Properties of the Electron

The electron is pretty darned important.  If it wasn't for the electron, we wouldn't have ice cream in the summer.  It is pretty small.  It has a charge.  It behaves like a particle sometimes, and then at other times it behaves like a wave.  I guess that holds for all of the fundamental particles, but the electron was the first fundamental particle to be discovered.

In the following series of experiments, we will not only study the property of electrons, but we will learn about a number of ground-breaking (Nobel Prize-winning) experiments which changed the way we understand matter.

• SAFETY

• References

• Background

• Experiment 1: Effects of E and B on the Electron.

• Experiment 2: Bragg Diffraction of Electrons.

Safety: you will be using very delicate equipment and very high voltages.  Before you turn on any power supply, be sure that you and your partner check the circuits until you are certain you have it right.  Then, do not try to move any wires without turning off all of the power supplies.  Treat the Cathode Ray Tubes with extreme care, they are fragile, and we do not have extras.

References: any introductory text for the properties of an electron in magnetic and electric fields, and any modern physics text for Bragg scattering.

Background: your introduction should stress the importance of these experiments and the breakthroughs they allowed.

Experiment 1 - The effect of E and B: measuring the charge and the mass of the electron

In a series of experiments performed just prior to the turn of the century, J. J. Thomson was able to show that electrons behaved as particles of mass m that carried a fixed amount of (negative) electrical charge e. These particles moved in trajectories governed by the laws of electricity, magnetism and classical mechanics. Thomson received the Nobel Prize for this work in 1906. (J. J. Thomson's son, G. P. Thomson, received the Nobel Prize in 1937 for experiments that showed that electrons could exhibit wave-like properties as well.) Thomson was not able to measure the charge or mass separately but only the ratio of these two quantities, e/m.

A. Basic E&M theory

In order to do all of the exercises in Experiment 1, you will need to use the basic equations relating potentials to electric fields, and EM fields to forces on charges:

In addition, because of the way the apparatus is setup, we will approximate that the electric and magnetic fields that we produce are constant over the area of interest.

B. Experimental Apparatus

• Deflection Tube (pictured below)
• 5000V variable power supply (e.g. Tel-Atomic kV 813 Power Supply)
• Standard DC Power Supply
• Multimeter

In this exercise you will repeat some of Thomson's experiments using apparatus similar to his. This apparatus consists of an evacuated clear glass bulb containing an electron "gun" similar to that found in a modern TV picture tube. This gun is a heated filament that emits electrons that are then accelerated toward a metal plate kept at a positive potential relative to the filament. A narrow slit in the plate allows the electrons to pass through, and they eventually strike a flat mica sheet on which there is printed a centimeter scale. Metal electrodes above and below this mica sheet allow us also to apply electric fields to the region.

Apparatus for Measurement of e/m for electron

The glass bulb is supported between two large coils of wire. The arrangement of these two coils (suggested by Helmholtz) is such that the separation between them is equal to one-half of their average diameter. A current through these coils will create a magnetic field B which is substantially uniform over the central region between the coils. 

You need two power supplies: You can use the high-voltage power supply for both the accelerating potential (anode potential, V_a) and the deflecting potential (V_d). If you have access to two separate high-voltage supplies, then use two separate ones for those two voltages. Any regular low-voltage DC power supply can be used for the Helmholtz Coils. 

1. Notice that each Helmholtz coil has a terminal labeled A and one labeled Z. Connect the Z terminals to one another and the A terminals to the DC power supply.  You will want to include the multimeter in the circuit to give you a better value for the current through the coils.  Important note: never exceed 1A of current in the Helmholtz coils for an extended period of time (not more than a few minutes).
2. The filament in the electron gun should be connected to the two 6.3V terminals on the TEL-Atomic power supply, or to a low-voltage DC power supply set at 6.3 volts. The plus and minus terminals on the variable part of the power supply should be attached across the filament and anode, to supply the accelerating voltage from the filament toward the anode. In addition, the same voltage (or the second high voltage power supply) should be attached across the parallel deflection plates.  Make sure that appropriate banana plug connections are used so that no end of a plug is left exposed.  The connections described are pictured below with a single power supply powering both the deflection and accelerating voltages.

3. Notice that if we want to observe the deflection due to only the magnetic field, we don't want any potential difference across the parallel plates.  In that case you would want to detach the deflection plates (the terminals coming from the top and bottom of the large part of the bulb) from the + and - connections on the power supply, and simply short the two plates together.  You will want to do this at various times in the lab as noted in the procedure.

C. Calibration of E and B:

Exercise 1.1: Determine theoretically for this apparatus the magnitude of E as a function of voltage applied: considering these are parallel plates (thus a constant field), and each square between them is 1 cm, what should the theoretical relationship be for E vs. V for this apparatus? Give the answer only in terms of V and numerical quantities.

The magnitude of magnetic field, B, from the coils is given by the manufacturer as

(1.2) , where

N = number of turns of wire in each coil (320)
r = mean radius of coils (0.068m)
I = current in Amperes

D. The Effects of Electric and Magnetic Fields on a Charge

1. Once you are certain the circuit is hooked up correctly, turn on the high-voltage power supply and slide the 5kV voltage control up.  When the voltage exceeds about 1500 Volts, you will begin to see the blue glow of the electron beam (did you know electrons were blue? Why are they blue?). (The electron beam is easier to see in a darkened room.) If you are using a single high-voltage power supply for both the acceleration and deflection, you will notice that as you increase the accelerating voltage, you are also increasing the deflection voltage, hence the deflection does not vary! (At some point try to get your hands on a second power supply and vary them separately, and you will see you can vary the deflection. 

Exercise 1.2: You observed this above, now show theoretically that the deflection does not vary if the accelerating voltage and deflection voltage are changed at the same rate.  (Hint: You need to show that the y-position of the beam as a function of x is independent of the accelerating/deflection potential. You might do this by finding the horizontal velocity of the electron being accelerated by the potential V using energy conservation, and then finding the resulting displacement in the horizontal direction after a certain deflection in the vertical direction due to the electric force.)

Exercise 1.3: From your derivation from Exercise 1.2, and by measuring the actual experimental deflection, show that the equation you derived from exercise 1.1 is true (or not). That is, find the correct, experimental calibration relating E and V (still assume that E is constant).

Exercise 1.4:  What is the theoretical shape of the path of the electrons passing between the parallel plates? Plot the theoretical trajectory (use the equation you derived) using 3000 V (using Matlab, Excel or anything else). Does it look like what you actually see (take a picture of the actual)?

2. Turn off the power supplies and disconnect the parallel plates from the voltage source as described in Section B.3.  Turn it back on to see the horizontal electron beam.  
3. Now turn on the Tesla Coil power supply and observe the effect of the magnetic field on the moving electrons. Vary the accelerating voltage as well as the magnetic field. Observe the effect of reversing the direction of the magnetic field by reversing the direction of the current through the Helmholtz coils.
4. Check your knowledge of the direction of the magnetic field and the direction of the force it exerts on a moving negatively charged particle. Exercise 1.5: What is the direction of the magnetic field between the Helmholtz coils for a given direction (CW or CCW) of the current? What is the direction of the force that this field exerts on a moving electron?

Exercise 1.6: Recall that a charged particle moving in only a magnetic field moves in a circle.  Using Newton's second law for uniform circular motion, and a determination of the speed due to the accelerating potential (Exercise 1.2), show the following result:

(1.3)

The voltage V is read directly from the high voltage power supply. For a given current, the magnetic field B may be computed from Equation 1.2, above. All we need is a value for R in order to be able to compute the ratio of electronic charge to electronic mass.  The electron beam begins at the origin.  The grid allows you to follow its coordinates as it travels.  A little geometrical analysis will show that, for circles passing through the origin and the point (x,y), the radius R of that circle is given by

(1.4)   

If you are not comfortable with it, you should derive it.

E. Measuring e/m

We would like to measure e/m.  Do it in a way similar to the following: 

1. For a given high voltage V, adjust the coil current I so that the electron beam passes through the point: x = 10, y = + 2.6.
2. Keeping this same voltage, reverse the magnetic field and alter its strength until the beam passes through the point: x = 10, y = - 2.6.
3. Use the average value of these two currents when computing the magnetic field in Equation 1.2.
4. Repeat this process for five different values of the accelerating voltage and compute five different values for e/m. Average the values and find the 90 % confidence interval for the five values.
5. Compare your final value to the accepted value within confidence limits.

F. Crossed Electric and Magnetic Fields

1. Reconnect the parallel plates to deflect the electrons with both an electric and magnetic field.
2. A combination of electric and magnetic fields applied at right angles forms what is known as a velocity selector and is a common occurrence in mass spectrometers and particle accelerators. The physical principle is quite simple: an electric field E exerts a force qE on a charged particle while a magnetic field exerts a force qvB on the same particle. When these two forces are equal in magnitude but oppositely directed, the particle will move in a straight line. Its velocity will be given by v = E/B (set qE=qvB!).
3. You can approximate this condition over a narrow portion of the electron beam. Adjust the potential difference applied to the electrodes above and below the beam and the magnetic field until you see the beam move in a straight line across the middle portion of the tube (where the magnetic field is most nearly uniform).  Use the approximate value of speed which you had calculated earlier from energy conservation (using your value of e/m) to get an approximate value of E you are using in the velocity selector.  Does this value of E match what you had calibrated in Exercise 1.3?

Experiment 2: Diffraction of Electrons

The purpose of this experiment is to demonstrate that electrons have wave properties. Specifically, we wish to show

1. that electrons can produce diffraction effects when scattered from a Carbon target,
2. that the pattern so produced follows the same Bragg equation that applies to x­radiation, and
3. that the results confirm the de Broglie hypothesis - namely, that all material particles have a wavelength which is inversely related to momentum.

A. Theory:

The de Broglie wavelength of a material particle is

(2.1)   

where h is Planck's constant. For electrons accelerated through a potential difference V, the velocity v can be obtained from the classical expression

(2.2)   

and substituted into the de Broglie relation obtaining:

(2.3)   

If a wave is diffracted off of Bragg planes, it will bounce off the Bragg planes randomly in all directions, creating a fundamental diffraction ring at an angle theta from the original path, given by the relationship governing constructive interference (derived in Physics 251).

(2.4)  

where d = the interatomic spacings, D is the ring diameter, and L is the pathlength from the carbon target at the gun aperture to the luminescent screen. Combining this with the previous relation,

(2.5)   

If you vary V, then D varies (everything else is constant) so the relation can be verified by means of a graph related to these variables.

B. Apparatus:

The electron diffraction tube, TEL.555, comprises a 'gun' which emits a narrow converging beam of electrons within an evacuated clear glass bulb on the surface of which is deposited a luminescent screen. Across the exit aperature of the 'gun' lies a micro­mesh nickel grid onto which has been vaporized a thin layer of graphitized carbon. The beam penetrates through this carbon target to become diffracted into two rings. Note that now your electrons are green, while earlier they were blue - why do you think this is?

Connect the tube as shown in the following diagram:

• Click here for a "real" diagram rather than a block diagram.
• Filament Voltage ( V_F ). . . 6.3 V ac (on the 813 Power Supply)
• Anode Voltage ( V_A ). . . 2500 - 5000 V dc (on the 813 Power Supply)
• External Bias ( V_B ). . . 0 - 50 V dc (on the 801 Power Supply)
• Anode Current ( I_A ). . . 0.15 mA at 4000 V ( 0.20 mA max.)
• Note: V_A and V_F  are on the same, 813, power supply.  Use the ground on the 813 for the common ground for all the supplies.
• If you are using the diffraction tube with the "Universal Stand":  Connect C5 on the universal stand to the negative on the HV supply. F4 goes to the positive on the heater supply and the low voltage bias. G7 goes to the positive lead on the HV supply. F3 goes to the negative on the heater supply.

C. Experiment:

1. Measure the pathlength from the Carbon target at the 'gun' exit aperture to the luminescent screen as accurately as possible.
2. Switch on the heater supply and wait one minute for the cathode temperature to stabilize.
3. While watching the meter which reads the anode current, gradually begin to increase the anode voltage. The anode current should never be allowed to exceed 0.2 mA. Higher anode voltages can be achieved without exceeding the limit by increasing the external bias control which also helps in focusing the electron beam. Set the voltage to 2500 volts for now.
   

4. Two prominant rings about the central spot are observed. Variation in the anode voltage causes a change in diameter.
5. Measure and tabulate the ring diameter, D, for both the inner and the outer rings at different anode voltages, say from 2500 volts up to 5000 volts in steps of 500 volts.
6. Plot a graph of V^-½ versus D for each ring. From the slopes together with equation (5) determine the value of the two spacings d in Carbon which would yield these results.

7. Optional: look up the spacings in carbon to see if these are correct (hint, they are 0.12 nm and 0.21 nm).
8. In your conclusions, be sure to say a little bit about how your results address the stated objectives.