Page 466 - Numerical Methods for Chemical Engineering

Scattering theory 455

between the segment lengths
1 1
A 1 B 1 =− (k i · R 21 ) A 2 B 2 =− (k s · R 21 ) k =|k i |=|k s | (9.100)
k k
If E s (θ) is the common amplitude of the two scattered waves (we neglect any polarization
effects), the scattered electric fields emitted from each electron vary as a function of position
and time as
i(k s ·r−ωt) i(k s ·r−ωt+ϕ 21 )
E 1 (r, t; q) = E s (θ)e E 2 (r, t; q) = E s (θ)e (9.101)
where ϕ 21 is the phase lag due to the path-length difference A 2 B 2 − A 1 B 1 ,
(A 2 B 2 − A 1 B 1 )
ϕ 21 = (2π) = k(A 2 B 2 − A 1 B 1 ) (9.102)
λ
Substituting for the path lengths,
1

ϕ 21 = k − [(k s · R 21 ) − (k i · R 21 )] =−[(k s − k i ) · R 21 ] =−(q · R 21 ) (9.103)
k
The total electric field from both scattered waves is
E tot (r, t; q) = E 1 (r, t; q) + E 2 (r, t; q) = E s (θ)e i(k s ·r−ωt) [1 + e iϕ 21 ] (9.104)

Thus, the net intensity from the two scattered waves at the detector is
2
2
I tot (r D , t; q) =|E tot (r, t; q)| = 2|E s (θ)| [1 + cos ϕ 21 ] (9.105)
In the decoherent limit q → 0,ϕ 21 → 0, there is no interference between the waves, and
the time-averaged intensity at the detector is
2
[I tot (q)] decoh = I tot (q → 0) = 4|E s (θ)| (9.106)
Thus, the structure factor – the ratio of the observed scattered intensity to that in the
decoherent limit – for this two-electron system is
I tot (q) 1 1
S(q) = = [1 + cos ϕ 21 ] = {1 + cos[−(q · R 21 )]} (9.107)
I tot (q → 0) 2 2
From S(q), we obtain information about the relative electron position R 21 .
We now extend this analysis to the case with N scattered waves emitted from electrons at
the positions {R 1 , R 2 ,..., R N }. The phase angle ϕ j of the scattered wave at q from electron
j is

ϕ j =−(q · R j ) (9.108)
and the phase lag between electrons j and m is

ϕ mj = ϕ m − ϕ j =−(q · R mj ) R mj = R m − R j (9.109)
Thus, the electric field at the detector arising from all N scattered waves is
N
i(k s ·r−ωt)
E tot r D , t; q = E s (θ)e e iϕ j (9.110)
j=1
We assume that the sample is so small, or the interaction is so weak, that the scattered waves