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

Copyright c© 2018 Cameron Jorgensen

All Rights Reserved

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

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.

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.

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

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

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

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

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.

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].

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

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].

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.

8 Chapter 2 Introduction of Permutation Entropy

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

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

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

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

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.

14 Chapter 3 Methods

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

15

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.

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.

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.

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.

19