Results of computations are presented to show the variations of coefficients of four different Legendre series, one for each of the four scattering functions needed in describing directional dependence of the radiation scattered by a sphere. Values of the size parameter (x) covered for this purpose vary from 0.01 to 100.0. An adequate representation of the entire scattering function vs scattering angle curve is obtained after making use of about 2x + 10 terms of the series. It is shown that a section of a scattering function vs scattering angle curve can be adequately represented by a fourier series with less than 2x + 10 terms. The exact number of terms required for this purpose depends upon values of the size parameter and refractive index, as well as upon the values of the scattering angles defining the section under study. Necessary expressions for coefficients of such fourier series are derived with the help of the addition theorem of spherical harmonics.
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The number in the parenthesis represents the power of 10 by which the preceding number is to be multiplied, e.g., 5.0 (− 01) = 5.0 × 10−1. If there is no parentheses after the number, the power of 10 is equal to zero.
Table II
Values of the Normalized Coefficients of the Legendre Series for Scattering Functions of a Sphere; x = 10.0, m = 1.342
k
Λk(1)(x,m)
Λk(2)(x,m)
Λk(3)(x,m)
Λk(4)(x,m)
1
1.0645
9.3551(−1)
9.0259(−1)
−1.0109(−1)
2
2.1853
1.9286
2.0843
4.4926(−2)
3
2.7321
2.7926
2.6642
−9.0608(−2)
4
2.5467
2.4535
2.4983
2.9276(−2)
5
2.2937
2.3913
2.3778
1.3266(−1)
6
2.1454
2.2061
2.0681
4.6171(−2)
7
2.0520
2.0194
2.1048
2.9736(−1)
8
2.1552
2.0738
2.0710
1.6766(−1)
9
2.3562
2.1007
2.2183
2.9973(−1)
10
2.5434
2.3038
2.4925
2.8820(−1)
11
2.9173
2.4815
2.6113
2.7102(−1)
12
3.1462
2.7860
3.1150
3.5453(−1)
13
3.4761
3.1150
3.2254
2.8577(−1)
14
3.8971
3.5953
3.8084
3.1660(−1)
15
3.9568
3.9086
4.0649
3.7816(−1)
16
4.4225
4.7123
4.4791
6.1476(−2)
17
4.1765
4.5631
4.5421
2.5024(−1)
18
3.8141
5.0155
4.5404
−3.7044(−1)
19
3.0307
4.0056
3.5374
−7.8945(−1)
20
2.2171
2.5007
2.4057
−8.7630(−1)
21
1.4823
1.3086
1.1182
−9.0760(−1)
22
3.4957(−1)
−1.2688(−1)
1.0509(−1)
−5.4768(−2)
23
1.4843(−1)
7.2065(−2)
9.5872(−2)
−2.4685(−2)
24
4.5331(−2)
2.0767(−2)
2.9265(−2)
−7.0836(−3)
25
1.1477(−2)
4.5711(−3)
7.0955(−3)
−1.4658(−3)
26
2.5031(−3)
8.4961(−4)
1.4629(−3)
−2.4298(−4)
27
4.8173(−4)
1.3824(−4)
2.6389(−4)
−3.3629(−5)
28
8.3323(−5)
2.0143(−5)
4.2478(−5)
−3.9757(−6)
29
1.3134(−5)
2.6727(−6)
6.1969(−6)
−4.0650(−7)
30
1.9069(−6)
3.2724(−7)
8.2970(−7)
−3.5526(−8)
31
2.5508(−7)
3.7323(−8)
1.0310(−7)
−1.3215(−9)
32
3.0920(−8)
3.7615(−9)
1.1234(−8)
−1.1240(−11)
33
3.4798(−9)
3.5833(−10)
1.1529(−9)
−5.5487(−14)
34
3.5894(−10)
3.1542(−11)
1.0902(−10)
1.8598(−15)
35
3.3466(−11)
2.5254(−12)
9.3547(−12)
6.5504(−17)
Table III
Values of Fn(1)(x,m,μ′,μ) for a Few Selected Values of μ′; x = 10.0, m = 1.342, μ = 0.0
n
μ′ = cos10°
μ′ = cos30°
μ′ = cos50°
μ′ = cos70°
μ′ = cos90°
1
2.0907(−01)
3.3096(−01)
6.6961(−01)
1.3142
3.2897
2
4.5249(−02)
2.6843(−01)
8.0138(−01)
2.0044
5.8812
3
−5.4385(−02)
1.1697(−01)
5.7854(−01)
1.5084
5.2537
4
5.1981(−02)
5.5943(−02)
2.6323(−01)
9.3576(−1)
4.4934
5
2.7793(−02)
2.1202(−03)
1.6559(−01)
5.2051(−1)
3.9230
6
−1.1307(−02)
2.6341(−02)
3.8726(−02)
2.8169(−1)
3.5697
7
−3.5766(−03)
−5.8576(−03)
2.3252(−02)
8.4788(−2)
3.3050
8
9.5099(−04)
7.8603(−03)
−4.9093(−02)
−3.1264(−2)
3.1581
9
2.2385(−04)
3.0818(−02)
−5.0069(−02)
−1.0552(−1)
3.0507
10
−4.0067(−05)
−4.2322(−02)
−7.6546(−02)
−2.3122(−1)
2.9468
11
−7.8530(−06)
−2.9079(−02)
−6.5303(−02)
−2.2998(−1)
2.8587
12
9.4096(−07)
2.1328(−02)
−5.0866(−02)
−3.7047(−1)
2.7589
13
1.6304(−07)
1.0994(−02)
−7.8184(−03)
−3.1891(−1)
2.5997
14
−1.2774(−08)
−4.3674(−03)
−8.6115(−03)
−3.7280(−1)
2.5051
15
−2.0120(−09)
−2.0076(−03)
−1.1672(−02)
−2.9478(−1)
2.2414
16
9.9630(−11)
4.2701(−04)
5.1686(−02)
−1.9189(−1)
2.0696
17
1.4065(−11)
1.8425(−04)
3.1464(−02)
−8.8012(−2)
1.7402
18
−4.3836(−13)
−1.9455(−05)
−1.6651(−02)
1.4685(−2)
1.3664
19
−4.7748(−14)
−7.6247(−06)
−1.0440(−02)
3.7958(−2)
1.0059
20
1.2755(−15)
4.0059(−07)
9.6949(−04)
1.2373(−1)
6.1834(−1)
21
4.4328(−17)
7.3321(−08)
7.0878(−04)
8.6092(−2)
3.9090(−1)
22
−4.0331(−18)
−1.2211(−08)
3.4031(−05)
1.6949(−2)
9.1213(−2)
23
−2.3747(−20)
8.4767(−10)
4.1930(−05)
7.5101(−3)
3.6892(−2)
24
5.3215(−21)
3.5734(−10)
1.2702(−05)
2.2149(−3)
1.0902(−2)
25
4.2733(−22)
6.3490(−11)
2.7864(−06)
5.3111(−4)
2.6850(−3)
26
2.1317(−23)
8.2559(−12)
4.9833(−07)
1.0870(−4)
5.7139(−4)
27
8.3048(−25)
8.8035(−13)
7.6418(−08)
1.9535(−5)
1.0753(−4)
28
2.7288(−26)
8.0988(−14)
1.0354(−08)
3.1462(−6)
1.8215(−5)
29
8.0002(−28)
6.6859(−15)
1.2692(−09)
4.6126(−7)
2.8151(−6)
30
2.1641(−29)
5.0887(−16)
1.4336(−10)
6.2327(−8)
4.0105(−7)
Table IV
Values of N(μ′ = cos θ′, μ = cos θ) for Three Different Values of θ; x = 100.0, m = 1.342
θ′/θ
30°
60°
90°
0
1
1
1
10
45
47
48
20
83
82
84
30
111
115
117
40
117
144
150
50
116
171
170
60
117
190
190
70
115
190
206
80
117
192
215
90
118
189
217
100
117
190
215
110
118
193
206
120
118
190
190
130
118
173
170
140
118
148
150
150
113
118
117
160
84
83
84
170
47
47
48
180
1
1
1
Tables (4)
Table I
Values of the First Five Normalized Coefficients of Legendre Series for the Scattering Function M2 (x,m,Ө); m = 1.342
The number in the parenthesis represents the power of 10 by which the preceding number is to be multiplied, e.g., 5.0 (− 01) = 5.0 × 10−1. If there is no parentheses after the number, the power of 10 is equal to zero.
Table II
Values of the Normalized Coefficients of the Legendre Series for Scattering Functions of a Sphere; x = 10.0, m = 1.342
k
Λk(1)(x,m)
Λk(2)(x,m)
Λk(3)(x,m)
Λk(4)(x,m)
1
1.0645
9.3551(−1)
9.0259(−1)
−1.0109(−1)
2
2.1853
1.9286
2.0843
4.4926(−2)
3
2.7321
2.7926
2.6642
−9.0608(−2)
4
2.5467
2.4535
2.4983
2.9276(−2)
5
2.2937
2.3913
2.3778
1.3266(−1)
6
2.1454
2.2061
2.0681
4.6171(−2)
7
2.0520
2.0194
2.1048
2.9736(−1)
8
2.1552
2.0738
2.0710
1.6766(−1)
9
2.3562
2.1007
2.2183
2.9973(−1)
10
2.5434
2.3038
2.4925
2.8820(−1)
11
2.9173
2.4815
2.6113
2.7102(−1)
12
3.1462
2.7860
3.1150
3.5453(−1)
13
3.4761
3.1150
3.2254
2.8577(−1)
14
3.8971
3.5953
3.8084
3.1660(−1)
15
3.9568
3.9086
4.0649
3.7816(−1)
16
4.4225
4.7123
4.4791
6.1476(−2)
17
4.1765
4.5631
4.5421
2.5024(−1)
18
3.8141
5.0155
4.5404
−3.7044(−1)
19
3.0307
4.0056
3.5374
−7.8945(−1)
20
2.2171
2.5007
2.4057
−8.7630(−1)
21
1.4823
1.3086
1.1182
−9.0760(−1)
22
3.4957(−1)
−1.2688(−1)
1.0509(−1)
−5.4768(−2)
23
1.4843(−1)
7.2065(−2)
9.5872(−2)
−2.4685(−2)
24
4.5331(−2)
2.0767(−2)
2.9265(−2)
−7.0836(−3)
25
1.1477(−2)
4.5711(−3)
7.0955(−3)
−1.4658(−3)
26
2.5031(−3)
8.4961(−4)
1.4629(−3)
−2.4298(−4)
27
4.8173(−4)
1.3824(−4)
2.6389(−4)
−3.3629(−5)
28
8.3323(−5)
2.0143(−5)
4.2478(−5)
−3.9757(−6)
29
1.3134(−5)
2.6727(−6)
6.1969(−6)
−4.0650(−7)
30
1.9069(−6)
3.2724(−7)
8.2970(−7)
−3.5526(−8)
31
2.5508(−7)
3.7323(−8)
1.0310(−7)
−1.3215(−9)
32
3.0920(−8)
3.7615(−9)
1.1234(−8)
−1.1240(−11)
33
3.4798(−9)
3.5833(−10)
1.1529(−9)
−5.5487(−14)
34
3.5894(−10)
3.1542(−11)
1.0902(−10)
1.8598(−15)
35
3.3466(−11)
2.5254(−12)
9.3547(−12)
6.5504(−17)
Table III
Values of Fn(1)(x,m,μ′,μ) for a Few Selected Values of μ′; x = 10.0, m = 1.342, μ = 0.0
n
μ′ = cos10°
μ′ = cos30°
μ′ = cos50°
μ′ = cos70°
μ′ = cos90°
1
2.0907(−01)
3.3096(−01)
6.6961(−01)
1.3142
3.2897
2
4.5249(−02)
2.6843(−01)
8.0138(−01)
2.0044
5.8812
3
−5.4385(−02)
1.1697(−01)
5.7854(−01)
1.5084
5.2537
4
5.1981(−02)
5.5943(−02)
2.6323(−01)
9.3576(−1)
4.4934
5
2.7793(−02)
2.1202(−03)
1.6559(−01)
5.2051(−1)
3.9230
6
−1.1307(−02)
2.6341(−02)
3.8726(−02)
2.8169(−1)
3.5697
7
−3.5766(−03)
−5.8576(−03)
2.3252(−02)
8.4788(−2)
3.3050
8
9.5099(−04)
7.8603(−03)
−4.9093(−02)
−3.1264(−2)
3.1581
9
2.2385(−04)
3.0818(−02)
−5.0069(−02)
−1.0552(−1)
3.0507
10
−4.0067(−05)
−4.2322(−02)
−7.6546(−02)
−2.3122(−1)
2.9468
11
−7.8530(−06)
−2.9079(−02)
−6.5303(−02)
−2.2998(−1)
2.8587
12
9.4096(−07)
2.1328(−02)
−5.0866(−02)
−3.7047(−1)
2.7589
13
1.6304(−07)
1.0994(−02)
−7.8184(−03)
−3.1891(−1)
2.5997
14
−1.2774(−08)
−4.3674(−03)
−8.6115(−03)
−3.7280(−1)
2.5051
15
−2.0120(−09)
−2.0076(−03)
−1.1672(−02)
−2.9478(−1)
2.2414
16
9.9630(−11)
4.2701(−04)
5.1686(−02)
−1.9189(−1)
2.0696
17
1.4065(−11)
1.8425(−04)
3.1464(−02)
−8.8012(−2)
1.7402
18
−4.3836(−13)
−1.9455(−05)
−1.6651(−02)
1.4685(−2)
1.3664
19
−4.7748(−14)
−7.6247(−06)
−1.0440(−02)
3.7958(−2)
1.0059
20
1.2755(−15)
4.0059(−07)
9.6949(−04)
1.2373(−1)
6.1834(−1)
21
4.4328(−17)
7.3321(−08)
7.0878(−04)
8.6092(−2)
3.9090(−1)
22
−4.0331(−18)
−1.2211(−08)
3.4031(−05)
1.6949(−2)
9.1213(−2)
23
−2.3747(−20)
8.4767(−10)
4.1930(−05)
7.5101(−3)
3.6892(−2)
24
5.3215(−21)
3.5734(−10)
1.2702(−05)
2.2149(−3)
1.0902(−2)
25
4.2733(−22)
6.3490(−11)
2.7864(−06)
5.3111(−4)
2.6850(−3)
26
2.1317(−23)
8.2559(−12)
4.9833(−07)
1.0870(−4)
5.7139(−4)
27
8.3048(−25)
8.8035(−13)
7.6418(−08)
1.9535(−5)
1.0753(−4)
28
2.7288(−26)
8.0988(−14)
1.0354(−08)
3.1462(−6)
1.8215(−5)
29
8.0002(−28)
6.6859(−15)
1.2692(−09)
4.6126(−7)
2.8151(−6)
30
2.1641(−29)
5.0887(−16)
1.4336(−10)
6.2327(−8)
4.0105(−7)
Table IV
Values of N(μ′ = cos θ′, μ = cos θ) for Three Different Values of θ; x = 100.0, m = 1.342