Multipactor in Low Pressure Gas and in ... - of Richard Udiljak

Figure 5.1: The geometry used in the considered model.
Eo is the field amplitude at the outer conductor, and introducing the
notation Λ = qEoRo/m, Eq. (5.1) can be written:
r ′′ = − Λ
sin ωt (5.2)
r
The relation between the field amplitude and the voltage amplitude is
given by Uc = EoRo ln (Ro/Ri). Since the field is inhomogeneous and
stronger near the centre conductor, there will be a net average force that
slowly, compared to the fast harmonic oscillations, pushes the electron
towards the outer conductor. This force is called the ponderomotive or
Miller force [10] and it tends to push the electrons away from regions with
high amplitudes of the RF electric field. By separating r(t) according
to r(t) = x(t) + R(t), where x(t) is the fast oscillating motion and R(t)
the slowly varying motion (the time averaged position), an approximate
solution of Eq. (5.2) can be derived (see paper E) where the position
and velocity of the electron are given by:
r(t) ≈ Λ
ω2 �
sin(ωt)
+
(5.3)
and
�
C1(t − C2) 2 + Λ2
2ω 2 C1
C1(t − C2) 2 + Λ2
2ω 2 C1
r ′ (t) ≈ 1
�
C1(t − C2) +
R(t)
Λ
�
cos (ωt)
ω
, (5.4)
71