Pc(ks,ki) = VI(F(ks,ki))1 = VA A A in .NET

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Pc(ks,ki) = VI(F(ks,ki))1 = VA A A
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Integration of (7.3.25) over scattered directions gives
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The integration of Pc over directions gives the power contained in the coherent wave. Summation of the integration of P and Pc then gives, from (7.3.26) and (7.3.24)
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+ n~ J drlfl 2 COS(ki . r)47r v
drlfl 2 cos(ki .r)(g(r)
As clear from (7.3.25), the phase matrix of the coherent wave is a Dirac delta function in the forward direction in the limit of large V. Its power as given by (7.3.26) is nonconvergent with large V and also depends on the shape of V. Purely forward scattering does not affect radiative transfer, which describes the redistribution of radiative energy in different directions. This further justifies the exclusion of the coherent wave in the phase matrix.
Second-Order Solution
Next, we show that energy conservation is obeyed if we include second-order scattering amplitude in the forward direction. In the second-order solution, the exciting field is
E j = exp(~ki . rj)
~ exp(iklrj - rzl)
exp(~ki . rz)
Putting (7.3.28) into (7.3.3) gives the N-particle scattering amplitude as
F(k s , ki ) =
exp(iki . rz) exp( -ik s . rz) .-
+Lf exp(-~ks rz)Lf
~ exp(iklrj - rzl)
exp(~ki rj) (7.3.29)
j"" The forward scattering amplitude to second order is
F ( (ki,k i ) = Nf
j,i' j=l
2exp(iklrj - rzl)
Ir J
) exp(-iki' (rz-rj ) (7.3.30)
where F(2) is the sum of the first-order term and the second-order term. Taking the average using P2(rj, rz) as given by (7.3.8c)
(F(2)(k i , ki )) = N f
dr j
expi~kl~ r-I rzl)
(7.3.31 )
3.2 Scattering by Collection of Clusters
Using the property that g(r) = g( -r) gives
(F(2)(k i , ki ))
N2 2 N 1 + -21 V N 1 + 1,11 2
j drj j d'F exp(iklrj _I rll) cos(k 1_ 1
rj -
(rl - 'Fj))g(rl - rj)
exp(ikr) r cos(k i . r)g(r)
To verify optical theorem, we take the imaginary part of (7.3.32) 47r Im{ (F(2) (l';i, k
ki ))}
2 + n of 2
V 47r = -nolm{f}
j' dr 47r sin(kr) cos(k k
(-k-) (7.3.33) cos i ' r r Comparing the right-hand sides of (7.3.33) and (7.3.27) shows that they agree with each other. Thus
2 d+ n o r 47r sin( kr) k
r)[g(r) - 1]
j dO s (p(k s, ki ) + pc(k s, ki ))
47r Im{ (F(2)(k i , ki ))} k V
[1.F(1)(~,kilI2 + I(F(1)(~,ki))12]
Thus, to apply the optical theorem, the N-particle forward scattering amplitude has to be calculated to the second order so that
"'e =
l' 1m
jdn IF (ks,ki)1 HS-'-----------'---'---V
1 [47rlm{(F (2 )(ki ,ki ))} - V jdOsl(F(1)(ks,ki))12]
Scattering by Collection of Clusters
We next consider scattering by collection of clusters. Let each cluster be labeled as a primary scatterer, and let the point scatterers within each cluster be labeled as secondary scatterers. Then the phase function and extinction coefficients depend on gs, which is the pair distribution function among secondary scatterers within a primary scatterer as well as the pair function between clusters gpo
... _-------- -----.".
,/""i; /
............~.....i/ x
.. . .. .. ...
: :
------ - , i~
. e. ::e ~/'
. e
/,/i ( .!
Figure 7.3.2 Clustered point particles are randomly distributed in a cubic box of size L with volume V. The clusters are randomly distributed, and within each cluster of size Ie the particles are randomly distributed.
Consider a volume element V as defined in Section 1. The volume contains N p primary scatterers (clusters), each of which consists of N s secondary point scatterers (Fig. 7.3.2). The N p clusters are centered at r a , a = 1,2, ... ,Np , and within each cluster a the secondary scatterers are centered at raj with respect to the center of the ath cluster, j = 1,2, ... ,Ns ' Thus
N = NsNp
is the total number of particles in volume V. Then N no = - = Nsn p (7.3.37) V is the number of particles per unit volume and n p = Np/V is the number of clusters per unit volume. Note that Nand N p are large numbers in V while N s does not have to be large. Then, from (7.3.11), the first-order collective scattering amplitude is