thermal-expansion anisotropy
A cubic crystal is like a uniformly inflating balloon: heat it and it grows by the same fraction in every direction, so a single number alpha describes it. But most ceramic crystals are not cubic. In a crystal built from stacked layers or aligned chains, the bonds along one axis differ from the bonds across it, so the crystal grows by different amounts in different directions. That directional difference is thermal-expansion anisotropy — the crystal has more than one alpha, one for each independent crystal axis.
The size of the effect ranges from mild to extreme. In alpha-alumina (hexagonal) the expansion along the c-axis is only a little larger than across it (roughly 9 versus 8 x 10^-6 /K) — barely anisotropic. Quartz is strongly anisotropic and also jumps at its phase change. The extreme cases are spectacular: aluminium titanate (Al2TiO5) and graphite actually expand along one crystal axis while contracting along another over some temperature ranges, so alpha is positive in one direction and negative in another. This happens because the bonding and the packing are so different in different directions that heating stretches some bonds while letting the structure fold or pucker along others.
Anisotropy matters most because of what it does inside a polycrystalline ceramic, where thousands of grains point in random directions. On cooling from the firing temperature, a grain trying to shrink a lot along one axis is clamped by neighbours shrinking little, and vice versa, so every grain boundary carries a residual stress set by the difference in alpha. Push that mismatch far enough and the ceramic cracks itself along its grain boundaries — the microcracking treated in the next entry. Engineers exploit this deliberately: aluminium titanate's huge anisotropy makes its bulk expansion nearly zero (because the microcracks open and close to absorb the mismatch), giving outstanding thermal-shock resistance.
In aluminium titanate a single crystal expands strongly along two axes but contracts along the third on heating. A dense polycrystal of it therefore riddles itself with grain-boundary microcracks on cooling, and its measured bulk CTE drops close to zero — near 1 x 10^-6 /K — far below any of its single-crystal axis values.
Extreme single-crystal anisotropy can drive a polycrystal to microcrack, which paradoxically gives it a near-zero and highly shock-tolerant bulk expansion.
Anisotropy is a property of the single crystal; in a fine-grained, randomly oriented polycrystal the measured bulk CTE is an average — but that average hides internal grain-to-grain stresses that can still fracture the material.