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Direct Determination of the Phase of a Structure Factor by X-Ray Imaging

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Direct Determination of the Phase of a Structure Factor by X-Ray Imaging ! Direct!determination!of!the!phase!of!a!structure!factor! by!X5ray!imaging! ! ! ! Maria!Civita ! Supervisor:!Prof.!Dr.!Ian!K.!Robinson! ! ! ! ! ! ! ! ! ! ! ! London!Centre!for!Nanotechnology! University!College!London! ! March!2015! Introduction AtypicalX-raydiffraction experiment is performed by illuminating the sample with an X-ray beam and by collecting the resulting diffraction patterns in the far field. In this configuration we are only able to collect part of the information contained in the transmitted wave: its intensity can be recorded with the use of a detector while the phase is lost. This is what we usually call “the phase problem”. In crystallography, the missing phases are usually derived by self consistency with known physical properties of the crystal, that the electron density is real and mostly confined to the cores of the atoms in the unit cell. The development of computational "direct methods" in the 1950’s led to a revolution in crystallography because it allowed direct inversion of diffraction patterns to atomic-resolution real- space images of the crystal structure [1, 2]. Interference methods can be used to measure phases experimentally. The magnitude of one struc- ture factor is predicted to become modulated in a characteristic way when a second Bragg peak is simultaneously excited [3]. The shape of the interference is determined by the relative phases of the two refections involved, and implicitly by that of the difference reflection [4, 5], which are all cou- pled through the dynamical theory of X-ray diffraction [6]. This has been developed into a practical method for measuring "triplet phases" comprising of the sums of the three phases involved. When enough triplets are known, the individual phases can be deduced and the structure solved [7, 8]. A Bonse-Hart interferometer was used by Hirano and Momose to measure the change in phase of an X-ray beam transmitted through a diamond crystal when a Bragg reflection was excited [9]; as we will show in this Thesis, the effect can be used to measure structure-factor phases. A more direct interference method was recently proposed by Wolf [10] in which the mutual coherence function is evaluated be- tween the direct beam and a single Bragg reflected beam. Since this contains their relative phases, the unknown phase of the Bragg reflection can be extracted. This method has not yet been demonstrated 1 for X-rays. In this Thesis work a new method, related to that of Wolf [10], in which the phase of the direct beam transmitted through a crystal is shown to change whenever a Bragg peak is generated will be proposed and demostrated. Since only one diffracted beam is involved, its phase relative to the incident beam is directly determined, without the need for decoding of triplet combinations [7, 8]. The reflected beam needs to be strong enough to influence the forward beam, so the diffraction has to be at least at the beginning of the dynamical regime [6]. The novelty of our method is to measure the phase of the forward beam, using the powerful phase sensitivity of the new X-ray ptychography method [11]. By imaging the phase of the crystal under investigation by ptychography [12] we can accurately measure the phase shift of the beam as it is transmitted through the crystal. This phase shift, which is sensitive to the X-ray refractive index and thickness, is found to change when a Bragg peak is generated inside the crystal. In Chapter 1 an introduction to X-ray microscopy will be presented starting from a more general overview of modern microscopy. The main aspects of the dedicated optics and traditional setups will be discussed. Chapter 2 will describe the phase problem in the X-ray Coherent Diffraction Imaging (XCDI) technique and will provide an overview on the most important phase retrieval methods. In Chapter 3 the attention will be focused on Ptychography. Here different phase retrieval algo- rithms will be discussed. At the end of this chaper the artifacts introduced in the reconstructed phase will also be discussed, proposing methods that can be used for their correction. Chapter 4 will discuss the kinematical and dynamical diffraction theories, after a brief introduction on X-ray cristallography. In Chapter 5 we will discuss the experimental implementation of our phase shift retrieval method by showing the results obtained on a gold nanocrystals sample. Chapter 6 will show how we designed and produced another set of Si and InP samples in order to continue our experimental investigations. In Chapter 7 the last results obtained on our Si and InP samples will be presented and discussed. Chapter 8 will present the conclusions that we could draw after comparing all our experimental results. 2 Acknowledgements The completion of this research project would have not been possible without the contribution of many people who supported me during these past few years. The first person that I would like to thank is my supervisor, Prof. Ian Robinson, who has always believed in this project and whose passion for the research activity has inspired me throughout all these years. His understanding and patience have helped me many times and his humanity and generosity gave me an example that I will try to follow for the rest of my life. Another very important person that I would like to thank is Dr. Graeme Morrison, who has always been there for me with good advices and smart suggestions. I really had the best time with him and Dr. Malcolm Howells at the TwinMic beamline, where we went several times to perform experiments on structured illumination. My deepest gratitude goes to Dr. Ana Diaz, beamline scientist at the cSAXS beamline at the Swiss Light Source synchrotron, whose dedication was extremely important in achieving our first experimental results. I would also like to thank Dr. Ross Harder, beamline scientist at the APS 34-IDC beamline, for his support during the many experiments that we conducted with him. I also had a great time together with my fellow team members. Thanks to them I have many good memories that will always put a smile on my face. Last but not least, I would like to thank my family. My husband Guido who has always been there for me with his love and support, giving me the strenght to overcome the many difficulties that I encountered during these years. My mother Luigia who has always put her needs aside so that I could pursue my dreams. Thank you mom for being the strongest person I know! 3 Contents 1 X-Ray microscopy 7 1.1 Introductiontomodernmicroscopy. 7 1.2 X-ray microscopes . 9 1.2.1 Focusing devices . 10 1.2.2 Transmission X-ray Microscope . 12 1.2.3 Scanning Transmission X-ray Microscope . 12 1.2.4 Structured Illumination Microscopy . 15 2 Coherent X-ray Diffraction Imaging 17 2.1 ThephaseprobleminCXDI.................................. 17 2.2 Phase retrieval methods . 19 2.2.1 Oversampling . 19 2.2.2 Iterativealgorithms .................................. 22 2.3 CoherenceofX-raysources .................................. 24 3Ptychography 27 3.1 TheoreticalprinciplesofPtychography . 28 3.2 PtychographicIterativeEngine(PIE) . 29 3.3 Extended Ptychographic Iterative Engine (ePIE) . 32 3.4 Difference Map method . 33 3.5 Artifacts introduced in the reconstructed phase . 35 3.5.1 Phasewrapping..................................... 35 4 3.5.2 Phase ramps . 42 4Diffraction of X-rays by crystals 48 4.1 Introduction to X-ray Crystallography . 48 4.2 Kinematical diffraction . 49 4.2.1 TheEwaldsphere ................................... 57 4.3 Dynamical diffraction . 62 4.3.1 Description of the crystallographic structure . 65 4.3.2 Maxwell’s equations solution . 67 5 First experimental results: gold nanoscrystals 78 5.1 Ptychography on gold nanocrystals . 78 5.1.1 Experimental setup . 78 5.1.2 Data analysis . 82 5.1.3 Theoreticalbackground . 87 6 Design and preparation of new samples 92 6.1 Sample’s design . 92 6.2 Cleanroomproduction..................................... 95 7 Si and InP: experimental results 101 7.1 Si samples . 103 7.1.1 Si pillar: 4x4 microns . 107 7.1.2 Si pillar 4x8 micron . 112 7.2 InPsamples...........................................114 7.2.1 InP:{111}reflection ..................................115 7.2.2 InP:{220}reflection ..................................119 7.2.3 InP:{200}reflection ..................................121 8 Conclusions 124 8.1 Structure factor . 124 8.2 Phase of the scattered beam . 128 5 8.3 Structure factor phase . 131 6 Chapter 1 X-Ray microscopy 1.1 Introduction to modern microscopy The concept at the basis of a microscope is to use a system of lenses to focus a beam of visible light in order to obtain magnified images of small samples. This basic idea has been developed in many different ways throughout the years because the resolution that one can get from a microscope is limited by the wavelength of the radiation being used to illuminate the samples. Furthermore there are cases where the samples are opaque to visible light so that one need to use a different radiation in order to get not only a higher resolution but also a better penetration. One first example is the Transmission Electron Microscope (TEM) whose concept follows the same principles at the basis of the light microscope but uses a beam of electrons instead of light. On one side the source’s much lower wavelength leads to a dramatic improvement in terms of resolution which makes it possible to see objects to the order of a few angstrom, but on the other hand the TEM has also limitations. In fact unless the sample is very thin, electrons are either absorbed or scattered within the object rather than transmitted. For this reason other research has been consucted in order to develop new electron microscopes which can be capable of examining relatively thick (also called bulk) specimens [13].
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