Lecture -- Lattice Vectors

[@afafsgdchdhdg]

Take my bow. This is the best explanation of reciprocal lattice vectors that I have seen

[@empossible1577]

Very happy to hear it helped you!

[@baigao6062]

this is the best explanation for reciprocal lattice I have ever seen (as someone who is difficult to accept the definition of it). Thank you so much Prof. Rumpf

[@empossible1577]

You are very welcome! Thank you!!

[@abhishekjoshi1677]

Dear Dr. Rumph you are a life saviour , i am a taking a course in condensed matter physics, you lectures which are highly visual gave a geometric picture on what is going on. Please keep them coming.

[@empossible1577]

That is great to hear these are helping you! Thank you!!!

[@SeyedAgent47]

It was phenomenally good bor TNX

[@shiqinghuang2981]

Dear Dr. Rumpf, these are the best tutorials I have ever seen! I am studying acoustic metamaterials, but I focus on in the fields of acoustic. Will all the theories work in the range of sound? Looking forward for your reply.

[@empossible1577]

As far as I know, all of the phenomena that EM people investigate manifests itself acoustically as well. Some obvious exceptions are things dealing with polarization. But lattice vectors, band gaps, self-collimation, dispersion, etc., are the same for all waves. Great to hear the lectures are helping you!

[@okeokomo9613]

Hello, Where does the 2pi come from? at minute 4:45.

[@empossible1577]

It is very convenient to calculate a cosine function as cos(kz), where k essentially controls how quickly the cosine function oscillates. If the wavelength is lam, then k=2*pi/lam. Periodic structures can be described with cosine functions according to their grating vector K. In this case K=2*pi/L where L is the period of the grating. Reciprocal lattice vectors are grating vectors so it is convenient to define them the same way with the 2*pi term. Also, there is a very intimate relationship between wave vectors k and grating vectors K and the 2*pi needs to be there to be consistent.

[@okeokomo9613]

@@empossible1577 Thank you!

[@sayanjitb]

@@empossible1577 In your words, reciprocal lattice vectors are grating vectors, then here why did you denote grating vectors as K instead of T?

[@empossible1577]

@@sayanjitb Both notations are used in the literature and I want to connect them for you. Grating vectors are most often seen when discussing diffraction gratings. Lattice vectors are seen mostly in the context of photonic crystals, metamaterials and solid state electromagnetics.

[@amithasanarpon2527]

Dear Dr. Rumpf, I am an undergrad student in nuclear engineering. I am trying to learn the finite difference method for PDEs. Since you are a specialist on this topic, I want to ask for some advice. This is why I am commenting on the latest video instead of on a video related to FDM. My actual target is to learn solving systems of 3D time-dependent multigroup neutron diffusion equations. (Neutrons are sorted into some groups according to their energies. Neutron fluxes are the dependent variables of the system. They depend on x, y, z, and time. Fluxes from each group act as a source for other groups. A PDE is formulated for each of the groups according to neutron diffusion theory.) As of now, I have self-studied ODEs, PDEs, and some numerical methods for ODEs like FWD Euler, BWD Euler, Trapezoidal method, RK4, 3/8th Rule, etc. I have written solvers for each method in Python. I have also studied a bit of curve fitting. Now I am trying to get into FDM. I found your channel and really like your style of teaching. But the content related to FDM is scattered across many videos and lecture series. Also, my intended field of study is not EM so I'd like to learn the FDM method as a general mathematical tool for solving PDEs rather than learning to apply it to a particular problem. Therefore I am asking for some guidance regarding how to best utilize the content on this channel to learn the FDM method as a general mathematical tool for solving PDEs. Where should I start and which videos should I go through? Please take a minute and give me some advice, thanks.

[@empossible1577]

Thank you! Sounds like you have already done quite a bit. I think the best presentation of FDM that I have is in my Computational Methods class. See lectures 6b, 6c, and all of Topic 7. From there, you can go into my FDTD course or CEM course. empossible.net/academics/emp4301_5301/ empossible.net/academics/emp5304/ empossible.net/academics/emp5337/ If you are completely new to time-domain finite-difference, here is an online course and the first half is completely free. The benefit of this one is that it is meant for the complete beginner. empossible.thinkific.com/courses/1D-FDTD

[@amithasanarpon2527]

Thanks for the reply. Dr. Rumpf. I have gone through topics 5a, 6a, 6b, and 6c. Now I have started topic 7a. I plan to go through 7a-7f and then watch Lecture 9 of Computational Electromagnetics playlist. I will post an update comment once I finish. Thanks again for taking the time to reply to me.

[@empossible1577]

@@amithasanarpon2527 Excellent! The method I teach for implementing finite-difference method is pretty neat. It has allowed my team to do some incredible things. Just keep in mind that it may not be the best when your grid size becomes huge.

[@MagicTech777]

How did you calculate the 2D reciprocal lattice vectors? And what exactly is the matrix representation that you have shown? 14:17

[@empossible1577]

Calculating the reciprocal lattice vectors is summarized on Slide 23 around 13:45 in the video.

[@empossible1577]

The equations at 14:17 are similar to those at 13:45, but specifically for two-dimensional lattices where there are only two reciprocal lattice vectors and those vectors have no z component. The "matrices" are actually column vectors containing two components of the translation vectors. The upper-left equation says that to calculate reciprocal lattice vector T1 you would calculate as T1_x = 2*pi/|t1xt2| * t2_y T1_y = - 2*pi/|t1xt2| * t2_x

[@MagicTech777]

@@empossible1577 Yeah got it. Thanks 👍

[@athul_c1375]

This video really helped Can you make a playlist It's hard to find other videos on this topic

[@athul_c1375]

I am a physics student what is this course called in your website?

[@athul_c1375]

kzbin.info/aero/PLLYQF5WvJdJWBrjspvP1xKjiy-3959SIm

[@empossible1577]

Happy to help! I recommend accessing the course material through the course website. The information is organized, you can download the notes, get links to the latest version of the videos, etc. empossible.net/academics/emp6303/ The material you are watching now is in Topic 4. Hope this helps!!

[@wabidemeke3067]

a bit silly question but here you go, you said that the reciprocal lattice not what we can touch but it just talk about the symmetry of direct lattice. Then what is the purpose of representing already symmetry direct lattice into another symmetry reciprocal lattice vector? I have noticed that this representation give computation benefits in Fourier space representation. Is that it? or something I am missing?

[@empossible1577]

I think you got it! The reciprocal lattice vectors are grating vectors (i.e. 2 pi divided by the spacing of planes). This means there is a very intimate relationship between the reciprocal lattice and periodic structures and waves analyzed in Fourier space. While the math takes a bit to understand, it is quite elegant. Keep watching the series of lectures in Solid State Electromagentics. I also recommend accessing the videos through the course website. It has the latest version of the notes (error corrections), links to the latest videos, MATLAB implementations and other resources. empossible.net/academics/emp6303/

[@wabidemeke3067]

Thanks. I will do that.

[@sayanjitb]

You mentioned that non-primitive lattice vectors are an integral linear combination of primitive translational vectors, but I learnt that when lattice vectors are expressed as non-integral linear combinations of primitive vectors are termed as non-primitive lattice vectors. Is it right?

[@empossible1577]

I don't think a non-integer combination of primitive lattice vectors is any kind of lattice vector. I am not even sure what meaning such a vector would have because it would no longer align with the lattice or describe any of its repeating planes.