Okay, so we finished off on this slide the last day. And what we had covered was the theory behind nuclear magnetic resonance transitions. And we had said that nuclei, if they have a spin value greater than 0, this gives them an angular momentum and they have a charge. So when you place something like that in a magnetic field, then it obtains a magnetic moment. And if we have the example here on the bottom right hand corner of the field in that direction, and we call the field, magnetic field we put it into B0, then we say this spinning charge precesses around like we show here. So here is precessing with, this is i the spin quantum number is a half and it can precess with the magnetic field like that and this magnetic field direction, and that's a plus a half, or it can align itself against to make the next field. So you would flick that over and it would precess against the magnetic field. And this precession frequency here which we call omega 0 there in that diagram and as you can see that's proportional to the magnetic field strength. Directly proportional to the magnetic field strength and then the proportionality constant is that gamma. And we said that that's the magnetogyric ratio for the particular nucleus involved. So different nuclei will have different values for that parameter. Another point is that you create the energy levels in a spectroscopy using the magnetic field. And therefore this calls this, as we see here, this precession at a certain frequency. And the way you induce a transition then, this gives you two energy levels, I'll pick the one where it's aligned along the field and then other one where it's aligned against the field. And if you come along with a frequency that's equal to this precession frequency here, then you'll cause it to flip over. And what you record that then is you record that in your spectrum as a peak. So it's different to the other spectroscopes that we talked about in new climatic resonance you use a field to create the energy levels. So using that field then we can use different fields, we can spread these apart by different amounts. So it's unique in that aspect compared with the other spectroscopies we talked about. Right, so here we have another demonstration of that and here we have a situation. So here on the left, you have no applied field and now you have no magnetic field. So the magnets, they're not magnetized. They can pick up any spin, random orientations and now here is when you apply the field. So here it be B0 is equal to 0 and here B0 is greater than 0. So now you have when it's aligned with the magnetic field plus a half or from that it's called the alpha spin state. And then you have the minus a half state on top of the higher energy state, and that's also called the beta spin state. So what you can do then is you can imagine what you come along then is with your electromagnetic radiation to fill that gap. So there's an energy gap here, this is energy. So that's an energy between the two gaps. So you need electromagnetic radiation of frequencies corresponding to that gap. And what you generally find is that that falls into the radio frequency range of the electromagnetic spectrum. So again I have talked about the job with the magnet is to create the energy levels. And the energy gap, as we just talked about, is proportional to magnetic field strength. And then in practice, which we're not going to go into in any detail, you have to use field super-conducting magnets. And the idea is, in high resolution NMR, is to make that field as big as possible, because that gives you higher resolution. Okay, so here's another way of looking at that. So here you have the proton, and here you have the energy plus the density magnetic field strength. So if you can vary the magnetic field strength from zero down here to some undefined value up here. As you can see by we take two values here, 7, magnetic fields are generally given in the unit tesla. So here you have a 7 tesla magnetic field and you can see that as you increase the field, the gap between the plus a half and the minus a half spin states is varied, it's opening up if you like. So a 7 tesla, you can work under that gap as 300 MHz, that's the alarm or precession frequency that you get for a 7 tesla field. And then as you go up to 14, you'd get 600, 600MHz. So again we have this equation here, this is a delta E, so that's the energy gap. And you obtained that if we look at the equation we had already is your omega 0. Remember we said it's proportional to the magnetic field, and the proportionality constant was gamma. Now omega 0 is a frequency, but it's in radians, it's called radians per second. And for those of you who have done some mathematics, you know there's two pi. When we talk about nu, that's seconds to minus 1, that's cycles per second. So we are talking about, there are 2 pi radians in a cycle, think of it as a circle. So the nu therefore, nu is equal to omega 0 divided by 2 pi, okay? So now two ways of frequency, omega 0 is in radians per second. The cycle nu is in cycles per second, so you can see now if we have nu is equal to omega 0 over 2 pi. Then we can, using this relationship here, we can say that nu is equal to gamma over 2 pi B0 because we just divided across by 2 pi. And now we know from right up here, we know from Planck's relationship that delta E is equal to H nu. So, what we can say there for that h nu is equal to gamma h B0 over 2 pi or we just derive that relationship there. Okay, so the key thing is that the frequency is proportional to the magnetic field. And as we see here, a field of 7 T, I just plug in these values into this equation will produce an energy gap in kJ per mol of T 1.2 x 10 to the minus 4 and that corresponds to 300 MHz or 300 by 10 to the 6th Hertz. And that's in, to go back to our first lecture, that's in the radio wave part of the electromagnetic spectrum. Okay, so just to bring that home, we'll do a little example. So what we're given here is a very simple calculation, but it might bring it's own to the relationship between the radiofrequency and the magnetic field. So we're told that the magnetogyric ratio of the proton is 26.7522 x 10 to the 7. And the units there are teslas minus 1, seconds minus 1. And now you're asking us to calculate the magnetic field needed to satisfy the resonance condition for protons in a 550 megahertz radiofrequency field. So you have to calculate the magnetic field corresponding to the proton resonating at that value. You simply use the relationship we just developed. You say that nu is equal to gamma over 2 pi times the magnetic field. So therefore, what you have, in this case, you're given nu and your unknown is the magnetic field. So you just rearranged that equation, so B0 is equal to nu times 2 pi and you divide by the magnetic gyro ratio gamma. So if you plug the values in, you get 2 times pi by the radiofrequency which is 550.0MHz. So that's 550 by 10 to the 6. And that's in Hertz, so that we know is seconds minus 1. So keep your dimensions and then you divide it by the magnetic gyro ratio, which is given up here, so that's 26.75 22 x 10 to the 7th and that's the units up here is Tesla -1 seconds -1. So now we cross out the seconds -1 and we're left with tesla -1 below the line. So if you bring that up above the line it becomes tesla, so, if you work it out on your calculator you should get a value of 12.9 to one decimal place tesla. All right, so let's go back to our Jump on. Right, so just a little bit about just to emphasize to you that we're not talking about one proton. We're talking about a moles of something here. So we're talking about moles of protons or 6.023 by 10 to the power of 23 protons. And I don't want to go into this too much, but what you'll have like is, because of this separation, here we have our alpha state. And here we have our beta state. So these are the energy levels for the proton. Now what you won't have is you'll have an Avogadro number of protons and here we just, we haven't got an Avogadro number but just to show you this is more than one. So we'll have them filling up, they'll fill up both levels, they won't all go in, they won't all have the energy of the lower and the ground state. Because this gap here is, the energy gap is quite small so energetically it's favorable for them to occupy both levels. So let's say initially equal and then, What you'll find though because you have this energy, the alpha say is a bit lower in energy is you'll find that there's slightly more that occupy the lower level. So you have a slight favor of the lower alpha state. So what you have really, is you have a net population, population difference. And this means that when you come along with your h-nu that you will get an absorption. If you didn't have this difference in populations then you wouldn't get any absorption. I don't want to dwell on this too much but just to make sure that you realize this, we're just not talking only about one single proton here. We're talking about an Avogadro number, more than that number of protons and that fill up the levels like this and that you'll have a slight excess in the lower state. The theory behind this is called a Boltzmann Distribution which will favor the lower energy state. But as I say, all I want you to emphasize at this stage is that you have more than one proton involved.