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ABSTRACT
Rebuilding phonon mean free path (MFP) spectra from experimental data is integral to phonon MFP spectroscopy. However, being based on effective thermal conductivity, the current integral equation for this precludes the use of certain heat sources of convenient shapes, such as a cylindrical nanoline. Herein, to enable using diverse specimens exhibiting a ballistic effect, we develop a ballistic thermal resistance-based integral equation, utilizing the ease and accuracy of the modified ballistic–diffusive equations demonstrated in the companion paper. The availability of more diverse shapes of specimens will enhance further development and widen use of phonon MFP spectroscopy.
Nomenclature
| A | = | area (m2) |
| C | = | volumetric specific heat (J/m3∙K) |
| D | = | characteristic length of a heater, density of states |
| f | = | distribution function |
| ħ | = | reduced Planck constant |
| j | = | polarization index |
| k | = | thermal conductivity (W/m∙K) |
| n | = | the unit vector normal to the boundary |
| q | = | heat flux (W/m2) |
| = | nondimensional heat flux | |
| q | = | heat flux vector (W/m2) |
| r | = | coordinate in r direction |
| = | nondimensional coordinate in r direction | |
| r | = | position vector |
| s | = | distance along the propagation direction |
| = | nondimensional distance along the propagation direction | |
| S | = | suppression function |
| t | = | time |
| T | = | temperature (K) |
| u | = | specific internal energy (J/m3) |
| v | = | group velocity (m/s) |
| v | = | group velocity vector (m/s) |
| x | = | coordinate in x direction |
| z | = | coordinate in z direction |
| = | nondimensional coordinate in z direction |
Greek symbols
| χχ | = | a parameter defined as Λ/D |
| ϕ | = | mean free path spectrum of a bulk medium or polar angle |
| = | normalized phonon mean free path spectrum | |
| Λ | = | mean free path |
| μ | = | cosine of an angle θ |
| θ | = | angle |
| τ | = | relaxation time |
| τ1 | = | size parameter |
| ω | = | angular frequency |
| Ω | = | solid angle |
| = | direction vector |
Subscripts
| b | = | ballistic |
| e | = | emitting |
| eff | = | effective value |
| F | = | Fourier’s law |
| i | = | incident |
| m | = | diffusive |
| max | = | maximum value |
| n | = | normal |
| R | = | based on ballistic thermal resistance |
| w | = | wall or boundary |
| ω | = | spectral property in terms of angular frequency |
