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All pages in this lab. Note To print Full Lab Write-up go to lower left side and click on Printable Version and print, Note Pre-Labs printed separately

I. Hall Effect in a Plasma

II. Pre-Labs must be printed separately. HAL Pre-Lab .PRINT, FILL THIS OUT, Get it Signed by 111-Staff, turn in your Signed Pre-Lab Sheet with your report, to the 111-Lab Staff.

III. Hall Effect Circuit Diagrams

IV. Hysteresis

Reprints and other information can be found on the Physics 111 Library Site

Before The Lab

View the Hall Effect in a Plasma video. Discuss pre-lab questions with an instructor, and get the Hal Pre Lab Questions and Staff Sign Off Sheets.....PRINT, FILL THIS OUT, Get it Signed, and Turn it in with your report.

Discuss the Physics about this experiment with the faculty or the GSI's in the 111-Lab before starting.

You should keep a laboratory notebook. The notebook should contain a detailed record of everything that was done and how/why it was done, as well as all of the data and analysis, also with plenty of how/why entries. This will aid you when you write your report.

Introduction

In low-density plasmas, such as the positive columns of glow discharges, the Hall effect is large and easily observable. In this experiment the Hall voltage across a helium discharge column is determined as a function of magnetic field, discharge current, and gas pressure. Electron drift velocities and densities are inferred from measurements of electrical parameters. A measurement of the resistance permits evaluation of the collision frequency of the electrons. The electron temperature can be calculated.

Prerequisite Reading Materials

Index for Reading Materials [1]

S. C. Brown, Introduction to Electrical Discharges in Gases, Wiley, New York, (1965).
R. N. Franklin, Plasma Phenomena in Gas Discharges, Clarendon Press, Oxford (1976), page 48.
Golant et al., Fundamentals of Plasma Physics, Wiley, New York (1980), Chapter 7.
M. N. Hirsh, and J. J. Oskam, eds., Gaseous Electronics, Vol 1 "Electric discharges", Academic Press, New York (1978). Chapter 2. A good book on Plasma Discharge structures.
C. Kittel, Introduction to Solid State Physics, Wiley, New York (1971), 4th Ed., pp. 287-289.
C. Kittel and H. Kroemer, Thermal Physics, Freeman, SFO (1980).
W. B. Kunkel, "Hall Effect In A Plasma", America Journal of Physics 49,733(1981)
Lorrain, Paul, & Corson, Dale R. "Electromagnetic Fields and Waves"; Section 7.3, pp. 299-301.
Nedospasov, A.V. "Striations"; Soviet Physics Uspekhi: Vol. 11, No. 2, Sept. 1968, pp. 174-187.
Pekarek, L. "Ionization Waves (Striations) in a Discharge Plasma"; Soviet Physics Uspekhi:Vol. 11, No. 2, Sept. 1968, pp. 188-208.
"Introduction to Plasma Physics and Controlled Fusion" , Chen, Francis f., Chapter 1 & Chapter 5;Vol. 1, 2nd ed.; Plenum Press (1984).

The Hall Effect

To measure the Hall effect in a conductor we apply both electric and magnetic fields. The analysis is simplest if we take care to apply them perpendicular to each other. In the most common Hall effect geometry, we measure the current that flows in the direction of the applied electric field. We confine the carriers in the direction perpendicular to the electric field, establishing a boundary condition that this transverse current is zero. To analyze the magnitudes of the various currents and voltages that are developed in steady-state, we use some simple force balance considerations.

In thinking about the electrons in a plasma, we should be careful about the distinction between the velocity of individual electrons, $\overrightarrow {v}$ , and the drift velocity of an ensemble of electrons $\overrightarrow {\Delta v}\equiv <\overrightarrow {v}>$ . Before we apply any fields, the electrons are moving randomly at a high speed that reflects their temperature. The average velocity of the ensemble is zero. When we apply the longitudinal electric field, the ensemble will move in the direction of the applied force, at a drift velocity that is in general much slower than the mean thermal velocity. To determine the drift velocity, we consider what happens to an electron during its motion between collision events. During this time it experiences the electric force,

$\overrightarrow {F}_{E}=q \overrightarrow {E}$ ,

where q=-e for our problem. If the velocity of the electron just after its last collision is $\overrightarrow {v}$ , then its velocity just before the next collision is $\overrightarrow {v}+\overrightarrow {\Delta v}$ , where

$\overrightarrow {\Delta v}= \frac{q}{m\tau}\overrightarrow {E}$ .

In the equation above, $τ$ is the mean free time. It is also common to relate the drift velocity to the collision frequency, $ν$ , which is just reciprocal of $τ$ .

Next, we average $\overrightarrow {v}+\overrightarrow {\Delta v}$ , over many collisions. As we assume that the initial velocity is random, the average velocity is just given by $\overrightarrow {\Delta v}$ .

The effect of the magnetic field on the electrons is a little bit trickier. In the absence of an electric field, the velocities are random, and the B field has no net effect. In other words, electrons are equally likely to be curving one way as the other. However, in the presence of drift induced by the electric field, there will be a time-averaged magnetic force.

$\overrightarrow {F}_{M}=q \overrightarrow {\Delta v} \times \overrightarrow {B}$ .

Let's choose coordinates such that the magnetic field is along the z-axis and the electric field is along the x-axis. Under such conditions, electrons will drift in the x-direction under the influence of the applied field. The force caused by the magnetic field will therefore be in the y-direction. Note that neither E nor B will yield a force in the z-direction. As stated above, in the most common Hall geometry, we confine the electrons so that the net current in the y-direction is zero. As a result of this boundary condition, an electric field develops that balances the magnetic force on the drifting electrons. The quantitative study of the Hall effect is all about the ratio of this electric field to the applied B field.

The magnetic force on the drifting electrons is obtained by substituting the drift velocity into the Lorentz force equation,

$F_y = -\frac {q^2E_xB_z} {m \nu}$ .

To maintain a condition of zero current flow in the y-direction, an electric field given by,

$E_y = \frac {qE_xB_z} {m_q \nu}$ ,

must be established. Since physicist's love dimensionless quantities, we should consider the ratio,

$\frac{E_y}{E_x} = \frac {qB_z} {m_q \nu_q}$ ,

which is known as the Hall angle. Notice that familiar combination of parameters appears here. Yes, qB/m is just the cyclotron frequency, which is 1.8x10^11 Hz for a magnetic field of 1 Tesla. A nice expression for the Hall angle is then,

Θ H a l l = ω c τ

.

The analysis of the Hall effect that we have done is valid when the cyclotron frequency is smaller than the collision frequency. In this regime, the electron's motion is interrupted by collision before it can complete a cyclotron orbit.

To learn as much as we can about conduction in the plasma we should measure the longitudinal current as well as the Hall angle. The electric current density $\overrightarrow {j}$ is given by,

$\overrightarrow{j} = q n \overrightarrow {\Delta v}$ .

The resistance of the plasma, which we call $η$ , is the ratio of this current to the longitudinal electric field, or,

$\eta = \frac{j_x}{E_x}$ .

In the longitudinal (unconfined) direction, the current is determined by balancing the force due to the applied electric field and the frictional force. This balancing act yields the following expression for the resistance,

$\eta = \frac{m\nu}{q^2n}$ .

If we can measure both the Hall angle and the resistance, this gives us two independent observables. In general, a conductor under study has four independent parameters: the density, charge, mass, and scattering rate of the mobile charged species. In the case of our plasma, the current is carried by electrons, and therefore the mass and charge are known to us. In this case our two independent observables are sufficient to determine the remaining two parameters - the density and collision rate of the electrons.

Hall Effect in Glow Discharge Columns

In the discussion above we assumed that the density of electrons was uniform in space. For homogeneous systems in thermal equilibrium this is a reasonable assumption. However, this is not case for the plasma. The free electrons in the plasma are produced in ionizing collisions with gas atoms or molecules confined at a low pressure in a long narrow discharge tube. At the low current densities under consideration here, recombination of the charged particles takes place almost exclusively at the tube walls. This means that the electrons (and positive ions) produced in the gas must find their ways to the walls before they disappear. The resulting electron density distribution is not uniform, but has a maximum in the center of the tube, and falls nearly to zero at the walls. When a transverse magnetic field is applied the electrons are redistributed to counteract the Lorentz force, and a Hall field is set up. This Hall field is not uniform across the plasma. Therefore, the Hall voltage, which is the integral of the Hall field, is not simply the product of the Hall field and separation of the electrodes. However, it turns out that Hall voltage that you actually measure is, in fact, exactly one-half the expected value, for the case of slab geometry. The proof of this can be found in the book by Golant et al (SEE Prerequisite Reading Materials).

Armed with this knowledge, you can use the measured Hall voltage and longitudinal current density to obtain the electron density and scattering rate. As a quantitative example, we can operate a glow discharge at a current density of 10 A/m² in a mixture of 1% argon in helium at p = 30 torr pressure and find a Hall field E_H = 300 V/m when B = 100 gauss, and an ohmic field E_o = 5000 V/m. This tells us that the electron drift speed is u_e = 30,000 m/sec, and therefore the free electron density is approximately n_e = 2 x 10¹⁵ m^-3. The gas density calculated from the ideal gas law, on the other hand, is N_g = 10²⁴ m^-3, so that the degree of ionization n_e/N_g is only about 2 x 10^–9. Note: Measure the distance between the pole pieces!

While we have learned a lot, notice that we don't obtain any direct information about the thermal velocity of the electrons in the plasma. However, we can make some indirect inferences. We know the electron collision frequency and the density of the atoms that the electrons are colliding with. According to kinetic theory, these are related by the scattering cross section, that is,

$\nu \cong N_g \sigma v$ .

In our experiment using He gas the cross section is approximately equal to 3.8 x 10^–20 m². Using this value for cross section and the expression above, the mean electron speed and hence the mean electron energy, or temperature can be estimated. The collision rate equation gives the average speed and the temperature is related to the mean square speed. For the MB distribution the mean square speed is not equal to the square of the average speed and you should take this factor into account. See Kittle for discussion on this point.

If you do your calculations correctly, you should obtain some rather large electron speeds and temperatures. The electron temperature T_e can exceed 10,000K. You might be wondering if you should keep your hands off the glass tube that encloses the plasma. In fact, the glass is only barely warm to the touch. What is this telling about the nonequilibrium state in the plasma? A hint is that it should be related to the efficiency of energy transfer from the electrons to the He atoms. Can you explain why the thermal contact between the free electrons and the background gas is so weak? It can also be shown that the energy in random motion of the free electrons has an upper bound given by

$Energy = \left ( \frac {1}{2} \right ) m \left \langle v^2 \right \rangle < \left ( \frac {1}{2} \right ) M_g \Delta v^2$ (8)

where $Δ v$ denotes again the drift speed and M_g is the mass of the gas atom or molecule. The upper limit is reached when electron energy loss due to inelastic collisions (excitation and ionization) is negligible compared to that caused by elastic collisions. A substitution of the mass of the helium atom and the electron drift speed of our example into eq. (8) yields an energy upper limit of 18 eV.

Although 1 eV is very much higher than the thermal energy of the gas, it is much lower than most ionization energies. The ions produced in our gas mixture are primarily argon, which has an ionization energy of 15.8 eV. How can ionization take place when the mean electron energy is much lower than the energy required to liberate an electron from the gas atoms? You may have noticed that in the steady state, a local rate balance between ionization and recombination at an electron temperature of 10,000K and at the densities quoted here, the degree of ionization should be very high rather than the very low level observed. The explanation is that the loss of ions in these discharges at low gas pressures is controlled by transport to and recombination at the relatively cool walls. This is a second and equally important process causing pronounced deviations from equilibrium conditions.

The matter of energy balance and of charged particle production, transport, and loss are thoroughly discussed in the texts on ionization phenomena in gases. The steady state condition for the column of a glow discharge is a delicate balance between several non equilibrium processes. It is not surprising that such columns display a variety of oscillations and non-uniformities that sometimes interfere with our Hall effect observations and therefore deserve some attention.

Instabilities, Oscillations, and Striations

It is characteristic of all plasmas that they can support many types of waves. Some of these waves grow spontaneously from thermal fluctuations to large amplitudes. In that case they are called instabilities. Most instabilities in our discharge are driven by the electric current, somewhat like whistle tones are excited by air streaming through pipes. A small change in current or in gas density can change the oscillatory behavior completely. Therefore, pay attention and don’t change the gas pressure or flow in the middle of taking a set of data.

You should have read the articles on Striations

Observations reveal that oscillation frequencies are in the multi-kilohertz range, with amplitudes sometimes in excess of a few volts. It is therefore important that measuring devices be protected by low-pass filters consisting of RC input circuits with about 0.2 sec time constants. Quiescent (oscillation-free) operation is a necessity for all measurements. You may have to invest some time finding a stationary state, by varying the current, gas pressure, and flow rates. Patience is needed. The gas is a mixture of helium with 1% of argon and 0.1% of nitrogen. The argon is most easily ionized and supplies most of the free electrons, while the nitrogen is added to make the entire mixture relatively insensitive to contamination by air and other residual impurities. A small flow rate is required to keep the mixture constant. Too rapid a flow causes turbulence in the gases.

Under many operating conditions, particularly at the lower pressures, a striking stationary structure appears in the visible glow consisting of alternating bright and dim regions or striations. These can be considered as large-amplitude ionization waves. The amplitude modulation of the luminous intensity, which may approach 100%, is much larger than the axial variation of electron density which rarely exceeds 20%. It turns out to be primarily caused by the variation in the tail of the electron energy distribution which is responsible for most of the visible radiation. The effect on our Hall voltage is therefore only minor, and we can ignore the presence of stationary striations for our purposes. The effect on E_o is also minor if the distance between probes is made large enough to permit averaging over several wavelengths of these striations.

Equipment Notes

The vacuum and gas feed system including the line between the inflow needle valve and the high-pressure regulator is good enough to permit pumping down to 5 mm of mercury pressure, and may hold that pressure in a sealed-off condition but does not have to for the experiment to work properly. Glow discharges are sensitive to contamination, especially by organic vapors. Considerable flushing is required, for several minutes, before stable and reproducible discharge conditions can be established. The color of the glow is sensitive to gas composition and can be used as a convenient indicator of stationary. Although the system is tight, it is desirable to keep the gas pressure upstream of the input needle valve slightly above atmospheric pressure. This has the disadvantage of operating with a large pressure drop across the input needle valve, thereby making the pressure in the experiment very sensitive to the setting of this valve. On the other hand, it offers the possibility of making accurate adjustments in the discharge tube pressure by making small changes in the regulated upstream gas pressure setting.

How to set a gas pressure: 1st pump out the system with the course pump out valve open, on the left side of rack, then open the gas inlet valve on the right side of rack one turn, with the main gas cylinder tank closed. When the vacuum comes down close the main course pump out valve and open all the way the needle valve on the pump out side of the system (NOTE: these two valves are in parallel). Now open the cylinder tank main valve and adjust the pressure meter to read above 1 lb. Time to adjust the gas inlet valve on the left side of rack to the lowest pressure you want for the first pressure set point. Now adjust the pump out needle valve for the other pressures you want for the experiment.

Acceptable discharge conditions -- the oscilloscope showing no fluctuations larger than 0.1 V -- can be produced at pressures between 3 torr and 35 torr, with currents between 0.5 and 2 mA. At the low-pressure end of this range stationary striations are very pronounced and can interfere with the measurements. So keep the current low if possible. Some modest stationary striations (relatively small intensity modulation) are visible over most of the range but are mild enough not to distort the reading of E_o and E_H by more than 10%. The conditions can be kept reproducible by maintaining a very small flow rate which replaces the gas content of the tube about every five seconds.

It is essential that the discharge power supply have a floating ground connection so that the Hall probes can be kept close to ground potential by means of the potentiometer bridge arrangement. See the Circuit Diagrams. The reason for this requirement is the following: the effective contact impedance between a small probe and a low density plasma is several megohms. It is also nonlinear, increasing rapidly when drawing primarily ion current from the plasma. The leakage resistance of standard cable connectors and feed-through insulators is between 10⁹ and 10¹⁰ ohms. If either the cathode or anode were at ground potential, the probes and Hall measuring circuits would be at positive or negative potentials of about one kilovolt, and unwanted probe currents would be quite large. The biasing potentiometer, which adjusts the potential of the probes, needs adjustment each time a discharge parameter is changed. When this precaution is taken, good linear relationships are found between the Hall probes and the B-field up to at least 200 gauss in either direction. Observed deviations from the linear relationship given in eq. (4) at higher magnetic fields are presumably caused by magnetically induced distortions of the discharge. When the field is too strong the visible appearance of the glow between the pole faces is noticeably changed, indicating excessive deviations from axial symmetry.

Procedure

Turning on the system

Check all electrical and gas flow connections. All gas valves must be closed, all electrical power off.
Turn on pump.
Open pump-out valve (see Fig. 2). Vacuum gauge should go down towards 2 torr. If not, call a Staff person and ask for help.
Open main valve and coarse valve on He supply tank. ASK IF YOU DON'T UNDERSTAND WHAT EACH KNOB OR HANDLE DOES, OR WHAT THE GAUGES MEASURE. The high-pressure gauge should read between 100 and 2,000 lb/in2 (psi) pressure. If the tank pressure is below 50 lb/in2, get a new bottle. (Call for assistance from the staff)
Close pump-out valve. Low pressure in discharge tube should not rise rapidly. If it does, there is a leak. Get help.
Open outflow valve 1 turn.
Open "Gas In" needle valve slowly 1/4 turn, or until pressure rise is very noticeable. At the same time, adjust regulator valve - set to the blue mark on the left-hand regulator gauge.
Adjust pump-out needle valve to obtain steady conditions between 10 and 30 torr pressure. Once a small flow rate is established it is best not to touch the input valve again, but to do all regulation with the outflow needle valve. It permits much finer adjustment because the pressure drop across it is much smaller.
After a few minutes of steady flow, turn on high voltage to about 2,500 volts. There should be a glow, purplish pink in color, and the meter should read about 1 mA.
Turn on oscilloscope, to monitor fluctuations in plasma potential by means of the grounded probe. Under proper conditions oscillations should be much smaller than 0.1 volt, perhaps 50 mV peak-to-peak. If oscillations are too large, change pressure, flow rate, high voltage setting, etc. until quiescent operation is found. If unsuccessful, turn discharge off for one minute, then start it again. The start-up voltage may have to be higher than the operating voltage, particularly at the higher pressures. Repeat the process a few times if necessary. If oscillations are still too large, get help.
If all is well, turn on probe circuits with the voltmeter range setting at 100V or 1000V. Leave magnet power off. DO NOT USE THE DVM
Short out voltmeter inputs with cables and the junction box provided, when you adjust the zero on readouts. Be sure you understand how the range switch relates to the voltmeter scale. It is a good idea to connect a 6V or 9V battery provided in the junction box, across the inputs to check the meter.
Adjust discharge ground with the potentiometer, so that probe #2 floats near ground potential. NOTE: (The "zero adjust" does not change the probe potential. The probe floats near ground. Adjust so that the plasma potential is the same as the probe.) Use the voltmeter between the probe output #4 (through the 10 ohm series resistance) and ground to monitor 1/10 of this voltage, first at 100V, then at 10V, and finally at the 1V setting. If conditions are steady, this probe floating potential will not drift by more than a few volts (a fraction of one volt on our meter) in several minutes. Every change in conditions requires a potentiometer adjustment, however.

Measurements

V_o vs. I_d: The purpose of these first measurements is to illustrate some of the interesting properties of a plasma discharge. For a range of discharge tube pressures between 15 and 30 torr, measure the discharge voltage as a function of discharge current. The voltage is given by the potential between probes 2 and 3. Note that there is a 10¹⁰ ohm resistor in series with probe 3, to keep current flow around 10^–8 A. Before you take data, play around with the high voltage, discharge current, gas pressure, and gas flow, in order to get a feel for the limits on the parameters. When taking data, a typical procedure is to set the pressure to 15 torr, set the high voltage somewhat higher than it takes to keep the discharge going properly, set the current, and record it and the voltage. Then increase the current by some amount, and again record the current and voltage. Repeat until you have enough points to plot a curve. Now change the pressure to 20 torr, and repeat. Continue until 4 or 6 plots are obtained. Think about what you are doing, and what other data you might need. What do you need to know to go from current to current density, from power supply voltage to discharge tube voltage, etc. E
B vs. I_m: Using a gauss meter model 5180 with the HIGH VOLTAGE SUPPLY OFF, measure the magnetic field strength as a function of magnet current for both directions of magnet current flow (measure both the FORWARD and REVERSE directions of magnet current flow – just flip the COIL POLARITY switch into the correct position).

1st = Select Auto Zero. To select AUTO ZERO operation, press the ZERO pushbutton. Unit automatically returns to normal operation. 2nd = Select Auto Range. To select AUTO RANGE operation, press the SHIFT pushbutton followed by the RANGE pushbutton. Press the SHIFT pushbutton followed by the RANGE pushbutton to exit Auto Range mode. Manual Range. Also; To select MANUAL RANGE operation, press the RANGE pushbutton. Press the UP (5) and DOWN (6) arrow pushbuttons to select ranges. Press the RANGE pushbutton to return to normal operation. For a complete Manual See [Gausmeter] For an interactive Manual see [HAL 5180Manual.exe] See the Circuit Diagrams for more details from the schematic. Plot B as a function of I_m. Does the magnet display significant hysteresis? See the Hysteresis page.[| Hysteresis]

Is the B-I relationship linear? In this experiment, errors owing to hysteresis are small compared to other errors. Do not spend too much time calculating and explaining them or the phenomenon of hysteresis.

E_H vs. B. For a range of pressures between 15 and 30 torr measure the Hall field between probes 1 and 2 as a function of magnetic field. Take data for the full range of magnet current. Remember to keep the probes near ground potential by adjusting the potentiometer. Plot your results for each pressure. Is E_H linearly dependent on B? For the linear parts of your data (usually B less than 300 gauss) calculate quantities describing the electron gas (v_e, n_e, $ν e$ , $\left \langle \sigma v \right \rangle$ , T_e, etc.). Are your results reasonable? How do these quantities change with pressure? Explain your results?

Glow Discharge Structure

Below shows the Voltage -Current curve of self-sustainng glow discharge. Notice the Normal Glow Region which is linear. This is the region where probes 2 and 3 are located. The voltage here should be linear and constant.

Shutting off the system

Shut off probe circuits and oscilloscope. Turn off magnet power and discharge power (high voltage).
Open pump-out valve.
Close main valve on helium tank.
Open inflow valve wide, until the high-pressure regulator gauge goes to zero. Then close the inflow valve.
Close outflow valve. Turn off pump.
Shut off all valves not closed already.

The system is now shut down, but call for help if there are any problems. The main thing to be careful of is putting a pressure greater than atmospheric in the discharge tube. There is an automatic relief valve, but if it fails to operate -- as it has in the past -- the glassware explodes.

Hall Effect tube and magnet electronic diagram

Hall Effect Magnet Power Circuit Schematic

Hall Effect in a Plasma

From Physics 111-Lab Wiki

Contents

Before The Lab

Introduction

Prerequisite Reading Materials

The Hall Effect

Hall Effect in Glow Discharge Columns

Instabilities, Oscillations, and Striations

Equipment Notes

Procedure

Turning on the system

Measurements

Glow Discharge Structure

Shutting off the system

Hall Effect tube and magnet electronic diagram

Hall Effect Magnet Power Circuit Schematic

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