NMR

I: Spin Coherences and Echoes.- 1 Introductory Remarks.- 2 Isolated Spins in Inhomogeneous Fields.- 2.1 Two-Pulse Hahn Echo.- 2.2 Three-Pulse Hahn Echoes.- 2.3 Gradient-Recalled Echo.- 2.4 Multiple Echoes.- 3 Rotary Echoes.- 3.1 Signal at ?? = ?1.- 3.2 Signal at ?=?0.- 4 Solid Echoes of Dipolar-Coupled Spins.- 4.1 Two-Pulse Dipolar Solid Echoes.- 4.1.1 Systems of Two Equivalent Spins 1/2.- 4.1.2 Approximate Treatment of Multi-Spin 1/2 Systems.- 4.2 Three-Pulse Dipolar Solid Echoes.- 5 Solid Echoes of I =1 Quadrupole Nuclei.- 5.1 Two-Pulse Quadrupolar Solid Echo.- 5.2 Three-Pulse Quadrupolar Solid Echoes.- 6 Dipolar and Quadrupolar Magic Echoes.- 6.1 Principle.- 6.2 The "Magic Sandwich" Pulse Sequence.- 6.3 Dipolar-Coupled Two-Spin 1/2 Systems.- 6.4 Mixed Echoes.- 7 Coherence Transfer of J-Coupled Spins.- 7.1 Two RF Pulses.- 7.1.1 Correlated Two-Dimensional Spectroscopy.- 7.1.2 Echo Formation.- 7.1.3 Spin-Echo Correlated 2D Spectroscopy.- 7.1.4 Homonuclear J Resolved 2D Spectroscopy.- 7.2 Three RF Pulses.- 7.2.1 Longitudinal-Magnetization Transfer Echo.- 7.2.2 Scalar-Order Transfer Echo.- 7.2.3 Zero-Quantum Coherence-Transfer Echo.- 7.2.4 Single-Quantum Coherence-Transfer Echoes.- 7.2.5 Double-Quantum Coherence-Transfer Echo.- 7.3 Multiple-Quantum Coherence Based Spectroscopy.- 7.3.1 Double-Quantum Filtered Correlated Spectroscopy (DQF-COSY).- 7.3.2 Double-Quantum/Single-Quantum Correlated Spectroscopy.- 7.4 Discrimination of Coherence-Transfer Echoes.- II: Molecular Motion.- 8 Survey.- 9 Categorization of Relaxation Phenomena.- 9.1 General Remarks.- 9.1.1 Observables Subject to Relaxation.- 9.1.2 Spin Interactions Subject to Fluctuations.- 9.1.3 The Autocorrelation and the Intensity Functions.- 9.2 Limits and Definitions for Spin-Lattice Relaxation.- 9.3 Limits and Definitions for Transverse Relaxation.- 9.3.1 Irreversibility and Spin-System Size.- 9.3.2 Irreversibility and Molecular Motion.- 9.3.3 Classification of Transverse Relaxation.- 10 Spin-Relaxation Functions.- 10.1 The Homonuclear Bloch Equations.- 10.2 Solutions for Laboratory-Frame Experiments.- 10.3 Solutions for Rotating-Frame Experiments.- 11 Perturbation Theory of Spin Relaxation.- 11.1 Iterative Approximation.- 11.2 The Master Equation.- 12 Spin-Lattice Relaxation.- 12.1 Laboratory-Frame Spin-Lattice Relaxation by Dipolar Coupling.- 12.1.1 Reduced Dipolar Correlation and Intensity Functions.- 12.1.2 S-Spin-Equilibrium Limit.- 12.1.3 Like-Spin limit.- 12.2 Laboratory-Frame Spin-Lattice Relaxation by Other Interactions.- 12.2.1 Scalar Coupling.- 12.2.2 Quadrupolar Coupling (I=1).- 12.2.3 Chemical-Shift Anisotropy.- 12.3 Rotating-Frame Spin-Lattice Relaxation by Dipolar Coupling.- 13 Transverse Relaxation.- 13.1 Motional-Averaging Limit.- 13.1.1 Single-Quantum Coherences of Dipolar-Coupled Spin Pairs.- 13.1.2 Single-Quantum Coherences of (I =1) Quadrupole Nuclei.- 13.1.3 Multiple-Quantum Coherences.- 13.2 Local-Field Theory.- 13.2.1 The Anderson/Weiss Ansatz.- 13.2.2 The Second Moment ??2?.- 13.2.3 Partial Motional Averaging.- 14 Examples of Autocorrelation Functions.- 14.1 Isotropic Continuous Rotational Diffusion.- 14.2 Discrete-Coupling Jump Models.- 14.2.1 Two-State Jump Model.- 14.3 Reorientation Mediated by Translational Displacements.- 14.3.1 Diffusion on Rugged Surfaces.- 14.3.2 Lévy-Walk Surface Diffusion.- 15 Field-Cycling NMR Relaxometry.- 15.1 Laboratory-Frame Experiments.- 15.1.1 Field-Cycling Magnets.- 15.1.2 The Switching Intervals.- 15.2 Spin-Lock Adiabatic Field-Cycling Imaging Relaxometry.- 15.2.1 Adiabatic Variation of the Effective Field.- 15.2.2 Spin-Lock Field-Cycling Laboratory-Frame Imaging Relaxometry.- 15.2.3 Spin-Lock Field-Cycling Rotating-Frame Imaging.- 16 Field-Cycling Relaxometry in Biosystems.- 16.1 Fluctuations in Proteins.- 16.2 Fluctuations in Lipid Bilayers.- 16.3 Deuteron T1 Frequency Dispersion of Protein Solutions.- 16.4 Critical Water Contents.- 16.5 Proton Relaxation in Tissue.- 17 The Dipolar-Correlation Effect.- 17.1 Outline of Attenuation Mechanisms and Time Scales.- 17.1.1 The Motional-Averaging Contribution to Echo Attenuation.- 17.1.2 The Residual-Coupling Contribution to Echo Modulation.- 17.2 Density-Operator Formalism for Equivalent Two-Spin 1/2 Systems.- 17.2.1 The Modified Primary Echo.- 17.2.2 The Modified Stimulated Echo.- 17.3 The Dipolar-Correlation Quotient.- 17.3.1 Exponential Correlation Function.- 17.3.2 Correlation Function for Liquid-Crystal Director Fluctuations.- 17.4 Applications of the Dipolar-Correlation Effect.- 17.4.1 Macroscopic Order.- 17.4.2 Short-Range Order and Polymers.- 18 Survey of NMR Diffusometry.- 18.1 The Diffusion Propagator.- 19 Main-Field Gradient NMR Diffusometry.- 19.1 The Principle.- 19.2 Pulsed-Gradient Spin-Echo (PGSE) Diffusometry.- 19.2.1 The Displacement-Correlation Function.- 19.2.2 The Mean Squared Phase Shift.- 19.2.3 The Echo-Attenuation Factor for Anomalous Diffusion.- 19.2.4 The Echo-Attenuation Factor for Ordinary Diffusion.- 19.2.5 Direct Evaluation of the Mean Squared Displacement.- 19.3 Steady-Gradient Spin-Echo (SGSE) Diffusometry.- 19.3.1 Echo-Attenuation Factors.- 19.3.2 Relaxation-Compensated Pulse Sequences.- 20 Reciprocal- vs Real-Space Representations.- 20.1 The Generalized Reciprocal-Space Formalism.- 20.2 The Real-Space Representation.- 20.2.1 The Longitudinal-Magnetization Grid.- 20.2.2 One-Dimensional Real-Space Evaluation.- 20.2.3 Two-Dimensional Real-Space Evaluation.- 20.2.4 Geometrical Confinements.- 21 RF-Field-Gradient NMR Diffusometry.- 21.1 Magnetization-Grid Rotating-Frame Imaging.- 21.1.1 Rapid MAGROFI Diffusometry.- 21.1.2 Experimental Aspects of MAGROFI Diffusometry.- 21.2 Comparison of B0 and B1 Gradient Methods.- 22 Examples for Anomalous Self-Diffusion.- 22.1 Anomalous Diffusion in Lacunar Systems.- 22.2 Reptation/Tube Model.- 22.2.1 The Doi/Edwards Limits.- 22.2.2 Evaluation Formula for PGSE/SGSE Experiments.- 23 Exchange.- 23.1 Equation of Motion for Discrete Spin Environments.- 23.1.1 Interpretation of the HMM Matrices.- 23.1.2 HMM Solutions in Terms of Eigenvalues.- 23.1.3 Two-Environment-Exchange Model.- 23.2 Two-Dimensional Exchange Spectroscopy.- 23.2.1 Matrix Formalism.- 23.2.2 Exchange Between Two Environments.- 23.2.3 Comparison with 2D Exchange NQR Spectroscopy.- III: Localization and Imaging.- 24 Survey.- 25 Fundamentals of NMR Imaging.- 25.1 Slice Selection by Soft Pulses.- 25.1.1 Approximation for Small Tip Angles.- 25.1.2 Frequently Used Pulseshapes.- 25.1.3 Refocusing of the Coherences.- 25.1.4 Variation of the Slice Width and Position.- 25.2 Phase Encoding.- 25.3 (Larmor) Frequency Encoding.- 25.4 Two- and Three-Dimensional Fourier Imaging.- 25.4.1 2DFT Imaging.- 25.4.2 3DFT Imaging.- 25.4.3 Gradient-Recalled Spin-Echo Imaging.- 25.4.4 Echo-Planar Imaging.- 26 Parameter-Weighted Contrasts.- 26.1 Contrast Parameters of Conventional Images.- 26.2 Contrasts in Gradient-Echo Tomography.- 26.3 Relaxation-Weighted Contrasts.- 26.4 Functional Tomography.- 26.5 Diffusive Attenuation and "Edge Enhancement".- 27 Relaxation-Dispersion Maps.- 28 Frequency-Offset Maps.- 28.1 MRSI Pulse Sequences.- 28.2 Theory of MRSI.- 28.3 Post-Detection MRSI Data Processing.- 28.4 Post-Detection Correction of Frequency-Offset Artifacts.- 28.5 Spectroscopic Maps and Shift-Selective Images.- 29 Gradient-Pulse Moments and Motions.- 29.1 Bipolar Gradient Pulses.- 29.2 Velocity-Compensated Gradient Pulses.- 30 Velocimetry and Velocity Maps.- 30.1 Phase Encoding of the Velocity.- 30.2 Mapping of Velocity Fields.- 30.3 Typical Applications.- 31 Diffusivity Maps.- 32 Resolution.- 32.1 Field of View, Spectral Width, Velocity Range.- 32.2 Digital Resolution.- 32.3 Physical Resolution Limits.- 32.3.1 Spatial In-Plane Resolution.- 32.3.2 The Sensitivity Limit.- 33 Multi-Stripe/Plane Tagging.- 33.1 DANTE Pulse Combs.- 33.2 Imaging Pulse Scheme and Applications.- 34 Rotating-Frame Imaging.- 34.1 Nutation Frequency Encoding.- 34.2 Multi-Dimensional Representations.- 34.3 Rapid Rotating-Frame Imaging.- 35 Imaging of Solid Samples.- 35.1 Experimental Strategies of Materials Imaging.- 35.1.1 The Reservoir of Contrast Parameters.- 35.1.2 The Spatial-Resolution Problem.- 35.2 Magic- and Mixed-Echo Phase-Encoding Imaging.- 35.3 Magnetic Resonance Force Microscopy.- 36 Slice-Selective Homonuclear Spin-Locking.- 36.1 Review of Slice-Selection Principles.- 36.2 Theory of Slice Selection by Spin-Locking.- 36.2.1 Liquid-State Limit.- 36.2.2 Solid-State Limit.- 36.3 Variation of the Slice Width and Position.- 37 Homonuclear Localized NMR.- 37.1 The Homonuclear VOSY/VOSING Family.- 37.1.1 Double-Quantum Volume-Selective Spectral Editing.- 37.1.2 Cyclic Polarization Transfer Volume-Selective Spectral Editing.- 37.1.3 Volume-Selective Relaxometry, Diffusometry, and Velocimetry.- 37.2 The Homonuclear LOSY Pulse Sequence.- 38 Cross-Polarization Principles.- 38.1 Categorization of Cross-Polarization Techniques.- 38.2 Spatially Selective HH-Matching.- 38.3 Adiabatic J Cross-Polarization.- 38.3.1 Adiabatic Level-Crossing Condition.- 38.3.2 The Principle of Adiabatic J Cross-Polarization.- 39 Single-Transition Operator Theory of Cross-Polarization.- 39.1 Weakly Coupled AX Spin Systems.- 39.1.1 Laboratory-Frame Hamiltonian.- 39.1.2 Transformation to the Tilted Doubly-Rotating Frame.- 39.1.3 Single-Transition Operator Representation.- 39.1.4 Solution of the Liouville/von Neumann Equation.- 39.1.5 The Cross-Polarized Magnetizations.- 39.1.6 HH-Matched Resonant Cross-Polarization.- 39.1.7 HH-Mismatch Losses of Resonant Cross-Polarization (VJCP).- 39.1.8 Off-Resonance Losses of HH Cross-Polarization (JCP-LOSY).- 39.1.9 Resonant Adiabatic J Cross-Polarization (AJCP).- 39.1.10 Adiabatic J Cross Polarization Localized Spectroscopy (AJCP-LOSY).- 39.2 From Two-Spin to Multi-Spin Systems and Solids.- 40 Proton-Detected Localized 13C NMR.- 40.1 Heteronuclear MQF-VOSING Spectroscopy.- 40.2 Cyclic Cross-Polarization Localized Spectroscopy (CYCLCROP-LOSY).- 41 Heteronuclear Imaging.- 41.1 Proton-Detected 13C Imaging.- 41.1.1 Multiple-Quantum Edited Hydrocarbon Maps.- 41.1.2 Cross-Polarization Edited Hydrocarbon Maps.- IV: Analytical NMR Toolbox.- 42 Miscellaneous Formulae and Rules.- 42.1 Some Algebraic Symbols.- 42.2 The Delta Function.- 42.3 Fourier Transforms.- 42.4 Spherical Harmonics.- 42.5 Classification of Operators.- 42.6 Spin-Operator Relations.- 42.6.1 Spin-Operator Representations.- 42.6.2 Ladder Operators.- 42.6.3 Spherical Spin Operators.- 42.6.4 Relations for Spin-1/2 Operators.- 42.6.5 Relations for Spin-1 Operators.- 42.6.6 Two-Spin Systems.- 42.6.7 Single-Transition Operators.- 42.6.8 An Instructive Exarhple.- 43 Rules for Traces.- 44 Commutator Algebra.- 44.1 General Operators.- 44.2 Spin Operators.- 45 Exponential and Trigonometric Operators.- 46 Spin Hamiltonians.- 46.1 Zeeman Interaction.- 46.2 RF Irradiation.- 46.3 Internal Spin Interactions.- 46.3.1 Chemical-Shift Interaction.- 46.3.2 Dipolar, Scalar, and Indirect Couplings.- 46.3.3 Quadrupolar and Spin-Rotation Couplings.- 47 The Density Operator.- 47.1 Definition of the Density Operator.- 47.2 Thermal Equilibrium.- 47.3 Evolution of the Density Operator.- 47.3.1 Segmented Treatments.- 47.3.2 Average Hamiltonians.- 48 Unitary Transformations in NMR.- 48.1 Diagonalization of a Matrix.- 48.1.1 The Eigenvalue Problem.- 48.1.2 Transformation Matrices.- 48.1.3 The Inverse Matrix.- 48.2 Coordinate Systems.- 48.3 Euler Angles.- 48.4 Transformation of the Chemical-Shift Tensor.- 48.5 Transformations by Single-Spin Operators.- 48.5.1 Transformation to the Rotating Frame.- 48.5.2 RF Pulses and Unitary Transformations.- 48.5.3 Precession.- 48.6 Transformations by Bilinear Spin-1/2 Operators.- 48.7 Equations of Motion in the Rotating Frame.- 48.7.1 Classical Precession Equation.- 48.7.2 Liouville/von Neumann Equation.- 48.7.3 Time Dependent Schrödinger Equation.- 48.7.4 Heisenberg Equation.- 48.8 Rotating-Frame Hamilton Operators.- 48.9 Dipolar Hamiltonian in the Tilted Rotating Frame.- 48.10 Quadrupolar Hamiltonian in the Tilted Rotating Frame.- 48.11 Spin-Spin Coupling in the Doubly-Rotating Frame.- 49 Irreducible Spherical Tensor Operators.- 49.1 Rotational Transformation of IST Operators.- 49.2 Commutation of IST and Spin Operators.- 49.3 Analytical Form of IST Operators.- 49.4 Wigner/Eckart Theorem.- 49.5 Selection Rules for Stationary Nuclear Moments.- 49.6 IST Representation of the Quadrupolar Hamiltonian.- 50 Derivation of Basic NMR Spectra.- 50.1 Pake Spectrum.- 50.1.1 Dipolar Coupling.- 50.1.2 Quadrupolar Coupling.- 50.2 Chemical-Shift Anisotropy Spectrum.- 50.3 A2, AB, and AX Spectra.- 51 Product Operators for Spins.- 51.1 Simple Example of a Product Operator Basis Set.- 51.2 Orthogonality.- 51.3 Matrix Interpretation.- 51.3.1 Example 1: In-phase Single-quantum Coherences.- 51.3.2 Example 2: Multiple-quantum Coherences.- 51.3.3 Example 3: Longitudinal Scalar Order.- 51.3.4 Summary of the Interpretations of Cartesian Product Operators.- 51.4 Hamiltonians and Applicability Limits.- 51.5 Evolution Rules for Cartesian Product Operators.- 51.6 Evolution Rules for Spherical Product Operators.- 52 Spin Operators for I = 1 Quadrupole Nuclei.- References.