Stochastic Processes in Magnetic Resonance

The following sections are included:

• INTRODUCTION

• CONTENTS

The following sections are included:

• Liouville's Original Equation in Classical Mechanics

• Wave Functions and Expectation Values in Quantum Mechanics

• Level Shift Operators on Hilbert space

• A Free Two Level System
  • Possible results of measurements
  • The level-shift operators

• The Quantum Mechanical Density Matrix
  • Motivation for defining a density matrix
  • Formal definition of the density matrix

• The Liouville - von Neumann equation of motion
  • The equation of motion of the density matrix
  • The Liouville - von Neumann equation in Liouville space
  • The formal structure of the Liouville operator

• Density Matrices for Continuously Irradiated S=½ spins
  • The density matrix of a single free S=½ spin
  • Irradiating an ensemble of free spins (S = ½)
  • Calculating the density matrix by manipulation of operators
  • Calculation with matrix representations
  • Analytical calculation in Liouville space
  • Matrix representations in Liouville space

• Suggested References

The following sections are included:

• The Phenomenological Bloch Equations for Magnetic Resonance

• Applying the Bloch Equations
  • No relaxation (T₂,T₁→ ∞)
  • No irradiation (ω₁ =0)
  • Steady state CW irradiation

• Quantum Mechanical Treatment of Elementary Magnetic Resonance
  • Approximate transition probabilities
  • An exact quantum mechanical description

• Basic Concepts in Stochastic Processes

• Approximate Treatments of Relaxation in Simple Cases
  • Autocorrelation functions in time dependent perturbation theory
  • The master equation for populations

• The Density Matrix Relaxation Theory
  • A semi-classical derivation of a master equation for the density matrix
  • Generalizations of the equation

• An Operator Form for the Master Equation

• Suggested References

The following sections are included:

• The Classical Rate Equations

• The Quantum Mechanical Density Matrix Formalism
  • The equation of motion for the density matrix
  • Approximations for slow and fast exchange

• Invariance Properties of Exchanging Systems
  • Invariance of the total spin I_z projection
  • Symmetries of the exchanging molecule
  • Symmetries of the exchange process

• Suggested References

The following sections are included:

• Relaxation Due to Random Modulations of the Motion
  • Random modulation of a classical oscillator
  • Solution for the case of Markovian modulation
  • Cumulant expansion for the relaxation function
  • Relaxation behaviour as a function of the modulation rate

• The Stochastic Liouville Equation - a Relaxation Function Approach
  • The stochastic Liouville equation in classical mechanics
  • The stochastic Liouville equation in quantum mechanics

• The Stochastic Liouville Equation - a Distribution Function Approach
  • Derivation of the equation in classical mechanics
  • Derivation of the equation in quantum mechanics
  • Modification of the equation for a correct equilibrium solution

• Suggested References

The following sections are included:

• The Method of Cumulants
  • General form of the solution
  • Solving with the cumulant expansion for Gaussian modulation
  • Cumulant expansion for a Poisson process

• Discretization of the Stochastic Parameter
  • A finite difference approach to a diffusion-type SLE
  • Application to rotational diffusion
  • Application to CIDEP

• The Method of Eigenfunctions
  • Solution of the SLE in the general case
  • Solution for rotational diffusion

• Numerical Methods of Solution in the Eigenfunctions Method
  • The general computational scheme
  • Using the Lanczos algorithm for diagonalization
  • Using the conjugate gradients technique for inversion

• Suggested References

The following sections are included:

• Doublet EPR - isotropic rotational diffusion in isotropic solvents
  • Axially symmetric g-tensor - non-saturated lines
  • Asymmetric secular g-tensor - non-saturated lines
  • Axially symmetric g-tensor and hyperfine tensor - non-saturated lines
  • Asymmetric g-tensor - saturated line

• Doublet EPR - more general rotational diffusion problems
  • Anisotropic rotational diffusion in isotropic solvents
  • Rotational diffusion in anisotropic media

• EPR of photoexcited triplets undergoing rotational diffusion
  • Photoexcited triplet states - time dependence
  • Line shape theory for the general case
  • Rate and Anisotropy Dependence of Lineshapes

• NMR line shapes for rotational diffusion in symmetrical systems
  • Rotational diffusion combined with discrete jumps
  • NMR line shapes for rotational diffusion in a symmetric potential

• Suggested References

The following sections are included:

• Multiple Resonance Methods
  • Saturation and relaxation in the motional narrowing regime
  • Electron-electron double resonance (ELDOR)
  • Electron-nuclear double resonance (ENDOR)
  • Saturation and relaxation in the slow motional regime

• Two Dimensional Magnetic Resonance
  • Calculation of the density matrix in a multiple pulse experiment
  • Multiple pulse experiments on chemically exchanging systems

• Suggested References

The following sections are included:

• Magnetic Resonance Spectroscopy: General

• Special Experimental Techniques
  • Time-resolved EPR
  • Direct-detection (CW) and Fourier transform (pulsed) EPR
  • Multiple resonance methods: electron-nuclear and electron-electron double resonance (ENDOR and ELDOR)

• Suggested References

The following sections are included:

• APPENDIX A ANGULAR MOMENTUM IN QUANTUM MECHANICS
  • The Angular Momentum Operators
  • Angular Momentum as the Generator of Rotations
  • Matrix Representations for Angular Momentum
  • Suggested References

• APPENDIX B ROTATION MATRICES
  • Suggested References

• APPENDIX C ANGULAR MOMENTUM COUPLING COEFFICIENTS
  • Suggested References

• APPENDIX D IRREDUCIBLE SPHERICAL TENSORS
  • Suggested References

• APPENDIX E REPRESENTATIONS OF FINITE GROUPS
  • Suggested References

• Subject Index