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[[Phonons]] can scatter through several mechanisms as they travel through the material. These scattering mechanisms are: [[Umklapp scattering|Umklapp phonon-phonon scattering]], phonon-impurity scattering, phonon-electron scattering, and phonon-boundary scattering. Each scattering mechanism can be characterised by a relaxation rate 1/<math>\tau</math> which is the inverse of the corresponding relaxation time.

All scattering processes can be taken into account using [[Matthiessen's rule]]. Then the combined relaxation time <math>\tau_{C}</math> can be written as:

:<math>\frac{1}{\tau_C} = \frac{1}{\tau_U}+\frac{1}{\tau_M}+\frac{1}{\tau_B}+\frac{1}{\tau_{ph-e}}</math>

The parameters <math>\tau_{U}</math>, <math>\tau_{M}</math>, <math>\tau_{B}</math>, <math>\tau_{ph-e}</math> are due to Umklapp scattering, mass-difference impurity scattering, boundary scattering and phonon-electron scattering, respectively.

==Phonon-phonon scattering==
For phonon-phonon scattering, effects by normal processes (processes which conserve the phonon wave vector - N processes) are ignored in favor of Umklapp processes (U processes). Since normal processes vary linearly with <math>\omega</math> and umklapp processes vary with <math>\omega^2</math>, Umklapp scattering dominates at high frequency.<ref name=Mingo>{{Cite journal
| last = Mingo | first =  N
| year = 2003
| title = Calculation of nanowire thermal conductivity using complete phonon dispersion relations
| journal = Journal reference: Phys. Rev. B Phys Rev B
| volume = 68
| pages = 113308
| arxiv = cond-mat/0308587
|bibcode = 2003PhRvB..68k3308M |doi = 10.1103/PhysRevB.68.113308 }}</ref> <math>\tau_U</math> is given by:

:<math>\frac{1}{\tau_U}=2\gamma^2\frac{k_B T}{\mu V_0}\frac{\omega^2}{\omega_D}</math>

where <math>\gamma</math> is [[Grüneisen parameter|Gruneisen anharmonicity parameter]], μ is [[shear modulus]], V<sub>0</sub> is volume per atom and <math>\omega_{D}</math> is [[Debye frequency]].<ref name = ZouBalandin>{{Cite journal
| last = Zou
| first = Jie
| coauthors = Balandin, Alexander
| year = 2001
| title = Phonon heat conduction in a semiconductor nanowire
| journal = Journal of Applied Physics
| volume = 89
| issue = 5
| pages = 2932
| doi = 10.1063/1.1345515
| url = http://www.ndl.ee.ucr.edu/jap-zou-1.pdf
|bibcode = 2001JAP....89.2932Z }}</ref>

==Mass-difference impurity scattering==
Mass-difference impurity scattering is given by:

:<math>\frac{1}{\tau_M}=\frac{V_0 \Gamma \omega^4}{4\pi v_g^3}</math>

where <math>\Gamma</math> is a measure of the impurity scattering strength. Note that <math>{v_g}</math> is dependent of the dispersion curves.

==Boundary scattering==
Boundary scattering is particularly important for low-dimensional [[nanostructures]] and its relaxation time is given by:

:<math>\frac{1}{\tau_B}=\frac{V}{D}(1-p)</math>

where D is the dimension of the system and p represents the surface roughness parameter. The value p=1 means a smooth perfect surface that the scattering is purely specular and the relaxation time goes to ∞; hence, boundary scattering does not affect thermal transport. The value p=0 represents a very rough surface that the scattering is then purely diffusive which gives:

:<math>\frac{1}{\tau_B}=\frac{V}{D}</math>

This equation is also known as Casimir limit.<ref name=Casimir>{{Cite journal
| last = Casimir | first =  H.B.G
| year = 1938
| title = Note on the Conduction of Heat in Crystals
| journal = Physica, 5
| volume = 6
| bibcode = 1938Phy.....5..495C
| issue = 6
| doi = 10.1016/S0031-8914(38)80162-2
| pages = 495
}}</ref>

==Phonon-electron scattering==
{{mergefrom| lattice scattering|date=September 2011}}
Phonon-electron scattering can also contribute when the material is lightly doped. The corresponding relaxation time is given as:

:<math>\frac{1}{\tau_{ph-e}}=\frac{n_e \epsilon^2 \omega}{\rho V^2 k_B T}\sqrt{\frac{\pi m^* V^2}{2k_B T}} \exp \left(-\frac{m^*V^2}{2k_B T}\right)</math>

The parameter <math>n_{e}</math> is conduction electrons concentration, ε is deformation potential, ρ is mass density and m* is effective electron mass.<ref name = ZouBalandin /> It is usually assumed that contribution to thermal conductivity by phonon-electron scattering is negligible.

==References==
{{Reflist}}

[[Category:Condensed matter physics]]
[[Category:Scattering]]