What Is a Crystal Lattice? Order That Reaches All the Way Down

A crystal lattice is defined by a unit cell, the smallest arrangement of atoms that, repeated by simple translation in three directions, generates the entire structure. Everything about the crystal follows from the cell's dimensions, its angles and what sits inside it. Mathematics constrains the possibilities severely, since only a limited number of ways exist to fill space by repetition, and the result is seven crystal systems and fourteen distinct lattice types, with two hundred and thirty possible symmetry arrangements in total. That is a remarkably small number for something that covers every crystalline substance in existence. The constraint also rules things out, and the classic case is five-fold symmetry, which cannot tile a plane by pure repetition, which is why no ordinary crystal has five-fold faces. The discovery in the 1980s of quasicrystals, which show five-fold diffraction patterns and are ordered without being periodic, was resisted strongly because it appeared to contradict this, and it turned out to extend the definition of crystal rather than break the geometry, eventually earning a Nobel Prize.

Many familiar properties are direct consequences of the internal arrangement rather than separate facts to memorise:

• •External shape, since faces develop along planes of dense atomic packing and the angles between corresponding faces of a given mineral are constant regardless of the crystal's size
• •Cleavage, meaning the tendency to split along particular planes, which occurs where the bonding between layers is weakest, and which is why mica peels into sheets and halite breaks into cubes
• •Hardness differences, including the striking case of graphite and diamond being the same element arranged differently
• •Optical behaviour, with birefringence and colour effects arising from how the arrangement interacts with light travelling in different directions
• •Electrical properties, since the lattice determines how electrons move and whether a material conducts, insulates or behaves as a semiconductor
• •Melting behaviour, since a crystalline solid has a sharp melting point where the ordered arrangement collapses, unlike glass which softens over a range

Atoms are far too small for light microscopy, and the solution was to use radiation with a wavelength comparable to the spacing between atoms. X-rays passing through a crystal are scattered by the electrons around each atom, and because the atoms are regularly spaced, the scattered waves reinforce each other in specific directions and cancel in others, producing a pattern of spots whose positions and intensities encode the arrangement. Working backwards from that pattern to the structure is the technique of X-ray crystallography, developed from 1912 onward, and it has been among the most productive experimental methods in science. It established the structures of salts, metals, minerals and then of enormously complex biological molecules, including the double helix, whose determination rested on diffraction images produced by Rosalind Franklin, and the structures of penicillin, insulin and vitamin B12 determined by Dorothy Hodgkin. Related methods using neutrons and electrons extend the approach, and electron cryo-microscopy has recently displaced crystallography for many biological structures because it avoids the requirement to grow a crystal at all.

Perfect lattices are rare and would generally be less useful than imperfect ones. Point defects include missing atoms, extra atoms squeezed into gaps, and foreign atoms substituted into the structure, and the last of these is what produces most gemstone colour, since pure corundum is colourless while traces of chromium make ruby and traces of iron and titanium make sapphire. Line defects called dislocations allow planes of atoms to slip past each other progressively rather than all at once, which is why metals bend rather than shattering, and controlling dislocation movement by introducing obstacles is the basis of nearly all metal strengthening, including alloying and work hardening. Grain boundaries, where regions of different orientation meet, affect strength, corrosion and conductivity, and materials engineering is largely about controlling their size and arrangement. Deliberate introduction of foreign atoms, called doping, is what turns silicon into a useful semiconductor and therefore what makes every electronic device possible, so the entire industry rests on the controlled imperfection of a lattice.

A unit cell repeated by translation generates the whole solid, and geometry permits only seven systems and fourteen lattice types, which is why no ordinary crystal has five-fold faces. External shape, cleavage planes, hardness and optical behaviour all follow from the internal arrangement. X-rays scattered by regularly spaced atoms reveal it. Defects matter more than perfection, since dislocations let metals bend and doping makes semiconductors work.