Table of contents for Understanding NMR spectroscopy / James Keeler.

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Contents
1 What this book is about and who should read it 1
1.1 How this book is organized 2
1.2 Scope and limitations 3
1.3 Context and further reading 3
1.4 On-line resources 4
1.5 Abbreviations 4
2 Setting the scene 5
2.1 NMR frequencies and chemical shifts 5
2.1.1 Chemical shift scales 6
2.1.2 Conversion from shifts to frequencies 7
2.1.3 The receiver reference frequency and the
o.set frequency 8
2.2 Linewidths, lineshapes and integrals 9
2.3 Scalar coupling 10
2.3.1 Tree diagrams 10
2.4 Weak and strong coupling 12
2.5 The basic NMR experiment 14
2.5.1 Heteronuclear NMR and broadband decoupling 15
2.6 Frequency, oscillations and rotations 17
2.6.1 Motion in a circle 17
2.6.2 Frequency 17
2.6.3 Angular frequency 18
2.6.4 Phase 19
2.6.5 Representation using complex numbers 21
2.7 Photons 21
2.8 Moving on 22
2.9 Further reading 22
2.10 Exercises 23
3 Energy levels and NMR spectra 25
3.1 The problem with the energy level approach 26
3.1.1 Wavefunctions and mixed states 26
3.1.2 Energy levels in NMR 27
3.2 Introducing quantum mechanics 28
3.2.1 Wavefunctions 28
3.2.2 Operators 29
3.2.3 Eigenfunctions and eigenvalues of operators 30
3.2.4 Measurement 30
3.2.5 Hamiltonians and angular momentum 31
3.2.6 Eigenfunctions and eigenvalues of ^ Iz 31
3.2.7 Eigenvalues for the one-spin Hamiltonian 32
3.2.8 Summary 33
3.3 The spectrum from one spin 33
3.3.1 Energy levels 33
3.3.2 The Larmor frequency 34
3.3.3 Writing the energies in frequency units 35
3.4 Writing the Hamiltonian in frequency units 36
3.5 The energy levels for two coupled spins 37
3.5.1 Introducing scalar coupling 39
3.6 The spectrum from two coupled spins 40
3.6.1 Multiple quantum transitions 42
3.7 Three spins 43
3.7.1 The Hamiltonian and energy levels 43
3.7.2 Single quantum spectrum 44
3.7.3 Multiple quantum transitions 44
3.7.4 Combination lines 46
3.8 Summary 46
3.9 Further reading 47
3.10 Exercises 48
4 The vector model 51
4.1 The bulk magnetization 51
4.1.1 Axis systems 53
4.1.2 The equilibrium magnetization 53
4.2 Larmor precession 54
4.3 Detection 55
4.4 Pulses 56
4.4.1 Rotating frame 57
4.4.2 Larmor precession in the rotating frame 59
4.4.3 The e.ective field 59
4.4.4 The e.ective field in frequency units 60
4.4.5 Summary 61
4.5 On-resonance pulses 62
4.5.1 Hard pulses 63
4.6 Detection in the rotating frame 64
4.7 The basic pulse-acquire experiment 64
4.7.1 Spectrum with several lines 65
4.8 Pulse calibration 66
4.9 The spin echo 67
4.9.1 180. pulses as refocusing pulses 68
4.9.2 How the spin echo works 68
4.10 Pulses of di.erent phases 70
4.11 O.-resonance e.ects and soft pulses 71
4.11.1 Excitation of a range of shifts 73
4.11.2 Selective excitation 73
4.11.3 Selective inversion 75
4.12 Moving on 75
4.13 Further reading 76
4.14 Exercises 77
5 Fourier transformation and data processing 81
5.1 How the Fourier transform works 82
5.1.1 Mathematical formulation of the Fourier transform 85
5.2 Representing the FID 86
5.3 Lineshapes and phase 87
5.3.1 Absorption and dispersion lineshapes 88
5.3.2 Phase 90
5.3.3 Phase correction 92
5.3.4 Frequency dependent phase errors 92
5.4 Manipulating the FID and the spectrum 94
5.4.1 Noise 94
5.4.2 Sensitivity enhancement 96
5.4.3 The matched filter 98
5.4.4 Resolution enhancement 98
5.4.5 Defining the parameters for sensitivity and
resolution enhancement functions 100
5.4.6 'Lorentz-to-Gauss' transformation 102
5.4.7 Other weighting functions 102
5.5 Zero filling 103
5.6 Truncation 104
5.7 Further reading 105
5.8 Exercises 106
6 The quantum mechanics of one spin 109
6.1 Introduction 109
6.2 Superposition states 110
6.3 Some quantum mechanical tools 111
6.3.1 Dirac notation 111
6.3.2 Normalization and orthogonality 112
6.3.3 Expectation values 113
6.3.4 The x- and y-components of angular momentum 115
6.3.5 Matrix representations 115
6.4 Computing the bulk magnetization 116
6.4.1 The ensemble average 117
6.4.2 Populations 119
6.4.3 Transverse magnetization 120
6.5 Summary 121
6.6 Time evolution 123
6.6.1 Free evolution 123
6.6.2 E.ect of free evolution 126
6.7 RF pulses 127
6.7.1 E.ect on the components of angular momentum 128
6.7.2 E.ect on the components of the bulk magnetization 130
6.8 Making faster progress: the density operator 130
6.8.1 Introducing the density operator 131
6.8.2 Calculating the components of the bulk
magnetization 132
6.8.3 Equilibrium density operator 133
6.8.4 Time evolution of the density operator 133
6.8.5 Representing the density operator using a
basis of operators 134
6.8.6 The equilibrium density operator - again 136
6.8.7 Summary 137
6.9 Coherence 138
6.10 Further reading 140
6.11 Exercises 141
7 Product operators 145
7.1 Operators for one spin 145
7.1.1 Hamiltonians for free precession and pulses 146
7.1.2 Rotations 147
7.2 Analysis of pulse sequences for a one-spin system 149
7.2.1 Pulse-acquire 149
7.2.2 The spin echo 151
7.3 Speeding things up 152
7.3.1 90. and 180. pulses 152
7.3.2 Diagrammatic representation 153
7.3.3 The 1 - 1 sequence 154
7.4 Operators for two spins 155
7.4.1 E.ect of coupling 156
7.5 In-phase and anti-phase terms 158
7.5.1 In-phase terms 159
7.5.2 Anti-phase terms 160
7.5.3 Observable terms 162
7.6 Hamiltonians for two spins 163
7.7 Notation for heteronuclear spin systems 163
7.8 Spin echoes and J-modulation 164
7.8.1 Spin echo in a homonuclear spin system 165
7.8.2 Spectra from a J-modulated spin echo 168
7.8.3 Spin echoes in heteronuclear spin systems 170
7.9 Coherence transfer 172
7.10 The INEPT experiment 173
7.10.1 Why the experiment was developed 173
7.10.2 Analysis of the INEPT experiment 174
7.10.3 Decoupling in the INEPT experiment 176
7.10.4 Suppressing the signal from the equilibrium
magnetization on the S spin 176
7.11 Selective COSY 178
7.12 Coherence order and multiple-quantum coherences 180
7.12.1 Raising and lowering operators: the classification
of coherence order 180
7.12.2 Generation of multiple-quantum coherence 182
7.12.3 Evolution of multiple-quantum coherence 182
7.13 Summary 184
7.14 Further reading 185
7.15 Exercises 186
8 Two-dimensional NMR 189
8.1 The general scheme for two-dimensional NMR 190
8.1.1 How two-dimensional spectra are recorded 191
8.1.2 How the data are processed 191
8.2 Modulation and lineshapes 192
8.2.1 Cosine amplitude modulated data 193
8.2.2 Sine amplitude modulated data 194
8.2.3 Mixed cosine and sine modulation 195
8.3 Axes and frequency scales in two-dimensional spectra 196
8.4 COSY 196
8.4.1 Overall form of the COSY spectrum 197
8.4.2 Detailed form of the two-dimensional multiplets 198
8.4.3 Phase properties of the COSY spectrum 203
8.4.4 How small a coupling can we detect with COSY? 204
8.4.5 The problem with COSY 205
8.5 Double-quantum filtered COSY (DQF COSY) 206
8.6 Double-quantum spectroscopy 210
8.6.1 Detailed analysis of the pulse sequence 211
8.6.2 Interpretation and application of doublequantum
spectra 212
8.6.3 INADEQUATE 213
8.7 Heteronuclear correlation spectra 215
8.7.1 Normal or inverse correlation 215
8.8 HSQC 216
8.8.1 Coupled or decoupled acquisition 218
8.8.2 Suppressing unwanted signals in HSQC 218
8.8.3 Sensitivity 219
8.9 HMQC 219
8.10 Long-range correlation: HMBC 222
8.10.1 Suppressing one-bond peaks in HMBC spectra 224
8.11 HETCOR 227
8.12 TOCSY 228
8.12.1 TOCSY for two spins 229
8.12.2 TOCSY for more extended spin systems 232
8.13 Frequency discrimination and lineshapes 233
8.13.1 Obtaining cosine and sine modulated data 234
8.13.2 P- and N-type selection: phase-twist lineshapes 235
8.13.3 The States-Haberkorn-Ruben method 237
8.13.4 The TPPI or Redfield method 238
8.13.5 The States-TPPI method 241
8.13.6 Phase in two-dimensional spectra 242
8.14 Further reading 244
8.15 Exercises 245
9 Relaxation and the NOE 247
9.1 What is relaxation? 248
9.1.1 Behaviour of individual magnetic moments 249
9.1.2 Local fields 250
9.1.3 Coming to equilibrium with the lattice 252
9.1.4 Transverse relaxation 253
9.1.5 Summary 254
9.2 Relaxation mechanisms 255
9.2.1 The dipolar mechanism 256
9.2.2 Chemical shift anisotropy 256
9.2.3 Relaxation by paramagnetic species 257
9.3 Describing random motion - the correlation time 257
9.3.1 The correlation function 259
9.3.2 The spectral density 262
9.3.3 Motional regimes 264
9.3.4 Transverse relaxation and the spectral density
at zero frequency 265
9.3.5 Summary 265
9.4 Populations 266
9.4.1 The z-magnetization in terms of populations 266
9.4.2 Relaxation in terms of populations 268
9.5 Longitudinal relaxation behaviour of isolated spins 270
9.5.1 Estimating the rate constant for longitudinal
relaxation 272
9.5.2 Making a quick estimate of the relaxation
rate constant 273
9.5.3 How long do I have to leave between experiments? 273
9.6 Longitudinal dipolar relaxation of two spins 275
9.6.1 Energy levels and transition rates 275
9.6.2 Rate equations for the populations and zmagnetizations
276
9.6.3 Relaxation rate constants 279
9.6.4 Cross relaxation in the two motional regimes 280
9.7 The NOE 281
9.7.1 The transient NOE experiment 282
9.7.2 The steady-state NOE experiment 286
9.7.3 Heteronuclear steady-state NOE 287
9.7.4 Two-dimensional NOESY 288
9.7.5 The NOE in more extended spin systems 292
9.8 Transverse relaxation 293
9.8.1 Di.erent contributions to transverse relaxation 295
9.8.2 Relaxation by random fields 295
9.8.3 Transverse dipolar relaxation of two spins 297
9.8.4 Transverse cross relaxation: ROESY 298
9.9 Homogeneous and inhomogeneous broadening 301
9.9.1 Describing inhomogeneous broadening: T_
2 304
9.9.2 Measuring the transverse relaxation rate constant 305
9.10 Relaxation due to chemical shift anisotropy 305
9.10.1 Specifying the CSA 305
9.10.2 Relaxation rate constants due to CSA 306
9.11 Cross correlation 307
9.11.1 Cross correlation in longitudinal relaxation 307
9.11.2 Cross correlation in transverse relaxation 310
9.12 Summary 312
9.13 Further reading 312
9.14 Exercises 314
10 Advanced topics in two-dimensional NMR 321
10.1 Product operators for three spins 322
10.1.1 Interpretation of the product operators for
three spins 323
10.1.2 Evolution due to o.sets and pulses 325
10.1.3 Evolution of couplings 325
10.2 COSY for three spins 327
10.2.1 Structure of the cross-peak multiplets 328
10.3 Reduced multiplets in COSY spectra 332
10.3.1 COSY for a three-spin system containing
one heteronucleus 332
10.3.2 Determining the relative signs of the passive
coupling constants 335
10.3.3 Measuring the size of the passive coupling
constants 336
10.3.4 Reduced multiplets in homonuclear spin systems 338
10.4 Polarization operators 339
10.4.1 Construction and interpretation of polarization
operators 339
10.4.2 Free evolution 340
10.4.3 Pulses 341
10.4.4 Small flip angle COSY 342
10.5 ZCOSY 347
10.6 HMBC 349
10.7 Sensitivity-enhanced experiments 351
10.7.1 Sensitivity-enhanced HSQC 352
10.8 Constant time experiments 356
10.8.1 Constant time COSY 356
10.8.2 Constant time HSQC 359
10.9 TROSY 361
10.9.1 Line-selective transfer 362
10.9.2 Implementation of line-selective 180. pulses 364
10.9.3 A TROSY HSQC sequence 365
10.9.4 Processing the TROSY HSQC spectrum 366
10.10 Further reading 369
10.11 Exercises 370
11 Coherence selection: phase cycling and field gradient pulses 373
11.1 Coherence order 374
11.1.1 Possible values of the overall coherence order 375
11.1.2 Evolution of operators of particular coherence
orders 376
11.1.3 The e.ect of pulses 377
11.1.4 Observables 378
11.2 Coherence transfer pathways 379
11.2.1 Coherence transfer pathways in heteronuclear
experiments 380
11.3 Frequency discrimination and lineshapes in twodimensional
spectra 381
11.4 The receiver phase 383
11.5 Introducing phase cycling 387
11.5.1 Selection of a single pathway 388
11.5.2 Combining phase cycles 391
11.6 Some phase cycling 'tricks' 392
11.6.1 The first pulse 393
11.6.2 Grouping pulses together 393
11.6.3 The final pulse 394
11.6.4 High-order multiple-quantum terms 394
11.6.5 Refocusing pulses 394
11.7 Axial peak suppression 395
11.8 CYCLOPS 396
11.9 Examples of practical phase cycles 397
11.9.1 COSY 397
11.9.2 DQF COSY 398
11.9.3 Double-quantum spectroscopy 399
11.9.4 NOESY 399
11.9.5 HMQC 399
11.10 Concluding remarks concerning phase cycling 400
11.10.1 Summary 400
11.10.2 Deficiencies of phase cycling 400
11.11 Introducing field gradient pulses 401
11.11.1 The spatially dependent phase 403
11.11.2 Selection of a single pathway using two gradients 405
11.11.3 The spatially dependent phase in heteronuclear
systems 406
11.11.4 Shaped gradient pulses 407
11.11.5 Dephasing in a field gradient 407
11.12 Features of selection using gradients 409
11.12.1 Selection of multiple pathways 409
11.12.2 Obtaining absorptionmode lineshapeswhen
gradients are used in t1 409
11.12.3 Refocusing pulses 410
11.12.4 180. pulses in heteronuclear experiments 411
11.12.5 Phase errors due to gradient pulses 411
11.12.6 Selection of z-magnetization 412
11.13 Examples of using gradient pulses for coherence
pathway selection 413
11.13.1 DQF COSY 413
11.13.2 HMQC 414
11.13.3 HSQC 417
11.14 Advantages and disadvantages of coherence selection
with gradients 418
11.15 Suppression of zero-quantum coherence 419
11.15.1 The z-filter 420
11.15.2 Implementation of z-filters in two-dimensional
experiments 422
11.15.3 Zero-quantum dephasing 422
11.16 Selective excitation with the aid of gradients 424
11.16.1 The double pulsed field gradient spin echo 425
11.16.2 The DPFGSE NOE experiment 425
11.17 Further reading 427
11.18 Exercises 429
12 How the spectrometer works 433
12.1 The magnet 433
12.1.1 Shims 434
12.1.2 The lock 435
12.2 The probe 436
12.3 The transmitter 436
12.3.1 Power levels and 'dB' 437
12.4 The receiver 438
12.5 Digitizing the signal 439
12.5.1 The analogue to digital converter 439
12.5.2 Sampling rates 440
12.5.3 Mixing down to a lower frequency 441
12.6 Quadrature detection 441
12.7 The pulse programmer 442
12.8 Further reading 443
12.9 Exercises 444
A Some mathematical topics 445
A.1 The exponential and logarithms 445
A.2 Complex numbers 447
A.2.1 The complex exponential 448
A.3 Trigonometric identities 449
A.4 Further reading 450
Index 451

Library of Congress Subject Headings for this publication:

Nuclear magnetic resonance spectroscopy -- Textbooks.