Bragg Diffraction Using Microwaves
Joshua Webster
Partners: Billy Day & Josh Kendrick
PHY 3802L
11/24/2013
Webster
Lab 4: Bragg Diffraction
Abstract
The following experiment was conducted to place an emphasis on the importance of the
Bragg equation and its implications in science. Microwave radiation was used to experimentally
determine the value for the Bragg reflection angle, which was determined to be 25.51ᵒ. This
angle was then used to calculate the plane spacing of a Cenco steel chrome ball lattice and was
determined to be 3.3 ± 0.1 cm. Using the direct measurement of the plane spacing, the theoretical
value for the Bragg reflection peak angle was determined to be approximately 26ᵒ. The crystal
plane spacing of a Cenco steel chrome ball lattice was determined to be 3.2 cm by direct
measurement.
1
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Lab 4: Bragg Diffraction
Table of Contents
Abstract ........................................................................................................................................... 1
Introduction ..................................................................................................................................... 3
Background ..................................................................................................................................... 4
Experimental Techniques................................................................................................................ 6
Diagrams and Images .................................................................................................................. 6
Data ................................................................................................................................................. 9
Analysis......................................................................................................................................... 12
Discussion ..................................................................................................................................... 15
Conclusion .................................................................................................................................... 16
Appendix ....................................................................................................................................... 17
References ..................................................................................................................................... 19
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Lab 4: Bragg Diffraction
Introduction
Diffraction is when a wave encounters an obstacle and continues to propagate.
Diffraction patterns can be observed with constructive and destructive interference. This report
deals with the case of Bragg diffraction. Bragg diffraction occurs when electromagnetic radiation
hits a crystal lattice barrier and scatters. The Bragg equation, or Bragg’s law, allows the
calculation of either: the wavelength of radiation, the lattice spacing, or the angle of incidence.
When two of the three variables are known the other can be easily obtained. This relation was
realized by Sir William Lawrence Bragg, who along with his father (Sir William Henry Bragg)
received the Nobel Prize in 1915 for their proposed equation, which confirmed the existence of
real particles at the atomic scale.1
Usually x-ray radiation is used in Bragg diffraction experiments that are intended to study
the structure of crystalline solids however, in the case of the experiment described in this report
microwave radiation was used. The reason x-rays are normally used is because the wavelength is
on the same order as the lattice spacing of the crystalline solids being examined, and this is a
general rule for Bragg diffraction. Instead of conducting experiments on the atomic scale, our
experiment utilizes a crystal-like lattice of steel balls inside a foam cube. Experimentation on this
larger scale justifies the use of microwave radiation as the source. A microwave transmitter is
used to project the microwave radiation signal, and a receiver is used to “catch” any signal that is
incident of the lattice cube at a specified angle. In this report, the term “plane” is referred to with
some number in front. The numbers in front denote the exact plane, and can be thought of as the
steps in the x, y, and z directions of the lattice of steel balls. Jumping straight into an example,
the 100 plane would be 1 step in the x direction, another step, etc. forming a straight line. The
110 plane would be one step in the x direction and one step in the y, forming the most basic
diagonal.
Experiments were conducted in this lab to show the importance and possible uses of
Bragg’s law. The following sections of this report consist of the Backround, Experimental
Techniques, Data, Analysis, Discussion, Conclusion, Appendix, and References.
1
(Bragg's Law, 2013)
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Lab 4: Bragg Diffraction
Background
The derivation of Bragg’s law is remarkably simple. It can be accomplished by simply
analyzing the geometry of the incident radiation on the lattice plane and by knowing a little
trigonometry. To begin the derivation we must introduce a diagram of the lattice plane with the
angles formed by the incident radiation shown.
Diagram 1: The above diagram shows the geometry that defines the Bragg equation.2
In Diagram 1, the incident radiation has a wavelength ( ), lattice spacing ( ), and an
angle of incidence that is equal to the angle of diffraction ( ). From the diagram, a relation is
evident:
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅̅̅
( )
( )
Also,
̅̅̅̅
( )
(
)
For constructive interference, the path difference is equal to an integer number of wavelengths,
.
̅̅̅̅
̅̅̅̅
̅̅̅̅̅̅
(
)
We can now combine equations 1.1 and 1.2 (since they are equivalent) to formulate the Bragg
equation:
( )
2
(Kimmel)
4
(
)
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Lab 4: Bragg Diffraction
The theoretical plane separation can be calculated using the following formula:
( )
√
S is the value that is measured directly on the cube with a ruler, X, Y and Z represent the plane
values (i.e. 100 plane is X = 1, Y = 0, Z = 0).
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Lab 4: Bragg Diffraction
Experimental Techniques
Diagrams and Images
Diagram 2: Shown above, the Bragg Reflection Cube Set, composed of five layers of 1.9 cm
thick polyethylene foam that is virtually invisible to microwaves. The layers have holes to
accommodate 125 steel chrome balls that act as scattering centers.3
Diagram 3: Shown above is the Bragg Reflection Cube laboratory setup with its various
components labeled. DMM stands for Digital Multi-Meter, which was not used in our setup. The
function generator generates a signal that the transmitter then broadcasts towards the cube, which
at specific angles is picked up by the receiver and made visible by the oscilloscope.
3
(Central Scientific Company, 1994)
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Lab 4: Bragg Diffraction
Diagram 4: Shown above is the receiver and the reflection cube. The grazing angle is the angle
at which the microwaves encounter the lattice and are reflected, and is the complement to the
incidence angle.
Diagram 5: The diagram above shows the “atomic” planes of the Bragg reflection cube.
For the initial setup of the experiment, the power supply of the modulator was plugged
into the wall outlet and then connected to the transmitter. The LED light on the transmitter was
lit indicating that the unit was functional. The intensity switch was changed from off to 30x,
which corresponds to the lowest level of amplification. The battery indicator light on the receiver
was lit indicating that the battery did not need replacement.
The foam Bragg reflection cube was already placed on the alignment disk, and the arrow
was pointing to 0ᵒ. It was checked that the alignment was proper. Since everything was basically
already setup, the transmitter was on the stationary arm about 50-60 cm from the turntable
(where it needed to be). The receiver was on the rotatable arm and at a distance of about 35 cm
from the turntable. Then, the transmitter and receiver were positioned in a straight line to be
directly facing one another, and the reflection cube was aligned for the 100 plane to be parallel
with the transmitter (the source of incident radiation). The polarization angles of both the
transmitter and receiver were adjusted to be the same (e.g. the horns had the same orientation).
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Lab 4: Bragg Diffraction
The variable sensitivity knob on the receiver was adjusted so that the meter reading
would be midscale. In the case of no deflection of the meter, the amplification was increased by
raising the intensity. Each meter reading must be multiplied by its respective intensity setting for
comparison to other readings.
The oscilloscope was connected to the output of the receiver via channel 1, and also to
the scope output of the modulator via channel 2. The trigger on the oscilloscope was set on
channel 2. The frequency, amplitude, and bias of the modulating signal were adjusted by the
controls on the modulator in an effort to optimize the signal being displayed on the oscilloscope.
These settings were very close to the “Typical Equipment Settings”4. The purpose of the
modulator is to provide a triangular wave output with a variable frequency (0.4 - 4 kHz),
amplitude (0 - 6 V peak-to-peak), and bias. It allows input of a signal (e.g. a microphone, music
device, etc.) that can then be transmitted to a speaker. Once the optimal settings have been
achieved, the modulator was not adjusted for the rest of the experiment as this would change the
parameters of the signal and could not easily be set back to proper settings.
After the setup was complete, we were then ready to begin recording data. The reflection
cube was rotated (by the turntable) one degree clockwise, and the receiver arm was rotated two
degrees clockwise. The grazing angle of the incident radiation beam, the meter readings (from
both the oscilloscope and the receiver), and the intensity setting were recorded. The oscilloscope
gave measurements in voltage, and the receiver gave measurements in current. The oscilloscope
was set to average over 128 scans so as to provide more stable results for the voltage. At each
successive angle the oscilloscope was set to reacquire the signal in order to discard any data that
might still be stored from the previous 128 scans. Real values for both the current and the voltage
are just the recorded values multiplied by the intensity setting. After the data for the first rotation
was recorded, the cube and receiver were rotated again by the same amount of degrees in the
same directions as before. Data was recorded from -10 to 55ᵒ for the 100 plane. One important
aspect to note is that the first voltage peak appeared at approximately 3 degrees as opposed to 0
(in theory), so this offset must be applied to the data. The propagation of this result will be
shown in the Analysis section. All of the data is recorded in Table 1.
Data was also recorded for the 110 plane, however it was completely off from theoretical
expectations and was completely unusable. This is most likely due to the increasing complexity
of the higher planes, in which a more reliable “crystal” would be mandatory.
At the end of the experiment we set the 100 plane back in place, and connected a cell
phone to the modulator in order to transmit the signal to the receiver which could be heard
through a speaker that was connected to it. The angle at which the signal being heard was
optimal indicated the maximum peak.
4
(PASCO Scientific, 1992)
8
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Lab 4: Bragg Diffraction
Data
Table 1: This table lists all the data collected during the laboratory experiment. The Real
Voltages and Real Currents are just their respective values (Voltage and Current) multiplied by
the intensity multiplier value, and the same goes for their uncertainties. All uncertainties listed
are associated with the device measurement uncertainties. Theoretically, a peak is to occur at 0
degrees however, the data recorded reflects a peak at approximately 3 degrees. Therefore, there
must be an offset of -3 degrees present in any calculations that follow (i.e. subtract 3 degrees
from the peak angle).
Grazing Voltage Δvoltage Current
Angle (⁰)
(mV)
(mV)
(mA)
-10
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
1000
238
296
314
294
272
264
266
282
290
272
226
174
120
94
640
324
336
17
1.8
0.8
0.76
0.56
1.64
8.9
13.3
2.32
0.28
4.8
140
30
5
2
1
2
2
2
2
2
1
2
2
1
1
2
50
8
4
1
0.1
0.08
0.04
0.08
0.04
0.2
0.3
0.16
0.04
0.24
50
0.4
0.24
0.4
0.42
0.4
0.39
0.38
0.38
0.4
0.4
0.38
0.26
0.18
0.09
0.06
0.4
0.15
0.08
0
0
0
0
0
0
0
0
0
0
0
0.03
Δcurrent
(mA)
Real
Intensity Voltage
(Multiplier) (mV)
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.05
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
1
30
30
30
30
30
30
30
30
30
30
30
30
30
30
3
3
1
1
1
1
1
1
1
1
1
1
1
1
1
9
1000
7140
8880
9420
8820
8160
7920
7980
8460
8700
8160
6780
5220
3600
2820
1920
972
336
17
1.8
0.8
0.76
0.56
1.64
8.9
13.3
2.32
0.28
4.8
140
∆Real
Voltage
(mV)
30
150
60
30
60
60
60
60
60
30
60
60
30
30
60
150
24
4
1
0.1
0.08
0.04
0.08
0.04
0.2
0.3
0.16
0.04
0.24
50
Real
Current
(mA)
0.4
7.2
12
12.6
12
11.7
11.4
11.4
12
12
11.4
7.8
5.4
2.7
1.8
1.2
0.45
0.08
0
0
0
0
0
0
0
0
0
0
0
0.03
∆Real
Current
(mA)
0.02
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.15
0.06
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
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Lab 4: Bragg Diffraction
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
420
210
30
160
810
830
180
16.8
13.6
16.4
5.44
1.2
0.32
0.32
0.32
0.32
0.4
0.36
0.36
0.36
0.48
0.48
0.32
0.48
2.08
0.8
0.72
2.64
0.8
0.32
0.48
0.32
8
10
2
20
20
20
10
0.4
0.2
0.4
0.12
0.08
0.001
0.001
0.001
0.001
0.04
0.04
0.04
0.04
0.08
0.08
0.001
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.12
0.04
0
0.02
0.38
0.4
0.04
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.02
0.01
0.01
0.01
0.08
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
10
420
210
30
160
810
830
180
16.8
13.6
16.4
5.44
1.2
0.32
0.32
0.32
0.32
0.4
0.36
0.36
0.36
0.48
0.48
0.32
0.48
2.08
0.8
0.72
2.64
0.8
0.32
0.48
0.32
8
10
2
20
20
20
10
0.4
0.2
0.4
0.12
0.08
0.001
0.001
0.001
0.001
0.04
0.04
0.04
0.04
0.08
0.08
0.001
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.12
0.04
0
0.02
0.38
0.4
0.04
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.02
0.01
0.01
0.01
0.08
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
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Lab 4: Bragg Diffraction
Graph 1: The graph below plots the data from Table 1. Grazing angles below 3 degrees have
been left out, because the voltages had unexpected peaks. It is important to note that the peak
angles must be offset by -3 degrees to reflect the initial peak at 3 degrees instead of 0.
Voltage vs. Grazing Angle
9000
8000
7000
Voltage (mV)
6000
5000
4000
Series 1
3000
2000
1000
0
3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55
Grazing Angle (ᵒ)
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Lab 4: Bragg Diffraction
Analysis
From Graph 1, we can see that the values start out at the highest peak, which is slightly
less than 9000 mV (8700 mV to be exact). This first peak value at 3 degrees will not be used in
the calculations to follow however, the result of this peak being at approximately 3 degrees will
be applied to the other peak angles. The voltage drops to around 0 at 12 degrees. The next peak
is approximately 24 degrees (with an offset of -3 the peak is approximately 21 degrees) with a
value of around 420 mV, and the final (but second largest) peak is approximately 29 degrees
(offset by -3 it is approximately 26 degrees) with a value of around 830 mV. These offsets will
be taken into account in the calculations that follow. The reason I say that the peaks appear
approximately at an angle is because we cannot actually know the definitive peak values given
the data recorded. We can however, find the uncertainty in the angles based off of the uncertainty
in the voltage values. We can do this by finding a quadratic fit for the four data points in Table 1
that make up the third peak. Mathematica was used to determine the fit. The results are shown
below, with the angles scaled (i.e. 1 represents the angle 28).
Since we have obtained the quadratic equation which describes our data points we can
now find the maximum (determined below), which is found to occur at a voltage of 901 mV with
an angle of 28.51 degrees. This would put the error in our measurement at
,
so ± 0.49 degrees.
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Lab 4: Bragg Diffraction
The separation for the 100 plane can be easily found just by measuring the distance from
one steel ball to the next. Our measured value for S was 3.2 cm. For the 100 plane, the
theoretical separation would be given by equation 2:
( )
√
√
The spacing can also be found experimentally using the data previously recorded and Bragg’s
law:
( )
13
(
)
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Lab 4: Bragg Diffraction
Solving for d,
( )
Using the estimated peak angle with
degrees:
and remembering our initial peak started at 3
(
)
Now finding the error propagation we can use equation A.1 from the Appendix:
√(
)
(
)
(
)
(
)
Which simplifies in our case to,
(
√(
Where
√(
)
)(
(
( )
√(
)
(
))
comes from the error of the sine function: |[
( )
) (
)
)
(
)
(
)]|.
The experimentally determined value for the plane spacing of 3.3 ± 0.1 cm is in good
agreement with our measured value. The measured value for the plane spacing can be used to
calculate the theoretical values for the angle in which the Bragg peak should occur. Solving
equation 1.3 for :
( )
(
)
(
14
(
)
)
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Lab 4: Bragg Diffraction
Discussion
The theoretical results in this experiment seem to be in good agreement with the values
that were obtained by measurement. There could be some slight error in the angle that could be
due to incident radiation that was not part of the experiment (noise), even though this was cut
down by evaluating oscilloscope readings over 128 scans. This could be corrected or at least
limited by conducting the experiment inside a kind of Faraday cage or possibly underground.
What other families of planes might you expect to show diffraction in a cubic crystal? Would you
expect the diffraction to be observable with this apparatus? Why?
Other families of planes that would be expected to show diffraction in a cubic crystal
would be the 111 plane or the 101 plane. These planes would be hard to observe with this
apparatus, because of the small crystal size.
Suppose you did not know beforehand the orientation of the “inter‐atomic planes” in the crystal.
How would this affect the complexity of the experiment? How would you go about locating the
planes?
The complexity of the experiment would definitely be increased. The crystal could be
oriented to produce maximum transmission which would indicate a 100 plane. Then proceeding
as was done in this experiment by taking data for various angles until an idea of the spacing
could be determined.
What limit is imposed on the wavelength by the Bragg reflection equation? How could you
increase the numbers of orders observed?
The limit imposed on the wavelength in the Bragg equation is the plane spacing. The
wavelength of the incident radiation must be the same order of magnitude as the plane spacing in
order for Bragg diffraction to occur. This is the reason why, in our experiment, microwave
radiation was used. To increase the numbers of orders observed either the wavelength of the
radiation or the size of the crystal would need to change. Also, the lattice centers must be highly
ordered to produce constructive and destructive interference.
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Lab 4: Bragg Diffraction
Conclusion
This experiment provided promising results that were in agreement with values obtained
by direct measurement and theory. The plane spacing was determined to be 3.2 cm by direct
measurement, and was calculated (using the experimentally determined value for the Bragg
reflection peak angle of 25.51ᵒ) to be 3.3 ± 0.1 cm. Using the direct measurement of the plane
spacing, the theoretical value for the Bragg reflection peak angle was determined to be
approximately 26ᵒ.
16
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Lab 4: Bragg Diffraction
Appendix
A.1 Formula for the propagation of errors:
Given a function, , with variables , , and . The uncertainty in is the square root of the sum
of the squares of the partial derivatives of with respect to each variable, and each partial
derivative is multiplied by the square of it’s uncertainty.
√(
)
(
)
(
A.2 Schematic for the microwave transmitter:
17
)
(
)
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Lab 4: Bragg Diffraction
A.3 Schematic for microwave receiver:
18
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Lab 4: Bragg Diffraction
References
Bragg's Law. (2013, September 23). Retrieved November 9, 2013, from Wikipedia:
http://en.wikipedia.org/wiki/Bragg%27s_law
Central Scientific Company. (1994, June). Bragg Reflection Cube Set No. 36860 Operating
Instructions. Retrieved November 10, 2013, from
http://www.physics.fsu.edu/courses/Fall13/phy3802L/exp3802/optics/cenco_bragg.pdf
Kimmel, R. A. (n.d.). Derivation of Bragg's Law. Retrieved November 10, 2013, from Penn
State: https://www.e-education.psu.edu/matse201/node/582
PASCO Scientific. (1992, May). Microwave Modulation Kit. Retrieved November 15, 2013,
from PASCO: http://www.physics.fsu.edu/courses/Fall13/phy3802L/exp3802/optics/01202960C.pdf
PASCO Scientific. (1999, April). Instruction Manual and Experiment Guide for the PASCO
Scientific Model WA-9314B. Retrieved November 10, 2013, from Microwave Optics:
http://www.physics.fsu.edu/courses/Fall13/phy3802L/exp3802/optics/012-04630F.pdf
19