Spectroscopic Investigations of Magnetic Anisotropy in Molecular Nanomagnets
ECMM workshop on Magnetic Anisotropy Karlsruhe, 11th‐12th October 2013 Prof. Dr. Joris van Slageren Institut für Physikalische Chemie
Dr. Shang‐Da Jiang 1. Physikalisches Institut
Universität Stuttgart Germany Contents 2 1. Introduction 4. Inelastic Neutron Scattering 1.1 Magnetic Anisotropy 4.1 Theoretical background and Experimental Considerations 1.2 Magnetic Anisotropy in Transition Metal Clusters 4.2 Examples 1.3 Magnetic Anisotropy in f‐Elements
5. Electronic Absorption and Luminescence 2. Single‐crystal SQUID measurements 5.1 Electronic Absorption (f elements)
2.1 Motivation 5.2 Luminescence (f elements) 2.2 Magnetic susceptibility tensor 5.3 Magnetic circular dichroism (transition metal clusters, f elements). 2.3 Determination
3. High‐frequency EPR spectroscopy 3.1 Theoretical background and Experimental Considerations 3.2 Examples 3.3 Frequency Domain EPR Contents Literature 3 A. Abragam/B. Bleany – Electron Paramagnetic Resonance of Transition Ions, 1970 N.M. Atherton – Principles of Electron Spin Resonance, 1993. R. Boča – Theoretical Foundations of Molecular Magnetism, 1999 R. Carlin –Magnetochemistry, 1986 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 O. Kahn –Molecular Magnetism, 1993 F.E. Mabbs/D.J. Machin – Magnetism and Transition Metal Complexes, Chapman and Hall, London, 1973 K.R. Lea, M.J.M. Leask, W.P. Wolf, J. Phys. Chem. Solids, 23, 1381 (1962) H. Lueken – Magnetochemie, 1999 H. Lueken – Course of lectures on magnetism of lanthanide ions under varying ligand and magnetic fields, http://obelix.physik.uni‐ bielefeld.de/~schnack/molmag/material/Lueken‐kurslan_report.pdf 2008 D.J. Newman/Ng (Ed.) –Crystal Field Handbook A.J. Orchard –Magnetochemistry, 2003 1. Introduction 2. Single Crystal Magnetometry 3. High‐Frequency EPR Spectroscopy 4. Inelastic Neutron Scattering 5. Electronic Absorption and Luminescence Ch. 1. Introduction Section 1.1 Magnetic anisotropy 5
Magnetic anisotropy. The highly simplified, hand‐waving explanation • Magnetic anisotropy means that the magnetic properties (of the molecule) have an orientational dependence. • This means that the response to an external magnetic field depends on the direction in the molecule along which the field is applied, e.g., g‐value anisotropy, hard/easy axes of magnetization.
• Susceptibility becomes anisotropic. M = χ H xx 00 χ 00 yy 00zz • This also means that the magnetic moment of the molecule prefers (lower potential energy) to lie along a certain direction, e.g., double well picture in transition metals. ΔE = |D|S2 Energy (arb. u.) (arb. Energy
Hard plane
-90 0 90 180 270 Angle between easy axis and magnetic moment Easy axis Ch. 1. Introduction Section 1.1 Magnetic anisotropy 6 Magnetic anisotropy. The highly simplified, hand‐waving explanation • 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 between 3d transition metals and f‐elements is that the crystal field and exchange interactions are much weaker in the latter, while spin‐orbit coupling can be much stronger.
Lueken99, Lueken08 Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 7
A highly simplified, hand‐waving explanation. • Most free ions have an orbital angular momentum (except d5 high‐spin). • In most complexes, the orbital angular momentum appears to have disappeared. (quenching of the orbital moment).
• This is due to the crystal field splitting of the d‐orbitals ("t2g‐eg").
Lueken99, Atkins‐PC Ch. 1. Introduction 8 Section 1.2 Magnetic anisotropy in Transition Metal Clusters x x
Quenching of the orbital moment. z y • Orbital angular momentum is generally pictured as a circular motion of electrons around the nucleus. d • The (classical) orbital angular momentum is dxz xy related by 90o rotn. about x axis lrp x x
• Translated into quantum mechanics, we can picture orbital angular momentum as being due to a circular motion of the electron through y y degenerate orbitals, that are related by a rotation.
• For example: dx2-y2 dxy o • dxz / dxy (x axis) related by 45 rotn. about z axis
• dxy / dx2‐y2 (z axis) x x • px / py (z axis).
y y
px py related by 90o rotn. about z axis Mabbs/Machin Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 9
Quenching of the orbital moment. x x • In transition metal complexes, the degeneracy of the d‐orbitals is lifted by the interaction with the ligands (crystal/ligand field splitting). z y • As a result, the circular motion is no longer possible and the orbital dxz dxy angular momentum disappears. related by 90o rotn. about x axis
x x • For Oh and Td complexes, only in certain cases an orbital angular momentum is retained y y
orbital contribution? No Yes No dx2-y2 dxy ground state type E T A related by 45o rotn. about z axis • Importantly, in lower symmetry, fewer orbitals are degenerate: x x
y y
px py related by 90o rotn. about z axis
Mabbs/Machin Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 10 g‐Value anisotropy. • Spin orbit coupling reintroduces orbital angular momentum (2nd order perturbation). • Quantitative formula: ˆˆ ml00 l m gg ee2;2 ggE Λ m EEm 0 • = x, y, or z; m is the excited d‐orbital.
• For dxy ‐ dx2‐y2 mixing induced by lz: 22ii 8 ggzz e 2 ge EExy 22 EE xy xy x22 y
1 • > 0 for d gz = g|| < ge. • Anisotropy must correspond to point group symmetry.
Mabbs/Machin; Atherton Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 11
Types of anisotropy. Dxx 00 D 00D • In transition metals we have three kinds: yy 1. Zero‐field splitting: We can express the D tensor in terms of Λ. 00Dzz mlˆˆ00 l m 3 1 22 DDEDD22zz,| xx yy | D ; D Λ m EEm 0
2. g‐anisotropy: gxx 00 mlˆˆ00 l m g 00g yy gg ee2;2 ggE Λ m EE m 0 00gzz • Relation between ZFS and g value anisotropy (this relation does not always hold).
ggxy Dg z 22 Egg() 4 xy
3. Anisotropic, antisymmetric exchange interactions Mabbs/Machin; Atherton; Abragam/Bleaney Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 12
Zero‐Field Splitting in Transition Metal Clusters. • The cluster zero‐field splitting is a linear combination of: • single‐ion zero‐field splittings, • dipolar spin‐spin interaction (usually a minor contribution)
si DDDSiiijijdd iij
• In addition anisotropic and antisymmetric exchange interactions lead to energy splittings in zero field.
Factors of Importance
1. Magnitude and sign of projection coefficients di. 2. Orientation of single ion zero‐field splitting. 3. Magnitude and sign of single ion zero‐field splitting.
Bencini/Gatteschi Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 13
Zero‐Field Splitting in Transition Metal Clusters.
1. Magnitude and sign of projection coefficients di.
• Projection coefficients di < 1. • For each pairwise coupling D decreases. • In the end the energy barrier is only linearly dependent on the spin of the ground state:
N 21/ S SI 2 SSii(2 1) i EDSmax SI di 21/S SS(2 1) i1
For S = n x Si Waldmann, IC,2007; Sessoli, ICA, 2008: "Waldmann's dire prediction" (Hill, DT, 2009) Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 14
Zero‐Field Splitting in Transition Metal Clusters.
• Example: Mn19: S = 83/2 but very small D. Why? 2. Orientation of single ion zero‐field splitting.
DDsi d Sii Ako, Powell, ACIE, 2006 i
[Mn6O4Br4(Me2dbm)6] [Mn6O2(sao)6(O2CPh)2(EtOH)4] Aromí, JACS, 99 Milios, JACS, 07 S = 12 S = 12 D = 0.009 cm‐1 D = ‐0.43 cm‐1 Ch. 1. Introduction Section 1.2 Magnetic anisotropy in Transition Metal Clusters 15
Energy scales in Transition Metals. • Example Mn3+, d4 HS
1G
3P, 3F
1F
1D 1D, 1S 1I, 3D, 1G ZFS 5 5 3F, 3G T2 A1g ms = 0 3H, 3P D ms = ±1
3D
5 5 E E B1g 5D ms = ±2
Ligand Field D4h Spin‐Orbit Coupling Electron repulsion Ligand Field Oh elongated Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 16
Energy scale of interactions‐ Example Tb3+ 7 FJ 0 1 2
3
4
describe ground doublet as 5 effective spin 1/2 with very anisotropic g
6
electron repulsion spin‐orbit coupling crystal‐field splitting Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 17
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. Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 18
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). Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 19
Crystal field splitting • Going from a free ion to a compound, the symmetry around the f‐ element ion is lowered, leading to splitting of the states. mJ • Group theory can be exploited to predict the way in which the terms split (symmetry aspect). • 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). Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 20
Crystal field splitting: Stevens operator equivalents • We can write the electrostatic potential created by the ligands as an expansion in terms of spherical harmonics.
• The resulting operators 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. ˆˆˆ • xJyJzJrxyz,,,(1) JJ • To take into account the noncommutation between the angular momentum operators, we have 1 ˆˆ ˆˆ to form symmetrized products, e.g. xyJJJJ2 x yyx • We end up with a Hamiltonian of the form f‐electrons
246 ˆ qqˆˆ qq qq HLF BOBOBO22 44 66 qqq246
d‐electrons q • The parameters Bk are taken as free parameters
Lueken; Boča Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 21
Crystal field splitting: Stevens operator equivalents • A table of common Stevens operator equivalents and the related spherical harmonics 53zr22 YOJJJ20ˆ 31ˆ 2 0216 r 2 z 2 15 xiy shift operators YO22ˆ 1 Jˆˆ2J2 222 2 ˆˆˆ 32 r J JiJx y 4224 93530zzrr 3 2 YO40ˆ 35Jˆˆ4 25 30JJ 1J2 6JJ 1 3J2J 1 04256 r 4 zz 4 xiy 44315 ˆ 1 ˆˆ44 YO44JJ 512 r 4 2 • The properties of the angular momentum operators. ˆ 2 J |Jm JJ JJ( 1)| Jm ˆ JJmzJ| JJ mJm | ˆ JJm+J| JJ(1)-( mmJJ 1) | Jm J 1 ˆ JJm | JJ JJ(1)-( mmJJ 1) | Jm 1
Lueken; Abragam, Bleany. Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 22
Crystal field splitting: Symmetry • Furthermore, the CF Hamiltonian must have the same symmetry as the complexed ion. • Hence, with increasing symmetry, more terms must be zero by symmetry ˆ 00ˆˆˆ 00 00 • For example in D4d symmetry, the crystal field Hamiltonian reads: HCF BOBOBO22 44 66 • 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; Newman/Ng; C. Görller‐Wallrand Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 23
Crystal field splitting: Kramers Theorem • Whatever the symmetry, for an odd number of electrons, all microstates are doubly degenerate, in the absence of a magnetic field. • This is known as Kramers' theorem, and ions with half integer angular momentum ground states are called Kramers ions. • This degeneracy has major implications for spin dynamics,t in tha quantum tunnelling cannot be induced by the crystal field.
H. Lueken; Abragam/Bleaney Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 24
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. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 25
Crystal field splitting: Crystal field ground states • The shape of the electron distribution is different for the different CF states. • Hence by choosing the ligands, one can influence the CF state energies. • Hence different ground states can be obtained. • The ground state and the energy splittings can be influenced by judicious choice of the ligands
• Note that the mJ states are the eigenstates only in C∞v, D∞h and D4d symmetries.
J. Sievers, Z. Phys. B, 45, 289 (1982), J.D. Rinehart, Chem. Sci., 2, 2078 (2011); Lueken Ch. 1. Introduction Section 1.3 Magnetic anisotropy in f‐Elements 26
Magnetic and spectroscopic measurements • The magnetic susceptibility is the thermal average of the contributions due to the occupied microstates. • Luminescence spectroscopy allows direct determination of the splitting of the ground Russell‐Saunders multiplet. • Absorption/MCD spectroscopy allows determination of CF splittings of excited states: beyond Russell‐Saunders coupling. (intermediate coupling: CF and SOC at same level) • Far infrared/inelastic neutron scattering allows investigation of intramultiplet excitations.
300K 1. Introduction 2. Single Crystal Magnetometry 3. High‐Frequency EPR Spectroscopy 4. Inelastic Neutron Scattering 5. Electronic Absorption and Luminescence Ch. 2. Single Crystal Magnetometry 28
Contents 2.1 Motivation 2.2 Magnetic susceptibility tensor 2.3 Determination (0) Large enough crystal (1) Faceindex (2) Define the experimental space (3) Transformation matrix (4) Mount the crystal (5) Three orthogonal rotations (6) Fitting the tensor elements (7) Symmetry and equivalent considerations (8) Diagnalization (9) Express in abc space (10) ground state determination Ch. 2. Single Crystal Magnetometry Section 2.1 Motivation 29
• Determination of the molecular principal axes • magneto-relations eff ˆ • Possibility of ground state determination ggJJJ// 2 J zz z
80 0.25
0.20 60 T along easy m K T along media m -1 m -1
0.15 T along hard / molemu m -1 40 along easy m linear fit 0.10 / emu mol -1 T
m 20
0.05
0 0.00 0 2 4 6 8 10 12 14 16 T / K R. Sessoli,et al, Angew Chem Int Ed 2012, 164, 1238; S.‐D. Jiang, X.‐Y. Wang,unpublished; Ch. 2. Single Crystal Magnetometry Section 2.2 magnetic susceptibility tensor 30
• Molecular tensor relation: † mq q m p q ij p ij AA p • In the paramagnetic limit, crystal behavior is the sum of all the molecules’ one † in the unit cell, so is tensor; Cry mq q m p q ij ij p ij AA p q q • In the paramagnetic limit, the molecular tensor can be fully determined only when the molecule symmetry is not lower than the crystal symmetry:
(a) Only one symmetrically independent molecule in the unit cell; (b1) The molecules are related by inversion center (P-1 space group) or (b2) The molecule is at highest symmetry position;
Neumann’s Principle: the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 31
(0) large enough single crystal • The background of the rotator provided by Quantum Design MPMS‐XL SQUID is around 10‐4 emu; • The crystal should be larger than 0.5 mg; (1) Face index Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 32
(2) Define the experimental space • b as X axis; • (001) is XY plane; • Z follows right hand rule;
(3) Transformation matrix • Derive the abc unit vector into the same length! • High school geometry game • Complicated! a 041.978.1 X b 0093.11 Y c 18.1271.040.2 Z Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 33
(4) Mount the crystal • L shaped Cu-Be support: small background; • Fixed with Apiezon N-grease;
b axis
Rotation axis
This abc‐XYZ relation is not the one defined before
15
1.7 K 2.0 K z rotation 2.3 K 2.6 K Ch. 2. Single Crystal Magnetometry 3.1 K 3.6 K = 90 4.1 K 4.8 K 5.5 K 6.4 K
-1 7.4 K Section 2.3 Determination 10 8.6 K 34 10.0 K 12 K 13 K 16 K
(5) Three orthogonal rotations / emu mol
m 5
30 0 1.7 K x rotation 2.0 K 2.3 K -90 0 90 180 270 2.6 K 3.1 K 25 = 90 3.6 K 4.1 K / 4.8 K 5.5 K 6.4 K 7.4 K 8.6 K 20 10.0 K -1 12 K 13 K 16 K 15
10 / emu mol / emu m
5
0 -90 0 90 180 270 /
40
1.7 K 35 2.0 K 2.3 K 2.6 K 3.1 K 3.6 K 30 y rotation 4.1 K 4.8 K 5.5 K = 0 6.4 K 7.4 K
-1 8.6 K 25 10.0 K 12 K 13 K 16 K 20
15 / emu mol m
10
5
0 -90 0 90 180 270 / Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 35
(6) Fitting the tensor elements
T cossin cossin xzxyxx HM 0 sinsin yzyyyx sinsin cos zzzyzx cos
Surface fitting with two variables
Curve fitting with one variables (three functions parallelly) Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 36
(7) Symmetry and equivalent considerations • Neuman’s principle: b of monoclinic system is always one of the principal axes • equivalent positions
25 x rotation T = 3 K y rotation 20 H = 0.1 T z rotation fitting 61.1043.140.8
-1 15
96.122.143.1 10 / emu mol / m 7019.1596.161.10 5
0 -90 0 90 180 270 tensor in XYZ space Angle(for x and y, for z) / Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 37
(8) Diagonalization Eigenvectors in abc {‐0.0164857,‐0.0593218,0.0664437} 61.1043.140.8 Fitting results 96.122.143.1 {‐0.102119,0.00984252,‐0.0168336} 7019.1596.161.10 {0.0249881,‐0.0625164,‐0.0452197} a 041.978.1 X Diagonalization b 0093.11 Y c 18.1271.040.2 Z 54.23 {‐0.577552,‐0.108217,0.809149} 96.0 {‐0.104866,‐0.973128,‐0.204999} 82.0 {‐0.80959,0.20325,‐0.550684} eigenvalues eigenvectors in XYZ Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 38
(9) expressed in abc space
pseudo inversion Ch. 2. Single Crystal Magnetometry Section 2.3 Determination 39
(10) Ground state determination • Curie Law: Population is not changed a lot in the temperature range • The first excited state is high enough.
80 0.25 1 2 SSgT 1 0.20 60 T along easy 8 m K T along media m -1 m -1
0.15 T along hard emu / mol m -1 40 along easy m linear fit 0.10 / emu mol -1 T
m 20
0.05 ggJJJeff 2 ˆ 0 0.00 // J zz z 0 2 4 6 8 10121416 T / K Ch. 1. Introduction 40 Magnetic Anisotropy Spectroscopic techniques Field‐independent: high‐frequency Crystal‐Field Splitting electron paramagnetic resonance
Field‐independent: Zero‐Field Splitting inelastic neutron scattering
Field‐dependent: electronic absorption, g‐value Anisotropy luminescence
Transition Metal Clusters f‐Elements
Molecular Nanomagnets 1. Introduction 2. Single Crystal Magnetometry 3. High‐Frequency EPR Spectroscopy 4. Inelastic Neutron Scattering 5. Electronic Absorption and Luminescence Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 42 Basics of EPR • The Zeeman term describes the interaction of the spin with a magnetic field. • In the absence of other interactions the field is chosen along the z‐axis. ˆ Zeeman ˆ • The electronic Zeeman term then has the form: Η gBSBz • Accordingly, the energies of the spin states depend on m : S EgBm BS
Note: remember Ŝz |Sms ImI> = ms |Sms ImI>. • g = 2.00235… for a free electron.
1 α 5 EgB 2 B EgB 2 B 3 EgB 2 B 1 EgB 2 B 1 Energy Energy EgB 2 B EgB 3 1 β 2 B EgB 2 B 5 EgB 2 B Magnetic Field Magnetic Field
For S = 1/2 For S = 5/2 Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 43 Basics of EPR
• For S = 1/2, the energy difference between the two mS levels is: EgB B • If the energy of the electromagnetic radiation matches the energy difference, radiation can be absorbed. The spin interacts with the magnetic field of the electromagnetic radiation.
• This is called the resonance condition: hgB B • For technical reasons, in EPR conventionally the frequency is kept constant, while the field is swept.
• The EPR selection rule is therefore ΔS = 0, ΔmS = ±1 (perpendicular mode)
S = ½
mS = ½
mS = -½ Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 44 Basics of EPR – Selection rules. • Which transitions can be observed?
• Typically microwave magnetic field b1 external magnetic field B0. • What about the Zeeman interaction of
b1 with the spin? ˆ Zeeman ˆ Η gbSBxy1,
• What is the action of Ŝx and Ŝy on the spin state |Sms ImI> ?
• It is useful to make combinations of Ŝx and Ŝy, called shift operators: ˆˆ ˆˆˆˆ SSiSSSiSx yxy
• The shift operators change the mS quantum number by ±1: ˆ S|+sSm SS( 1)- mss ( m 1) | Sm s 1 ˆ S|-sSm SS( 1)- mss ( m -1) | Sm s-1
• The EPR selection rule is therefore ΔmS = ±1 • In addition ΔS = 0. Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 45 Basics of EPR – High‐frequency EPR • Conventional EPR uses 9 GHz radiation frequency. • High‐frequency EPR with frequencies up to 1000 GHz possible. 9.74 GHz • High‐frequency means high field, which gives increased g‐value resolution. 108 GHz • This is usually not of interest in molecular magnetism 217 GHz E z y x 325 GHz
437 GHz
B Field
Photosystem I, light‐induced P700+•, A. Angerhofer in O.Y. Grinberg (Ed.), Very High Frequency EPR Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 46 Zero‐Field Splittings: Axial ZFS ˆ ˆˆˆˆ22221 ˆˆ 22 • Spin Hamiltonian H ZFS DSzxyz E() S S DS2 E () S S
• The D parameter lifts the degeneracy of the mS levels.
• For D <0,themS =±S levels are lowest in energy.
• For D >0,themS =0ormS = ±½ are lowest in energy. • D can be 0‐102 cm‐1.
• For S = 1, energy gap between mS = 0 or mS = ±1 equals D. S = 1 z m = +1 S mS = 0 mS = ±1
mS = 0
mS = ±1 mS = 0 mS = -1 D < 0 D > 0 Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 47 Large Zero‐Field Splittings.
mS = 1
D Energy mS = -1 X Band W Band
mS = 0
• Only ΔMS = ±1 allowed. Field •At X‐band microwave quantum too small. •At higher frequencies transitions can be observed. •Large ZFS ions with integer spin are therefore traditionally called EPR silent Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 48 High‐Field limit S = 1, D = ‐1 cm‐1 = ‐30 GHz, ν = 150 GHz, T = 300 K
θ = 0: B0 || z θ = 90: B0 || z Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 49 High‐Field limit S = 1, D = ‐1 cm‐1 = ‐30 GHz, ν = 150 GHz, T = 300 K Absorption First derivative
Forbidden transition Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 50 High‐Field limit 9.448 GHz
• [Ni(5‐methylpyrazole)6](ClO4)2 • S = 1, D = 0.72 cm‐1 = 21.6 GHz, T = 100 K
34.112 GHz
178.114 GHz
Collison, J. Chem. Soc., Faraday Trans., 1998, 3019 Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 51 High‐Field limit •HFEPR allows the determination of the sign of D
At low temperature, and B0 // z: •For D < 0 the low field line remains. •For D > 0, the high field line remains. S = 3, D > 0 B0//z, S = 3, D < 0
400 300 300 200 200 100 100 0
-100 0 energy [GHz] energy [GHz] -200 -100
-300 -200
-400 -300
0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 magnetic field [mT] magnetic field [mT] Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 52 High‐Field limit
• For gmB H >> D and uniaxial anisotropy there are 2S resonance lines. • The resonance fields are given by:
• B// =(ge/g//)[B0+(2MS‐1)D] B =(ge/g)[B0‐(2MS‐1)D/2]
• Spacing: 2D/gμB for B//zD/gμB for Bz • Example: D negative
2D/gB B//
D/g kBT< B Ch. 3. High‐Frequency EPR Section 3.2 Examples 53 Example [Fe4(L)2(dpm)6] • L = 2‐(bromomethyl)‐2‐(hydroxymethyl)‐1,3‐propanediol. • Antiferromagnetic exchange coupling leads to S = 5 ground state. MS = −5, −4, …, 4, 5. • Lines with large spacing at low fields, lines with small spacing at high field. • That means that D < 0. ‐1 0 ‐5 ‐1 • From fit (black): D = −0.432 cm , B4 = 2 ×10 cm . 0 2 0 0 • B4 is a higher order ZFS term. HZFS = D Ŝz + B4 Ô4 . [Accorsi, JACS, 2006] Ch. 3. High‐Frequency EPR Section 3.2 Examples 54 Example Ni(PPh3)2Cl2 S = 1, D = +13.20 cm‐1 = 396 GHz, E = 1.85 cm‐1, g = 2.20 (isotropic), T = 4.5 K |+1> + |‐1> 2E= 3.7 cm‐1 |+1> ‐ |‐1> D= 13.2 cm‐1 |0> J. Krzystek, Inorg. Chem., 41, 4478 (2002) Ch. 3. High‐Frequency EPR Section 3.2 Examples 55 Example Ni(PPh3)2Cl2 S = 1, D = +13.20 cm‐1 = 396 GHz, E = 1.85 cm‐1, g = 2.20 (isotropic), T = 4.5 K Rhombic anisotropy: x‐ and y‐axis different, θ and φ both important. θ = 0, φ = 0: B0 || z θ = 90, φ = 90: B0 || y θ = 90, φ = 0: B0 || x z θ y x φ J. Krzystek, Inorg. Chem., 41, 4478 (2002) Ch. 3. High‐Frequency EPR Section 3.2 Examples 56 Example Ni(PPh3)2Cl2 S = 1, D = +13.20 cm‐1 = 396 GHz, E = 1.85 cm‐1, g = 2.20 (isotropic), T = 4.5 K Rhombic anisotropy: x‐ and y‐axis different, θ and φ both important. yz‐plane xz‐plane xy‐plane J. Krzystek, Inorg. Chem., 41, 4478 (2002) Ch. 3. High‐Frequency EPR Section 3.2 Examples 57 Example Ni(PPh3)2Cl2 • B not much larger than D: no high‐field simplification. • Fictitious 2000 GHz spectrum goes up to 80 T. Ch. 3. High‐Frequency EPR Section 3.2 Examples 58 Example Ni(PPh3)2Cl2 • B not much larger than D: no high‐field simplification. • Fictitious 2000 GHz spectrum goes up to 80 T. Ch. 3. High‐Frequency EPR Section 3.1 Theoretical Background and Experimental Considerations 59 Where can I do high‐frequency (ν > 95 GHz) EPR on molecular nanomagnets? • France: LNCMI Grenoble (Barra) • Germany: IFW Dresden (Kataev) • Germany: HLD Dresden (Zvyagin) • Germany: Uni Stuttgart (Van Slageren) • Italy: CNR Pisa (Pardi) • USA: HMFL Tallahassee (Hill, Krzystek, Ozarowski). • Japan: Kobe (Nojiri) Ch. 3. High‐Frequency EPR Section 3.3 Frequency domain methods 60 Frequency Domain Magnetic Resonance FDMR: No field required EPR: External field required S = ½ mS mS = ½ mS = ‐½ 0 Van Slageren, Top. Curr. Chem. 2012 Ch. 3. High‐Frequency EPR Section 3.3 Frequency domain methods 61 Monochromatic sweepable sources vs interferometer (FTIR) based methods Monochromatic sweepable sources • synthesizer + multipliers or backward‐wave oscillators (+ multipliers) + high resolution, easier below 10 cm–1. – limited range Interferometer • Mercury lamp or synchrotron +easy to obtain ultra broad band spectrum, easier at higher frequencies > 25 cm–1 – field/frequency not independent. Mn12ac Van Slageren, Top. Curr. Chem. 2012; Schnegg, HZB Berlin Ch. 3. High‐Frequency EPR Section 3.3 Frequency domain methods 62 III Example 1. [Mn 6O2(Me2Bz)2(Et‐sao)6(EtOH)4] (ΔE = 84 K) • S = 12. D = –0.43 cm‐1 from magnetisation. ΔE = 84 K (record). • Powder pellet sample. 6 sharp magnetic resonance lines • Oscillating baseline due to interference within pellet • Single scan takes ca. 30 seconds 654321 100 10-1 5K 10K TransmissionTransmission 15K 20K 10-2 30K 120 GHz 40K 300 GHz d 46810 Wavenumber (cm-1) Carretta, Van Slageren et al., PRL 2008, Pieper, Van Slageren et al. PRB 2010 Ch. 3. High‐Frequency EPR Section 3.3 Frequency domain methods 63 III Energy (meV) Example 1. [Mn 6O2(Me2Bz)2(Et‐sao)6(EtOH)4] (ΔE = 84 K) 0.50 0.75 1.00 1.25 • Giant spin model (ground state only). 100 2 0 0 • H =DŜz + B4 Ô4 ‐1 • D = ‐0.362 cm 5 K -1 10 K 0 ‐6 ‐1 10 • B4 = ‐6.08 x 10 cm 15 K 20 K Transmission 30 K 0 40 K 100 -10 )) -1-1 6 -1 -20 10 10 K 5 Expt. -30 Fit 4 10-2 3 Transmission Energy (cmEnergy (cm -40 2 -50 4.0 6.0 8.0 10.0 1 -1 -60 Wavenumber (cm ) -12 -8 -4 0 4 8 12 M S Carretta, Van Slageren et al., PRL 2008, Pieper, Van Slageren et al. PRB 2010 Ch. 3. High‐Frequency EPR Section 3.3 Frequency domain methods 64 III Example 1. [Mn 6O2(Me2Bz)2(Et‐sao)6(EtOH)4] (ΔE = 84 K) • Comparison with HFEPR • Single crystal. • D = –0.360(5) cm–1 0 –6 –1 • B4 = –5.7(5) 10 cm • 7 Frequencies x 7 T sweep.... Datta, Dalton Trans. 2009 Ch. 3. High‐Frequency EPR Section 3.3 Frequency domain methods 65 Example 2. [Ni(PPh3)2Cl2] • Read off D and E from single scan. 1.0 0.7 |+1> + |-1> 2E= 3.7 cm-1 0.4 |+1> - |-1> 0.2 Transmission D= 13.2 cm-1 0.1 0 5 10 15 20 25 30 Frequency (cm-1) |0> Vongtragool, IC, 2003 Ch. 3. High‐Frequency EPR Section 3.3 Frequency domain methods 66 + – Example 3. (NBu4) [Ln(Pc)2] • Bruker 113v FTIR, Ln = Ho. Divide by 6T, 10K spectrum 0 10 6T 5T 4T 3T 2T 1T 0T 10-1 Transmission 10-2 20 40 60 80 100 Wavenumber / cm-1 Divide by 0T, 5K spectrum Marx, Dörfel, Moro, Waters, Van Slageren, unpublished 1. Introduction 2. Single Crystal Magnetometry 3. High‐Frequency EPR Spectroscopy 4. Inelastic Neutron Scattering 5. Electronic Absorption and Luminescence Ch. 4. Inelastic Neutron Scattering Section 4.1 Theoretical background and experimental considerations 68 Introduction • Neutrons can have energies in the same range as the microwave/THz electromagnetic radiation used in EPR. • However, the neutron wavelength is much shorter, and can be of the order of bond distances. • Some data on the neutron: • Mass m = 1.674927351(74)×10−27 kg −3 • Magnetic moment μ = –1.04187563(25)×10 μB. • Spin s = ½. hh9.044605 • De Broglie wavelength : [Å] p 2[meV]mE E • For an energy of E = 25 cm–1 ≈ 3 meV, λ = 5 Å 2 • Rather than the wavelength, we can deal with the wave vector k Ch. 4. Inelastic Neutron Scattering Section 4.1 Theoretical background and experimental considerations 69 Introduction • Because the wavelength is much shorter than for photons of the same energy, we have to consider momentum conservation in addition to energy conservation. • Energy conservation: the energy change of the neutron is taken up by the sample: ΔE = ħω = Ef – Ei. • Momentum conservation: Δ • Neutrons are detected at different angles. • Time of arrival corresponds to kinetic energy • Plot of S(Q,ω). • Selection rules ΔS = 0, ±1; ΔmS = 0, ±1 E Sample Ch. 4. Inelastic Neutron Scattering Section 4.1 Theoretical background and experimental considerations 70 Introduction • Determining the nature of the excitaton from the Q‐dependence. • Magnetic excitation: ΔS 1: → , → 0 I Q ΔS 0: → , → I • Phonons: Q 1 , , ∝ 1 ⁄ • Generally confine to low Q to focus on magnetic transitions E Q Ch. 4. Inelastic Neutron Scattering Section 4.1 Theoretical background and experimental considerations 71 Example: Mn6 • Compare INS with FDMR T = 1.5 K = 6.7 Å T = 12.0 K T = 17.0 K 6 654321 5 100 4 3 10-1 Intensity (a.u.) 2 5K 10K 1 TransmissionTransmission 15K 20K 10-2 30K 0 40K -4-2024681012 Energy loss (cm-1) 46810 Wavenumber (cm-1) Pieper, PRB, 2010 Ch. 4. Inelastic Neutron Scattering Section 4.1 Theoretical background and experimental considerations 72 FDMR vs HFEPR vs INS HF Cavity EPR HFEPR/FDMRS INS Selection rules ∆MS = ±1, ∆S = 0 ∆MS = ±1, ∆S = 0 ∆MS = 0, ±1, ∆S = 0, ±1 Sample quantity Few mg 50 – 200 mg 1 g Resolution 10-2 cm-1 10-2 cm-1 0.5 cm-1 Intensity 1.0 EPR 0.8 INS 0.6 0.4 0.2 Boltzmann populations Intensity determined by determined Intensity 0.0 0 4 8 1216202428 Temperature (K) Ch. 4. Inelastic Neutron Scattering Section 4.1 Theoretical background and experimental considerations 73 Example: Mn6 • Take advantage of the ΔS = 0, ±1 selection rule of INS T = 1.5 K T = 12.0 K = 6.7 Å = 5.0 Å T = 6.0 K T = 17.0 K T = 18.0 K 2.0 6 5 1.5 4 3 1.0 Intensity (a.u.) 2 Intensity (arb.u.) 0.5 1 0 0.0 -4-2024681012 0 4 8 12 16 20 -1 Energy loss (cm ) Energy loss (cm-1) Pieper, PRB, 2010 Ch. 4. Inelastic Neutron Scattering Section 4.1 Theoretical background and experimental considerations 74 Example: Mn6 • The Q dependence reveals the nature of the spin excitation ΔS 1: → , → 0 I Q ΔS 0: → , → I = 5.0 Å T = 6.0 K Q T = 18.0 K 2.0 1.5 1.0 Intensity (arb.u.) 0.5 0.0 0 4 8 121620 Energy loss (cm-1) Pieper, PRB, 2010 1. Introduction 2. Single Crystal Magnetometry 3. High‐Frequency EPR Spectroscopy 4. Inelastic Neutron Scattering 5. Electronic Absorption and Luminescence Ch. 5. Electronic Absorption and Luminescence Section 5.1 Electronic Absorption 76 Lanthanides • f‐Orbitals are buried deep within the electron cloud • ff‐transitions are Laporte‐forbidden (u ↔ u) • Hence, the extinction coefficients are very small (ε ~ 1 M–1 cm–1), cf. dd 102, CT 104. • On the other hand, the absorption bands are very narrow, and split due to the CF splitting of the excited state. Eu3+ Bünzli in Lanthanide Luminescence: Photophysical, Analytical and Biological Aspects, Springer Ser Fluoresc (2010), DOI 0.1007/4243_2010_3 Ch. 5. Electronic Absorption and Luminescence Section 5.1 Electronic Absorption 77 Lanthanides • Excitations can be electric dipole, magnetic dipole, or electric quadrupole • In actinides, extinction coefficients are larger. • Judd‐Ofelt theory describes the absorption intensity of ED transitions. • Parameters Ω are phenomenological. Bünzli in Lanthanide Luminescence: Photophysical, Analytical and Biological Aspects, Springer Ser Fluoresc (2010), DOI 0.1007/4243_2010_3 Mills, Nature Chem. 2011 Ch. 5. Electronic Absorption and Luminescence Section 5.2 Luminescence 78 Lanthanides • Many lanthanides are strongly luminescent. Natrajan, EUFEN2, COST CM 1006, Dublin 2013 Ch. 5. Electronic Absorption and Luminescence Section 5.2 Luminescence 79 Lanthanides • The splitting of the luminescence band yields information on the crystal field splitting of the ground state. mJ H.‐D. Amberger, many publications; Görller‐Wallrand, many publications; Boulon, ACIE, 2013 Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 80 Magnetic Circular Dichroism • The intensity of an absorption band due to an electronic transition between two states is proportional to the square of the electric dipole matrix element: 2 I XG • In MCD, the difference in absorption between left‐ (σ‐)and right‐(σ+)‐circularly polarised light is measured: 1 MCD 2 • No MCD without field. K Applied magnetic field can: • Lift magnetic degeneracies of the ground and excited states. X • Change the population of the levels via a new Boltzmann distribution. • Mix the levels G and X with other electronic levels. G Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 81 Magnetic Circular Dichroism • This leads to three contributions to the MCD spectrum: • the A‐term, due to Zeeman splitting of the ground and/or excited degenerate states, • the B‐term, due to field‐induced mixing of states, • the C‐term, due to a change in the population of molecules over the Zeeman sublevels of a paramagnetic ground state. fE() C0 B BA10 B fE() EEkT B • Δε is MCD extinction coefficient. • E = hν. • γ is bunch of constants including dielectric permittivity. • μB is Bohr magneton. • f(E) is lineshape function. Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 82 Magnetic Circular Dichroism. Example 1 [Cr8F8Piv16] • UV/Vis spectrum gives energies of excited states. • Spin‐allowed transitions stronger than spin‐forbidden. • Exchange coupling enhances intensity of spin‐forbidden transitions. • Resolution typically not enough to resolve all transitions. 0.4 0.2 Absorption 0.0 30000 25000 20000 15000 Wavenumber (cm-1) Piligkos, Van Slageren, Dalton Trans. 2010 Ch. 5. Electronic Absorption and Luminescence Section 5.1 Theoretical background and experimental considerations 83 Magnetic Circular Dichroism. Example 1 [Cr8F8Piv16] • MCD signal can be both positive and negative. • This often leads to much better resolution. • Spin forbidden transitions are often pronounced. 4 4 2 2 4 2 4 2 2 T1 T2 T1 E T1 T2 T2 T1 E 0.4 5 0.2 0 Absorption -5 MCD 0.0 corrected MCD / mdeg 30000 25000 20000 15000 30000 25000 20000 15000 Wavenumber (cm-1) UV/Vis Piligkos, Van Slageren, Dalton Trans. 2010 Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 84 Magnetic Circular Dichroism. Example 1 [Cr8F8Piv16] • MCD splitting of absorption bands is related to ZFS. 2 2 2 • H = d Ŝz + e (Ŝx ‐Ŝy ). • d = ‐0.364 cm‐1; e = 0.119 cm‐1. • cf. d = ‐0.334 cm‐1 (strong exchange) • cf. d = ‐0.24 cm‐1; e = 0.032 cm‐1 (microscopic, neglecting dipolar). 4 2 4 2 2 T1 T2 T2 T1 E 5 0 -5 MCD corrected MCD / mdeg 30000 25000 20000 15000 Piligkos, Van Slageren, Dalton Trans. 2010; Bradley, Dalton Trans. 2008 Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 85 S = 2 Magnetic Circular Dichroism. Example 1 [Cr8F8Piv16] S = 1 • MCD intensity is related to the magnetization of the sample. S = 0 • It gives information about the magnetic properties of the cluster. E • MCD intensity vs. B or vs. B/T shows more than just a step 4 at the level crossing from S = 0 to S = 1. B 2 • No MCD intensity is expected for the S = 0 state. / M 0 B 80 Cr F Piv MCD in poly(dimethylsiloxane) mull 8 8 16 60 60 = 415 nm 50 40 40 T = 1.6 K 20 30 T = 2.8 K 1.6 K T = 5.0 K 20 T = 10 K 0 3.0 K 5.2 K 10 = 415 nm -1 10 K = 24096 cm -20 Rotation (mdeg) Rotation (mdeg) 0 4A -> 4T 2 1 -40 -10 -60 0.0 0.5 1.0 1.5 2.0 2.5 -12 -8 -4 0 4 8 12 H/2kT B / T Piligkos, Van Slageren, Dalton Trans. 2010 Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 86 60 Axial fit (E = 0) Magnetic Circular Dichroism. Example 1 [Cr8F8Piv16] 50 experiment fit • MCD curves cannot be simulated by summing 40 contributions due to different spin states. 30 20 • Mixing between spin states occurs. rotation / mdeg 10 • Rhombic term (E) necessary. 0 -10 • Spin Hamiltonian: 0.0 0.5 1.0 1.5 2.0 2.5 222 B / 2kT ˆˆ ˆ ˆ ˆ B H 2JdSSij izSeSS ix y BSgB ij i i experiment ˆ 60 fit lSxx M yz • MCD intensity i 50 2 • . ˆ 40 NlSiy y M xz sin d d ES4 00 i 30 i ˆ 20 lSzz M xy i 10 S = 3/2 rotation / mdeg g = 1.98 0 D = ‐0.58 cm‐1 -10 0.00.51.01.52.02.5 E / D = 0.1 B / 2kT ‐1 B J = ‐3.5 cm Rhombic fit (E ≠ 0) Piligkos, Van Slageren, Dalton Trans. 2010 Ch. 5. Electronic Absorption and Luminescence Section 5.1 Theoretical background and experimental considerations 87 – Magnetic Circular Dichroism. Example 2 [Ln(Pc)2] • Anions in EtOH:MeOH glass. • N N Transitions based on ligand. N • MCD bands have derivative lineshapes: N N A‐term intensity. N • Lowest‐energy transition has opposite N N MCD sign for Tb3+ and Dy3+. - 1.0 [DyPc ] , 1.6 K, 7 T 3000 2 [TbPc ]-, 1.6 K, 7 T 2 0.8 0.6 0 0.4 Absorption 0.2 MCD intensity (mdeg) MCD intensity -3000 0.0 15000 20000 25000 30000 35000 10000 20000 30000 40000 -1 Wavenumber (cm-1) wavenumber (cm ) UV/Vis MCD Krivokapic, Waters, Van Slageren, unpublished Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 88 0/– Magnetic Circular Dichroism. Example 2 [Ln(Pc)2] • Hysteresis is observed for all complexes in spite of ligand based nature of transitions. 0 • For [DyPc2] no hysteresis is observed in powder magnetisation measurements. 400 MCD on frozen solution 1000 magnetisation on powder 2 800 emu) -2 DyPc2- 200 600 -1 14599cm 1 400 T=1.6K 200 0 0 0 -200 MCD (mdeg) -200 -1 -400 MCD intensity (mdeg) -600 -400 moment magnetic (10 -800 -2 hyst_685nm_fast_0.973 T/min -1000 -1 0 1 -2 -1 0 1 2 External Field (T) Field (Tesla) ‐ 0 [DyPc2] [DyPc2] Krivokapic, Waters, Van Slageren, unpublished; Gonidec, JACS, 2010 Ch. 5. Electronic Absorption and Luminescence Section 5.3 Magnetic Circular Dichroism 89 Magnetic Circular Dichroism. Example 3 [C(NH2)3]5[Ln(CO3)4] (1‐Ln) with Ln = Dy, Er • Crystal field split ff transitions Rechkemmer, Van Slageren, unpublished Ch. 6. Some words on relaxation of the magnetization Section 6.1 Spin‐Lattice relaxation 90 The fundamental difference between magnetization dynamics in single‐ion and polynuclear species • In polynuclear TM clusters, Arrhenius behaviour over many orders of magnitude; sharp transition to tunnelling • In f‐elements generally curved Arrhenius plots. Different temperature regime. • The reason is that in single‐ion systems, there are many relaxation pathways with different field‐ and temperature dependences (direct process, tunnelling, Raman, Orbach, multistep direct process). • Hence a straight line in the Arrhenius plot is not necessarily expected [Mn3O(Me‐salox)3(2,4′‐bpy)3(ClO4)], Yang, Tsai, IC, 2008 [Dy(hfac)3(PyNO)]2, Yi, CEJ, 2012 Ch. 6. Some words on relaxation of the magnetization Section 6.1 Spin‐Lattice relaxation 91 The fundamental difference between magnetization dynamics in single‐ion and polynuclear species The first f‐element "single‐molecule magnets". • Relaxation parameters ΔE/kB (K) τ0 (s) ref • Tb/– 331 6.3 x 10–8 Ishikawa, J. Am. Chem. Soc., 125, 8694 – 8695 (2003) • Dy/– 40.3 6.3 x 10–6 Ishikawa, J. Am. Chem. Soc., 125, 8694 – 8695 (2003) • Tb/0 590 1.5 x 10–9 Ishikawa, Inorg. Chem, 43, 5498 – 5500 (2004) • Dy/0 – • Many substituted (terbium) double deckers known • Strong correlation between ΔE and τ0. Michael Waters PhD Thesis University of Nottingham, 2013 Ch. 6. Some words on relaxation of the magnetization Section 6.1 Spin‐Lattice relaxation 92 Basic Spin‐Lattice Relaxation Mechanisms in dilute systems • assume dilute systems. energy conservation in both steps • A further mechanism is quantum tunnelling 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. 6. Some words on relaxation of the magnetization Section 6.1 Spin‐Lattice relaxation 93 General formulas (not considering tunnelling) • non‐Kramers ions: 1 1373 TR1 dO coth Rrexp 1 RT R0 2kT00 kT • Kramers ions: 1 13975 TR1 dO coth Rrexp 1 RTR R00R T 2kT00 kT k Abragam/Bleaney