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Since the first demonstrations of nuclear magnetic resonance (NMR) in condensed matter in 1946, the field of NMR has yielded a continuous flow of conceptual advances and methodological innovations that continues today. Much progress has been made in the utilization of solid-state NMR to illuminate molecular structure and dynamics in systems not controllable by any other way. NMR deals with time-dependent perturbations of nuclear spin systems and solving the time-dependent Schrodinger equation is a central problem in quantum physics in general and solid-state NMR in particular. This theoretical perspective outlines the methods used to treat theoretical problems in solid-state NMR as well as the recent theoretical development of spin dynamics in NMR and physics. The purpose of this review is to unravel the versatility of theories in solid-state NMR and to present the recent theoretical developments of spin dynamics.

As front-line theories to control spin dynamics in solid-state nuclear magnetic resonance, the average Hamiltonian theory (AHT) and Floquet theory (FLT) have assumed great prominence and influence since the development of multiple pulse sequences and the inception of magic-angle spinning (MAS) methods in the 1960s [

Floquet theory dissimilar to AHT, is not restricted to stroboscopic observation, yield a more universal approach for the description of the full time dependence of the response of a periodically time-dependent system [

Historical overview of the first observations of NMR: Normally, credit for NMR first observation should go to Rabi and co-workers in 1939 who used a beam of silver atoms [

• Otto Stern, USA: Nobel Prize in Physics 1943, “for his contribution to the development of molecular ray method and his discovery of the magnetic moment of the proton”.

• Isidor I. Rabi, USA: Nobel Prize in Physics 1944, “for his resonance method for recording the magnetic properties of atomic nuclei”.

• Felix Bloch, USA and Edward M. Purcell, USA: Nobel Prize in Physics 1952, “for their discovery of new methods for nuclear magnetic precision measurements and discoveries in connection therewith”.

• Richard R. Ernst, Switzerland: Nobel Prize in Chemistry 1991, “for his contributions to the development of the methodology of high resolution nuclear magnetic resonance (NMR) spectroscopy”.

• Kurt Wüthrich, Switzerland: Nobel Prize in Chemistry 2002, “for his development of nuclear magnetic resonance spectroscopy for determining the three-dimensional structure of biological macromolecules in solution”.

• Paul C. Lauterbur, USA and Peter Mansfield, United Kingdom: Nobel Prize in Physiology or Medicine 2003, “for their discoveries concerning magnetic resonance imaging”.

An important landmark to describe the effect of time-dependent interactions and the accompanying improvements was the introduction of average Hamiltonian theory in solid-state NMR. Since its formal inception in 1968 by John Waugh, the average Hamiltonian theory has become the main tool to study the dynamics of spin systems subject to an RF perturbation and the most popular theoretical method in NMR. Its popularity stems from its excellently simple conceptual structure and computational elegance. AHT is a mathematical formalism that allows us to analyze how pulse sequences affect internal spin interactions. The rise of AHT in solid-state NMR began with the time-reversal experiments in dipolar-coupled spin systems [

The basic understanding of AHT involves considering a time dependent Hamiltonian H ( t ) governing the spin system evolution, and describing the effective evolution by an average Hamiltonian H ¯ within a periodic time ( T ) . This is satisfied only if H ( t ) is periodic ( T ) and the observation is stroboscopic and synchronized with period ( T ) . Two major expansions (Baker-Cambell-Hausdorff and Magnus) and an exact computation including the diagonalization of the time evolution operator defined the average Hamiltonian. This technique has been widely used in the NMR literature in the development of multiple pulse sequences and in the context of both decoupling and recoupling experiments. AHT is especially convenient in the derivation and analysis of pulse sequences that incorporate a block of rf irradiation that is repeated many times. The AHT set the stage for stroboscopic manipulations of spins and spin interactions by radio-frequency pulses and also explains how periodic pulses can be used to transform the symmetry of selected interactions in coupled, many-spin systems considering the average or effective Hamiltonian of the RF pulse train [

In 1883, M. Gaston Floquet proved a remarkable theorem that asserts the existence of a periodic unitary transformation that maps a system of normal differential equations with periodic coefficients into a system of differential equations with constant coefficients [

Floquet Hamiltonian for multi-spin systems cannot be calculated exactly, and approximate methods such as AHT should therefore be used [

The Floquet-Magnus expansion was developed nearly a decade and half ago by Casas, Oteo, and Ros [

The intuitive origins of Fer expansion date in the seminal 1958 Fer paper [

Setting an infinite sequence (a1, a2, a3,∙∙∙) ( u 1 , u 2 , u 3 , ⋯ ), the n t h partial sum σ n is the sum of the first n terms of the sequence,

σ n = ∑ l = 1 n u l (1)

A series is convergent if the sequence of its partial sums { σ 1 , σ 2 , σ 3 , ⋯ } become closer and closer to a given number when the number of their terms increases. Mathematically speaking, a series converges, if there exists a number p such that for any arbitrarily small positive number ξ there is a large integer N such that for all n ≥ N

| σ n − p | ≤ ξ (2)

If the series is convergent, the unique number p is called the sum of the series. The Magnus and some of its equivalent such as Fer expansions have been applied to a wide range of problems in time-dependent quantum mechanics. Exponential time-dependent perturbation theories such as the Magnus expansion or Fer expansion, have proven useful in the treatment of a variety of problems in non-relativistic quantum dynamics. Until in the 1980’s, very little was known about the convergence of exponential perturbation theory. In the original version, Magnus stated its convergence criterion in terms of the eigenvalues of the exponent itself. However, several groups have reported that application of the Magnus expansion in the Schrodinger representation to some problems of spectroscopy interest gave results which were less adequate [_{c}. Several results on the radius of convergence r_{c} in terms of the H amiltonian have been obtained in the literature. Pechukas and Light and Karasev and Mosolova obtained a radius r_{c} = log2 = 0.693∙∙∙ [_{c}= 0.577. Blanes et al. obtained the improved bound radius of r_{c} = 1.086 [_{c} = 1.086∙∙∙) previously obtained [_{c} = 2) and this new analytic bound was in agreement with the numerical estimate of the convergence radius such as no accuracy was lost in the bound [_{c} = π was derived by Moan but in the context of the conventional Magnus expansion for real matrices [

Recoupling schemes have all been extensively treated with Floquet theory inconjunction with the Van Vleck Transformation [_{F} after the first few terms. The appropriateness of the FME and FE are well related to the problem of convergence. This problem has played a pivotal role in the field of solid-state NMR and spin dynamics [

This calculated radius, ξ = 0.60275 , of the convergence of the symmetric Fer expansion by Zanna is smaller than the calculated radius for the classical Fer expansion [

Using the FME and FE approaches, many problems can be attacked in other fields of physics beyond the scope of NMR. It is important to remember that these considered methods have recently found new major areas of applications such as topological materials [

1) the ME has been used as an alternative to conventional perturbation theory for quantum fields to graph rules for functions of the time-evolution operator where normal products and Wick theorem were used. This was useful in the treatment of infrared divergences for some quantum electrodynamics process such as the scattering of an electron on an external potential or the bremsstrahlung of one hard photon [

2) an extension of the ME has been applied to the context of Connes-Kreimer’s Hopf algebra approach to perturbative renormalization of quantum field theory showing that the generalized MEallows to solve the Bogoliubov-Atkinson recursion [

3) in the field of high energy physics, ME has also found applications such as to heavy ion collisions. ME is applied in collision problems when the use of unitary approximation scheme is necessary such as the unitary of the time evolution operator imposing some bound on the experimentally observable cross sections [

4) the problem in neutron oscillations which is closely related to solar neutrino problem. As neutrinos with different masses propagate with different velocities, the mixing allows for flavor conversion corresponding to neutrinos oscillations [

The introduction of FME and FE as theoretical approaches to control the spin dynamics in the field of nuclear magnetic resonance are new exploratory and developmental researches which is a significant addition to the existing theoretical framework of AHT and FT. QFT is the basic mathematical language used to describe and analyze the physics of elementary particles. The theory by itself is an abstract representation for constructing quantum mechanics models of subatomic particles in particle physics and quasiparticles in condensed matter physics. The application of the FME and FE approaches as intuitive approaches in simplifying calculations to solve some specifics problems in the field of high energy physics and QFT such as those outlined in the above paragraph is of major interest. It is worth noting that, the FME has the advantage of having the unitary character of the evolution operator which is preserved at all orders of approximation while the FE has an advantage over the ME that only an evaluation of nested commutators is required in the calculation of the Hamiltonian [

To summarize, our descriptions for all four theories suggest that the Fer expansion is advantageous over the other three theories (AHT, FLT, and FME) in calculation of higher-order corrections. As explained above, while the AHT and FLT are common in solid-state NMR, both the FME and Fer expansion are relatively newcomer although the mathematical formalism has been known for several decades [

The author thank the CUNY Office Assistant Oana Teodorescu for reading and for editing the manuscript. He acknowledges the support from the CUNY GRANT CCRG# 1517 and the CUNY RESEARCH SCHOLAR PROGRAM-2017-2018. He also acknowledges the mentee’s student Francesca Serrano for helping in editing the manuscript. The contents of this paper are solely the responsibility of the author and do not represent the official views of the NIH.

The author declares that there is no conflict of interest regarding the publication of this paper.

Mananga, E.S. (2018) Theoretical Perspectives of Spin Dynamics in Solid-State Nuclear Magnetic Resonance and Physics. Journal of Modern Physics, 9, 1645-1659. https://doi.org/10.4236/jmp.2018.98103