Preface ....................................................... xxi
Preface to the First Edition ................................ xxiii
Introduction .................................................... l
Part l Nuclear Magnetism ....................................... 3
1 Matter ....................................................... 5
1.1 Atoms and Nuclei ........................................ 5
1.2 Spin .................................................... 5
1.2.1 Classical angular momentum ....................... 6
1.2.2 Quantum angular momentum ......................... 6
1.2.3 Spin angular momentum ............................ 7
1.2.4 Combining angular momenta ........................ 8
1.2.5 The Pauli Principle .............................. 9
1.3 Nuclei .................................................. 9
1.3.1 The fundamental particles ........................ 9
1.3.2 Neutrons and protons ............................ 10
1.3.3 Isotopes ........................................ 11
1.4 Nuclear Spin ........................................... 12
1.4.1 Nuclear spin states ............................. 12
1.4.2 Nuclear Zeeman splitting ........................ 14
1.4.3 Zero-spin nuclei ................................ 14
1.4.4 Spin-1/2 nuclei ................................. 15
1.4.5 Quadrupolar nuclei with integer spin ............ 15
1.4.6 Quadrupolar nuclei with half-integer spin ........ 15
1.5 Atomic and Molecular Structure ......................... 15
1.5.1 Atoms ........................................... 15
1.5.2 Molecules ....................................... 16
1.6 States of Matter ....................................... 17
1.6.1 Gases ........................................... 17
1.6.2 Liquids ......................................... 17
1.6.3 Solids .......................................... 19
2 Magnetism ................................................... 23
2.1 The Electromagnetic Field .............................. 23
2.2 Macroscopic Magnetism .................................. 23
2.3 Microscopic Magnetism .................................. 25
2.4 Spin Precession ........................................ 26
2.5 Larmor Frequency ....................................... 29
2.6 Spin-Lattice Relaxation: Nuclear Paramagnetism ......... 30
2.7 Transverse Magnetization and Transverse Relaxation ..... 33
2.8 NMR Signal ............................................. 36
2.9 Electronic Magnetism ................................... 36
3 NMR Spectroscopy ............................................ 39
3.1 A Simple Pulse Sequence ................................ 39
3.2 A Simple Spectrum ...................................... 39
3.3 Isotopomeric Spectra ................................... 42
3.4 Relative Spectral Frequencies: Case of Positive
Gyromagnetic Ratio ..................................... 44
3.5 Relative Spectral Frequencies: Case of Negative
Gyromagnetic Ratio ..................................... 46
3.6 Inhomogeneous Broadening ............................... 48
3.7 Chemical Shifts ........................................ 50
3.8 J-Coupling Multiplets .................................. 56
3.9 Heteronuclear Decoupling ............................... 59
Part 2 The NMR Experiment ..................................... 63
4 The NMR Spectrometer ........................................ 65
4.1 The Magnet ............................................. 65
4.2 The Transmitter Section ................................ 66
4.2.1 The synthesizer: radio-frequency phase shifts ... 67
4.2.2 The pulse gate: radio-frequency pulses .......... 68
4.2.3 Radio-frequency amplifier ....................... 69
4.3 The Duplexer ........................................... 69
4.4 The Probe .............................................. 70
4.5 The Receiver Section ................................... 72
4.5.1 Signal preamplifier ............................. 73
4.5.2 The quadrature receiver ......................... 73
4.5.3 Analogue-digital conversion ..................... 74
4.5.4 Signal phase shifting ........................... 76
4.6 Overview of the Radio-Frequency Section ................ 76
4.7 Pulsed Field Gradients ................................. 77
4.7.1 Magnetic field gradients ........................ 78
4.7.2 Field gradient coils ............................ 79
4.7.3 Field gradient control .......................... 80
5 Fourier Transform NMR ....................................... 85
5.1 A Single-Pulse Experiment .............................. 85
5.2 Signal Averaging ....................................... 86
5.3 Multiple-Pulse Experiments: Phase Cycling .............. 89
5.4 Heteronuclear Experiments .............................. 90
5.5 Pulsed Field Gradient Sequences ........................ 91
5.6 Arrayed Experiments .................................... 91
5.7 NMR Signal ............................................. 93
5.8 NMR Spectrum ........................................... 96
5.8.1 Fourier transformation .......................... 96
5.8.2 Lorentzians ..................................... 96
5.8.3 Explanation of Fourier transformation .......... 100
5.8.4 Spectral phase shifts .......................... 102
5.8.5 Frequency-dependent phase correction ........... 103
5.9 Two-Dimensional Spectroscopy .......................... 105
5.9.1 Two-dimensional signal surface ................. 105
5.9.2 Two-dimensional Fourier transformation ......... 105
5.9.3 Phase twist peaks .............................. 107
5.9.4 Pure absorption two-dimensional spectra ........ 109
5.10 Three-Dimensional Spectroscopy ........................ 114
Part 3 Quantum Mechanics ..................................... 119
6 Mathematical Techniques .................................... 121
6.1 Functions ............................................. 121
6.1.1 Continuous functions ........................... 121
6.1.2 Normalization .................................. 122
6.1.3 Orthogonal and orthonormal functions ........... 122
6.1.4 Dirac notation ................................. 122
6.1.5 Vector representation of functions ............. 123
6.2 Operators ............................................. 125
6.2.1 Commutation .................................... 126
6.2.2 Matrix representations ......................... 126
6.2.3 Diagonal matrices .............................. 129
6.2.4 Block diagonal matrices ........................ 129
6.2.5 Inverse ........................................ 130
6.2.6 Adjoint ........................................ 130
6.2.7 Hermitian operators ............................ 131
6.2.8 Unitary operators .............................. 131
6.3 Eigenfunctions, Eigenvalues and Eigenvectors .......... 131
6.3.1 Eigenequations ................................. 131
6.3.2 Degeneracy ..................................... 131
6.3.3 Eigenfunctions and eigenvalues of Hermitian
operators ...................................... 132
6.3.4 Eigenfunctions of commuting operators: non-
degenerate case ................................ 132
6.3.5 Eigenfunctions of commuting operators:
degenerate case ................................ 132
6.3.6 Eigenfunctions of commuting operators:
summary ........................................ 133
6.3.7 Eigenvectors ........................................... 134
6.4 Diagonalization ....................................... 134
6.4.1 Diagonalization of Hermitian or unitary
matrices ....................................... 135
6.5 Exponential Operators ................................. 135
6.5.1 Powers of operators ............................ 135
6.5.2 Exponentials of operators ...................... 136
6.5.3 Exponentials of unity and null operators ....... 136
6.5.4 Products of exponential operators .............. 137
6.5.5 Inverses of exponential operators .............. 137
6.5.6 Complex exponentials of operators .............. 137
6.5.7 Exponentials of small operators ................ 137
6.5.8 Matrix representations of exponential
operators ...................................... 138
6.6 Cyclic Commutation .................................... 138
6.6.1 Definition of cyclic commutation ............... 138
6.6.2 Sandwich formula ............................... 139
7. Review of Quantum Mechanics ................................ 143
7.1 Spinless Quantum Mechanics ............................ 143
7.1.1 The state of the particle ...................... 143
7.1.2 The equation of motion ......................... 144
7.1.3 Experimental observations ...................... 144
7.2 Energy Levels ......................................... 145
7.3 Natural Units ......................................... 146
7.4 Superposition States and Stationary States ............ 147
7.5 Conservation Laws ..................................... 148
7.6 Angular Momentum ...................................... 148
7.6.1 Angular momentum operators ..................... 149
7.6.2 Rotation operators ............................. 149
7.6.3 Rotation sandwiches ............................ 151
7.6.4 Angular momentum eigenstates and eigenvalues ... 152
7.6.5 The angular momentum eigenstates ............... 154
7.6.6 Shift operators ................................ 154
7.6.7 Matrix representations of the angular
momentum operators ............................. 156
7.7 Spin .................................................. 157
7.7.1 Spin angular momentum operators ................ 157
7.7.2 Spin rotation operators ........................ 158
7.7.3 Spin Zeeman basis .............................. 158
7.7.4 Trace .......................................... 159
7.8 Spin-1/2 .............................................. 160
7.8.1 Zeeman eigenstates ............................. 160
7.8.2 Angular momentum operators ..................... 160
7.8.3 Spin-1/2 rotation operators .................... 160
7.8.4 Unity operator ................................. 161
7.8.5 Shift operators ................................ 161
7.8.6 Projection operators ........................... 161
7.8.7 Ket-bra notation 162
7.9 Higher Spin ........................................... 162
7.9.1 Spin I = 1 ..................................... 163
7.9.2 Spin I = 3/2 ................................... 164
7.9.3 Higher spins ................................... 165
Part 4 Nuclear Spin Interactions ............................. 169
8 Nuclear Spin Hamiltonian ................................... 171
8.1 Spin Hamiltonian Hypothesis ........................... 171
8.2 Electromagnetic Interactions .......................... 172
8.2.1 Electric spin Hamiltonian ...................... 173
8.2.2 Magnetic spin interactions ..................... 176
8.3 External and Internal Spin Interactions ............... 177
8.3.1 Spin interactions: summary ..................... 177
8.4 External Magnetic Fields .............................. 177
8.4.1 Static field ................................... 179
8.4.2 Radio-frequency field .......................... 179
8.4.3 Gradient field ................................. 181
8.4.4 External spin interactions: summary ............ 181
8.5 Internal Spin Hamiltonian ............................. 182
8.5.1 The internal spin interactions ................. 182
8.5.2 Simplification of the internal Hamiltonian ..... 185
8.6 Motional Averaging .................................... 186
8.6.1 Modes of molecular motion ...................... 186
8.6.2 Molecular rotations ............................ 186
8.6.3 Molecular translations ......................... 187
8.6.4 Intramolecular and intermolecular spin
interactions ................................... 189
8.6.5 Summary of motional averaging .................. 190
9 Internal Spin Interactions ................................. 195
9.1 Chemical Shift ........................................ 195
9.1.1 Chemical shift tensor .......................... 196
9.1.2 Principal axes ................................. 197
9.1.3 Principal values ............................... 198
9.1.4 Isotropic chemical shift ....................... 198
9.1.5 Chemical shift anisotropy (CSA) ................ 198
9.1.6 Chemical shift for an arbitrary molecular
orientation .................................... 200
9.1.7 Chemical shift frequency ....................... 201
9.1.8 Chemical shift interaction in isotropic
liquids ........................................ 201
9.1.9 Chemical shift interaction in anisotropic
liquids ........................................ 203
9.1.10 Chemical shift interaction in solids ........... 204
9.1.11 Chemical shift interaction: summary ............ 206
9.2 Electric Quadrupole Coupling .......................... 206
9.2.1 Electric field gradient tensor ................. 207
9.2.2 Nuclear quadrupole Hamiltonian ................. 208
9.2.3 Isotropic liquids .............................. 209
9.2.4 Anisotropic liquids ............................ 209
9.2.5 Solids ......................................... 210
9.2.6 Quadrupole interaction: summary ................ 210
9.3 Direct Dipole-Dipole Coupling ......................... 211
9.3.1 Secular dipole-dipole coupling ................. 213
9.3.2 Dipole-dipole coupling in isotropic liquids .... 215
9.3.3 Dipole-dipole coupling in liquid crystals ...... 216
9.3.4 Dipole-dipole coupling in solids ............... 216
9.3.5 Dipole-dipole interaction: summary ............. 217
9.4 J-Coupling ............................................ 217
9.4.1 Isotropic J-coupling ........................... 219
9.4.2 Liquid crystals and solids ..................... 221
9.4.3 Mechanism of the J-coupling .................... 222
9.4.4 7-coupling: summary ............................ 223
9.5 Spin-Rotation Interaction ............................. 223
9.6 Summary of the Spin Hamiltonian Terms ................. 224
Part 5 Uncoupled Spins ....................................... 229
10 Single Spin-1/2 ............................................ 231
10.1 Zeeman Eigenstates .................................... 231
10.2 Measurement of Angular Momentum: Quantum
Indeterminacy ......................................... 232
10.3 Energy Levels ......................................... 233
10.4 Superposition States .................................. 234
10.4.1 General spin states ............................ 234
10.4.2 Vector notation ................................ 234
10.4.3 Some particular states ......................... 235
10.4.4 Phase factors .................................. 237
10.5 Spin Precession ....................................... 238
10.5.1 Dynamics of the eigenstates .................... 239
10.5.2 Dynamics of the superposition states ........... 240
10.6 Rotating Frame ........................................ 241
10.7 Precession in the Rotating Frame ...................... 245
10.8 Radio-frequency Pulse ................................. 247
10.8.1 Rotating-frame Hamiltonian ..................... 247
10.8.2 x-pulse ........................................ 248
10.8.3 Nutation ....................................... 251
10.8.4 Pulse of general phase ......................... 252
10.8.5 Off-resonance effects .......................... 253
11 Ensemble of Spins-1/2 ...................................... 259
11.1 Spin Density Operator ................................. 259
11.2 Populations and Coherences ............................ 261
11.2.1 Density matrix ................................. 261
11.2.2 Box notation ................................... 261
11.2.3 Balls and arrows ............................... 262
11.2.4 Orders of coherence ............................ 263
11.2.5 Relationships between populations and
coherences ..................................... 263
11.2.6 Physical interpretation of the populations ..... 264
11.2.7 Physical interpretation of the coherences ...... 265
11.3 Thermal Equilibrium ................................... 266
11.4 Rotating-Frame Density Operator ....................... 268
11.5 Magnetization Vector .................................. 269
11.6 Strong Radio-Frequency Pulse .......................... 270
11.6.1 Excitation of coherence ........................ 271
11.6.2 Population inversion ........................... 273
11.6.3 Cycle of states ................................ 274
11.6.4 Stimulated absorption and emission ............. 275
11.7 Free Precession Without Relaxation .................... 276
11.8 Operator Transformations .............................. 279
11.8.1 Pulse of phase Øр = 0 .......................... 279
11.8.2 Pulse of phase Øр = π/2 ........................ 279
11.8.3 Pulse of phase Øр = π .......................... 279
11.8.4 Pulse of phase Øр = Зπ/2 ....................... 279
11.8.5 Pulse of general phase Øр ...................... 280
11.8.6 Free precession for an interval r .............. 280
11.9 Free Evolution with Relaxation ........................ 281
11.9.1 Transverse relaxation .......................... 281
11.9.2 Longitudinal relaxation ........................ 283
11.10 Magnetization Vector Trajectories .................... 285
11.11 NMR Signal and NMR Spectrum .......................... 287
11.12 Single-Pulse Spectra ................................. 289
12 Experiments on Non-Interacting Spins-1/2 ................... 295
12.1 Inversion Recovery: Measurement of T1 ................. 295
12.2 Spin Echoes: Measurement of Т2 ........................ 298
12.2.1 Homogenous and inhomogenenous broadening ....... 298
12.2.2 Inhomogenenous broadening in the time domain ... 299
12.2.3 Spin echo pulse sequence ....................... 299
12.2.4 Refocusing ..................................... 302
12.2.5 Coherence interpretation ....................... 303
12.2.6 Coherence transfer pathway ..................... 305
12.3 Spin Locking: Measurement of T1p ...................... 305
12.4 Gradient Echoes ....................................... 306
12.5 Slice Selection ....................................... 307
12.6 NMR Imaging ........................................... 309
13 Quadrupolar Nuclei ......................................... 319
13.1 Spin I = 1 ............................................ 319
13.1.1 Spin-1 states .................................. 319
13.1.2 Spin-1 energy levels ........................... 320
13.1.3 Spin-1 density matrix .......................... 321
13.1.4 Coherence evolution ............................ 323
13.1.5 Observable coherences and NMR spectrum ......... 325
13.1.6 Thermal equilibrium ............................ 326
13.1.7 Strong radio-frequency pulse ................... 326
13.1.8 Excitation of coherence ........................ 328
13.1.9 NMR spectrum ................................... 328
13.1.10 Quadrupolar echo .............................. 331
13.2 Spin I = 3/2 ......................................... 334
13.2.1 Spin-3/2 energy levels ......................... 335
13.2.2 Populations and coherences ..................... 336
13.2.3 NMR signal ..................................... 338
13.2.4 Single pulse spectrum .......................... 339
13.2.5 Spin-3/2 spectra for small quadrupole
couplings ...................................... 341
13.2.6 Second-order quadrupole couplings .............. 342
13.2.7 Central transition excitation .................. 343
13.2.8 Central transition echo ........................ 345
13.3 Spin I = 5/2 .......................................... 345
13.4 Spins I = 7/2 ......................................... 349
13.5 Spins I = 9/2 ......................................... 350
Part 6 Coupled Spins ......................................... 353
14 Spin-1/2 Pairs ............................................. 355
14.1 Coupling Regimes ...................................... 355
14.2 Zeeman Product States and Superposition States ........ 356
14.3 Spin-Pair Hamiltonian ................................. 357
14.4 Pairs of Magnetically Equivalent Spins ................ 359
14.4.1 Singlets and triplets .......................... 359
14.4.2 Energy levels .................................. 360
14.4.3 NMR spectra .................................... 362
14.4.4 Dipolar echo ................................... 363
14.5 Weakly Coupled Spin Pairs ............................. 363
14.5.1 Weak coupling .................................. 363
14.5.2 AX spin systems ................................ 364
14.5.3 Energy levels .................................. 364
14.5.4 AX spectrum .................................... 365
14.5.5 Heteronuclear spin pairs ....................... 366
15 Homonuclear AX System ...................................... 369
15.1 Eigenstates and Energy Levels ......................... 369
15.2 Density Operator ...................................... 370
15.3 Rotating Frame ........................................ 375
15.4 Free Evolution ........................................ 376
15.4.1 Evolution of a spin pair ....................... 376
15.4.2 Evolution of the coherences .................... 377
15.5 Spectrum of the AX System: Spin-Spin Splitting ........ 378
15.6 Product Operators ..................................... 381
15.6.1 Construction of product operators .............. 382
15.6.2 Populations and coherences ..................... 383
15.6.3 Spin orientations .............................. 386
15.7 Thermal Equilibrium ................................... 389
15.8 Radio-Frequency Pulses ................................ 391
15.8.1 Rotations of a single spin pair ................ 392
15.8.2 Rotations of the spin density operator ......... 393
15.8.3 Operator transformations ....................... 395
15.9 Free Evolution of the Product Operators ............... 397
15.9.1 Chemical shift evolution ....................... 399
15.9.2 J-coupling evolution ........................... 400
15.9.3 Relaxation ..................................... 405
15.10 Spin Echo Sandwich ................................... 405
16 Experiments on AX Systems .................................. 409
16.1 COSY .................................................. 409
16.1.1 The assignment problem ......................... 409
16.1.2 COSY pulse sequence ............................ 411
16.1.3 Theory of COSY, coherence interpretation ....... 411
16.1.4 Product operator interpretation ................ 415
16.1.5 Experimental examples .......................... 418
16.2 INADEQUATE ............................................ 418
16.2.1 13C isotopomers ................................ 418
16.2.2 Pulse sequence ................................. 423
16.2.3 Theory of INADEQUATE ........................... 424
16.2.4 Coherence transfer pathways and phase
cycling ........................................ 429
16.2.5 Two-dimensional INADEQUATE ..................... 431
16.3 INEPT ................................................. 436
16.3.1 The sensitivity of nuclear isotopes ............ 436
16.3.2 INEPT pulse sequence ........................... 437
16.3.3 Refocused INEPT ................................ 440
16.4 Residual Dipolar Couplings ............................ 443
16.4.1 Angular information ............................ 443
16.4.2 Spin Hamiltonian ............................... 443
16.4.3 Orienting media ................................ 444
16.4.4 Doublet splittings ............................. 446
17 Many-Spin Systems .......................................... 453
17.1 Molecular Spin System ................................. 453
17.2 Spin Ensemble ......................................... 454
17.3 Motionally Suppressed J-Couplings ..................... 454
17.4 Chemical Equivalence .................................. 455
17.5 Magnetic Equivalence .................................. 458
17.6 Weak Coupling ......................................... 461
17.7 Heteronuclear Spin Systems ............................ 462
17.8 Alphabet Notation ..................................... 463
17.9 Spin Coupling Topologies .............................. 464
18 Many-Spin Dynamics ......................................... 467
18.1 Spin Hamiltonian ...................................... 467
18.2 Energy Eigenstates .................................... 468
18.3 Superposition States .................................. 469
18.4 Spin Density Operator ................................. 470
18.5 Populations and Coherences ............................ 471
18.5.1 Coherence orders ............................... 471
18.5.2 Combination coherences and simple coherences ... 471
18.5.3 Coherence frequencies .......................... 472
18.5.4 Degenerate coherences .......................... 473
18.5.5 Observable coherences .......................... 473
18.6 NMR Spectra ........................................... 475
18.7 Many-Spin Product Operators ........................... 477
18.7.1 Construction of product operators .............. 477
18.7.2 Populations and coherences ..................... 478
18.7.3 Physical interpretation of product operators ... 480
18.8 Thermal Equilibrium ................................... 481
18.9 Radio-Frequency Pulses ................................ 481
18.10 Free Precession ...................................... 482
18.10.1 Chemical shift evolution ...................... 482
18.10.2 J-coupling evolution .......................... 483
18.10.3 Relaxation .................................... 485
18.11 Spin Echo Sandwiches ................................. 485
18.12 INEPT in an I2S System ............................... 488
18.13 COSY in Multiple-Spin Systems ........................ 491
18.13.1 AMX spectrum .................................. 492
18.13.2 Active and passive spins ...................... 493
18.13.3 Cross-peak multiplets ......................... 494
18.13.4 Diagonal peaks ................................ 496
18.13.5 Linear spin systems ........................... 497
18.14 TOCSY ................................................ 497
18.14.1 The ambiguity of COSY spectra ................. 497
18.14.2 TOCSY pulse sequence .......................... 499
18.14.3 Theory of TOCSY ............................... 499
Part 7 Motion and Relaxation ................................. 507
19 Motion ..................................................... 509
19.1 Motional Processes .................................... 509
19.1.1 Molecular vibrations ........................... 509
19.1.2 Local rotations of molecular groups ............ 510
19.1.3 Molecular flexibility .......................... 510
19.1.4 Chemical exchange .............................. 510
19.1.5 Molecular rotations ............................ 511
19.1.6 Translational motion ........................... 512
19.1.7 Mechanical motion .............................. 513
19.2 Motional Time-Scales .................................. 513
19.3 Motional Effects ...................................... 514
19.4 Motional Averaging .................................... 515
19.5 Motional Lineshapes and Two-Site Exchange ............. 516
19.2 Slow intermediate exchange and motional broadening .... 518
19.5.2 Fast intermediate exchange and motional
narrowing ...................................... 520
19.5.3 Averaging of J-splittings ...................... 523
19.5.4 Asymmetric two-site exchange ................... 524
19.5.5 Knight shift ................................... 525
19.5.6 Paramagnetic shifts ............................ 527
19.6 Sample Spinning ....................................... 527
19.7 Longitudinal Magnetization Exchange ................... 529
19.7.1 Two-dimensional exchange spectroscopy .......... 529
19.7.2 Theory ......................................... 532
19.7.3 Motional regimes ............................... 539
19.8 Diffusion ............................................. 539
20 Relaxation ................................................. 543
20.1 Types of Relaxation ................................... 543
20.2 Relaxation Mechanisms ................................. 543
20.3 Random Field Relaxation ............................... 545
20.3.1 Autocorrelation functions and correlation
times .......................................... 545
20.3.2 Spectral density ............................... 548
20.3.3 Normalized spectral density .................... 549
20.3.4 Transition probabilities ....................... 550
20.3.5 Thermally corrected transition probabilities ... 551
20.3.6 Spin-lattice relaxation ........................ 552
20.4 Dipole-Dipole Relaxation .............................. 556
20.4.1 Rotational correlation time .................... 556
20.4.2 Transition probabilities ....................... 557
20.4.3 Solomon equations .............................. 561
20.4.4 Longitudinal relaxation ........................ 564
20.4.5 Transverse relaxation .......................... 565
20.5 Steady-State Nuclear Overhauser Effect ................ 566
20.6 NOESY ................................................. 570
20.6.1 NOESY pulse sequence ........................... 570
20.6.2 NOESY signal ................................... 570
20.6.3 NOESY spectra .................................. 573
20.6.4 NOESY and chemical exchange .................... 575
20.6.5 Molecular structure determination .............. 576
20.7 ROESY ................................................. 577
20.7.1 Transverse cross-relaxation .................... 577
20.7.2 Spin locking ................................... 578
20.7.3 Transverse Solomon equations ................... 578
20.7.4 ROESY spectra .................................. 580
20.7.5 ROESY and chemical exchange .................... 582
20.7.6 ROESY and TOCSY ................................ 583
20.8 Cross-Correlated Relaxation ........................... 584
20.8.1 Cross-correlation .............................. 584
20.8.2 Cross-correlation of spin interactions ......... 585
20.8.3 Dipole-dipole cross-correlation and angular
estimations .................................... 586
20.8.4 TROSY .......................................... 590
Part 8 Appendices ............................................. 597
Appendix A: Supplementary Material ............................ 599
A.l 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.l.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.ll.l 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.11.10 Selection of a single pathway II .............. 639
A.11.11 Dual pathway selection ........................ 640
A.11.12 Internal phases I ............................. 641
A.ll.13 Internal phases II ............................ 642
A.11.14 Nested phase cycles I ......................... 644
A.11.15 Nested phase cycles II ........................ 645
A.11.16 Different ways of constructing phase cycles ... 648
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 Exercis ........................................ 693
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