spin dynamics2.4 spin precession 27 2.5 larmor frequency 30 2.6 spin-lattice relaxation: nuclear...
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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
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