3.4 Scattering by a Group of Spherical Objects
47
×
2 p + 1
2 p − 1
α(m, n, −μ, ν, p − 1),
where
α(m, n, μ, ν, p) =
2 p + 1
2
( p − m − μ)!
( p + m + μ)!
1
−1
P
m
n (x)P
μ
ν (x)P
m+μ
p
(x)dx.
In these expressions r l, j , θ l, j , φ l, j are the spherical coordinates at the center of the
lth particle in the jth coordinate system. From the summation theorems it follows
that
M
3
mn (l) =
∞
ν=0
ν
μ=−ν
[AO
mn
μν (l, j)M
1
μν ( j) + B O
mn
μν (l, j)N
1
μν ( j)],
(3.62)
N
3
mn (l) =
∞
ν=0
ν
μ=−ν
[B O
mn
μν (l, j)M
1
μν ( j) + AO
mn
μν (l, j)N
1
μν ( j)].
(3.63)
Let us derive expressions for the scattered field from (3.58) taking into account
(3.62) and (3.63); we obtain
E s (l, j) = −
∞
n=1
n
m=−n
i E mn [ p
l, j
mn N
1
mn + q
l, j
mn M
1
mn ],
(3.64)
H s (l, j) = −
k
μω
∞
n=1
n
m=−n
E mn [q
l, j
mn N
1
mn + p
l, j
mn M
1
mn ],
(3.65)
where
p
l, j
mn = −
∞
ν=1
ν
μ=−ν
[a
l
μν A
μν
mn (l, j) + b
l
μν B
μν
mn (l, j)],
q
l, j
mn = −
∞
ν=1
ν
μ=−ν
[a
l
μν B
μν
mn (l, j) + b
l
μν A
μν
mn (l, j)],
A
μν
mn =
E μν
E mn
AO
μν
mn = i
ν−n (2μ + 1)(n + m)!(ν − μ)!
(2n + 1)(n − m)!(ν + μ)!
AO
μν
mn ,
B
μν
mn =
E μν
E mn
B O
μν
mn = i
ν−n (2μ + 1)(n + m)!(ν − μ)!
(2n + 1)(n − m)!(ν + μ)!
B O
μν
mn .
Substituting (3.8), (3.59), (3.64) into (3.58), we obtain