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Heat capacity fitter: Debye and Einstein temperatures from measured Cp(T), with the electronic γ and the low-temperature C/T against T² plot

Paste or open measured heat capacity against temperature and fit the Debye and Einstein models, with or without an electronic γT term. Get the Debye or Einstein temperature with its standard error, γ, the fit quality of all four models side by side, residuals, and the classic low-temperature C/T against T² line. Download the fit as CSV, image, PDF or Word.

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  1. Data
  2. Model
  3. Fit
  4. θ and γ
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Measured heat capacity

The fit keeps the high-temperature limit at 3nR (Dulong-Petit), so give n, the atoms per formula unit (2 for NaCl, 1 for copper). Relative weighting suits data that span several decades, as heat capacities do at low temperature. Measured Cp is fitted as if it were Cv; above about room temperature Cp is a few per cent higher. The example data are made up.

Fit

All four models

θ is fitted by least squares with γ solved exactly at each θ; the ± values are one standard error from the fit's curvature, scaled by the scatter of the residuals. The Debye model is usually much better at low temperature, where only long-wavelength phonons are excited (C ∝ T³); the Einstein model falls off too fast there.

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What it does

The heat capacity of a solid rises from zero at absolute zero to about 3R per mole of atoms at high temperature (the Dulong-Petit limit, about 24.9 J/(mol·K)). How fast it rises depends on how stiff the crystal is, which is summed up in one number: the Debye temperature θ_D (or, in the simpler Einstein picture, the Einstein temperature θ_E). In metals, conduction electrons add a small term γT that dominates at the lowest temperatures.

This tool fits your measured heat capacity with:

  • the Debye model, where atoms vibrate as sound waves up to a maximum frequency, giving C ∝ T³ at low temperature;
  • the Einstein model, where every atom vibrates at one frequency, which falls off exponentially at low temperature (and is often a good description of optical phonons);
  • each with or without the electronic γT term.

It reports the fitted θ (and γ) with standard errors, the fit quality of all four combinations side by side, a residual plot, and the classic C/T against T² straight line at low temperature, whose intercept is γ and slope gives θ_D independently.

How to use it

  1. Paste two columns, temperature in kelvin and heat capacity, or open a CSV, TXT or Excel file. A header row is fine.
  2. Pick the units of the heat capacity. For J/(g·K), type the molar mass so the tool can convert to J/(mol·K).
  3. Give n, the atoms per formula unit (1 for an element, 2 for NaCl, 5 for SrTiO₃). The fit keeps the high-temperature limit at 3nR.
  4. Choose the main model, whether to include γT, and the weighting. Relative weighting (the default) treats each point as equally precise in per cent, which suits data spanning several decades.
  5. Optionally narrow the fit range, and set the upper temperature for the low-temperature line (it should be well below θ_D/50 for the T³ law to hold).

Worked example

The page opens with made-up data from 4 to 300 K. The Debye fit with γT and relative weights gives Debye θ = 340.24 ± 0.39 K, γ = 0.701 ± 0.012 mJ/(mol·K²), with residuals of about 0.8 % rms, close to the 1 % noise put into the made-up data. The Einstein model fits far worse (60 % rms on its own, 39 % with γT) because its heat capacity collapses too quickly at low temperature. The low-temperature line through the six points at or below 15 K gives θ_D = 341.31 ± 0.23 K from the slope and γ = 0.721 ± 0.012 mJ/(mol·K²) from the intercept, agreeing with the full fit.

Tips

  • Leave out γT for insulators. If the fitted γ comes out negative, the data do not support an electronic term.
  • Real solids are not perfect Debye solids: θ_D fitted at different temperatures often differs by 10 % or more. Try narrowing the fit range to see how θ drifts.
  • Cp measured at constant pressure is a little above Cv at high temperature; the difference is ignored here.

FAQ

What do the ± values mean? One standard error from the least-squares curvature, scaled by the scatter of your residuals. They do not include systematic errors such as addenda or sample mass errors.

Why does the Einstein model still matter? Optical phonons in compounds behave much like Einstein oscillators. A Debye term plus Einstein terms is a common model for complex solids; this tool fits one at a time.

Is this suitable for publication? It is a careful fitting tool for study and checking. For publication, also report how the data were measured and corrected.

Where the numbers come from

  • Debye (1912) and Einstein (1907) models of the lattice heat capacity, the electronic term γT and the low-temperature law C/T = γ + βT² with β = 12π⁴nR/(5θ_D³): standard textbook results, for example Kittel, C., Introduction to Solid State Physics, chapters 5 and 6, and Ashcroft, N. W. and Mermin, N. D., Solid State Physics, chapter 23.
  • Checked by fibub against SciPy (quad for the Debye integral and curve_fit for the fits, with the same weights) on the made-up example data; R is the exact CODATA value.
  • The example data are made up: generated from a Debye model with θ = 340 K, γ = 0.7 mJ/(mol·K²) and 1 % random noise. They are not measurements of any material.

Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.

Last updated 2026-11-09.

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