The gradient coils are used to spatially encode the positions of the MRI spins by varying the magnetic field linearly across the imaging volume such that the Larmor frequency varies as a function of position.
- Think "main magnetic field".
A Helmholtz pair is two coaxial loops of current carrying wire placed symmetrically about the origin, such that the ratio of their separation and their radii gives optimal field homogeneity at the origin (ratio = 1). More than two loops / coils are used in creating the main magnetic field, but the principle of adding coils at specific positions is continued to increase homogeneity. The current goes round the same way in both loops. Helmholtz coils are also used in some RF coil designs.
- Think "z gradient".
A Maxwell pair is like a Helmholtz but the current goes opposite ways round the loops. This creates a magnetic field that varies with position. The distance between the loops follows a different convention also.
- Think "x or y gradient".
Circular loops will do in creating a linearly varying magnetic field in the z-direction, because they may be placed around the magnet bore. The x and y gradients however pose a challenge: how does one place a loop around the magnet in a direction orthogonal to the z-axis? A solution is the Golay coil, which uses a different shape. Conceptually, it's the same as a Maxwell pair in terms of creating the linearly varying magnetic field, but the calculation of that field is more involved. More complicated designs are used these days, based on the Golay coil, e.g. "fingerprint coils".