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 19 [#x4a85709]
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 [#m7c191a2]
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 [#rac44700]

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 [#za78f62b]
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 [#a9bea3b9]

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.

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 [#c5892e4d]

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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