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APPLICATIONS OF X-RAY DIFFRACTION
by
Cameron Jorgensen
A senior thesis submitted to the faculty of
Brigham Young University - Idaho
in partial fulfillment of the requirements for the degree of
Bachelor of Science
Department of Physics
Brigham Young University - Idaho
December 2018
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Copyright c© 2018 Cameron Jorgensen
All Rights Reserved
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BRIGHAM YOUNG UNIVERSITY - IDAHO
DEPARTMENT APPROVAL
of a senior thesis submitted by
Cameron Jorgensen
This thesis has been reviewed by the research committee, senior thesis coor-dinator, and department chair and has been found to be satisfactory.
Date Richard Datwyler, Advisor
Date David Oliphant, Senior Thesis Coordinator
Date Stephen McNeil, Committee Member
Date Todd Lines, Department Chair
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ABSTRACT
APPLICATIONS OF X-RAY DIFFRACTION
Cameron Jorgensen
Department of Physics
Bachelor of Science
Powder x-ray diffraction is method of bombarding a mineral sample with x-
rays and at certain angles the x-rays will be diffracted into a detector. The
attempt in this experiment was to see if there was a chance to use an alternate
method than the diffraction patterns to identify a mineral sample.The alter-
nate method to normal diffraction patterns was using Permutation Entropy.
Further development showed that the PE method would be more appropriately
associated with pure and impure samples, not mineral identification. The data
suggested it was inconclusive using PE method. The results suggested the PE
needs to be tested further to decide the effectiveness between pure and impure
mineral samples.
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ACKNOWLEDGMENTS
I would like to thank Brigham Young University - Idaho for being available
for me to do my research here. I would also like to thanks many faculty
for their endless help, including: Richard Datwyler, Quinn Norris, Stephen
McNeil, David Oliphant, Lance Nelson to name a few that were essential.
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Contents
Table of Contents xi
List of Figures xiii
1 Introduction of X-ray Diffraction 11.1 Diffraction for minerals . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Applications of Diffraction . . . . . . . . . . . . . . . . . . . . . . . . 31.3 This Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Introduction of Permutation Entropy 52.1 Introduce Permutation Entropy . . . . . . . . . . . . . . . . . . . . . 52.2 Applications of PE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Methods 93.1 Philips X-ray X’pert Diffraction Machine . . . . . . . . . . . . . . . . 93.2 Python Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Future Method Changes . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Results 154.1 Individually Useless . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Proper Results Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Original Hypothesis was Incomplete . . . . . . . . . . . . . . . . . . . 17
Bibliography 19
xi
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List of Figures
1.1 Examples of different planes in the same set of atoms . . . . . . . . . 21.2 A tray with powder in it. . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Diffraction for quartz . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 A brief example of the PE for a small data set . . . . . . . . . . . . 6
3.1 Arm aperture with circular motion . . . . . . . . . . . . . . . . . . . 93.2 The Philips X’Pert X-ray Diffraction Machine . . . . . . . . . . . . . 103.3 Where the sample goes . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Diffraction scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1 Diffraction scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 different scans comparison . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Curve comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
xiii
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Chapter 1
Introduction of X-ray Diffraction
X-ray Diffraction (XRD) is when x-rays get diffracted by the spacing between atoms
that create a crystal. Because x-rays wavelength are on the same magnitude of the
spacing between the atoms they are the photon of choice. This can be done to any
substance. Usually the focus is on identification of mineral samples. At specific angles
the x-rays will diffract and when those angles are noted the combination of the angles
is what is used to identify the mineral. For example, in a pure copper sample, there
will be diffraction angles near 43 and 50 degrees. At any other angles this would not
be a copper scan, it would be a different mineral.
1.1 Diffraction for minerals
Diffraction is a principle where waves will bend around physical obstacles. This can be
used on any scale whether it is on the scale of kilometers or angstroms. This research
is on the scale of angstroms. If one sends x-rays at mineral sample, the x-rays can
diffract between the atoms of the material. Specifically the distance between atoms
can result in a specific angle that gets diffracted out. This concept is described with
1
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2 Chapter 1 Introduction of X-ray Diffraction
Braggs Law.
Braggs Law was created by Lawrence and William Bragg in the early 1900s,
winning the Nobel prize in 1913 with the discovery of the equation. They discovered
this ability to see x-rays bend through materials at specific angles. The equation for
diffraction
2 d sin(θ) = n λ
is what the basis of mineral identification is based on. where d is this and lambda is
this...etc. Over the many years since then, known materials have been used with this
XRD method to help determine what material is there based on the angles found.
A crystal is any atomic structure that repeats itself. The idea of a crystal and all
of its different atomic structures are more of a solid state physics lesson, but I would
like to briefly introduce a few of those topics here. The theta mentioned previously
will be based upon which plane of atoms the equation wants to deal with [1]. This is
best done with an image see Figure 1.1.
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θFigure 1.1 Examples of different planes in the same set of atoms
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1.2 Applications of Diffraction 3
1.2 Applications of Diffraction
While not a commonly used method in today’s world for identification, XRD had
applications for many experiments in the early twentieth century. After the discovery
of Bragg’s Law in 1913, the very next year Max Von Laue used the law to discover that
x-rays diffract inside a crystal. This work went on to spur the race for identifying
the mineral based on its diffraction. This database of all the different diffraction
patterns are now incredibly large and have an endless number of catalogs for different
materials.
1.3 This Experiment
Initially the goal of this experiment was to see if there were any abnormal patterns.
Whether this meant identifying specific minerals or just being able to distinguish
from a pure sample to an impure sample, I was not sure but wanted to find an
additional application of XRD. Originally it was thought that I would just be looking
at different diffraction patterns and seeing if there was a difference between multiple
scans of two different coppers. I noticed that there was nothing special about these
scans, I attempted to see the percent difference between different scans of the same
sample. This was a new look at the data. At this point there was not enough data
to work with, so additional scans were done with soil and pure quartz.
Figure 1.2 A tray with powder in it.
Sample preparation was an impor-
tant focus of this experiment. The cop-
per had two sides, a rough side and a
smooth side, so it remained a solid crys-
tal.The rough side of the copper was es-
sentially sandblasted to be made rough.
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4 Chapter 1 Introduction of X-ray Diffraction
The quartz and soil were both able to
grind down to a powder. When a sample
can be ground into a powder as opposed
to remaining a solid crystal, this allows every possible orientation of the crystal to
be scanned. When every orientation is there the scan will show where the diffracted
x-rays come in large concentration bunches. These will be shown as peaks, because
there will be a lot of x-rays at that angle. As you can see in the graph for quartz.
Figure 1.3 Diffraction for quartz
It was not just a fluke that they
made it into the detector. If the powder
had every possible orientations then this
would be an ideal sample. If the sam-
ple cannot be ground into powder then
it can still be scanned it just needs to be
rotated so there are a maximum possible
orientations.
Because the copper had known peaks
already [2]. I decided to focus the scans
on the two known peaks separately. Quartz and soil were scanned on a full range of
angles to see where the peaks are in general. Dirt and quartz were both scanned from
0 to 80 degrees. Whereas the copper was scanned from 40 to 42 and 49 to 51, the
ranges of the two known peaks [2].
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Chapter 2
Introduction of Permutation
Entropy
2.1 Introduce Permutation Entropy
Permutation Entropy (PE) is the analysis of a time series to see how much ’random-
ness’ is in that series [3]. As the name suggests it is the order or organization of the
entropy of a system. If that organization turns out to be predictable, there can be
value in using that method for analysis.
This was done with reference to the angle in which the x-ray detector was located.
At each angle step the arm detector counted how many x-rays were diffracted. When
the scan finished there is a long list of counts that we applied PE to. PE is calculated
by taking an ordered data set and dividing it in groups called symbols of length n
and looking at the probabilities of the different possible orderings of the values within
the symbol. For symbols of length 3, the values can be ordered in one of six ways:
Low (L), Middle (M), High (H); L,H,M; M,L,H; M,H,L; H,L,M; H,M,L; By counting
the number of occurrences of each ordering, the probability of each ordering can be
5
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6 Chapter 2 Introduction of Permutation Entropy
Figure 2.1 A brief example of the PE for a small data set
determined. The PE is calculated as the Shannon entropy of the orderings. If the
data set is completely random, each ordering should have the same probability. If the
sample is monotonically increasing, like a natural log curve, (completely non-random)
than L,M,H should have 100% probability. Using permutation entropy, it should be
possible to separate completely random data from data containing patterns. This
experiment was n = 3. A more clear example is shown in Figure 2.1
The equations to calculate PE are
Hn = −n!∑j=1
p′j log2 (p′j)
and
hn = − 1
1 − n
n!∑j=1
p′j log2 (p′j)
Hn is the PE while hn is the PE per symbol. The symbol is the term used because
it is unit-less. Because PE for data set could be a wide range of numbers, using the
PE per symbol helps create a scale that is more comparable from one data set to the
next. When using a set of 3 data points, like done in this experiment, the hn of a
completely random sample is 1.292. If the sample is predictable then the hn comes
out to be 0 [5].
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2.2 Applications of PE 7
2.2 Applications of PE
In other work there have been specific applications of PE for data sets. Some specific
examples include the van-der-Pol oscillator, the Lorentz system and a logistic system.
These applications were used to see that other n values were useful than just the 3
that was used in this experiment. In each of the different applications it was just a
measurement of the complexity of the situations. [?]
This experiment was to see if using PE was a more convenient or efficient way of
determining whether the sample scanned was a mineral sample with lots of different
mineral types or if it was a pure sample of one mineral type.
The hypothesis was that a pure sample would be completely predictable thus
giving an hn near 0 and that the hn of impure substances would be 1.292.
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8 Chapter 2 Introduction of Permutation Entropy
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Chapter 3
Methods
3.1 Philips X-ray X’pert Diffraction Machine
The parameters of this experiment are extensive. The Philips machine is a precise
and accurate diffraction machine.
Figure 3.1 Arm aperture with circularmotion
It has an aperture receiver that will
rotate around a sample in a semicircle,
like a planet goes around the sun, detect-
ing the x-rays that are being diffracted
by the sample. While it rotates around
the sample there is an emitter positioned
to the side. As seen in 3.1.
This emitter is stationary and posi-
tioned on the left and the receiver is on
the right in 3.1. There are several parameters which can be adjusted to try and
make the scan as useful as possible. The angle in which the arm starts and stops,
the amount of angle the arm will rotate around the sample at each interval (this is
9
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10 Chapter 3 Methods
usually pretty small), the time in which the arm will stop and collect x-rays at each
interval, are several parameters that are most important for this work. For the copper
samples, because of the known angles for the diffraction from previous work [2] Those
two scans were scanned each with ranges of 42 - 44 degrees and 49 - 51 degrees with
steps in the angle of 0.001 degrees. The time at each angle was 4 seconds. With 4
seconds at each angle step and over a wide range of angles these scans can take a
long time! The quartz and soil were scanned from 12 - 65 degrees with steps of 0.1
degrees for 5 seconds at each step.
Figure 3.2 The Philips X’Pert X-rayDiffraction Machine
To have an ideal scan, the crystals in
the sample need to be in as many ori-
entations as possible in order to see all
of the possible angles of diffraction for
the sample. Usually this is done by tak-
ing the sample, grinding it into a pow-
der and then scanning the powder while
the powder rotates and is also tilted for
all possible orientations. Notice in Fig-
ure 3.3a and Figure 3.3b where the sam-
ple would be. This is only helpful in the
event one can spare grinding the sample
into a powder. In this case for the copper
it could not be ground into a powder and
was left as an entire solid crystal sample.
This still allowed for diffraction angles to be found as the sample spun and was bom-
barded by x-rays.
The different sample types would not be comparable to each other. Taking the
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3.2 Python Code 11
(a) An example of the tray with-out the powder
(b) This is where the tray thatholds the sample would go
Figure 3.3 Where the sample goes
quartz and comparing it to the soil sample was not part of our hypothesis. The bulk
of the work was done between multiple scans of the same sample. Two scans were
performed on the copper at the theta range where the peaks were known. Angles
around 42 and 49 degrees each got three scans for comparison. To be able to use
the scans and compare them they needed to have the exact same step size, time and
range for each of the scans in order for their arrays in python to match in length.
Three scans of a pure quartz sample were taken, with all of its peaks, to use as a
reference against the single peak scans of the copper.This was part of the hypothesis
that there was a difference in PE from a single peak to lots of different peaks.
3.2 Python Code
Initially there was a halt in the work in trying to figure out how to input the infor-
mation from the scans of the XRD to a python code of another computer. There are
functions in python called parent and child tags. These tags are essential in help-
ing extract the data from the file. In doing so the XML file had the list of counts.
The code for this experiment was written specifically for this experiment and could
work with any set of numbers as long as the numbers were a one dimensional array
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12 Chapter 3 Methods
of the counts for the x-rays. It was implemented that the parameters of each scan
was automatically found for each individual scan. This is one reason why scans from
different types of materials would not be beneficial for analysis. It is an important
piece of information to know that the file is formatted as an XRDML file. Which is
just specific for this machine. Its not the common thing to run into. But in the case
that it is, it still works similar to XML it just needs XRDML in all the child tags.
The array of counts were inside the XML file. Once those were extracted they
were placed in a plot function to create a graph of the diffraction. This was done to
ensure that the information extracted was correct.
The information was not conclusive by itself. Percent difference needed to be
done to show that the information provided could be interpreted as useful. A per-
cent difference was used to determine the significance of the scans comparisons with
each other. For each of the copper scans, analysis was focused in those scans for
comparison.
It was advised that the diffraction patterns may have had too much background
noise. To help with the noise an average was taken. In a more detailed explanation
the data was smoothed by taking the data point, the two successive and two previous
and replacing the original data point with that average of those five numbers. For
example, if the set of data points was [2,3,1,5,9] I would take their average (4) and
replaced it where 1 is at. Then I would move to location of 5 and take an average for
that data point and so on and so forth.
Once this was done it was plotted on a graph again, but as a percent difference.
The two data sets were the different scans of the copper. Those percentage differences
are shown here
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3.3 Future Method Changes 13
(a) The percent differ-ence diffraction patternnear 42 degrees for therough side of copper
(b) The percent differ-ence diffraction patternnear 42 degrees for thesmooth side of copper
(c) The percent differ-ence diffraction patternnear 49 degrees for thesmooth side of copper
Figure 3.4 Diffraction scans
3.3 Future Method Changes
The machine was not faulty during use, but in the event the machine is not operating
at maximum efficiency another machine could be beneficial. This machine was handed
down to the school after a decade of use [4]. It should be noted that the machine would
give extremely high counts and sometimes low counts. It seemed to be temperamental
in that way, and could not control that variable.
Filtering is a mathematical approach to be able to change the data for a particular
curve or data set.It might have been noticed that there was a method for smoothing
out the noise. In its basic form this is a form of filtering. It could be suggested
that for future applications of this work that a different type of filtering would be
beneficial.
Understanding that this is an idea of application in its most fundamental form
and a lot of unknowns that have yet to be explored, mass spectroscopy is an under-
standable and alternate method in which the PE could be applied instead of diffrac-
tion patterns. Mass spectroscopy is the tool used to identify materials, specifically
their atomic masses. Because the mass spectroscopy could be in a similar pattern to
diffraction that could be used also.
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14 Chapter 3 Methods
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Chapter 4
Results
4.1 Individually Useless
Because the data for the diffraction sets came out very clear the scans were a success,
They are shown in 4.1a , 4.1b, and 4.1c.
Comparing the scans to themselves to make sure of their accuracy showed that the
highest percentage difference from one to the next was about 5% . This is sufficient
for use. Now that the data has been confirmed to be useful we applied PE. It is
a good idea to look at some of the scans of the copper compared to one another.
Something that is interesting that I did not focus on with this research is the trend
(a) The diffraction pat-tern peak near 42 degrees
(b) The diffraction pat-tern of copper near 49
(c) Diffraction patternfor the soil sample
Figure 4.1 Diffraction scans
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16 Chapter 4 Results
(a) six different scans of smoothand rough copper at the same an-gle of near 43
(b) Six scans of smooth andrough copper at the same angle ofnear 50
Figure 4.2 different scans comparison
that the brown, red, and orange graphs have compared to the blue, green, and purple.
For some reason those curves all rise earlier than the others, or in other words their
centers are at different locations. The brown red and orange are smooth sides and
blue green and purple are all rough sides.
After the percentage difference was evaluated, using the percent difference in the
PE method was done. Since we wanted to see the applications of XRD it was used
on the original scans to see what was produced. According to our results, the hn
of quartz, copper of 43, copper of 50, and soil were: 1.291, 1.291, 1.291, and 1.262
respectively. A perfectly random sample would have an hn of 1.292.
Recall that the lower the value of PE the more predictable it is, the soil sample
had the lowest value of PE. That means it had the most pattern in its graph trends.
It may have to do with the fact that the soil had less background x-rays or that the
pure samples used too much time in their scans. But if soil can be more predictable
it may be worth investigating multiple soil types.
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4.2 Proper Results Filtering 17
(a) A comparison of nine curvesof copper near 43
(b) A comparison of nine curvesof copper near 50
Figure 4.3 Curve comparison
4.2 Proper Results Filtering
After some filtering, the data was smoothed out, but sadly this did not change any of
the results. Some other applications of smoothing could be extended to the percent
difference that was found. The reason this percentage data would be beneficial is
because the curve would not come back to the baseline.
Other attempts of filtering would be encouraged at this point as well. Other forms
of filtering are available for getting rid of the general noise but filtering methods that
could do it would be Fourier Transform filter or a Gaussian filter. It was decided that
the background noise was over bearing to the PE applications.
4.3 Original Hypothesis was Incomplete
There could be several reasons as to why the soil sample was more predictable than
the rest of the diffraction scans. At this point it would be only speculation. I think
that because the copper was performed at such a small range and back ground had
much more time to pick up stray x-rays created a lot more background radiation.
This affected the baseline for our data sets.
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18 Chapter 4 Results
Another idea that could cause this is due to its diverse nature. Soil has a lot of
different minerals in it, quartz and copper do not. They are only one mineral. This
impure combination could be convenient for this particular set of data. Notice in
Figure 4.3a there is on spot in particular that all the scans almost overlap on each
other flawlessly. That would be worth further investigation.
Or it was just a fluke. Since the data was pretty consistently 1.291.
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Bibliography
[1] Glazer, A M, Crystallography., Oxford University Press, 2007.
[2] Wyckoff, RWG, Crystal Structures., American Minerlogist, 78, 1104-1107, 1963.
[3] Bandt, C. Pompe, B. Permutation Entropy: A Natural Complexity Measure for
Time Series, Rev. Lett, 88, 2002.
[4] Johnson, Jacob, Qualifications of Phillips XPert MPD Diffractometer for Appli-
cations in Xray Diffraction and Reflection., BYUI Catalog, 2.
[5] Riedl, M., Mller, A. Wessel, N, Practical considerations of permutation entropy:
A tutorial review., Eur. Phys. J. Spec Top, 249-262, 2013.
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