Solid-State NMR Interactions 1 Introduction

Interactions in solid-state NMR can cause broadening of lineshapes, making the extraction of NMR parameters, and even determining the number of resonances present, difficult. These interactions, such as dipolar (Section 2.10.3) and quadrupolar couplings (Section 2.12), have an orientation dependence, and unless all molecules/crystals in the sample have the same orientation, will result in line broadening. In solution-state NMR, however, the rapid molecular motion and tumbling means that the anisotropic parts of these interactions are ‘averaged out’ to zero, and acquired spectra often have sharp, narrow lineshapes, reflecting just the isotropic or averaged components of the interactions. Whilst lineshape broadening may be problematic in the acquisition of solid-state NMR spectra (in particular, those with spin I > 1/2 having quadrupolar coupling contributions), several techniques may be employed to remove the anisotropic broadenings caused by these interactions by averaging their orientation dependence to zero. Such techniques include magic-angle spinning (MAS), double rotation (DOR) NMR and multiple-quantum (MQ) MAS experiments, outlined in Sections 2.11, 2.13 and 2.14, respectively.

The interactions that affect solid-state NMR spectra can be described by a Hamiltonian, H, expressed for general interactions as

H = I!. R!. X , (2.31)

where I is a spin angular momentum operator, R is a second-rank Cartesian tensor describing the interaction and X is an additional spin operator or the magnetic field. Each of the terms may be expanded in matrix form

H = (IxIyIz) Rxx Rxy Rxz Ryx Ryy Ryz Rzx Rzy Rzz Xx Xy Xz . (2.32)

H = !HZ + Hrf+ HQ + HCS+ HD+ HJ , (2.33)

where HZ is the Hamiltonian of the Zeeman interaction, Hrf the radiofrequency pulse, HCS the chemical shielding, HQ is the quadrupolar coupling, HD the dipolar coupling and HJ the scalar coupling. It is worth noting that the interactions above can be divided into two distinct categories: external (HZ and Hrf) and internal (HQ, HCS, HD and HJ), where it is the internal interactions that contribute to the observed line broadening in NMR spectrum. It can be difficult to express this easily for a powder distribution of crystallites, where all possible crystallite orientations are simultaneously present. Interactions are usually expressed in the principal axis system (PAS) - a frame where the interaction tensor is diagonal, as shown schematically in Fig. 2.13. This allows us to define the magnitude and asymmetry of the interaction, while the rotation back to the laboratory frame provides the orientation dependence for each individual crystallite.

Much of the work in this thesis will focus on the study of quadrupolar nuclei (I > 1/2) where the quadrupolar coupling will have the largest effect on the spectral lineshapes (after the Zeeman interaction). Whilst other interactions also contribute to lineshape broadening, their effects are substantially smaller, and as such, only a brief overview of these interactions will be given in this thesis.

2.10.2. Chemical Shift Anisotropy

As described previously, in an NMR experiment, the field experienced by a spin is not the same as the applied field, B0, as the electrons around the nucleus generate their own small electric field. The shielding constant, σ, was previously introduced as an isotropic scalar component. However, for solids, σ is anisotropic and, therefore, represented by a tensor (i.e., it is orientation dependent), and described in the PAS by

σ11 0 0

0 σ22 0

0 0 σ33

This shielding is usually described in terms of three parameters, the isotropic shielding, σiso, the shielding anisotropy, ΔσCS and asymmetry, ηCS, where each is

given by σiso = 1 3 (σ11+ σ22+ σ33) (2.35a) ΔσCS = σ33– σiso (2.35b) η_CS = σ11 – σ22 σ₃₃–σ_iso , (2.35c)

such that 0 ≤ ηCS ≤ 1.In the laboratory frame, with B0 defined along the z axis, the

secular approximation can be applied and the chemical shift Hamiltonian reduces to

HCS = γ B0σ33Iz , (2.36)

i.e., dependent upon the σ33 component of the laboratory frame tensor. This can be

determined by considering a rotation from the PAS

σ33 = σiso + ( ∆σCS 2 ) [(3 cos2θ–1) ηCS (sin 2_{θ cos 2φ)] ,} (2.37)

where θ and φ specify the crystallite orientation. The different crystallites will, therefore, have a different σ33 and, therefore, ultimately a different chemical shift,

resulting in broadened lineshapes. In solution, rapid tumbling motion averages the anisotropy, leaving just the isotropic component, σiso.

2.10.3. Dipolar Coupling

The dipolar interaction is a through-space magnetic spin-spin coupling, decreasing in magnitude with the cube of the inverse distance between the spins. Each nucleus produces a small localised magnetic field which can interact with the dipole moments of other nearby nuclei. The dipolar coupling interaction is orientation dependent, and is therefore anisotropic. Similar to the CSA, this interaction is averaged to zero in solutions due to rapid molecular tumbling.

For two spins, j and k, separated by a distance, rjk, the dipolar coupling constant in the PAS, ωDPAS, is given by

ω_DPAS ₌ –µ0 4π . γ_jγ_kħ r_jk3 , (2.38)

where µ0 is the permittivity of free space. It can be seen that ωDPAS is inversely proportional to the separation cubed, and this coupling constant (in the PAS) is used to determine the magnitude of the dipolar interaction for a crystallite/molecule included at an angle θjk to B0 by x y z 11 22 33

(a)

(b)

Figure 2.13: (a) Crystallite in the standard laboratory frame of reference and (b) crystalline in the PAS.

ωD = ωDPAS (3 cos2θjk –1) ,

(2.39)

where θ describes the orientation of the internuclear vector to B0. Not only, therefore, is the dipolar coupling constant, ωD, dependent upon the spin-spin separation, it is also has an orientation dependence proportional to 3cos2 θ – 1. It is important to remember, however, that the dipolar interaction is not only present between two spins, but between many more spins in a sample, and this contributes significantly to the line broadening observed in NMR spectra, especially for nuclei with high γ and high abundance.

2.10.4. Scalar Coupling

The scalar (or J) coupling is a through-bond electron-mediated spin-spin coupling on the order of 1 – 103 _{Hz, making this a relatively small interaction. The interaction is} described by a tensor with both isotropic and anisotropic components, where the latter is orientation dependent. The magnitude of the coupling depends upon the number of bonds between the two spins, and generally decreases with distance. Experimentally, the anisotropy of the J coupling is not normally observed directly since it cannot be distinguished from the substantially larger dipolar coupling. Isotropic J couplings can be resolved if they are greater than the inherent linewidths.