# Debye model The **Debye model** is a description of the [[Heat_capacity|heat capacity]] of a crystalline solid in which the thermal energy is carried by [[Phonon|phonons]] — quantized [[Sound|sound]] waves — whose frequencies run from zero up to a cutoff, the Debye frequency, chosen so that the number of vibrational modes equals three per [[Atom|atom]]. Peter Debye published it in 1912.[^debye1912] The model reproduces both ends of the measured curve: at high [[Temperature|temperature]] the heat capacity settles on the [[Dulong–Petit_law|Dulong–Petit]] value of 3R per mole, about 24.9 J/(mol·K), and at low temperature it falls as the cube of the temperature, the Debye T³ law, which the [[Einstein_solid|Einstein model]] of 1907 had not been able to give.[^einstein1907] Everything in the model is fixed by one material parameter, the Debye temperature θ_D, which runs from about 105 K for soft, heavy [[Lead|lead]] to 2,230 K for [[Diamond|diamond]].[^spec-m14] In the microsim below the reader slides the reduced temperature T/θ_D and picks a material preset (Pb 105 K, Cu 343 K, Al 428 K, Si 645 K, diamond 2,230 K); the plot draws the Debye curve `C_v = 9·N·k·(T/θ_D)³·∫₀^{θ_D/T} x⁴·eˣ/(eˣ − 1)² dx` against the Einstein curve and the flat 3Nk plateau, and reads out the low-temperature T³ coefficient. The equation answers one question: what fraction of the classical 3Nk a solid can store at a given temperature. On the [[Materials_science]] flagship this article serves the *Heat capacity* section of Part III — Fundamentals › Properties, where the shared solid-thermal sim (C36) first appears; its siblings [[Thermal_expansion]] and [[Thermal_conductivity_and_resistivity]] take the same vibrating lattice and ask how it stretches and how it carries heat. ## Derivation A crystal of N atoms has 3N vibrational degrees of freedom. In the Debye model every one of them is a [[Standing_wave|standing wave]] of sound in an elastic continuum, with angular frequency ω = v·k for a wavevector of magnitude k and a sound speed v. Counting the allowed wavevectors in a box of volume V gives the number of modes per unit frequency, `g(ω) = 3·V·ω²/(2·π²·v³)`, where the factor 3 counts one longitudinal and two transverse polarizations and v is an average speed defined by `3/v³ = 1/v_L³ + 2/v_T³`. A continuum has no upper limit on ω, but a lattice of atoms does: the model cuts the spectrum off at the frequency ω_D that makes the total mode count equal 3N, `∫₀^{ω_D} g(ω) dω = 3N`, which gives `ω_D³ = 6·π²·v³·N/V`. Each mode is a [[Quantum_harmonic_oscillator|quantum harmonic oscillator]], and in thermal equilibrium at temperature T its mean energy above the [[Zero-point_energy|zero-point]] value is `ħω/(e^{ħω/kT} − 1)`: the quantum ħω times the [[Bose–Einstein_statistics|Bose–Einstein]] occupation, the same statistics Planck had used for radiation.[^likharev-ch2] Here k is the [[Boltzmann_constant|Boltzmann constant]] and ħ the reduced [[Planck_constant|Planck constant]].[^openstax-v2-appc] The [[Internal_energy|internal energy]] of the lattice is the sum over modes, `U = ∫₀^{ω_D} g(ω)·ħω/(e^{ħω/kT} − 1) dω`. Substituting x = ħω/kT and writing the cutoff as a temperature, `θ_D = ħ·ω_D/k`, turns the integral into `U = 9·N·k·T·(T/θ_D)³·∫₀^{θ_D/T} x³/(eˣ − 1) dx`. Differentiating with respect to T gives the heat capacity at constant volume that the microsim plots: `C_v = 9·N·k·(T/θ_D)³·∫₀^{θ_D/T} x⁴·eˣ/(eˣ − 1)² dx`. The integral has no closed form, so the sim carries it as a baked lookup table of 512 values of the Debye function over T/θ_D.[^spec-m14] *Try:* drag T/θ_D from 2 down to 0.05 and watch the plateau give way to the T³ fall; switch presets and the same curve is relabelled in kelvin. ## Debye's derivation Debye's own treatment starts from the solid as a continuous elastic body.[^debye1912] Such a body has a vibration spectrum whose mode density grows as ω², and the one concession he made to atomic structure was the cutoff: a body of N atoms cannot have more than 3N independent modes, so the continuum spectrum must stop where that count is reached. From the cutoff he obtained the function of θ_D/T that now carries his name and the low-temperature coefficient of T³ expressed through the elastic constants and the [[Density|density]] of the solid. That made θ_D a prediction rather than a fit: room-temperature elastic constants give a number to compare with heat capacities measured near 20 K. He compared the resulting curve with the low-temperature specific-heat measurements then available and with the empirical Nernst–Lindemann formula, a sum of two Einstein terms that had been used to patch the Einstein model's too-fast fall, and the single Debye curve replaced it.[^debye1912] Lattice dynamics, which keeps the atoms discrete and uses the real vibration spectrum, contains the Debye model as its long-wavelength limit; the two are compared under *Debye frequency* below. ## Another derivation The counting can be done directly in wavevector space. Take a cube of side L with periodic boundary conditions. The allowed wavevectors form a cubic grid with spacing 2π/L, so each mode of one polarization occupies a volume (2π/L)³ = 8π³/V of k-space. The modes with |k| ≤ k_D fill a sphere of volume (4/3)·π·k_D³, so their number per polarization is `V·k_D³/(6·π²)`. Setting that equal to N gives the Debye wavevector `k_D = (6·π²·n)^{1/3}`, with n = N/V, and ω_D = v·k_D. For [[Copper|copper]], with about 8.5×10²⁸ atoms per cubic metre,[^cu-density] k_D = 1.71×10¹⁰ m⁻¹ and the shortest phonon wavelength the model allows, 2π/k_D, is 0.37 nm, a little more than one atomic spacing. A wave shorter than the distance between atoms has no atoms to wave. A second route to the heat capacity goes through the free energy rather than the energy. Each oscillator contributes `k·T·ln(1 − e^{−ħω/kT})` (plus its zero-point term) to the Helmholtz free energy; summing with g(ω), differentiating once for the [[Entropy|entropy]] and once more for C = T·(∂S/∂T) returns the same Debye integral.[^likharev-ch2] The route matters for the [[Third_law_of_thermodynamics|third law]]: the Debye entropy vanishes as T³ at absolute zero, as the law requires, where classical equipartition (a constant 3Nk) does not. ## Temperature limits At high temperature x = ħω/kT is small for every mode, the integrand x⁴·eˣ/(eˣ − 1)² tends to x², and the integral becomes (θ_D/T)³/3. The heat capacity then reaches `C_v = 3·N·k`, or 3R = 24.9 J/(mol·K) per mole of atoms, the value Petit and Dulong found empirically in 1819 from the specific heats of thirteen solid elements.[^dulong1819] This is the [[Equipartition_theorem|equipartition]] result for 3N oscillators with two quadratic terms each,[^openstax-v2-ch2] and the first quantum correction is `C_v/(3·N·k) ≈ 1 − (1/20)·(θ_D/T)²`: at T = θ_D the exact Debye value is 0.952 of the plateau, and at 2θ_D it is 0.988.[^derived-debye] At low temperature the upper limit θ_D/T goes to infinity and the integral becomes the constant 4π⁴/15, so `C_v = (12·π⁴/5)·N·k·(T/θ_D)³ ≈ 234·N·k·(T/θ_D)³`, or `C_v ≈ 1,944·(T/θ_D)³ J/(mol·K)` per mole. For copper (θ_D = 343 K) at 10 K this gives 0.048 J/(mol·K); for lead (θ_D = 105 K) at the same 10 K it gives 1.68 J/(mol·K), where the exact Debye integral gives 1.65 because T/θ_D = 0.095 is already at the edge of the T³ region.[^derived-debye] The T³ law holds where the excited phonons are much longer than the lattice spacing: below about θ_D/20 the exact integral and the T³ form differ by less than half a percent, and at θ_D/10 by three percent.[^derived-debye] The table gives the reduced heat capacity computed for this article from the Debye integral.[^derived-debye] | T/θ_D | 0.05 | 0.10 | 0.20 | 0.30 | 0.50 | 1.0 | 2.0 | |---|---|---|---|---|---|---|---| | C_v/(3Nk), Debye | 0.010 | 0.076 | 0.369 | 0.608 | 0.825 | 0.952 | 0.988 | | C_v/(3Nk), T³ law | 0.010 | 0.078 | 0.623 | 2.10 | 9.74 | 77.9 | 623 | In a metal the [[Electron|electrons]] add a term linear in T, `C_el = γ·T`, predicted by the [[Free_electron_model|free-electron model]], that overtakes the phonon T³ term below a few kelvin; there C/T plotted against T² is a straight line with intercept γ and a slope set by θ_D.[^openstax-v3-ch9] The sim omits this electronic term (ILLUSTRATIVE, stated in its HUD). ## Debye versus Einstein [[Albert_Einstein|Einstein]]'s 1907 model gave all 3N oscillators the same frequency, ω_E, so that `C_v/(3·N·k) = (θ_E/T)²·e^{θ_E/T}/(e^{θ_E/T} − 1)²` with θ_E = ħω_E/k.[^einstein1907] It explained the central fact, that heat capacity falls toward zero at low temperature, but it falls too fast: exponentially, as e^{−θ_E/T}, where measurements fall as T³. The Debye model corrects this by keeping the low-frequency modes. However cold the crystal, some sound waves are long enough to be excited, and their ω² density makes the fall a power law. The two models can be matched at high temperature. The Einstein expansion is `1 − (1/12)·(θ_E/T)²` and the Debye expansion `1 − (1/20)·(θ_D/T)²`, so they agree to that order when `θ_E = (3/5)^{1/2}·θ_D ≈ 0.775·θ_D`.[^derived-debye] With that choice the curves are indistinguishable above about 0.5 θ_D and part company below: at T = 0.1 θ_D the Debye value is 0.076 of the plateau and the Einstein value 0.026.[^derived-debye] That gap, opening as the reader drags T/θ_D toward zero with both curves on screen, is the sim's central picture. Einstein's form survives as a good description of optical phonon branches, whose frequencies really are nearly equal. ## Debye temperature table Because the Debye curve is universal in T/θ_D, one number per material sets its whole heat-capacity curve. The sim's presets, and the room-temperature heat capacity the model predicts from each, are:[^spec-m14][^derived-debye] | Material | θ_D (K) | ν_D = ω_D/2π (THz) | C_v(300 K)/3R | |---|---|---|---| | Lead | 105 | 2.2 | 0.99 | | Copper | 343 | 7.1 | 0.94 | | Aluminium | 428 | 8.9 | 0.91 | | Silicon | 645 | 13.4 | 0.80 | | Diamond | 2,230 | 46.5 | 0.17 | The ordering follows ω_D = v·(6π²n)^{1/3}: stiff bonds and light atoms give fast sound and a high θ_D, so diamond, which is [[Carbon|carbon]] at its stiffest, sits at the top, while soft, heavy lead sits at the bottom and behaves classically at room temperature. Diamond at 300 K stores only a sixth of the Dulong–Petit value, which is why diamond was the classic exception to Petit and Dulong's rule and the test case Einstein chose for his 1907 formula.[^einstein1907] A tabulated θ_D is a fit: the value fitting the T³ region is not exactly the one fitting the curve near θ_D, because the real phonon spectrum is not the Debye ω² law (see below), so tables state the range their θ_D refers to. ## Extension to other quasi-particles The Debye integral is the general form for any gas of bosonic excitations with linear dispersion ω = v·k in three dimensions; only the cutoff is specific to phonons. Remove the cutoff and the same integral, with two polarizations instead of three and v = c, is [[Planck's_law|Planck's law]] for [[Black-body_radiation|black-body radiation]]:[^planck1901] the [[Photon|photon]] gas has U ∝ T⁴ and C ∝ T³ at every temperature, and the phonon T³ law is the same integral evaluated before the cutoff matters. The rule generalizes: quasi-particles with dispersion ω ∝ k^s in d dimensions give a low-temperature heat capacity C ∝ T^{d/s}. Magnons, the quantized spin waves of a ferromagnet, have ω ∝ k² and therefore C ∝ T^{3/2}, Bloch's law of 1930.[^bloch1930] Electrons in a metal are fermions, not bosons, and follow the [[Fermi_gas|Fermi gas]] result C = γ·T instead.[^openstax-v3-ch9] ## Extension to liquids A liquid has no lattice and no static shear rigidity, yet its heat capacity near the melting point is close to the solid's 3k per atom and falls toward 2k as it is heated. The phonon theory of liquid thermodynamics, developed by Bolmatov, Brazhkin and Trachenko in 2012, explains this with Debye's own counting.[^bolmatov2012] A liquid supports longitudinal sound at all frequencies but transverse (shear) waves only above the Frenkel frequency ω_F ≈ 1/τ, where τ is the time between molecular rearrangements. Taking the Debye density of states for the longitudinal branch and for the transverse branches only in the window ω_F < ω < ω_D reproduces the measured heat capacities of many liquids, including [[Mercury_(element)|mercury]], [[Argon|argon]] and [[Water|water]], with no free parameters beyond the [[Viscosity|viscosity]] that sets τ.[^bolmatov2012] As the temperature rises, τ shortens, ω_F climbs toward ω_D, the transverse modes disappear, and the heat capacity falls from 3k toward 2k per atom; the line in the phase diagram where ω_F reaches ω_D, the Frenkel line, separates liquid-like from gas-like behaviour even in a [[Supercritical_fluid|supercritical fluid]].[^bolmatov2012] ## Debye frequency The cutoff is the one place where the atoms enter the model, so its definition, its relation to θ_D and its limits deserve a section of their own. ### Definition The Debye frequency is the highest phonon frequency the model admits, `ω_D = v·(6·π²·N/V)^{1/3}` for an isotropic solid with a single effective sound speed v; the corresponding ordinary frequency is ν_D = ω_D/2π and lies in the terahertz range for every solid, from 2.2 THz for lead to 46.5 THz for diamond.[^derived-debye] ### Relation to Debye's temperature The Debye temperature is the same quantity in kelvin, `θ_D = ħ·ω_D/k`.[^likharev-ch2] For copper, θ_D = 343 K gives ω_D = 4.49×10¹³ rad/s, ν_D = 7.1 THz and a top phonon energy ħω_D = 29.6 meV, using the CODATA values of ħ and k.[^nist-codata][^derived-debye] Read the other way, at T = θ_D the thermal energy kT equals the energy of the highest phonon, so above θ_D every mode is excited and the solid is classical. ### Debye's derivation The textbook form of Debye's count takes standing waves in a cube of side L with fixed faces, whose wavevector components are positive multiples of π/L. The modes with |k| ≤ k_D then fill one octant of a sphere, containing `(1/8)·(4/3)·π·k_D³/(π/L)³ = V·k_D³/(6·π²)` modes per polarization, the same count as with periodic boundary conditions; setting it equal to N and multiplying by the sound speed gives ω_D.[^likharev-ch2] Debye's paper reached the same ω² spectrum and the same cutoff for an elastic body of any shape.[^debye1912] ### Polarization dependence An elastic solid carries one longitudinal and two transverse waves with different speeds, v_L > v_T. Counting each branch separately gives every branch the same k_D but its own cutoff frequency, ω_L = v_L·k_D and ω_T = v_T·k_D. The single-speed form used in the sim replaces the three by one average, `3/v³ = 1/v_L³ + 2/v_T³`, which leaves the low-temperature T³ coefficient exactly right because that coefficient depends only on the sum of 1/v³ over branches.[^likharev-ch2] The slower transverse waves dominate the sum, so the low-temperature heat capacity of a solid is mostly a property of its shear stiffness. ### Derivation with the actual dispersion relation The sound-wave law ω = v·k holds only for long waves. In a one-dimensional chain of atoms of mass m joined by springs of stiffness K, the exact dispersion is `ω(k) = 2·(K/m)^{1/2}·|sin(k·a/2)|`, with k confined to the first Brillouin zone, |k| ≤ π/a. The true maximum frequency is 2·(K/m)^{1/2}, reached at the zone edge, whereas Debye's straight line ω = v·k with v = a·(K/m)^{1/2} reaches the zone edge at π·(K/m)^{1/2}, too high by a factor π/2.[^likharev-ch2] The Debye model overestimates the top of the spectrum because the real curve flattens toward the zone boundary. This is why an effective θ_D fitted to data drifts with temperature and why full lattice dynamics, which uses the real ω(k), is needed for a precise fit; but the T³ law is untouched, since at low temperature only the straight part of ω(k) is occupied. ### Alternative derivation A rougher argument fixes the same scale without any counting. The shortest wave a chain of spacing a can carry has wavelength 2a, so k_max = π/a and ω_max ≈ π·v/a. In three dimensions with n = 1/a³ atoms per unit volume the exact Debye count gives k_D = (6π²)^{1/3}/a = 3.90/a, against π/a = 3.14/a from the wavelength argument: the two agree to about 25 percent.[^derived-debye] The lesson carried into the sim's readout is that θ_D is set by two things only, how fast sound travels and how closely the atoms are packed. ## See also - [[Einstein_solid]] — the single-frequency model the sim draws for comparison - [[Dulong–Petit_law]] — the 3R plateau at the top of the sim's plot - [[Phonon]] — the quantized sound waves the model counts - [[Heat_capacity]] — the quantity itself, on the Energy and Physics flagships - [[Specific_heat_capacity]] - [[Black-body_radiation]] — the same integral without a cutoff - [[Free_electron_model]] — the γT term the sim omits - [[Third_law_of_thermodynamics]] ## References [^debye1912]: Debye, P. (1912). "Zur Theorie der spezifischen Wärmen." *Annalen der Physik* 344 (14): 789–839. https://doi.org/10.1002/andp.19123441404 [^einstein1907]: Einstein, A. (1907). "Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme." *Annalen der Physik* 327 (1): 180–190. https://doi.org/10.1002/andp.19063270110 [^dulong1819]: Petit, A.-T.; Dulong, P.-L. (1819). "Recherches sur quelques points importants de la Théorie de la Chaleur." *Annales de Chimie et de Physique* 10: 395–413. Pre-DOI journal; no DOI exists. [^planck1901]: Planck, M. (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum." *Annalen der Physik* 309 (3): 553–563. https://doi.org/10.1002/andp.19013090310 [^bloch1930]: Bloch, F. (1930). "Zur Theorie des Ferromagnetismus." *Zeitschrift für Physik* 61 (3–4): 206–219. https://doi.org/10.1007/BF01339661 [^bolmatov2012]: Bolmatov, D.; Brazhkin, V. V.; Trachenko, K. (2012). "The phonon theory of liquid thermodynamics." *Scientific Reports* 2: 421. https://doi.org/10.1038/srep00421 [^likharev-ch2]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 2, "Principles of physical statistics" (pp. 29–72): harmonic-oscillator statistics, the Planck distribution and the Debye theory of the specific heat of solids (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^openstax-v2-ch2]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 2, "The Kinetic Theory of Gases", §2.3 Heat Capacity and Equipartition of Energy: `C_v = (d/2)·R` for d quadratic degrees of freedom, p. 97. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-v2-appc]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Appendix C, "Fundamental Constants", pp. 719–720. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-v3-ch9]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 9, "Condensed Matter Physics" (pp. 393–440), §9.4 Free Electron Model of Metals (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^nist-codata]: National Institute of Standards and Technology. "CODATA Internationally Recommended Values of the Fundamental Physical Constants." *The NIST Reference on Constants, Units, and Uncertainty.* https://physics.nist.gov/cuu/Constants/ [^cu-density]: Derived for this article: n = ρ·N_A/M with the handbook density and molar mass of copper (8.96 g/cm³, 63.55 g/mol) gives 8.5×10²⁸ m⁻³; density against *University Physics Volume 1* (2016), Chapter 14, Table 14.1 (page to pin), molar mass against *University Physics Volume 2*, Appendix F, pp. 727–728. [^derived-debye]: Computed for this article from the Debye integral, the Einstein function and the constants of [^nist-codata], with the θ_D presets of [^spec-m14]; the values are not printed in a Portal Book. [^spec-m14]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M14: sim concept, control (T/θ_D), presets θ_D (Pb 105, Cu 343, Al 428, Si 645, diamond 2,230 K) and the `bake_lut.py --fn debye_cv` bake (512 f32 values). The presets are the commonly tabulated low-temperature-fit values; cross-check against Likharev, Part SM, Chapter 2 (page to pin). ## Further reading - Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*, Chapter 2. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*, Chapters 1–2 (temperature, heat, kinetic theory). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*, Chapter 9 (condensed matter). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 ## External links - The Wikipedia pair's *External links* section is the place to look for archived copies of the 1912 paper; the open texts above are the Portal Book sources of this page. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Debye_model.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Debye model* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Debye_model.html" data-title="Debye model"></div> *Built from `MICROSIM_GUIDE/specs/sims/Debye_model.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Debye_model) : [Wikitube](https://en.wikitube.io/wiki/Debye_model) · pinned revision [1371769308](https://en.wikipedia.org/w/index.php?oldid=1371769308) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M14 · sim pending (matter/Debye_model).*