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Electron Diffraction
Purpose
To verify the wave nature of electrons by observing electron diffraction; to verify qualitatively predictions about the
relationship between electron momentum and wavelength; to use electron diffraction to determine the spacing between atomic
planes in a solid
Equipment
Electron diffraction tube; vernier caliper; computer with Excel (optional)
Introduction
1.
Describe the pattern of dots produced on a screen when a beam of light passes through an ordinary diffraction grating.
Your description should also tell how the dots are oriented relative to the direction of the slits in the grating.
2.
Predict how the pattern will look if a second grating is stacked behind the first so the slits of the two gratings are
perpendicular.
3.
Was your prediction correct? Describe what actually happens when two gratings are stacked with their slits
perpendicular.
4.
Predict what will happen when several gratings are stacked with their slits oriented in different directions.
5.
Was your prediction correct? Describe the pattern formed by 10 - 12 stacked gratings.
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6.
How would the pattern look if there were millions or billions of gratings oriented in all directions?
7.
How would the pattern look if there were millions of gratings with different slit spacings?
You have found that the diffraction pattern formed by a beam of electrons passing through a material should form
concentric circles around the bright central spot formed by the original beam. Because most materials contain planes of atoms
with different spacings in different directions, there are many circles of different diameters that form a repeating pattern for
different diffraction orders. In this experiment, you’ll observe diffraction rings caused by a particular material (which at this
point is unknown). You’ll verify qualitatively the predicted relationship between electron momentum and wavelength, and
you’ll arrive at an estimated value for the spacing between the set of planes responsible for one of the diffraction circles.
Theory
8.
Write the equation describing the relationship between groove spacing, diffraction angle, and wavelength for a
transmission diffraction grating. Include a diagram.
9.
Suppose the diffraction pattern is formed on a screen a distance D from the grating, and the first-order spots are a
distance y away on either side of the central maximum. Rewrite the equation from 8 in terms of y and D, and solve it for the
spacing d. (Note: there are two ways to do this, and you should consider both. One is to write the sine function entirely in
terms of y and D, and the other is to write the sine function in terms of the inverse tangent.)
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10.
Write the deBroglie relationship between the momentum p and wavelength  of a particle.
11. Write the definitions of kinetic energy K and magnitude of momentum and use those to find magnitude of momentum in
terms of kinetic energy and mass m. If the particle has a charge of magnitude q and has acquired its kinetic energy as a result of
acceleration through potential difference V, give a final result in terms of q, m, and V.
12.
Combine the results of 10 and 11 to get an equation giving  in terms of q, V, and m.
You have found (using the variables defined)  
h
. Shortly you’ll be shown how to use this to get a “formula”
2mqV
for finding the wavelength of electrons whose kinetic energy is given in eV.
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Procedure
13. Turn on the electron diffraction tube. Set the accelerating potential at about 5 kV and make sure the beam is positioned
to produce a clearly defined diffraction pattern. (This may be done as a demonstration.) Describe the resulting diffraction
pattern. Does it match the pattern predicted in the Introduction? Refer to Questions 5 - 7 in formulating your response.
14.
Predict how the diameter of the circles will change as the accelerating potential increases. Explain your answer.
(In the actual experiment, you'll be asked to adjust the accelerating potential. When changing the voltage, it may
be necessary to adjust the horizontal and vertical position controls to keep a good pattern on the screen. You should
also turn the intensity down as the voltage is increased. Keep the pattern bright enough that you can see the circles
without difficulty, but it should not be too bright. An intense electron beam will actually burn the phosphor on the
tube, causing irreparable damage.)
15. Increase the accelerating voltage and describe how the diameter of the circles changes. State clearly whether your
prediction in 14 was correct or not.
16. Each group in the class will be asked to measure the diameter of the first order diffraction circle for two or three different
voltages. Each diameter is 2y in the diffraction pattern described in Question 9. Each person in the group should make an
independent measurement, and then average the group's values for each voltage. When everyone is done, the class data will be
tabulated on the chalkboard. Fill in the tables below with your own data and the summarized class data.
Your group’s data:
Enter one value for D
Enter your assigned V
values and the values
of d measured by each
group member
D (cm)
V (kV)
2y (cm)
Class data:
V (kV)
2y (cm)
5
6
7
8
9
10
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17. You have two equations that can be used to determine the distance d between the planes of atoms that are responsible for
producing the diffraction pattern:

1.227
nm
K ( eV )
d

sin 
where the sine is determined by using y and D directly or by using those quantities to find the angle via the arctan function.
Note that since the equation for  gives the units of nm, d will also be given in nm. Use the tabulated data to compute a value
of d for each of the measurements and then find an average and standard deviation for the entire data set. You may start Excel
and create a spreadsheet to do these calculations. (This is highly recommended.) Tabulate your values of d and give the
average and standard deviation in the box at the bottom of the page. The final value for d should have three significant digits,
and its standard deviation must have the corresponding (correct) number of digits.
Final Result: ___________________
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Homework Questions
1.
While we didn't check the deBroglie relationship for the electron wavelength quantitatively, this experiment provides
good qualitative evidence for the wave nature of electrons and for the prediction of how the wavelength depends on the
electron's momentum. Please explain, using experimental observations to support your explanation.
2.
We can't really check the accuracy of the value obtained for d because we don't know what the nature of the target is, and
we don't know the exact distance from the target to the screen. However, we can decide whether the value of d is reasonable.
Calculate an expected value for d using the properties of aluminum and compare that with your experimental result.
3.
The standard deviation you found tells how the individual values of d are spread out. However, there is a large source of
uncertainty in the experiment because the exact value of D isn’t known. Suppose the correct value of D is somewhere between
17.0 and 19.0 cm. Then we would give the datum D as 18.0  1.0 cm. Find the resulting uncertainty in d and give your result
in the form of d  uncertainty. (This uncertainty will be larger than the standard deviation found when averaging the individual
d’s, so you can ignore it for this question.)
4.
Use your value of d to find the diameter of the bright second order diffraction circle for V = 10 kV.
5.
Describe how the diffraction circles would change if protons were used instead of electrons. Everything else would
remain the same. Explain your reasoning. (You should be able to give a quantitative answer to this question--by what factor
will the sizes of all the circles change?)
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