Technische Universitat¨ Munchen¨ Fakult¨atf¨urMathematik Lehrstuhl f¨urAnalysis Radiation for the Analysis of Molecular Structures with Non-Crystalline Symmetry: Modelling and Representation Theoretic Design Dominik J¨ustel Vollst¨andigerAbdruck der von der Fakult¨atf¨urMathematik der Technischen Universit¨at M¨unchen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Martin Brokate Pr¨uferder Dissertation: 1. Univ.-Prof. Gero Friesecke, Ph. D. 2. Univ.-Prof. Dr. Rupert Lasser 3. Prof. Richard D. James, Ph. D. University of Minnesota, Minneapolis, USA (schriftliche Beurteilung) Die Dissertation wurde am 25.06.2014 bei der Technischen Universit¨atM¨unchen einge- reicht und durch die Fakult¨atf¨urMathematik am 05.10.2014 angenommen. Dedicated to my late mother Abstract X-ray crystallography is the main tool for the structural analysis of molecules today. In this dissertation, an extension of the principles of this method to a general class of highly symmetric structures is studied by introducing a criterion for the design of suitable radiation. The solutions to these design equations are explicitly determined for a class of nanotube-like structures. Under certain conditions, a generalization of the von Laue condition of X-ray crystallography can be formulated. Tools include harmonic analysis and representation theory. Zusammenfassung Die R¨ontgenstrukturanalyse ist heutzutage die zentrale Methode zur Bestimmung der Struktur von Molek¨ulen. Die vorliegende Arbeit behandelt die Verallgemeinerung der Prinzipien, die dieser Methode zugrunde liegen, auf eine allgemeine Klasse von hochgradig symmetrischen Strukturen. Ein theoretisches Kriterium f¨urdas Design strukturadap- tierter elektromagnetischer Strahlung wird formuliert und f¨ureine Klasse von Struk- turen explizit gel¨ost.Unter bestimmten Bedingungen kann die von-Laue-Bedingung der Kristallstrukturanalyse verallgemeinert werden. Dabei werden Methoden der harmonis- chen Analysis sowie der Darstellungstheorie verwendet. Contents Acknowledgement v Introduction 1 Part 1. Analysis of Molecular Structures by Plane Wave Diffraction 7 Chapter 1. Scattering of Plane Waves 11 1. Plane Wave Radiation 12 2. Scattering of Plane Waves 16 3. X-ray Crystallography 21 4. Coherent Diffraction Imaging 26 Chapter 2. Reconstruction from Intensity Measurements 29 1. The Phase Problem 30 2. Phase Retrieval 39 Motivating Example: A Nanotube 47 Part 2. Radiation Design for Non-Crystalline Structures 51 Chapter 3. Radiation Design 55 1. Scattering of Time-Harmonic Radiation 56 2. Reconstruction and Design 69 3. The Design Equations for Abelian Design Groups 75 Chapter 4. Nanotube Structures and Twisted Waves 87 1. Solution of the Design Equations { Twisted Waves 87 2. Scattering of Twisted Waves { the Twisted von Laue condition 98 Chapter 5. Symmetry-Adapted Waves 109 1. The Structure of Abelian Design Groups 109 2. Wigner-Projections and the Zak Transform 116 3. Symmetry-Adapted Waves for Abelian Design Groups 123 iii iv CONTENTS Chapter 6. Radiation Design for Abelian Design Groups 135 1. The Scalar Wave Transform 136 2. The Generalized von Laue Condition 147 3. Phase Retrieval 151 Chapter 7. Radiation Design for Compact Design Groups 153 1. The Design Equations { Characters and Matrix Coefficients 154 2. The Wave Transform 159 Outlook 167 Appendix A. Fourier analysis 171 Appendix B. Maxwell's equations 177 Appendix C. Crystallography 183 Bibliography 189 Nomenclature 195 Acknowledgement I would like to thank my supervisor Prof. Gero Friesecke for giving me the possibility to work on this project, for his support of my research, and for sharing many of his mathematical and non-mathematical insights during the last years. Thanks to Prof. Richard D. James, who started this project together with Prof. Friesecke, for his many ideas, and for his questions that always induced a deeper under- standing. Thanks to Prof. Rupert Lasser for being my TUM Graduate School mentor, and for teaching me many of the mathematical concepts during my studies that were crucial for this dissertation. Thanks to my colleagues and friends Yuen Au Yeung, Bertram Drost, Michael Fauser, Felix Henneke, Christian Mendl, David Sattlegger, and Andreas Vollmayr for helpful discussions on different topics related to my work. Thanks to Frauke B¨acker for her help in administrative matters, and for many pleasant chats. Thanks to my family { to my mother Traudl, my father J¨urg,my brother Martin, my sisters Susi and Ela, and my niece Lisa { and to Gitti and Franzi for all the little and big things they did to support me. Finally, I thank Hedi for her emotional support, for her love, and for the little pushes she gave me at times to keep me going. v Introduction X-ray Crystallography. Before the invention of X-ray crystallography, structural analysis of molecular structures at the atomic scale was limited to theoretical considera- tions. This situation changed dramatically, when in 1912, Max von Laue, Walter Friedrich and Paul Knippig first demonstrated the diffraction of X-rays by crystals [FKL12] in Wil- helm R¨ontgen's laboratory in Munich [Ewa62]. Von Laue's team was awarded the 1914 Nobel Prize for Physics for this discovery. Not only did their experiment clarify the na- ture of electromagnetic radiation, it also was quickly realized that the highly structured diffraction patterns bear the possibility to reconstruct the atomic structure of a crystal. electron density screen/detector of a unit cell single polychromatic crystal x-ray source Figure 0.1. Visualization of the classic von Laue method of X-ray Crystal- lography. Polychromatic X-rays are diffracted by a single crystal, resulting in a structured peak pattern on a screen. The electron density of a unit cell of the crystal can be recovered (up to the phase problem) from the measured intensity of the outgoing radiation. When measuring the intensity of the scattered radiation, a part of the information contained in the field is lost. This problem is called the phase problem and makes the reconstruction a mathematically challenging task. William Lawrence Bragg and his father William Henry Bragg solved the first simple crystal structures in the following years [Bra13], earning them the 1915 Nobel Prize for Physics for their contributions. The list of Nobel laureates related to X-ray crystallog- raphy is long. Most notably, Herbert Hauptman and Jerome Karle won the 1985 Nobel Prize for Chemistry for their mathematical work on the phase problem [HK53] that led 1 2 INTRODUCTION to the reconstruction of more complex structures and made X-ray crystallography the central tool for the structural analysis of biomolecules, and Dan Shechtman won the 2011 Nobel Prize for Chemistry for his discovery of quasicrystals [SBGC84] that was motivated by X-ray diffraction data. This illustrates the significance of X-ray crystallography for the natural sciences. The major drawback of X-ray crystallography is its exclusive applicability to crystal structures. Because of this restriction, a lot of effort is put in the crystallization of structures that naturally do not form crystals [GEA11]. A method to analyze the structure of macromolecules without the need to crystallize them is therefore of great interest. Coherent Diffraction Imaging. In 1999, the possibility of the extension of X-ray diffraction methods for the analysis of non-crystalline samples was first demonstrated by Jianwei (John) Miao et al. [MCKS99]. The method is based on an observation made by David Sayre [Say52-2]. Coherent Diffraction Imaging (CDI) uses highly brilliant third generation X-ray sources to illuminate a general sample. The continuous diffraction pattern is then used to recon- struct a projection of the sample. The time gap between the idea and its realization has two reasons { the need of high-quality sources and detectors on the one hand, and for algorithms and computational power to solve the corresponding high-dimensional phase problem for reconstruction on the other hand. projection of the screen/detector electron density general highly brilliant structure x-ray source Figure 0.2. Visualization of Coherent Diffraction Imaging. Monochro- matic X-rays are diffracted by a specimen. A projection of the electron density of the sample can be recovered (up to the phase problem) from the measured intensity of the outgoing radiation. Even though CDI is a very promising method, it currently only achieves a resolution of a few nanometers and is hence not useful for the analysis of molecular structures. The resolution of the reconstruction depends on the quality of the source and the resolution of the detector, as well as on the energy of the used radiation. For the analysis of INTRODUCTION 3 biomolecules, this last point is crucial, because at high energies the specimen is destroyed. Methods like femtosecond diffractive imaging [CEA06] try to avoid the destruction of the specimen by a short exposure time. Many other methods related to CDI were proposed, e.g. Ptychography [TEA08], Fresnel coherent diffractive imaging [WEA06] or massively parallel X-ray holography [MEA08], just to name a few. All these methods have one thing in common. They do not use any structural informa- tion, even though, many interesting structures in biology and nanotechnology are highly symmetric. The idea of this dissertation is to exploit these symmetries by designing new kinds of radiation for the analysis of molecular structures. The von Laue Condition and Radiation Design. X-ray Crystallography achieves atomic resolution for crystals with a few hundred atoms per unit cell with little technical requirements [US99]. For simple crystals, atoms can even be resolved using a simple X- ray tube. The reason for this superiority in resolution is of theoretical nature and lies in the intimate relationship between the periodic structure of a crystal and the translation invariance of plane wave radiation. It leads to constructive interference of the outgoing waves in a discrete set of directions and, more importantly, to destructive interference else. As a consequence, on a screen in the far-field, a highly structured peak pattern is produced by the outgoing radiation (see Figure 0.1).