Please, cite this online document as:
Sykora S., Vasini E.M.,
Nuclear Magnetization Evolution During the Switching Time in Field Cycling NMR,
14th MRPM, Gainesville (FL, USA), February 18-22, 2018.
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 predicting 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 must 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.
In stationary magnetic fields, the evolution of nuclear magnetization is in general treated by means of the phenomenological Bloch equations , which find a somewhat deeper quantum-mechanical justification in the framework of a density matrix evolution  under the influence of stochastic Hamiltonian terms related to molecular motions.
When the magnetic field undergoes fast and ample variations, however, the matrix terms in Bloch equations become time-dependent. This is further complicated by the fact that even the relaxation times T1 and T2, due to their intrinsic dependence on the field strength B, are usually strongly time dependent. So far, despite various attempts , there did not emerge any general theory suitable to handle these problems; the best one can do is a brute-force numeric integration.
This, by itself, would not be much of a problem, but it tends to obscure important qualitative insights. There are, however, some yet unpublished special cases in which explicit solutions can be found and which turn out to be both illuminating and important in actual practice. In this presentation, we want to focus on those.
Probably the most striking case of the use of varying magnetic fields in NMR occurs in Field Cycling NMR Relaxometry (FC-NMR, a variety of the more general Variable Field NMR, or VF - NMR). This technique suitable to measure the longitudinal relaxation times over a very broad range (up to 5 decades) of Larmor frequencies and (therefore) field strengths . In this method, the static magnetic field B field is switched during each experiment (scan) between several field values (zero, polarization, relaxation, and acquisition fields). During each field-switching interval, the value of B follows an actively controlled ramp function. The total switching times of such ramps depend upon the particular FC-NMR instrument technology and can vary from tens of milliseconds (mechanical shuttles, up to about 100 Tesla/s) to less than one millisecond (main current switching, up to about 1000 Tesla/s). What the nuclear magnetization does during the switching ramps is therefore of top importance for the technique. So far, very few (if any) theoretical insights regarding this topic had been ever presented, except for a preliminary and qualitative talk by one of us at a 2009 FFC meeting .
We show that in the case of a linear variation of the magnetic field strength, with no variation in its direction, and under the assumption of a constant, field-independent T1, there exists an explicit solution for the temporal evolution of the longitudinal magnetization T1. Though this is indeed a very special case (yet it can be realized experimentally), it provides important insights into the matter and elucidates the counter-intuitive experimental observation that it is relatively easy to measure FC NMRD profiles of very fast relaxing samples even when their relaxation times T1 are an order of magnitude shorter than the field switching periods.
We also show how to extend these explicit solutions to more realistic cases of samples with arbitrary NMRD profiles, and to cases (hardware's) in which the switching ramp B(t) is not linear. Apart from providing new insights, such solutions make it possible, for example, to make realistic estimates of the expected sample magnetization at the beginning of the relaxation period and/or to dynamically optimize the acquisition times of NMRD profiles.
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