Electron Paramagnetic Resonance Spectroscopy: Fundamentals

Grenoble Sciences
Preface
Contents
Fundamental constants - Unit conversions
Chapter 1 - The electron paramagnetic resonance phenomenon
	1.1 – What is a spectroscopy experiment?
		1.1.1 – Exchange of energy between matter and radiation
		1.1.2 – Spectroscopic techniques
	1.2 – Magnetic spectroscopy techniques
	1.3 – Diversity of paramagnetic centres
		1.3.1 – Electrons have two magnetic moments
		1.3.2 – Paramagnetic atoms
		1.3.3 – Paramagnetic molecules
	1.4 – Principle of electron paramagnetic resonance experiments
		1.4.1 – Reduced angular momenta
		1.4.2 – Interaction between a paramagnetic centre and a magnetic field
		1.4.3 – EPR transitions
		1.4.4 – Expression of the absorption signal
	1.5 – Basic EPR spectrometry instrumentation
		1.5.1 – A few orders of magnitude
		1.5.2 – Simplified description of an EPR spectrometer
		1.5.3 – Tuning the spectrometer
	1.6 – Points to consider in applications
		1.6.1 – Electronic and nuclear paramagnetism
		1.6.2 – Importance of paramagnetic centres
		1.6.3 – Continuous wave EPR and pulsed EPR
		1.6.4 – Some observations
	Complement 1 – Magnetic moment created by a rotating point charge
	Complement 2 – Why are B_2 and O_2 paramagnetic molecules?
	Complement 3 – How magnetic field modulation affects the signal detected
	References
	Exercises
	Answers to exercises
Chapter 2 - Hyperfine structure of a spectrum in the isotropic regime
	2.1 – The various origins of spectral features in EPR
	2.2 – Hyperfine interactions
		2.2.1 – The nuclear magnetic moment
		2.2.2 – Hyperfine interactions between unpaired electrons and nuclei
		2.2.3 – The isotropic regime
	2.3 – EPR spectrum for a centre which interacts with a single nucleus in the isotropic regime
		2.3.1 – Expression of the energy levels for the centre
		2.3.2 – EPR spectrum for S = ½ and I = ½
		2.3.3 – EPR spectrum for S = ½ and I = 1
		2.3.4 – General case
	2.4 – EPR spectrum for a centre which interacts with several nuclei in the isotropic regime
		2.4.1 – Hyperfine interactions with several equivalent nuclei
		2.4.2 – Interaction with non-equivalent nuclei
	2.5 – Important points for applications
		2.5.1 – Importance of hyperfine interactions
		2.5.2 – Free radicals
		2.5.3 – Transition ion complexes
	Complement 1 – The paramagnetic radical probes technique
	Complement 2 – “Pascal’s triangles“
	References
	Exercises
	Answers to exercises
Chapter 3 - Introduction to the formalism of the space of spin states. The Hamiltonian operator
	3.1 – Introduction
	3.2 – Space of spin states associated with an angular momentum
		3.2.1 – Construction of linear operators from J – Specific bases of E_J
		3.2.2 – The scalar product
		3.2.3 – Representation of an operator by a matrix
		3.2.4 – Eigenvectors and eigenvalues of an operator
		3.2.5 – Application to a centre characterised by J = ½
		3.2.6 – How can we use the formalism of the space of spin states associated with an angular momentum?
	3.3 – Spin states and allowed energy levels for a paramagnetic centre placed in a magnetic field
		3.3.1 – Interaction between a centre with an isotropic magnetic moment and a field B
		3.3.2 – When the magnetic moment is anisotropic
	3.4 – Transition probabilities and allowed transitions
	3.5 – Possible spin states and allowed transitions in the presence of hyperfine interaction
		3.5.1 – Determining the energy levels
		3.5.2 – Allowed transitions
	3.6 – Points to consider in applications
		3.6.1 – Why is a spin Hamiltonian used in EPR and magnetic spectroscopies?
		3.6.2 – What does the spin Hamiltonian do?
		3.6.3 – Looking back to the procedure described in chapters 1 and 2
	Complement 1 – Diagonalisation of H_zeeman  in any basis
	Complement 2 – The principle of perturbation theory
	References
	Exercises
	Answers to exercises
Chapter 4 - How anisotropy of the g and A matrices affects spectrum shape for radicals and transition ion complexes
	4.1 – Introduction
	4.2 – The g matrix
		4.2.1 – How the molecule’s symmetry properties affect the g matrix
		4.2.2 – The principal values of the g matrix
	4.3 – Shape of the spectrum produced by an ensemble of paramagnetic centres in the absence of hyperfine interaction
		4.3.1 – Variation in gʹ values with the direction of B
		4.3.2 – Shape of the EPR spectrum depending on the nature of the sample
		4.3.3 – Notes on the spectra produced by polycrystalline powders or frozen solutions
	4.4 – How anisotropic hyperfine interaction affects the shape of the EPR spectrum
		4.4.1 – The hyperfine matrix A
		4.4.2 – Expression of the resonance field in the presence of anisotropic hyperfine interaction
		4.4.3 – Effect of g and A matrix anisotropy on the shape of the powder
	4.5 – How molecular movements affect the spectrum: isotropic and very slow motion regimes
		4.5.1 – A hypothetical experiment
		4.5.2 – Effects of rotational Brownian motion of paramagnetic molecules
	4.6 – Points to consider in applications
		4.6.1 – Spectrum for a single crystal
		4.6.2 – Powder spectrum for centres of spin ½
		4.6.3 – Spectra for transition ion complexes
		4.6.4 – Spectra for free radicals
		4.6.5 – The EPR spectrum contains additional information
	Complement 1 – Splitting of the energy levels for the electrons in an octahedral complex
	Complement 2 – Possible values of gʹ when the g matrix is rhombic
	Complement 3 – Expression for the density of resonance lines for a centre with axial symmetry
	Complement 4 – Expression giving the energy levels for any direction of B when the g and A matrices are anisotropic
	Complement 5 – An example of a study of a single crystal: identification of the site of Ti^3+ fluorescence in LaMgAl_11O_19
	References
	Exercises
	Answers to exercises
Chapter 5 - Spectrum intensity, saturation, spin-lattice relaxation
	5.1 – Introduction
	5.2 – Spectrum intensity at thermal equilibrium
		5.2.1 – Absorption signal and intensity of a resonance line
		5.2.2 – Expressions for the absorption signal and the intensity of the spectrum for a powder or a frozen solution
		5.2.3 – Intensity of the spectrum produced by a single crystal
		5.2.4 – Intensity of the resonance lines and of the spectrum in the presence of hyperfine interactions
	5.3 – Signal saturation 
		5.3.1 – Saturation of an EPR transition
		5.3.2 – Expression for the absorption signal in the saturated regime
		5.3.3 – Significance of the saturation phenomenon
	5.4 – Spin-lattice relaxation
		5.4.1 – The various spin-lattice relaxation processes
		5.4.2 – How can the spin-lattice relaxation time T_1 be measured?
		5.4.3 – Relaxation phenomena and EPR spectroscopy in practice
	5.5 – Points to consider in applications
		5.5.1 – Intensity of the resonance lines and the spectrum
		5.5.2 – Use of spin-lattice relaxation
	Complement 1 – Fermi’s golden rule
	Complement 2 – Expression of the intensity factor for an axially symmetric centre of spin ½
	Complement 3 – Homogeneous and inhomogeneous lines
	References
	Exercises
	Answers to exercises
Chapter 6 - The zero-field splitting term. EPR spectrum for paramagnetic centres of spin greater than 1/2
	6.1 – Introduction
	6.2 – The zero-field splitting term
		6.2.1 – The D matrix
		6.2.2 – The D and E parameters
	6.3 – Definition and general characteristics of “high-field” and “low-field” situations
		6.3.1 – The energy levels for a centre with axial symmetry for the canonical directions of the magnetic field
		6.3.2 – “High-field” and “low-field” situations
	6.4 – General properties of the spectrum in the high-field situation
		6.4.1 – Energy levels and allowed transitions
		6.4.2 – Intensity of the resonance lines and the spectrum
	6.5 – Shape of the powder spectrum in the high-field situation
		6.5.1 – Expression for the resonance field in axial symmetry
		6.5.2 – Shape of the spectrum in axial symmetry
		6.5.3 – The“half-field” line for S = 1
		6.5.4 – Shape of the spectrum in “rhombic” symmetry
	6.6 – EPR spectrum for complexes of half-integer spin in the low-field situation. Kramers doublet
		6.6.1 – Case of a complex with axial symmetry
		6.6.2 – Generalisation to a complex of any geometry
	6.7 – EPR spectrum for integer-spin complexes in the low-field situation
	6.8 – Points to consider in applications
		6.8.1 – Organic molecules in a triplet state
		6.8.2 – Transition ion complexes in the high-field situation
		6.8.3 – Transition ion complexes in the low-field situation
		6.8.4 – Spin-lattice relaxation for centres of spin greater than ½
	Complement 1 – Intensity of the resonance line at high temperature in the high-field limit
	Complement 2 – Shape of the low-field spectrum for a centre of spin S = 1
	References
	Exercises
	Answers to exercises
Chapter 7 - Effects of dipolar and exchange interactions on the EPR spectrum. Biradicals and polynuclear complexes
	7.1 – Introduction
	7.2 – Origin and description of intercentre interactions
		7.2.1 – The true nature of exchange interaction
		7.2.2 – Phenomenological description of exchange interaction
		7.2.3 – “Anisotropic components” of exchange interaction
		7.2.4 – Magnetic dipolar interaction
	7.3 – Effects of weak intercentre interactions on the spectrum
		7.3.1 – Effects of the dipolar interactions
		7.3.2 – Effects of the exchange interaction
		7.3.3 – General case
	7.4 – Effects of strong exchange interaction on the spectrum Biradicals and polynuclear complexes
		7.4.1 – Introduction
		7.4.2 – Construction of equivalent Hamiltonians for a pair of paramagnetic centres
		7.4.3 – Equivalent Hamiltonians and EPR spectra for a few typical pairs
	7.5 – How intercentre interactions affect the intensity of the spectrum and the relaxation properties
		7.5.1 – EPR spectrum intensity
		7.5.2 – Relaxation properties
	7.6 – Points to consider in applications
		7.6.1 – How weak intercentre interactions affect the spectra
		7.6.2 – How strong exchange interaction affects the spectrum
		7.6.3 – Dynamic effects of intercentre interactions
	Complement 1 – Equivalent Hamiltonian for a trinuclear complex
	References
	Exercises
	Answers to exercises
Chapter 8 - EPR spectrum for complexes of rare earth and actinide ions
	8.1 – Rare earth ions
		8.1.1 – Magnetic moment of free rare earth ions
		8.1.2 – Hyperfine interaction with the nucleus
	8.2 – Complexes of rare earth ions: effect of interaction with ligands
		8.2.1 – Expression describing the interaction of electrons in the 4f subshell with ligands
		8.2.2 – Effects of interaction with ligands on the ground multiplet
	8.3 – The EPR spectrum for complexes of rare earth ions with half-integer J values
		8.3.1 – Introduction
		8.3.2 – Expression for the effective parameters
		8.3.3 – The case of cations in an S state
		8.3.4 – Application: analysing the data obtained for ethyl sulfates
		8.3.5 – Spin-lattice relaxation for complexes with half-integer spin
	8.4 – The EPR spectrum for complexes of rare earth ions with integer J values
		8.4.1 – Intradoublet transitions
		8.4.2 – Transitions between singlets
	8.5 – Actinide complexes
		8.5.1 – Introduction
		8.5.2 – Comparison of spectra for complexes of trivalent rare earth and actinide cations
		8.5.3 – Complexes of high valence actinides: example of cations with a 5f^1 configuration
	8.6 – Points to consider in applications
		8.6.1 – Comparison of the EPR characteristics of transition ion and rare earth ion complexes with a half-integer spin.
		8.6.2 – Interpreting spectra for rare earth complexes
		8.6.3 – Actinide complexes
	Complement 1 – Rare earth elements and actinides: etymological considerations
	References
	Exercises
	Answers to exercises
Chapter 9 - How instrumental parameters affect the shape and intensity of the spectrum. Introduction to simulation methods
	9.1 – Introduction
	9.2 – How field sweep and field modulation affect the shape of the spectrum
		9.2.1 – Effects of modulation at the level of the sample
		9.2.2 – Effects of magnetic field modulation and sweep at the level of the detection chain
	9.3 – How the power and frequency of the radiation affect the spectrum. The temperature parameter
		9.3.1 – Effect of the power and frequency of the radiation
		9.3.2 – Temperature-related effects
		9.3.3 – Case study: seeking the origin of line splitting in an EPR spectrum
	9.4 – Simulating spectrum saturation
		9.4.1 – Simulating saturation of a homogeneous Lorentzian line
		9.4.2 – Simulating saturation of an inhomogeneous line
		9.4.3 – Simulating saturation of a powder spectrum
	9.5 – Introduction to numerical simulation of the EPR spectrum
		9.5.1 – Why simulate a spectrum?
		9.5.2 – How can the spectrum be numerically calculated?
		9.5.3 – The linewidth problem
	9.6 – Points to consider in applications
		9.6.1 – How should the modulation and sweep parameters be selected?
		9.6.2 – How can a saturation curve be simulated?
		9.6.3 – How can an EPR spectrum be simulated?
	Complement 1 – Some properties of the convolution product
	Complement 2 – Quantitative analysis of the saturation curve for an inhomogeneous line
	Complement 3 – Quantitative study of relaxation broadening
	Complement 4 – Using standard samples in EPR spectroscopy
	Complement 5 – Numerical simulation software
	References
	Exercises
	Answers to exercices
Appendix 1 - Expression of the magnetic moment of a free atom or ion
	First step: electrostatic interactions to which electrons are subjected
		1 – Microstates
		2 – (L, S ) terms
	Second step: magnetic interactions to which electrons are subjected
		1 – Multiplets
		2 – Expression for the magnetic moment
Appendix 2 - Expression of g and A matrices given by ligand field theory for a transition ion complex
	1 – Electrostatic interactions in free ions
	2 – Electrostatic interactions with ligands
	3 – Magnetic interactions
		3.1 – Effects of spin-orbit coupling and interaction with a magnetic field
		3.2 – Effects of hyperfine interactions
		3.3 – The case of complexes of cations in an S state
Appendix 3 - Dipolar interactions between a nuclear magnetic moment and electron spin magnetic moments
	1 – The dipolar matrix T
		Case where the φ(r) orbital is spherically symmetric relative to the nucleus
	2 – Principal axes and principal values of the T matrix
		2.1 – φ(r) is an atomic orbital centred at O
		2.2 – φ(r) is centred at a point C which is remote from the nucleus
	3 – Case with several unpaired electrons
Appendix 4 - Some properties of angular momentum operators. Spin coupling coefficients and equivalent operators. Application to Landé’s formula and to dipolar hyperfine interactions.
	1 – Definition of coupled bases and spin coupling coefficients
		1.1 – Product bases and coupled bases
		1.2 – Construction of the matrices representing operators defined from J1 and J2 in the coupled basis
		1.3 – Spin coupling coefficients
		1.4 – Application to the energy of multiplets. Demonstration of the Landé formula
	2 – Calculation of the dipolar components of the hyperfineinteraction within an (L, S) term
Appendix 5 - The notion of spin density
	1 – Definition
	2 – Spin density in a mononuclear complex
	3 – Spin density in a dinuclear complex
Appendix 6 - Example of calculation of the spin-lattice  relaxation time T_1: the direct process
Appendix 7 - Matrix elements of operators defined from components of an angular momentum
Glossary
Index