Thermal Expansion
Length change ΔL = α·L·ΔT, strain and the change in 633 nm waves, with room-temperature expansion coefficients of optical glasses, crystals and mount metals, and silicon's α(T) integrated from 120 to 1500 K.
α itself changes with temperature; over large ranges use the mean α of the range (Si: 2.92 × 10⁻⁶/K from 300 to 400 K vs 2.57 at 300 K, Okada & Tokumaru 1984; the silicon section below integrates α(T)).
ΔL (expansion)
0.0710 mm
Strain ε
7.100e-4
ΔL / λ (at 633nm)
112.2 waves
Silicon: α(T) integrated from T₀ to T₀ + ΔT
α(T₀) → α(T₀ + ΔT)
2.510 → 3.178
10⁻⁶/K at 293.1 K and 393.1 K
Mean α over the range
2.874
10⁻⁶/K, ∫α dT / ΔT
ΔL of Si with α(T)
0.0287 mm
L · expm1(∫α dT)
ΔL with α frozen at T₀
0.0251 mm
-12.7 % vs α(T)
Silicon α(T), Okada & Tokumaru 1984 (±0.2 × 10⁻⁶/K)
- α(T)
- T₀, T₀ + ΔT
Reference CTE Values (×10⁻⁶/K)
| Material | Type | α (10⁻⁶/K) | Notes | Source |
|---|---|---|---|---|
| ULE 7972 | Glass | 0.03 | Corning titania-silica; mean α specified as 0 ± 0.03, 5–35 °C (limit shown) | Corning |
| Zerodur | Glass Ceramic | 0.05 | Expansion class 1: mean α 0 ± 0.05, 0–50 °C (limit shown); classes 0.007–0.1 | SCHOTT |
| Fused Silica | Glass | 0.55 | at 300 K | Crystran |
| Invar 36 | Metal | 0.9 | Fe-36Ni near 293 K, ± 20 % (fit slope; mean 250–300 K 1.2); heat treatment changes it (Kaye & Laby: 0.13) | NIST |
| Diamond | Crystal | 1 | at 293 K | Crystran |
| Si | Semiconductor | 2.6 | at 20 °C, increases with T (Okada–Tokumaru fit: 2.51 at 20 °C, 2.57 at 300 K) | Crystran |
| Pyrex (Borosilicate) | Glass | 2.8 | at 293 K | Kaye & Laby |
| Sapphire ∥ c | Crystal | 5.6 | 5.0 perpendicular to the c-axis | Crystran |
| SF11 | Glass | 6.1 | High-index flint; mean −30 to +70 °C (N-SF11: 8.5) | SCHOTT |
| Ge | Semiconductor | 6.1 | at 298 K | Crystran |
| BK7 | Glass | 7.1 | Standard optical glass | Crystran |
| ZnSe | II-VI | 7.1 | at 273 K | Crystran |
| MgF₂ ∥ c | Crystal | 13.7 | 8.9 perpendicular to the c-axis | Crystran |
| Stainless 304 | Metal | 15.4 | at 293 K (Kaye & Laby, austenitic: 14.1) | NIST |
| Copper | Metal | 16.5 | Heat sinks, mirrors; at 293 K | Kaye & Laby |
| CaF₂ | Crystal | 18.85 | Large expansion, needs care | Crystran |
| Aluminum 6061-T6 | Metal | 22.6 | Common mount material; at 293 K | NIST |
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
ΔL = αLΔT with one room-temperature α, good over tens of kelvin. Silicon: Okada & Tokumaru's α(T) (120–1500 K, ±0.2 ×10⁻⁶/K; 2.57 at 300 K, 4.56 at 1500 K) integrated in closed form, ΔL = L·expm1(∫α dT). Crystals from Crystran; glasses from SCHOTT, Corning (ULE and Zerodur: the specified limit) and Kaye & Laby; metals at 293 K from Kaye & Laby or the slope of NIST's fits. Alloys vary with heat treatment (Invar 0.13–1.2).
- Okada Y., Tokumaru Y. (1984). Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K. J. Appl. Phys. 56, 314–320. doi:10.1063/1.333965
- Crystran. Handbook of Infrared and Ultraviolet Optical Materials, data sheets: thermal expansion of fused silica, N-BK7, sapphire, CaF₂, MgF₂, Si, Ge, ZnSe and diamond. crystran.com
- SCHOTT optical glass data sheets (2017-01-20): SF11 α 6.1, N-SF11 8.5 ×10⁻⁶/K (−30 to +70 °C). refractiveindex.info
- SCHOTT. ZERODUR technical details: CTE grades, mean α (0 to 50 °C) 0 ± 0.100 (class 2), 0.050 (class 1), 0.020 (class 0), 0.010, 0.007 ×10⁻⁶/K. schott.com
- Corning (2016). ULE Corning Code 7972 Ultra Low Expansion Glass, product information: mean CTE 0 ± 30 ppb/°C, 5 to 35 °C. web.archive.org
- Kaye & Laby Tables of Physical and Chemical Constants, §2.3.5 Thermal expansion (NPL, archived web edition): α at 293 K of Cu 16.5, borosilicate (Pyrex) 2.8, Invar 0.13, austenitic stainless 14.1 ×10⁻⁶/K. web.archive.org
- NIST Cryogenic Technologies Group. Material properties of 6061-T6 aluminum, Invar (Fe-36Ni) and 304 stainless: fits of (L − L₂₉₃)/L₂₉₃ over 4–300 K, whose slopes at 293 K give 22.6, 0.9 and 15.4 ×10⁻⁶/K. trc.nist.gov