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

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Chapter 2 Overview of the Theoretical Background 17 parameters acting on the doublet (2.15) are |Γ + 3 〉〈Γ + 3 | − |Γ − 3 〉〈Γ − 3 | ∼ O 0 2 (2.20) |Γ + 3 〉〈Γ − 3 | + |Γ − 3 〉〈Γ + 3 | ∼ O 2 2 (2.21) i [ |Γ − 3 〉〈Γ + 3 | − |Γ + 3 〉〈Γ − 3 | ] ∼ T xyz . (2.22) By constructions, O2 0 and O2 2 are real operators, i.e., they are time reversal invariant, while T xyz is an imaginary operator, i.e., it is time-reversal-odd. This corresponds to the definition of the order parameters used in Table 2.3. 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 the conduction electrons like the usual exchange interaction. Based on symmetry consideration we can tell the possible order parameters, but we may pose the question: which multipolar moment will order? It depends on microscopic details of the system which multipolar interaction will be relevant. In the next Section we discuss the nature of the interactions between the multipoles. 2.4 Interactions Between the Multipoles We saw that the single-site ionic degrees of freedom are in general highly reduced at low temperatures, but usually not all of them are quenched, and the static magnetic behavior depends on the ion-ion interactions. The ground state usually an ordered state which is a consequence of the interactions. The first attempt to understand the magnetic behavior was the Heisenberg exchange Hamiltonian H exch = JS 1 · S 2 (2.23) where S 1 and S 2 are the spins of two ions and J is the spin-spin coupling constant. The effective coupling between the spins may arise from either direct exchange, or superexchange, or from RKKY-type indirect exchange. However, the spin-only form (2.23) can be used for d-electron systems only for which the orbital degrees of freedom are quenched. In f-electron systems this Hamiltonian does not work well, because the spin exchange is highly anisotropic, and the interactions between the higher order multipoles are equally important. The first interaction type which may come to mind is the classical (direct) electric and magnetic multipole interactions. The direct dipole-dipole
Chapter 2 Overview of the Theoretical Background 18 interaction between spins S 1 and S 2 has the form H dd = g 2 µ 2 B ( S1 · S 2 − 3 (S ) 1 · r)(S 2 · r) . (2.24) r 3 r 5 We may get the classical electric interactions by the expansion of the Coulomb interaction term e 2 /r ij between two electrons at different i and j sites as a power series of ri k rj k′ /R k+k′ +1 allowing only even k and k ′ values (R is the distance between the two ions, r i and r j are the distances of the two electrons from the ion sites i and j, respectively). We may conclude, that the order of magnitude of the classical dipoledipole interaction is not unimportant in f-electron systems 11 , however it cannot be the primary cause of magnetism. This follows also from the Bohr-van Leeuwen theorem, which states that magnetic ordering is absent in classical statistical mechanics. Direct exchange derived from the Coulomb interaction is a quantum mechanical effect which follows from the antisymmetrization of the wave functions and the Pauli principle, though the Coulomb interaction (H Coulomb = e 2 /|r 1 − r 2 |) is spin-independent. Direct exchange for orthogonal orbitals is ferromagnetic: this gives rise to Hund’s first rule. There are many situations, as in the case of f-electron systems, when the inter-ionic distance is large compared to the orbital radius and direct overlap can be neglected. In these cases we need a different mechanism of interaction: the RKKY-type interaction. This can be visualized as follows: the localized spin makes a perturbation in the distribution of the conduction electrons, and the spin of another ion feels this perturbation, and its alignment will be affected by the spin of the former ion. For magnetic dipoles, this interaction also has the form (2.23) with distance-dependent coupling constant J(r). In an insulator the indirect interaction is short range, and it is mediated by nonmagnetic atoms lying between magnetic ions. This is called superexchange which is important in systems like MnF 2 , FeF 2 or LaTiO 3 . The interaction is usually antiferromagnetic, but when it occurs between ions with different orbital states, it can be also ferromagnetic. For the sake of completeness we also mention double exchange which operates in mixed valent metals and gives a mechanism of strong ferromagnetism in manganites. Described in the previous Section, f-shells can possess a number of multipolar degrees of freedom in addition to the magnetic dipoles. The mechanism 11 We expect this because of the relatively large value of the total angular moment J in f-electron systems.
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Chapter 4 PrFe 4 P 12 Skutterudite
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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 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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