Inelastic Neutron Scattering (INS) Philip L.W. Tregenna-Piggott. Department of Chemistry University of Bern Switzerland

Inelastic Neutron Scattering (INS) Philip L.W. Tregenna-Piggott Department of Chemistry University of Bern Switzerland

Seminar Overview Introduction to INS - Principles and Instrumentation - Advantages and Disadvantages Over Other Techniques - Selection Rules for Magnetic INS Applications - Ground State Zero-Field-Splittings of non-kramers Ions - Exchange Coupling in Molecular Clusters

The Technique Provides a Measure of Energy and Momentum Transfer Between the Neutron and Sample Incoming Neutrons Sample k r r k Q r Scattered Neutrons Energy Transfer Momentum Transfer : : 2 h hω = 2m r hq = h ( 2 2 k k ) r r ( k k )

Schematic of a Time-of-Flight (TOF) INS Spectrometer One measures the time it takes for neutrons to travel from the sample to a position on the detector bank

Schematic of a Triple Axis Spectrometer From http://www.ill.fr Incoming white neutron beam Incident energy selection Scattering by the sample Final energy selection Allows the measurement of inelastic excitations at a given point in reciprocal space

Principal Facilities for INS in Europe Facility Organisation Location TYPE Description of Instruments ILL Institut Laue- Grenoble, High-flux http://www.ill.fr/index_sc.html Langevin France reactor ISIS Rutherford Appleton Oxford, UK Pulsed neutron facility http://www.isis.rl.ac.uk/instruments /index.htm Laboratory SINQ Paul Scherrer Institute Near Villigen, Switzerland Continous spallation http://sinq.web.psi.ch/sinq/instrume nts.html neutron source. BENSC Berlin Neutron Scattering Centre Hahn-Meitner- Institut, Berlin, Germany Cold neutron guides http://www.hmi.de/bensc//instrume ntation/instrumentation_en.html The list is by no means complete. See also: http://www.neutron-eu.net/en/index.php

Variation of INS Intensity with Q, differs for Magnetic and Phonon Transitions Intensity Intensity Q Magnetic Transitions Q Phonon Transitions

In which energy range is INS applicable? general: Ability to detect low energy citations limited by the width the elastic line Momentum transfer increases ith increasing energy transfer: vours detection of phonon rather an magnetic transitions - Resolution becomes poorer with increasing energy transfer - Practical range of magnetic INS: ~.5 to ~5 cm -1, depending on sample, and instrument configuration

Advantages of INS Over Other Techniques -Direct Determination of Energy Levels in Zero Magnetic Field -Change in Intensity and Energy as a Function of Q Carries Useful Information Concerning the Wavefunctions Disadvantages of INS Over Other Techniques -Several Grams of Sample Must be Used -Often Necessary to Use Fully Deuteriated Samples

Background due to incoherent scattering from protons often precludes the observation of magnetic transitions Deuterium Neutron Scattering Cross-sections of Hydrogen and Deuterium Coherent Incoherent Hydrogen

Magnetic Scattering Selection Rules Powdered Sample, Unpolarised Neutron Beam + + + + + + + + = i y e y f f y e y i i x e x f f x e x i i z e z f f z e z i i INS S g k L S g k L S g k L S g k L S g k L S g k L kt E Q I ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ exp ) ( M L M S S L,,, Basis Basler, Tregenna-Piggott, J.A.C.S., 123, 3377 (21). Carver, Tregenna-Piggott, Inorg. Chem., 42, 5711 (23). S,M S Basis ( ) 2 2 2 ˆ ˆ ˆ 2 exp ) ( i f i f i z f i INS s s s kt E Q I Ψ Ψ Ψ Ψ Ψ Ψ + + + = M S = ±1, S = ±1,

Seminar Overview Introduction to INS - Principles and Instrumentation - Advantages and Disadvantages Over Other Techniques - Selection Rules for Magnetic INS Applications - Ground State Zero-Field-Splittings of non-kramers Ions - Exchange Coupling Parameters in Molecular Clusters

Ground State Zero-Field-Splittings of non-kramers Ions Case Studies 1. [Mn(OD 2 ) 6 ] 3+ 2. [V(OD 2 ) 6 ] 3+ 3. [Fe(OD 2 ) 6 ] 2+ 4. [Fe(imidazole) 6 ] 2+ 5. [Cr(OD 2 ) 6 ] 2+ INS is complementary to other spectroscopic techniques, for the characterisation of transition metal complexes. The main advantage of INS is that it affords the direct determination of the low-lying electronic structure in zero magnetic field.

Why is the Characterisation of Integer-Spin Transition Metal Complexes Difficult? Kramers Ions are Amenable to Study by Electron Paramagnetic Resonance: S N Energy hν N S Magnetic Field Intensity Magnetic Field

In Non-Kramers Transition Metal Ions, the Zero-field Splitting is generally large compared to hν: Energy D 3D hν 1 1 2 Values of D in S = 2 systems are in the range 2 to 6 cm -1 2 Magnetic Field along z r

1. Determination of Ground State Zero-Field-Splittings of non-kramers Ions using INS OF Inelastic Neutron Scattering Spectra of Cs[Mn III (OD 2 ) 6 ](SO 4 ) 2 6D 2 O α β γ ~ 2 > 2±1> D = -4.524(1) cm -1 E =.276(1) cm -1 1 I II intensity [arb. units] ~ 2±2> 1.4 K 15 K 3 K II' I' γ' β' α' α β γ I II -2-1 1 2 energy transfer [mev] sler, Tregenna-Piggott, J.A.C.S., 123, 3377 (21). 1 mev ~ -

2. Spectroscopic Studies of [V(OH 2 ) 6 ] 3+ cation States of the d 2 electronic configuration in octahedral and trigonal fields 6 4 Energy 2-2 -4-6 -8-1 Inelastic Neutron Scattering Electronic Raman Spectroscopy Optical Spectroscopy -12 High-Field EPR 2 4 6 8 1 12 Trigonal Field / cm -1 14 16 18 2 Tanabe-Sugano Diagram for the Energies of states of the 3 T 1g term as a function of a positive trigonal -1

igh-field EPR and Inelastic Neutron Scattering Studies of the [V(OH 2 ) 6 ] 3+ cation Intensity 115 GHz B=B z B x =B y = Intensity [C(ND 2 ) 3 ][V(OD 2 ) 6 ][SO 4 ] 2 T = 5 K 9.5 9.55 Field / Tesla 9.6 9.65 High-Field EPR Spectrum of [V(OH 2 ) 6 ] 3+ Doped Caesium Aluminium Sulphate -8-6 -4-2 2 4 Energy Transfer / cm -1 Inelastic Neutron Scattering Spectrum of [V(OH 2 ) 6 ] 3+ in Guanidinium Vanadium Sulphate 6 8 1 Data modelled with an S=1 Spin Hamiltonian D / cm -1 = 4.8713(1) g = 1.95188(3) g = 1.8662(4) A / cm -1 =.988(4) A / cm -1 =.8(6)

Experimental and Theoretical Electronic Raman Profiles of C(NH 2 ) 3 [V III (OH 2 ) 6 ](SO 4 ) 2 a g a g e g e g A, 1 3 E A, A, 1 3 A, 1, A, M s

Mössbauer Study of Cs[Fe II (OH 2 ) 6 ](PO 4 ) Data Collected and Analysed by Eckhard Bill 2K T= 4.2K, H= 7T 16K 8K T= 4.2K, H= 3T Intensity (arb. units) 7K 6K 55K Intensity (arb. units) T= 4.2K, H= T 5K 45K 4.2K -4-3 -2-1 1 Velocity (mms -1 ) 2 3 4-4 -3-2 -1 1 2 3 4 Velocity (mm s -1 ) Shows phase transition beautifully, but does not readily yield low temperature electronic structure

Inelastic neutron data of CsFe(D 2 O) 6 PO 4 Intensity exp. sim. 1 + sim. 2 (1:1) sim. 2 sim. 1-3 -2-1 1 2 3 Energy transfer [mev] 1 K INS Spectrum of a powdered sample of CsFe(D 2 O) 6 PO 4 Simulated spectrum consists of a sum of two spectra, each calculated using an S=2 Spin Hamiltonian with the following parameters: Species 1: D=11.999 cm -1 E=2.1117 cm -1 Species 2: D=12.121 cm -1 E=1.346 cm -1 irect determination of energies of low-lying states by INS

HFMF-EPR data of CsFe(D 2 O) 6 PO 4 HFEPR spectrum (5K, 284.9973 GHz) of a powdered sample of CsFe(D 2 O) 6 PO 4 fitted using two S=2 Spin Hamiltonians: Species 1: Intensity exp. * sim. 1 + sim. 2 (1:1) D=11.999 cm -1 E=2.1117 cm -1 g x =2.22,g y =2.22, g z =2.32 sim. 2 Species 2: sim. 1 D=12.121 cm -1 E=1.346 cm -1 g x =2.32,g y =2.22, g z =2.32 1 2 3 4 5 Field [Tesla] EPR data simulated with zero-field-splitting parameters from INS. The EPR experiments then yield the g values

25 2 States of the 5 T 2g (O h ) term as a function of the trigonal field, with λ set to 1 cm -1 Energy 15 1 5 5 1 15 2 Trigonal field [cm -1 ] J=1 ground state perturbed by trigonal field:- Model Data with S=1 spin-hamiltonian 5 A 1g (D 3d ) ground term perturbed by trigonal field:- Model Data with S=2 spin-hamiltonian

Inelastic neutron data for Fe(II)(Imidazole) 6 (NO 3 ) 2 Energy [cm -1 ] 8 4 II I II' I' e e a Normalised Intensity 1..8.6.4 1.5K exp. 15K exp. 3K exp. 5K exp. 1.5K calc. 15K calc. 3K calc. 5K calc. Intensity 5K 3K 15K 1.5K I' I II.2. 1 2 3 4 Momentum transfer Q [Angstrom -1 ] Experimental and calculated Q dependence of the 19.4 cm -1 peak at various temperatures. Ground state energies ( 5 A) fitted using the ligand field Hamiltonian: 5-4 -2 2 4 Energy Transfer [cm -1 ] INS spectra (incident wavelength 2.2 Angstrom) of a powdered sample of deuterated Fe(Im) 6 (NO 3 ) 2 at various temperatures. 6 8 28 π 1 3 3 Hˆ = Dq Y4 + ( Y4 Y4 ) 3 7 14 π + Y2 + λ ˆ L Sˆ 3 5 operating in the 5 D manifold

Magnetisation data 5.5 5. exp. calc. µ-effective [Bohr Magnetons] 4.5 4. 3.5 3. 2.5 Magnetisation data reproduced using the ligand field Hamiltonian: ˆ 28 π 1 3 3 H = Dq Y4 + ( Y4 Y4 ) 3 7 14 π + Y ˆ ˆ 2 + λ L S 3 5 k ( ˆ L + 2 Sˆ ) β B operating in the 5 D manifold using the parameters obtained from INS. Orbital reduction factor, k =.95. 5 1 15 2 Temperature [K] 25 3

HFMF-EPR data 2 e 15 Energy [cm -1 ] 1 a Energy [cm -1 ] 1-1 5 1 8 6 4 B [Tesla] 2 2 4 6 8 1 B [Tesla] Calculated energies of the lower three energy levels as a function of field directed parallel and normal to the three-fold axis. 2 4 6 Resonant field [Tesla] Experimental and calculated resonant field positions for HFMF-EPR. 8 1 G. Carver, P.L.W. Tregenna-Piggott, Inorg. Chem. 23, 42, 5771-5777

Inelastic Neutron Scattering Spectra of (ND 4 ) 2 [Cr II (OD 2 ) 6 ](SO 4 ) 2 297 K ~ 2 > 25 K β γ 2 K α } 2±1> 15 K Intensity 1 K 75 K 3 K I II }~ 2±2> 1 K 1.5 K II I γ β α α β γ III -8-6 -4-2 2 4 6 8 Energy Transfer (cm -1 ) Dobe, Tregenna-Piggott, Chem. Phys. Lett. 362, 387, (22)

High-Field EPR Spectra and Simulations of (ND 4 ) 2 [Cr II (OD 2 ) 6 ](SO 4 ) 2 * 25 K 285GHz Five Parameters Required to Model the EPR Data with an S=2 Spin- Hamiltonian. With INS, the Zero-Field Transitions are Obtained Directly Intensity (arb units) 5 K Simulation * 95GHz Simulation 2 4 6 Field (T) 8 1 12

Variation of the zero-field-splitting parameters D and E with temperature, obtained by least-squares refinement of the eigenvalues of: ˆ ˆ 2 1 2 2 H = D ( 1) [ ˆ ˆ S ] z S S + + E Sx S y 3 to the energies of the INS transitions. -2. -2.1.25 D / cm -1-2.2-2.3 E / cm -1.2.15-2.4.1 5 1 15 2 Temperature / K 25 5 1 15 2 Temperature / K 25

Variable Temperature X-band EPR Spectra and Simulations of (ND 4 ) 2 [Cr II (OD 2 ) 6 ](SO 4 ) 2 * * 275K 25K 225K 2K Intensity 175K 15K 12K 7K Sim 4 8 Field (G) 12 16

Variation of Cr-O bond distances with temperature Cr-O(9) 2.35 2.3 Cr Low Temperature Bond Length (Å) 2.25 2.2 2.15? 2.1 Cr-O(8) 2.5 Cr-O(7) 5 1 15 2 Temperature (K) 25 3 Cr High Temperature Unable to account for the data using static model

Jahn-Teller Potential Energy Surface of [Cr(OD 2 ) 6 ] 2+ in Cubic Symmetry Cr 6 4 2 1 5 15 1 5-2 15 1-5 -15 5-1 -5-2 -1-15 -15-1 -5-1 -2-1 -5 2 5 1 Q θ -15-5 -5-15 -2-2 Cr -4-6 -2-15 -1-5 -15-1 Cr -6-4 -2 2 4 6 Q ε

Calculated Potential Energy Surface of [Cr(OD 2 ) 6 ] 2+ in (ND 4 ) 2 [Cr II (OD 2 ) 6 ](SO 4 ) 2 at 1 K 6 Q θ 4 2 15 1 5-5 -1-2 -25-15 -5 5 1 15 25 2-5 -1-15 Potential energy Cr Cr Cr -2-4 -6-2 -3-25 -15-1 -2-6 -4-2 2 4 6 Q ε Angular distortion co-ordinate Degeneracy of Distorted Configurations Lifted by Anisotropic Strain

H = H +H ph + H JT + H st = 1 1 τ U = 1 1 θ U = 1 1 U ε ( ) ( ) [ ], 3.5 3 2 3 2 2 2 2 τ θ ε θ ε θ ε θ ω U Q Q Q A Q Q P P H ph + + + + = h ( ) ( ) ( ) ε ε θ θ θ ε ε ε θ θ U Q Q U Q Q A Q U Q U A H JT 2 2 2 2 1 + + + = ε ε θ θ U e U e H st + = Q θ ~ 2 z 2 x 2 y 2 and Q ε ~ x 2 y 2. Hamiltonian for the System P.L.W. Tregenna-Piggott, Advances in Quantum Chemistry, 144, 461 (23)..

ψ Q θ ψ Q ε ψ ψ Cr-O Bond Lengths Eigenvalues and Eigenvectors I INS Timescale ~ 1-14 -1-13 E exp kt i ψ f µ ψ i 2 INS Spectrum Input Rate of Inter-state Relaxation Density Matrix Method Timescale < 1-9 EPR Spectrum

Theoretical INS Spectra. Orthorhombic Strain and 1Dq Diminish with Increasing Temperature 297 K 25 K 297 K 2 K 25 K 15 K 2 K Intensity 1 K Intensity 15 K 75 K 1 K 3 K 75 K 1 K 3 K 1 K II I γ β α α β γ 1.5 K III 1.5 K -8-6 -4-2 2 4 6 8 2 4 6 8 1 Energy Transfer (cm -1 ) Energy Transfer (cm -1 )

Theoretical and Experimental Cr-O bond lengths 2.4 2.35 2.3 Cr-O(9) Theory Experiment Bond Length (Å) 2.25 2.2 2.15 2.1 Cr-O(8) 2.5 Cr-O(7) 5 1 15 2 Temperature (K) 25 3

Theoretical EPR Spectra as a Function of Temperature: Fast Inter-State Relaxation ν = 285 GHz Potential energy Cr Cr Cr Angular distortion co-ordinate

Spin-Hamiltonian Parameters of the System Depend on the Experimental Technique Used to Determine Them

Seminar Overview Introduction to INS - Instrumentation - Advantages and Disadvantages Over Other Techniques - Selection Rules for Magnetic INS Applications - Ground State Zero-Field-Splittings of non-kramers Ions - Exchange Coupling in Molecular Clusters

Exchange Coupling in Molecular Clusters Case Studies 1. [(NH 3 ) 5 Cr(OH)Cr(NH 3 ) 5 ]Cl 5 3H 2 O 2. [Mn 12 O 12 (OAc) 16 (H 2 O) 4 ] 3. MnMoO 4 INS has played a pivotal role in the elucidation of exchange interactions in molecular clusters. The variation of the energies and intensities of the transitions with Q, provides a wealth of information that cannot be obtained with other techniques.

1. Acid Rhodo Complex Bis-µ-hydroxo-pentamine chromium(iii) Hex = 2 Jab Sa Sb E W.E. Hatfield et al. 1973 3 12 Jab 2 6 Jab 1 S= 2 Jab

[(NH 3 ) 5 Cr(OH)Cr(NH 3 ) 5 ]Cl 5 3H 2 O Luminescence at 7 K first cited tate lowest level of 2 E 4 A 2 D C B A 5 B 1 S = 2 C B D 7 B 2 S = 3 A round state 4 A 2 4 A 2 splitting 5 A 1 S = 2 3 B 2 S = 1 1 A 1 S = Hans U. Güdel and Jim Ferguson, Australian Journal of Chemistry 26(3), 55-11, (1973),

[(ND 3 ) 5 Cr(OD)Cr(ND 3 ) 5 ]Cl 5 D 2 O Inelastic Neutron Scattering 3-ax, λ = 2.34 Å E 3 12 J ab 18 / 2 j ab A 2 1 S = A B C 6 J ab 27 / 2 j ab 2 J ab 13 / 2 j ab B H ex = 2 J ab S a S b + j ab (S a S b ) 2 C 2 J ab = 3.84 ±.12 mev + j ab =.21 ±.23 mev ns U. Güdel, Albert Furrer, Molecular Physics 33(5), 1335-44, (1977),

Variation of Intensity of INS transition A with Momentum Transfer scillatory behaviour due to interference effects, which are direct product of metal-metal distance in the cluster Albert Furrer, Hans U. Güdel Phys. Rev. Lett. 39, 657, (1977),

H = H H =. Paramagnetism, Superparamagnetism, and Ferromagnetism M ramagnet H H = H H = uperaramagnet M H H = H H = M erromagnet H

Single Molecule Magnets (SMMs) Molecular Clusters with Large Spin Ground States and Ising Type Anisotropy Slow Relaxation of the Magnetisation at Low Temperatures Intermediate Behaviour Between Classical and Quantum Magnets

he Prototype SMM [Mn 12 O 12 (OAc) 16 (H 2 O) 4 ] OAc = acetate Mn 4+ Mn 3+

he Prototype SMM [Mn 12 O 12 (OAc) 16 (H 2 O) 4 ] {Mn 12 O 12 } core 8Mn III S = 2 4 Mn IV S = 3/2 S = 1 R. Sessoli et al, J. Am. Chem. Soc. 1993, 115, 184.

Energy Barrier to Magnetisation Relaxation H = Barrier height: = S 2 D (integer S) Mn 12 : S = 1 D ~ -.5 cm -1 = 5 cm -1

Slow Magnetisation Relaxation At low temperature... H = H // easy axis H =

Resonant Quantum Tunnelling H = H magnetisation hysteresis loops H = nd/gµ B steps due to quantum tunnelling through the energy barrier (D. Gatteschi, R. Sessoli Angew. Chem. Int. Ed., 42, 268 (23))

Magnetic Relaxation 1 2 1 1 τ (s) 1 1-1 1-2 DC AC 1-3 1-4 2 4 6 8 1 1/T (1/K) Thermally Activated Relaxation Arrhenius Expression: τ = τ exp ( /kt) Quantum Mechanical Tunnelling

Mn 12 - Acetate Inelastic Neutron Scattering Instrument IN5 (ILL), l = 5.9 Å Roland Bircher and Hansueli Güdel, Unpublished Results See also: I. Mirebeau, M. Hennion, H. Casalta, H. Anders, H.U. Gudel, A.V. Ivadova, and A. Caneschi, Phys. Rev. Lett. 83, 628 (1999).

[Mn 12 O 12 (OCH 3 COO) 16 (H 2 O) 4 ] 2CH 3 COOH 4H 2 O H ZFS = D (S Z2 1 / 3 S(S+1)) + B 4 O 4 + B 44 O 4 4 O 4 = 35S Z4 (3S(S+1) 25)S Z2 + 3S 2 (S+1) 2 6S(S+1) O 44 = 1 / 2 (S +4 + S 4 ) D =.457(2) cm 1 B 4 = 2.33(4) 1 5 cm 1 B 44 = ±3.(5) 1 5 cm 1

INS Study of MnMoO 4 Edge-sharing Mn(II)O 6 octahedra tetramers Tetramers are connected by Mo(VI)O 4 tetrahedra Exchange pathways: within tetramers: Mn-O-Mn between tetramers: Mn-O-Mo-O-Mn Ferromagnetic alignment of the spins within the clusters. Antiferromagnetic ordering of the clusters at 1.7 K.

TOF INS Spectrum of MnMoO 4 instrument: FOCUS at PSI incident wavelength: 4.75 Å temperature: 1.5 K 4 peaks on neutron energy loss side S. T. Ochsenbein et al, Phys. Rev. B 68, 9241 (23)

Exchange Interactions of MnMoO 4 Isotropic quantum mechanical model for the intracluster interactions Mn 1 J J' 3.692 Å 13.9 14.13 3.417 Å 95.73 Mn 4 O 1 Mn 3 Mn 2 Hˆ = 2 J i j Si S j J =.4 cm -1, J = -.15 cm -1 S = 1 Ground state

Exchange Interactions of MnMoO 4 Molecular field model for the intercluster interaction Molecular Field at Cluster: gµ B H mol = -2J inter S cluster z, where z = number of neighbours on opposite sublattice J inter =-.36 cm -1 (average intercluster interaction)

Variation in Energies as a Function of Q easured spectra along (hh): eak positions and intensities vary as a function of the scattering vector Q spin wave dispersion neutron counts k = h = l = 1.5 h = l = 1.4 h = l = 1.3 h = l = 1.2 h = l = 1.1 h = l = 1. h = l =.9 h = l =.8 h = l =.7 h = l =.6 h = l =.5 -.5..5 1. 1.5 energy transfer (mev) 2. 2.5

Dispersion curves along (h h) Dispersion revelations contain detailed information on the inter-cluster exchange interactions. T. Ochsenbein, H. U. Güdel, P. S. Häfliger, A. Furrer, B. Trusch, J. Hulliger, work in progres

Acknowledgements Ground State Zero-Field-Splittings of non-kramers Ions Group of Philip L.W. Tregenna-Piggott, Department of Chemistry, University of Bern Graham Carver Theory of Vibronic Coupling, Spectroscopy of Imidazole Complexes Christopher Dobe Cr(II) Tutton Salt Project Christopher Noble Density Matrix Treatment of Relaxation in JT Systems Markus Thut INS and EPR Study of Mn(III) hexa-aqua Exchange Coupling in Molecular Clusters Group of Hansueli Güdel, Department of Chemistry, University of Bern Roland Bircher INS Studies of Single Molecule Magnets Stefan Ochsenbein Spin-Wave Dispersion in MnMoO 4 Reto Basler INS Study of Mn(III) hexa-aqua Funded by the Swiss National Science Foundation