Chemical Shift

The shielding interaction describes the effect of the local electronic environment on the magnetic field experienced by individual nuclear spins. The presence of the strong B0 field causes the electrons to orbit about the field, hence inducing a secondary magnetic field, opposing the external field. This secondary field has the effect of shielding the atomic nuclei from the external field. The magnitude of this shielding effect, and hence the local magnetic field strength, will vary between different nuclear sites due to differences in the distribution of electrons across a molecule. Consequently nuclei experiencing different shieldings will precess at different frequencies, producing an NMR signal with components corresponding to a range of resonance offsets and hence leading to an NMR spectrum comprised of peaks at a range of frequencies.

2.4. INTERNAL INTERACTIONS 31

In practice, the frequency at which a peak appears in an NMR spectrum is expressed as a chemical shift which is defined relative to a reference frequency. The isotropic chemical shift,δiso, is defined in terms of the measured frequency, ω, and the reference

frequency, ωref, as:

δiso=

ω−ωref

ωref

×106 (2.69)

The factor of 106 _{is required to express the chemical shift in term of parts per million}

(ppm) of ωref. Expressing the frequency in this way removes the dependence on the

strength of the B0 field, allowing results obtained from experiments using different

magnet field strengths to be compared.

The shielding interaction is described by the shielding tensor, ˜σ, a second rank, Cartesian tensor. The Hamiltonian describing the shielding interaction is:

ˆ

Hσ =γIˆ·σ˜·Bˆ0 (2.70)

Expressed in terms of spherical tensors, the shielding interaction Hamiltonian becomes:

ˆ

H_σP AS =AP AS₀₀ Tˆ00+AP AS20 Tˆ20+AP AS22 Tˆ2−2+AP AS2−2 Tˆ22 (2.71)

where the spatial terms are expressed in terms of components of the Cartesian shielding tensor as: AP AS₀₀ =−γ r 1 3 σ P AS xx +σP ASyy +σzzP AS (2.72) AP AS₂₀ =γ r 1 6 2σ P AS zz −σxxP AS−σyyP AS (2.73) AP AS₂±2 =γ 1 2 σ P AS xx −σyyP AS (2.74)

The A00 spherical tensor is unaffected by the rotation and corresponds to the

from the PAS frame to the LAB frame using equations 2.51 and 2.50: ALAB₂₀ = 2 X m=−2 AP AS_2m d2_m0(βP L) exp(−imαP L) =AP AS₂₀ d2₀₀(βP L) +AP AS22 d220(βP L) exp(−2iαP L) +AP AS2−2 d2−20(βP L) exp(2iαP L) =AP AS₂₀ 1 2(3 cos 2_{θ−1) +A}P AS 2±2 r 3 8sin 2_β P L(exp(−2iαP L) + exp(2iαP L)) (2.75) In the NMR spectrum of a powder, consisting of many crystallites with random orien- tations, the angular dependence in the shielding causes an anisotropic broadening of the spectral lines.

In practice it is the chemical shift, rather than the shielding, which is directly mea- sured in an NMR experiment. A corresponding chemical shift tensor, ˜δ may therefore be defined such that the the components of ˜δ are related to the the shielding tensor components as:

δαβ =

σαβ(ref)−σαβ

1−σαβ(ref)

(2.76)

The isotropic chemical shift,δiso, chemical shift anisotropy,δaniso(CSA) and asymme-

try η are defined in terms of the elements of this Cartesian chemical shift tensor:

δiso= 1 3 δ PAS xx +δPASyy +δPASzz (2.77) δaniso=δzzPAS−δiso (2.78)

η= δ

PAS

xx −δPASyy

δaniso

(2.79)

These three parameters contain information on all of the principal components of the chemical shift tensor, and may be used to completely describe the form of the shape of the CSA pattern that appears in a static solid-state NMR experiment. By convention [86], the components of the chemical shift tensor are ordered such that

|δ_zzPAS−δiso| ≥ |δxxPAS−δiso| ≥ |δPASyy −δiso| (2.80)

2.4. INTERNAL INTERACTIONS 33 (a)

δ

δ

η.δ

δ

δ

δ

δ

Figure 2.3: (a) overlapping peaks due to different orientations of crystallites (blue) contributing to an anisotropic chemical shift powder pattern. (b) schematic spectrum showing positions of the principal components of the chemical shift tensor, the location of the isotropic chemical shift, and the measurement corresponding to the chemical shift anisotropy (for η= 0.5).

NMR experiment performed on a single crystal. In this case, all chemically identical nuclei within the sample will be aligned with the same orientation with respect to the magnetic field. The local electronic distribution at each of these sites will therefore be the same, and hence a single peak at a specific chemical shift will appear in the NMR spectrum. Considering the case for a powder comprising many crystallites at random orientations with respect to the field, a large number of peaks at a range of different chemical shifts will now be present in the spectrum, due to the differences in the total magnetic field strength at each nuclear site. The result is a broad line formed from these many overlapping peaks, as shown in figure 2.3a.

chemical shift. Figure 2.3b shows the conventions used when discussing CSA in later chapters. The isotropic chemical shift is located at the average of the three principal component values. The chemical shift anisotropy is used to define the maximum sepa- ration of the principal tensor components from the isotropic value. The asymmetry, η is related to the measurementη·δ_aniso in the figure. In the case of an axially symmetric tensor, (δxx=δyy) the asymmetry has a value of zero.