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

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Contents 1 Introduction 3 2 Overview of the Theoretical Background 5 2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Crystal Field Theory . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Stevens Equivalents . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Crystal Field Splitting . . . . . . . . . . . . . . . . . . 11 2.3 Local Order Parameters . . . . . . . . . . . . . . . . . . . . . 14 2.4 Interactions Between the Multipoles . . . . . . . . . . . . . . . 17 2.5 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Review of f-electron Systems . . . . . . . . . . . . . . . . . . 25 2.6.1 NpO 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.2 Pr-filled Skutterudites . . . . . . . . . . . . . . . . . . 29 2.6.3 URu 2 Si 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Octupolar Ordering of Γ 8 Ions 36 3.1 Octupolar Moments in the Γ 8 Quartet State . . . . . . . . . . 37 3.2 The Octupolar–Quadrupolar Model . . . . . . . . . . . . . . . 41 3.2.1 The Case of Zero Magnetic Field . . . . . . . . . . . . 42 3.2.2 The Case of H‖(111) Magnetic Field . . . . . . . . . . 46 3.3 The Effect of Magnetic Field with Arbitrary Direction on the Octupolar Ordering . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.1 Ground State Calculation . . . . . . . . . . . . . . . . 51 3.3.2 Field Induced Multipoles . . . . . . . . . . . . . . . . . 53 3.3.3 Connection with the Mean-Field Results . . . . . . . . 55 3.4 Symmetry Classification of Multipoles in External Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.1 Symmetry Related Expansion of the Helmholtz Potential: Method I. . . . . . . . . . . . . . . . . . . . . . . 59 3.4.2 Inclusion of the Time Reversal: Method II. . . . . . . . 64 3.4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1
CONTENTS 2 4 PrFe 4 P 12 Skutterudite 68 4.1 Crystal Field Model . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.1 Γ 1 –Γ 4 Level Scheme, Order Parameters . . . . . . . . . 69 4.1.2 Interaction Induced Ordering . . . . . . . . . . . . . . 71 4.2 The Effect of External Magnetic Field . . . . . . . . . . . . . 72 4.3 Mean-Field Results . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Tetrahedral Environment . . . . . . . . . . . . . . . . . . . . . 77 4.5 Another Pr Compound: PrBa 2 Cu 3 O 6 . . . . . . . . . . . . . . 79 5 URu 2 Si 2 System 84 5.1 Order Parameters and Crystal Field Schemes . . . . . . . . . . 85 5.2 External Magnetic Field . . . . . . . . . . . . . . . . . . . . . 87 5.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3.1 Low-Field Octupolar Phase . . . . . . . . . . . . . . . 91 5.3.2 High-Field Quadrupolar Phase . . . . . . . . . . . . . . 93 5.4 Stress Induced Antiferromagnetism . . . . . . . . . . . . . . . 95 6 Conclusion 98 A Appendix 102 B Appendix 106 C Appendix 108 Acknowledgement 110 Bibliography 111
Page 1: Ph.D. THESIS Multipolar ordering in
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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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