Wiedemann Franz LawEdit
The Wiedemann–Franz law is a cornerstone result in solid-state physics that ties together how metals conduct heat and electricity. It states that, at a given temperature, the ratio of a metal’s thermal conductivity κ to its electrical conductivity σ is proportional to that temperature: κ/(σ T) ≈ L, where L is the Lorenz number. In many common metals near room temperature, this proportionality is remarkably close to a universal value L0, given by L0 = π^2 k_B^2/(3 e^2) ≈ 2.44×10^-8 W Ω K^-2. For readers who want to trace the idea back to its origins, it is named after Gustav Wiedemann, who observed the behavior in the 1850s, and Rudolf Franz, who helped formulate the empirical relation. Wiedemann–Franz law Gustav Wiedemann Rudolf Franz
The law emerged from early attempts to understand how electrons carry both charge and heat in metals. In the 19th century, models of metals as a gas of free electrons growing from the Drude model gave physicists a framework to connect electrical conduction with thermal transport. In modern language, the law is understood as a consequence of the degenerate electron gas described by the Fermi gas and the ways electrons scatter off impurities, lattice vibrations, and other electrons. The relationship is most transparent when the electrons are in a regime where their transport is governed by a single characteristic timescale for both charge and energy transport. For readers exploring the theory in depth, see the connections to the Drude model and the refinement provided by Arnold Sommerfeld’s quantum treatment of conduction.
The law
Mathematical statement
In its simplest form, the Wiedemann–Franz law asserts that κ = L σ T for metals in regimes where electrons dominate transport and scattering processes affect heat and charge transport similarly. The best-known constant in this relation is the Lorenz number L0, which follows from the quantum statistics of fermions. The practical upshot is that, once σ is known, κ can be estimated up to the jurisdiction of the temperature T and the applicability of the underlying assumptions. See Lorenz number for the continuum of related concepts and definitions.
Physical interpretation
The intuition rests on electrons that carry both charge and energy. If the scattering mechanisms that impede electrons do not distinguish too strongly between dissipating electrical current and distributing energy as heat, then the same carriers mediate both processes with comparable efficiency. In the idealized picture, Fermi-Dirac statistics and a degenerate electron gas produce a nearly universal ratio between heat and charge transport when temperature remains moderate and the lattice plays a secondary role. Link this to the broader theory of metallic transport by exploring free-electron model and Sommerfeld–Drude theory.
Experimental status
In many simple metals such as Copper, Silver, and Aluminum, measurements at moderate temperatures show κ/(σ T) very close to L0, validating the law as a practical rule of thumb for engineering rough estimates and sanity checks in design work. The precision of the law improves when phonon (lattice) contributions to heat flow are small or can be separated from the electronic part, since the Wiedemann–Franz law specifically concerns the electronic contribution κ_e to the total κ. For a more detailed discussion of these separations, see discussions of κ_e and κ_l in the context of the law.
Limitations and deviations
Temperature and scattering
The empirical accuracy of the law is best in metals where electrons are strongly degenerate and scattering has a similar impact on charge and energy transport. At very low temperatures, the lattice contribution κ_l becomes small but electron-electron and impurity scattering can drive κ away from the simple κ = L σ T form. At higher temperatures, phonon scattering becomes more important and can introduce deviations from the simple expression, even in good metals. The result is a range of temperatures over which the law holds only approximately.
Complex and strongly correlated materials
In materials where electrons experience strong correlations, non-Fermi-liquid behavior, or unconventional scattering physics, the Wiedemann–Franz law can fail. Examples include certain oxides, heavy-fermion compounds, and some high-temperature superconductors materials, where the electron method that underpins the simple law does not capture all the relevant transport channels. These deviations are active areas of research and are used as diagnostic tools to identify when more elaborate theories are required. See Heavy fermion and Non-Fermi liquid for discussions of how these systems challenge the classical picture.
Practical misapplications
Because κ includes both electronic and lattice contributions, applying the law to the total κ without accounting for κ_l can lead to misleading conclusions in materials where lattice heat transport is nontrivial. In thermoelectric contexts, engineers explicitly separate the electronic and lattice parts to understand and optimize device performance, since a high κ_l can mask the relationship the law would predict for the electronic component alone. See thermoelectric materials and Seebeck coefficient for related concepts.
Relevance to technology and policy
From a practical, results-oriented engineering perspective, the Wiedemann–Franz law provides a reliable bridge between two fundamental transport properties, enabling quick checks and cross-validation of measurements in metals. It guides material selection and design in applications where heat management and electrical performance must be understood together, such as in power electronics, heat exchangers, and metal-based thermoelectric concepts that rely on a careful balance between κ_e and κ_l. In thermoelectrics, designers aim to suppress κ_l while maintaining adequate electrical conductivity to maximize performance, a strategy that leverages the same physical intuition the law embodies while acknowledging its limits in complex materials. See thermoelectric materials for context on how transport principles translate into device performance.
In the broader public and policy discourse, a pragmatic stance toward fundamental transport laws emphasizes evidence and testable predictions. The law’s strength lies in its clear, falsifiable content and its well-documented domain of applicability, while its failures in certain materials highlight the value of ongoing research into electronic structure, scattering mechanisms, and advanced transport theory. See discussions of Fermi liquid theory and deviations in non-Fermi liquid systems for deeper theoretical context.