Ph.D. THESIS Multipolar ordering in f-electron systems

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Chapter 1 Introduction Orbital ordering phenomena in f-electron systems have been extensively studied in the past two decades. The highly degenerate f-shells of rare earth and actinide ions support a great variety of local degrees of freedom: magnetic, quadrupolar and octupolar at least. This leads to variegated and complex phase diagrams and magnetic properties. Quadrupolar ordering was found to explain the properties of many Ce, Tm and Pr-based rare earth compounds like CeB 6 , TmTe, and PrPb 3 . The detailed theory of CeB 6 contains the first full symmetry classification of multipoles, including the effect of an external magnetic field [5]. In earlier studies, orbital ordering was usually understood to mean just quadrupolar ordering, but it has become clear that orderings of higher multipoles are also realized in some f-electron compounds. Recent experimental observations on NpO 2 indicate that the primary order parameter of the 25K phase transition is purely octupole. Multipoles of still higher order, namely hexadecapoles and triakontadipoles may play a role in the physics of Pr-filled skutterudites and URu 2 Si 2 . The physics of multipolar orderings is far from being fully explored yet. One may think that higher order multipoles are irrelevant because their interaction is much weaker than ordinary dipolar interaction. This seems to be suggested by the multipole expansion in classical electrodynamics. This is, however, misleading. The leading interaction term between the multipoles has quantum mechanical origin, it is mediated by electrons in wide bands like the usual indirect exchange interaction, thus the interactions between multipoles with different rank are equally important. Many f-electron systems order magnetically. However, quite a few f- electron systems have phase transitions which are thermodynamically as strong as the magnetic transitions, but the low temperature phase is not magnetically ordered. In these cases, we associate the phase transition with 3
Chapter 1 Introduction 4 the ordering of a multipolar moment. Often it is called ”hidden order” because, in contrast to magnetic order which is easy to detect, multipolar order is not easily seen. Therefore, theoretical investigations of the behavior of multipolar models and the consequences for the magnetic properties are important. My thesis work is organized as follows. In Chapter 2, a general overview of the f-electron systems and the relevant theoretical concepts will be given. Part of my motivation was to understand the properties of the compounds NpO 2 , PrFe 4 P 12 , and URu 2 Si 2 . I give a short review of the experimental results and previous theories in Section 2.6. Motivated by the findings on NpO 2 I introduced a Γ 8 lattice model of octupolar ordering (Chapter 3). I used this example to develop a description of the field induced coupling of multipolar moments (Section 3.4). I describe the Γ 1 –Γ 4 quasi-quartet model of the PrFe 4 P 12 skutterudite in Chapter 4. Finally, I present a new model of the temperature–magnetic field phase diagram of URu 2 Si 2 in Chapter 5. In this model, the so-called hidden order is interpreted as octupolar order. Chapter 6 is the conclusion, and the last parts include the Appendix, the Acknowledgment and the Bibliography.
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Page 3: CONTENTS 2 4 PrFe 4 P 12 Skutterudi
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Chapter 3 Octupolar Ordering of Γ
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Chapter 4 PrFe 4 P 12 Skutterudite
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Chapter 5 URu 2 Si 2 System We desc
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Chapter 5 URu 2 Si 2 System 86 with
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Chapter 5 URu 2 Si 2 System 88 ∆
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Chapter 5 URu 2 Si 2 System 90 fiel
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Chapter 5 URu 2 Si 2 System 92 Ther
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Chapter 5 URu 2 Si 2 System 94 As w
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Chapter 5 URu 2 Si 2 System 96 8 T=
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Chapter 6 Conclusion One of the aim
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Chapter 6 Conclusion 100 ordering.
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Appendix A Appendix Stevens Operato
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APPENDIX A. APPENDIX 104 becomes as
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Appendix B Appendix Numerical Param
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Appendix C Appendix Here I list som
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Acknowledgement I would like to exp
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BIBLIOGRAPHY 112 [15] P. Santini an
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BIBLIOGRAPHY 114 [51] F. Bourdarot,
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Ph.D. THESIS Multipolar ordering in f-electron systems

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