Lecture 18. Eigenstates & Eigenvalues


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Lecture 18. Eigenstates & Eigenvalues
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Martin, Rachel
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UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 18. Molecular Structure & Statistical Mechanics -- Eigenstates & Eigenvalues. 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: 0:01:19 Matrix Representations 0:05:21 Zeeman Basis 0:11:31 Raising and Lowering Operators 0:17:01 Superpositions 0:21:16 Spin Operators and Eigenstates 0:23:46 Pulsed NMR 0:29:05 NMR Probes 0:31:09 Nutation Curves (Solenoid) 0:36:41 Spin-Lattice Relaxation (T1) 0:44:10 Inversion Recovery (T1) 0:47:11 Relaxation Along the Z-Axis
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the last time we ended with I'm talking about some of operators that are involved in in animal and you know here were still worse still equilibrium we still have stuff aligned along the axis but so this is the starting point for for being able to understand and marched across the from the physical chemistry perspective of site ended with us
last time but let's go back to read make sure that that everything makes sense so we have an operator that that corresponds to the mechanization along the and forest and one-half nucleus that means it's either tonsillitis one-half of a down the state's recalled often data in and the island states of survives have values that correspond to their spin on a number so we know postulated half here and if we operate in this if we operators operator unexciting state we just yet that quite a number back and the original state so OK not so surprising this looks similar to things he did last quarter just gets a little bit different systems I think it would make a lot of sense actually teach anymore first one reflecting upon a mechanics because yet the the idea states in the operators that we are using a very simple and it's easy to see what they do so hopefully this maybe even clarifies some days that there were challenging last quarter are excellent that means clearly written this down in a very general way what that means is that if we operate by Zia and Alpha we get 1 of if we operated on data we get minus one-half data and you remember these things make up the North normal said offer does not equal minus Bader they are in fact orthogonal to each other so if we take the inaugural oath alfalfa Baden-Baden we're going to get 1 and if we have a matrix Olmert that looks like this was an overlapping and role it looks like this where we have to have these states that gives a 0 so we've seen these things before in context where it's really easy to visualize what is so in this case what's the space that were integrating over its Spanish faces the two-dimensional Hilbert space that has to do with these 2 operators so there isn't really an easy way to visualize it but fortunately mathematically it's pretty simple going to think we have going on we've got minus one-half and we know that often data to each other so it's easy to set up the the on overlapping roles a case announced we want to make matrix representations for Our operators so far the only 1 we really know is also had a separate major Solomon OK so a matrix element is we we have a the 2 states that were looking at with the operator sandwiched in between them and this is a little bit different way of approaching the problem but it's similar to what we've already done before in the context of talking about her fury so we made matrix representations for operators In a spacer in a context where we can easily visualize what the operator does we made a matrix that way also now we have a situation where we don't have any easy spatial representation for it and we just have to do it mathematically but we're doing the same thing so we're gonna make matrix representations for our operators OK so if we have the alfalfa matrix element that means that 1st we operate eyes on Alpha and we get of course have health and then we can pull the scalar out of that and so we just get one-half alfalfa and that gives us a house if we do the same thing for the matrix element for Alpha Beta criticize the operator the 1st thing we have to do is operate ICI data which gives us minus one-half data again you can pull the constant out in front and you can see that this matrix element is a 0 and for all our Analyses of and more operators In this is essentially what we're going do and there are cases where we're not looking at things that right in states and Morgan have to figure out how to write stuff in terms of operations that can trivially you'll find it via the item for so are able to take another look at I see in the Hambletonian so here is my matrix frisee and so the way I got that is the arm the factor 2 you have been sloppy but dropping my age bars but they they are there in there the factor to comes from the fact that the eigenvalues posturings one-half so Poland the matrix and then we said that the alfalfa matrix only demise is 1 and then for alpha-beta 0 the same thing forbade alpha and beta beta it's minus 1 year again with an 1 half it's everybody OK with how I
got there prior to announce look at the Hambletonian so for an animal experiments the Hambletonian is I don't know maybe not and so that means I'm just going to do the same thing but I have some more were constants in front and it and we can make the matrix elements the same way so here again I'm I'm operating the Hambletonian on the cat 1st pulling out whatever Constance I have and then taking the overlap integral what's left and so these are the answers and we know if we look at the major representation of 2 operators if diagonal on the same basis than they knew so this is really powerful in quantum in general because in our view that operators use that gives you an important information about the system itself if stuff commutes with the Hambletonian and energy is conservative in the news this for a lot of things but OK so so far were just talking about you know when we say this is the Hambletonian This is a Man Hambletonian rates we just have spins aligned with the new and within against the main magnetic field and so we can see the energy difference for that you know we know how to operate I see on the states but we haven't learned anything about the actual anymore experiment because for that we need to be able to apply pulses we need to look at stuff the ex-wife played had we do that so you remember from your homework assignment a long time ago we did that you know you proved that the angular momentum operators don't with each other in fact they have this sector connotation relationship here were calling alliance to develop because we're talking about a spin and not I'm not an actual angular momentum from something rotating but the math is the 2nd so we know that these things don't with each other so I X and Y which had the spin operators that we need for looking at our system in the ex-wife planes those who don't commute with icy and that means they don't commit with the Hambletonian and it's not obvious how to work with them so we need to define some other operators To look at that so here's what the majors representations of IXI why are just for reference and if you don't understand how I got that that is perfectly understandable were at the minute but I want to share with the answers before we get there OK so in order to get this result we need to define something called the raising and lorry operators so what the raising and lowering operators do In the raising over the the state of the system but here here's what the defined as I think the so we've got I plus as I X plus ii y and I minuses IX minus ii White and so we've we've kind of alluded to before alike were getting were getting real imaginary parts of our signal in the ex-wife plane you can start to see how this fits together with the the formalism that using OK here's what I close my mind is due so you have this collection of constants sound foreign relating to Ellen M. that is your right in value and then the Eigen stated Of raising operator is the state you have 1 so few so if it's if you started with minus one-half it's going to go to Moscow hostilities if you operate I lost get Alpha if you operate by plus alpha there's no state that's higher than that in the system that's defined so that the interview 0 that's not really true for spins that have I greater than half right so we has been 1 we can operator plus and minus 1 and will get the item valued times the state with 0 and then if we operate I plus an 0 we do that again and get must 1 1st 1 one-half system we only have 2 choices upper down so you can either reason lower and if you operate the reason for that if you operate the raising operator on the highest state you get 0 and equivalent for the lorry
operators OK so it's right that explicitly so I plus operated on offer gives 0 I plus on beta gives you Alpha here I have dropped the constants so united value in front of us I will try to go back and put that in the for post the slides I minus on offer gives you again here constants times data and I minus invaders 0 so why did these operators exist where we can use them they have a bunch of users in 1 mechanics actually but in this particular context what we want is we don't know how to deal with IXI why because if we if we have our mechanization quantise along 1 of those acts you know know now it's not the at the Eigen states and IXI wire something else not there there some sort of linear combination of the there's a month island states but we don't know how to measure that we don't know how to operate on it you know they're not if you measure you know the almost spin states often in Baidoa when the quantized along I Y you'll just get often data with 50 percent probability in that experiment doesn't tell you anything we need a way to deal with it so these operators are defined in terms of IXI why in a way that they give us something that we know how to take that we know how to find the idea value of that is going to give us a the finances in Bernie deferring of steel to any a caseloads finding the matrix elements a bypass and I minus in this basis so again were stolen examinations Williams than one-half so if we operate I plus on Alpha we have 0 now if we operate I plus that gives us our constant times Alpha and then we take the integral of alfalfa which gives us blind and again actually times the item value again operating I plus and offer gives this 0 so that 1 0 if we operate by plus on beta that gives us a constant times but then the unit will be the health of his 0 so that the matrix for any loss and we can do the same thing for I either so we operate a minus an offer that gives us beta but the integral been alpha 0 I -minus operated on data 0 by minus operated on Alpha gives us a constant times Bader but in the end of they have been give us 1 and so we get these matrix for I might as so then these are just convenient operators that we can work with in the Zaman basis and they give us a matrix representation that makes sense in our the use the definition of these things in terms of IAC's my wife to enable us to work with the I been states of those operators which again what we want to do that because that's a signal that we can actually measure in the experiment OK so here's how these work and I am I'm gonna let you use this to verify the matrix representations for IXI white it's tedious but it's good practice so they can just go through and operate these things and you know like I showed you in the in the previous couples lies and once you work it through once but I think you'll be pretty comfortable dealing with these kinds of operators and you know what the answers are because interview earlier in a lecture and here again it's tedious so if you if you do 1 of them and you think you totally get it that's good enough but if you need extra practice both so again here I believe the answer is that you get so now we really taking art the knowledge that we learned from looking at group theory and being able to make matrix representations of operators and have fun and now we can apply that to a quantum mechanical system worthy the transformations that we're doing are not obvious you can't really visualize it in Cartesian space because it doesn't live in Cartesian space lives in space but because we have these skills of the it'll put operators in terms of of matrices we can use all that same formalism to do stuff for it's not so easy to visualize so now we are hopefully that the point of being able to do that becomes clear so we we practiced on systems where it's easy to verify the answer because you can visualize it now we can do these things were full but more abstract OK as I have sort of hinted at along the way we can have spins in a similar position of alpha and beta we don't have to have everything just in 1 nite instead of the other and so that again this is wary the basic textbook picture Aymara goes wrong you get this idea that everything is either in the office stated beta state while it's not you can you can have the superposition OK so you can have a wave function for your stance again it's it's a funny way in function it's not you know it's not as a function in the sense that we use looking at it's a probability mass on either Alpha Beta or something combination of the 2 so odds Wednesday can have a superposition where we have some amounts of alpha and beta How much as described by these constants and we can write that down as a vector so in that location Alpha and years later why is that useful to be able to do because we have all of our operators were not in terms of vacancies the and these things are analyzed as I said when we talked about how to do the matrix elements and again it's kind of hard to picture pictures these functions and how the orthogonal to each other their instant space it is it is pretty abstract but but they are so the are orthogonal and the normalized alright so now we get to what are the United States of IX my wife and I'm not gonna prove this year's as to how those are the how we get to those of the United States just because you know there's a limit to how many I a mark or don't we can do about that this is what they are so if we have any idea state for plus acts so that's you're you're spinning quantized along the positive x-direction it has this particular item stated often data so are constants for each of them is what is 1 of the scariest so now if work in this state the ax component sharp and wires in wines you're not so we can measure along the axial were in this class excited state every time we measure allowing access organized at that value if we measure along wines eh it's going to be ill-defined it's not shop in that case OK so again we're going to apply matrices that that we have further that we've been writing down talk radio operator on the cat right students stayed in vector notation and then multiply the appropriate matrix by it so here is what we get for I IX operated on the plus state and that gives us if we simplify the constants that gives us a half the text which makes sense right up so we expect to get the original state back as the when we said tonight in state and then we get it set its value of it's been quite a number which is possible in half so so that's 1 of these things let's look at it the value
for us -minus so similarly we can write it out and again I'm not going to show you how we get this as the United state workers and look at the result fans here it is in vector notation we can also operate Iwai on it and we see that we get minus one-half -minus I of sorry wife and this is what's detected in a typical Baltimore experiments so if we do even some complicated pulse sequences were we flipped dispensed through all kinds of gymnastics and they can do different things at the end we have to end up with -minus y as United States because of what we can detect OK so I'm gonna show you how to operate some of these things on our on our spin states and look at what a realistic and more experiment does so here are rotation operators for or X or wide receiver and this is rotating about the X wires Accies used by some angle beta and what we can do this it should look familiar because they're the same as the rotation matrices that we've been making for rotating some physical object about an axis right so if we have not so bad as the angles of we have an operator Rx Pirates too that rotates the mechanization 90 degrees about the axis and the rotation operator commutes with the angular momentum operator about the same axis so or executes with IX but it wasn't me with my wife and I see and so for a different angle momentum operator we have this kind of a relationship so there's a lot of math and it's a little abstract but stick with me because we're getting it back to power this actually works in the real pollster Mark Berman OK so we want to apply a poles With phase X so we have our stands there aligned along the z axis there the quantized along in the alpha and beta stage and we want to put them into a state that we can measure so we applied the next fall and that's going to take us from whatever our starting state was to final state and so we're going to do that by operating operator on the initial spin state and so that means you and take whatever state we started in rated and vector notation and then multiply the rotation matrix for the polls by it and again this is called the the polls propagator and this been is the flip angle the policy so Hi over to an example of that we're talking about operates so was back up and talk about that for a minute this is something that is it's traded in your book and kind of a hand weighty way and I wanted to really show you how it works it's so it's important understanding are it's so we talk about the help also marks the works say that we have our spins along as z axis and then we apply a pulse on residents so we have the right amount of energy and it flips the mechanization to the XY well how I get it to actually be exactly perpendicular With the main magnetic field right see you can imagine it save its offer residents a little bit or if the pollsters isn't strong enough we can take that part way down and we will see a very strong signal and because we can only measure the production along the x y plane if it's too strong saying it rotates it farther down there were a few weaker signal there too in fact we can rotate 180 degrees and just invert the mechanization relative to how it was at the beginning and then we will see anything because it'll just be along the negative easy access so how do we know that Oracle's is actually maybe repulsed the answers we typically measure this experimentally you can you can calculated in and get close but we optimizes experimentally the flip angle depends on the mutation frequency of the we are so that is you can imagine the RF as you so it's an oscillating electric field in enough you in a direction for final the main magnetic field but we can also imagine it as inducing oscillations in the mechanization so we started 1 if we apply a pirate tuples we took that this way if we go too far and give it a 180 degree pulsar goes like this and you can imagine if we're looking at the signal for that if we get this a solitary behavior and so we can express that frequency in frequency units in and so were were measuring the strength of the magnetic field in a way but in frequency units and that tells us about you how much power we have to flip these pulses so the slip it is that field strength kilohertz times the the time the length of the polls so you know we have an angular frequency times at time and so that gives us an angle what's with the without that looks like so here's a rotation matrix for but of full painful Bader and again notice how it looks just like the rotation matrix that we were there we used for looking at physical rotations of molecules in a particular importance system so these are things that we have learned our are definitely applicable to this system that's a bit 1 extract so I cannot see what this looks like so we have our Magna's Asian actor initially 1 the we turn on the polls we have calculated the flip angle and the mutation frequency To be exactly right so that 292 repulsed and that's going to give us all right position in the miner's wife I did state it also picks up a phase factor which is this extra little he did over for which we shouldn't worry about right now but this tells us about your house how we can experimentally make spins flips you as sat on the side conversations it's very distracting it's distracting to me and I think it's distracting other people too OK so let's look at
experimentally how you do this so this is something that I think that my lab does we build and more grossly we build the RF circuits that produce these radio frequency pulses that foot the spins and it turns out that there are lots of experiments that that we can do that you have to develop special hardware to to do and grad students and actually fewer undergrads in my life have I have worked on this so you know that means that we work in a machine shop we build electronics it's so it's pretty interesting stuff so here's what the problem looks like so it's it's really it's really long because it inside the magnets so when you see pictures of mom it's really going to be a model out of here too to run the experiment if you're just a casual user Noble the stuff you don't see what's actually doing most of the interesting stuff so inside the magnet there is a little oil that is the thing that delivers the pulses to the sample it also listens to the signal it comes back and I know that's that's just representative the conductor here but on another devices is long because the that oil has to be located in the very center of the magnetic field so it's hit the inside of the magnet the actual the business and it is relatively simple so we have this this parallel resonant circuit so that's the on the conductor in parallel with the tune capacity called CT by adjusting the variable capacitor we can change the resonant frequency and the circuit and then you notices a subtle capacitor in series with that that's the match pasta we can adjust that to 0 out the imaginary part of the the incoming RF so that matches the signal the 50 or so we have to have the pulse that's that's coming in impedance matched to the below so this is how we experimentally deliberately
are also and so what I want to show you is we've been talking about mutation frequencies and how we can measure that here are some the experimentally measured for some real approach so 1 thing you notice is that we have 3 of these so we're looking at protons carbon and nitrogen so when we're doing a multidimensional animal experiment here we talk about the OK we can look at proton carbon nitrogen phosphorus for every nucleus that we're looking at we have to have a separate channel the probe you need a separate circuit 2 the Leopold interact with them and particularly when we get into talking about you know OK we're going to really look at Proton and we look at Proton and the couple carbon you have to have 2 channels To be able to interact with that means that they need to there at circuits and they're all coupled so why am I showing you this just to give you a feel for we can talk about all the stuff in theory and you know it's me it works out really nicely but it does it just doesn't give you a feel for what you actually do so and so yeah you get you're getting that flavor for with in the case of NMR because that's what my lab does if you if I did something else some you then you might be hearing more about IRA spectroscopy or something like that but again this is so this is how you experimentally measure the full panel the pulses in having a signal sequence he like we have the the RF field strength at some constant value so here for year for parts and it's this is what the 1 refers to suffer Proton we had 132 kilohertz carbon it's 71 . 4 nitrogen 86 . 2 that frequency refers to the frequency at which the magnification is going around and around in these assignments and in that case it's a measure of the amplitude of the field of being applied and it's it's 1 of these things were the units a very weird strange to think of a magnetic field in units of frequency but we do this in a moral time so we're talking about the main magnetic field and frequency units like we said we usually say that we have a 500 megahertz magnet in a rather than an 11 . 7 Tesla magnet which would be the appropriate sigh SI unit for a magnetic field the reason you letters were saying that the mutation that the procession frequency for protons in that Magnus 500 megahertz and that's something that's convenient to talk about in terms of the March sinking here we're talking about the amplitude of the RF fields that were applying not in units of of Tesla something else but In in terms of How much can we actually influence stems and so again that the mutation frequency time is the time that the polls applied gives you the flip angle so if you look at these plots in the case of the proton each 1 of these steps is . 5 microseconds so if we apply the polls 4 . 5 microseconds to see the 1st .period doesn't have the 9 musician much all only get a weak signal the 2nd 1 after no 1 microsecond tips but more and then we go up to 90 degrees and new and so on as as the mechanization of around around I want to point out that if experimentally of stuff perfect this should look like a perfect sine wave and mechanization should go around around forever in there should be no limits that's not how things actually work so it turns out that Europe oil is not perfect they using to apply the the field and if you look at you know especially via the proton channel here you can see if you look at the the 3rd or the 4th maximum the overall amplitude is a little bit lower listing during indicate that's because stuff starts lose coherence as you apply the field for longer and longer it's not perfect 1 reason for that is that your quite it is not your infinitely long solenoid where the magnetic field is the same in all parts of it it's higher in the middle and falls off toward the end and we actually can do things to try to make it better when you were engineering these things so into effort for a solenoid for example we just have oil that's literally round on a cylindrical form which sometimes we do use you can make it stretched in the center and squished on the edges to try to even up the magnetic field that's that's something that we do so here is
a I'm kind of a funky looking coil that was that was built in my lab and you can see that 1 of the things I want properties it has is that it has a really nice magnetic field right in the center of this plot is the magnetic field as a function of distance from the inside oil so right in the center it has a really nice magnetic fields and it falls offered quickly at the entrance and that gives us these very nice-looking mutation curves now they look so nice because in that experiment the sample is restricted to only been in the region where the choir looks perfect OK so we talked about how to deal with our stand operators you got some homework as far as applying the raising and lowering operators say you know which is just to give you the experience of working with you have to apply these operators to stuff in and be able to make this work we really that to pollster Nevada and how we actually see a signal now I wanna talk a little bit about relaxation and relate not to actual experimental factors OK so we're going with this is we've talked about it when you you have your perfect navy repulsive Paulson envisioned the it's and it's going to relax back to equilibrium and end up better off along z so so far all I've told you about how the process works is that it's not mission of are system does not stand out a packet of RF and come back to equilibrium and does something else White said .period about that OK so I put the sample in the magnet so you know I go to the liquids machine for playful landmark tube the top magnet and once that there How long does it take for my spins to align 1 z 40 think anybody would take a guess 20 minutes I mean if you're looking at like Silicon 29 or you know maybe a carbon that's not close to anything at all like if it's delicate carbonation a perfect diamond that might be good guess if it's so if it's a liquid it's a couple seconds but here's what it's not it's not nanoseconds so when you put your sample and you no 1 guessed that people often make is well I have 500 megahertz magnets so take 1 over 500 megahertz and that's how long it takes the to align it's not it's independent of the mutation frequency it's a different effect so what's happening is you have all your little stands in there and they get bumped into by other arm other molecules and so and so when the molecules move other little oscillating fields and they get bumped and they eventually end of aligning with the field and that process takes a little while but it depends on the the it depends on the standard depends on the local chemical environment here what's actually causing the relaxation and it's on the order of half a 2nd for a few seconds for her some typical samples so you hear about this when you're talking about Holston more in kind of a practical context because you know that you have go calls anyway for the men's issue like that if you wait long enough for the 2nd standard not going to get very much signals so if I only wait for it to come halfway back in and pulse again I'm going to get a smaller signal was at the time of course that's not what we wanna do we want to see elaborate a time and added many skin so we have to wait long enough for that mechanization to come back so this relaxation that we're talking about is called longitudinal relaxation and that's a lot this is the direction and so were D coupling the interaction in the XY plane from this relaxation at this point in the discussion and that's that's very reduced services this is what relates to the other spins losing energy to the surroundings and and coming back to equilibrium so here's what that looks like here's the the functional form but that so we have our Evans III is a function of time -minus and not and not as the initial value that depends on -minus T over this constant T-1 61 is a longitudinal relaxation constant use relaxation along the and notice that this is just an exponential decay there's no a solitary component here were just talking about the relaxation is the direction OK so let's talk about what causes it so in organic molecules are proteins and things like that a lot of what causes it is natural rotation so in the context of other kinds of molecular motions we've sent a bunch of times that number from groups of free-standing all the time methyl groups have a carbon which could be see 13 usually it's not but it has 3 Proton doesn't have a nice
strong local magnetic field and their standing around this is something that can cause relaxation another thing that can cause it is segmental motions of we have a a chain here again stuff rotates freely about single bonds so in this particular liquid-crystal faces a spectrum that I took all of these chains working around in that causes relaxation another thing that can cause relaxation is chemical shift anisotropy so if we have and an isotropic chemical shift we have an electron distribution that's that's shaped you know the interim like say it's it's shaped like a football as that moves around there's a locally changing little magnetic fields that can induce relaxation in you're right things there's also dipole dipole relaxation so that's like you again we're talking about we've talked about by poor coupling as school students act like bar magnets may interact with each other with 1 a work you dependence they can also relax each other so that's why I said that if we're talking about something like like if we want to come up with an example of something that has a long relaxation time on the order of 20 minutes or something that would be a nucleus that's very isolated it doesn't have any of these mechanisms for relaxations of something that would take a really long time to relax would be like I see 13 the carbon that natural abundance and environment so see 13 is only 1 per cent natural abundance so that 1 little see 13 in a sea of C 12 is going to take a really really long time to relax because it has none of these mechanisms that doesn't have any magnetically active nuclei nearer to interact and have give energy to its environment through the lattice vibrations and things like that will take a lot longer this can be a huge pain for some samples of people interested in looking at so for instance if you have an organic molecule that has a bunch of protons and problems if it has a lot of Quaternary problems that attaching protons they can take a really long time to relax and you waste all your time experimentally because you're really have to wait for the mechanization relax all the way back along and we'll have a few nuclei year sample literally stubborn about us and yet it takes a really long time OK so here's a pulse sequence for measuring so what's talk about pulse sequences have you and I have you seen any of this in inorganic chemistry so raise your hand if this looks familiar OK How many musicians do we have a lot of people playing music quite a few yeah OK so and Markel sequences are like musicals cost so this particular 1 is only 1 dimensional we were talking about protons or Aussie 13 or in 15 year 1 kind of place at a time .period when we look at pulse sequences that are more realistic organist see a whole bunch of lines for you the protons are doing 1 thing the carbon they're doing something else in the and 15 they're getting something else and they're all synchronized and it's a lot like a musical score and so you know like a musical score the special-education on we're not going to get into it too much and it's it's fun but basically what we need to know about this is that if we have a pipe calls that's 180 degree polls that's written as a pulse of longer duration and a lot of times it's it the square is open a pirate to polls is written as shorter duration and a lot of times the square is black and then the free induction decays is clear In this single-handed Arrow Tower it means that were going in this experiment but we're not just going to do it once we're gonna repeated over and over again and we're gonna make cows longer each time and that's sort of culinary experiments and I here's what the results of that experiment look like for an organic molecules so this is called the inversion recovery experimental to show you any sort of finger pointing out explanation why it's called that so the pirate to bolster beginning in works it and now I wait some time tower and it starts to relax back to the equilibrium position but then before it gets there I posted again and detected so you know at the at the very beginning it's going to be almost all the way 1 minus the axis so all give a strong signal rifles about going the extraction but the next time they make towel but wonders has a little bit more time to recover before a measure and then as we get to the point where it's if it crosses through 0 the democracy in signal light poles and that's gonna start coming back so we will see this logarithmic dependents were we start with a negative signal and then it slowly comes back the models off because it's never going to get higher than the original equilibrium value rates so here's
what that looks like for this organic molecules so we have you for these different carbon atoms that are labeled heroes we seen In spectrum number 1 here that's that's our our time that were waiting everything is down along minus the axis for everything is inverted and then as we waited longer and longer times some of these things start to recover and we see that as you would expect from what I just said the CH 3 and C H 2 recover 1st these are things that have a lot of motion there permeated the fat polar interactions with the protons and then the things that are in the final ring that have less mechanisms for relaxation and take a little bit longer to relax all what's quick therefore this for now
what what I want you to get about to understand about this is OK what causes the longitudinal relaxation Nunavut that conceptually water the molecular factors causing it and also I would like you to understand conceptually the experiment that that we do measured as the inversion recovery experiments and you should practice operating matrix operators on the southern states and that is that they have a good weekend


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