1 Diffraction of Microwaves

Transcription

1 1 Diffraction of Microwaves 1.1 Purpose In this lab you will investigate the coherent scattering of electromagnetic waves from a periodic structure. The experiment is a direct analog of the Bragg diffraction of x-rays by crystalline materials. In 1912 Max von Laue realized that a crystalline solid might be thought of as a three-dimensional diffraction grating for x-rays. However, the grating in this case is comprised of a series of point reflectors rather than the series of lines that form an optical diffraction grating. When an electromagnetic wave enters a crystal it is scattered by each of the atoms in the crystal. While the scattering occurs in all directions, the combination of all the scattered waves from all of the point scatterers produces regions of constructive and destructive interference. The result is a sequence of maxima and minima in the scattered wave and the locations of the maxima and minima are determined by the wavelength of the incoming wave and the spacing between the atoms in the crystal. The resulting pattern is called the Bragg diffraction pattern after W. L. Bragg who first derived the refraction rule and won the Nobel Prize in physics in 1915 for his work with x-rays to study the structure of crystals. In our lab we model the x-ray experiment by using microwaves. These waves have wavelengths in the centimeter range and are less dangerous to use than x-rays. Of course, since the scattering centers must be spaced apart by distances commensurate with the wavelength of the electromagnetic wave that illuminates the structure, we will use a Styrofoam crystal that has metal spheres embedded in it. The spheres are arranged to form a simple, cubic crystal structure. In this lab you will: Measure the wavelength of the microwaves to be used Locate the peaks in the diffraction pattern produced by the styrofoam crystal Use the wavelength you determined to estimate the spacing between the metal spheres in the crystal. 1.2 Apparatus microwave transmitter and receiver (Pasco WA pair) an aluminum mirror a meter stick and tape

2 the Styrofoam crystal translating table lab stands, rods, clamps 1.3 Procedure Part 1: Measuring the wavelength of the microwaves In order to measure the wavelength of the microwaves we can set up the apparatus to produce a set of standing waves and then move the receiver and measure the distance between peaks in the standing wave. Tape the meter stick to the table and place the transmitter at on the stick. Place the mirror at the other end of the meter stick and and tape in in place also. Place the receiver about 5 cm from the transmitter as shown in Figure 1. Make sure that the transmitter and receiver horns are at the same height. Figure 1: Top view of the standing wave setup Turn on the transmitter and receiver and move the transmitter until the receiver reads a minimum. This is a node in the standing wave. Record the location of the transmitter on the meter stick. Slowly move the transmitter toward the mirror recording the position of each node as you pass by. Try to locate as many nodes as you can since you will determine the wavelength of the microwaves from these measurements. (You should find more than 5 nodes.) From this information you should be able to estimate the wavelength of the microwaves. Make sure that you also note the uncertainty in your node locations.

3 1.3.2 Part 1: Analysis Calculate λ, the wavelength of the microwaves. Be sure to include a calculation of the uncertainty in your value along with notes about the accuracy of your measurements Part 2: Measuring the Bragg diffraction pattern The location of constructive interference maxima from a crystal can be found by a simple analysis of the scattering of waves from two of the points in the crystal. As shown in Figure 2, constructive interference occurs when 2d sinθ = mλ (1) where m is the order number of an intensity maximum, d is the spacing between the scattering plane and λ is the wavelength of the radiation. Figure 2: Construction of the Bragg interference maximum For a regular set of point scatterers that make up a crystal there are a number of planes that can be passed through the scatterers. For each possible crystal structure there are a number of unique planes that can be defined. Since the cubic crystal is used in this experiment is limited in the number of scatterers we will only investigate three planes. These are shown in Figure 3. The planes in a crystal are identified by a crystallographic convention that numbers the plane according to the components of the normal vector to the plane. For example, [100] denotes a plane who s normal points along the horizontal (ˆx) axis. This notation is called the Miller index for the plane. The general Miller index can be written as [hkl] and for this case the separation between planes is given by a d hkl = (2) h2 + k 2 + l 2 For a complete set of Miller indices, see the appendix.

4 Figure 3: Three planes in a cubic crystal Procedure Place the Styrofoam crystal in the center of the translating table. Make sure that the white dots on both faces of the Styrofoam block are aligned. Place the microwave receiver and emitter on opposite sides of the table as shown in Figure 4. (You will have to use the extension arms on the tables to mount the pair.) Figure 4: Top view of the Bragg diffraction setup Adjust the positions of the transmitter and receiver so that they are equidistant form the crystal and that they are at the same height as the crystal center. Turn on the transmitter and receiver and remove the Styrofoam crystal. Turn the reciever to the lowest setting that does not peg the meter. As you carry out your measurements you may have to adjust the setting in order to measure the current in the receiver.

5 Place the Styrofoam crystal on the table and point the needle on the crystal so that the [100] plane is parallel to the microwave beam (see Figure 5, left panel). Figure 5: Orientations of the Styrofoam crystal for the Miller planes Figure 6: Top view of the Bragg diffraction setup Take a reading with the receiver at θ = 0 o. Then move both of the arms in 1 o steps (See Figure 6) taking readings as you go. There should be a peak in the received signal at θ = 0 o and a larger value. Make sure you move the arms up to θ = 60 o. (You should be able to predict the location of the peak from equations (1) and (2).) Plot your values as you go so you can be sure to catch the locations of the maxima. After finding the [100] peak, rotate the Styrofoam crystal in order to examine the peaks produced by the [110] and [210] planes. The appropriate orientations of the Styrofoam crystal are shown in Figure 4. For each orientation take data from 0 o to 60 o. After completion of the measurements, carefully disassemble the Styrofoam crystal and measure the distance between the spheres. This is the value of a shown in Figure 3 and used in equation 2.

6 1.3.5 Part 2: Analysis Plot the receiver data as a function of angle for each orientation of the Styrofoam crystal. Calculate the value of a for each peak in your plots. Compare your estimates of a with those you measured directly.

7 2 Appendix Figure 7: A diagram of the Miller indices and planes they represent

8 Figure 8: Typical equipment setup for Part 1 Figure 9: Typical equipment setup for Part 2