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Lecture 17. NMR (Pt. II)


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Title Lecture 17. NMR (Pt. II)
Title of Series Chem 131B: Molecular Structure & Statistical Mechanics
Part Number 17
Number of Parts 26
Author Martin, Rachel
License CC Attribution - ShareAlike 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/18925
Publisher University of California Irvine (UCI)
Release Date 2013
Language English

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Subject Area Chemistry
Abstract UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 17. Molecular Structure & Statistical Mechanics -- NMR -- Part 2. 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: 0:02:54 Zeeman Effect 0:06:46 High Field Magnets for NMR/MRI 0:09:09 Nuclear Zeeman Effect 0:11:23 Nuclear Spin Hamiltonian 0:13:28 Relative Sizes of Interactions 0:19:01 Pulsed NMR 0:21:49 Protons Absorbing in a Predictable Region 0:37:58 Spin Quantum Number 0:40:29 Angular Momentum Operators 0:41:40 Eigenstates and Eigenvalues 0:43:35 Zeeman Basis
the morning everybody is my like working with the think of again on a couple of announcements 1st and I II unfortunately have to cancel office hours today I have a dentist appointment and I don't think I can promise to get back in time so sorry about that but I just I thought it might work better I'm not I'm not sure who knows how long it takes also I'm kind 2nd I think nobody wants this cold so you will just builders have office hours tomorrow and as always if you want questions please post stuff on the Facebook page also I just want to comment about the outcome the key council a lot of people are going that's really great on people asking questions and participating in that's that's really cool but I just want mentioned most people are doing what they're supposed to but I have noticed that you it is a large group of people in there and there is some years inappropriate side conversations and stuff going on during the question-and-answer session please don't do this it said the it's really great to have everybody there and it's good to ask questions and participate in the discussion but when everybody's talking among themselves during the discussion or you know meaning in an obtrusive manner it's so it's not great wondered when the speaker is making jokes about people leaving halfway through like that's really not so call these people visitors UCI we want to give them a great impression and again most people are doing exactly what they're supposed to just you know make sure that that you were 1 of those people I'm considering have any questions before we get started talking about an hour a couple people
came in trade ghastly stuff was sitting up and I didn't have time to answer right then so I know there are questions of anyone ask MPs on wanting to end the cantonment wherever I mean you can succumb under my office door you can get into your TA anything you want on that's another thing that I wanted to mention I have most of the extra credit similar things great and I know that there's a stack of the Heather Allen ones that power in my office somewhere that I look for supersonic underscore that sorry about that I will look for for it also I'm a little behind the regrade the planned to catch up on stuff this weekend and I was kind 6 0 I'm sorry about that I will get it done recently I'm many other questions OK we'll talk about tomorrow OK so last time
we left off talking about the Zaman a fact which is that the condition where anything that that has a non-zero spin so electrons and some atomic nuclei have the the generosity of the state's broken in magnetic fields so if we have our article spins and there's no applied magnetic fields there just all random orientations and they all have the same energy and if we put the sample in a magnetic field then and now we have a quantification axis and the DeGeneres is broken and I want to put this appearances from an organic chemistry work and this is the explanation that you see pretty often where you have all your little stands on random orientation and you Britain magnetic fields and they all magically either go into the alpha or beta state with upper down that's not actually what happens they don't they don't all have to pick 1 or the other of the states in fact a lot of them are in different superposition states there still random distribution of durable orientations of the spins but what this means is that's your conversation axis so if we measure values of the of the the spending we're going to be able to measure states that are either in the alpha beta state and the rest of the right to be well-defined the thing that is correct about this is that alpha and beta have different energies as opposed to the condition where there's no magnetic field and they're all generous and also this change in energy for the different spin states is really small and later on toward the the end of the end of our discussion we get into talking about Bolton distributions and start moving interstate Mac will see exactly how small the energy differences and it's really amazing that that the skin and all column and depends on these very small population differences and when we're looking at a typical and simple most of the nuclei we are not giving us any signal so there are almost equal numbers of spins in the often estates and most of the murders canceling each other out and it is amazing that it works at all so OK since the energy difference between the alpha and beta seems really small we want to maximize as much as we can In order to get more cynical and that is 1 of the reasons why people like to have bigger and bigger and more magnets there's another reason also that has to do with chemical shift dispersion and being able seperate out nuclei that in chemically different environment so that having a higher field magnet gives you both greater sensitivity in greater resolution and here's a plot of what that looks like so the energy differences between the spin states for a particular nuclear electrons are determined by the strength of of the not the main magnetic fields and so here just some pictures of instruments that that we have UCI that we have 300 megahertz instruments we also have a 600 and there is an 800 megahertz magnet which is the large 1 here OK so
just to show you some of the high and instruments that that people use the this thing that looks like it lands on Mars is the Oxford 900 megahertz magnet but it just has a bunch of fancy packaging that the this you the platform around the top and everything is just just for show but you know it is important make really big magnets to To get higher resolution at the end the state of the chemical shift and then the lower picture is a Highfield MRI scanner for medical diagnostics and the same thing applies there suing imaging instead of looking at local differences in the magnetic field from the local chemical environment of the nuclei what we're looking at is essentially all water and magnetic field gradients are applied in order to make a parent compulsion differences that are spatially encoded and it's desirable to have bigger and bigger magnets for that too because the larger the magnetic field behavior signallers and if we apply larger gradients we can get finer and finer resolution but the problem with that is that the magnetic fields and print particularly the effort we have to you start to actually interact with your brain that at these levels so we have to be careful about applying to much power in hiddink tissue up on and also if you apply very strong magnetic field gradients it can actually induced electrical signals in your brain and you see flashes of light and itself it's kind of interesting but not what most people want to experience when they go in for so this picture of brain is actually my brain and my head it's stand at UC Berkeley was a post-hoc because 1 of my friends does this kind of research and so on you know I got to experience fun things like :colon ingredients of really high and seeing flashes of light in there so you know it's it's need enemies research instruments people usually Highfields but for the ones that are actually in the clinic yet to be a little bit careful because random 6 people were not are not interested in experiencing these things OK
so back to talking about the same in fact let's put this in terms of quantum mechanical things that we've seen before OK so we mentioned that spinner is called Alpha and stand down as cultivator Alford does not equal minus data we have gotten into this 1 we're talking about the year doing term symbols and looking at the electronic states the individual electrons are interchangeable like those for nuclei as well you know if you have an elephant abated they don't cancel each other out except in the sense that if you have equal numbers of American cinema signal so this energy difference the difference between made an offer again is is directly proportional to be not so this damn I hear is the dark magnetic ratio which is we know we can do it it has to do with the structure of the nucleus we can take it as pretty much a fundamental physical constant for a particular kind of nuclear so that's something that that we look up so for a particular take the nucleus whether it's a proton Hirsi 13 hour or whatever we have to startling magnetic ratio we have effective H-bomb and the not so if we want to increase our our signal at this point really all we can do is increase the strength of magnetic field it turns out there are other things you can do to increase the polarization difference we can use what's called hyper forestation and if we have time we'll talk a little bit more about that later but In terms of traditional Ahmadi PR techniques for increasing the sensitivity all you have is increasing the number of spends all were making a magnetic field beer N again its nuclear magnetic resonance so the residents condition is that you're the energy of the RF that you put in has to be equal to the energy difference between these 2 states for Iraq to single OK so here's our nuclear spin Hambletonian just like we talked about an electronic spectroscopy wouldn't treat nuclei and the electron separately and so here we worried about the nuclear spin Hambletonian and so were going to ignore the electrons except as a time-averaged local magnetic field that the nuclear icy and this is why and is useful to chemists we have these local magnetic fields letter that depend on the distribution of electrons around the nucleus which of course are primarily due to electrons in the chemical bonds and that's what enables us to find out things about structures so if you go back in the early early literature 50 years ago physicists unit discovered and you know they they discovered the effect and and they were really excited about it in the original paper where this is described they're kind of speculating about what it's useful for and they said Well maybe it would be useful as a really accurate means of measuring the strength of magnetic fields except that there's the scrappy thing called the chemical shift where proton doesn't just behave like a proton it's different depending on the chemical environment that it's so that makes it less useful and of course that's the whole reason that this is useful as an analytical technique because we do have differences in local chemical environment that they have to do with the molecular structure so the lesson there is near the application that you think might be most useful for something isn't necessarily what will end up being used for of lucky you publish something people in different fields picking up and find that other stuff to do with that and we also it's good to do basic research you know you never know what's what applications things will have OK so we're talking about our our spinning Hambletonian there are all kinds of terms that go on in here and a graphical representation of what the different interactions are in a market OK so we notice we have different plots for solids and liquids so in organic chemistry and I'm pretty sure you've mostly just seen solutions data Margaret that's that's most of what we're going to talk about the camp too but will talk about solids little bit because they have a lot of interesting effects that are not present in solution and also that's what that's what I do so I needed to hear about sold Cinemark right so in this this Hambletonian for the new Europe nuclear spins we have all these different terms and here the size of the circles is proportional to the relative sizes of the interaction so it's just to give you an idea so the 1st term is Zaman interaction so that's that has to do with what kind of nucleus is and how big is that the magnetic field and that is under normal experimental conditions that's almost always going to dominate so that the next term here is the IRS the radio frequency pulse so again remember we put our stands in the big magnetic field and the lineup but that's boring that doesn't give us a signal we have to change our conversation access to get them to release of energy that we can measure and that's done with the frequency fields and I have to tell you some details about how we do that and you equally for solids and liquids this is the next most important terms India the Hambletonian did you talk about perturbation theory classes so who knows what I'm talking about my separation .period sorta is OK so you can think about the an Aymara Hambletonian here as you're your unperturbed term is the Zaman interaction and then the first-order operation to that is the arrest and then we have all this other stuff going on OK so the next thing involved is some the dipole interaction and so this is the spatial interaction between the nuclear spends so we can treat them like little magnets and these little by polls interact with each other through space and that interaction goes as 1 of our Cuban also has an orientation to tenants and you can imagine that this is really useful in solving molecular structures here we have an orientation dependence and we haven't distance dependence for these little by polls and insults data Amara this is in fact we get a lot of our structural information but it also makes the specter of more complicated notice that it's not in liquids that's because solution the molecules are tumbling isotropic leader moving around really fast on the time scale of the experiments and so anything that has an orientation dependence is going get averaged out OK so the next thing down here is the Kunkel shift so for solids this is quite a bit smaller than the diet for interaction but for liquids this is the next largest terming the Hambletonian the chemical shift gears again this interaction between the nuclear spin and the local magnetic field that there is a result of interactions with the electrons and were treating the electrons as just smeared out time-averaged magnetic field but the nuclear notice that the chemical shift is larger for solids and is for liquids that's because there is an isotropic part in an isotropic part and again in liquids everything is moving around really fast and gets averaged out in solids that isn't true OK so the next item down or is the quarter polar interaction which is you can be quite large and solids and what that is is the interaction that's due to nuclei weighed only exists for nuclei that has been greater than half and in liquids this is also ever suspend so nuclear what's been greater than half include deuterium nitrogen 14 lots of medals lots of things like sodium will see some examples that later on but again we don't have to to worry about in liquids and then the last small interaction here is the day coupling that's the scalar coupling it's this interaction between nuclei that is transmitted through bonds and as the name implies it's a scalar so it stays unchanged regardless of the motions of the molecule and so it is there in both solid and liquid and it's something that we can use to tell us something about the structures of the the molecules as you will most likely seen in organic chemistry OK so that's kind of an overview of what the terms in the Hambletonian look like him will see this picture again as we go through the different interactions
let's go through and talk about how this experiment works OK so if we have our posted a more experiments this is a little bit different from other types of spectroscopy so again if you open up your organic chemistry book depending on which wanted it might have an explanation Panama that's not quite right so a lot of them I was horrified to discover recently have this picture where you put in the frequency polls and your spin state goes from Alpha to beta and then a photon gets submitted new detected that's not actually how it works and that's that's analogous to other types of spectroscopy that's but that's not really what's going on in a mock so remember we talked about what happens when you have some excitation you put energy into its system and they're all these different mechanisms by which it can relax back some of which we can measure and some which we cannot in any more the relevant relaxation mechanisms are all kinds of other things other than your system spitting out an photon that's that's not really in a his business that stimulated emission is not really important effect here so instead what we see is really deliver a 90 degree polls and put our conservation access into the XY plane and we see this free induction decay wherever we have the the name musician relaxing back to the equilibrium position after we released by the polls and has this dependence on because we're detecting any in the ex-wife so we get a decaying exponential convoluted with the co-signed function and he again are a Fourier transforms so the FIT has this kind of a functional form and then the Fourier Transform about is a already in which we approximate with his 1st term and there is an inverse relationship between the length of the FID in the time domain and the width of the Lawrence Ian In the frequency domain so if we have a signal that takes a long time to die away that's going to give us a nice their alliance if it dies away quickly then we have a broad peaks were going to talk about the things that might dictate that a little bit later on as a result of this week a spectrum so you again it's Gates and completely different mechanism from the CW case where we sweep the frequency and see how the sample responded different at at different energy levels for putting in a pollster exciting the whole thing at the same time and taking the free transport OK so the information that you get is on the basic level largely independent of whether you're doing CW were pulsed NMR except that in the in the polls case it works a lot better but the information that we're getting in the chemical sense is essentially the same so here
were were just looking at the protons but this this holds true for any kind of nuclear is that has a non-zero spin we conceded a mock signal so protons in a particular kind of chemical environment are going to have a characteristic chemical shift and so this tells us a lot about what kinds of functional groups are present in the molecule and what kinds of structure we have and so this table is something that I'm sure you've seen before in organic chemistry works and these are useful things to know it's good to know where different types of of protons show up roughly In terms of chemical although when I say it's good to know that means of there likely to be exam questions were you have to sketch the spectrum of some molecule and I will give you some kind of basic rudimentary chemical shift table but it's good to have a general idea about how this stuff works so I here in organic chemistry you get a really complicated specter only have to figure out the structure of molecules for Petercam I'm likely to have you do it the other way or do you molecule and you have to predict what the unmarked spectrum looks like because that's you know that's really what it's about what I understand how the spectroscopy works the so here is a spectrum of a molecule and you can see the on methyl groups show up between 1 and 2 ppm as we expected and then on the on the medical that's attached to the oxygen is is has increased chemical shift so everybody remember what the chemical shift is from organic chemistry and not realize not going over this but I think it's review for everyone is that is that true yeah OK so I will just say it's has to be defined relative to some reference that's usually I TMS detrimental slightly so it's just a silicon atom with methyl groups all around it that is defined as being 0 ppm so you know year if you if you go measured animosity from without having a reference if you have an old instrument like the 1 in my lab you will get this access skin however it's basically years have a frequency scale and the PPM scale is parts per million so it's it's kind of like a percentage of a million and that is relative to the main magnetic field and relative to what do do the protons and TMS generally the other references that you can use for different things but that's a standard for a lot of organic molecules OK so that's the the chemical shift from kind of a practical perspective ahead we want use this to to see what what structure of molecules have what's look at it a little bit more as far as work comes from so what we're looking at here is the electron clouds around a particular spin and the electrons are making all local magnetic field depending on their distribution and that's so that causes the nuclei to see this local effects that neither adds to her surprise from the main magnetic fields so here's here's a molecular model of glycine deceit and see this and I'm showing USC 13 spectrum just just to remind everybody that we don't have to hold hands all the time there are lots of other nuclei that give interesting and more spectrum and if we look at the molecular model and your picture the electron clouds it's it's really clear that the the carbon that's attached to the that that's the Carbondale it's attached to oxygen is going to have a very different distribution electrons bend to the methylene and so here I've labeled these at that the 2 carbons in red and blue schematically just 2 to indicate that this has the same general trend as as proton so so methyl carbons are going to be methyl or aliphatic carbons are going to be as you have lower power use of chemical shifts and the things that are attached to something like a carbon your going to be at higher chemical should values just isn't part constructed a case that
typically what people do with this in a synthetic context is get more or less a fingerprint of a molecule have one-dimensional proton spectra of you may get more and more messy and organic chemists are really good at looking at these things and pulling out structures and so on the final Professor it teaches a graduate and more class that's all about this kind of stuff so it's all about know how to interpret really complex specter and get structures of organic molecules I also teach a graduate of our class that is all about Hambletonian sending you how do you write your own pulse sequences in and done really developing the spin physics they're very different skills you we and we have joked that we couldn't pass each other's finals which you may or may not be true but I there really are very different ways to approach it and what I'm going to try to give you in this class is a little bit of the physical chemist perspective on an unmarked so you don't lose sight of the fact that you can use this to solve the structures of molecules it's fantastically usefulness and that a context but there's a whole field of Adamov research where we do something else OK so Back to talking
about chemical shift look at this 0 what this looks like a the solid-state so so far we've talked about chemical shift is Though it's just a number so we have a different distribution of electrons around the nuclei and as a result of that they experience a magnetic field that's added to or subtracted from the main magnetic field in the shop a different place on this spectrum well that's only true if you're molecules are moving around really quickly and the timescale of the experiment and averaging out orientation effects if we have something that's in a solid so say we have a protein in a crystal and let's say it's a single crystals that hasn't really well defined orientation if we look at a Kerviel carbon in the protein backbone if we look at that double bonds between carbon and the oxygen and think about the local field that the carbon is experiencing as a result of those electrons if it stands still we can easily imagine that this is not isotropic so that carbon sees a different local magnetic field in the x y and z directions and you'll see a signal for each of those and it gives this funny line shapes that and that's that's called chemical shift anisotropy again it's not really that liquids we only see the the isotropic value which is essentially the average value but in solids is is really important and as with many these things it's a double-edged sword it contains a lot of information so we can fit this 1 shape and get very detailed information about exactly how that Kerviel's oriented relative to the rest of the protein is certainly relative to the main magnetic field this is really useful in context like looking at a peptide in membrane proteins where you want to get the relative orientation of that carbon deal with respect to the memory however if you have a whole protein worth of wine shapes that look like this in their overlapping that saw a little bit harder to deal with because it's difficult to separate out the signals because the overlapping and a lot of solid-state methods development is about how we deal with this you know putting in these interactions selectively during the times that we want to see them and I can be done either with selective labeling involving putting see 13 in specific places in the sample or it can be done spectroscopic OK so chemical shifts in as I alluded to in the previous slide is not turned in solid that's not a number at tenser and so we can show up and you know we can make matrix representations of the Zaman a fact which Uribe admitted the began export mn are chemical shift cancer in 3 dimensions and you don't really have to worry about this except on the conceptual level and I can ask you do anything with it but I do want you to know that it exists and that there's there's more to the picture than just the solution state idea where we have just the isotropic value right here some pictures of actual chemical shift tenses depending on the shape of the the electron that the electron density around the nucleus and you can see it look really different depending on whether you have temporal leader Oberly ellipsoid or if you have something that is Central symmetric versus something that's completely asymmetric and so there is this orientation dependence that can be fantastically useful or it can be a nuisance if you have a bunch of these things on top of each other OK so that's sort of the rundown of the the chemical shift and everything that's that's associated with that we will come back to it and and talk about it some more what's talk about what's going back to organic chemistry picture of structural elucidation with Annamari so if we're talking about protons are common around 50 year anything like this there are some features that tell us something about the structure so the number of signals is the 1st thing that gives us a clue about what's going on that tells us about the number of chemically an equivalent nuclei the position of the signals chemical shift tells us exactly what functional groups are present the intensity of the signals if we integrate the area under all the peaks tells us about the relative number of protons we have to be very careful about using that former head of nuclei things proponents and the reason is because mechanization gets transferred from proton to see 13 in the course of a lot of the experiments that people typically use and so you can't just take a C 13 spectrum under typical experimental conditions and assumed that its quantitative because you're also going to be seeing information about which carbons are close to the protons and others commercial protons that is a good assumption you can you can integrate things and find the relative numbers of them the last thing that's important is the spin spin splitting so this entailed in the solution context mostly in the coupling and this can be again between a committee Homer nuclear 100 there certainly between protons or if you depending on how you the experiment can be between protons and the 13 per continent 15 and there's so that gives you information about connectivity of chemical environments and I will also add if we're talking about solids type or couplings are very important in learning about the structure please give us long-range distances OK so let's look at some practical examples In the goal here is to tie together what you already know from organic chemistry you have a look at these spectra in a practical way with the underlying physical chemistry concepts of what's going on and I know that's not happening please feel free to ask questions breaks don't forget some typical examples so it's just a reminder in order for protons to give a different and more signals they have to be chemically in public so profound that are occupying sites that are the same in molecule or that look the same when things emotionally averaged would will show up at the same place so for this particular molecule the methyl protons see on their labeled in blue we have free rotation around single bonds in solution here everything is is ice probably aberration all those methyl groups show up in the same place the the same thing for the 2 methylene protons here now again this is something that wouldn't necessarily be true the solid if we had this molecule crystallized and things were really rigid it's possible that in the way the particular crystal structure worked out that some of these problems could be closer to other things than than others and we would see splitting in solution that's definitely not going to happen and you have to assume that everything is moving freely OK so the number of animosity Knowles is given equal to the number of complete accord in the number of chemically equivalent types of protons in compound so here just some examples where you have different numbers of of different kinds of protons and and the again here some examples that are going to give slightly more complicated spectra and will revisit some of these molecules as we we talk about drawing these kinds of spectra yourself and here again you have to take into account the rigidity of the molecule so in this cycle propane With the chlorine in 1 site you know that this thing can flex very much so those are the ones on the bottom are not equivalent to the ones on the top even if they otherwise looks metric but OK so the intensity of the signals also tells you something since assuming that we're talking about protons and you can't just measure the height you have to integrate because peaks might have different widths even in the same spectra you you you can have here again the on the peak with depends on the relaxation time and that can be different for different types of protons in the same sample we will see how that works so again this gives you a ratio not so not an absolute number of protons that we have but it does give us a good relative idea of how many of each type there are in the sample OK so getting back to the quantum mechanical underpinnings of of the stuff we've mostly been talking about spin half and I'm sure that so that's pretty much what you've seen in your previous work on these things there also nuclei that has been greater than half and I alluded to this little bit talking about the quarter action and these things are important and we are going to do some problems pertaining to them later on in the class so for example In organic chemistry you assume that if protons on a molecule or decorated figure not to exceed a signal from the deuterium and that's true if you're looking at the proton resonant frequency so 1 thing that's nice about Anwar is that it's incredibly specific in terms of the resonant frequency of new player if you if you're looking at protons you're not going to see interference from other kinds of nuclei except indirectly from the day couplings of coupling is strong enough it turns out that the day coupling between deuterium and anything else that you can see is sufficiently weak that you often don't have to to worry about it but deuterium is a perfectly fine and nucleus with spin 1 and in my lab for instance we look at all the time Lin and lots of animal let's do that so just to tick give us a more general way here's the spin quantum number 4 a nucleus so it's the same as well as other types of angler momentum that we've looked at there is an overall angular momentum and there's also is the component of the England momentum so again the math works out just like orbital angular momentum and and other things that you've seen in physics context but here we're talking about nuclear spin so what is nuclear spin electron spin on nobody really knows it's so it's a it's intrinsic property Of these objects that happens to obey the same mathematical formalism as spinning charges but you it's really convenient to understand how do the math but that doesn't necessarily mean we we understand it OK so again if we go back to the specter if he have to stay if we have the spins a greater than a half we need to worry about the quarter poor interaction we can look at our arms our nuclear angular momentum In the same way as some of these other things we've seen electron anger rent and you've seen this before the cycling commutation relationship between angular momentum operators that was that came up in a homework assignment and so previously we will really use it for anything it was just an example of finding commentators on things like that but we needed to to do look at Matrix representations of operators well now we're Oregon use it for something so these values of the spin angular momentum as you can imagine pretty useful in Amara because we have no UIC is the guiding value of the Zaman interactions so there the item values plus minus 1 half that so that corresponds to the United States being Elfenbein and accent I Y are what we can measure in the ex-wife plane so it's going to review here angular momentum operators because we're about to use them OK so as I said the Eigen states of icy are specified by these quantum numbers and you can write them as a cat like this and that's useful were talking about nuclei that have spins greater than one-half so forest in 1 house she there's only 2 states and you can call up and down the alpha and beta the ones for nuclei with larger values of I don't have nicknames so you have to to represent them using this kind of a cat so if we operate I see on this state with that With the values of Eleanor we get em back as the item value and the United States is this original state and so in the same month basis here again our sample aligned along 0 and we have well-defined values of ICE that we can measure here I can states are alpha and beta and here's what does look like if we right now as these cats with values of L & M OK so all that means is that if we measure if we have our spins aligned in magnetic field long eyes at Wolong icy and we measure the values of icy we're going to get off and they're in in some well-defined ratio that depends on the roles of populations if we measure IX Orion we so were not quantized line that access so we'll get random proportions of of the states OK I also want to point out that the Hambletonian and I 0 both diagonal in the Zaman basis and that means they commute so I have a little bit of an animation failing your son is going to put everything out and talk about it right so here is our major for presentation for eyes so are spends India were in the Zaman and so what that means is we have this sort this beach borrower to out front which show it has to do with the particular energy values but more important as is working with the United States are at this point so everything is in the at the Alpha Beta spin state and no again the 1 half has been pulled out in front the since the values of November customers one-half and so we have values on the diagonal and nothing off the diagonal which tells us that everything is either an alpha beta and we know what our Hambletonian this is endemic in Omega not times spicy and so that means if we measure the energy novell so we get back one-half demo not Alpha and similarly for beta we might as well have them not beta this is the same thing that we've already seen and so we can use that to construct the matrix representation Of the Hambletonian so we're just applying these operators the same way that we have before with things that were more concrete you now applying to the spin states and so we can make these matrix representations of both icy and Hambletonian and since they are both diagonal on this basis they can be with each other and if that's not 100 per cent clear that's fine we spend more time talking about it next next time I just wanted to introduce it so that people have something to think about fervor for the next class of I'm going to post this lecture plus some practice problems for the end Paris later today is anybody have any questions before request yes because the people in the region the appeals of the theory that the research was so that's a really good question OK so that the pulse is a frequency fields that's essentially producing a magnetic field that's orthogonal too the main magnetic field and that should be really weak trade because like my my big magnetic fields is enough we talk about and frequency of the infrequency units in my lab it's 500 megahertz the RF fields I began to give a typical values may be 140 kilohertz so it should be weighing weaker than the main magnetic fields so it's amazing that there's anything the only reason that it does anything is because it's on residents saw nuclei of possessing about the magnetic field really fast and that applied field that you're adding is following around for thousands of revolutions and so exactly and so you tip only the ones that are on residents and so that's why we don't randomly see carbon signals when we're looking at a proton spectra has has the carvings on the way different frequency and they're not interacting with that are out there you the the president right and you control the frequency that you apply that that an experimental parameters thinking that you control and 1 of the main thing is that we do in my lab is billed probe circuits to apply forces and in different ways and change the experimental conditions also embezzled about problem around right we're done for today the next time
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