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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. Plasma Physics Laboratory, UAIC IASI, Romania Page 2 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 = 2v|| / c = 2mev|| / 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|| Plasma Physics Laboratory, UAIC IASI, Romania Page 3 (monoenergetic electron beam) and the same gyroperiod Tc = 2me / 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 Plasma Physics Laboratory, UAIC IASI, Romania Page 4 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. Plasma Physics Laboratory, UAIC IASI, Romania Page 5 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 = 2mev|| / 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 2mv|| / e) we can obtain v||. The cathode fall voltage Ucf can be estimated as: Ucf = mv||2/2e. Plasma Physics Laboratory, UAIC IASI, Romania Page 6 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. Plasma Physics Laboratory, UAIC IASI, Romania Page 7 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 = 2mv|| / e . Consequently, the axial velocity v|| of the electrons accelerated in the cathode fall is: v|| = etg 2m 2.8107 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). Plasma Physics Laboratory, UAIC IASI, Romania Page 8 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. Plasma Physics Laboratory, UAIC IASI, Romania Page 9 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. Plasma Physics Laboratory, UAIC IASI, Romania Page 10 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. Plasma Physics Laboratory, UAIC IASI, Romania Page 11