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

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Chapter 2 Overview of the Theoretical Background 19 of kinetic exchange, superexchange, and RKKY interaction can be generalized to include multipolar interactions. It follows that multipolar couplings are not necessarily weaker than dipolar couplings. A microscopic study of Ce-Ce interactions gave the result that octupolar, quadrupolar and dipolar couplings are of the same order of magnitude [6]. On the experimental side, it is known that quadrupolar ordering happens before magnetic ordering in many Ce, Tm and Pr compounds. We will discuss in detail the case of NpO 2 in which octupolar ordering is the leading instability in Chapter 3. We are going to discuss which form of inter-site interactions is allowed by symmetry considerations. The local order parameters are the multipoles whose single-site symmetry classification for cubic environment was shown in Table 2.3. The most general form of the interaction Hamiltonian between lattice sites i and j is H ij = 3∑ k,l=1 7∑ + k,l=1 J kl ij (m)J k i J l j + 5∑ k,l=1 J kl ij (Q)Q k i Q l j J kl ij (O)O k i O l j + . . . (2.25) where Ji k , Q k i and Oi k are the dipolar, quadrupolar and octupolar momentum components at a site i. Jij kl (m), Jij kl (Q) and Jij kl (O) are the magnetic, quadrupolar and octupolar coupling constants, respectively 12 . The lattice Hamiltonian is H int = ∑ ij H ij (2.26) the sum of all pair interactions. H int is invariant under the symmetry operations of the lattice. Each term of multipoles with different orders in Hamiltonian (2.26) must be an invariant (basis element for the identity representation of the symmetry group of the lattice). This gives a stringent restriction for the form of (2.26). In the absence of external magnetic field, time reversal is an additional symmetry. This is manifest in the form of (2.25) in which odd (even) multipoles are coupled with only odd (even) multipoles 13 . While H int has the full symmetry of the lattice, the pair interaction H ij has only the symmetry of a two-site cluster (atoms i and j), which is a lower symmetry. For instance, even in a cubic system, the pair interactions have 12 We think the effects of the conduction electrons included into these coupling constants. 13 This would, in principle, allow dipole-octupole interaction which we did not include in Hamiltonian (2.25).
Chapter 2 Overview of the Theoretical Background 20 only tetragonal symmetry. Let us examine this situation through the case of the quadrupolar coupling term. We saw in the previous Section that in cubic environment the five dimensional representation spanned by the quadrupolar operators splits into a doublet and a triplet Γ 3g (2) ⊕ Γ 5g (3). This means that we expect two independent quadrupolar coupling constants, and we may think that the quadrupolar interaction term has the following form H quad ij = Q 3 [3O 2 2,iO 2 2,j + O 0 2,iO 0 2,j] + Q 5 [O xy,i O xy,j + O yz,i O yz,j + O zx,i O zx,j ] . In this expression, both terms are a cubic invariant. Such a form is often used, but strictly speaking, it is only an approximation. The correct form of the Γ 3 part of the quadrupolar interaction, for example, is the sum of pair interaction terms each of which has only tetragonal symmetry with the axis of the pair as the tetragonal fourfold axis H Γ3 = Q 3 ∑ i [ O 0 2,iO 0 2,i+ẑ + 1 4 (3O2 2,i + O 0 2,i)(3O 0 2,i+ˆx + O 0 2,i+ˆx) + 1 ] 4 (3O2 2,i − O2,i)(3O 0 2,i+ŷ 0 − O2,i+ŷ) 0 . (2.27) It is the sum over all orientations of the pair which is manifestly cubic. Leaving such complications aside, the interaction (2.25) must contain dipolar, quadrupolar, octupolar, etc., terms: H ij = H dip + H quad + H oct = J 4 [J x,i J x,j + J y,i J y,j + J z,i J z,j ] + Q 3 [3O 2 2,iO 2 2,j + O 0 2,iO 0 2,j] + Q 5 [O xy,i O xy,j + O yz,i O yz,j + O zx,i O zx,j ] + O 2 T xyz,i T xyz,j + O 4 [T α x,iT α x,j + T α y,iT α y,j + T α z,iT α z,j] +O 5 [T β x,iT β x,j + T β y,iT β y,j + T β z,iT β z,j] + . . . (2.28) As we have discussed in Section 2.3, the dimensionality and the nature of the local Hilbert space decides which order parameters are supported by the subspace, i.e., which multipolar operators have to be included in the Hamiltonian. Finally note that since the electric multipoles can be coupled to the lattice distortion (or displacement), there is a further indirect interaction type between the electric multipoles mediated by the phonons. This virtual phonon process is analogous of the indirect exchange between the multipoles mediated by the conduction electrons. This type of interaction does not exist
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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 88 ∆
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Chapter 5 URu 2 Si 2 System 90 fiel
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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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