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in terms of 3D spatial Fourier transform
N o Δ
V
p = 1
e j k · r p t
n e ( k , t
)
(9)
of the electrons 5
of the microscopic density function n e ( r , t
)=∑ p δ ( r r p (
t
))
in volume Δ V .
V can then be expressed 6
The scattered field spectrum for volume
Δ
as
r e
r 2 |
2
2
2
|
( ω ) |
=
E i |
|
( k ,
ω ) |
Δ
E s
n e
V
(11)
in terms of the electron density frequency spectrum
ˆ
1
Δ
r
c )
r
c + τ )
2
e j ωτ
n e ( k , t
|
( k ,
ω ) |
V
( k , t
n e
d
τ
n e
(12)
which simplifies as
N o ˆ
2
e j ωτ
e j k · Δ r
2
|
n e ( k ,
ω ) |
=
d
τ
≡|
n te ( k ,
ω ) |
(13)
2
for independent electrons. We also have an identical expression
describing the
density spectrum independent ions in the same volume in terms of ion displacements
|
n ti ( k , ω ) |
Δ r .
2
2
While neither
are accurate representations of the density
spectra of electrons and ions in a real ionosphere (because of the neglect of collective effects),
it turns out that an accurate model for
|
n te ( k ,
ω ) |
nor
|
n ti ( k ,
ω ) |
2
|
n e ( k ,
ω ) |
can be expressed as a linear combination
2
2
of
|
n te ( k ,
ω ) |
and
|
n ti ( k ,
ω ) |
given by
2
2
2
2
= |
j
ω o + σ i |
|
n te ( k ,
ω ) |
+ | σ e |
|
n ti ( k ,
ω ) |
2
|
n e ( k ,
ω ) |
,
(14)
|
ω o
+ σ e
+ σ i |
2
|
ω o
+ σ e
+ σ i |
2
j
j
where
σ e , i denote the AC conductivities of electrons and ions in the medium. This result
can be derived (e.g., Kudeki & Milla, 2011) by enforcing charge conservation (i.e., continuity
equation) in a plasma carrying quasi-static macroscopic currents
σ e , i E forced by longitudinal
polarization fields 7 E produced by the mismatch of thermally driven electron and ion density
fluctuations n te ( k , t
)
and n ti ( k , t
)
. Furthermore, Nyquist noise theorem (e.g., Callen & Welton,
1951) stipulates that the required conductivities are related to the thermal density spectra via
relations
2
k 2 e 2
ω
2
|
n te , i ( k ,
ω ) |
=
2 KT e , i Re
{ σ e , i ( k ,
ω ) }
.
(15)
5
Here
δ ( · )
's denote Dirac's deltas utilized to highlight the trajectories r p
(
t
)
of individual electrons.
6
This expression can be generalized as a soft-target radar equation
ˆˆ |
2 /2
E i |
η o
d
2
r e |
2
P r
=
A r 4
π
n e
( k ,
ω ) |
dV
(10)
r 2
4
π
π
for backscatter ISR's having a scattering volume defined by the beam pattern associated with the
effective area function A r
( r )
.
7
Note that it is the response of individual particles to the quasi-static E that produces the mutual
correlations in their motions.
 
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