Nuclear Spin Interactions in the Solid State

3.1.1. Basic Concepts

Atomic nuclei are comprised of protons and neutrons. In case when the number of protons and neutrons are both even, the nuclei are NMR inactive. On the other hand, when nuclei have an odd number of neutrons and/or protons, the nuclei are said to be NMR active. Each NMR active nuclei possesses a magnetic moment:

𝜇̂ = 𝛾Î (3.1)

here  is the gyromagnetic (or magnetogyric) ratio which is a constant for a given nucleus (T-1 _s-1_{) and Î is a spin angular momentum. Î is a vector quantity; its direction and magnitude} are quantised, and its units are ℏ (Planks constant). The spin angular momentum is characterised by its spin quantum number 𝐼 which is a constant for a given nucleus. For example, 𝐼 =1

2 for

1_H, 13_C, 15_{N, and 𝐼 = 1 for} 2_{H and} 14_{N; nuclei with 𝐼 = 0 are spin inactive,} e.g. 12_{C and} 16_{O, and cannot be used for NMR spectroscopy.}

When a sample is placed in a static magnetic field, B0, the spins align themselves in such a way that there is a small net magnetisation, M, generated. This net magnetisation precesses around the magnetic field at Larmor frequency:

𝜔0= −𝛾𝐵0 (3.2)

This net magnetisation can be manipulated by using radio-frequency (RF) pulses. The most important part of the NMR experiments is the nuclear spin interaction Hamiltonian, which describes the energy of interactions of the nuclear spins with their environment (external interactions) and with each other (internal interactions within the sample). The overall Hamiltonian describing the energy of a nuclear spin can be written as:

ℋ̂ = ℋ̂ + ℋ𝑍 ̂ + ℋ𝑅𝐹 ̂ + ℋ𝐷𝐷 ̂ + ℋ𝐶𝑆 ̂ + ℋ𝑄 ̂ 𝐽 (3.3)

ℋ̂ represents the Zeeman interaction, the interaction between the nuclear spin and the 𝑍

applied magnetic field of the NMR experiment; ℋ̂ describes effects of RF pulses applied in 𝑅𝐹

the NMR experiment on the nuclear spin; ℋ̂ describing dipolar coupling between the 𝐷𝐷

33 the electron density surrounding the nucleus (which also interacts with the applied magnetic field) with the nucleus; ℋ̂ is the electric quadrupole coupling interaction, which is only relevant 𝑄

for spins with I > 0, and ℋ̂ is the scalar or J-coupling, an electron-mediated through-bond 𝐽

coupling between nuclear spin interaction. The dipolar, shielding, quadrupole and scalar coupling are all anisotropic interactions in general, i.e. their magnitude depends on the orientation of the molecule containing the nuclear spin, with respect to the applied magnetic field. The work outlined in this thesis is mainly concerned with the shielding and dipolar interactions. These interactions are described in further detail in the following sections.

3.1.2. Chemical Shielding

In NMR spectroscopy, the main interaction is between the observed nuclei and the magnetic field. The distribution of electrons around the nucleus is not spherical, it is anisotropic, and hence different orientations of the electron distribution will interact with the applied magnetic field (B0) differently. The electrons are said to be shielding a nucleus from the magnetic field or exerting a local magnetic field on the nucleus. The shielding effect from core electrons leads to an observable chemical shift in the NMR spectrum. This means that different nuclei which have different chemical environments will have different chemical shifts, for example, glycine molecule has two different carbon atoms, and as a result, they will be separated in the 13_C NMR spectra (Figure 3.1.).

Molecular tumbling in the solution-state averages out the anisotropy of the electron distributions. This allows the molecule to be treated as being spherically symmetric, giving rise to one chemical shift in the NMR spectrum (Figure 3.1. (a)). However, the lack of tumbling in the solid-state affords a line-shape that is obtained as a superimposition of all molecular orientations with respect to the magnetic field. This superposition gives a very broad peak in the NMR spectrum with a characteristic powder pattern (Figure 3.1. (b)). This orientation dependence is called chemical shift anisotropy (CSA). Each differently oriented nucleus will experience a slightly different degree of chemical shielding and will exhibit a difference in chemical shift. As a result, the NMR spectrum will display a broad line that represent all possible orientations of all the nuclei within the sample studied. The shape of the line will further depend on the symmetry of the nuclei.

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Figure 3.1. Simulated 13_{C NMR spectra of} -glycine using SIMPSON simulation package.147 _{(a) A}

solution-state NMR spectrum showing isotropic chemical shifts of carbonyl C=O and C signals at 178 and 42 ppm, respectively. (b) The powder line-shape of the carbonyl C=O and C nuclei.

3.1.3. Dipolar Interaction

As it was mentioned before, any nuclear spin possesses a magnetic moment (Equation 3.1), that generates a small, local magnetic field. If two spins are close to one another, they can interact directly. This through-space interaction is known as dipole-dipole or dipolar coupling. This interaction will depend on the distance between the two nuclei, the orientation of the intermolecular vector with respect to B0 and the nuclei type. The dipolar coupling between two spins I and S can be written as:

𝑑 = ℏ (

)

𝛾

𝛾

where 𝜇0 is the vacuum permeability, 𝛾𝐼 and 𝛾𝑆 are gyromagnetic ratios of spins I and S,

respectively, 𝑟 is a distance between spins I and S, and 𝜃 is an angle between the internuclear vector and the applied magnetic field, Bo. The dipolar coupling between spins can be homo- and heteronuclear, where spins I and S are the same species or different, respectively. The dipolar coupling is a function of an internuclear separation, therefore it can be employed as a useful probe for measuring the internuclear distances in a system. This approach is widely adopted in the studies of different materials, including organic, inorganic and biological structures. Furthermore, this spatial separation and interaction of the spins is exploited by multiple NMR experiments used in this thesis.

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3.1.4. Quadrupolar Interaction

Nuclear spins with 𝐼 > 1/2 are referred to as quadrupolar nuclei. These spins possess a nuclear electric quadrupolar moment, that arises from an asymmetric distribution of a charge in the nucleus.141,148 _{Quadrupolar nuclei will interact not only with external magnetic field of the} NMR experiment and all local magnetic fields, but also with any electric field gradient (EFG) present at the nucleus. The electron density surrounding a quadrupolar nucleus typically provides an electric field gradient unless the nucleus is at a site of cubic symmetry.

The work carried out in this thesis is predominantly centred around interactions concerning dipolar and shielding interactions as discussed previously. Experiments involving quadrupolar nuclei were used in this work for performing relaxation experiments on a reference sample and will be discussed in more detail in the Section 4.2.2.