= Debye T-cubed test
{c}
{title2=$C_{V,\mathrm m}^{\rm lattice}\simeq\beta T^3$}
The <Debye T-cubed test> asks whether a bulk three-dimensional <crystal>'s lattice <molar heat capacity> approaches a cubic law well below its <Debye temperature> $T_D$. It tests the low-<frequency> <acoustic phonon branches> in the <Debye model>, rather than asserting that every contribution to a real sample's <heat capacity> is cubic. <Peter Debye>'s continuum description makes a striking connection: a thermal measurement can reveal the low-<frequency> vibrational <density of states> without resolving individual vibrations.<Statistical Physics by David Tong/3 Quantum Gases>[Statistical Physics by David Tong; §3.3.1, The Debye Model: low-temperature heat capacity]{ref}
For a monatomic bulk model, the <Debye model> has vibrational <density of states> $g(\omega)=9N\omega^2/\omega_D^3$ up to the <Debye frequency> $\omega_D$, containing $3N$ <normal modes>. With $x=\hbar\omega/(k_BT)$, the thermal vibrational <internal energy> becomes
$$
U_{\rm lattice}=9Nk_BT\left(\frac{T}{T_D}\right)^3
\int_0^{T_D/T}\frac{x^3}{e^x-1}\,dx.
$$
The reduced <Planck constant> is $\hbar$, the <Boltzmann constant> is $k_B$, and $N$ counts <chemical atoms>. For $T\ll T_D$, replace the upper limit by infinity. The <integral> is $\pi^4/15$, as in the <Stefan–Boltzmann law> derivation. Differentiating at fixed <volume (thermodynamics)> gives, per <mole> of <chemical atoms>,
$$
C_{V,\mathrm m}^{\rm lattice}\simeq\beta T^3,
\qquad \beta=\frac{12\pi^4R}{5T_D^3},
\qquad [\beta]=\mathrm{J\,mol^{-1}\,K^{-4}},
$$
where $R$ is the <molar gas constant>. The cubic exponent comes from three-dimensional <acoustic phonon branches> with linear low-<wavevector> <dispersion relations>. The coefficient encodes their propagation speeds. Formula-unit normalization in a compound requires its number of <chemical atoms> per formula unit; do not silently substitute moles of formula units for moles of <chemical atoms>.
For a simple <metal> in its nonsuperconducting state, well below both <Debye temperature> and <electron Fermi temperature>, include <electronic heat capacity>:
$$
C_{p,\mathrm m}(T)\simeq\gamma T+\beta T^3,
\qquad \frac{C_{p,\mathrm m}}{T}\simeq\gamma+\beta T^2.
$$
This approximation also requires negligible <difference between constant-pressure and constant-volume heat capacities> at the measurement precision, and no significant magnetic, nuclear or <Schottky anomaly> contribution in the fit interval. Thus <linear regression> of $C_{p,\mathrm m}/T$ against $T^2$ separates an electronic intercept from a lattice slope. For an insulating <crystal> with no relevant low-<energy> magnetic excitations, the electronic term can instead be negligible; a <superconductivity> transition changes the electronic response.<Statistical Physics by David Tong/3 Quantum Gases>[Statistical Physics by David Tong; §§3.3.1 and 3.6.3: phonon and electronic low-temperature contributions]{ref}
After subtracting <calorimeter addenda>, Brando's second <gold> sample, $72.6\,\mathrm{mg}$, gave a linear $C_{p,\mathrm m}/T$ versus $T^2$ plot in the nonsuperconducting state. The reported electronic intercept was $7.0\times10^{-4}\,\mathrm{J\,mol^{-1}\,K^{-2}}$; deviations from reference <heat capacities> were below $2\%$.<Development of a Relaxation Calorimeter for Temperatures between 0.05 and 4 K by Manuel Brando>[§IV.B and Fig. 9: separate validation sample, molar electronic intercept and reference-comparison inset; plotted temperatures extend to about 3 K]{ref} This supports the combined linear-plus-cubic description over the plotted interval. Because the platform subtraction used reference <gold>, it is also an instrument validation with a shared material standard, rather than a wholly independent determination of the <Debye model> from an uncalibrated instrument.
\Image[/debye-normal-metal-intercept-test.png]
{title=The <Debye T-cubed test> in a normal <metal>: lattice slope and electronic intercept}
{description=Original constructed model with $\gamma=7.00\times10^{-4}\,\mathrm{J\,mol^{-1}\,K^{-2}}$ and $\beta=4.00\times10^{-4}\,\mathrm{J\,mol^{-1}\,K^{-4}}$. It represents a stipulated monatomic normal <metal>, not digitized <gold> measurements. Solid curves apply only over the chosen $1$–$3\,\mathrm K$ fit interval, with negligible other terms and $C_p-C_V$. Dashed portions illustrate extrapolation; error bars are assigned one-<standard deviation> uncertainties, not scatter measured in the experiment.}
For a worked original inference, stipulate values of $y=C_{p,\mathrm m}/T$ equal to $(0.001100,0.002300,0.004300)\,\mathrm{J\,mol^{-1}\,K^{-2}}$ at $T=(1,2,3)\,\mathrm K$. Assign independent $u(y)=0.000020\,\mathrm{J\,mol^{-1}\,K^{-2}}$ and negligible <temperature> error. Equal-weight <linear regression> on $x=T^2$ gives
$$
\gamma=(7.00\pm0.20)\times10^{-4}\,\mathrm{J\,mol^{-1}\,K^{-2}},
\qquad \beta=(4.000\pm0.035)\times10^{-4}\,\mathrm{J\,mol^{-1}\,K^{-4}}.
$$
The points sit exactly on the stipulated line; these <standard uncertainties> come from the assigned input precision, not an estimate from zero residual scatter. With $R=8.314462618\,\mathrm{J\,mol^{-1}\,K^{-1}}$, inversion gives
$$
T_D=\left(\frac{12\pi^4R}{5\beta}\right)^{1/3}=169.38\,\mathrm K,
\qquad \frac{u(T_D)}{T_D}=\frac{u(\beta)}{3\beta},
\qquad T_D=(169.4\pm0.5)\,\mathrm K.
$$
At $2\,\mathrm K$, the electronic and lattice <molar heat capacities> are respectively $0.0014$ and $0.0032\,\mathrm{J\,mol^{-1}\,K^{-1}}$. Their equality occurs at $\sqrt{\gamma/\beta}=1.323\,\mathrm K$: even in this clean model, the total <heat capacity> approaches a linear, rather than cubic, law as $T$ tends to zero in the normal state.
A straight line over a narrow interval is a useful diagnostic, not a universal proof. Inspect residuals and repeat the fit over several upper and lower <temperature> cutoffs; curvature can signal higher lattice terms, an unremoved <Schottky anomaly>, or a faulty <calorimeter addenda> subtraction. Correlated <thermometer calibration> and sample-amount errors affect both axes and can dominate the illustrative <linear regression> uncertainty. Low-dimensional samples, additional low-<energy> excitations and proximity to a <phase transition> need other models. The test is valuable precisely because its failures can reveal physics beyond the bulk acoustic approximation.
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