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Contents •xv
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
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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
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
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Contents •xvii
19.5.1 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 TRO