Crystallography (II) – Unit Cell

So we have talked about lattice in the previous part. One should know that lattice is actually composed of the smallest possible regular (repetitive) array of unit cells, which is defined as the smallest building blocks of the lattice. First, let us inspect what the unit cell of the crystal structure from Figure 6 is:

Figure 1. Identification of unit cell from crystal structure in Figure 6 from https://andreaslm.wordpress.com/2015/06/21/crystallography-i-basis-and-lattice/
Figure 1. Identification of unit cell from crystal structure in Figure 6 from https://andreaslm.wordpress.com/2015/06/21/crystallography-i-basis-and-lattice/

Based on the repetition pattern of the arranged atoms in the lattice, we find that the unit cell in this crystal structure is hexagonal unit cell. Remember that repetition of the smallest possible pattern plays role here.

To elaborate more, how the unit cell build the lattice, let us assume we have unit cell formed as it is illustrated in the Figure 2a As we stack this unit cell, in the sense of arranging the unit cell with the closest neighboring unit cell and so on, eventually the lattice is produced, as we can see from Figure 2b.

Figure 2a. Unit cell
Figure 2a. Unit cell
Figure 2b. Lattice
Figure 2b. Lattice

Ok, now let us dive to the real example, a NaCl crystal structure, Figure 3. The gold atoms are Na and green atoms are Cl. There are three unit cells identified in this crystal structure and from our understanding of the relation between between unit cell and lattice, there must be only one unit cell composing the lattice. It means that we need to examine the possible options out of three unit cells and eliminate the other two.

Figure 3. NaCl crystal structure with its possible identified unit cells. Source: http://minerva.mlib.cnr.it/mod/book/view.php?id=269&chapterid=77
Figure 3. NaCl crystal structure with its possible identified unit cells. Source: http://minerva.mlib.cnr.it/mod/book/view.php?id=269&chapterid=77

All the three unit cells have the same possibility to arrange the NaCl crystal structure by put the respective unit cell one to each other. To determine which is the real unit cell, we need to consider:

  1. The symmetry of unit cell –> having higher symmetry is more likely to be real unit cell.
  2. The volume of unit cell –> having smaller volume is more likely to be real unit cell.

Basically, there are two different shapes of unit cell in here: A being parallelogram, B and C being square. The unit cell A has 2-fold axis of rotation, meaning that if you pick one corner and rotate it 360° (either clockwise or counterclockwise), it only fit every 180° rotation to its corresponding shape (Figure 4a). The unit cell B and C has 4-fold axis of rotation, as it firs perfectly when it is rotated 90° (Figure 4b and 4c).

Figure 4a. 2-fold rotation of parallelogram, unit cell A
Figure 4a. 2-fold rotation of parallelogram, unit cell A
Figure 4b. 4-fold rotation of square, unit cell B
Figure 4b. 4-fold rotation of square, unit cell B
Figure 4c. 4-fold rotation of square, unit cell C
Figure 4c. 4-fold rotation of square, unit cell C

We also discover that according the volume of the unit cell (remember, lattice is composed of the smallest possible pattern), the rank of compactness is unit cell C being the most dense with minimum space between atoms followed by unit cell A and unit cell B for having the widest empty space among the atoms.

Based on the two established criteria, we can conclude that unit cell C is the real unit cell composing lattice and thus the crystal structure of NaCl.

The figures and concept I made in here are based on these sources:

http://departments.kings.edu/chemlab/animation/untolat.html

http://minerva.mlib.cnr.it/mod/book/view.php?id=269&chapterid=77

Crystallography (I) – Basis and Lattice

Finally, I can make a note for myself as a continuation from last post. I hope you will find this useful.

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Have you ever wondered how atom arrangement in solid? Then welcome! If you interest with this, I recommend you to study further in crystallography, a branch of physics investigating the arrangement of atoms in the crystalline solids. With the aid of transmission electron miscoscopy (TEM), the possibility of having resolution up to nanometer scale, hence seeing the real atom is realized, and even down to Angstrom scale with the help of aberration-corrected technique in the TEM. An example of how atoms look like:

The crystal structure of the mineral cordierite taken from approximately 200 Angstrom thick using high resolution TEM. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 - frontispiece, available from http://www.doitpoms.ac.uk/tlplib/crystallography3/intro.php
Figure 1. The crystal structure of the mineral cordierite taken from approximately 200 Angstrom thick using high resolution TEM. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 – frontispiece, available from http://www.doitpoms.ac.uk/tlplib/crystallography3/intro.php

The atoms are represented as a white spots arranged in 6-fold rings, while the black area corresponds to hollow channels through the structure. The distance between the black spots is approximately 9.7 Angstrom. If we take only one 6-fold rings, compare it with its vicinity first, and observe throughly the whole figure, do you notice something? Yes, It looks like all the atoms have the same size and they repeat themselves in regular manner, where, in this case in x-and y-direction. Of course, when it comes to bulk structure, the arrangement of atoms will have the same behaviour in x-, y- and z.

Here we deal with two important concept. First concept is what we call translational symmetry, defined as the invariance of a system under any translation, as it is illustrated in Figure 2. With this characteristics, the size of the atoms will not change as it translates from one place to another.

The shape of the circle is maintained as it experiences translation symmetry from point A to point B
Figure 2. The shape of the circle is maintained as it experiences translation symmetry from point A to point B

Generally, the number of atom which exist in here is not limited by how many atoms and whether it is identical or different atoms, as it is depicted in Figure 3. It can be only one atom (Cu), two identical atoms (Si), two different atoms (GaAs or GaN) or up to \textit{n} number of atoms (protein crystals). No matter how many atoms exist in here, as long as it follows the translational symmetry, this/these what composes/compose basis.

The number of atom and its size is maintained as it experiences translation symmetry from point A to point B
Figure 3. The number of atom and its size is maintained as it experiences translation symmetry from point A to point B

The second part is what we called as lattice. In mathematical point of view, lattice is actually specific coordinates in space. The reason why these atoms are arranged regularly is because of they sit on the lattice. By observing the lattice points, we can identify what pattern these atoms are forming. Once more, the arrangement of atoms in lattice is required to be a repeatable. So, how to recognize the lattice then? This flash animations of Lattice Point Game is good starting point. Here I put my own result (selected lattice is highlighted with yellow color) in Figure 4

Identifying brick pattern
Figure 4. Identifying brick pattern, taken from http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php

In the same way, we can identify the lattice pattern of Figure 1, and in the Figure 5a we recognize it has triangular pattern. Furthermore, as we observe other than recognized lattice pattern carefully, we notice that there is another lattice pattern. Figure 5b shows two different arrangements of lattice, one with triangular with black dots on it (red) and one without the black dots (green). Each of them forms different orientation of triangle.

Identifying lattice pattern
Figure 5a. Identifying lattice pattern, taken from Figure 1. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 – frontispiece, available from from http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php
Two lattice patterns
Figure 5b. Two lattice patterns, taken from Figure 1. Source: A. Putnis, Introduction to Mineral Sciences, Cambridge University Press, 1992 – frontispiece, available from from http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php

Now, by putting basis (atom/atoms) on lattice, the crystal structure is formed. As we can see from Figure 6, the crystal structure will have same size of atoms everywhere (fulfilling criteria of the translational symmetry) and they are arranged periodicaly (lattice).

Lattice + Basis=Crystal structure

Figure 6. Combination of basis and lattice forms crystal structure
Figure 6. Combination of basis and lattice forms crystal structure

That is all for the first part. The concept introduced in here is heavily influenced by these resources:

http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php

http://www.physics.iisc.ernet.in/~aveek_bid/PH208/Lecture%202%20Crystal%20lattice.pdf

http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_1/basics/b1_3_1.html

I want to return to the basic matter

The last two weeks, I have encountered two papers discussing about the Miller indices in the hexagonal structure (wurtzite). I spent almost total one hour to understand it. I can’t believe I took that long time! My conscious of comprehension in crystallography lied to me. That’s it, conscious is not the same with facts. That is the reality.

My first time in learning crystallography was six years ago. Well, I just memorized the basic of crystallography and it looked simple that time. One of them was Miller indices. I knew it but did not understand what it was actually mean.

Then I took course in 2011 and recently 2014. Part of these courses dealt with Miller indices. Again, it was just passing my mind. Nothing so special about this section that really memorable to me.

Then last week, I read a paragraph in one paper. The authors of that paper used selected area diffraction technique (TEM) to analyze the crystal structure and the lattice constant. It was strange number at the beginning and after spending an hour, I finally discovered where that number came from. Note: One hour!

In the next few posts, I think I want to write short note about the crystallographic, which I am now trying to appreciate. Though it is bit basic, but I discover that it is really helpful to me. I find out this website is really amazing to re-visit the concept of crystallography.

Getting rid of prejudice

Living together with other person under the same roof is not simple at all. There are many things to consider in order to maintain such conducive moment, so that one from home can give great impression as a start in the beginning of the day and when the day is ended, one can relax peacefully.

As most people out there, I believe that living together with other people can be found to be difficult. People are different one to each other. It does not matter whether they come from the same country (the same culture) or other countries (the distinct culture), rules are needed to retain what must be done or what are prohibited.

With the specified rules being agreed by these people, then the rest is up to the will to execute the rules. For me, this is what I call “external rules”. If there are external rules, there must be also internal rule. The internal rule is simple: reflection. I think this is more important than the external rules which can be changed over time and place. Let’s say that I need to move from one house to another or migrate from one country to another. They have completely different “external rules”. But the internal rule is still the same: reflection.

I define “reflection” as the act that someone must take to reflect why something still goes wrong after the external rules have been well executed. This kind of attitude also can avoid being judgmental and prejudiced, which are very important factor in living together with other people. Once someone’s brain is filled with judging and prejudice, one can be having negative influence in their surrounding. The people who are living around can feel it, regardless how smart they can hide it.

I am reminded with the Words from Matthew:

Why do you look at the speck of sawdust in your brother’s eye and pay no attention to the plank in your own eye?

All I can say is that, getting rid of prejudice can be started with reflection. I realize that this is very important by the experience I had last week.

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For the past two months, I have been living alone until one new guy came and started to live with me. The first thing was, after he came, my room got some really weird smell. Then, I judged directly at him as a cause for this smell. Being uncomfortable with the situation I had that time, I was thinking what to say to him.

Fortunately enough, I tried to reflect first before telling him what was in my mind. After some examinations, I began to believe that the real source for this smell came from my room. It was not easy at the beginning to acknowledge that “this is my fault, not him”. I was ashamed to myself for having bad impression to someone who has not done any mistake to me. Since then, I realize that reflecting on why this happened is really important to find out the actual problem.

I think this is also can be applied not only in the apartment, but also in the society. Getting rid of prejudice is somehow similar to denying your own opinion, which is not very comfortable for me. Well, being humble to yourself and other people is one key to have and sustain serenity among human being.

Plasma Optics (IV)

To start this section, we need the wave equation expression, which we have derived before (The Wave Equation):

waveEq

where u is wave function and c is the speed of light in medium, defined by

wave func and speed of light

Meanwhile, we know that the relationship between refractive index n has connection with the speed of light in medium by this expression:

n1

We have the speed of light, dielectric function and magnetic permeability in free space, c_0, \epsilon_0, \mu_0, respectively. For nonmagnetic material, the value of \mu=\mu_0, so that we have the simplified expression:

n2

From this equation, we can say that negative dielectric function will give imaginary value of refractive index. In the other way around, positive dielectric function will give positive refractive index. We hope to come back to this section in the next section.

Let us come back to the expression of the wave equation and wave function. We can get the dispersion relation for electromagnetic waves by inserting equation wave function u into wave equation,

disp_rel1

disp_rel2

The last equation is dispersion relation and it tells us two important things:

  • If the dielectric function, \epsilon is real and positive, the wavevector k is real and positive.
  • If the dielectric function, \epsilon is real and negative, the wavevector k is imaginary.

Ok, now we can apply these conditions to the wave function. As a note, absorption is negligible in the following case

First condition is when the dielectric function, \epsilon is real and positive, we will have real and positive value of wavevector k. Let’s say that the value of the k is 1, then

waveEq_pos

 

The result of plotting the wave function with real and positive k as a function of distance is shown below:

Figure 1. This is how light will propagate through the material (grey colored) if the dielectric function is positive
Figure 1. This is how light will propagate through the material (grey colored) if the dielectric function is positive

Figure 1 shows us how light will propagate through the material (grey colored). Since light can propagate through material, it will become transparent. This happens when the dielectric function is positive, i.e. incoming light has wavelength smaller than the plasma wavelength of the respected material (see Plasma Optics (III)), as we expected. Since there is no absorption from the material (from the assumption), the amplitude of light propagating inside the material has the same value of the light propagate outside of the material, and it will look transparent in the infinity. This will be different if we assume that the material has ability to absorb the light, where it will be only reaching certain distance.

The second condition is when the dielectric function, \epsilon is real and negative, we will have imaginary value of wavevector k. Let’s say that the value of the k is \sqrt{-1}, then

waveEq_neg1 waveEq_neg2

The result of plotting the wave function whose the value of k is imaginary as a function of distance is shown below:

Figure 2. This is how light will be reflected through the material (grey colored) if the dielectric function is negative
Figure 2. This is how light will be reflected through the material (grey colored) if the dielectric function is negative

Figure 2 shows us how light will be reflected through the material (grey colored). Physically speaking from the figure 2, the light is not reflected, but rather absorbed. Therefore the material becomes solid. This happens when the dielectric function is negative, i.e. incoming light has wavelength higher than the plasma wavelength of the respected material (see Plasma Optics (III)), as we expected. We can say that with the negative value of dielectric function, the wave of light is damped directly in the surface of the material. This is not the same with the first condition (with the assumption where the material has absorption coefficient), the light can propagate to some certain distance before it is vanished.

How big is the influence of the absorption of the material to the light propagation? We will discuss this in the next section.

Plasma Optics (III)

First, we need to remember that the relation between \omega and f is linear, as described from \omega=2\pi f. We also know that f and \lambda are inversely proportional, which we can see from \lambda=\frac{c}{f}. This means that the behaviour of wavelength and frequency is opposite to each other. When the value of wavelength is high, we will get low value of frequency and vice versa.

The implications of dielectric function, from Plasma Optics (II)

diel func as plasma freq

are described as follows:

  • When incoming light interacting with the material has frequency lower than plasma frequency of the material, such that \omega<\omega_p, we will get negative value of the dielectric function. It implies the light is reflected from the surface of the material when the wavelength of the incident light is higher than the plasma wavelength of the material.
  • When incoming light interacting with the material has frequency higher than plasma frequency of the material, such that \omega>\omega_p, we will get positive value of the dielectric function. It implies the light is propagated through the surface of the material when the wavelength of the incident light is lower than the plasma wavelength of the material.

The above statements agree with the illustration given in the Figure 1 from Plasma Optics (I).

Figure 1. Visible wavelength, with group of alkali metal wavelength (155 - 362 nm)
Figure 1. Visible wavelength, with group of alkali metal wavelength (155 – 362 nm)

 

If the value of \omega_p=1 and the incoming light has \omega varying from 0 to 2, the response of the dielectric function is given in the figure 2.

Figure 2. The response of dielectric function to the omega of incoming light varying from 0 to 2
Figure 2. The response of dielectric function to the omega of incoming light varying from 0 to 2

How can negative dielectric function give rise to the reflected light? How can positive dielectric function give rise to the propagated light? We will try to find out in the next section.

Plasma Optics (II)

Two limits for \epsilon(\omega,K): \epsilon(\omega,0) and \epsilon(0,K). The first one refers to the collective excitations of the Fermi sea, which is related to the plasma, and the latter describes the electrostatic screening of the electron with electron, lattice and impurity interactions in crystals.

To obtain plasma frequency \omega_p, we can proceed with the equation of motion of a free electron in an electric field:

free el motion

x is the position of electron. Assuming electron moves according to the behaviour of simple planar wave, we need to introduce e^{-i\omega t} on both side,

with e

Further, we have polarization: the dipole moment per unit volume,

polarization

Since we are dealing with polarization, we can’t forget electric field, E, which both of them expressed in the electric displacement, D,

almost

That is, we find the characteristic of each material depicted by specific mass, m. Thus, this is the plasma frequency \omega_p,

plasma freq

The dielectric function can be re-written as follows

diel func as plasma freq

In the next section, we will deal with the implication of this equation.

The main concept from Kittel’s book: Introduction to solid state physics, 8th edition.