SPIN DYNAMICSBasics of Nuclear Magnetic Res

öinance

Malcolm H.Levitt

Preface

xxi

Introduction

i

Part 1 Nuclear Magnetism

3

1

Matter

5

1 .1 Properties of Nuclei

51.2 Spin

61 .2 .1 Classical angular momentum

61 .2 .2 Quantum angular momentum

61 .2 .3 Spin angular momentum

7

1 .2 .4 Combining angular momenta

9

1 .2.5 The Pauli principle

1 0

1 .3 Atomic and Molecular Structure

101 .3.1 The fundamental particles

10

1 .3.2 Neutrons and protons

1 1

1 .3.3 The atomic nucleus

121 .3.4 Atoms

15

1 .3.5 Molecules

16

1 .4 States of Matter

17

1 .4.1 Gases

17

1 .4.2 Liquids

18

1 .4.3 Solids

19

Notes

20Further Reading

21

Exercises

21

2

Magnetism

23

2.1 The Electromagnetic Field

23

2.2 Macroscopic Magnetism

23

2 .3 Microscopic Magnetism

25

2.4 Spin Precession

27

2 .5 Larmor Frequency

3 0

2 .6 Spin-Lattice Relaxation: Nuclear Paramagnetism

3 2

2 .7 Transverse Magnetization and Transverse Relaxation

3 6

2 .8 NMR Signal

39

2 .9 Electronic Magnetism

39

Notes

4 1

Further Reading

4 1

Exercises

42

3

NMR Spectroscopy

43

3.1 A Simple Pulse Sequence

43

3.2 A Simple Spectrum

43

3.3 Isotopomeric Spectra

47

3.4 Relative Spectral Frequencies - Case of Positive Gamma

49

3.5 Relative Spectral Frequencies - Case of Negative Gamma

5 1

3 .6 Inhomogeneous Broadening

53

3.7 Chemical Shifts

56

3 .8 Multiplet Structure

61

3.9 Heteronuclear Decoupling

65

Notes

67Further Reading

67

Exercises

67

Part 2 The NMR Experiment

69

4

The NMR Spectrometer

71

4.1 The Magnet

714.2 The Transmitter Section

734 .2 .1 The synthesizer: r .f. phase shifts

734 .2 .2 The pulse gate. r.f. pulses

744 .2.3 R.f. amplifier

764.3 The Duplexer

7 64.4 The Probe

774.5 The Receiver Section

8 04 .5 .1 Signal preamplifier

804 .5 .2 The quadrature receiver

804 .5 .3 Analogue-digital conversion

824 .5 .4 Signal phase shifting

844.6 Overview

85Notes

S 6Further Reading

87

5

Fourier Transform NMR

89

5.1 A Single-Pulse Experiment

895.2 Signal Averaging

905.3 Multiple-Pulse Experiments : Phase Cycling

935.4 Heteronuclear Experiments

9 55.5 Arrayed Experiments

965.6 NMR Signal

9 85.7 NMR Spectrum

10 15.7.1 Fourier transformation

101

5 .7.2 Lorentzian peakshapes

101

5 .7 .3 Explanation of Fourier transformation

10 55 .7 .4 Spectral phase shifts

10 85 .7 .5 Frequency-dependent phase correction

1095.8 Two-dimensional Spectroscopy

11 05 .8 .1 Two-dimensional signal surface

11 0

5 .8 .2 Two-dimensional Fourier transformation

11 1

5 .8 .3 Phase twist peaks

1125 .8 .4 Pure absorption two-dimensional spectra

1145.9 Three-dimensional Spectroscopy

119Notes

120Further Reading

12 1Exercises

122

Part 3 Nuclear Spin Interactions

12 5

6

Review of Quantum Mechanics

127

6.1 Functions

1276 .1 .1 Continuous functions

1276.1 .2 Normalization

1286.1 .3 Orthogonal and orthonormal functions

1286.1 .4 Dirac notation

1286.1 .5 Vector representation of functions

129

6.2 Operators

13 16 .2 .1 Commutation

1326 .2 .2 Matrix representations

1336 .2 .3 Diagonal matrices

13 56 .2 .4 Block diagonal matrices

135

6 .2 .5 Inverse

1366 .2 .6 Adjoint

136

6 .2 .7 Hermitian operators

137

6 .2 .8 Unitary operators

1376.3 Eigenfunctions and Eigenvalues

13 8

6 .3 .1 Eigenequations

13 86 .3 .2 Degeneracy

138

6.3 .3 Eigenfunctions and eigenvalues of hermitian operators

1386.3 .4 Eigenfunctions of commuting operators :

non-degenerate case

1396.3 .5 Eigenfunctions of commuting operators : degenerate case

1396 .3 .6 Eigenfunctions of commuting operators : summary

1406.4 Exponential Operators

1406 .4 .1 Powers of operators

1406 .4 .2 Exponentials of operators

14 16 .4 .3 Exponentials of unity and null operators

14 16 .4 .4 Products of exponential operators

1426 .4.5 Inverses of exponential operators

1426 .4 .6 Complex exponentials of operators

1426 .4.7 Exponentials of small operators

14 36 .5 Cyclic Commutation

1436 .5.1 Definition of cyclic commutation

14 36 .5 .2 Sandwich formula

14 36.6 Spinless Quantum Mechanics

14 56 .6 .1 The state of the particle

1466 .6 .2 The Equation of motion

1466 .6 .3 Experimental observations

1476.7 Energy Levels

1486.8 Natural Units

1496.9 Superposition States and Stationary States

1496.10 Conservation Laws

1506.11 Angular Momentum

15 16 .11 .1 Angular momentum operators

1526 .11 .2 Rotation operators

1526 .11 .3 Rotation sandwiches

1546 .11 .4 Angular momentum eigenstates and eigenvalues

15 66 .11 .5 The angular momentum eigenstates

15 76 .11 .6 Shift operators

15 86 .11 .7 Matrix representations of the angular momentum

operators

15 96.12 Spin

1606 .12 .1 Spin angular momentum operators

1606.12.2 Spin rotation operators

16 16 .12 .3 Spin Zeeman basis

1626 .12 .4 Trace

1626.13 Spin-1/2

1636 .13 .1 Zeeman eigenstates

1636 .13 .2 Angular momentum operators

1636 .13 .3 Spin-1/2 rotation operators

164

6 .13 .4 Unity operator

16 46 .13 .5 Shift operators

1646.13.6 Projection operators

16 56 .13 .7 Ket-bra notation

165

Notes

166Further Reading

166Exercises

167

7

Nuclear Spin Hamiltonian

169

7.1 Spin Hamiltonian Hypothesis

1697.2 Electromagnetic Interactions

1707.2 .1 Electric spin Hamiltonian

1717.2 .2 Magnetic spin interactions

1747.3 External and Internal Spin Interactions

1767.4 External Magnetic Fields

1767.4 .1

Static field

1777.4 .2 Transverse field

1787.4 .3 External spin interactions : summary

18 07.5 Internal Spin Hamiltonian

18 07.5 .1 The internal spin interactions

18 07.5 .2 Simplification of the internal Hamiltonian

18 37.6 Motional Averaging

1847 .6 .1 Modes of molecular motion

1847 .6 .2 Molecular rotations

1847 .6 .3 Molecular translations

18 67 .6 .4 Intramolecular and intermolecular spin

interactions

1877 .6 .5 Summary of motional averaging

1887.7 Chemical Shift

1927 .7 .1 Chemical shift tensor

1937 .7 .2 Secular approximation

1947 .7.3 Isotropic chemical shift

1947 .7.4 Chemically shifted Larmor frequency

1967.7.5 Influences on the chemical shift

1977 .7.6 Anisotropic liquids

1987 .7.7 Chemical shift interaction in solids

1997.7.8 Chemical shift interaction : summary

2007.8 Electric Quadrupole Coupling

20 17.8 .1 Isotropic liquids

20 17.8 .2 Anisotropic liquids

2027.8 .3 Solids

2027.8 .4 Quadrupole interaction : summary

2027.9 Direct Dipole-Dipole Coupling

2037.9 .1 Secular direct-direct coupling

2057.9 .2 Isotropic liquids

2077.9 .3 Anisotropic liquids

2087.9 .4 Solids

2097.9 .5 Dipole-dipole interaction: summary

210

7.10 J-Coupling

21 1

7.10 .1 Isotropic J-coupling

212

7.10 .2 Liquid crystals and solids

214

7.10.3 Mechanism of the J-coupling

21 57 .10 .4 J-coupling : Summary

2167.11 Spin-Rotation Interaction

216

7.12 Summary of the Spin Hamiltonian Terms

217Notes

218Further Reading

219Exercises

220

8

Spin Systems in Isotropic Liquids

223

8.1 Molecular Spin System

2238.2 Spin Ensemble

2248.3 Motionally Suppressed J-couplings

2258.4 Chemical Equivalence

2268.5 Magnetic Equivalence

2298.6 Weak Coupling

2338.7 Heteronuclear Spin Systems

2348.8 Alphabet Notation

2358.9 Spin Coupling Topologies

236Notes

237Further Reading

237Exercises

237

Part 4 Uncoupled Spins-1/2

239

9

Single Spin-1/2

241

9.1 Zeeman Eigenstates

2419.2 Measurement of Angular Momentum . Quantum

Indeterminacy

2429.3 Energy Levels

2439.4 Superposition States

2449.4 .1 General spin states

2449 .4.2 Vector notation

2459 .4 .3 Some particular states

2459 .4 .4 Phase factors

2489.5 Spin Precession

2489.5 .1 Dynamics of the eigenstates

2499 .5.2 Dynamics of the superposition states

25 19.6 Rotating Frame

2529.7 Precession in the Rotating Frame

256

9 .8 Radiofrequency Pulse

2589 .8.1 Rotating-frame Hamiltonian

2589 .8.2 x-Pulse

2609 .8.3 Nutation

2639 .8.4 Pulse of general phase

2639 .8.5 Off-resonance effects

265Notes

269Further Reading

270Exercises

270

10

Ensemble of Spins-1/2

273

10.1 Spin Density Operator

27310.2 Populations and Coherences

275

10 .2.1

Density matrix

275

10 .2 .2

Box notation

276

10 .2 .3

Balls and arrows

276

10 .2 .4

Orders of coherence

277

10 .2 .5

Relationships between populations and

coherences

278

10 .2 .6

Physical interpretation of th e

populations

278

10 .2 .7

Physical interpretation of the coherences

28010.3 Thermal Equilibrium

28110.4 Rotating-Frame Density Operator

28410.5 Magnetization Vector

28510.6 Strong r.f. Pulse

286

10 .6 .1

Excitation of coherence

288

10 .6 .2

Population inversion

289

10 .6 .3

Cycle of states

290

10 .6 .4

Stimulated absorption and emission

29 110.7 Free Precession Without Relaxation

29210.8 Operator Transformations

295

10 .8 .1

Pulse of phase Or = 0

296

10.8 .2

Pulse of phase ¢p = 7/2

296

10.8 .3

Pulse of phase OF = n

296

10.8 .4

Pulse of phase ßy = 37/2

296

10.8 .5

Pulse of general phase (/)y

297

10.8 .6

Free precession for an interval r

29710.9 Free Evolution with Relaxation

298

10 .9 .1

Transverse relaxation

298

10 .9 .2

Longitudinal relaxation

30010.10 Magnetization Vector Trajectories

303

10.11 NMR Signal and NMR Spectrum

30510.12 Single-Pulse Spectra

307

Notes

31 0Further Reading

31 2

Exercises

31 2

11

Experiments on Non-Interacting Spins

315

11.1 Inversion-Recovery : Measurement of Tl

31511.2 Spin Echoes: Measurement of T2

31811 .2 .1

Homogenous and inhomogenous broadening

31 811 .2 .2

Inhomogenous broadening in the time domain

31 911 .2 .3

Spin echo pulse sequence

32011 .2 .4

Refocussing

32311.3 NMR Imaging

325Notes

334Further Reading

335Exercises

335

Part 5 Coupled Spins-1/2

337

12

Homonuclear AX System

339

12.1 Weakly Coupled Spin-Pair Hamiltonian

34 012 .2 Zeeman Product States and Superposition

States

34112.3 Energy Levels

34212.4 Total Angular Momenta

34312.5 Density Operator

34412.6 Rotating Frame

34712.7 Free Evolution

35012.7.1

Evolution of a spin pair

35012 .7.2

Evolution of the coherences

35 112 .8 Spectrum of the AX System: Spin-Spin Splitting

35212.9 Product Operators

35512 .9 .1

Construction of product operators

35612.9 .2

Populations and coherences

35812.9 .3

Spin orientations

36112 .10 Thermal Equilibrium

36412 .11 Radiofrequency Pulses

36612 .11 .1

Rotations of a single spin pair

36712 .11 .2 Rotations of the spin density operator

36812.11 .3 Operator transformations

37 112 .12 Free Evolution of the Product Operators

37312 .12 .1

Chemical shift evolution

37512 .12 .2 I-Coupling evolution

37612 .12 .3

Relaxation

381

12.13 Spin Echo Sandwich

381Notes

384Further Reading

384Exercises

384

13

Experiments on AX Systems

387

13.1 COSY

38713.1 .1

The assignment problem

38713.1 .2

COSY pulse sequence

38813.1 .3

Theory of COSY: coherence interpretation

39013 .1 .4

Product operator interpretation

39513 .1 .5

Experimental examples

39813.2 INADEQUATE

39913 .2.1

13C Isotopomers

39913 .2.2

Pulse sequence

40313 .2.3

Theory of INADEQUATE

40513 .2 .4

Coherence transfer pathways and phase cycling

41013 .2 .5

Two-dimensional INADEQUATE

41213.3 INEPT

41613 .3 .1

The sensitivity of nuclear isotopes

41 613 .3 .2

INEPT pulse sequence

41 813 .3 .3

Refocussed INEPT

42113.4 AX Systems in Weakly Oriented Liquids

42513 .4 .1

Angular information

42513 .4 .2

Spin Hamiltonian

42613 .4 .3

Bicelles

42713 .4 .4

Doublet splittings

42913 .4 .5

Why not orient the molecules more strongly?

431Notes

431Further Reading

432Exercises

432

14

Multiple Spin-1/2 Systems

435

14.1 Spin Hamiltonian

43514.2 Energy Eigenstates

43614.3 Superposition States

43814.4 Spin Density Operator

43814.5 Populations and Coherences

43914.5 .1

Coherence orders

43914.5.2

Combination coherences and simple coherences

44014 .5.3

Coherence frequencies

44014 .5.4

Degenerate coherences

44214 .5.5

Observable coherences

44314.6 NMR Spectra

444

14.7 Multiple-Spin Product Operators

44614.7.1

Construction of product operators

44 714.7.2

Populations and coherences

44714 .7 .3

Physical interpretation of product operators

45014.8 Thermal Equilibrium

45 014.9 Radiofrequency Pulses

45 114.10 Free Precession

45 114.10 .1

Chemical shift evolution

45 214.10 .2

J-Coupling evolution

45314.10 .3

Relaxation

45 414.11 Spin Echo Sandwiches

45 514.12 INEPT in an I2S System

45714.13 COSY in Multiple-Spin Systems

46014 .13 .1 AMX spectrum

46 114 .13.2

Active and passive spins

46314 .13 .3

Cross-peak multiplets

46314 .13 .4 Diagonal peaks

46614 .13.5

Linear spin systems

46614.14 TOCSY

46714 .14 .1 The ambiguity of COSY spectra

46714.14 .2 TOCSY pulse sequence

46914.14 .3 Theory of TOCSY

470Notes

474Further Reading

475Exercises

475

Part 6 Motion and Relaxation

477

15

Motion

479

15.1 Motional Processes

47915 .1 .1

Molecular vibrations

47915 .1 .2

Local rotations of molecular groups

48015 .1 .3

Molecular flexibility

48015 .1 .4

Chemical exchange

48015 .1 .5

Molecular rotations

48115 .1 .6

Mechanical rotation of a solid

48215.1 .7

Translational motion

48315 .2 Motional Timescales

48515.3 Motional Effects

48615.4 Motional Averaging

48715.5 Motional Lineshapes and Two-Site Exchange

48815 .5 .1

Slow intermediate exchange and motionalbroadening

490

15 .5 .2

Fast intermediate exchange and motional

narrowing

493

15.5 .3

Averaging of J-splittings

496

15.5 .4

Asymmetric two-site exchange

497

15.5 .5

Knight shift

499

15 .5 .6

Paramagnetic shifts

50015.6 Longitudinal Magnetization Exchange

500

15 .6 .1

Two-dimensional exchange

spectroscopy

501

15 .6 .2

Theory

504

15 .6 .3

Motional regimes

509Notes

510Further Reading

511Exercises

51 1

i6

Relaxation

513

16.1 Types

of Relaxation

51316.2 Relaxation Mechanisms

51416.3 Random Field Relaxation

515

16 .3 .1

Autocorrelation functions and correlation

times

516

16 .3 .2

Spectral density

519

16 .3.3

Normalized spectral density

520

16 .3.4

Transition probabilities

520

16 .3.5

Thermally corrected transition probabilities

522

16 .3.6

Spin-lattice relaxation

52316.4 Dipole-Dipole Relaxation

527

16 .4.1

Rotational correlation time

528

16 .4.2

Transition probabilities

528

16 .4 .3

Solomon equations

533

16.4 .4

Longitudinal Relaxation

536

16.4 .5

Transverse relaxation

53716.5 Steady-State Nuclear Overhauser Effect

53816.6 NOESY

543

16.6 .1

NOESY pulse sequence

543

16.6 .2

NOESY signal

543

16.6 .3

NOESY spectra

546

16.6 .4

NOESY and chemical exchange

548

16 .6 .5

Molecular structure determination

54916.7 ROESY

550

16 .7 .1

Transverse cross-relaxation

550

16 .7 .2

Spin locking

55 1

16 .7 .3

Transverse Solomon equations

552

16 .7 .4

ROESY spectra

554

16.7.5

ROESY and chemical exchange

55616.7.6

ROESY and TOCSY

55616.8 Cross-Correlated Relaxation

55 716 .8 .1

Cross-correlation

55 716 .8 .2

Cross-correlation of spin interactions

55 916 .8 .3

Dipole-dipole cross-correlation and angula r

estimations

56 016 .8.4

TROSY

56 4Notes

56 8Further Reading

56 9Exercises

57 0

Part 7 Appendices, Symbols and Answers to Exercises

571

17

Appendices

573

17.1 Rotations and Cyclic Commutation

57317.2 Rotation Sandwiches

57517.3 Spin-1/2 Rotation Operators

57617.4 Full Quadrupolar Interaction

57717.5 Secular Approximation

57817.6 J-Couplings and Magnetic Equivalence

58317 .7 Quadrature Detection and Spin Coherences

58517.8 Strong Coupling

58817.9 Spin Echo Sandwiches

59417.9 .1

Short duration limit

59517.9 .2

Long duration limit

59617.9 .3

Two spin echo sequences

59717.9 .4

Heteronuclear spin echo sequences

59817.10 Phase Cycling

60017.10 .1

Coherence transfer pathways

60 117.10 .2

Coherence transfer amplitudes

60217.10 .3

Coherence orders and phase shifts

60317.10 .4

The pathway phase

60517.10 .5

A sum theorem

60617.10 .6

Pathway selection I

60717.10 .7

Pathway selection II

60917.10 .8

Pathway selection III

61 117.10 .9

Selection of a single pathway I

61217.10 .10

Selection of a single pathway II

61 317.10 .11

Dual pathway selection

61417.10 .12

Internal phases I

616

17.10 .13

Internal phases II

61717 .10 .14

Nested phase cycles I

61 862017.10 .15

Nested phase cycles II

17 .10 .16

Suppressing receiver artefacts I

622

17 .10 .17

Suppressing receiver artefacts II

625

17 .10 .18

Different ways of constructing phase cycles

625

17 .10 .19

Pulsed field gradients

62517.11 Bloch Equations

62617.12 Chemical Exchange

627

17 .12 .1

The incoherent dynamics

628

17 .12 .2

The coherent dynamics

629

17 .12 .3

The spectrum

630

17 .12 .4

Longitudinal magnetization exchange

63117.13 Solomon Equations

63317 .14 Cross-Relaxation Dynamics

635Notes

636Further Reading

636

List of Symbols

639

Answers to the Exercises

657

Index

669