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B

ACHELOR

T

HESIS

P

HYSICS AND

A

STRONOMY

The Van Hove singularities and

their relation to the Boson Peak

Author

Mirjam BRUINSMA 10441204

Supervisor

Edan L

ERNER

Assessor

Philippe C

ORBOZ

Faculty of Science Amsterdam

Size: 15 EC

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Abstract

Crystalline solids can vibrate with different frequencies. The number of states per interval of frequency (the Density of States) of highly ordered systems was theoretically predicted by Debye, who treated the vibrations of interact-ing atoms as phonons. Debye’s theory turned out to be correct for finite sam-ples of elastic vibrations, but for infinite samsam-ples - which are suitable models for periodic crystalline solids - Elliot Montroll analytically demonstrated that there would be an excess of vibrational modes. This excess results in infinite peaks of the Density of States or of its derivative: the Van Hove singularities. The singularities happen at frequencies where the group velocity of the waves within the system is zero, caused by minimums, maximums or saddle points in the dispersion relation. In this paper we will both explain the nature of the singularities and we will discuss papers that examine the equivalence of the low frequency Van Hove singularity to the ’Boson Peak’ - a singular peak in the Density of States of highly disordered solids. Several researches show that the Boson Peak does originate in the acoustic Van Hove singularity and that the shift of the peak to lower frequencies is caused by both the density and the degree of order in the system.

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1

Dutch summary

Alle vaste stoffen om ons heen zijn grofweg in te delen in twee groepen. Allereerst zijn er de extreem geordende materialen, ook wel kristallen genoemd. In een kristal vormen alle atomen een perfect rooster. Voorbeelden van kristallen zijn keukenzout, sneeuwvlokjes en metalen. Door de overzichtelijke structuur van een kristal is het relatief makkelijk om de eigenschappen van het materiaal vast te stellen.

Figure 1: Een simpel kristal-rooster

De andere groep vaste stoffen wordt geken-merkt door hun ongeordende structuur. Ex-treme voorbeelden hiervan zijn glas en zand. De chaos binnen het systeem maakt het moeilijker om eigenschappen te verklaren. Om deze reden is er nog een hoop onbekend over ongeor-dende materialen. Een manier om on-geordende materialen beter te leren be-grijpen is door te kijken of bepaalde eigen-schappen van de vaste stof voortkomen uit eigenschappen die het materiaal zou hebben wanneer het wel geordend zou zijn. Dit is ook de opzet van dit pa-per.

De atomen in een kristal staan meestal niet

stil, maar bewegen met een bepaalde frequentie. Je kunt dit voor je zien als een rooster van knikkers die met elkaar verbonden zijn via springveren. De werk-ing van sprwerk-ingveren in het knikkerrooster komt overeen met de wisselwerkwerk-ing van de krachten van atomen in een echt kristal. Wanneer je een tik tegen een knikker geeft, beginnen alle knikkers te vibreren en dus zal het hele rooster met een bepaalde frequentie heen en weer gaan bewegen. In een kristal vindt een soortgelijke trilling plaats, die bij bepaalde frequenties niet uitdooft. Deze frequenties worden bepaald door de structuur van het rooster en de krachten tussen de deeltjes. Rondom sommige frequenties lijkt het kristal een oneindig aantal manieren te hebben om te trillen. Deze oneindige pieken worden de singulariteiten van Van Hove genoemd.

In ongeordende materialen bestaat een soortgelijk fenomeen, genaamd de Bo-son Peak. Dit is geen oneindige piek zoals de singulariteiten van Van Hove dat wel zijn, maar de Boson Peak laat wel zien dat het ongeordende materi-aal rondom sommige frequenties op meer manieren kan trillen dan we zouden verwachten. Dit heeft grote gevolgen voor de warmtecapaciteit van het mate-riaal.

In dit paper zullen we eerst proberen te begrijpen waar de singulariteiten van Van Hove vandaan komen, om vervolgens te kijken of dit fenomeen ten grond-slag ligt aan het vormen van de Boson Peak in niet-geordende materialen.

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Contents

1 Dutch summary 2 2 Introduction 4 3 Lattice structure 7 3.1 Periodic lattice . . . 7 3.2 Periodic function . . . 8 3.3 Reciprocal space . . . 9 3.3.1 Brillouin Zone . . . 10 3.3.2 Boundary conditions . . . 11 4 Crystal dynamics 13 4.1 The history of crystal dynamics . . . 13

4.1.1 Einstein . . . 13

4.1.2 Debye . . . 13

4.2 The equation of motion . . . 15

4.3 Density of States by Montroll . . . 20

4.3.1 Singularity in the acoustic branch . . . 21

4.3.2 Singularity in the optical branch . . . 22

4.3.3 The singularities plotted . . . 23

4.4 Density of States in general . . . 24

5 Order to disorder 27 5.1 Taraskin et al. . . 28 5.2 Chumakov et al. . . 30 5.3 Tong et al. . . 32 6 Conclusion 34 7 Acknowledgement 34

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2

Introduction

In the field of solid state physics, it is important to distinguish ordered and dis-ordered solids (Tong et al., 2015). Highly dis-ordered systems are the most stable solids. In these solids the atoms are arranged in a regular crystal lattice (Ziman, 1972), see figure 1. This lattice is invariant under a translation through its basic vectors, which are given by the structure of the crystal. Due to the translational symmetry in a crystal lattice, it is possible to reduce the system of a crystal to a much smaller system with a finite number of atoms, and by examining a repe-tition of this system caused by periodic boundary conditions, deduce a theory about the physical properties of the material (Ziman, 1972).

Where crystals are highly ordered, amorphous solids like glass and granular assemblies are highly disordered. There is still a lot unknown about the dy-namics of disordered structures. Soft matter physicists often approach this problem by making a model of a crystal, slowly changing the structure to a more disordered structure and then looking at how the properties of the system change. The philosophy behind it is that in order to understand disordered sys-tems, one can try to understand the physical properties of counterpart crystals, i.e. crystals that have a similar structure as the disordered material, and then find similar properties in models of less ordered systems (Tong et al., 2015). The latter will also be the approach of this literature study. First the Van Hove singularities in ordered systems will be explained, after which the relation to a similar event in disordered systems, called the Boson Peak, will be discussed.

Figure 2: A simple ordered lattice. The lines between the particles represent the spring like interactions between the particles.

The Van Hove singularities occur in a property of the system called the Density of States. This property tells something about the way a system can move due to interactions between particles. The forces of the particles in a crystal lattice interact similar to particles connected by a spring.

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When a crystalline material is in equilibrium, the highly ordered lattice struc-ture causes the net force on each particle to be zero (Kosevich, 2005). When the solid is not in mechanical equilibrium, the particles move around their equi-librium position. This movement is periodic with a certain frequency. Crystal dynamics describes the motion of the atoms around their idealized equilibrium position (Ziman, 1972). Due to the oscillating particles, the whole system oscil-lates. The system can not oscillate with every frequency - the structure of the system allows only certain frequencies to be occupied. If we want to find the permitted frequencies, first an equation of the motion of the crystal needs to be derived. This can be found by looking at the energy within the system. The total potential energy of the system is the sum of the potential energy between every pair of interacting particles, given by the force constants in the system (Montroll, 1947). The kinetic energy is the sum of the kinetic energy of every particle, depending on its position relative to its equilibrium position. The two formulae for the energies can be combined by the Lagrange formula, to even-tually derive the permitted frequencies (Montroll, 1947).

The calculations for the number of permitted frequencies (so called ’states’) is not the same for every similar sized frequency range. For instance, the number of states in the low frequency range can be bigger or smaller than the number of states at a higher frequency range. The distribution of the possible frequencies is described by the Density of States (DoS). To be more precise, the DoS de-scribes the number of states per interval of frequency or energy (Ziman, 1972). The DoS is determined by the topological structure of the system (Van Hove, 1953). Finding the DoS can thus tell a lot about the structure of the system. Also, the DoS determines a great part of the thermodynamics of the material (Van Hove, 1953). This makes the DoS an important function in solid state physics.

For a long time, DoS calculations were done on models of crystals with a finite number of particles. These calculations showed a smooth distribution of states (Van Hove, 1953). However, this never gave a good representation of real crys-talline materials, since the number of atoms in such materials is so big that the boundaries of the material should not influence the behavior of the particles inside the system, which means that they act as if the system consists of an infinite number of particles (Kittel, 1996). A more accurate model would be a system with an infinite number of particles. This can be simulated by applying periodic boundary conditions on a finite sized system.

Using this method, soon singularities were found in the DoS of crystals (Van Hove, 1953). At those frequencies where the singularities occur, the DoS itself or its derivative to the frequency seems to be infinite. In 1947 Montroll showed through exact calculations in a two dimensional lattice that the DoS contained two logarithmic singularities at certain frequencies. In addition to this dis-covery, in 1953 Leon van Hove came with a more general explanation for the singularities and its nature.

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In disordered systems, there are not such singularities in the DoS. Neverthe-less, there is a similar excess of states at low frequencies called the Boson Peak (Chumakov et al., 2011). In this paper we will try to see if there is a relation between the Van Hove singularities in ordered systems and the Boson Peak in disordered systems. In order to do this, we will first look at the structure of crystals. With this information, we can derive equations for the dynamics of the crystal, which brings us to the Density of States and its singularities. Know-ing where the sKnow-ingularities come from and what their nature is, we can try to compare them to the Boson Peak.

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3

Lattice structure

Crystal lattices are highly symmetric. The translational symmetry of the sys-tem provides a convenient method to find the properties of a syssys-tem. This method uses the fact that for every crystal basic vectors exist along which trans-lations can be performed without changing the system. For now, we will look at a system of N by N particles with interatomic spacing a.

3.1

Periodic lattice

To reduce the problem of finding the system’s properties, it is convenient to think of the system as it were made out of a certain number of identical "blocks", each consisting a certain number of particles. These "blocks" are subtended by all basis vectors (a1+a2+ ..., see figure 2) and are called unit cells. If the unit

cell consists of only one particle, the lattice is called a Bravais Lattice. The unit cell characterizes the whole crystal.

Figure 3: A simple two dimensional Bravais square lattice. The unit cell is spanned by the basic vectors a1and a2.

Each point in the lattice can be reached by adding a sum of basic vectors to the coordinates of any other point in the lattice. These points are called lattice sites. The translation vector in a three dimensional bravais lattice is given by

l=l1a1+l2a2+l3a3 (1) where l1, l2and l3 are integers. Instead of the unit cell where the basic

vec-tors form the boundaries, often the Wigner-Seitz cell is used. This cell can be found by choosing a symmetric centre in the original unit cell and drawing the translational vectors to the nearest equivalence lattice sites. The normal bisec-tor lines (in two dimensions) or planes (in three dimensions) of these vecbisec-tors

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form the boundaries of the Wigner-Seitz cell.

Figure 4: A simple Wigner-Seitz cell in a two dimensional Bravais lattice.

3.2

Periodic function

Because of the symmetry in the lattice, the behavior of two particles that are exactly a translational vector apart from each other is the same. This means that many properties of the lattice are periodic functions as well. If we would choose a random periodic function f in a one dimensional lattice, the period-icity would imply

f(x+l) = f(x) (2)

The solution for such periodic functions is given by the exponential Fourier series (Ziman, 1972). f(x) =

n Anei( 2πn a )x (3)

Where An are coefficients, a is the interatomic spacing and n is an integer. We

can rewrite the formula using g= 2πna . f(x) =

g

Ageigx (4)

Here, g can be seen as the translation in reciprocal space, which will be ex-plained later (see section 2.3). Since f(x) = f(x+l), one can deduct that eigl=1. This means that

gl= 2πn

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The three dimensional case is similar to the one dimensional case f(r+l) = f(r) (6) with f(r) =

g Ageig·r (7)

The translation vector in reciprocal space times the translation vector of the crystal lattice is now given by

g·l= 2πn1 a1 l1a1+ 2πn a2 l2a2+2πn a3 l3a3=×integer (8)

and thus, as expected, eigl =1. It can be concluded that the properties are the same despite the translation, as long as the translation occurs along a sum of the basic vectors.

3.3

Reciprocal space

As stated in the previous section, g is the translation in the reciprocal lattice. This lattice is build by the basis vectors

b1= aa3 a1·a2×a3 b2= a3×a1 a1·a2×a3 b3= a1×a2 a1·a2×a3

The reciprocal lattice is a convenient way to visualize the possible frequencies of the system, since coordinates in the reciprocal lattice correspond to possible wave vectors that define the frequency. Now, the translation in reciprocal space is given by

g=2πn1b1+2πn2b2+2πn3b3 (9) with n1, n2and n3being integers. And thus, to demonstrate the periodicity of

the reciprocal lattice

g·l=2πn1l1+2πn2l2+2πn3l3=×integer (10)

which means that indeed

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For purposes that will become clear in the next section, we’ll need to know the volume of a unit cell in reciprocal space. A unit cell in the reciprocal lattice is spanned by the vectors 2πb1, 2πb2and 2πb3and thus the volume of the unit

cell is ()3(b1·b2×b3) = ()3(a2×a3) · (a3×a1) × (aa2) (a1·a2×a3)3 = (a2×a3) · [a3· (a1×a2)a1a1· (a1×a2)a3] (a1·a2×a3)3 = () 3 a1·a2×a3 = ()3 Vcell (12)

Where Vcellis the volume of a unit cell in the crystal lattice.

3.3.1 Brillouin Zone

An example of a periodic function is the wave function. The wave function describes the state of a particle depending on wave vector k and is a solution to the Schrödinger equation. Because of the symmetry within the system, the state of the particle also describes the state of the whole system. Since all atoms are in a periodic well, the wave function is given by ψk(r) =ek·r(Ziman, 1972).

Using this with equation 6 gives

ψk(r+l) =eik·(r+l)=eik·lψk(r) (13)

To show the consequences of the periodicity of the system, we can change the wave vector by a translation vector in reciprocal space. The new wave vector is now given by k0 =g+k. This gives

ψk(r+l) =eik

0·l

ψk(r) =ei(g+k)·lψk(r) =eig·leik·lψk(r) =eik·lψk(r) (14)

Which is the same solution as it was for the old wave vector k in equation 13. This demonstrates that more than one wave vector can account for the same state and thus frequency.

Figure 5: The Brillouin zone in the reciprocal lattice in one dimension (Ziman, 1972).

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Every wave function has a whole host of possible wave vectors, that differ by the vectors of the reciprocal lattice. Every state can be characterized by a wave vector that lies in the first unit cell. Because of symmetry, often the Wigner-Seitz cell is used as a unit cell. This means that every state can be characterized by a wave vector that is closer to the origin than to any other lattice point in reciprocal space. For example, in one dimension the characterizing wave vectors are within the range−π

a <k≤ πa. This first unit cell or ’reduced zone’

in a reciprocal lattice is called the Brillouin zone (see figure 4).

3.3.2 Boundary conditions

Because real solids exist of so many particles, the boundaries of the solid do not influence the behavior of most of the particles. This means that we can treat the lattice far from the boundary as if the solid has an infinite number of particles. Until now, we already implied this by talking about perfect transla-tional symmetry, since that is only possible for infinite systems. If we would not assume this, the modes of the system would vanish at the boundaries. The boundaries would then reflect all waves making all the modes plane waves. An infinite number of particles however would be impossible to examine be-cause it would make the number of possible states infinite as well. To solve this problem, Born-von Karman thought of a system consisting a finite sample of particles with the boundaries stuck together, making particle 1 equal to particle N+1. In one dimension a crystal of L cells would then make a circle

ψk(x+La) =ψk(x) (15)

Using equation 12, it can be seen that

eikLa =1 (16)

In other words, kLa=×integer. The wave vector is thus defined by

k= 2πn

La (17)

Where n is the integer.

We are only interested in the Brillouin Zone where −π

a < k ≤ πa, so n can

have the values−1

2L < n ≤ 12L. This makes a total of L wave vector values

that are all La apart. In two dimensions, the translational Born-von Karman lattice looks like a torus. The three dimensional Born-von Karman lattice is not physically possible, but mathematically it is possible to use a system that has periodic boundary conditions in all three dimensions. In three dimensions, the translation in different directions is given by L1a1, L2a2and L3a3. This implies that

eik(L1a1)=eik(L2a2)=eik(L3a3)=1 (18)

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k= 2πn1 L1 b1+ 2πn2 L2 b1+ 2πn1 L3 b3 (19)

This vector has exactly the same value as the translation vector g, only divided by the lengths of the crystal lattice, making it correspond to the reduced wave vectors. The total number of allowed wave vectors within the Brillouin Zone thus is as many as there are unit cells in the reciprocal lattice, which is the same as there are unit cells in the crystal. This number can be calculated by the volume of the whole lattice in reciprocal space, divided by the unit volume of one cell in reciprocal space (see equation 12).

Nk=

Vcell

()3

Z Z Z

d3k (20)

The integration is possible because the wave vectors are evenly distributed in reciprocal space and the great amount of particles makes the wave vector den-sity high.

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4

Crystal dynamics

Knowing the structure of a system allows one to make hypotheses about the dynamics of the system. The events that we are trying to understand in this thesis - the Van Hove singularities - are measurements that deviate from the-ories that were thought to describe the motion of particles within the system. Although these theories are proven to not be exact, they are still a good approx-imation of how the particles in the system oscillate. Current research is still per-formed on measurements that deviate from these theories. For instance, when referring to an ’excess of states’ like the Boson Peak, one means the number of states above the number that was predicted by Debye.

4.1

The history of crystal dynamics

In the beginning of the 20th century, the heat capacity that was measured in crystals did not match with the classical theoretical predictions (Simon, 2013).

4.1.1 Einstein

To overcome this problem, Einstein came up with an approximate model for crystal dynamics (Einstein, 1907). In his model every atom acted like an in-dependent harmonic oscillator in its well that was created by the forces of the particles surrounding it. All the oscillating particles would have the same frequency, called the Einstein frequency (ωE) (Simon, 2013). Because Einstein

didn’t account for the wells to change due to the movement of the surrounding particles, this model was only sufficient for relatively high temperatures where it is justified to assume the particles vibrate independently.

4.1.2 Debye

In 1912, Peter Debye came with a more accurate model that coupled the atoms and treated their vibrations like sound waves; an idea that is known as phonons in the quantum theory. According to this theory, low energy excitations are not just oscillations of a single atom, as Einstein’s theory predicted, but are col-lective modes advancing through the system. The dispersion relation - the formula that relates the frequency to the wave vector - in a two dimensional model would be equal to that of an acoustic phonon and thus be linear (Zi-man, 1972):

ν∝ vs|k| (21)

Where vsis the velocity of the sound wave in the material.

Treating the vibrations like sound waves also meant that there had to be a min-imum and maxmin-imum wavelength. As shown in figure 5 for a one dimensional lattice, the maximum wavelength can not be longer than two times the length

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Figure 6: A few possible normal modes in a system of 10 particles.

of the system in each dimension: λmax = 2∗L. The minimum wavelength can

not be less than two times the space between two particles: λmin= 2∗a. The

maximum possible frequency, corresponding to the minimum wave length, is called the Debye frequency.

Around the same time as Debye unfolded his theory, Born-von Karman came with an additional theory about the boundary conditions for the vibrations in a crystal (Van Hove, 1953). The theory stated that, as written in section 2.3.2, a cyclic lattice can be used to simulate the infinite repetition of a finite sample in a material. This assumption was tested on crystals with finite samples of elastic vibrations. The findings for the Density of States (DoS) matched with Debye’s theory both at low frequencies and at the maximum frequency, since it continuously tended to zero at the maximum frequency found by Debye. At the intermediate frequencies however, this theory gave a few non singular maximums that were not predicted by Debye. For a long time, the cause for the singularities in the DoS was unknown. A few decades later, the subject of crystal dynamics gained sudden attention when Montroll analytically demon-strated through exact calculations on an infinite system, that the DoS of an two dimensional crystal has two logarithmically infinite peaks that are inherent to the periodicity of the system. To understand the nature of these singularities, the dynamics of crystals have to be understood more precisely.

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4.2

The equation of motion

Given a square lattice of N by N particles, which each lie a distance a apart from their nearest neighbors. The atoms that are the next nearest are the diagonal neighbors in figure 6. All atoms have mass M and are labeled by the parameters l and m, which both are integers that range between−1

2N and 12N.

Figure 7: A bravais square lattice. The indices l and m define the particles direction in respectively the x and y-direction. The letters u and v define the displacement of a particle in respectively x- and y-direction (Montroll, 1947). The translation vector in this lattice is given by

l=lax+may (22)

The kinetic energy of the lattice is given by the equation

K= 1 2M 1 2N

l,m=−1 2N (˙u2lm+ ˙v2lm) (23)

where ulmand vlmare the displacements of particle lm.

The potential energy is given by the sum of the potential energy that is con-tained in every interaction between two particles. For simplicity, we will use indices i and j instead of lm to indicate the particles.

U(r_1, r_2, ..) =

N2

i>j=1

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The restriction i>j is to prevent interactions to be counted twice. If the system is in equilibrium, the net force on each particle is zero.

Fi = ∂U ∂di 0 =0 (25)

Where diis the displacement vector of particle i in all directions.

Since the total potential energy of the system depends on the displacement of the atoms and we can assume their displacements are small, a Taylor series can be made of the potential energy around the point of stable equilibrium.

U'U0+ N2

i di " ∂U ∂ri # 0 +1 2 N2

i>j=1 didj " ∂U ∂ri∂rj # 0 +... (26)

In this context we can choose the potential energy in equilibrium state to be zero, making the first term unimportant. The second term is also equal to zero, leaving us with only the second order derivative. This can be written as the harmonic approximation U(r_1, r_2, ..) = N2

i>j=1 1 2(didj) 2c(a ij) (27)

where c(aij)is a function of the distance between the particles i and j in

equi-librium. For nearest neighbors this distance in equilibrium is a, for next nearest neighbors this (diagonal) distance is√2a. From now on we will use α and γ for respectively c(a)and c(

√ 2a)

2 . Since only the forces of nearest and next nearest

neighbors will be taken in account, the potential energy is now given by U(r_1, r_2, ..) = 1 2αnearest

(didj) 2+ γ

next nearest (didj)2 (28)

Using the notations and indices as in figure 6, this gives

U=1 2α

l,m[(ul,m−ul+1,m) 2+ (v l,m−vl,m+1)2] +γ

l,m [(ul,m−ul+1,m+1+vl,m−vl+1,m+1)2 + (ul,m−ul+1,m−1−vl,m+vl+1,m−1)2] (29)

Be aware that only the interactions with neighbors on the right and/or top are used in this equation. Again, this prevents the interactions to be included twice due to the summation.

Until now the summation over all particles was taken to obtain the energies for the whole system. To explain the derivation of the formula describing the dynamics of the system more easily, we will now look at the equation for a

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single particle lm and later use this equation to sum over all particles again. The potential energy of particle lm follows from equation 29, summing over all neighbors and next neighbors of this particle:

Ulm=1 2α[(ul−1,m−ul,m) 2+ (u l,m−ul+1,m)2+ (vl,m−1−vl,m)2 + (vl,m−vl,m+1)2] +γ[(ul−1,m+1−ul,m+vl−1,m+1−vl,m)2 + (ul,m−ul+1,m+1+vl,m−vl+1,m+1)2 + (ul−1,m−1−ul,m+vl,m−vl−1,m−1)2 + (ul,m−ul+1,m−1+vl+1,m−1−vl,m)2] (30)

To get an equation that combines the potential energy and kinetic energy to describe the motion of the system, Lagrange’s equation can be used:

∂t ∂(K−U) ∂ ˙d ! = ∂(K−U) ∂d (31)

For particle lm we find

−Mu¨l,m=α(2ul,m−ul+1,m−ul−1,m) +γ(4ul,m−ul+1,m+1−ul+1,m−1−ul−1,m−1−ul−1,m+1 −vl+1,m+1+vl+1,m−1−vl−1,m−1+vl−1,m+1) (32) and −Md 2v l,m dt2 =α(2vl,m−vl,m+1−vl,m−1) +γ(4vl,m−vl+1,m+1−vl−1,m+1−vl−1,m−1−vl+1,m−1 −ul+1,m+1+ul−1,m+1−ul−1,m−1+ul+1,m−1) (33)

As discussed in section 2.3.2, our two dimensional lattice network forms a torus, which means

ul,N+1=ul,1

uN+1,m=u1,m

vl,N+1=vl,1

vN+1,m =v1,m

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To find periodic solutions for the displacement, we need

φ1=ak1= 2πn1 N φ2=ak2= 2πn2 N (35)

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Where k1 and k2are the reduced wave vectors and φ1 and φ2 are the phase

differences of successive atoms in the two dimensions. The integers n1 and

n2 extend over the range−12N ≤ n ≤ 12N. Later on, (φ1, φ2)-space will be

used instead of reciprocal space. The idea is the same, only the length of the ’Brillouin zone’ changes to 2π instead of2πa . The periodicity of the system now gives solutions for particle lm

ul,m= A1e2πνt+lφ1+mφ2

vl,m= A2e2πνt+lφ1+mφ2

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where ν is the frequency of a certain vibration of the system and A is the am-plitude in either direction. Substitution of equation 36 in equationS 32 and 33 gives two equations for the amplitudes A1and A2.

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4Mπ2ν2A1+2αA1(1−cos(φ1)) +4γA1(1

−cos(φ1)cos(φ2)) +4γA2sin(φ1)sin(φ2) =0

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4Mπ2ν2A2+2αA2(1−cos(φ2)) +4γA2(1

−cos(φ1)cos(φ2)) +4γA1sin(φ1)sin(φ2) =0

For practical matters we will use

B(φ1, φ2) =(1−cos(φ1)) +(1−cos(kx)cos(ky)) (39)

D(kx, ky) =4γsin(kx)sin(ky) (40)

This can be calculated by substitution, or by realizing that the determinant of the coefficients of A1and A2must vanish in the following matrix:

B(kx, ky) −2ν2M D(kx, ky) D(kx, ky) B(ky, kx) −2ν2M (41)

This matrix contains all solutions. Before finding these solutions, we will cal-culate the maximum frequency, knowing that at the maximum frequency

δν δφ1 =0 δν δφ2 =0 This gives νL = r + 2M (42)

as the maximum frequency. If we define τ as a parameter for the force constants within the system equal to 8γ+4α , f as the frequency fraction of the maximum

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frequency (ν = f νL), cj as cos(φj)and sj as sin(φj), the solutions (Montroll,

1947) of the matrix in equation 41 are given by

f2=1 4[(1−c1c2) + (1−τ)(2−c1−c2)] ±1 4 q 2(c2 2−1)(c21−1) + (1−τ)2(c2−c1)2 (43)

This gives two possible dispersion relations, defined by the±sign, depending on whether the positive or negative square root is taken. These two differ-ent solutions are called the +branch (or optical branch) and the -branch (or acoustic branch), referring to optical and acoustic phonons (Ziman, 1972). In optical phonons, adjacent atoms move in opposite direction, where as these atoms move together in acoustic phonons. The velocity of acoustic phonons corresponds to the sound velocity in the material(Simon, 2013). The disper-sion relation of the acoustic branch is about the same as Debye’s predictions until the frequency approaches the Debye frequency. The optical branch was not predicted by Debye, but does look similar to Einstein’s prediction, having a weak dispersion in some wave vector regions and not tending to zero when the wave vector is zero.

What is important to notice, is that we used solutions for the displacement in two directions caused by the normal modes of the system to obtain the disper-sion relation. A more general solution would be

d=Aei(kr−ωt) (44)

When the direction of the amplitude is along the direction of the wave vector (and the direction of propagation), the wave function is longitudinal. If it is perpendicular to the direction of propagation, the wave function is transverse. Both can happen for the optic branch as well as for the acoustic branch when the wave vector is in the direction of high symmetry, in this two dimensional Bravais lattice being the x or y direction. Therefore often is spoken about the TA branch (transverse acoustic), TO branch (transverse optic), LA branch (lon-gitudinal acoustic) and LO branch (lon(lon-gitudinal optic), which all have different dispersion relations. In reality the wave vector is often not exactly in the direc-tion of high symmetry. In this case the wave funcdirec-tions are not easily defined into purely longitudinal or purely transverse waves (E.Y.Tsymbal, 2005). Because φ1and φ2each extend over N possible values in every direction, the

total number of possible frequencies is N2for every degree of freedom, which makes 2N2in total. Since most of the time N is not very small, it is hard to calculate every frequency unless you are only searching for frequencies within a small frequency range. Because of this, it is common to work with the Density of States (Montroll, 1947).

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4.3

Density of States by Montroll

In the last section we found the relation between the frequency fraction and the phase differences φ1 and φ2 for two different branches. To get a better

understanding of these results, it is useful to look at the solutions in reciprocal space, or, in this case, (φ12)-space.

Figure 8: A curve of constant frequency in a unit cell in (φ12)-space (Montroll,

1947).

Each branch has its own independent dispersion relation and thus its own re-ciprocal lattice. In figure 7 a curve is drawn for a random branch in a cell of the reciprocal lattice. All the points on that curve account for the same constant frequency (fνL). The number of points on this line is equal to the number of

normal modes with the same frequency. All points that lie inside of this curve are of frequencies less than the constant frequency of the curve. The fraction of the frequencies less than this frequency, so the fraction of points inside the curve, can be calculated for the plus (optical) or minus (acoustic) branch with the equation lim N→∞ N±(f νL) N2 = 1 2 Z Z F±12)≤ f 12 (45)

where the limit to infinity is taken for the number of particles, since in actual materials the number is so big that the points within the reciprocal unit cell are dense enough to be able to take the integral. Now this fraction can be defined by the frequency distribution function (g±(νL)) times the frequency range we

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g±(ν) = lim N→∞ δ δν hN±(f νL) N2 i = 1 2ν L δ δ f Z Z F±12)≤ f 12 (46)

The total fraction of normal modes less than fνLis the sum of the optical and

acoustic branch divided by the total number of modes, being 2N2, as discussed

in the last paragraph. The fraction is now given by N(f νL) 2N2 = 1 2 hN+ N2 + N− N2 i (47) which shows for the frequency distribution function that

g(f νL) = 1

2[g+(f νL) +g−(f νL)] (48) Now that these functions are determined, calculations can be made by setting the force constants of the system. As an example, Montroll used a system where τ = 13. With this assumption he could calculate the frequencies using equation 43. He found maximum frequencies of

f+2(max) =1 f−2(max) =2

3

He continued to calculate more frequencies of both branches and drew their curves separately in the Brillouin zone. Because in this case every quarter of the Brillouin zone accounts for the same fraction of frequencies inside the curve, one can choose to look at only one quarter of the Brillouin zone.

4.3.1 Singularity in the acoustic branch

Because of the periodicity of the system, the curve of every frequency should intersect the boundaries of the Brillouin zone perpendicular. If we would look at every transverse and longitudinal branch apart, the curves would not inter-sect perpendicular. But because they always appear together, the different type of waves of a certain frequency within the optical or acoustic branch can be combined, and it would intersect the boundaries perpendicular (Ziman, 1972). For the acoustic branch in figure 8 it can be seen that for every frequency this constriction is met, except for one frequency, where f2= 13. When calculating the frequency distribution function, Montroll indeed found a discontinuity at this frequency. The exact calculations can be found in Montroll’s article (Mon-troll, 1947). νLg−(f νL) =        24 f π2(2−3 f2)K( 2 f2 2−3 f2) for 0≤ f2≤ 13 8 π2fK( 2−3 f2 3 f2 for 13< f2≤ 23 0 for 23< f21 (49)

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Figure 9: The curves of different frequencies of the acoustic branch in (φ12

)-space (Montroll, 1947).

Where K(...) is an elliptic integral.

4.3.2 Singularity in the optical branch

Figure 10: The curves of different frequencies of the optical branch in (φ12

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When looking at the curves of the optical branch in figure 9, one can see that there are two curves for every frequency, each accounting for a different branch. All curves intersect the boundaries perpendicular, but for one frequency the two branches intersect. Again, this is the same frequency at which Montroll’s calculations showed a discontinuity in the frequency distribution function.

νLg+(f νL) =              4 f π2 q 1−43f2K(f r 2−3 f2 3−4 f2) for 0≤ f2≤ 23 8 f π2(1− f2)K( f 1− f2 q f22 3) for 23 < f2≤ 34 8 π2 q f2−2 3 K(1− ff 2q 1 f2−2 3 ) for 34 < f2≤1 (50)

4.3.3 The singularities plotted

The plot of the frequency distribution functions in figure 10 shows the two sin-gularities.

Figure 11: The singularities in the Density of States in respectively the acoustic and optical branch (Montroll, 1947).

In each branch there is also a discontinuity at f2= 23, but this disappears when the two branches are combined (see figure 11).

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Figure 12: The two singularities in the DoS, formed by combining the DoS of the optical and acoustic branch (Montroll, 1947).

4.4

Density of States in general

As seen above, in a two dimensional Bravais lattice the optical and acoustic branches of the spectrum cause singular peaks in the frequency distribution function. These singularities are implied by the periodicity of the lattice and also occur in three dimensional lattices with more than one atom per unit cell. To demonstrate this, Van Hove examined a crystal with Z atoms per cell in p dimensions. We will use q= k as our wave vector, with

q=

p

α=1

lαbα (51)

where lαare integers and b are the basic vectors of the reciprocal lattice. Due

to the periodicity, we know

ν(q+ p

α=1 lαbα) =ν(q) (52) (see section 2.2).

For every wave vector of the system, there are pZ independent plane wave vi-brations. Therefore, the frequency function consists of pZ branches. The com-bined volume of the unit cells of every branch in a three dimensional reciprocal space is now

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(3Z)()3(bb2×b3) =

(3Z)()3

Vcell

(53) Using equation 53 to derive an equation similar to equation 20 and knowing that q= k, the Density of States can be written as

g(ν)= Vcell

pZ

Z Z Z

dpq (54)

Where the integral runs over the part of the unit cell in reciprocal space where

νν(q) ≤ν+dν. We can rewrite this expression as

g(ν)= Vcell

pZ

Z Z

dSνdq⊥ (55)

Here, Sνis the line (in two dimensions) or surface (in three dimensions) in one

reciprocal cell where the frequency is constant (ν(q) =ν) and qis

perpendic-ular to that line or surface. This equation remains valid under every translation in reciprocal space, since the volume element dpqis invariant. To simplify the equation, we can use the following derivation for each branch

dq⊥ = δνq δqα (56)

The denominator in this equation,

δνq

δqα

, is called the group velocity and de-scribes the velocity of the energy propagation. The actual system consists of more than one branch, so we have to sum over the branches. This gives

g(ν) =Vcell pZ

Z 1 q ∑αp(δqδνα) 2dSν (57)

From this equation it is easy to see that singularities occur when the group velocity is zero, which happens when all the derivatives δν

δqα vanish. We can expand around such a critical point as a Taylor series, as ν(q)is a continues function. The first derivation of the critical point with respect to the wave vector is zero, since δνc

δqα =0. The Taylor series is then given by:

νq=νc+w1(q1−q1,c)2+w2(q2−q2,c)2+... (58)

Where w is the second order derivative of νcwith respect to the wave vector.

When w1and w2are both negative, νcis a local maximum. When they are both

positive, νcis a local minimum. And if one of them is positive and the other

one negative, the critical point is a saddle point. In one dimension, the singu-larities only occur at frequencies where the dispersion relation has a maximum or minimum. In three dimensions, if all second order derivatives are negative, the critical point is a maximum. If all second order derivatives are positive, it’s a minimum and in the two other cases where two of them are positive or two

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of them are negative, the critical point is a saddle point (Van Hove, 1953). Now that the nature of the singularities are known, the question remains why these saddle points are inherent to the periodicity of the system. To answer that question, think of a point in a two dimensional Brillouin zone where the frequency is maximum (see figure 12).

Figure 13: A saddle point in the dispersion relation caused by the periodicity in reciprocal space.

Because of the periodicity, that same maximum exists at the same place in ev-ery other cell of the reciprocal lattice. If you walk along a path of one absolute maximum to another, there has to be a minimum somewhere on that path. The existence of this minimum is independent of the path that is taken and, since the dispersion relation is continues, all the minimums of the different paths between the two maximums together form a path. This path also crosses the absolute minimum in the Brillouin zone and the corresponding minimums in other reciprocal cells. But if you walk along a path of one absolute minimum to another, there has to be a maximum somewhere on that path. At this point, the dispersion relation shows a minimum in one direction and a maximum in the other direction, together forming a saddle point.

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5

Order to disorder

As stated before, another great focus in solid state physics are highly disor-dered systems. The properties of these amorphous materials differ from crys-talline counterparts. A highly ordered system can be evolved into an amor-phous material by changing the structure. Even though many efforts have been put into distinguishing the properties of amorphous solids, a lot is still unknown about the phase in between crystals and glasses. Simply characteriz-ing the systems by defincharacteriz-ing the magnitude of order is not enough, since greatly ordered systems can sometimes behave like amorphous solids and vice verse (Tong et al., 2015).

Figure 14: The Boson Peak on the left and the acoustic Van Hove singularity on the right (Chumakov et al., 2011).

At low temperature, the properties of amorphous solids differ from crystalline solids in their excess of specific heat and a region in the thermal conductivity where there is a weak temperature dependence (Chumakov et al., 2011) (Gold-ing et al., 1986). These properties are caused by an excess of vibrational states at low energy, called the "Boson Peak". For a long time physicists thought that the Boson Peak was caused by features that go beyond acoustic waves. Ana-lytical and computer models now show that it is possible that the excess does originate in acoustic waves (Chumakov et al., 2011). Models show that the Bo-son Peak appears as a counterpart of the acoustic Van Hove singularity, i.e. the lower Van Hove singularity, shifted towards lower frequencies when the struc-ture is made less ordered (Chumakov et al., 2011).

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5.1

Taraskin et al.

In 2001, Taraskin et al. attempted to link the Boson Peak to the lowest Van Hove singularity by examining disordered systems where the atoms are positioned like a crystal, but the springs between the atoms, characterized by spring con-stant κ, are distributed according to a certain probability distribution ρ(κ). This kind of disorder is easiest to work with, since it is the simplest disorder that actually makes the system behave like an disordered solid. Because the system is so similar to crystalline reference structures, there is a well developed ana-lytical approach to this problem, called the Coherent Potential Approximation (CPA). This method reduces the problem to one with a known solution, of an effective crystal. CPA can be used when the eigenstates are not localized, i.e. when the probability density of a mode is not limited in a region of the system that is significantly smaller than the whole system. This is the case in Tarask-ins study since the vibrational modes in the low-frequency range, around the Boson Peak and lower, are found not to be localized (Taraskin et al., 2001). In the effective crystal, the complex spring constant of the springs between particles is given by ˜κ = ˜zκ0, where κ0 is the average spring constant in the

real crystal and ˜z the dimensionless effective spring constant. For the disor-dered system, a distribution is chosen for z∈ [z− ,z+ ]. For very low energies (e→0), Taraskin found that the real part of the dimensionless effective spring constant ( ˜z0) slowly approaches a constant value, while the imaginary part ( ˜z00) approaches zero, under the constraint that the disorder is not bigger than a certain critical number. When the critical disorder is exceeded, the system be-comes unstable.

Once the effective spring constant ˜z(e →0)is known, the DoS for the crystal and the disordered system (when e→0) can be calculated with the formulae:

gcryste e ˜z0  =χD e ˜z0 D2−1 (59) and gdise (e→0) = 1 ˜z0g cryst e e ˜z0  = 1 ˜z0χD e ˜z0 D2−1 (60) With D the space dimensionality and χDbeing the coefficient in the Debye law

(gcryste (e) =χDeD2−1) (Taraskin et al., 2001). Both equations obey the Debye law

(ge = 2ωgω), but the coefficient of the disordered DoS is bigger since ˜z0 < 1.

The reduced DoS (gω = DωD−1) of the reference crystal deviates from

De-byes prediction with increasing frequency and eventually forms the Van Hove singularity. The reduced DoS of the disordered system does the same, but be-cause of the bigger coefficient the excess forms at lower frequencies, forming the Boson Peak.

Taraskin then continued to examine the states in the disordered system (|di) by analyzing them in terms of bare states (|k, βi) in the reference crystal, charac-terized by their wave vector (k) and branch number (β). A state with energy e

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in the disordered system is build of different bare states with different weights. These weights are given by the spectral density:

Ak(e) = 1 ND D

d |hk, β|d|k, β|di |2δ(eed) E (61) The crystalline states that participate most in a certain disordered state of en-ergy e can be characterized by the most probable bare enen-ergy ecrystmax(e, )and/or

average bare energy:

<ecryst(e, ) >= 1

gdis

e

k

ekAk(e) (62)

Using the CPA method, it can be shown that when the system is stable, emaxcryst(e, )

and<ecryst(e, ) >are close to each other as long as the system is not close to

the critical disorder. From calculating<ecryst(e, ) >, Taraskin concluded that

at lower frequencies e was lower than<ecryst(e, ) >and at higher frequencies ewas greater than<ecryst(e, ) >.

Figure 15: This plot of the energy of the disordered system state against the most probable bare energy shows the level-repelling effect for two different degrees of order. These two different values for show that the effect increases with increasing disorder (Taraskin et al., 2001).

This level-repelling effect explains why the peak in the DoS shifts downwards at increasing disorder. Furthermore, the calculations show that the states in the region of the Boson Peak mainly originate from the bare states in the region around the acoustic Van Hove singularity. The disorder also causes the bare states that build a certain disordered state to be of different branches. Around the Boson Peak, the dominating branch is also the branch that causes the acous-tic Van Hove singularity in the reference crystal (Taraskin et al., 2001).

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acoustic Van Hove singularity. Also it shows that there is a correlation be-tween the shift of the excess of states to lower frequencies and disorder. For a few years it was thought that there was a direct causal link between the two, until further research.

5.2

Chumakov et al.

In later studies, often was stated that the differences in the thermodynamics and atomic dynamics of glasses and crystals was not caused by disorder, but by the fact that glasses almost always have lower densities than crystals (Chu-makov et al., 2016). A lower density would mean a lower sound velocity. The sound velocity determines properties of the elastic medium and thus results in different dynamics of the system. According to many physicists this is why in glasses the excess of states is observed at lower energies than in crystals (Chu-makov et al., 2014). The statement that density rather than disorder causes the difference in properties is also confirmed by the rare cases in which glasses have a higher density than their counterpart crystals. To demonstrate this, Chumakov et al. examined the properties of SiO2. This material can exist as

different polymorphs, meaning it can take the form of different crystalline and glass structures. This makes it possible to analyze the atomic dynamics of the material as a function of local structure and density.

Figure 16: A comparison of the (reduced) Density of States for ambient glass and α-quartz on the left, and densified silica glass and α-cristobalite on the right (Chumakov et al., 2014).

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When SiO2is used in studies to compare its crystal structure to the glass

struc-ture, often α-quartz and silica glass are used to show that the peak in the DoS shifts to lower frequencies with increased disorder. But the density of the glass is a lot loer than the density of the crystal. When the SiO2crystal -cristobalite is

compared to densified silica glass, which has a higher density than the crystal, the results are opposite, making it look like disorder shifts the peak of the DoS to higher frequencies (see figure 15). This shows that it is actually the density that causes the differences of the results.

Figure 17: A comparison of the Density of States for materials that have more or less the same densities (Chumakov et al., 2014).

To look at the effects of density, Chumakov et al. compared SiO2 glass with

a SiO2crystals of almost the same density. The results (figure 16) show many

similarities: the DoS of the crystal and the glass with the same density look similar, the peaks of the reduced DoS are almost at the same energies and the specific heat is more or less the same. The total number of states in the region of peaks is almost the same for both structures. Furthermore, the fraction of states in this energy region are for both structures about the same as the fraction of acoustic states in the whole energy range of the crystal.

Despite the great similarities, there still remain a few differences which can be assigned to the disorder. These differences are the smearing out of the DoS features in the glass, and the fact that for the glass the peaks of the DoS and

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reduced DoS aren’t observed at the same energy.

All together, Chumakovs study seems to prove that the nature of the acoustic Van Hove Singularity indeed is similar to that of the Boson Peak. The smearing out of the DoS caused by disorder remains a puzzle. Chumakov hints on the explanation that this can be caused by the fact that the Brillouin Zone of a disordered system has a different width than that of a crystal, but to make this conclusion further research has to be done.

5.3

Tong et al.

To investigate whether the order itself determines changes in the Density of States, Tong et al. investigated slightly disordered systems at small size poly-dispersity. The systems he examined are still highly structured as crystals, but exist of atoms of different size and thus show deviating properties. Slowly increasing the polydispersity (η), and by that increasing the disorder, allows to monitor several quantities that change due to the disorder. To measure the degree of order, Tong uses the more often used average coordination number z, which stands for the average number of interacting neighbors that atoms in the system have. The strength of the disorder is now given by the spatial fluctuation of this coordination number δz

δz= v u u t1 N N

i=1 (zi−z)2 (63)

The polydispersity where this parameter shows a transition from the ordered to a disordered structure is called the critical polydispersity (ηc).

Figure 18: The reduced DoS for different polydispersities at constant packing fraction φ=0.91 (Tong et al., 2015).

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When determining the (reduced) DoS, Tong measured the system at a constant packing fraction (the fraction of the volume that is taken by particles) to some-what exclude the role of density on the shifting of the DoS. The Boson Peak in the reduced DoS - the peak on the left in figure 17 - indeed flattens out with decreasing disorder but constant packing fraction, while at the same time the two Van Hove singularities arise at higher frequencies. The polydispersity at which the Boson Peak completely disappears is ηBP.

Figure 19: The values of the critical polydispersity and the polydispersity where the Boson Peak disappears in one plot (Tong et al., 2015).

Figure 18 shows an comparison of ηc and ηBP at different packing fractions.

The line through those points shows ηc =ηBP. This implicates that the Boson

Peak arises as a result of disorder itself. With this result Tong concludes that it is the interplay between structural order and density that determines place and shape of the Boson Peak.

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6

Conclusion

Before discussing further research, we will summarize our findings. First we have seen that the Van Hove singularities occur where the group velocity is zero, i.e. at frequencies where the relation between frequency and the wave vector form a minimum, saddle point or maximum. In a two dimensional bra-vais lattice, the Density of States shows two singularities: one in the optical branch and one in the acoustic branch. For higher dimensions, the number of singularities in every branch is related to the number of different saddle points. This number always equals the dimension minus one.

The researches of Taraskin, Chumakov and Tong clearly show that there is a relation between the acoustic Van Hove singularity and the Boson Peak, since we know that the acoustic Van Hove singularity and Boson Peak both occur in the same branch. Furthermore, the level-repelling effect shows that the wave functions within the Boson Peak come from the wave functions around the acoustic Van Hove singularity. Even though this confirms the hypothesis that there is a relation between the singularity and the Boson Peak, this still does not answer the question why the Boson Peak arises from the acoustic Van Hove singularity. Although we have seen that density and disorder both play a role in this event, a lot of research has to be done to determining how they interplay. In order to answer that question it is necessary to explain more aspects of the Boson Peak. From Chumakov’s research we can see that, under constant den-sity, the Boson Peak arises at almost the same energy as the acoustic Van Hove singularity, but at the same time it is much more smeared out. Also, for crys-tals the peak of the DoS is at the same energy as the peak of the reduced DoS, where as for glasses the energy of the two peaks deviate. These behaviors of the Boson Peak can not be explained with our current knowledge and thus need further exploring.

An other matter that needs discussion is the parameter for disorder. In Tong’s research ηcis defined as the polydispersity at which the system makes the

tran-sition from an ordered to a disordered system. It is the coordination number z that is used to mark this transition. Since we still do not know a lot about the transition and how to quantify disorder, we need to be aware that our order parameter z may be less accurate than assumed.

7

Acknowledgement

I would like to thank my supervisor for this project, Edan Lerner, for the op-portunity and for his patience. Furthermore, I also would like to thank Robbie Rens, who is doing a PhD with Edan Lerner. I could always count on his help and support whenever I got stuck.

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References

Chumakov, A., Monaco, G., Fontana, A., Bosak, A., Hermann, R., Bessas, D., Wehinger, B., Crichton, W., Krisch, M., Ruffer, R., Baldi, G., Carini Jr., G., Carini, G., D’Angelo, G., Gilioli, E., Tripodo, G., Zanatta, M., Winkler, B., Milman, V., Refson, K., Dove, M., Dubrovinskaia, N., Dubrovinsky, L., Ked-ing, R., and Yue, Y. (2014). Role of disorder in the thermodynamics and atomic dynamics of glasses. Physical Review Letters, 112(2).

Chumakov, A., Monaco, G., Han, X., Xi, L., Bosak, A., Paolasini, L., Chernyshov, D., and Dyadkin, V. (2016). Relation between the boson peak in glasses and van hove singularity in crystals. Philosophical Magazin, 96(7-9). Chumakov, A., Monaco, G., Monaco, A., Grichton, W., Bosak, A., and Rüffer,

R. (2011). Equivalence of the boson peak in glasses to the transverse acoustic van hove singularity in crystals. Physical Review Letters, 106(225501).

Einstein, A. (1907). Die plancksche theorie der strahlung und die theorie der spezifischen wärme. Annalen der Physik, 327(1).

E.Y.Tsymbal (2005). Introduction to Solid State Physics. University of Nebraska-Lincoln.

Golding, B., Graebner, J., and Allen, L. (1986). The Thermal Conductivity Plateau in Disordered Systems. Springer Berlin Heidelberg.

Kittel, C. (1996). Introduction to solid state physics. New York: Wiley.

Kosevich, A. (2005). The Crystal Lattice: Phonons, Solitons, Dislocations, Superlat-tices, Second Edition. WILEY-VCH Verlag GmbH and Co. KGaA.

Montroll, E. (1947). Dynamics of a square lattice i. frequency spectrum. The journal of chemical physics, 15(575).

Simon, S. (2013). The Oxford solid state basics. Oxford University Press.

Taraskin, S., Loh, Y., Natarajan, G., and Elliott, S. (2001). Origin of the boson peak in systems with lattice disorder. Physical Review Letters, 86(7).

Tong, H., Tan, P., and Xu, N. (2015). From crystals to disordered crystals: a hidden order-disorder transition. Scientific Reports, 5(15378).

Van Hove, L. (1953). The occurrence of singularities in the elastic frequency distribution of a crystal. Physical Review Letters, 89(1189).

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