Bloch–Grüneisen temperature
For typical three-dimensional metals, the temperature-dependence of the electrical resistivity "ρ(T)" due to the scattering of electrons by acoustic phonons changes from a high-temperature regime in which "ρ ∝ T" to a low-temperature regime in which "ρ ∝ T" at a characteristic temperature known as the Debye temperature. For low density electron systems, however, the Fermi surface can be substantially smaller than the size of the Brillouin zone, and only a small fraction of acoustic phonons can scatter off electrons. This results in a new characteristic temperature known as the Bloch–Grüneisen temperature  that is lower than the Debye temperature. The Bloch–Grüneisen temperature is defined as "2ħvk/k", where "ħ" is the Planck constant, "v" is the velocity of sound, "ħk" is the Fermi momentum, and "k" is the Boltzmann constant. When the temperature is lower than the Bloch–Grüneisen temperature, the most energetic thermal phonons have a typical momentum of "kT/v" which is smaller than "ħk", the momentum of the conducting electrons at the Fermi surface. This means that the electrons will only scatter in small angles when they absorb or emit a phonon. In contrast when the temperature is  higher than the Bloch–Grüneisen temperature, there are thermal phonons of all momenta and in this case electrons will also experience large angle scattering events when they absorb or emit a phonon. In many cases, the Bloch–Grüneisen temperature is approximately equal to the Deybe temperature (usually written formula_1), which is used in modeling specific heat capacity. However, in particular circumstances these temperatures can be quite different. The theory was initially put forward by Felix Bloch and Eduard Grüneisen. The Bloch–Grüneisen temperature has been observed experimentally in a two-dimensional electron gas and in graphene. Mathematically, the Bloch–Grüneisen  model produces a resistivity given by: formula_2. Under Bloch's original assumptions for simple metals, formula_3. For formula_4, this can be approximated as formula_5 dependence. In contrast, the so called Bloch–Wilson limit, where formula_6 works better for s-d inter-band scattering, such as with transition metals. The second limit gives formula_7 at low temperatures. In practice, which model is more applicable depends on the particular material.