# Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit

## Abstract

Scattering of normally incident longitudinal and transverse acoustic waves by a randomly rough surface of an elastically isotropic solid is analyzed within the small-perturbation approach. In the limiting case of a large correlation length L compared with the acoustic wavelength, the specularity reduction is given by 4η ^{2}k ^{2} ,where η is the rms roughness and k is the acoustic wave vector, which is in agreement with the well-known Kirchhoff approximation result often referred to as Ziman's equation [J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960)]. In the opposite limiting case of a small correlation length, the specularity reduction is discovered to be proportional to η ^{2}k ^{4}L ^{2} , with the fourth power dependence on frequency as in Rayleigh scattering. Numerical calculations for a Gaussian autocorrelation function of surface roughness connect these limiting cases and reveal a maximum of diffuse scattering at an intermediate value of L . This maximum becomes increasingly pronounced for the incident longitudinal wave as the Poisson's ratio of the medium approaches 1/2 as a result of increased scattering into transverse and Rayleigh surface waves. The results imply that thermal transport models using Ziman's formula are likely to overestimate the heat flux dissipationmore »

- Authors:

- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

- Publication Date:

- Research Org.:
- Energy Frontier Research Centers (EFRC) (United States). Solid-State Solar-Thermal Energy Conversion Center (S3TEC); Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1370990

- Alternate Identifier(s):
- OSTI ID: 1181141

- Grant/Contract Number:
- SC0001299; FG02-09ER46577

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review. B, Condensed Matter and Materials Physics

- Additional Journal Information:
- Journal Volume: 91; Journal Issue: 13; Related Information: S3TEC partners with Massachusetts Institute of Technology (lead); Boston College; Oak Ridge National Laboratory; Rensselaer Polytechnic Institute; Journal ID: ISSN 1098-0121

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; solar (photovoltaic); solar (thermal); solid state lighting; phonons; thermal conductivity; thermoelectric; defects; mechanical behavior; charge transport; spin dynamics; materials and chemistry by design; optics; synthesis (novel materials); synthesis (self-assembly); synthesis (scalable processing)

### Citation Formats

```
Maznev, A. A. Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit. United States: N. p., 2015.
Web. doi:10.1103/PhysRevB.91.134306.
```

```
Maznev, A. A. Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit. United States. doi:10.1103/PhysRevB.91.134306.
```

```
Maznev, A. A. Tue .
"Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit". United States. doi:10.1103/PhysRevB.91.134306. https://www.osti.gov/servlets/purl/1370990.
```

```
@article{osti_1370990,
```

title = {Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit},

author = {Maznev, A. A.},

abstractNote = {Scattering of normally incident longitudinal and transverse acoustic waves by a randomly rough surface of an elastically isotropic solid is analyzed within the small-perturbation approach. In the limiting case of a large correlation length L compared with the acoustic wavelength, the specularity reduction is given by 4η2k2 ,where η is the rms roughness and k is the acoustic wave vector, which is in agreement with the well-known Kirchhoff approximation result often referred to as Ziman's equation [J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960)]. In the opposite limiting case of a small correlation length, the specularity reduction is discovered to be proportional to η2k4L2 , with the fourth power dependence on frequency as in Rayleigh scattering. Numerical calculations for a Gaussian autocorrelation function of surface roughness connect these limiting cases and reveal a maximum of diffuse scattering at an intermediate value of L . This maximum becomes increasingly pronounced for the incident longitudinal wave as the Poisson's ratio of the medium approaches 1/2 as a result of increased scattering into transverse and Rayleigh surface waves. The results imply that thermal transport models using Ziman's formula are likely to overestimate the heat flux dissipation due to boundary scattering, whereas modeling interface roughness as atomic disorder is likely to underestimate scattering.},

doi = {10.1103/PhysRevB.91.134306},

journal = {Physical Review. B, Condensed Matter and Materials Physics},

number = 13,

volume = 91,

place = {United States},

year = {2015},

month = {4}

}

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#### Figures / Tables:

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*