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Bragg Diffraction Using Microwaves

Joshua Webster

Partners: Billy Day & Josh Kendrick

PHY 3802L

11/24/2013

Webster Lab 4: Bragg Diffraction

1

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.


2

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


3

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)


4

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)


5

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).


6

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)


7

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).


8

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)


9

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 Angle (⁰)

Voltage (mV)

Δvoltage (mV)

Current (mA)

Δcurrent (mA)

Intensity (Multiplier)

Real Voltage

(mV)

∆Real Voltage

(mV)

Real Current

(mA)

∆Real Current

(mA)

-10 1000 30 0.4 0.02 1 1000 30 0.4 0.02

-5 238 5 0.24 0.01 30 7140 150 7.2 0.3

-4 296 2 0.4 0.01 30 8880 60 12 0.3

-3 314 1 0.42 0.01 30 9420 30 12.6 0.3

-2 294 2 0.4 0.01 30 8820 60 12 0.3

-1 272 2 0.39 0.01 30 8160 60 11.7 0.3

0 264 2 0.38 0.01 30 7920 60 11.4 0.3

1 266 2 0.38 0.01 30 7980 60 11.4 0.3

2 282 2 0.4 0.01 30 8460 60 12 0.3

3 290 1 0.4 0.01 30 8700 30 12 0.3

4 272 2 0.38 0.01 30 8160 60 11.4 0.3

5 226 2 0.26 0.01 30 6780 60 7.8 0.3

6 174 1 0.18 0.01 30 5220 30 5.4 0.3

7 120 1 0.09 0.01 30 3600 30 2.7 0.3

8 94 2 0.06 0.01 30 2820 60 1.8 0.3

9 640 50 0.4 0.05 3 1920 150 1.2 0.15

10 324 8 0.15 0.02 3 972 24 0.45 0.06

11 336 4 0.08 0.01 1 336 4 0.08 0.01

12 17 1 0 0.01 1 17 1 0 0.01

13 1.8 0.1 0 0.01 1 1.8 0.1 0 0.01

14 0.8 0.08 0 0.01 1 0.8 0.08 0 0.01

15 0.76 0.04 0 0.01 1 0.76 0.04 0 0.01

16 0.56 0.08 0 0.01 1 0.56 0.08 0 0.01

17 1.64 0.04 0 0.01 1 1.64 0.04 0 0.01

18 8.9 0.2 0 0.01 1 8.9 0.2 0 0.01

19 13.3 0.3 0 0.01 1 13.3 0.3 0 0.01

20 2.32 0.16 0 0.01 1 2.32 0.16 0 0.01

21 0.28 0.04 0 0.01 1 0.28 0.04 0 0.01

22 4.8 0.24 0 0.01 1 4.8 0.24 0 0.01

23 140 50 0.03 0.01 1 140 50 0.03 0.01


10

24 420 8 0.12 0.02 1 420 8 0.12 0.02

25 210 10 0.04 0.01 1 210 10 0.04 0.01

26 30 2 0 0.01 1 30 2 0 0.01

27 160 20 0.02 0.01 1 160 20 0.02 0.01

28 810 20 0.38 0.08 1 810 20 0.38 0.08

29 830 20 0.4 0.02 1 830 20 0.4 0.02

30 180 10 0.04 0.01 1 180 10 0.04 0.01

31 16.8 0.4 0 0.01 1 16.8 0.4 0 0.01

32 13.6 0.2 0 0.01 1 13.6 0.2 0 0.01

33 16.4 0.4 0 0.01 1 16.4 0.4 0 0.01

34 5.44 0.12 0 0.01 1 5.44 0.12 0 0.01

35 1.2 0.08 0 0.01 1 1.2 0.08 0 0.01

36 0.32 0.001 0 0.01 1 0.32 0.001 0 0.01

37 0.32 0.001 0 0.01 1 0.32 0.001 0 0.01

38 0.32 0.001 0 0.01 1 0.32 0.001 0 0.01

39 0.32 0.001 0 0.01 1 0.32 0.001 0 0.01

40 0.4 0.04 0 0.01 1 0.4 0.04 0 0.01

41 0.36 0.04 0 0.01 1 0.36 0.04 0 0.01

42 0.36 0.04 0 0.01 1 0.36 0.04 0 0.01

43 0.36 0.04 0 0.01 1 0.36 0.04 0 0.01

44 0.48 0.08 0 0.01 1 0.48 0.08 0 0.01

45 0.48 0.08 0 0.01 1 0.48 0.08 0 0.01

46 0.32 0.001 0 0.01 1 0.32 0.001 0 0.01

47 0.48 0.08 0 0.01 1 0.48 0.08 0 0.01

48 2.08 0.08 0 0.01 1 2.08 0.08 0 0.01

49 0.8 0.08 0 0.01 1 0.8 0.08 0 0.01

50 0.72 0.08 0 0.01 1 0.72 0.08 0 0.01

51 2.64 0.08 0 0.01 1 2.64 0.08 0 0.01

52 0.8 0.08 0 0.01 1 0.8 0.08 0 0.01

53 0.32 0.08 0 0.01 1 0.32 0.08 0 0.01

54 0.48 0.08 0 0.01 1 0.48 0.08 0 0.01

55 0.32 0.08 0 0.01 1 0.32 0.08 0 0.01


11

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.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

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

Vo

ltag

e (

mV

)

Grazing Angle (ᵒ)

Voltage vs. Grazing Angle

Series 1


12

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.


13

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:

( ) ( )


14

Solving for d,

( )

Using the estimated peak angle with and remembering our initial peak started at 3

degrees:

( )

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 :

( ) ( )

(

) (

)


15

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.


16

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.


17

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:


18

A.3 Schematic for microwave receiver:


19

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/012-

02960C.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

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Bragg Diffraction Using Microwaves - · PDF file24.11.2013 · Bragg Diffraction Using Microwaves Joshua Webster Partners: Billy Day & Josh Kendrick PHY 3802L 11/24/2013