Here I am, finally writing the acknowledgments of my thesis. The first person I am deeply indebted to is my theses (both MSc and PhD) supervisor, Jürgen Härtwig. The fact that, even during the more stressful period, you have been able to sustain me with the right words and a smile is priceless! It has been a real pleasure to work with you during almost five years and to travel in four different continents together. Danke schön Herr Härtwig.

Thanks to Prof. Simon Connell, with whom, after creating a nice professional relationship, I am also managing to create a great familiar one. What I found impressive about Simon is the fact that, whatever you chat with him about (could be nuclear physics, diamond synthesis, the rules of rugby, how to prepare a perfect braai or fixing a broken car) he is always able to engross you in an amazing way. I feel very lucky that I will be able to continue to listen to you for many years to come! But one thing is sure, I will never ever think that rugby is more fun to watch than soccer Another person at the ESRF which I have rarely seen not smiling is José Baruchel. Thank you very much José for many things: for allowing me to come to the ESRF as a Master student, for finding some time in your very busy schedule to discuss the results of our experiments on the ferroelectric crystals, for accepting to be the president of this jury and especially for the nice words you had for me during our last meeting in your office and that I will never forget.

The other part of the joint collaboration between my university in Turin and the ESRF, which allowed me to come to Grenoble as a Master student, is due to Prof. Raffaello Garfagnini. Thank you very much for allowing me to start this adventure in Grenoble, which changed the rest of my life so deeply.

The other Professor at Turin University which I am indebted to is Prof.

Carlo Lamberti. Unfortunately we didn't succeed in spending a lot of time together, but every time we met you have been able to, in a balanced way, compliment me for the results achieved and motivated me for the future challenges.

I would like to thank Prof. Brian Tanner and Prof. Tilo Baumbach for accepting the task of reviewing my thesis. Thanks also to Pierre Bastie for accepting to be part of the thesis jury on such short notice.

A special thanks goes to Tamzin Lafford. As I have told you Tamzin, you have been a kind of Marry Poppins for me. You arrived with many new ideas which improved considerably the working conditions at the beamline.

Moreover, your dedication in helping me with essentially all my tasks has been fundamental for the final achievement of my thesis, not to mention your help in correcting the manuscript. But even more than this, how can I thank you for the patience which you showed listening to all the rehearsals of my presentations, the wonderful advice you gave me for my job interviews and, icing on the cake, your great muffins and delicious cakes you brought us for our delight? Well, I try with this THANK YOU TAMZIN Another very important person has been Claudio Ferrero. The amount of nights we spent together, working on convolution and deconvolution code has been enormous. But during all this time, you always have managed to nicely switch between work and pleasant discussions about many different subjects. Thanks Claudio, even though I will never be able to talk or understand the dialect from my home town, il piemontese. This five years spent in Grenoble would not have been the same without wonderful friends and colleagues, which I will briefly list here: Hedi, Harry, Rafael, Aymeric, Peter, Michael, David, Sebastien, Xavier, Federico, Genziana, Pierre-Jean, Valentina, Iulia, Min, Camille, Ali, Eliza, Grazie ai miei genitori, che con il loro supporto morale ed economico mi hanno permesso di arrivare fino a qua ed alle mie sorelle, per essere sempre disponibili ad ascoltarmi ogniqualvolta ho avuto bisogno di qualcuno a cui parlare.

And last, but not least, to my wonderful wife Bibi, to whom this thesis is dedicated. You have been there for me during the hardest period of my thesis, the writing-up. Unfortunately, I will not be there for you in the same hard period, and of that I am deeply sorry; but I hope time will show us that the decision I took was the good one. To conclude with a happy thought, I remember that in my Master thesis acknowledgements, I wrote that I found my diamond in South Africa. Well, I am happy to say that this diamond will now be with me forever and, with God's will, it will allow us to generate even more little diamonds in the future Chapter 1

Introduction

The exceptional properties of synchrotron light sources have been and increasingly are exploited in very different disciplines, from archaeology to chemistry, from material science to biology, from medicine to physics. Among these properties it is important to mention the high brilliance, continuum spectrum, high degree of polarization, time structure, small source size and divergence of the beam, the last resulting in a high transversal coherence of the produced radiation. This high transversal coherence of the synchrotron sources has permitted the development of new techniques [START_REF] Lengeler | Coherence in X-ray physics[END_REF], e.g. phase contrast imaging [2, 3], X-ray photon correlation spectroscopy [4,5] and coherent X-ray diffraction imaging (CXDI) [6,7].

The importance of X-ray coherence is striking if considering that the words coherence or coherent appear approximately 200 times in the so-called "ESRF Science and Technology Programme -2008 -2017" 1 , where the future challenges and necessary upgrades of the ESRF are presented. Therefore, it is of primary importance for many applications that the optical elements preserve this property, disturbing as little as possible the wavefront. At the same time it is fundamental to have methods able to characterize the coherence of the radiation.

Many different methods to characterize the coherence of hard X-rays have been investigated and they make use of, e.g. a wedge-shaped silicon crystal and measuring the visibility of the Pendellösung fringes, [8], a Fresnel biprism composed of two diamonds [START_REF] Lang | Production of synchrotron X-ray biprism interference patterns with control of fringe spacing[END_REF], a round fiber or slits [START_REF] Kohn | Direct Measurement of Transverse Coherence Length of Hard X Rays from Interference Fringes[END_REF], a system of double slits [START_REF] Leitenberger | Interferometric measurements with hard X-rays using a double slit[END_REF], a double pinhole [12] [17] double-grating interferometer [13]. We have been using two other methods, one based on the partial Talbot effect, [14] and another based on a system of bi-lenses [15].

With the new fourth generation sources, e.g. the X-ray free-electron lasers (XFEL), the coherence of the beam will be pushed even further since the radiation produced will be transversally almost fully coherent, reaching a degree of transverse coherence of 95%, [16]. At the same time, the heat-load on the optical elements will also increase considerably, especially for machines such as the European XFEL. Different solutions are proposed to overcome the heat-load problem at the E-XFEL. One is to use mirror optics which will allow to de-focus the beam and to have an enlarged footprint on the monochromator, therefore reducing the heat load. Another solution, in the case when beam expansion is not possible or not desirable, is to use diamond crystals.

In fact, among all the outstanding properties that diamond exhibits, three in particular make this material ideal for use as optical elements in light sources such as synchrotrons and X-ray free-electron lasers (XFEL):

• thermal conductivity at room temperature (23 Wcm -1 K -1 );

• low expansion coefficient at room temperature 1.1 × 10 -6 K -1 ;

• low linear absorption coefficient for X-rays (7.5 cm -1 at 8 keV);

The combination of these three properties is summarized in a figure of merit, obtained dividing the conductivity by the absorption and expansion coefficients. In tab.1.1 are compared the thermal properties of the most used materials for X-ray optical elements, showing that diamond is by orders of magnitudes the best material.

There are three principal issues that could impede the success of diamond as X-ray optical elements: the crystal perfection, the size and the processing technologies. Natural diamonds are not suitable for two reasons: they possess a higher number of defects and greater strain due to the complicated growth process inside the Earth; and the cost is also prohibitive, since they are mostly used in jewellery. In contrast, synthetic diamonds seem to have the necessary characteristics.

Recently, two more possible applications of diamond crystals have been reported. The first is to use diamonds as element in an X-ray free-electron laser oscillators (XFELO) [18], by using high X-ray reflectivity of diamonds. The second is to use a diamond crystal between two undulators for generating highly monochromatic X-rays through the so-called self-seeding effect, [19].

My thesis work will consist essentially of three parts:

• characterization of high quality crystals foreseen as X-ray optical elements, in particular those made out of diamond;

• study of the temperature dependent behavior of the domain in periodically poled ferroelectrics crystals, which are possible only thanks to the coherence of the beam • diamond related results, obtained in the framework of external collaborations More specifically, in chapter 2, I will introduce the necessary theoretical basis of synchrotron radiation generation, some notions of the kinematical and dynamical theories of X-ray diffraction and an introduction to the formalism describing coherence.

The experimental techniques used will be presented in chapter 3: high resolution Xray diffraction, X-ray diffraction topography, grazing incidence diffraction and X-ray reflectivity.

In chapter 4, I will show the results obtained with diamond crystals of different crystalline quality using the characterization techniques presented.

Chapter 5 will be dedicated to the results obtained from polishing tests of high quality silicon crystals and beryllium windows.

In chapter 6 I will present an example of applications that make use of the transversal coherence of a synchrotron beam: the investigation by coherent X-ray section topography of ferroelectric domain behaviour as a function of temperature in periodically-poled Rb:KTP.

INTRODUCTION

Chapter 7 will be a short introduction to studies which are not directly related with the main topic of my thesis but that have in common diamond as the material and X-ray topography as the characterization technique used.

Finally, in chapter 8 I will draw conclusions and comment on the outlook for the future.

Chapter 2

Theoretical basis

In this chapter, I introduce the theoretical basis necessary to understand the experimental techniques used and the data obtained in the rest of the thesis. Firstly, some of the properties of synchrotron radiation are presented, together with a historical overview.

Afterwards, the fundamentals of kinematical and dynamical theories of X-rays diffraction are summarized. Finally, the necessary formalism to describe the phenomenon of coherence is introduced.

Synchrotron radiation

Short historical overview on synchrotron radiation

Synchrotron radiation is an electromagnetic wave generated by the acceleration of relativistic charged particles (moving at relativistic velocity) on curved trajectories in magnetic fields. An historical review of synchrotron radiation is available in [START_REF] Blewett | Synchrotron Radiation -Early History[END_REF]. I present here the more important events.

In 1898, Alfred Liénard published a paper in'L' Éclairage Électrique' entitled 'Champ électrique et magnétique produit par une charge électrique concentrée en un point et animée d'un mouvement quelconque' where he derived, directly from Maxwell's equations, the so-called Liénard-Wiechert potentials. By using this potential, Liénard was able to calculate the energy lost by electrons moving in a circular trajectory in a homogeneous magnetic field. This theory can be considered the basis of synchrotron radiation, which was experimentally discovered about 50 years later.

THEORETICAL BASIS

The first observation, fig. 2.1, of radiation emitted by relativistic charged particles, took place on the 24th April 1947, at the General Electric Research Laboratory, where a 70

MeV electron synchrotron was in use [START_REF] Elder | Radiation from Electrons in a Synchrotron[END_REF]; the radiation was thus called synchrotron radiation. The bright spot visible in the picture (visible thanks to a transparent vacuum chamber and a mirror) was bluish-white when the energy of the electrons was 70 MeV, and changed to yellow at 40 MeV, red and weaker at 30 MeV and disappeared below 20

MeV. This phenomenon is related to the fact that the critical energy of the radiation is proportional to the energy of the electron beam. Initially, this radiation, produced by the bending magnets (magnetic dipoles), was a nuisance for high energy physicists, because it limits the maximum energy achievable by the accelerated particles, especially for light particles such as electrons and positrons.

In fact, the first synchrotrons were designed exclusively for particle physics. Starting from the 1970's, a parasitic use of the synchrotron radiation began; those were the so-called first generation sources, e.g. SURF -USA, Frascati -Italy, INS -Japan, DESY -Germany, ACO -France.

The first storage ring designed for using synchrotron radiation, second generation source, was the Synchrotron Radiation Source in Daresbury -UK, followed by oth-

Synchrotron radiation

ers (NSLS -USA, Photon Factory -Japan, BESSY -Germany, LURE -France).

The next step, that brought the synchrotron community towards the third generation machine, was the idea of adding, in the straight sections of the ring, so-called Insertion Devices (ID). Inside the insertion devices, the electrons (or positrons) follow a periodic undulating trajectory in the magnetic field produced by a number of permanent magnets with alternating pole directions. This results in emitted beams much more intense than that from a single magnetic dipole. The European Synchrotron Radiation Facility, France, with its 6.04 GeV storage ring, was the first third-generation high energy synchrotron in the world (1994), quickly followed by the 7 GeV Advanced Photon Source, USA in 1996, the 8 GeV Spring8 in Japan in 1997 and only recently (2009) by Petra III, Germany with an energy of 6 GeV. These synchrotrons are, up to now, the four high energy synchrotron radiation sources in the world.

The next step is the fourth generation sources: these are light sources which exceed the performance of third generation synchrotrons by at least one order of magnitude in an important parameter such as brightness, coherence or pulse duration [22].There are three general categories [START_REF] Talman | Accelerator x-ray sources[END_REF]:

Storage Ring Based [24], where very low emittance is achieved in a much bigger circumference and by increasing considerably the number of bending magnets [25]. The newest PETRA III synchrotron sits between the old third generation sources and the new fourth generation.

Energy Recovery Linac ERL which is also based on obtaining an electron beam with a very small emittance. This is achieved by having the electrons circulating only once into a transport loop where the different undulators for the production of X-rays are installed. In order to accelerate the electrons, a linear accelerator is used. The main problem with this scheme is the large amount of energy needed to accelerate the electrons, that are then damped after one circuit of the ring.

This difficulty has been overcome by the concept of energy recovery. After one circuit, the electrons enter the linac again, but this time out of phase with the RF cavities. In this way the electrons are decelerated by the linac and they return most of their energy to the cavities themselves [START_REF] Bilderback | Energy recovery linac (ERL) coherent hard x-ray sources[END_REF]. At the moment there are two main projects to build ERL for light sources: one is at Cornell University, USA

THEORETICAL BASIS

and the other is at the High Energy Accelerator Research Organization KEK, Japan.

X-ray Free Electron Laser XFEL which can be considered in a simplistic way as a very long undulator, [START_REF] Feldhaus | X-ray free-electron lasers[END_REF]. The photons produced inside this undulator will interact with the electrons themselves. This organizes the electrons inside the bunches into microbunches, that are periodically separated by a period length equal to the undulator fundamental harmonic. The bunched electrons produce coherent radiation. This effect is called Self-Amplified Spontaneous-Emission and is at the basis of X-ray lasing. The first such machine, the Linear Coherent Light Source, (LCLS), was inaugurated in 2009 in Stanford, USA, using part of the already existing linear accelerator. The next will be the Spring-8 Compact SASE Source, Japan in 2011, followed by the European-XFEL, Germany in 2015.

Synchrotron radiation properties

Some of the main properties of synchrotron radiation at a third generation light source are briefly described here. First of all, three quantities that are important for characterizing and comparing different synchrotron sources are introduced: flux, brightness and brilliance, [START_REF] Morrison | Potential opportunities for materials research at the advanced photon source[END_REF].

Flux is defined as the number of photons per second in 0.1% of the spectral bandwidth.

The brightness is the flux over the emission angle. The brilliance is the brightness divided by the source area. As a formula:

= # photons [s] [mrad 2 ] [mm 2 ] [0.1% bandwidth]
(2.1)

In order to have a very high brilliance, it is therefore important to have a very high flux, very well collimated, coming from a small source. All these properties are achieved at third generation synchrotrons.

The high flux derives directly from the total power radiated by an accelerated relativistic particle on a circular path:

P = 2 3 e 2 c R 2 β 4 γ 4 (2.2)
where R is the radius of the circular path, e the charge of the electron, c the speed of light, β is the ratio between the speed of the particle and the speed of light and γ is the

Synchrotron radiation

Lorentz factor. It increases for high speed particles (therefore, high energy synchrotron)

and for small radii of curvature. Remembering that γ = E m 0 c 2 , it is possible to see that the emitted power strongly decreases with the mass of the particle. This is the reason why only electrons and positrons are used in synchrotron light sources.

The high collimation of the photon beam is the result of the Lorentz transformation of the power emitted by accelerated relativistic particles. In the frame of the moving electron, the power distribution is the same as the pattern of an oscillating electric dipole. In the laboratory frame, the intensity distribution is transformed into a forwardpeaked distribution, with semi-opening angle ≈ The source size of the photon beam is given by the convolution of the sizes (σ x,y ) and divergences (σ x,y ) of the electron beam with the intrinsic radiation characteristics of a single particle. The emittance (defined as the product of size and divergence of the electron beam) is therefore an important parameter. In a synchrotron, the horizontal emittance is larger than the vertical emittance. This is due to larger betatron oscillations in the plane of the ring. These oscillations are caused by the fact that the electrons, by emitting photons, lose a part of their energy, therefore changing the orbit radius. The energy lost by the electrons is recovered in the Radio Frequency (RF)

THEORETICAL BASIS

cavities. The vertical emittance is produced by the small inaccuracy in the magnetic fields or alignment of the magnetic quadrupoles.

The emittance of a beam is proportional to the square of the energy of the stored electrons and to the third power of the bending angle (in the approximation of small bending angles), [START_REF] Raoux | Neutron and synchrotron radiation for condensed matter studies[END_REF]. This has important consequences, e.g. for the new light source Petra III. The storage ring used is bigger than any existing one, with a circumference of 2304m. Therefore the bending angles are much smaller and consequently the emittance is smaller, ≈ 1 [nm rad] compared to the 3 -3.9 [nm rad] values for Spring8 and the ESRF.

Moreover, the small source size and the possibility of having long source-to-sample distances play a major role also in another important property of third generation synchrotrons: i.e. the high degree of transversal coherence. This point will be discussed later in the chapter, 2.3.

As previously mentioned, the main characteristic of third generation synchrotron is the presence of insertion devices in the straight sections. This results in an increase in the brilliance of the radiation produced compared to that from a bending magnet.

The idea of using insertion devices in order to increase the brightness comes from theoretical work by Ginzburg in 1947, [START_REF] Ginzburg | On emission of micro waves and their absorption in air[END_REF] and Motz in 1951, [START_REF] Motz | Applications of the Radiation from Fast Electron Beams[END_REF]. In 1953 Motz and co-workers verified this idea experimentally , building an undulator that produced radiation with a wavelength varying from millimeters to hundreds of nm using a linear accelerator at Stanford University. The idea was to make the electron oscillate rapidly in the orbit plane using an array of permanent magnets. This device was inserted in the straight section of the Chasman-Green cell and for that reason they are called Insertion Devices (ID).

Two different types of ID are used: wigglers and undulators. In a wiggler, the amplitude of the oscillations is rather large and the cones of the emitted radiation do not superpose.

Therefore, the intensities of the radiation fields are added incoherently. This results in an increased photon flux by a factor 2N, where N is the number of magnetic dipoles. If the oscillations are small enough, then it is possible to add the radiation fields emitted by the electron coherently: this is the case for an undulator. To define what is meant by "small enough", a dimensionless parameter is used:

K = eB 0 λ u 2πmc

Synchrotron radiation

where λ u is the spatial period of the oscillations and B 0 is the maximum magnetic field.

For a wiggler K 1, whilst K ≈ 1 for an undulator. is the fact that the interference produced by adding coherently the electromagnetic fields gives a "quasi-monochromatic" wave (also called pink beam) and the presence of higher harmonics. This is just the opposite of what happens with a wiggler, where the incoherent sum gives a broad spectrum or white beam. There is no sharp border between these two devices. Depending on parameters there is a continuous transition between those limiting cases.

The very high brilliance is one of the main properties of synchrotron radiation. Just as

THEORETICAL BASIS

a term of comparison, the brilliance of an X-ray tube is about 10 orders of magnitude smaller than the brilliance for an undulator at the ESRF. Other important properties of the radiation produced at third generation sources are the high degree of polarization (σ-polarized in the plane of the orbit, slightly elliptical out of it), and the pulsed structure, allowing e.g. stroboscopic studies of acoustic vibrations in LiNbO 3 piezoelectric crystals, [START_REF] Zolotoyabko | Visualization of 10 µm surface acoustic waves by stroboscopic x-ray topography[END_REF].

X-ray diffraction

The first X-ray diffraction experiment was performed in 1912 by Friedrich, Knipping and Laue [START_REF] Friedrich | Interferenzerscheinungen bei Röntgenstrahlen[END_REF]. The idea that the X-ray would be diffracted by a crystal came to Laue during a meeting he had with Ewald [START_REF] Laue | Concerning the Detection of X-ray Interferences[END_REF], whose thesis subject was "To find the optical properties of an anisotropic arrangement of isotropic resonators". During this meeting von Laue asked Ewald: "what is the distance between such resonators". Ewald's answer was: "about one thousandth of the wavelength of light". From previous work done by Sommerfeld, the wavelength of the radiation was also supposed to be of that order of magnitude and Laue immediately realized that, if crystals were made of a periodic array of atoms, they should diffract 1 .

Two main theories exist in order to explain the diffraction of the X-ray by a crystal:

the kinematical and the dynamical theories of X-ray diffraction.

Kinematical theory of X-ray diffraction

The basis of the kinematical theory was given in the first paper written after the discovery of diffraction, [START_REF] Friedrich | Interferenzerscheinungen bei Röntgenstrahlen[END_REF]. It states that the amplitude diffracted by the crystal is simply the sum of the amplitudes of the waves diffracted by each atom, considering the phase shifts due to the different optical paths but neglecting the interaction of the incident wave with the crystal itself and is equivalent to the first Born approximation in scattering theory. This means that the amplitude of the incident wave is constant inside the crystal, which is a clear violation of the conservation of energy. Therefore

X-ray diffraction

the kinematical theory is valid only as an approximation; the relevant approximations will be shown later.

Following Authier and Malgrange [START_REF] Authier | Diffraction Physics[END_REF], we describe a crystal as a three dimensional, infinite, periodical array of electron densities ρ 0 (r) multiplied by a shape function y(r), that takes into account the finite size of the crystal (y(r) = 1 inside and 0 outside the crystal). The amplitude of the diffracted wave is:

A(∆K) = ∞ -∞ ρ 0 (r)y(r)e -2πi∆K•r dτ (2.3)
where ∆K is the diffraction vector defined as the difference between the incoming and

diffracted wavevectors 1 , ∆K = K h -K o .
The integration is over all space and dτ is a volume element. The amplitude A(∆K) can be considered as the Fourier transform of the electron density distribution. By defining the structure factor as:

F h = ∞ -∞ ρ 0 (r)e 2πih•r dτ (2.4)
with h being the reciprocal lattice vector, we can rewrite the amplitude as:

A(∆K) = 1 V h F h Y (∆K -h) (2.5)
where V is the volume of the unit cell, Y (∆K -h) is the Fourier transform of the shape of the crystal and the summation is over all the reciprocal lattice vectors h. It is important to notice that Y (∆K -h) is proportional to the volume of the crystal itself, i.e. for larger and larger crystals the diffracted amplitude will diverge. This is, of course, not possible since it violates the conservation of energy. The kinematical theory is therefore only a first order approximation, valid for thin crystals. In order to define what is thin, it is necessary to introduce a more general theory, i.e. the dynamical diffraction theory.

Dynamical theory of X-ray diffraction by a perfect crystal

The main purpose of this section it is not to give a full description of dynamical theory; many excellent books, e.g. [START_REF] Zachariasen | Theory of X-ray Diffraction in Crystal[END_REF][START_REF] Authier | Dynamical Theory of X-Ray Diffraction[END_REF], and papers, e.g. [START_REF] Batterman | Dynamical Diffraction of X Rays by Perfect Crystals[END_REF], are already available on the matter. The idea is simply to introduce a few concepts that will be used in the rest of 1 I follow the crystallographic definition of wavevector k = 1 λ , omitting the factor 2π

THEORETICAL BASIS

the thesis. There are three different classical approaches to the dynamical diffraction theory for perfect crystals (not taking into consideration the more general quantum field theoretical approach). The first is the one proposed by C. G. Darwin [START_REF] Darwin | The theory of X-ray reflexion[END_REF]. His theory assumes that at every lattice plane, two waves are generated: a partially transmitted wave S and a partially reflected wave T , similarly to the optics of classical light at plane interfaces. It is possible to express this by writing the following recurrent formulae [START_REF] Authier | Dynamical Theory of X-Ray Diffraction[END_REF] S n = iqT n + (1 -iq 0 )e -iφ S n+1 (2.6a)

T n+1 e iφ = (1 -iq 0 )T n -iqe -iφ S n+1 (2.6b)
where:

q = N d hkl λr e |F hkl | sin θ B φ = d hkl λ
with N is the number of scattering unit per unit volume, d hkl is the interplanar spacing of the lattice plane with Miller index hkl, λ the wavelength, r e the classical electron radius, F hkl the structure factor, θ B is the Bragg angle; q 0 and q follow the same expression as q but replacing F hkl with F 000 and F hhh , respectively.

The second theory was proposed by P.P. Ewald [START_REF] Ewald | Zur Begründung der Kristalloptik. III. Die Kristalloptik der Röntgenstrahlen[END_REF]. In his theory, Ewald, had a microscopic approach: he supposed that the crystalline lattice is formed of a threedimensional array of electric dipoles. When excited by an external electromagnetic field, each dipole radiates a spherical wave. These spherical waves radiated by each dipole will also excite the other dipoles, producing what Ewald called a wavelet. He then used Hertz vector potentials to describe the electromagnetic field inside the crystal.

I will not enter into details of this formulation.

The third and most commonly used formulation was made by M. von Laue [START_REF] Laue | Die dynamische Theorie der Röntgenstrahlinterferenzen in neuer Form[END_REF]. Laue used a macroscopic approach based on Maxwell's equations in a medium with a periodic dielectric constant. This is the theory followed by Batterman and Cole [START_REF] Batterman | Dynamical Diffraction of X Rays by Perfect Crystals[END_REF] and part of the formulation I will use follows from this paper. Ewald's and Laue's theories have been proved to be equivalent if the same physical model is chosen for the electromagnetic resonators, i.e. point dipoles or atoms [START_REF] Wagenfeld | Ewald's and von Laue's dynamical theories of X-ray diffraction[END_REF].

X-ray diffraction

The problem, as stated by Laue, is to solve Maxwell's equations in a medium with a periodic dielectric constant. The dielectric susceptibility in such a medium is defined as

χ(r) = 1 - r e λ 2 πV h F h e -2πih•r.
(2.8)

F h are the complex and wavelength dependent structure factors, defined as:

F h = n (f + ∆f + i∆f ) n e 2πih•rn (2.9)
where f is the atomic scattering factor, ∆f and ∆f are the Hönl corrections to the atomic scattering factor and are necessary to include the effects of anomalous dispersion, r n is the position of atom n and the summation is over all atoms in the unit cell.

In order to solve Maxwell's equations, we use as Ansatz for the electric and magnetic fields a sum of triply periodic plane waves (Bloch waves):

A = e 2πik 0 •r h A h e 2πih•r ≡ h A h e 2πik h •r (2.10)
where A can be one of the four fields (E, D, B or H). The two definitions are equivalent if we consider that k h = k o + h, but they give two explicit views:

• sum of plane waves with periodically modulated amplitude

• wavefield of infinite plane waves with different wavevectors k h

By solving Maxwell's equation with this Ansatz, we obtain for the electric field

E K 2 E m -k 2 m E m[⊥km] K 2 + n P mn χ m-n E n = 0 (2.11)
that is a homogeneous system of equations of infinite order. The m and n are two reciprocal lattice points, k m are the wavevectors in the crystal, K is the wavector in the vacuum, E m[⊥km] is the component of E m perpendicular to k m and P mn is the amplitude polarization factor defined as the scalar product of the polarization unit vector e m and e n .

The system of infinite equations has non-trivial solutions if and only if the determinant is equal to zero. It is possible to make some qualitative considerations in order reduce the number of equations to a finite number:

THEORETICAL BASIS

• the second part of the system of equations is approximately constant (due to the conservation of energy)

• E m[⊥km] is, to a good approximation, equal to E m • if K 2 ≈ k 2 m then this implies that E m is large
With these simple considerations, it appears clear that only waves with wavevectors (reciprocal space points) that end close to the Ewald sphere are strong waves.

In the case of X-rays with not too high energies and/or not for crystals with large cell parameters like protein crystals, the radius of the sphere is quite small and by consequence often only two lattice points are close to the Ewald sphere. In this case, the system of equations becomes of order two, and the indices m and n take the value o (for the direct beam direction) and h (for the diffracted beam)

     [K 2 (1 + χ o ) -k 2 o ]E o + K 2 P oh χ hE h = 0 K 2 P oh χ h E o + [K 2 (1 + χ o ) -k 2 h ]E h = 0 (2.12)
where P oh is the polarization factor and is equal to 1 for electric vectors perpendicular to the plane of incidence (σ polarization) and cos 2θ B for electric vectors parallel to the plane of incidence (π polarization). Imposing the condition that the determinant needs to be equal to zero to obtain non-trivial solutions, we obtain the dispersion relation

(k 2 -k 2 o )(k 2 -k 2 h ) = K 4 (P oh ) 2 χ h χ h (2.13)
where k = K 2 (1 + χ 0 ) is the average wavevector in the material. The solution of 2.13 is a complex surface in reciprocal space. A section of the real part of this complex surface is shown in fig. 2.4, plotted in red. The points o and h are reciprocal lattice points, and the vector h is the reciprocal lattice vector. In all the dispersion surface drawings I will consider only the σ polarization, being the synchrotron radiation σ polarized in the plane of the orbit.

The considerations so far apply to an infinitely extended crystal. For a finite crystal we have to apply boundary conditions for the wavevectors and for the amplitudes. For the wavevectors, the continuity of the tangential components is required; the boundary condition for the amplitudes will be discussed later on, sec.2.2.2.3.

X-ray diffraction

The continuity of the tangential components of the wavevectors can be graphically represented by plotting on the dispersion surface fig.

h o La Lo K k A 1 A 2 A 3 A 4 n e M h n e K o K h k h k o K h K o K o Figure 2
.4: Section of the real part of three dimensional complex dispersion surface for the two beam case and one polarisation. M is the excitation point, A 1 to A 4 are the tiepoints. In the inset: direct space representation.

The only wavefields that may be strong are the whose tie points are close to the Lorentz point. It is possible to make an approximation, i.e. to ignore all the terms above the second order in 2.13. These approximations are valid except for "extreme diffraction conditions" (grazing incidence, grazing exit diffraction, or Bragg angle close to 90 • ).

With this approximation, the dispersion relation can be written as

ξ o ξ h = 1 4 K 2 (P oh ) 2 χ h χ h (2.14)

THEORETICAL BASIS

where ξ o and ξ h are (k -k o ) and (k -k h ), respectively. This is a second order equation, and, in terms of the dispersion surface plotted in fig. 2.4, is equivalent to considering only the tiepoints A 1 and A 2 , i.e. the solutions close to the Laue and Lorentz points. The dispersion surface consists of two branches: the one closer to the Laue point (the crossing point of the two spheres of radius the wavevector in the vacuum K) is often called the α branch, while the other, further from the Laue point is called the β branch. These wavefields are coherent because they were all produced by one incoming plane wave. They will interfere inside the crystal producing two different interference effects.

La Lo A 1 A 2 ξ o (1) 2θ B towards o towards h ½Kχ 0 α β k h (α) k o (β) k o (α) k h (β) ξ h (1) ξ h (2) ξ o (2) T o T h n e M M' A' 1 A' 2 n e K h h K od K o Ψ h Ψ o
The first one is produced by wavevectors that have the same tiepoints but different directions. This interference will produces a standing wave field whose periodicity is

X-ray diffraction

the lattice plane distance, when the Bragg condition within the crystal is fulfilled. Depending on the branch of the tiepoint, the maxima of the standing wave field are either between or on the lattice planes. This leads to large differences in the photoelectric absorption and results in the so-called anomalous absorption, discovered by Borrmann in 1941 [START_REF] Borrmann | Über Extinktionsdiagramme der Röntgenstrahlen von Quarz[END_REF] (therefore it is also called Borrmann effect) and explained by Laue in 1949, [START_REF] Laue | Die Absorption der Röntgenstrahlen in Kristallen im Interferenzfall[END_REF] and by the same Borrmann in 1950, [START_REF] Borrmann | Die Absorption von Röntgenstrahlen im Fall der Interferenz[END_REF].

The second interference effect is produced by wavefields with the same direction but different tiepoint. Since the relative length differences between the wavectors are small (≈ |χ o |K ≈ 10 -5 10 4 µm -1 ), the beating frequency Λ of the interference field is of the order of tens of µm. This effect is called Pendellösung effect. The maximum value of Λ corresponds to the case that the surface normal passes through the Lorentz point, shown in fig. 2.5 with the dashed arrow passing through M' (since in this case the length differences will be the smallest). This value is called Pendellösung distance Λ 0 and is defined as

Λ 0 = λ γ o |γ h | P √ χ h χ h (2.
γ o = sin(α + ϑ B ) γ h = sin(α -ϑ B ) (2.16)
where α is the angle between the diffracting lattice planes and the surface of the crystal and is called asymmetric angle. It is equal to zero for symmetric Bragg case and 90

•
for the symmetric Laue case.

The Pendellösung distance Λ 0 is also used to distinguish the range of applicability of the kinematical theory of diffraction by introducing the parameter A = πt/Λ 0 , where t is the thickness of the crystal. The approximations used in the kinematical theory hold in the case A 1, i.e. thin crystal.

Bragg and Laue cases

In the coplanar diffraction one distinguishes between two main scattering geometries:

THEORETICAL BASIS

the Bragg case (or reflection geometry), where the diffracted beam exits from the entrance surface;

the Laue case (or transmission geometry), where the diffracted beam is directed towards the inside of the crystal and it exits from the opposite surface.

La Lo A 1 A 2 ξ o (1) 2θ B towards o towards h ½Kχ 0 α β k h (α1) k o (α2) k o (α1) k h (α2) ξ h (1) ξ h (2) ξ o
(2) One of the consequences of this is that, for the Bragg case, there is a range of excitation points such that there is no intersection of the normal to the surface with any of the real part of the dispersion surfaces. This implies an imaginary solution resulting in an interference total reflection range. The width of the total reflection range is The penetration depth is the depth at which the intensity inside the crystal falls to 1/e of the incoming intensity. In Bragg diffraction, this is a function of ∆θ (the difference between the angle of incidence and the Bragg angle) and is plotted in fig. 2.8 for a Si 111 symmetric reflection, E= 14.413 keV. For angles far away from the interference total reflection range, the penetration depth tends asymptotically towards t abs

T o T h n e K h n e h K o Ψ o Ψ h
w h = 2|P | √ χ h χ h sin 2θ B |γ h | γ o (2.17
t abs = 2 µ 0 1 γ 0 + 1 γ h (2.18)
which is controlled only by the linear absorption coefficient µ 0 and by the geometry.

In the region of interference total reflection, the absorption is dominated by extinction.

In the middle of this region, it assumes the smallest value, and is equal to

t ext = Λ 0 2π (2.19)

THEORETICAL BASIS

where Λ 0 is the Pendellösung length defined in 2.15 (in the Bragg case it is also called extinction distance). As a result of anomalous transmission, the curve is asymmetric. The absorption is larger in the higher angle region than in the lower angle region. Even though the experiments carried out by Borrmann were in transmission geometry the physics behind them is the same in the reflection case. The lower angle side of the curve is produced by wavefields that belong to the α branch of the dispersion surface, whilst the higher angle side are created by the β branch and which undergo anomalous absorption.

Asymmetric reflections

From formula 2.17, it is possible to see that one of the possible ways to act on the width of the reflection is to change the ratio between the direction cosines. The asymmetry factor b is defined as as

b = γ h γ o (2.20)

X-ray diffraction

By comparing the insets in fig. 2.5 and fig. 2.6 it is possible to see that γ o is always positive whereas γ h is positive in the Laue case but is negative in the Bragg case. In the case of the symmetric reflections, the value of b is either +1 or -1. Consider now an asymmetric Bragg case in grazing incidence, i.e. the angle between the incoming beam and the surface is smaller than the angle between the diffracted beam and the surface (the inverse result would be obtained for a grazing exit case).

The dispersion surface for such a geometry is shown in fig. 2.9. In the case of a divergent monochromatic wave with divergence w o , the diffracted wave will have a divergence of w h . The relationship between the two divergences will simply be:

w h = b w o (2.21)
The Liouville theorem tells us that phase space is conserved. Therefore, if we consider the incoming beam to have a finite size L o , then the size of the diffracted beam must be such that

w o L o = w h L h which implies that L h = Lo b La Lo o h

Range of interference total reflection n e h k h k o

Figure 2.9: Section of the real part of a three dimensional complex dispersion surface for the two beam case, asymmetric reflection. In the bottom right, a sketch of an asymmetric reflection in real space.

THEORETICAL BASIS

This property of asymmetric reflections is very important and will be used in the rest of the thesis, e.g. to reduce the angular instrumental function at ID18, 3.1 or to obtain X-ray topographs in the quasi-plane wave approximation 4.2.2. Nevertheless, equation 2.21 is an approximation valid only for monochromatic waves. In the real case of polychromatic radiation there is an additional term to be considered. ∆θ out is defined as the divergence of the diffracted beam and ∆θ in as the divergence of the incoming beam. The relation between the two is given by the following formula:

∆θ out ≈ γ o γ h ∆θ in + ∆λ λ 1 -γ 2 h 1 2 -1 -γ 2 o 1 2 γ h (2.22)
From this formula and from the definition of the direction cosines 2.16 it is possible to see that, in the case of polychromatic radiation, there is always an increase of the divergence due to the contribution of the second term in the equation, except in the symmetric Bragg case where this term is zero.

This also has an important consequence for coherence-preservation optics. With only one crystal, the only way not to increase the divergence of the beam, and therefore not to diminish the coherence of it, is by using symmetric Bragg reflections. With any other geometry, it is necessary to use more than one crystal in a non-dispersive configuration in order not to spoil the coherence of the beam.

Concerning the penetration depth in the case of asymmetric reflections, there will be two different asymptotes for the lower and higher angle sides of fig.2.8. For example, for grazing incidence geometry, the penetration depth will be smaller at lower incidence angles and greater at higher incident angles, and vice-versa for grazing exit.

Reflectivity curves in Bragg case

The boundary conditions for the wavectors allow fixing the possible wavefields excited by the incoming beam. By introducing the boundary conditions for the electric fields it is possible to determine the absolute values of the intensities of the diffracted and transmitted waves, and not only their ratios. The reflectivity curve of a crystal is defined as the ratio between the incident and the diffracted beam intensities. We will discuss only the Bragg case, since this is the geometry we used in the High Resolution Diffraction (see 4.2.1). For a plane and parallel crystal plate the boundary conditions at the two surfaces are the continuity of the tangential component of the electric field. Since the

X-ray diffraction

index of refraction is very close to 1, it is possible to approximate the continuity of the tangential component with the continuity of the electric field vector itself. By imposing these boundary conditions it is possible to obtain the formula for the reflectivity curve, which is presented in the Zachariasen [START_REF] Zachariasen | Theory of X-ray Diffraction in Crystal[END_REF] (see also [START_REF] Fatemi | Explanatory Notes on W. H. Zachariasen's "Theory of X-Ray Diffraction in Ideal Crystals[END_REF] for the complete derivation in the Laue case), but also in a slightly more compact form in [START_REF] Caciuffo | Monochromators for x-ray synchrotron radiation[END_REF]:

R h (∆θ) = (2.23) |b|P 2 |χ h | 2 (sin 2 (av) + sinh 2 (aw)) Q + U sinh 2 (aw) -V sin 2 (av) + 1 2 U 2 -|q| 2 sinh |2aw| + 1 2 V 2 -|q| 2 sin |2av| with z = 1 -b 2 χ 0 + b sin(2θ B )∆θ q = bχ h χ h v + iw = q + z 2 a = πk 0 t 0 γ 0 Q = |q + z 2 | U = Q + |z| 2 V = Q -|z| 2
where t 0 is the crystal thickness.

This formula is valid for any thickness of the sample. It is possible to obtain a simplified formulation by considering a semi-infinite crystal. By following [START_REF] Batterman | Dynamical Diffraction of X Rays by Perfect Crystals[END_REF][START_REF] Fingerland | Some properties of the single-crystal rocking curve in the Bragg case[END_REF], we introduce the normalized angular coordinate η:

η = -b∆θ sin(2θ B ) -1 2 χ 0 (1 -b) |P oh | |b| √ χ h χ h (2.24)
The term normalized angular coordinate come from the fact that, for these coordinates, the region of interference total reflection goes from -1 to +1. ∆θ 0 is the complex refraction correction. This is the difference between the Bragg angle calculated with the kinematical theory and the middle of the reflection range and is defined by

∆θ 0 = - χ 0 (1 -b) 2 sin(2θ B ) (2.25)
Only in the Laue symmetric case (b=1) the refraction correction is zero.

In a semi-infinite crystal the wavefields which propagate inside the crystal are absorbed before reaching the exit surface. In this approximation it is possible to neglect one wavefield and to obtain a simplified formula for the reflectivity curve Fig. 2.11 shows a comparison between the reflectivity curves obtained using the general formula 2.24 and the approximated one for semi-infinite crystals, 2.26.

R h = |b| χ h χ h η ± (η 2 -
It is possible to see that increasing the thickness results in two effects: an increase in the period of the oscillations and a decrease of their amplitudes. In this particular case, for thicknesses larger than 500µm, the calculations obtained using the two formulations are almost equivalent, with discrepancies that are of the order of 3×10 -4 . Therefore, in the case of "thick" samples, i.e. the silicon crystals described in this thesis, the simpler formulation has been preferred whilst for the low absorbing diamonds the more general formulation has been used.

X-ray diffraction

THEORETICAL BASIS

Coherence

The notion of coherence is widespread in many areas of physics, from optics to quantum mechanics, from neutron to X-ray scattering [50]. I will briefly introduce here the main concepts of coherence of light. A light source is said to be coherent if it has the possibility of forming interference fringes. This is possible if the phase difference of the electromagnetic field between all pairs of points inside a region is constant with time. In the case of perfectly coherent light, this region extends to the entire space. For The longitudinal coherence length (also called temporal coherence length) is related to the non-monochromaticity of the source. Every source produces wave trains whose length is finite. This is equivalent (via Fourier transform) to saying that every source produces radiation which has a certain spectral bandwidth. The longitudinal coherence is defined as:

partially
l l = λ 2 ∆λ (2.27)

Coherence

It is possible to measure the longitudinal coherence length by using, for example, a

Michelson interferometer. If the optical path length between the two rays is equal to or larger than the transverse coherence length, no interference is recorded.

The transverse coherence length (also called spatial coherence length) is related to the fact that the source size has a finite extension is space. The transverse coherence length is defined as:

l t = λL S (2.28)
where L is the distance from the source and S is the source size. The effect of an extended source size is that, in a Young's double slit experiment, if the distance between the slits is equal to or larger than the longitudinal coherence length, no interference is recorded.

The coherence of light can also be treated, in a more general way, as the study of the randomness of the light through a more statistical approach. See [START_REF] Born | Principles of Optics -Sixth Edition[END_REF] for a more complete description. It is possible to define the mutual coherence function as:

Γ 12 (τ ) =< V 1 (t + τ )V * 2 (t) > (2.29)
where V 1 and V 2 represent the electromagnetic field at positions P 1 and P 2 , respectively, and τ is the time delay at which the correlation between the fields is calculated. If the two points coincide we obtain Γ 11 (τ ), that is the self-correlation function; if the delay τ is set to zero, Γ 11 (0) is simply the intensity I 0 . The normalized version of the mutual coherence function is the complex degree of coherence, which is defined as:

γ 12 (τ ) = Γ 12 (τ ) √ I 1 I 2 (2.30)
The quantity γ 12 (τ ) has values between zero and one. If it is equal to zero (or one), the radiation is incoherent (or coherent). For all other values, the radiation is partially coherent. If we assume the light to be quasi-monochromatic, i.e. the spectral bandwidth is small compared to the frequency of the radiation, we find that the visibility of the fringes is proportional to the complex degree of coherence. In the case I 1 = I 2 , the visibility is equal to the complex degree of coherence. This is an important conclusion, which is used to determine the degree of spatial coherence of an X-ray beam after an optical element, by using either the Talbot effect [14] or an interferometer based on refractive bi-lenses [15].

THEORETICAL BASIS

The wavefront can be perturbed by imperfections in the optical elements. For example, low quality beryllium windows [START_REF] Snigirev | On the requirements to the instrumentation for the new generation of the synchrotron radiation sources. Beryllium windows[END_REF] create speckles that deteriorate the wavefront [START_REF] Vartanyants | Origins of decoherence in coherent X-ray diffraction experiments[END_REF].

Wang et. al. [START_REF] Wang | High-efficiency frequency converters with periodically-poled Rb-doped KTiOPO4[END_REF] simulated the effect of roughness in beryllium window and measured a decrease of the spatial coherence for rms values above 0.15 µm. The requirements on the roughness become more stringent in the case of a mirror; due to the grazing angle of incidence, the maximum acceptable roughness is of the order of 0.15 Å.

In the case of monochromator crystals, vibrations can cause an increase of the virtual source size. As an example, this has been measured by Diaz et al. [13] using a double-grating interferometer and comparing a channel-cut monochromator with a double crystal monochromator. The results showed that vibrations of the double crystal monochromator results in an increased effective source size.

In our analysis we decided to work with a simpler approach to coherence in diffraction, and we always separate the transverse and longitudinal coherence lengths. A more complete approach on the preservation of the coherence in diffraction was done by Yamazaki and Ishikawa [START_REF] Yamazaki | Propagation of X-ray coherence for diffraction of perfect crystals[END_REF]. It requires solving the linear time-dependent Takagi-Taupin equations the result obtained is that the temporal coherence and the spatial coherence of the beam mix in X-ray diffraction and that the coherence is modified in the direction of the diffraction vector.

Chapter 3

Experimental techniques

In this chapter I will present the X-ray experimental techniques used to characterize the different samples, basing myself especially on books written by the following authors:

Bowen and Tanner [START_REF] Bowen | High Resolution X-Ray Diffractometry and Topography[END_REF], Authier, [START_REF] Authier | Dynamical Theory of X-Ray Diffraction[END_REF], Als-Nielsen and McMorrow [START_REF] Als-Nielsen | Elements of Modern X-Ray Physics[END_REF], Daillant and Gibaud [START_REF] Daillant | X-Ray and Neutron Reflectivity: Principles and Applications[END_REF] and Holý, Pietsch and Baumbach [START_REF] Holý | High-Resolution X-ray Scattering from thin films and multilayers[END_REF]. Each technique has different characteristics, such as sensitivity to bulk and/or near-surface defects, local or integrated information, strain sensitivity etc. Therefore, the combined use of complementary techniques is very useful and results in a deeper understanding of the crystal quality.

High Resolution X-ray Diffractometry

Introduction

High Resolution X-ray Diffractometry (HRXRD) includes the measurement of rocking curves of a sample, i.e. the ratio of the diffracted and the incident intensities as a function of the angular deviation of the sample from the Bragg angle. The term high resolution is referred to the condition of having a small resolution area in the reciprocal space, i.e. a small divergence of the beam and a small spectral width. HRXRD is one of the standard methods of characterizing the quality of crystals. It is widely applied, for example, to the study of epitaxial structures of compound semiconductors, where performances of the devices are strongly correlated with the quality of the epilayers, e.g. GaAs and AlAs/GaAs on silicon [START_REF] Tapfer | Structural investigation by X-ray diffraction of GaAs epilayers and AlAs / GaAs superlattices grown on Si by MBE[END_REF] or Al x Ga 1-x N/AlN/GaN [START_REF] Lafford | Influence of alloy composition and interlayer thickness on twist and tilt mosaic in Al[END_REF]. It has been demonstrated that in the case of large mismatch between the different layers it is possible to resolve the presence of a single buried monolayer [START_REF] Tapfer | Monolayer resolution by means of x-ray interference in semiconductor heterostructures[END_REF].

EXPERIMENTAL TECHNIQUES

In our case, we used HRXRD to characterize the quality of crystals foreseen for use as X-ray optical elements. We assume that the presence of defects in the crystals (or on their surfaces) gives rise to a broadening of the rocking curve. This broadening may be well below an arc second. We had two different situations. In the first we wanted to study the quality of the surface of silicon crystals after different surface treatments whilst in the second we were interested in the contributions to the rocking curve from bulk and surface defects in synthetic diamonds.

In the first case, we used an X-ray tube with a simple double crystal configuration using an asymmetrically cut silicon monochromator (to obtain a good collimation) was enough to obtain the desired resolution. The obtained results will be shown in chapter 5.

For the second condition, due to the impossibility of having a perfect reference diamond crystal to be used as monochromator, an extraordinary effort was made to reduce the instrumental function as much as possible, both in angular and in wavelength spread.

This has been achieved by using the Inelastic Scattering beamline ID18, 3.1. The energy of the X-ray beam was 14.413 keV, which corresponds to the 57 Fe nuclear resonance transition. We assume the experimental rocking curve of a high quality single crystal to be the result of the convolution of three curves, see fig. 3.2: the reflectivity curve defined in 2.2.2.3, the instrumental broadening/apparatus function, which takes into account all contributions from the optical elements that condition the beam impinging on the

High Resolution X-ray Diffractometry

sample and the broadening due to defects, which is the effect produced by the presence of bulk and surface defects. Thanks to the very high monocromaticity of the beam delivered at ID18 ( ∆λ λ ≈ 10 -7 ), we can concentrate only on the angular convolutions, ignoring the wavelength spread. Therefore, in order to be able to isolate the function describing the broadening due to defects, it is necessary to determine the apparatus function with high precision.

Determination of apparatus function.

It is possible to describe the effect of the instrumental broadening with a convolution integral:

e(t) = -∞ ∞ f (τ )i(t -τ )dτ def = (f * i)(t) (3.1)
The function f (t) represents the theoretical function one is attempting to measure;

the function e(t) is the measured curve, that is a weighted average by the function i(t) of the true curve f (t). The instrumental broadening can be either calculated or measured experimentally. We decided to follow both methods and to compare the The curve obtained (corresponding to the curve after the 7th crystal in figure 3.3) has a FWHM of 0.037 arc seconds. It is important to highlight the fact that this estimation is done without taking into consideration any imperfection in the system. It is therefore to be considered as a theoretical lower limit of the real instrumental function.

In order to measure experimentally the apparatus function it would be necessary to sample the incoming beam with a δ-shaped curve, obtaining in response the instrumental function itself, as it is possible to see from 3.1 and from fig. 3.2. This is of course impossible, but it is possible to approach the situation. Firstly, one has to use a crystal (analyser) the perfection of which is such that we can ignore the broadening due to defects. Secondly, by measuring the incoming beam with different reflections with de-

High Resolution X-ray Diffractometry

creasing Darwin widths of that "perfect" sample, in our case of silicon, it is possible to approach step by step a δ function. The analysis for this experimental determination of the instrumental function has been developed in two independent ways: in the first, the instrumental broadening has been modelled by a curve with free parameters and fitted; in the second, direct deconvolution algorithms have been used. One of the most serious problems in both methods has been simply the fact that the instrumental broadening by the optical system installed at the beamline ID18 is indeed very small, thanks to its high beam collimation and energy resolution. To extract tiny broadenings from a series of measured rocking curves is quite a complicated task.

Model and fit approach

The instrumental function has been described by the sum of a Gaussian and a Lorentzian function (similar to a pseudo Voigt, often used in the description of the diffraction peak shape, for example by Ida et al. [START_REF] Ida | Extended pseudo-Voigt function for approximating the Voigt profile[END_REF]):

app(∆θ) = A L 2 π ω 2 4 ∆θ 2 + ω 2 + A G 1 √ 2πσ e -∆θ 2 2σ 2 (3.2)
where A L and ω are the area and the FWHM of the Lorenztian, respectively, and A G , σ are the area and standard deviation, respectively, of the Gaussian.

EXPERIMENTAL TECHNIQUES

The code was written in Mathematica [64] and uses the Global Optimization 6. The first attempt to solve this problem was to include the Fraunhofer diffraction from the slits as an additional broadening of the theoretical reflectivity curve. The results

were not satisfactory, above all for slits opened less than 100 µm. We supposed that other effects (such as diffuse scattering, small angle scattering or not ideal shape profile of the slits) dominate for narrow apertures. Since these effects are more complicated to simulate, we decided to include the diffraction from the slits into the apparatus function. So we worked with three different apparatus functions for the three slits size

High Resolution X-ray Diffractometry

EXPERIMENTAL TECHNIQUES

4 2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 Θ ΘB " R a.u.
A L ω A G σ FWHM FW20%M FW2%M

High Resolution X-ray Diffractometry

Direct deconvolution methods

A problem with every deconvolution algorithm is that, in general, equation 3.1 need not have any solution at all [START_REF] Jones | The problem of error in deconvolution[END_REF]. In fact, the presence of statistical noise in the experimental data makes the equation irreversible, and the deconvolution becomes an ill-posed problem. To overcome this problem approximations are always necessary. I wrote a code using Mathematica that uses two different algorithms: the Burger-Van Cittert algorithm, [START_REF] Burger | Wahre und scheinbare Intensitätsverteilung in Spektrallinien[END_REF] and the Richardson-Lucy algorithm, [START_REF] Richardson | Bayesian-Based Iterative Method of Image Restoration[END_REF][START_REF] Lucy | An iterative technique for the rectification of observed distributions[END_REF], in order to deconvolve the instrumental broadening. The theory of the two methods, together with some results and limitations, are presented in Appendix A.

Comparison of the results obtained.

In fig. 3.8 the results obtained with all the different methods and for a slits size of 200 µm are presented. This is also the slits size that gave the best results when analyzing the diamond crystal. The reason for this is simply because for the wider slits the footprint

EXPERIMENTAL TECHNIQUES

on the sample was ≈ 1.5 mm for the 400 reflection and ≈ 3.4 mm for the 111 reflection, therefore measuring at the same time the high and low quality regions of the crystal.

For the narrower slits, the Fraunhofer diffraction from the slits covers the "broadening due to defects" effects we are interested in measuring.

As discussed previously, the direct calculation of the instrumental broadening is the narrowest curve, giving a theoretical lower limits in the case no imperfections are present in the system. The results obtained with the Van Cittert deconvolution algorithm depend strongly on the different theoretical reflections, i.e. a much wider curve is obtained when deconvolving the Si 333 rocking curve respect to the one obtained from the Si 555. This effect is not present in the results obtained using the Richardson-Lucy algorithm, where the apparatus functions obtained are not influenced by the width of the reflectivity curve. Finally, the results obtained with the model-and-fit approach compare well with the direct deconvolution result, and therefore are the ones used for the rest of the analysis.

Recently we performed a last experiment in order to obtain an even more accurate estimation of the apparatus function. This was done by using a different energy (23.880 keV, corresponding to the nuclear resonance of 119 Sn) in order to be able to excite even higher order reflection, 777, 888 and 999 together with the quasi-forbidden 222 reflection. The idea behind the experiment was the same of the previous one, i.e approaching a δ-shaped function in order to be able to record only the instrumental broadening function. The FWHM for the allowed reflections, 777, 888 and 999 are respectively 0.063", 0.062" and 0.033" whilst for the 222 forbidden reflection I obtained a value of 0.025" by using the experimentally determined structure factor by Alkire et al. [START_REF] Alkire | Determination of the absolute structure factor for the forbidden (222) reflection in silicon using 0.12 Å γ rays[END_REF] of 1.456. Fig. 3.9 shows the comparison of the experimental data obtained with the different reflections together with the apparatus function obtained for a slits size of 200 µm. This graph demonstrates that, even though the theoretical curves are narrower and narrower, the experimental rocking curve is almost constant, i.e. we are directly measuring the apparatus function. Moreover the agreement between the experimental data and the apparatus function determined previously is very good, considering that the experimental conditions for the two cases were not the same.

To conclude, we are confident that the apparatus function determination is accurate and will allow a correct estimation of the the broadening due to defects for the diamond samples.

High Resolution X-ray Diffractometry

Si222

EXPERIMENTAL TECHNIQUES

X-ray topography

X-ray topography is an imaging technique based on diffraction and described by the dynamical diffraction theory. It is used in the characterization of crystal defects and macroscopic deformations. The first topographic experiments were made in reflection geometry by Berg in 1931 [START_REF] Berg | Über eine röntgenographische Methode zur Untersuchung von Gitterstörungen an Kristallen[END_REF] and Barrett in 1945 [START_REF] Barrett | A new microscopy and its potentialities[END_REF] and in transmission geometry by Ramachandran in 1944, [START_REF] Ramachandran | X-ray topographs of diamond[END_REF], (who also was the first to use the name topography for this technique). Bond and Andrus [START_REF] Bond | Structural imperfections in quartz crystals[END_REF] were the first to use a double crystal configuration followed by Lang [START_REF] Lang | Direct Observation of Individual Dislocations by X-Ray Diffraction[END_REF], when for the first time individual dislocations became visible in X-ray topographs. It must be remarked that the name is somewhat misleading, because it suggests this technique to be surface sensitive (the Greek origin of the name is in fact topo-, "place", and graphia, "writing", and is often used to represent surfaces shapes and features), whereas it is often used to probe the bulk of a crystal. In fact, it is the topography of the crystal lattice planes that is examined. An alternative name for the technique is X-ray diffraction imaging.

A spatially extended X-ray beam, impinging on the crystal is diffracted and the diffracted beam/beams is/are recorded on a two dimensional detector (e.g. a photographic film or a CCD camera) giving a 2D image/images of the crystal. Different X-ray topography techniques have been developed. It is possible to classify them depending on three basic elements [START_REF] Härtwig | X-ray diffraction imaging (X-ray topography) -An Overview of Techniques and Applications[END_REF]: 1) the source geometry and characterisitics, together with the eventual optical elements (slits, other crystals, etc...); 2) the sample properties and geometry, e.g. transmission or reflection; 3) the detector, together with possible optical elements. We can define:

Limited beam techniques: the size of the incoming beam is reduced in order to avoid the overlap of too many Borrman triangles (see 2.7) generated at the entrance surface of the crystal. As a rule of thumb the beam needs to be much narrower than the base of the Borrmann triangle, e.g. in the symmetrical Laue case, is equal to 2t sin(θ B ), where t is the thickness of the crystal. Typical values of 10 to 20 µm are used. The beam can be limited either in one or two directions, obtaining respectively section and pin-hole topographs. An advantage of these techniques is, in the case of section topography, the possibility of spatially separating on the detector intensities arising from different depths in the crystal.

Moreover, a section topograph of a low absorbing perfect crystal will show Kato

X-ray topography

fringes, [START_REF] Kato | A study of pendellösung fringes in X-ray diffraction[END_REF] produced by the interference between the wavefields created from the two different branches of the dispersion surface. This is a manifestation of the Pendellösung effect explained in section 2.2.2 in the case of incoming spherical waves. The fringes are very sensitive to lattice distortion, as shown e.g. by Patel [START_REF] Patel | X-ray anomalous transmission and topography of oxygen precipitation in silicon[END_REF] in the study of oxygen precipitation in silicon. In the framework of this thesis, section topography was the technique used for the visualization of ferroelectric domains in chapter 6.

Extended beam techniques: the size of the incoming beam is larger than in the limited beam techniques. At a synchrotron the distance from the source to sample can be large, the size of the beam (especially at wiggler or bending magnet beamlines) can go up to few cm, e.g. at ID19 or BM05. In the laboratory, the size of the beam is small; one way to obviate this problem has been found by Lang, [START_REF] Lang | The projection topograph: a new method in X-ray diffraction microradiography[END_REF]. A narrow beam (typically of the order of 100-250 µm) impinges on the sample, which can translate synchronously with the detector. In this way, the image of the entire crystal can be recorded. This method is called Lang or traverse topography. The characterization of high quality crystals was done using extended beam techniques at different beamlines, (see chapters 4.2.2 and 5).

Integrated wave techniques the angular source size is larger than the width of the reflectivity curve of the crystal, defined in 2.17 and/or the energy bandwidth of the incident beam is larger than the Darwin width defined in energy scale,

w λ h = λw h cot θ B .
Concerning the angular source size, in a laboratory source it is in general bigger than the angular acceptance of the crystal; at a synchrotron beamline, it could be smaller than the crystal acceptance e.g. at ID19 the divergence of the beam is ≈ 0.05 arc seconds. Regarding monochromaticity, in both laboratory and synchrotron configurations, it is possible to be in integrated wave conditions. Examples of integrated wave techniques are Lang topography, white beam topography, and in some cases, monochromatic beam topography.

Monochromatic plane wave techniques: contrary to integrated wave conditions,

the angular source size and the energy bandwidth are smaller than the reflectivity curve width in angular and energy space respectively. Special effort is needed to obtain these conditions, but the strain sensitivity achieved is much higher

EXPERIMENTAL TECHNIQUES

compared to integrated wave techniques. Moreover, the quantitative analysis of monochromatic plane wave topographs is easier than for the integrated wave techniques.

After defining the different types of techniques, the sources of inhomogeneities in the images is now discussed. Defects inside the crystal produce long range strain fields which distort the perfect lattice structure. This leads to local differences in the diffracted intensity and consequently to "contrast" in the image at the detector. Following Baruchel and Härtwig [START_REF] Baruchel | X-Ray Topography[END_REF] it is possible to distinguish three types of contrast:

Structure factor contrast due to differences in the phase or modulus of the structure factor. This contrast is fundamental to the visualization of ferroelectric domains in chapter 6;

Orientation contrast arises when different parts of the crystal are misoriented with respect to one another. If the misorientation in a given part is larger than the beam angular source size (or the equivalent wavelength spread is larger than the bandwidth of the incident beam), then the misoriented part will not satisfy the Bragg condition and will not diffract, and a loss of intensity is registered by the detector (fig. 3.10a). If the wavelength spread of the incident beam is sufficiently large (e.g. white beam), then all misoriented parts will diffract, because there will always be a wavelength to fulfil the Bragg condition in the misoriented part.

Consequently, differently oriented parts of the crystal will diffract in different directions. Thus, the images of these different crystal regions (e.g. different grains) may separate or overlap, if the sample to detector distance is large enough, leading to local losses or gains of intensity (fig. 3.10b). This type of contrast can occur also around lattice defects like dislocations, as shown by Dudley et al. [START_REF] Dudley | Assessment of orientation and extinction contrast contributions to the direct dislocation image[END_REF].

Extinction contrast is produced by the local variations of the scattering power, e.g. around a defect. In order to interpret this contrast it is necessary to use the dynamical diffraction theory for deformed crystal. The parameter that describes the effect of the distortion of the lattice on diffraction is the effective misorientation δθ:

δθ( r) = θ( r) -ϑ perfect B = - λ sin 2ϑ B ∂( h • u( r)) ∂s h (3.3)

X-ray topography

δθ = ∆d d tan ϑ B + (n t • n g )∆ϕ (3.4)
where n t and n g are the unit vectors normal to the incidence plane and parallel to the defect misorientation axis, respectively. This equation permits the derivation of a powerful tool for characterizing defects, the contrast extinction rule. A defect whose displacement vector field is perpendicular to the diffraction vector gives no contrast on the image. It is possible to define three different types of images produced by the extinction contrast in the case of integrated wave (see fig. 3.11):

• the direct image, consisting of additional intensity that originates from regions with an effective misorientation larger than the Darwin width of the perfect crystal, 2.17. It is visible under the conditions of integrated wave (or section topography) and low absorption (µ 0 t 2). It arises from the kinematical diffraction of the X-rays that are outside the wavelength range

EXPERIMENTAL TECHNIQUES

(∆λ/λ = cot θ B w h ) participating in the dynamical diffraction in the perfect crystal.

• the dynamical image, consisting of dips in the intensity. This is due to the disruption of the transmission created by the defect (it can be imaged as a shadow produced by the defect). This contrast is the predominant in case of high absorption (µ 0 t 6), since the direct image is completely absorbed.

• the intermediate image, produced by the interference between the wavefields existing in the crystal and a new wavefield created by interbranch scattering in the region of the defect itself. This interference produces an oscillatory contrast. This contrast is visible only in the case of low and intermediate absorption (µ 0 t 6). All the topography experiments in this thesis were carried out at synchrotron radiation sources. The properties of synchrotron radiation, such as high intensity, continuous spectrum, narrow angular collimation and small source size, opened new possibilities compared to conventional sources, see Tanner and Bowen [START_REF] Tanner | Synchrotron X-radiation topography[END_REF] for a review. These include:

X-ray topography

• shorter exposure time, allowing real time experiments, [START_REF] Bordas | Use of synchrotron radiation for Xray topography of phase-transitions[END_REF], [83], [START_REF] Gastaldi | Ultra-high vacuum heating camera for in situ synchrotron radiation X-ray topographic studies[END_REF] and [START_REF] Buffet | Real-time and in situ solidification of Al-based alloys investigated by synchrotron radiation: a unique experimental set-up combining radiography and topography techniques[END_REF].

• the possibility of having larger sample-to-film distances whilst maintaining micrometer image resolution, allowing room for bulky apparatus such as cryostats [START_REF] Tanner | Cryogenic X-ray topography using synchrotron radiation[END_REF], [START_REF] Baruchel | Synchrotron radiation topographic study of the thick ferromagnetic-fan interface in MnP[END_REF] and furnaces [START_REF] Maccormack | Application of X-ray synchrotron topography to in situ studies of recrystallization[END_REF]. Moreover, the possibility of changing the sampleto-film distance gives the possibility of studying focusing/defocusing effects for dislocations [START_REF] Zontone | New Features of Dislocation Images in Third-Generation Synchrotron Radiation Topographs[END_REF], dynamic deformation such as vibrations [START_REF] Matsouli | A study of the x-ray focusing effect of a magnetoacoustically excited FeBO 3 crystal using synchrotron radiation diffraction imaging[END_REF] and static deformation, e.g. created in a bonded wafer [START_REF] Härtwig | Xray Diffraction Topography at a Synchrotron Radiation Source Applied to the Study of Bonded Silicon on Insulator Material[END_REF]. I have also used this possibility to determine the bending radius of diamonds bent by femtosecond laser ablation [START_REF] Balling | Bending diamonds by femtosecond laser ablation[END_REF].1

• the very simple possibility of collecting more than one topograph at the same time in a short time, allowing e.g. the determination of the type of dislocations present in a crystal and the associated Burgers vector from one single exposure [START_REF] Tuomi | Use of synchrotron radiation in x-ray diffraction topography[END_REF].

Two techniques using a wide and homogeneous beam are often employed for synchrotron X-ray topography: white beam topography (single crystal) and monochromatic beam topography (double crystal), 3.12. Parts of the crystal with different d-spacings will diffract with a different wavelength but in the same direction and with the same intensity onto the detector. Moreover, all parts of the crystal diffract, including strongly misoriented regions. Some drawbacks include the heat load in case of highly absorbing material, leading to deformations and sometimes even the destruction of the sample (a possible solution is the use of a chopper or to some extent filters to remove the softer X-rays, [START_REF] Barrett | The present status of the ESRF diffraction topography beamline: new experimental results[END_REF]) and the limited sensitivity for weak distortions (the sensitivity to effective misorientation in this technique is in the order of 10 -3 -10 -6 radians).

Using in addition a monochromator crystal (or a system of several crystals) that diffracts the radiation coming from the source in a narrow energy band, it is possible to obtain higher strain sensitivity, e.g. by working on the flanks of steep rocking curves. This technique is called double crystal or monochromatic beam topography.

In order to avoid artefacts on the topographs that are not due to the sample, the monochromator crystals must be of very high quality. It is possible to distinguish two different configurations for monochromatic beam topography: the non-dispersive (+n,n) and the dispersive (+n, ±m) configurations, where n and m represent two different reflections and the signs are defined according the DuMond notation [START_REF] Dumond | Theory of the Use of More Than Two Successive X-Ray Crystal Reflections to Obtain Increased Resolving Power[END_REF].

The difference between the two arrangements (in the synchrotron case) is shown in

X-ray topography

source detector The main advantage of non-dispersive schemes is that the rocking curve obtained is narrower and has steeper flanks compared to the dispersive curve. Recording topographs on the flanks of the rocking curve allows high strain sensitivity since, in this region, a small change in the effective misorientation gives rise to large changes in the recorded intensity. The main drawback is that very few kinds of crystal are available with sufficiently high quality, and this restricts the use of this technique to a limited number of substances. In the case of synchrotron sources, there are suitable solutions. The limited narrow local band on the second crystal that fulfils the Bragg condition may be extended by rocking the crystal and integrating the topograph on the detector. The penalty for the larger reflecting crystal area is the reduced strain sensitivity, as for the integrated wave techniques introduced before.

1 st crystal 2 nd crystal a) λ 1 λ 2 λ 3 1 st crystal 2 nd crystal (different material/reflection) λ 1 λ 3 λ 2 b)
Another possibility is to correct for dispersion by bending the monochromator. For an experimentally-defined bending radius, the dispersive configuration becomes non-

EXPERIMENTAL TECHNIQUES

dispersive. In this way, high strain sensitivity is achieved for unmatched sample and monochromator materials. This technique is also called curved collimator topography, introduced at laboratory source [START_REF] Jenichen | Checking plane-wave approximation for Double Crystal Topography with curved samples[END_REF][START_REF] Köhler | High-Resolution X-ray Topography[END_REF] and later adapted to synchrotron radiation [START_REF] Altin | X-ray diffraction topography using a diffractometer with a bendable monochromator at a synchrotron radiation source[END_REF] with the construction of an instrument. This instrument was recently relocated from ID19 to BM05. In this occasion a large part of the electronics and the mechanics was improved/changed and I have been strongly involved (together with D. Van Brussel and P. Van Vaerenbergh) in the reinstallation and the commissioning of the instrument.

It is also possible to use a perfect crystal as the first crystal, choosing a set of planes whose d-spacing is close to that of the reflecting planes in the sample and obtaining a quasi-non-dispersive setup. Moreover, by using an asymmetric reflection, it is possible to obtain the low divergence required for quasi-plane wave topography, [START_REF] Ishikawa | Synchrotron Plane Wave X-Ray Topography of GaAs with a Separate (+, +) Monochro-Collimator[END_REF][START_REF] Ishikawa | X-ray topography with highly collimated beam at photon factory[END_REF]. Another possibility is to use multiple crystal system in order to obtain a highly monochromatic beam, therefore limiting the possible dispersion [START_REF] Hashizume | A Multiple Crystal System for High Strain-Sensitivity X-Ray Topography and Its Applications[END_REF].

The advantages of the dispersive and non-dispersive double crystal techniques with respect to white beam topography are: a higher strain sensitivity can be achieved (in the range of about 10 -4 -10 -7 for the dispersive configuration, up to about 10 -8 or even better for the non-dispersive), the lower heat load and the fact that images are easy to simulate. Drawbacks are the more demanding requirements of the experiment (e.g. sample orientation is necessary, sensitivity to mechanical and temperature drifts) and the fact that only one diffraction can be recorded at one time.

A powerful extension of double crystal topography is a technique called Rocking Curve

Imaging. Here, the crystal is rotated step by step over the entire rocking curve, and at each step an image is recorded. In this way, a series of topographs is obtained with each image belonging to a different working point (angular position) on the rocking curve. The strain fields are therefore mapped in a more complete way. Moreover, by using CCD cameras as detectors and PC-based analysis, it becomes easy to analyze the image series pixel by pixel and to extract local parameters such as the local rocking curve maxima (which reflect the local Bragg angles), the rocking curve widths or their integral reflectivities, [START_REF] Lübbert | µm-resolved high resolution X-ray diffraction imaging for semiconductor quality control[END_REF]. Taking those series for different sample positions (rotating around the normal of the reflecting lattice planes) it is possible to separate the lattice parameter and the lattice inclination contributions to the effective misorientation.

X-ray surface sensitive techniques.

X-ray surface sensitive techniques.

Two techniques have been used in order to characterize the surface roughness and crystalline quality of the sub-surface layer of high quality diamond crystals: X-ray Reflectivity (XRR) and Grazing Incidence Diffraction (GID). Both techniques take advantage of the fact that the index of refraction of X-rays in matter is less than one and may be written in the form:

n = 1 -δ -ıβ (3.5)
with:

δ = 2πρ e r e k 2 β = µ 0 2k (3.6)
where k is the wavevector, ρ is the electron density of the material and µ 0 is the linear absorption coefficient. For X-rays, δ and β are of the order of 10 -6 and 10 -9 , respectively. The fact that the refractive index is less than 1 implies that there exists a critical angle α c below which X-rays undergo external total reflection. The value of the critical angle is obtained by using the Snell-Descartes law. In X-ray optics, the convention for the beam angles is to measure them between the beams and the surface, while in geometrical optics it is between the beams and the normal to the surface,3.14.

This implies that the sine functions in the Snell-Descartes law are converted into cosines, giving:

n 1 cos α i = n 2 cos α t (3.7)
where α i,t are respectively the incident and transmitted angles and n 1,2 are the indices of refraction of the different media. Total reflection occurs when α t is equal to zero.

By substituting the refractive indices for air (to a good approximation, n air = 1) and for the medium from 3.5, neglecting the absorption and using the Maclaurin expansion of the cosines, it is possible to obtain the critical angle α c :

α c ≈ √ 2δ (3.8)
Typical values for α c are of the order of milliradians, i.e. grazing incidence. The penetration depth Λ as a function of the angle of incidence α i (for the specular condition, α i = α f ) can be written [START_REF] Dosch | Depth-Controlled Grazing-Incidence Diffraction of Synchrotron X Radiation[END_REF]:

EXPERIMENTAL TECHNIQUES

α i α r α t n n 1 = 1 n 2 = 1-δ-iβ K i K r K t
Λ(α ı ) = λ 4 √ 2π (α 2 c -α 2 ı ) + (α 2 ı -α 2 c ) 2 + (2β) 2
(3.9) Fig. 3.15 is a plot of the penetration depth of 7 keV X-rays in diamond.

For angles below α c , an evanescent wave is produced inside the sample fig. 3.16. This wave propagates parallel to the surface, and is exponentially damped in the perpendicular direction. The minimum depth of this wave is ≈ λ/4πα c , therefore it depends only on the material and wavelength used. It is of the order 3 nm for diamond at 7 keV X-ray energy. Around the critical angle, the penetration depth increases abruptly and above the critical angle it follows the Beer-Lambert law. This phenomenon enables studies of the quality of the crystal at different penetration depths.

X-Ray Reflectivity (XRR)

XRR is insensitive to the degree of long-range order of the material investigated (e.g. crystalline, amorphous). It is sensitive to changes in the local electron density (including surface and interface roughness) and not to strain fields, which are exploited in X-ray topography. XRR was pioneered by Kiessig [START_REF] Kiessig | Investigations into the total reflexion of roetgen rays[END_REF] and developed by Parrat [START_REF] Parratt | Surface Studies of Solids by Total Reflection of X-Rays[END_REF]. In the case of a perfectly smooth, infinite surface with an infinitely thick bulk and zero absorption coefficient, the reflectivity is one below the critical angle and decays proportional to q -4 = sin(α i ) 4πλ -4 for larger angles. If the sample is rough, the decay is stronger. A first theoretical model to take into account the roughness has been developed by Nevot and Croce, [START_REF] Névot | Sur l'étude des couches superficielles monoatomiques par réflexion 'rasante' (spéculaire ou diffuse) de rayons X, par la méthode de l'empilement 'sandwich[END_REF], assuming the roughness to have a Gaussian distribution. A second model based on the distorted wave Born approximation (DWBA) has been formulated by Sinha [START_REF] Sinha | X-ray and neutron scattering from rough surfaces[END_REF].

X-ray surface sensitive techniques.

Moreover, if the sample is made of layers of different electron densities, interference fringes (called Kiessig fringes) are obtained in the reflectivity curve above the critical angle. These fringes are more pronounced if the difference in electron density between the layers is large. The distance between fringes is inversely proportional to the thickness of the layer. These effects are used to build multilayer monochromators, where many layers of materials of alternating high and low electron densities are stacked on each other, giving high intensity peaks at larger angles compared to typical critical angles. These peaks have a broader bandwidth than that obtained using crystal monochromators ( ∆λ λ ≈ 10 -1 -10 -3 compared to ∆λ λ ≈ 10 -4 -10 -5 ), see fig. 3.17, allowing a higher photon intensity at the price of a lower energy resolution. Geometrical In our experiment on the study of diamond surfaces after polishing, three kinds of XRR scans were performed:

specular scans -the detector angle is swept at twice the rate of the sample angle whilst maintaining the condition ω = 0.5 * 2θ (called ω -2θ scans). In this type of scan, the scattering vector q is perpendicular to the surface of the sample, and only the electron density differences in this direction are probed;

X-ray surface sensitive techniques.

off-specular scans -longitudinal diffuse -the detector is again moved at twice the angular speed of the sample (ω -2θ), but the sample is initially positioned with an offset θ off compared to the specular scan. It is possible to subtract the average of two longitudinal diffuse scans collected at ∆θ ± θ off from the data obtained in a specular scan, obtaining the so-called true specular.

transverse diffuse scans -in this case the detector is kept at a fixed position whilst the sample is rocked. These scans allow the scattering vector parallel to the surface to be probed in order to determine in-plane correlation of the roughness.

Moreover, transverse diffuse scans allow the separation of roughness and composition grading which is impossible in specular reflectivity. Characteristic peaks are present on both sides of the specular peak, called Yoneda peaks [START_REF] Yoneda | Anomalous Surface Reflection of X Rays[END_REF]. They appear when either the incoming or the exit beam angles with respect to the sample surface are equal to the critical angle.

Grazing Incidence Diffraction (GID) in non-coplanar geometry.

GID combines the surface and the depth-controlled sensitivity of X-ray at grazing incidence together with Bragg diffraction from planes approximately perpendicular to the sample surface. The technique was introduced by Marra et al. [START_REF] Marra | X-ray total-external-reflection-Bragg diffraction: A structural study of the GaAs-Al interface[END_REF] and used in a depth-controlled study by Dosch et al. [START_REF] Dosch | Depth-Controlled Grazing-Incidence Diffraction of Synchrotron X Radiation[END_REF]. The geometry used is shown in fig. 3.18.

The incidence and exit beams are kept at a small angle around the critical angle of total reflection. Lattice planes perpendicular to the crystal surface are used for Bragg diffraction (non-coplanar diffraction geometry). Exploiting such a geometry gives access to the momentum transfer almost parallel to the crystal surface, therefore probing the crystalline structure close to the surface, e.g. possible mosaicity near the surface, or sub-surface distortion due to the crystal processing. This can complement the measurements of the strain components perpendicular to the crystal surface measured by high-resolution diffraction.

The angle between the incident beam and the surface of the crystal is kept constant during a scan; this implies that the penetration depth is also constant. By correctly implementing the rotations of the sample (around the surface normal) and of the detector, it is possible to collect transverse and radial scans of the reciprocal lattice point corresponding to the diffracting lattice planes. By repeating this at different incident

EXPERIMENTAL TECHNIQUES

E 0 E h E s θ B Figure 3
.18: Grazing incidence diffraction, E 0 , E s , and E h are the incident, specular and diffracted waves respectively, and θ B is the Bragg angle, from [START_REF] Stepanov | Dynamical x-ray diffraction of multilayers and superlattices: Recursion matrix extension to grazing angles[END_REF].

angles, it is possible to study the crystalline structure at different depths and, e.g.

obtain a quantitative measure of the thickness of the region which is distorted due to polishing.

Chapter 4

Characterization of single crystal diamonds

In this chapter, I will first present the characteristics that make diamond an interesting material for 3rd and 4th generation synchrotron sources. Afterwards, I will show some of the results obtained from the characterization of single crystal diamonds, mostly using X-ray related techniques but also, in some cases, compared with alternative techniques such as Atomic Force Microscopy (AFM) and optical profilometer. Some of the results presented here have been published in different papers, [START_REF] Burns | Diamonds for X-ray optical applications at 3rd and 4th generation X-ray sources -art. no. 67050K[END_REF][START_REF] Burns | HPHT growth and x-ray characterization of high-quality type IIa diamond[END_REF][START_REF] Van Vaerenbergh | High Heat Load Diamond Monochromator Project at ESRF[END_REF], of which I am co-author.

Diamond

Properties

Diamond is one of the allotropes of carbon, some of the others being e.g. graphite, graphene and fullerene. At room temperature, diamond is metastable; the thermodynamically stable form is graphite. At high temperature and ambient pressure is possible to graphitize diamonds, from 600 • C to 800 • C in the presence of oxygen or a metallic catalyst [START_REF] John | The oxidation of (100) textured diamond[END_REF], and from 950 • C or higher in vacuum [START_REF] Davies | Graphitization of Diamond at Zero Pressure and at a High Pressure[END_REF]. The diamond crystal is formed by carbon atoms bounded by four equidistant nearest neighbours in a tetrahedral symmetry due to the sp 3 orbital. In comparison, in graphite, the bonding between carbon atoms is sp 2 , resulting in layers of graphite weekly bonded by van der Waals forces. The diamond unit cell consists of two interpenetrating face-centered

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

cubic lattices, one at (0,0,0) and the second shifted at ( 1 4 , 1 4 , 1 4 ). The lattice parameter at room temperature is between (0.356683 ± 0.00001) nm and (0.356725 ± 0.00003) nm [START_REF]Properties of Natural and Synthetic Diamond[END_REF]. Two different crystallographic forms of diamond exist: the cubic and the hexagonal (also known as lonsdaleite). The difference between these two forms is the result The pictures of the two structures were generated using the software Vesta, [START_REF] Momma | VESTA: a three-dimensional visualization system for electronic and structural analysis[END_REF] Diamond has the highest hardness of all natural materials (only man-made aggregated diamond nanorods [START_REF] Dubrovinskaia | Aggregated diamond nanorods, the densest and least compressible form of carbon[END_REF] and ultrahard fullerite [START_REF] Blank | Ultrahard and superhard phases of fullerite C60: Comparison with diamond on hardness and wear[END_REF] are harder than diamond). This explains the origin of the name, adamas (Greek) that means the "untameable" or the "unconquerable".

Synthesis of diamonds

The first commercially successful synthetic diamond was synthesised in Stockholm, Sweden, in 1953 using High Pressure and High Temperature (HPHT). The discovery was kept secret [START_REF] Lundblad | High pressure synthesis of diamond in Sweden in 1953[END_REF]. One year later, there were two reported syntheses of diamond at the General Electric Research Laboratory: one by H.M. Strong and the second by H.T.

Hall. The results were quickly published in Nature [START_REF] Bundy | Man-made Diamonds[END_REF] and the method of growing man-made diamond was patented. Those first diamonds were tiny (1×0.3 mm) and hardly reproducible. After further experiments, it became clear that to synthesize

Diamond

diamond, a metal "solvent/catalyst" bath was necessary. In 1971, R.H Wentorf Jr.

showed the possibility of growing bigger diamonds using the so called "thermal gradient method" [START_REF] Wentorf | Some Studies of Diamond Growth Rates[END_REF]. This method is best done in a large volume belt press system and will be described in more detail in the next section. With the HPHT method, single crystal diamonds with more than 1 cm maximum dimension can be grown and routinely 5-7 mm diamonds with low nitrogen content are grown by companies such as Element Six (formerly De Beers Industrial Diamond) and Sumitomo Electric Industries. An alternative method, the "pressure gradient" method, makes use of a different type of high pressure system, called split-sphere BARS press (from the Russian acronym of press-free high-pressure setup "split sphere") to grow large size single crystal, see e.g. [START_REF] Pal'yanov | High-pressure synthesis of highquality diamond single crystals[END_REF].

Another technique used to grow synthetic diamond is Chemical Vapour Deposition (CVD) [START_REF] Lee | CVD diamond films: nucleation and growth[END_REF], [START_REF] Ferro | Synthesis of diamond[END_REF]. It is the most widely used technique for depositing many materials in the semiconductor industry, including a wide range of dielectrics and many metals and metal alloys. The technique generally involves the growth of a solid material from the gas phase using a reactive gas mixture, which supplies the necessary active species (carbon from methane in the case of diamond) onto a controlled surface (or substrate). The substrate does not have to be diamond but in order to grow single crystal CVD diamond (instead of polycrystalline diamond) it is necessary to use a diamond substrate with high quality surface finish. Numerous different methods have been developed; the main difference consists in which technique is used to activate the gas-phase carbon-containing precursor molecules (e.g. hot filament, DC-plasma, RFplasma, microwave-plasma, electron cyclotron resonance-microwave plasma). The main advantages of CVD techniques are the low pressure necessary (consequently a lower cost in terms of equipment); the possibility of growing large diamonds; and the possibility of growing diamond on different substrates. Unfortunately, until now, the quality of the bulk material does not meet the requirements for use as X-ray optical elements, even though recently great improvements have been made by using high quality IIa HPHT single crystals as seeds, [START_REF] Martineau | High crystalline quality single crystal chemical vapour deposition diamond[END_REF]. In the next section, the HPHT method will be described. Up to now, this method facilitates synthesis of diamonds with the highest structural quality. • the CVD method at low pressure;

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

• shock wave synthesis, where very high pressure and temperature are achieved by detonation;

• HPHT direct conversion;

• HPHT catalytic synthesis.

Shock wave synthesis produces very small diamonds (of the order of a few µm), not suitable for application as optical elements. Direct conversion of graphite to carbon faces a high kinetic barrier, so that it takes place at pressures above 13 GPa and temperatures around 3000 K [START_REF] Bundy | Direct Conversion of Graphite to Diamond in Static Pressure Apparatus[END_REF][START_REF] Sung | Graphite diamond transition under high pressure: A kinetics approach[END_REF] but these conditions are difficult and expensive to achieve. In order to overcome the kinetic barrier, a metal solvent-catalyst bath is used. This allows conversion at lower temperature and pressure, typically around 1550

Diamond

to 1600 K and 5 to 6 GPa [START_REF] Burns | The Properties of Natural and Synthetic Diamond[END_REF]. This is the method used to grow high quality synthetic diamonds and attention will now be focused on this specific technique. • carbon source: could be graphite but higher quality crystals are obtained using a fine diamond powder as the carbon source;

• a growth medium of metal solvent-catalyst. As mentioned before, the liquid metal solvent is necessary to enable the process to proceed at lower pressure and temperature. Alloys of metals such as Fe-Ni, Mn-Ni and Co-Fe are generally used;

• a diamond seed: as usual in crystal growth, it is necessary to use a very high quality seed crystal in order to grow high quality diamond. The seed must also be embedded to avoid it floating away, and it must be in contact with the bottom of the metal bath.

All the components are placed inside a heater and the heater is filled with a pressure medium, e.g. NaCl. The cell as formed is compressed in a high pressure vessel. A thermal gradient (20 to 50 K) is produced inside the heater between the carbon source

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

and the seed (e.g. in a simple cylindrical heater the temperature is higher in the middle and it decreases along the axis of the cylinder). The driving force for the reconstitution of diamond is provided by the difference in solubility of carbon in the metal bath at different temperatures (as usual in crystal growth from solution). The carbon source is kept at a higher temperature, so that diamond dissolves in the bath whilst the seed is kept at a lower temperature. This generates a supersaturation of the carbon that deposits on the diamond seed homoepitaxially, fig. 4.4. The growth parameters (pressure, temperature gradient) must be extremely stable.

Fluctuations may produce growth defects that may not be tolerable for a crystal foreseen for X-ray optics applications. The growth rate of the diamond is also related to the temperature gradient: larger gradients give rise to higher growth rates, but metallic inclusions are easily incorporated in the diamond, destroying the quality of the crystal.

A compromise between high quality and feasible growth time has to be found. Typical growth rates for a good crystal are around 2 to 5 mg/h and typical growth times are of the order of 200 to 500 hours, [START_REF] Burns | Growth of high purity large synthetic diamond crystals[END_REF].

Defects in diamonds

During the growth process, numerous crystallographic defects may be formed inside the crystal, destroying the necessary perfection. These defects could be zero-, one-, two-

Diamond

or three-dimensional, respectively atomic impurities, dislocations, stacking faults and inclusions. Among a large variety of possible atomic impurities in diamond, the most common are nitrogen and boron. A classification of natural diamonds has been made

and it applies perfectly also to synthetic crystals, [START_REF]Properties of Natural and Synthetic Diamond[END_REF].

• Type Ia: high concentration of nitrogen, from 10 up to 3000 ppm, in aggregated form; two subcategories are the IaA with nitrogen clusters made by neighbouring pairs of substitional atoms and the IaB, believed to be four substitional nitrogen atoms neighbouring a vacancy. These kinds of impurities do not absorb visible light, so they are typically transparent under optical illumination. Almost 98% of natural diamonds are type Ia.

• Type Ib: 25 to 800 ppm nitrogen atoms, in single substitutional form; these impurities absorb radiation in the UV region, with a tail also in the violet-blue, giving the characteristic yellow-brown colour. They are quite rare in nature but are the most common synthetic diamonds produced because of the higher growth rate.

• Type IIa: very low concentration of nitrogen, normally less than 10 ppm; they are optically transparent and very rare in nature albeit some of the finest historical stones, such as the Cullinan and Koh-i-Noor, are type IIa.

• Type IIb: in this case the classification is not due to the nitrogen but to the boron impurities, up to 50 ppm; this kind of impurity gives absorption in the red/IR region, giving to the stone a characteristic blue colour. They are very rare in nature but it has been discovered that they show interesting semiconducting properties

The concentration of N impurities in synthetic diamond strongly depends on the crystallographic orientation of the growth sector, [START_REF] Woods | Cathodoluminescence, optical absorption and x-ray topographic studies of synthetic diamonds[END_REF]. For example, growth in the 111 direction contains more N than growth in the 100 direction, and much more than in the 110 , fig.4.5. At the boundaries between these growth sectors, the lattice is slightly deformed due to the difference in the diamond lattice parameter, which for single substitutional nitrogen (also called C or P1 defect) is [START_REF] Burns | HPHT growth and x-ray characterization of high-quality type IIa diamond[END_REF] ∆a 0 a 0 = (0.12 ± 0.03) × C N (4.1) where C N is the concentration of nitrogen impurities expressed as an atomic fraction.

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

Due to the lower nitrogen content, type IIa diamonds are less affected by this kind of deformation. Consequently, it is accepted that for the highly demanding application of X-ray optical elements, type IIa crystals of very high quality are the preferred material.

To produce type IIa diamond, it is necessary to add to the metal solvent/catalyst elements with high affinity for N, such as Ti, Zr and Al. These elements are called "nitrogen getters" and forming nitrides such as TiN and ZrN in the solvent catalyst, they lower the concentration of nitrogen impurities inside the diamond. • Edge dislocations may be thought of as the result of adding a half-plane of atoms mid way through the crystal.

• Screw dislocations can be thought of as the result of cutting through the crystal and slipping one half with respect to the other by a lattice vector parallel to the edge of the cut.

It is possible to define a vector, called the Burgers vector, which represents the magnitude and direction of the lattice distortion arising from a dislocation. The Burgers vector and the dislocation line are parallel for a screw dislocation and perpendicular for an edge dislocation. In a mixed dislocation, the angle will be between 0 • and 90 • .

Stacking faults are planar defects and are regions within the crystal where the regular stacking sequence has been interrupted. Two types of stacking fault are possible in fcc

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

crystals: intrinsic stacking faults (ABCABABC instead of ABCABCABC), which can be pictured as vacancy agglomeration and extrinsic stacking faults (ABCABACAB), which can be thought as a layer of interstitial atoms added to the sequence. Inclusions are normally due to metal from the solvent catalyst bath that remains trapped inside the crystal during the growth process. The size of the inclusions can vary from 1 µm to 1 mm and their shape is normally plate or rod-like, with rounded corners. The strain in the crystal due to this kind of defect may be very large, so that inclusions must be completely avoided for a diamond to be use as an X-ray optical element. A greater concentration of defects is present near the seed (e.g. dislocations, stacking faults) [START_REF] Pal'yanov | High-pressure synthesis of highquality diamond single crystals[END_REF] and [START_REF] Pal'yanov | Highquality synthetic diamonds for SR application[END_REF]. The dislocations radiate outwards from the seed, following roughly the growth direction of the respective growth sector. Therefore the upper part of the crystal possesses quite a wide central zone almost defect-free. Moreover, as is usual in crystal growth, crystals with a lower density of defects may be grown using a defect-free diamond as the seed [START_REF] Sumiya | Crystalline perfection of high purity synthetic diamond crystal[END_REF].

Characterization of synthetic diamonds. Experimental results.

In chapter 3 I have introduced the different methods used to characterize synthetic diamonds, here I will present a selection of the results obtained.

High Resolution X-ray Diffractometry

After the determination of the experimental apparatus function discussed in 3.1 it is now possible to estimate the rocking curve broadening for different diamond samples due to the presence of defects. This has been done by using a similar procedure (model and fit) presented for the determination of the apparatus function, equation 3.2.

Table 4.1 shows an overview of different typical diamond plates, selected from all the samples analyzed. In order to quantify the broadening of the rocking curve, we decided to extract the full width at different peak heights, namely 50%, 20% and 2% of the maximum intensity. This allows one to study also the influence of defects on the wings of the rocking curve. For comparison, the Darwin widths of the intrinsic reflectivity curves of diamond for the different reflections, sample thicknesses and the energy used (14.413 keV) are given together with the white beam topographs. 1.000" 2.954" ∆FW50%M 0.04" ± 0.06" 0.86" ± 0.06" 0.37" ± 0.06" 0.43" ± 0.06" ∆FW20%M 0.08" ± 0.09" 1.71" ± 0.09" 0.74" ± 0.09" 0.68" ± 0.09" ∆FW2%M 0.28" ± 0.15" 3.25" ± 0.15" 2.61" ± 0.15" 11.9" ± 0.15" The results obtained show a negligible broadening for sample 1, IIa HPHT diamond, 100 oriented in a region close to the centre of the plate. For the Ib sample and the CVD diamond the situation is worse, with a greater broadening recorded at all levels.

For sample 4, a type IIa HPHT plate 111 oriented, the results are different. The bulk quality seems good (the theoretical rocking curve is broader for the 111 reflections than for the 400, therefore the increase of 0.43 arcseconds is relatively smaller), but the full width at 2% of the maximum is very large. We think this is related to a much worse quality of the surface of the plate, since the 111 surface is the most difficult to polish.

The surface quality of the crystal has been characterized also using X-ray techniques, and the results are presented in section 4.2.4.

X-ray Topography

In our case, a better adapted method to control the quality of the different diamond crystals is X-ray topography, introduced in section 3. the fact that the signal recorded is proportional to the X-ray intensity whilst for the films, an additional digitalisation step of the transmitted light is required and this is an additional source of error in the cases when a quantitative analysis is performed. In the case where the plate is cut in a different crystallographic direction, e.g. 110, the region closer to the seed is clearly distinguishable from the opposite side, see fig. 4.10;

White Beam Topography (WBT)

the seed was clearly on the bottom part of the topograph while the 100-cubic growth sector, defect free, is at the top.

Double crystal Topography

The main drawbacks for the white beam are the high background, the limited sensitivity to weak distortion and the sensitivity only to lattice inclination. This problems can be overcome by using a reference crystal (or a set of crystals) before the sample. The double contrast can be understood by assuming that at the boundaries between the two regions the lattice planes are distorted. After cutting out a crystal plate from the crystal, the internal stresses relax, leaving the surface stress free. This leads to a rather complicated strain field with two strongly strained regions close to the outcrops

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

of the growth sector boundary through the surfaces, as shown in fig. 4.12, from [START_REF] Härtwig | Investigation of quartz crystals with non-homogeneous distributions of impurity atoms by X-ray methods[END_REF],

and which are responsible for the double contrast.

X-ray topography in the non-dispersive configuration.

In the ideal configuration, the crystal conditioner is a perfect crystal of the same material and diffracts from the same atomic planes as the sample. Unfortunately this is not always possible. The problem of having a flawed reference crystal is that, since the distance from the source is very large, every defect on the first crystal is imaged together with the defects from the second. Fig. 4. 13 shows two white beam topographs (220 type reflections) of two 111 oriented type IIa HPHT diamond crystals. These two crystals have being used for a mechanical feasibility test of a double diamond high-heat load monochromator, [START_REF] Van Vaerenbergh | High Heat Load Diamond Monochromator Project at ESRF[END_REF].

Since it was only a feasibility test, the quality of the two crystals was not very high.

In fact, in both plates are present a high number of defects, mostly stacking faults and dislocation bundles. We note the very strong deformation present in the second crystal (vertical double line with interference fringes on both sides), that is similar to micropipes which are normally found in SiC. We suppose it is a superdislocation with a large Burgers vector but, as distinct from a micropipe, without being hollow.

Characterization of synthetic diamonds. Experimental results.

First crystal

Second crystal 2 mm In order to overcome the problem of dispersion in double crystal topography when a good reference crystal of the same material is not available, there are different solutions.

One is to obtain a beam with high monocromaticity and small divergence using multiple crystal systems, [START_REF] Hashizume | A Multiple Crystal System for High Strain-Sensitivity X-Ray Topography and Its Applications[END_REF][START_REF] Petroff | Synchrotronradiation plane-wave topography .1. Application to misfit dislocation imaging in III-V heterojunctions[END_REF]. We concentrated on two other possible solutions: to use a perfect crystal and to slightly bend it in order to correct for dispersion [START_REF] Jenichen | Checking plane-wave approximation for Double Crystal Topography with curved samples[END_REF][START_REF] Köhler | High-Resolution X-ray Topography[END_REF][START_REF] Altin | X-ray diffraction topography using a diffractometer with a bendable monochromator at a synchrotron radiation source[END_REF] or to use a set of planes in the beam conditioner whose d-spacing is not too different from that of the reflection in the sample [START_REF] Ishikawa | Synchrotron Plane Wave X-Ray Topography of GaAs with a Separate (+, +) Monochro-Collimator[END_REF]. I will now present some of the results obtained with the two methods. Finally, it is not possible to discern whether the inhomogeneity of the diffracted intensity for the topographs near the maximum of the rocking curve is due to the small dispersion or to the fact that the sample is slightly bent.

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS 2mm

Figure 4.15: Part of a series of topographs in a non-dispersive arrangement taken at ID19 using the beam coming from a bendable. E= 20 keV, 800 reflection of the bendable silicon crystal, 444 reflection for the diamond, Bragg geometry. The diamond sample is the same as the one in fig. 4.11. The step size between two images is 0.2" and the arrow shows the projection of the incident and diffracted wavevectors. By combining all the topographs collected at different angular positions it is possible to obtain a map of local rocking curves pixel by pixel. This method is called Rocking Curve Imaging RCI and has been developed by Lübbert and co-workers, [START_REF] Lübbert | µm-resolved high resolution X-ray diffraction imaging for semiconductor quality control[END_REF]. One of the requirements for the RCI is that the topographs are collected with a CCD camera.

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

In order to analyze the series of images I have used the program Visrock, developed by Lübbert and Baumbach, [START_REF] Lübbert | Visrock: a program for digital topography and X-ray microdiffraction imaging[END_REF] 1 Visrock analyzes the series of images to obtain different maps related to the local rocking curve pixel by pixel, such as angular position of the maximum or FWHM of the rocking curve. It is possible to perform a refined estimation of the nitrogen content difference between the different growth sectors using a similar series of topographs as the one shown in fig. 4.17 (with a smaller step size and an extended angular range), using the angular position of the centre of mass of each pixel.

For example, the difference between the growth sectors shown in yellow in fig. 4.17 and the large part of the diamond shown in orange is of ≈ 0.0002 • . This can be transformed in a difference of d-spacing of ≈ 3.5 × 10 -6 . This is similar to the value obtained by the previous estimation, but now it is obtained pixel by pixel. Moreover, this value

Characterization of synthetic diamonds. Experimental results.

is similar to what was found by Lang and co-workers, [START_REF] Lang | On the dilatation of synthetic type Ib diamond by substitutional nitrogen impurity[END_REF] on a type Ib sample. By using equation 4.1, we obtain a difference of concentration of (29 ± 7) ppm, which is a plausible value for Ib diamonds.

A second example of the potentiality of RCI is illustrated in fig. 4.18, which shows an example of the results obtained with a similar series of topographs of the IIa diamond shown in fig. 4.16 (with smaller step size and a broader scan). The rocking curve collected in one pixel belonging to the more perfect region, together with the integral rocking curve of the entire crystal, are compared with the theoretical reflectivity curve.

For the single pixel rocking curve, there are very small deviations from the ideal reflectivity curve, which is a confirmation of the high quality of this diamond plate in this region.

Quantitative analysis of the effective misorientation in a high quality type IIa diamond

It is possible to determine the effective misorientation defined in 3.4 using X-ray topography. Nevertheless, it is important to mention that to "back calculate" the origin of the deformation from the topograph is an inverse problem and is not possible in a general way due to the missing information related to the phase, which is lost when recording the diffracted intensities, [START_REF] Lerche | Direct backward calculation from x-ray double crystal topographs[END_REF]. An exception is given if it is possible to apply the dynamical theory for perfect crystal to locally deformed crystals, [START_REF] Härtwig | Hierarchy of dynamical theories of x-ray diffraction for deformed and perfect crystals[END_REF]. This is an approximation that holds only if the effective misorientation is small inside the "area of influence" [START_REF] Kubêna | Investigation of the growth striations in silicon by X-ray topography[END_REF], which is represented with the red regions for the Bragg and Laue case in fig. 2.7.

We used two different methods in order to determine the effective misorientation in this approximation:

• analyzing a topograph collected on the steepest part of the flank of the rocking curve, where the strain sensitivity is highest;

• collecting a series of topograph for several working points along the rocking curve and recording the angular positions of the maximum intensity in the different regions of the sample.

The first method was introduced by Bonse in 1958 [START_REF] Bonse | Zur Rontgenographischen Bestimmung Des Typs Einzelner Versetzungen In Einkristallen[END_REF]. If we assume that the changes in δθ in the region of interest are so small that the working point shifts in the linear

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

Characterization of synthetic diamonds. Experimental results.

part of the flank it is possible to directly relate the intensity variation on the topograph to the variation of the effective misorientation simply by

δI x I x max = k δϑ (4.2)
where δI x is the relative change of intensity, I x max is the X-ray intensity on the top of the rocking curve, k is the slope of the rocking curve normalized to one (the superscript x is to indicate X-ray). It is clear from equation 4.2 that for larger slopes the intensity variation corresponding to the same misorientation will be larger (or similarly, a same intensity variation will corresponds to smaller effective misorientation). One of the advantages of the bent monochromator is the possibility to change the strain sensitivity by changing the pair of reflections used for the monochromator and the sample. In order achieve this, a series of silicon bendable monochromators with different orientation are available. Fig. 4.19 shows three topographs of a high quality type IIa HPHT diamond, 100 oriented, see [START_REF] Burns | Diamonds for X-ray optical applications at 3rd and 4th generation X-ray sources -art. no. 67050K[END_REF]. The left image is a white beam transmission topograph, reflection 220. There are clearly visible scratches on the bottom left part of the plate, the rest of the plate is almost defect-free, with mainly three isolated dislocations barely visible. The other two topographs are collected using the bendable monochromator, in Bragg geometry. The middle one was taken at 12 keV, using the 444-reflection for the monochromator (Darwin width w h = 1") and the diamond 115-reflection (w h = 1.02").

The slope of the rocking curve, in the steepest point is about 1.2" -1 . If we assume that it is possible to detect intensity variation of 1%, this implies a theoretical detection limit of ≈ 4 × 10 -8 . In this topograph, the dislocations are now well visible, but the rest of the plate seems still homogenous. The right topograph was taken at 20 keV using Si-800 and C*-800 reflections (Darwin widths of 0.31" and 0.25" respectively).

The slope on the flanks is about 4" -1 , and the detection limit is ≈ 1.2 × 10 -8 . With this enhanced strain sensitivity it is possible to see inhomogeneities in the region which before were homogenous; moreover, the image width of the single dislocations is now much larger, up ≈ 1 mm from the core of the dislocation.

The topographs can be collected with either CCD cameras or with films. In the latter case, an intermediate conversion is necessary in order to obtain the X-ray intensity that stroke the film. If the film is not overexposed, the darkening S is proportional to the X-ray intensity. During the digitalization, optical light is transmitted through the film and is recorded by a CCD camera. The visible light is absorbed by the film following the Beer-Lambert law of absorption, S(x, y) = log(I L 0 /I L (x, y)), where I L 0 is the intensity of light transmitted at a point of the film that is unexposed and I L (x, y)

is the intensity at a point (x, y) of the image. These considerations will bring to the following

δI x I x max = I x -I x av I x max = log I L 0 I L -log I L 0 I L av log I L 0 I L max = log I L av I L log D R (4.3)
where the subscript av stands for the average intensity in the region we want to study (if possible -in a perfect reference region) and D R is the dynamic range. This is defined as the ratio of the light intensities in an unexposed part of the film, I L 0 , and in the region we are interested, measured on the topograph collected on the top of the rocking curve, I L max . The estimation of the dynamic range can be source of errors, e.g. if the angular step size used to collect the topographs is too large one could jump over the top of the rocking curve, resulting in a wrong I L max . Finally, by joining equations 4.2 and 4.3 we obtain:

δϑ = log I L av I L k log D R (4.4)

Characterization of synthetic diamonds. Experimental results.

It is now important to determine the value of k, which is the slope of the normalized rocking curve. As defined previously in section 3.1, the rocking curve is given by the convolution of the reflectivity curves of the two crystals, i.e. the bent silicon and the diamond crystal.

Uschmann and co-workers [START_REF] Uschmann | X-ray reflection properties of elastically bent perfect crystals in Bragg geometry[END_REF] have calculated the X-ray reflection properties of elastically bent crystals in Bragg geometry. They based their calculations on the Takagi-Taupin theory [START_REF] Takagi | Dynamical theory of diffraction applicable to crystals with any kind of small distortion[END_REF][START_REF] Taupin | Dynamic theory of x-ray diffraction in crystals[END_REF] for deformed crystals. It is necessary to understand whether for our experimental conditions it is necessary to calculate the reflectivity curve for the bent crystal or if the perfect flat crystal one is a sufficient good approximation. This is possible by estimating the bending radius of the silicon crystal. After Chukhovskii et al. [START_REF] Chukhovskii | Nondispersive Bragg diffraction in the general case of two cylindrically bent crystals[END_REF], the general equation for the non-dispersive setting for a polychromatic and divergent wave and two curved crystals is

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

tan θ B,2 tan θ B,1 = 1 - L 0,2 R 2 sin θ 0,2 1 + 2L 0,1 R 1 sin θ 0,1 1 + L 0,1 R 1 sin θ 0,1 (4.5)
where the index 1 and 2 represent the first and second crystal, R are the curvature radii (positive if the first crystal is convex and the second is concave), L 0 is the source to crystal distance, θ B and θ 0 are respectively the Bragg and incident angles, allowing asymmetric reflections to be considered. In our case we assume the diamond crystal to be flat, i.e. R 2 is infinite. From 4.5 is possible to obtain the radius of curvature of the first crystal to obtain a non-dispersive setting

R 1 = L 0,1 sin θ 0,1   1 + 1 1 - tan θ B,2 tan θ B,1   ‡ (4.6)
If we substitute in the formula typical experimental values for the distances and the Bragg angles, we obtain a bending radius of the order of few kilometres at ID19 and some hundredths of meters at BM05. Therefore in the first approximation, we decided to use the reflectivity curve of the flat crystal to calculate the theoretical rocking curve, and to calculate the slope on the flanks. Fig. 4.20 shows a comparison between the theoretical calculation (Si880 and C* 660 reflections, E=20 keV) and the experimental data. The data have been obtained from a series of topographs collected with a PCO camera on a high quality type IIa diamond and integrating the intensity in a ROI of ≈ 0.5×0.5 mm 2 , chosen in a part of the sample without defects. The agreement is quite good on the low angle side of the rocking curve but is worse on the high angle side. We attributed this difference to the fact that the first crystal is bent. In order to confirm this, we decided to calculate the rocking curve using the reflectivity curve of a bent silicon crystal. dislocation is much larger on the left topograph than on the right one. In addition, whilst in the topograph taken on the lower strain sensitive condition the intensity is quite homogenous on most of the crystal plate, on the topograph taken on the high strain sensitive side the image is more inhomogenous.

In the second method, a series of topographs at different angular positions is recorded.

The scan needs to be large enough in order to cover the entire rocking curve. By using the software Visrock, the angular position of the center of mass of the rocking curve of each pixel is obtained. This map represents directly the effective misorientations.

In both methods it is possible to separate the two components -∆d/d and ∆ϕ -of 3.4 by collecting the topographs at two opposite azimuthal positions, i.e. by rotating about the surface normal of 180 • between the two positions. This method was introduced by Bonse [START_REF] Bonse | Elastic strain and coloration pattern in natural quartz crystals[END_REF] in the study of natural quartz crystal, and has been used among others by Kikuta [154] on silicon crystals, by Yoshimura et al. [START_REF] Yoshimura | Measurement of local variations in spacing and orientation of lattice plane of synthetic quartz[END_REF] on synthetic quartz, by Lang et al. [START_REF] Lang | On the dilatation of synthetic type Ib diamond by substitutional nitrogen impurity[END_REF] on Ib diamond crystals and recently by Macrander et al. [START_REF] Macrander | Strain mapping with parts-per-million resolution in synthetic type-Ib diamond plates[END_REF] also on Ib diamond crystals. The idea behind this method is that, when changing the azimuthal positions of the crystal by 180 • , the regions where the effective misorientation is composed only of lattice spacing variation will present the same contrast, whilst for the tilted regions the contrast will be inversed.

By calling Θ 0 (x, y) and Θ 180 (x, y) the angular positions of the maximum for the pixel (x, y) and the azimuthal angles 0 • and 180 • respectively, we obtain that the d-spacing variation can be written as:

∆d d (x, y) = cot ϑ B Θ 0 (x, y) + Θ 180 (x, y) 2 (4.7)
whilst the angular tilt can be expressed as ∆ϕ(x, y) = Θ 180 (x, y) -Θ 0 (x, y) 2 (4.8)

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

It is possible now to compare the results obtained with the two methods on a high quality diamond sample. Fig. 4.24 shows the quantitative misorientation map obtained from a topograph collected on the steepest part of the flank of the rocking curve, obtained using the 880 reflection of the silicon crystal and the 660 reflection of the diamond, at 20 keV. The slope used in the calculation was 5.8" -1 , resulting in a theoretical detection limit of ≈ 0.810 -8 . The image was collected on a film due to the very low intensity of the reflection, and the dynamic range was estimated to be ≈ 24. A CCD camera was also used to collect the series of topographs at different angular positions, but the pictures had a very small dynamic range (≈ 5) and noisy data. Therefore the results obtained are worse and will not be presented. this case is smaller, with a slope of the rocking curve of ≈ 2" -1 . At the same time, the intensity of the reflections is higher than in the high strain sensitive configuration discussed previously, and therefore it was possible to collect images with a good statistic using the CCD camera. Therefore the values obtained with the two methods are in good agreement and are a proof of the validity of the method. It is therefore possible to select the method that better suits the experimental conditions. It is finally important to highlight the incredibly high quality of the diamonds studied. In regions of up to 1 × 1 mm 2 the quality is comparable to the one obtained for silicon crystals, [START_REF] Bonse | X-ray measurement of minute lattice strain in perfect silicon crystals[END_REF], and are therefore perfectly suited to be used as optical elements at the beamlines.

In fig. 4.27 it is possible to see the improvements in the quality of the synthetic diamonds used as phase plates at the ESRF. On the top row, two Ib diamonds used in the past as phase plates, the left one at BM28 and the right one at ID24; on the bottom row, two of the recently purchased plates, now in use at the beamlines ID06 and ID20.

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

Coherence measurements

In 2.3 the theoretical basis of the coherence of electromagnetic waves is presented. In this section I will show the results obtained using the Talbot effect in the characterization of the wavefront deformations related to the use of diamonds, especially when used as diffracting elements but also in transmission, as filter or window. The Talbot effect is the self-imaging of periodic structures under coherent illumination. It was discovered in 1836 by Henry Fox Talbot [START_REF] Talbot | Facts relating to optical science[END_REF] and subsequently explained by Lord Rayleigh [START_REF] Rayleigh | On copying diffraction-gratings, and on some phenomena connected therewith[END_REF] to be a manifestation of Fresnel diffraction, and it has been demonstrated for hard X-rays by Cloetens et al. [START_REF] Cloetens | Fractional Talbot imaging of phase gratings with hard x rays[END_REF]. When a coherent wave impinges on a periodic object, e.g. a diffraction grid, images of the object are formed at periodic distances from the object itself. The periodic distance is called the Talbot distance D and is given by:

D T = 2a 2 λ (4.9)
where a is the (lateral) period of the object and λ the wavelength of the light. For distances D equal to integer multiples of half the Talbot distance (also called D R , replication distance), an image of the object may be registered, with a translation equal to a/2 in the detector plane for the odd integers and no translation for even integers, see fig. 4.28.

In the case of a phase gratings, the results are different: no image is recorded at a distance equal to the replication distance (and its integer multiple) but images of the grid (with a contrast that depends on the phase shift created by the grating) are recorded at a distance equal to D T /4 (and odd integer multiples of this).

For a perfectly coherent illumination the pattern reproduces itself at each replication distance. In the case of a partial coherent beam, the contrast of the image at every distance becomes weaker and weaker at every repetition distance. It is possible to model this by introducing a damping factor S(zf ), [START_REF] Cloetens | Fractional Talbot imaging of phase gratings with hard x rays[END_REF]: The Talbot effect for an absorbing gratings of period a, duty cycle 0.1 (90% is opaque, 10% is transparent). Indicated are the positions of the screen where the maximum contrast with the same period as the grid are collected. The image is a re-elaboration of the one in [START_REF] Case | Realization of optical carpets in the Talbot and Talbot-Lau configurations[END_REF].

I m = S(
size, it is possible to obtain an estimate of the angular source size α 0 for the partially coherent beam as:

α 0 = a π 2(z 2 k -z 2 0 ) ln I 1 (z 0 ) I 1 (z k ) (4.11)
where z k and z 0 are distances from the screen with z k = z 0 + D r and the I 1 (z) are the Fourier coefficients for the frequency 1/a of the image collected at the position z.

Experimental results

The experiments were performed at the long beamline ID19. The source size is 30×120 µm 2 (V×H) and the distance from the source is 150 m, resulting in an angular source size of ≈ 0.2×0.8 µrad 2 and a transversal coherence length of ≈ 290×70 µm 2 at 0.6 Å. The choice of ID19 was taken especially considering the high degree of spatial coherence at the sample position. The grating we used was a two-dimensional grid, of period ≈ 6 µm, made of silicon. Fig. 4.29 shows an SEM image of the grid, taken using the secondary electrons with the in-lens detector. It shows that the differences of thickness in the grid are about 1.8 µm. At the energy we performed the experiment, 20.5 keV, the difference in absorption is negligible, ≈ 0.1%. Therefore the grid can be considered as a pure phase grid. The beam was produced by a U32 undulator with a gap of 17 mm, using the third harmonic. From the beam energy and the period of the grid is possible to calculate the Talbot distance, using equation 4.9. The results is D T ≈ 1.2 m. The detector was a FReLoN camera [START_REF] Labiche | FRELON camera: Fast REadout LOw Noise[END_REF]. The pixel size after the optics was 0.7 µm resulting in a field of view of 1.4×1.4 mm 2 . Images were collected at distances of 300 mm and 900 mm from the sample, being respectively D T /4 and 3D T /4, the positions where the contrast is maximum. Possible contributions from higher harmonics (mostly 333 reflection of the ninth undulator harmonics at 61.5 keV)

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

were reduced (e.g. by a factor ≈ 70 for the third harmonic) by detuning the second crystal of the monochromator ≈ 0.5" (≈ 20 % of the Si 111 Darwin width). From equation 4.11 it is possible to see that it is possible to obtain the angular source size (in the approximation of a Gaussian distribution for beam) by using the coefficients of the Fourier transform of the image of the grid (at the spatial frequency corresponding to the inverse of the period) for two distances, in our case D T /4 and 3D T /4. We collected a dataset of three images for each position: a darkfield image without the beam, a flatfield image with the beam but without the grid and finally a fullfield image, with the beam and the grid. In order to correct for "hot" pixels in the camera, which give fixed white spots, the darkfield was substracted from both the other images.

Characterization of synthetic diamonds. Experimental results.

Afterwards, to correct for the presence of inhomogeneities in the beam, the resulting images are divided, fullfield/flatfield. In order to measure the increase in angular source size (resulting in a smaller transverse coherence length) arising from imperfections in the diffracting crystals, it is necessary to measure the angular source size of the beam impinging on the sample. This was done simply by collecting the two images in transmission without any sample. The values obtained were 0.3×1.2 µrad, slightly larger than the theoretical value (0.2×0.8 µrad). This is related to the presence of optical elements (monochromator, beryllium windows and diamond filter) that disturb the wavefront.

To analyze the data I used a program written by Rafael Klünder (which is an improved version of a previous one by Peter Cloetens and Pierre Bluet). One of the improvements was the possibility of correct small misalignments of the grid with respect to the camera.

Also, the two corrected images can be slightly misaligned, and a translation correction was made in order to have them overlapping exactly. Another point in the analysis that was improved compared to the previous versions was the leakage correction. Leakage is a problem that arises when using a Fast Fourier Transform of a signal which is not periodic in the sample window. This results in a smearing out of the amplitude of the real frequency into adjacent frequencies. In order to correct for this effect, two processes are effectuated on the data. First, the window is automatically cut pixel All the data were collected using horizontal geometry, i.e. the diffraction vector was horizontal. Moreover, the beam was monochromatized by a double silicon monochromator, diffracting in the vertical plane. Therefore, the geometry was a dispersive in both the horizontal and vertical direction, resulting in a diagonal band diffracting at each angular position of the sample. For all the samples there is a considerable increase of the source size, from a factor of two to a factor of more than 4. The biggest values are for the type Ib diamonds and charged with diamond powder is used. The polishing process is strongly anisotropic:

the wear rate strongly depends on the crystallographic orientation of the surface being polished and on the direction of the polishing [START_REF] Grillo | The polishing of diamond[END_REF]. This is well known by gem polishers, who categorize diamond surfaces in "hard" and "soft" directions. The softer directions are the 100 on {110} and {100} surfaces (with the former being the softest)

whilst the hardest are all directions on a {111} surface. It has been recently shown using molecular dynamics simulations that during mechanical polishing of diamonds the crystal undergoes an sp 3 to sp 2 order-disorder transition resulting in the formation of an amorphous layer [START_REF] Pastewka | Anisotropic mechanical amorphization drives wear in diamond[END_REF]. The growth rate of this layer depends on orientation and the difference originates in the way the crystal bonds are broken. At the same time, the Therefore, alternative methods to the mechanical polishing need to be considered in order to obtain well-polished {111} surfaces. One method that avoids polishing is by cleaving the crystal. Cleavage is the tendency of a crystal to split along some definite crystallographic orientations. Diamond cleaves preferentially on the {111}

{111}

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

surfaces. A theoretical explanation has been recently given [START_REF] Telling | Theoretical Strength and Cleavage of Diamond[END_REF] relating it to a much lower strength and energy for fracture on the {111} planes compared to other planes.

Unfortunately, the cleaved surfaces normally appear not to be atomically smooth, but to present terraces, [START_REF] Van Bouwelen | Wear by friction between diamonds studied by electron microscopical techniques[END_REF][START_REF] De Theije | Oxidative etching of cleaved synthetic diamond 111 surfaces[END_REF]. Other alternative methods of polishing diamonds, such as chemo-mechanical, electrical discharge or thermo-chemical polishing, are presented in the reviews by Malshe, et al. [START_REF] Malshe | A review of techniques for polishing and planarizing chemically vapor-deposited (CVD) diamond films and substrates[END_REF] and by Chen and Zhang [START_REF] Chen | On the polishing techniques of diamond and diamond composites[END_REF].

The thermo-chemical polishing was investigated by our group at the ESRF, in collaboration with the University of Johannesburg. The results obtained have been submitted

to the proceedings of the XTOP 2010 conference [START_REF] Bussone | Investigation of surface and sub-surface damage in high quality synthetic diamonds by X-ray reflectivity and grazing incidence diffraction[END_REF]. Polishing using a hot-metal plate was introduced by Grodzinsky in 1953 [START_REF] Grodzinski | Diamond technology production methods for diamond and gem stones[END_REF] and has been recently developed by Yoshikawa [START_REF] Yoshikawa | Development and performance of a diamond-film polishing apparatus with hot metals[END_REF]. Different metals have been tested for the polishing wheel but the highest polishing rate is obtained using very pure iron. By pressing a diamond on a heated metal plate, carbon atoms are diffused from the diamond to the plate itself.

This process will remove atoms from the diamond surface and leaves it flatter by measuring angular positions of the symmetric 111 reflection, after having carefully adjusted the surface for the XRR data collection.

In our experiments we wanted to study both the roughness of the surface and the depth of the deformed layer created (if any) during the polishing process, using respectively X-ray reflectivity and Grazing Incidence X-ray Diffraction (in-plane). It needs to be pointed out that our understanding of the technical requirements to perform the experiments increased after each experimental session. For example, the requirements on the crystal plates are very stringent: the miscut angle needs to be smaller than the critical angle in order to be able to perform GID studies. For this reason, we decided to work at lower energy (7 keV for XRR, 8.5 keV for GID) in order to maximize the critical angle. To work at even lower energies would have required the beam path to be under vacuum to minimize absorption. We decided to avoid this since it would limit the number of samples that is possible to analyze due to the long time to switch from atmospheric to low pressures. depends on the objective; it can be 4 µm, 2.2 µm and 0.5 µm for the magnifications of 2.5x, 5x, 50x, respectively. In order to cover the whole surface of each diamond, since they are larger than the field of view of the objective, a microstitching interferometry technique is used. This consists of taking sub-apertures but overlapping them, and reconstructing the topography through the computational assemblage of the image fields obtained by VSI. The Hot Metal polished faces have deformed profiles, with a rms roughness of almost 1 µm. The cleaved surface shows a deeply terraced profile, with height differences of several µm. Sample 3 surfaces are curved, and have height differences across the face of ≈1 µm. This is due probably to small instabilities of the goniometer head during the polishing process; improvements in this direction are already being made by our collaborators in South Africa.

Optical surface profile and AFM results

Micro roughness analyses have also been performed, over four regions of approximately 100×130 µm 2 . On the polished side the roughness is ≈20 nm for sample 1, ≈100 nm for sample 2, and ≈5 nm for sample 3. The polished side of sample 3 is locally smoother than sample 1 and 2, and has some characteristic grooves, due to the lapping process.

The roughness on the cleaved surface is almost the same for all samples, the value being close to ≈6 nm locally. AFM results confirm VSI analysis. In

Sample preparation

The original carbon surface of diamonds can be rapidly polluted by airborne contaminants, or by contact with other material surfaces. Metal traces derived from the polishing process, and particulates in the atmosphere could be responsible for the presence of thin layers on the diamond surfaces which are then evidenced in the common surface investigation methods. In order to minimise surface contamination, before the XRR and GID analysis, the diamonds were cleaned in a boiling (≈90 • C) acid mixture of (HCl:HNO 3 ) in a molar ratio 1:1 for ten minutes and an acid preparation (H 2 SO 4 :HNO 3 ) with a 3:1 molar ratio at boiling temperature (≈250 • C), and subsequently stored in deionised water-filled hermetic ampoules. The acid etch applied is a modified process based on [START_REF] Maguire | Investigation of the Band Structure of Diamond Using Low-Energy Electrons[END_REF] and leaves an oxygen terminated surface [START_REF] Thoms | Production and characterization of smooth, hydrogen-terminated diamond C(100)[END_REF]. During the experiment, the samples were exposed to normal atmosphere for several minutes, before the X-ray data collection which was carried out in a helium atmosphere. The contamination in air to a level significant for the sensitivity of the techniques used cannot be discounted.

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

Grazing Incidence Diffraction

For the GID measurements an energy of 8.5 keV was used (the reason for this was that at 7 keV, 2θ B for the 220 reflection is close to 90 • , resulting in an impossibility of the detector, attached to the ν rotation of a six circle diffractometer, to move out of the plane of incidence). The incident beam was limited to 0.5×0.7 mm 2 (V×H). The scattering geometry was vertical, i.e. sample mounted vertically, in order to exploit the smaller vertical divergence of the beam. Detector slits where 0.4×5 mm 2 . The data were collected using a NaI scintillator counter (Cyberstar). In order to study the depth of the distorted region due to the polishing, we collected GID scans at different incidence angles. Two types of scans have been recorded, namely radial and transverse scans together with reciprocal space maps. The radial scans are sensitive to the dspacing changes whilst the transversal scans measure the mosaic spread. 1 (surface contribution) and a constant value (background). In addition the normalized areas under each curve were calculated and compared to assess the contributions arising from bulk and surface. The analysis of these data was performed by Genziana Bussone as part of her Master's thesis. The in-plane grazing incidence diffraction analysis shows clearly the difference between the two faces of sample 1. The cleaved face is locally smooth; in fitting the curves, the contribution of the background intensity is insignificant compared to that of the bulk.

At greater penetration depths an increase of the background, attributed to defects in the bulk, is observed. On the Hot Metal polished surface, the polishing process produces a deformed surface layer which gives high background and surface contributions at small penetration depths. Both background and surface contributions decrease as penetration depth increases, as the higher quality bulk starts to dominate the measurement. In reflection -202 radial scans, a strong asymmetry of the surface peak can be observed for depths less than 4 µm, the diffraction from the surface layer being displaced from that of the contribution from the bulk. It is possible to obtain the strain directly from the displacement of the surface peak. The strain calculation indicates a compressivelystrained, near-surface region a few µm deep. The strain calculation for the cleaved face, considering the same reflection, shows a region a few micrometers deep ( 4 to 7

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

µm) under a smaller compressive stress. At greater penetration depths, more than 7 µm, all data tend toward bulk behaviour. In contrast, the 0-22 reflection of both faces shows approximately bulk behaviour for all penetration depths. Thus it appears that the surface strain is anisotropic, as shown in fig. 4.37. of the Hot Metal polished face, the FWHM values are higher than those at greater depths (tending to bulk), consistent with a deformed surface region. For the cleaved face, the FWHM values vary less with penetration depth, that is, the mosaic spread at the surface is no greater than in the bulk.

X-ray Reflectivity

Measurements of specular and transverse diffuse reflectivity curves have been performed in order to obtain information about the roughness, density, thickness and the interfaces of thin layered structures apparent in the samples. The true specular reflectivity was obtained by subtracting the average forward-diffuse scatter ±0.12 • either side of the specular peak, from the specular XRR data. The fitting was performed by Tamzin Lafford using the Bede REFS software [START_REF] Wormington | Characterization of structures from X-ray scattering data using genetic algorithms[END_REF].

Characterization of synthetic diamonds. Experimental results.

1.E-10

1.E-08

1.E-06

1.E-04

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00 -0.25 -0.15 -0.05 0.05 0.15 0.25

Normalised intensity q x [Å -1 ]
-Hot Metal Polished face -Cleaved face q z = 0.062 q z = 0.124 q z = 0.248 q z = 0.031 q z = 0.046 q z = 0.062 q z = 0.062 q z = 0.124 q z = 0.248 Sample 3 Chapter 5

Improvements to polishing of silicon crystals and beryllium windows

In this chapter I will present results obtained in the characterization of materials, mainly high quality single crystal silicon and ultra high purity beryllium foils, used as optical elements that should not disturb the wavefront, spoiling the coherence properties of the synchrotron beam.

Improvements and quantification of silicon crystals processing at the ESRF Crystal Laboratory

Thanks to the huge investment made in the last decades by the electronic industry, [START_REF] Zulehner | Historical overview of silicon crystal pulling development[END_REF], it is possible to obtain silicon crystals with a very high quality and a residual strain of the order of 10 -8 , [START_REF] Bonse | X-ray measurement of minute lattice strain in perfect silicon crystals[END_REF].

In order to be used as optical element, the crystal needs to be cut into the foreseen shape and the surfaces need to be polished with high quality. This is done in different steps that, for what concerns the ESRF, are often made at the Crystal Laboratory:

• cutting the plates from the silicon float zone wafer with diamond saw (76 µm grain);

• rough lapping using silicon carbide (SiC) powder (37 µm) mixed with water;

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

• chemical etching for 3 to 5 minutes in a bath made of a mixture of nitric acid (HNO 3 ) and hydrofluoric acid (HF) in a ratio 1:4 and soft lapping with SiC of 17 µm;

• mechanical polishing (also called optical polishing since it results in mirror-like surfaces), using a slurry of diamond powder with 1 µm grain size and alcohol.

For this process the polishing plate is covered by a silk cloth.

It is known that processing of crystals induces lattice deformations in the crystal, [START_REF] Buck | Crystallographic damage to silicon by typical slicing, lapping and polishing operations[END_REF][START_REF] Zhang | Towards a deeper understanding of plastic deformation in mono-crystalline silicon[END_REF], in the form of a cold deformed layer. This layer creates an elastic strain in the crystal underneath whose strength decreases with depth. The thickness of the deformed layer is proportional to the dimension of the grain used. Therefore, each step is required to remove the deformed layer from the previous step but at the same time, it creates a new thinner deformed layer. One way of removing a deformed layer is to chemically etch it away; however, as it will be shown in the results, etching results in a modulation of the surface, called "orange peel". Depending on the applications, this modulation may be tolerated or not, e.g. for small angle scattering it needs to be minimised or avoided.

Another way of removing a deformed layer is to use a mechano-chemical polishing, which uses very small grains and leaves a flatter surface compared to the etching. At the ESRF Crystal Laboratory this consists of two steps. In the first step it is used a slurry of colloidal silica (Grace Davison LUDOX R P-W30) with grain size dispersed around an average value of 30 nm and a pH of 9. The second step is done by replacing the pad used in the previous step with a polyurethane embossed pad named ÅKERS SUPRA5 and using the same slurry as before.

The removal rate of this technique is rather low (typically of the order of 1 µm per hour) and therefore it can be used only crystals which have a very small (few µm thick) deformed layer.

In order to find the critical points of the actual procedure, we performed different polishing tests.

First test -Localization of the main problem.

Normally at the Crystal Laboratory, the way to inquire whether a crystal is well polished or not is by inspection via optical microscopy. Fig. 5.1 shows three images, each showing

Improvements and quantification of silicon crystals processing at the ESRF Crystal Laboratory

the state of the surface after a removal of respectively 9 µm, 13 µm and 19 µm. Many spot-like features are visible on the first image; these are indentations related to the lapping step. In the second image, most of the indentations appear to be removed.

Finally in the last picture, no more indentations are visible. At this stage, the crystal was considered as ready for the following final step, the mechano-chemical polishing. In order to verify whether this procedure was accurate enough we made a series of 111 oriented silicon samples. The reason to work with 111 oriented samples is twofold: first, as shown previously in section 4.2.4, this is the preferred direction for Bragg monochromator. Second, it allows for the collection of non-dispersive X-ray topographs with the standard double Si111 monochromator of BM05. The description of the different samples is shown in tab.5.1. The three etching steps were done in order to obtain a rough idea of how thick is the deformed layer due to the lapping and mechanical polishing. The etching reaction is

Step/Sample Description

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

composed of two steps [START_REF] Robbins | Chemical Etching of Silicon[END_REF][START_REF] Steinert | Study on the Mechanism of Silicon Etching in HNO3-Rich HF/HNO3 Mixtures[END_REF]:

3 Si + 4 HNO 3 → 3 SiO 2 + 4 NO + 2 H 2 O SiO 2 + 6 HF → H 2 SiF 6 + 2 H 2 O
the first step is oxidation of silicon by nitric acid followed by dissolution of SiO 2 by the hydrofluoric acid. The samples have been characterized using different methods, namely High Resolution X-ray Diffraction, X-Ray Topography, Optical Profilometer and AFM.

The HRXRD measurements have been carried out initially with a laboratory source first and then again afterwards at ID18. For the laboratory source setup, a crystal with an asymmetry angle of 9.5 • from the 111 direction has been prepared by the crystal lab. This crystal used in grazing incidence geometry (with asymmetry factor b=-0.2 for Cu-K α1 radiation) act as a monochromator and as a collimator, producing a radiation with a divergence 5 times smaller than the acceptance of the sample. In order to avoid contaminations of the Cu-K α2 radiation, a double set of slits has been used at the exit of the X-ray tube.

An example of the normalized recorded rocking curves of the 111 reflection for the series of samples is shown in fig. 5.2. It is obvious (and expected) the large difference

Improvements and quantification of silicon crystals processing at the ESRF Crystal Laboratory

between the rocking curve for the lapped (sample 1) and optical polished (sample 2) samples, due to the bad surface status. But already after the first etching (sample 3), the different rocking curves become indistinguishable and they agree perfectly with the theoretical values.

A slight difference is recorded if the rocking curves are measured with higher order reflection (444) using the highly monochromatized beam produced at ID18, E=21.541 keV (corresponding to the 151 Eu nuclear resonance transition) with ∆λ λ ≈ 10 -7 , see fig. 5.3. In this case it is possible to discern a small difference in the tails between samples 3 -4 and sample 5. The difference between the samples appears to be even clearer when comparing X-ray topographs of the different samples, fig. 5.4. The topographs are collected at BM05 on the flanks of the rocking curve, non-dispersive configuration at 20 keV. The strain detection limit for these conditions is ≈ 1.4 × 10 -7 , obtained from the slope of the rocking curve, which is 0.35" -1 and by considering that it is possible to detect intensity variation of the order of 1%. In the lapped sample 1, the deformed layer creates a grainy structure of deformations. In sample 2 scratches coming from the optical polishing are clearly visible, together with strong indentations, black spot-like contrasts, coming from the lapping. For the second series, we decided to add one extra polishing step with a granularity between the lapping at 17 µm grain size and the optical polishing at 1 µm, using an intermediate slurry with silicon carbide of 9 µm. After this extra step, the four samples have been polished like for the previous series, obtaining an estimate thickness removal of 14 µm, 29 µm, 56 µm and 73 µm.

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

The two series of samples have been characterized using Rocking Curve Imaging, [START_REF] Lübbert | µm-resolved high resolution X-ray diffraction imaging for semiconductor quality control[END_REF].

We collected 101 images along the rocking curve with a step size of 0.2", using the symmetric 111 reflection at E=20 keV. The images have been analyzed using the Visrock software, [START_REF] Lübbert | Visrock: a program for digital topography and X-ray microdiffraction imaging[END_REF]. The FWHM maps appear to show all the deformations in the crystal and we decided to use them.

Some of the results obtained are presented in fig. 5.6. For both series it is clear the complete disappearance of indentations spots in the topographs for removal of the order of 30 µm and above. No particular difference was noticed between the series of samples with the intermediate lapping with 9 µm grain and the one without this step.

Since the extra step brings technical complications (related to the limited number of polishing machines available at the Crystal Lab) we decided to not pursue this solution any longer.

These results confirmed what was obtained with the previous rough estimation made via etching. It is necessary to remove at least 30 µm of material during the optical polishing. In order to be on the safe side, we decided that for all the future crystals polished at the Crystal Laboratory at least 40 µm needs to be removed.

Finally, we decided to also optimize the next polishing step, the mechano-chemical polishing.

Third test -Optimization of the MCP.

An important parameter for the Mechano Chemical Polishing step is the pH of the slurry used. Using the same philosophy as before a series of samples were prepared in an identical way (lapping, optical polishing with removal thickness of at least 40 µm).

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

Sample 41 was put aside, as witness of the optical polishing. The other samples were mechano-chemical polished in three different ways:

• samples 42 and 43 using pure Grace Davison LUDOX R P-W30, ph 9, for respectively 12 and 24 hours;

• samples 44 and 45 using a mixture of LUDOX R P-W30 and GIGAPUR detergent, with a final pH of 12, for respectively 12 and 24 hours;

• samples 46 and 47 using a mixture of LUDOX R P-W30 and KOH, final pH 10.21 , for respectively 12 and 24 hours.

The layer removed was measured using the Fogale Low Coherence Interferometric Sensor. The values obtained vary between 2 µm and 20 µm, but they are not directly correlated with the polishing time or the pH used. It seems that other factors may influence the removal speed, e.g. the quality of the pad or of the slurry may change with time. The samples were characterized using Rocking Curve Imaging at BM05 and using optical profilometer and AFM at the Metrology Lab. From the X-ray topographs collected, we established that in the case the removal is below 3 µm, scratches are still visible. For removal above 10 µm, the scratches disappeared, but "clouds" of little dots appear in the topographs. We conclude that a removal between 3 µm and 10 µm seems optimal, but further tests to obtain a better quantification are necessary. The influence of the pH value on the quality of the crystals was not very clear. For what concerns the roughness, better values (between 3 Å at 50X and 9 Å at 2.5X) were obtained with higher pH (10.2 and 12) compared to the pH 9 (rms-roughness between 4-5 Å at 50X

and 17 Å at 2.5X).

Coherence measurements of the polished samples

The set of samples was also used to measure the transversal coherence of the X-ray beam after diffraction. In this way it is possible to verify the influence of the surface quality on the coherence. Two different experiments have been performed: the first one at ID19, using the Talbot effect already discussed in 4.2.3 for the diamond samples and the second at ID06 using an interferometer based on bi-lens [15].

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

Original topography

Vertical coherence map the transverse coherence length l t , see eq.2.28, then each lens is illuminated coherently and will generate two waves which will interfere at a distance z 1 > z f d/A eff , with A eff the absorption limited effective aperture of the lens. The authors demonstrated that

Coherence measurements of the polished samples

Coherence measurements of the polished samples

it is possible to measure the value of the source size from the visibility of the fringes using the following formula: at ID06 without any sample, which corresponds to a source size of 67 µm, which is much bigger than the expected value. A possible explanation is vibrations of the monochromator. Nevertheless, by comparing the relative increases in angular source size with respect to the reference values for both ID19 and ID06, we see that the behaviour is similar.

S = Λz 0 z f + z 1 - ln V 3.56

Coherence measurements of the polished samples

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

Improvements on the polishing of beryllium windows

Most beamlines at synchrotron radiation sources use beryllium for windows separating regions with high-vacuum and normal air pressure. For a good performance it is important that the window does not deform the wavefront. In order to obtain this, it is essential to minimise fluctuation in the bulk material density (no pores or voids, no density fluctuations) as well as to guarantee geometrically smooth surfaces.

1 notch 2 notches 3 notches 400 μm In collaboration with P. Theveneau from ESRF and the company Brush Wellmann we performed a study of the quality of the material and of the polishing of Be plates of the ultra high purity class IF-1 R produced by eletrofusion.

The tests were performed at ID19 and BM05 collecting phase sensitive X-ray radiographs of the three Be plates (≈ 500 µm thick) by using a large sample to camera distance after different polishing steps. Moreover, the geometrical surfaces were characterized by using a Micromap Promap 512 optical profiler at the Metrology Laboratory.

In order to classify the three samples, each one had a different number of notches on the corner, respectively one, two and three notches. The results of the first test are shown in fig. 5.14. This test was performend at ID19, using a FReLoN camera with 0.7 µm resolution and a field of view of 1.4 × 1.4 mm 2 . The energy of the beam was 20.5 keV and the sample-to-detector distance 0.9 m. conclude, the sample processing should still be improved, in order to have surface modulations of less than 0.1 µm and rms roughness smaller than about 20 nm.

Chapter 6

X-ray coherence exploited in imaging of ferroelectric domains

In this chapter, I will present an example of applications exploiting the transversal coherence of the X-ray beam: the investigation by coherent X-ray section topography of ferroelectric domain behaviour as a function of temperature in periodically-poled rubidium-doped potassium titanyl phosphate (Rb:KTP).

Before entering into the details of the experiment, I will introduce briefly some concepts about ferroelectricity, together with the main applications of periodically-poled ferroelectric crystals. Afterwards, I will describe the furnace that we developed and characterized at the ESRF and that allowed the investigation. Finally I will present the results obtained, together with simulations and a model proposed to describe the behaviour of the domains at high temperature.

Ferroelectricity

Ferroelectricity is the property of some dielectrics to have a permanent polarization in the absence of an external electric field. The discovery of ferroelectric crystals by J.

Vasalek dates back to 1920 [START_REF] Valasek | Piezo-Electric and Allied Phenomena in Rochelle Salt[END_REF]. By modifying the electric field on a plate of Rochelle salt (potassium sodium tartrate) he recorded a hysteresis loop in the polarization, with the presence of a residual polarization at zero electric field. The name ferroelectricity comes by analogy with the properties of ferromagnetic materials, although the physical natures of the phenomena are very different. By analogy with ferromagnetic crystals,

X-RAY COHERENCE EXPLOITED IN IMAGING OF FERROELECTRIC DOMAINS

ferroelectric crystals possess a temperature, the Curie temperature T c , where a phase transition from the ferroelectric to the paraelectric state occurs.

All ferroelectric crystals are non-centrosymmetric and they have one polar axis. They belong to a larger class of crystals, called pyroelectrics, which are polarized if the temperature is changed.

Barium titanate (BaTiO 3 ) is an example of a ferroelectric crystal. This crystal undergoes different phase transitions with temperature. Above 393 K, it has a perovskite structure: this is a centrosymmetric cubic structure. Therefore the crystal is paraelectric, since all the charges in the lattice are compensated. At lower temperatures, the crystal is subject to different structural changes, becoming tetragonal at 393 K, orthorhombic at 278 K and finally rhombohedral below 183 K. In all these structures, the crystal lattice is no longer centrosymmetric, and the charges are not compensated due to a displacement of the cations Ba 2+ and Ti 4+ with respect to the O 2 -. This displacement is in three different crystallographic directions: [START_REF] Ishikawa | X-ray topography with highly collimated beam at photon factory[END_REF] for the tetragonal, [START_REF] Yoneda | Anomalous Surface Reflection of X Rays[END_REF] for the orthorhombic and [START_REF] Marra | X-ray total-external-reflection-Bragg diffraction: A structural study of the GaAs-Al interface[END_REF] for the rhombohedral structure.

In fig. The pictures of the structures were generated using structural data from [START_REF] Buttner | Structural parameters and electron difference density in BaTiO 3[END_REF] and the software Vesta. The difference between the two structures is very small, and the best way to see it is to focus on the slight bending of the Ti-O bonds upwards in the tetragonal structure.

to depolarize the crystal, it is necessary either to increase its temperature above T c or to repeat the hysteresis cycle with smaller and smaller electric field strengths until the original position O is reached.

The applications of ferrolectric crystals are very wide-ranging: among others, they can be used as tunable capacitors, non-volatile memory, sensors, e.g. in ultrasound machines, or in infrared cameras. Moreover, some ferroelectrics present very high nonlinear optical properties that make them very useful as optical devices, for example in laser optics as frequency converters via quasi-phase matching (QPM). This particular application was foreseen for the crystal we have been studying, therefore the basis of the theory will be introduced here.

Ferroelectric crystals used for non-linear optics applications.

When electromagnetic radiation propagates in a medium, the electric field produces a polarization density inside the medium. This polarization can be expressed as a Taylor series expansion [START_REF] Xu | Ferroelectric materials and their applications[END_REF]

P i (t) = j 0 χ j E j (t) + j,k 0 χ (2) j,k E j (t)E k (t) + j,k,l 0 χ (3) j,k,l E j (t)E k (t)E l (t) + • • • (6.1)
where the subscript indicates the Cartesian component of P (t), the instantaneous polarization; E(t), the instantaneous electric field; χ the linear susceptibility; and χ (n) are the non-linear susceptibility of order n.

When the intensity of the electric field is weak, the non-linear terms may be neglected but in the case of high intensities, e.g. lasers, they cannot, resulting in non-linear optical effects.

One of the non-linear effects widely used in laser technology is the so-called Second Harmonic Generation (SHG), also known as frequency doubling. This phenomenon was observed for the first time by Franken and co-workers in 1961 [START_REF] Franken | Generation of Optical Harmonics[END_REF]. In fig. 6.3 is shown a diagram of SHG. An intense incident beam is passed through a non-linear crystal. Inside the crystal, there will be two polarization waves, corresponding to linear and quadratic susceptibilities. Therefore, at the exit of the crystal there will be a wave composed of two wavelengths: the fundamental, which has the same wavelength as the pump-wave; and the SHG wave, with a wavelength that is half of the incident wavelength. In the picture, the two waves are ideally separated by a prism. SHG is extremely sensitive to the phase of the waves. If the phase of the two waves is not matched, the intensity of the frequency-doubled radiation produced is very small. In order to be phase matched, the phase velocities of the pumping and the frequencydoubled radiations need to be equal. This implies that n(ω) = n(2ω). The waves are said to be perfectly matched and the intensity of the SHG-radiation will increase inside the medium, at the same time reducing intensity from the pump wave, in socalled (pump depletion). Nevertheless, in the normal dispersion region, n(ω) < n(2ω), resulting in different phase velocities. The result of having different phase velocities for the two waves is depicted in fig. 6.4.

The graph represents the value of the second harmonic electric field, E S H as a function of the distance travelled, z, by the wave in the crystal. The black dash-dotted line indicates the case of perfect phase matching, whilst the green dashed line shows the case of zero phase matching. In the small inset are plotted the photon momenta for the different cases. In the case of zero phase matching, the strength of the electric field increases until the radiation has travelled a distance called the coherence length 1 , defined as l c = π/∆k where k is the photon momentum. This point corresponds to the first half of the polygon plotted in green in the inset. Starting from this point, E S H decreases since the energy is flowing back from the frequency-doubled wave to 1 not to be confused with the coherence length defined in 2.3

X-RAY COHERENCE EXPLOITED IN IMAGING OF FERROELECTRIC DOMAINS

the pump wave. After a distance z = 2l c the SHG wave disappears, with all the energy reconverted into the pump wave. This phenomenon repeats periodically, with small maxima for odd multiples of l c and minima for even multiples. In the inset is presented a geometrical approach to the phenomenon. In the case of perfect phase match, the arrows are all aligned, producing a linear increase of E S H with the distance. If there is a phase shift of ∆k, represented with a rotation of the photon momentum, the intensity increases until it reaches a maximum and decreases to zero, green curves. Finally, if quasi-phase match is implemented, it is possible to achieve an increase in the SHG radiation.

One way to match the phases is the so-called quasi phase matching. This method was proposed in theoretical work by Armstrong and co-workers, [START_REF] Armstrong | Interactions between Light Waves in a Nonlinear Dielectric[END_REF] and consists of alternating slabs of crystals with each slab being a symmetric inversion of the previous one with respect to the polar direction. If the length of the slabs is matched to the coherence length, it is possible to obtain the red curve in 6.4. Among the different methods used to achieve this, we are interested in the periodic poling of ferroelectric crystals.

Periodic poling is a method to obtain inversed domains in a crystal with widths which match the coherence length. The method was proposed by Yamada and co-workers, [START_REF] Yamada | First-order quasi-phase matched LiNbO 3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation[END_REF]. It consists in applying to a single-domain ferroelectric crystal an external electric 6.3 Visualization of ferroelectric domains using X-ray field higher than the coercive field. The field is applied to periodic electrodes which are deposited using lithographic techniques on the c+ polar face of the crystal. In this way, it is possible to obtain a periodic structure of inverted domains. The width of the period is matched to the coherence length, which is calculated via the dispersion relation. When the two domains have the same width, i.e. duty-cycle 50:50, the quasiphase matching is the highest and is said to be of first order. Domain widths are typically of the order of 3 µm to 40 µm.

Visualization of ferroelectric domains using X-ray

There are several methods of visualizing inverted ferroelectric domains, among others polarization microscopy, surface etching, powder or liquid crystal decorations, atomic force microscopy and X-ray topography techniques [START_REF] Soergel | Visualization of ferroelectric domains in bulk single crystals[END_REF]. I will concentrate only on the last. First of all, we need to consider that for the 180 • inverted domains, there is no lattice mismatch between the domains at the domain walls, unlike for e.g the 90 • ones.

Nevertheless, since the polarity is inversed, the same diffraction spot will be hkl for one domain type and hkl for the other, composing a Friedel pair. In a Friedel pair, the moduli of the structure factors are the same whilst the phases are inverted, therefore no intensity variation between domains is expected within the diffracted beam under normal conditions.

One way to visualize the inverted domains is to work in anomalous scattering conditions, i.e. at an energy close to an absorption edge of one of the elements. If one of the atoms in the lattice is scattering anomalously, then the amplitude of the factor structure is different for each component of the Friedel pair. This method has been used by Nizeki and Hasegawa [START_REF] Niizeki | Direct Observation of Antiparallel 180 • Domains in BaTiO 3 by X-Ray Anomalous Dispersion Method[END_REF]. They visualized inverted domains in a BaTiO 3 crystal using CrK α radiation and the 00±3 Friedel pair.

Another method of visualizing inverted ferroelectric domains is by a combination of Bragg and Fresnel diffraction [START_REF] Rejmánková-Pernot | Phase Retrieval by Combined Bragg and Fresnel X-Ray Diffraction Imaging[END_REF], and it is the method we used for our experiments.

This method requires a large transverse coherence length of the incident X-ray beam, and can therefore be performed only at third generation synchrotron sources. As previously stated, the phase of the inverted domains is different from each reflection of the Friedel pair. A phase jump ∆φ = φ hkl -φ hkl is present. Therefore a periodically poled ferroelectric crystal can be seen as a phase grating by the incoming X-ray beam if set in

X-RAY COHERENCE EXPLOITED IN IMAGING OF FERROELECTRIC DOMAINS

Bragg condition for a given reflection. As it has been shown in 4.2.3, a periodic object illuminated with a coherent beam will produce a periodic pattern in both the direction of the periodicity and in the direction of propagation of the beam.

This method has been successfully used to solve the structural matching between inverted domains in potassium titanyl arsenate (KTA) and potassium titanyl phosphate (KTP), [START_REF] Pernot-Rejmánková | Periodically poled KTA crystal investigated using coherent X-ray beams[END_REF][START_REF] Pernot-Rejmánková | Structural matching of ferroelectric domains and associated distortion in potassium titanyl phosphate crystals[END_REF]. It has also been successfully applied to in situ studies of the behavior of rubidium titanyl arsenate (RTA) under an electric field [START_REF] Walker | X-ray imaging investigation of periodically electroded rubidium titanyl arsenate, RbTiOAsO 4 , under an applied electric field[END_REF]. In the experiments we performed, we wanted to study the behaviour of domains at temperatures close to and above the Curie temperature of KTA and rubidium doped KTP (Rb:KTP) crystals.

In fact, the process of transition from one stable polarization state to another with opposite direction of the macroscopic electrical dipole remains poorly understood and related studies on KTP and isomorphic crystal are limited. However, its understanding is crucial for obtaining highly efficient QPM devices.

In order to perform the experiment, we developed a furnace whose main characteristic is to preserve the coherence of the incoming X-ray beam.

Coherence preserving furnace

The furnace was developed in the Sample Environment Laboratory of the ESRF by B.

Gorges and H. Vitoux. The results of the characterization of the furnace that I report here have been also published in an article, [START_REF] Pernot | A furnace for coherent beam transmission topography applied to ferroelectric crystals[END_REF], of which I am a co-author. The challenge in the design of the furnace concerns the facts that the crystals we want to study are, in general, optically transparent and thermal insulators, and the experiments needs to be performed in transmission geometry. If the sample is heated only from below, a very large thermal gradient would be produced, provoking distortions that would conceal the effects we are looking for. Therefore, an alternative way of heating the sample was proposed, consisting of two black-body radiator plates located near the sample. The radiators need to have low absorption for X-ray in the 15-50 keV range, be chemically homogenous, with very small thickness variation, i.e. very well polished, with good heat conductivity and high irradiation power. By considering all these requirements, a good solution seemed to be to use single silicon crystal plates of 150 µm thickness.

Coherence preserving furnace

Another problem is the fixing of the sample in order to keep it free of strain when heated. The use of glue or a clamp can introduce distortion when the sample is heated due to the different thermal expansion coefficients of different materials. In order to avoid this, the sample is simply rested on a boron nitride support between the silicon plates. A thermocouple is in contact with the lower part of the sample through a hole in the boron nitride support. Moreover, to avoid deleterious contact between the heater, made out of nickel, and the silicon, a boron nitride framework was also fabricated. In (the strain field related to the different domains tends to disappear), while the middle and top parts of the crystal are still in the ferroelectric state (still strongly distorted). Finally, the last test was intended to check whether the furnace preserved the coherence of the beam. In order to do so, we measured the coherence of the beam using the Talbot effect as explained in 4.2.3. We measured the value of the angular source size:

Coherence preserving furnace

(i) without the furnace; (ii) in transmission with the furnace at room temperature;

(iii) in transmission with the furnace at 800 • C. We obtained the following results: (i) 0.80 ± 0.02, (ii) 0.98 ± 0.02, (iii) 0.99 ± 0.02. A slight increase in the value of the angular source size was measured. Nevertheless, the transversal coherence length of the beam with the furnace is ≈40 µm and does not depend on the temperature. This value is enough for the visualization of inverted ferroelectric domains through the coupling of Bragg and Fresnel diffraction.

X-RAY COHERENCE EXPLOITED IN IMAGING OF FERROELECTRIC DOMAINS

6.5 Visualization of inverted domains in KTA and Rb:KTP crystals.

We performed two separated experiments, both at the ESRF beamline BM05. In the first experiment we studied a KTA crystal whilst in the second one we studied a Rb:KTP crystal.

The KTA sample was about 3 × 3 × 0.4 mm 3 with a poling period of 39µm. The Curie temperature for KTA is 880 • C [START_REF] Bierlein | Linear and nonlinear optical properties of flux-grown KTiOAsO[sub 4[END_REF]. In order to index the diffraction spots I used a software called Orient Express1 , see fig. 6.8. .8: Indexation of the KTA crystal using the software Orient Express. In green, the reflections which were analyzed because they showed a maximum contrast of the periodic pattern. Some spots are due to the silicon crystal plates and therefore are not taken into account in the indexation process.

Unfortunately, the experiment was not very successful, due to a severe distortion of the crystal under heating, fig6.9. We understood that the calibration made for the thermal gradient was not correct at higher temperatures (above 700 • C ) and that we needed to improve the design of the furnace if we wanted to study samples at such high temperatures.

X-RAY COHERENCE EXPLOITED IN IMAGING OF FERROELECTRIC DOMAINS

Thanks to the technical help of colleagues from the Sample Environment Lab, we made two improvements. First, we added a small aluminium cylinder with two holes (one for the incoming and one for the diffracted beam) to be used as an additional infrared mirror. Second, we enclosed the whole furnace inside a larger dome, again with entrance and exit holes for the beam, closed with Kapton R foils, 6.10. By using a primary pump, we evacuated the air inside the dome, then we replaced the air with nitrogen (in order to avoid oxidation of the nickel components of the furnace) and finally we evacuated These improvements allowed us to perform the experiment on a Rb-doped KTP crystal. The results of this experiment have been published in the Journal of Applied Crystallography, [START_REF] Masiello | Investigation by Coherent X-Ray Section Topography of Ferroelectric Domain Behaviour as a Function of Temperature in Periodically-Poled Rb:KTP[END_REF].

Investigation by coherent X-ray section topography of ferroelectric domain behaviour as a function of temperature in periodically-poled Rb:KTP.

Rb-doped KTP is one of the most attractive ferroelectric non-linear optical materials for quasi-phase matched devices due to its limited domain broadening, which grants 6.6 Ferroelectric domain behaviour with temperature in Rb:KTP high pattern fidelity, and to its high resistance to optical damage, especially in the bluegreen region [START_REF] Wang | High-efficiency frequency converters with periodically-poled Rb-doped KTiOPO4[END_REF][START_REF] Jiang | Rb-doped potassium titanyl phosphate for periodic ferroelectric domain inversion[END_REF]. The sample was obtained from a 3mm thick single domain c-cut flux-grown Rb:KTP crystal. The poling was done using an aluminium grating with period Λ = 38.83 µm with a duty cycle of 25%:75% (Al:insulator). The thickness of the sample was then reduced to 500 µm by mechano-chemical polishing. The structure of the Rb:KTP crystal used is shown in fig. 6.11. The structural data were obtained from a small section of the crystal (210 x 80 x 120 µm 3 ) using a Gemini kappa-geometry CCD Diffractometer (Oxford Diffraction Ltd., UK) with Mo-Kα radiation (0.707 Å).

The refinement of these data confirmed the structure as that accepted for pure KTP, namely orthorhombic in space-group Pna21. The refined percentage of Rb replacing K was 0.3%ś0.1%. The Curie temperature was estimated to be approximately the same as for KTP, Tc (≈ 935 • C, [START_REF] Yanovskii | Ferroelectric phase transitions and properties of crystals of the KTiOPO 4 family[END_REF]) given the very low content of Rb and the linear relationship between T c and the amount of Rb suggested by Thomas et al., [START_REF] Thomas | Crystal structures of RbTiOAsO 4 , KTiO(P 0 .58,As 0 .42)O 4 , RbTiOPO 4 and (Rb 0 .465,K 0 .535)TiOPO 4 , and analysis of pseudosymmetry in crystals of the KTiOPO 4 family[END_REF]. The fundamental harmonic of reflection used in the experiment was the 162 as obtained by using the software Orient Express. By knowing the d-spacing of the 162 reflection

X-RAY COHERENCE EXPLOITED IN IMAGING OF FERROELECTRIC DOMAINS

together with the distances from the sample to the film and from the topograph to the direct beam spot, it is possible to obtain the wavelength of the fundamental harmonic by simply applying Bragg's law. We obtained a value of 0.62 Å for the first harmonic.

Nevertheless, considering the sample thickness and absorption, we estimated that most of the fundamental harmonic was absorbed. Therefore the spot recorded corresponds to the 12 2 4 reflection with a wavelength of 0.31 Å. The phase shift between adjacent domains was calculated using the structural data and the software DABAX included in the package XOP, [START_REF] Del Río | XOP 2.1: A new version of the X-ray optics software toolkit[END_REF]. We obtained a phase shift ∆φ of 25.77 • . The phase shift was calculated assuming ( [START_REF] Pernot-Rejmánková | Structural matching of ferroelectric domains and associated distortion in potassium titanyl phosphate crystals[END_REF]) that the atom P1 is the pivot atom for the inversion of the adjacent domain. Due to this phase shift, the crystal acts as a phase grating for the diffracted beam, therefore producing, via theTalbot effect, a periodic pattern in both the propagation distance and the detector plane. The simulation of the topographs was made using the same software used by Rejmankova et al., [START_REF] Rejmánková-Pernot | Phase Retrieval by Combined Bragg and Fresnel X-Ray Diffraction Imaging[END_REF]. The Fourier coefficients of the intensity profile recorded at a distance D from a sample can be expressed as: The observed behaviour can be explained by assuming the presence of an internal electric field, E bi , built in during the thermal treatment of the crystal. This field is due to concentration gradients of ionized defects created mainly by out-diffusion of potassium and oxygen atoms when heated, as shown by Kugel and Rosenman [START_REF] Kugel | Domain inversion in heat-treated LiNbO[sub 3] crystals[END_REF] for another ferroelectric crystal, LiNbO 3 (in that case, only the diffusion of oxygen has been considered) and by Angert et al. for KTP, [START_REF] Angert | Ferroelectric domain reversal in KTP crystals by high-temperature treatment[END_REF]. The depth into which the built-in 6.6 Ferroelectric domain behaviour with temperature in Rb:KTP electric field penetrates increases with the heat treatment time and temperature, though the magnitude of the field is reduced. At the same time, the coercive field, E c necessary to switch the domains becomes weaker as the temperature is increased, [START_REF] Urenski | Polarisation reversal and domain anisotropy in flux-grown KTiOPO 4 and isomorphic crystals[END_REF]. This results in E bi > E c in near surface regions and the domains are switched consecutively. ) at high temperature a built-in electric field is created, represented in cyan. This field attempts to switch the polarity in the areas near the sample surfaces corresponding to a c + face as-poled. This results in the "saw tooth" domains represented in c). At higher temperature, the inverted domains disappear on the originally c --dominated surface, while they increase on the opposite surface (d).

X-RAY COHERENCE EXPLOITED IN IMAGING OF FERROELECTRIC DOMAINS

Conclusions

The study of the high temperature behaviour of ferroelectric domains in a periodicallypoled Rb:KTP crystal from room temperature to around the Curie temperature has been performed in situ using synchrotron radiation. This has been possible thanks to the transversal coherence of the beam and is an example of the studies which are possible to perform only at a third generation synchrotron source.

The simulations confirm the validity of the atomic model presented by Pernot et al [START_REF] Pernot-Rejmánková | Structural matching of ferroelectric domains and associated distortion in potassium titanyl phosphate crystals[END_REF] that consider the atom P(1) as the pivotal atom between inverted domains. The obtained data have been successfully modelled in terms of the spontaneous polarization inversion due to ionized defect gradients created by out-diffusion of potassium and oxygen ions when the sample is heated close to T c in a low vacuum.

The results obtained are in agreement with ex situ work published by Kugel and Rosenman [START_REF] Kugel | Domain inversion in heat-treated LiNbO[sub 3] crystals[END_REF] and Angert et al. [START_REF] Angert | Ferroelectric domain reversal in KTP crystals by high-temperature treatment[END_REF].

Chapter 7

Further diamond related studies

In this chapter I will introduce three studies which resulted in three different published papers that I have co-authored. These studies are not directly related with the main topic of my thesis but they all have in common the used material (diamond) and the characterization techniques used (X-ray Topography).

Bending diamonds by femtosecond laser ablation

The subject of the first paper, [START_REF] Balling | Bending diamonds by femtosecond laser ablation[END_REF], is the possibility of producing statically bent diamonds by using femtosecond laser ablation; such a bent crystal could be used as an extraction device for high energy beam, e.g. at the Large Hadron Collider (LHC) in Geneva. It was proven that charged particles incident on a single-crystal with small angles respect to crystallographic directions and having low transverse momentum with respect to the axis of the crystal are channelled inside the lattice. This effect persists even if the crystal is slightly bent, such that the particles may be deviated from their original direction of motion as in a dipole magnet. Since the fields are very strong, this results in bending power equivalent to a magnetic field of several thousand Tesla. Diamond has proven to be resistant to the harsh environment close to the particle beam, but being rigid and brittle, mechanical bending has been proven to be almost impossible. Laser ablation seems a promising technique to bend the crystals. Two high quality HPHT type IIa diamond samples were characterized before and after laser ablation by white beam X-ray topography in order to verify whether the ablation destroys the quality of the crystal. One sample presented trenches only on one surface while the

FURTHER DIAMOND RELATED STUDIES

other on both surfaces. A bent crystal in Laue geometry focuses a polychromatic beam.

If the crystal is cylindrically bent, the radius of curvature for meridional focusing and in the case of a large source to sample distance is R ≈ 2q/ cos(θ B ), where q is the focusing distance, [START_REF] Mocella | Bent crystals in Laue geometry: dynamical focusing of a polychromatic incident beam[END_REF]. By collecting topographs at different sample to film distances and by comparing the dimension of the topographs, we obtained an estimation of the bending radius of the crystals. For the crystal with trenches on both sides, no difference in the size of the topographs was recorded at the different distances, meaning that the sample was flat. For the other sample instead, focusing of the beam (in both vertical and horizontal direction) was obtained if the trenches were upstream, and defocusing for downstream trenches, see fig. From the widths and heights of the topographs at the different positions it was possible to obtain the focal lengths in the two directions, (84.0 ± 0.8) cm and (97.2 ± 0.5) cm respectively. This results radii of curvatures of (171 ± 2) cm in the horizontal direction and (200±2) in the vertical. Furthermore, by converting the distance between adjacent trenches in each picture to local curvature, it is found that the maximum possible non-7.2 X-ray topography of diamond using forbidden reflections: which defects do we really see?

uniformity in relative curvature across the crystal is about 3%.

7.2 X-ray topography of diamond using forbidden reflections: which defects do we really see?

The second paper, [START_REF] Shiryaev | X-ray topography of diamond using forbidden reflections: which defects do we really see[END_REF], was written in collaboration with A. Shiryaev and colleagues from Russia. The subject of the paper was to compare the contrast obtained on synthetic diamond in allowed and forbidden reflections. In the diamond structure (space group F d3m), the X-ray reflections from the 222 planes are forbidden because of symmetry restrictions. However, the thermal anharmonicity of atomic vibrations, and the covalent character of bonding in crystals like diamond, Si and Ge, lead to distortion of the shape of the electron cloud which, in contrast with an ideal crystal, deviates from spherical. These reflections are sometimes called quasi-forbidden. The latter sample was very interesting since it presents two regions with different types of nitrogen impurities, as it is possible to see from fig. 7.2 in the central part of the plate, the N is single substitutional (C defects) while in the bottom part, close to the seed it form nitrogen pairs (A defects). We collected X-ray topographs of this sample at room temperature and also, using a furnace, at temperature of about 1000 K, using a FReLoN camera with 3 µm resolution. We measured the ratio of the diffracted intensity between two regions with different nitrogen defects. For the allowed 111 reflections, this ratio is constant with temperature, but for the 222 reflections we see that above 873 K it decreases, therefore suggesting that point defects can influence the intensity of a forbidden reflection.

Crystal growth and perfection of large octahedral synthetic diamonds

Crystal growth and perfection of large octahedral synthetic diamonds

The third and last paper, [START_REF] Khokhryakov | Crystal growth and perfection of large octahedral synthetic diamonds[END_REF] was written in collaboration with F. Khokhryakov, I.

Kupriyanov and other Russian colleagues. In this work we characterized octahedral diamond crystals grown by the temperature gradient method at T=1550±20 • C and at a pressure P=5.7 ±0.1 GPa, using a split-sphere (BARS) apparatus, using Ni 0 • 7 Fe 0 • 3 alloy as solvent catalyst. The fact that the source of most of the defects is in the region close to the seed is a result which is common to most of the plates studied. This could be related to a low quality of the seed crystal or of its surface finishing.

CONCLUSIONS

Johannesburg. An idea of the quality jump between the "old" (produced in the last decade of the last century) and "new" material is given by the comparison of phase We pushed the possibilities of HRXRD to quantify sub-arsecond broadening of rocking curves of high quality crystals (in particular of diamond) that is due to the presence of residual strains in the bulk and the subsurface region. This was performed in two different steps. First, by implementing different numerical methods either to calculate the instrumental function or to determine it from measurements. The final result was compared with a direct measurement of the instrumental function by using high order e.g. 999 and forbidden 222 reflections. Second, once we could determine the apparatus function with sufficient reliability, we were able to determine the broadening of the rocking curves due to the presence of defects. The results obtained show that, in the case of high quality type IIa diamonds, this broadening is practically negligible. For lower quality crystals (bulk and/or surface quality), a broadening varying from 0.4" to 0.9" at the FWHM level was measured.

Several X-ray topography techniques were extensively used to characterize a big variety of diamond plates. As a first, but very important step, this characterization was simply qualitative in order to assess the presence and distribution of extended defects in the crystal plates. This is of great importance for the crystal growers to be able to optimise the growth parameters and to reduce or even avoid the creation of defects. At the same time, we showed the possibility of obtaining semi-quantitative estimates of the stronger strain fields, e.g. related to metallic inclusions in the crystal.

A special effort was undertaken in order to measure the local effective misorientation in high quality type IIa diamonds. This was achieved by analysing topographs with selected non-dispersive and quasi non-dispersive double-crystal setups, but mainly by using an instrument with bendable monochromators . We showed that in regions up to 0.5×0.5 mm 2 the effective misorientation in a crystal is 4×10 -8 . This value is comparable to those routinely obtained for float-zone silicon crystals.

CONCLUSIONS

The quality of ultra-high-purity beryllium foils was studied in cooperation with the producer to check and help optimise the bulk quality and the polishing procedure.

After some polishing steps no phase contrast was visible on the radiograph recorded at 125 cm from the sample. Already this is a very promising result. In future it could be necessary to make tests with much larger propagation distances, or to make ultimate tests directly at the beamlines where those plates will be used.

In order to exploit the coherence of the beam in the study of ferroelectric domains as a function of temperature, we developed, in collaboration with the Sample Environment Laboratory, a special furnace. This furnace allows the performance of experiments in transmission geometry while preserving the coherence of the beam. Moreover, it allows the heating of optically transparent and thermally insulating samples without creating a large thermal gradient across the crystals. The furnace was tested with different crystals and the coherence properties were established using the Talbot effect.

The main motivation of the investigation of the ferroelectric inverted domains as a function of temperature was to improve the understanding of the transition from one electrical polarization state to the opposite one. This will help in the fabrication of highly efficient QPM devices.

Imaging the domains was performed using combined Bragg-Fresnel imaging. The white beam section topographs obtained at different temperatures, from room temperature to 930 • C were correctly simulated. The modelling of the results obtained at high temperature is based on the assumption that an internal electric field (built in) is created on the c + face during heating. The penetration depth of this field into the crystal increases with the heat treatment time and temperature, though the magnitude of the field is reduced. Moreover, the coercive field necessary to switch the polarization state decreases with temperature. This results in an inversion of the polarization on the c + surface, producing head-to-head domains which are energetically unfavourable.

The system therefore relaxes into a "saw tooth" structure, as shown in the experimental data.

Finally, I presented three different studies which were not directly related with the main subject of the thesis but that have in common diamond as the material used and X-ray topography as one of the characterization techniques used. The first on the possibility of bending diamonds using laser ablation, the second on the investigation of the type of defects which are possible to record using forbidden reflections and the third on the characterization of large octahedral synthetic diamonds grown with the split-sphere (BARS) method.

In conclusions, X-ray characterization techniques have been successfully applied to the study of the materials quality of interest for X-ray optical applications, resulting in improvements of the bulk quality of the diamonds and of the surface treatment processing of the silicon. Moreover, X-ray Bragg Fresnel imaging was efficaciously applied to the understanding of ferroelectric domain behaviour as a function of temperature. Van Cittert deconvolution Si555 slits 0.04 mm

IX

Riassunto.

Le proprietà eccezionali delle sorgenti di radiazione di sincrotrone sono state e sono tuttora utilizzate in maniera sempre più frequente in discipline molto diverse, che spaziano dall'archeologia alla chimica, dalla scienza dei materiali alla biologia, dalla medicina alla fisica. Tra queste proprietà, è importante sottolineare la brillanza della radiazione, lo spettro continuo, l'alto grado di polarizzazione, la struttura temporale e la ridotta dimensione della sorgente e della divergenza del fascio. Quest'ultime proprietà risultano in una grande coerenza trasversale della radiazione prodotta. La coerenza della luce di sincrotrone ha permesso lo sviluppo di nuove tecniche, ad esempio l'imaging basato sul contrasto di fase, la spettroscopia a correlazione di fotoni e la diffrazione X coerente. È dunque di primaria importanza che le ottiche possano preservare questa proprietà, perturbando il meno possibile il fronte d'onda. La tesi è costituita da tre parti. Nella prima parte, viene presentato il lavoro effettuato all'interno del gruppo ottico dell'ESRF relativo alla caratterizzazione di cristalli di diamante sintetico di alta qualità previsti per applicazioni nelle ottiche a raggi X. Questa caratterizzazione è stata effettuata utilizzando diverse tecniche complementari basate sui raggi X, quali la diffrattometria ad alta risoluzione, la topografia, la diffrazione ad incidenza rasante, la riflettomettria e misure di preservazione della coerenza utilizzando l'effetto Talbot. Nella seconda parte, vengono esposti i risultati ottenuti nello studio del comportamento di domini ferroelettrici ad alta temperatura in un cristallo periodicamente polarizzato. In questo tipo di misure, basato sulla diffrazione di Bragg-Fresnel, è necessario un'alta coerenza della luce di sincrotrone. Nella terza parte vengono illustrati i risultati ottenuti nella caratterizzazione di diamanti per applicazioni differenti dalle ottiche a raggi X.

Rèsumè.

Les propriétés exceptionnelles des sources de rayonnement synchrotron ont été et sont de plus en plus exploités dans des disciplines très différentes, allant de l'archéologie à la chimie, de la science des matièraux a la biologie, de la médecine à la physique. Parmi ces propriétés, il est important de mentionner la haute brillance, le spectre continu, le haut degré de polarisation, la structure temporelle, la petite taille de la source et la petite divergence du faisceau. Ces dernières propriètèes entraînant une forte cohérence transversale du rayonnement produit. La cohérence du faisceau a permis le développement des nouvelles techniques comme par exemple, l'imagerie à contraste de phase, la spectroscopie à corrélation des photons et l'imagerie par diffraction des photons cohérents. Par conséquent, il est de première importance que les éléments optiques puissent préserver cette propriété, perturbant le moins possible la front d'onde. Ce travail de thèse se constitue de trois parties. Dans la premiere partie, je vais présenter le travail effectué au sein du groupe optique de l'ESRF dans la caractérisation des cristaux de diamant synthétique de haute qualité prévus pour des applications aux optiques pour les rayons X. Cet caractérisation a été effectué en utilisant différentes techniques rayons X complémentaires, telles que la diffractométrie à haute résolution, la topographie, la diffraction en incidence rasante, la réflectométrie et des mesures de préservation de cohérence en utilisant l'effet Talbot. Dans la deuxième partie, je expose les résultats obtenu dans l'étude du comportement à haute température des domaines ferroélectriques dans un cristal périodiquement polarisé. Dans cet type d'étude, basé sur la diffraction de Bragg-Fresnel, est nécessaire un haute cohérence du faisceau. Dans la troisième partie, je présents des résultats obtenu dans la caractérisation des diamants prévus pour des applications autres que les optique rayons X.

Figure 2 . 1 :

 21 Figure 2.1: The General Electric 70 MeV synchrotron; the synchrotron radiation light is indicated by the arrow.

Figure 2 . 2 :

 22 Figure 2.2: The effect of relativity on the angular distribution of the emitted beam.

Figure 2 . 3 :

 23 Figure 2.3: The different types of radiation in synchrotron facilities: a) bending magnet; b) wiggler; c) undulator

  fig.2.4. The intersections of the circles with radius K (wavevector in the vacuum) and k (average wavevector in the medium) are called Laue (La) and Lorentz (Lo) points respectively.

Fig.2. 5 ,

 5 Fig.2.5, shows a close-up of this region. Due to the fact that the radius of the circles (spheres) is large compared to the difference between the wavectors in vacuum and the average wave vectors in the material, it is possible to approximate them with straight lines (planes), indicated with T h and T o for the sphere of radius k. We can see that, in the two-beam case, each tiepoint corresponds to a wavefield, composed of two plane waves, one with wave vectors directed towards h and the other towards o.

Figure 2 . 5 :

 25 Figure 2.5: Zoom of the region around the Laue and Lorentz point, Laue case

15 )

 15 γ h and γ o are the cosines of the angles Ψ h and Ψ o (see the inset in fig.2.5) defined between the normal to the surface and the diffracted and incident wavevectors respectively. It is possible to show that γ h and γ o are equal to

Figure 2 . 6 :

 26 Figure 2.6: of the region around the Laue (La) and Lorentz (Lo) points, Bragg case.

Figure 2 . 7 :

 27 Figure 2.7: Areas of influence in the Laue and Bragg geometries, symmetrical cases.

Figure 2 . 8 :

 28 Figure 2.8: Penetration depth z e in a Si111 crystal at 14.413 keV in a symmetric reflection geometry. The dashed horizontal line shows the penetration depth due to absorption, the vertical one the position of the kinematic Bragg angle. In the Bragg range the penetration depth is dominated by extinction.

Figure 2 . 12 :

 212 Figure 2.12: a) the longitudinal coherence, related to the bandwidth of the source; b) the transverse coherence, related to the angular source size of the source.

Figure 3 . 1 :

 31 Figure 3.1: ID18 experimental configuration used for High Resolution X-ray Diffractometry.

Figure 3 . 2 :

 32 Figure 3.2: Pictorial example of a rocking curve assumed to be the convolution of three distinct curves: the reflectivity curve, apparatus function and a function describing in an integral way the broadening due to defects. Experimental data were obtained on a high quality IIa HPHT crystal, 400 reflection, E=14.413 keV.

  results obtained. The calculation of the instrumental broadening has been made by using a program written by Franco Cembali, an external collaborator. This program calculates the instrumental broadening by taking into account all the optical elements present at the beamline (slits, monochromators, Compound Refractive Lenses -CRL).In fig.3.3 the calculations of the angular instrumental function after each crystal are plotted.

Figure 3 . 3 :

 33 Figure 3.3: The angular instrumentation function calculated after each crystal of the configuration shown in 3.1.

Figure 3 . 4 :

 34 Figure 3.4: Comparison of the Si 555 reflectivity curve, E=14.413 keV, with the Fraunhofer diffraction from rectangular slits opened respectively 200 µm and 40 µm.

  0 package (Loehle Enterprises, Naperville, IL, USA) to minimize the difference between experimental data and the convolution of the instrumental function with the theoretical reflectivity. Due to the large number of convolution integrals required in this procedure, the code is quite slow and has problems in dealing with curves with more than ≈ 250 points. The first result we obtained during the analysis of the data was the dependence of the apparatus function on the size of the vertical slits. The Fraunhofer diffraction from the slits becomes important in such a high resolution experiment of sub-arcseconds distortions, fig.3.4.

  used: 700 µm, 200 µm and 40 µm. Another problem was that different sets of parameters may give similar goodnesses of fit. An example of this is visible in fig.3.5; three fitting functions, shown in fig.3.5.b) (together with the respective parameters), fit equally well the experimental data collected, fig.3.5.a) (the inset shows a zoom in the region indicated). This problem was solved by comparing the results obtained for the different fit parameters with a broader scan (in this case ±25 arc seconds), fig.3.5.c),d) and e) which could not be analyzed directly due to the limit on the maximum number of points the fitting code accepts. In this particular case, the best solution is the one shown in fig.3.5.e), corresponding to the blue fit parameters. Finally, a series of instrumental broadening functions was obtained for the different reflections used, namely 111, 333, 444 and 555 symmetric Bragg reflections. By extrapolating the results obtained towards a "zero-width curve", e.g. for the 200 µm slits size 3.6, we obtained the final three instrumental broadening functions for the different vertical slits sizes used during the experiment, shown in fig.3.7.

  Figure 3.5: a) comparison between the experimental data (Si 111 reflection) with the results obtained from the convolution of the reflectivity curve with three different apparatus functions obtained using the fitting method. In the inset, zoom of the region indicated with a black rectangle, showing that the three solutions are equally good. b) the three apparatus functions, plotted on a log scale, together with the fit parameters of each one c),d),e) comparison between the experimental data collected on a broader range (± 25 arc seconds) with the three functions shown in b). The best results is obtained with the blue curve, as shown in e).

Figure 3 . 6 :

 36 Figure 3.6: Extrapolation of the instrumental broadening towards a "zero width" curve, slits size 200 µm. The points represent the values of the FWHMs for the instrumental broadening corresponding to the different reflections (111, 333, 444 and 555).

Figure 3 . 7 :

 37 Figure 3.7: Plot of the final apparatus functions obtained for the three different vertical slits sizes, together with the fit parameters and the full widths at different heights (50%, 20%, 2%).

Figure 3 . 8 :

 38 Figure 3.8: Comparison of the instrumental broadening functions obtained with the different methods.

Figure 3. 9 :

 9 Figure 3.9: Comparison of the rocking curves obtained with high order reflections (777, 888 and 999) and with the 222 forbidden reflection together with the apparatus function estimated from the previous experiment.

Figure 3 . 10 :

 310 Figure 3.10: Misorientation contrast. a) the misorientation in the middle part is larger than beam divergence and/or equivalent wavelength spread b) the misorientation is smaller than beam divergence and/or equivalent wavelength spread

Figure 3 . 11 :

 311 Figure 3.11: Extinction contrast: formation of the three types of image around a defect: (di) direct, (i) intermediate and (dy) dynamical images.

Figure 3 . 12 :

 312 Figure 3.12: Schema of the two techinques used: a) white beam topography, b) monochromatic beam topography

fig.3. 13 .

 13 fig.3.13. The source produces a divergent and polychromatic beam which impinges on the crystal at a range of angles (e.g. ≈ 0.3 µrad at BM05). For each angle there is a wavelength satisfying the Bragg condition. All the different wavelengths that are diffracted by the first crystal impinge on the second crystal at different angles. If the second crystal is of the same material as the first, set in a way that the same reflection is excited and with antiparallel diffraction vectors, then all the wavelengths diffracted by the first crystal are diffracted by the second, and the arrangement is said to be non-dispersive (fig.3.13a). All other configurations are dispersive, i.e. the two crystals are made of different materials and/or different reflections are used or from the same

Figure 3 . 13 :

 313 Figure 3.13: Example of two different configurations: a) non-dispersive (+n,-n), b) dispersive (+n,-m).

Figure 3 . 14 :

 314 Figure 3.14: Scheme of the incoming, reflected and transmitted beam together with the beam angles at the boundary between vacuum and a material with index of refraction smaller than 1.

Figure 3 . 15 :

 315 Figure 3.15: Penetration depth in diamond as a function of the incident grazing angle, E= 7 keV

Figure 3 . 16 :

 316 Figure 3.16: The evanescent wave created in total reflection, and its penetration depth Λ

Figure 3 . 17 :

 317 Figure3.17: Reflectivity curve of tungsten-carbon multilayer for Cu Ka X-rays, (a) linear scale and (b) logarithmic scale. From[START_REF] Underwood | X-ray microprobe using multilayer mirrors[END_REF]

Figure 4 . 1 :

 41 Figure 4.1: Differences between diamond (cubic) and lonsdaleite (hexagonal).The pictures of the two structures were generated using the software Vesta,[START_REF] Momma | VESTA: a three-dimensional visualization system for electronic and structural analysis[END_REF]

Figure 4 . 2 :

 42 Figure 4.2: The P,T phase diagram of carbon, from Bundy [129].

Figure 4 . 3 :

 43 Figure 4.3: Scheme of a HPHT system for growing diamonds, from Sumiya et al.[START_REF] Sumiya | [END_REF]

Figure 4 . 4 :

 44 Figure 4.4: A Scanning Electron Micrograph (SEM) of a synthetic diamond, from Kanda [135]; the diamond seed is clearly visible on the bottom part of the stone.

Figure 4 . 5 :

 45 Figure 4.5: Concentration-dependence of nitrogen in the different growth sector. The colour density indicates the N concentration; N concentration in 111 > 100 > 113 > 110 , from [114].

Fig. 4 .

 4 Fig.4.6 shows a comparison between type Ib and type IIa diamond specimens. It is possible to observe in the optical image the pale yellow colour of the Ib and the transparency of the type IIa. This effect is due to the nitrogen impurity, which is a deep donor and absorbs blue light and so it makes the diamond appear yellow. Also clearly visible are the different growth sectors with different nitrogen concentrations (as illustrated in fig.4.5). These are also clearly visible in the UV image: the nitrogen, excited by the UV light, emits in the green-yellow region, except for the 110 sector, where the concentration is much lower. The different growth sectors are also visible in

Figure 4 . 6 :

 46 Figure 4.6: Comparison between type Ib and type IIa synthetic diamonds.

4. 2

 2 Characterization of synthetic diamonds. Experimental results.

  2, since it allows us to obtain local information about the defect distribution in the crystal.All the results presented here have been collected at two beamlines at the ESRF: BM05 and ID19. It is impossible to present all the topographic results obtained during my PhD, since a lot of them have been assessment of a industrially sensitive nature of4. CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDSpurchased high quality IIa diamonds. I will concentrate on a selection of interesting cases, showing the differences between crystals of different quality but also between the results obtained using different strain sensitive techniques. The X-ray topographs were recorded either on photographic films or using a CCD camera. For the films, we typically used Kodak Industrex SR45 films or AGFA Structurix D3, with a grain size of approximately 3 µm. For the CCDs we used either the FReLoN camera (developed at the ESRF,[START_REF] Labiche | FRELON camera: Fast REadout LOw Noise[END_REF]) or a PCO.2000 camera 1 . The reason of having two different cameras is simply the fact that the FReLoN camera is too heavy to be mounted on some of the existing diffractometers. In both cases, the X-rays are first converted to optical light by scintillator crystals, then, by using a set of lenses, the image is focused on the CCD chip. It is therefore possible by changing the optics to change the spatial resolution of the image (e.g. with the PCO, two sets of optics produce an estimated pixel size of 3 µm and 10 µm). All the different optics have been conceived and developed in the Detector Group of the ESRF.The main advantage of films over CCD cameras is the combination of a large field of view (e.g. 13×18 cm in our case) with a high spatial resolution (1 to 3 µm in our case), allowing, for example, the collection of the different Laue spots in a white beam experiment. Nevertheless, when the field of view required is not large, the CCD cameras allow Rocking Curve Imaging, i.e. a combination of High Resolution Diffractometry and Topography. An additional advantage of CCD cameras over photographic films is

Fig. 4 .

 4 Fig.4.7 shows an example of a transmission WBT of a high quality type IIa diamond.The different inhomogeneities in the images are related to defects such as dislocations, stacking faults and inclusions but also, in the case of the equal thickness fringes, simply related to the fact that the edge are bevelled and confirm the high local crystal quality (interferences are not perturbed).

Fig. 4 .

 4 Fig.4.8 shows a zoom of fig.4.7, where a clearly visible set of dislocations is present. The conditions for the formation of the three types of extinction images (direct, dynamical

Figure 4 . 7 :Figure 4 . 9 :

 4749 Figure 4.7: Example of a white beam topograph of a IIa type diamond, 220 reflection

Figure 4 . 10 :

 410 Figure 4.10: White beam 111 topograph, projection of the diffraction vector shown with an arrow, of a IIa high quality diamond crystal, together with a diagram showing approximately where the plate was extracted. From the defects, it is possible to determine that the seed was in the lower part of the plate.

Figure 4 . 12 :

 412 Figure 4.12: Scheme of the boundary between two different growth sectors with different N concentrations, showing the source of the double contrast in fig.4.11.

Figure 4 . 13 : 14 Fig. 4 .

 413144 Figure 4.13: White beam topography in transmission geometry of two diamond plates, 111 oriented. The reflections used were of the {220} family, and the projection of the diffraction vectors are indicated by the arrows. Both plates present a large number of defects, which are clearly visible in the double crystal topographs shown in fig.4.14

Fig. 4 .

 4 fig.2.8. For the 220 reflection at 13 keV the extinction distance t ext is ≈ 4 µm whilst the absorption depth t abs is ≈ 1.2 mm, i.e. more than the sample thickness. The large absorption depth allows a perspective view of all the defects in the bulk. On the top of the rocking curve, e.g. the first topograph on the second row, only the surface layer is visible since the absorption is dominated by the extinction.

Figure 4 . 17 :

 417 Figure 4.17: Map of the angular center of mass positions of the pixel by pixel rocking curve obtained using the diamond shown in fig.4.15 using the software Visrock.

Figure 4 . 18 :

 418 Figure 4.18: Top: Rocking Curve Imaging maps of the type IIa diamond shown in fig.4.16; left: angular position of the center of mass, and right: value of the FWHM calculated pixel by pixel. Bottom: comparison between the rocking curve obtained from the pixel indicated with the arrow, the average rocking curve of the entire crystal and the theoretical reflectivity curve.

Figure 4 .

 4 Figure 4.19: Left, white beam topographs in Laue geometry, 220 reflection, of a high quality type IIa diamond. Middle and right, double crystal topographs collected with the bendable monochromator diffractometer. The detection limits for the two topographs are of about ≈ 4 × 10 -8 for the middle one and ≈ 1.2 × 10 -8 for the right one.

1 Figure 4 . 20 :

 1420 Figure 4.20:Comparison between the experimental and the theoretical rocking curves, together with their derivatives. The experimental rocking curve is different respect the theoretical one in the high angles side. The experimental data are obtained from a RCI series using a ROI of ≈ 0.5×0.5 mm 2 in a region of the sample without defects.

Figure 4 . 21 : 1 Figure 4 . 22 :

 4211422 Figure 4.21: Comparison between the theoretical rocking curve for flat and bent silicon crystal for two different radii of curvature. 880 reflections, E= 20 keV, crystal thickness 2 mm.

4. 2 Figure 4 . 23 :

 2423 Figure 4.23: X-ray topographs in a non-dispersive setup collected with the bendable monochromator; left on the low angle side of the rocking curve, right on the high angle side. The difference in strain sensitivity, as estimated by the rocking curve in fig.4.22 is well visible, for example in the width of the single dislocation image on the left side of the plate. The arrow represent the projection of the incoming and diffracted wavevectors.

Figure 4 . 24 :

 424 Figure 4.24: Effective misorientation map of a IIa diamond crystal, bent silicon 880, diamond 660 reflections, 20 keV. The values in the colour bar are expressed in 10 -8 .

  The maps of d-spacing variations and tilt are shown in fig.4.26. The values obtained for the d-spacing variation in a similar region of interest of 0.5 × 0.5 mm 2 is about 8 × 10 -8 ; in the same region the tilt variations are of the order of 4 × 10 -8 . 0 degrees 180 degrees

Figure 4 . 25 :

 425 Figure 4.25: Maps of the angular position of the center of mass of the rocking curve pixel by pixel, obtained using Visrock. Si 440 and C* 440 reflection, E = 20 keV. The color bar is expressed in arcseconds. The roundish structured features presented in the images are phase contrast artefacts present in the direct beam.

Figure 4 . 26 :

 426 Figure 4.26: Separated maps of d-spacing variations and tilt. In order to vizualize the high quality part of the crystal the contrast has been exaggerated. The values are expressed in 10 -8 .

  Figure 4.28:The Talbot effect for an absorbing gratings of period a, duty cycle 0.1 (90% is opaque, 10% is transparent). Indicated are the positions of the screen where the maximum contrast with the same period as the grid are collected. The image is a re-elaboration of the one in[START_REF] Case | Realization of optical carpets in the Talbot and Talbot-Lau configurations[END_REF].

Figure 4 . 29 :

 429 Figure 4.29: SEM picture of the Talbot grid.

Fig. 4 .

 4 Fig.4.30 shows the configurations used to measure the transverse coherence of the beam.

Figure 4 . 30 :

 430 Figure 4.30: Talbot configurations used to measure the transverse coherence after optical elements: a) in transmission (filters, windows) b) in diffraction (monochromators, beam splitter). From [162].

  by pixel until the amplitude of the coefficient of the Fourier transform at the spatial frequency 1/a is maximised. Second, the signal is multiplied by a function, called the Hanning truncation function defined as H(x) = cos 2 πx 2w (4.12) Where w is the width of the as-cut window. The user defines a Region of Interest (ROI) in the image. After the corrections described previously, a Fourier transform of the data in both the vertical and the horizontal direction is performed. The angular source size is obtained by inserting in equation 4.11 the intensities of the Fourier-transformed coefficient at the frequency 1 a for each propagation distance. With the configuration shown in fig.4.30, it is possible to measure the coherence of the beam either in transmission or in diffraction. A summary of the results in diffraction is shown in tab.4.2, since in transmission the coherence appears to be always preserved.

4. 2

 2 Characterization of synthetic diamonds. Experimental results.

Figure 4 . 31 :

 431 Figure 4.31: Example of ROIs and ROI displacement, shown on top of a diffraction image recorded at 300 mm from the sample. The coherence is calculated using all the pixels inside the ROI and the value is recorded in the top-left region of size equal to the displacement. Sample: 100 oriented, HPHT IIa crystal, reflection 400, and horizontal diffraction.

Figure 4 . 32 :

 432 Figure 4.32: Example of the local coherence map of the sample shown in fig.4.31. ROI size 200×200 pixels, ROI-displacement 5 pixels.

  surfaces are the most interesting if the crystal is to be used as Bragg monochromator, due to the largest structure factor and consequently largest integrated intensity of the 111 reflection, see tab.4.3 where the different integrated intensities have been calculated for a diamond sample 1 mm thick and at an energy of 12 keV. The structure factors are expressed in units of Thomson scattering length (r e = 2.82 × 10 -15 m). The integrated reflectivities are obtained multiplying the reflectivity curve by itself and they have been normalized to the integrated intensity obtained for the 111 reflection. Moreover, a large structure factor results also in a small extinction depth, making the 111 reflection more sensitive to sub-surface distortion than any other symmetric reflection.

  . The system implemented at the iThemba Labs in Johannesburg uses an iron wheel heated to around 810-850 • C, 4.33. The load on the samples (pressing them onto the hot metal plate) was 0.1 N and the polishing time was 4 hours.

Figure 4 . 33 :

 433 Figure 4.33: The Hot-metal polishing apparatus.

Fig. 4 .Figure 4 . 34 :

 4434 Fig.4.34 shows the optical surface profiles for both surfaces of two diamonds (sample 1 and sample 3, sample 2 being very similar to sample 1).These maps are obtained by Vertical Scanning Interferometry (VSI) using a WYKO NT9300, working with filtered white light at ≈550 nm. The vertical resolution is ≈0.5 nm rms (root mean square) for a highly rough surface. The lateral resolution

  samples 1 and 2, several µm of roughness characterize the Hot Metal polished face, and the cleaved face appears locally smooth on a single terrace. Sample 3 was lapped on both faces, and 4.2 Characterization of synthetic diamonds. Experimental results. some characteristic grooves are visible on the unpolished face. The opposite face was analyzed before and after the Hot Metal polishing process 4.35. Polishing appears to smooth out the grooves a little.

Figure 4 . 35 :

 435 Figure 4.35: AFM analysis of sample 3. Surface before (on the left) and after (on the right) the Hot Metal polishing process. In the image on the right the roughness is ≈ 5 nm over a 15×15 µm 2 area.

Figure 4 . 36 :

 436 Figure 4.36: Example of radial scans for the sample 1, reflection -202, E= 8.5 keV. It is possible to notice, at small penetration depth a broad peak related to a layer under compressive strain.

Fig. 4 .

 4 Fig.4.36 shows an example of radial scans collected for sample 1, reflection -202.The penetration depths are estimated from the angle of incidence using eq.3.9. The experimental data are a combination of bulk, surface and background contributions to intensity. A higher, narrow peak originates from the bulk, while a low, broad peak originates from the sample surface. The curve has been modelled as the sum of a Pseudo Voigt function (for the contribution from the bulk), a split-Lorentzian function

Figure 4 . 37 :

 437 Figure 4.37: Comparison of the strain estimations for the cleaved and Hot Metal polished face of sample 1 as a function of the penetration depth.

Figure 4 . 38 :

 438 Figure 4.38: Comparison of the FWHM of the peak for the transversal scans, for the Hot Metal polished and cleaved faces of sample 1 for two directions (-202 and 0-22) as a function of the penetration depth.

Figure 4 . 39 :

 439 Figure 4.39: True specular reflectivity of sample 1, Hot Metal polished and cleaved surfaces.

Sample 1 Figure 4 . 40 :

 1440 Figure 4.40: Transverse diffuse scattering of sample 1, Hot Metal polished and cleaved surfaces.

Figure 4 . 41 :

 441 Figure 4.41: Transverse diffuse scattering of sample 3, Hot Metal polished surface. There is very little specular reflectivity, indicating either a very rough or a highly curved surface.

Figure 5 . 1 :

 51 Figure 5.1: Optical microscopy of silicon crystals after different polishing times and consequently different thicknesses removed, measured using an infrared interferometer.

Figure 5 . 2 :

 52 Figure 5.2: Normalized rocking curves of the different samples from the first polishing series. Cu -K α1 radiation, 111 reflection.

Figure 5 . 3 :

 53 Figure 5.3: Normalized rocking curves of the different samples from the first polishing series. E = 23.871 keV, 444 reflection.

Figure 5 . 4 :

 54 Figure 5.4: Double crystal topographs of the different samples from the first polishing series. E = 20 keV, 111 reflection. For geometrical reasons, the image appears compressed in the vertical direction by a factor 10.12 (=1/ sin(θ B ). The arrow represent the

Figure 5 . 5 :

 55 Figure 5.5: Roughness root mean squared values of the different samples from the first series. The increase in roughness at low spatial frequency (5X) is clear for the etched samples.

5. 1 Figure 5 . 6 :

 156 Figure 5.6: Rocking Curve Imaging -FWHM maps of the samples from the second polishing test; in red are indicated the estimated removed thicknesses.

Together

  Angular source size [μrad]

Figure 5 . 7 :

 57 Figure 5.7: Example of results obtained with one of the samples from the silicon polishing testsample 21. The coherence map was calculated using a ROI of 20 x 20 pixels and a ROI's displacement of 10 pixels, see fig.4.31 for the meaning of ROI and ROI's displacement.

FrequencyFigure 5 . 8 :Figure 5 . 9 : 123 5Figure 5 . 10 :

 5859123510 Figure 5.8: Histogram of the vertical coherence angular source size map of sample 21, ROI 100x100 points.

Fig. 5 .

 5 Fig.5.10a) shows the operating principle of the interferometer. Two identical parallel planar CRLs are transversally separated by a distance d. Each CRL focuses the incoming beam at a distance z f = F/(1 -F/z 0 ), where z 0 is the distance from the source to the lenses and F = R/2N δ is the focal length of the CRL, R being the radius of curvature of one parabolic surface, N the number of lenses and δ the decrement of refraction index, see eq.3.6. If the distance between the lenses d is smaller than than

Figure 5 . 11 :

 511 Figure 5.11: The experimental configuration used for the experiment at ID06.

Fig. 5 .Figure 5 . 12 :

 5512 Fig.5.12 shows an example of recorded interference fringes. The different coloured regions represent the different ROI that have been analyzed. I wrote a code in Matlab to analyze these data and to obtain the value of the source size. The program analyzes the data inside the ROI and calculates the visibility of each fringe. The final visibility for each ROI is then calculated as the average of the single fringe visibility, using the standard deviation as the error. The results obtained for the same samples at ID19 using the Talbot effect and at ID06 are shown in fig.5.13. It is important to notice the big angular source size measured

Figure 5 . 13 :

 513 Figure 5.13: Angular source size measured at ID19 and ID06 using the Talbot effect and an interferometer based on bi-lenses.

Figure 5 . 14 :

 514 Figure 5.14: X-ray radiographs of three beryllium plates, flat field corrected. It is possible to see large inhomogeneities distributed on most of the plates. The straight feature on the left of the images is one of the edges of the samples. The number of notches was used to identify the samples.

Figure 5 . 15 :

 515 Figure 5.15: X-ray radiographs of one beryllium plate after the first polishing. The corner of the sample is visible in the image. In the zoom on the right it is possible to clearly see polishing grooves.

Figure 5 . 16 :

 516 Figure 5.16: X-ray radiographs of one beryllium plate after the second polishing. In the zoom on the right it is now impossible to distinguish any features related to the Be window.

Figure 5 . 17 :

 517 Figure 5.17: Profilometer maps of one of Be plates after the second polishing process, obtained at three different magnifications.

134 6. 2 Figure 6 . 1 :

 134261 Figure6.1: Cubic and tetragonal structures of BaTiO 3 . The pictures of the structures were generated using structural data from[START_REF] Buttner | Structural parameters and electron difference density in BaTiO 3[END_REF] and the software Vesta. The difference between the two structures is very small, and the best way to see it is to focus on the slight bending of the Ti-O bonds upwards in the tetragonal structure.

Figure 6 . 2 :

 62 Figure 6.2: Hysteresis cycle of a ferroelectric crystal. P S and P r are the saturation and residual polarizations, E c is the coercive field.

6. 2 Figure 6 . 3 :

 263 Figure 6.3: Second Harmonic Generation (SHG) scheme. A monochromatic wave of frequency ω impinges a non-linear optical crystal. At the exit of the crystal, a combination of the pump-wave and the frequency-doubled wave is obtained (symbolically separated by a prism).

Figure 6 . 4 :

 64 Figure6.4: Plot of the strength of the electric field of the SHG radiation as a function of the distance travelled inside the crystal. In the inset is presented a geometrical approach to the phenomenon. In the case of perfect phase match, the arrows are all aligned, producing a linear increase of E S H with the distance. If there is a phase shift of ∆k, represented with a rotation of the photon momentum, the intensity increases until it reaches a maximum and decreases to zero, green curves. Finally, if quasi-phase match is implemented, it is possible to achieve an increase in the SHG radiation.

Figure 6 . 5 :Figure 6

 656 fig.6.5 are shown the different parts of the furnace, together with a photograph of it.

Finally, inFigure 6

 6 Figure 6.7: a) BaTiO 3 crystal at room temperature; the crystal appears strongly distorted due to the presence of 90 • ferroelectric domains. b) T=120 • C, the crystal is undergoing phase transition. The bottom part of the crystal, which is closer to the heater, is slightly warmer and is already in the paraelectric phase. The middle and the top parts are still ferroelectric and therefore deformed. c) T=125 • C, the crystal is almost completely paraelectric, except at the top right corner. The arrow represents the projection of the diffraction vector.

1 Figure 6

 16 Figure6.8: Indexation of the KTA crystal using the software Orient Express. In green, the reflections which were analyzed because they showed a maximum contrast of the periodic pattern. Some spots are due to the silicon crystal plates and therefore are not taken into account in the indexation process.

Figure 6 . 9 :

 69 Figure 6.9: White beam topographs of the 184 reflection of the KTA sample, λ ≈ 0.29 Å, film-to-sample distance 120 cm. The image at 700 • C appears very blurred, due most probably to strong distortion and vibrations of the crystal

Figure 6 . 10 :

 610 Figure 6.10: a) furnace for the experiment with the KTA crystal; b) improved furnace, with an internal infrared mirror (not visible in the picture) and the external dome.

Figure 6 . 11 :

 611 Figure 6.11: Structure of the Rb:KTP crystal. Due to the very low level of doping, there are no Rb atoms displayed, even though from the structure we know that 0.3% of the K is replaced by Rb.

I

  m (D) = I m,coh (D) γ(λDm/a) R(m/a) (6.2) where I m,coh (D) are the Fourier coefficients corresponding to the illumination by a transversal fully coherent beam, γ(λDm/a) is the degree of coherence between two points separated by (λDm/a), R(m/a) is the resolution of the detector for the spatial frequency m/a, and a is the period of the grating. D is the defocusing distance, defined asD = (d 1 d 2 ) (d 1 + d 2 ) (6.3)where d 1 is the distance from the source to the sample and d 2 from the sample to the film. Using the Zernike-Van Cittert theorem[START_REF] Born | Principles of Optics -Sixth Edition[END_REF], it is possible to express the degree of coherence as a function of the angular source size as γ(λDm/a) = e -(απDm/2a) 2 / ln 2 (6.4) for an angular Gaussian beam profile of width α.In fig.6.12 are shown the experimental topographs together with the simulations obtained at room temperature and at different defocusing distances. The agreement 6.6 Ferroelectric domain behaviour with temperature in Rb:KTP between the data and the simulations is very good, and confirms the validity of the theory used.

Figure 6 . 12 :

 612 Figure 6.12: Part of the section topographs of 2 12 4 reflection, recorded at different defocusing distances, compared with the simulations made over 8 periods. λ = 0.31 Å, point spread function of the film being 2 µm. Small absorption (µ 0 t ≈ 0.7). h is the projection of the diffraction vector onto the film.

Figure 6 . 13 :

 613 Figure 6.13: Parts of the section topographs of the 2 12 4 reflection showing six periods, recorded at defocusing distance of 117 cm and at different temperatures (as read by the thermocouple), together with the simulations obtained using the domain models shown on the right of the corresponding simulations. The model profiles are drawn over one period, where black represents the inverted domain part of the period, and white the non-inverted one.

Fig. 6 .Figure 6 . 14 :

 6614 Fig.6.14 shows the model used to interpret the results obtained. Fig.6.14.a) presents the situation at low temperature. The inverted domains are represented in blue while the non-inverted ones are in white. The arrows represent the polarization, and the left surface corresponds to the predominately c + face. In fig.6.14.b) the built-in electric field is created, in light blue, near the c + faces. Under the condition E bi > E c , the domains are switched near the surfaces. This would produce head-to-head domain walls which are energetically unfavourable. A compromise between electrostatic and elastic energies is necessary, and the system relaxes into the "saw toothThese results confirmed domain structure, as shown in fig.6.14.c). By increasing the temperature, the inverted domains disappear from the initially c --dominated face of the sample, and become wider at the opposite face 6.14.d).

Figure 7 . 1 :

 71 Figure 7.1: White beam topographs of a laser ablated diamond at different sample to film distances. Focussing of the beam is obtained for trenches upstream. Projection of the diffraction vectors are represented by the arrows.

Figure 7 . 2 :

 72 Figure 7.2: On the left, X-ray topograph in transmission geometry, obtained by rocking the crystal and integrating the diffracted intensity, E = 20keV. a) 400 reflection, b) 222 reflection. On the right, IR maps showing the concentrations of C and A defects.

Figure 7 . 3 :

 73 Figure 7.3: Zoom of the white beam topography of a Ib synthetic diamond. The projection of the diffraction vector is represented by the arrows.

  plates shown in fig.4.27. An impressive example of the increased efficiency of the new material used in an application as X-ray optical elements may be that of beam-splitter monochromator at the ESRF beamline ID14. Recently, just by replacing an old type Ib plate by a new type IIa one, a tenfold increase of the flux at the sample position was obtained.

  fig.A.6, where the two methods are compared together with the Fraunhofer diffraction profile for the experiemental slits size.

Figure A. 6 :

 6 Figure A.6: Comparison between the deconvolved functions from the 555 reflection, nominal slits size 40 µm with both methods and the Fraunhofer diffraction from 32 µm slits.

Table 1 . 1 :

 11 Figures of merit for different material for X-ray optics

	or a

Η a.u. Rh a.u. Θ " Figure 2.10: Comparison

  between reflectivity curves with and without absorption. The case with absorption is a Si 111 reflection at 14.413 keV. The x-scale on the bottom is the normalized coordinate range η whilst the one on the top is the angular deviation from the Bragg angle, ∆θ.

	1.0484	1.5484	2.0484	2.5484	3.0484	3.5484	4.0484	4.5484	5.0484	5.5484	6.0484
	1.0
										no absorption
										with absorption
	0.8
	0.6
	0.4
	0.2
	0.0
	5	4	3	2	1	0	1	2	3	4	5

Si 111 14.413 keV thickness 20 Μm Si 111 14.413 keV thickness 100 Μm Si 111 14.413 keV thickness 500 Μm "Thick" Si 111 14.413 keV

	1.0	1.0484	1.5484	2.0484	2.5484	3.0484	3.5484	4.0484	4.5484	5.0484	5.5484	6.0484
	0.8
	0.6
	0.4
	0.2
	0.0
		5	4	3	2	1	0	1	2	3	4	5

Η a.u. Rh a.u. Θ " Figure 2.11: Reflectivity

	curves of various thicknesses calculated using equation 2.24
	(red, blue, green) compared to the one for semi-infinite crystals, 2.26 (black).

  coherent light, it has finite size. It is common to define two different coherence lengths that determine this region: the longitudinal and transverse coherence lengths,

	fig.2.12.

Si333 Van Cittert Si444 Van Cittert Si555 Van Cittert Fit model Direct calculation Si333 Richardson Lucy Si444 Richardson Lucy Si555 Richardson Lucy

Table 4 .

 4

1: Widths of the broadening due to defects curve at different heights of the peaks for different types of diamonds, together with the Darwin widths w h . The errors are estimated from the values obtained for the apparatus function at the same height.

Table 4 . 2 :

 42 Angular source size for the reference beam and for different diffracting diamonds.

	4. CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS
	sample	reflection	α 0 [µrad]	S [µm]
	reference	-	0.32 ± 0.05	48 ± 8
	diamond Ib (100)	022	1.7 ± 0.2	255 ± 30
	diamond Ib (111)	111	0.91 ± 0.13 137 ± 20
	diamond IIa (100)(top plate)	400	0.65 ± 0.12 98 ± 18
	diamond IIa (100)(middle plate)	400	1.4 ± 0.2	210 ± 30
	diamond IIa (100)(top plate)	022	1.3 ± 0.15 195 ± 23
	diamond IIa (111)	0 22	1.4 ± 0.3	210 ± 45
	diamond IIa (111)	111	0.85 ± 0.15 128 ± 23

4.2 Characterization of synthetic diamonds. Experimental results.

	hkl	F hkl	integr. refl.	t ex [µm]	t abs [µm]
	111	17.67	1	2.21	630
	220	16.00	0.60	3.99	1029
	400	12.45	0.36	7.25	1456
	333	7.52	0.21	15.58	1890

Table 4 . 3 :

 43 Structure factor, integrated reflectivity, extinction and absorption depth values for 1 mm thick diamond at 300 K calculated using the XCRYSTAL tool in XOP, E = 12 keV.

4.2 Characterization of synthetic diamonds. Experimental results.

		Sample 1	Sample 2	Sample 3
	dimensions [mm 3 ]	4.5×5.8×0.69	6.0×7.6×0.69	4.9×5.9×0.78
	first surface	as-grown + HMP	HMP	lapped + HMP
	miscut ( )	0.23 •	0.02 •	5.20 •
	miscut (⊥)	0.04 •	0.02 •	5.20 •
	second surface	cleaved	cleaved	lapped
	miscut ( )	≈ 0 •	not measured	not measured
	miscut (⊥)	0.40 •	not measured	not measured

Table 4 .

 4

4: Samples parameters. The miscuts were measured in two orthogonal directions, indicated with and ⊥.

Table 5 . 1 :

 51 Series of samples for the calibration of the optical polishing process

available at http://www.esrf.eu/AboutUs/Upgrade/documentation/purple-book/

this publication together with others that arise during my work at the ESRF but that are not part of the main subject of the thesis will be presented in chapter 7.

see http://www.pco.de/sensitive-cameras/pco2000/ for details about this camera

Visrock is available from this link: http://www.esrf.eu/computing/scientific/VISROCK/MAIN. htm.

‡ in the Altin et al.,[START_REF] Altin | X-ray diffraction topography using a diffractometer with a bendable monochromator at a synchrotron radiation source[END_REF] a similar formula is obtained, but is present a typing error, θB,1 and θB,2 are inverted. Moreover, the convention for the sign is opposite.

a split function consists of two functions with different FWHMs but same height, allowing asymmetric peaks to be treated[START_REF] Howard | An Evaluation of Some Profile Models and the Optimization Procedures used in Profile Fitting[END_REF]

The value of pH 10.2 has been obtained after private discussion with colleagues from Max Planck Institute for solid state research in Stuttgart and with prof. Yasunaga from Tokai University

available from http://www.ccp14.ac.uk/ccp/web-mirrors/lmgp-laugier-bochu/

Acknowledgements

Series of topographs in a quasi non-dispersive arrangement taken at BM05 using the beam coming from an asymmetric reflection. E= 13 keV, 531 reflection from the silicon collimator, 220 reflection for the diamond. The sample is a high quality IIa type HPHT diamond. The defects that clearly appear on the top of the rocking curve (darker pictures) are coming from the direct beam and are phase contrast images probably related to a low quality beryllium window. The projection of the incoming and diffracted wavevectors is represented by the arrow.

mm

Old crystals New crystals

CHARACTERIZATION OF SINGLE CRYSTAL DIAMONDS

for the symmetric Laue reflections, whilst the smaller values are obtained compared to the symmetric Bragg (for the same plate).

Recently, I've extended the main program, in collaboration with Rafael Klünder. By inserting two parameters, ROI size and displacement (fig. 4.31), the software calculates the coherence over the entire image and produces a map of vertical and horizontal angular source sizes. In order for the program to analyze the data correctly, a lower limit on the ROI size of ≈ 150×150 pixels (100×100 µm 2 ) has been used. There is no theoretical lower limit for the smaller displacement, but the time of calculation increases quadratically; in practice a displacement of 25 pixels gives a good compromise between calculation time and the resolution obtained on the coherence map.

The idea behind this extension to the original program is to try to measure the local effect of crystalline defects on the coherence properties of the diffracted beam. Moreover, by simply making a histogram of results "pixel by pixel", it is possible to obtain statistically valid values, such as the mean angular source size and standard deviation. All the diamonds tested increased the angular source size, partially spoiling the coherence of the beam. Since the quality of the synthetic diamond bulk has been shown in the previous sections to be in some cases very good, we think that the problem still resides mainly in an imperfect finishing of the diamond surface or could be related to a small bending of the sample (some samples are rather thin, varying from 200 to 700 µm). Efforts for the future are concentrated in this direction: to improve the polishing process, in some cases by finding alternative methods, and to characterize not only the surface profile but also the damaged sub-surface layer.

Surface sensitive techniques.

Polishing diamonds is not an easy task, since diamond is among the hardest of materials. Nevertheless, diamonds for jewellery have been polished for centuries, and are still, using the so-called scaife technique. In this technique, a cast iron wheel (called a scaife)

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

After the first etching (removal of ≈ 10 µm), there are still scratches and indentations visible. After the second etching (total removal of ≈ 20 µm), all the scratches are removed, implying that they are located in the upper part of the deformed layer.

Nevertheless, indentations are still visible, i.e. they are located deep in the layer. Finally, with the third etching (total removal of ≈ 30 µm), no defects are visible in the topographs. This implies that the thickness of the deformed layer is between 20µm and 30 µm.

The roughness measurement obtained using the optical profilometer and the AFM at the Metrology Laboratory are shown in fig. 5 From this first polishing test we started to doubt whether the removal of material done by the optical polishing was too small, leaving the crystal with a thick deformed layer and with the indentations produced during the lapping. Therefore we decided to make a second test in order to calibrate the removal of the deformed layer by the optical polishing.

Second test -Calibration of the optical polishing.

In the second polishing test, we try to define how much material is necessary to remove in order to completely eliminate the indentations after the optical polishing.

We decided to procede in two different ways, producing two different series of sample.

In the first series (samples 21 to 24) we increased the time of optical polishing for each crystal, starting with all the crystals, then removing sample 21 and continuing with the rest and so on. We obtained therefore four samples with a different thickness of material removed, namely 19 µm, 31 µm, 66 µm and 80 µm. The removed thicknesses were measured using a Fogale Low Coherence Interferometric Sensor, which accuracy is of ±1 µm.

Coherence measurements of the polished samples

Talbot effect measurements

As in the previous case for the diamond samples 4.2.3, the beam energy was 20.5 keV, the grid period 6 µm and the Talbot distances 300 and 900 mm. An example of the results obtained for the sample 21 (optical polished with a removal of 19 µm is shown in fig. 5.7.

All the deformations that are visible in the topograph (on the left) are visible also in the calculated coherence maps. It is clear that the indentations, seen here as elongated deformations in the vertical direction, disturb considerably the coherence, increasing the vertical angular source size from 0.3-0.5 µrad to 1-2 µrad.

It is important to notice that the scratches appear as dark lines in the coherence map,

i.e. as if they improve the coherence. In reality, the calculated values of source size on the scratches are zero, which has no physical meaning. The program fails to correctly calculate the coherence locally in the neighbourhood of the scratches, and returns a null value.

In order to have an estimation of the angular source size after diffraction from each silicon crystal a statistical analysis was performed on the obtained 2D maps. In fig. 5.8

is shown an example of histogram corresponding to sample 21 and a ROI of 100x100 pixels.

The null values are automatically removed. The mean values and standard deviations of every map for some of the samples are compared in fig. 5.9.

The general behaviour of the angular source sizes measured with the different samples is in good agreement with the previous X-ray Topography characterization, i.e. less deformations visible on the topographs implies a smaller angular source size.

IMPROVEMENTS TO POLISHING OF SILICON CRYSTALS AND BERYLLIUM WINDOWS

It is clear from these images that the samples present a high number of inhomogeneities and it is not of the necessary quality to be used as window for synchrotrons.

The situation improved after the first polishing process, as shown in fig. 5.15, which show the phase sensitive radiograph of one of the three plates (the other two being very similar to this one). This test was performed at BM05 using the PCO camera with a resolution of 3 µm and a field of view of 6×6 mm 2 and a sample to detector distance of 125 cm. The large inhomogeneities are not visible any longer but strongly anisotropic polishing grooves are visible.

Finally, after a second polishing, no more features are visible on the radiographs, see fig. 5.16 (same conditions as for the previous test).

The values of the rms-roughness (expressed in µm) maesured with the optical profilometer at the Metrology Laboratory for the three beryllium samples and for the first and second polishing are shown in tab.5.2.

The roughness appears almost constant for the three magnifications after the first polishing and is slightly smaller after the second polishing, especially at the high frequency (magnification 50X). Also from the profilometer maps it is possible to see that the polishing process is anisotropic, with straight parallel lines which are visible at all magnifications, see fig. By comparing the X-ray radiographs and the data obtained with the optical profiler it is possible to conclude that a clear improvement of the surface is visible between the first and the second polishing. No beam inhomogeneities have been recorded with the X-ray radiographs; nevertheless it is important to highlight the fact that the detector to sample distance used is still very small compared to the window to detector distance at the beamlines. Some inhomogeneities at much larger distance can still appear. To

Chapter 8

Conclusions

The main goal of this PhD was twofold.

On the one hand the task of characterizing different materials of interest to be used as X-ray optical elements (diamond, silicon and beryllium). This was done in collaboration with the producers and/or with the people involved in the finishing processes. The main purpose was to improve the bulk and/or surface quality in order to meet the stringent requirements for the materials to be used as X-ray optical elements which preserve the transverse coherence of the beam.

On the other hand, the purpose was to use this extraordinary property of coherence in a synchrotron beam to investigate, for the first time with in situ visualization methods, the behaviour of ferroelectric domains in a periodically-poled rubidium-doped potassium titanyl phosphate (Rb:KTP) single crystal as the Curie temperature T c is approached.

Characterization of high quality crystals was performed with a set of different X-ray based methods (and supplemented by other methods, e.g. optical) on many different beamlines and in collaboration with many scientists and technical staff. Here a main task was not only to use the complementary methods, but to push their limits and to obtain semi-quantitative and, above all, quantitative results concerning crystal defects and crystal quality.

One of the main results regarding the diamonds is the considerable improvement of the bulk quality (and to some degree also of the crystals dimension) of the synthetic diamonds. This was only possible thanks to the longstanding collaboration between the ESRF, Element Six in South Africa and on the Isle of Man and the University of

We studied the influence of the quality of the diamond crystal on the transverse coherence of the beam and we found that the coherence of the diffracted beam is never preserved, with increases of the angular source size of the order of two to five times the values obtained for the reference beam. This influence was attributed to the presence of remaining residual strains in the bulk of the plates, small local bendings (samples were rather thin) and of a deformed sub-surface layer due to the polishing of the crystal.

Therefore, a quantification of the deformation due to the polishing was attempted using surface-sensitive X-ray techniques. This is, to our knowledge, the first time that results on the crystalline quality of diamonds in this deformed layer have been reported.

We found that, in the case of a 111 diamond polished by the Hot Metal Polishing method, a layer ≈4 µm thick is compressively strained and that this strain is anisotropic.

Moreover, the "mosaic spread" measured in the deformed layer is larger than in the bulk.

In the case of a cleaved sample, terraces are present on the surface of the diamond and neither strain nor "mosaicity" of the sub-surface layer is recorded. Some improvements to the polishing apparatus are now being completed by our colleagues in South Africa and new experiments to study further the subsurface damage with X-ray techniques are planned for the near future.

An important task for the ESRF Crystal Laboratory was (and still is) the improvement, and in particular the quantification of the different steps of the polishing technology. In collaboration with the Crystal and the Metrology Laboratories of the ESRF, the surface quality of silicon crystals was studied using different X-ray-based and complementary (optical, AFM) characterization techniques. The main results obtained are, firstly, that the thickness of the deformed layer due to the lapping procedure is of the order of 30 µm. Therefore, the removal of at least 40 µm by optical polishing is necessary.

Secondly, the layer thickness necessary to be removed by mechano-chemical polishing seems to be between 3 µm and 10 µm. Further tests are necessary and will be done in future. In this way a crystal surface with an undetectable (by our double crystal topography methods) deformed sub-surface layer is obtained. Moreover, the coherence preservation properties of various silicon crystals with different surface qualities were tested using two independent methods: using the Talbot effect and by bi-lens-based interferometry. The results obtained are similar and are consistent with those of the X-ray topography.

Appendix A

Apparatus function determination -Direct deconvolution methods

In order to deconvolve the apparatus fucntion, I wrote a code in Mathematica that uses two different deconvolution algorithms: the Burger-Van Cittert algorithm [START_REF] Burger | Wahre und scheinbare Intensitätsverteilung in Spektrallinien[END_REF] and the Richardson-Lucy algorithm [START_REF] Richardson | Bayesian-Based Iterative Method of Image Restoration[END_REF][START_REF] Lucy | An iterative technique for the rectification of observed distributions[END_REF].

A.1 Burger -Van Cittert

By using the commutative property of the convolution, it is possible to write the convolution integral 3.1 as

since now the unknown is the instrument broadening function that is broaden by the known reflectivity curves of decreasing widths. It is possible to write A.1 as a Fredholm integral equation of the second kind

An approximate solution of A.2 is given by the following iterative algorithm:

A. APPARATUS FUNCTION DETERMINATION -DIRECT DECONVOLUTION METHODS

where

It is possible to write A.3 by using the binomial coefficient, defined in the multiplicative form as

Let's clarify the ideas behind this deconvolution algorithm: in the first iteration, the experimental data e is taken as the first approximation to the instrumental broadening function i 1 . Convolving i 1 with the reflectivity curve f yields e 1 , the first approximation of e. The second approximation i 2 is then obtained by adding a correction to i 1 which is the difference between e and the e 1 and so on and so forth.

Hill and Ioup, [START_REF] Hill | Convergence of the van Cittert iterative method of deconvolution[END_REF] studied the necessary convergence conditions for the Van Cittert iterative method. Among others, it is required that the used function is even, with a maximum on the origin and, with continuous derivatives everywhere if the derivatives are continuous at the origin. All these requirements are fulfilled in our case. There is one peculiarity considering the requirements that the maximum has to be in the origin; the origin is not always, as one could think, the position of the maximum of the experimental data. In fact, in some cases, the theoretical curve and the experimental data have been shifted in order to obtain a more stable solution.

Nevertheless, the algorithm fails to converge to a good solution in many cases.

V

A. APPARATUS FUNCTION DETERMINATION -DIRECT DECONVOLUTION METHODS

The main problem with the Van Cittert is the fact that the first approximation of the instrumental broadening is the experimental curve itself. This can work for curves where the broadening is considerable (e.g. when the slits are closed to a value smaller than 100 µm, the Fraunhofer diffraction from the slits become important). In fact, the results obtained for example in fig.A.2 are in good agreement with the Fraunhofer diffraction calculated from 32 µm slits, very close to the nominal value of 40 µm. This is a systematic nuisance in deconvolution problems, and it is well stated in [START_REF] Jones | The problem of error in deconvolution[END_REF]: it is difficult to deconvolute a narrow curve from a broader one. Nevertheless, this has been possible by using the Richardson-Lucy algorithm.

A.2 Richardson -Lucy

In [START_REF] Lucy | An iterative technique for the rectification of observed distributions[END_REF] the convolution is seen as a particular type of a more general probabilistic theory. where P (x|ξ) is the probability that x will fall in (x, x + ∆x) if ξ = ξ. In our case, the probability function is the theoretical reflectivity, while φ and ψ are respectively the experimental data and the instrumental broadening function.

Bayes' theorem for conditional probabilities states that:

where P (A), P (B) are the probabilities a priori of A and B, while P (A|B) is the conditional probability of realizing A given B.

By using this theorem and by imposing the normalization of P (x|ξ) we obtain:

By guessing a first value for ψ, we can obtain an estimation of Q(ξ|x) , that we can use in the first part of A.8 to obtain an improved estimation of ψ by using φ, an estimation of φ (in our case the experimental data).

VI

A.2 Richardson -Lucy

We obtain the following recursive algorithm: