Study of the plasma decay with
microwaves
Contents
1 Introduction 2
2 Diffusion 2
3 Volume recombination 3
4 Determining the dominant process 3
5 Resonator method for the electron concentration determination 4
6 Experimental set-up 5
6.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1
1 Introduction
Plasma is a state of matter, which, apart from neutrals, also contains charged particles:
ions and electrons. Naturally the concentration of these particles is one of the fundamental
plasma characteristics. In order to be sustained, plasma needs the energy to be continuously
supplied - when the energy flow is interrupted, plasma starts to decay. The decay
process manifests as a gradual drop in the concentration of charged particles. The drop
is caused by two principles:
• diffusion and following recombination on the walls
• volume recombination
2 Diffusion
Starting with the continuity equation for electron density
∂n
∂t
+ divΦ = 0 (1)
while the term divΦ can be rewritten as
divΦ = −div grad(Dn) = − 2
(Dn) (2)
with D being the diffusion coefficient. Therefore
∂n
∂t
− 2
(Dn) = 0 (3)
Assuming that D is constant in the whole volume (D = f(x, y, z)), the solution can be
expressed in terms of eigen-functions nj of the D 2n operator. The concentration then
equals
n(r, t) =
j
aj(t)nj(r) (4)
Eigen-functions must satisfy the relation
2
nj +
1
Dτj
= 0 (5)
Substituting into the original equation, the solution can be expressed as the superposition
of so called diffusion modes, where each diffusion mode has certain relaxation time τj
n(r, t) =
j
nj(r) exp −
t
τj
(6)
If only the first diffusion mode is considered (i.e. the slowest mode of decay) we get
n(r, t) = n0(r) exp −
D
Λ2
t (7)
where Λ2 = τ0D is the diffusion length. Furthermore, if the shape of the discharge tube is
cylindrical, and the length-to-radius ratio is many times larger than one, it can be treated
as a one-dimensional problem.
n(x, t) = n0(x) exp −
D
Λ2
t (8)
2
In this case, the radial profile of concentration is
n0(x) = const · J0
x
Λ
(9)
and the diffusion length is directly proportional to the radius of the discharge tube r0
Λ ≈
r0
2.405
(10)
where the numerical factor 2.405 is the first order of a Bessel function of the first kind J0.
3 Volume recombination
The number of particles, that recombine per unit of volume and time is proportional
to the product of electron and positive ion concentrations
nr = α · ni · n (11)
The linear factor α is known as the recombination coefficient. The temporal change of
concentration in neutral plasma (ni = n) is then
dn
dt
= −αn2
(12)
4 Determining the dominant process
Generally speaking, the recombination losses take over at higher pressures, while the
diffusion losses are significant at lower pressures. Which process is actually dominant, can
be determined from the electron concentration profile in time n = f(t).
• Diffusion- losses are characterised by the time development.
n(t) = n0 exp −
D
Λ2
t (13)
in other words, the function ln n = f(t) is linear; the diffusion constant D can be
determined from its linear coefficient.
• In the case of recombination, the temporal evolution of charged particles can be
expressed as
1
n(t)
=
1
n0
+ αt (14)
meaning, that the inverse of charged particle concentration 1
n = f(t) is linear here;
where the recombination coefficient α can be obtained from its linear coefficient.
Using this approach, allows us to either determine the recombination coefficient or the
diffusion coefficient, but never both (the other effect is neglected altogether, which can
lead to inaccurate results). Given that both processes exist simultaneously and both
are taken into account, we get the new differential equation for the time evolution of
concentration:
dn(t)
dt
= −
D
Λ2
n(t) − αn2
(t) (15)
3
Solving this differential equation yields:
n(t) =
1
c · exp (tD/Λ2) − αΛ2/D
(16)
The measured data can be ultimately fitted with this improved equation, getting the
parameters D, α and the constant c. Adding the initial condition n(0) = n0, we may
calculate the n0 as well.
5 Resonator method for the electron concentration deter-
mination
This method was originally introduced by Slater in 1946 and is based on the measurement
of the complex high-frequency conductivity of plasma. Filling of the resonator
with plasma induces changes both in the resonator eigenfrequency ω and the quality of
the resonator Q. The calculation of these changes (as functions of the degree of resonator
filling) was modelled by the perturbation method, which yields the result:
1
Q
− 2i
ω
ω
=
1
ε0ω
V σE2dV
V E2dV
(17)
where V is the volume of the whole cavity, V is the volume of the plasma-filled part,
σ the conductivity of plasma and E is the amplitude of the electric field. The electrical
conductivity is in general a complex quantity
σ =
ne2
m
ν
ω2 + ν2
− i
ω
ω2 + ν2
(18)
For the electron concentration n determination, the imaginary part of the equation
(17) with the eigenfrequency shift ω term is used. Provided that the collision frequency
is lower than the angular frequency of the resonator, the imaginary part of conductivity is
ν < ω ⇒ ν2
ω2
⇒ σi = −
ne2
m
1
ω
(19)
The shift in resonance frequency (with and without plasma) is then
2 ω
ω
=
e2
ε0mω2
V n(r)E2(r)dV
V E2(r)dV
(20)
In the cylindrical resonator, usually the mode TM010 is excited:
Er = 0
Eϕ = 0
Ez(r) = E0 · J0(2.405 r/R)
where E0 is the amplitude of high-frequency electric field at the axis of the resonator with
radius R and the factor 2.405 is the first order of the Bessel function of the first kind J0.
Since the discharge tube region occupies only the vicinity of the resonator axis, the
E(r, ϕ, z) can be replaced with constant E = 0.8·E0 where 0.8 is an appropriate numerical
factor. Taking the E0 from the numerator out of the integral, we get
2 ω
ω
=
e2
ε0mω2
(0.8)2E2
0 V n(r, z, ϕ)dV
E2
0 V J2
0 (2.405 r/R)dV
(21)
4
The integral in the numerator can be expressed as
V
n(r, z, ϕ)dV = ¯nV (22)
where ¯n is the mean value of the electron concentration inside the discharge tube volume
V . Following with the denominator integration
V
J2
0 2.405
r
R
dV =
2π
0
h
2
−h
2
R
0
J2
0 2.405
r
R
rdr dz dϕ = 2πh · I(R) (23)
where h is the discharge tube length and
I(R) =
R
0
J0(2.405
r
R
)
2
r dr =
R2
2
{[J0(2.405)]2
0
+ [J1(2.405)]2
0.271
} =
R2
2
· 0.271 (24)
This yields following relation of the eigenfrequency shift and the eigenfrequency of the
unperturbed case.
2 ω(t)
ω
=
0.64
0.271
e2
ε0mω2
V ¯n(t)
V
(25)
From this equation, the mean electron density (in the discharge tube with radius R )
as a function of time t can be expressed
¯n(t) =
0.271
0.64
V
V
2 ω(t)
ω
ε0mω2
e2
=
0.271
0.64
R2
R 2 f(t)
8π2ε0mf
e2
(26)
6 Experimental set-up
K
VN M A
GM
X
X1
2
TRIG
V
R
Figure 1: The diagram of the experimental set-up in this laboratory.
The high-frequency energy from the source K with tunable frequency passes through
the attenuator and precision wave meter V to the resonator (R =40 mm). The discharge
5
tube (R =9 mm) passes through the resonator, aligned with the axis. The transmitted
signal is rectified with diode and guided to the input of an oscilloscope.
The high voltage from the source VN is keyed via the electronic pentode trigger M and
applied to the discharge tube through the cathode resistance R. The pentode is controlled
by the rectangular pulse generator GM, which is also used as the trigger for the oscilloscope.
The discharge current is measured with an ampermeter and the voltage across the
discharge tube can be observed at the second channel of the oscilloscope.
A combination of the rotary vane and the diffusion pump is used to evacuate the
discharge tube, with the pressure measured by the Pirani gauge. The pressure of the
working gas in the discharge tube (helium) can be adjusted with a dispenser which consists
of two valves with a reservoir in between.
In reality the whole experimental set-up is needs a lot of space which might cause
confusion during measurement. For better orientation and reference please see the guiding
photo Fig.2.
6.1 Measurement
Let the empty resonator have eigenfrequency f0. This value increases to f1 after the
ignition of a discharge. As soon as the discharge is turned off, the plasma starts to decay
and the eigenfrequency gradually decreases towards the initial value f0. This sequence is
periodically repeated and thus, with appropriate synchronisation, the extinction of plasma
can be observed on the oscilloscope,.
The resonator eigenfrequency decreases with time elapsed from the moment of turning
the discharge off. At certain moment t the fixed frequency f of the high-frequency source
will be equal to the eigenfrequency, which will manifest as a maximum in the transmission
of the high-frequency energy to the resonator and therefore a peak in the signal detected
by the oscilloscope.
The frequency f is tuned in the range (f0; f1). From the oscillograph, the time t is
determined and from the difference f = f − f0 the electron concentration ne(t) can be
calculated. Afterwards these functions need to be plotted:
• 1/n = f(t)
• ln n = f(t)
in order to fit parameters α, D and determine, which process (recombination/diffusion) is
dominant. Furthermore both coefficients and the value of n0 will be determined also by
fitting the equation (16). Finally results of the former and the latter approach should be
compared. The measurement is to be performed at various pressures (200 Pa, 1000 Pa a
5000 Pa).
6
Figure 2: The actual appearance of the resonator method experiment. 1 - the resonator
with discharge tube. 2 - the VN source, 3 - the pentode trigger, 4 - the shunt resistor, 5 the
rectangular pulse generator, 6 - the pressure dispenser, 7 - Pirani gauge, 8 - the high
frequency source K with the the wave meter, 9 - oscilloscope. The rotary vane and the
diffusion pump are not visible.
7
Figure 3: The discharge current, resonator eigenfrequency and the magnitude of the signal
transmitted through the resonator as a function of time.
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