# Basics

When considering what the chemical shift is, how it arises and what we can do with it, it is useful to begin by considering nuclear angular momentum. Magnetic nuclei all possess an intrinsic angular momentum, referred to as nuclear spin. Its magnitude is determined by the spin quantum number, I, following the equation

magnitude of angular momentum = [I (I+1)]½ ħ

where I can be of integer or half-integer value, i.e. I = 0, ½, 1, ⅔, 2, etc. It follows that nuclei with spin quantum number 0 do not posses any intrinsic angular momentum. The nuclei most commonly used in protein NMR (1H, 15N and 13C) all have spin quantum number ½. The spin angular momentum is a vector quantity (i.e. it has a particular direction as well as a certain magnitude) and in spin-½ nuclei it is quantised into two states. Normally, these two states are of equal energy and are thus also equally populated. However, in a magnetic field, the degeneracy of these two states is lifted – the energy of one is slightly lowered and that of the other slightly raised. As a consequence the two energy states are now no longer populated equally, but according to the Boltzmann distribution. The energy difference between the two states is proportional to the strength of the magnetic field and the nuclei’s gyromagnetic ratio (an intrinsic constant for each nucleus). The energy difference generally lies within the radio-frequency range. Thus, by applying radio-frequency pulses it is possible to move nuclei between energy levels (or rather to change the angular momentum from one direction to another).

In a 14.1 T magnet the energy levels in a 1H nucleus are split by (and the nucleus is said to resonate at) approximately 600 MHz. (Hence a 14.1 T magnet is colloquially referred to as a 600 MHz magnet.) However, every 1H nucleus within a molecule has a slightly different electronic environment which means that it experiences a slightly different magnetic field. This produces very small differences in the energy gap seen for different 1H nuclei in the same molecule. These energy (or frequency) differences are on the levels of ten to hundreds of Hz. Rather than measuring these small frequency differences in Hz, a more convenient measure is used, referred to as the chemical shift, δ, given by

δ = 106 (ν – νref) / νref

and measured in parts per million (ppm). ν is the frequency of interest and νref is a convenient reference frequency. The advantage of using the chemical shift is firstly, that it removes the need to distinguish between MHz frequencies that only differ by several tens of Hz and secondly, that it becomes independent of the magnetic field used. Thus the chemical shift of a 13C nucleus in methanol measured at 9.4 T (400 MHz) is the same as that measured at 14.1 T (600 MHz) or that at 18.8 T (800 MHz).

Rather unusually, the chemical shift axis is drawn from right to left (i.e. back-to-front compared to normal graphs). Small or negative chemical shifts are said to be ‘upfield’ and large chemical shifts are referred to as being ‘downfield’.