes the reflectivity of the surface without adsorbate which is nothing else than the Fresnel reflectivity (see equations 2.12 and 2.13). Rσ is the reflectivity with adsorbate on the surface. σ indicates the polarisation of the radiation (s or p), and

ǫ ǫ^➐✔i ǫ^➐➐ (2.8)

16

describes the dimensionless dielectric constant of the reflecting substrate. For absorbing media, ǫ is complex. The absolute value of the wave vector is:

k 2π⑦λ . (2.9)

ζ and ζ^➐ are defined as follows:

ζ cot❼Θ➁ , (2.10)

ζ^➐ ➺

ǫ❼ζ²✔1➁ ✏1 . (2.11)

The differential reflectivity ∆R (see equation 2.7) describes the sensitivity of the reflection system. To calculate values for ∆R the equations 2.5 and 2.6 were multiplied with the reflected intensity according to the Fresnel equations. The Fresnel reflectivity R0s and R0p are given for s polarisation by

r0s

1✏ζ^➐⑦ζ

1✔ζ^➐⑦ζ (2.12)

and for p polarisation

r0p

1✏ζ^➐⑦❼ǫζ➁

1✔ζ^➐⑦❼ǫζ➁ (2.13)

according to [8]. The reflection factor of the intensity is given by

R0σ ❙r0σ❙² . (2.14)

Equations 2.5 and 2.6 allow the prediction of the band shape which will be covered later. Equation 2.6 reproduces the band shape for 0.1 ML of CO a Pt(111) in good agreement with the band shape reported in [24] using the same values.

The Brewster angle Θ_B is of great importance for the layout of the optical system.

Here, the absorption for the p-polarised radiation is the highest and also the sign of the absorption changes around this angle (see section 2.2.2) for dielectric sub- strates. Due to the minimum in the reflection Rp a measurement at Θ ΘB leads to a loss of the signal in the final spectra. The most common definition of ΘB is

ΘB arctan❼n^➐⑦n➁ . (2.15)

ndescribes the refractive index of the incidence medium andn^➐that of the medium where the radiation is reflected on. The relationship between ǫ and n is given by

n ➸

ǫ . (2.16)

Equation 2.15 is only valid ifnandn^➐are real and for this case the reflectivity of the p-polarised radiation is zero at ΘB. Formula 2.15 can be derived from Snell’s law, which is valid only for realn andn^➐. These types of media show a minimum of the reflectivity Rp. In the following section the Brewster angle, in some literature also called pseudo Brewster angle, in case of complex ǫ, is determined as the minimum of Rp and calculated numerically from equation 2.13.

2.2.2 Analysis of the reflectivity

The knowledge of how a surface of a particular material with an adsorbate on it interacts with infrared radiation is very important for the design of a IRAS setup.

One major output of the following plots is the optimum angle of incidence. This optimum angle depends on the sensitivity ∆R, reflectivity R, polarisation, orien- tation of the dipole moment on the surface, and the substrate where the infrared radiation is reflected on. One has to consider that the equations in chapter 2.2.1 describe only fields hitting the surface under a fixed angle Θ. In reality, radiation hits the surface under Θ✖κwhereκdescribes the half opening angle of the focused beam. In section 2.1 it was shown that there is an optimum or limiting value forκ for maximum intensity at the sample. Therefore considerations of the acceptable half opening angleκof the beam incident on the surface are made. Another factor playing a role in the design of the reflection setup, which is also important for the interpretation of spectra, is the band shape of the absorption bands shown in the following part.

For the diagrams presented below, arbitrary values were used to calculate the polarisability with the model of the Lorentzian oscillator (see equation 2.4). It is important to note that only small values for the polarisability are relevant. The values αe, αv, ν0 and γ were assumed to be αe 0,αv 0.3 ų and γ 5 cm^✏1. Ns

was assumed to be 0.6✕10¹³cm^✏2. To consider the orientation of the dipole mo- ment in the calculationsα_Õ αandα_Ù 0 were used for parallel orientation of the dipole. For the perpendicular orientation of the dipole moment the polarisability was set to α_Õ 0 and α_Ù α.

To get a solid base for the optical layout different materials were chosen. Ma- terials with an absorption coefficient ǫ^➐➐ close to zero are represented by SiO₂, TiO2 and Si. The absorbing medium is Fe3O4. The values for ǫ and n are shown in table 2.1.

Sensitivity

The sensitivity ∆R R0 ✏R describes the difference in reflectivity between the bare surface and the covered surface. Due to the adsorbate on the surface, the re- flectivity of the substrate changes compared to the bare surface. To get maximum signal, it is advisable to measure at incidence angles where ∆R has a maximum or minimum. Optimising ∆R is therefore an important criterion for the design of an IRAS system. Using different sample materials will most probably require a compromise between the different materials concerning the optimum incidence angle.

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material ˜ν [cm^✏1] ǫ ǫ^➐✔ǫ^➐➐ i n SiO₂ 1700 1.6✔0.004 i 1.3✔0.001 i

3500 2✔0.0003 i 1.4✔0.0001 i TiO2

1700 4.66✔0.33 i 2.2✔0.006 i 3500 5.64✔0.002 i 2.4✔0.0005 i

Si 1700 11.7 3.4

3500 11.8 3.4

Fe3O4

1700 10.4✔8.7 i 3.5✔1.3 i 3500 7.3✔7 i 2.9✔1.2 i

Table 2.1:ǫ and n of different materials which are used for the calculations. The values are form reference [25].

The following plots show different signs of ∆R depending on the polarisation of the radiation and the orientation of the dipole moment µ. In case of p-polarised radiation, ∆R changes its sign close to the Brewster angle ΘB. The sign of ∆R determines the direction of the peaks in the final spectra. Values less than zero predict negative peaks in the infrared spectra, positive values indicate absorption bands. If ∆R is negative, the reflectivity R is increased by the adsorbate com- pared to the reflectivity R0 of the bare surface. For positive values of ∆R, the reflectivity R0 is bigger thanR due to the adsorbate on the surface. Reflectivity for the s-polarised radiation shows always an increase of the reflectivity due to the adsorbate, where R0❅R.

The calculations in this section are based on the equations from chapter 2.2.1.

Each of the figures 2.4, 2.6 and 2.7 show two plots of ∆R versus incidence angle Θ at different wavenumbers. The solid line (curve (1)) shows the behaviour of p-polarised radiation with the dipole moment oriented parallel (indicated by Õ) to the surface. The p polarisation with the dipole moment oriented perpendicular to the surface is represented by the dashed line. The dotted curve (curve (3)) stands for s polarisation and is only influenced by the dipole moment, oriented parallel to the surface.

(a) (b)

Figure 2.4: Sensitivity ∆R on TiO₂ as function of the incidence angle Θ for differ- ent wavenumbers. For p-polarised light, the plots show the signal from molecules with the dipole moment oriented Õ and Ù to the surface.

For s-polarised light, the plots show the signal from molecules with the dipole moment oriented Õto the surface. Table 2.1 shows the material parameters used to calculate the curves.

Figure 2.4 (a) shows the sensitivity ∆R for TiO2 with an adsorbate at ˜ν 1700 cm^✏1. Both ˜ν and ˜ν0 were set to be equal. Here, ∆R has a peak in good approximation. In case of Fe3O4 it is not the peak position any more. This case will be covered later. The polarisability of the adsorbate was calculated with the values stated in section 2.2.2. As one can see, the highest sensitivity is shown by curve (2). Here, the dipole moment µ is oriented perpendicular to the surface, excited by p-polarised radiation. This minimum, with a value of about -7.6✕10^✏5 is located at 83.8 °. Following the curve from this minimum to lower angles of incident Θ, a zero crossing appears. This is the point of band inversion where the absorption band changes its sign. The inversion takes place at the Brewster angle Θ_B of 65 °. Continuing to lower Θ’s, curve p-polÙshows a maximum of 2.1✕10^✏5, at 47.6 °. At both extreme values of Θ (0 ° and 90 °) the sensitivity ∆Rp✏polÙ is zero.

∆R for s and p polarisation withµ oriented parallel to the surface (curve (1) and (2)), is about ✏1.2✕10^✏5 at Θ 0. The curve for p-pol Õ increases with increas- ing incidence angle, again crossing zero at the same Brewster angle, and reaches a maximum of 1.3✕10^✏5 at Θ 83.7 °. This is very similar to the p-polarised radiation with the dipole perpendicular to the surface, but opposite sign, and less signal❙∆R❙: The factor between ΘpÙmin and ΘpÕmax is✏5.6. In all three cases, ∆R is zero at Θ = 90 °. The values for the s polarisation in figure 2.4 (a) are always negative, with a minimum of ✏1.6✕10^✏5 at Θ 60.4 °. The values of ∆R and the important values of Θ for p polarised radiation can be found in table 2.2 and 2.3.

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Figure 2.4 (b) shows the sensitivity on TiO2 for a wavenumber of 3500 cm^✏1. Here, the sensitiviy values increase by an average factor of 4.4 compared to fig- ure 2.4 (a). Due to the strong dispersion the Brewster angle shifts by about 2 ° to 67.1 °. Also the angles of the extrema of ∆R are shifted. The maximum of 1.1✕10^✏4 and the minimum of ✏3.5✕10^✏4 for p-pol Ù are located at incidence angles of 49.2 ° and 84.3 °, respectively. The shift of the minimum above the zero crossing is, with 0.4 °, smaller than the shift of the maximum of about 1.7 ° below the zero crossing. A similar behaviour is also shown by p-pol Õand s-pol Õ. The maximum of p-pol Õ above the zero crossing is shifted by✔0.5 ° to 84.1 °, and the minimum of s-pol Õshifts by -4.5 ° to 56.2 °.

Figure 2.5: Reflectivity of TiO2 at 1700 cm^✏1 for s and p polarisation plotted over the incidence angle Θ.

Figure 2.5 shows the Fresnel reflectivity on TiO2 at ˜ν 1700 cm^✏1. Both reflec- tivities, start at about Rs Rp 0.13. Rs increases steadily to 1 at Θ 90 °.

P polarised radiation shows a different behaviour. Rp first decreases to 0 at the Brewster angle of 65.1 °. After ΘB, the reflectivity increases dramatically to 1 at Θ 90 °. The effect of the adsorbate on the reflectivity is in the order of 10^✏4 and not visible in the plot.

Figure 2.6 (a) shows the sensitivity ∆R for SiO₂ at ˜ν = 1700 cm^✏1 with an adsor- bate. The refractive index nSiO2 is smaller than nT iO2. This results in a smaller Brewster angle of 52 ° for SiO2. On TiO2, the Brewster angle ΘB is 65 °. Also the maximum sensitivity at grazing angle shifts to lower angles of θ 82.7 ° on SiO₂ compared to TiO2. The maximum and minimum values for ∆Rp can be seen in table 2.2.

(a) (b)

Figure 2.6: Sensitivity ∆Ron SiO2as function of the incidence angle Θ for different wavenumbers. For p-polarised light, the plots show the signal from molecules with the dipole moment oriented Õ and Ù to the surface.

For s-polarised light, the plots show the signal from molecules with the dipole moment oriented Õto the surface. Table 2.1 shows the material parameters used to calculate the curves.

(a) (b)

Figure 2.7: Sensitivity ∆R on Si as function of the incidence angle Θ for different wavenumbers. For p-polarised light, the plots show the signal from molecules with the dipole moment oriented Õ and Ù to the surface.

For s-polarised light, the plots show the signal from molecules with the dipole moment oriented Õto the surface. Table 2.1 shows the material parameters used to calculate the curves.

Figure 2.7 shows the same calculation for silicon. Here, the index of refraction nSi 3.4 is higher than that of SiO2 and TiO2. Both frequencies are to small to excite electron-hole pairs in Si (hν ❅ 1.12eV). Therefore it is an ideal dielectric.

22

The most obvious difference one can see, is the ratio between the minimum of the p polarisation, with the dipole oriented perpendicular to the surface, and the maximum of p-polÕ(∆RpÙmin/∆RpÕmax). Here, almost a ratio of✏13 is in between the two extreme values. Whereas for TiO2, the factor is about✏6.5, which is almost half. For SiO2 the ratio of about ✏4 is even less than for TiO2 or Si. In contrast to the two materials, TiO2 and SiO2, the Brewster angle and the points of the extreme values do not shift significantly due to the almost frequency-independent dielectric constant of Si. The important values, for the layout of the system, are ΘpÕmax = 85.7 °, ΘpÙmin = 85.8 ° and the Brewster angle ΘB 73.7 °. The values for ∆RpÙmax, ∆RpÙmin and RpÕmax can be found in table 2.2. The maximum of curve (2) in the figures 2.7 (a) and (b) is ΘpÙmax = 55.5 °. The reflectivity of Si at

˜

ν 1700 cm^✏1 is about 0.3 at an incidence angle of 0 °.

For ǫ tending to infinity the surface selection rules have to be considered.

(a) (b)

Figure 2.8: Sensitivity ∆R on Fe3O4 as function of the incidence angle Θ for dif- ferent wavenumbers. For p-polarised light, the plots show the signal from molecules with the dipole moment oriented Õ and Ù to the sur- face. For s-polarised light, the plots show the signal from molecules with the dipole moment oriented Õto the surface. Table 2.1 shows the material parameters used to calculate the curves.

Up to this point, substrates, where the imaginary part ofǫ tends to zero, were dis- cussed. The aim of the infrared reflection system is to measure also on substrates withǫ^➐➐①0. Here, the absorbing material Fe3O4 was chosen as substrate for model calculations.

Figure 2.8 shows the sensitivity, plotted over the incidence angle Θ, for wavenum- bers of (a) ˜ν ν˜0 1700 cm^✏1 and (b) ˜ν ν˜0 3500 cm^✏1. Here, the peak position of ∆R does not match with the resonance frequency any more (see figure 2.14 (b)).

Each plot considers s and p polarisation and the orientation of the dipole moment parallel (Õ) or perpendicular (Ù) to the surface. Figure 2.8 is very similar to figure 2.7 (Si). Both materials feature a ǫ^➐ of about 11. The most obvious difference to graph 2.7 is, that the zero crossing of the two curves for p polarisation is not at the same angle any more. This is connected to the different behaviour of RpÙ and RpÕ

around the Brewster angle. Here, the reflectivityRp, shown in figure 2.9, does not reach zero any more at ΘB, due to the nonzeroǫ^➐➐. The zero crossing in figure 2.8 (a) for p-pol Ùis located at about 76.4 °. ΘpÕmax = 85.7 ° and ΘpÙmin = 86 ° with values of 8.8✕10^✏6 and -9.7✕10^✏5, respectively. The maximum of curve (2) in figure 2.8 (a) is ΘpÙmax = 57.8 °. The angle values of figure 2.8 (b) are very similar again with higher shifts for angles below the Brewster angle. ΘpÙmin decreases from 86 ° to 85.5 °.

Figure 2.9 shows the reflectivity on Fe3O4 for ˜ν = 1700 cm^✏1. The reflectivity for both polarisations at Θ = 0 ° is 0.36. Rs increases with increasing angle of in- cidence to 1. For p polarisation, the reflectivity decreases to a minimum of about 0.027 at 74.7 °. After the minimum Rp increases to 1. For ˜ν = 3500 cm^✏1 the (pseudo-) Brewster angle shifts to 72.3 °, compared to 74.7 ° at ˜ν = 1700 cm^✏1. Here, R0p is 0.031.

Figure 2.9: Reflectivity of Fe3O4 at 1700 cm^✏1 for s and p polarisation plotted over the incidence angle Θ.

For the layout of the optical setup, the most important factor from the discussion above is the sensitivity for p-polarised radiation and perpendicular orientation of the dipole moment. For all of the previous discussed materials, it shows the highest sensitivity values.

24

Fe3O4

Si TiO2

SiO2

Figure 2.10: Comparison of the sensitivity ∆R for perpendicular dipole moment orientation at 1700 cm^✏1.

material ∆R_pÙmax ∆R_pÙmin ∆R_pÕmax SiO2

1700 cm^✏1 3.2✕10^✏6 ✏6.3✕10^✏5 1.6✕10^✏5 3500 cm^✏1 2.3✕10^✏5 ✏2.6✕10^✏4 6.8✕10^✏5

∆R₃₅₀₀/∆R₁₇₀₀ 7.1 4.2 4.3 TiO2

1700 cm^✏1 2.1✕10^✏5 ✏7.6✕10^✏5 1.3✕10^✏5 3500 cm^✏1 1.1✕10^✏4 ✏3.5✕10^✏4 5.1✕10^✏5

∆R₃₅₀₀/∆R₁₇₀₀ 5.4 4.6 3.9 Si

1700 cm^✏1 5.8✕10^✏5 ✏1.1✕10^✏4 8.8✕10^✏6 3500 cm^✏1 2.5✕10^✏4 ✏4.8✕10^✏4 3.7✕10^✏5

∆R₃₅₀₀/∆R₁₇₀₀ 4.3 4.3 4.2 Fe₃O₄

1700 cm^✏1 7.4✕10^✏5 ✏9.7✕10^✏5 8.8✕10^✏6 3500 cm^✏1 2.4✕10^✏4 ✏3.4✕10^✏4 4.3✕10^✏5

∆R₃₅₀₀/∆R₁₇₀₀ 3.3 3.5 4.9

Table 2.2: Maximum and minimum values of the sensitivity ∆R for p-polarisation and different dipole moment orientations.

Figure 2.10 shows the comparison of the sensitivity for TiO2, Si, SiO2and Fe3O4

with p-polarised radiation and a perpendicular dipole moment orientation at a wavenumber of ˜ν = 1700 cm^✏1. Here, the minima are all located between 83.8 ° (SiO2) and 86 ° (Fe3O4) (see table 2.2 and 2.3). Another important limitation is set by the Brewster angle. The range of ΘBis between 59.5 ° (SiO2) and 74.7 ° (Fe3O4).

The reflectivity at the Brewster angle and around ΘB is very small causing a low intensity on the detector, leading to low signal-to-noise ratios. Another factor is the

sign change at the zero crossing. Overshooting the zero crossing will decrease the signal again. So the lower limit for Θ is set by the highest value of the zero crossing at an angle of 76.4 ° observed on Fe3O4. All these criteria together result in a cone angleκ☎ ✖7 ° of the focussing optics, emphasising rays of higher angle of incidence where the substrates show higher reflectivity for p polarisation. In combination with the results of section 2.1 (κ☎ ✖19 °) this will require a compromise between intensity on the signal area and angle of incidence for better reflectivity.

material ΘB [°] ΘpÙmax [°] ΘpÙmin [°] Θ_pÕmax [°]

SiO2

1700 cm^✏1 52 38.5 82.8 82.2

3500 cm^✏1 54.9 40.3 82.6 82

Θ3500✏Θ1700 2.9 1.8 -0.19 -0.07 TiO2

1700 cm^✏1 65.1 47.6 83.8 83.7

3500 cm^✏1 67.2 49.3 84.3 84.1

Θ₃₅₀₀✏Θ₁₇₀₀ 2 1.7 0.4 0.5

Si

1700 cm^✏1 73.7 55.5 85.8 85.7

3500 cm^✏1 73.8 55.6 85.8 85.7

Θ₃₅₀₀✏Θ₁₇₀₀ 0.06 0.07 0.02 0.02 Fe₃O₄

1700 cm^✏1 74.7 57.9 86 85.7

3500 cm^✏1 72.3 55.5 85.5 85

Θ3500✏Θ1700 -2.4 -2.3 -0.6 -0.7

Table 2.3: Brewster angle ΘB, maximum and minimum values of the incidence angle Θ for p-polarisation and different dipole moment orientations.

Fe3O4

Si

TiO² SiO2

Figure 2.11: Comparison of the sensitivity ∆R multiplied by cos❼Θ➁ for perpen- dicular dipole moment orientation at 1700 cm^✏1.

26

Figure 2.11 shows the sensitivity ∆R multiplied with cos❼Θ➁. The factor cos❼Θ➁ takes the intensity loss due to grazing incidence into account. Here, Fe3O4 and Si show higher sensitivity at lower angles of incidence than above the Brewster angle. The exact maximum and minimum values for ∆R cos❼Θ➁ and Θ can be seen in table 2.4. Although figure Figure 2.11 shows benefits for the sensitivity on Fe3O4, Si and TiO2 at an incidence angle of about 40 °, this case was not further considered due to the mechanical restrictions in the chamber requiring a grazing incidence of the beam on the sample.

material ∆R_pÙmax cos❼Θ➁ Θmax[°] ∆R_pÙmin cos❼Θ➁ Θmin[°] zero crossing [°]

SiO2 2.5✕10^✏6 36.3 ✏1.1✕10^✏5 76 52

TiO2 1.5✕10^✏5 43 ✏1.2✕10^✏5 78.5 65.2

Si 3.6✕10^✏5 48.3 ✏1.2✕10^✏5 82 73.7

Fe3O4 4.4✕10^✏5 49.7 ✏9.2✕10^✏6 83 76.4

Table 2.4: Maximum and minimum values of ∆R cos❼Θ➁, Θ and the zero cross- ing for ˜ν = 1700 cm^✏1, p-polarisation and perpendicular dipole moment orientations.

Based on the figures 2.4 and 2.6–2.8, one can see that the sensitivity increases for higher wavenumbers compared to lower wavenumbers. Most of the ratios

∆R3500/∆R1700 in table 2.2 are close to the squared ratio of the wavenumbers,

❼3500⑦1700➁² 4.2. The most significant deviation is shown by ∆RpÙmax of SiO2

(see table 2.2). This is related to the rather large change of the dielectric constant with the wavenumber (see table 2.1). The influence of changing ǫ has a smaller effect on ∆R above the Brewster angle than below. The maximum and minimum angles above the Brewster angle are influenced less than the angles below the Brewster angle and ΘB itself.