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Beim Aufsatzwettbewerb der Peter-Wust-Gesellschaft von 1988 ergab sich für mich zum ersten Mal die Gelegenheit, mich mit dem Humanitätsbegriff Peter Wusts auseinanderzusetzen. Nach ihm tritt der Mensch schon als Kind mit dem Gottesgeschenk der Humanität, dem großen Menschheitsideal, in die Welt ein. Die Hauptaufgabe besteht darin, die eigene Kindheit durch alle Gefahren, mit denen die Freiheit droht, glücklich bis zum Tode hinüber zu retten. So schreibt Peter Wust: "Bewahrt er [der Mensch] die Kindheit, die in ihm schlummert, durch alle Gefahrenzonen der Willens- und Freiheitsoszillation hindurch, dann hat er das Palladium der Humanität glücklich heimgeholt. Verliert er diese Kindheit im Kampfgebiet des Lebens, dann war alles umsonst; dann war der Naturegressus des Geistes für ihn ein Fluch; denn er hat ihn zum absoluten Egressus aus dem Reich ewiger Liebe und Güte geführt." Über diesen Begriff der Humanität und das "Erbgute" des Menschen sowie ihre Bewahrung durch das ganze Leben sprach ich mehrmals mit Frank-Werner Veauthier, zu dessen Ehren ich meine Gedanken zusammenstelle.

Selbstdiffusionsuntersuchungen in relaxiertem Si-Ge in Abhängigkeit vom Ge-Gehalt y und der Temperatur T wurden mittels des Radiotracer-Verfahrens durchgeführt. Als Sonden für die Ge-Selbstdiffusion wurden 71Ge-Atome (Halbwertszeit t½ = 11 d) verwendet, die am Institut für Strahlen- und Kernphysik (ISKP) der Universität Bonn oder am Massenseparator ISOLDE am Europäischen Kernforschungszentrum (CERN) in Genf implantiert wurden. Als Sonden für die Si-Selbstdiffusion wurden 31Si-Atome (Halbwertszeit t½ = 2,6 h) verwendet, die am IGISOL-Beschleuniger der Universität Jyväskylä implantiert wurden. Die bestimmten 71Ge- und 31Si-Diffusionskoeffizienten D unterscheiden sich für gleiche Zusammensetzungen kaum, nehmen mit zunehmendem Ge-Gehalt zu und folgen Arrhenius-Gesetzen D = D0exp(-H/kT). Beim Übergang von Si-reichem zu Ge-reichem Si1-yGey findet ein Wechsel des Diffusionsmechanismus vom indirekten Zwischengitteratom-Mechanismus zum Leerstellen-Mechanimus statt. Dieser Wechsel macht sich als deutlicher Knick bei y = 0,35 in den Kurven D(y) bei festem T, H(y) und D0(y) bemerkbar. Bei dieser Zusammensetzung findet in der ungeordneten Legierung Si-Ge ein Übergang von einem festen und harten (Si-ähnlichen) Kristallgitter zu einem weichen und lockeren (Ge-ähnlichen) Kristallgitter statt. Zur Messung der Diffusion kurzlebiger Isotope (Halbwertszeiten < 1 d) war es nötig, eine Vakuumkammer so umzubauen, daß in ihr die Implantation der kurzlebigen Radioisotope (z. B. 31Si), die Diffusionstemperung und die anschließende Schichtenteilung durch Ionenstrahlzerstäubung on- beamline und in-situ durchgeführt werden können. In dieser neuartigen Apparatur konnte zusätzlich die Diffusion von 11C (t½ = 20 min) in Ge gemessen werden. Die Ergebnisse für H und D0 legen die Annahme eines Leerstellen-Mechanismus für die C-Diffusion in Ge nahe., Self-diffusion in Si-Ge alloys was studied as a function of Ge conent y and temperature T using radiotracer techniques. 71Ge isotopes (half-life 11 d) were used a probes for Ge self-diffusion. The isotopes were implanted at Institut für Strahlen- und Kernphysik (ISKP) at Bonn University or at the mass separator ISOLDE at the European Organization for Nuclear Research (CERN) in Geneva, Switzerland. 31Si isotopes (half-life 2.6 h) were used as probes for Si self-diffusion. The isotopes were produced and implanted at the IGISOL-type accelerator of Jyväskylä University, Finland. For given compoisitions, the 71Ge and 31Si diffusion coefficients D only differ slightly, increase with increasing Ge-content, and follow Arrhenius laws D = D0exp(-H/kT). With increasing Ge content a change-over from indirect interstitialcy-diffusion on the Si-side to vacancy-mediated diffusion on the Ge-side was observed. This change-over is indicated by a break at y = 0,35 in the curves of D(y) at constant T, H(y), and D0(y). At this composition, in the disordered alloy Si-Ge a transition from a rigid Si-type lattice to a floppy Ge-type lattice takes place. For measuring the diffusion of short-lived radioisotopes with half-lives < 1 d (e.g., 31Si) it was necessary to construct a set-up, in which implantation of the radioisotopes, diffusion annealing, and subsequent serial sectioning by ion-beam sputtering can be performed on beamline and in situ. In this novel set-up the diffusion of 11C (half-life 20 min) in Ge was measured. The results for H and D0 showed evidence for vacancy-mediated diffusion of C in Ge.

Die vorliegende Arbeit leistet einen Beitrag zur Aufklärung der Selbst- und Fremddiffusion in amorphen, kovalent gebundenen Materialien. Hierzu wurden mit Hilfe von Radiotracer-Verfahren Diffusionsuntersuchungen an a-C, an zwei amorphen Siliziumnitriden, a-Si3N4 und a-Si4N3, sowie an a-Si29B9C41N21 durchgeführt. Radiotracer-Experimente zur Messung der Selbstdiffusion in amorphen, kovalent gebundenen Materialien waren bisher aus zwei Gründen äußerst schwierig: i) Die meisten dieser Materialien, z.B. die amorphen Elementhalbleiter a-Si und a-Ge, kristallisieren, bevor messbare Selbstdiffusion einsetzt. ii) Amorphe, kovalent gebundene Materialien bestehen vorwiegend aus Elementen, deren Radioisotope für konventionelle Radiotracer-Experimente ungeeignet sind, da deren Halbwertszeiten, t1/2, entweder extrem lang [z.B. bei 14C (t1/2 = 5736 a)] oder sehr kurz [z.B. bei 11C (t1/2 = 20,38 min) oder 31Si (t1/2 = 2,6 h)] sind. Sog. Precursor-Keramiken sind in den letzten Jahren aufgrund ihrer hohen thermischen Beständigkeit und ihrer hervorragenden mechanischen Eigenschaften in den Mittelpunkt des Interesses gerückt. Sie basieren auf den Elementen Silizium, Kohlenstoff und Stickstoff und werden durch Pyrolyse präkeramischer Verbindungen hergestellt. Bei ihrem Herstellungsprozess durchlaufen sie amorphe Zwischenzustände, die bis zu hohen Temperaturen (1400-1600°C) thermisch stabil sind. Dadurch ist bei a-Si-(B-)C-N-Keramiken der Temperaturbereich, der für Selbstdiffusionsuntersuchungen unterhalb ihrer Kristallisationstemperatur zur Verfügung steht, sehr groß. Sie bilden somit ein ideales Materialsystem, um Selbstdiffusionsuntersuchungen an amorphen, kovalent gebundenen Materialien durchzuführen. Diffusionsuntersuchungen mittels konventioneller Radiotracer-Verfahren erfolgen in folgenden Schritten: (i) Aufbringen der Radiotracer-Atome auf die Probenoberfläche, (ii) Diffusionstemperung der Probe, (iii) Schichtenteilung der Probe, (iv) Messung der Radioaktivität der Schichten. Dabei sind nur Radioisotope sinnvoll einsetzbar, deren Halbwertszeit größer als etwa 1 d ist. Im Rahmen der vorliegenden Arbeit ist es gelungen, eine Anlage zur Messung der Diffusion von kurzlebigen Radioisotopen mit Halbwertszeiten zwischen 10 min und 1 d - z.B. 11C und 31Si - derart weiter zu entwickeln, dass mit ihr Diffusionsmessungen systematisch und zuverlässig durchgeführt werden können. Das neue Verfahren beruht darauf, dass die Schritte (i) bis (iii) in einer einzigen transportablen Anlage durchgeführt werden. On-line- und In-situ-Diffusionsexperimente wurden mit dem neuen Verfahren am Isotope Separator On-line Device (ISOLDE) des Europäischen Kernforschungszentrums (CERN) in Genf und am On-Line Ion Guide Isotope Separator (IGISOL) der Universität Jyväskylä in Finnland durchgeführt. Fremddiffusionsuntersuchungen wurden im Rahmen dieser Arbeit mittels konventioneller Radiotracer-Verfahren durchgeführt. Zum einen wurde 71Ge (t1/2 = 11,2 d) verwendet, das am Massenseparator der Universität Bonn sowie an ISOLDE erhältlich ist. 71Ge wurde als Radioisotop gewählt, da es - wie Si - ein Element der vierten Hauptgruppe des Periodensystems ist und in Analogie zu kristallinen Halbleitern erwartet wurde, dass es in amorphen, kovalent gebundenen Materialien über einen indirekten Mechanismus diffundiert, der dem Si-Diffusionsmechanismus ähnelt. Diese Vermutung sollte durch einen direkten Vergleich der 31Si- mit der 71Ge-Diffusion in a-Si29B9C41N21 untermauert werden. Zum anderen wurde das nur an ISOLDE erhältliche Radioisotop 195Au (t1/2 = 186 d) zu Fremddiffusionsuntersuchungen verwendet, von dem aus früheren Diffusionsuntersuchungen an amorphen Elementhalbleitern und a-Si28C36N36 bekannt war, dass es ein Kandidat für direkte Diffusion in amorphen, kovalent gebundenen Materialien ist., This work aims at revealing the basic properties of diffusion in amorphous, covalent solids, namely amorphous (a-)C, a-Si3N4, a-Si4N3, and a-Si29B9C41N21, the latter of which is an amorphous, polymer-derived ceramic. For this purpose, self-diffusion and foreign-atom-diffusion measurements were performed by means of radiotracer techniques, in which the radiotracer atoms were implanted into the specimens [1]. These investigations continue a series of previous studies which have been concerned with the diffusion of 195Au and 71Ge in amorphous, covalent solids with simpler structures, such as the elemental semiconductors a-Si and a-Ge [2-5] or the boron-free ceramic a-Si28C36N36 [6-8]. Until recently self-diffusion measurements on amorphous, covalent solids with the aid of radiotracer techniques were rather difficult, since (i) self-diffusion below the crystallization temperatures of such solids is very slow and thus in most cases not measurable at all and (ii) the atoms constituting these solids only possess extremely long-lived (e.g., 14C: half-life t1/2 = 5736 a) or quite short-lived (e.g., 31Si: t1/2 = 2.6 h, 11C: t1/2 = 20.4 min, or 13N: t1/2 = 9.96 min) radioisotopes, which are not appropriate for diffusion measurements. However, polymer-derived amorphous ceramics, a novel class of lately emerged materials [9], are excellent candidates for investigating self-diffusion in amorphous, covalent solids. Whereas in amorphous elemental semiconductors crystallization sets in prior to self-diffusion (e.g., in a-Si at about 630°C [10]), in amorphous ceramics this is not the case. For instance, in a-Si28C36N36 considerable self-diffusion occurs well below its crystallization temperature (about 1400°C) [6], which in B-containing amorphous ceramics is even higher (about 1600°C) [11-15]. For this reason, the major part of this thesis is devoted to diffusion in the ceramic a-Si29B9C41N21. The problem concerning the short half-lives of radioisotopes being of interest for measuring self-diffusion in amorphous, covalent materials could be solved by developing a novel technique for measuring diffusivities of short-lived radioisotopes in solids (Sec. 3). With the aid of this technique, the diffusion of 11C in a-Si29B9C41N21, a-C, a-Si3N4, and a-Si4N3 as well as the diffusion of 31Si in a-Si29B9C41N21 were studied. 71Ge and 195Au were chosen as additional foreign-atom diffusers. Their half-lives (71Ge: t1/2 = 11.2 d, 195Au: t1/2 = 186 d) are convenient for diffusion measurements with conventional radiotracer techniques. 71Ge is a close relative of Si and thus, in analogy to crystalline semiconductors [16-18], was expected to diffuse via a mechanism similar to that of Si. By contrast, 195Au is known to undergo an interstitial-like, direct diffusion in the amorphous elemental semiconductors; therefore, it was supposed to be a candidate for direct diffusion in amorphous, covalent solids.

Inverse problems can usually be modelled as operator equations in infinite-dimensional spaces with a forward operator acting between Hilbert or Banach spaces—a formulation which quite often also serves as the basis for defining and analyzing solution methods. The additional amount of structure and geometric interpretability provided by the concept of an inner product has rendered these methods amenable to a convergence analysis, a fact which has led to a rigorous and comprehensive study of regularization methods in Hilbert spaces over the last three decades. However, for numerous problems such as x-ray diffractometry, certain inverse scattering problems and a number of parameter identification problems in PDEs, the reasons for using a Hilbert space setting seem to be based on conventions rather than an appropriate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, non-Hilbertian regularization and data fidelity terms incorporating a priori information on solution and noise, such as general Lp-norms, TV-type norms, or the Kullback–Leibler divergence, have recently become very popular. These facts have motivated intensive investigations on regularization methods in Banach spaces, a topic which has emerged as a highly active research field within the area of inverse problems. Meanwhile some of the most well-known regularization approaches, such as Tikhonov-type methods requiring the solution of extremal problems, and iterative ones like the Landweber method, the Gauss–Newton method, as well as the approximate inverse method, have been investigated for linear and nonlinear operator equations in Banach spaces. Convergence with rates has been proven and conditions on the solution smoothness and on the structure of nonlinearity have been formulated. Still, beyond the existing results a large number of challenging open questions have arisen, due to the more involved handling of general Banach spaces and the larger variety of concrete instances with special properties. The aim of this special section is to provide a forum for highly topical ongoing work in the area of regularization in Banach spaces, its numerics and its applications. Indeed, we have been lucky enough to obtain a number of excellent papers both from colleagues who have previously been contributing to this topic and from researchers entering the field due to its relevance in practical inverse problems. We would like to thank all contributers for enabling us to present a high quality collection of papers on topics ranging from various aspects of regularization via efficient numerical solution to applications in PDE models. We give a brief overview of the contributions included in this issue (here ordered alphabetically by first author). In their paper, Iterative regularization with general penalty term—theory and application to L1 and TV regularization, Radu Bot and Torsten Hein provide an extension of the Landweber iteration for linear operator equations in Banach space to general operators in place of the inverse duality mapping, which corresponds to the use of general regularization functionals in variational regularization. The L∞ topology in data space corresponds to the frequently occuring situation of uniformly distributed data noise. A numerically efficient solution of the resulting Tikhonov regularization problem via a Moreau–Yosida appriximation and a semismooth Newton method, along with a δ-free regularization parameter choice rule, is the topic of the paper L∞ fitting for inverse problems with uniform noise by Christian Clason. Extension of convergence rates results from classical source conditions to their generalization via variational inequalities with a priori and a posteriori stopping rules is the main contribution of the paper Regularization of linear ill-posed problems by the augmented Lagrangian method and variational inequalities by Klaus Frick and Markus Grasmair, again in the context of some iterative method. A powerful tool for proving convergence rates of Tikhonov type but also other regularization methods in Banach spaces are assumptions of the type of variational inequalities that combine conditions on solution smoothness (i.e., source conditions in the Hilbert space case) and nonlinearity of the forward operator. In Parameter choice in Banach space regularization under variational inequalities, Bernd Hofmann and Peter Mathe provide results with general error measures and especially study the question of regularization parameter choice. Daijun Jiang, Hui Feng, and Jun Zou consider an application of Banach space ideas in the context of an application problem in their paper Convergence rates of Tikhonov regularizations for parameter identifiation in a parabolic-elliptic system, namely the identification of a distributed diffusion coefficient in a coupled elliptic–parabolic system. In particular, they show convergence rates of Lp-H1 (variational) regularization for the application under consideration via the use and verification of certain source and nonlinearity conditions. In computational practice, the Lp norm with p close to one is often used as a substitute for the actually sparsity promoting L1 norm. In Norm sensitivity of sparsity regularization with respect to p, Kamil S Kazimierski, Peter Maass and Robin Strehlow consider the question of how sensitive the Tikhonov regularized solution is with respect to p. They do so by computing the derivative via the implicit function theorem, particularly at the crucial value, p=1. Another iterative regularization method in Banach space is considered by Qinian Jin and Linda Stals in Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces. Using a variational formulation and under some smoothness and convexity assumption on the preimage space, they extend the convergence analysis of the well-known iterative Tikhonov method for linear problems in Hilbert space to a more general Banach space framework. Systems of linear or nonlinear operators can be efficiently treated by cyclic iterations, thus several variants of gradient and Newton-type Kaczmarz methods have already been studied in the Hilbert space setting. Antonio Leitao and M Marques Alves in their paper On Landweber–--Kaczmarz methods for regularizing systems of ill-posed equations in Banach spaces carry out an extension to Banach spaces for the fundamental Landweber version. The impact of perturbations in the evaluation of the forward operator and its derivative on the convergence behaviour of regularization methods is a practically and highly relevant issue. It is treated in the paper Convergence rates analysis of Tikhonov regularization for nonlinear ill-posed problems with noisy operators by Shuai Lu and Jens Flemming for variational regularization of nonlinear problems in Banach spaces. In The approximate inverse in action: IV. Semi-discrete equations in a Banach space setting, Thomas Schuster, Andreas Rieder and Frank Schopfer extend the concept of approximate inverse to the practically and highly relevant situation of finitely many measurements and a general smooth and convex Banach space as preimage space. They devise two approaches for computing the reconstruction kernels required in the method and provide convergence and regularization results. Frank Werner and Thorsten Hohage in Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data prove convergence rates results for variational regularization with general convex regularization term and the Kullback–Leibler distance as data fidelity term by combining a new result on Poisson distributed data with a deterministic rates analysis. Finally, we would like to thank the Inverse Problems team, especially Joanna Evangelides and Chris Wileman, for their extraordinary smooth and productive cooperation, as well as Alfred K Louis for his kind support of our initiative.