In this work, a 3-terminal capacitance dilatometer was set up and used for measurements of the thermal expansion and magnetostriction of novel superconducting iron pinictides and related materials. In particular, \re~with R\,=\,La, Ce, Pr, Sm, Gd, \laf~and Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ have been investigated. The data on polycrystalline \laf~are the first published thermal expansion data on this material. The lattice effects at the structural and the magnetic phase transition have been investigated and the phase diagram upon F-doping has been studied. A main result is the observation of a previously unknown fluctuation regime for the doping level $\mathnormal{x}\leqslant$ 0.04 over a large $T$ range above the structural transition temperature \ts. The absence of any structural anomalies in the normal state of the superconducting \laf~samples with $\mathnormal{x}\geqslant$ 0.05 corroborates the discontinuous character of the phase boundary not only for the magnetism but also for the structural degrees of freedom. Similarly, the presence of high-temperature fluctuations is found for all \re~undoped materials under study. The discussion of the probable origin of the fluctuations as well as the definition of the structural transition temperature \ts~are done. The low temperature features shown by the thermal expansion data for \re~are caused by the onset of long range magnetic order of the $4f$-moments and their different configurations. In particular, \pr, which has a very pronounced anomaly associated with Pr-ordering exhibits a large magnetostriction at low temperatures. By discussing this effect along with the magnetization, resistivity and other measurements, it is found that this large magneto-elastic effect may originate from the correlations between the momentum from Fe$^{3+}$ and Pr$^{3+}$. Last, the thermal expansion of Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ 122 single crystals is investigated. Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ is one of the first materials where single crystals are available. The thermal expansion results on the undoped compound with $x=0$ show a large anomaly at the combined magnetic and structural transition which is far sharper than that for polycrystalline systems. Upon doping, both transitions are suppressed and their splitting is visible in the thermal expansion data. The high precision thermal expansion and magnetostriction results presented in this work are among the first data on the novel family of iron-based superconductors. A valuable insight in the respective ordering phenomena and the thermodynamic properties is provided.

In this work, a 3-terminal capacitance dilatometer was set up and used for measurements of the thermal expansion and magnetostriction of novel superconducting iron pinictides and related materials. In particular, \re~with R\,=\,La, Ce, Pr, Sm, Gd, \laf~and Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ have been investigated. The data on polycrystalline \laf~are the first published thermal expansion data on this material. The lattice effects at the structural and the magnetic phase transition have been investigated and the phase diagram upon F-doping has been studied. A main result is the observation of a previously unknown fluctuation regime for the doping level $\mathnormal{x}\leqslant$ 0.04 over a large $T$ range above the structural transition temperature \ts. The absence of any structural anomalies in the normal state of the superconducting \laf~samples with $\mathnormal{x}\geqslant$ 0.05 corroborates the discontinuous character of the phase boundary not only for the magnetism but also for the structural degrees of freedom. Similarly, the presence of high-temperature fluctuations is found for all \re~undoped materials under study. The discussion of the probable origin of the fluctuations as well as the definition of the structural transition temperature \ts~are done. The low temperature features shown by the thermal expansion data for \re~are caused by the onset of long range magnetic order of the $4f$-moments and their different configurations. In particular, \pr, which has a very pronounced anomaly associated with Pr-ordering exhibits a large magnetostriction at low temperatures. By discussing this effect along with the magnetization, resistivity and other measurements, it is found that this large magneto-elastic effect may originate from the correlations between the momentum from Fe$^{3+}$ and Pr$^{3+}$. Last, the thermal expansion of Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ 122 single crystals is investigated. Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ is one of the first materials where single crystals are available. The thermal expansion results on the undoped compound with $x=0$ show a large anomaly at the combined magnetic and structural transition which is far sharper than that for polycrystalline systems. Upon doping, both transitions are suppressed and their splitting is visible in the thermal expansion data. The high precision thermal expansion and magnetostriction results presented in this work are among the first data on the novel family of iron-based superconductors. A valuable insight in the respective ordering phenomena and the thermodynamic properties is provided.

Es werden Fragestellungen aus zwei aktuellen Problemkreisen der elektronisch hochkorrelierten Materialien untersucht. Dem unkonventionellen supraleitenden Zustand sowie dessen Wechselspiel mit magnetischen Effekten gelten die Arbeiten am Schwere-Fermionen-System UBe13 sowie der Dotierungsreihe (UTh)Be13. Sogenanntes Nicht-Fermiflüssigkeits-Verhalten steht im Zentrum der Untersuchungen an der Schwere-Fermionen-Verbindung CeNi2Ge2. Der Schwere-Fermionen-Supraleiter U1-xThxBe13 zeigt neben einem nichtmonotonen Verlauf der Übergangstemperatur in den supraleitenden Zustand Tc(x) einen zweiten Phasenübergang Tc2 < Tc im Konzentrationsbereich 0,019 < x < 0,0455. Als Ursache dieses Übergangs werden sowohl eine mit dem supraleitenden Zustand koexistierende magnetische Ordnung (Spindichtewelle) als auch eine Änderung des supraleitenden Zustands selber diskutiert. Hier konnte mittels dilatometrischer Untersuchungen gezeigt werden, dass der Phasenübergang bei Tc2 eine Vorläuferstruktur im Bereich x < 0,019 besitzt. Die aus diesem Ergebnis folgende Zuordnung charakteristischer Linien im T-x-Diagramm von U1-xThxBe13 schließt gängige Szenarien, die sich für T < Tc2 ausschließlich auf die Änderung des supraleitenden Zustands beziehen, aus. Manche Schwere-Fermionen-Verbindungen zeigen bis zu tiefsten Temperaturen keinen Übergang in einen kohärenten Fermiflüssigkeits-Zustand. Als Ursache dieses Nicht-Fermi-flüssigkeits-Verhaltens wird u.a die Ausbildung kritischer Spinfluktuationen diskutiert. Diese magnetischen Fluktuationen werden in Nähe eines quantenkritischen Punkts (QKP) erwartet, für den bei T=0 als Funktion eines Kontrollparameters ein magnetischer Phasenübergang auftritt. Die Vorhersagen des Konzepts eines "nearly antiferromagnetic Fermi liquid", für die Temperaturabhängigkeiten verschiedener Messgrößen von Systemen nahe eines QKP können an der Verbindung CeNi2Ge2 überprüft werden. Während bei nicht allzu tiefen Temperaturen in der vorliegenden Arbeit eine Übereinstimmung mit den Vorhersagen gefunden wurde, muss die Anwendbarkeit des Konzepts für CeNi2Ge2 bei tiefsten Temperaturen in Frage gestellt werden.

Es werden Fragestellungen aus zwei aktuellen Problemkreisen der elektronisch hochkorrelierten Materialien untersucht. Dem unkonventionellen supraleitenden Zustand sowie dessen Wechselspiel mit magnetischen Effekten gelten die Arbeiten am Schwere-Fermionen-System UBe13 sowie der Dotierungsreihe (UTh)Be13. Sogenanntes Nicht-Fermiflüssigkeits-Verhalten steht im Zentrum der Untersuchungen an der Schwere-Fermionen-Verbindung CeNi2Ge2. Der Schwere-Fermionen-Supraleiter U1-xThxBe13 zeigt neben einem nichtmonotonen Verlauf der Übergangstemperatur in den supraleitenden Zustand Tc(x) einen zweiten Phasenübergang Tc2 < Tc im Konzentrationsbereich 0,019 < x < 0,0455. Als Ursache dieses Übergangs werden sowohl eine mit dem supraleitenden Zustand koexistierende magnetische Ordnung (Spindichtewelle) als auch eine Änderung des supraleitenden Zustands selber diskutiert. Hier konnte mittels dilatometrischer Untersuchungen gezeigt werden, dass der Phasenübergang bei Tc2 eine Vorläuferstruktur im Bereich x < 0,019 besitzt. Die aus diesem Ergebnis folgende Zuordnung charakteristischer Linien im T-x-Diagramm von U1-xThxBe13 schließt gängige Szenarien, die sich für T < Tc2 ausschließlich auf die Änderung des supraleitenden Zustands beziehen, aus. Manche Schwere-Fermionen-Verbindungen zeigen bis zu tiefsten Temperaturen keinen Übergang in einen kohärenten Fermiflüssigkeits-Zustand. Als Ursache dieses Nicht-Fermi-flüssigkeits-Verhaltens wird u.a die Ausbildung kritischer Spinfluktuationen diskutiert. Diese magnetischen Fluktuationen werden in Nähe eines quantenkritischen Punkts (QKP) erwartet, für den bei T=0 als Funktion eines Kontrollparameters ein magnetischer Phasenübergang auftritt. Die Vorhersagen des Konzepts eines "nearly antiferromagnetic Fermi liquid", für die Temperaturabhängigkeiten verschiedener Messgrößen von Systemen nahe eines QKP können an der Verbindung CeNi2Ge2 überprüft werden. Während bei nicht allzu tiefen Temperaturen in der vorliegenden Arbeit eine Übereinstimmung mit den Vorhersagen gefunden wurde, muss die Anwendbarkeit des Konzepts für CeNi2Ge2 bei tiefsten Temperaturen in Frage gestellt werden.

In der vorliegenden Arbeit wird die Einkristallzüchtung nach dem Czochralskiverfahren von GdCa4O(BO3)3 (GdCOB) beschrieben. Aus insgesamt 18 Zuchtversuchen, bei denen auch die Ziehgeschwindigkeit zwischen 1 und 3mm/h variiert wurde, wurden erfolgreich nahezu perfekte Einkristalle gewonnen. In einigen Kristallen traten jedoch auch Risse oder Einschlüsse auf. Diese enthielten neben Iridium vom Tiegelmaterial auch andere Phasen des Gd2O3–B2O3–CaO-Systems, sowie P und Yb, deren Herkunft unklar ist. Als Hauptziehrichtung wurde die kristallographische b-Achse gewählt, ergänzt durch einige Experimente in der c-Richtung. In den drei kristallographischen Hauptrichtungen wurden die thermischen Ausdehnungskoeffizienten von GdCOB bestimmt. Diese können in zwei nahezu lineare Bereiche unterteilt werden: von Zimmertemperatur bis etwa 850° C und von 850 bis 1200° C, wobei die Koeffizienten im Hochtemperaturbereich deutlich höher sind (unter 850° C: alpha_a=11.1, alpha_b=8.6, alpha_c=13.3 10^-6/K, oberhalb 850° C: alpha_a=14.1, alpha_b=11.7, alpha_c=17.8 10^-6/K). Daraus ergibt sich, dass ein Phasenübergang höherer Ordnung vorliegen muss. Als mögliche Ursache wurde mittels HT-Raman Spektroskopie ein Ordnungs-Unordnungs-Übergang identifiziert, während dessen die BO3-Gruppen in der Struktur leicht rotieren. Weitere Untersuchungen mittels thermodynamischer Methoden führten zu schwachen, aber eindeutigen Signalen, die diesem Effekt ebenfalls zuzuordnen sind. Obwohl das Material ein vielversprechender Kandidat für piezoelektrische Anwendungen im Hochtemperaturbereich ist, wurde dieser Effekt bisher unzureichend beschrieben. Dieses Verhalten, kombiniert mit den anisotropen thermischen Ausdehnungskoeffizienten, könnte eine der Ursachen für das Vorkommen von Rissen in den Kristallen während der Synthese darstellen. Spektroskopische Untersuchungen ergaben einen großen Transparenzbereich von 340 bis 2500nm (29 400–4000 cm^-1), was für optische Anwendungen von großer Bedeutung ist. / In a series of 18 growth experiments, GdCa4O(BO3)3 (GdCOB) single crystals were successfully grown by the Czochralski method. They have a well-ordered structure, as revealed by single crystal structure analysis. Although the main growth direction was along the crystallographic b-axis, some experiments were conducted using the cdirection. Pulling velocities were varied between 1 and 3mm/h. Except for a few crystals with cracks or elongated "silk-like" inclusions consisting of multiphase impurities, most of the obtained crystals are of good quality. Those inclusions contain iridium, deriving from the crucible, P and Yb with unclear source, and other phases from the system Gd2O3–B2O3–CaO. Thermal expansion coefficients of GdCOB were determined in the directions of the crystallographic axes and found to be approximately linear in two temperature ranges: from 25° C to around 850° C, and from 850 to 1200° C, with the latter range showing significantly higher coefficients (below 850° C: alpha_a=11.1, alpha_b=8.6, alpha_c=13.3 10^-6/K, and above 850° C: alpha_a=14.1, alpha_b=11.7, alpha_c=17.8 x10^-6/K). This sudden increase of thermal expansion coefficients indicates a phase transition of higher order. An order-disorder transition in form of the rotation of BO3-triangles in the structure was made tentatively responsible for this transition, as revealed by HT-Raman spectroscopy. This transition was also detected by DSC-methods but appeared to result in very weak effects. Although the material is thought to represent a promising candidate for high temperature piezoelectric applications (noncentrosymmetric space group Cm), this effect of change in specification has not been described and it is unknown whether it has influence on the piezoelectric properties. Furthermore, this characteristic behaviour in combination with anisotropic coefficients may be the reason for the development of cracks during cooling of crystals, making the growth difficult. Spectroscopic investigation revealed a wide transparency range from 340 to 2500nm (29 400–4000 cm^-1) of GdCOB, which is a very important property for optical applications.

No description available.

In der vorliegenden Arbeit wird die Einkristallzüchtung nach dem Czochralskiverfahren von GdCa4O(BO3)3 (GdCOB) beschrieben. Aus insgesamt 18 Zuchtversuchen, bei denen auch die Ziehgeschwindigkeit zwischen 1 und 3mm/h variiert wurde, wurden erfolgreich nahezu perfekte Einkristalle gewonnen. In einigen Kristallen traten jedoch auch Risse oder Einschlüsse auf. Diese enthielten neben Iridium vom Tiegelmaterial auch andere Phasen des Gd2O3–B2O3–CaO-Systems, sowie P und Yb, deren Herkunft unklar ist. Als Hauptziehrichtung wurde die kristallographische b-Achse gewählt, ergänzt durch einige Experimente in der c-Richtung. In den drei kristallographischen Hauptrichtungen wurden die thermischen Ausdehnungskoeffizienten von GdCOB bestimmt. Diese können in zwei nahezu lineare Bereiche unterteilt werden: von Zimmertemperatur bis etwa 850° C und von 850 bis 1200° C, wobei die Koeffizienten im Hochtemperaturbereich deutlich höher sind (unter 850° C: alpha_a=11.1, alpha_b=8.6, alpha_c=13.3 10^-6/K, oberhalb 850° C: alpha_a=14.1, alpha_b=11.7, alpha_c=17.8 10^-6/K). Daraus ergibt sich, dass ein Phasenübergang höherer Ordnung vorliegen muss. Als mögliche Ursache wurde mittels HT-Raman Spektroskopie ein Ordnungs-Unordnungs-Übergang identifiziert, während dessen die BO3-Gruppen in der Struktur leicht rotieren. Weitere Untersuchungen mittels thermodynamischer Methoden führten zu schwachen, aber eindeutigen Signalen, die diesem Effekt ebenfalls zuzuordnen sind. Obwohl das Material ein vielversprechender Kandidat für piezoelektrische Anwendungen im Hochtemperaturbereich ist, wurde dieser Effekt bisher unzureichend beschrieben. Dieses Verhalten, kombiniert mit den anisotropen thermischen Ausdehnungskoeffizienten, könnte eine der Ursachen für das Vorkommen von Rissen in den Kristallen während der Synthese darstellen. Spektroskopische Untersuchungen ergaben einen großen Transparenzbereich von 340 bis 2500nm (29 400–4000 cm^-1), was für optische Anwendungen von großer Bedeutung ist. / In a series of 18 growth experiments, GdCa4O(BO3)3 (GdCOB) single crystals were successfully grown by the Czochralski method. They have a well-ordered structure, as revealed by single crystal structure analysis. Although the main growth direction was along the crystallographic b-axis, some experiments were conducted using the cdirection. Pulling velocities were varied between 1 and 3mm/h. Except for a few crystals with cracks or elongated "silk-like" inclusions consisting of multiphase impurities, most of the obtained crystals are of good quality. Those inclusions contain iridium, deriving from the crucible, P and Yb with unclear source, and other phases from the system Gd2O3–B2O3–CaO. Thermal expansion coefficients of GdCOB were determined in the directions of the crystallographic axes and found to be approximately linear in two temperature ranges: from 25° C to around 850° C, and from 850 to 1200° C, with the latter range showing significantly higher coefficients (below 850° C: alpha_a=11.1, alpha_b=8.6, alpha_c=13.3 10^-6/K, and above 850° C: alpha_a=14.1, alpha_b=11.7, alpha_c=17.8 x10^-6/K). This sudden increase of thermal expansion coefficients indicates a phase transition of higher order. An order-disorder transition in form of the rotation of BO3-triangles in the structure was made tentatively responsible for this transition, as revealed by HT-Raman spectroscopy. This transition was also detected by DSC-methods but appeared to result in very weak effects. Although the material is thought to represent a promising candidate for high temperature piezoelectric applications (noncentrosymmetric space group Cm), this effect of change in specification has not been described and it is unknown whether it has influence on the piezoelectric properties. Furthermore, this characteristic behaviour in combination with anisotropic coefficients may be the reason for the development of cracks during cooling of crystals, making the growth difficult. Spectroscopic investigation revealed a wide transparency range from 340 to 2500nm (29 400–4000 cm^-1) of GdCOB, which is a very important property for optical applications.

We characterize the impact of high-energy, 172 nm vacuum ultraviolet photons on the molecular weight and the glass transition temperature of poly(methyl methacrylate). We found that the molecular weight is reduced strongly on the surface of the exposed samples with a continuous transition towards the unexposed bulk material being located below the modified region. The glass transition temperature was found to be significantly lowered in the exposed region to well below 50°C compared to that of the 122°C of the bulk region. We could use this material contrast to selectively reflow the top surface of the exposed samples only. This allowed us to create ultra-smooth micro-optical structures by postprocessing without influencing the overall geometry that is required for the optical functionality.

In the past years, more concerns are focused on the nuclear waste management due to the very long half-lives of various actinides produced in Light Water Reactors (LWRs). Sodium Fast Reactors (SFRs) are thus becoming more attractive since they are known to be very efficient to transmute long-lived radionuclides present in spent fuel. However, the current simulation tools (thermal-hydraulics code with point kinetics) and safety assessment methods are not as mature as for LWR applications and need to be enhanced. This thesis aims at filling the gap in safety analysis of SFR cores to reach a standard similar to LWR applications by applying multi-physics modelling. In contrast to LWRs, the reactivity in SFRs is affected by three main feedback: the Doppler broadening reactivity effect, the sodium density change reactivity effect and the thermal expansion of several mechanical components of the reactor. In this thesis, the thermal-hydraulic system code ATHLET is coupled with the three-dimensional neutron-physics code PARCS for transient analysis. Developed at GRS, ATHLET was recently upgraded for sodium coolant properties. The nodal diffusion codes PARCS, developed at the University of Michigan, can solve the multi-group diffusion equation in hexagonal geometry. While both codes already have the main features to simulate SFRs, the development of models dedicated to the thermal expansion effect of reactivity is necessary. The latter has three main origins i.e. the core axial thermal expansion effect (caused by the fuel and the cladding axial thermal expansion), the core radial thermal expansion effect (caused by the diagrid thermal expansion), the control rod displacement due to the thermal expansion of the Control Rod Drive Lines (CRDLs), the strongback and the reactor vessel. Thus, the three main new developments achieved in the scope of this work are: - Development of a method to generate homogenized multi-energy-group neutron macroscopic cross sections (needed by PARCS) for SFR applications which consider not only the Doppler temperature and sodium density but also the core axial and radial thermal expansion. - Development of a three-dimensional core radial thermal expansion model and its implementation in PARCS. A core axial thermal expansion model has already been developed for PARCS prior to this work. - Development of a module in ATHLET for modelling the control rod displacement as a result of the influence of the reactor structures thermal expansion. The parametrized homogenized multi-energy-group neutron macroscopic cross section libraries for PARCS applications are generated with the Monte Carlo reactor-physics code Serpent. For all materials contained in fuel assemblies, a three-dimensional model is used while the SPH method is applied to materials contained in non-fuel assemblies (e.g. control rods, etc.). The cross section libraries are collapsed into a 12-energy-group structure. Furthermore, a dedicated module was successfully developed and implemented within the core simulator KMACS (developed at GRS). The core radial thermal expansion effect is implemented in PARCS using a coordinate transformation of the diffusion equation from the expanded state to the nominal geometry. The core radial thermal expansion depends on the diagrid temperature. It is calculated by ATHLET and transferred to PARCS by the extended interface between both codes. The modelling of the control rod displacement as a result of the reactor structures thermal expansion is performed by a module linked to ATHLET. The strongback, the reactor vessel and the CRDLs are modelled as heated structures in ATHLET, which calculates their respective temperature. The module can compute the thermal expansion of each structure as well as the total control rod banks displacement. The new techniques are verfied on a selected case study, the ASTRID core design. First, full core criticality simulations are performed with the Monte Carlo reactor-physics code Serpent (considered as reference calculations) and with PARCS. Good agreement between the two codes is achieved in terms of multiplication factors and power distribution. This allows to conclude that the developed method for neutron cross section libraries can be used for SFR applications. The newly implemented core radial expansion model in PARCS is successfully verified on the ASTRID core with the standalone version of PARCS. Then, various transient simulations are performed in order to separately analyse the different contributions to the reactivity by: the Doppler broadening effects, the sodium density change effect, the core radial and axial thermal expansion effect and the control rod displacement effect. It is demonstrated that the core power responses are plausible which allows the conclusion that all the different thermal expansion models are properly implemented. Furthermore, the presented simulations show very different core power responses. It appears that the effect of the sodium density change on reactivity is a parameter that is strongly heterogeneous (depending on the core location). This shows the importance of using a three-dimensional neutron kinetics model rather than a point-kinetic model for transient simulations with thermal-hydraulic codes. Moreover, the time-scale of the various effects are ranging from few seconds to several hundred seconds. While the Doppler broadening, the sodium density change, as well as the core axial and radial thermal expansion effects on reactivity are fast, the thermal expansion of the strongback and the vessel only appears after several hundred seconds. This emphasizes the importance of considering all thermal expansion effects in addition to the usual thermal-hydraulic feedback parameters (e.g. fuel temperature, coolant density etc.) to be able to compute the core behavior realistically.:Contents Abstract II List of Figures VII List of Tables X List of Acronyms XI Acknowledgments XIII 1 Introduction 1.1 Sodium cooled fast reactors 1.1.1 Fast reactor development 1.1.2 Comparison of sodium fast reactor and pressurized water reactor designs 1.1.2.1 Neutron spectrum 1.1.2.2 Breeding 1.1.2.3 Partitioning and Transmutation 1.1.2.4 Control of the reactivity in the core 1.1.2.5 Coolant properties 1.1.2.6 Reactivity feedback 1.1.2.7 Comparison summary 1.2 Objectives and structure of the thesis 1.2.1 Objectives 1.2.2 Structure of the thesis 2 State of the art of Sodium Fast Reactor safety assessment 2.1 Relevant safety events to consider for Sodium Fast Reactors 2.2 Major reactivity feedback mechanisms 2.3 State of the art of safety analysis methods for Sodium Fast Reactor 3 Methods and codes for safety assessment of sodium cooled fast reactors 3.1 Neutronics core calculations 3.1.1 Core calculations with the diffusion code PARCS 3.1.2 Generation of nodal few-group cross sections with the Monte Carlo code Serpent 3.1.3 Core simulator KMACS 3.2 Thermal-hydraulics simulations with the system code ATHLET 3.3 Coupled three-dimensional thermal-hydraulics / neutronics calculations 4 Development of three-dimensional thermal expansion models 4.1 General calculation approach proposed for safety assessment 4.2 Thermal expansion in solids 4.3 Model for generating nodal few-energy-group cross sections for deterministic core analysis 4.3.1 Energy group structure 4.3.2 Full-scale three-dimensional fuel assembly models in Serpent 4.3.3 Two-dimensional non-fuel assembly models in Serpent 4.3.4 Super homogenization method for non-multiplying media 4.3.5 Automated creation of Serpent models for parametrized cross section generation with KMACS 4.4 Core radial thermal expansion effect 4.4.1 Description of the core radial thermal expansion phenomenon 4.4.2 Coordinate transformation of the diffusion equation 4.4.3 Implementation of the coordinates transformation in PARCS 4.4.4 Adapted cross section parametrization scheme for the core radial expansion model 4.4.5 Diagrid model in ATHLET and temperature transfer 4.5 Core axial thermal expansion effect 4.5.1 Description of the core axial thermal expansion phenomenon 4.5.2 Implementation of a core axial thermal expansion model in PARCS 4.5.3 Appropriate cross section parametrization scheme 4.6 Control rod displacement due to reactor structures thermal expansion effects 4.6.1 Modelling scheme 4.6.2 Strongback model in ATHLET 4.6.3 Vessel model in ATHLET 4.6.4 Control rods drive lines ATHLET model 5 Verification on a case study 5.1 Description of the ASTRID reactor 5.2 Full core models 5.2.1 Full core Serpent reference models of the ASTRID core 5.2.2 Three-dimensional neutron kinetics model of ASTRID core in PARCS 5.2.3 Generation of appropriate few-group cross sections 5.2.4 Thermal-hydraulic model in ATHLET and ATHLET-PARCS feedback mapping 5.3 Verfications of the radial core expansion model 5.4 Assessment of the Doppler and sodium density effects 5.4.1 Assessment of the Doppler effect 5.4.2 Assessment of the sodium density effect 6 Coupled three-dimensional thermal-hydraulics/neutron-physics transient simulations with ATHLET-PARCS 6.1 Description of the models and transient simulations 6.2 Simulation 1: Doppler effect 6.2.1 Description 6.2.2 Results 6.3 Simulation 2: Sodium density effect 6.3.1 Description 6.3.2 Results 6.4 Simulation 3: Doppler and sodium density effects 6.4.1 Description 6.4.2 Results 6.5 Simulation 4: Core radial thermal expansion effect 6.5.1 Description 6.5.2 Results 6.6 Simulation 5: Doppler, Sodium density and core radial thermal expansion effects 6.6.1 Description 6.6.2 Results 6.7 Simulation 6: Core axial thermal expansion effect 6.7.1 Description 6.7.2 Results 6.8 Simulation 7: Doppler, Sodium density and core axial thermal expansion effects 6.8.1 Description 6.8.2 Results 6.9 Simulation 8: Doppler effect, Sodium density effect, core radial thermal expansion effect and core axial thermal expansion effect 6.9.1 Description 6.9.2 Results 6.10 Simulation 9: Doppler effect, Sodium density effect, core radial thermal expansion effect, core axial thermal expansion effect and control rod displacement due to reactor structures thermal expansion effect 6.10.1 Description 6.10.2 Results 6.11 Preliminary conclusions of the test calculations 7 Conclusion and outlook for future developments 7.1 Summary and conclusions 7.2 Suggestions for future work Appendices A The Boltzmann equation B Macro-group structure Bibliography

We present a phase-field crystal model for solidification that accounts for thermal transport and a temperature-dependent lattice parameter. Elasticity effects are characterized through the continuous elastic field computed from the microscopic density field. We showcase the model capabilities via selected numerical investigations which focus on the prototypical growth of two-dimensional crystals from the melt, resulting in faceted shapes and dendrites. This work sets the grounds for a comprehensive mesoscale model of solidification including thermal expansion.