Table I
Functions of partial derivatives of Legendre polynomials for the arguments cos45° and cos135°.
| πn | | | [xπn−(1−x2)πn′] |
---|
n | 45° | 135° | |πn| | 45° | 135° | |xπn−(1−x2)πn′| |
---|
1 | + | + | 1.00000 | + | − | 0.707107 |
2 | + | − | 2.12132 | | | 0.000000 |
3 | + | + | 2.25000 | − | + | 3.71231 |
4 | + | − | 0.88388 | − | − | 8.75000 |
5 | − | − | 1.40625 | − | + | 10.2751 |
6 | − | + | 3.24827 | − | − | 3.93750 |
7 | − | − | 3.33594 | + | − | 9.47413 |
8 | − | + | 1.34240 | + | + | 22.4297 |
9 | + | + | 1.73584 | + | − | 24.4708 |
10 | + | − | 4.08278 | + | + | 9.77539 |
11 | + | + | 4.15320 | − | + | 16.6891 |
12 | + | − | 1.68654 | − | − | 39.6808 |
13 | − | − | 2.01479 | − | + | 42.1323 |
14 | − | + | 4.77520 | − | − | 17.0503 |
15 | − | − | 4.83564 | + | − | 25.1135 |
16 | − | + | 1.97304 | + | + | 59.8836 |
17 | + | + | 2.26038 | + | − | 62.6863 |
18 | + | − | 5.37978 | + | + | 25.5262 |
19 | + | + | 5.43353 | − | + | 34.5959 |
20 | + | − | 2.22347 | − | − | 82.6596 |
21 | − | − | 2.48214 | − | + | 85.7742 |
22 | − | + | 5.92320 | − | − | 35.0542 |
23 | − | − | 5.97209 | + | − | 45.0301 |
24 | − | + | 2.44868 | + | + | 107.747 |
25 | + | + | 2.68582 | + | − | 111.145 |
Table II
Angular-distribution functions for γ=45°, m=1.20.
α | R(i1*) | I(i1*) | i1 | R(i2*) | I(i2*) | i2 |
---|
0.2 | +0.001013 | −0.000001 | 0.051027 | +0.000715 | −0.0000007 | 0.06512 |
0.4 | +0.007857 | −0.000043 | 0.046174 | +0.005545 | −0.000030 | 0.043075 |
0.6 | +0.025111 | −0.000469 | 0.036308 | +0.017672 | −0.000331 | 0.0331242 |
0.8 | +0.054805 | −0.002435 | 0.0030094 | +0.038402 | −0.001718 | 0.0014778 |
1.0 | +0.094395 | −0.008223 | 0.0089786 | +0.065663 | −0.005779 | 0.0043450 |
1.2 | +0.136381 | −0.020546 | 0.0190218 | +0.093745 | −0.014321 | 0.0089933 |
1.4 | +0.167337 | −0.040285 | 0.0296245 | +0.112525 | −0.027581 | 0.0134226 |
1.6 | +0.171219 | −0.063527 | 0.0333512 | +0.110047 | −0.041755 | 0.0138538 |
1.8 | +0.135021 | −0.079980 | 0.0246276 | +0.077239 | −0.047434 | 0.0082158 |
2.0 | +0.055816 | −0.074763 | 0.0087049 | +0.014424 | −0.031080 | 0.0011740 |
2.2 | −0.053737 | −0.034657 | 0.0040888 | −0.062910 | +0.018133 | 0.0042865 |
2.4 | −0.167118 | +0.042274 | 0.0297155 | −0.125911 | +0.096587 | 0.0251827 |
2.6 | −0.253578 | +0.141001 | 0.0841832 | −0.148017 | +0.179698 | 0.0542002 |
2.8 | −0.291572 | +0.235259 | 0.140361 | −0.120291 | +0.230836 | 0.0677549 |
3.0 | −0.272122 | +0.299063 | 0.163489 | −0.056424 | +0.218230 | 0.0508079 |
3.2 | −0.195367 | +0.312168 | 0.135617 | +0.011928 | +0.129627 | 0.0169456 |
3.4 | −0.069974 | +0.260859 | 0.0729434 | +0.047866 | −0.016924 | 0.0025776 |
3.6 | +0.085292 | +0.138697 | 0.0265116 | +0.030934 | −0.174877 | 0.0315389 |
3.8 | +0.240266 | −0.051242 | 0.0603535 | −0.028909 | −0.290381 | 0.0851570 |
4.0 | +0.353152 | −0.292316 | 0.210165 | −0.096760 | −0.326441 | 0.115926 |
4.2 | +0.374442 | −0.545443 | 0.437715 | −0.130112 | −0.277121 | 0.0937250 |
4.4 | +0.266634 | −0.739932 | 0.618594 | −0.100089 | −0.170292 | 0.0390172 |
4.6 | +0.036968 | −0.786568 | 0.620057 | −0.011844 | −0.049719 | 0.0026123 |
4.8 | −0.248189 | −0.616442 | 0.441599 | +0.101184 | +0.052318 | 0.0129754 |
5.0 | −0.476315 | −0.220166 | 0.275349 | +0.200872 | +0.125020 | 0.0559797 |
5.2 | −0.526660 | +0.321224 | 0.380556 | +0.260135 | +0.170947 | 0.0968930 |
5.4 | −0.340663 | +0.841011 | 0.823351 | +0.268141 | +0.197622 | 0.110954 |
5.6 | +0.030398 | +1.159775 | 1.346002 | +0.231506 | +0.208507 | 0.0970702 |
5.8 | +0.451312 | +1.165652 | 1.562427 | +0.163643 | +0.194039 | 0.064430 |
6.0 | +0.756392 | +0.849189 | 1.293251 | +0.069331 | +0.139057 | 0.0241438 |
6.2 | +0.805878 | +0.318155 | 0.750662 | −0.068498 | +0.048468 | 0.0070411 |
6.4 | +0.592957 | −0.254059 | 0.416144 | −0.229663 | −0.047307 | 0.054983 |
6.6 | +0.187647 | −0.706643 | 0.534556 | −0.400894 | −0.103084 | 0.171343 |
6.8 | −0.273845 | −0.971783 | 1.019353 | −0.520115 | −0.091679 | 0.278924 |
7.0 | −0.685516 | −1.041616 | 1.554898 | −0.524968 | −0.019408 | 0.275968 |
Table III
Angular-distribution functions for γ=90°, m=1.20.
α | R(i1*) | I(i1*) | i1 | R(i2*) | I(i2*) | i2 |
---|
0.2 | +0.001019 | −0.000001 | 0.051038 | −0.000002 | | 0.0114 |
0.4 | +0.008051 | −0.000043 | 0.046482 | −0.000022 | | 0.0948 |
0.6 | +0.026522 | −0.000470 | 0.037039 | −0.000172 | +0.000001 | 0.07296 |
0.8 | +0.060494 | −0.002449 | 0.0036655 | −0.000376 | +0.000008 | 0.06542 |
1.0 | +0.110820 | −0.008342 | 0.0123507 | −0.002297 | +0.000072 | 0.055281 |
1.2 | +0.174620 | −0.021229 | 0.0309428 | −0.005896 | +0.000415 | 0.043494 |
1.4 | +0.243662 | −0.043191 | 0.0612366 | −0.013095 | +0.001824 | 0.0317481 |
1.6 | +0.306277 | −0.073393 | 0.099192 | −0.025789 | +0.006410 | 0.0370616 |
1.8 | +0.350288 | −0.107768 | 0.134316 | −0.044911 | +0.018620 | 0.0023637 |
2.0 | +0.365395 | −0.140802 | 0.153339 | −0.067358 | +0.045078 | 0.0065691 |
2.2 | +0.343280 | −0.167416 | 0.145869 | −0.082603 | +0.090423 | 0.0149996 |
2.4 | +0.277061 | −0.182087 | 0.110181 | −0.075027 | +0.149007 | 0.0278321 |
2.6 | +0.161106 | −0.179817 | 0.0582893 | −0.035572 | +0.201933 | 0.0420423 |
2.8 | −0.003973 | −0.145926 | 0.0213102 | +0.028441 | +0.228469 | 0.0530070 |
3.0 | −0.203612 | −0.064471 | 0.0456144 | +0.099540 | +0.218957 | 0.0578504 |
3.2 | −0.402152 | +0.073246 | 0.167091 | +0.162407 | +0.176524 | 0.0575368 |
3.4 | −0.553464 | +0.249591 | 0.368618 | +0.209672 | +0.108996 | 0.0558424 |
3.6 | −0.629716 | +0.421166 | 0.573923 | +0.239560 | +0.022049 | 0.0578752 |
3.8 | −0.635232 | +0.548361 | 0.704219 | +0.250188 | −0.082873 | 0.0694620 |
4.0 | −0.586240 | +0.618340 | 0.726022 | +0.233722 | −0.205372 | 0.0968036 |
4.2 | −0.486831 | +0.636463 | 0.642090 | +0.174914 | −0.336643 | 0.143923 |
4.4 | −0.327028 | +0.604867 | 0.472811 | +0.061691 | −0.452388 | 0.208461 |
4.6 | −0.097661 | +0.509732 | 0.269364 | −0.099945 | −0.516983 | 0.277260 |
4.8 | −0.188771 | +0.322613 | 0.139714 | −0.280785 | −0.502450 | 0.331296 |
5.0 | +0.479975 | +0.021237 | 0.230827 | −0.439960 | −0.403487 | 0.356367 |
5.2 | +0.692830 | −0.370832 | 0.617530 | −0.543872 | −0.239042 | 0.352938 |
5.4 | +0.767905 | −0.768125 | 1.179694 | −0.579371 | −0.038926 | 0.337186 |
5.6 | +0.715267 | −1.073497 | 1.664003 | −0.549714 | +0.174213 | 0.332536 |
5.8 | +0.589311 | −1.240035 | 1.884974 | −0.457935 | +0.388861 | 0.360917 |
6.0 | +0.432722 | −1.275692 | 1.814638 | −0.298446 | +0.592004 | 0.439539 |
6.2 | +0.226313 | −1.204679 | 1.502469 | −0.063404 | +0.753511 | 0.571799 |
6.4 | +0.017369 | −1.046495 | 1.095453 | +0.232843 | +0.828564 | 0.740734 |
6.6 | −0.280869 | −0.755115 | 0.649086 | +0.547008 | +0.781760 | 0.910366 |
6.8 | −0.589914 | −0.284289 | 0.428819 | +0.819019 | +0.604380 | 1.036067 |
7.0 | −0.787433 | +0.347302 | 0.740669 | +0.994636 | +0.320013 | 1.091709 |
Table IV
Angular-distribution functions for γ=135°, m=1.20.
α | R(i1*) | I(i1*) | i1 | R(i2*) | I(i2*) | i2 |
---|
0.2 | +0.001026 | −0.000001 | 0.051051 | −0.000726 | +0.000001 | 0.06526 |
0.4 | +0.008245 | −0.000043 | 0.046798 | −0.005841 | +0.000030 | 0.043412 |
0.6 | +0.027984 | −0.000471 | 0.037834 | −0.019876 | +0.000333 | 0.0339517 |
0.8 | +0.066571 | −0.002463 | 0.0044377 | −0.047459 | +0.001746 | 0.0022554 |
1.0 | +0.129054 | −0.008461 | 0.0167266 | −0.092490 | +0.006019 | 0.0085907 |
1.2 | +0.219091 | −0.021917 | 0.0484810 | −0.158150 | +0.015706 | 0.0252580 |
1.4 | +0.337482 | −0.046133 | 0.116022 | −0.246020 | +0.033540 | 0.0616509 |
1.6 | +0.483646 | −0.083475 | 0.240881 | −0.357048 | +0.062271 | 0.131361 |
1.8 | +0.656737 | −0.136567 | 0.449954 | −0.491880 | +0.106070 | 0.253196 |
2.0 | +0.854142 | −0.210777 | 0.773986 | −0.648374 | +0.172346 | 0.450092 |
2.2 | +1.066980 | −0.313412 | 1.236673 | −0.817005 | +0.269458 | 0.740105 |
2.4 | +1.278106 | −0.447000 | 1.833363 | −0.981272 | +0.398636 | 1.121806 |
2.6 | +1.469762 | −0.603881 | 2.524873 | −1.126976 | +0.549870 | 1.572433 |
2.8 | +1.631986 | −0.770739 | 3.257415 | −1.251747 | +0.712089 | 2.073940 |
3.0 | +1.762492 | −0.939183 | 3.988443 | −1.357057 | +0.885794 | 2.626235 |
3.2 | +1.857560 | −1.108707 | 4.679761 | −1.432223 | +1.078262 | 3.213913 |
3.4 | +1.905073 | −1.281859 | 5.272464 | −1.454402 | +1.285824 | 3.768628 |
3.6 | +1.884050 | −1.454130 | 5.664138 | −1.407098 | +1.487602 | 4.192884 |
3.8 | +1.774348 | −1.605293 | 5.725277 | −1.291687 | +1.661404 | 4.428718 |
4.0 | +1.574030 | −1.703509 | 5.379512 | −1.116296 | +1.798304 | 4.480014 |
4.2 | +1.305929 | −1.728068 | 4.691667 | −0.881110 | +1.896135 | 4.371684 |
4.4 | +0.994851 | −1.688921 | 3.842179 | −0.581014 | +1.944132 | 4.117227 |
4.6 | +0.634928 | −1.608263 | 2.989643 | −0.221939 | +1.924769 | 3.753993 |
4.8 | +0.199835 | −1.477573 | 2.223156 | +0.179284 | +1.829065 | 3.377623 |
5.0 | −0.307226 | −1.254182 | 1.667360 | +0.609686 | +1.658186 | 3.121297 |
5.2 | −0.832841 | −0.914601 | 1.530120 | +1.062420 | +1.409478 | 3.115363 |
5.4 | −1.324701 | −0.493413 | 1.998289 | +1.522715 | +1.072940 | 3.469861 |
5.6 | −1.786343 | −0.037889 | 3.192455 | +1.961800 | +0.645595 | 4.265452 |
5.8 | −2.240756 | +0.455405 | 5.228381 | +2.349630 | +0.141611 | 5.540815 |
6.0 | −2.662967 | +1.017288 | 8.126268 | +2.664873 | −0.414021 | 7.272962 |
6.2 | −2.993911 | +1.633631 | 11.63225 | +2.909834 | −0.999920 | 9.466976 |
6.4 | −3.189029 | +2.232432 | 15.15366 | +3.062991 | −1.601280 | 11.94601 |
6.6 | −3.297066 | +2.781873 | 18.60947 | +3.129470 | −2.217448 | 14.71066 |
6.8 | −3.326364 | +3.293384 | 21.91108 | +3.077565 | −2.833674 | 17.50112 |
7.0 | −3.240988 | +3.777106 | 24.77053 | +2.881455 | −3.415512 | 19.96851 |
Table V
Additional angular-distribution functions for m=1.20.
γ | R(i1*) | I(i1*) | i1 | R(i2*) | I(i2*) | i2 |
---|
α=1.06 |
45 | 0.10726 | −0.01113 | 0.01163 | 0.07440 | −0.007810 | 0.005597 |
90 | 0.1288 | −0.01134 | 0.01673 | −0.003107 | −0.031244 | 0.059668 |
135 | 0.1531 | −0.01155 | 0.02357 | −0.1099 | 0.008231 | 0.01215 |
α=2.02 |
45 | 0.04588 | −0.07248 | 0.07358 | 0.006965 | −0.02773 | 0.038176 |
90 | 0.3650 | −0.1438 | 0.1539 | −0.06945 | 0.04875 | 0.007200 |
135 | 0.8750 | −0.2197 | 0.8138 | −0.6649 | 0.1806 | 0.4747 |
α=3.91 |
45 | 0.3108 | −0.1793 | 0.1287 | −0.06854 | −0.3212 | 0.1079 |
90 | −0.6143 | 0.5937 | 0.7299 | 0.2453 | −0.1483 | 0.08214 |
135 | 1.6744 | −1.6678 | 5.585 | −1.2024 | 1.7414 | 4.478 |
α=4.26 |
45 | 0.3568 | −0.6140 | 0.5042 | −0.1294 | −0.2492 | 0.07883 |
90 | −0.4458 | 0.6325 | 0.5988 | 0.1469 | −0.3746 | 0.1619 |
135 | 1.2167 | −1.7218 | 4.445 | −0.7980 | 1.9166 | 4.310 |
Table VI
Comparison of theoretical and experimental values of the polarization ratio.
γ | ρ135 | ρ90 | ρ45 | ρ45 |
---|
Exptl | Theor | Exptl | Theor | Exptl | Theor | Corrected |
---|
1.06 | 0.52 | 0.515 | 0.00041 | 0.000578 | 0.50 | 0.481 | 0.48 |
2.02 | 0.585 | 0.583 | 0.043 | 0.0468 | 0.44 | 0.1111 | 0.107 |
3.91 | 0.76 | 0.802 | 0.117 | 0.1125 | 0.79 | 0.838 | 0.83 |
4.26 | 0.85 | 0.970 | 0.22 | 0.270 | 0.62 | 0.1563 | 0.133 |
Table VII
Comparison of theoretical and experimental values of the dissymmetry (135°/45°).
γ | D1 (vertically polarized) | D2 (horizontally polarized) |
---|
Exptl | Theor | Ds | Exptl | Theor | Ds |
---|
1.06 | 1.56 | 2.03 | 1.09 | 1.61 | 2.17 | 1.15 |
2.02 | 13.6 | 110.6 | 8.1 | 17.3 | 580 | 5.9 |
3.91 | 11.6 | 43.4 | 27.1 | 11.5 | 41.5 | 29.4 |
4.26 | 10.3 | 8.73 | 30.8 | 14.1 | 55.8 | 71.5 |
Table VIII
Comparison of values of α calculated from various sources.
From electron microscope | From ρ45 | From ρ45 (corr) | From ρ135 | From ρ90 | From D1 | From D2 |
---|
1.06 | Below 0.40 | 1.00 to 1.10 | 1.00 to 1.25 | 0.99 | 0.87 | 0.84 |
2.02 | 2.09 or 1.49 | 2.02 | 2.03 | 2.00 | 1.71 | 1.64 |
3.91 | 3.93 | 3.91 | 3.74 | 3.89 | 4.19 | ⋯ |
4.26 | 3.99 | 4.28 | 4.04 | 4.20 | 4.21 | ⋯ |