Lattice specific heat and the Debye temperature At sufficiently high temperature, the specific heat of a crystalline metal approaches a constant (the Dulong–Petit limit). The characteristic temperature that marks this transition scale is called the:

Introduction / Context:
The temperature dependence of specific heat reveals how lattice vibrations (phonons) are excited. The Debye model improves on classical Dulong–Petit by predicting low-temperature T^3 behavior and recovery of the classical limit at high temperature.

Given Data / Assumptions:

• Crystalline metal approximated by a 3D elastic continuum with a cutoff frequency (Debye frequency).
• At low T, only low-frequency phonons are excited; at high T, the phonon spectrum is saturated.

Concept / Approach:
Debye introduced a characteristic temperature θD (Debye temperature) corresponding to the highest vibrational frequency (ωD) of the solid via k_B θD = ħ ωD. The specific heat at constant volume approaches 3R per mole when T ≫ θD because all vibrational modes are equally populated (equipartition). Below θD, quantum effects suppress high-frequency mode occupation, reducing specific heat.

Step-by-Step Solution:
Recognize the high-T constant limit is the Dulong–Petit limit (≈ 3R).Identify θD as the characteristic scale separating quantum and classical regimes.Therefore, the temperature referred to is the Debye temperature.

Verification / Alternative check:
Experimental plots of C_v/T versus T show saturation toward 3R beyond material-specific θD values (e.g., Al, Cu have θD of order a few hundred kelvin).

Why Other Options Are Wrong:
Curie temperature: Magnetic order transition (ferromagnets). Neel temperature: Antiferromagnetic order. Transition temperature: Nonspecific term. Kondo temperature: Scale for magnetic impurity scattering in metals, not lattice specific heat saturation.

Common Pitfalls:
Confusing magnetic transition temperatures with lattice vibrational scales; assuming saturation occurs at a universal temperature rather than material-dependent θD.

Final Answer:
Debye temperature