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

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Chapter 2 Overview of the Theoretical Background 27 Figure 2.1: Temperature dependence of the linear susceptibility of NpO 2 . The open points are the measured result of [14]. shown in Fig. 2.1 rises to a small cusp at the transition temperature, and stays almost constant below [22, 52]. First, the observations were ascribed to antiferromagnetic ordering. However, neutron diffraction experiments did not detect magnetic order [23], and Mössbauer measurement has given an upper limit 0.01µ B for the ordered moment. Np 4+ ions have the configuration 5f 3 , the corresponding Hund’s rule ground state set belongs to J = 9/2. Let us immediately observe that for an odd number of electrons the symmetry classification of the electron states has to come from the theory of double groups. Without going into the details, we recall the result that Γ J=9/2 = Γ 6 +2Γ 8 where Γ 8 is a four-dimensional (Γ 6 is a two-dimensional) irrep of the cubic double group. The experimentally established splitting scheme is schematically shown in Fig. 2.2. The |Γ (1) 8 〉 ground state quartet is well separated from the other states. The relevant local Hilbert space is the four dimensional |Γ (1) 8 〉. Its fourfold degeneracy can be understood as combined of a twofold Kramers and a twofold non-Kramers degeneracy. Resonant X-ray scattering measurement [17] found long-range order of the Γ 5 electric quadrupole moments. However, quadrupolar ordering alone cannot resolve the Kramers degeneracy: there should remain magnetic moments at low temperatures giving rise to a Curie susceptibility, and this is in contrast with the observations. Furthermore, muon spin relaxation shows that local magnetic fields, with a pattern suggestive of magnetic octupoles, appear below the 25K transition temperature [53]. Octupolar order can resolve the Kramers degeneracy. In NpO 2 there exists only one phase transition, so the question is whether quadrupolar and octupolar order can appear at the
Chapter 2 Overview of the Theoretical Background 28 Γ 6 J = 9/2 Γ 1 2 Γ 8 8 Figure 2.2: The schematic representation of the splitting of the J = 9/2 ten-dimensional multiplet in cubic environment. same time. It turns out that this is possible because they can have the same Γ 5 symmetry 18 . The current understanding is that the primary order parameters of the 25K transition are the Γ 5 octupolar moments. Γ 5 quadrupolar order is induced by the primary ordering. It is a peculiarity of the fcc structure that an ”anti” alignment is preferably four-sublattice with moments aligned in the (111), (111), (111) and (111) directions. A realistic description of NpO 2 will have to be based on the triple-⃗q four-sublattice order. However, for many aspects of the behavior it is only important that the ordering involves Γ 5 octupoles supported by a Γ 8 subspace, whatever the relative orientation of nearest neighbor moments is. The preference for aligning moments along (111) directions, and the coupling to Γ 5 quadrupoles follows. It is the simplest to consider a ferro-octupolar model which is interesting in its own right. In H = 0 magnetic field its mean-field solution is actually equivalent to that of the four-sublattice problem. Our main interest, however, is the study of the effects of an external magnetic field. Octupoles are magnetic, but they appear only in the non-linear magnetic response. Crystal field anisotropy makes the nature of the field induced multipoles complicated. This question is of basic importance for all multipolar models; our most systematic study was carried out in the context of the 18 For this, see Table 2.3. Note that the irreps of order parameters are derived from the products of the irreps of the states and so do not belong to double groups.
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Page 3 and 4: CONTENTS 2 4 PrFe 4 P 12 Skutterudi
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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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