This is Eric's logbook for Nano Japan project from 2012.6.3-2012.7.27. 

#contents

* Daily schedule [#daily]
- 09:00-10:00 Discussion (1)
- 10:00-12:00 Solve the problem and write the progress on pukiwiki
- 12:00-13:30 Lunch
- 13:30-14:30 Discussion (2)
- 14:30-16:30 Solve the problem and write the progress on pukiwiki
- 16:30-17:30 e-mail report and problems

* Goal of the project. [#goal]

- Consider the time evolution of a special vibration (radial breathing mode RBM) of a single wall carbon nanotube after applying the local force at t=0  whose functional shape is a Gaussian in the real space. The time evolution is given by differential equations and solution should be given by animation gif generated by Povray and Giam software. (by R. Saito 2012.5.18)

- Keywords: Coherent phonon, radial breathing mode, carbon nanotubes, Differential equation, Fourier Transform, Povray and Giam, Mathematica 

* Schedule for discussion [#discussion]

|Time and Day|Mon|Tue|Wed|Thu|Fri|h
|09:00-10:00|Sato|Nugraha|Tapsanit|Tatsumi|Nugraha|
|13:30-14:30|Saito|Simon|Hasdeo|Simon|Tapsanit|

- Please note that:
All appointment should be done by communicating each other on what kind of subjects will you teach.
--If Eric-san asks you what he needs, please answer it with CC to Saito.
-- If you do not have any materials to be taught, please give him an easy question to solve.
-- If Eric-san could not answer this question, it means that your question is bad (needs some knowledge).
-- If Eric-san can answer within 2 mins, it means that you gave improper question (the answer is too trivial).

* Questions and Answers [#QA]
- This section is for posting questions from Eric-san and answers from other group members.
- Please list here with some simple reasons or details.
- For every problem, give a tag double asterisks (**) in the code so that it will appear in the table of contents.
- For the answer, give a tag triple asterisks (***) in the code below the problem in order to make a proper alignment.
- List from new to old.

** Q: About Brillouin Zone [#Q-BZ]
Hey guys! So I believe this is where I should post questions. I don't have any specific questions today, but was just trying to understand Brillouin zones and Bloch's theorem, which appear a lot in Saito-sensei's book on Carbon Nanotubes.

From google I have an ok idea of what Brillouin zones are, but am still trying to figure out how they relate to the energy gap of the cell such as on page 28 of the CN book. I was also wondering how on page 47 the Brillouin zone can be a long line segment instead of a polygon.

If anybody has any quick words of advice for how to understand Bloch's theorem that would be great as well! It's a little bit intimidating and I'm not sure if there's a simple way to begin to understand it.

Thanks so much and see you guys in a couple of weeks!

Eric

*** Answer (by Tapsanit) [#A-BZ]
Please consider my answers one by one in order to understand the Brillouin zone.
- What is the Bloch'theorem written in page. 17 of Saito's book?
-- A: The electron in solid can behave as wave and its wave function which is denoted by $\Psi(r)$ should satisfy the Bloch's theorem due to the periodicity of the atoms in solid. $\Psi$ is sometimes called the Bloch wave function. It can be written as the phase factor $\exp(ikr)$ times the periodic function in real space $u(r)$, i.e., $u(r+T) = u(r)$; $\Psi(r) = \exp(ikr)u(r)$. It is recommended to  check that $\Psi(r) = \exp(ikr)u(r)$ satisfies the Bloch's theorem.

- What is the Brillouin zone? (solved in the discussion)
-- A: We must understand the reciprocal lattice before answering this question, because the Brillouin zone is defined from the reciprocal lattice.

- What is the reciprocal lattice that define the Brillouin zone?
-- A: As mentioned above, the electron in solid is a kind of wave whose wave function satisfies the Bloch's theorem. This means that it has the wavevector with amplitude and direction just like normal wave. The dimension  of wavevector is 1/[length] which is a reciprocal of length in real lattice. Then, reciprocal lattice is constructed to obtain the wavevector of the electron. Please note that there are many wavevectors possibly obtained from the reciprocal space. The unit vectors of the reciprocal lattice $(\vec{b}_1},\vec{b}_2,\vec{b}_3)$ are obtained from the unit vectors of the real lattice $(\vec{a}_1,\vec{a}_2,\vec{a}_3)$ by following relations ($\cdot$ denotes  dot product of two vectors):
$\vec{b}_i\cdot\vec{a}_j = 2\cdot\pi\delta_{ij}$, where $\delta_{ij} = 1$ if $i = j$ and  $\delta_{ij} = 0$ if $i \neq j$. 
It is recommended to try using this relations by deriving the unit vectors of reciprocal lattice, $\vec{b}_1$ and $\vec{b}_2$, of graphene in Eq. (2.23) of Saito-sensei's book using $\vec{a}_1$ and $\vec{a}_2$ in Eq. (2.22).

** Q: About copy machine [#Q-copymachine]
Can I use a copy machine or printer? 
*** A: (not solved) [#A-copymachine]
Yes, and they are across the hall from my room.

** Q: About library [#Q-copymachine]
I would like to know where the library is and how to use it. 
*** A: (not solved) [#A-copymachine]
The library is on the 7th floor. To check out a book I write information on a special wooden card and place it where the book had been. I also give the slip inside of the book to the librarian, and I return the book to the librarian.

* Report [#report]

This part is basically written by Eric-san. Any other people can add this.
Here the information should be from new to old so that we do not need to scroll.

**June 13 [#a9bea3b9]
** July 20 [#m4fdb092]
This morning I continued to write and prepare my presentation, abstract and poster. I then submitted my final version of the abstract to the RQI. Also, for the mathematica animations that we've been trying to create for the coherent phonon, I defined a new function A(t) to be the integral of Q(x,t) over all of x, at a specific point t is time. The plot of t is a function of time is exactly that of a damped harmonic oscillator which was displaced when considering a heaviside theta force for time. I showed this to Nugraha-san and Hasdeo-san, and they agreed that it looked good. We examined the freqency of oscillations and found them to be in the THz range, which is what we are looking for.

This morning Tapsanit-san taught me about the refractive index and the complex refractice index. Materials have a refelctivity R=I$_{r}$/I$_{o)$, where I$_{R}$ is the intensity of the reflected light. They have a similar transmission coefficient T=I$_{t}$/I$_{o)$. Light slows down in a meduium, and we define the refractice index n=c/v. We define the absorption coefficient $\alpha$ so that intensity I(z)=I$_{o}$e^(-$\alpha$z). 
** July 19 [#df3357d7]

Today I continued to work on my presentation, poster and RQI abstract. In the afternoon I practiced my presentation for the Mt. Zao meeting. We played ping pong and drank tea in the afternoon. I continued to think about the CP problem.

** July 18 [#ya0149c1]

Today I worked on my power point slide for Thursday, on my RQI abstract and on my RQI poster. I also continued to consider the Mathematica animations we're making, and why they don't always work. Hasdeo-san and Nugraha-san helped me figure out the physical parameters for the carbon nanotube, obtained from Hasdeo-san's data, and I worked to try to include this in the presentation.

** July 17 [#s4e745b2]
Today was spent writing. In the morning I coninued to type up the report of everything I have been working on this summer. I have completed an introduction and acknowledgments section and have a decent 1st draft of background information on the nanotube. I am just beginning to discuss the actual project and our approach, although I will have to go back and re organize my explanations of concepts like the reciprocal lattice.

This afternoon I continued to do this, and also began to work on my powerpoint presentation for thursday. The presentation should be clean, concise and explain the important points of our project without telling viewers about boring steps of the process or unimportant side tracks. Sentences should not exist on the powerpoint slides.

** July 13 [#i022d706]

Today I continued to write my reports, and to work on an abstract and prepare a poster for the RQI. Nugraha-san also gave me some material about the wave equation, and I tried to classically derive the Klein-Gordan equation by assumung that the nanotube is a string with tensile strength T and mass density $\rho$.

** July 12 [#m31e2133]

Today I continued to write the reports and think about the coherent phonon. Hasdeo-san showed me how to derive $\omega$ as a function of q beginning from the Klein-Gordan equation.

** July 11: [#e68d0b5e]

Today I met with Kono-sensei, Sarah-sensei and Matherly-sensei. It was really fun seeing them and we had a great lunch - all you can eat curry for 680 yen believe it or not. I showed Kono-sensei what I've been working on.

We also coninued to think about the Klein-Gordan equation. In the afternoon Hasdeo-san helped me to Fourier transform the equation to show how the RBM mode might be connected to the equation. It was a very interesting day and I continued to write my summer report late in the evening.

For dinner I ate some delicious seafood with Saito-sensei, Otsuji-sensei, Kono-sensei and many others. I had some great conversation with Matherly-sensei and some Russian theoreticians from Otsuji-sensei's lab.

**July 10: [#a685dcb6]
Today I continued to consider the klein-gordan differential equation (telegraph equation) that we thought about yesterday. The solutions appear pretty good. I made more elaborate graphs and animations on mathematica, and I?ve begun to format the mathematica programs as nicely as possible so that Nugraha-san can use them when I finish. As the graphs get more elaborate I?m thinking about the actual physical parameters of the nanotube. For instance, how many units cells there are, and the shape of the Gaussian forces in each unit cell. I have a couple of days to go but the progress with this model is promising, we should be able to have models that look good by the end of the week. I am beginning to consider how the speed c changes throughout the unit zones, but this requires some more analysis. Hopefully I will be able to work on the question a little bit more by myself, and then talk about it with Nugraha-san. If we can graph w(q) in the reciprocal lattice and c = 2Aq, then it should be possible to graph c as a function of x in the real lattice, by finding the relationship between q and x. I have a couple of ideas I must try to plot, but this should solve the specific problem of how c changes.

However, I do not know why the klein-gordan equation seems to yield an appropriate solution to this problem. I also do not know how the remarkable NDSolve function on mathematica works, but this is ok. Hopefully it will be possible to do analysis to find the klein-gordan function as a solution for this problem. 

This afternoon I went to the very useful Mathematica seminar. I learned many important ways to code that I did not know about, and I learned about some really cool graphical features that exist. I still have a couple of questions but I will be able to email the presenter if it is necessary. After the presentation I tidied up all of the papers in my folders and put them in a binder. It is still not well organized but the entropy is much decreased. Saito-sensei gave me a mechanics problem to solve if I have time, and I?m continuing to work on my projects. 


**July 9 [#b95a3b62]
After an unforgettable weekend in Hiroshima and Kyoto I have returned to work. Early this morning I skyped with Stanton-sensei. We talked about how NanoJapan was going and he said that he'd send me some more papers and helpful data this evening. He also said that I should feel free to email and skype him more often, a few times a week or even every day, because the time I have left is quickly diminishing.

Next I talked with Saito-sensei who reminded me of the increasing importance of writing up what I have been doing. Nugraha-san will continue to work on this subject when I leave, so it is important that I create a README file describing all of the programs I have made (mostly in Mathematica). Also, I will write a sort of thesis on what I have been working on. It will take time to make this and I should start now, by writing for a few hours in the mornings.

After going to lunch, I prepared some more questions for the Mathematica Q&A period which is tomorrow morning.

In the afternoon I talked again with Saito-sensei and showed him the NDSolve function on Mathethmatica, and the way in which I had modified our differential equations. This was very productive and thought provoking. If we add a -u(x,t) term to the right side of the wave equation we find a solution (subject to boundary conditions) that makes physical sense. It oscillates and dies off in time. These results were interesting and I went to the library to search for more books on waves and these related differential equations. It turns out the equation which we were using is called a Telegram, or Klein-Gordon function. 

The next step is to investigate this function further by analysis. Saito-sensei quickly wrote on the whiteboard how the lattice, with atoms connected by springs, may yield this solution. I must examine his mathematics which I did not fully understand, and try to provide a physical explanation for this function especially by regarding the phonon dispersion. 

Next, I must create mathematica programs to mimic the conditions of the nanotube. It is possible that our new equation is accurate and that coherence could eventually be observed by considering the telegraph equation.


**July 4 [#u832e234]
This morning Tapsanit-san came and reviewed with me cutting lines, the nanotube unit cell, and periodic boundary conditions. From this knowledge I will be able to set periodic boundary conditions and find numerical lengths for the nanotube and the different unit cells. There will be one gaussian force in each unit cell. Hasdeo-san has said that he used a (10, 10) nanotube so it will be relatively easy to calculate the unit cell length.

Yesterday evening Nugraha-san told me about numerical solutions to differnetial equations. I discovered that Mathematica can provide reasonably good numerical solutions to our differential equations. However, I must learn more about this function on mathematica before I can use it properly, so I will ask about it next week.

This afternoon Hasdeo-san came and taught me about the saddle point method. It is complicated but I followed his very good explanation. We did not look at the proofs for the saddle point method only how to use it. After learning about this, Hasdeo showed me how to take the inverse fourier transform of our function and we found a solution for u(x,t). The solution seemed reasonable because it had a term for Gaussian decay multiplied by sin(x^2 t K). However, when we graphed our solution it didn't make any physical sense. There was an asymptote and x=0 with very rapid oscillations moving inward toward the asymptote. We checked our math that we did not make a mistake. It is possible that we made a mistake but did not catch it, or that the approximations used in the saddle point method are not valid for this problem.

**July 3 [#i1a0bfcc]

Today I learned many things about the coherent phonon project and about Mathematics. In the morning Saito-sensei and Nugraha-san taught me about excitons and the finer details of the project. Kono-sensei?s group has observed the coherent phonon in CN that have been excited by light and through solving the wave equation we are trying to show that this is classically possible. Very importantly, because of the excitons (electron hole pairs) we must consider a force of one Gaussian centered I each unit cell of the carbon nanotube. This is because of quantum mechanics and I must assume this to begin the project.I showed Saito-sensei the rough draft of my presentation for Kono-sensei for next week. He and Nugraha-san gave me a lot of advice for how to improve the presentation, and I spent much of the morning creating a better power point. 

To find a solution of the wave equation I should use fourier and inverse fourier transforms, which requires me to learn about complex integration. Nugraha-san taught me much of this in the afternoon. He was a great teacher and I learned and incredible amount of information from him. He taught me about analytic function, the Cauchy-Gaursat theorem, the Residue theorem, the Cauchy Integral formula, the Laurent series. After learning this complex analysis we then spent an hour considering the phonon project and the actual physical parameters to use in the calculations.


**July 2 [#rep20120702]
Today Saito-sensei has returned from his travels. Meeting with him was very helpful and I made progress on my project, and I see where I must go next. Last week I found a way to solve the wave equation using fixed boundary conditions (only considering the sin term of the series). With Saito-sensei's suggestions I included the cosine terms into the equation and believe I have created a mathematica program to solve the wave equation for an applied force, with periodic boundary. I created the mathematica program today late in the morning so I still need to check that its solutions make physical sense and to try using a wide variety of applied forces. However, the mathematica wave program is almost complete.

This afternoon Saito-sensei continued to help me with the project. I must focus my attention on the coherent phonon + radial breahting mode, which is the theme of the project. I can begin to consider the phonon dispersion of a Brillouin zone based on the data that Hasdeo-san gave me. I can consider the variety of applied forces that I have talked about with Saito-sensei and the others. Hopefully I can create a model of the coherent phonon on mathematica before I leave for Kyoto.

To Do:
- Prepare a report/powerpoint presentation on what I have done so far. This presentation will be given to Kono-sensei when he visits next week, and a first draft should be shown to Saito-sensei tomorrow morning before he leaves for Taiwan.
- Make travel plans for Shikoku and the south
- Edit for English Tatsumi-san's work when he gives it to me. 

**June 29 [#rep20120629]

This morning Nugraha-san came and helped me to consider my project. I am currently trying to finish a mathematica program to solve and animate the solution to the inhomogeneou wave equation. I have already done this for the 'D'Alambert's' solution plus a term to account for the force. However, the animations produced are not so satisfactory and i would like to solve the wave equauation by using fourier series. Online I found detailed notes from a math source which solves the inhomogenous heat equation with fourier series and gives the general idea of how to solve the wave equation, but does not provide a final formula. I tried to follow the math to create a fourier series solution to the wave equation but was unsuccessful, however, if I read through the mathematics notes carefully I can probably figure it out.

Nugraha-san also gave me some papers to read about the coherent phonon/RBM. YOu have already mentioned this but he also said that it would be desireable to make a mathematica program to output a carbon nanotube with its diameter based on the n,m chirality. If I approximate the nanotube as a cylinder and do not think about graphics this is relatively straightforward, and I will try to make this this weekend.

Tapsanit-san came in during the afternoon and taught me about cutting lines, the K1, K2 vectors of the carbon nanotube, and the boundary conditions of the nanotube in the T and Ch directions.

Finally, I was considering the problem of how the velocity of the wave changes as a function of q. At Tatsumi-san's recommendation I asked Kuramoto-sensei about this problem, who said that I could not use the simple wave equation that I've been using, and instead I must consider the Lagrangian and the Hamiltonian. This seems like a difficult problem and I will think about it only after examining a constant velocity case. However, it is interesting and it would be nice to get to this point.

**June 28 [#rep20120628]

This morning Tatsumi-san and Ominato-san came and helped me on my problem of waves. The problem is as follows: We are looking for solution(s) to the 1D wave equation that consist of a force that is applied and creates a disturbance. The disturbance will then propogate in either direction along the string as a function of time, with periodic boundary conditions.

I have studied several solutions to the wave equation, such as Professor Stanton-sensei's in his paper, a similar solution I found on a math courses internet website, and another solution based on fourier series. Upon inputting forces, (and initial positions and velocities usually zero) I am able to animate/plot a wide variety of functions. However, the solutions generated don't make physical sense based on the scope of the carbon nanotube problem. The problem is also difficult because the velocity of the waves is a function of q and $\omega$. This, along with the requirement of periodic boundary conditions are currently stumping me. I have never studied Green's function and am learning about Fourier Transforms, however for the present I don't know enough to be able to solve all these problems analytically. Ominato-san and Tatsumi-san helped me try to derive some solutions this morning, but we did not find anything new.

This afternoon Hadeo-san gave me data he had calculated for the RBM q vs $\omega$ plot, and I worked on calculating a parabolic regression based on the method of least squares. I built a mathematica program to quickly calculate the coefficients to the parabolic regression based on input data. However, I still haven't figured out how to export the data (which I put into microsoft excel) into a mathematica matrix. I don't feel like typing in the 800 data points by hand... but once I find how to import data we will generate a parabolic regression. Additionally, because I converted the data into excel it's easy to get a regression by using excel's own regression operation, which presumably uses the method of least squares. However, I'd like to do the calculation in mathematica because it's more legit, and a good excersize.

I think that it is time to move further along in the problem of RBM. The full understanding of waves is still alluding me. I was thinking that it may be good to read more about RBD and coherence in general, because I still don't have a solid mathematical understanding of these concepts. Also, I think we're trying to model some experimental data. Perhaps it will be helpful to look at this data, because it will give me a better idea of what types of waves to search for.

**June 27 [#rep20120627]

This morning Tapsanit-san came and continued to teach me about the graphene lattice. We went over the Ch and T vectors, and how they defined a unit cell of the carbon nanotube. We then derived the K1 and K2 vectors, which are important because they help to define the Brillouin zone of the nanotube. Tapsanit-san gave me the task of deriving the values we talked about for a zig zag nanotube. (He showed me an armchair tube this morning). I am close to understanding the Brillouizn zone of the nanotube, and based on the chirality.

For the rest of the day I considered the problem of the equation of motion with the inhomogenous force term. The solution Hasdeo-san and Nugraha-san helped me find yesterday afternoon proved to be incorrect, and I continued to try to solve the equation. Later in the day I read through Professor Stanton-sensei's solution to this problem presented in the paper that he sent to me: Coherent Acoustic Phonons in GaAs... (equation 7) with the help of Hasdeo-san we followed Stanton-sensei's solution for an impulsive and displacive force... and the solutions we found don't make physical sense (the wave is never returned to it's original position, even when the force is very quickly equal to zero). Additionally, the solution provided which we graphed was given in terms of Green's function. The Green's functions for the two cases provided by Stanton-sensei were given, but I do not know how to derive the Green's function for other intitial forces.

Also in the afternoon Hasdeo-san reviewed the Method of Least Squares with me, and tomorrow I will use this method to fit a parabola for the RBM phonon mode, which he has calculated on Fortran.

I then reconsidered solving the equation of motion by hand. Assuming I did the fourier transform of the initial equation correctly (which I should re-check with others tomorrow), the inverse fourier transform to find the solution to the wave does not seem to converge. The situation becomes more complicated when we consider that c and $\omega$ are both functions of q. In this case I think the fourier transform I have been working with is incorrect, but I do not yet know how to solve this problem.

I've been learning a ton but am starting to get a little worried that I won't have much to talk about at the Kyoto mid term meeting. This wave problem is getting a little bit stale and I'm eager to solve it and move on to further parts of the project.

**June 26 [#rep20120626]

Today I continued to consider the problem of solving for the equation of motion Q(x,t). In the morning Nugraha-san continued to help me think about this problem, and then I worked on it. I attacked the problem by considering the Fourier series transformation of the equation of motion, isolating the fourier transform of Q, and then transforming the equation back into real space to find a solution for Q. After having learned about Fourier transforms over the last few days it was relatively simple to isolate the fourier transform of Q in terms of the transform of an applied force, w and q. However, the inverse fourier transform is proving tricky. Mathematica did not yield a solution, and I was unable to complete the problem by pencil and paper analysis.

Additionally, I have been learning about delta functions, and tried among other things to model forces as a delta function (of time) multiplied by another function of space (a Gaussian function of x, for instance. 

An hour ago I brought the problem to Nugraha-san and Hasdeo-san, and we agreed that the necessary integration was difficult to solve analytically and could not be solved fully by Mathematica. However, we split the integral into smaller subsections, some of which we evaluated by Mathematica and some by hand. Hasdeo-san and Nugraha-san were able to obtain a final answer on Mathematica that included an error function but was graphable on MAthematica. Good news!!! However, I still need to go back through our work and make sure there are no errors. I also need to examine the plot/animation of the function we found on mathematica, to make sure that it makes sense. Finally, I need to include constants such as alpha (describing the width of the Gaussian) into our integration, because we generally assumed the constants to equal one.

I considered a few other small problems and read a little bit, but trying to solve this equation is basically all that I did today.

**June 25 [#rep20120625]

This morning I skpyed with Professor Stanton-sensei, and he gave me some helpful tips on solving the differential equations of motion. He also gave mentioned to me the solutions to the equation of motion in his chapter on the coherent phonon in GaAs. Later I read through this section in the chapter. I also talked to Nugraha-san, who said that it would be helpful to derive for myself equations 2, 4 and 7 in Stanton-sensei's chapter. I worked on this problem but was unable to derive the solution. I believe that I need to understand Green's function and/or the laplacian transformation. Perhaps I can learn these soon. 

Once I understand the solution of Q(t) I can expand to a solution Q(x,t). It is my goal to be able to do this as quickly as possible, and definitely before the end of the week. I did make a mathematica plot of Stanton-sensei's equation  corresponding figure 9 in the chapter (implusive and displacive forces).

Today I also reconsidered the problem of approximating the solution to the equation of motion for an applied force as a series of polynomials of different order all multiplied by a Gaussian with respect to time (as proposed by Saito-sensei in week 1), each term with a different coefficient found by a recursion formula. Earlier I had graphed the solution to Q(t) by considering by plugging in the derived coefficiets, and the result was strange. Part of the problem was that I had neglected to include the homogenous solution -Sin[x] to the answer. However, my results were still strange. The behavior of the series was very different with 10 terms, 20 terms, or 50 terms, and the 50 term result was strange. I don't know if the series will converge to an answer that matches the messy (but accurate) solution given by the DSolve function of mathematica, but more likely there is a mistake in my derivation of the recursion formula (although I have checked my work several times). 

I also worked a little bit on Hasdeo-san's homework of building a program that uses the Runga-Katta method, but did not complete the program. Today was a little bit frustrating because although I worked hard I didn't solve any problems, like solving the wave equation. Hopefully I can learn about the necessary mathematics soon because it would be good to be able to understand and analyze coherent motion the week before Kyoto, so I can think about it and then present it to the other NanoJapan people.


**June 22 [#rep20120622]

Saito-sensei and Nugraha-san explained to me the method of Fourier transforms this morning. The method is very useful for solving differential equations, especially the wave equation with an applied force. Nugraha-san worked me through how to find the fourier transform of u as a function of the fourier transform of F, and c, q and $\omega$. However, I still have not found a formula for u. Next week it is very important that I find a solution for u(x,t) and begin to graph this equation.

We ate a delicious lunch of soba noodles and tempura.

In the afternoon Tap-san taught me more about fortran and worked me through how to create a program to evaluate a simple mathematical function and display its results in command prompt. He then continued to teach me about the reciprocal lattice, and the vectors Ch and T. We talked about the periodic boundary conditions of the Brillouin zones, and values of k1 and k2 in the Ch and T directions, respectively. He also showed me how the area of a graphene hexagon is a1 x a2 which I had not yet realized.

To do:
- Solve the wave equation for an applied force. Study fourier transforms to understand how this is done, and use mathematic to help with specific calculations.
- Read the chapter on Fourier transforms given to me by Nugraha-san.
- Complete the fortran assignment for Hasdeo-san. (doing the Runga-Kutta method)
- Read Tap-san's thesis and check the english.


**June 21 [#rep20120621]

Tatsumi-san and Ominato-san taught me about LaGrange's equation, which is a generalized form of Newton's law F=ma. The equation is really neat because it holds under any coordinate system, and perhaps once I understand it better I can use Lagrange's equation in my calculations.

I went to Japanese class, and then finished preparing for my presentation. Presenting was fun and a good learning experience. There were some mistakes in my presentation and ways I can improve my style of presentations, but it was good practice so next time I can improve. Towards the end of the presentation Saito-sensei presented to us the problem of finding the wave equation solution that is a seried of sin and cos functions, but doing this when a force is applied.

Now that I have presented I will develop a schedule for the rest of my NanoJapan project:

Weeks:

June 25-29: Solve 1D wave equation for an applied force. Do this using Fourier series and appropriate initial conditions, animate in mathematica. Look for coherence

July 2-6: Consider the wave equation solution as it relates to the nanotube. For instance, consider when wave speed c is a function of k. Input numerical values of constants into the equations, so that more realistic model can be developed. Prepare for Kyoto presentation.

Kyoto, Midterm meeting

July 9-13: Prepare for presentation to Kono-sensei when he visits. Find some coherent motion. Analyze the coherent motion and experimet with a variety of applied forces. Expand wave equation to consider not only the 1D case.

July 16-20: Begin to write up results. Hopefully I'll have some interesting results relating to coherent motion. Expand 1D results to a 3D lattice if possible.

July 23-27: Write up results. Enjoy the last week in Sendai.

**June 20 [#rep20120620]

This morning Tapsanit-san continued to teach me about the interaction of light with matter. If a molecule that is vibrating at a certain frequency is hit by E&M radiation of the same frequecny, then the light will be absorbed. We derived a formula for polarization, and derived the real and imaginary parts of epsilon, the complex relative dialectric constant.

In the afternoon Hasdeo-san continued to teach me about Fortran and how to make graphs. Very importantly, he taught me how to solve differential equations numerically using Fortran. We made a program to solve a simple differential equation using Euler's method, and then discussed the more exact Runga-Katta method. Hasdeo-san also spent time helping me think about my Mathematica wave animations, and how I can model a wave propogating due to a Gaussian force.

To Do:
- Prepare for the presentation tomorrow!!!
- Fortran programming assignment for Hasdeo next week: Find solution to equation by Runga-Katta method.
- Understand to mathematics of waves.

**June 19 [#rep20120619]
Today was a very productive day, and heavy on the math. My brain is very tired.

First thing in the morning Saito-sensei explained to me how to access the \\flex server from my laptop so that I can view previous group member's presentations as I prepare my own for thursday. Saito-sensei also showed me where group photos are kept.

Nugraha-san then came and gave me tips on how to prepare a presentation for Thursday. He also talked to me more about the mathematics of waves. He explained the difference between phase velocity and group velocity, and described that phase velocity $v_p=\omega/k$ while $v_g = \partial \omega / \partial k$. He also gave me 4 photocopied chapters about waves, the Schrodinger equation and gaussian wavepackets. Although some of the material was more advanced than necessary there was a ton of useful information contained in the chapters and I spent most of the day studying them. I do not yet have a strong understanding of the Schrodinger wave equation and its solutions, or of Gaussian wavepackets, but my understanding of them is greatly increased. Towards the end of the day I began to graph Gaussian wavefunctions (and their Real and Imaginary parts) on Mathematica. 

Florian-san also came in during the afternoon and talked to me more about POV-Ray. He also gave me photocopied solutions he had made to several differential equations with applied forces. I also showed Florian the Carbon-Nanotube source code that I had found on the mathematica website.

To Do:
- Prepare a powerpoint presentation for Thursday
- Continue to study Gaussian wavefunctions from Nugraha-san's chapters, and continue graphing them on mathematica.
- If I have time, improve the mathematica carbon nanotube source code so that I can make an animation where radius changes as a function of time.


**June 18 [#rep20120618]
Week three begins. Professor Stanton-sensei skyped me in the morning, and sent me a paper of his about the coherent phonon that solves the equation of motion given an applied force. The paper is complicated for me because it discusses quantum mehcanical effects, and electron phonon interactions which I have not learned about, however sections of it are very useful to me. 

I continued to work on animating functions of applied forces in mathematica. While the mathematics I'm using seems to be correct, the waves created still do not behave as I imagine that they physically should. I still need to fully understand the solutions to the wave equation, and when discussing the equations to Saito-sensei in the afternoon I realized that I should understnad the solution well enought to be able to explain it to others.

When I became tired of thinking about differential equations, I began to think about the animations I will eventually create. I found some source code on the Mathematica website that gives an animation of a carbon nanotube. The animation has an interface so that the user may vary the length of the chain that is displayed, and the number of hexagonal pieces along the circumference. I modified the program so that the user could also vary the radius of the tube, but I would like to go further so that the radius of the tube can be set to change as a mathematical function of time. For instance, when I find an equation to describe the radius of a nanotube it would be neat to be able tp present it on an actual nanotube itself. Just an idea that I had. POV-Ray software would be ideal but probably much harder to work with.

I met again with Saito-sensei who taught me that the speed of a phonon is the slope of the optical/acoustic mode curve, which I had not previously realized. I can use this relation to eventually describe the speed of my waves as a function of x, the wavevector. We continued to talk about mathematica and solving the equations of motion, and I now have a more direct understanding of how I can go about understanding the problem that has been given to me.

To Do:
- Understand the solution of the wave equation well enough so that I can explain it confidently to others.
- Solve the equation of motion for the force given to me by Saito-sensei. Begin to vary the speed c as a function of x.
- Begin to prepare a powerpoint presentation for Thursday. I'm presenting what I have been working on, and what my motivations are and what I have learned so far.

**June 15 [#rep20120615]

This morning I continued to think about how to create an animation of 1D waves by using Mathematica. I found a database of Mathematica Demo's that already provided many animations involving waves, in addition to several animations that provided pictures of graphene and carbon nanotubes. One of the animations described a string that had been pulled back to a certain point. Other animations described Gaussians or traveling waves. [#wcd68afa]

Next, Nugraha-san came in and continued to teach me about waves. He wrote the Schrodinger equation and defined the Hamiltonian. In the Hamiltonian, he defined V(x)=(1/2)k*w^2*x^2 = the potential of a harmonic oscillator. He provided a solution to this wave equation however this weekend it is my task to find the coefficients. Nugraha-san also explained to me that coherence is the when the superposition of waves is such that the overall wave created in constant in space and time. This is what I will attempt to model in the nanotube.

The lab ate a delicious lunch of soba and vegtable tempura.

After lunch Tapsanit-san continued to teach me the theory of optics. We reviewed the complex refractive index n~, the extinction coefficient K and the absorption coefficient $\alpha$. The reflectivit R is a function of K and n. We also defined the dielectric constant e=n^2. We discussed dipoles and the polarization P which is the amount of dipoles per unit volume. Tapsanit-san gave me three problems to do which I completed.

Next Ominato-san came to go over the solutions to the problems he had given me last week. Some of the problems were long and difficult, and his solutions were very good. We solved the differential equations for the equation of motion of electrons, and related this to current. Ominato-san also taught me about the Hall effect. He then taught me about more about band gaps, and mentioned what a topological insulator is.

To do:
-Make a good animation of waves. With periodic boundary conditions. I have a short presentation next week, so I need to prepare.
-Solve Nugraha-san's wave problem.

**June 14 [#rep20120614]
This morning I continued to work on developing a code in Mathematica that solves the 1 dimensional wave equation, and then showing the wave in a Mathematica animation. I am closer to doing this but the waves I generate are not always bounded, and further study of the mathematics of the wave equation is needed.

Next Tatsumi-san came and explained to me how the  wavevector k, was related to the energy in the case of light. We found that 1eV corresponds to k=8.06*10^(3) 1/cm. Tatsumi-san also gave me the assignment of completeing one of the problems in Saito-sensei's Raman Spectroscopy book, writing the solution in LaTeX, and then uploading it the website. However, LaTeX was not working on my laptop, so after struggling with it Nugraha-san eventually reinstalled LaTeX in the afternoon. I can also use LaTeX on emacs, but it is faster and more convenient to use on my laptop.

I helped to make lunch, and in the afternoon I listened to Tatsumi-san's presentation on the Quantum Dot and carbon nanotubes. I learned about the band gap's of carbon molecules and saw some topics from quantum mechanics. In the afternoon I continued to study the solution to the wave equation. At 4:30 I played ping pong and then had tea.

**June 13 [#rep20120613]

First I put the numerically calculated constants in the series we derived for the solution to the EQM with a Gaussian force. This solution died of very quickly as a function of time, and was near zero after about one oscillation. In contrast, the solution given by the DSolve function of mathematica did not appear to die off. I should understand this difference, and perhaps there is an error in the derived EQM.

Then morning Tapsanit-san taught me about the refractive index and the complex refractice index. Materials have a refelctivity R=I$_{r}$/I$_{o)$, where I$_{R}$ is the intensity of the reflected light. They have a similar transmission coefficient T=I$_{t}$/I$_{o)$. Light slows down in a meduium, and we define the refractice index n=c/v. We define the absorption coefficient $\alpha$ so that intensity I(z)=I$_{o}$e^(-$\alpha$z). 

We defined the complex refractive index n~ = n +iK, where iK is the extinction coefficient, and then we derived that $\alpha$ = 2Kw/c = 4$\pi$ K/ $\lam$. K is the extinction coefficient and alpha is the absorption coefficient.

**June 12 [#c5892e4d]
In the afternoon I spent more time studying the solution of the wave equation, and how to find the coefficients of the wave. It seems that the wave equation and its coefficients are an application of Fourier series. I began to make an animation on Mathematica that will model a vibrating string. 

Then I accompanied my fellow lab-mates to the grocery story, and bought inexpensive groceries. I learned a lot about cooking to prepare for the party that the lab hosted in the tea room. The food was delicious and the conversation was excellent, I am very happy to be in Saito-sensei's lab.

To do: 
-Complete the animation on mathematica of the vibrating wave
-Once this is done, make a POV-ray animation
-Complete the problems from Ominato-san
-Think about the solution to the equation of motion that we approximated for the Gaussian. Why doesn't it match? Type up the derivation and perhaps I will find a mistake.

**June 12 [#rep20120612]

This morning Nugraha-san taught me more about computer programming in Fortran. He walked me through the process of making a program to calculate the optical and acoustic modes of a 1D phonon. He then showed me how to make a .dat file to plot the data in table form, and how to graph the function in xmgrace. Later in the morning Saito-sensei taught me how to modify the axis of the graph so that characters (like Pi/a) are displayed instead of units. The way to do this is to make the graph, go to plot, axis properties, special. Change a tab to 'numbers and characters', and insert characters in the appropriate window. For the Greek letters there is a special code, which I cannot remember but could be looked up online. 

Also in the morning I compared a gaussian force (f=Ae^(-t^2/(a^2)) to a polynomial force (f=t^2(t-2b)^2) and graphed these two functions to see how similar they were. I integrated each the total area of the function and set that equal to 1. I then calculated the relationship between the coefficients. The graph of these two functions side by side on the interval from 0 to 2b were very similar. The polynomial function is easier to work with because it yields a much simpler solution to the equation of motion x'' + w^2x = f(t). 

Next Saito-sensei taught me about the equation for a wave which is Qxx - c^2*Qtt= f(x,t). f(x,t) is the applied force and Q is a function of x and t. In the afternoon I solved the wave equation for f(x,t)=0 (with help from online notes from the University of British Columbia math department.) However, I also need to solve for the coefficients of the solution, which can be done using the initial conditions of the problem. 

In the afternoon Florian-san also taught me more about differential equations relating to springs, and photocopied some of his notes so that I may use them as reference. He also taught me a lot about how to use POV-Ray animation software, and now I've seen how to make objects move in animations. Florian has a lot of code and information compiled on his website and I may use this in order to help me create my own animations.

To Do:
-Solve for the coefficients of the solution to the wave equation.
-Consider when a force is applied to the string. Look for solutions to the wave equation force different force functions, eventually working up to our approximation of a Gaussian function.
-Finish the problems given by Ominato-san
-Type up the derivation of the recursion formula used to find coefficients in our approximation of a Gaussian. Plug the solution using these coefficients into a computer and see if it matches reality.
-Type up notes from today.

**June 11 [#rep20120611]

Today Saito-sensei taught me how to find a solution for the differential equation of a Gaussian. I learned a lot today about Gaussian functions, and about using LaTeX and Mathematica.

Saito-sensei taught me how to use the equation feature of LaTeX so that my equations will be numbered sequentially. He also found a function in Mathematica that converts an equation in Mathematica into LaTeX code, which will really help speed things up in the future. 

The Mathematica function is: TeXForm[f(t)] 
Make sure to always use a capital letter X. Insert the function inside the brackets and LaTeX code will be created. Also, TeXForm[%] may be used if the TeXForm command is in the same section as the function with the % sign taking the place of the function.

On LaTeX I created a summary of the solution we found and include the recursion formulas that we found for the coefficients:

Here we have the equation of motion for an applied force, which we set to be a Gaussian:
\[ \ddot{Q} - k\dot{Q}+\omega^2Q = F(t) \]
\[ \ddot{Q} - \omega^2Q = A e^{-\frac{t^2}{\alpha^2}} \]

We Assume a solution of form:
\[ \Sigma_{n}^{\infty} C_{n}t^{n}e^{-\frac{t^2}{\alpha^2}}+A_{n}\sin (\omega t)+B_{n}\cos(\omega t) \]

For Odd Coefficients we have where m=2n+1, m grater than or equal to 1:
\[ C_{2m+1}=\frac{1}{4m^{2}+2m}[C_{2m-1}(\frac{8m-2}{\alpha^2})-\omega^2C_{2m-1}-\frac{4}{\alpha^2}C_{2m-3}] \]

And for Even Coefficients where m=2n, m greater than or equal to 2:
\[ C_{2m}=\frac{1}{4m^{2}-2m}[C_{2m-2}(\frac{8m-6}{\alpha^2})-\omega^2C_{2m-2}-\frac{4}{\alpha^2}C_{2m-4}] \]

Saito-sensei and I also talked about how best to model the applied force. A Gaussian force is ideal but very complicated mathematically and also takes a long time to approach its peak. (It never truly starts and zero, and the closer to zero is starts the longer it takes to reach its peak.)

Saito-sensei suggested modeling the force as either a polynomic equation of degree 4, or something of the form f(t)=t*E^(-t^2). We talked about these approaches and I will study these functions more. Additionally, we began to think about the wave as a function Q(x,t) which satisfies the wave equation. 

To Do:
-The Fortran assignment for Hasdeo-san
-The problems for Ominato-san
-Think about the wave equation for Saito-sensei, and the solutions for various types of forces
-Continue to document my work on LaTeX
-Check the accuracy of the coefficients generated by the recursion method by plotting this functions next to the Gaussian. (Although we will probably not end up using this Gaussian)


**June 8 [#rep20120608]
This morning Nugraha-san came and we talked a little bit about the reciproical lattice. We then attacked the problem of the differential equation given by a Gaussian force. After working for a while we did not find a satisfying solution, and it's possible that for this equation a simple solution doesn't exists. If we had more boundary conditions it's possible that the solution would be simpler.

We then talked a little bit about Fortran and Nugraha-san showed me a website from Boston University that had a lot of good Frotran tutorials.

In the afternoon Tapsanit-san came in and we constructed the Brillioun zone of both the square lattice and of graphene. We obtained the symmetry points (Gamma, M, K and K') in the Brillioun zone. We compared the area of the Brillioun zone with that of the square lattice.

Finally, we looked at the Fortran task given by Hasdeo-san and we created a program that gives a cascading array of integers, as desired. Tapsanit-san printed out some pages on optical processes that I can read this weekend if I have time.

To Do:
-Complete the fortran programming assignments given by Hasdeo-san.
-Solve the diffeq for a parabolic force, and then consider the Gaussian case
-Learn LaTeX
-Solve the problems given by Ominato-san
-If I have time, work through the BU Fortran tutorial 
-If I have time, read the pages given by Tapsanit-san

**June 7 [#rep20120607] 
In the morning Tatsumi-san and Ominato-san explained to me the reciprocal lattice of graphene. First we derived the unit cell of graphene. We derived the reciprocal lattice and found then found the Brillouin zone of graphene, which is also a hexagon but rotated by 90 degrees.

To somewhat review what Tapsanit-san taught me yesterday we then talked about the wavevector (k) in the reciprocal lattice, and frequency as a function of k. We showed that this satisfied Bloch's theorem and described the periodic boundary conditions. Then, we talked about the acoustic mode vs the optical mode. We showed that the number of modes will increase if we increase the number of atoms in a unit cell. In the problem where two different types of masses are connected with identical springs, the lower valued function is the 'acoustic mode', where all the atoms vibrate in the same direction at the same value of time. The higher valued function is the optical mode, where some of the atoms are pushed towards each other while other atoms are pushed away from each other. 

At 10:30 I accompanied Hasdeo-san to Japanese class where I learned the ta form of verbs. For instance: "Kyoto e itta koto ga arimaska?" = "Have you ever been to Kyoto?". Iie, arimasen. Watashi wa Kyoto ni ikitai.

In the afternoon I went to the group meeting where Simon-san presented his animations and some differential equations. His POV-Ray animations were very cool and also really helped to illustrate the acoustic vs. optical vibrational modes. Tomorrow we made plans to discuss some of the differential equations he had been working with, because I need to know them as well.

Later in the afternoon, Ominato-san and Tatsumi-san returned to teach me more physics. Ominato-san had written a long sequence of very good problems some of which we worked through. We defined the Drude Model of electron conduction, and wrote the equation of electron motion including a frictional term:
\[ m\frac{dv}{dt}=e(E+\frac{1}{c}v X H)-(\frac{m}{T})v \]

We derived v(t) when H=0, and then for the steady state solution (dv/dt=0)
We then Derived $\sigma$ (the conductivity) based on the relation: 
\[ j=nev= \sigma E \]

Next, we derived v(t) for a more general case and put into matrix form the equation:
\[ j= \sigma E \]

Ominato-san explained the classical Hall effect and we breifly talked about special relativity. The energy of a particle with mass is given by 
E=+/- sqrt((cp)^2+(mc^2)^2) were E is a parabolic function of p. However in graphene electron behave as if they have no mass and E=+/- cp. 

To Do:
-Finish the problems given by Ominato-san, and rewrite good solutions for the problems we went over this afternoon.
-Consider the differential equation given by Saito-sensei, study his email and study the equation in more detail.
-Finish the graphs for Nugraha-san.
-Finish the graphs for Hasdeo-san and the cascading number Fortran program.

I learned a tremendous amount of theory today. My understanding of the lattice has made good progress over the last several days.


** June 6 [#rep20120606] 
- Tapsanit-san explained the Brillouin zone of a 1D lattice, as well as 2D and 3D square lattices. We discussed what k (the wave vector) is and constructed the reciprocal lattice. We then talked about how Bloch's wavefunction has periodic boundary conditions. We have a finite number of k, equal to N, the number of atoms. On Friday we will construct the Brillouin zone of graphene. Later in the morning I fixed the format of the equations I'd posted on PukiWiki and finished the graph of the different modes of phonon frequency as a function of k. I did this graph on SciDAVis and am not entirely happy with the axis format, so I should do it on Xmgrace too.

In the afternoon Hasdeo-san explained further about Fortran programming language. After working through a program he made he gave me several assignments to do. Hasdeo-san also helped me learn to navigate the directories of command prompt.

To Do:
1. Get a correctly formatted w(k) graph to Nugraha-san AND learn about/explain which is the optical mode and which is the acoustic mode.
2. Solve Saito-sensei's differential equation by learning about Fourier transforms.
3. Solve Hasdeo-san's problems: 
   - plot the cascading list of integers
   - graph two fxns (different parabolas) using Fortran
   - output fxn data in column form

** June 5 [#rep20120605] 
- Nugraha-san explained the problem of an infitie 1D series of masses connected by springs. We solved for the value of w^2, although there was a small error in my calculation. In excel we graphed the value of w^2 as a function of k, the wavevector. In the afternoon we graphed the (incorrect) data generated in excel into an xmgrace graph. Nugraha-san also explained how to use the simpler SciDAVis software. 

Also, Florian-san showed me how to use the POV-Ray software. We installed it on the desktop machine and I worked through the basic tutorial.

- The homework is: by friday complete a graph of w(k) for the 1D phonon problem, determine the max/min for each branch, and interpret this physically. Which branch is optical and which is acoustic? 

Also, solve the diff eq for a Gaussian.

If I have time consider the infinite 1D series of spring mass problems, but in the case that all masses are the same, but the spring constants are different.


** June 4 [#rep20120604]
- Sato-sensei showed me around the campus and installed Xming on the desktop.
- Hasedo-sensei introduced me to Fortran language and we wrote a very simple program.
- Solved the differential equation for f(t)=(bt)/a=x''+w^2x
   Solution is: f(t)=(b/(aw^2))(t-(1/w)sin(wt))
   Could look into the eqn and new initial conditions for t>a

to do: 
- Solve the same diff eq for f(t)=Ae^(-t^2/a^2)
- Clean up the solution for the reciprocal lattice from last week

** Task from Tapsanit (Bloch's theorem and reciprocal lattice) [#task1]
+ Check that the $\Psi(\vec{r})= e^{i\vec{k}\cdot\vec{r}} u(\vec{r})$ satisfies the Bloch's theorem.
-- Answer:
Bloch's theorem is:
\[
T_{\vec a_i}\Psi = e^{(i\vec{k}\cdot\vec{a_i})}\Psi
\]
Here $T_{\vec a_i}$ is a translation operator: $T_{\vec a_i} \Psi(\vec{r})= \Psi(\vec{r}+\vec{a}_i)$.
Then, also note that $u(r)$ is a periodic function. We have $u(\vec{r} + \vec{a}_i) = u({\vec r})$ because $\vec{a}_i$ is a unit lattice vector. Thus:
\[
T_{\vec{a}_i} \Psi(\vec{r}) = \Psi(\vec{r} + \vec{a}_i)
\]
\[
= e^{(i\vec{k} \cdot (\vec{r} + \vec{a}_i))} u(\vec{r} + \vec{a}_i)
\]
\[
= e^{i\vec{k} \cdot \vec{a}_i} [e^{i\vec{k}\cdot \vec{r}} u(\vec{r})]
\]
\[
=e^{i\vec{k} \cdot \vec{a}_i}\Psi\left(\vec{r}\light)
\]
The Bloch theorem is satisfied.
+ Derive the unit vectors of reciprocal lattice, $\vec{b}_1$ and $\vec{b}_2$, of graphene in Eq. (2.23) of CN book using $\vec{a}_1$ and $\vec{a}_2$ in Eq. (2.22).
-- Answer: Here we use
\[ \vec{a}_i = {\rm unit~vector~of~real~lattice} \]
\[ \vec{b}_i = {\rm unit~vector~of~reciprocal~lattice} \]
\[ \vec{a}_i\cdot\vec{b}_i = 2\pi \] 
\[ \vec{a}_i\cdot\vec{b}_j = 0 \quad {\rm if} \quad i \neq j \]
From Eq. (2.22) of CN book: 
\[
 \vec {a}_1 = \left(\frac{\sqrt{3}}{2}a,\frac{a}{2}\right), \quad
 \vec {a}_2 = \left(\frac{\sqrt{3}}{2}a,-\frac{a}{2}\right)
\]

\[ \vec{a}_1\cdot\vec{b}_1 = a_{1x}b_{1x}+a_{1y}b_{1y}=b_{1x}\frac{a\sqrt{3}}{2}+b_{1y}\frac{a}{2}=2\pi
\]
\[ \vec{a}_2\cdot\vec{b}_1 = a_{2x}b_{1x}+a_{2y}b_{1x}=b_{1x}\frac{a\sqrt{3}}{2}-b_{1y}\frac{a}{2}=0
\]

\[b_{1y}\frac{a}{2}=b_{1x}\frac{a\sqrt{3}}{2}
\]
\[b_{1x}\frac{a\sqrt{3}}{2}+b_{1x}\frac{a\sqrt{3}}{2}=2\pi
\]

\[b_{1x}=\frac{2\pi}{a\sqrt{3}}
\]
\[b_{1y}=\frac{2\pi}{a}
\]

\[
 \vec {b}_1 = \left(\frac{2\pi}{a\sqrt{3}},\frac{2\pi}{a}\right), \quad
 \vec {b}_2 = \left(\frac{2\pi}{a\sqrt{3}},\frac{-2\pi}{a}\right)
\]
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