Chapter 2 NMR Theory
2.5 Chemical Shift
Every term describing the spatial dependency of the second rank tensor is now time dependent. The integral of equation 2.57 over a full rotor period
ALAB2,0(t)d t=0. (2.58)
Equation 2.58 is interpreted as the time dependent part of the second rank tensor being averaged to zero over a full rotor cycle. However, the evolution during the free induction decay is often sampled more than once over a rotor cycle, and may therefore not be completely averaged to zero. This leads to the observation of spinning sidebands, if the spinning frequency is low compared to the magnitude of an interaction such as the dipolar coupling or chemical shielding.
The behaviour of the nucleus in reaction to an external magnetic field is the primary interest in NMR. However, the electrons which surround every nucleus are also affected by the external magnetic field of the NMR spectrometer. The electrons react to produce a secondary field which modifies the the total mag- netic field experienced by the nucleus in a manner which depends on the local electron distribution. The interaction between this secondary magnetic field and the nucleus is called theshielding interaction and the frequency shift it causes in a NMR spectrum is called thechemical shift. When considering the shielding interaction it is helpful to decompose it into two components.
Firstly, all electrons affected by the external magnetic field are subject to the Lorentz force. For bound electrons this induces a current as they move within their orbitals. This motion causes a secondary field which opposes the external field at the centre of motion. The total magnetic field which is experienced by the nucleus is therefore reduced by the electron motion. This is called a shield- ing interaction and is known as the diamagnetic contribution. The strength of the interaction depends on the distance between the electrons and the nucleus. The dependence is of the order of 1/r3, where r is the radial distance between the nucleus and electrons. Consequently, the core electrons being closest to the nucleus contribute more than valence electrons which are further from the nu- cleus. This effect therefore tends to be fairly constant for a particular type of nucleus. However, the diamagnetic current at any one nucleus are affected by the surrounding atoms which each generate their own currents. The total effect of these diamagnetic currents on the NMR frequency is observed by changing the Larmor frequency of a given nucleus depending its local environment. This is the primary effect which causes the chemical shift.
Secondly, the external magnetic field distorts the ground state electron or- bitals which determine the electron density distribution around a nucleus. This distortion of the ground state can be described as a perturbation of ground state by orbitals of a higher energy excited electronic state. These excited electronic states can be paramagnetic and therefore cause the resulting ground electronic state in an external magnetic field to slightly paramagnetic. The result of such a paramagnetic state is to create a secondary magnetic field which supports the external field and is therefore said to deshield the nucleus.
Importantly both these components are affected by changes in the lo- cal electronic density distribution and geometry. The resulting chemical shift is therefore highly dependent on the local electronic environment. This makes the chemical shift extremely valuable to an NMR spectroscopist as it is diagnostic of the local electronic density and hence structure.
of an external magnetic fieldB0is written
HC S =−γˆI·σ·B0, (2.59)
whereσ is a second rank Cartesian tensor called the chemical shielding tensor. The chemical shielding tensor can be diagonalised by choosing the correct prin- cipal axis frame. In this PAF the diagonal elements are frequently expressed as the isotropic shift,δiso, anisotropy,∆, and asymmetry,η, which are defined as follows σiso = 1 3 σx x+σy y+σzz (2.60) ∆ = σzz−σiso (2.61) η = σx x+σy y ∆ . (2.62)
These diagonal elements are commonly ordered and labeled as σzz − σiso≥σx x−σiso ≥σy y−σiso. This is called the Haeberlen convention.
A typical sample used in solid state NMR is powdered. Such a sample contains many crystallites which have all possible orientations relative to the ex- ternal magnetic fieldB0. Crystallites at each orientation give rise to a frequency
shift. When the resonances of all crystallites at all orientations are co-added this yields a broad resonance called a powder pattern.
NMR experiments do not measure the chemical shift directly. Rather, the a reference compound is used and the frequencies of all resonances are mea- sured relative to a specific resonance in spectrum of the reference. Therefore any chemical shifts quoted are offset frequencies relative to specific resonance in a specific compound and calculated using the following equation
ν−νr e f
νr e f
whereδiso is the isotropic chemical shift in units of parts per millionorppm. There are more interesting theory and applications of the chemical shield-
ing which are discussed by Duer  but these are outside the scope of this work.
The anisotropies giving rise to a powder patterns which are recorded in static NMR experiments contains much information. However, in a spectrum con- taining more than a couple of resonances the patterns tend to overlap which ob- scures any useful information. The chemical shielding interaction,σ, is a second rank tensor and as discussed in section 2.4 any anisotropic components arising from the interaction can be averaged to zero over a rotor period by spinning the sample around the magic angle. For each resonance where the MAS frequency is smaller than the anisotropy this results in the observation of a series spin- ning sidebands separated by the spinning frequency where the solid resonances would have been.