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Magnetism of F-Element-Complexes

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Magnetism of F-Element-Complexes Magnetism of f‐Element‐Complexes Winter School in Theoretical Chemistry 2014 Theoretical f‐Element Chemistry December 15th–18th, 2014 Prof. Dr. Joris van Slageren Institut für Physikalische Chemie Universität Stuttgart Germany f‐Element Magnetism: Scope of the lectures Pure f‐Metals 2 • Helical magnetic ordering • Indirect/RKKY exchange • Magnetostriction Alloys • Samarium‐Cobalt, Neodymium‐Iron‐Boron • Large coercivity/magnetic field Rare Earth Garnets A3B2X3O12 • Magnetic bubble memory • Magneto‐optics • Compensation temperature f‐Element complexes • Crystal Field splitting • Magnetization dynamics • Unusual magnetic states f‐Element Magnetism 3 Contents 1. Electronic Structure (at low energies) 1.1 Theoretical description 1.2 Experimental determination 1.3 Actinides 2. Static magnetic properties 2.1 Theoretical description 2.2 Experimental determination 2.3 Magnetic couplings 3. Dynamic magnetic properties 3.1 Theoretical description 3.2 Experimental determination 3.3 Examples Literature f‐Element Magnetism and Crystal Field theory 4 H. Lueken – Magnetochemie. R. Boča–Theoretical Foundations of Molecular Magnetism. A. Abragam/B. Bleany – Electron Paramagnetic Resonance of Transition Ions. D.J. Newman/Ng (Ed.) –Crystal Field Handbook C. Görller‐Wallrand, K. Binnemans in Handbook on the Physics and Chemistry of Rare Earths, Vol. 23, http://dx.doi.org/10.1016/S0168‐1273(96)23006‐5 K.R. Lea, M.J.M. Leask, W.P. Wolf, J. Phys. Chem. Solids, 23, 1381 (1962) http://obelix.physik.uni‐bielefeld.de/~schnack/molmag/material/Lueken‐ kurslan_report.pdf Recent reviews Woodruff, Winpenny, Layfield, Chem. Rev. 2013; Luzon, Sessoli, Dalton Trans., 2012; Sorace, Benelli, Gatteschi, Chem. Soc. Rev. 2011 1. Electronic Structure 2. Static magnetic properties 3. Dynamic magnetic properties Ch. 1. Electronic Structure Section 1.1 Theoretical description 6 Energy scale of interactions • Electrons experience the following interactions: • attraction to the nucleus (Coulomb) • repulsion by other electrons • spin‐orbit coupling • crystal field (Coulomb) • magnetic field (Zeeman) • The main difference with transition metals is that the crystal field and exchange interactions are much weaker H. Lueken, Course of lectures on magnetism of lanthanide ions under varying ligand and magnetic fields, RWTH Aachen, 2008 http://obelix.physik.uni‐bielefeld.de/~schnack/molmag/material/Lueken‐kurslan_report.pdf Ch. 1. Electronic Structure Section 1.1 Theoretical description 7 Energy scale of interactions‐ Example Tb3+ 7 For ions with odd number of unpaired electrons, FJ the minimum degeneracy due to the crystal field 0 1 splitting is twofold (Kramers theorem). 2 3 E/cm–1 mJ 4 2 +6 5 0 –6 –2 6 B /T 01z electron repulsion spin‐orbit coupling crystal‐field splitting magnetic field Ch. 1. Electronic Structure Section 1.1 Theoretical description 8 Coupling of angular momenta (Russell‐Saunders coupling) • In d‐ and f‐ions, often several unpaired electrons are present. • Their angular momenta can be coupled together. • Because angular momenta are vector quantities, more than one result is possible (triangle condition, Clebsch‐Gordan series): • The total orbital angular momentum L = l1+l2, l1+l2−1, …, |l1−l2|. • The total spin angular momentum S = s1 + s2, s1+s2−1, …, |s1−s2|. • The magnetic quantum numbers must fulfill: mL = ml1 + ml2, mS = ms1 + ms2. • The two total angular momenta can couple to a total electronic angular momentum (Spin‐Orbit coupling) • J = L+S, L+S−1, …, |L−S|. • The corresponding g‐factor is J(1)(1)(1)(1)(1)(1) J LL SS JJ LL SS gg g Jl2(JJ 1) s 2( JJ 1) JJ(1)(1)(1) LL SS 1 2(JJ 1) Ch. 1. Electronic Structure Section 1.1 Theoretical description 9 Coupling of angular momenta (Russell‐Saunders coupling) • The ground state can be found by using the three Hund's rules: 1. The term with maximum spin multiplicity lies lowest 2. For a given spin multiplicity, the term with maximum orbital angular momentum lies lowest. 3. For atoms with less than half filled shells J = |L − S| is lowest in energy For atoms with more than half filled shells J = L + S is lowest in energy. 2S+1 Term symbol: The electronic state is denoted by a term symbol LJ with S, P, D, F, G, H for L = 0, 1, 2, 3, 4, 5 Example f2 • Two electrons: Maximum spin is S = 1, hence 2S+1 = 3 –3 –2 • Maximum M = 5, hence maximum L = 5 L –1 • J = |L – S| = 4 0 3 • Ground term H4 +1 +2 +3 Ch. 1. Electronic Structure Section 1.1 Theoretical description 10 Beyond Russell‐Saunders • In the Russell Saunders coupling scheme L, S, J are all good quantum numbers. • In the intermediate coupling scheme multiplets with different L, S but the same J mix: L,S are no longer good quantum numbers, but J is. • In J‐mixing also the mixing between multiplets with different J is considered Ch. 1. Electronic Structure Section 1.1 Theoretical description 11 Dieke Diagram • Because f‐electrons are little influenced by metal‐ ligand bonding, a general scheme of energy levels in lanthanides can be constructed (Dieke diagram). • To a first approximation, the energy difference between multiplets that differ in J only is due to spin‐orbit coupling. • The energy difference between groups of multiplets that differ in L and or S is due to electron repulsion. • Originally obtained by analysis of the optical 3+ (absorption/luminescence) spectra of Ln :LaCl3 3+ and Ln :LaF3. • The thickness of the lines indicates the crystal field splitting. D. Chen, Nano Energy, 1, 73 (2012); W.T. Carnall, J. Chem. Phys., 90, 3443 (1989); G.H. Dieke, Spectra and Energy Levels of Rare Earth Ions in Crystals, Wiley, New York (1968) Ch. 1. Electronic Structure Section 1.1 Theoretical description 12 The Hamiltonian for the free ion • Energies and eigenfunctions are calculated by diagonalizing the Hamilton matrix. • The Hamiltonian contains the usual kinetic energy and Coulomb terms, but also spin‐ orbit coupling and Zeeman splitting: 222NN NN ˆ 2 Ze e Hr iiii sl 2mrrii11iij iji 1 • This can be rewritten in terms of orthogonalized operators as ˆ k kkk HEAVE Ff k 427 f A SO LL 1 GG GR tT i mM k pP k k2,4,6 ikk2,3,4,6,7,8 0,2,4 2,4,6 C. Görller‐Walrand, K. Binnemans in Handbook on the Physics and Chemistry of Rare Earths, Vol. 23, http://dx.doi.org/10.1016/S0168‐1273(96)23006‐5 Ch. 1. Electronic Structure Section 1.1 Theoretical description 13 The Hamiltonian for the free ion • Meaning of terms 1 EAVE 1. Spherical contribution to energy. 2. Electrostatic repulsion between electrons of 4fn configuration. k most important 2 Ffk Fk radial integrals; f angular part. k2,4,6 k 3. ζ4f is SOC constant, ASO is angular part. 3 4 fSOA 41LL two‐particle two‐ and three‐particle operators operating within corrections 4fn configuration: change of basis functions 5 GG2 6 GR7 k 7. Three‐particle correction for n>2 7 tTi i2,3,4,6,7,8 8. spin‐spin and spin‐other‐orbit interactions k 8 mMk 9. electrostatic correlated spin‐orbit interactions. k0,2,4 k 9 pkP k2,4,6 C. Görller‐Walrand, K. Binnemans in Handbook on the Physics and Chemistry of Rare Earths, Vol. 23, http://dx.doi.org/10.1016/S0168‐1273(96)23006‐5; Wybourne Ch. 1. Electronic Structure Section 1.1 Theoretical description 14 The Hamiltonian for the free ion C. Görller‐Walrand, K. Binnemans in Handbook on the Physics and Chemistry of Rare Earths, Vol. 23, http://dx.doi.org/10.1016/S0168‐1273(96)23006‐5 Ch. 1. Electronic Structure Section 1.1 Theoretical description 15 Crystal field interaction • The negatively charged ligand electrons have a repulsive interaction with the electrons (d or f) of the metal ion: the total energy of the molecule is raised. • This interaction has a qualitative aspect related to the symmetry of the complex. • Symmetry considerations allow to understand into how many components a given set of orbitals or states split as a consequence of the crystal field. Tsukerblat 4.2 http://en.wikipedia.org/wiki/Crystal_field_theory Ch. 1. Electronic Structure Section 1.1 Theoretical description 16 Crystal field interaction • This interaction also has a quantitative aspect: how big are the splittings. • In order to calculate this, we must consider the Coulomb interaction between metal and ligand electrons. The potential energy is given by: NN eq V c k ik11rRik here the index i runs of the metal electrons, the index k over the ligand electrons. • The magnitude of the splitting can be obtained from crystal field theory (purely Coulombic interaction of point dipoles) or ligand field theory (taking into account σ‐bonding) (energy aspect). Tsukerblat 4.2 Ch. 1. Electronic Structure Section 1.1 Theoretical description 17 Crystal field splitting: quantitative • The electron density ρ(R) = ρ(Θ,Φ) due to the ligands creates an electrostatic potential at the position r of the f‐electron, leading to a change in potential energy of the electron: ()R ()R Ve()r d or as a Hamiltonian Heˆ d ρ(R) Rr R LF Rr R R–r • This we can expand in terms of spherical harmonics: (unperturbed eigenfunctions are the spherical harmonics) R r Spherical harmonics Charge multipoles Lueken; Atkins; Wikipedia Ch. 1. Electronic Structure Section 1.1 Theoretical description 18 Crystal field splitting: quantitative • This we can expand in terms of spherical harmonics: (unperturbed eigenfunctions are the spherical harmonics) k ˆ qk k HArCLF k q , kqk0 q ()R AqkeCd 1, kq Rk 1 R geometrical factor k r average distance of electron to nucleus 1 2 kk4 CYqq,, 21k spherical harmonics Lueken; Atkins; Wikipedia Ch. 1. Electronic Structure Section 1.1 Theoretical description 19 Crystal field splitting: quantitative‐ Stevens notation k ˆ qk k • In principle the sum over k is infinite.
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