Nuclear magnetic moments interact with the local magnetic field, which will not necessarily be identical to the applied magnetic field. The applied field can induce motion of electrons in orbitals, which gives rise to an additional magnetic field at the nucleus. This extra field δB is proportional to the applied field, so it is common to write:
where σ is a dimensionless constant known as the shielding constant (usually positive though it can be negative).
The value of this constant for a nucleus depends upon the details of the electron density in the vicinity of the nucleus, so the value of the shielding constant is different for nuclei of the same isotope in different chemical groups.
As a consequence of the way we have defined the extra field above, we can write the total local field at any given spot in the following way:
and consequently the Larmor frequency of any particular nucleus is given by the following expression:
where σ is the shielding constant for that particular nucleus.
Thus we can clearly see that nuclei of the same element will come into resonance at different frequencies if their local environments differ.
Resonance frequencies are most usually expressed as a quantity known as the chemical shift, which is related to the difference between the resonance frequency ν of the nucleus under consideration and a standard reference frequency νº obtained at that same field strength. The chemical shift has the symbol δ, units parts per million (ppm), and is given by the following relation:
The standard reference for protons is the resonance frequency of the protons in trimethylsilane, Si(CH3)4 (also abbreviated to TMS).
The advantage of the chemical shift scale is that the values are independent of the strength of the applied field, B, as it appears in both the numerator and the denominator of the expression for δ.
Note that as the shielding, σ, gets smaller, the chemical shift, δ, gets larger. Nuclei with a large chemical shift are thus often described as strongly deshielded. By convention, NMR spectra are normally plotted with δ increasing from right to left.