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   2mrrii11iij iji  1

• This can be rewritten in terms of orthogonalized operators as ˆ k kkk HEAVE Ff k 427 f A SO  LL 1  GG   GR   tT i  mM k   pP k k2,4,6 ikk2,3,4,6,7,8 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. k2,4,6 k 3. ζ4f is SOC constant, ASO is angular part. 3  4 fSOA

41LL two‐particle two‐ and three‐particle operators operating within n  corrections 4f configuration: change of basis functions 5 GG  2  6 GR 7 k 7. Three‐particle correction for n>2 7  tTi i2,3,4,6,7,8 8. spin‐spin and spin‐other‐orbit interactions k 8  mMk 9. electrostatic correlated spin‐orbit interactions. k0,2,4 k 9  pkP k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; Wybourne Ch. 1. Electronic Structure Section 1.1 Theoretical description 14

The Hamiltonian for the free ion

• 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 ik11rRik 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 , kqk0   q ()R AqkeCd 1,   kq Rk 1  R geometrical factor k r average distance of electron to nucleus  1 2 kk4 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. HArCLF  k q , kqk0 • However it turns out that only the terms with even k cause a splitting, and that the maximum value of k is 6 for f‐electrons, leading to: 246 ˆ 000224262qqq HLF ArC00 ArC 2 q ArC 44 ArC 66 qqq246   • The first term is spherically symmetric and does not lead to a splitting. • Furthermore, the CF Hamiltonian must have the same symmetry as the complexed ion.

• For example in D4d symmetry, the crystal field Hamiltonian reads: ˆ 02 2 04 4 06 6 H LF ArC204060 ArC ArC

Lueken; Newman/Ng (Ed.), Crystal Field Handbook Ch. 1. Electronic Structure Section 1.1 Theoretical description 20

Crystal field splitting: quantitative • A Table of nonzero CF parameters in different symmetries, ±means parameters with both +q and –q are nonzero.

k |q| D2h D3h D4h D∞h D2d D4d C2v C3v C4v C∞v C2h C3h C4h C2 S4 C1 2 0 + + + + + + + + + + + + + + + + 2 1 ± 2 2 + + ± ± ± 4 0 + + + + + + + + + + + + + + + + 4 1 ± 4 2 + + ± ± ± 4 3 + ± 4 4 + + + + + ± ± ± ± ± 6 0 + + + + + + + + + + + + + + + + 6 1 ± 6 2 + + ± ± ± 6 3 + ± 6 4 + + + + + ± ± ± ± ± 6 5 ± 6 6 + + + + ± ± ± ±

H. Lueken‐Magnetochemie; Newman/Ng (Ed.)‐Crystal Field Handbook; C. Görller‐Wallrand in Handbook on the Physics and Chemistry of Rare Earths, Vol. 23 Ch. 1. Electronic Structure Section 1.1 Theoretical description 21

Crystal field splitting: Wybourne vs Stevens notation qk • The Stevens notation Ak r suggests that the CF parameters can be factorized into a lattice part q k Ak and a radial part r . • This turns out to be practically impossible. • Hence, the Stevens notation can be simplified, as proposed by Wybourne, who used the notation • These two notations differ by a constant factor

qk Ark  kq,  k Bq

H. Lueken‐Magnetochemie; Newman/Ng (Ed.)‐Crystal Field Handbook; C. Görller‐Wallrand in Handbook on the Physics and Chemistry of Rare Earths, Vol. 23 Lea, Leask, Wolf, J. Phys. Chem. Solids, 23, 1381 (1962) Ch. 1. Electronic Structure Section 1.1 Theoretical description 22

Crystal field splitting: Stevens operator equivalents • If we consider the ground multiplet only (i.e., only applicable for excitations within the ground multiplet or magnetic (resonance) properties), then we can simplify the calculation drastically.

k • The operators Cq , act on each f‐electron separately, hence for each state |JMJ>, the corresponding wavefunction must be determined. This is rather complicated. • In order to reproduce the splittings, we can replace the coordinates in the spherical harmonics by angular momentum operator components, which have simple properties (see spin Hamiltonians). ˆˆˆ • xJyJzJrxyz,,,(1) JJ  • To take into account the noncommutation between the angular momentum operators, we have 1 ˆˆ ˆˆ to form symmetrized products, e.g. xyJJJJ2  x yyx • We end up with a spin Hamiltonian of the form

k ˆˆqqˆ HBOJ kk xyz,, kJq0,2, ,2 0

Lueken; Boča Ch. 1. Electronic Structure Section 1.1 Theoretical description 23

Crystal field splitting: Stevens operator equivalents • A table of common Stevens operator equivalents and the related spherical harmonics 53zr22 YOJJJ202ˆ 31ˆ  0216 r 2 z 2 xiy 222215 ˆ 1 ˆˆ YOJJ222 2  32  r 4224 93530zzrr 3 2 YOJJJJJJJJ40422ˆ 35ˆˆ 25 30 1 6 1 3  1 04256 r 4 zz  4 xiy 4444315 ˆ 1 ˆˆ YOJJ44 512 r 4 2 

Lueken; Abragam, Bleany. Ch. 1. Electronic Structure Section 1.1 Theoretical description 24

Crystal field splitting: Stevens operator equivalents • The crystal field and operator equivalent differ by a constant factor. • This constant accounts for the orbital angular momentum contribution to J, hence depends on J but is different for different ions, even if J is the same (because L is different). • The constant also depends on k, and on whether the CF Hamiltonian is formulated in terms of k k spherical harmonics Yq or Racah tensors Cq .

• For k = 2, 4, 6, the constants are denoted as

αJ, ϐJ, γJ

• This approach is only valid for the ground LSJ multiplet

Lueken Ch. 1. Electronic Structure Section 1.1 Theoretical description 25

Crystal field splitting: Crystal quantum number q q • A crystal field interaction term Bk Ôk will mix mJ states only if mJ – mJ' = q. • We can define a new quantum number, the crystal quantum number μ to designate a group of

states satisfying mJ = μ (mod q), i.e. mJ = μ + n q (n = integer). • For even numbers of electrons: For odd numbers of electrons:

• For odd numbers of electrons, there is a one‐to‐one correspondence with the irreducible representations that the states belong to. • For even numbers of electrons, the groups of states can contain states belonging to two different irreducible representations, which can be distinguished using 0+, 0– and 1+, 1–. • ±μ states are degenerate

Wybourne; Goerller‐Wallrand Ch. 1. Electronic Structure Section 1.1 Theoretical description 26

The concept of effective Hamiltonians • We can devise a Hamilton operator that reproduces the energies of the levels: an effective Hamiltonian. • In this process, information about the physical origin of the levels and their energies is lost. • Clearly the wavefunctions will not be the same. • This may or may not be disadvantageous. • Many phenomena can no longer be described: you cannot calculate the Big Bang with a spin Hamiltonian. • The group of levels that are considered in the calculation are the space in which the calculation is performed.

after O. Waldmann, http://obelix.physik.uni‐bielefeld.de/~schnack/molmag/ecmm/ecmm‐sat‐2013‐waldmann.pdf Ch. 1. Electronic Structure Section 1.1 Theoretical description 27 Spin Hamiltonians • Spin is a property of a single electron. • We can treat the electronic ground multiplets of orbitally nondegenerate ions as spins: effective spin • We can treat a single doublet within the ground state as S = 1/2: fictitious spin.

• We can reproduce the energy levels by means of Spin Hamiltonians. • They exploit the (spin) angular momentum properties.

• In general: H Zee BBSg

• For isotropic g and field along z‐axis: H  gBzBsˆ

Second rank zero‐field splitting: ˆ ˆˆˆˆ22221 ˆˆ 22 Η ZFSDS z E S x  S y DS z 2 E S  S 

Isotropic exchange coupling: ˆ ˆˆ (watch out for sign and presence of factor 2) Η Ex 2J SS 1 2

Crystal field splitting of Russell‐Saunders multiplet: f‐electrons Here we can discuss if we can consider this a spin Hamiltonian.

246 To my mind, as soon as you limit the space to a single RS multiplet, use the mJ ˆ qqˆˆ qq qq states as basis (rather than one‐electron wavefunctions) as well as Stevens H CFB 22OBOBO 44 66 qqq246   operators, you are using an effective/fictitious spin and hence a Spin Hamiltonian Not everyone will agree. d‐electrons Ch. 1. Electronic Structure Section 1.2 Experimental determination 29

(High‐Frequency) EPR, Far‐Infrared, Inelastic Neutron Scattering, UV/Vis/NIR absorption, Magnetic Circular Dichroism, Luminescence, Electronic Raman

E/cm–1 4 I13/2 2 +mJ Luminescence 0 Abs, MCD FIR, INS Raman

–mJ –2 4 I15/2 01 Ch. 1. Electronic Structure Section 1.2 Experimental determination 30

EPR FIR, INS El. Abs., MCD, Raman, luminescence

http://www2.lbl.gov/MicroWorlds/ALSTool/EMSpec/EMSpec2.html Ch. 1. Electronic Structure Section 1.2 Experimental determination 31

(High‐Frequency) EPR

• Magnetic‐dipole transitions between projections of the angular momentum (mJ), with the selection rule ΔmJ = ±1. • Often within ground (quasi)doublet. • Magnetic field modulation gives 1st derivative of the absorption spectrum, enabling determination of principal g‐tensor components from a powder spectrum E/cm–1 H Zee  BBSg 2 gggxxxyxz +m  J gggg yxyyyz  gggzx zy zz • After suitable rotation of coordinate system 0

gxx 00  gg 00 –mJ yy  –2 00gzz 01 Ch. 1. Electronic Structure Section 1.2 Experimental determination 32

(High‐Frequency) EPR • Example: Ln(trensal). Ln = ErIII (1), DyIII (2) • W‐Band EPR (95 GHz) 4 6 • Ground multiplets I15/2 (Er) and H15/2 (Dy). • Effective g values give information on composition of ground doublet

E/cm–1

2 +mJ

–mJ –2 01 E. Luccaccini, Chem. Commun., 50, 1648 (2014) Ch. 1. Electronic Structure 33 Section 1.2 Experimental determination 33

Far‐Infrared and Inelastic neutron scattering

• Example: (NBu4)[HoPc2). • Transition frequencies allow direct determination 21S  of crystal field splitting fo the ground multiplet. LJ 5 • Ground multiplet I8 (not a Kramers ion). • Eigenstates from CASSCF.

±(0.34x±5+0.34x±6)

±(0.35x±1+0.34x±4)

±(0.41x±8+0.35x±5)

±(0.44x±7+0.40x±8)

±(0.40x±5+0.35x±3)

±(0.46x±6+0.35x±4)

R. Marx, J. van Slageren, Chem. Sci., 5, 3287 ‐ 3293 (2014). Ch. 1. Electronic Structure Section 1.2 Experimental determination 34

Luminescence • Example: Sm(trensal) • Transition frequencies allow determination of all Hamiltonian parameters

Flanagan, Riley, Inorg. Chem. 41, 5024 (2002) Ch. 1. Electronic Structure Section 1.3 Actinides 35

Actinides vs Lanthanides • For trivalent actinides, the Coulomb (electron‐electron) interaction is about 60% of that of the lanthanide ion with the same electronic configuration, • the spin‐orbit coupling constant ζ is about twice as large, • the crystal field splitting is more than double that in the corresponding lanthanide k q Configuration, Ion ionic radius max(F ) ζ max(|Bk |) (nm)a

f3: U(III)b 0.1025 40 x 103 1.6 x 103 1.3 x 103 f2: U(IV)c 0.089 43 x 103 1.8 x 103 2.8 x 103 f1: U(V)d 0.076 0 1.6 x 103 13 x 103

f3: Np(IV)e 0.087 45 x 103 2.1 x 103 3.5 x 103 f2: Np(V)f 0.075 49 x 103 2.2 x 103 15 x 103 f1: Np(VI)g 0.072 0 2.4 x 103 45 x 103

f3: Nd(III)g 0.0995 72 x 103 0.88 x 103 0.7 x 103 f2: Pr(III)g 0.101 68 x 103 0.75 x 103 0.6 x 103 f1: Ce(III)g 0.103 0 0.64 x 103 1.0 x 103 H. Lueken, Magnetochemie; Liddle, Van Slageren in Lanthanides and Actinides in Molecular Magnetism, Layfield, Murugesu (Eds.) Ch. 1. Electronic Structure Section 1.3 Actinides 36

Actinides vs Lanthanides • The electron‐electron interaction increases slightly with OS and atomic number • The spin orbit coupling constant increases with atomic number • The crystal field splitting is very strongly dependent on oxidation state (as indeed for the transition metals) and also increases with atomic number. k q Configuration, Ion ionic radius max(F ) ζ max(|Bk |) (nm)a

f3: U(III)b 0.1025 40 x 103 1.6 x 103 1.3 x 103 f2: U(IV)c 0.089 43 x 103 1.8 x 103 2.8 x 103 f1: U(V)d 0.076 0 1.6 x 103 13 x 103

f3: Np(IV)e 0.087 45 x 103 2.1 x 103 3.5 x 103 f2: Np(V)f 0.075 49 x 103 2.2 x 103 15 x 103 f1: Np(VI)g 0.072 0 2.4 x 103 45 x 103

Actinides vs Lanthanides Radial extension of the f orbitals is larger for An than for Ln: • This leads to larger crystal field splittings • Potentially stronger exchange interactions between the f‐ion and other ions. • Spin‐phonon coupling becomes stronger, which leads to faster spin‐lattice relaxation.

H. Lueken, Magnetochemie; Liddle, Van Slageren in Lanthanides and Actinides in Molecular Magnetism, Layfield, Murugesu (Eds.) http://hudry.weebly.com/research.html; Los Alamos Science 2000, 26, 364‐381 Ch. 1. Electronic Structure Section 1.3 Actinides 38

Typical actinide behaviour • Actinides are often between pure RS and pure jj coupling.

Edelstein, Lander, Handbook Handbook on the Physics and Chemistry of Rare Earths, Vol. 20 1. Electronic Structure 2. Static magnetic properties 3. Dynamic magnetic properties Ch. 2. Static magnetic properties Section 2.1 Theoretical description 40

Microscopic origin of magnetism 1. Ring currents in atoms induced by the applied field: diamagnetism 2. Circular motion of unpaired electrons around nucleus: orbital angular momentum 3. Intrinsic angular momentum of electron: spin. Ch. 2. Static magnetic properties Section 2.1 Theoretical description 41

B‐ and H‐fields

• In vacuum B = μ0 H. • In matter, things are more complicated: there can be electron motions (bound currents) through the atoms, ions or molecules that compose the sample. • These currents create individual magnetic moments that can be aligned in an external magnetic field.

• These lead to a magnetization of the sample and hence a change of the flux density: B = μ0(H+M) Ch. 2. Static magnetic properties Section 2.1 Theoretical description 42

Macroscopic magnetic quantities: the magnetization • The magnetization M is defined as the magnetic moment of a sample divided by its volume. • The magnetization is proportional to the H‐field: M = χH  xxxyxz • The magnetic susceptibility is a 3x3 matrix in its most general form:    yxyyyz   zx zy zz • The magnetization is also the first derivative of the energy with respect to the H‐field.

• At zero temperature (E0 is the energy of the ground state): 1 E ()H S = 1 MH() 0 VH • At finite temperatures, the magnetization of the sample is the thermal average over all thermally occupied states:

EkTnB/ S = 0  MHen () MHT(,) n eEkTnB/ T = 0 K T > 0 K n • The susceptibility is the derivative of the magnetization with respect to the field (hence, the second derivative of the energy with respect to the field). M ()H  H

• In the linear regime (high T, low H), χ = M/H. Ch. 2. Static magnetic properties Section 2.1 Theoretical description 43

EkTnB/ NeAn  Derivation n Susceptibility of free ions (small applied fields) M  eEkTnB/ n • Van Vleck equation: (0) (1) 2 (1) power series expansion of EWnn  BWBW n n  (0) Zeeman coefficients (1) 2 (2) WkTnB/ (0) ()/2WkTWe x EkT// W(0) kTBW  nn for B/kT<<1ex1 : eenn 1n  MM n kT 00 N A (0) E HB eWkTnB/ E μ B   n   n n B WWB(1) 2 (2) st nn n • The 1 order Zeeman coefficients: insert into M: (0) (1) (1) WBWBWkTe(1)2()/ (2)  (1) 2 WkTnB/ WnHnnlsn ˆ 2ˆ  nnn nz ezz  MN n A WkT(0) / nd e nB • The 2 order Zeeman coefficients: n 2 ˆ In zero external field, the magnetization of a paramagnet is zero, mlsn ez2ˆ z hence: (2) (0) BW()/2(1) 2 kTWe (2) WkTnB/ Wn  (0) (0)  nn n WWmn MN A (0) eWkTnB/ n

• Magnetic susceptibility (considering RS ground multiplet): Ng22 • TIP = temperature independent paramagnetism, 0 AJ B Van Vleck paramagnetism TJJT(1)0 3k  • Leads to straight line with positive slope in χT.     TIP Curie law

Lueken; Kahn; Mabbs/Machin Ch. 2. Static magnetic properties Section 2.1 Theoretical description 44

Susceptibility of free ions • The susceptibilities of lanthanide compounds at room temperature are often very close to the free ion values. • Exceptions are Sm3+, Eu3+ (low lying excited J‐multiplets).

• Note: the effective magnetic moment μeff is defined as (neglecting TIP) 3k eff TgJJ  B J(1) B 0 N A

 Ng22 TJJ 0 AJ B (1) 3k    Curie law

Lueken Ch. 2. Static magnetic properties Section 2.1 Theoretical description 45 Free ions Typical uranium behaviour 0 • U(VI) f . TIP only (example UF6) • U(V) f1. • No electron‐electron repulsion. 2 2 • F5/2 ground state F7/2 excited state.

• EPR observed if ground doublet contains mJ = ±½ (μ = ±½).

• μeff = 2.0 –2.3 μB at rt • U(IV) f2. • Total CF splitting of the order of kT at room temperature • non‐Kramers ion.

• Ground microstate can be mJ = 0.

• μeff = 2.2 –4.0 μB at rt • U(III) f3. • EPR can sometimes be observed.

• μeff = 3.1 –3.7 μB at rt

Edelstein, Lander, Handbook Handbook on the Physics and Chemistry of Rare Earths, Vol. 20; Benelli, CR, 2002 Ch. 2. Static magnetic properties Section 2.2 Experimental determination 46 SQUID magnetometry • The most common method to study the temperature dependence of magnetic properties is SQUID magnetometry. • SQUID = superconducting quantum interference device. • The magnetized sample is moved through a superconducting coil. This induces a current. • The SQUID element is a superconducting loop with two tunnel (Josephson) junctions. It converts the current into a voltage, which is amplified. • Hence the magnetic moment is measured, this can be converted to susceptibility by χ ≈ M/H Ch. 2. Static magnetic properties Section 2.2 Experimental determination 47 Magnetic properties of lanthanide complexes  Ng22 TJJ 0 AJ B (1) • Decrease of χT to lower T due to CF splitting. 3k    Curie law 21S  • Analysis: The space considered is the Russell‐Saunders ground multiplet. LJ • A suitable Hamiltonian is then 246 ˆ qqˆˆ qq qq ˆ H BO22 BO 44 BO 66 BJ gBJ qqq246  

q • The operators Ôk are functions of the angular momentum operators to even powers. Ch. 2. Static magnetic properties Section 2.2 Experimental determination 48

Effective charge description of crystal field splitting 246 ˆ qqˆˆ qq qq H B22OBOBO 44 66 • For low symmetries one has to fit up to 27 CF parameters. qqq246   • It would be nice to have a description in terms of parameters that are specific to each ligand. • The assumption is that the effect is additive. • This has led to various models, such as the superposition model, angular overlap model, lone‐pair effective charge model. • For the last, one uses the standard Stevens notation, using spin operators. k ˆ qq2 HaArOLFkkk   kqk2,4,6 • The geometric factor can be calculated:

N 2 q 4 q ZeY  ,  Acq 1 ikii kkq k 1 21k  i1 Ri • One now takes the distances and charges to be effective distances and charges. • In a second step, a horizontal displacement factor can be taken into account

Newman, Gaita Arino Ch. 2. Static magnetic properties Section 2.2 Experimental determination 49

Caveat • Magnetometry is not a good way to determine crystal field splitting parameters

R. Marx, J. van Slageren, Chem. Sci., 5, 3287 ‐ 3293 (2014). Ch. 2. Static magnetic properties Section 2.2 Experimental determination 50

CASSCF based methods to calculate the CF splitting • Use experimental crystal structure without optimization. • 1st step: perform CASSCF calculation. • 2nd step: include spin‐orbit coupling via RASSI. • 3rd step: project lowest levels resulting energy spectrum onto an effective angular momentum corresponding to the Russell‐Saunders ground multiplet (single_aniso in MOLCAS). • No dynamical correlation (CASPT2 or similar). • Accuracy appears to be of the order of 20‐30%. Note that the absolute accuracy is of the order of 10‐20 cm‐1, or a few meV.

Chibotaru, Ungur; R. Marx, J. van Slageren, Chem. Sci., 5, 3287 ‐ 3293 (2014). Ch. 2. Static magnetic properties Section 2.1 Theoretical description 51

Crystal field splitting. Magnetic anisotropy • Due to the crystal field, the molecule responds differently to magnetic fields applied in different directions in the molecule. • This is called magnetic anisotropy. • The magnetic susceptibility becomes anisotropic. • This can be measured directly by single‐crystal susceptibility measurements • The molecular anisotropy corresponds to the crystal anisotropy only when • there is one crystallographically independent molecule in the unit cell. • there is only one orientation of the molecule in the unit cell • the molecule is on the highest symmetry position.

• The z‐axis is chosen as the molecular Cn axis with highest n>2 • In monoclinic systems, the b‐axis must be one of the principal axes. N. Ishikawa, Inorg. Chem. 2003, Luzon, Ch. 2. Static magnetic properties Section 2.2 Experimental determination 52

Single‐crystal DC susceptibility. • Measurement of the magnetic susceptibility as a function of angle for three orthogonal orientations.

 8.40 1.43 10.61     1.43 1.22 1.96    10.61 1.96 15.7019

 tensor in XYZ space

after S.‐D. Jiang, http://obelix.physik.uni‐bielefeld.de/~schnack/molmag/ecmm/ecmm‐sat‐2013‐slageren.pdf Ch. 2. Static magnetic properties Section 2.2 Experimental determination 53

Single‐crystal DC susceptibility. xx xy xz  x x 00   rot 22 00 • Fit to  cos   sin  2 cos sin yx yy yz y y  (with cyclic permutations    of αβγ over xyz) . zx zy zz 00 z z • Diagonalization of the χ tensor gives principal axes and directions

after S.‐D. Jiang, http://obelix.physik.uni‐bielefeld.de/~schnack/molmag/ecmm/ecmm‐sat‐2013‐slageren.pdf Ch. 2. Static magnetic properties Section 2.1 Theoretical description 54 Easy axis of magnetization in dysprosium complexes

• Assume mJ = ± J= ± 15/2 ground Kramers doublet

• Electron density distribution depends on mJ microstate.

• Ligand electron density can stabilize or destabilize the mJ = ± J doublet. • Considering the space of a ground doublet , the anisotropy is expressed as g tensor anisotropy.

• For the mJ = ± 15/2 doublet in dysprosium (gxx, gyy, gzz) = (0, 0, 20) is expected.

• The gzz = 20 comes from 2 gJ = 4/3 ∙ 15/2 ∙ 1.

Sievers, Z. Phys. B 1982; Rinehart Chem. Sci. 2011 Ch. 2. Static magnetic properties Section 2.1 Theoretical description 55 Easy axis of magnetization in dysprosium complexes • Electron density of an mJ state of a given lanthanide ion can be calculated:

n7 JM   2 J M 7 JkJ JkJ k33i k 3 3 0  r rkY 1211 k 0 k  k 0,2,4,6 4 MMJM0 0 000i1 0 4  ii 4

• This electron density interacts with the crystal field potential m k 41  qYm  , VrY,, km nk n n CF k   k 1 kmk2,4,6 21k  n Rn • From this the energy of the mJ ground doublet as a function of the orientation (α,β) can be calculated:

2 , EV15,,,sindd CF  15  22 00 

Sievers, Z. Phys. B 1982; Chilton Nature Commun. 2013 Ch. 2. Static magnetic properties Section 2.1 Theoretical description 56 Easy axis of magnetization in dysprosium complexes • The minimum in energy gives the orientation of the easy axis. • This information can then be used to design complexes where the maximum magnetic moment state is stabilized

Chilton Nature Commun. 2013 Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 57 Spin‐spin interactions • From the susceptibility, the nature of the interactions can be learned

Ferromagnetic coupling: • χT increases

Antiferromagnetic coupling: • χT decreases

Ferrimagnetic coupling • As AF coupling but two different spins • Moments do not cancel

• Note the distinction between (anti‐)ferromagnetic ordering (long range) and (anti‐)ferromagnetic coupling • A further caveat: in f elements, most of the decrease in χT will come from the CF splitting, not from magnetic couplings Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 58

Spin‐spin interactions • For AF systems, χ vs T is most • For F systems, χT vs T is most • For Ferri systems, χT vs T is useful. useful. most useful. • The position of the maximum • Decrease to low T due to • The position of the is very sensitive to fit saturation/ Zero field splitting/ minimum is sensitive to fit parameters intermolecular interactions parameters.

0.20 10 17 9 )

-1 16 0.15 8 ) –1

-1 15

K mol K 7 3 14

0.10

6 mol K 3 13 T (cm / emu mol / emu

 5  0.05 (cm 12 T

4  11 3 0.00 10 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 T / K Temperature (K) T (K)

III [Cr8F8Piv16] [Ni4(MeOH)4Cl4(hmp)4] [Fe 4(acac)6(Br‐mp)2] Van Slageren, Chem. Eur. J., 8, 277 (2002) E.C. Yang, Inorg. Chem., 45, 529 (2006) Schlegel, Chem. Eur. J., 16, 10178 (2010) Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 59

Microscopic origin of spin‐spin interaction magnetic dipolar interaction charge transfer mechanism of superexchange • Potential interaction energy between two • Ion X has spin up  magnetic dipole moments is: • Covalent interaction leads to partial donation     dip 0  312rr   from L to X. V 3212 4 rr• This donated e‐density must have spin down • When both magnetic dipole moments are (Pauli exclusion principle). parallel (e.g., because of the application of • Remaining e‐density on L has then spin up. a magnetic field), then the potential energy • Spin up e‐density donated to ion Y. is:    13cos 2  • Therefore ion Y has spin down. V dip  012 4 r3 • All possible cases are summurized in • For two magnetic dipole moments of μ = 1 Goodenough‐Kanamori rules.

μB each at a distance of r = 1 Å, the interaction energy is Vdip ≈ 10–23 J ≈ 1 K.

Kahn; Orchard Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 60

Magnetic coupling. • Couplings typically of the order of a few cm–1 maximum.

3– – [{[(Me3Si)2N]2Ln(THF)}2(N2 )] (Ln = Gd, Dy)

• Strong magnetic coupling between lanthanide and triply reduced N2 bridge. • Arrhenius plot much more linear. • Clear hysteresis observed at modest temperatures and sweep rates.

Rinehart, Nature Chem., 2011 Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 61

Magnetic coupling. Example: [(MeC5H4)3U]2(μ‐N2C6H4) 1 2 • Uranium(V) has f configuration, hence no electron‐electron interactions. Ground multiplet F5/2.

• The symmetry around U is approximately C3v, hence lowest nonzero q in the CF operator is q =3. • Therefore the values for crystal field quantum number are μ = ±1/2 (two states) and μ = ±3/2 (one state). • Behaviour of para and meta compunds is very different.

Rosen, Edelstein, JACS, 1990 Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 62

Magnetic coupling. Example: [(MeC5H4)3U]2(μ‐N2C6H4) • Due to the absence of an EPR signal the ground doublet of U was supposed to be μ = ±3/2

(intra‐doublet transition has ΔmJ > 1, hence not allowed; also g┴ = 0). • Thermally isolated doublet can be treated as effective spin S' = ½. ˆˆ ˆ ˆ H 2JSSzz12  g Bz B S z 1  S z 2 • Susceptibility is given by  Ng22 1  1 AB 3 J /kTB 21kTB  e

• Fit gives large value for J. • Interaction pathway because of resonance structure or spin polarization

Rosen, Edelstein, JACS, 1990 Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 63

Magnetic coupling. Example: (cyclam)M[(Cl)U(Me2Pz)4]2 • Subtraction method • U‐Zn‐U data show thermal dependence of magnetic susceptibility due to crystal field splitting. • Subtracting these data from U‐Cu‐U and U‐Ni‐U gives magnetic behaviour of TM ion plus effects of magnetic coupling. • In U‐Ni‐U ferromagnetic interactions appear to be present. • This is supported by the orbitals of the unpaired electrons of Ni being orthogonal to those on U.

Kozimor, Long, JACS, 2007 Ch. 2. Static magnetic properties Section 2.3 Magnetic couplings 64

Toroidal ground states. Example: [Dy3(OH)2L3Cl(H2O)5]Cl3.Solvent. HL = o‐Vanillin

• The ground state of Dy3 is nonmagnetic due to the coplanar arrangement of the local easy axes of the dysprosium ions. • States are doubly degenerate, where each of the components has opposite chirality. • This particular arrangement of the magnetic moments has been called a toroidal magnetic moment.

Tang, Powell, Sessoli, Angew, 2006; Luzon, Sessoli, PRL, 2008 1. Electronic Structure 2. Static magnetic properties 3. Dynamic magnetic properties Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 66 Energy barrier chronology‐ the case for f‐elements

• Arrhenius law: τ = τ0 exp (ΔE / kBT). 8 –7 4 • Assuming τ = 10 yrs = 3 x 10 s, τ0 = 10 s, T = 300 K, ΔE must be 10 K. • Lanthanides very promising

1993

2007

2003

Rinehart, CS, 2011; Web of Science: Topic=(lanthanide* single molecule magnet*) Woodruff, CR, 2013 Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 67 Difference between clusters and single‐ion systems • In clusters, the system goes up many steps on one side, then down many steps on the other side. • The detailed mechanism of the individual steps becomes less important. • It is usually assumed that the energy is given out/taken up as a phonon of exactly the same energy (direct process). • In single ion systems, the system goes up one (maybe two) steps before going down again. • The mechanism of the individual steps becomes important.

0 -10 12 -2 3 -3 4 -4 5 -5

6 -6 )) –1–1 7 -7 8 -8

9 -9

10 -10 Energy (cmEnergy (cm E = DS2 angle between easy axis and magnetic moment Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 68

Basic Spin Lattice Relaxation Mechanisms in dilute systems • The main mechanism is modulation of the crystal field splitting by lattice vibrations (phonons). • This generates an oscillating electric field. • Through spin‐orbit coupling this electric field results in an oscillating magnetic field. • The oscillating magnetic field can cause spin flips: spin‐lattice relaxation. • Energy needs to be conserved. • Important is that the phonon density of states at low frequencies is quite small. • It is assumed that the phonons have long wavelengths compared to the lattice constants (acoustic phonons)

Orbach in Electron Paramagnetic Resonance, Geschwind (Ed.), 1972 Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 69

Spin‐Lattice relaxation by modulation of the ligand field • The magnetic ion experiences an electric field due to the ligands • Moving the ions will change this electric field. • stress: the force that is applied per unit area: • strain: the deformation that results • Because the effect of strain on the crystal field potential is difficult to calculate, we can expand the CF potential in a series in powers of the strain.

(V(0) is the static CF potential, ε is the fluctuating strain caused by lattice vibrations) Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 70

Phonons • For a diatomic chain we get two dispersion curves • The higher‐frequency branch is always called the optical branch, the lower always the acoustic branch

‐

Kittel – Solid state physics; Burns Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 71

Phonons • we have described displacements of atoms from equilibrium in terms of travelling waves. • for each k value there are 3z modes with 3z different frequencies • An arbitrary displacement of an atom involves all 3zN different modes. • The motion of the atoms can be best described using normal modes: suitable linear combinations of atomic motions that are decoupled from each other. • Each normal mode is quantized:

1 Enkk 2  kk n 0,1, 2,...

• The nkth excited state of the kth normal mode can be viewed as a state of nk identical excitations, each with energy ħω k

• These excitations or quanta we call phonons. • Definition: a phonon is a quantum of excitation in a normal mode

• When exciting a normal mode from nk to nk+1 phonons, a phonon is created or emitted

• When deexciting a normal mode from nk to nk‐1 phonons, a phonon is annihilated or absorbed Burns – Solid State Physics Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 72

Phonons • Example solid argon (MW ~40) , xenon (MW ~130). • Expected maximum acoustic phonon frequency for LnPc2 perhaps 400 GHz?

Batchelder.D.N., B. C. G. Haywood, D. H. Saunderson, J. Phys. C 1971, 4, 910. B. J. Palmer, D. N. Batchelder, D. H. Saunderson, J. Phys. C 1973, 6, L313. Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 73

Phonon density of states • For small k, ω ~ k. V 2 • The density of states g(ω) is then gD() () 2 22v v is the sound velocity • For N primitive unit cells, there are 3N acoustic modes. 333 • This leads to a maximum acoustic mode frequency: D  6/vN V • This is called the Debye frequency. • The real density of states does follow a quadratic dependence at low frequencies.

Burns – Solid State Physics Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 74

Basic Spin‐Lattice Relaxation Mechanisms in dilute systems • assume dilute systems energy conservation in both steps

ħω

0 Direct Van Vleck Raman Orbach 2nd order Raman (1st order Raman) • Note that the figure was changed compared to literature. It seems to be more logical this way Bertini in Handbook of Electron Spin Resonance, Misra in Multifrequency Electron Paramagnetic Resonance Orbach in Electron Paramagnetic Resonance, Geschwind (Ed.), 1972; Abragam and Bleaney Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 75

The direct process • The induced transition rate due to a perturbation is 2 2 01   2  2 wiHjfij   0  together with the crystal potential expansion, VV  V V  this gives: 2 wiVjf 2  2 1  ij 2 

• Considering the phonon energy density, we find for T1:  2 133 1  TV1  5  coth  22 kT0 • ρ is the density of the sample • υ is the speed of sound in the material, e.g. υ = 103 ms–1. • |V(1)| here is the matrix element between the two relevant spin states. • ω is the energy difference between the two states expressed as a frequency.

• Very low phonon density of states at microwave energies

Abragam and Bleaney Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 76

The two‐phonon Orbach process • System has excited spin state at an energy below the Debye energy. • The ion can absorb a phonon, exciting it from spin state |b> to |c>. • The ion can then emit a phonon with a different frequency, causing it to relax to |a>. • Note that Orbach prefers to consider the transition to c and back virtual (two phonons once) rather than two consecutive direct transitions (one phonon twice). • Neglecting the direct process, the rate equations for the change in population of |a> and |b> are: dn dn abwn  wn  wn  wn dtac a ca c dt bc b cb c dn() n • Assuming w ≈ w ≡ w , we can write abwn  n ac bc ↑ dt  ab • From which

2 13311 Δ Tw1  45 V  2exp/1  kTB 0 

13 • In the limit Δ >> kT0 (first step limiting): TkT10exp  / B  • 143 For rare earths: TkT10~10 exp / B 

Abragam and Bleaney; Orbach • In molecular magnetism Δ~30 K, T0~10 K Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 77

The two‐phonon Raman process

• Lattice vibrations at frequencies ω1 and ω2 combine to give a crystal potential fluctuating at frequency ω = | ω1 – ω2 |. Frequencies higher than Debye frequency. • If this frequency corresponds to the frequency difference between |a> and |b>, a transition can occur. • First order Raman occurs if the CF potential has matrix elements between |a> and |b>. Transition probability: Δ 2 wVf 2  2 2 12 

• Second order Raman: virtual transition from |b> to |c>, followed by virtual transition from |c> to |a>. Transition probability (Δ is energy of |c>):

(1) (1) 2 2 11VV 2 2 wf 2   

Abragam and Bleaney; Orbach Ch. 3. Dynamic magnetic properties Section 3.1 Theoretical description 78

Formulas describing the relaxation of the magnetization

 1 3    137  non‐Kramers ions TR1 dOrR coth R exp  1 RT0 2kT00  kT  1 13975    TR1 dOrRR coth R exp  1 RTR00  T Kramers ions 2kT00  kT k Ch. 3. Dynamic magnetic properties Section 3.2 Experimental determination 79

0 0.6

AC susceptibility. 0.4 ' -5  ) -1 • M =  H;  = dM / dH. 0.2

-10 • If applied magnetic field oscillates, Energy (cm 0.0

the dynamic (ac) magnetic -15 0.3 1 Hz susceptibility is measured: 10 Hz -20 0.2 M(t) = MAC() exp(i t) 100 Hz -4 -2 0 2 4 1000 Hz M S 0.1 • If the magnetisation dynamics of the studied material is not fast 0.0 compared to the oscillation frequency, a phase lag of the susceptibility 2 4 6 8 10 12 occurs. T (K) " • AC = ' + i " with ' =  cos  the real or in‐phase component, and  " =  sin  the imaginary or out‐of‐phase component  ' • Maximum in " when  = 1. Ch. 3. Dynamic magnetic properties Section 3.2 Experimental determination 80

AC susceptibility. • χ' and χ" can be expressed in terms of the = χ isothermal and adiabatic susceptibilities χT and χAd: S    ()TS 1 22 S    () TS 1 22  • In the presence of a distribution of relaxation times α, the formulas get longer     • A plot of χ" as a function of χ' is called a Cole‐Cole or Argand plot.

1    1sin/2           TTS  122 12 sin /2     1 cos / 2          TS  122    12 sin /2   2 ST ST           TS 2  2tan 122 4tan  1  Ch. 3. Dynamic magnetic properties Section 3.3 Examples 81

AC susceptibility. Example: [Dy5(OiPr)13] • Out‐of‐phase signal up to over 40 K (1500 Hz) • Thermal relaxation follows Arrhenius law:

EkT/ B E 1  00e ln ln kTB

• Energy barrier ΔE = 528 K. • However, linear regime very limited.

• In fact, ΔE and τ0 are strongly correlated (example substituted double deckers).

Blagg, Angew. Chem. Int. Ed., 51, 1601 (2012); Waters, Van Slageren, unpublished Ch. 3. Dynamic magnetic properties Section 3.3 Examples 82

AC susceptibility. [Tb(H2O)9]3(C2H5SO3) • No CF state at 25 K, hence Orbach process cannot be operative Ch. 3. Dynamic magnetic properties Section 3.3 Examples 83

Magnetic Hysteresis. Example: (NBu4)[HoPc2]

• Hysteresis only in dilute samples (NBu4)[Ho0.02Y0.98Pc2] • Steps due to quantum tunnelling. • Quantum tunnelling due to interaction with nuclear spin.

1.8 K Mn12ac 2.0 K 2.2 K 2.4 K 2.6 K • Why is tunnelling so much more important than in transition metal systems? 4.2 K -3 0 3 • Because it is a single particle system. Magnetization

Magnetic Field / T

Ishikawa, JACS, 2005 Ch. 3. Dynamic magnetic properties Section 3.3 Examples 84

Quantum tunnelling • Quantum tunnelling is typically very important in single ion SMMs. • Therefore, usually no magnetic hysteresis with appreciable coercivity is observed. • Tunnelling can be induced by transverse magnetic fields, nuclear spins, or the crystal field (the last not for Kramers ions). Ch. 3. Dynamic magnetic properties Section 3.3 Examples 85

The importance of magnetic coupling • Virtually all Ln‐based SMMs suffer from fast relaxation close to zero field, due to tunnelling. • This is linked to the single ion nature of the magnetic properties. • Multispin systems do not show such efficient tunnelling. • Achieving strong magnetic couplings in lanthanides is challenging.

Long, JACS, 2012 Ch. 3. Dynamic magnetic properties Section 3.3 Examples 86

The first An SMM – • [U(Ph2BPz2)3] Ph2BPz = diphenylbis(pyrazolyl)borate. • ligand electron density above and below the plane perpendicular to the pseudo‐symmetry axis.

–1 –9 • Ueff = 20 cm , τ0 = 1x10 s • Deviation from straight line was attributed to tunnelling. • However, properties unchanged in dilute solution. • What cause tunnelling? 238U has f3 and I = 0. • Idealized three‐fold symmetry was stressed.

Rinehart, J. Am. Chem. Soc., 2009 Ch. 3. Dynamic magnetic properties Section 3.3 Examples 87

Field induced An SMMs

• [U(I)3(THF)4]

• [U(N(SiMe3)2)3] TMS • [U(BIPM )(I)2(THF)]

• Idealized symmetries Cs, C3v, C1 respectively • All are field‐induced SMMs. • Slow relaxation persists in solution.

Moro, Liddle, Van Slageren et al., ACIE, 2013 Ch. 3. Dynamic magnetic properties Section 3.3 Examples 88

Quantum tunnelling in SMMs based on depleted uranium(III) TMS • Example [{U(BIPM )(Cl)(μ‐Cl)(THF)}2] • Slow relaxation but no hysteresis in dilute conditions • Uranium(III) is a Kramers ion (5f3). CF cannot induced tunnelling • U‐238 has no nuclear spin. No hyperfine interactions to induce tunnelling • Intermolecular interactions weak in dilute conditions. • Where does the tunnelling come from?

Moro, Liddle, Van Slageren et al., Nat. Chem. 2011 Ch. 3. Dynamic magnetic properties Section 3.3 Examples 89

Actinide Single‐Molecule Magnets • Future for 5f‐3d systems? 90

The End