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A lattice-friction stress for dislocation glide, associated with Rudolf Peierls and Frank Nabarro. A simple model gives ${\tau }_P\sim 3G\exp (-2\pi w/b)$, where $w$ is core width and $b$ the Burgers vector magnitude. Spreading slip over many atomic rows reduces the periodic energy barrier; narrower cores are harder to move in this model.

A Peierls transition, named for Rudolf Peierls, is a metal-insulator transition caused by a Peierls distortion of an initially metallic chain.^[ref] A modulation at $2k_F$ mixes states near opposite Fermi points and can open a band gap at the Fermi energy. At half filling with one atomic orbital per chemical atom, this modulation doubles the original lattice period and alternates strong and weak bonds. In a weak-coupling zero-temperature description, the Peierls energy gain is proportional to $-({\delta }^2/t)\ln (t/|\delta |)$ whereas the elastic cost is proportional to ${\delta }^2$, where $t>0$ is the undistorted hopping strength and $\delta$ its small alternating change. The logarithm permits a distortion at arbitrarily weak coupling in the ideal model. Finite temperature, finite size, interchain coupling and thermal fluctuations matter for a real transition; strictly one-dimensional reasoning alone does not establish finite-temperature long-range order.

Rudolf Peierls introduced an argument explaining how a two-dimensional Ising model can sustain magnetic order at low temperature. A reversed island of spins has a boundary whose energy cost grows with its length. The number of possible boundaries also grows, but sufficiently low temperature suppresses their Boltzmann weights strongly enough for order to survive. The Peierls argument makes the competition between energy and the number of fluctuations explicit, helping explain why dimensionality changes collective magnetic behavior.^[ref]

A physicist associated with the Peierls transition and the Peierls argument.