Chapter 1: Introduction

Published:05 Jun 2024
Nuclear Magnetic Relaxation and Molecular Dynamics, Royal Society of Chemistry, 2024, ch. 1, pp. 115.
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This introductory chapter recapitulates the basic concepts of nuclear magnetic relaxation. This includes the elementary physics of spins. Based on the Bloch equation for the time evolution of magnetizations, transverse and longitudinal spin relaxation times (or rates) are defined. As far as relaxation processes in general are concerned, the Wiener/Khinchin theorem plays a central role. It relates two crucial functions, namely autocorrelation functions and spectral densities. These are the information carriers of molecular dynamics, in the sense that they reflect the characteristics of the fluctuations of the spin interactions due to thermal motions.
Materials and systems, such as fluidfilled porous media, biological tissues, polymers, liquid crystals and any heterogeneously structured matter in general, are classified as ‘complex’, and molecular dynamics in such media tends to be more complicated than just unrestricted translational and isotropic or anisotropic rotational diffusion. For example, molecules may reside in environments of different mobilities, so that exchange processes are part of the fluctuations of the participating particles. Steric and topological constraints can constrain molecular reorientations. Adsorbate molecules can be permanently or temporarily bound to the internal surfaces of porous media. Molecular entities coupled by energetic or entropic elasticity can result in distributions of collective modes. Related questions of interest are whether molecular dynamics are rotational or translational in nature, whether molecular reorientations are restricted or not, whether the statistics are Markovian or not, whether spininteraction partners are homo or heterospecies, and so on. Experimentally, it is therefore desirable to measure spin relaxation on as wide a range of frequency and temperature scales as is technically and thermodynamically feasible, and physically reasonable with respect to theoretical limits. The purpose of this book is to show how nuclear magnetic relaxation studies can reveal and specifically characterize such scenarios. For complementary comparisons with other, nonnuclear magnetic relaxation techniques the reader may find it useful to refer to ref. 1.
1.1 Basic Concepts, Relationships, and Definitions
A note on the term ‘spin coherence’ introduced above. Coherence does not necessarily mean that all spins have the same Larmor precession phase throughout. Precession phases can be dephased by magnetic field inhomogeneities or gradients, but the coherence property can still be present. This type of dephasing is reversible and can be restored by refocusing the coherences in the form of spin echoes (see, for example, Section 2.6). On the other hand, there are irreversible processes that eventually destroy the spin coherences. These are, in particular, transverse relaxation and translational diffusion in inhomogeneous fields.
Note that the ‘hightemperature’ attribute of the limit in eqn (1.22) is extremely well justified near room temperature and at the usual magnetic flux densities of laboratory magnets: the populations of spinup and spindown states differ by less than a few parts per million from each other. For example, a population difference of 1 ppm means that for one million spins, the lower level of the Zeeman scheme in Figure 1.2 is populated on average with only one additional spin compared to the upper level at thermal equilibrium.
1.2 Bloch’s Equation
The Bloch equation in the laboratoryframe version given in eqn (1.27) describes the evolution of the magnetization vector with respect to Larmor precession, as well as transverse and longitudinal relaxation. With respect to Larmor precession and transverse relaxation, complete motional averaging of dipolar or quadrupolar socalled local fields is assumed, as it is expected for lowviscous liquids. Chapter 4 provides a detailed discussion of this important issue. Effects due to chemical shift and indirect spin–spin couplings^{3 } are not accounted for in eqn (1.27). As far as the transverse magnetization components are concerned, the strict validity of eqn (1.27) is therefore limited to liquids with magnetically equivalent nuclei in the absence of indirect spin–spin couplings.
1.3 The Wiener/Khinchin Theorem
The Wiener/Khinchin Fourier relation given in eqn (1.31) is often taken as a definition of the spectral density. However, it is more of a law. In principle, one can independently measure both the correlation function G(t) and the spectral density J(ω) of arbitrary stochastic processes. It turns out that the datasets corresponding to them obey the Wiener/Khinchin relation, given that the fluctuations obey stochastic rules. This is definitely the case for molecular dynamics at thermal equilibrium. Each partner of this conjugate function pair contains, in principle, all the information of the molecular fluctuations. Therefore, it is a matter of experimental accessibility which of the two counterparts will be the primary object of measurement.
In essence, and in the present context, the meaning of the autocorrelation function $G(t)$ is the probability that molecular orientations and/or positions are unchanged after a time interval t. The autocorrelation function can thus be considered as a kind of fingerprint of the type of molecular dynamics. Different dynamical processes, such as rotational and translational diffusion, steric or topological constraints on reorientations or displacements, and superpositions of elastic modes of polymers or liquidcrystalline domains, are examples of the wide scope addressed when talking about molecular dynamics.
As a standalone function—not just the Fourier transform of the autocorrelation function—the spectral density $I(\omega )$ is the distribution of angular frequencies in the random ‘noise’ that reflects thermal molecular fluctuations. Thus, $I(\omega )d\omega $ is the fraction that contributes between ω and ω + dω.
The main strategy of NMR relaxometry is to predict expressions for $G(t)orI(\omega )$ based on model calculations or computer simulations. Fits of the model theory to the experimental data can then verify or rule out the model. In the latter case, there will be feedback on how the model should be modified to achieve a better match. A common triplet of terms and a popular title of introductory lectures that best characterizes what we are dealing with in this book is the pun ‘fluctuation, correlation, relaxation’. This is the golden thread, so to speak, that we will pursue in the following chapters.
More details on correlation functions of spin interactions and the conjugate spectral densities, and how these functions can be calculated for certain statistical model scenarios of molecular dynamics, are outlined in Chapter 6. It is shown that the properties of these functions can be traced back to specific equations of motion for the processes of interest, including any dynamical constraints that may be relevant. Correlation functions and spectral densities in principle characterize Brownian motions, as far as possible. Therefore, the main goal of NMR relaxometry techniques is to gain experimental access to the complete time evolution of the temporal correlations of spin interactions. A number of typical applications can be found in ref. 4, for example. As far as translational diffusion is concerned, a fieldcycling NMR relaxometry technique is described in detail in Chapter 7, which, in combination with fieldgradient NMR diffusometry,^{5,6 } probes a time scale hardly accessible by nonNMR techniques.
In most cases, NMR relaxometry techniques primarily provide information about the spectral density of the fluctuations contributing to relaxation. However, instead of referring to the frequency domain, a representation via correlation functions, i.e. in the time domain, is more descriptive and reflects the character of the fluctuations more vividly. Therefore, we will prefer to discuss correlation functions instead of spectral densities whenever possible and appropriate.
1.4 Some More Introductory Remarks
Secular spin interactions are primarily responsible for shifts or splittings of spin energy levels, and thus for all spectral variations of nuclear magnetic resonance lines. Essentially, NMR spectroscopy in both liquids and solids is based on the secular parts of the spininteraction Hamiltonians that are reduced, to some degree, by motional averaging.^{3 } In a collective form, secular spin interactions are also responsible for socalled local fields. This phenomenon, and motional averaging of secular spin interactions, will be the subject of Chapter 4. Another effect, for which secular spin interactions are largely or even dominantly responsible, is the relaxation of transverse magnetization components or, more generally, the attenuation of spin coherences. Transverse relaxation will be discussed in more detail in Section 5.2.
On the other hand, nonsecular spin interactions are responsible for spin–lattice relaxation, i.e. relaxation of the longitudinal magnetization component. The fundamental origin of spin–lattice relaxation is the induction of spin transitions by fluctuations of the nonsecular terms of the spininteraction Hamiltonians. This class of processes is discussed in detail in Section 5.1.
In Chapter 2, the main nuclear magnetic relaxation methods are discussed. Some of these techniques are predominantly based on nonsecular spin interactions, while others are grounded in the secular part of the couplings. Taken together, this repertoire of techniques covers a wide range of the conjugate time and frequency variables that can be studied to elucidate molecular dynamics in complex materials.
This definition of a vector with three independently determinable components does not conflict with the quantum mechanical rule that different observables cannot be simultaneously quantified if their operators do not have common eigenfunctions. That is, the corresponding operators must commute. In the present case, the dipole moment is based on the spin vector operator $ I \u2192 = \mu \u2192 /(\u210f\gamma )$ , whose components do not commute, as is well known: [I_{x}, I_{y}] = iI_{z}, [I_{y}, I_{z}] = iI_{x} and [I_{z}, I_{x}] = iI_{y}. However, the expectation value vector above refers to an ensemble of dipoles. It does not specify the components of a single dipole.