Nuclear Magnetic Resonance and Relaxation

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ISBN: 9780521018111 | 0521018110
Cover: Paperback
Copyright: 8/22/2005

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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.

Preface

xix

1 Introduction

(19)

1.1 Historical overview

(3)

1.1.1 The interaction of radiation with matter

(1)

1.1.2 Nuclear magnetic resonance

(1)

1.1.3 The first observations of NMR

(1)

1.1.4 Pulsed NMR

(1)

1.1.5 Scope of this book

(1)

1.2 Nuclear magnetism

(4)

1.2.1 Magnetic moments

(1)

1.2.2 Nuclear magnetic moments

(1)

1.2.3 Energy splitting and resonance

(1)

1.2.4 Uses of NMR

(2)

1.3 Two views of resonance

(4)

1.3.1 The damped harmonic oscillator

(1)

1.3.2 Time response

(1)

1.3.3 Frequency response

(1)

1.3.4 Linearity and superposition

(1)

1.3.5 Fourier duality

(1)

1.4 Larmor precession

(5)

1.4.1 Spin equation of motion

(1)

1.4.2 Precession

(1)

1.4.3 Quantum versus classical treatment

(1)

1.4.4 Behaviour of real systems — relaxation

(2)

1.5 Typical signal size

(3)

1.5.1 The induced voltage

(1)

1.5.2 Initial conditions

(1)

1.5.3 A typical system

(1)

1.5.4 Other considerations

(1)

2 Theoretical background

(26)

2.1 Paramagnetism and Curie's Law

(4)

2.1.1 Paramagnetism

(1)

2.1.2 Calculation of magnetisation

(1)

2.1.3 Curie's Law

(1)

2.1.4 Magnetic susceptibility

(1)

2.1.5 Conditions for linearity

(1)

2.2 Relaxation

(2)

2.2.1 Equilibrium states

(1)

2.2.2 The relaxation process

(1)

2.2.3 Spin interactions

(1)

2.2.4 Spin–lattice and spin–spin relaxation

(1)

2.2.5 The Bloch equations

(1)

2.3 Quantum and classical descriptions of motion

(4)

2.3.1 The Heisenberg equation

(2)

2.3.2 Equation of motion for μ

(1)

2.3.3 Evaluation of commutators

(1)

2.3.4 Expectation values

(1)

2.4 The rotating frame

(3)

2.4.1 The equivalence principle

(1)

2.4.2 Transformation to the rotating frame

(1)

2.4.3 The fictitious field

(1)

2.5 Rotating and oscillating fields

(4)

2.5.1 Rotating and counter-rotating components

(1)

2.5.2 The effective field

(1)

2.5.3 Resonant transverse field

(1)

2.5.4 90° and 180° pulses

(2)

2.6 The dynamic magnetic susceptibility

(9)

2.6.1 Complex susceptibility

(2)

2.6.2 Dynamic magnetic susceptibility

(2)

2.6.3 The transverse relaxation function

(3)

2.6.4 Response to linearly polarised field

(2)

3 Detection methods

(37)

3.1 The CW method

(8)

3.1.1 Frequency and time domains

(1)

3.1.2 Detection of CW NMR

(1)

3.1.3 Q meter detection

(3)

3.1.4 Saturation

(3)

3.2 The pulsed NMR spectrometer

(7)

3.2.1 Outline of a pulsed NMR system

(2)

3.2.2 Pulse programmer

(1)

3.2.3 The transmitter

(1)

3.2.4 The receiver

(2)

3.2.5 Display of signals

(1)

3.3 Magnets

(2)

3.3.1 Basic requirements

(1)

3.3.2 Homogeneity

(1)

3.3.3 Time stability

(1)

3.3.4 Maximum field

(1)

3.3.5 Large volume magnets

(1)

3.4 The sample probe and noise considerations

(12)

3.4.1 Introduction

(1)

3.4.2 Electrical noise

(1)

3.4.3 The NMR probe

(4)

3.4.4 Signal-to-noise ratio

(3)

3.4.5 Recovery time

(1)

3.4.6 The Ernst angle

(2)

3.5 Refinements to the CW method

(8)

3.5.1 Uses of CW NMR

(1)

3.5.2 The Robinson oscillator

(2)

3.5.3 Lock-in detection of CW NMR

(1)

3.5.4 Modulation distortion

(4)

4 Classical view of relaxation

(41)

4.1 Transverse relaxation in solids

(5)

4.1.1 Local magnetic fields

(1)

4.1.2 Solution of equations of motion

(2)

4.1.3 Relaxation times

(1)

4.1.4 Shape of the FID

(1)

4.2 Motion and transverse relaxation

(6)

4.2.1 Averaging of the local fields

(2)

4.2.2 The equations of motion

(2)

4.2.3 Gaussian distribution of phases

(2)

4.2.4 The autocorrelation function

(1)

4.3 Consequences of the model

(6)

4.3.1 The correlation time

(2)

4.3.2 Exponential relaxation

(2)

4.3.3 Gaussian relaxation

(1)

4.3.4 Condition for exponential or Gaussian decay

(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

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Nuclear Magnetic Resonance and Relaxation

Summary

Table of Contents

Supplemental Materials