LAPTAG Plasma Laboratory Manual

Transcription

1 LAPTAG Plasma Laboratory Manual Bob Baker Richard Buck Walter Gekelman Bill Layton Joe Wise Version II

2 Introduction This laboratory manual and the experiments it contains will hopefully give the reader a much clearer understanding of plasma. A brief review of the important physical concepts is included. It is intended that this introduction into the world of plasma will arouse curiosity and create an enthusiasm for further study. The study of plasma is not only very significant in the world of science, but also has tremendous impacts on our society, both present and future. The following are just a couple examples that illustrate its great importance: Plasma is used to create nuclear fusion -- the energy source that could possibly solve the world s energy crisis forever. Plasma is responsible for the continued advancement of computer technology by the fabrication of microprocessors. Simply put, a plasma is an ionized gas." A plasma is sometimes referred to as the fourth state of matter (with the first three being a solid, a liquid, and a gas). However, not all ionized gases are plasmas. In order to truly be considered a plasma it must be a quasineutral gas of charged and neutral particles which exhibit collective behavior." (Chen, pp. 3) To better understand the above statement, an examination of the terms is required. Collective behavior means that the group of particles in a plasma act as one. Everything that affects one particle affects all the others. Some or all of the particles are ionized into free electrons (with a negative charge) and ions (with a positive charge). Their motion creates electric fields, current, and magnetic fields which affect all of the particles. In addition to the collective behavior of particles, a plasma must also be quasineutral. In a plasma the ion density (n i ) and electron density (n e ) are equal, and are therefore referred to as the plasma density (n). The overall electric charge of the sample is neutral, and the electric potential is uniform throughout. If an electric potential is created somewhere in the plasma, the electrons (due to their low mass and mobility) will surround and shield the potential, thereby preventing it from disrupting the uniformity in the rest of the plasma. The thickness of this shield, called the Debye length, determines if the collection of particles can truly be considered a plasma or if it is simply an ionized gas. 2

3 Chapter I Turning on the Device UCLA HIGH SCHOOL PLASMA DEVICE Turning on the Chamber 1) Check the status of the plasma chamber: 3

4 Front view of plasma chamber 2) All vacuum pumps should be turned on and running. (Mechanical Pump, Turbo Pump) Mechanical pump 4

5 3) The ionization pressure gauge controller should show a pressure in the mid 10-7 torr range. The Convectron Gauge should read less than 50 mtorr Ionization gauge showing pressure in chamber (with no added gas) 5

6 Making a Plasma: We make a plasma out of an inert gas such as Argon, or Helium. Gas tank gauge (right) and line gauge (left) The tank valve and the line valve must be completely open. In the picture below the supply tank pressure valve is on the right. It is 2000 when the tank is new and will read zero when the tank is empty. If the tank is empty notify Walter, and he will get another tank. The line pressure to the chamber is indicated by the gauge on the left. It should read 40 psi. The gas that becomes plasma is bled into the chamber by a precision valve. The final chamber pressure before making plasma is 3-4X10-4 Torr. The valve is shown in the picture below: Slowly turn the valve CCW to an ultimate reading of about 128. Turn it slowly while watching the pressure. The is a time delay between turning and visible change on the ionization gauge controller. 6

7 Turning on the RF source. The source is very simple to use. A picture of it is below: Turn the power supply on. The supply will go through a boot-up proceedure (about 10 seconds). Next turn the RF on. Raise/Lower the power adjust to go to about 40 Watts for ion acoustic wave experiments. You should see a blue glow in the chamber from the plasma. You may now turn on the magnetic field power supply to Amps. The plasma should get brighter. Note the RF supply is connceted to the antenna with vacuum capacitors. They are are in a plastic box and should not be touched since they are at high voltage. Touching things in the RF box could result in a burn. Capacitors (don t touch!) 7

8 Plasma Turn-Off Procedure: 1) Turn off the RF Source (press White On-Off button) 2) Turn off the Magnetic Field current. 3) Close the bleeder valve to 113 4) Turn off the oscilloscope. 5) Disconnect the Langmuir probe from the battery. 6) CLEAN UP AREA Keeping a Log: All faculty and students using the LAPTAG Plasma Device must make an entry into the LAPTAG Journal. The journal is kept on the table with the software manuals. The following items should be clearly entered: 1) Date and Time 2) Name of operator (This is a person trained and qualified to run the machine) 3) Names of students, visitors, etc. w/ school affiliations 4) A report on the initial state of the machine, i.e. pressure, cleanliness, everything was properly turned off and prepared for the next group. This information helps us monitor machine use and assures that the machine is turned off properly and ready for use by the next group. This time should be used to verify that the machine is ready to turn on. Pay particular attention to the pressure, water flow, any unusually noises from the turbo-pump, etc. Note that the turn on procedure was successful and that everything is running as expected. 5) Give a brief description of the experiment you plan to run. Identify a naming scheme for your files that will be generated. Be sure to give the path for where the files are stored. Note any unusual observations and offer any suggestions for further experimentation. 6) Carefully note the machine parameters for your experiment. Remember, we may need to verify data that was collected at a later date. 7) Carefully note the shut-down procedure. This assures that everything is in the proper state for the next group. It also helps us determine if there was any unauthorized use of the machine. 8

9 Wave Experiments: Introduction: When a tuning fork vibrates, the prongs of the fork move back and forth, colliding with the air molecules around them. While the prongs move forward, they cause the air molecules to bounce off with a large velocity. These molecules would be compressed together, causing an area of high pressure. While the prongs move backward, they cause the air molecules to bounce off with a small velocity. The space between molecules increases, causing an area of low pressure. These variations of high and low pressure will spread throughout the gas, as a result of collisions between the molecules. This wave, when it has a frequency within the range of 20 16,000 Hz, can be detected by the human ear and perceived as sound. The speed of a sound wave propagating through a neutral gas is given by the formula 1. v s = γ KT M where M is the mass of the molecule, K is Boltzmann s constant, T is the temperature of the gas, and γ is a thermodynamic constant. The temperature of the gas is related to the average velocity of the molecules. As the temperature of the gas increases, the molecules on average move faster, and thus collide more often. An increase in collision rate results in the wave propagating faster. This fact accounts for the KT term in the velocity formula. The dependence of the speed of the wave on mass is a result of the fact that molecules with larger masses undergo smaller accelerations during collisions, and thus achieve lower velocities, decreasing the rate of collision. This decrease slows down the propagation of the wave. An ion acoustic wave in plasma is similar to a sound wave in a neutral gas. The vibrations of a sound wave propagate due to forces that result from collision of molecules. In plasma, in which collisions do not occur, the electric forces between ions transmit the vibrations of the wave. Ion acoustic waves are low frequency plasma waves due to the involvement of massive ions. The velocity of an ion acoustic wave is given by the formula 2. v ion _ sound = KTe + γkti M ion 9

10 where T e is the electron temperature, T i is the ion temperature, M is the mass of the ion, and γ is a constant of order three. When the ions become bunched together, two processes spread them apart. First, the thermal motion of the ions, which is randomly directed, tends to cause them to disperse. The hotter the ions, the quicker they will spread out and propagate the wave. This process accounts for the KTi term in the velocity formula. Second and more importantly, the electric fields of the ions will spread the ions out. However, the electrons in the plasma tend to shield the electric fields caused by the bunching of ions, and so a reduced field is available to disperse the ions. The degree to which the electrons are effective at shielding depends on their temperature. The electrons try to shield the ions and if the shielding were perfect there would be plasma wave. However, a potential of order KT e /e leaks out between the electron and ion clumps. This process accounts for the KTe term in the velocity formula. The velocity formula s mass dependence is the same as that seen for sound. Often in laboratory plasma, the ion temperature is so low that the KT i term in the velocity formula can be neglected. The formula is then given by 3. v ion _ sound = KT M e ion The dispersion relationship for an ion acoustic wave is given in the figure shown below. ω = ω pion ω = 2πf k = 2π λ In the first region of the graph, where k is small, the wave is a constant velocity wave. The slope of the curve in this region is precisely the velocity of the wave given by formula 3. The characteristics of 10

11 the wave at these wavelengths are derived from a treatment of the plasma as a fluid. In the second region of the graph, where k is large, the wave becomes a constant frequency wave, however it is very highly damped and is difficult to see. This occurs when the frequency of the wave is equal to the ion plasma frequency ( ω pion = 2 ne ε M 0 ion ). At this point, the wave s characteristics are no longer derivable from the treatment of the system as plasma because in this region the wavelength has approached the scale of the Debye length (see appendix). If one raises the frequency such that the wavelength would be smaller then the Debye length the wave ceases to exist. This is because that at scales below λ D, we are no longer dealing with a plasma. Since these waves are collective plasma phenomena, no plasma-no wave! 11

12 Ion Acoustic Wave Experiment: Set up the Langmuir probe. Langmuir probe and grid access Check that probe is not touching the screen in the chamber. Position the probe at least 3 cm away from the copper grid. Inside view of copper grid 12

13 Identify the two leads from the probe. One connects to the center of the probe; the other connects to the sheath of the probe, which is itself grounded on the plasma chamber. Probe bias circuit Put three 22.5 v batteries in series. Connect center lead to the negative pole of the batteries. From the positive pole of the last battery connect a 1 kilo-ohm resistor. Then connect the other end of the resistor to the ground lead of the probe. Battery bias of probe. Put a DC voltmeter across the resistor and take a reading. It should be zero. 13

14 oscilloscope Ch 1 AM502 Differential amp + terminal Function Output Pulse Out Function Generator Probe bias Batteries Probe Grid Using provided clips, connect preamplifier across the resistor in the Langmuir probe bias circuit. Connect preamplifier to preamplifier power supply. Send signal from power supply to the oscilloscope s channel 3. This connection will allow the oscilloscope to measure fluctuations in the Langmuir probe current that occur due to variations in potential at the probe tip. These variations are caused by ion acoustic waves. I. Experiment 1: Determine the Saturation Current of the Langmuir Probe I = V/R (saturation current) V: Get from the DCVM R: The resistor on the Langmuir Probe bias II. Experiment 2: Determine the Ion Density ( ion particles/cubic centimeter ) PV = STP 14

15 P 1 = P 2 n 1 n 2 a) Find n 2. b) Find the number of particles in a cubic centimeter. c) Find the number of ions in a cubic centimeter. d) We know that (for argon): I saturation = (density of ion) (charge of electron) (assumed speed of ions) (area of probe) Or I saturation = (n) (1.6 x ) (3.3 x 10 5 ) (.3π) 2 (For helium, the assumed speed of the ions is 10.4 x 10 5 ) e) Use this to find n ion after substituting I saturation from experiment 1. f) % ionization = n ion / n gas III. Experiment 3: Measure the speed of the ion wave. 1. Connect resistor on Langmuir probe to filter then amplifier (gain of 5K / 100 button pushed in - don t change the red calibration) then to the oscilloscope (input). _ Oscilloscope 2. Turn on oscilloscope - Line On press down- green light 3. Function Generator should be on and left on when you are finished. It should be connected to the wave generator antenna - the screen in the plasma machine- via a BNC connector to oscilloscope input 1. Do a self test from the display. 3) Function type TRIG 15

16 4) Sweep/modulate BURST, trig is nothing, MO/SWP 5) Trigger at the rear of function generator goes to the input 4 of the oscilloscope 6) Change the frequency with MODIFY buttons top right. Set frequency to 100,000 Hz. 4. Oscilloscope 1) Turn on line button (green light) 2) Auto Scale Press under setup top right 3) Expand a) Menu- Time Base Full Dial - Expand Wave Forms b) Delay- from screen use right button use dial to translate wave to middle of screen. c) Press Channel Menu with screen button change channel to 2 d) Press screen button v/div sue dial to change scale to fit screen 5. Measure the wavelength use fλ = c Oscilloscope display of input (into plasma) and output (I A wave) signals. 4) Move Langmuir probe so that the probe signal is 180 out of phase 5) from it s initial setting that will give 1/2 λ. 6) Find the speed of the ion acoustic wave. 16

17 Te 6. Find the Temperature. csound = 98. X10 5 µ T e is in volts, and µ is the ratio of the mass of the gas to that of a proton. What the Ion Acoustic Data Looks Like: Phase Velocity: Track the position of a maxima (or minima) for small time intervals. The 4 traces above were taken from an oscilloscope. The probe was closest to the grid (from where the waves originate) in the uppermost trace. The probe was moved 5 mm away and the next (blue) trace was recorded. The furthest trace (green) is 10 mm away. The bottom trace is the input signal to the grid. Examination of these traces allows us to calculate the phase velocity of the ion acoustic wave. By identifying one point on each curve (here the position of the second maxima) one can calculate the x velocity using v = t. In this example we arrive at v 10 mm = = cm / sec. Compare this to the 5 µ s speed of sound in air! Note there is room for error analysis here. Use uncertainties in positioning the probe, and reading the time off the graph to calculate the experimental uncertainty in the speed. Since the kte T e v = γ γ ' = MI M' I ion acoustic speed depends on the electron temperature T' e = electron Temp in ev Mass of Ion M' I = = 40 for Ar Mass of a proton 17

18 Then using what we found for v, we get an electron temperature of T e = 1.67 e.v = 19,300 degrees K. Note in this formula γ = c p, the ratio of specific heats (ask your teacher what they are!) and for reasons cv we won t spell out here we assume it is 1.0. Another way to plot the wave data is shown in the picture below: ion wave 6 4 dx = 1.4 cm ion wave amplitude red/blue 4.4 usec apart speed = 1.4 cm/4.4 X10-6 = 3.18 X107 m/sec = 31,800,000 m/sec position (cm) Here we have a plot of the amplitude of the received signal as a function of position at two different times 4.4 microseconds apart. To get the curve shown above we first record the wave as a function of time at each position 0-30 cm. We then select a timestep from this data file and plot the amplitude verses position. Next we select another timestep (4.4 µsec later) and do the same thing. We used the same type of data collected to make the figure above. This was done in another data at different plasma conditions so the sound speed is different. Note the blue curve is what the waveform looks like as a function of position in the chamber at a frozen moment in time. It is displaced because it moved to the right and is smaller because it is damped. What do you think causes the damping (or signal loss)? 18

19 If we make a plot of the signal at every position in a line away from the grid (starting near the center of the grid) as a function of time after the wave was launched we get a space time diagram. Such a diagram is shown in the next figure (Note here we call the distance from the grid the z direction) Note that there are two sets of lines. The set that is associated with the wave is tilted with respect to the time axis. The slope of this line is dz/dt and is precisely the ion acoustic speed. (Note it is 30% higher in this experiment). Compare this and the previous figures and make sure you understand how they are related. The green lines are the wave maxima so that this is a wave burst of 5 cycles. The set of vertical lines is not a plasma wave. Note they are vertical which implies the speed associated with them is infinite! This part of the data is called direct pickup. The very sensitive amplifier have detected the signal in the air of the room radiated from the wave generator and recorded them as well. These signals travel at the speed of light and seem vertical on the timescale shown here. This is a good example of what can occur in any experiment, a strange signal that must be tracked down and explained. 19

20 Sound Waves in Air The above experiment to determine the speed of the ion-acoustic wave was designed to make use of the LAPTAG plasma machine and to introduce the methods used in determining an important property of a plasma. However, the same experimental procedure can be used to measure more familiar sound waves and with slight modifications, a fairly inexpensive piece of apparatus can be constructed to simulate the plasma experiment with sound waves. (Details on the construction of the sound wave apparatus will be presented in the appendix.) The sound waves are launched by a small speaker, mounted at the end of a long plastic tube. A small microphone is mounted at the end of a moving brass tube that can be moved to different positions along the axis of the plastic tube. The sound itself consists of three or four cycles of a sound wave gated into pulses that can be repeated at regular time intervals. (This method of gating a few cycles into short repeated pulses is exactly the way the waves were produced in the plasma chamber. This procedure prevents the development of chamber resonance that can completely obscure the determination if the arrival of the desired pulse.) The basic apparatus is illustrated below. With the microphone positioned fairly near the speaker, a sound pulse is launched and the time for the pulse to arrive at the speaker is measured. (This time measurement can be measured with an oscilloscope that has been set to trigger on the sound source or it can be measured using the more elaborate apparatus illustrated above.) The original position of the microphone and then is moved to a new position. Again the time for the sound pulse to arrive at the microphone is noted and the position of the microphone is noted. Several data points can be obtained by repeating these measurements and this data can be used to compute the speed of sound. 20

21 The graph below illustrates sound data of the amplitude of the wave received at the microphone at many different positions along the axis of the tube. The red graph is all the data received at a particular time and the blue graph is the amplitude vs. position of the sound wave exactly 100 microseconds later. The slight differences in the shape of the wave forms are due to damping and tube resonance. However, it is clear that the first wave maximum advanced 3.5 cm along the tube in 100 microseconds so the speed of sound can be computed to be 350 meters/sec. If we make a plot of the signal at every position in a line away from the microphone (starting near the microphone) as a function of time after the sound wave was launched we get a space time diagram. Such a diagram is shown in the next figure. This figure is exactly analogous to the plot shown for the ion acoustic wave earlier. However, there are significant differences that you might also like to consider. Why isn t there a vertical set of lines suggesting an infinite speed that we identified were actually due to electromagnetic noise? More important, what is the meaning of the lines that are sloped the wrong way suggesting moving backward in time or what? Since this plot involves sound waves and since sound waves should be very familiar to you, you attempt to understand this plot, its similarities and differences with the ion acoustic wave plot should help you to understand more about both experiments. 21

22 When the plot above is studied carefully you will note that the wave packet not only changes slope from plus to minus as time passes, but it also decreases in amplitude. This should be fairly easy to explain. However, look very carefully at the plot. Notice the very faint lines of the wave packet that seem to have a different slope than the major lines. (You can see this packet starting at about 400 microseconds.) We now believe we understand the reason for these lines and they were an unexpected finding in what we thought would be a very routine experiment 22

23 Appendix I External Potentials in Plasma and Debye Length The potential of a positively charged sphere in a vacuum is given by the formula: φ = Q 4πε o x Where Q is the total charge of the sphere, ε o is the permittivity of free space, and x is the distance from the center of the sphere. As can be seen in the formula, the potential falls off as 1/x. If the same sphere is placed in plasma, then the potential created by the sphere falls off much faster due to a phenomenon called Debye shielding. Before the sphere is placed in the plasma, the plasma is electrically neutral. There are an equal number of ions and electrons. If the temperature of the electrons and ions in the plasma is zero, then they lack any thermal motion. The electrons will be attracted to the positive charge of the sphere, and will move towards the surface. The ions will remain relatively still due to their large mass. As electrons are lost to the sphere, the plasma as a whole will become positively charged. This will continue until the force of attraction between electrons and sphere is balanced by the force of attraction between electrons and the plasma. At this point the electrons, on average, will not move. The plasma immediately surrounding the sphere has a higher electron density than plasma further away. Therefore, far away electrons and ions do not experience any electrical effects, because electrons in the region of the sphere shield out the electrical potential of the sphere. If the electrons have a finite temperature, then they have a random thermal motion. Each electron has an associated angular momentum with respect to the sphere due to its thermal velocity. Since the electrostatic force between electron and sphere is an example of a central force, angular momentum is conserved. As electrons are drawn closer to the sphere their speed and path change in such a manner that their angular momentum is constant. Due to this constrained path, it is unlikely that electrons will deposit themselves on the surface of the sphere and they instead will form a cloud around the sphere. Their motion is analogous to the orbits of satellites around the sun. Some electrons form closed orbits like planets and are confined within the cloud. At the edge of the cloud, electrons with thermal energy, KT e, larger than the electrostatic potential energy due to the sphere will be able to escape and follow an open 23

24 orbit. Once electrons break free from the cloud, the total charge of the cloud is no longer equal to the charge of the sphere. Thus when the electrons have a finite temperature, the influence of the potential due to the sphere extends farther into the plasma. Potentials on the order of KT e /e tend to leak out into the plasma. Calculating the thickness of the cloud, called the sheath, and the potential due to the sphere outside of the cloud requires solving the differential equation: 1. ε 0 2 φ = en 2 i n x ( ) e where e is the electron charge, n i is the ion particle density, and n e is the electron particle density. This equation (Poisson s equation) is a differential form of Gauss s Law. It relates the potential to the charge density. The electron particle density is given by: eφ 2. ne = n 0 exp( ) kte In the above formula n o is the plasma density very far away from the sphere. The e φ term in the above formula shows that the electron density is high in areas with large positive electric potential and low in areas with large negative electric potential. The KT e term shows that as the thermal energy of the electrons increase, the electrons are able to penetrate farther into areas with large negative potential before being turned around, thus increasing the electron density in these areas. The ion density is assumed to be uniform due to the large mass of the ions and their low temperature: n = n Poisson s equation stated in terms of the electron and ion density formulas is: 3. ε 0 2 Φ x eφ kt = en 2 0(exp e 1 constant, k = 1.38X10-23 Joules/degree Kelvin i o ) e = 1.6X10-19 Coulombs. K is Boltzman s This is a nonlinear equation due to the φ in the exponent. Using a Taylor expansion, the equation is given by: 4. ε φ eφ 1 φ = e + en... x KTe KTe This technique of expanding the exponential into a polynomial is valid only if Abs (εφκτε )<< 1. Thus, this equation is only valid in regions where the thermal energy of the electrons is greater than the potential energy, which corresponds to areas located outside of the sheath. Since each successive term in our 24

25 polynomial contributes less than the previous term, nonlinear terms can be dropped, and to a high degree of accuracy 5. εo 2 d φ = 2 dx n0e KT 2 e φ. The solution to this equation is: where 6. φ = φ exp x 0 λ D φo is the potential on the surface of the sphere, andλ D is the Debye length given by the formula: 7. λ D = 0 ε KT e 2 ne The idea of the Debye length provides a criterion for whether a phenomenon involving collections of charged particles should be considered within the domain of plasma physics. If the scale of the system to be considered is greater than many of Debye lengths, then the system is considered plasma. If the scale of the system is smaller than the Debye length, then the system needs to be described using techniques from the general arsenal of electrodynamics. This is due to the fact that the charged particles of the system contribute substantially to the electric and magnetic fields present in the system and thus the system can not be considered plasma. If we move one Debye length away from the center of the sphere, the potential drops by a factor of 1/exp = If we move three Debye lengths, the potential is now exp -3 = times its original value. 25

26 Figure 1, above, shows how the potential due to a charged sphere falls off in plasma as compared to how it decreases in a vacuum. The Debye length is a measurement of the distance that is needed to shield plasma from an external potential or from an internal potential that results from the concentration of charged particles. Increasing the temperature of the electrons increases the thickness of the sheath, meaning potentials leak farther into the plasma. Increasing the electron density, decreases the Debye length, since more electrons are now available to participate in shielding. The idea of the Debye length provides a criterion for whether a phenomenon involving collections of charged particles should be considered within the domain of plasma physics. If the scale of the system to be considered or particular characteristics of the system are smaller then the Debye length, than the system may be described as a bunch of charged particles in external electric and magnetic fields. When there are many Debye lengths in a system it is then considered to be a plasma. In this case the particles themselves make a major contribution to whatever electric and magnetic fields may be present. As an example, a wave propagating through a collection of charged particles with a wavelength smaller than the Debye length, would not be affected by the charged particles. It would behave as a wave in vacuum (such as light) does. 26

27 The Langmuir Characteristic Curve I Langmuir Probes The simplest diagnostic tool for exploring plasma is a Langmuir probe. The probe can consist of the tip of a wire or a small sphere or disc. If the probe is placed in the plasma it can be biased relative to the plasma chamber wall. As the bias of the probe is changed, the amount of current collected by the probe changes. From these measurements, characteristics of the plasma such as electron temperature and plasma density can be derived. The electrons in plasma are moving considerably faster than the ions due to the differences in their masses. Consequently, if a probe at the plasma potential is placed in the chamber, it will collect current due to the fact that a larger number of the electrons collide with it than ions. If the probe is biased positive relative to the plasma potential, the ions that were previously able to make it to the probe are now repelled. However, this does not appreciably increase the current since the ion contribution was minimal to begin with and since the probe was already collecting almost all the electrons heading towards it already. If the probe is gradually brought negative, electrons of higher and higher energy are repelled. This causes a decrease in the electron current 1. The larger the electron temperature, the more negatively biased the probe must be in order to repel the most energetic electrons. Thus, the width of the transition region in figure 2 is dependent upon electron temperature. Eventually the probe is brought to a potential where it stops collecting current. At this point, electrons and ions are collected at equal rates. This potential is the same as the potential of the chamber wall, and is referred to as the floating potential. 2 The characteristics of the sheath formed by the probe at this point are similar to those formed by a plane sheath as discussed in the previous section. The ions at this point are required to enter the sheath region with a velocity greater than the critical drift velocity. If the potential of the probe is decreased further, the probe reaches a point where it will not collect additional ions due to Debye shielding; thus, ion current is saturated. The ion saturation current is considerably smaller than the electron saturation current. E 1 KT The electron current in the transition region is given by the formula Ielectron = ea 1 mkt ne e 2π de e E. min 2 If a probe is placed in plasma and is not biased, the probe is said to float. The probe collects current because there are more electrons colliding with it than ions. The probe becomes charged increasingly negative, and it continues to do so until the rates at which elections and ions collide with the probe are equal. At this point the potential of the probe is referred to as the floating potential. 27

28 If the probe has a surface area, A, and the ions entering the sheath have a drift velocity uo = ( KTe Mi) 1 2, then the ion saturation current is given by: 17. I n ea KT e ion sat = s M i The density n s is the plasma density at the edge of the sheath. In order for the ions to have exactly the critical velocity, the potential at the sheath s edge has to be: 18. φ s 1 e 2 KT / e relative to the plasma potential. Since the electron density is equal to the plasma density at the edge of the sheath: 19. n n = n s i 0 eφ exp and using 18 KT ni = n 1 exp n 2 = e As mentioned, when the bias is very probe collects only ions, this is the ion saturation current. It is given by the following formula: 20. I n ea KT e ion sat Mi Once again, A is the area of the probe in cm 2, T e is the electron temperature, M i is the mass of the ion (we use Argon), and e is the charge of the electron. The ion saturation current for Argon (mass=40*mass-proton) is 21. Iion sat 15. X10 14 A Te The ion saturation current is in Amperes and the electron temperature is in electron volts (e.v). 1 e.v. = 11,600 degrees C. 28

29 The following are experimental measurements of the Langmuir characteristic current voltage (I-V) curve; 29

30 30

31 Plasma Potential and Sheaths The process of Debye shielding ensures that throughout plasma the electrical potential is relatively uniform. However, this is not true at the boundaries between plasma and the chamber walls confining the plasma. When plasma is created, electrons and ions collide and are collected by the wall of the chamber. The frequency at which electrons hit the chamber wall is considerably higher than the frequency at which ions collide, due to the differences in ion and electron mass. Therefore as the plasma is created, the wall builds up a negative charge. As the walls of the chamber become increasingly negative, it repels all but the most thermally energetic electrons. The charged wall creates a Coulomb (or negative electrical) barrier for electrons that adjusts its height until the rate at which electrons and ions collide with the wall is equal. Since the wall is charged negatively, the plasma itself has a net positive charge. The plasma, therefore, has a positive potential relative to the potential of the wall. If the potential of the plasma is labeled as zero, the wall potential will consequently be negative. Most of the variation in potential between wall and plasma occurs in a layer which is about 10 Debye lengths in width. This layer is referred to as the sheath. Critical Ion Drift Velocity In plasma, the walls of the chamber continuously collect ions. These ions are required to enter the sheath surrounding the wall with a critical drift velocity, u o, which is given by: 8. u 0 = KTe M ion Assuming that the ions are cold, and thus have no thermal motion, the drift velocity of the ions is a result of a slight potential difference between the sheath and the pre-sheath region. The necessity for this velocity to be larger than a particular critical velocity is a requirement based on the characteristics of the potential within the sheath. The kinetic energy of an ion as it enters the sheath is EMBED Word.Picture.8 M iu o 2. As it moves towards the wall, its velocity, u () x, and kinetic energy increase, due to the change in potential as a function of position. Energy conservation requires: 2 31

32 9. Solving for u gives: Mu i ( x) = Mu i 0 eφ( x) 2 2 2eφ 10. ux ( )= u0 M i 1 2 Since the rate at which ions leave the plasma is equal to the rate at which ions enter the sheath an ion continuity equation can be used to relate the ion density inside the sheath, to the ion density in the plasma: 11. n o uo = n i( x) u( x) Solving for n i and using ( 10 ), the ion density inside the sheath in terms of the drift velocity is: 12. ni( x)= n0 1 2eφ 2 Mu 0 12 / The electron density everywhere is given by equation 2, and in order to determine the potential in the sheath, solutions need to be found for Poisson s equation 1. This equation winds up being nonlinear and only has solutions for values of u o above the critical value mentioned above (equation 10). A physical interpretation for the requirement that ions must move faster than the critical velocity can be seen as a result from Poisson s equation. The fact that the potential inside the sheath must be decreasing in order for it to be acting as a potential barrier for electrons, requires the left-hand side of equation (1) to be negative. This requires the ion density to be larger than the electron density inside the sheath. The electron density is dependent on the thermal energy of the electrons, the ion density and drift velocity are dependent on the electron temperature. 32

33 Two experiments involving sound that simulate the LAPTAG Plasma Ion Acoustic wave experiment. Experiment: "The speed of sound vs. frequency in a small diameter tube, resonance method." Introduction: The speed of sound through an unbounded homogeneous medium is essentially independent of the frequency of the source. However, when sound is forced to travel through a small diameter tube, it is possible that the speed might change as the wavelength of the sound begins to approximate the diameter of the tube. The purpose of this experiment is to investigate the speed of sound in a small diameter (4 cm) tube over a wide range of frequencies ( ,000hz) which will produce wavelengths considerably larger and tube at a particular frequency and the distance between nodes will be measured. Apparatus and Experiment Design: The essential physical apparatus to be used in this experiment is illustrated below: Sliding Brass Tube Condenser Microphone Small High Frequency Speaker Plastic "resonance" tube A schematic diagram of all of the electrical apparatus is illustrated below: Black Red Shield The audio oscillator drives the high frequency speaker (perhaps through a matching transformer and input filter capacitor) which produces sound waves at a known frequency inside of the long plastic tube. A small condenser microphone can be moved to various positions inside of the plastic tube by adjusting the sliding brass tube. The output of the microphone is attached to an oscilloscope. (Note that the condenser 33

34 microphone requires a bias battery with the plus attached to the red lead. The signal is from the black lead and the shield is common to the battery negative and microphone ground.) With the oscillator set at a particular frequency the microphone should be positioned at a node as near the speaker as the sliding brass tube will allow. The microphone is then moved away from the speaker as successive nodes are located and measured. Since the nodes will be successive half wavelengths apart, the wavelength of the sound is easily found and with the frequency, the speed of the sound in the tube can be calculated. Suggested Procedure: 1. The length of the tube and the high frequency nature of the speaker will prevent meaningful measurements from being made below 1000 hz. It is suggested that you begin making measurements at 1000 hz. Move the microphone as near the speaker as possible and with the output of the speaker set to produce a good output measurement, move the microphone away from the speaker until the smallest signal is indicated on the scope. Note the position of the microphone with the sliding grommet on the brass tube and then move the tube outward until the next output minimum is achieved. Measure the distance the microphone has moved. Repeat for several nodes. 2. Return the microphone to near the speaker, set the oscillator at a higher frequency and repeat the above. 3. Be advised that interesting things begin to happen at certain higher frequencies. Precautions: 1. The small speaker could be damaged if too large a signal were placed across it at a very low frequency. It is best to begin with a frequency that is clearly audible (1000hz) and adjust the volume so you can hear it but it does not have to be very loud for the experiment to work. The scope is quite sensitive and your only objective is to experience easily measured differences between the nodes and antinodes. A filter capacitor can be inserted either between the speaker and the transformer or at the input of the transformer to prevent low frequencies from damaging the speaker. 2. Although the condenser microphone draws essentially no current, when you are finished with the experiment, disconnect the bias battery by displacing one of the AA cells so contact is no longer made. The best bias voltage is about 4.5 volts and under no circumstances should it be more than 9 volts. Comments: Using resonance to measures the wavelengths of the sound in the tube usually assumes the distance between the nodes is half a wavelength. This assumes standing waves have been set up in the tube which may not always be the case if the length of the tube isn t some whole multiple of the wavelength produced. Think carefully about what conditions are necessary to establish resonance in the tube as you attempt to make sense of your data. Is the speed of sound the same for all frequencies or can you detect some "dispersion" effect? 34

35 Experiment: "Measuring the speed of sound using short duration sound pulses." Introduction: If you performed the experiment to measure the speed of sound using the resonance method, you might have observed that as you approached higher frequencies there came a point in which it was impossible to make measurements. As the wavelength of the sound generated gets smaller and approaches the diameter of the tube, transverse resonances are excited making it very difficult to locate nodes and antinodes. In this experiment we will use essentially the same apparatus as in the resonance method experiment only this time we will use short duration sound pulses and measure time it takes for these pulses to travel a measured distance. The sound pulses will be produced by an oscillator which can gate two or three cycles of a particular sound frequency into a short duration pulse. Since the particular sound frequency is produced for such a short duration, resonances will not have time to build up in the tube. Apparatus and experiment design: Review of experimental concept behind the resonance method: In the resonance method, a continuous sine wave was presented to the speaker which produced standing waves in the closed tube. The small microphone was moved from node to node and the distance between nodes was measured. Since the nodes are half a wavelength apart, the wavelength of the sound waves can be determined and with the frequency of the continuous wave, the velocity of the sound can be computed using: velocity = frequency x wavelength. Oscillator set to produce continuous sine waves Scope set to measure amplitude of signal. Trigger not required. Microphone is moved from node to node as the distance moved is noted. Overview of the experiment to determine sound speed with the pulse method: In this experiment, the oscillator will be set to produce short "gated" sound pulses. As this gated pulse is initiated, the scope will be triggered and the time from the initiation of the pulse to its arrival at the microphone will be measured. Knowing the distance between the speaker and the microphone and the time required for the pulse to reach the microphone, the velocity of sound can be computed by a simple application of velocity 35

36 = distance/ time. Since we will be using a dual trace oscilloscope, it will be instructive to observe the signal at the speaker as well as the signal received at the microphone so the time required for the sound to travel from speaker to microphone can be easily determined. The basic apparatus setup is illustrated on the next page. Function generator set for gated sonic pulse at a particular frequency. Oscilloscope set to observe original signal and signal received at microphone. Original pulse also triggers scope Trig. HZ CH 1 CH 2 Here the distance between the speaker and the microphone is to be measured. Suggestions and details on performing the pulsed method experiment: 1. Become familiar with the function generator and the oscilloscope. Since the settings required to perform this experiment are more complex than in the resonance method, it is probably best to become familiar with both the function generator and the assorted ways the scope can be used before making the actual measurements. Rather than starting with the sound apparatus fully wired, it might be more instructive to wire the function generator directly into the oscilloscope and see if you can produce and observe a gated sonic pulse. There is hardly anything you can do to damage the apparatus so begin with the following equipment set up and fool around until you can observe a clean gated pulse. 36

37 Trig. CH 1 CH 2 Think about what you are doing, read the assorted knobs and "ports" in the function generator and the oscilloscope and you should get the hang of what you are expected to do. Attempt to get about three cycles of the sound frequency to be gated into a single sound burst and adjust the triggering and sweep on the oscilloscope to have this gated burst of sound to appear on the left side of the screen. Play and learn! 2. A few details about the speaker, sound tube and microphone apparatus. Here is a sketch of the apparatus to be used with the function generator and oscilloscope to measure the speed of sound: High frequency speaker Condenser microphone Matching transformer Bias Battery The output impedance of the function generator is too high to drive the speaker directly without some distortion so a matching transformer is inserted between the generator and the speaker. The following schematic shows the basic hookup: Function generator 37

38 The small condenser microphone requires a small bias battery. The microphone actually draws very little current but to prevent running the batteries down between sessions, remove one of the batteries when shutting down after a run. The schematic showing the wiring of the bias battery and the oscilloscope is shown below: Black Condenser microphone Red Shield Performing the experiment: Note: The essential character of this experiment is the same as measuring the speed of assorted waves in the plasma chamber. However, the advantage here is that the apparatus is almost impossible to ruin and if you make a mistake, there will be no dire consequences. After you have become familiar with the use of the function generator and the oscilloscope the produce gated sonic bursts, set them to produce a burst of about three cycles of a 1 khz wave. Now wire the output of the function generator to the input of the matching transformer to the speaker. (You should hear something from the speaker. The only thing you might do wrong here is to give this high frequency speaker to low a signal--avoid using sound frequencies of less than about 500 hz.) Also wire the output of the function generator to the input of channel 1 of the oscilloscope and the trigger input. Fool around until a good signal appears in the upper part of the oscilloscope screen. With the condenser microphone properly biased (remember to disconnect the battery when you finish experimenting today) connect the output of the microphone to channel 2 of the oscilloscope. Fool around until you see a second trace of the received signal on the lower part of the oscilloscope screen. Since the signal from the microphone will be much smaller than the signal directly from the function generator, the gain on channel 2 will have to be much higher than on channel 1. Also, since the sound burst has to move some distance before reaching the microphone, the received signal will be displaced to the right on the screen. You are now ready to experiment. Moving the microphone probe toward and away from the speaker will displace the signal seen on channel 2 to the left and right. You can measure the time difference between the original signal and the received signal using the time scale on the oscilloscope. Using your brain and a ruler, you can figure out how to measure the distance between the speaker and the microphone. You are ready to perform experiments on the speed of sound in the small tube. Very interesting and unexpected results have been obtained in the past--look for them. 38