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

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Chapter 2 Overview of the Theoretical Background 31 Figure 2.5: Experimentally investigated temperature-magnetic field phase diagram of PrFe 4 P 12 skutterudite [37]. moments. Pr 3+ ions have the 4f 2 configuration and the ninefold degenerate J = 4 Hund’s rule ground state multiplet. The main component of the crystal field is cubic, but there is also a tetrahedral component [38]. While recent investigations indicate that the inclusion of the tetrahedral component gives rise to interesting features, it is a good approximation to take a cubic crystal field which splits J = 4 multiplet according to Γ J=4 = Γ 1 ⊕ Γ 3 ⊕ Γ 4 ⊕ Γ 5 . (2.33) The corresponding eigenstates are listed in Appendix A. The order of the crystal field levels is difficult to derive from experiments. Analyzing the anisotropy of the magnetization curves above the ordered phase where single-ion results are valid, it was concluded that the likely possibilities are: a Γ 1 ground state and a low-lying exited state Γ 4 (the Γ 1 –Γ 4 scheme); or the Γ 1 –Γ 5 scheme; or the Γ 3 –Γ 4 scheme [37]. At the first sight, the assumption of a Γ 3 ground state looks plausible. We have shown in (2.19) that the Γ 3 doublet carries the Γ 3g quadrupolar moments, so antiferro-quadrupolar ordering can give a unique ground state. The assumption of the Γ 3 ground state (see the left part of Fig.2.6) is also compatible with the symmetry analysis of the structural distortion accompanying the antiferro-quadrupolar ordering [39]. However, the field dependence of the transition and other magnetic properties remain unexplained. As mentioned before, magnetization data also allow that the ground state be the Γ 1 singlet. Of course, a singlet carries no moment (either quadrupolar or anything else), so it cannot be taken alone. We analyzed the two allowed possibilities for the two lowest levels: Γ 1 –Γ 4 and Γ 1 –Γ 5 , and concluded that
Chapter 2 Overview of the Theoretical Background 32 Γ 1 Γ 5 Γ 5 Γ 3 Γ 4 Γ 4 Γ 3 Γ 1 Figure 2.6: The proposed order of the J = 4 crystal field levels, arising from the splitting of the ninefold degenerate free ion state in cubic environment, for modelling the PrFe 4 P 12 skutterudite in previous works (left) and in our work (right). only Γ 1 –Γ 4 gives a useful model (see the right part of Fig. 2.6). Taking the two levels as approximately degenerate defines a four-dimensional local Hilbert space, which supports 15 order parameters, including the required Γ 3g quadrupoles. We developed the mean-field theory of the Γ 1 –Γ 4 quasiquartet model, and showed that it gives a satisfactory explanation of the experimental findings (shown in Fig. 2.4 and Fig. 2.5). These results are described in Chapter 4. 2.6.3 URu 2 Si 2 The metallic actinide system URu 2 Si 2 can be envisaged as an array of U 4+ ions, i.e., 5f 2 shells, embedded into a conduction electron Fermi sea. We assume that the 5f 2 shells are sufficiently localized, and have the L = 5, S = 1, J = 4 Hund’s rule ionic configuration. Like other systems considered by us, URu 2 Si 2 has a mysterious phase transition. Its basic characteristics are shown in Fig. 2.7. The large λ- anomaly of the specific heat at T N = 17.5K can be shown to belong to the full-scale ordering of a localized degree of freedom 20 . It does have a signature in magnetic properties 21 as shown by the λ-anomaly of the non-linear suscep- 20 The entropy change associated with the large T ≤ 20K is of O(k B ln2). Though the f-electrons of URu 2 Si 2 have itinerant aspects, the phase transition does not belong to some weak-coupling density wave. 21 Careful inspection shows that the linear susceptibility has a change of slope ∆∂χ/∂T > 0 at T = 17.5K. Though this is not a weak feature, in Fig. 2.7 it is masked by large single-
Page 1 and 2: Ph.D. THESIS Multipolar ordering in
Page 3 and 4: CONTENTS 2 4 PrFe 4 P 12 Skutterudi
Page 5 and 6: Chapter 1 Introduction 4 the orderi
Page 7 and 8: Chapter 2 Overview of the Theoretic
Page 31: Chapter 2 Overview of the Theoretic
Page 37 and 38: Chapter 3 Octupolar Ordering of Γ
Page 69 and 70: Chapter 4 PrFe 4 P 12 Skutterudite
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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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