Heat Capacity of Optical Materials
Specific heat c_p(T) of Si, Ge, GaAs, CaF₂, LiF, MgO, sapphire and diamond from the Debye model scaled to the measured room-temperature value, volumetric heat capacity, and the heat to warm a part from T₁ to T₂.
Choosing a material fills its measured specific heat at 298.15K, Debye temperature and density. Enter another material's values (its c_p at room temperature) to use the same model for it.
Specific Heat c_p(T)
0.7143 J/(g·K)
714.3 J/(kg·K) at 300 K
Volumetric ρ·c_p
1.664 MJ/(m³·K)
heat per unit volume and kelvin
Heat T₁ → T₂
1387.4 J
m ∫ c_p dT
Mean c_p over T₁–T₂
0.7708 J/(g·K)
20.0 °C → 200.0 °C
Specific heat of the presets, scaled Debye model
- Silicon
- Germanium
- GaAs
- CaF₂
- LiF
- MgO
- Sapphire (Al₂O₃)
- Diamond
Preset Materials
| Material | c_p (J/(kg·K)) | at (K) | θ_D (K) | Density (g/cm³) |
|---|---|---|---|---|
| Silicon | 712.4 | 298.15 | 645 | 2.330 |
| Germanium | 321.5 | 298.15 | 373 | 5.330 |
| GaAs | 323.9 | 298.15 | 344 | 5.317 |
| CaF₂ | 878.5 | 298.15 | 510 | 3.180 |
| LiF | 1612.1 | 298.15 | 732 | 2.639 |
| MgO | 920.7 | 298.15 | 945 | 3.580 |
| Sapphire (Al₂O₃) | 778.9 | 300 | 990 | 3.970 |
| Diamond | 509.5 | 298.15 | 1860 | 3.515 |
Model
C_V/(3Nk_B) = D(x) = (3/x³) ∫₀ˣ t⁴eᵗ/(eᵗ − 1)² dt, x = θ_D/T
c_p(T) = c_p(T_ref) · D(θ_D/T)/D(θ_D/T_ref)
Q = m ∫ c_p dT
The Debye curve gives the shape, the measured value its scale. One θ_D serves all temperatures, and the harmonic model has no c_p − c_V or anharmonic term, so above room temperature it runs low: silicon at 1000 K comes out 8 % below the measured value. One θ_D also drifts away from room temperature: sapphire is 5 % low at 200 K and 21 % low at 100 K, MgO runs 8–9 % high from 500 to 1000 K, and GaAs is 2 % low at 500 K and 9.5 % at 1000 K (against Piesbergen's 1963 data it is within 2.4 % from 100 to 273 K but 24 % low at 50 K). Diamond uses its caloric θ_D of 1860 K, the value that fits its measured c_p above 140 K, rather than the elastic 2250 K: within 1 % from room temperature to 1000 K, but more than 5 % high below about 120 K. Glasses have no preset (no cited θ_D).
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
c_p(T) = c_p(T_ref)·D(θ_D/T)/D(θ_D/T_ref) with the Debye function D and one Debye temperature; Q = m∫c_p dT. Harmonic, so it runs low above room temperature (Si at 1000 K: 8 % below Desai's value) and misses the dip of θ_D near 40 K; sapphire is within 1.1 % of JANAF from 300 to 1000 K (5 % low at 200 K, 21 % at 100 K), MgO 8–9 % above JANAF from 500 to 1000 K. Diamond takes its caloric θ_D of 1860 K (Pässler 2021), not the elastic 2250 K: within 0.9 % of the SGTE c_p from 298 to 1000 K. GaAs: 2 % low at 500 K, 9.5 % at 1000 K against Chatillon et al. 1990. No presets for glasses.
- Debye P. (1912). Zur Theorie der spezifischen Wärmen. Ann. Phys. 344, 789–839. doi:10.1002/andp.19123441404
- Råsander M., Moram M. A. (2015). On the accuracy of commonly used density functional approximations in determining the elastic constants of insulators and semiconductors. J. Chem. Phys. 143, 144104. Table VII: experimental Debye temperatures (Landolt–Börnstein): Si 645, Ge 373, GaAs 344, LiF 732, CaF₂ 510, MgO 945 K. doi:10.1063/1.4932334
- Archer D. G. (1993). Thermodynamic properties of synthetic sapphire (α-Al₂O₃), Standard Reference Material 720, and the effect of temperature-scale differences on thermodynamic properties. J. Phys. Chem. Ref. Data 22, 1441–1453. Debye temperature 990 K (§2). doi:10.1063/1.555931
- Chase M. W. (1998). NIST-JANAF Thermochemical Tables, 4th ed. J. Phys. Chem. Ref. Data Monograph 9. MgO (cr), table Mg-008: c_p 37.106 J/(mol·K) at 298.15 K, 45.544 at 500 K; α-Al₂O₃, table Al-096: 79.416 J/(mol·K) at 300 K; CaF₂, table Ca-014: 68.588 J/(mol·K) at 298.15 K; LiF, table F-021: 41.815 J/(mol·K) at 298.15 K. janaf.nist.gov
- Desai P. D. (1986). Thermodynamic properties of iron and silicon. J. Phys. Chem. Ref. Data 15, 967–983. Si c_p 20.007 J/(mol·K) at 298.15 K (Table 4), 26.569 at 1000 K (Table 5). doi:10.1063/1.555761
- Pässler R. (2021). Efficient Debye function interpolation formulae: sample applications to diamond. Recent Prog. Mater. 3(4), 042. Fig. 2: caloric θ_D of diamond 1857 ± 12 K from 140 to 700 K. doi:10.21926/rpm.2104042
- Dinsdale A. T. (1991). SGTE data for pure elements. Calphad 15, 317–425. Gibbs energies GDIAC (diamond: c_p 6.1197 J/(mol·K) at 298.15 K) and GHSERGE (Ge: 23.347 J/(mol·K)). doi:10.1016/0364-5916(91)90030-N
- Chatillon C., Ansara I., Watson A., Argent B. B. (1990). Re-assessment of the thermodynamic properties and phase diagrams of the Ga-As and In-As systems. Calphad 14, 203–214. GaAs Gibbs energy GB3GAAS: c_p 46.855 J/(mol·K) at 298.15 K. doi:10.1016/0364-5916(90)90021-Q
- Piesbergen U. (1963). Die durchschnittlichen Atomwärmen der AIIIBV-Halbleiter AlSb, GaAs, GaSb, InP, InAs, InSb und die Atomwärme des Elements Germanium zwischen 12 und 273 °K. Z. Naturforsch. 18a, 141. Tab. 3: the GaAs preset is within 2.4 % from 100 to 273 K, 24 % low at 50 K. doi:10.1515/zna-1963-0206
- Crystran. Handbook of Infrared and Ultraviolet Optical Materials, data sheets: density of Si, Ge, CaF₂, LiF, MgO and sapphire. crystran.com