The equilibrium state of nuclear magnetization is quite easy to derive from the spin Hamiltonian and from basic principles of statistical physics and thermodynamics. Somewhat less straightforward is the behavior of nuclear magnetization in imposed magnetic fields which vary either in magnitude or in direction or both. In such non-equilibrium conditions, one has to bring into the picture the dynamics of the spin system as described by its Larmor frequencies and its longitudinal and transverse relaxation times, all of which are themselves functions of the varying magnetic field.
Some of the possible limit situations which can arise have been analyzed and are described in the literature. These certainly include (i) the formation of a new equilibrium after a sudden jump in the magnitude of the external magnetic field (without changing its direction), (ii) a slow rotation of the external field (adiabatic lock) and (iii) under certain special conditions, a slow change in both the magnitude and the direction of the field (adiabatic switching).
Some of the phenomena arising in these contexts have been actively exploited in MRI and observed in fast-field-cycling (FFC) NMR relaxometry. In the latter discipline they may be of crucial importance when switching the external field down to what are the typical values of local magnetic fields within the sample and/or the background magnetic fields in the laboratory ("earth field" in common jargon). Even more striking phenomena are observed/expected when switching across the null field (field inversion) but they have never been pursued due to their apparent complexity.
This presentation concerns a unified treatment of the behavior of nuclear magnetization in variable magnetic fields, describes explicit solutions available for some special cases, and uses numeric simulations to illustrate others. The theory explains, first of all, how comes that is it possible to use the FFC-NMRD method to measure relaxation times over an order of magnitude shorter than the shortest available field-switching times. It also provides a method to optimize the FFC-NMR signal intensity by using appropriate field switching waveforms.
Even more interesting is the possibility to apply the theory to field switching down to nominal values which are close to zero, and even to main-field zero-crossing into "negative" values. The resulting complex phenomena depend strongly upon the sequence timing (including the switching rates and waveforms) which, in principle, is fully under our control, and upon the static and dynamic characteristics of the local magnetic fields inside a sample. Their quantitative analysis should therefore permit us to study the latter. Though this part of the theory is still under development, the perspectives are very encouraging.