NMR: the toolkit

NMR: the toolkit P. J. Hore Reader in Physical Chemistry, Physical and Theoretical Chemistry Laboratory and Corpus Christi College, Oxford University

J. A. Jones Royal Society University Research Fellow, Clarendon Laboratory and Corpus Christi College, Oxford University

S. Wimperis Reader in Magnetic Resonance, School of Chemistry, Exeter University

SUB Gottingen 217 959 113 2005 A 20878

OXFORD UNIVERSITY PRESS

7

Contents Preamble Part A 1

The vector model 1.1 Introduction 1.2 Bulk magnetization 1.3 The rotating frame 1.4 Pulses 1.5 Free precession 1.6 Tx and T2 relaxation 1.7 Spin echoes

2

Fourier transform NMR 2.1 Introduction 2.2 Detection of the NMR signal 2.3 Two-dimensional NMR 2.4 Pure phase two-dimensional NMR

7 7 7 9 11

3.1 3.2 3.3 3.4

Product operators I Introduction Product operators for one spin Product operators for two coupled spins Spin echoes

13 13 13 15 18

Product operators II 4.1 Introduction 4.2 INEPT 4.3 Multiple-quantum coherence 4.4 Multi-spin systems 4.5 DEPT

20 20 20 22 24 25

3

4

5

2 2 2 2 3 4 5 5

Two-dimensional NMR Introduction COSY DQF-COSY NOESY TOCSY HMQC HSQC "5.8 Three- and four-dimensional NMR Appendix 5.1 NOE, cross relaxation, and the Solomon equations

5.1 5.2 5.3 5.4 5.5 5.6 5.7

Phase cycling and pulsed field gradients 6.1 Introduction 6.2 Coherence transfer pathways 6.3 Phase cycling 6.4 Pulsed field gradients

.

28 28 28 30 32 33 34 36 37 37 40 40 40 42 43

PartB 7

Quantum mechanics 7.1 Introduction 7.2 Ket and bra vectors 7.3 Operators 7.4 Angular momentum 7.5 Free precession 7.6 Radiofrequency pulses Appendix 7.1 Vectors and matrices Appendix 7.2 Operators

45 45 45 46 47 49 50 51 53

8

Density matrices 8.1 Introduction 8.2 The density operator 8.3 Solving the Liouville-von Neumann equation 8.4 Ensemble averages 8.5 Application to NMR 8.6 Connection to product operators Appendix 8.1 Pure and mixed states, and coherence Appendix 8.2 Matrix diagonalization and matrix exponentials Appendix 8.3 The matrix exponential of lx Appendix 8.4 The rotating frame

55 55 55 56 57 59 60 61 62 64 65

9

Weak coupling and equivalence 9.1 Introduction 9.2 Density operators in two-spin systems 9.3 /coupling 9.4 Weak coupling: a brute force approach 9.5 Weak coupling: a more cunning approach 9.6 Equivalent spins 9.7 Evolution of multiple-quantum coherence 9.8 TOCSY Appendix 9.1 Direct products Appendix 9.2 Matrix representations of two-spin operators Appendix 9.3 Operator commutators

67 67 67 68 69 71 72 73 74 75 76 77

10 Strong coupling 10.1 Introduction 10.2 Free induction decay 10.3 Spin echoes

78 78 78 81

Bibliography

83

Index

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84