B. C. Sanctuary

Relaxation equations for multispin systems. II

The Redfield equations are derived quite generally for multispin systems of arbitrary number of spins with arbitrary spin magnitudes. The spin lattice coupling Hamiltonian is assumed to arise from random fluctuations only. The extreme narrowing limit is not taken.

Rotation operator approach to spin dynamics and the Euler geometric equations

The rotation operator approach proposed previously is applied to spin dynamics in a time‐varying magnetic field. The evolution of the wave function is described, and that of the density operator is also treated in terms of a spherical tensor operator base. It is shown that this formulation provides a straightforward calculation of accumulated phases and probabilities of spin transitions and coherence evolutions. The technique focuses, not on the rotation matrix, but on the three Euler angles and its characteristic equations are equivalent to the Euler geometric equations long known to describe the motion of a rigid body. The method usually depends on numerical calculations, but analytical solutions exist in some situations. In this paper, as examples, a hyperbolic secant pulse is solved analytically, and a Gaussian‐shaped pulse is calculated numerically.

The Wei–Norman Lie‐algebraic technique applied to field modulation in nuclear magnetic resonance

By implementing the Wei–Norman Lie algebra approach, this paper focuses on the development of a new method of solution to the quantum Liouville equation for NMR field modulation. First, the general method is reviewed and discussed for systems of any finite dimensional Lie algebra. The theory is then applied to arbitrary time‐dependent Zeeman interactions [or SU(2) Hamiltonians] in which the complete arbitrary spin I≥1/2 density operator problem reduces to a Riccati equation. Novel exact analytical solutions of the complete spin density operator for a class of temporal field modulations are obtained. In particular, a spherical tensor operator basis is used to expand the density operator, and the solutions retain the physically appealing form of Wigner rotations with time‐modulated rotation angles which are special functions with well known analytical properties. The exact solutions include the frequency swept hyperbolic secant pulse shapes, as well as any exponentially modulated amplitude pulse. In contras...

Relaxation of anisotropically oriented I=3/2 nuclei in the multipole basis: Evolution of the second rank tensor in the double quantum filtered nuclear magnetic resonance experiment

The relaxation of an I=3/2 spin system in an anisotropic environment characterized by a finite residual quadrupolar splitting ωq is modeled by analytically solving for the density operator from Redfield’s relaxation theory. The resulting equations are cast into the multipole basis in order to describe the tensorial components of the spin density matrix. Included in the relaxation matrix are off‐diagonal elements J1 and J2, which account for anisotropic systems with ωq values less than the width of the resonant line. With the Wigner rotation matrices simulating hard pulses, the response to an arbitrary pulse sequence can be determined. An analytical expression for the response to the double quantum filtered (DQF) pulse sequence (π/2)−(τ/2)−π−(τ/2)−θ−δ−θ−AQ for θ=π/2 is presented, showing explicitly the formation of a second rank tensor owing only to the presence of a finite ωq. This second rank tensor displays asymptotic behavior when the (reduced) quadrupole splitting is equal to either of the off‐diagona...

Magnetic multipoles in time dependent fields

The effects of time dependent magnetic vector fields on magnetic multipoles is presented. It is shown that a single rapid pulse causes the qth spherical component of the kth multipole in the rotating frame to evolve as the associated Legendre function Pkq(cosϑ) at resonance. The initial state for multispin systems which is prepared by such a pulse is given.

A theoretical model for phase transitions in lipid monolayers

The statistics of Guggenheim are used in conjunction with a ten‐state conformational model (originally formulated for phase transitions in lipid bilayers) to predict gel–fluid and fluid–vapor phase boundaries for lipid monolayers. The theoretical model is a model of interacting dimers on a two‐dimensional lattice with each dimer representing a lipid molecule with two acyl chains and one polar head. The acyl chains are assumed to interact via anisotropic van der Waals interactions. The interaction between the polar heads is simulated by an attractive isotropic interaction with coupling constant K0. The analysis of Costas and Sanctuary is then combined with this model so as to obtain self‐consistently the fraction of occupied and vacant lattice sites for the monolayer at different pressures. Surface pressure‐area isotherms and coexistence curves are obtained for fixed van der Waals interactions and several values of K0. It is shown that a fluid–vapor phase boundary occurs for values of K0 above a critical v...

Dynamics of single spin systems under arbitrary amplitude modulated fields

Using a spherical tensor operator basis a general method of solution to the time evolution of the spin density matrix for a spin of arbitrary magnitude I is given. Rather than using time ordering techniques, we present an integral equations approach for calculating the effects of arbitrary pulse shapes. The method is shown to provide a rapidly converging perturbation expansion which is useful in explaining many pulse shapes. In particular, a simple recipe for the calculation of the magnetization results from this technique. The successive terms in the perturbation expansion avoid multiple commutators such as are encountered in the Magnus expansion of the propagator. These terms are given simply as integrals of the pulse shape function. The examples of the Bloch–Siegert problem and the exact dynamics of a spin interacting with a rotating rf field combined with an isotropic relaxation and a regeneration mechanism are presented in the context of the method of transformations in Liouville space.

Non‐Lorentzian NMR Spectra at low densities

When the weak coupling approximation is not applicable to the derivation of nuclear magnetic relaxation rates, the transverse and longitudinal components of the magnetization decay nonexponentially. In this paper, the frequency domain is studied for a typical case of gases at low density. This example illustrates the non‐Lorentzian behavior expected and gives a clear realization of the rotating frame approximation.

Initial slips and transient effects in relaxation phenomena

For a set of observables, it is shown how the solution of a master equation can approximate the solution of the generalized master quation at long times, but with an aproximate initial slip. The slip phenomena is discussed theoretically and then made more concrete by the exact and approximate solution of a simple but realistic system, namely the initial time dependence for the relaxation of a nuclear spin.

SELECTIVE EXCITATION OF SINGLE AND MULTIPLE QUANTUM TRANSITIONS FOR SPIN 7/2 IN NQR

The dynamics of spin 7/2 in pure NQR is considered. The orientation of the applied RF field is assumed arbitrary and η ≠ 0. For this situation various selective pulses are considered and analytical results obtained. Shaped pulses are also treated. A key feature of this work is to construct the interaction representation from which an analytical calculation, using the computer program ‘Maple’, can be obtained in a tractable form. The results presented are general for all half integer spins of magnitude higher than 1/2.