Microwave Bragg Diffraction
Goals: In this experiment, you will try to determine the structure of several macroscopic
crystals using Bragg diffraction of microwaves. You will need to understand the basic ideas
involved in diffraction of waves by a crystal and the microwave equipment used.
Introduction:
To the first approximation, in a crystal, planes of atoms are assumed to act like mirrors
giving specular (mirror like) reflection. When the reflections from different parallel planes
interfere constructively, the amplitude of the outgoing wave increase. This is similar to thin film
interference. The equation for the conditions required for maxima in the outgoing waves is
called the Bragg equation and given
by:
(1)
Where the n is an integer, is the wavelength of the waves, d is the separation of the atomic
planes, and is the angle between direction of travel of the wave and the atomic plane (NOT the
normal to the plane). See Figure 1.
Figure 1 Derivation of the Bragg equation. The incoming plane waves are in
phase, but the outgoing reflected waves are not in phase because of the two path
lengths differ by a distance 2d sin , where d is the separation of the atomic
planes. Constructive interference occurs if this path lengths differs by an integer
number of wavelengths, . This yields the Bragg equation n = 2 d sin . How
could this equation be rewritten in terms of the angle 1 between the incident and
reflected beams? Note that 1 is easily measured in the lab and does not involve
the crystal..
For further details about Bragg diffraction, please see most introductory solid state text
such as Introduction to Solid State Physics 4 by Charles Kittel (1971, John Wiley and Sons, New
York, QC176 .K5 )
Equipment Procedure for Microwave Bragg Diffraction
To Start system:
1.
Match transmitter and receiver horn heights
2.
Point horns at each other ( about 30 cm apart)
3.
On CENCO 3 cm microwave transmitter
INTERNAL modulation
Klystron voltage ~ 2:00 o’clock
4.
On 415D SWR Meter
Gain (outer dial ) and gain vernier ( inner dial) arrows straight up
Range SWR DB (outer dial) 40, Expand (inner dial) Normal
Input selector XTAL 200 6
Power AC
5.
Maximize output of the transmitter by adjusting klystron voltage to peak SWR meter
reading. Adjust separation distance as needed.
6.
Allow a bit of warm-up time.
7.
Note higher SWR scale is more sensitive
8.
The SWR meter measures power, P, in terms of the meter reading of attenuation, A,
from a standard power P0. You can set the standard power, P0, with the gain knobs to an
arbitrary value on the scale.
or
. Thus the power
associated a meter reading of 60 dB is 1/10 the power associated with a reading of 50
dB, Note dB is decibel.
One way to measure the wavelength
1.
Point transmitter horn at the center of reflecting plane. This creates a standing wave field.
2.
Point receiver horn at reflecting plane.
3.
Move the receiver nearer or further from the plane to maximize SWR reading
4.
With the receiver at a maximum, adjust position of transmitter horn to maximize SWR reading
5.
With the receiver at a maximum, measure distance from receiver to reflecting plane.
6.
Move the receiver forward or backward ( relative to plane) and find additional maxima.
Measure these positions. ( more positions=better data = better wavelength)
7.
While moving the receiver do not change its angle relative to the plane
8.
Observe the effect of your body and hand movements on readings
To measure Bragg diffraction
1.
Point transmitter horn at the center of the turntable. You may wish to turn up the gain to near
maximum for these measurements.
2.
Place receiver horn on pivot arm pointing directly into the transmitter. The turntable should be
approximately centered between the horns
3.
With no “crystal” in place check tuning and determine the zero angle.
4.
With no “crystal in place measure the “background” microwave signal as the receiver and pivot
arm are rotated about the center of the turntable. (Adjust SWR meter scale as needed.)
5.
Install “crystal”
6.
With the receiver at zero degrees, rotate the crystal 360° and measure and record SWR readings.
7.
With the receiver horn at other angles (every 5° or 10°), rotate the crystal 360° and measure and
record SWR readings.
8.
With the crystal and receiver set at angles where there is a strong signal, carefully rotate the
crystal to maximize the signal, then rotate the receiver’s horn to maximize the signal. Iterate
this process as needed to find the optimum angles.
9.
Once the optimum crystal and horn angles are found, you can rotate the horn about the crystal
and find the angle at which the signal drops in half. This gives you a measure of the angular
resolution associated with the receiver’s angle. A similar method can be used to determine the
angular resolution associated with the crystal’s angle.
10.
Correct measurements for background as needed.
11.
Turn off system when done!
OR
USE the computer to collect the preliminary data.
a.
The program’s name is BRAGG.
b.
c.
d.
You need to have the 5 volt power supply on. This puts a voltage across the potentiometer
attached to the rotating horn arm, and produces a signal related to angular position of the horn.
Straight through is considered 0°
After starting the BRAGG program, you need to calibrate the angular position. Start this process
by pressing the “Z” key.
e.
To collect data, press”C”.
i.
Slowly spin the crystal. Use 1/4 revolution per second as a starting point. Use the
response time of the system to optimize this angular frequency.
ii.
Move the horn arm very slowly, approximately 20° to 160° and back in 40 seconds
(Count)
f.
To plot the data press “P”. Normal scaling is 0° to 180°. To get the negative angles rescale the
x axis.
This works reasonably well to get approximate positions of diffraction peaks. Final
determination should be by iterative manual tuning of crystal angle and horn angle.
g.
z.
Note the angular position near straight through are dominated by direct microwaves and might
as well be skipped.
Analysis Notes for Microwave Bragg Diffraction
To help make sense of what could prove to be a very large and complex data set, it might be useful to
calculate the expected results. If one example took a measurement every 5°( 72 angles) for he receiver horn and
for each of these every 5° of the crystal, one would take 722 = 5184 SWR readings. It is suggested that you
figure out what is important before you start measuring.
Make reasonable guesses about the wavelength, , of the microwaves ( a few centimeters) and
separation of the scattering centers (several centimeters, but probably not the same as the wavelength). Assume
a square crystal structure. With these assumptions find the various angles for which the Bragg condition,
, is met. Note that d is distance between scattering planes and this is not necessarily the
distance between closest scattering centers. Once you know what should happen, one can see what part of the
huge potential data set is actually useful.
1.
At what range(s) of angles ( both of the receiver and the crystal) will the Bragg condition most likely be
met?
2.
Is the angle of the receiver horn about the center of the turntable, important and crucial to the analysis?
3.
Is the angle of the crystal important and crucial to the analysis?
4.
Are there any symmetries in the expected results that could be useful in the analysis?
5.
How would the results change is the crystal is rectangular rather than square in nature?
Questions:
1.
What does the 415D SWR meter measure?
2.
What does SWR stand for?
3.
What is a dB or decibel?
4.
Is the reading of the SWR meter an absolute or relative measurement?
5.
Is the dB scale linear?
6.
How is what is measured by the SWR meter related to the intensity of the microwaves?
7.
How is what is measured by the SWR meter related to the amplitude of the electric field of the
microwaves?
8.
9.
How does a klystron work?
Why is the received signal larger if the klystron is modulated.
10.
Describe the standing wave microwave field used to measure the wavelength of the microwaves and
how this field is created?
In the standing wave field, what is the relationship between the distance between maxima and the
wavelength of the microwaves?
In the section title “To Start System” step #5 it is suggested to, “Adjust separation distance as needed.”
Why?
What is the energy (in eV) of a single microwave photon used in this experiment?
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
In a crystal with scattering centers forming a cube with a side of length, d, in two dimensions what
scattering planes exist?
For Bragg diffraction associated with different sets of scattering planes, what determines the intensity?
How will you determine the angular width of the diffraction pattern? Is this in terms of the angle of the
crystal orientation or the receiver?
For the Bragg condition,
, can one safely and wisely set d to the distance between the
first and third ( and fifth and ...) planes?
What is the angle of divergence of the microwaves from the transmitter? If you remove the ”horn” from
the transmitter, how does this change?
What is the spacial resolution of the detector? Does removing the horn from the transmitter change
this? Does it change the intensity of the signal received?
Do microwaves go above, below, or around the crystal and get detected?