Sommerfeld Free-Electron Heat Capacity

Derivation

Sommerfeld Free-Electron Heat Capacity

D-254 Home PU-303 Threads energy · chance · matter Depends on fermi-dirac-distribution, Density of States and Van Hove Singularities

Statement

For a three-dimensional gas of non-interacting electrons in the degenerate limit \(k_{\mathrm B}T \ll \varepsilon_{\mathrm F}\), the Sommerfeld expansion of the Fermi-Dirac occupation applied to the internal energy yields an electronic heat capacity that is linear in temperature, \(C_{\mathrm{el}} = \gamma T\) with \(\gamma = \tfrac{\pi^2}{3}k_{\mathrm B}^2\, g(\varepsilon_{\mathrm F})\), equivalently \(C_{\mathrm{el}} = \tfrac{\pi^2}{2}Nk_{\mathrm B}\,(T/T_{\mathrm F})\).

Why it matters

Classical equipartition assigns every electron \(\tfrac{3}{2}k_{\mathrm B}\), predicting a temperature-independent electronic heat capacity of order \(\tfrac{3}{2}Nk_{\mathrm B}\). Measured electronic heat capacities of metals are smaller than this by a factor of order \(T/T_{\mathrm F}\sim 10^{-2}\) at room temperature, and they vanish linearly as \(T\to 0\). The Sommerfeld result resolves this discrepancy: only the thin shell of states within \(\sim k_{\mathrm B}T\) of the Fermi surface can be thermally excited, so a fraction \(\sim T/T_{\mathrm F}\) of the electrons each absorb \(\sim k_{\mathrm B}T\).

The linear term \(\gamma T\) is the experimental fingerprint of the Fermi surface. It dominates the lattice \(\propto T^3\) contribution below a few kelvin, so plotting \(C/T\) against \(T^2\) isolates \(\gamma\), from which the density of states at the Fermi level — and, through many-body enhancement, effective masses — is extracted.

Assumptions

Non-interacting electrons (independent-particle picture).With electron-electron and electron-phonon interactions the sharp single-particle occupation is replaced by a quasiparticle distribution; \(\gamma\) acquires mass-renormalisation factors \((1+\lambda)\) and the clean linear law is only the leading Fermi-liquid term.

Degenerate limit \(k_{\mathrm B}T \ll \varepsilon_{\mathrm F}\).If dropped, the chemical potential drifts far from \(\varepsilon_{\mathrm F}\), the Sommerfeld series diverges, and one must integrate the full Fermi-Dirac form or cross over to the classical Maxwell-Boltzmann gas.

Smooth density of states near \(\varepsilon_{\mathrm F}\).The expansion needs \(g(\varepsilon)\) to be analytic over the \(\sim k_{\mathrm B}T\) window; a van Hove singularity or band edge at \(\varepsilon_{\mathrm F}\) invalidates the Taylor step and makes \(C_{\mathrm{el}}\) non-linear.

Temperature-independent density of states and fixed particle number.The chemical potential must be re-solved from number conservation as \(T\) rises; neglecting its shift keeps the correct leading term but spoils the higher-order coefficients and the entropy.

Derivation

\[ U(T) = \int_{0}^{\infty} \varepsilon\, g(\varepsilon)\, f(\varepsilon)\, d\varepsilon, \qquad N = \int_{0}^{\infty} g(\varepsilon)\, f(\varepsilon)\, d\varepsilon \]

Total internal energy and number are occupation-weighted sums over the density of states \(g(\varepsilon)\); \(f(\varepsilon)=\big[e^{(\varepsilon-\mu)/k_{\mathrm B}T}+1\big]^{-1}\) is the Fermi-Dirac distribution (prior result). A

\[ \int_{0}^{\infty} H(\varepsilon) f(\varepsilon)\, d\varepsilon = \int_{0}^{\mu} H(\varepsilon)\, d\varepsilon + \frac{\pi^2}{6}(k_{\mathrm B}T)^2 H'(\mu) + \frac{7\pi^4}{360}(k_{\mathrm B}T)^4 H'''(\mu) + \cdots \]

Sommerfeld expansion: for any smooth \(H(\varepsilon)\) with \(H(0)\) finite and \(H\) growing slower than \(e^{\varepsilon/k_{\mathrm B}T}\), integrate against the "step-plus-corrections" structure of \(f\). Legitimate because \(-\partial f/\partial\varepsilon\) is sharply peaked at \(\mu\) with even moments \(\int(\varepsilon-\mu)^{2n}(-\partial_\varepsilon f)d\varepsilon = a_n (k_{\mathrm B}T)^{2n}\). C

\[ -\frac{\partial f}{\partial\varepsilon}=\frac{1}{k_{\mathrm B}T}\frac{e^{x}}{(e^{x}+1)^2},\quad x=\frac{\varepsilon-\mu}{k_{\mathrm B}T};\qquad \int_{-\infty}^{\infty}\frac{x^{2}\,e^{x}}{(e^{x}+1)^2}\,dx=\frac{\pi^{2}}{3} \]

The coefficient \(\pi^2/6\) in Step 2 follows from writing \(I=\int H(-\partial_\varepsilon f)\,d\varepsilon\), Taylor-expanding \(H\) about \(\mu\), and using this standard even moment. The lower limit is safely extended to \(-\infty\) because \(-\partial_\varepsilon f\) is exponentially small for \(\varepsilon<0\) when \(\mu\gg k_{\mathrm B}T\). C

\[ N = \int_{0}^{\mu} g(\varepsilon)\, d\varepsilon + \frac{\pi^2}{6}(k_{\mathrm B}T)^2\, g'(\mu) \]

Apply Step 2 with \(H=g\). Since \(N\) is fixed, the \(T\)-dependent term forces \(\mu\) to shift with temperature. B

\[ 0 = \int_{\varepsilon_{\mathrm F}}^{\mu} g(\varepsilon)\, d\varepsilon + \frac{\pi^2}{6}(k_{\mathrm B}T)^2 g'(\varepsilon_{\mathrm F}) \;\Rightarrow\; \mu(T) \approx \varepsilon_{\mathrm F}-\frac{\pi^2}{6}(k_{\mathrm B}T)^2\frac{g'(\varepsilon_{\mathrm F})}{g(\varepsilon_{\mathrm F})} \]

Subtract the \(T=0\) statement \(N=\int_0^{\varepsilon_{\mathrm F}}g\,d\varepsilon\) from Step 4, approximate \(\int_{\varepsilon_{\mathrm F}}^{\mu}g\,d\varepsilon\approx g(\varepsilon_{\mathrm F})(\mu-\varepsilon_{\mathrm F})\), and evaluate \(g'\) at \(\varepsilon_{\mathrm F}\) to leading order. This is the \(O(T^2)\) chemical-potential shift. B

\[ U(T) = \int_{0}^{\mu}\varepsilon\, g(\varepsilon)\, d\varepsilon + \frac{\pi^2}{6}(k_{\mathrm B}T)^2\Big[g(\mu)+\mu\, g'(\mu)\Big] \]

Apply Step 2 with \(H(\varepsilon)=\varepsilon g(\varepsilon)\), so \(H'(\mu)=g(\mu)+\mu g'(\mu)\). B

\[ \int_{0}^{\mu}\varepsilon\, g\, d\varepsilon = \int_{0}^{\varepsilon_{\mathrm F}}\varepsilon\, g\, d\varepsilon + \varepsilon_{\mathrm F} g(\varepsilon_{\mathrm F})\,(\mu-\varepsilon_{\mathrm F}) \]

Split the energy integral at \(\varepsilon_{\mathrm F}\); over the narrow interval \([\varepsilon_{\mathrm F},\mu]\) the integrand \(\varepsilon g(\varepsilon)\approx \varepsilon_{\mathrm F}g(\varepsilon_{\mathrm F})\) is constant to the order kept. B

\[ U(T)-U(0) = \varepsilon_{\mathrm F} g(\varepsilon_{\mathrm F})(\mu-\varepsilon_{\mathrm F}) + \frac{\pi^2}{6}(k_{\mathrm B}T)^2\big[g(\varepsilon_{\mathrm F})+\varepsilon_{\mathrm F}g'(\varepsilon_{\mathrm F})\big] \]

Combine Steps 6 and 7 and evaluate the \(O(T^2)\) bracket at \(\varepsilon_{\mathrm F}\). Now insert the shift from Step 5: \(\varepsilon_{\mathrm F}g(\varepsilon_{\mathrm F})(\mu-\varepsilon_{\mathrm F}) = -\tfrac{\pi^2}{6}(k_{\mathrm B}T)^2\varepsilon_{\mathrm F}g'(\varepsilon_{\mathrm F})\). C

\[ U(T)-U(0) = \frac{\pi^2}{6}(k_{\mathrm B}T)^2\, g(\varepsilon_{\mathrm F}) \]

The two \(\varepsilon_{\mathrm F}g'(\varepsilon_{\mathrm F})\) terms cancel exactly. The chemical-potential shift precisely removes the \(g'\) dependence, leaving only \(g(\varepsilon_{\mathrm F})\) — the excess energy depends on the density of states at the Fermi level alone. B

\[ C_{\mathrm{el}} = \left(\frac{\partial U}{\partial T}\right)_{N,V} = \frac{\pi^2}{3}k_{\mathrm B}^2\, g(\varepsilon_{\mathrm F})\, T \]

Differentiate Step 9 with respect to \(T\); the \(T^2\) becomes \(2T\), giving the linear law. A

\[ g(\varepsilon_{\mathrm F}) = \frac{3N}{2\varepsilon_{\mathrm F}}=\frac{3N}{2k_{\mathrm B}T_{\mathrm F}} \;\Rightarrow\; C_{\mathrm{el}} = \frac{\pi^2}{2}\,N k_{\mathrm B}\,\frac{T}{T_{\mathrm F}} \]

For the free-electron gas \(g(\varepsilon)\propto\varepsilon^{1/2}\) (prior density-of-states result) gives \(g(\varepsilon_{\mathrm F})=\tfrac{3N}{2\varepsilon_{\mathrm F}}\); substitute and define \(T_{\mathrm F}=\varepsilon_{\mathrm F}/k_{\mathrm B}\). A

Result

\[ \boxed{\,C_{\mathrm{el}} = \gamma\, T,\qquad \gamma = \frac{\pi^2}{3}k_{\mathrm B}^2\, g(\varepsilon_{\mathrm F}) = \frac{\pi^2}{2}\frac{N k_{\mathrm B}}{T_{\mathrm F}}\,} \]

Reading. The electronic heat capacity rises linearly from zero as the metal is warmed. Its slope \(\gamma\) (the Sommerfeld coefficient) is a direct measure of the density of single-particle states at the Fermi energy. Physically, only electrons within an energy shell \(\sim k_{\mathrm B}T\) of \(\varepsilon_{\mathrm F}\) — a fraction \(\sim T/T_{\mathrm F}\) of the total — are thermally active, and each carries \(\sim k_{\mathrm B}T\) of excess energy, so \(U-U_0\sim g(\varepsilon_{\mathrm F})(k_{\mathrm B}T)^2\) and \(C\sim k_{\mathrm B}^2 g(\varepsilon_{\mathrm F})T\).

Units check. \([g(\varepsilon_{\mathrm F})]=\mathrm{J^{-1}}\) (states per joule), \([k_{\mathrm B}^2]=\mathrm{J^2\,K^{-2}}\), \([T]=\mathrm K\), so \([\gamma T]=\mathrm{J^{-1}\cdot J^2 K^{-2}\cdot K}=\mathrm{J\,K^{-1}}\), a heat capacity. In the second form \([N k_{\mathrm B}]=\mathrm{J\,K^{-1}}\) and \(T/T_{\mathrm F}\) is dimensionless. Consistent.

Limiting cases

\(T\to 0\): \(C_{\mathrm{el}}\to 0\) linearly, satisfying the third law (unlike the classical \(\tfrac32 Nk_{\mathrm B}\)).
\(T\ll T_{\mathrm F}\) (metals at ordinary \(T\)): \(C_{\mathrm{el}}/(\tfrac32 Nk_{\mathrm B}) = \tfrac{\pi^2}{3}(T/T_{\mathrm F})\ll 1\) — heat capacity strongly suppressed below the classical value.
\(k_{\mathrm B}T \gtrsim \varepsilon_{\mathrm F}\): the expansion fails; the gas becomes non-degenerate and \(C_{\mathrm{el}}\to \tfrac32 Nk_{\mathrm B}\) (classical equipartition).
Low-\(T\) total: \(C = \gamma T + \beta T^3\); the electronic term dominates the phonon \(T^3\) term below \(T^\ast=\sqrt{\gamma/\beta}\), typically a few kelvin.

Breaks when

Fermi level at a van Hove singularity or band edge. If \(g(\varepsilon)\) is non-analytic within \(k_{\mathrm B}T\) of \(\varepsilon_{\mathrm F}\), the Taylor step (2) is invalid and \(C_{\mathrm{el}}\) is no longer linear — one sees enhanced, temperature-dependent \(\gamma(T)\).
Non-degenerate / high-temperature regime, \(k_{\mathrm B}T\gtrsim\varepsilon_{\mathrm F}\). The Sommerfeld series diverges; the chemical potential leaves the band, and the crossover to the classical ideal gas must be treated with the full Fermi-Dirac integral.
Strong correlations / non-Fermi-liquid behaviour. Near a quantum critical point or in heavy-fermion and some cuprate systems, \(C_{\mathrm{el}}/T\) diverges logarithmically or as a power law, so no constant \(\gamma\) exists.
Gapped or superconducting state. Opening an energy gap \(\Delta\) at \(\varepsilon_{\mathrm F}\) replaces the linear law with an exponential \(\sim e^{-\Delta/k_{\mathrm B}T}\) (s-wave) or a power law (nodal gaps).

Failure modes

Forgetting the \(\mu(T)\) shift. Dropping Step 5 leaves a spurious \(\varepsilon_{\mathrm F}g'(\varepsilon_{\mathrm F})\) term; the exact cancellation in Step 9 is missed and the coefficient comes out wrong.
Using equipartition. Assigning \(\tfrac32 k_{\mathrm B}\) per electron overestimates \(C_{\mathrm{el}}\) by \(\sim T_{\mathrm F}/T\sim 100\) at room temperature.
Confusing \(g(\varepsilon_{\mathrm F})\) with \(N/\varepsilon_{\mathrm F}\). The free-electron result is \(g(\varepsilon_{\mathrm F})=\tfrac{3N}{2\varepsilon_{\mathrm F}}\); dropping the factor \(\tfrac32\) gives a \(33\%\) error in \(\gamma\).
Per-spin vs total density of states. Failing to include both spin orientations halves \(g(\varepsilon_{\mathrm F})\) and hence \(\gamma\).
Keeping the \(\tfrac{\pi^2}{6}\) but writing \(\tfrac{\pi^2}{6}\) instead of \(\tfrac{\pi^2}{3}\) in \(C\). The derivative of \(T^2\) supplies the factor 2; a common slip leaves \(\gamma\) too small by half.
Treating \(\gamma\) as measuring the bare band mass. The measured \(\gamma\) is enhanced by electron-phonon and electron-electron interactions, so \(\gamma_{\mathrm{exp}}=(1+\lambda)\gamma_{\mathrm{band}}\).

Discussion

The physical heart of the result is the exact cancellation in Steps 8-9. Naively both the number and energy integrals produce \(O(T^2)\) terms proportional to \(g'(\varepsilon_{\mathrm F})\), yet the requirement that particle number be conserved forces \(\mu\) to drift by exactly the amount that annihilates the \(g'\) contribution to the energy. What survives depends only on \(g(\varepsilon_{\mathrm F})\): the heat capacity is a pure probe of how many states sit at the Fermi surface, blind to the slope of the band. This is why \(\gamma\) is such a clean experimental quantity.

Because \(C_{\mathrm{el}}=\gamma T\) while the Debye lattice contribution is \(\beta T^3\), the standard low-temperature analysis plots \(C/T = \gamma + \beta T^2\) against \(T^2\): the intercept gives \(\gamma\) and the slope gives \(\beta\) (hence the Debye temperature). Comparing the measured \(\gamma\) with the free-electron prediction defines the thermodynamic effective mass \(m^\ast/m = \gamma_{\mathrm{exp}}/\gamma_{\mathrm{free}}\), which lumps together band-structure curvature and many-body renormalisation.

The same Sommerfeld machinery gives the electronic entropy \(S_{\mathrm{el}}=\gamma T\) (identical coefficient, since \(C=T\,\partial S/\partial T\) with \(S\propto T\)) and, through the analogous expansion of the grand potential, the \(T^2\) correction to the chemical potential and the Pauli paramagnetic susceptibility \(\chi_{\mathrm P}=\mu_0\mu_{\mathrm B}^2 g(\varepsilon_{\mathrm F})\). The ratio \(\chi_{\mathrm P}/\gamma\) (the Wilson ratio) is a fixed number for free electrons and a sensitive diagnostic of correlations when it deviates.

At the next order the expansion yields a \(T^3\ln T\) electronic term and \(O(T^2)\) corrections to \(\gamma\) that carry information about \(g''(\varepsilon_{\mathrm F})\); in a genuine Fermi liquid these are reorganised into quasiparticle interaction (Landau) parameters, and the leading linear \(\gamma T\) survives with a renormalised coefficient. The Sommerfeld derivation is thus the \(T=0\) fixed-point statement of Fermi-liquid thermodynamics: interactions dress the quasiparticles but preserve the linear law until the Fermi-liquid description itself breaks down.

Common misconceptions. The linear-in-\(T\) heat capacity does not mean electrons individually gain energy linearly in \(T\); it reflects that the number of thermally excited electrons grows \(\propto T\) while each gains \(\propto T\), and the product \(\propto T^2\) in \(U\) differentiates to \(\propto T\). Also, \(\gamma\) is not "the heat capacity of one electron" — it is a collective property fixed by the Fermi-surface density of states, not by the total electron count alone.

Worked examples

Sommerfeld coefficient of copper (per mole)

Copper: one conduction electron per atom, \(\varepsilon_{\mathrm F}=7.00\ \mathrm{eV}\), so \(T_{\mathrm F}=\varepsilon_{\mathrm F}/k_{\mathrm B}=8.12\times10^{4}\ \mathrm K\). Take \(N=N_{\mathrm A}=6.022\times10^{23}\) electrons per mole. A

\[ \gamma = \frac{\pi^2}{2}\frac{N_{\mathrm A}k_{\mathrm B}}{T_{\mathrm F}} = \frac{\pi^2}{2}\,\frac{R}{T_{\mathrm F}} \]

Symbols first; \(N_{\mathrm A}k_{\mathrm B}=R=8.314\ \mathrm{J\,mol^{-1}K^{-1}}\). A

\[ \gamma = \frac{\pi^2}{2}\cdot\frac{8.314}{8.12\times10^{4}}\ \mathrm{J\,mol^{-1}K^{-2}} = 5.05\times10^{-4}\ \mathrm{J\,mol^{-1}K^{-2}} \]

Insert numbers: \(\pi^2/2=4.935\). A

\[ \gamma_{\text{free}} \approx 0.51\ \mathrm{mJ\,mol^{-1}K^{-2}} \]

Reading. The measured value for copper is \(\gamma_{\mathrm{exp}}=0.695\ \mathrm{mJ\,mol^{-1}K^{-2}}\), giving \(m^\ast/m\approx1.37\) — modest enhancement from band structure and electron-phonon coupling, confirming copper is a good nearly-free-electron metal. Units. \(\mathrm{J\,mol^{-1}K^{-2}}\), correct for a molar \(\gamma\).

Where electronic and lattice heat capacities are equal in potassium

Potassium: \(\gamma = 2.08\ \mathrm{mJ\,mol^{-1}K^{-2}}\), Debye temperature \(\Theta_{\mathrm D}=91\ \mathrm K\). The lattice term is \(C_{\mathrm{ph}}=\beta T^3\) with \(\beta=\tfrac{12\pi^4}{5}R\,\Theta_{\mathrm D}^{-3}\). B

\[ \beta = \frac{12\pi^4}{5}\,\frac{R}{\Theta_{\mathrm D}^{3}}; \qquad C_{\mathrm{el}}=C_{\mathrm{ph}} \Rightarrow \gamma T = \beta T^3 \Rightarrow T^\ast=\sqrt{\gamma/\beta} \]

Set the two contributions equal; symbols before numbers. B

\[ \beta = \frac{12\pi^4}{5}\,\frac{8.314}{(91)^3}\ \mathrm{J\,mol^{-1}K^{-4}} = 2.57\times10^{-3}\ \mathrm{J\,mol^{-1}K^{-4}} \]

\(\tfrac{12\pi^4}{5}=233.8\), \((91)^3=7.54\times10^{5}\). So \(\beta=2.57\ \mathrm{mJ\,mol^{-1}K^{-4}}\). A

\[ T^\ast=\sqrt{\frac{\gamma}{\beta}}=\sqrt{\frac{2.08\times10^{-3}}{2.57\times10^{-3}}}\ \mathrm K = \sqrt{0.809}\ \mathrm K \]

Ratio of the two molar coefficients (units \(\mathrm K^2\) under the root). A

\[ T^\ast \approx 0.90\ \mathrm K \]

Reading. Below about \(0.9\ \mathrm K\) the electronic term dominates the specific heat of potassium; above it phonons take over. This is why \(\gamma\) is measured in the sub-kelvin regime by plotting \(C/T\) vs \(T^2\). Units. \(\sqrt{(\mathrm{J\,mol^{-1}K^{-2}})/(\mathrm{J\,mol^{-1}K^{-4}})}=\sqrt{\mathrm K^2}=\mathrm K\). Correct.

Problems

Show that the fraction of conduction electrons thermally excited at temperature \(T\) is of order \(T/T_{\mathrm F}\), and use it to argue \(C_{\mathrm{el}}\sim Nk_{\mathrm B}(T/T_{\mathrm F})\).
Solution
Electrons within \(\sim k_{\mathrm B}T\) of \(\varepsilon_{\mathrm F}\) can be excited. Their number is \(\Delta N\approx g(\varepsilon_{\mathrm F})\,k_{\mathrm B}T = \tfrac{3N}{2\varepsilon_{\mathrm F}}k_{\mathrm B}T = \tfrac32 N\,(T/T_{\mathrm F})\), so the excited fraction is \(\sim T/T_{\mathrm F}\). Each gains \(\sim k_{\mathrm B}T\), so \(U-U_0\sim \Delta N\,k_{\mathrm B}T\sim Nk_{\mathrm B}T^2/T_{\mathrm F}\), and \(C=\partial U/\partial T\sim Nk_{\mathrm B}(T/T_{\mathrm F})\), matching the exact \(\tfrac{\pi^2}{2}Nk_{\mathrm B}(T/T_{\mathrm F})\) up to the \(O(1)\) factor.
Silver has \(\varepsilon_{\mathrm F}=5.49\ \mathrm{eV}\). Compute the free-electron molar Sommerfeld coefficient and the ratio \(C_{\mathrm{el}}/(\tfrac32 R)\) at \(T=300\ \mathrm K\).
Solution
\(T_{\mathrm F}=5.49\ \mathrm{eV}/k_{\mathrm B}=6.37\times10^{4}\ \mathrm K\). \(\gamma=\tfrac{\pi^2}{2}R/T_{\mathrm F}=4.935\times8.314/6.37\times10^{4}=6.44\times10^{-4}\ \mathrm{J\,mol^{-1}K^{-2}}\approx0.64\ \mathrm{mJ\,mol^{-1}K^{-2}}\). At \(300\ \mathrm K\), \(C_{\mathrm{el}}=\gamma T=0.193\ \mathrm{J\,mol^{-1}K^{-1}}\). Ratio to classical: \(C_{\mathrm{el}}/(\tfrac32 R)=\tfrac{\pi^2}{3}(T/T_{\mathrm F})=3.29\times(300/6.37\times10^{4})=1.55\times10^{-2}\), i.e. the electronic heat capacity is about \(1.5\%\) of the equipartition value.
Starting from \(U-U_0=\tfrac{\pi^2}{6}(k_{\mathrm B}T)^2 g(\varepsilon_{\mathrm F})\), derive the electronic entropy \(S_{\mathrm{el}}(T)\) and verify \(S_{\mathrm{el}}=C_{\mathrm{el}}\).
Solution
\(C_{\mathrm{el}}=\partial U/\partial T=\tfrac{\pi^2}{3}k_{\mathrm B}^2 g(\varepsilon_{\mathrm F})T=\gamma T\). Then \(S_{\mathrm{el}}=\int_0^T \tfrac{C_{\mathrm{el}}}{T'}dT'=\int_0^T \gamma\,dT'=\gamma T\). Hence \(S_{\mathrm{el}}=\gamma T = C_{\mathrm{el}}\); both vanish linearly as \(T\to0\), consistent with the third law. The Helmholtz free energy contribution is \(F_{\mathrm{el}}=U-U_0-TS_{\mathrm{el}}=\tfrac12\gamma T^2-\gamma T^2=-\tfrac12\gamma T^2\).
A metal shows \(C/T = 1.35 + 12.0\,T^2\) in \(\mathrm{mJ\,mol^{-1}K^{-2}}\) (with \(T\) in K). Extract \(\gamma\), \(\beta\), the Debye temperature, and the crossover temperature \(T^\ast\).
Solution
Intercept: \(\gamma=1.35\ \mathrm{mJ\,mol^{-1}K^{-2}}\). Slope: \(\beta=12.0\ \mathrm{mJ\,mol^{-1}K^{-4}}=1.20\times10^{-2}\ \mathrm{J\,mol^{-1}K^{-4}}\). From \(\beta=\tfrac{12\pi^4}{5}R/\Theta_{\mathrm D}^3\): \(\Theta_{\mathrm D}^3=233.8\times8.314/1.20\times10^{-2}=1.62\times10^{5}\), so \(\Theta_{\mathrm D}=(1.62\times10^5)^{1/3}=54.6\ \mathrm K\). Crossover: \(T^\ast=\sqrt{\gamma/\beta}=\sqrt{1.35/12.0}=\sqrt{0.1125}=0.335\ \mathrm K\).
Estimate the ground-state (zero-point) total energy \(U_0=\tfrac35 N\varepsilon_{\mathrm F}\) and compare the \(T=300\ \mathrm K\) thermal excess \(U-U_0\) with it for copper (\(\varepsilon_{\mathrm F}=7.00\ \mathrm{eV}\)), per mole.
Solution
\(U_0=\tfrac35 N_{\mathrm A}\varepsilon_{\mathrm F}=0.6\times6.022\times10^{23}\times7.00\times1.602\times10^{-19}\ \mathrm J=4.05\times10^{5}\ \mathrm{J\,mol^{-1}}\). Thermal excess: \(U-U_0=\tfrac{\pi^2}{6}(k_{\mathrm B}T)^2 g(\varepsilon_{\mathrm F})=\tfrac12\gamma T^2\) with \(\gamma=0.51\ \mathrm{mJ\,mol^{-1}K^{-2}}\): \(U-U_0=0.5\times5.1\times10^{-4}\times(300)^2=23.0\ \mathrm{J\,mol^{-1}}\). Ratio \((U-U_0)/U_0=23.0/4.05\times10^5=5.7\times10^{-5}\): even at room temperature the thermal energy is a tiny fraction of the degenerate ground-state energy, confirming deep degeneracy.