Fixed point theory of iterative excitation schemes in NMR

Iterative schemes for NMR have been developed by several groups. A theoretical framework based on mathematical dynamics is described for such iterative schemes in nonlinear NMR excitation. This is applicable to any system subjected to coherent radiation or other experimentally controllable external forces. The effect of the excitation, usually a pulse sequence, can be summarized by a propagator or superpropagator (U). The iterative scheme (F) is regarded as a map of propagator space into itself, U n+1=F U n . One designs maps for which a particular propagator U or set of propagators {U} is a fixed point or invariant set. The stability of the fixed points along various directions is characterized by linearizing F around the fixed point, in analogy to the evaluation of an average Hamiltonian. Stable directions of fixed points typically give rise to broadband behavior (in parameters such as frequency, rf amplitude, or coupling constants) and unstable directions to narrowband behavior. The dynamics of the maps are illustrated by ‘‘basin images’’ which depict the convergence of points in propagator space to the stable fixed points. The basin images facilitate the optimal selection of initial pulse sequences to ensure convergence to a desired excitation. Extensions to iterative schemes with several fixed points are discussed. Maps are shown for the propagator space S O(3) appropriate to iterative schemes for isolated spins or two‐level systems. Some maps exhibit smooth, continuous dynamics whereas others have basin images with complex and fractal structures. The theory is applied to iterative schemes for broadband and narrowband π (population inversion) and π/2 rotations, MLEV and Waugh spin decoupling sequences, selective n‐quantum pumping, and bistable excitation.

[1]  R. Tycko,et al.  Composite sequences for efficient double-quantum Excitation over a range of spin coupling strengths , 1985 .

[2]  A. J. Shaka,et al.  Spatial localization of NMR signals in an inhomogeneous radiofrequency field , 1985 .

[3]  E. Schneider,et al.  Composite pulses without phase distortion , 1985 .

[4]  A. Pines,et al.  Iterative schemes for broad-band and narrow-band population inversion in NMR , 1984 .

[5]  A. Pines,et al.  Spatial localization of NMR signals by narrowband inversion , 1984 .

[6]  A. J. Shaka,et al.  Spatially selective radiofrequency pulses , 1984 .

[7]  A. Pines,et al.  Broadband population inversion in solid state NMR , 1984 .

[8]  D. Suter,et al.  Composite pulse excitation in three-level systems , 1984 .

[9]  Richard R. Ernst,et al.  Product operator formalism for the description of NMR pulse experiments , 1984 .

[10]  R. R. Ernst,et al.  Improvement of pulse performance in N.M.R. coherence transfer experiments , 1983 .

[11]  A. J. Shaka,et al.  Composite pulses with dual compensation , 1983 .

[12]  Richard R. Ernst,et al.  Composite pulses constructed by a recursive expansion procedure , 1983 .

[13]  A. Pines,et al.  Broadband population inversion by phase modulated pulses , 1983 .

[14]  R. Tycko,et al.  Broadband Population Inversion , 1983 .

[15]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[16]  A. J. Shaka,et al.  Evaluation of a new broadband decoupling sequence: WALTZ-16 , 1983 .

[17]  A. J. Shaka,et al.  An improved sequence for broadband decoupling: WALTZ-16 , 1983 .

[18]  J. Marsden,et al.  Introduction to Dynamics , 1983 .

[19]  Michael Mehring,et al.  Principles of high-resolution NMR in solids , 1982 .

[20]  A. Redfield,et al.  Nuclear Magnetism: Order and Disorder , 1982 .

[21]  John S. Waugh,et al.  Theory of broadband spin decoupling , 1982 .

[22]  Ray Freeman,et al.  Supercycles for broadband heteronuclear decoupling , 1982 .

[23]  M. Levitt.  Symmetrical composite pulse sequences for NMR population inversion. II. Compensation of resonance offset , 1982 .

[24]  J. Waugh.  Systematic Procedure for Constructing Broadband decoupling Sequences , 1982 .

[25]  David M. Doddrell,et al.  Proton‐polarization transfer enhancement of a heteronuclear spin multiplet with preservation of phase coherency and relative component intensities , 1982 .

[26]  Malcolm H. Levitt,et al.  Symmetrical composite pulse sequences for NMR population inversion. I. Compensation of radiofrequency field inhomogeneity , 1982 .

[27]  Ray Freeman,et al.  Broadband heteronuclear decoupling , 1982 .

[28]  R. R. Ernst,et al.  Low-power multipulse line narrowing in solid-state NMR , 1981 .

[29]  R. Freeman,et al.  Composite pulse decoupling , 1981 .

[30]  Ray Freeman,et al.  Compensation for Pulse Imperfections in NMR Spin-Echo Experiments , 1981 .

[31]  R. Helleman.  Nonlinear dynamics; Proceedings of the International Conference, New York, NY, December 17-21, 1979 , 1980 .

[32]  Warren S. Warren,et al.  Theory of selective excitation of multiple‐quantum transitions , 1980 .

[33]  J. Eckmann,et al.  Iterated maps on the interval as dynamical systems , 1980 .

[34]  Geoffrey Bodenhausen,et al.  Multiple-quantum NMR , 1980 .

[35]  Gareth A. Morris,et al.  Enhancement of nuclear magnetic resonance signals by polarization transfer , 1979 .

[36]  Ray Freeman,et al.  NMR population inversion using a composite pulse , 1979 .

[37]  S. Vega.  Fictitious spin 1/2 operator formalism for multiple quantum NMR , 1978 .

[38]  G. Drobny,et al.  Fourier transform multiple quantum nuclear magnetic resonance , 1978 .

[39]  I. Good,et al.  Fractals: Form, Chance and Dimension , 1978 .

[40]  R. R. Ernst,et al.  Selective excitation and detection in multilevel spin systems: Application of single transition operators , 1977 .

[41]  A. Pines,et al.  Operator formalism for double quantum NMR , 1977 .

[42]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[43]  Haeberlen Ulrich,et al.  High resolution NMR in solids : selective averaging , 1976 .

[44]  T. Hashi,et al.  Excitation and Detection of Coherence between Forbidden Levels in Three-Level Spin System by Multi-Step Processes , 1975 .

[45]  J. Eberly,et al.  Optical resonance and two-level atoms , 1975 .

[46]  J. Waugh,et al.  Advances In Magnetic Resonance , 1974 .

[47]  W. Rhim,et al.  Analysis of multiple pulse NMR in solids. II , 1973 .

[48]  Alexander Pines,et al.  Proton‐enhanced NMR of dilute spins in solids , 1973 .

[49]  Peter Mansfield,et al.  Symmetrized pulse sequences in high resolution NMR in solids , 1971 .

[50]  Alexander Pines,et al.  Time-Reversal Experiments in Dipolar-Coupled Spin Systems , 1971 .

[51]  E. D. Trifonov,et al.  Applications of Group Theory in Quantum Mechanics , 1969 .

[52]  I. Bialynicki-Birula,et al.  Explicit solution of the continuous Baker-Campbell-Hausdorff problem and a new expression for the phase operator , 1969 .

[53]  U. Haeberlen,et al.  Coherent Averaging Effects in Magnetic Resonance , 1968 .

[54]  U. Haeberlen,et al.  Approach to High-Resolution nmr in Solids , 1968 .

[55]  R. Wilcox.  Exponential Operators and Parameter Differentiation in Quantum Physics , 1967 .

[56]  John C. Light,et al.  On the Exponential Form of Time‐Displacement Operators in Quantum Mechanics , 1966 .

[57]  E. Hahn,et al.  Nuclear Double Resonance in the Rotating Frame , 1962 .

[58]  Richard Phillips Feynman,et al.  Geometrical Representation of the Schrödinger Equation for Solving Maser Problems , 1957 .

[59]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[60]  C. Bloch.  Theory of Nuclear Level Density , 1954 .