Results of computational simulation to characterize

optimal structures for AlIr and MgZn quasicrystals

Justin Richmond-Decker

Engineering Physics 2012, Cornell University

Thesis for graduation with honors

Prepared for: Department of Applied & Engineering Physics

Completed on: May 18, 2012

For graduation on: May 27, 2012

Project Advisor: Christopher L. Henley, LASSP

Faculty Advisor: Richard V. E. Lovelace, A&EP

i

TABLE OF CONTENTS

List of Figures iii

List of Tables iii

I. Introduction and Background 1

A. Quasicrystals 1

B. Context for Project 1

II. Magnesium-Zinc Quasicrystal 2

A. General Information 2

1. Physical Characteristics 2

2. Tiling Components 2

3. Component Energies 6

B. Finding a Tile Hamiltonian 6

C. Tie-Line 9

D. Computer Implementation of MgZn 10

1. Computer Used 10

2. Files and Programs Used 10

a. MgZn Atom Files 10

b. Tiling Files 11

c. Relaxation 11

d. Displaying the Structure 12

e. Tiling Database 12

3. Acquiring Data for the Fit 13

4. Analyzing the Data 13

E. Fit Results 14

1. Standard Fit 14

2. Cutoff Fit 15

F. Redecoration of Lozenge and Boat Tiles 16

1. The Purpose of Redecoration 16

2. Lozenge Redecoration 17

3. Boat Redecoration 21

G. Optimal RT Tilings and the MgZn Supertiling 23

ii

1. Optimal MgZn Tilings 23

2. Supertiling 24

a. Definition and Characteristics 24

b. Determining Supertilability 25

c. The Simplest Supertile Approximant 26

d. Supervertices 27

III. Aluminum-Iridium 28

A. General Information 28

1. AlIr As a Simplified Model 28

2. Structural Characteristics 28

3. Pair Interactions 29

B. Computer Implementation of AlIr 30

1. Similarities to MgZn 30

2. Building an AlIr Atom File 30

3. Visualization of Clusters 31

C. Uniform Backgrounds and the Interaction Matrix 31

1. A Fixed Background Fit 31

2. Results of the Fixed Background Fit 32

D. Changing Orientations 33

E. Corner Clusters 35

F. Pair Fit 36

1. Two Equivalent Fits 36

2. Analysis of LSQ Fit Coefficients 38

3. Pair Fit Interaction Matrix 39

IV. Future Work to Be Done 42

V. References 43

iii

List of Figures

Figure 1 RT Tiling 3

Figure 2 Edges 4

Figure 3 Vertices 5

Figure 4 Pair Potentials 7

Figure 5 Tie-Line 9

Figure 6 Lozenge and Boat Tiles 17

Figure 7 Pure Lozenge Tiling 18

Figure 8 Lozenge Decorations 19

Figure 9 RLT6 Tiling 20

Figure 10 Pure Boat Tiling 21

Figure 11 Boat Decorations 22

Figure 12 B2H2 Tiling 23

Figure 13 Supertiling Example 24

Figure 14 Smallest Supertiling 26

Figure 15 Supervertices 27

Figure 16 Corner Cluster 36

Figure 17 Fit Correlation 39

List of Tables

Table 1 Tiling Database 12

Table 2 Standard Fit 15

Table 3 Cutoff Fit 16

Table 4 Pair Types 29

Table 5 Fixed Background Singular Values 32

Table 6 Pair Occurrences 34

Table 7 Two Pair Fits 38

Table 8 Interaction Matrix 40

Table 9 Interaction Matrix Standard Deviations 40

Table 10 Interaction Matrix Singular Values 40

1

I. Introduction and Background

A. Quasicrystals

A quasicrystal is a structure which is ordered, but not periodic. They may possess many

different types of symmetry, but can't be characterized by a repeating unit cell as a true crystal

can. Quasicrystals are metal alloys, which when properly cooled, develop into structures with

these complex forms.

Many quasicrystals can be represented in geometrical terms, such as a two-dimensional

tiling. This allows us to consider larger scale characteristics of these structures without looking at

each individual atom. An important problem for a quasicrystal is to determine the characteristics

of the lowest energy structures. These quasicrystals can have different energies depending on

their configuration, and in a physical system at sufficiently low temperatures, the quasicrystal

naturally relaxes to such a low energy structure.

To find these low energy characteristics from a theoretical standpoint, we must simulate

the creation and interactions of these quasicrystals using computers. Since it would take im-

mense storage space and computational power to account for each atom in an actual quasicrys-

tal, we represent a quasicrystal with a periodic unit cell called an approximant. Though this

does not allow for the true aperiodicity of the quasicrystal, the approximant is sufficient to ex-

hibit many of the characteristic properties we are interested in.

B. Context for Project

In this project, we are looking into the two structures magnesium-zinc (MgZn) and

aluminum-iridium (AlIr). In the past couple of decades, much has been learned and predicted

about the ordering of structures like this. With the recently improved computational speed, we

are now able to easily simulate the atomic interactions of these structures. For both of these

materials, we wish to understand the orderings that produce the lowest energy structures.

Over the past several years, Marek Mihalkovic and Chris Henley have studied these

two materials. Coming into this project, we understand quite well the low level ordering of

these structures (i.e. the rectangle and triangle tiles for MgZn and the clusters for AlIr). In this

2

project, we are trying to understand the higher level ordering of the structures (how these tiles

or clusters like to be arranged next to each other).

II. Magnesium-Zinc Quasicrystal

A. General Information

1. Physical Characteristics

The first type of quasicrystal we dealt with was the Magnesium-Zinc (MgZn) quasicrys-

tal. The MgZn quasicrystal we are looking at is periodic in one direction. The structure takes the

form of identical quasiperiodic layers stacked on top of each other. Each of these layers is

thick. Having this periodicity in one direction allows us to think of the structure in two dimen-

sions, rather than three. The MgZn quasicrystal is decagonal, meaning it has 10-fold local sym-

metry.

We represent a MgZn quasicrystal with a rectangle-triangle (RT) tiling. The triangles are

54o-54

o-72

o, and the two edge lengths of the rectangles correspond to the two edge lengths of the

triangles. These two shapes can be used to tile all space aperiodically. The position of every atom

in the structure is determined by the position of the rectangles and triangles. The Zn atoms lie

along the edges and corners of these shapes, while the Mg atoms lie inside them. An example of

such a tiling is shown in Figure 1. To make it easier to visualize the structure, the tilings are dis-

played as fat rhombi (two triangles) and hexagons (two triangles with a rectangle between them).

See section II.D.2.d for an explanation of the display of these RT tilings.

3

Figure 1: An RT tiling for an MgZn quasicrystal

In this project, a two dimensional RT tiling will represent an approximant of period

in the direction, with the RT tiling spanning the and directions. At each vertex,

two Zn atoms are spaced apart in the direction. These Zn atoms are at the same height

for every vertex. So, the vertex Zn atoms form two equally spaced planes. The Zn atoms along

the edges lie on one of the two planes equally spaced between these vertex planes. Each trian-

gle contains one Mg atom, and each rectangle contains four Mg atoms arranged in a tetrahe-

dron.

2. Tiling Components

For each triangle, the structure has one Mg atom and two Zn atoms, and for each rectan-

gle, the structure has four Mg atoms and three Zn atoms. The number of each atom can be related

to the number of each shape by Equation 1.

4

Equation 1

There are two edge lengths in the MgZn quasicrystal. We call the shorter one the edge

and the longer one the edge. The edge is long, and the edge is long. There

are seven different edge types that are inequivalent by any symmetry operation (i.e. rotation and

mirror flips). Four of them are edges, and three are edges. These edges are shown in Figure

2.

Figure 2: Images of the 7 distinct edge types in an RT tiling. Figure by C. Henley.

There are 10 possible inequivalent vertex types that can be made with the RT tiling

(shown in Figure 3). Each of them is named by the angles that meet at the vertex in order. We

5

use the numbers , , to represent angles of , , , respectively. By looking

at each vertex, we see that there are several edges associated with it. For example, the rectangle

vertex contains two edges and two edges.

Figure 3: Images of the 10 distinct vertex types in an RT tiling. Figure by C. Henley.

Considering only the RT tiling aspect of the MgZn quasicrystal, we now have 19 dif-

ferent component counts that make up a structure: the two tile counts and , the seven

edge counts through , and the ten vertex counts through . However, these counts

6

have several interdependencies. For example, and are the same in every structure. Also,

. These dependencies become very important when trying to fit the energies

of the components. In fact, it turns out there are only eight independent counts in a structure.

3. Component Energies

Each tiling component determines the atoms that are placed around it. Since the energy of

a structure comes from the interaction of its atoms, the tiling components each contribute some

energy to the structure. If the arrangement of atoms in a component can relax well, the compo-

nent will have a favorable (positive) energy contribution, but if the atoms are too tightly packed

or too sparse, the component will have an unfavorable (negative) contribution.

From the beginning of this project, we hypothesized that the rectangle-rectangle edges

( , ) and the zig-zag edge ( ) were energetically unfavorable. In the rectangle-rectangle

edges, the magnesium atoms are too tightly packed. seems to be less favorable than its part-

ner because the symmetry of doesn't allow the internal magnesium atoms to move apart

when they are relaxed.

Just like edges, certain vertices are more favorable than others. We believe the star

vertex to be very good, because the five-way rotational symmetry places each zinc atom at a

happy distance from its neighbors. Meanwhile, the rectangle vertex contains two edges

and two edges, so we hypothesize it to be extremely unfavorable. Similarly, the , , and

vertices should also be quite unfavorable because of the prevalence of the three unfavorable

edges.

B. Finding a Tile Hamiltonian

The main goal of the MgZn project is to determine what characterizes the best (i.e. lowest

energy) MgZn structures. For this project, the energy of a MgZn structure is defined by a set of

pair potentials. This means that all the energy comes from pair interactions of the individual at-

oms. The pair potentials consist of three lists of data, representing the interaction energy of the

three types of pairs (Zn-Zn, Zn-Mg, Mg-Mg), from a distance of to . The plot of these

potentials is given in Figure 4.

7

Figure 4: The pair potentials for MgZn. Plot of a data file created by M. Mihalkovic.

As the plots show, these potentials diverge at zero distance, and maintain an oscillating

decay as the distance increases. Each potential well represents a happy distance for the corre-

sponding pair. So, a Zn-Zn pair is happiest at a distance of , a Zn-Mg pair is happiest at

, and a Mg-Mg pair is happiest with a separation of . Notice that these distances

correlate with the relative sizes of the atoms in the pairs. The bigger atoms prefer to be farther

apart.

The magnitude of the pair interactions decays quickly, and is almost negligible past a dis-

tance of for all three pair types. This implies that the energy of a structure is mostly deter-

mined by interactions within this distance, which is about the length of a edge.

8

Because of the short scale of interaction distances, we can find a tile Hamiltonian, which

is a fit of the component counts of a structure to its energy. We want to be able to assign an ener-

gy value to each of the component counts. With such a Hamiltonian, we will be able to deter-

mine which components are giving positive contributions to the energy (unfavorable), and which

give negative contributions (favorable). It follows that the best structures will use the most fa-

vorable components and the fewest unfavorable ones.

Specifically, we will need to determine the energies and component counts of several

MgZn structures. Given these data, we can perform a linear least squares fit to find the best value

for the energy of each component. Since there are only eight degrees of freedom in the compo-

nent counts, it is not obvious which choice of components will give the most meaningful fit.

A good choice is to use the tile counts and , along with the counts of some unfa-

vorable components, such as the edges , , and . Just these five values allow us to calculate

the counts of the rest of the edges, as shown in Equation 2.

Equation 2

However, there are still three degrees of freedom among the vertices that are not determined by

these values (because there are eight total, as mentioned in II.A.2), but we can add some of the

vertex types to the fit as well, which will improve the accuracy.

The total energy of any one of our approximants will be negative, and larger structures

will be more negative, proportional to the size. So, the coefficients of the shape counts will be

negative, to account for the size of the structure. Meanwhile, the coefficients of the unfavorable

components should be positive, to identify the bad effect they have on the structures. This is the

type of Hamiltonian we will be looking for.

9

C. Tie-Line

Depending on the ratio of Mg to Zn in a MgZn quasicrystal, the possible RT tilings will

differ. The Mg and Zn atoms have different interaction energies and different contributions to a

structure. For this reason, it is meaningless to simply compare the energies of two structures if

they have different percentages of magnesium. This comparison is only valid for two structures

with the same ratio of Mg to Zn. However, we can make a tie-line, which is essentially a linear

interpolation of the expected energy of a structure with a given ratio. We define a value ,

the percentage of atoms that are magnesium, which is given by Equation 3.

Equation 3

The tie-line will connect the lowest energy structures known for a given value of .

This tie-line will give us a good guess of what the best energy structures are for a given ratio,

even if we have not looked at any such structures. The tie-line for the MgZn tiling database used

in this project is shown in Figure 5.

Figure 5: The tie-line created from our MgZn database. The two endpoints are the bounds for all possible MgZn RT til-

ings. Smallest Mg% belongs to the Laves phase, a pure triangle tiling, and largest Mg% belongs to a pure rectangle tiling.

10

The two endpoints on the left and right are structures of pure triangles (Laves phase)

and pure rectangles, respectively. The possible values of range from to .

The actual tie-line probably looks much more rounded. Since we don't have data about struc-

tures with , our database only gives a limited picture of the true tie-line. Our

tie-line is a good reference point for how energetically good a structure is, but it can’t neces-

sarily tell us if a structure will be stable or not. To determine this, we use density functional

theory (DFT) analysis, which is useful for the tile redecorations in II.F.2 and II.F.3.

D. Computer Implementation of MgZn

1. Computer Used

The computer used to do all computations for this project was a Lenovo G460 with an

Intel i3 processor, running Windows 7. We used Oracle VM VirtualBox to run Ubuntu on this

computer. Marek Mihalkovic provided us with a package in the Linux environment which

handled all simulations of the MgZn quasicrystals. Data was gathered in this VirtualBox envi-

ronment using these programs, along with Bash and AWK scripting. Outside of the Virtual-

Box, we used MATLAB to perform the data analysis.

2. Files and Programs Used

a. MgZn atom files

An atom file is a list of the atoms in a three-dimensional simulation cell, with implied

periodicity in all three directions. The first three lines in the atom file determine the size of the

simulation cell in the , , and directions, respectively. The fourth line is the number of at-

oms in the file. The rest of the file lists of the position and type of all the atoms in the structure.

Each position is given by three numbers, which represent the , , and z coordinates of the at-

om proportional to the size of the structure, so the numbers range from to , although

numbers outside this range will be interpreted accordingly by the relaxation program.

11

b. Tiling files

An RT-tiling is represented by a list of points, which represent the vertices of the tiling.

The points are represented by a pentagonal coordinate system, where the five unit vectors are

given by Equation 4.

Equation 4

Each point is represented by a length five vector of integers in this basis. The use of positive or

negative integers allows for the decagonal symmetry of the MgZn quasicrystal.

An unrelaxed atom file is made from a tiling file by the decort program. This pro-

gram identifies the rectangles and triangles created by the tiling file, and places the atoms ac-

cordingly, producing the corresponding atom file.

c. Relaxation

Given a decorated atom file, we can run a relaxation program on it, which perturbs the

positions of the atoms to decrease the energy of the structure. The atom positions in the unre-

laxed file are arbitrarily placed by the decoration program. The relaxation process adjusts the at-

oms into a more physically meaningful state. The program uses the information from the pair

potential file to minimize the energy of the structure.

The relaxation program runs in steps, with each step moving the atoms in a file to de-

crease the total energy. The distance these atoms move decays exponentially with the number of

steps. The average distance moved, maximum distance moved, and energy per atom are all print-

ed by the program for each step. At the end of the process a relaxed atom file is created, with the

same format as the unrelaxed file, but with the atom positions adjusted to their relaxed positions.

The user can adjust the number of steps run by the program. Increasing this number in-

creases the accuracy of the resulting relaxed file, but the relaxation process takes longer. The

cutoff distance can also be adjusted. When relaxing the structure, pair interactions beyond the

cutoff distance are ignored. The maximum cutoff is , because our pair potential file only

contains information up to this distance.

12

This relaxation program is the only way to determine the energy of a structure. The un-

relaxed energy can be found by simply relaxing the structure for zero steps. The relaxation

program gives the energy per atom, so to find the total energy we simply multiply this value by

the number of atoms in the structure.

d. Displaying the structure

We can use Xmgrace to display an atom file with the xmfig command, allowing us to

visualize the structure. The user can specify the dimensions of the area shown, as well as the

axis on which the image is projected. The show-fig command calls xmfig, and helps illus-

trate the tiling by connecting the vertices with black lines. Instead of showing every rectangle

and triangle, the shapes are combined into fat rhombuses and hexagons to make visualization

easier.

e. Tiling database

Along with the programs for decoration and relaxation, we started with a database of 159

tilings. These tilings fell into five different simulation cell sizes. The table below displays the

size of the approximants in my database, the number of tilings of each, as well as the correspond-

ing number of rectangles and triangles. The size is given as the total number of vertices

∑ . This value is related to the number of rectangles and triangles by the equation

. The value is the number of tilings of a given size. This number will add up to give

159 tilings over all five sizes.

Table 1: The size of our MgZn database

11 16 3 1

18 28 4 3

29 44 7 12

44 64 12 63

47 72 11 80

13

All 159 of these tilings are used to fit the components to the energies in our tile Hamilto-

nian. Of all the tilings, only the 18 and 44 sized tilings contained the edge , and only the 44

sized tilings contained the vertex . Also, the 44 tilings have the same ratio of Mg to Zn as the

11 tiling, because the 44-vertex cell is just a doubling of the 11-vertex cell in both directions.

It is also interesting to note that with the exception of 44, the sizes of these tilings

are related by a Fibonacci sequence, in that each one is the sum of the previous

two. This phenomenon comes up often when dealing with systems of decagonal symmetry.

3. Acquiring Data for the Fit

To start, we wrote two AWK scripts which read an atom file and print out the counts of

each type of edge, or each type of vertex. We also wrote a script to print the number of Mg and

Zn atoms in a structure, but these numbers are the same for all tilings of a certain size.

To get the data required for the fit, we needed to get a list of the component counts and

the energies for all the tilings. This was done using two relatively straightforward scripts, which

handled component counts and energies separately. The scripts simply looked at each tiling, and

decorated it into an atom file. At this point, the script would either print out the number of edges

and vertices, or relax the structure and print out the total energy. Using these scripts, we were

able to obtain a list of the component counts and energies for each tiling.

4. Analyzing the Data

Once we gathered the data, we used MATLAB to fit the coefficients. The component

counts we want to fit are stored in a matrix, where each row is a list of the counts, and there are

as many rows as there are tilings. The energies are simply stored in a vector.

Given these two arrays, it is very simple to fit the components in MATLAB by using the

linsolve function. To find the linear least square solution to the system, we use the command

fit = linsolve(counts, energies). The resulting vector will have the coefficients

for each tiling component in the fit.

14

MATLAB makes it very easy to change which component counts are used in the fit. It is

also easy to use different energy vectors (e.g. for relaxed and unrelaxed). This is important, be-

cause we performed several different fits throughout the project: we changed which component

counts were used to find the most accurate and meaningful fits, and we used different sets of en-

ergies with and without a relaxation cutoff.

E. Fit Results

1. Standard Fit

The form of the fit we chose was to use the two tile components ( ), the three unfa-

vorable edges ( ), and two vertices ( ). The hope is that the coefficients of the tile

components will be negative, to account for the fact that total energy is proportional to the struc-

ture size. The coefficients of the edge components should be positive, in this way representing

an energy cost for the presence of each of these edges. We could perform a fit with just

, but since this only constrains five of the eight degrees of freedom, we add

the two vertices ( ) to improve the accuracy of the. Adding a third vertex as well ( , spe-

cifically), to constrain the last degree of freedom, had an existent but negligible effect on the ac-

curacy of the fit, so we don’t include it in our fit.

The fit in Table 2 was obtained using the 159 database tilings, each relaxed for 100 steps

with no relaxation cutoff. The standard deviations were calculated by splitting the tilings into

100 different subsets of around 80 tilings, and calculating the fits for each of these subsets, then

finding the standard deviation of this data. The unrelaxed fit is also included Table 2, as a refer-

ence.

15

Table 2: Results of the standard fit for MgZn tile Hamiltonian

Comp. Unrelaxed Relaxed

-0.6039(1) -0.6201(4)

-0.8010(4) -0.8987(15)

0.4325(13) 0.1337(45)

0.0307(3) 0.0305(11)

0.1279(7) 0.1026(26)

0.0063(8) 0.0146(29)

0.0237(6) 0.0181(17)

We can see that the fit is of the form we were hoping for, regarding the sign of each co-

efficient. The coefficients of the tile components and are negative, to account for the

size of the tiling. The edge coefficients are positive, to represent an energy cost for these com-

ponents. We see that the rectangle-rectangle edges and have a significant energy cost.

The zig-zag edge also has a positive edge cost, as expected from section II.B. The less posi-

tive and vertices are included to provide a more accurate fit, and the physical meaning for

their coefficients is less obvious.

2. Cutoff Fit

To determine the effect of second neighbor interactions, we also performed a cutoff fit, in

which each atom only interacts with atoms within the cutoff range. By comparing this fit with

the original fit, we can determine how important the energy contributions from second neighbor

interactions is. Here, we used a cutoff of , again relaxing each of the 159 tilings for 100 steps.

16

Table 3: Results of the cutoff fit

Comp. Unrelaxed Relaxed

-0.6111(0) -0.6285(4)

-1.2756(1) -1.3799(14)

0.4249(1) 0.1522(34)

0.0000(0) 0.0045(9)

0.0001(1) 0.0043(18)

0.0001(1) 0.0088(20)

0.0000(1) 0.0158(21)

By looking at the difference in the relaxed cutoff fit as compared to the relaxed stand-

ard fit, we can see that the second neighbor interactions are important to the positive energy

contributions of and . In the cutoff fit, it appears that these edges do not have a strongly

positive energy contribution, meaning the majority of their energy costs must be attributed to

the second neighbor interactions.

F. Redecoration of Lozenge and Boat Tiles

1. The Purpose of Redecoration

When trying to characterize the best structures of the MgZn quasicrystal, we operated

under the assumption that such structures can be represented by pure RT tilings. As it turns out,

there are two other possibly competitive shapes that can be made by the inclusion of the thin

rhombus in the tilings. These two shapes are the lozenge tile and the boat tile. The lozenge tile is

a fat hexagon, and can be made by two fat rhombi with a thin rhombus nested between them. The

boat tile can be made by three fat rhombi with a thin rhombus nested among them. The shapes

are shown in Figure 6 with their component rhombi.

17

Figure 6: The two new tiles to redecorate

The thin rhombus in its default decoration is inherently unfavorable. The Zn atoms are

simply too close together. However, it is conceivable that we could come up with a new decora-

tion for these two tiles that could possibly produce structures more stable than the best pure RT

structures.

Our hypothesis was that the pure RT tilings are best, and to test this we tried to create the

best possible decorations of the lozenge and boat tiles. Then, we placed these tiles into structures

where they fit quite nicely. If the best structures we can make with boats or lozenges are unsta-

ble, then it is very likely that pure RT tilings are in fact the best.

2. Lozenge Redecoration

Using a naive approach, there are thousands of possible ways to decorate the interior of

the lozenge tile. It would be unreasonable to test all these possibilities, so we use educated as-

sumptions to simplify the process. To get an idea of the best lozenge decorations, we used a pure

lozenge tiling, and compared the energies of each decoration to the tie-line. We know from deal-

ing with the RT tilings that the Mg atoms inside the fat rhombus are very happy (especially when

relaxed). The distance between the two Mg atoms is in a potential well, and both Mg atoms are at

good distances from the Zn atoms on the edges and vertices of the fat rhombus. So, in redecorat-

ing the lozenge, we filled in the top and bottom sections with three Mg atoms, each spaced the

same as they are in the fat rhombus. This structure is shown in Figure 7.

18

Figure 7: Pure lozenge tiling with assumed optimal magnesium atoms

Using the simplifying assumption that the set of six Mg atoms on the top and bottom is

the happiest arrangement of atoms, only the horizontal middle section needs to be determined.

By trying a few types of decorations, we determined that at least one and at most two atoms

should fit in this middle section. With three Zn atoms in the middle strip, the atoms are much too

tightly packed, but with nothing in the middle, the structure is much too empty. In both of these

cases, the energies are much too high to be competitive with the RT tilings.

The two best decorations we found with respect to the tie-line were the Mg6Zn9 lozenge

(with two Zn atoms on the left and right), and the Mg7Zn7 lozenge (with one Mg atom in the cen-

ter). Both decorations are shown in Figure 8.

19

Figure 8: The two best lozenge decorations we found

The Mg6Zn9 lozenge was slightly above our tie-line, fitting in with the highest energy RT

tilings in my database. The Mg7Zn7 lozenge was actually below our tie-line, meaning the latter

seems to be the superior structure for the pure lozenge tiling. However, we analyzed the Mg7Zn7

structure using high-precision DFT relaxation, and determined that it was unstable by

, or . This means this structure could not be maintained in a physical

MgZn quasicrystal.

Though these two structures seem energetically decent (as compared to the RT tilings in

our database), the reason they are unstable is most likely because they are too empty in the cen-

ter. The coordination number (number of close neighbors) of the atoms in these two lozenge

decorations is lower than it should be. Unfortunately for the lozenge, it seems that its best struc-

tures are too empty, and trying to place any more atoms inside makes the structure too full.

Although our best lozenge decorations were unstable in the pure lozenge tiling ( ), this

does not necessarily mean any RT tiling containing a lozenge will be unstable. To test this, we

tried placing the Mg6Zn9 lozenge and the Mg7Zn7 lozenges in an tiling, shown in Figure 9.

20

Figure 9: The RLT6 tiling used to test the lozenge decorations

The reason we suspected this tiling to be very good is that contains no zig-zag edges, a

feat which is impossible with pure RT tilings. It also contains a very high density of star vertices

(which is one of the happiest vertices) than is possible with a pure RT tiling.

The tiling with the Mg7Zn7 was slightly above the tie-line, but when replaced with

the Mg6Zn9 lozenge, the structure fell below the tie-line by the small margin of .

This is important however, because the atom composition (atom percentage of Mg) of this struc-

ture is in the range of the RT tilings in my database. The fact that it is below the tie-line in this

range proves that it is competitive with the pure RT tilings. However, the DFT analysis showed

the Mg6Zn9 structure to be unstable by . However, this is still much better

than the instability of the pure Mg7Zn7. The best tiling we could find using the

lozenge tile was unstable, if only just, meaning that the lozenge tile is not a competitive alterna-

tive to the best RT tilings in a physical MgZn system, and most likely could never be observed.

21

3. Boat Redecoration

The boat tile is not as nicely symmetric as the lozenge tile, making it more difficult to

work with geometrically and computationally. As in the case of the lozenge tile, there are too

many possible decorations to test all of them, so we used simple arguments to narrow them

down. Using the RT decoration rule, the top of the boat contains three fat rhombuses. Since the

fat rhombuses love to have two Mg atoms in the middle, and have a Zn atom in between them,

we let the top of the boat remain how it likes to be in the RT decoration.

Indeed, we found that replacing the two Zn atoms on the top with Mg made the structure

too full (because the close Mg-Mg interactions are highly unfavorable). Moreover, replacing the

Mg pairs with Zn atoms was also unfavorable (presumably because the Zn-Zn pairs are too far

apart). After this simple but important process, we concluded with more certainty that this ar-

rangement of the top of the boat was very good, if not optimal. After this, we only had to deter-

mine the arrangement of atoms along the bottom of the boat. The following image shows the

tiling we used to test the decorations of the boat tile.

Figure 10: A pure boat tiling with assumed optimal top atoms

22

The leftover space in the bottom of the boat leaves room for three or four atoms to fit.

When trying decorations for the bottom of the boat, we found that the boat likes to have Mg at-

oms on the left and right bottom, because they interact more favorably with the Zn along the

edges. This only leaves the middle bottom atom to determine.

With these two Mg atoms on the left and right, we found that putting two Zn atoms in the

middle (on top of each other) was bad, because it made the structure too full, forcing unfavorable

interactions. The best two structures we found had either a single Mg atom or a Zn atom in the

center. These two decorations were the Mg9Zn12 boat and the Mg8Zn13 boat, respectively. Both

of these decorations are shown in Figure 11.

Figure 11: The two best boat decorations we found

The pure boat tiling using the Mg8Zn13 boat was above our tie-line, while the

Mg9Zn12 boat was actually below it. However, we found that the Mg9Zn12

structure is unstable by about , or .

Since we found that the lozenge tile benefitted from being placed in a structure where it

formed several star vertices, we tried to do something similar with the boat tile. The best exam-

ple we found was the tiling decorated with the Mg8Zn13 boat tiles, as shown in Figure 12.

23

Figure 12: The B2H2 tiling used to test our redecorations

The energy of this structure falls almost exactly on our RT tie-line. However, DFT calculations

showed that the structure was unstable by .

After searching for optimal decorations, and taking advantage of favorable tilings, we

were still unable to create a stable structure with the boat or lozenge tiles. This does not conclu-

sively prove that these tiles are necessarily unstable, but it does show that a structure with these

tiles is unlikely to be energetically competitive with the best RT tilings.

G. Optimal RT Tilings and the MgZn Supertiling

1. Optimal MgZn Tilings

Using the tile Hamiltonian we fitted, we ran zipper simulations to build RT tilings which

were energetically optimal based on the fitted coefficients. We found that in these optimal til-

ings, five of the ten vertex types never occur. That is, never occur. Equivalent-

ly, these structures only contain the vertices . This means that this optimal class

of structures does not contain either of the rectangle edges or .

24

After analyzing a few of these structures, we noticed larger scale order to the structure.

These structures have several star vertices ( ) quite close together, and connecting the star verti-

ces gave rise to similarities throughout the structures. To clarify this larger scale order, we creat-

ed a supertiling to simplify the common characteristics of optimal MgZn tilings.

2. Supertiling

a. Definition and characteristics

After noting common properties of optimal MgZn structures, we defined a supertiling of

three supertiles with the following properties:

No rectangle-rectangle edges are allowed.

All zig-zag edges come in groups of three.

All supertiling vertices (supervertices) are at a star vertex.

We call these supertiles , , and . The and tiles always come in pairs, denoted as and

, respectively. An example of the supertiling is shown Figure 13, drawn on top of the corre-

sponding RT tiling. This is a 76-tiling, and was able to be perfectly covered by our supertiles.

Figure 13: An example supertiling showing all supertiles.

Figure by C. Henley, for Dec. 2011 paper1

We spent a while trying to supertile various approximants of the RT tiling. The success

of these attempts varied depending on the size of the approximant. In some cases, the approxi-

mant can be mostly covered with supertiles, leaving a few defective tiles, as was the case with

25

the 47-tiling. In other cases, only certain sections of the structure can be supertiled, while the

pattern breaks down elsewhere. Of the tilings we looked at, only the 76-tiling was able to be

perfectly supertiled.

b. Determining supertilability

It turns out there is an easy way to determine if it is impossible to perfectly supertile a

certain sized approximant. The numbers of supertiles in any periodic supertiling are constrained

by the relation , or

. This relation is a consequence of the edg-

es of each supertile. The size and shape of each supertile is defined by the RT tiling shapes it

contains. The supertile contains nine triangles and one rectangle, the supertile contains 12

triangles and three rectangles, and the supertile contains 16 triangles and one rectangle. Com-

bining these relations, we get the dependencies in Equation 5.

Equation 5

All possible supertilings will contain at least one of each type of supertile (see Figure 15

in Section II.G.2.d). If the number of rectangles and triangles is such that any of the above equa-

tions does not give a positive integer value, the approximant cannot possibly be supertiled. How-

ever, if all the above equations do give positive integer values, it is not necessarily the case that

the approximant can be supertiled. It is necessary to show that these numbers of supertiles can

actually be arranged in such a way to fill the approximant.

When dealing with MgZn, we have mostly been looking at sizes related to each other

by a Fibonacci sequence. These approximants have vertices,

with each number being the sum of the previous two. It turns out that in a sequence like this,

only every third approximant has dimensions that can be filled by the supertiles. In this case,

we found that the 76-tilings and the 322-tilings could possibly supertiled. While the other sizes

could not, it turns out that doubling these approximants creates the possibility of a supertiling.

While the approximants did not have the correct area, the

26

approximants all have areas that can be filled with supertiles. We have yet

to determine what proportion of the theoretically supertileable approximants can actually be

supertiled. It seems the only way to know if an area can be supertiled is to actually construct

the supertiling.

c. The simplest supertile approximant

The smallest possible combination of supertiles satisfying

is:

. This gives an approximant with 38 vertices. We were able to con-

struct a supertiling with an approximant of this size, as shown below. The thick black line repre-

sents the simplest rectangular unit cell.

The structure can also be thought of as long vertical slabs placed next to each other.

Each slab self-enforced periodicity in the vertical direction. However, they fit next to each oth-

er in the horizontal direction even if one of them is flipped upside down. This means that a

structure of these slabs can consist of the two orientations placed next to each other at random,

creating a structure which is aperiodic in the horizontal direction.

Figure 14: left - The unit cell for the smallest supertiling. right - the stacking of supertile slabs

27

d. Supervertices

Each star vertex in the supertiling is a supervertex. We determined there to be 11 distinct

types of supervertices in this supertiling. Many angles of the supertile corners have irrational an-

gles with other supertile corners, which requires that every vertex contain at least one corner

from all three types of tiles. The 11 vertex types are described by their corners, with being the

corner of the tile with the smallest angle, and being the corner with the largest angle. These

vertex types are listed in Figure 15.

Figure 15: Image showing all supervertices

These supervertices are all shaped very irregularly, so it is often difficult to see how

they go together to make a larger structure. Still, having an image of the supervertices can be

useful when trying to manually build supertilings, as it gives a visual listing of how a supertil-

ing can be expanded from its current state.

28

III. Aluminum-Iridium

A. General Information

1. AlIr As a Simplified Model

Aluminum-Palladium-Manganese (AlPdMn) is a quasicrystal which has been success-

fully made and studied experimentally. In this project, we consider a theoretical model for an

Aluminum-Iridium (AlIr) quasicrystal. The AlIr quasicrystal is a simplified model of AlPdMn,

and has never been produced. These structures fall into a general class of structures in which

freely rotating clusters within icosahedral shells are arranged in a cubic ordering. Understand-

ing the order of AlIr should help us to understand the order of quasicrystals such as AlPdMn.

2. Structural Characteristics

The structure of the Aluminum-Iridium quasicrystal has clusters of 10 aluminum atoms

with an iridium in the center, arranged in a cubic lattice, each surrounded by an icosahedral shell

of iridium and aluminum atoms. At the center of each 10-cluster is an iridium atom. The ratio of

the overall structure is Al11Ir4. The AlIr quasicrystal also contains 9-clusters of aluminum, and a

real AlIr quasicrystal will most likely have 10-clusters and 9-clusters mixed together with rough-

ly equal ratio. However, in this project, we have only been focusing on the 10-clusters.

The 10-clusters can each be arranged in 12 orientations. These orientations are most

easily defined by the apex atom, which can point in any of the 6 directions corresponding to

the faces of a cube. Each of these directions has two similar but distinct orientations (the /

orientations) which have mirror symmetry. We call the orientations { , , , , , ,

, , , , , }, where the 1-orientations point in the positive x-direction, the 2-

orientations in the positive y-direction, the 3-orientations in the positive z-direction, and the 4,

5, and 6-orientations point antiparallel to the first three. The orientation is exactly opposite

of the orientation.

29

3. Pair Interactions

We hypothesized that the atomic interactions within the AlIr quasicrystal can be approx-

imated by interactions of nearest neighbor clusters, adjacent to each other in the x, y, or z direc-

tion. By analyzing the symmetry of the cluster pairs, we concluded there are 28 distinct pairs,

inequivalent by any symmetry operation (such as rotations and mirror flips).

Table 4 gives the assigned pair number for each of the 144 configurations. This table as-

sumes that the two clusters are adjacent in the -direction, with the row-cluster on the left of the

column-cluster. The table is helpful for understanding pair equivalences, as pairs with the same

number are equivalent. We used similar tables to identify pairs adjacent in the and -directions

as well. The pair number is given by the table. If the number for the pair is , we call this pair

.

Table 4: Numbering of the pair types, with adjacency in the x-direction

1 2 3 4 5 3 6 2 7 8 5 7

9 10 11 9 12 13 5 14 11 5 15 13

16 17 18 19 11 20 3 21 22 7 13 23

4 2 7 1 5 7 8 2 3 6 5 3

24 25 17 24 10 21 2 26 17 2 14 21

16 21 20 19 13 18 3 17 23 7 11 22

27 24 16 28 9 16 1 24 19 4 9 19

9 14 13 9 15 11 5 10 13 5 12 11

19 17 22 16 11 23 7 21 18 3 13 20

28 24 19 27 9 19 4 24 16 1 9 16

24 26 21 24 14 17 2 25 21 2 10 17

19 21 23 16 13 22 7 17 20 3 11 18

If a pair type has multiplicity , it means that when permuting an adjacent pair of clus-

ters through all 144 possible orientation configurations of a pair, of these configurations will

be equivalent to the pair type. These pair types have multiplicities of 2, 4, or 8. Eight pair types

have multiplicity 2, eight have multiplicity 4, and twelve have multiplicity 8, adding up to a total

of 144. The multiplicities are listed in Table 6.

If the system is in fact simply governed by nearest neighbor interactions, we should be

able to get a linear fit which assigns energy coefficients to each of the 28 pair types. Such a fit

30

would tell us which pair types are more favorable, and therefore more likely to occur in an actual

AlIr structure.

Since the direction a cluster is pointing also represents the direction of the farthest pro-

truding Al atom, we expect that this direction will have an important effect on the interaction

with other such clusters. We hypothesized that pairs in which the two clusters are pointing to-

wards each other would be unfavorable, because the protruding Al atoms are forced too close

together.

B. Computer Implementation of AlIr

1. Similarities to MgZn

Many of the computational programs and ideas used are identical to those used in the

implementation of MgZn. Atom files are used the same way, now representing aluminum and

iridium atoms instead of magnesium and zinc atoms. The relaxation program is identical, now

using a new set of pair potentials representing the three interactions Al-Al, Al-Ir, and Ir-Ir. Da-

ta is gathered using the Bash environment, and analyzed in MATLAB. The only things that

differ about AlIr are the programs we use to create structures and to visualize them.

2. Building an AlIr Atom File

The orientation of a cluster is enough to define the position of all of its atoms. To create a

atom file, we use a program called mkstructure which requires information about the dimen-

sions of the approximant, and the orientations at each point. A command to make a structure

would be of the form:

mkstructure 223 1a,4b,2a,4a,3b,2b,6a,1a,2b,3a,4b,6b > xyz.rtv

The first string of three numbers determines the dimensions of the approximant, in the

x, y, and z direction, respectively. The locations of the cluster centers must be stored in a file

specific for the set of dimensions used. This file determines the order in which the clusters are

placed in the structure. The string of orientations separated by commas lists all the orientations

in the structure. The order of placement of these clusters corresponds to the order of cluster

31

centers defined in the file.

The atoms of a cluster are placed in consideration of the cluster center and the orienta-

tion. If the vectors represent the positions of each cluster atom relative to its center, rep-

resents the center of the cluster, and the matrix represents a three-dimensional rotation

matrix corresponding to the orientation of the cluster, the cluster Al atoms can be thought

of as the set in Equation 6.

Equation 6

⋃⋃

In this way, the atom file is filled in with the positions of all cluster Al atoms. The

cluster center atoms and icosahedral shell atoms have defined positions regardless of the orien-

tations of the clusters.

3. Visualization of Clusters

Because we are dealing with three-dimensional clusters, the two-dimensional projec-

tions of the Xmgrace program aren't as helpful in visualizing the arrangement of atoms in a

cluster. Instead, we use a program called envi to create a .xbs file which can be used to dis-

play a three dimensional model of the environment around an atom, which can be rotated along

any axis. The range of the displayed environment can be adjusted. The program is especially

helpful in visualizing the changing of cluster orientations before and after relaxation, as dis-

cussed in section III.D.

C. Uniform Backgrounds and the Interaction Matrix

1. A Fixed Background Fit

The first method we tried involved permuting an adjacent pair of clusters through all 144

possible pair configurations, while holding the background clusters constant. This method was

also used by Woosong Choi to get a fit for the CaCd4 quasicrystal with tetrahedral clusters.2 In

32

our case, the permuted pairs are adjacent in the -direction, and the total size of the approximant

is , allowing for two clusters between repeated instances of the interacting pair.

We begin by calculating the relaxed energies for all 144 configurations and storing them

in a energy matrix. This energy matrix cannot give us isolated information about the

relative energies of the pairs, because different pairs will also interact differently with the con-

stant background clusters. However, by subtracting out the mean total value of the matrix, and

then subtracting out the mean of each row and column, we eliminate the contribution from the

background clusters, and are left with an interaction matrix of pair energies. Because this interac-

tion matrix eliminates interactions from the background environment, the matrix should be the

same regardless of the background, assuming pair interaction energies are dominant.

2. Results of the Fixed Background Fit

When we actually performed this fit, our results did not meet expectations. We used uni-

form backgrounds, in which all clusters other than the permuting pair had the same orientation.

We relaxed each structure for 200 steps. It turns out that for uniform backgrounds shifted by an

/ flip, the interaction matrices had similar singular values, but element-wise, they were very

different. And for backgrounds whose directional orientations differed, the interaction matrices

differed significantly. Since it is difficult to judge the similarities of two matrices, we

compared the singular values. The table below shows these singular values for the uniform back-

ground orientations

Table 5: Singular values for four different fixed background fits,

showing discrepancy caused by different backgrounds

0.6717 0.6717 0.4597 0.4597

0.3829 0.3829 0.2646 0.2646

0.2693 0.2693 0.2180 0.2180

0.1678 0.1678 0.1525 0.1525

0.1374 0.1374 0.1291 0.1291

0.0886 0.0886 0.0985 0.0985

0.0707 0.0707 0.0801 0.0801

0.0353 0.0353 0.0544 0.0544

0.0290 0.0290 0.0394 0.0394

0.0272 0.0272 0.0206 0.0206

0.0124 0.0124 0.0011 0.0011

33

To make sure that these results were not caused by bugs in the decoration or relaxation

codes we calculated the unrelaxed interaction matrices, and found them to be identical regardless

of the background. We concluded that something other than nearest neighbor interactions causes

the interaction matrices to differ greatly depending on the background. Henley posited that some

of the clusters must be changing orientations during relaxation. This was confirmed for both uni-

form background structures and randomly generated structures, using the method described in

III.D.

D. Changing Orientations

To determine if and when the orientations were changing, we used a script called pmi-

analyze which can take a relaxed or unrelaxed atom file and print out a list of the orientations

of each cluster in the same order they would be input to the mkstructure program. If the

script cannot identify the orientation of a cluster as one of the 12 standard ones, it returns a value

0 for that cluster. For this process, we used 900 randomly generated structures, where

each of the 27 clusters was randomly assigned one of the 12 orientations. Each structure was re-

laxed for 300 steps. We did not perform this process with structures because the

amount of computing time required to get sufficient data at this size was too high.

We found that in 121 out of these 900 cases, all the orientations were the same after re-

laxation. However, in all the others, at least one cluster in the structure was no longer recogniza-

ble as its initial unrelaxed orientation. 418 of the structures relaxed in such a way that at least one

zero-orientation cluster was present.

To get an idea of which orientation pairs might be more favorable than others, we can

look at the relative occurrences of these pairs in the unrelaxed and relaxed structures. Pairs oc-

curring less often in the relaxed structures as compared to the unrelaxed structures are likely less

favorable than those that would occur more. To do this, we take the 482 structures whose relaxed

cluster orientations were fully determined, and simply average the counts of each of the pairs in

the unrelaxed and relaxed cases. The average counts per structure are shown in the

table below, along with the ratio of relaxed count to unrelaxed count.

34

Table 6: Table showing the multiplicities, and occurrences per structure before and after relaxation

Pair Mult. Unrel. Count Rel. Count Rel./Unrel.

1 4 2.1908 2.1370 0.9754

2 8 4.5684 4.2698 0.9346

3 8 4.5560 4.9544 1.0874

4 4 2.2116 2.2532 1.0188

5 8 4.6680 4.9958 1.0702

6 2 1.1868 1.0498 0.8846

7 8 4.1992 4.3486 1.0356

8 2 1.1162 0.9710 0.8699

9 8 4.5850 4.8216 1.0516

10 4 2.2946 2.5768 1.1230

11 8 4.7926 5.0580 1.0554

12 2 1.0912 0.6640 0.6084

13 8 4.6846 4.8258 1.0301

14 4 2.1826 2.4814 1.1369

15 2 0.8050 0.4564 0.5670

16 8 4.4730 4.5394 1.0148

17 8 4.3692 4.5768 1.0475

18 4 2.3278 2.1244 0.9127

19 8 4.6764 4.4730 0.9565

20 4 2.2448 2.4938 1.1109

21 8 4.3070 4.1120 0.9547

22 4 2.2158 1.9128 0.8633

23 4 2.1826 2.0248 0.9278

24 8 4.3070 4.0540 0.9412

25 2 1.2780 1.2034 0.9416

26 2 1.2074 1.2614 1.0447

27 2 1.1244 1.1078 0.9852

28 2 1.1536 1.2532 1.0863

Looking at this table, we can see that the pair with the highest ratio of 1.1369 is

which is the pair equivalent to a and a cluster adjacent in the x-direction. These clusters

both point in the x-direction, so the pairs formed by two clusters pointing in the same direction

down their connecting axis seems to be favorable. The similar pair , which is two clusters

adjacent in the x-direction also has a good ratio of 1.123. This seems to show that the protruding

atom of a cluster fits nicely into the opposite end of another cluster.

35

The worst ratio of relaxed occurrences to unrelaxed occurrences was 0.567 for . This

pair is equivalent to a and a cluster adjacent in the x-direction, meaning that the two clus-

ters are pointing towards each other. The similar pair , with a and a 6a adjacent in the x-

direction, also has a bad ratio of 0.6084. This seems to support the hypothesis that pairs that

point towards each other are too cramped, and therefore energetically unfavorable.

It is important to note that no pair types always disappear during relaxation, even after

complete relaxation. This means that the phenomenon of changing orientations cannot be a sim-

ple two body effect, and it depends on all the surrounding clusters. Given that there are twelve

orientations allowed for each cluster, having to account for all the neighboring clusters makes

this phenomenon rather difficult to characterize. As a result, we have no way of predicting the

orientation changing behavior; we can only run tests and make general observations, like which

pairs relax away the most.

E. Corner Clusters

When we relaxed these AlIr structures, several of them ended up with zero-clusters,

where the orientation identifying program was unable to recognize it as belonging to one of the

twelve standard orientations. To try to figure out what the cause of this was, we used the envi

program (introduced in III.B.3) to look at some of these zero-clusters. Upon full relaxation, the

clusters that cannot be identified seem to have the same shape as the standard clusters, but are

oriented along a ⟨ ⟩ axis. Rather than pointing along one of the six cube faces, these clusters

now point towards one of the eight cube corners.

36

Figure 16. A view of a corner cluster, looking down the z-axis. Protruding

atom is colored in. The cluster points in the (1,-1,1) direction.

This adds a new element to this AlIr system. The clusters don't always stay in one of the

12 determined orientations, and can relax to point in any of eight new directions. Accounting for

this would require allowing for 400 configurations for an adjacent pair of clusters to be in. Theo-

retically, it is possible to get a fit that would include these eight corner directions as well, but

gathering and interpreting the results would be difficult.

F. Pair Fit

1. Two Equivalent Fits

For a given structure, we can get a count of each of the pair types, and we can get its re-

laxed energy, so if we have this information for several structures, we can fit the pair counts to

the energy to get a pair Hamiltonian. For this fit, we used the 482 structures with no

zero-clusters during relaxation, relaxed 300 steps. We subtracted out the average energy, and at-

tribute it to the clusters in the structure. In this case, there are 27 clusters, and the average energy

was per structure, which gives us a value of per cluster.

37

We performed two types of fits. The first was a simple MATLAB fit using the linear

solver: fit = linsolve(counts, energies). This gave us a fit with only 24 nonzero

coefficients, implying that only 24 degrees of freedom exist among the 28 pair counts. The con-

straints removing these degrees of freedom do not affect the results of our fit, once converted to

an interaction matrix.

Then, we performed a least squares fit using the following formula, where is the matrix

of counts, is the vector of energies, and is the set of pair coefficients, which we want to find

to minimize the value ‖ ‖ . Here, represents the pseudoinverse of . To solve this sys-

tem, we use Equation 7.

Equation 7

This formula gives us the least squares fit of all 28 pair counts. To find the standard devi-

ation, we performed each fit for 100 randomly generated subsets of these structures, with each

subset having around 241 structures. The standard deviation among the 100 sets of coefficients is

taken as the standard deviation of the full fit. The coefficients for each pair type are given in Ta-

ble 7.

The two fits have different coefficients, but they are both fitting the same data, so they

should be equivalent. To see this, we will convert these fits to an interaction matrix, as discussed

in III.F.3.

38

Table 7: Coefficients and uncertainty for two different pair fits (all units in eV)

Pair MATLAB fit MATLAB std. LSQ fit LSQ std.

1 -0.0782 0.0144 -0.0248 0.0055

2 0.0210 0.0242 0.0444 0.0058

3 -0.0327 0.0143 -0.0334 0.0039

4 -0.0865 0.0141 -0.0331 0.0054

5 -0.0987 0.0127 -0.0507 0.0045

6 -0.0203 0.0125 0.0414 0.0066

7 -0.0191 0.0146 -0.0198 0.0042

8 0 0.0137 0.0617 0.0065

9 -0.0252 0.0160 0.0145 0.0048

10 -0.0577 0.0268 -0.0480 0.0049

11 0.0043 0.0164 -0.0101 0.0042

12 0.0358 0.0231 0.0700 0.0098

13 0.0211 0.0150 0.0067 0.0047

14 -0.0417 0.0269 -0.0321 0.0052

15 0 0.0213 0.0342 0.0091

16 0.0265 0.0151 0.0176 0.0043

17 0.0137 0.0136 -0.0253 0.0039

18 0.1277 0.0298 0.0646 0.0059

19 0.0260 0.0150 0.0170 0.0047

20 0.0230 0.0291 -0.0400 0.0053

21 0.0637 0.0147 0.0247 0.0048

22 0.0995 0.0290 0.0364 0.0051

23 0.1022 0.0299 0.0391 0.0053

24 -0.0191 0.0264 -0.0040 0.0038

25 0.0944 0.0464 0.0795 0.0064

26 0 0.0481 -0.0149 0.0068

27 0 0.0204 0.0452 0.0065

28 -0.0396 0.0211 0.0055 0.0060

2. Analysis of LSQ Fit Coefficients

The standard deviations in Table 7 show that these fits, especially the MATLAB fit, are

not very precise. Yet the LSQ fit coefficients can still give us a good idea of which pairs are

more favorable than others. For example, looking at the pairs and , which had the highest

and lowest ratios of relaxed to unrelaxed counts respectively, we see that the LSQ fit coefficients

support these findings. The coefficient for is , which is largely negative, and

supports our assumption that this pair is favorable. The coefficient for is , which is

39

largely positive and supports our assumption that this pair is unfavorable. If we plot the LSQ fit

coefficients in sorted order alongside the corresponding relaxation ratios, we can see a noticeable

correlation between pairs with high energy contributions and pairs which often relax away.

Figure 17: Correlation plot showing how pairs with higher fit coefficients are more likely to relax away

3. Pair Fit Interaction Matrix

We can show the equivalence of the LSQ fit and the MATLAB fit by putting each of

these fits into a matrix where each entry corresponds to the pair represented by the

two orientations. If we build this matrix, and then subtract out the total mean, as well as the

means of each row and column, we get an interaction matrix which is equivalent for both fits,

shown in Table 8. The standard deviations are also converted to this interaction matrix form,

using the same method of random subsets.

40

Table 8: The interaction matrix resulting from the pair fits (units in eV)

2a 3a 1a 5a 6a 4a 2b 3b 1b 5b 6b 4b

2a -0.023 0.048 -0.031 -0.031 -0.038 -0.031 0.056 0.048 -0.018 0.076 -0.038 -0.018 3a 0.015 -0.046 -0.010 0.015 0.082 0.007 -0.038 -0.030 -0.010 -0.038 0.046 0.007 1a 0.007 -0.035 0.054 0.006 -0.010 -0.051 -0.031 0.015 0.026 -0.018 0.007 0.029 5a -0.031 0.048 -0.018 -0.023 -0.038 -0.018 0.076 0.048 -0.031 0.056 -0.038 -0.031 6a -0.014 0.071 -0.035 -0.014 -0.046 0.015 0.048 -0.023 -0.035 0.048 -0.030 0.015 4a 0.007 0.015 -0.051 0.006 0.007 0.054 -0.031 -0.035 0.029 -0.018 -0.010 0.026 2b 0.034 -0.014 0.007 -0.006 0.015 0.007 -0.023 -0.014 0.006 -0.031 0.015 0.006 3b 0.015 -0.030 0.007 0.015 0.046 -0.010 -0.038 -0.046 0.007 -0.038 0.082 -0.010 1b 0.006 -0.035 0.026 0.007 -0.010 0.029 -0.018 0.015 0.054 -0.031 0.007 -0.051 5b -0.006 -0.014 0.006 0.034 0.015 0.006 -0.031 -0.014 0.007 -0.023 0.015 0.007 6b -0.014 -0.023 0.015 -0.014 -0.030 -0.035 0.048 0.071 0.015 0.048 -0.046 -0.035 4b 0.006 0.015 0.029 0.007 0.007 0.026 -0.018 -0.035 -0.051 -0.031 -0.010 0.054

Table 9: The standard deviations of the interaction matrix (units in eV)

2a 3a 1a 5a 6a 4a 2b 3b 1b 5b 6b 4b

2a 0.002 0.004 0.003 0.003 0.003 0.003 0.004 0.004 0.001 0.006 0.003 0.001 3a 0.001 0.004 0.001 0.001 0.008 0.001 0.003 0.003 0.001 0.003 0.003 0.001 1a 0.000 0.003 0.004 0.001 0.001 0.004 0.003 0.001 0.002 0.001 0.001 0.003 5a 0.003 0.004 0.001 0.002 0.003 0.001 0.006 0.004 0.003 0.004 0.003 0.003 6a 0.001 0.007 0.003 0.001 0.004 0.001 0.004 0.002 0.003 0.004 0.003 0.001 4a 0.000 0.001 0.004 0.001 0.001 0.004 0.003 0.003 0.003 0.001 0.001 0.002 2b 0.004 0.001 0.000 0.000 0.001 0.000 0.002 0.001 0.001 0.003 0.001 0.001 3b 0.001 0.003 0.001 0.001 0.003 0.001 0.003 0.004 0.001 0.003 0.008 0.001 1b 0.001 0.003 0.002 0.000 0.001 0.003 0.001 0.001 0.004 0.003 0.001 0.004 5b 0.000 0.001 0.001 0.004 0.001 0.001 0.003 0.001 0.000 0.002 0.001 0.000 6b 0.001 0.002 0.001 0.001 0.003 0.003 0.004 0.007 0.001 0.004 0.004 0.003 4b 0.001 0.001 0.003 0.000 0.001 0.002 0.001 0.003 0.004 0.003 0.001 0.004

Performing a singular value decomposition of this interaction matrix gives an idea of how

many terms we need to describe the effective interactions. The singular values for the pair fit

interaction matrix (Table 8) are listed in Table 10.

Table 10: Singular values for the pair fit interaction matrix

1 0.3034

2 0.1694

3 0.1073

4 0.0984

5 0.0604

6 0.0596

7 0.0408

8 0.0217

9 0.0136

10 0.0119

11 0.0032

41

Firstly, it is obvious that the pair fit singular values (Table 10) differ greatly from all sets

of singular values in the fixed background fits (Table 5). This essentially eliminates any possi-

bility that these fits are correlated. And as we determined, the quality of the fixed background

fits suffered because the phenomenon of changing orientations meant the backgrounds were not

always fixed. So, no similarities should be expected between these two types of fit.

The rate at which the singular values drop off is a good measure of how clean the fit is.

If most of the value of ∑

is contained within the first few singular values, we can significant-

ly reduce the number of parameters used in an effective Hamiltonian of the system. In this case,

however, the singular values do not drop off very quickly, so it is likely that we could not get an

easily representable fit of this system. This contrasts the results that Woosong Choi found with

the tetrahedral clusters in the CaCd6 quasicrystal, in which the singular values for a fixed back-

ground fit dropped off very quickly.2

42

IV. Future Work to Be Done

We successfully constructed a tile Hamiltonian for the MgZn quasicrystal, and found

higher order structuring in its optimal configurations. We hit a bit of a roadblock with our super-

tiling, and haven’t had much success covering rectangular approximants with these supertiles. If

we really wish to understand the higher order behavior of this material, we could look more

closely at this supertiling, or possibly define a new one. It is possible, however, that the optimal

structures of MgZn do not exhibit order which allows for a successful supertiling.

There is still a significant amount of work that can be done with the AlIr project. We

were able to get an interaction matrix for a pair fit, but the precision was low. Perhaps if the

structure sizes are increased (say to ), or the number of relaxation steps is increased

(until fully relaxed, if possible), and data is gathered again, the resulting fit could be significantly

better. With the computational power available to us, this was not a feasible endeavor. Howev-

er, the preliminary results seem reasonable, and there could definitely be hope for a nice effec-

tive Hamiltonian of the AlIr structure.

43

V. References

1. Mihalkovic, M., et al. (2011) Ab-initio tiling and atomic structure for decagonal ZnMgDy

quasicrystal.

2. Choi, W. and C. Henley. Effective interaction and orientational ordering of tetrahedral Cd4

in CaCd6 quasicrystal 1/1 approximant.