Environmental Engineering Reference
In-Depth Information
4.5.3
Cyclotron Resonance
One more interesting effect takes place when electrons are in a gas in a strong
magnetic field
ω
τ
1 and in an alternating electric field of high frequency
H
1. Then the behavior of electrons may have a resonant character. Indeed,
according to (4.133) for the conductivity tensor, we have for its components in the
directions parallel
ωτ
Σ
k
and perpendicular
Σ
?
to the magnetic field
1
C
i
ωτ
Σ
k
D
Σ
C
ω
2
τ
,
0
2
H
1
ω
2
C
2
i
ωτ
ω
τ
H
Σ
?
D
Σ
C
ω
2
τ
.
(4.143)
0
2
H
ω
2
C
2
i
1
ωτ
The conductivity is seen to have a resonance (so-called cyclotron resonance) at
ω
D
ω
H
of the width
Δ
ω
1/
τ
, and the components of the conductivity ten-
sor are
0
/2 at the resonance.
The cyclotron resonance has a simple physical explanation. In a magnetic field
an electron travels in a circular orbit with cyclotron frequency
Σ
k
D
i
Σ
?
D
Σ
H
, and if a harmon-
ic electric field is applied in the plane of the circular orbit, the electron continuously
receives energy from the field if the frequency of the electric fields coincides with
the electron cyclotron frequency. Similar to motion in a constant electric field, the
electron is accelerated until it collides with atoms. Hence, in both cases the conduc-
tivities are of the same order of magnitude and are expressed in terms of the rate
1/
ω
τ
ω
of collisions between the electron and atoms. If the field frequency
ω
dif-
fers from the cyclotron frequency
H
, the conductivity is considerably lower, since
the conditions of interaction between the electron and the field are not optimal.
We can analyze the cyclotron resonance from the standpoint of energy absorp-
tion. The power per unit volume absorbed by a plasma is
p
ω
D
j
E
.Ifwetakethe
x
-
axis to be in the direction of the electric field, then the specific power absorbed by
the plasma is
D
Σ
C
Σ
xx
E
2
2
p
xx
"
#
,
p
0
2
1
i
ωτ
1
C
i
ωτ
C
ω
τ
C
ω
τ
D
ωτ
C
2
H
2
H
1
ω
2
2
2
i
1
ω
2
2
C
2
i
ωτ
(4.144)
0
E
2
is the specific absorbed power for the constant electric field.
Absorption of energy by a plasma is accompanied subsequently by electron-atom
collisions that lead to energy transfer to the atoms.
In the region of the cyclotron resonance, with
where
p
0
D
Σ
ωτ
1,
ω
τ
1, and (
ω
H
ω
H
)
τ
1, (4.144) yields
"
#
.
p
0
2
1
C
ω
τ
p
D
(4.145)
2
H
1
ω
2
2
