Download Magnetic confinement of an electron beam

Alexandru Ioan Cuza
University
Faculty of Physics
Magnetic confinement of an electron beam
Practicum manual and documentation
C. Costin and L. Sirghi
Iaşi, ROMANIA
Magnetic confinement of an electron beam
The present manual and documentation gives technique details on construction and
usage of an experimental device demonstrating the effect of magnetic field on motion of
plasma electrons.
Level of the practicum High School students (demonstration) / Bachelor students
Goal of the practicum
-
Demonstration of the magnetic field effect on the movement of plasma electrons and
the magnetic confinement of plasma.
-
The students have the possibility to directly see the shape taken by the negative glow
of an electrical discharge when placed in a magnetic field and to explain this shape by
the helical trajectories of electrons moving in a linear magnetic field.
-
The students can study the relation between the magnetic field strength and the
parameters of the helical trajectory: radius and pitch.
-
The students can determine the velocity gained by the fast electrons accelerated in the
cathode fall of the luminescent glow discharge.
Theoretical aspects
An electron moving in a homogeneous and stationary linear magnetic field exhibits a
helical trajectory as shown in fig.1.
Fig.1. Electron’s helical trajectory in a linear magnetic field. For simplicity, when the electron is
in the origin of the coordinates system, its velocity is in the plane yOz, resulting thus the perpendicular
velocity (v) along Oz axis.
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Let’s consider that the only force that acts on the electron is the Lorentz force generated
by the presence of the magnetic field:
 
FL  e v  B ,
where e is the electron electric charge, v is the electron velocity and B is the magnetic
field strength.
Decomposing the electron velocity in two components, one parallel (v||) and one
perpendicular (v) to the magnetic field lines, the module of the Lorentz force can be written
as FL  ev B . This force is always perpendicular to the magnetic field lines and to the
velocity vector v and that makes it to react as a centripetal force that force the electron to
have a circular trajectory in the plane that is perpendicular to the magnetic field lines. Along
the magnetic field lines the resulting force on the electron is zero so the electron movement is
a translation with constant velocity v||. The combination of the circular movement
perpendicularly on the magnetic field lines and the translation movement along the magnetic
field lines results in a helical trajectory depicted in fig.1. This trajectory is characterized by
two geometrical parameters:
i) the gyroradius, also called Larmor or cyclotron radius, which is the radius of the
circular movement (ac) and it can be obtained equalizing the Lorentz force FL  ev B with
the centripetal force Fcp  me v 2 / a c , obtaining the expression:
ac = mev / eB;
ii) the pitch, which is the electron’s displacement along the magnetic field lines during
one period of the circular movement, marked with L in fig.1. This distance is:
L = v||Tc = 2v|| / c = 2mev|| / eB,
where me is the electron mass, Tc is the period of the circular movement (gyroperiod) and c
is the gyrofrequency given in radian/second.
If we can create a slightly divergent monoenergetic electron beam (cylindrically
symmetric), centred on the axis of a discharge tube, with v|| >> v (parallel and perpendicular
directions with respect to the tube’s axis), when applying a magnetic field B along the tube’s
axis, the glow region created by the electron beam will exhibit spindle shape, with regular
radial constrictions (fig.2). Those electrons existing in the same time in the same point on the
tube’s axis will move each on his own helical trajectory but after a gyroperiod they will all be
focused again on the tube’s axis because they all have the same axial velocity v||
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(monoenergetic electron beam) and the same gyroperiod Tc = 2me / eB (fig.2). Thus, the
spindle length is equal to the pitch’s length L of each individual helical trajectory. Due to the
cylindrical symmetry of the electron beam, the diameter of the spindle is four times the mean
electron Larmor radius D = 4ac = 4mev / eB, where v is the mean velocity of the electrons,
perpendicularly on the magnetic field lines (fig.3).
Fig.2. A slightly divergent monoenergetic electron beam produces spindle shaped glow region in the
presence of a magnetic field.
z
D = 4ac
x
Fig.3. The diameter of the spindle is four times the mean electron Larmor radius due to the cylindrical
symmetry of the electron beam.
Experimental set-up
The experimental set-up is schematically shown in fig.4 and a photo is given in fig.5. The
discharge tube is made of Pyrex glass, 61 cm length and 5 cm inner diameter. The cathode is
an empty cylinder made of aluminum (cavity cathode), isolated on the exterior with a glass
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tube so that only the internal surface of the cathode is active for the discharge. The discharge
tube is placed on the axis of a pair of Helmholtz coils so that a homogeneous axial magnetic
field can be created along the tube’s axis. The anode role is played by the grounded vacuum
system in order to keep only the negative glow in the discharge tube (the positive column
follows the anode and it is separated in the lateral tube, fig.6).
Fig.4. Scheme of the experimental set-up.
Fig.5. Photo of the experimental set-up.
Fig.6. Position of the negative glow
and the positive column in the
discharge tube.
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For a gas pressure below 10-1 torr, the cathode fall is larger than the inner diameter of the
cathode (in air) and the discharge regime is not that of a cavity cathode. The discharge burns
at high voltage (2-3 kV) and low current ~2-5 mA. Due to the special shape of the cathode,
the secondary electrons extracted from the cathode surface by ion bombardment are
collimated on the tube’s axis. They are accelerated in the cathode fall (which is the order of
kV), resulting thus a focused quasi monoenergetic electron beam that creates the negative
glow of the discharge. The energy of the electron beam corresponds to the cathode fall
voltage Ucf. When applying the magnetic field, the negative glow is spindle shaped with the
length of the spindle
L = 2mev|| / eB
and the diameter
D = 4mev / eB.
Practicum
The vacuum system is turned on so that the gas pressure in the discharge tube to be below
-1
10 torr. The power supply is turned on and the voltage is increased till the discharge burns at
high voltage (2-3 kV) and low current ~2-5 mA. If the discharge burns at lower voltages it is
necessary to decrease the gas pressure until the required regime is obtained. Once the
discharge is stable, the Helmholtz coils are supplied. We slightly increase the current through
the coils (applying thus the magnetic field on the discharge) till the first spindle is formed,
when we can see the first focusing point in the negative glow. We measure the current value
through the coils I and the spindle length L (the distance between the beginning of the
negative glow and the first focusing point). We continue to measure the spindle length for
different increased values of the current through the coils. When two or more spindles are
formed, we measure their total length and divide it to the number of spindles in order to
obtain L and to diminish its measuring error. Knowing the constant of the Helmholtz coils
(kH) we calculate the magnetic field corresponding to all the measured values of the current
through the coils (B = kHI). If we do not know this constant we can directly measure the
magnetic field by using a teslameter. We plot the graph L = f(1/B) in order to obtain a linear
dependence. We fit the curve with a linear function with the intercept equal to zero and from
the slope of the linear function (which is 2mv|| / e) we can obtain v||. The cathode fall voltage
Ucf can be estimated as:
Ucf = mv||2/2e.
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Finally we can estimate the fraction of the discharge voltage that is distributed on the cathode
fall:
f = Ucf / Udisch.
The same procedure used to determine v|| from L = f(1/B) is valid to determine v from D
= f(1/B). In practice, it is not easy to measure D due to its weak dependence on B (small value
of v).
Experimental results
An example of typical measurements is presented in the followings. The measured
values of I and L as well as the calculated values of B and 1/B are given in the Table 1. The
discharge voltage and current intensity were Udisch = 3 kV and Idisch = 5 mA, respectively. The
constant of the Helmholtz coil is kH = 25 G/A.
Nr.
I
B
1/B
L
Det.
(A)
(G)
(1/T)
(cm)
1
1.27
31.75
314.96
30.5
2
1.61
40.25
248.45
25.0
3
1.73
43.25
231.21
23.0
4
1.89
47.25
211.64
20.5
5
2.21
55.25
181.00
19.0
6
2.48
62.00
161.29
16.0
7
2.87
71.75
139.37
14.5
8
3.13
78.25
127.80
13.0
9
3.43
85.75
116.62
11.7
10
3.70
92.50
108.11
11.0
11
4.25
106.25
94.12
10.0
12
4.54
113.50
88.11
9.0
13
5.18
129.5
77.22
7.8
14
6.00
150.00
66.67
7.0
Table 1. Example of experimental data.
The dependence of the pitch length L on 1/B is plotted in Fig. 7. This dependence is fitted
with a linear function with the intercept equal to zero.
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30
25
L (cm)
20
15
10

5
0
0
50
100
150
200
250
300
350
1/B (1/T)
Fig. 7. The dependence of the pitch length L on 1/B.
The calculated slope of the linear function is tg = 0.1, which in the same time is
tg = 2mv|| / e .
Consequently, the axial velocity v|| of the electrons accelerated in the cathode fall is:
v|| = etg 2m  2.8107 m/s.
Once the axial velocity known, it is possible to estimate the voltage drop that accelerated the
electrons in the cathode fall:
Ucf = mv||2/2e  2.23 kV.
The fraction of the discharge voltage that is distributed on the cathode fall is:
f = Ucf / Udisch  0.74.
In Fig. 8 are shown the photos of the negative glow of the discharge, for four different values
of the magnetic field strength, B4 > B3 > B2 > B1 = 0 G. The arrows indicate the points where
the negative glow is focused on the tube’s axis (each arrow shows one end of a spindle).
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B1 = 0 G
B2 = 50 G
B3 = 80 G
B4 = 120 G
Fig. 8. The negative glow of the discharge for different magnetic field strengths. The arrows indicate the electron
focusing points.
Construction of discharge tube and magnetic coils
A photo of the discharge tube is presented in Fig. 4. A technical drawing of the
discharge tube is presented in Fig. 9. The main body of the discharge tube is made from a
Pyrex glass tube with the length of 650 mm and inner diameter of 50 mm. The outer diameter
is 53 mm. At a distance of 150 mm from one end the tube is connected to a tube ended with a
standard male conical joint 24/40 for connection to the vacuum system.
Fig. 9. Technical drawing of the discharge tube. Dimensions are shown in mm.
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At the tube ends there are fixed by Torseal adhesive paste two non magnetic stainless steel
rings with circular channels for rubber O-rings in order to seal the tube by two Plexiglas
flanges. Each flange is fixed at the end of the tube by four screws. The left flange has a
vacuum feedthrough on which is fixed the aluminum hallow cathode. The outer surface of the
cathode is insulated by a small glass tube (inner diameter of 11mm and length of 100 mm.
The main component of the discharge tube is the cathode. Its construction assures a good
emission of a slightly divergent electron beam. Figure 10 shows a photo of the cathode
mounted on a Plexiglas flange. A technical drawing with a cross section of the cathode is
shown in Fig. 11. The cathode is made from one peace of aluminum with the dimensions (in
mm) shown in Fig. 11.
Fig. 10 Photo of the hallow cathode mounted on the Plexiglas flange.
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Fig. 11 Technical drawing representing a cross section of the hallow cathode made from one aluminum piece.
The dimensions are shown in mm.
The magnetic coils are made of copper wire 3.6 mm in diameter wounded on supports
with inner diameter of 210 mm, outer diameter of 400 mm and length of 83 mm. Each coils
has N = 414 turns. The coils are placed at a distance of 60 mm (see Fig. 12) on the same axis
and connected in series at a power supply that allow dc currents as intense as 6 A at a
maximum voltage of 32 V.
Fig. 12 Technical drawing representing a cross section of the coils used to generate an axial magnetic field along
the discharge tube axis. The dimensions are shown in mm.
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