Nuclear Magnetic Resonance and Relaxation

by Brian Cowan

• ISBN13: 9780521018111
• ISBN10: 0521018110
• Format: Paperback
• Copyright: 2005-08-22
• Publisher: Cambridge University Press

Summary

This book provides an introduction to the general principles of nuclear magnetic resonance and relaxation, concentrating on simple models and their application. The concepts of relaxation and the time domain are particularly emphasised. Some relatively advanced topics are treated, but the approach is graduated and all points of potential difficulty are carefully explained. An introductory classical discussion of relaxation is followed by a quantum-mechanical treatment. Only when the the principles of relaxation are firmly established is the density operator approach introduced; and then its power becomes apparent. A selection of case studies is considered in depth, providing applications of the ideas developed in the text. There are a number of appendices, including one on random functions. This treatment of one of the most important experimental techniques in modern science will be of great value to final-year undergraduates, graduate students and researchers using nuclear magnetic resonance, particularly physicists, and especially those involved in the study of condensed matter physics.

Table of Contents

Preface	xix
1 Introduction	1	(19)
1.1 Historical overview	1	(3)
1.1.1 The interaction of radiation with matter	1	(1)
1.1.2 Nuclear magnetic resonance	1	(1)
1.1.3 The first observations of NMR	2	(1)
1.1.4 Pulsed NMR	2	(1)
1.1.5 Scope of this book	3	(1)
1.2 Nuclear magnetism	4	(4)
1.2.1 Magnetic moments	4	(1)
1.2.2 Nuclear magnetic moments	5	(1)
1.2.3 Energy splitting and resonance	5	(1)
1.2.4 Uses of NMR	6	(2)
1.3 Two views of resonance	8	(4)
1.3.1 The damped harmonic oscillator	8	(1)
1.3.2 Time response	8	(1)
1.3.3 Frequency response	9	(1)
1.3.4 Linearity and superposition	10	(1)
1.3.5 Fourier duality	11	(1)
1.4 Larmor precession	12	(5)
1.4.1 Spin equation of motion	12	(1)
1.4.2 Precession	13	(1)
1.4.3 Quantum versus classical treatment	14	(1)
1.4.4 Behaviour of real systems — relaxation	15	(2)
1.5 Typical signal size	17	(3)
1.5.1 The induced voltage	17	(1)
1.5.2 Initial conditions	17	(1)
1.5.3 A typical system	18	(1)
1.5.4 Other considerations	19	(1)
2 Theoretical background	20	(26)
2.1 Paramagnetism and Curie's Law	20	(4)
2.1.1 Paramagnetism	20	(1)
2.1.2 Calculation of magnetisation	21	(1)
2.1.3 Curie's Law	22	(1)
2.1.4 Magnetic susceptibility	23	(1)
2.1.5 Conditions for linearity	23	(1)
2.2 Relaxation	24	(2)
2.2.1 Equilibrium states	24	(1)
2.2.2 The relaxation process	24	(1)
2.2.3 Spin interactions	25	(1)
2.2.4 Spin–lattice and spin–spin relaxation	25	(1)
2.2.5 The Bloch equations	25	(1)
2.3 Quantum and classical descriptions of motion	26	(4)
2.3.1 The Heisenberg equation	26	(2)
2.3.2 Equation of motion for μ	28	(1)
2.3.3 Evaluation of commutators	29	(1)
2.3.4 Expectation values	29	(1)
2.4 The rotating frame	30	(3)
2.4.1 The equivalence principle	30	(1)
2.4.2 Transformation to the rotating frame	31	(1)
2.4.3 The fictitious field	32	(1)
2.5 Rotating and oscillating fields	33	(4)
2.5.1 Rotating and counter-rotating components	33	(1)
2.5.2 The effective field	34	(1)
2.5.3 Resonant transverse field	35	(1)
2.5.4 90° and 180° pulses	35	(2)
2.6 The dynamic magnetic susceptibility	37	(9)
2.6.1 Complex susceptibility	37	(2)
2.6.2 Dynamic magnetic susceptibility	39	(2)
2.6.3 The transverse relaxation function	41	(3)
2.6.4 Response to linearly polarised field	44	(2)
3 Detection methods	46	(37)
3.1 The CW method	46	(8)
3.1.1 Frequency and time domains	46	(1)
3.1.2 Detection of CW NMR	47	(1)
3.1.3 Q meter detection	48	(3)
3.1.4 Saturation	51	(3)
3.2 The pulsed NMR spectrometer	54	(7)
3.2.1 Outline of a pulsed NMR system	54	(2)
3.2.2 Pulse programmer	56	(1)
3.2.3 The transmitter	57	(1)
3.2.4 The receiver	58	(2)
3.2.5 Display of signals	60	(1)
3.3 Magnets	61	(2)
3.3.1 Basic requirements	61	(1)
3.3.2 Homogeneity	61	(1)
3.3.3 Time stability	61	(1)
3.3.4 Maximum field	62	(1)
3.3.5 Large volume magnets	62	(1)
3.4 The sample probe and noise considerations	63	(12)
3.4.1 Introduction	63	(1)
3.4.2 Electrical noise	64	(1)
3.4.3 The NMR probe	65	(4)
3.4.4 Signal-to-noise ratio	69	(3)
3.4.5 Recovery time	72	(1)
3.4.6 The Ernst angle	73	(2)
3.5 Refinements to the CW method	75	(8)
3.5.1 Uses of CW NMR	75	(1)
3.5.2 The Robinson oscillator	76	(2)
3.5.3 Lock-in detection of CW NMR	78	(1)
3.5.4 Modulation distortion	79	(4)
4 Classical view of relaxation	83	(41)
4.1 Transverse relaxation in solids	83	(5)
4.1.1 Local magnetic fields	83	(1)
4.1.2 Solution of equations of motion	84	(2)
4.1.3 Relaxation times	86	(1)
4.1.4 Shape of the FID	87	(1)
4.2 Motion and transverse relaxation	88	(6)
4.2.1 Averaging of the local fields	88	(2)
4.2.2 The equations of motion	90	(2)
4.2.3 Gaussian distribution of phases	92	(2)
4.2.4 The autocorrelation function	94	(1)
4.3 Consequences of the model	94	(6)
4.3.1 The correlation time	94	(2)
4.3.2 Exponential relaxation	96	(2)
4.3.3 Gaussian relaxation	98	(1)
4.3.4 Condition for exponential or Gaussian decay	98	(2)
4.4 Spin echoes	100	(4)
4.4.1 Recovery of lost magnetisation	100	(1)
4.4.2 Effect of a 180° pulse	101	(1)
4.4.3 Formation of a spin echo	102	(1)
4.4.4 72 and 7*2	102	(2)
4.5 Diffusion and its measurement	104	(6)
4.5.1 Echo relaxation function	104	(2)
4.5.2 Diffusion autocorrelation function	106	(2)
4.5.3 Cubic echo decay	108	(1)
4.5.4 Rapid diffusion	108	(2)
4.6 Measuring relaxation times	110	(14)
4.6.1 Measurement of 7*2	110	(1)
4.6.2 Measurement of T2	111	(3)
4.6.3 Pulse errors	114	(3)
4.6.4 Measurement of T1	117	(2)
4.6.5 Techniques when T1 is long	119	(2)
4.6.6 Accurate adjustment of the spectrometer	121	(1)
4.6.7 Data processing and curve fitting	122	(2)
5 Quantum treatment of relaxation	124	(33)
5.1 Introduction	124	(1)
5.1.1 Relaxation and resonance	124	(1)
5.1.2 The equilibrium state	124	(1)
5.1.3 The present approach	125	(1)
5.2 Expectation values of quantum operators	125	(7)
5.2.1 Expectation values: statement of the problem	125	(1)
5.2.2 Systems in equilibrium	126	(3)
5.2.3 Time independence of equilibrium quantities	129	(2)
5.2.4 Systems not in equilibrium	131	(1)
5.3 Behaviour of magnetisation in pulsed NMR	132	(2)
5.3.1 The Hamiltonian	132	(1)
5.3.2 Some approximations	133	(1)
5.3.3 90° and 180° pulses	133	(1)
5.4 The relaxation functions of NMR	134	(9)
5.4.1 Relaxation of longitudinal and transverse magnetisation	134	(2)
5.4.2 Complex form for transverse relaxation	136	(1)
5.4.3 The interaction picture	137	(5)
5.4.4 The relaxation functions	142	(1)
5.5 The dipolar and other interactions	143	(14)
5.5.1 The dipole—dipole interaction	143	(1)
5.5.2 The dipolar Hamiltonian	144	(2)
5.5.3 Like spins	146	(1)
5.5.4 Electron—nucleus interactions	147	(1)
5.5.5 Quadrupole interactions: classical picture	148	(2)
5.5.6 Quadrupole interactions: quantum expressions	150	(2)
5.5.7 Chemical shifts: basic ideas	152	(2)
5.5.8 Isotropic chemical shift	154	(1)
5.5.9 Anisotropic chemical shift	155	(2)
6 Dipolar lineshape in solids	157	(50)
6.1 Transverse relaxation: rigid lattice lineshape	157	(10)
6.1.1 Introduction	157	(2)
6.1.2 A pair of spins	159	(4)
6.1.3 Pake's doublet	163	(2)
6.1.4 Gypsum monocrystal	165	(1)
6.1.5 Three and more spins	166	(1)
6.2 A class of solvable systems	167	(14)
6.2.1 Local field models	167	(1)
6.2.2 The relaxation function	168	(5)
6.2.3 A model interaction	173	(2)
6.2.4 Magnetically dilute solids	175	(3)
6.2.5 Dilute solids in n dimensions	178	(3)
6.3 The method of moments	181	(17)
6.3.1 Rationale for the moment method	181	(4)
6.3.2 Formal expressions for the moments	185	(2)
6.3.3 Calculation of dipolar moments	187	(1)
6.3.4 Special shapes and their moments	188	(3)
6.3.5 A second look at magnetically dilute solids	191	(1)
6.3.6 Real systems	192	(1)
6.3.7 Widom's Theorem	193	(5)
6.4 Memory functions and related methods	198	(9)
6.4.1 The memory equation	198	(1)
6.4.2 Laplace transformation	199	(1)
6.4.3 Moment expansions	200	(2)
6.4.4 Relations between the moments of F(t) and Φ(t)	202	(1)
6.4.5 Gaussian memory function	202	(2)
6.4.6 Exponential memory function	204	(3)
7 Relaxation in liquids	207	(47)
7.1 Transverse relaxation: moments	207	(8)
7.1.1 Relaxation in liquids	207	(1)
7.1.2 Interaction picture	208	(2)
7.1.3 Moments and motion	210	(1)
7.1.4 Descriptions of motion	211	(1)
7.1.5 Semiclassical expressions for moments	212	(2)
7.1.6 Conclusions	214	(1)
7.2 Cumulant expansion treatment of relaxation	215	(10)
7.2.1 Introduction to the method	215	(3)
7.2.2 Evaluation of the terms	218	(3)
7.2.3 Transverse correlation functions	221	(1)
7.2.4 Longitudinal correlation functions	222	(1)
7.2.5 Properties of Gm(t)	223	(2)
7.3 The relaxation times	225	(9)
7.3.1 Adiabatic T2	225	(1)
7.3.2 Longitudinal relaxation	226	(1)
7.3.3 Non-adiabatic T2	227	(2)
7.3.4 Behaviour of the relaxation times	229	(3)
7.3.5 The frequency shift	232	(2)
7.4 Dipolar J(ω) and G(t)	234	(12)
7.4.1 Rotation of a diatomic molecule	234	(3)
7.4.2 Rotation of polyatomic molecules	237	(1)
7.4.3 Relaxation by translational diffusion	238	(4)
7.4.4 Low frequency behaviour	242	(3)
7.4.5 High frequency limiting behaviour	245	(1)
7.5 Some general results	246	(8)
7.5.1 Scaling treatment of relaxation	246	(2)
7.5.2 The T1 minimum	248	(1)
7.5.3 Moments and transverse relaxation – revisited	249	(3)
7.5.4 T1 sum rules and moments	252	(2)
8 Some case studies	254	(37)
8.1 Calcium fluoride lineshape	254	(13)
8.1.1 Why calcium fluoride?	254	(1)
8.1.2 Overview	255	(2)
8.1.3 Moment calculations	257	(1)
8.1.4 Abragam's approximation function	257	(3)
8.1.5 Experimental measurements on calcium fluoride	260	(2)
8.1.6 Theories of lineshape and relaxation	262	(3)
8.1.7 Engelsberg and Lowe's data	265	(1)
8.1.8 Postscript	266	(1)
8.2 Glycerol	267	(3)
8.2.1 Motivation	267	(1)
8.2.2 Relaxation mechanisms	267	(1)
8.2.3 Experimental measurements	267	(1)
8.2.4 High and low frequencies	268	(2)
8.2.5 The T1 minimum	270	(1)
8.3 Exchange in solid helium-3	270	(21)
8.3.1 Introduction	270	(1)
8.3.2 Measurements of T2	271	(2)
8.3.3 Zero point motion	273	(1)
8.3.4 Exchange	273	(5)
8.3.5 Relaxation	278	(2)
8.3.6 Traditional treatment	280	(1)
8.3.7 Spin diffusion	281	(1)
8.3.8 Moment calculations	282	(1)
8.3.9 The hydrodynamic limit	283	(1)
8.3.10 Spin diffusion measurements	284	(1)
8.3.11 Approximating the dipolar correlation functions	285	(3)
8.3.12 Comparison with experiment	288	(2)
8.3.13 Multiple spin exchange	290	(1)
9 The density operator and applications	291	(44)
9.1 Introduction to the density operator	291	(13)
9.1.1 Motivation and definition	291	(2)
9.1.2 Properties of the full density operator	293	(2)
9.1.3 Equation of motion – von Neumann's equation	295	(2)
9.1.4 The reduced density operator	297	(1)
9.1.5 Physical meaning of the reduced density operator	298	(1)
9.1.6 Properties of the reduced density operator	299	(1)
9.1.7 Equation of motion for the reduced density operator	300	(2)
9.1.8 Example: Larmor precession	302	(2)
9.2 Thermal equilibrium	304	(7)
9.2.1 The equilibrium density operator	304	(2)
9.2.2 Ensemble average interpretation	306	(1)
9.2.3 A paradox	306	(2)
9.2.4 Partial resolution	308	(1)
9.2.5 Macroscopic systems	308	(2)
9.2.6 External equilibrium	310	(1)
9.2.7 Spin systems	310	(1)
9.3 Spin dynamics in solids	311	(11)
9.3.1 The concept of spin temperature	311	(1)
9.3.2 Changing the spin temperature	312	(1)
9.3.3 The spin temperature hypothesis	313	(1)
9.3.4 Energy and entropy	314	(1)
9.3.5 Adiabatic demagnetisation	315	(1)
9.3.6 Dipole fields	316	(1)
9.3.7 High field case	317	(1)
9.3.8 FID	318	(2)
9.3.9 Provotorov equations	320	(2)
9.4 In the rotating frame	322	(6)
9.4.1 The basic ideas	322	(1)
9.4.2 Relaxation in the rotating frame	323	(4)
9.4.3 Spin locking	327	(1)
9.4.4 Zero-time resolution (ZTR)	327	(1)
9.5 Density operator theory of relaxation	328	(7)
9.5.1 Historical survey	328	(2)
9.5.2 Master equation for the density operator	330	(1)
9.5.3 Cumulant expansion of the density operator	331	(2)
9.5.4 An example	333	(2)
10 NMR imaging	335	(27)
10.1 Basic principles	335	(4)
10.1.1 Spatial encoding	335	(1)
10.1.2 Field gradients	336	(1)
10.1.3 Viewing shadows	337	(1)
10.1.4 Discussion	338	(1)
10.1.5 Spatial resolution	339	(1)
10.2 Imaging methods	339	(7)
10.2.1 Classification of methods	339	(1)
10.2.2 Sensitive point and sensitive line methods	340	(1)
10.2.3 Shaped pulses – selecting a slice	341	(3)
10.2.4 Back projection	344	(1)
10.2.5 Iterative reconstruction	345	(1)
10.3 Fourier reconstruction techniques	346	(6)
10.3.1 Fourier imaging in two dimensions	346	(1)
10.3.2 Fourier imaging in three dimensions	347	(2)
10.3.3 Fourier zeugmatography	349	(1)
10.3.4 Relaxation	350	(1)
10.3.5 Spin warp imaging	351	(1)
10.3.6 Two-dimensional spin-echo imaging	351	(1)
10.4 Gradient echoes	352	(4)
10.4.1 Creation of gradient echoes	352	(1)
10.4.2 Two-dimensional gradient-echo imaging	353	(1)
10.4.3 Three-dimensional gradient-echo imaging	354	(1)
10.4.4 Echo planar imaging	355	(1)
10.5 Imaging other parameters	356	(6)
10.5.1 The effect of relaxation	356	(1)
10.5.2 Imaging T2	357	(1)
10.5.3 Imaging T1	358	(2)
10.5.4 Imaging of diffusion	360	(1)
10.5.5 Studying fluid flow	360	(2)
Appendix A Fourier analysis	362	(13)
A.1 The real Fourier transform	362	(1)
A.2 The complex Fourier transform	363	(2)
A.3 Fourier transform relations	365	(3)
A.4 Symmetry properties of Fourier transforms	368	(1)
A.5 Dirac's delta function	368	(4)
A.6 Heaviside's step function	372	(1)
A.7 Kramers—Kronig relations	373	(2)
Appendix B Random functions	375	(10)
B.1 Mean behaviour of fluctuations	375	(3)
B.2 The autocorrelation function	378	(2)
B.3 A paradox	380	(1)
B.4 The Wiener—Kintchine Theorem	380	(2)
B.5 Quantum variables	382	(1)
B.6 Thermal equilibrium and stationarity	383	(2)
Appendix C Interaction picture	385	(4)
C.1 Heisenberg and Schriidinger pictures	385	(1)
C.2 Interaction picture	386	(1)
C.3 General case	387	(2)
Appendix D Magnetic fields and canonical momentum	389	(3)
D.1 The problem	389	(1)
D.2 Change of reference frame	389	(2)
D.3 Canonical momentum	391	(1)
Appendix E Alternative classical treatment of relaxation	392	(10)
E.1 Introduction	392	(1)
E.2 Equations of motion	392	(1)
E.3 Adiabatic T2	393	(3)
E.4 Treatment of T1	396	(2)
E.5 Non-adiabatic T2	398	(2)
E.6 Spectral densities	400	(2)
Appendix F Gm(t) for rotationally invariant systems	402	(2)
F.1 Introduction	402	(1)
F.2 Rotation of spherical harmonics	402	(1)
F.3 Simplification	403	(1)
Appendix G P(Ω, Ω0, t) for rotational diffusion	404	(3)
G.1 Rotational diffusion	404	(1)
G.2 General solution	404	(1)
G.3 Particular solution	405	(2)
Problems	407	(14)
References	421	(6)
Index	427