text: Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation rate is given by:
  where  is the characteristic length of the system and  represents the fraction of specularly scattered phonons. The  parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness , a wavelength-dependent value for  can be calculated using
  where  is the angle of incidence. An extra factor of  is sometimes erroneously included in the exponent of the above equation. At normal incidence, , perfectly specular scattering (i.e. ) would require an arbitrarily large wavelength, or conversely an arbitrarily small roughness. Purely specular scattering does not introduce a boundary-associated increase in the thermal resistance. In the diffusive limit, however, at  the relaxation rate becomes
  This equation is also known as Casimir limit.
  These phenomenological equations can in many cases accurately model the thermal conductivity of isotropic nano-structures with characteristic sizes on the order of the phonon mean free path. More detailed calculations are in general required to fully capture the phonon-boundary interaction across all relevant vibrational modes in an arbitrary structure.