Object-OrientedMagnetic Resonanc e

Classes and Objects ,

Calculations and Computation s

M . Mehring

V. A . Weberruß

Preface XII I

Notation XV

List of Graphical Symbols XXI

1 Motivation

Spin Physics 4

2 A Quick Tour 52.1 Classes and Objects in Hilbert Space 5

2 .1 .1

The Class of Hilbert States 52 .1 .2 The Class of Spin Operators 62 .1 .3 The Class of Propagators 9

2 .2 Classes and Objects in Liouville Space 1 02 .2 .1

The Class of Liouville States 1 02 .2 .2 The Class of Spin Superoperators 1 12 .2 .3 The Class of Superpropagators 1 2

3 The Objects in Hilbert Space 1 73 .1 The Discrete Hilbert Space of Spin States 1 7

3 .1 .1

Zeeman States 1 73 .1 .2

Hilbert State Vectors 1 93 .2 Operators I : Operators and Representations 2 1

3 .2 .1 The Two-Level System 2 13 .2 .2 The Three-Level System 263 .2 .3 The Multi-Level System 27

3 .3 Operators II : Sets of Independent Operators 3 13 .3 .1 The Two-Level System 31

3 .3 .2 The Three-Level System 3 23 .3 .3 The Multi-Level System 3 6

3 .4 Operators III : Rotations of Operators 4 33 .4.1

Spin Operator Rotations 453 .4.2 Tensor Operator Rotations 5 2

3 .5 Operators IV : Density Operator and Density Matrix 5 43 .5 .1

Ensembles of Spin-1/2 Particles 5 43 .5 .2 Ensembles of Spin-I Particles 60

3 .6 Operators V: Basis Changes 6 33 .6 .1 The Two-Level System 6 33 .6 .2 The Multi-Level System 6 5

3 .7 Operators VI : Spin Hamiltonians 6 83 .7 .1 The Zeeman Hamiltonian 6 83 .7 .2 The Quadrupole Hamiltonian 6 9

3 .8 Operators VII : Composite Spin Systems 7 03 .8 .1 Spin Operators of Two Spins I = 1/2 7 03 .8 .2 The Tensor Operators of Two Spins I = 1/2 7 33 .8 .3 The Density Operator of Two Spins I = 1 /2 7 33 .8 .4 Interaction Hamiltonians of Two Spins I = 1 /2 7 4

4 The Dynamics in Hilbert Space 7 74 .1 The Time Evolution 7 7

4 .1 .1

Object Dynamics in the Schrödinger Representation 7 84 .1 .2 Object Dynamics in the Heisenberg Representation 8 24 .1 .3 Object Dynamics in the Interaction Representation 8 4

4 .2 The State Representation 8 64 .2 .1 Time-Independent Perturbation Expansion 8 64 .2 .2 Time-Dependent Perturbation Expansion 8 84 .2 .3 Product Representation 9 04 .2 .4 Magnus Expansion 9 1

4 .3 Periodic Hamiltonians 9 24 .3 .1

Linearly and Circularly Polarized Excitations 9 24 .3 .2 An Introduction to the Average Hamiltonian Approach (AHA) 944 .3 .3 An Introduction to the Secular Averaging Approach (SAA) 9 5

4 .4 Periodic Excitations 9 84 .4 .1 Fundamental Circularly Polarized Excitations 9 84 .4 .2 Linearly Polarized Excitations 102

5 The Objects in Liouville Space 1095.1 The Liouville Space 10 9

5 .1 .1

Liouville States and Liouville Basis 11 05 .1 .2 Orthogonality and Completeness 11 15 .1 .3

Expectation Values 11 35 .2 Liouville Operators I : Superoperators 11 4

5 .2 .1

Definition 114

5 .2 .2 Matrix Elements 1145 .2 .3 Rotation Operations 116

5 .3 Liouville Operators II : Composite Spin Systems 11 85 .3 .1 The Two-Spin Density Operator : Basis Operators 11 85 .3 .2 The Two-Spin Density Operator : Time Evolution 11 95 .3 .3 The Liouville Matrix 12 3

6 The Way to Magnetic Resonance 12 96.1 Classes, Objects, and Functions 12 9

6 .1 .1

Objects and Functions in Hilbert Space 13 06 .1 .2 Objects and Functions in Liouville Space 13 1

6 .2 Pulse Sequences 13 56 .2 .1 Pulse Sequence Operators 13 56 .2 .2 The Delta Pulse Approximation 13 76 .2 .3 The Density Matrix Before the First Pulse 13 9

6 .3 Pulse Response Functions 14 16 .3 .1 Magnetic Resonance Response Functions 14 16 .3 .2 Fourier and Laplace Transformations 14 3

Magnetic Resonance 14 6

7 Spin Interactions and Spectra 14 77 .1 Hamiltonians 14 8

7 .1 .1

External Interactions 14 87 .1 .2 Internal Interactions (NMR) 14 97 .1 .3 Internal Interactions (ESR) 154

7.2 Spectra 15 67 .2 .1

Shift Interaction Spectra 15 77 .2 .2 Quadrupolar Spectra 16 17 .2 .3 Spin-Spin Interaction Spectra 16 3

7 .3 Rotations 1707 .3 .1

Sample Rotation 17 17 .3 .2 Sample Spinning 17 37 .3 .3 Molecular Reorientation 17 6

8 Relaxation and Decoherence 18 38 .1 Principles of Relaxation Measurements 184

8 .1 .1 The Spin-Lattice Relaxation 1858 .1 .2 Spin-Spin Relaxation 1908 .1 .3 Spin-Locking 190

8 .2 Relaxation in the Rapid Motion Limit 1928 .2 .1 Relaxation Rate and Memory Function 1928 .2 .2 Fluctuating Local Fields 1948 .2 .3 Relaxation Rates for Special Spin Interactions 19 78 .2 .4 Spin Fluctuations 202

8 .3 Relaxation in the Slow Motion Limit 204

8 .3 .1

Relaxation and Memory Effects . . . .

. 204

8 .3 .2 Rapid Motion Limit 206

8.4 Models of Molecular Motion 20 7

8 .4 .1

Isotropic Molecular Reorientations 20 8

8 .4 .2 Anisotropic Molecular Reorientations 21 1

8 .4 .3 Discrete Jump Models 21 5

9 Spin Echos 22 5

9 .1 The Hahn Echo in Inhomogeneous Fields 22 5

9 .1 .1 The Pulse Sequence of the Hahn Echo 22 7

9 .1 .2 The Response Function of the Hahn Echo 22 8

9 .1 .3 The Generalized Spin Echo Response Function 23 0

9 .1 .4 Phase Cycling 23 4

9 .2 The Rotary Echo 23 6

9 .2 .1 The Pulse Sequence of the Rotary Echo 236

9 .2 .2 The Response Function of the Rotary FID 23 7

9 .2.3 The Response Function of the Rotary Echo 239

9 .3 The Driven Echo 240

9 .3 .1

The Pulse Sequence of the Driven Echo . . : . : :. : . : ;: 240

9 .3 .2 The Response Function of the Driven Echo ,. . ;, . .: : 24 1

9 .4 The Stimulated Echo 247

9 .4.1 Pulse Sequence and Response Function 247

9 .4 .2 The Genuine Stimulated Echo 249

9.5 The Quadrupolar Echo 25 3

9 .5 .1 The Pulse Sequence of the Quadrupolar Echo 253

9 .5 .2 The Response Function of the Quadrupolar Echo 254

9 .5 .3 The Primary Quadrupole Echo 257

9 .5 .4 Separation of Magnetic and Quadrupole Echos 25 9

9 .5 .5 Multiple Quadrupole Echos 26 1

9 .6 The Solid Echo 264

9 .6 .1 The Pulse Sequence of the Solid Echo 26 4

9 .6 .2 The Response Function of the Solid Echo 26 5

9 .7 The Magic Echo 26 8

9 .7 .1 The Magic Echo Pulse Sequence 26 8

9 .7 .2 The Magic Echo Condition 26 9

9 .7 .3 The Magic Sandwich Superpropagator 27 0

9 .8 Echo Envelope Modulation 27 2

9 .8 .1 The Envelope Function of the Two-Pulse Echo 27 4

9 .8 .2 The Envelope Function of the Stimulated Echo 280

10 Double Resonance 28 910 .1 Double Resonance in Three-Level Spin Systems 289

10 .1 .1 The Boltzmann Equilibrium 29210 .1 .2 2-3 Saturation 29310 .1 .3 2-3 Inversion 29 310 .1 .4 Spin Alignment 294

10 .2 Double Resonance in Multi-Level Spin Systems 29610 .2 .1 The n-Level Population 29610 .2 .2 The z Magnetization 29710 .2 .3 The Inverse Spin Temperatures 297

10 .3 Electron Nuclear Double Resonance (ENDOR) 29810 .3 .1 Population and Polarization Dynamics 29910 .3 .2 Dynamic Nuclear Spin Polarization (DNP) 30 1

10 .4 Spin Echo Double Resonance (SEDOR) 30 2

10 .4 .1 The Spin Echo Response Function without I Pulse 30 2

10 .4 .2 The Spin Echo Response Function with I Pulse 30 3

10 .4 .3 The Spin Echo Response Function with Time Variation 30310 .5 Proton Enhanced Nuclear Induction Spectroscopy 304

10 .5 .1 Cross Polarization (CP) 30 510 .5 .2 Adiabatic Demagnetization and Cross Polarization 30810 .5 .3 Cross Polarization Dynamics 31010 .5 .4 Spin Decoupling Dynamics 31 3

10 .6 Pulsed ENDOR 31610 .6 .1 Davies and TRIPLE ENDOR 31610 .6 .2 Mims ENDOR 319

1 i Multiple-Pulse Experiments 32 311 .1 What are Multiple-Pulse Experiments? 32 311 .2 Carr-Purcell-Meiboom-Gill Multiple-Spin Echo Train 325

11 .2 .1 The Can-Purcell Pulse Sequence 32511 .2 .2 The Meiboom-Gill Pulse Sequence 327

11 .3 Chemical Shift Concertina 32 811 .3 .1 The Chemical Shift Concertina Pulse Sequence 32 811 .3 .2 Application of the Average Hamiltonian Theory 33 0

11 .4 The WAHUHA Four-Pulse Experiment 33 111 .4 .1 The WAHUHA Pulse Sequence 33 111 .4 .2 Application of the Average Hamiltonian Theory 33411 .4 .3 High-Resolution Solid State Spectra 33 5

11 .5 The Flip-Flop Lee-Goldburg (FFLG) Experiment 34 011 .5 .1 The Lee-Goldburg (LG) Pulse Sequence 34 011 .5 .2 The Flip-Flop Lee-Goldburg (FFLG) Pulse Sequence 34 2

11 .6 Advanced Multiple-Pulse Experiments 34 311 .6 .1 Eight-Pulse Cycles (HW-8 and MREV-8) 34 3

11 .6 .2 24-Pulse and 52-Pulse Cycles (BR-24 and BR-52) 34 6

11 .6 .3 Time Reversal Multiple-Pulse Cycles 348

12 Multiple-Quantum Spectroscopy 35 312 .1 Multiple-Quantum Transitions 354

12 .1 .1 Multiple-Quantum Transitions in Multi-Spin Systems 35412 .1 .2 Excitations by Strong Irradiation 35612 .1 .3 Double-Quantum Decoupling 35 8

12 .2 Time Domain Multiple-Quantum Spectroscopy 36 112 .2 .1 Multiple-Quantum Excitation, Evolution, and Detection 36512 .2 .2 Multiple-Quantum Spectra 37 112 .2 .3 Time Reversal Sequences 37 412 .2 .4 Generalized Multiple-Quantum Theory 37 512 .2 .5 Selective MQ Pumping 38 1

12 .3 Multiple-Quantum and Transient Sublevel ENDOR 38 312 .3 .1 Preparation for Sublevel ENDOR , 38 312 .3 .2 Multiple-Quantum ENDOR 38 512 .3 .3 Transient Sublevel ENDOR 38 7

13 Two-Dimensional Spectroscopy 39 513 .1 What is Two-Dimensional Spectroscopy? 39 513 .2 Principles of 2D Fourier Spectroscopy 397

13 .2 .1 Magnetic Resonance Line Shapes in 2D Spectroscopy 39 713 .2 .2 Quantum Evolution in 2D Spectroscopy 40 013 .2 .3 Skewed and Sheared 2D Spectra 403

13 .3 Separation of Interactions 40513 .3 .1 Spin-Spin versus Shift Interactions 40513 .3 .2 Correlation Spectroscopy (COSY) 40913 .3 .3 Exchange Spectroscopy 410

13 .4 Hyperfine Correlation Spectroscopy (HYSCORE) 41 313 .4 .1 The HYSCORE Response Function 41 313 .4 .2 The Case of S = 1/2 and I = 1/2 41613 .4 .3 The Case of S = 1/2 and I = 1 41 7

14 Spin Quantum Computing 42 114 .1 First Steps in Quantum Computing 422

14 .1 .1 The NOT Gate 42214 .1 .2 The CNOT Gate 42 214 .1 .3 The Toffoli Gate 42 314 .1 .4 The Quantum Bit 42 414 .1 .5 The Quantum Measurement 42 7

14 .2 Elementary Spin Quantum Gates 42 714 .2 .1 The Spin Implementation of the NOT Gate 42 914 .2 .2 The Spin Implementation of the '/NOT Gate 43 114 .2 .3 The Spin Implementation of the CNOT Gate 43 114 .2 .4 The Spin Implementation of the SWAP Gate 43514 .2 .5 The Spin Implementation of the Alternative Toffoli Gate 4361 4 .2 .6 The NMR Implementation by Cory et al 439

14 .2.7 The 2D NMR Representation of Quantum Gates 43 9

14 .3 Entangled Spin States 44 1

14 .3 .1 Two-Qubit Systems 44 114 .3 .2 Three-Qubit Systems 44 1

14 .3 .3 Two-Bit Entangled State by CNOT Operation 442

14 .4 Pseudo Pure and Mixed States 444

14 .4 .1 Mixed States 445

14 .4 .2 Pseudo Pure States 447

14 .5 The Implementation of the Deutsch Algorithm 45014 .5 .1 The Deutsch Algorithm 45014.5 .2 The NMR Implementation 453

14 .6 The Implementation of the Grover Search Algorithm 456

14 .6 .1 The Grover Search Algorithm 45614 .6 .2 The NMR Implementation 456

14 .7 Quantum Error Correction and Teleportation 458

Complementary Analytical and Numerical Methods 462

15 Analytical Methods 46 3

15 .1 The Floquet Approach (FA) 46415 .1 .1 The Floquet Theorem 464

15 .1 .2 The Floquet Strategy 464

15 .2 The Perturbation-Theoretical Approach (PTA) 46515 .2 .1 The Evolution Operator 466

15 .2 .2 The Density Operator 467

15 .2 .3 The Operator Modes 46815 .2 .4 The Decomposition Process 470

15 .3 The Secular Averaging Approach (SAA) 474

15 .3 .1 The Evolution Operator 474

15 .3 .2 The Decomposition Process 477

15 .4 Application: Linearly Polarized Excitations 48 1

15 .4 .1 The Total Evolution Operator in a Two-Level Spin System .

482

15 .4 .2 The Total Density Operator in a Two-Level Spin System 485

15 .4 .3 The Magnetization Vector in a Two-Level Spin System 487

15 .4 .4 A Numerical Analysis 49 1

15 .4 .5 Remarks on the SAA 499

15 .4 .6 Remarks on the FA 50 4

15 .5 FA, PTA, or SAA? 50 6

16 GAMMA 50 9

16 .1 Installation 51016 .2 Programming Structures 51 1

16.3 Classes, Objects, and Functions 51 3

16 .4 GNUPLOT 515

Appendix 520

17 Lists 52 117 .1 Objects 52 1

17 .1 .1 Tensor Operators 52 4

17 .1 .2 Hamiltonians 52 717 .2 Object Transformation 52 9

17 .2 .1 Rotations of Tensor Operators 53 1

17 .2 .2 Rotations of Hamiltonians 53 2

17 .3 Object Commutation 53 3

Bibliography 53 5

Index 54 9

About the Authors 557