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

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Chapter 2 Overview of the Theoretical Background 9 trostatic potential V can be expressed in terms of the spherical harmonics, V (r) = ∞∑ k∑ k=0 q=−k a q k rk Y q k (θ, φ) . (2.2) For example, for a tetragonal environment, the lowest order term is quadratic, and this is followed by a series of higher-order terms V tetr (r) = C 1 (3z 2 − r 2 ) + C 2 (x 4 + y 4 + z 4 − 6r 4 ) + ... (2.3) where r 2 = x 2 + y 2 + z 2 . 2.2.1 Stevens Equivalents V tetr (r) appearing in (2.3) should be put into the Schrödinger equation as a one-electron potential. However, we do not wish to solve the problem in full generality. Our main interest is in f-electron systems for which the relevant subspace is the (2J + 1)-dimensional J-eigenspace. It can be shown that within this Hilbert space, V tetr acts like V tetr (r) → c ′ 1[3J 2 z − J(J + 1)] + c ′ 2[O 0 4 + 5O 4 4] + ... (2.4) where the second term contains a complicated fourth order polynomial of J x , J y , J z (see Appendix A). Our point is the correspondence of the first term to the first term of (2.3). The general lesson is that for our purposes, operators expressed in terms of the cartesian coordinates x, y, z can be replaced by equivalent expressions of J x , J y , J z . However, to account for the fact that J x , J y , J z do not commute while x, y, z do, we always have to symmetrize the J components. The simplest example is xy → J x J y = (J x J y + J y J x ) (2.5) which happens to be the quadrupolar moment O xy . Here, and in the following, ”overline” means symmetrization. O xy is the Stevens equivalent of xy. Similarly, (2.4) is the Stevens equivalent of (2.3). It is a general consequence of the Wigner–Eckart theorem that Stevens replacements act equivalently to the original operators within a J-eigenspace. Introducing Stevens equivalents makes using a (J, J z ) basis very convenient. This is what we will usually do. We will use the Stevens equivalents of multipolar moments, and their intersite interactions, and diagonalize them in either single-site, or many-site (J, J z ) Hilbert spaces.
Chapter 2 Overview of the Theoretical Background 10 We return now to the general formulation of a crystal field Hamiltonian operating within the lowest (2J +1)-fold degenerate |LSJM J 〉 multiplet. The Wigner-Eckart theorem tells that in this subspace the spherical harmonics can be replaced by combinations of the components of the J operator, resulting in the equivalent operators (or Stevens operators) O q k (J). The matrix elements of these equivalent operators are proportional to the matrix elements of the original crystal field potential 〈JM J |Y q k (θ, φ)|JM ′ J〉 ∝ 〈JM J |O q k (J)|JM ′ J〉 . (2.6) Thus the crystal field potential can be written as H cf = ∑ k∑ k=2,4,... q=−k B q k Oq k (J) , (2.7) where B q k ∝ aq k 〈rk 〉. As we mentioned, the spherical harmonics of order k provide a (2k + 1) dimensional representation D (k) of the SO(3) rotational group, which is the group of all rotations in the three dimensional space. To determine the actual form of the H cf one has to decompose the representations D (k) according to the symmetry of the system. For instance, in the case of cubic symmetry, second order Stevens operator expressions do not occur in H cf because the splitting of D (2) does not contain the identity representation 4 . But it appears in the splitting of the D (4) and D (6) representations once in each, thus two terms of the Stevens operators are necessary to specify the potential H cub = c 4 [O 0 4 + 5O 4 4] + c 6 [O 0 6 − 21O 4 6] . (2.8) The expressions of the Stevens operator equivalents are listed in Appendix A. In the following, we will consider systems with not only cubic, but tetragonal symmetry, thus we give also the form of the tetragonal crystal field Hamiltonian H tetr = B 0 2O 0 2 + B 0 4O 0 4 + B 0 6O 0 6 + B 4 4O 4 4 + B 4 6O 4 6 , (2.9) which has five independent parameters. (This is the full expression replacing our previous (2.4)). There is a second order term in the Hamiltonian (see H tetr ) in contrast to the cubic case, because O 2 2 and O 0 2 become inequivalent in tetragonal environment, and O 0 2 gives a distortion along the c axis. 4 The splitting of D (2) in the cubic environment is Γ 3 ⊕ Γ 5 .
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Chapter 6 Conclusion One of the aim
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Appendix A Appendix Stevens Operato
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Appendix B Appendix Numerical Param
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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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