Spin Dynamics Basics of Nuclear Magnetic Resonance. Malcolm H. Levitt

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1 Spin Dynamics Basics of Nuclear Magnetic Resonance Second edition Malcolm H. Levitt The University of Southampton, UK John Wiley &. Sons, Ltd

2 Preface xxi Preface to the First Edition xxiii Introduction 1 Part 1 Nuclear Magnetism 3 1 Matter Atoms and Nuclei Spin Classical angular momentum Quantum angular momentum Spin angular momentum Combining angular momenta The Pauli Principle Nuclei The fundamental particles Neutrons and protons Isotopes Nuclear Spin Nuclear spin states Nuclear Zeeman splitting Zero-spin nuclei Spin-1/2 nuclei Quadrupolar nuclei with integer spin Quadrupolar nuclei with half-integer spin Atomic and Molecular Structure Atoms Molecules States of Matter Gases Liquids Solids 19

3 viii 2 Magnetism The Electromagnetic Field Macroscopic Magnetism Microscopic Magnetism Spin Precession Larmor Frequency Spin Lattice Relaxation: Nuclear Paramagnetism Transverse Magnetization and Transverse Relaxation NMR Signal Electronic Magnetism 36 3 NMR Spectroscopy A Simple Pulse Sequence A Simple Spectrum Isotopomeric Spectra Relative Spectral Frequencies: Case of Positive Gyromagnetic Ratio Relative Spectral Frequencies: Case of Negative Gyromagnetic Ratio Inhomogeneous Broadening Chemical Shifts J-Coupling Multiplets Heteronuclear Decoupling 59 Part 2 The NMR Experiment 63 4 The NMR Spectrometer The Magnet The Transmitter Section The synthesizer: radio-frequency phase shifts The pulse gate: radio-frequency pulses Radio-frequency amplifier The Duplexer The Probe The Receiver Section Signal preamplifier The quadrature receiver Analogue digital conversion Signal phase shifting Overview of the Radio-Frequency Section Pulsed Field Gradients Magnetic field gradients Field gradient coils Field gradient control 80

4 5 Fourier Transform NMR A Single-Pulse Experiment Signal Averaging Multiple-Pulse Experiments: Phase Cycling Heteronuclear Experiments Pulsed Field Gradient Sequences Arrayed Experiments NMR Signal NMR Spectrum Fourier transformation Lorentzians Explanation of Fourier transformation Spectral phase shifts Frequency-dependent phase correction Two-Dimensional Spectroscopy Two-dimensional signal surface Two-dimensional Fourier transformation Phase twist peaks Pure absorption two-dimensional spectra Three-Dimensional Spectroscopy 114 Part 3 Quantum Mechanics Mathematical Techniques Functions Continuous functions Normalization Orthogonal and orthonormal functions Dirac notation Vector representation of functions Operators Commutation Matrix representations Diagonal matrices Block diagonal matrices Inverse Adjoint Hermitian operators Unitary operators Eigenfunctions, Eigenvalues and Eigenvectors Eigenequations Degeneracy Eigenfunctions and eigenvalues of Hermitian operators Eigenfunctions of commuting operators: non-degenerate case Eigenfunctions of commuting operators: degenerate case Eigenfunctions of commuting operators: summary Eigenvectors 134

5 x 6.4 Diagonalization Diagonalization of Hermitian or unitary matrices Exponential Operators Powers of operators Exponentials of operators Exponentials of unity and null operators Products of exponential operators Inverses of exponential operators Complex exponentials of operators Exponentials of small operators Matrix representations of exponential operators Cyclic Commutation Definition of cyclic commutation Sandwich formula 139 Review of Quantum Mechanics Spinless Quantum Mechanics The state of the particle The equation of motion Experimental observations Energy Levels Natural Units Superposition States and Stationary States Conservation Laws Angular Momentum Angular momentum operators Rotation operators Rotation sandwiches Angular momentum eigenstates and eigenvalues The angular momentum eigenstates Shift operators Matrix representations of the angular momentum operators Spin Spin angular momentum operators Spin rotation operators Spin Zeeman basis Trace Spin-1/ Zeeman eigenstates Angular momentum operators Spin-1/2 rotation operators Unity Operator Shift operators Projection operators Ket-bra notation Higher Spin Spin 1= Spin 1 = 3/ Higher spins 165

6 Part 4 Nuclear Spin Interactions Nuclear Spin Hamiltonian Spin Hamiltonian Hypothesis Electromagnetic Interactions Electric spin Hamiltonian Magnetic spin interactions External and Internal Spin Interactions Spin interactions: summary External Magnetic Fields Static field Radio-frequency field Gradient field External spin interactions: summary Internal Spin Hamiltonian The internal spin interactions Simplification of the internal Hamiltonian Motional Averaging Modes of molecular motion Molecular rotations Molecular translations Intramolecular and intermolecular spin interactions Summary of motional averaging Internat Spin Interactions Chemical Shift Chemical shift tensor Principal axes Principal values Isotropic chemical shift Chemical shift anisotropy (CSA) Chemical shift for an arbitrary molecular orientation Chemical shift frequency Chemical shift interaction in isotropic liquids Chemical shift interaction in anisotropic liquids Chemical shift interaction in solids Chemical shift interaction: summary Electric Quadrupole Coupling Electric field gradient tensor Nuclear quadrupole Hamiltonian Isotropic liquids Anisotropic liquids Solids Quadrupole interaction: summary Direct Dipole Dipole Coupling Secular dipole dipole coupling Dipole dipole coupling in isotropic liquids 215

7 xii Dipole dipole coupling in liquid crystals Dipole dipole coupling in solids Dipole dipole interaction: summary 9.4 J-Coupling Isotropic J-coupling Liquid crystals and solids Mechanism of the J-coupling J-coupling: summary 9.5 Spin Rotation Interaction 9.6 Summary of the Spin Hamiltonian Terms Part 5 Uncoupled Spins Single Spin-1/ Zeeman Eigenstates Measurement of Angular Momentum: Quantum Indeterminacy Energy Levels Superposition States General spin states Vector notation Some particular states Phase factors Spin Precession Dynamics of the eigenstates Dynamics of the superposition states Rotating Frame Precession in the Rotating Frame Radio-frequency Pulse Rotating-frame Hamiltonian x-pulse Nutation Pulse of general phase Off-resonance effects 253 Ensemble of Spins-1/ Spin Density Operator Populations and Coherences Density matrix Box notation Balls and arrows Orders of coherence Relationships between populations and coherences Physical interpretation of the populations Physical interpretation of the coherences Thermal Equilibrium Rotating-Frame Density Operator 268

8 xiii 11.5 Magnetization Vector Strong Radio-Frequency Pulse Excitation of coherence Population inversion Cycle of states Stimulated absorption and emission Free Precession Without Relaxation Operator Transformations Pulse of phase Op = Pulse of phase (Pp = 7r/ Pulse of phase rbp = n Pulse of phase = 37r/ Pulse of general phase (b p Free precession for an interval r Free Evolution with Relaxation Transverse relaxation Longitudinal relaxation Magnetization Vector Trajectories NMR Signal and NMR Spectrum Single-Pulse Spectra Experiments an Non-Interacting Spins-1/ Inversion Recovery: Measurement of T Spin Echoes: Measurement of 7' Homogenous and inhomogenenous broadening Inhomogenenous broadening in the time domain Spin echo pulse sequence Refocusing Coherence interpretation Coherence transfer pathway Spin Locking: Measurement of T1p Gradient Echoes Slice Selection NMR Imaging Quadrupolar Nuclei Spin I = Spin-1 states Spin-1 energy levels Spin-1 density matrix Coherence evolution Observable coherences and NMR spectrum Thermal equilibrium Strong radio-frequency pulse Excitation of coherence NMR spectrum Quadrupolar echo 331

9 xiv 13.2 Spin 1= 3/ Spin-312 energy levels Populations and coherences NMR signal Single pulse spectrum Spin-3/2 spectra for small quadrupole couplings Second-order quadrupole couplings Central transition excitation Central transition echo Spin I = 5/ Spins I = 7/ Spins I = 9/2 350 Part 6 Coupled Spins Spin-1/2 Pairs Coupling Regimes Zeeman Product States and Superposition States Spin-Pair Hamiltonian Pairs of Magnetically Equivalent Spins Singlets and triplets Energy levels NMR spectra Dipolar echo Weakly Coupled Spin Pairs Weak coupling AX spin systems Energy levels AX spectrum Heteronuclear spin pairs Homonuclear AX System Eigenstates and Energy Levels Density Operator Rotating Frame Free Evolution Evolution of a spin pair Evolution of the coherences Spectrum of the AX System: Spin Spin Splitting Product Operators Construction of product operators Populations and coherences Spin orientations Thermal Equilibrium Radio-Frequency Pulses Rotations of a single spin pair Rotations of the spin density Operator 393

10 Operator transformations Free Evolution of the Product Operators Chemical shift evolution J-coupling evolution Relaxation Spin Echo Sandwich Experiments an AX Systems COSY The assignment problem COSY pulse sequence Theory of COSY: coherence interpretation Product Operator interpretation Experimental examples INADEQUATE C isotopomers Pulse sequence Theory of INADEQUATE Coherence transfer pathways and phase cycling Two-dimensional INADEQUATE INEPT The sensitivity of nuclear isotopes INEPT pulse sequence Refocused INEPT Residual Dipolar Couplings Angular information Spin Hamiltonian Orienting media Doublet splittings Many-Spin Systems Molecular Spin System Spin Ensemble Motionally Suppressed J-Couplings Chemical Equivalence Magnetic Equivalence Weak Coupling Heteronuclear Spin Systems Alphabet Notation Spin Coupling Topologies Many-Spin Dynamics Spin Hamiltonian Energy Eigenstates 468

11 xvi 18.3 Superposition States Spin Density Operator Populations and Coherences Coherence orders Combination coherences and simple coherences Coherence frequencies Degenerate coherences Observable coherences NMR Spectra Many-Spin Product Operators Construction of product operators Populations and coherences Physical interpretation of product operators Thermal Equilibrium Radio-Frequency Pulses Free Precession Chemical shift evolution J-coupling evolution Relaxation Spin Echo Sandwiches INEPT in an /2 S System COSY in Multiple-Spin Systems AMX spectrum Active and passive spins Cross-peak multiplets Diagonal peaks Linear spin systems TOCSY The ambiguity of COSY spectra TOCSY pulse sequence Theory of TOCSY 499 Part 7 Motion and Relaxation Motion Motional Processes Molecular vibrations Local rotations of molecular groups Molecular flexibility Chemical exchange Molecular rotations Translational motion Mechanical motion Motional Time-Scales Motional Effects Motional Averaging Motional Lineshapes and Two-Site Exchange 516

12 Slow intermediate exchange and motional broadening Fast intermediate exchange and motional narrowing Averaging of J-splittings Asymmetric two-site exchange Knight shift Paramagnetic shifts Sample Spinning Longitudinal Magnetization Exchange Two-dimensional exchange spectroscopy Theory Motional regimes Diffusion Relaxation Types of Relaxation Relaxation Mechanisms Random Field Relaxation Autocorrelation functions and correlation times Spectral density Normalized spectral density Transition probabilities Thermally corrected transition probabilities Spin lattice relaxation Dipole Dipole Relaxation Rotational correlation time Transition probabilities Solomon equations Longitudinal relaxation Transverse relaxation Steady-State Nuclear Overhauser Effect NOESY NOESY pulse sequence NOESY signal NOESY spectra NOESY and chemical exchange Molecular structure determination ROESY Transverse cross-relaxation Spin locking Transverse Solomon equations ROESY spectra ROESY and chemical exchange ROESY and TOCSY Cross-Correlated Relaxation Cross-correlation Cross-correlation of spin interactions Dipole dipole cross-correlation and angular estimations TROSY 590

13 XVlli Part 8 Appendices 597 Appendix A: Supplementary Material 599 A.1 Euler Angles and Frame Transformations 599 A.1.1 Definition of the Euler angles 599 A.1.2 Euler rotations: first scheme 599 A.1.3 Euler rotations: second scheme 600 A.1.4 Euler rotation matrices 601 A.1.5 Reference-frame orientations 601 A.1.6 Consecutive reference-frame transformations 602 A.1.7 Passive rotations 602 A.1.8 Tensor transformations 603 A.1.9 Intermediate reference frames 604 A.2 Rotations and Cyclic Commutation 604 A.3 Rotation Sandwiches 605 A.4 Spin-1/2 Rotation Operators 606 A.5 Quadrature Detection and Spin Coherences 608 A.6 Secular Approximation 611 A.7 Quadrupolar Interaction 614 A.7.1 Full quadrupolar interaction 614 A.7.2 First-order quadrupolar interaction 614 A.7.3 Higher-order quadrupolar interactions 615 A.8 Strong Coupling 615 A.8.1 Strongly-coupled Spin-1/2 pairs 615 A.8.2 General strongly coupled systems 620 A.9 J-Couplings and Magnetic Equivalence 621 A.10 Spin Echo Sandwiches 623 A.10.1 Short-duration limit 625 A.10.2 Long-duration limit 625 A.10.3 Two spin echo sequences 626 A.10.4 Heteronuclear spin echo sequences 627 A.11 Phase Cycling 629 A.11.1 Coherence transfer pathways 629 A.11.2 Coherence transfer amplitudes 630 A.11.3 Coherence orders and phase shifts 631 A.11.4 The pathway phase 632 A.11.5 A sum theorem 633 A.11.6 Pathway selection I 634 A.11.7 Pathway selection II 635 A.11.8 Pathway selection III 637 A.11.9 Selection of a single pathway I 638 A Selection of a single pathway II 639 A Dual pathway selection 640 A Internal phases I 641 A Internal phases II 642 A Nested phase cycles I 644 A Nested phase cycles II 645 A Different ways of constructing phase cycles 648

14 xix A.12 Coherence Selection by Pulsed Field Gradients 649 A.12.1 Field gradient dephasing 649 A.12.2 Pathway phase 651 A.12.3 Coherence transfer echoes 652 A.12.4 Pathway selection 652 A.12.5 Heteronuclear coherence transfer echoes 652 A.13 Bloch Equations 653 A.14 Chemical Exchange 654 A.14.1 The incoherent dynamics 655 A.14.2 The coherent dynamics 655 A.14.3 The spectrum 656 A.14.4 Longitudinal magnetization exchange 658 A.15 Solomon Equations 660 A.16 Cross-Relaxation Dynamics 662 Appendix B: Symbols and Abbreviations 665 Answers to the Exercises 681 Index 693

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