Atomic Structure & Interatomic Bonding

the equilibrium interatomic spacing

Left alone, two bonded atoms come to rest at one particular distance apart — not touching, not far, but a definite gap that is the same for every identical pair. This resting distance is the equilibrium interatomic spacing, usually written r0, and it is simply where the tug-of-war of attraction and repulsion ends in a draw. It is the natural bond length, the size of the bond.

You can find r0 in either of two equivalent ways: it is the separation at which the net interatomic force is exactly zero, or, seen on the bonding-energy curve, the position of the bottom of the valley where energy is lowest. For most solids r0 falls in the range of about 0.2 to 0.3 nanometres (a few tenths of a billionth of a metre). Add up r0 for all the bonds in a crystal and you get its lattice dimensions; that is why the spacing is what actually sets how big a unit cell is and, ultimately, a material's density.

The spacing is not perfectly fixed, and its small wobble matters enormously. Heat makes atoms vibrate around r0, and because the energy valley is lopsided (steeper on the squeeze side than the stretch side), the average distance grows slightly as temperature rises — that outward drift is thermal expansion. So the equilibrium spacing is both the resting point and, through the shape of the well around it, the origin of how materials swell when heated.

In common table salt the sodium and chlorine ions rest about 0.28 nm apart; in a metal like copper the atom centres sit about 0.26 nm apart. These sub-nanometre gaps, repeated billions of times, are what give a crystal its measurable size and its density.

Sum the bond lengths and you get the crystal's real size.

Bond length depends on which atoms and what bond, but only weakly on temperature: even heating a metal to near melting stretches r0 by just a percent or two. That is why thermal expansion, though critically important in engineering, is a small effect at the atomic scale.

Also called
bond lengthequilibrium separationr0鍵長平衡間距