The empirical rule that the resistivity of a metal is the sum of a temperature-dependent component due to lattice vibrations and a temperature-independent component due to impurities and defects.
Japanese name: マティーセンの規則
Matthiessen's rule appears in 1 practice question across 1 topic, including Electronic, magnetic and optical materials.
The electrical resistance of a metal arises from scattering of conduction electrons by lattice vibrations (phonons), impurity atoms, vacancies and dislocations, grain boundaries and so on. If each type of scattering can be regarded as occurring independently, the scattering rates add, so the resistivity is the sum of components, ρ = ρ_th(T) + ρ_i + ρ_d. This is Matthiessen's rule, found by Matthiessen in the 1860s from measurements of the resistance of alloys. Only the lattice-vibration component depends strongly on temperature; the impurity and defect components are nearly independent of it.
The resistivity due to lattice vibrations is nearly proportional to absolute temperature above about the Debye temperature, and at low temperature it falls steeply in proportion to T⁵. At liquid helium temperature (4.2 K), therefore, the lattice-vibration component almost vanishes and only the residual resistivity from impurities and defects remains. The ratio of resistance at room temperature to that at 4.2 K is called the residual resistivity ratio (RRR), and it is larger for metals that are purer and contain fewer defects. For niobium used in superconducting accelerator cavities, values such as an RRR of 300 or more are required as a quality indicator.
In solid-solution alloys, the impurity component increases greatly with composition; in disordered binary solid solutions it is roughly proportional to x(1 − x) for solute concentration x (Nordheim's rule). When ordering occurs, with atoms arranged regularly as in Cu₃Au, scattering decreases and resistance falls. On the other hand, when scattering differs greatly with electron energy or direction, or when there is a Kondo effect due to magnetic impurities, the components are no longer independent and Matthiessen's rule breaks down. It should be used as an approximate rule.
In copper for electric wire, even small amounts of impurities raise resistivity, so high purity and oxygen control are pursued. Conductivity is often expressed in %IACS, with annealed standard copper (resistivity 1.7241 µΩ·cm, about 58.0 MS/m) as 100%. Conversely, in alloys such as manganin and constantan, the temperature-independent residual component makes up most of the resistance, so their temperature coefficient of resistance is very small and they are used in precision resistors and strain gauges. In fine interconnects, scattering components from surfaces and grain boundaries are added and raise the resistivity.
Answer: C. About 3.5
By Matthiessen's rule, the total resistivity is the sum of a temperature-dependent lattice-vibration component and a temperature-independent impurity and defect component, ρ = ρ_th(T) + ρ_res. At 300 K it is 1.70 + 0.40 = 2.10 µΩ·cm and at 77 K 0.20 + 0.40 = 0.60 µΩ·cm, so the ratio is 2.10/0.60 = 3.5. About 8.5 is the ratio of the lattice-vibration components of pure copper alone (1.70/0.20), ignoring the residual resistivity. About 10.5 adds the residual resistivity only to the 300 K value, (1.70 + 0.40)/0.20, overlooking that it adds equally at both temperatures. Because the residual component makes up a larger share at lower temperatures, the ratio of resistance at room temperature to that at cryogenic temperature (the residual resistivity ratio) is used to evaluate the purity and defect content of metals.
It is the ratio of resistivity at room temperature (about 300 K) to that at cryogenic temperature (usually 4.2 K). At cryogenic temperature only the resistance from impurities and defects remains, so a larger RRR shows a purer metal with fewer defects.
In alloys, the temperature-independent resistance from solute atoms is large, so the temperature-dependent lattice-vibration component is relatively small. By Matthiessen's rule, the fraction of the total resistance that changes with temperature becomes small.
It is an approximate rule. When the scattering mechanisms cannot be regarded as independent, for example when the scattering strength changes greatly with electron energy or when magnetic impurities are present, deviations from the sum appear.
By Matthiessen's rule, the total resistivity is the sum of a temperature-dependent lattice-vibration component and a temperature-independent impurity and defect component, ρ = ρ_th(T) + ρ_res.
The resistivity of pure copper due to lattice vibrations is…Coercivity Curie temperature Indirect-bandgap semiconductor Soft magnetic material