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

updated 2012.6.1 by Nugraha

- 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

- 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

Time and Day | Mon | Tue | Wed | Thu | Fri |

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).

- 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.

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

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 &tex(): Error! The expression contains invalid characters.; should satisfy the Bloch's theorem due to the periodicity of the atoms in solid. &tex(): Error! The expression contains invalid characters.; is sometimes called the Bloch wave function. It can be written as the phase factor &tex(): Error! The expression contains invalid characters.; times the periodic function in real space &tex(): Error! The expression contains invalid characters.;, i.e., &tex(): Error! The expression contains invalid characters.;; &tex(): Error! The expression contains invalid characters.;. It is recommended to check that &tex(): Error! The expression contains invalid characters.; satisfies the Bloch's theorem.

- What is the Brillouin zone? (not solved)
- 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 &tex(): Error! The expression contains invalid characters.; are obtained from the unit vectors of the real lattice &tex(): Error! The expression contains invalid characters.; by following relations (&tex(): Error! The expression contains invalid characters.; denotes dot product of two vectors): &tex(): Error! The expression contains invalid characters.;, where &tex(): Error! The expression contains invalid characters.; if &tex(): Error! The expression contains invalid characters.; and &tex(): Error! The expression contains invalid characters.; if &tex(): Error! The expression contains invalid characters.;. It is recommended to try using this relations by deriving the unit vectors of reciprocal lattice, &tex(): Error! The expression contains invalid characters.; and &tex(): Error! The expression contains invalid characters.;, of graphene in Eq. (2.23) of Saito-sensei's book using &tex(): Error! The expression contains invalid characters.; and &tex(): Error! The expression contains invalid characters.; in Eq. (2.22).

Can I use a copy machine or printer?

I would like to know where the library is and how to use it.

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.

Note: Format didn't carry over from powerpoint. (Eric)

- Check that the &tex(): Error! The expression contains invalid characters.; 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 &tex(): Error! The expression contains invalid characters.; is a translation operator: &tex(): Error! The expression contains invalid characters.;. Then, also note that &tex(): Error! The expression contains invalid characters.; is a periodic function. We have &tex(): Error! The expression contains invalid characters.; because &tex(): Error! The expression contains invalid characters.; 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, &tex(): Error! The expression contains invalid characters.; and &tex(): Error! The expression contains invalid characters.;, of graphene in Eq. (2.23) of CN book using &tex(): Error! The expression contains invalid characters.; and &tex(): Error! The expression contains invalid characters.; 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)

\]a ?_1=(?3/2 a,a/2), a ?_2=(?3/2 a,(-a)/2) a ?_1?b ?_1=a_1x b_1x+a_1y b_1y=b_1x (a?3)/2+b_1y a/2=2? a ?_2?b ?_1= a_2x b_1x+a_2y b_1y=b_1x (a?3)/2-b_1y a/2=0 ?b_1y a/2 = b_1x (a?3)/2 ?b_1x (a?3)/2+b_1x (a?3)/2=2? ?b_1x=2?/(a?3) ? ?-b_1y a/2=0 ? b_1y=2?/a b_1=(2?/(a?3),2?/a)

Same process gives: b_2=(2?/(a?3),(-2?)/a)

- 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:
\[

- Who gave what? (XX mins)
- What is the home work?
- The answer of the home work should be put here.
- Evaluation of the answer by teacher
- Evaluation by Saito

- task 1
- task 2
- etc...

- task 4
- etc...