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Physical Review A

Theory of nonadiabatic spin transitions in time-dependent magnetic fields

B. C. Sanctuary

Published Sep 1, 1979• 11 citations
Publisher

Abstract

An angular momentum of fixed magnitude is studied in the presence of a time-dependent magnetic field by use of a density matrix expanded in multipole moments. The cases of resonant and nonresonant nonadiabatic processes are treated. Application to spin-state-selection experiments leads to oscillations in the magnetic-vector polarization consistent with experiment.

DOI
10.1103/physreva.20.1169

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An angular momentum of fixed magnitude is studied in the presence of a time-dependent magnetic field by use of a density matrix expanded in multipole moments. The cases of resonant and nonresonant nonadiabatic processes are treated. Application to spin-state-selection experiments leads to oscillations in the magnetic-vector polarization consistent with experiment.
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Nonadiabatic transitions in state-selection experiments on beams of metastable hydrogen atoms (2S122{S}_{\frac{1}{2}}2S21​​) display oscillations in the population of the F=1F=1F=1, $m=\ensuremath{-}1$ state R. D. Hight and R. T. Robiscoe, [Phys. Rev. A 17, 561 (1978)]. In this paper, theoretical calculations using exact and model solenoid fields averaged over a velocity distribution show that the population oscillations arise from the presence of at least two nonadiabatic regions in the solenoid. No oscillations occur theoretically if only a single nonadiabatic region exists, unless the unlikely assumption of a coherent hydrogen-atom beam is invoked.
We derive algebraic expressions for the atomic multipole moments mk,q(F){m}_{k,q}(F)mk,q​(F) describing the steady-state polarization of an atomic ensemble with angular momentum FFF in a static magnetic field of arbitrary direction and general multipole relaxation rates ${\ensuremath{\Gamma}}_{q}^{(k)}$. The longitudinal moments mk,0{m}_{k,0}mk,0​ are given in terms of truncated continued fractions, while the transverse moments ${m}_{k,q\ensuremath{\ne}0}$, representing the ensemble coherence, are given by products of truncated continued fractions. The special case of isotropic relaxation leads to particularly simple and elegant expressions. Our results are relevant for all domains of physics that consider the evolution of a spin system interacting with vector-type perturbations.
Spin dynamics in pulsed Nuclear Magnetic Resonance (NMR) spectroscopy are studied theoretically using spin density operator formulation. The theory exploits the use of irreducible spherical tensor (multipole) operators of spin angular momentum in developing alternative treatments to spin dynamical problems. Applications in spin decoupling and soft pulse dynamics are considered, giving new analytical solutions for the complete density matrix. The focus of the thesis, however, is on the development of two distinctly new methods of solution to the quantum Liouville equation for NMR field modulation. The first method derives a linear integral equation for the magnetization and provides a rapidly converging integral series solution which is useful in explaining many pulse shapes. The second method uses the Wei-Norman Lie algebra approach$ sp{ rm 96}$ which reduces the complete arbitrary spin I problem to a Riccati equation. Novel exact analytical solutions for a class of temporal field modulations are obtained. In contrast to all other existing formulations, the theory provides the first known examples of exactly soluble nonrectangular pulse shapes for all resonance off-set, all field amplitudes, all time and valid for all spin I $ geq$ 1/2. In addition, a new series expansion for the complete spin density matrix is obtained. The theory is applied to producing spin inversion profiles which show applicability to selective excitation.