Introduction

doi:10.1039/BK9781837671915-00001

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Nuclear Magnetic Relaxation and Molecular Dynamics

DOI:

https://doi.org/10.1039/9781837671915

Hardback ISBN:

978-1-83767-191-5

PDF ISBN:

978-1-83767-337-7

EPUB ISBN:

978-1-83767-338-4

Product Type: Textbooks

No. of Pages:

304

Publication date:

05 Jun 2024

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 fluid-filled 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 spin-interaction 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, non-nuclear magnetic relaxation techniques the reader may find it useful to refer to ref. 1.

1.1 Basic Concepts, Relationships, and Definitions

Let us first recapitulate some basic definitions and rules of the quantum mechanical background.² In nuclear magnetic resonance (NMR) experiments, the sample is placed in a magnet that produces an external magnetic flux density

{\vec{B}}_{0}

⁠. This vector is usually assumed to be aligned along the z direction of a Cartesian laboratory frame of reference (see Figure 1.1). The spin vector operator

\vec{I} = (\begin{matrix} I_{x} \\ I_{y} \\ I_{z} \end{matrix})

(1.1)

of the resonant nuclei is defined as a dimensionless quantity: that is, the components I_x, I_y and I_z must be multiplied by

ℏ

⁠, Planck’s constant divided by 2π, in order to obtain components of the dimension ‘angular momentum’.

Figure 1.1

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The basic setup of a nuclear magnetic resonance (NMR) experiment. A magnet (represented as a cross-section) produces the main magnetic field with flux density ${\vec{B}}_{0}$ aligned along the z axis of the laboratory (lab) reference frame. The sample (not shown) is placed in a radio frequency (RF) saddle coil at the middle of the magnet. The saddle coil is tuned by capacitors to the system’s carrier radio frequency and matched to the transmitter and the (phase-sensitive) receiver. The RF coil is used both to apply RF pulses to the sample and to receive NMR signals. The (linearly polarized) RF flux density $2 {\vec{B}}_{1}$ generated by the saddle coil oscillates perpendicular to ${\vec{B}}_{0}$ ⁠. The RF flux density can be split into two counter-rotating, circularly polarized components. Only one of these components is relevant to resonance, namely the one that rotates in the same sense as the Larmor precession. The magnitude of the RF flux density effective for magnetic resonance is therefore only B₁, half of the transmitted RF, as explained in Section 2.1.

There is no analytical expression for spin operators. Instead, they are defined by the eigenvalue equations

{\vec{I}}^{2} ψ = I (I + 1) ψ

(1.2)

and

I_{z} ψ = m ψ

(1.3)

The quantum mechanical wavefunctions ψ that satisfy and are referred to as eigenfunctions of the spin operators conjugated to the eigenvalues I(I + 1) and m, respectively. Depending on the particle species, the spin quantum number I can have half-integer or integer values, such as I = 1/2 (e.g. for protons), I = 1 (e.g. for deuterons), I = 3/2 (e.g. for sodium nuclei) and so on. The eigenvalues m of the spin operator I_z (also called magnetic spin quantum numbers) can take the 2I + 1 discrete values m = −I, −I + 1, ⋯, I − 1, I. For example, the eigenfunctions and magnetic quantum numbers for spins 1/2 are ψ = φ_+1/2, ψ = φ_−1/2 and m = −1/2, m = +1/2, respectively.

The analytic form of the eigenfunctions φ_m is not specifiable. Its information content is simply that the magnetic quantum number of this state has the value m. The vector operator of the magnetic dipole moment of nuclei of spin I is defined by

\vec{μ} = γ \vec{I} ℏ

(1.4)

where γ is the gyromagnetic ratio of the respective nuclide. γ is positive for protons, but negative for electrons and certain other nuclear species.

Solving the time-independent Schrödinger equation

H_{0} ψ = E ψ

(1.5)

for the Zeeman Hamiltonian

H_{0} = - \vec{μ} \cdot {\vec{B}}_{0} = - ℏ γ \vec{I} \cdot {\vec{B}}_{0} = - ℏ γ B_{0} I_{z}

(1.6)

of a magnetic dipole with the dipole moment vector operator

\vec{μ}

leads to the Zeeman energy eigenvalues

E_{m} = - ℏ γ m B_{0}

(1.7)

Obviously, this result is a consequence of the eigenvalue relation given in . Note that the eigenfunctions φ_m of the operator I_z are also eigenfunctions of the Hamiltonian of the Zeeman energy

H_{0}

⁠.

The Zeeman energy splitting is (see Figure 1.2)

Δ E = ℏ γ B_{0}

(1.8)

Magnetic resonance is achieved by irradiating the spin-bearing particles with energy quanta

Δ E = ℏ ω_{0}

⁠. This requires angular radio frequencies equal to the magnetic resonance frequency

ω_{0} = | γ | B_{0}

(1.9)

Figure 1.2

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Zeeman energy levels of isolated spins 1/2. The eigenvalues of the spin operator component I_z are given by the magnetic quantum numbers m = +1/2 and m = −1/2. They are referred to as ‘spin-up’, φ_+1/2, and ‘spin-down’, φ_−1/2, states, respectively. The Zeeman splitting is $Δ E = ℏ γ B_{0}$ ⁠, so that the angular resonance frequency results in ω₀ = γB₀. The signs in this scheme are chosen for positive values of the gyromagnetic ratio γ.

Assuming spins 1/2, a general solution of the time-dependent Schrödinger equation,

i ℏ \frac{\partial ψ}{\partial t} = H_{0} ψ

(1.10)

\begin{array}{l} ψ (t) & = & c_{+ 1 / 2} φ_{+ 1 / 2} \exp {- i \frac{E_{+ 1 / 2} t}{ℏ}} + c_{- 1 / 2} φ_{- 1 / 2} \exp {- i \frac{E_{- 1 / 2} t}{ℏ}} \\ = & \underset{dipole aligned along {\vec{B}}_{0} for pos . γ}{\underset{︸}{c_{+ 1 / 2} φ_{+ 1 / 2} \exp {+ i \frac{ω_{0} t}{2}}}} + \underset{dipole directed against {\vec{B}}_{0} for pos . γ}{\underset{︸}{c_{- 1 / 2} φ_{- 1 / 2} \exp {- i \frac{ω_{0} t}{2}}}} \end{array}

(1.11)

where c_+1/2 and c_−1/2 are constants obeying the normalization condition for the eigenstate probabilities, i.e. |c_+1/2|² + |c_−1/2|² = 1. We are already familiar with the quantities φ_+1/2 and φ_−1/2 as the time-independent eigenfunctions for the magnetic quantum numbers m = +1/2 and m = −1/2, respectively. Note that the magnitudes and the phase factors of the coefficients c_+1/2 and c_−1/2 can vary from particle to particle in a mixed ensemble. For now, we assume that all spins under consideration are subject to wavefunctions that are identical in this regard.

The expectation values of the Cartesian components 〈μ_x〉, 〈μ_y〉, 〈μ_z〉 of the vector operator of the magnetic dipole moment,

\vec{μ} = ℏ γ \vec{I}

⁠, are then calculated for this particular wavefunction (and in the absence of relaxation and spin coupling processes) as

\begin{array}{l} 〈 μ_{x} 〉 & = & ℏ γ 〈 ψ | I_{x} | ψ 〉 = ℏ γ c_{+ 1 / 2}^{⁎} c_{- 1 / 2} \cos (ω_{0} t) \\ 〈 μ_{y} 〉 & = & ℏ γ 〈 ψ | I_{y} | ψ 〉 = - ℏ γ c_{+ 1 / 2}^{⁎} c_{- 1 / 2} \sin (ω_{0} t) \\ 〈 μ_{z} 〉 & = & ℏ γ 〈 ψ | I_{z} | ψ 〉 = \frac{1}{2} ℏ γ (| c_{+ 1 / 2} |^{2} - | c_{- 1 / 2} |^{2}) \end{array}

(1.12)

where we have used the standard ladder operator relations

\begin{array}{l} I^{+} = I_{x} + i I_{y}, & I^{-} = I_{x} - i I_{y} \\ I_{x} = \frac{1}{2} (I^{+} + I^{-}), & I_{y} = \frac{1}{2 i} (I^{+} - I^{-}) \end{array}

(1.13)

and the orthonormality properties of the eigenfunctions

〈 φ_{+ 1 / 2} | φ_{+ 1 / 2} 〉 = 1, 〈 φ_{- 1 / 2} | φ_{- 1 / 2} 〉 = 1, 〈 φ_{+ 1 / 2} | φ_{- 1 / 2} 〉 = 0, 〈 φ_{- 1 / 2} | φ_{+ 1 / 2} 〉 = 0

(1.14)

Applying the ladder operators to the eigenfunctions of spins I = 1/2 yields

I^{+} φ_{- 1 / 2} = φ_{+ 1 / 2}, I^{+} φ_{+ 1 / 2} = 0, I^{-} φ_{- 1 / 2} = 0, I^{-} φ_{+ 1 / 2} = φ_{- 1 / 2}

(1.15)

Another condition used in the derivation of the expressions in is that expectation values of observables must be real, that is, the imaginary parts must vanish. This is the case for

c_{+ 1 / 2} c_{- 1 / 2}^{⁎} = c_{- 1 / 2} c_{+ 1 / 2}^{⁎}

(1.16)

(and for the conjugate complex form of this equation). The initial value of , and hence those of the initial values of the expectation values in , have been arbitrarily but conveniently assumed to be

〈 μ_{x} (0) 〉 = ℏ γ c_{+ 1 / 2}^{⁎} c_{- 1 / 2}

and 〈μ_y(0)〉 = 0. Based on the expectation values of the dipole components given in , an expectation value vector

〈 \vec{μ} 〉 = (\begin{matrix} 〈 μ_{x} 〉 \\ 〈 μ_{y} 〉 \\ 〈 μ_{z} 〉 \end{matrix})

(1.17)

can be formally defined.^† This vector has the same precession behavior as a classical magnetic dipole. Larmor precession is illustrated in this way in Figure 1.3.

Figure 1.3

Larmor precession of the expectation value vector of a nuclear dipole moment, 〈 μ → 〉 . The angular Larmor frequency vector is ω → L = − γ B → 0 . The gyromagnetic ratio γ is assumed to be positive. Based on the known coordinate transformation rules, the laboratory frame (x,y,z) shown in Figure 1.1 can be transformed into a rotating frame (x′,y′,z′), which rotates synchronously and in the same sense as the Larmor precession around the z ≡ z′ axis (for a formalism, see ref. 2, for example). Relative to this (resonantly) rotating frame, the external flux density B → 0 is effectively compensated. The precessing laboratory-frame vector, 〈 μ → 〉 , is transformed into the stationary rotating-frame vector 〈 μ → 〉 ′ .

View large Download slide

Larmor precession of the expectation value vector of a nuclear dipole moment, $〈 \vec{μ} 〉$ ⁠. The angular Larmor frequency vector is ${\vec{ω}}_{L} = - γ {\vec{B}}_{0}$ ⁠. The gyromagnetic ratio γ is assumed to be positive. Based on the known coordinate transformation rules, the laboratory frame (x,y,z) shown in Figure 1.1 can be transformed into a rotating frame (x′,y′,z′), which rotates synchronously and in the same sense as the Larmor precession around the z ≡ z′ axis (for a formalism, see ref. 2, for example). Relative to this (resonantly) rotating frame, the external flux density ${\vec{B}}_{0}$ is effectively compensated. The precessing laboratory-frame vector, $〈 \vec{μ} 〉$ ⁠, is transformed into the stationary rotating-frame vector $〈 \vec{μ} 〉'$ ⁠.

The time dependences of the components given in suggest that the expectation value vector

〈 \vec{μ} 〉

rotates on a cone about

{\vec{B}}_{0}

with the (angular) Larmor frequency

{\vec{ω}}_{L} = - γ {\vec{B}}_{0}

(1.18)

The magnitude of the Larmor frequency is obviously equal to the angular resonance frequency given in . Note that the sense of rotation of the Larmor precession changes with the sign of the gyromagnetic ratio.

The wavefunction given in is idealized because it suggests identical magnitudes and phase factors of the coefficients c_+1/2 and c_−1/2 for all members of the spin ensemble under consideration. More realistically, there will be distributions of these coefficients: that is, ensemble averages are needed. is thus converted into

\begin{array}{l} \bar{〈 μ_{x} 〉} & = & ℏ γ \bar{c_{+ 1 / 2}^{⁎} c_{- 1 / 2}} \cos (ω_{0} t) \\ \bar{〈 μ_{y} 〉} & = & - ℏ γ \bar{c_{+ 1 / 2}^{⁎} c_{- 1 / 2}} \sin (ω_{0} t) \\ \bar{〈 μ_{z} 〉} & = & \frac{1}{2} ℏ γ (\bar{| c_{+ 1 / 2} |^{2} - | c_{- 1 / 2} |^{2}}) \end{array}

(1.19)

where the bars indicate means. If the factors

\bar{c_{+ 1 / 2}^{⁎} c_{- 1 / 2}}

are finite, one speaks of spin coherences. That means non-vanishing transverse components imply the presence of such coherences. This is not the case at thermal equilibrium. The phase factors are then uniformly distributed so that

\bar{c_{+ 1 / 2}^{⁎} c_{- 1 / 2}} = 0

⁠. The equilibrium expectation value vector is thus reduced to

{\bar{〈 \vec{μ} 〉}}_{0} = (\begin{matrix} 0 \\ 0 \\ {\bar{〈 μ_{z} 〉}}_{0} \end{matrix})

(1.20)

When dealing with global magnetic properties of samples one usually refers to magnetizations. The magnetization vector is defined by

\vec{M} = \frac{1}{V} \sum_{j} 〈 {\vec{μ}}^{(j)} 〉

(1.21)

where V is the sample volume. The sum index j runs over the quantum mechanical expectation value vectors

〈 {\vec{μ}}_{j} 〉

of all magnetic dipoles in the sample. That is, the sum implies the ensemble averages of the components specified in , and it may include the possible presence of coherences. The magnetization thus has the dimension ‘dipole moment per volume unit’.

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.

In the high-temperature limit not far below room temperature, the mean thermal energy per particle (∝k_BT) is much larger than the Zeeman splitting energy, i.e.

k_{B} T ≫ Δ E = | γ | ℏ B_{0}

(1.22)

where k_B is the Boltzmann constant and T is the absolute temperature. The equilibrium magnetization

{\vec{M}}_{0}

⁠, assumed after a long time under steady-state conditions in this limit, is found to obey Curie’s law with great accuracy:

{\vec{M}}_{0} = \frac{n γ^{2} ℏ^{2} I (I + 1)}{3 k_{B} T} {\vec{B}}_{0}

(1.23)

where n is the number density of the spin-bearing particles. Note that the vector

{\vec{M}}_{0}

is aligned parallel to the quantizing flux density

{\vec{B}}_{0}

⁠. implies that the magnetization components transverse to

{\vec{B}}_{0}

vanish in thermal equilibrium, as suggested by . According to and , the magnitude of the equilibrium magnetization can also be expressed as

M_{0} = \frac{ℏ γ}{2 V} {\sum_{j} ({| c_{+ 1 / 2}^{(j)} |}^{2} - {| c_{- 1 / 2}^{(j)} |}^{2})}_{0}

(1.24)

Note that the ‘high-temperature’ 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 spin-up and spin-down 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.

The transverse magnetization, m_t, is often expressed as a complex quantity

m_{t} = M_{x} + i M_{y}

(1.25)

with the real and imaginary parts M_x and M_y, respectively. In this context, and as mentioned above, the term spin coherence is used, a term that refers to spin ensembles. The observation of a finite transverse magnetization means that the spins of the ensemble precess coherently, with correlated precession phases ensuring non-destructive superposition. Based on , the complex transverse magnetization is given by

\begin{array}{l} m_{t} & = & M_{x} + i M_{y} \\ = & \frac{1}{V} \sum_{j} (〈 μ_{x}^{(j)} 〉 + i 〈 μ_{y}^{(j)} 〉) \end{array}

(1.26)

where the sum includes all dipoles in the sample volume V.

1.2 Bloch’s Equation

‘Relaxation’ is generally defined as the evolution from a non-equilibrium state toward thermal equilibrium. If the components of magnetization, , are initially different from those of the equilibrium magnetization, , they will evolve toward the latter under the influence of fluctuating spin interactions. In principle, one has to distinguish between the relaxation of the longitudinal magnetization and that of the transverse magnetization. This difference is manifested in the laboratory-frame version of the Bloch equation for magnetic relaxation of the magnetization

\vec{M}

with the components M_x, M_y, M_z:

\frac{d \vec{M}}{d t} = \underset{\begin{matrix} Larmor \\ precession \end{matrix}}{\underset{︸}{γ \vec{M} \times {\vec{B}}_{0}}} \underset{transverse relaxation}{\underset{︸}{- \frac{M_{x}}{T_{2}} {\vec{u}}_{x} - \frac{M_{y}}{T_{2}} {\vec{u}}_{y}}} \underset{\begin{matrix} longitudinal \\ relaxation \end{matrix}}{\underset{︸}{- \frac{M_{z} - M_{0}}{T_{1}} {\vec{u}}_{z}}}

(1.27)

{\vec{u}}_{x}, {\vec{u}}_{y}, {\vec{u}}_{z}

are unit vectors along the three Cartesian coordinate axes of the laboratory frame, where

{\vec{B}}_{0}

is aligned along the positive z axis (see Figure 1.1). The longitudinal magnetization component M_z relaxes toward the Curie value M₀ with the spin–lattice relaxation time T₁. Instead of ‘spin–lattice relaxation time’, the synonymous term ‘longitudinal relaxation time’ is often used. However, the term spin–lattice relaxation is more common in the literature regardless of the existence of a ‘lattice’, i.e. the aggregate state of the system to which it applies. Relaxation of the transverse components M_x, M_y is characterized by the transverse relaxation time T₂. The reason for the disparity of the time constants T₁ and T₂ will become clear in Chapter 5 where the spin transitions relevant to magnetic relaxation will be described. Solutions of the vector differential equation given in are specified in Chapter 2 for the various experimental protocols considered in this book.

The Bloch equation in the laboratory-frame 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 so-called local fields is assumed, as it is expected for low-viscous liquids. Chapter 4 provides a detailed discussion of this important issue. Effects due to chemical shift and indirect spin–spin couplings³ 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.

The laboratory-frame version of Bloch’s equation represented by can be simplified by transforming it into a Cartesian frame x′, y′, z′ that rotates in the same sense as the Larmor precession about

{\vec{B}}_{0}

⁠. An illustration is provided in Figure 1.3. More details and further aspects of this transformation are given in and in the Appendix. Relative to the (resonantly) rotating frame of reference, the Bloch equation takes the reduced form

\frac{d \vec{M}'}{d t} = - \underset{˜}{R} \cdot (\vec{M}' - {\vec{M}}_{0}^{'})

(1.28)

where

M_{0}^{'} = M_{0}

⁠. In this version, all time dependences due to Larmor precession are removed. The relaxation rate matrix is defined by

\underset{˜}{R} = (\begin{matrix} {T_{2}}^{- 1} & 0 & 0 \\ 0 & {T_{2}}^{- 1} & 0 \\ 0 & 0 & {T_{1}}^{- 1} \end{matrix})

(1.29)

Expressed in Cartesian components, the rotating-frame magnetization vectors are

\vec{M}' = (\begin{matrix} M_{x'} \\ M_{y'} \\ M_{z'} \end{matrix}) and {\vec{M}'}_{0} = (\begin{matrix} 0 \\ 0 \\ M_{0} \end{matrix})

(1.30)

where

M_{z'} = M_{z}

⁠.

1.3 The Wiener/Khinchin Theorem

In Chapter 2, a number of NMR techniques are described that provide partially complementary experimental protocols for studying molecular fluctuations in the frequency or, equivalently, in the time domain. This equivalence is manifested by the Wiener/Khinchin theorem

J (ω) = \int_{- \infty}^{\infty} G (t) \exp {- i ω t} d t

(1.31)

It relates temporal autocorrelation functions G(t) to spectral densities J(ω) in the form of a pair of Fourier transforms. The conjugate variables are angular frequency ω and time t. Correlation functions of thermally fluctuating functions F(t) are defined as the ensemble average

G (t) = {〈 F (0) F^{⁎} (t) 〉}_{ensemble}

(1.32)

In the present context, the functions F(t) characterize spin interactions, as will be specified in Chapter 3. These functions depend on temporally fluctuating coordinates that define the instantaneous orientation of a spin-bearing molecule and/or its distance to the nearest neighbor. The asterisk in means conjugate complex, since F(t) can be a complex function. Correlation functions of stochastic processes tend to decay with time to zero, G(∞) → 0. A parameter that characterizes this decay is the correlation time. It can be defined by the relation

τ_{c} = \frac{1}{G (0)} \int_{0}^{\infty} G (t) d t

(1.33)

regardless of whether the correlation function is mono-, multi- or non-exponential.

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 the following we will frequently refer to normalized versions of G(t) and J(ω). They are symbolized by the calligraphic letters

G (t) and I (ω)

⁠, and are defined by the properties

G (0) = 1

(1.34)

and

\frac{1}{2 π} \int_{- \infty}^{\infty} I (ω) d ω = G (0) = 1

(1.35)

respectively. Denoted in this way, the Wiener/Khinchin theorem reads

\begin{array}{l} I (ω) & = & \int_{- \infty}^{+ \infty} G (t) \exp {- i ω t} d t \\ = & 2 \int_{0}^{+ \infty} G (t) \cos {ω t} d t \end{array}

(1.36)

The second line of indicates that we are assuming even correlation functions

G (t)

throughout. This time-reversal symmetry is justified for the stochastic processes of interest in the present context.

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 liquid-crystalline domains, are examples of the wide scope addressed when talking about molecular dynamics.

As a stand-alone function—not just the Fourier transform of the autocorrelation function—the spectral density $I (ω)$ is the distribution of angular frequencies in the random ‘noise’ that reflects thermal molecular fluctuations. Thus, $I (ω) d ω$ is the fraction that contributes between ω and ω + dω.

The main strategy of NMR relaxometry is to predict expressions for $G (t) or I (ω)$ 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 field-cycling NMR relaxometry technique is described in detail in Chapter 7, which, in combination with field-gradient NMR diffusometry,^5,6 probes a time scale hardly accessible by non-NMR 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

The total Hamiltonian to which spins are subjected in NMR is composed of the Zeeman term given in and a term representing the couplings and interactions of the spins:

H_{tot} = H_{0} + H_{int}

(1.37)

In the context of relaxometry, the most important examples of spin interactions are dipolar and quadrupolar couplings. Details are described in Chapter 3. In general, spin interactions depend on the structural and dynamical characteristics of the material hosting the spin-bearing molecules. This fact is the key to the overwhelming success of NMR techniques in elucidating structural and dynamic properties of materials.

There are two fundamental features to note in the context of spin interactions. First, the consequence of molecular dynamics due to thermal motions is that the Hamiltonians of spin interactions are randomly fluctuating operators:

H_{int} = H_{int} (t)

(1.38)

Second, spin-interaction Hamiltonians can be divided into secular and non-secular contributions:

H_{int} = H_{int}^{(\sec)} + H_{int}^{(non - \sec)}

(1.39)

The attribute ‘secular’ generally alludes to particularly slow variations of properties and effects. In this sense, the secular part of a spin-interaction Hamiltonian preserves spin states, or only allows the induction of spin transitions that do not alter the total Zeeman spin energy. This is in contrast to its non-secular counterpart, where fluctuations can induce spin transitions associated with energy transfer to or from the so-called lattice. The term ‘lattice’ refers collectively to the mechanical degrees of freedom of the molecular environment.

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 spin-interaction Hamiltonians that are reduced, to some degree, by motional averaging.³ In a collective form, secular spin interactions are also responsible for so-called 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, non-secular 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 non-secular terms of the spin-interaction 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 non-secular 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 $\vec{I} = \vec{μ} / (ℏ γ)$ ⁠, 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.