The quadrupolar interaction

For nuclei with spin > 1/2, another form of coupling exists, in addition to the scalar and dipolar coupling discussed above, known as the quadrupolar coupling. This interaction, unlike the dipolar interaction, is an electric rather than magnetic interaction. Spins which fall into this category, accounting for around three quarters of all NMR active nuclei, possess a nuclear electric quadrupole moment (eQ), in addition to a magnetic dipole and electric monopole moment (shown in Fig. 2.10), which interacts with the so- called electric field gradient (EFG) traversing the nucleus. The quadrupole moment (Q) of a given nucleus is a fixed quantity and can take positive (27_{Al, I = 5/2, Q = 14.66 fm}2_{) or negative (}17_{O, I = 5/2,}

Q = 2.558 fm2_{) values. These values arise from charge distribution within a given nucleus.}

Figure 2.10 Schematic representation of the expansion of charge distribution (say that in a nucleus) as a series of multipoles. Adapted from Duer.(165)

Importantly, the quadrupolar interaction is significantly larger (MHz) than the other internal interactions discussed in this chapter (kHz to Hz). However, the Zeeman interaction is (usually) still approximately ten times greater than the strongest quadrupolar interaction. Hence, the interaction can still be treated as a perturbation of the Zeeman energy. The Hamiltonian, in Cartesian coordinates, can be written as:

𝐻̂_𝑄 = 𝑒𝑄

2𝐼(2𝐼 − 1)ћ𝐼̂𝑉̃𝐼̂, (2.116)

𝑉̃ = (

𝑉𝑥𝑥 𝑉𝑥𝑦 𝑉𝑥𝑧

𝑉𝑦𝑥 𝑉𝑦𝑦 𝑉𝑦𝑧

𝑉𝑧𝑥 𝑉𝑧𝑦 𝑉𝑧𝑧

). (2.117)

For a nucleus at a site of cubic symmetry, the EFG is zero. However, local environments in real systems tend to lower the symmetry about a given nucleus. In order to describe the nature of the interaction, it is necessary to describe the anisotropy of the EFG. However, it is more convenient to describe the nuclear quadrupole coupling constant (𝐶𝑄), which is proportional to the anisotropy, with a low 𝐶𝑄 value indicative of high symmetry about the nucleus in question. The components of the tensor 𝑉̃, in the PAS, are described by the parameters 𝐶𝑄 and 𝜂𝑄:

𝐶_𝑄 =𝑒2𝑞𝑄 ℎ = 𝑒𝑄𝑉𝑧𝑧 ℎ , (2.118) and 𝜂_𝑄 =𝑉𝑥𝑥− 𝑉𝑦𝑦 𝑉𝑧𝑧 , (2.119) where |𝑉𝑥𝑥| ≤ |𝑉𝑦𝑦| ≤ |𝑉𝑧𝑧|. (2.120)

The asymmetry parameter, 𝜂𝑄, describes the strength of the field relative to three orthogonal directions, x, y and z. An 𝜂_𝑄 of zero represents an EFG that is axially symmetric. A value of one indicates high asymmetry.

The magnitude of the quadrupolar coupling is such that, for systems with small quadrupolar interactions, the interaction may be sufficiently treated as a first-order perturbation to the Zeeman energy. However, when the value of 𝐶𝑄 is large, higher order perturbations, specifically second-order, must be considered. Since, in all practical cases for NMR, the quadrupolar interaction is smaller than the total Zeeman energy, the quadrupolar Hamiltonian must be rotated from its PAS into the laboratory frame. Expressing the Hamiltonian in spherical tensor form, the expression, in the PAS, may be written as:

𝐻̂_𝑄𝑃𝐴𝑆 ₌ 2𝜋

2𝐼(2𝐼 − 1)(𝐴20𝑃𝐴𝑆𝑇̂20− 𝐴22𝑃𝐴𝑆𝑇̂2−2− 𝐴2−2𝑃𝐴𝑆𝑇̂22), (2.121)

with the spatial terms defined as:

𝐴₂₀𝑃𝐴𝑆 _{= √}3 2𝐶𝑄, (2.122) 𝐴₂₂𝑃𝐴𝑆 _{= 𝐴} 2−2 𝑃𝐴𝑆 ₌1 2𝜂𝑄𝐶𝑄. (2.123)

In the laboratory frame, the Hamiltonian becomes:

𝐻̂_𝑄𝐿₌ 2𝜋

2𝐼(2𝐼 − 1)(𝐴20𝐿 𝑇̂20− 𝐴𝐿21𝑇̂2−1− 𝐴𝐿2−1𝑇̂21+ 𝐴𝐿22𝑇̂2−2+ 𝐴2−2𝐿 𝑇̂22). (2.124)

Under a first-order perturbation, the secular approximation holds, hence only 𝐴20𝐿 terms need to be considered:

𝐴₂₀𝐿 _{= 𝐴} 20 𝑃𝐴𝑆_𝐷

002 + 𝐴22𝑃𝐴𝑆𝐷202 + 𝐴𝑃𝐴𝑆2−2𝐷−202 . (2.125)

Applying the relevant rotation matrices yields the following expression for the spatial component of the quadrupolar Hamiltonian in the laboratory frame:

𝐴₂₀𝐿 _{= √}3

2 𝐶_𝑄 2𝐼(2𝐼 − 1)

1

2[(3 cos2𝛽𝑃𝐿− 1) + 𝜂𝑄sin2𝛽𝑃𝐿cos 2𝛼𝑃𝐿]. (2.126)

Considering spatial and spin components, the first-order quadrupolar Hamiltonian is therefore:

𝐴₂₀𝐿 _{= √}3

2 𝐶_𝑄 2𝐼(2𝐼 − 1)

1

𝐸_𝑚(1)= ⟨𝑚|𝐻̂1_{|𝑚⟩. (2.128)}

Substitution of the first-order Hamiltonian yields:

𝐸_𝑚(1)= √3 2 𝐶_𝑄 2𝐼(2𝐼 − 1) 1 2(3𝑚2− 𝐼(𝐼 + 1))[(3 cos2𝛽𝑃𝐿− 1) + 𝜂_𝑄sin2_𝛽 𝑃𝐿cos 2𝛼𝑃𝐿]. (2.129)

Figure 2.11 Schematic representation of the perturbation to the Zeeman energy levels for a spin-1 nucleus, such as 14_{N. Both}

first- and second-order perturbations are considered. Note that the first-order perturbation does not affect the total transition energy (m = −1 ↔ m = 1), however the individual SQ transitions (m = −1 ↔ m = 0, m = 0 ↔ m = 1) have different energies. The second-order perturbation does affect the SQ transition energy. Note that these perturbations are exaggerated

to more suitably demonstrate the change to the energy levels.

For 14_{N, which has spin 1, there are three possible energy levels} (𝑚 = −1, 0, 1) _{and hence two possible}

transitions (𝑚 = −1 ↔ 𝑚 = 0, 𝑚 = 0 ↔ 𝑚 = 1) as shown in Fig. 2.11. The total transition energy

(𝑚 = −1 ↔ 𝑚 = 1) is unchanged relative to the total Zeeman energy. However, the individual SQ transitions have different energies relative to each other.

For the majority of quadrupolar nuclei, in which the value of 𝐶𝑄 is such that the second-order perturbation to the Zeeman energy must be considered, the energy may be written as:

𝐸_𝑚(2)= ∑ ⟨𝑛|𝐻̂

1_{|𝑚⟩⟨𝑚|𝐻̂}1_|𝑛⟩

𝐸_𝑛(0)− 𝐸_𝑚(0)

𝑚≠𝑛

. (2.130)

For second-order perturbations, the secular approximation no longer holds, hence all 𝐴2−𝑚,…,𝑚𝐿 terms must be considered. These terms can be calculated in the same manner as for the 𝐴₂₀𝐿 terms. Calculation of the second-order perturbation is performed by multiplying 𝐴2𝑚2 spatial terms. The multiplication of

𝐴2_{2𝑚. 𝐴} 2𝑚

2 _{yields zeroth-, second- and fourth-rank Wigner rotation matrices, which have important} implications for the effect of MAS upon quadrupolar lineshapes. Consequently, the second-order perturbation may now be written as:

𝐸_𝑚(2)= − ( 𝐶𝑄 4𝐼(2𝐼 − 1)) 2 ₂ 𝜔₀𝑚 [([𝐼(𝐼 + 1) − 3𝑚2]𝐷000(𝑄)) + ([8𝐼(𝐼 + 1) − 12𝑚2_{− 3]𝐷} 202(𝑄)) + ([18𝐼(𝐼 + 1) − 34𝑚2_{− 5]𝐷} 404(𝑄))], (2.131) where 𝐷₀₀0(𝑄)= −1 5(3 + 𝜂𝑄2), (2.132) 𝐷₂₀2(𝑄)= 1

28[(𝜂𝑄2− 3)(3 cos2𝛽𝑃𝐿− 1) + 6𝜂𝑄2sin2𝛽𝑃𝐿cos 2𝛼𝑃𝐿], (2.133)

𝐷₄₀4(𝑄)=1 8[(

1

140(18 + 𝜂𝑄2)(35 cos4𝛽𝑃𝐿− 30 cos2𝛽𝑃𝐿+ 3)) + (3

7𝜂𝑄sin2𝛽𝑃𝐿(7 cos2𝛽𝑃𝐿− 1) cos 2𝛼𝑅𝐿) + (1

4𝜂𝑄2sin4𝛽𝑃𝐿cos 4𝛼𝑃𝐿)].

(2.134)

Under a second-order perturbation, the energy levels presented in Fig. 2.10 are perturbed further, relative to the first-order case. The magnitude of the second-order perturbation is inversely proportional to 𝜔0, hence experimentally, the effects of the quadrupolar interaction are reduced by moving up in magnetic field strength, e.g., from 14.1 T to 20.0 T.

The zeroth-rank term, being isotropic, adds a second isotropic shift to the spectra of quadrupolar nuclei, alongside the isotropic chemical shift. This so-called isotropic second-order quadrupolar shift, 𝛿_𝑖𝑠𝑜𝑄 , is defined, in the ppm scale for a (𝑚 → 𝑚 − 1) transition, as:

𝛿_𝑖𝑠𝑜𝑄 = − (3 40) ( 𝑃_𝑄 𝜈0) 2_{[𝐼(𝐼 + 1) − 9𝑚(𝑚 − 1) − 3]} [𝐼2_{(2𝐼 − 1)}2_] × 106. (2.135)

When 𝐼 = 1 and 𝑚 = 0, 1, the expression is simplified to, 𝛿_𝑖𝑠𝑜𝑄 = (3 40) ( 𝑃_𝑄 𝜈₀) 2 × 106_{, (2.136)}

where 𝑃𝑄 is defined as:

𝑃_𝑄 = 𝐶_𝑄√1 +𝜂𝑄

2

3 , (2.137)

and is known as the quadrupolar product.

The second-rank term is anisotropic and hence is removed when the system is under the influence of MAS. Residual broadening due to the quadrupolar interaction is retained however, since the fourth- rank term also has an anisotropic dependence. It is not possible to average both second- and fourth-rank terms simultaneously by rotation about a single angle (i.e. under MAS). It is possible, however, to remove this contribution to the line broadening by spinning the sample at a second angle relative to the magic angle. This concept forms the basis of the double rotation (DOR) method,(168) which is not considered further in this thesis.

Chapter 3

: Experimental methods in solid-state NMR