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(a) Ion ACF ( λ B =3m)
α =0 . 0
α =0 . 5
α =1 . 0
α =90 . 0
Brownian
(b) Ion ACF ( λ B =0 . 3m)
α =0 . 0
α =0 . 5
α =1 . 0
α =90 . 0
Brownian
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
2
2.5
3
0
0.05
0.1
0.15
0.2
0.25
0.3
Time [ms]
Time [ms]
Fig. 6. Simulated single-ion ACF's at different magnetic aspect angles α for two radar Bragg
wavelengths: (a)
0.3 m. The simulation results (color lines) are
compared to theoretical ACF's computed using expression (51) of the Brownian-motion
approximation (dashed lines). Note that there is effectively no dependence on aspect angle
λ B =
3 m and (b)
λ B =
α
(from Milla & Kudeki, 2011).
where, assuming a Brownian-motion process with distinct friction coefficients
ν
and
ν
in
the directions parallel and perpendicular to B , the mean square displacements will vary as
e −ν τ
2 C 2
ν
r 2
Δ
=
ν τ
1
+
(46)
2
and
2 C 2
2 cos
2 γ )
Δ r 2
e −ν τ cos
=
(
2 γ )+ ν τ
( Ω τ
(47)
2
ν
+ Ω
KT / m and
tan 1
in which
qB / m are, respectively, the thermal
speed and gyrofrequency of the particles. Furthermore, simulated
γ
( ν
/
Ω )
, and C
Ω
r 2
Δ
match (46) and (47)
,
with
ν ν ν i / i ,
(48)
where
N e e 4
ln
Λ i
ν i / i =
(49)
o m i C i
12 π
3/2
is the Spitzer ion-ion collision frequency given by Callen (2006) and Milla & Kudeki (2011).
The simulations also indicate, in the case of oxygen ions,
r 2
r 2
C i τ
2
Δ
Δ
(50)
ν τ
ν τ < Ω i τ
for short time delays
1 and
1, in consistency with (46) and (47). Hence
(45) simplifies to
1
2 k 2 C i τ
e j k · Δ r i
e
2
. (51)
Evidently, the single-oxygen-ion ACF's are essentially the same as in collisionless and
non-magnetized plasmas because (a) the ions move by many Bragg wavelengths
λ B =
2
π
/ k
 
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