Light Diffraction through a Grating and
Electron Diffraction Through a Carbon Mesh
Discussion:
As we have discussed in class, all objects have wave-like characteristics. One method for demonstrating the wave-like character is to observe whether the object displays diffraction and interference as is passes through a grating or slit whose dimensions are of the order of the wavelength of the wave.
The existence of such phenomena conclusively demonstrated that light `is’ a wave in the 1600’s, thus tearing down Newton’s idea that light was a corpuscle. (We now know that both descriptions are correct!!!).
According to the de Broglie hypothesis of 1923, a particle of matter also has a wave-like character, with a wavelength dependent on its momentum. Since interference patterns can be observed when waves travel through a grating with dimensions on the order of the light’s wavelength, matter waves might also be expected to interfere under certain conditions. The existence of such phenomena conclusively demonstrates the necessity of a quantum-mechanical description of nature.
In these series of experiments, we will observe diffraction and interference effects with light and matter. From your measurements you will be able to to infer information about the slits and the structure of a graphite lattice.
For further discussion, see Hecht, Chapter 9 and 10, and Serway, Chapter 4.
Optical Diffraction and Interference Phenomena
This part of the laboratory will allow you to study a variety of optical interference effects. Examine the slit assemblies provided with the experimental setup to identify the location of the various apertures. Attach one of the slit assemblies to the vertical edge of the table with the C-clamp (be gentle!). Set the laser (a helium-neon laser operating at 633 nm) on the table across the room (so the beam will expand a bit before it passes through the slits), and position it to illuminate the slits at normal incidence. Project the patterns onto a sheet of graph paper near the end of the table. Measure the distance from the slide to this graph paper screen. The width and spacing of the slits are given on the slit assemblies.
A. Single Slit Diffraction Patterns
The purpose of this experiment is to examine the diffraction pattern formed by laser light passing through a single slit and verify that the positions of the minima in the diffraction pattern match the positions predicted by theory.
Measure the distance between the first-order AND second-order minima produced by the 0.02, 0.04, and 0.08 mm slits. It may be useful to trace the diffraction patterns directly onto the graph paper. Make a sketch of the diffraction pattern to scale in your notebook. In your report, calculate the widths of these slits using both the measurements for the first- and –second-order minima and compare your results to value on the slit wheel. Discuss the agreement between theory and experiment.
Does this distance between the minima increase or decrease when the slit width is increased? You might want to play around with the continuously variable slit.
B. Double Slit Interference Pattern
The purpose of this experiment is to examine the diffraction and interference patterns formed by laser light passing through two slits and verify that the positions of the maxima in the interference pattern match the positions predicted by theory.
Measure the distance between the first-order AND second-order minima produced by the double slits (0.04 mm slit width) spaced by 0.25 and 0.5 mm. Make a sketch of the interference patterns in your notebook.
Measure the distance between the first-order AND second-order minima produced by the double slits (0.08 mm slit width) spaced by 0.25 and 0.5 mm. Make a sketch of the interference patterns in your notebook.
In your report, calculate the spacing between slits using both the measurements for the first- and –second-order minima and compare your results to value given on the slit wheel. Discuss the agreement between theory and experiment.
Does the distance between the maxima increase, decrease, or stay the same when the slit separation is increased?
Does the distance between the maxima increase, decrease, or stay the same when the slit width is increased?
Does the distance to the first minima in the diffraction envelope increase, decrease, or stay the same when the slit separation is increased?
Does the distance to the first minima in the diffraction envelope increase, decrease, or stay the same when the slit width is increased?
C. Many Slit Interference Patterns
The purpose of this experiment is to compare the diffraction and interference patters formed by laser light passing through various combinations of slits.
Observe the qualitative changes that occur in the interference patterns as you move from one to five slits. Note in particular how the interference maxima become increasingly sharp as the number of slits increase. Sketch the three slit pattern.
Measure the spacing between maxima for all four grids. In your report, calculate the wavelength of the laser light from these data (use the slit spacings and widths given on the slit wheel). Express your final answer as an average of the five measurements with an error calculated from their standard deviation. Compare this result to the known wavelength of 633 nm.
Sketch the pattern observed when you illuminate the square, hexagonal, random dot, and random hole patterns.
Does the distance between the maxima increase, decrease, or stay the same when the number of slits is increased?
Does the sharpenss of the maxima increase, decrease, or stay the same when the number of slits is increased?
What diffraction patterns seem similar to that observed in the electron diffraction apparatus?
Electron Diffraction Through a Graphite Mesh
Principles of the Apparatus:
A Tel-Atomic electron diffraction tube is shown schematically in Fig. 2.
(Fig. 2: Schematic from Tel-Atomic manual.)
Electrons are boiled off a hot filament and accelerated across a potential difference V. By doing so these electrons acquire an energy eV where e is the charge of the electron. This energy can be reasonably assumed to be converted entirely to kinetic energy.
or
Using de Broglie’s famous proposed
we can conclude that the wavelength will be
The electrons with this wavelength are passed through a pure carbon film a few atomic layers thick and projected onto the phosphorous screen on the far end of the tube. The presence, shape, and size of the interference rings indicates the interaction of the beam with the diffraction grating formed by the carbon film. Using the common formula for interference patterns
we can measure to learn about d, the spacing of lines in the grating.
Experimental Procedure:
In this experiment, you will accelerate the electrons with a range of voltages and measure the interference patterns produced. A graph of the size of the patterns vs. V^-.5 should be linear, and the slope of the graph should contain information about the diffraction grating, assuming the wavelength to be well-known. If you observe several distinctive features that can be measured, measure them all, and graph them as separate lines on the same axes.
The diffraction patterns produced can be expected to be
(hand-drawn sketch of horizontal blobs)
if the grating consists of parallel vertical lines. A more likely structure for a crystal would be parallel vertical and horizontal lines. Such a pattern might be observed from X-rays passing through a cubic lattice structure such as NaCl.
(hand-drawn sketch of x-ray diffraction)
In the write-up of this lab, you will produce a graph like the one discussed above, and use the information on it to make an argument for or against a hexagonal structure for graphite (carbon) crystals like the one in the film in the Tel-Atomic tube. Why hexagonal? There are only three regular polygons that can fill a space: a triangle, a square, and a hexagon. We know that graphite consists of a set of two-dimensional crystals laid atop one another from its easy and clean shearing properties. And we know that carbon likes to form four bonds at 109.5 degrees from one another. So a square lattice, with four bonds at 90 degrees, is not an implausible arrangement for carbon to take. To form a triangular lattice, each carbon atom would have to form six bonds at 60 degree angles- an implausible overcrowding. But to form a hexagonal structure, each atom could form three bonds and a 120 degree angle- it seems less likely than the square matrix, but should be considered a possibility.
A hexagonal structure has not one but two characteristic dimensions that could act as a diffraction grating. While there are many directions that you could look across such a structure and see an irregularly-spaced structure of atoms, there are exactly two ways to look across and see a perfectly uniform periodic structure:
(sketch from Tel-atomic manual of hex crystal)
These are referred to as the 1 0 and 1 1 directions. (These definitions are explained in any Solid State physics textbook.) Look at a big sheet of hex paper if you’re not convinced by the sketch above- look for straight lines that are all equally spaced. (It doesn’t matter if the atoms within each line are not equidistant from their neighbors- the important thing is that the lines be equidistant from each other. Why?) Do a little geometry to convince yourself the ratio of one length to the other is about 1:1.7 . (What is it exactly?) It is easier to see the spacing of the two viable diffracting patterns in a square lattice like this one:
(sketch of a square lattice showing a side and a diagonal as repeating, identical grid lines.)
What is the ratio of the size of the 1 0 grating to the 1 1 grating.in the square lattice?
In your report, note the shape of the diffraction pattern observed. Discuss what you can learn about the carbon film from this shape. (There is an obvious answer and a not-so-obvious answer. Hint: what is a “crystal domain”?) Do the two structures represent different integer n’s in the equation n lambda = d sin theta ? If not, where are the higher-order structures? If graphite had a square structure and thus had lengths in a ratio of 1:1.4 , what would be the ratio of the slopes of the two lines on the graph you’ve drawn? What does their actual ratio signify?
Don’t forget to correct for the spherical nature of the projection screen when you convert your pattern sizes into thetas! Derive an equation to correct for this in an appendix to your report.
(sketch from manual showing spherical correction.)
Notes on the apparatus:
1. The cables should be all in place. If they are not, ask the TA for assistance. The on-off switch is on the back of the apparatus. Turn the knob all the way down before activating the box.
2. Let the heater warm up for a minute before turning up the accelerating voltage (knob on the front panel).
3. Make sure the screen is measuring kV, not amps. There are push-button controls for this just below and to the right of the accelerating voltage control knobs.
4. Keep an eye on the graphite film. If you are at a high voltage for a long time, it might overheat and puncture. If it begins to glow red, turn down the voltage for a minute.
5. Measure the size of the diffraction patterns with vernier calipers. It’s OK to touch them to the tube.
References:
1. Serway, Ch. 4. Discusses de Broglie hypothesis.
2. Thomson, G. P., 1928, Proc. Roy. Soc. 117, 600; 119, 651. Original papers on the electron diffraction experiment.
3. Tomomura, A. et al, 1989, Am. J. Phys, 57, 117. Neutron diffraction.
Note that big, fat, individually visible (to an electron microscope) sodium atoms have now been diffracted in the same manner as electrons! Can larger objects diffract? Think about it.