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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
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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
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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 timedependent 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 timedependent 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 timedependent 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
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wait forever before the the function comes back around to to where it was at the beginning of a case 100 to look at this we need to start with the complex form of the Fourier series so we can write that like this so this is the same thing we're looking at that before it's just we know what what you'd get if you allow these things to be complex numbers and if if you're confused about how I got from signs and coastlines to an exponential I definitely recommend checking up the Wikipedia page and the war from site OK so we can we can write down these coefficients In terms of this kind of mineral again and In downward defining Our angular frequency omega not in terms of the period tower and so were going to evaluate these coefficients and to do that we're substituting user doubly integration variable which I'm going to call you and I'm doing that just so that I don't get confused between the 2 in the central role in the TV in the sun which otherwise might cause some problems because we have to tell you that the Romans to get back into the the sum to get the answer OK so now I'm integrating that in terms of you N I'm going to say OK what happens in the period of dysfunction becomes very large that means that a major not as frequency gets very small they have an inverse relationship and so I'm going to call that Delta omega it's just so you know very small frequency OK so if I do that in plug my result back into the the original for the the complex version of the Fourier series then I get something looks like which I can evaluate and we're we're still looking at this in terms of our dummy variable you Hey it OK so let's take the limit of that as our a little frequencies increment ultimately goes to 0 this looks like the renowned some definition of an integral so we can just write this Fourier a series as an integral and this is an important result so this is for reasonable fear and it tells us something about 40 transformed Paris so that we can use so as you can see from these examples frequency and time Fourier Transform care they have an inverse relationship we we can use this kind of of an integral treatment to look at them so we usually call the 1 that's in the frequency domain big F and the 1 in the time domain small and and this is really useful because it allows us to go backandforth between frequency and time for reciprocal space which we get from a diffraction experiments and real space which posits the crystal structure this was a pretty amazing result for its time so Fourier worked this all out and if you look at his original work it seems kind of ill defined by our standards just because when he was doing it the definitions of a function and the directive of of a function and we know what it means for for things to be continuous were really not wellestablished but it has held up well that these are not used these results were later formalized by other people but you know the basic thing has has stood up well question I have a really and good and but he added so that the type of the DU should not be in the expert on that point OK so let's look at some Fourier Transform Paris and here you know you would have to work this out actually figure out what they are but on all the show you some examples so if we have a complex exponential like this the in the time domain the Fourier transform in the frequency domain gives us a delta function news so it's got a constant in front of it and then we have a delta function for something that looks like this kind of an exponential the Fourier transform Albert said it looks like that let's look at some examples there no more likely to come up in real problems so if we have a Gallician saviors a functions the Fourier transform of that is another Garcia I want to point out that they have reciprocal relationships so if you have a gassy and that's very narrow in the time domain its Fourier transform in the frequency domain will be really broad and vise versa and that's that's something that I knew you should remember as part of your intuition about these things the same thing in crystallography if you have we are really high resolution crystal structure that means that your data go out really far in space tourist reciprocal space so if you have if you have more spots extending out relief are case space that means that you're getting really really good resolution in real space thinking with NMR signal if we have a signal that goes on for a very long time in the time domain that corresponds to narrow peaks in the frequency domain so just to give you a visual idea what these looks for the wikis look like you're some Fourier transforms of different objects so you can you can do this in 2 dimensions are 3 dimensions were covered many dimensions you want really which show no costs that makes sense what we're talking about something like a crystal structure of threedimensional or multidimensional more spectra but so here are what the Fourier transforms look like for some standard looking objects so you can see it so you get so some rounding out and you get these periodic sort wiggles at sharp corners but you can see the relationships for the
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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 Xray 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 Xrays 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 twodimensional 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 lowresolution 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 onedimensional 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
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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 Xrays 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 highresolution 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
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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
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on to talking about polls tomorrow and we're going to go through different aspects of this and so on and talk about some things that aren't in your book so the description of them are in the book is a little bit like the organic chemistry view of it so there's a lot of descriptive stuff you know how do you tell the structures of molecules given another more spectrum and that's important it's it's a good thing to know how to do but I also wanted to want to get into a little bit more detail about how it actually works in the Quantum and spectroscopy sort of sense because this is he can then we should go into a little bit more detailed and the descriptive part but also I should point out that since I'm in a mosque across the best that's the that's the thing that you're going to happen to get a lot more detail on you know the If this were being taught by someone who desire spectroscopy for a living it would probably be different but that said that's the joy of having you know a bunch of different professors to do research for 4 classes are it's so let's look at how Colston works so depending on what sort of the the level of detail that was gone into it Europe organic chemistry class you might have heard different things about how works so 1 description is the CW description were you sweep the frequency and get different responses from the molecules when you put the nuclei in a magnetic fields but again you can do that it works but it's it's outdated that's not really what people do so In a typical in March the awaits conducted In a modern instrument we have our sample in the magnetic field and all the spins lineups either against or with the magnetic field and as will seal but later that's not strictly true but an approximation and when we pulse sample so we have our spends theirs and that nite musician vector of spends 20 with the magnetic field repulsed the sample and put them into the XY .period so we have spends Oriental like this now we have them oriented like that perpendicular to the field that means we change the quantification access to the system but for now we can just approximated as electors and so what happens if I release that polls that's keeping them perpendicular to the magnetic field it is that is is the main position vector is going to return to equilibrium it's going to come back to the starting point but of course it doesn't go straight back because the spends or processing about them magnetic field so it has this periodic sort of character as it comes back and of course my detector is in the it's White Plains so I have a detector along the X and Y axis because I'm collecting the real imaginary parts there are talking to each other and so on as I'm looking at this at the signal in the XY plane I'm going to detect a periodic function and so here I represented that as just perfect signs and cosigns because I'm neglecting the relaxation just for the sake of simplicity but you know hopefully what I want to get out of this at this point just that orator detectors X Y plane and it's picking up a signal so what that looks like with relaxation is you have that periodic function but now we've put an exponential decay onto it because of course you know I have maximum signal when I started in the XY plane and then it's the mechanization is going to go back up to equilibrium and that's where this acts as a natural decay comes from then by Fourier transform out so this is my time to signal if I Fourier transform that I get what's called Lawrence if I'm lucky sometimes he speaks look look more gasoline and shape but so here's my f of the and so as you'd expect a God an exponential decay and then a "quotation mark function start with maximum signal and then it decays and has this periodic character if I Fourier transform here's what function of frequency looks like it's it's called Valencia and in a realistic and experiment I can neglect this term and esthetic the 1st term and so this is the function for I animosity OK so you might ask why do we have to do all this Fourier Transform stuff we have a perfectly good function that we can't fit and we know what it's functional form as in the time domain why not just use that like 1 of my just analyze this coastline functions seems reasonably well well behaved the reason is that these things look messy will we have a hold bunch different frequencies like finish as difficult and more Burma for complicated molecule OK you have seen this before I'm just putting it in a little bit different but the same thing so here as a T is the animosity mold and the obtaining it from life by measuring it directly and then if we 48 transformed that we get a function of of new which is the the spectrum OK so let's say that we have a complicated molecule it's not even very complicated in this case we have 4 different frequencies going on here of course before looking at the nuclei better there in a real molecule Oregon have different chemical shifts different resonant frequencies depending on the local environment of course the local environment mostly depends on the magnetic fields around them which produced by the electrons which of course are in the chemical bonds so that's why tomorrow so useful to us as far as determining chemical environment of things but if we add up all those different frequencies you even in this case were only have 4 we get a time may signal that looks really messy and it would be extremely hard to interpret so that's why we don't do it if we Fourier transform this then we pick out the signals in the frequency domain and here there you know the Bronner's delta functions I Of course for a real signal they're going to have this in line shape and all have a win that's in inversely proportional to the length of the time signal but you here we're going users as perfect periodic
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functions so again here's another example that has 3 peaks and you can see the time signals a mass and you don't want to interpret so Fourier transforming it and looking at the frequency domain is the way to go OK so what I'm I do now is back up and talk about the the physical basis of anymore and how that works you know we've talked about the Fourier transform how we process the signals we're going to be coming back to 2 this fall but later I just wanted to introduce it does anybody have any questions about this part before we go 1 yes b so while the nor the 2nd from the Lawrence and that's it said it Peter I'm going to give you a a lame explanation all kind of think if there is a way to see the real explanation concisely for next time 0 it's so if it's basically off resonance with the effect that we're looking at so it doesn't have a very very big effect it does come back and bite you sometimes it is not it is not a good approximation that you can leave it alone in in all Anwar experiments just effort for most things that you do in the context of looking at structures of molecules that's fine approximation I feel like there's something else going to say about this OK yeah I just wanted to mention that you know skipping Chapter 9 but if you if you're interested in the crystallography stuff and you things about solids there's a bunch of stuff in there so I recommend reading interfere if you're interested but as far as what we have time to do that's basically immediate fervor for Chapter 9 and solids OK so let's back up and talk about the physical basis of animal and in fact but for the moment understand talk about the physics of electrons so we know that electrons have spin 1 half and we know that they behave as from eons and they pair up and and all these things what's back up and talk about how we know so the stronger lock experiment was the original demonstration of electrons having spent and this was done in 1922 so what the it is they had a bunch of silver atoms here if you remember you electron configurations from General Chemistry silver atoms have 1 unpaired electrons so you know the rest of the morale the rest of the electrons all in a closed shells so they're not really contributing anything to the fact we were just looking at the effect of that 1 unpaired valence electrons and what they did is the heated up a bunch of silver in a furnace so that they had a bunch of the Adams coming out at the vapor and then they around that theme of silver atoms through in in homogeneous magnetic field so we've got know magnet were it's it has sort of funny shape sequencing the field lines sticking out here they're not homogeneous N instead of having this being just kind of spread out like this and give you know give 1 spot that's messy it actually splits and so nobody understood this theoretically until 1925 so this experimental result was published and your head if you put these Adams through in homogeneous magnetic field you get 2 spots on the detector nobody knows why this was just sitting out there in literature and then all of this was explained in 1925 sorts look at the the classical and the quantum explanations for this OK so we have our potential energy for this autumn In classical terms and we can say before the Adams go through the electromagnet their magnetic moments are completely randomly distributed so were approximating the electrons looking like Wilbur magnets and see if there's no magnetic field there just randomly distributed in space and we can calculate the force which would be 0 off the field work homogeneous but it's not and so to do that we need to take the gradient and we can also look at the total moments relative to the position of the atom and so you hear what's so what's gamma that's the German ratio in this case of the electron and so what we see is that DLD team is perpendicular to L and the angular momentum Of this Adam is processing around magnetic field and so too find the force for any use some approximations reinstated new action new wire 0 so this thing is processing around the magnetic field which recalls the and so on X and Y components cancel each other out and newsy is basically constant because we're processing around the main magnetic field and so our course is just going in the direction and so when we go take the gradient you know we know that the Passover with respect to our X and Y on 0 and so the force that's causing a deflection of a little of electromagnets is parallel to the axis and proportional the move magnetic moment in the direction which makes sense and so we measure that deflection distance that you don't know where these spots are relative to straight through that's indirectly measuring angular momentum along the borders the magnetic moment and so here's what you get when you do the experiment and I wanted to show the original data because you know where a lot of time who were you talking about these classic physics experiments we look at really stylized presentations of of how it works and it makes it seem really beautiful it's not that way this kind of messy it's so it it's hard to interpret this as the original data the island the beam is actually splattered so here's without the magnetic field and with the magnetic field that square with a minimum so it's
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not nice and clean and perfect but you can call it that it's going on N if we look at this indeed using the quantum picture which it turns out that we need to is to figure out what's going on where know that measuring this deflection distance so how much these the position of these Adams deviates from the expected possible result that His measuring Elzy museum we need a quantum observable to go along with els e Lizzie angular amount and so we colored eyes eh so this is 1 of the consulates my mechanics right from from last quarter so for every quantum operator need observable to go along with it and this observable has to Nunda Jenna Ivan values it's enough increments of age so we have it's plus or minus H. Barbara too so if we look at this and compared the classical and quantum pictures center in the classical sense are energy is just minus new . and we can write this down as a quantum Hambletonian you aware of they were using age to represent the at the energy and an arms were replacing you with our quantummechanical operator and then we see that the magnetic moment is proportional to the angular momentum so now who is a regular Agreement of the World War II and we get this constant out Gamma which I said as the German integration and that's you know this is a fundamental constant that has to do with the particular type of spin that we're looking at this 1 is for electrons will see that nuclei have different job magnetic ratios and cement what what we get here we see that all when a magnetic fields applied we break generosity between the values of plus and minus 1 half for the electron spin so you can you can think about this year and the classical selfsustaining saying having the electron aligned with the fields is a lower energy state and having aligned against the field and again when we actually do this it doesn't mean that all the electrons either aligned with the against the field exactly there is still a distribution it's just that that's all you can measure can only measure these 2 state the the ID values of this particular Hambletonian and that's all we can measure we will talk a little bit more about what that means next time so we break generously we raise the energy of the apostate 1 half these constants times maybe not annually lower the lower energy state by the same amount and this this whole thing with the and this is a factor about splitting the energy levels for electrons nuclei that have been greater than 0 magnetic field was told the same effect OK so let's look at the Germanic ratio will be more closely so this has to do with the on the charge of electrons and its mass and so for a magnetic field along the z axis where inequality not here's the Hambletonian for that electron so we have this constant that has to some fundamental properties of electrons mn it depends on the knots depends on and we have this this observable eyes eh the Eigen values of eyes eh R and Sibelle times H box that's where we get the sort of plus or minus 1 half and here's the expression for Musee in this case and we can also write down the energy of interaction In terms of the diamond ratio the answer Bellvale you times each and magnetic field and 1 thing that you learned from this is that if we want to get a big energy separation between plus and minus onehalf Nunavut measure these effects it's going to work better if you have a big magnetic field and this is definitely related to why people want bigger and bigger and more amendments In the same thing goes to offer EPR tool .period although there are challenges in working with high field there what's so so this this tells you about the the basic affecting watch what's going on with the vehement interaction had we not the classical energies on 2 quantum observables in terms of things that we know how to deal with and we have been talking about angular momentum alive in the sense of unity didn't last quarter talking about or relying lamented you for electrons within Adams we also talked about in the context of occasional spectroscopy if you were willing to look at it for spins of of electrons in nuclei so here is a here is an example of how electron spin is is used in its interaction with playing fields so this is how a cyclotron works we have a beam of electrons that gets shot through a big magnetic field and these are usually made with arrays of of permanent magnets at this point and the magnetic field bans the path of these charged particles electrons and when you do that Xrays didn't minutes we really high energy Xrays so this is where I am a lot of these the the experiments for calling for a lot of the early highenergy physics experiments came from the first one was that an LDL in Berkeley and here's another demonstration of the electron Zaman effect from astronomy example sunspots have really strong magnetic fields and In this picture on the right you're seeing an unnatural example what's taking place in the sun spot of the month splitting this thing or explode into 3 it is from a transition into a P 4 electrons in within the sun OK so there's also works
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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 onehalf 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 onehalf 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 onehalf 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 upanddown states of the been have nucleus and we will see them a bunch I note that alSadr 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 onehalf if it's in the beta state it's in the state were exciting value was minus onehalf what that function is you know it's not something that's welldefined 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
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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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Metadaten
Formale Metadaten
Titel  Lecture 16. Fourier Transforms, NMR Intro 
Serientitel  Chem 131B: Molecular Structure & Statistical Mechanics 
Teil  16 
Anzahl der Teile  26 
Autor 
Martin, Rachel

Lizenz 
CCNamensnennung  Weitergabe unter gleichen Bedingungen 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nichtkommerziellen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben. 
DOI  10.5446/18924 
Herausgeber  University of California Irvine (UCI) 
Erscheinungsjahr  2013 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Chemie 
Abstract  UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 16. Molecular Structure & Statistical Mechanics  Fourier Transforms, NMR Intro  Part 1. Instructor: Rachel Martin, Ph.D. Description: Principles of quantum mechanics with application to the elements of atomic structure and energy levels, diatomic molecular spectroscopy and structure determination, and chemical bonding in simple molecules. Index of Topics: 0:01:33 Fourier Series 0:18:45 XRay Crystallography 0:23:07 Ion Channels 0:25:12 Pulsed NMR 0:28:48 Free Induction Decay 0:30:34 Fourier Transforms 0:32:31 3 Peaks of the NMR Signal 0:34:25 SternGerlach Experiment 0:41:55 Electron Zeeman Effect 0:47:02 Nuclear Zeeman Effect 