Table 1
Best-fit particle-surface phase functions P∂V(θ) modeled as double Henyey-Greenstein phase functions (H-G) for media with fractional-Brownian-motion surface roughness.
| v | H | σ | rms | C | w∂V | g1, ∂V | g2, ∂V |
| 10-2 | 10-3 | 10-3 | |||||
|
|
|||||||
| 0.2 | 0.4 | 3 | 2.234 | 1.64 | 0.195 | –0.9615 | 0.0203 |
| 4 | 2.286 | 1.63 | 0.180 | –0.9633 | 0.0149 | ||
| 5 | 2.664 | 1.56 | 0.293 | –0.9534 | 0.0023 | ||
| 6 | 2.467 | 1.63 | 0.252 | –0.9554 | 0.0202 | ||
| 0.6 | 3 | 2.215 | 1.65 | 0.186 | –0.9624 | 0.0201 | |
| 4 | 2.293 | 1.66 | 0.252 | –0.9544 | 0.0228 | ||
| 5 | 2.418 | 1.65 | 0.213 | –0.9606 | 0.0232 | ||
| 6 | 2.500 | 1.43 | 0.222 | –0.9610 | –0.0287 | ||
| 0.8 | 3 | 2.219 | 1.64 | 0.182 | –0.9630 | 0.0177 | |
| 4 | 2.254 | 1.72 | 0.203 | –0.9590 | 0.0334 | ||
| 5 | 2.326 | 1.61 | 0.260 | –0.9559 | 0.0130 | ||
| 6 | 2.231 | 1.52 | 0.160 | –0.9676 | –0.0090 | ||
|
|
|||||||
| 0.3 | 0.4 | 3 | 2.213 | 1.24 | 1.412 | –0.9194 | –0.0273 |
| 4 | 2.324 | 1.25 | 1.678 | –0.9130 | –0.0241 | ||
| 5 | 2.288 | 1.37 | 1.165 | –0.9248 | 0.0036 | ||
| 6 | 2.800 | 1.45 | 1.106 | –0.9232 | 0.0247 | ||
| 0.6 | 3 | 2.374 | 1.22 | 1.406 | –0.9210 | –0.0354 | |
| 4 | 2.230 | 1.17 | 1.641 | –0.9170 | –0.0447 | ||
| 5 | 2.083 | 1.21 | 1.363 | –0.9236 | –0.0366 | ||
| 6 | 2.309 | 1.32 | 1.298 | –0.9217 | –0.0059 | ||
| 0.8 | 3 | 2.321 | 1.15 | 1.780 | –0.9140 | –0.0481 | |
| 4 | 2.219 | 1.22 | 1.247 | –0.9251 | –0.0351 | ||
| 5 | 2.001 | 1.14 | 1.432 | –0.9228 | –0.0546 | ||
| 6 | 2.218 | 1.21 | 1.305 | –0.9256 | –0.0372 | ||
|
|
|||||||
| 0.4 | 0.4 | 3 | 2.799 | 1.06 | 4.084 | –0.8834 | –0.0441 |
| 4 | 2.613 | 1.10 | 2.335 | –0.9083 | –0.0434 | ||
| 5 | 2.243 | 1.18 | 1.781 | –0.9154 | –0.0212 | ||
| 6 | 2.686 | 1.26 | 1.933 | –0.9086 | 0.0007 | ||
| 0.6 | 3 | 2.554 | 1.05 | 4.292 | –0.8824 | –0.0453 | |
| 4 | 2.685 | 1.00 | 2.930 | –0.9037 | –0.0694 | ||
| 5 | 2.327 | 1.05 | 2.271 | –0.9129 | –0.0568 | ||
| 6 | 2.578 | 1.15 | 2.153 | –0.9114 | –0.0229 | ||
| 0.8 | 3 | 2.729 | 0.96 | 4.626 | –0.8796 | –0.0775 | |
| 4 | 2.666 | 1.00 | 3.907 | –0.8885 | –0.0633 | ||
| 5 | 2.281 | 1.02 | 2.296 | –0.9129 | –0.0647 | ||
| 6 | 2.573 | 0.50 | 2.751 | –0.9045 | –0.0662 | ||
|
|
|||||||
| 0.5 | 0.4 | 3 | 2.793 | 0.96 | 3.853 | –0.8961 | –0.0647 |
| 4 | 2.622 | 1.09 | 3.324 | –0.8972 | –0.0209 | ||
| 5 | 2.930 | 1.15 | 3.289 | –0.8929 | –0.0042 | ||
| 6 | 2.934 | 1.20 | 2.804 | –0.8974 | 0.0042 | ||
| 0.6 | 3 | 2.735 | 0.92 | 3.666 | –0.9012 | –0.0758 | |
| 4 | 2.792 | 0.95 | 3.833 | –0.8972 | –0.0651 | ||
| 5 | 2.585 | 0.99 | 3.169 | –0.9031 | –0.0539 | ||
| 6 | 2.811 | 0.98 | 3.781 | –0.8953 | –0.0543 | ||
| 0.8 | 3 | 3.201 | 0.86 | 6.533 | –0.8720 | –0.0862 | |
| 4 | 2.617 | 0.89 | 4.038 | –0.8993 | –0.0861 | ||
| 5 | 2.716 | 0.86 | 4.092 | –0.8987 | –0.0977 | ||
| 6 | 2.738 | 0.90 | 3.851 | –0.8996 | –0.0844 | ||
Notes. v, H, and σ are the volume density, Hurst exponent, and standard deviation, respectively, C is a scaling coefficient for intensity (each double H-G phase function is to be divided by the corresponding C in order to allow comparisons), and w∂V, g1, ∂V, and g2, ∂V are the double H-G parameters. The volume-element phase functions PV(θ) follow from Eq. (5). The entry shown in boldface results in the best fit displayed in Fig. 4.
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