Measurement of the first Townsend coefficient
Contents
1 Introduction 2
2 Experimental set-up 3
3 Measurement 4
4 Assignments 5
1
1 Introduction
The avalanche theory, originally proposed by J.S.Townsend, explains the
fundamentals of the ionization mechanism, necessary for the existence of
the self-sustained electric discharge. Let us have two parallel metal plates
and a homogeneous electric field E in the gap between them. The free
electrons in the gap are accelerated by this electric field and even at low
pressures there is a high probability that they will collide a collision with
neutral atoms and molecules of the gas during the flight. In the case of
inelastic collisions this neutral species may be ionized or excited.
Given that n is the number of electrons located at x, moving in the x-axis
direction, then there will be dn new electrons produced from their collisions
over the distance dx. It can be written
dn = nα dx (1)
where the linear coefficient α is known as the first first Townsend ionization
coefficient. It is the number of ionizing collisions one electron executes
over the unit of distance. Integration of (1) yields ln n = αx + const., which
can be rewritten as n = n0 exp(αx), where n0 is the number of electrons at
x = 0.
The Townsend coefficient α depends on the electric field intensity E and
the gas pressure p. The electric field intensity determines the acceleration
of an electron between collisions, therefore E is the measure of the electron
mean energy gain along its mean free path. The pressure p corresponds to
the particle concentration and effectively the length of the mean free path.
Naturally the ionization by the accelerated electron will happen only
if the energy of this electron is larger than the ionization potential of gas
molecules/atoms. The ratio E/p is therefore decisive, whether the collisions
will be ionizing. Furthermore, with the E/p fixed, the coefficient α is
proportional to the number of collisions per unit length. It can be written
as:
α = pf
E
p
(2)
The solution for number of ionizing collisions was simplified by Townsend,
assuming that each collision is ionizing, provided that the mean free path
of the electron (λe) is larger than the mean free path between two ionizing
collisions (λi): λe > λi. Defining N as the number of collisions over unit
distance, the electron that travels the unit distance λ (λ > λi) will perform
N exp(−λi/λe) ionizations. Since the first Townsend coefficient is defined as
the number of ionizing collisions per unit length, it can be written as
2
α = N exp −
λi
λe
(3)
.
From its definition, λi is the distance, that the electron must travel in
order to reach the ionization energy threshold of the surrounding gas Ui and
the electron energy is equal to the electric field intensity multiplied by the
distance travelled by the electron. This means that λi can be replaced as
λi = Ui/E. Furthermore the mean free path of an electron is equal to the
inverse of N (number of collisions per unit length) λe = 1/N. The equation
(3) then yields.
α = N exp −
N Ui
E
(4)
The collision number is linearly proportional to the pressure. In can be
used to express the number of collisions per unit length N as N = N0 · p,
where the number of collisions per unit length per unit pressure N0 is defined.
Substituting the term in (4) leads to:
α
p
= N0 exp −
N0 Ui
E
p (5)
The experimental results show, that even in the general case, the first
Townsend coefficient α (as a function of E) has the form:
α
p
= A exp −
B p
E
(6)
where A a B are certain constants, that satisfy Ui = B/A. Their
values can be determined experimentally and in effect also the function
α = f(E/p).
2 Experimental set-up
A sketch of the experimental setup is shown in Figure 1. The planar aluminum
cathode is illuminated by the mercury lamp. Photoelectrons are
accelerated by the homogeneous electric field and the resulting current is
collected at the grating of the anode. The position of the cathode can be
adjusted, changing the distance travelled by electrons (along which the ionization
occurs).
The discharge tube is pumped by the rotary oil vane, while the argon is
continuously admitted to the system. Setting the argon flow will therefore
set the pressure in the discharge tube. The pressure is monitored by the
Pirani gauge. A DC voltage is applied across the electrode gap. It must be
set appropriately, not allowing the formation of an self-sustaining discharge
3
Figure 1: Diagram of the experimental set-up.
(maximum intensity of the electric field 80-120 V/cm). The photo of the
actual experimental set-up is presented in Figure 2.
Figure 2: Photo of the experimental set-up.
3 Measurement
The electron beam is emitted from the cathode by the photoemission (UV
radiation) and accelerated by the homogeneous electric field. The electrons
passing through the gas ionize the neutrals. The coefficient α is measured
from the total current in the circuit as the function of the distance between
electrodes at constant electric field intensity E and pressure p. The total
4
current is related to the electrode distance as follows:
i = i0 exp(α x) (7)
Plotting the functions i = i0 f(x) and ln i = ln i0 + α x in graphs enables
the determination ofi0 and α. A typical behaviour of i = f(x) is presented
in Figure 3.
i
i = i 0
e x p ( αx )
x
i 0
Figure 3: Total current i as the function of the electrode distance x.
This measurement is repeated for different values E/p, which enables
the construction of the graph for ln α/p = f(p/E). This function has to be
linear according to (6), meaning both constants A and B can be determined.
Finally, the ionization potential of argon is calculated from these constants.
4 Assignments
1. Perform the measurement of i = f(x) at given gas pressure in the
discharge tube and 5 values of the electric field intensity between the
electrodes from the interval E ∈ [30 V/cm; 200 V/cm].
2. Plot the graphs for:
• i = f(x)
• ln i = g(x)
• ln α/p = f(p/E)
3. Determine the coefficients α, i0, A, B and Ui from the graphs.
4. Discuss the results and compare the measured value of Ui with litera-
ture.
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