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# Lecture 16. Fourier Transforms, NMR Intro

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OK so too they were going to move on from electronic spectroscopy and talk about 40 transforms a little bit of crystallography and get started on animal so somebody asked where are we in the book at this point so were skipping around what then happens and there is a special section on Fourier transforms at all remember where but it's written and sold us so if you check out the the reading assignments on the sole those words were a little behind relative to the scandal that posted that's OK that happens but on the prejudices of the foreign transforms as a it's like a little special math section we're
going to talk about crystallography briefly that is in Chapter 9 there's there's a whole chapter on solids and you can read it if you want it's interesting on most of what's in there is pretty descriptive it's nice to know that there is not 1 of the locker room within the terms of problems and then hopefully at the end will see if we get there were going to start talking about Panama and that is I believe Chapter 12 OK so let's get started talking about 40 series Emporia transplants so how many of you have seen for a series of Fourier transforms before quite a few OK so this will mostly be reviewed but
if not so as always for for such topics I recommend Wikipedia page and the wall from website on 40 transforms and for a series of new extra practice OK so a Fourier series is basically a bunch of signs and coastlines that we can use to approximate other functions and that this works best if you have a periodic function but it works pretty well for a lot of other things also the main thing is you want to be continuous In order for there the the function to converge but it actually does work pretty well for things that are piecewise continuous as long as the function and its 1st derivative has lost properties OK so here's what this looks like soft function whatever arbitrary hopefully periodic and continuous function convergence to these to this series as you take an infinite number of terms and it's just a combination of signs and coastlines and you know when you 1st look at the is it seems kind of strange that you could have something like a square wave that has a sharp corners and things and how that I have that approximated by signs science but it works pretty well together of large number of terms in the series OK so we have these coefficients 18 B and Europe just I've written down the definition of a lot so a is the coastline terms and the takes care of sign terms and of course you need both because we have odd and even functions and see the signs and coastlines in there in some cases 1 or the other is going to drop out and we can use these things too all kinds of functions even even things that are not periodic even things that you wouldn't think this would be a very good approximation to end this turns out to be really important for a lot of things in physical chemistry so when we talk about Panama you the explanation of that's given in for attacks treats it like other spectroscopy we say OK we sweep through the frequency and look at the response that's how the early animal experiments were done in a very very early on but the fields been around for 50 years a lot of stuff as has happened at this point basically everybody does Fourier transform so you know we're going to give up holes perturbed the system and get back a time-dependent signal which we're running it for you instrument put the frequency domain this also comes up and I are spectroscopy FTIR is it's very common he probably used to these instruments and organic lavender or bring a new research on as currency in a minute it's also really important in crystallography the information that you get from a diffraction pattern is in reciprocal spacing after Fourier transform into get spatial pattern of the crystals yes and I knew it was you could start up again this series at 1 1 L In also 8 starts at 0 and the starts at 1 yet the series about the series also 0 is a 8 0 is a separate terms because you have to divide by 2 otherwise sound yeah but otherwise it said go check out the Wikipedia page BIA listed separately because of suspected to their own kind so let's look at an example so here's a square wave it's something that doesn't really look like it's very well approximated by signs and "quotation mark because it has a won a sharp corners but it it is periodically over the interval were looking at and we can use this Fourier series to describe it pretty well OK so it's period the way this is defined as 2 key not so you know that just means we were looking at this function and if we if we wait to tee not the value of it comes back to to work started in time and we can also define and angular With respect to that and that's so that's going to be important for discussion of these kinds of time-dependent signals so here is our for a series unplugging India you now we've got a T and T not because that's what were something over and so we can find this to the expression for the the Fourier coefficients and I know I'm going a little bit faster but this is the the same thing that we just looked out words completed in for this function and evaluated because it's it's pretty simple to do and I will give you some practice problems on this and on an Aymara hopefully today tomorrow OK so we can look at the ABC series for this 1 and safe well that's not function so goes to 0 which as always is a really good thing to be able to do saves us a of time OK we still have to worry about the at the sign coefficients those are automatically and so we can evaluate that I and just evaluating it for this function here's what I get so we can in fact sum up a bunch of signed terms for this square wave and this picture on the right is what we get if we take the 1st 10 terms of this Fourier series so that you can see it starts to look pretty good and there's little wiggles around the edges of it's not perfect but it's it starts to look like a pretty reasonable approximation so we can use this for all kinds of things where it doesn't necessarily look like signs in Cosenza can be the best race just laughing at it OK so now let's look at what happens if the function is not periodic and of course this is kind of important because you know where we're talking organized talk about some applications in physical chemistry where the function is periodic over some time-dependent dies away so for instance no signal relaxes its going to die out but you know it is it is periodic over some time period but not not forever and there are a lot of applications and physics and chemistry and engineering where this is important OK so if the Fourier series isn't periodic that means that the period goes to infinity so you have to
shapes OK so here's a crystallography example so this is from a relatively recent Nature paper where are the people were looking at gold rods and they wanna see what is the surface look like at the crystal structure kind of love and here's the data on the top so they have these these patterns of dots there based on the diffraction from the the gold atoms and he's look really nice and shot especially if you used to looking at the protein crystallography data because we just have these individual atoms of the electron density around is really well defined and also gold atoms have a lot of electron density compared to things like carbon and nitrogen oxygen in the things that proteins are made of so we see these really beautiful struck spots that also tells us that there are crystal is really perfect if we had a lot of Mozier city in the crystal lighting in a lot of different a lot of service scattering the spatial positions of these things we would see the spots being kind of out and not as high resolution and that so many in fact in the middle picture you can see a little bit of that the via the spots are not as sharp as as they are in similar ones but we can see that this corresponds to these regular crystal lattices but it's not giving you direct picture you have to take the the Fourier transform and there's a reciprocal relationship there OK so let's go back to where we left off last time talking about X-ray crystallography and how we can use this to get structures of molecules so we just sigh with a really simple example world we have going on is just a lattice of of gold atoms and so that's pretty easy to visualize how you Fourier transform that the diffraction pattern see work but the structure looks for something like a protein molecule that's a lot harder the unit cells are large and complex the atoms that we're looking at don't have as much electron density so there are a few a lot of fun of things going on so in order to get the day in 1st we need to grow a crystal this particular 1 is Wise's I'm not so a crystal that I grew which anyone who's worked in a protein knows that places I'm is really easy to crystallize this this is something people uses a set up sample very often and then there's this is not my data needed that but when you have your your regular repeating lattice of protein molecules you can get X-rays diffraction and again the Fourier Transform relationship told us that if we have a lot of spots going really far out in the diffraction pattern your summer so that some were looking at you were taking images of this crystal and tilting the sample will bid so we have a two-dimensional map of the diffraction and then we get the 3rd dimension by taking spices and we told the the crystal if we have a lot of thought that here that means it's a really high resolution structure that something where we just see something where we just have a few spots in the middle that's lower resolution an example of a relatively low-resolution things this fiber diffraction pattern that was for you Watson and Crick DNA structure and 1 reason it's low resolution is because it's just a fiber it's one-dimensional so we see a lot of intensity out here In 1 dimension that's that's where the fiber excesses and then across the other way we don't see a lot of intensity so it's very asymmetric whereas for this thing we have a perfect crystal and so we see on a lot of intensity in all directions OK
so here's what that looks like in 3 dimensions so in order to get the idea this is the it's the electron density should be careful not call the structure yet in Oregon electron density map from this kind of diffraction pattern we have this density function and we have to some over K in L. a which are that's the reciprocal space taxis so it's the inverse of x y and z and so we take this this kind of Fourier transform and we are able to get spatial function having to do with the periodic crystal lattice from and again here are some examples of simple crystal lattices there's a lot of this stuff in Chapter 9 of course the ones that that you wanna look out for organic molecules protein crystals are going to be much more complicated than that we have different space Troops OK so again here's a here's a rundown of the the process so you have to wrestle with periodic lattice of of molecules we Divac X-rays at which we can do because the electrons have some way of character and because we have a repeating lattice we get these nice diffraction patterns and if we do with the great transform an electron density now and then you can see here the island how people go by analyzing wireframe thing is the actual electron density that that's the part that's experimental and then you have to kind you use your imagination and fill in the particular protein side chains Q then go ahead and get the structure N I just want show you example that that yeah that was used to solve the structure of an interesting molecule also some examples of animosity arches and get there too but here's 1 really can really see the electron density in the it's a nice high-resolution structure and it looks very convincing as far as what's in there I'm unfortunately not all of them were that high resolution and it's not as it's obvious what's going on but this particular examples from Roderick MacKinnon is no Nobel lecture he got the Nobel Prize for solving the structures of somebody's eye and so this particular 1 is a potassium channel it selectively what's potassium come through and keep sodium silicate got a selectivity filter and then it has these diagnosing where there is a specific size and you see the universe the part of the helices ,comma meals are oriented toward these islands and that can shovel the the ions through and this thing can undergo conformational changes depending on how much potassium is around it such that these helices RE the clothes like this were or they open up when the channel's working and it's transmitting potassium and so
again here's what that looks like we have it has to confirmation that Kony collapsed where this spot in the middle is shown together and there's no room for the iron stove to go through and it can be opened and conducting these things so it's just I just 1 show you an example where are looking at the structure of the molecule really tells us a lot about its function and you can see here in these electron density not yeah how you can measure directly from that electron density where the the irons and the parts of martial OK so let's move
for the nuclei on some nuclei have spin greater than 0 so in the end of context you know for organic molecules we mostly look at protons and see 13 and 15 which fortunately spin one-half that makes it easy to study some of these molecules of course the most common isotopes of carbon and nitrogen has been 0 carbon 12 and fought in and out yeah just mistaken 14 is not been era has been 1 but it's difficult to see in any case 13 and in the end and 15 than one-half which means they have to orientations with respect to the magnetic field and that that makes things easier to look at since protons are also spend 1 half we can use exactly the same treatment so there's nothing new here were just talking about nuclei instead of electrons but I do want to point out a couple of things just In terms of terminology so 1 is that we have to beat plus one-half I don't value which is being called Alpha here and minus 1 beta and we're going to use this throughout we talk about a more spin states so alpha and beta are just nicknames for the that the up-and-down states of the been have nucleus and we will see them a bunch I note that al-Sadr does not equal minus data and you know we call these things wave functions but Indiana wave functions this is kind of a funny description right because it's not you know where used to the wave functions being something that looks like a real function so we have the Herbie polynomials for harmonic oscillator a spherical harmonics for a rigid rotor the Myanmar wave functions are just their functions in the sense that it's a probability mass that's all on 1 value so if something is in the office state that just means that it's in the state were exciting values plus one-half if it's in the beta state it's in the state were exciting value was minus one-half what that function is you know it's not something that's well-defined come these other things it's it's kind of a delta function were all its probability masses on that 1 state yeah think I was point out is that organized talking about an armada nuclear magnetic resonance the resonant conditions is when H new our incoming radiation equals again H. part the not so we mostly think about the main magnetic field when we talk about anymore but that parts boring that's just lining up the nuclei the part that we need to opera turn them with is the frequency fields there were putting into foot spins and change the conversation axis so that's an
introduction to the conceptual part of an Aymara power works wouldn't talk about it a lot more next week but I I think Monday is a holiday disastrous that happy Presidents Day everyone else here Wednesday
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