Dipolar coupling

Nuclei placed within an external magnetic field will generate a secondary magnetic field. Such a field will interact directly with other fields from other nuclei, through space, with this interaction commonly referred to as the dipole – dipole interaction or dipolar coupling. Classically, this interaction can be described as the interaction between pairs of bar magnets. Note that this interaction is quite different from the indirect J coupling, which is mediated via electrons.

In solution-state NMR, the effect of dipolar couplings are not observed since the interaction is averaged to zero by fast molecular tumbling (since the dipolar coupling tensor is traceless and hence has only an anisotropic component). In the solid state however, for spin ½ nuclei, the magnitude of the interaction usually forms the major contribution to the observed line broadening. That being said, since the dipolar coupling is a direct interaction between nuclei, mediated through space, the interaction is intrinsically dependent upon the intermolecular separation between sites, and hence may be exploited to yield the structural constraints of the system.

If we now consider the quantum mechanical case for a simple isolated spin ½ pair, I and S, there exist four possible Zeeman transition states which in simplistic terms correspond to whether a spin is aligned with or against the external field. This is demonstrated in Fig. 2.6.

Figure 2.6 Energy level diagrams for a spin ½ pair interacting via a dipolar coupling for both the homonuclear and heteronuclear case. Note that transitions between Zeeman eigenstates correspond to coherence changes.

Transitions between the |𝛼𝛼⟩ and |𝛽𝛽⟩ energy level is referred to as double-quantum (DQ) coherence, whilst those between |𝛼𝛽⟩ and |𝛽𝛼⟩ correspond to a zero-quantum (ZQ) transition. All other possible transitions are referred to as single-quantum (SQ) coherence. Note the difference between homonuclear i.e. 1_{H –} 1_{H and heteronuclear i.e.} 1_{H –} 13_C, |𝛼𝛽⟩ _and |𝛽𝛼⟩ _{energy levels. In the homonuclear case these}

levels have essentially degenerate energy. However in the heteronuclear case the energy difference can be of the order of 100s of MHz, meaning that the dipole-dipole interaction (of kHz magnitude) is never sufficient to drive ZQ polarisation transfer between these states. In order to solve this problem, experimentally, one must employ double-resonance pulse sequences such as cross-polarisation (CP) MAS, which transfers polarisation from, say, protons to the lower gamma nuclei. Such methods will be discussed in detail in chapter 3.

The dipolar Hamiltonian, in Cartesian coordinates, is written as:

𝐻̂_𝐷= 2 ∑ 𝐼̂_𝑖𝐷̃𝑆̂_𝑗

𝑖<𝑗

, _(2.99)

where 𝐼̂𝑖 and 𝑆̂𝑗 represent the coupled spins. Since the dipolar coupling strength, for a 1H – 1H pair, is usually on the order of 10s of kHz, the Zeeman interaction is still the dominant interaction influencing the system. Hence, it is necessary to rotate from the dipolar PAS, aligned along the internuclear vector between two coupled sites, into the laboratory frame. For this interaction, only the 𝐴₂₀𝑃𝐴𝑆 term is non- zero, therefore, 𝐻̂_𝐷𝑃𝐴𝑆 _{= 𝐴} 20 𝑃𝐴𝑆_𝑇̂ 20, (2.100) where

𝐴₂₀𝑃𝐴𝑆 _{= √6𝑑}

𝐼𝑆, (2.101)

where 𝑑_𝐼𝑆 is defined as the dipolar coupling constant (in rad/s):

𝑑_𝐼𝑆 = −ћ (𝜇0 4𝜋)

1

𝑟3𝛾𝐼𝛾𝑆. (2.102)

Note the 𝑟3 dependence on the internuclear distance. Dividing by an additional factor of 2𝜋 is necessary to convert from radians to hertz.

By invoking the secular approximation, only 𝑚 = 0 terms in the laboratory frame need to be considered, therefore under static conditions:

𝐴₂₀𝐿 _{= 𝐴} 20 𝑃𝐴𝑆_𝐷 002 = √6𝑑𝐼𝑆{𝑒−𝑖𝛼𝑃𝐿0𝑑002 (𝛽𝑃𝐿)𝑒−𝑖𝛾𝑃𝐿0} = √6𝑑𝐼𝑆 1 2(3 cos2𝛽𝑃𝐿− 1), (2.103)

and under MAS,

𝐴𝐿_{20= √6𝑑} 𝐼𝑆 1 2(3 cos2𝛽𝑃𝑅− 1) 1 2(3 cos2𝛽𝑅𝐿− 1). (2.104)

The corresponding spin term is written as:

𝑇̂20=

1

√6(𝐼̂𝑧𝑆̂𝑧− 1

2(𝐼̂𝑥𝑆̂𝑥+ 𝐼̂𝑦𝑆̂𝑦)). (2.105)

The dipolar Hamiltonian, for a static experiment, may therefore be written (in the laboratory frame) as:

𝐻̂𝐷,ℎ𝑒𝑡= 𝑑𝐼𝑆

1

2( 3 cos2𝛽𝑃𝐿− 1)(2𝐼̂𝑧𝑆̂𝑧) (2.106)

for the heteronuclear case and

𝐻̂_{𝐷,ℎ𝑜𝑚𝑜}= 𝑑_𝐼𝑆1

for the homonuclear case. The matrix forms of the spin operators are as follows: 2𝐼̂_𝑧𝑆̂_𝑧 = ( 1 2 0 0 0 0 −1 2 0 0 0 0 −1 2 0 0 0 0 1 2) , (𝐼̂𝑥𝑆̂𝑥+ 𝐼̂𝑦𝑆̂𝑦) = ( 0 0 0 0 0 0 1 2 0 0 1 2 0 0 0 0 0 0) . (2.108)

In the case of heteronuclear dipolar coupling, the off-diagonal elements of the corresponding matrix are zero, since only the 2𝐼̂𝑧𝑆̂𝑧 spin term is considered in this case. For a pair of spin ½ nuclei, under the influence of a heteronuclear dipolar coupling, the spin eigenstates correspond to the Zeeman product states, shown in Fig. 2.6,  A typical lineshape, obtained under static conditions for a system under heteronuclear dipolar coupling, is known as a Pake doublet as shown in Fig. 2.7. The two horns represent two different crystallite orientations, both being perpendicular to the external field, 𝐵₀ (corresponding to the two different transitions having the opposite sign: two I spin and two S spin transitions for a heteronuclear I – S spin pair). The separation between the horns is equal to |𝑑𝐼𝑆|/2𝜋 in Hz. Importantly, this means that the contribution to the line broadening from heteronuclear dipolar coupling has an intrinsic orientation dependence, and hence the effect from this interaction can be fully removed by MAS.

Figure 2.7 Simulated NMR lineshape for a heteronuclear dipolar coupling between two spin – ½ nuclei, in this case |𝑑_2𝜋𝐼𝑆|= 6 kHz. The above pattern is often referred to as a Pake powder pattern.

Since the majority of the work presented in this thesis concerns itself with homonuclear 1_{H –} 1_{H dipolar}

coupling, this more complicated effect must also be considered in some detail. Returning to the homonuclear dipolar Hamiltonian, specifically the matrix representation of the (𝐼̂_𝑥𝑆̂_𝑥+ 𝐼̂_𝑦𝑆̂_𝑦) spin term, the off-diagonal elements are no longer necessarily non-zero. It is more convenient to express this term as a combination of so-called lowering and raising operators:

(𝐼̂_𝑥𝑆̂_𝑥+ 𝐼̂_𝑦𝑆̂_𝑦) ≡ (𝐼̂₋𝑆̂₊+ 𝐼̂₊𝑆̂₋). (2.109)

These terms are often referred to as a flip-flop term. Importantly the spin eigenstates for a pair of spin ½ nuclei are no longer simple Zeeman product states but rather a linear combination of Zeeman levels, as presented in Fig 2.8.

Therefore, in a real system where one considers a vast network of dipolar coupled protons, a number of degenerate eigenstates exist. This leads to a large range of different transition frequencies in the NMR spectrum, resulting in broadening of individual 1_{H resonances. In quantum mechanical terms, this}

means that the dipolar Hamiltonian no longer commutes with itself at different times. Indeed, this phenomenon has the significant consequence that homogeneous line broadening due to a homonuclear dipolar coupled network is only partially removed under magic angle spinning (MAS).