the coefficient of thermal expansion
If thermal expansion is materials grow when heated, then the coefficient of thermal expansion is the exact exchange rate: how much strain you get per degree. It is the number you look up to predict how far a part will move, and it is the single most important thermal figure for fitting parts together.
The linear coefficient alpha_l is defined by delta_L / L0 = alpha_l x delta_T — that is, fractional length change per degree. Units are per kelvin (1/K), and the numbers are tiny, usually 10^-6 to 25 x 10^-6 per K (often written as ppm per degree). Examples: fused silica about 0.5 x 10^-6, borosilicate (Pyrex) about 3 x 10^-6, steel about 12 x 10^-6, copper about 17 x 10^-6, aluminum about 23 x 10^-6, and many polymers 50 to 200 x 10^-6. For volume, the volumetric coefficient alpha_v is approximately 3 x alpha_l for an isotropic solid, because all three directions expand. Worked example: a 2 m aluminum rail, alpha = 23 x 10^-6, heated 40 degrees C grows delta_L = 23 x 10^-6 x 2 x 40, which is 1.84 mm.
CTE tracks bond strength inversely: high-melting, strongly bonded materials (tungsten, alumina, silica) have low CTE, while soft, low-melting ones have high CTE. This is exactly why Pyrex resists thermal shock (low CTE) while ordinary soda-lime glass cracks. Honest caveat: CTE is not a fixed constant — it varies with temperature and, in anisotropic crystals or composites, with direction, so a single catalog value is a mid-range approximation.
Aluminum (alpha about 23 x 10^-6) expands nearly twice as much as steel (about 12 x 10^-6) for the same heating, and about 46 times as much as fused silica (about 0.5 x 10^-6). That is why precision optics are mounted in low-CTE materials.
CTE spans a wide range; matching or minimizing it is the heart of thermal design.
The volumetric coefficient is about three times the linear one only for isotropic materials; anisotropic crystals expand differently along different axes, so alpha_v = 3 x alpha_l can be badly wrong for them.