A Bloch state in a crystal experiences a periodic lattice potential. Its band dispersion relation encodes the lattice response, so an effective mass allows a wavepacket's acceleration under a weak applied force to be described without explicitly following the microscopic periodic forces.
In one dimension the wavepacket group velocity and semiclassical equation for crystal momentum areDifferentiating the first equation with respect to time givesDefining yieldsThis band effective mass is controlled by band curvature, not the slope. A band minimum has positive ; a band maximum has negative Electron and is often more conveniently described with positive-mass electron holes. In three dimensions the inverse effective mass is generally a tensor, . The scalar formula assumes motion along a specified direction in an isolated band, weak fields and negligible interband transitions; the bare Electron mass has not physically changed.
Let denote the elementary charge, so an Electron's charge is . In the relaxation-time Drude model with positive transport effective mass,The steady drift velocity is , defining positive electron mobility . The current density is opposite to the Electron drift:By the definition of electrical conductivity, , soIts units are . The mobility definition alone establishes the result in the scalar linear-response regime; the relaxation-time equation supplies a microscopic approximation for .
Interpret “3 mm wire” as a circular wire of diameter , and take the current density as uniform. One conduction Electron per atom givesUsing , the drift velocity corresponding to a current magnitude of isIt points opposite to conventional electric current. A simple distance scale over the cycle is , but this treats the magnitude as constant. For sinusoidal mains, alternating-current drift excursion gives zero net displacement after a full cycle.
If is the usual root mean square current, and the displacement amplitude and total absolute drift distance per cycle areIf instead denotes peak current, these become and . The paper does not state peak versus RMS; the characteristic drift is the boxed on whichever current convention is chosen. It also does not explicitly say diameter; taking as the radius would reduce all these velocities and drift distances by four. None of these distances is the microscopic path travelled at roughly the Fermi velocity between scattering events; that path cannot be obtained from the supplied current alone.
In a normal metal, the electrical resistivity can often be approximated using Matthiessen's rule,Static impurities, defects and grain boundaries scatter conduction Electrons even as , giving the residual resistivity. Thermal lattice phonons add a temperature-dependent contribution through electron-phonon scattering. In a clean three-dimensional acoustic model, small-angle electron-phonon transport at low temperature gives at low ; once thermal phonon populations are approximately classical, the resistivity is approximately linear in . An electron-electron contribution proportional to can be relevant in some low-temperature ranges.
Schematic normal-metal resistivity with residual, low-temperature phonon and approximately linear high-temperature regimes
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Carbon in mild steel, together with associated disorder, strain and microstructural changes, adds scattering and generally raises the residual resistivity relative to purer iron. Thus a dilute alloy may have a similar broad phonon-dependent rise but a larger baseline. The sketch is a generic phonon-plus-disorder model, not measured iron data. Iron's magnetic scattering and phase changes, and real steel's phases and processing history, can add structure; the simple sum is approximate rather than an exact material law. The expected explanation is greater Electron scattering from the added carbon and resulting disorder.
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