a portion of the Fermi surface lying in the second (or higher) zones. As with the band structure itself, the Fermi surface can be displayed in an extended-zone scheme where is allowed to have arbitrarily large values or a reduced-zone scheme where wavevectors are shown modulo (in the 1-dimensional case) where a is the lattice constant. In the three-dimensional case the reduced zone scheme means that from any wavevector there is an appropriate number of reciprocal lattice vectors subtracted that the new now is closer to the origin in -space than to any . Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to form ground states where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states are superconductors, ferromagnets, Jahn–Teller distortions and spin density waves. The state occupancy of fermions like electrons is governed by Fermi–Dirac statistics so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi surface although the term is not generally used outside of condensed-matter physics. Experimental determination Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields , for example the de Haas–van Alphen effect (dHvA) and the Shubnikov–de Haas effect (SdH). The former is an oscillation in magnetic susceptibility and the latter in resistivity. The oscillations are periodic versus and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by Lev Landau. The new states are called Landau levels and are separated by an energy where is called the cyclotron frequency, is the electronic charge, is the electron effective mass and is the speed of light. In a famous result, Lars Onsager proved that the period of oscillation is related to the cross-section of the Fermi surface (typically given in Å−2) perpendicular to the magnetic field direction by the equation. Thus the determination of the periods of oscillation for various applied field directions allows mapping of the Fermi surface. Observation of the dHvA and SdH oscillations requires magnetic fields large enough that the circumference of the cyclotron orbit is smaller than a mean free path. Therefore, dHvA and SdH experiments are usually performed at high-field facilities like the High Field Magnet Laboratory in Netherlands, Grenoble High Magnetic