 Welcome to this course on NMR spectroscopy. NMR spectroscopy is one technique which has grown enormously ever since its discovery and has generated applications in all areas of science. Let me begin by showing you the pioneers who have made this technique and popular and brought it to the level where it is now. Several people have contributed to it and it is continuing to grow and we will try and see how the technique has developed starting from the very fundamental principles and taking to the level where it is today. If you see here the first pioneers where Otto Stern, I, Rabi, Felix Bloch, Ypressell all four rather got Nobel prizes in physics. It was in the realm of physics as it started. Otto Stern 1943, Rabi 1944, Felix Bloch and Ypressell together got it in 1952. Then it was followed by two more Nobel prizes in chemistry. Once in 1991 to Richard Ernst and again in 2002 to Kurt Wittrick and parallely the developments are going on in different areas. Those resulted in another Nobel Prize to Peter Mann's Field in Paul Otterberg in 2003 and this time it was in physiology and medicine. So you can see the technique has started from physics gone into chemistry, gone into biology and physiology and medicine. So it is all encompassing. Naturally therefore to study this technique in detail we need to have a certain combination of physics, chemistry, biology in our training which would imply that certain amount of knowledge of mathematics and physics and chemistry, biology are required. We will try and develop this course in that manner. What it requires is a certain knowledge of mathematics with regard to the vector algebra, linear algebra, matrix algebra, differential equations and then some quantum mechanics like Schrodinger equations, angular momentum, some knowledge of this is required. Nevertheless we will try and give this knowledge at appropriate places for those who have not studied this in the past. Applications to chemistry will obviously require proper knowledge of chemistry and when it entered the area of biology we need to understand the biological principles. When it entered physiology and medicine of course you also require to understand how the system works in our bodies. So for healthcare this has become an important technique. I will list the important milestones in this process so that you know exactly how the technique has evolved over the years. As I said it started in the realm of physics, then it entered in chemistry, then it entered in biology, then it entered in physiology and medicine and no stopping, keep counting. There may be more Nobel Prizes who knows as the applications pick up in different areas of science. Let me first give you the major milestones of this important technique. It started in 1922 with the discovery of nuclear magnetic moment which is the Stern-Gerlach experiment and this actually led to the Nobel Prize to Otto Verstern in 1943. And in 1929 Zach Rabi discovered the first magnetic resonance in the gas phase in his molecular beam experiments and that led to the Nobel Prize in 1944. Subsequently this continued and in the condensed phase the basic discovery was made in 1946 by two groups one at Stanford led by Felix Bloch, other one at MIT led by E. M. Prussell and they shared the Nobel Prize in 1952. In the 1950s a large number of major discoveries were made. The chemical shift you can see here I am saying coupling constants over Hauser effects, Penn-Echo each one of them is such an important technique. The revolutionized applications in various areas spin decoupling in 1960s cross polarization which led to applications in solid state then the major breakthrough came in 1966 which is the Fourier transform NMR. This laid the foundation for the development of two dimensional NMR, NMR imaging and multi-dimensional NMR in the subsequent years. The idea of this course is that the students who are taking it they must be able to interpret the NMR spectra unambiguously have a clear appreciation of the information content in those spectra and at the end of the day they must be able to develop techniques in case they have two for interpreting their results. Some problems they may come across in their systems and to understand and solve these problems one may have to develop some new techniques and the course is intended to prepare the students from that point of view. So then let us start from the very basics we will as I said we will start from the very basics and take you to the level where the technique has reached today by the end of the course. Nuclear magnetic resonance is a technique which is a consequence of some intrinsic properties of the nuclei. There are two intrinsic properties which are present simultaneously those are called nuclear spin angular momentum first one is nuclear spin angular momentum and the second thing is the nuclear magnetic moment. We will look at these things in some detail. Many nuclei possess an intrinsic angular momentum called spin angular momentum. Notice that this nomenclature spin angular momentum is somewhat of a misnomer because it may tend to give an impression that there is something which is spinning or something is rotating that was a concept which one had thought about because angular momentum deals with angular motions and therefore one thought that it is some kind of an angular motion which is happening. However this is not the case here this is an intrinsic property and this angular momentum is a vector and it obeys all the laws of angular momentum and that is why this nomenclature has survived and it is called as spin angular momentum. The basic laws of angular momentum seem to be applicable here it follows those laws and therefore this nomenclature has remained and so I would like to emphasize again this has no connection with any angular motion of the nuclei within themselves and this is a vector obviously which has a magnitude and an orientation space. The magnitude is given by this formula P is equal to h cross under root i into i plus 1 where i is a number which takes discrete values 0, half, 1, 3 by 2, 2, 5 by 2 etc. This comes as a result of nuclear theory structure of the nuclei and so and so forth we are not going to go into those details we take it that this i value takes discrete numbers 0, half, 1, 3 by 2, 2, 5 by 2 etc. So therefore it implies that the magnitude of P is quantized it is not continuous it can only have a certain values depending upon the value of i. i is thus a quantum number which is characteristic of a given nucleus and it is called its spin. Every nucleus therefore has a particular i value and that is called its spin. The nuclear structure terms this refers to the ground state of the nucleus. So therefore we always deal with the ground state of the nucleus and the i value there is called as the nuclear spin. Here h cross is equal to h by 2 pi where h is the Planck's constant. There are some empirical rules for calculation of i. What values i takes? How do we know which nucleus has what value? As I said this comes as the result of the nuclear theory but there are some empirical rules which have been found to be valid which allows us to calculate the i value for any given nucleus. If the atomic number is e1 and the isotope number is e1 then the spin i takes the value 0. If the mass number is odd then its spin will have a hard integer values such as half, 3 by 2, 5 by 2 etc. For odd atomic number and even mass number the nuclear spin value takes the integral values 1, 2, 3 etc. This is shown in the tabular form here. So on this you have the e1 atomic number e1 or odd and along the rows you have this e1 or odd mass number and you can see that if the mass number is e1 the spin will have the values 0 or 1, 2, 3 etc. If this is e1 e1 it is 0, it is e1 odd it is 1, 2, 3. If the mass number is odd regardless of what this atomic number is it is always half integral values, half 3 by 2, 5 by 2 etc. Here are a few examples of this. Take proton is atomic number is 1, mass number is 1, spin is half. Deuteron atomic number is 1, mass number is 2, spin is 1 it fits with that. If you take carbon 12 it is atomic number is 6, mass number is 12 so it has a spin 0 that is e1 e1. Carbon 13 atomic number is 6, mass number is odd which is 13 therefore it has a value of half. Nitrogen 14 atomic number 7, mass number 14 it is 1. Nitrogen 15 is 7, 15 this spin is half. F 19 is atomic number 9, mass number 19 it again has a spin of half. Phosphorus 31 atomic number 15, mass number 31 this is half again. Oxygen 17 atomic number 8, mass number 17 this is 5 by 2, silicon 29 this is 14, 29 and 3 by 2. I have listed here only a sum of those isotopes and elements which are of common use. By and large it is proton, deuterium, carbon 13, nitrogen 15, P31 are the ones which are used in chemistry and biology and things like O17 or silicon these are used in material science. Therefore they come in the area of materials chemistry or condensed matter physics. So we said the angular momentum magnitude is quantized as it is a vector it turns out that not only the magnitude of the angular momentum is quantized but also its orientation in space. What does it mean that for any vector you will have 3 components along the x, y or z axis and it turns out that one of the components of P takes discrete values as given by this equation PQ is equal to m h cross where Q is equal to x, y or z. One of them is quantized other can be any continuous value one of the components is quantized. The number m takes discrete values i, i minus 1, i minus 2 up to minus i for a given value of spin i that means there will be total of 2 i plus 1 values and this number m is called the magnetic quantum number and sometimes it is also called azimuthal quantum number. What does it mean conventionally the z component of the angular momentum Pz is taken to be quantized we could have chosen other ones also but it has been a convention to use the z component as the one which is quantized and therefore we always say it z component is only some discrete values. So for any spin i the orientation can be such that the z component can have any of the one of the 2 i plus 1 values. For i is equal to half 2 i plus 1 is 2 therefore there will be two orientations in space this is indicated here. These various arrows which are there these represent the orientations of the different spins some of or with the m is equal to plus half and some are with m is equal to minus half. In each case they are of different place but the component along the z axis for all of them is the same this is the z component which is here and the x components or the y components can be anything that is why you are having so many vectors here and similarly for the m is equal to minus half the z component is the same for all of these orientations which is minus half h cross and the x and the y components can be anything. So therefore we say one of the components is quantized the other components are not quantized. Let me show this more explicitly and you also notice here that all these vectors which are drawn they have a certain angle with respect to the z axis. I will show you why you can immediately guess it why it should be so but I will explicitly show in the next slide. This is the case for i is equal to half I have shown only one vector here this is the case for i is equal to 1 there are three orientations possible I have only shown one vector again there will be many vectors and for i is equal to 3 by 2 there will be four vectors. So what are these why are why is this an angle and what is the angle between them. So this can be seen by recognizing that the magnitude of this vector is h cross under root half into 3 by 2 i into i plus 1 and the z component here is half h cross the two values are not the same therefore the length of the vector and the z component are not the same. Therefore there is an angle in this which is dictated by the ratio of this the cosine theta is equal to half divided by under root half into 3 by 2 for the case i is i plus 1 this length of the vector is h cross under root 1 into 2. So this is root 2 and the z component for this is 1 the z component for this orientation is 0 and if I went to draw the z component here for this the it is minus 1. Similarly for i is equal to 3 by 2 I will have the magnitude 3 by 2 into 5 by 2 i into i plus 1 and the z component for this is 3 by 2 h cross because that is i z is 3 by 2 and for this it will be half h cross and for this it will be minus half h cross and for this it will be minus 3 by 2 h cross. So therefore all of them have different orientations in space because the z components are different compared to the magnitude of the angular momentum. Now we turn to the other important property of the nucleus which I mentioned in the beginning and that is the nuclei with non-zero magnetic moment. There is a magnetic moment and only those which have the magnetic moment of course are NMR active. Nuclear with non-zero i value have an intrinsic magnetic moment we represent as mu and whose magnitude is linearly proportional to the angular momentum. So mu is equal to gamma times p, p is here the magnitude and therefore it is equal to gamma h cross under root i into i plus 1. The gamma is a constant known as gyromagnetic ratio. This is characteristic of a nucleus can have positive or negative values. Once again this comes from the theory of nuclear structure and we simply take what is the result that is coming from there and p is a vector as we said earlier it can have certain orientations and therefore mu is also a vector. Now if we write this vector equation mu is equal to gamma times p now both these now are written as vectors. So it follows that depending upon whether gamma is positive or negative the magnetic moment is parallel to the angular momentum or anti-parallel to the angular momentum vector. That is an important factor and here are some of the values of certain nuclei of the magnetic moments and the magnetic ratio. In early days the aim of the NMR experiment was to measure the magnetic moments. People used to measure the magnetic moments of all the different kinds of nuclei. Of course they have measured it and here is the summary of some of those nuclei which are for relevance to us in the context of chemistry, biology and material science. Proton which has the 100 percent almost 100 percent natural abundance most abundant nuclei and this has a spin of half and it has a magnetic ratio 26.753. Deuterium has very low natural abundance 0.01 percent so and therefore this is extremely and this has a spin of 1 this also we saw earlier boron 11 is spin of 3 by 2 it is 81 percent natural abundance and it has the magnetic ratio as indicated here. Carbon 13 is 1.1 percent natural abundance the spin of half O17 is 0.037 percent abundance 5 by 2 F19 is 100 percent this is of course a very abundant nucleus silicon is 4.7 percent so these ones have a spin of half each and all of these P31 is 100 percent abundant spin of half NN15 is spin of half but 0.37 percent natural abundance. Therefore some of these nuclei are very abundant and therefore they will be the most sensitive nuclear from the point of observation of signaling NMR and for those which are low abundant nuclei some special things are required of course this will be discussed as we go along. And notice here for O17 the magnetic moment is negative. Likewise for N15 it is negative for silicon also it is negative for all others this the gyromagnetic ratio is positive and therefore magnetic moment and angular momentum are collinear and they have the same orientation whereas for these ones they are collinear but they have opposite orientation magnetic moment and angular momentum have opposite orientation. This will have an effect with regard to the interaction energies and we will see as we look at the interactions between the nuclei and between the nuclei and externally applied magnetic fields which is an integral part of the NMR spectrometer. Now we see what happens to these nuclear spins we talked about the nuclear spins in isolation the nuclear spins suppose we put them in a magnetic field they have a magnetic moment and therefore there will be certain interaction with the magnetic field. So what does the magnetic field do since it is a magnetic dipole it induces a torque it applies force on the two dimensions on the two edges of the magnetic dipole and which results in a torque. So it will try to orient this it will induce a motion in this nuclear magnetic moment and tries to cause a motion because of the torque. However since the magnetic moment has a definite orientation in space it cannot align itself with respect to the magnetic field axis typically the magnetic field is applied along the z axis. It cannot orient itself with respect to that it induces a motion which is called as a processional motion and that is called as a Larmer procession. The nucleus if it is oriented here it will move in this fashion it exhibits a motion in along this circle here and this is called as a processional motion. A nucleus which is oriented in this direction will also move and this will exhibit a motion in this circle along this circle the edge will represent a motion along this circle here this is a also the processional motion. Here I have considered two orientations because this is typically true for i is equal to half but if it is i is equal to 1 there will be three such motions three such orientations and all of them will move around the magnetic field axis with the particular speed this is called that is the angular frequency and this is the processional frequency. Now any such motion is described interaction is described this by this kind of an equation d mu by dt because it has to change with time magnitude does not change it is only the orientation that changes and that is given by this equation gamma times mu cross H naught this is the Larmer procession motion which is described by classical physics. Now any angular motion of this type is written by an equation of this type you have angular frequency if you have angular frequency for a vector mu then you write this d mu by dt is equal to gamma into omega times mu and here the motion is assumed to be in the anticlockwise sense this equation tells you that. Now both are representing the same where one case you actually have taken into account the field in the other case you represent it in terms of the frequency of motion. Now therefore if you compare this you will find that omega is equal to minus gamma H naught. So if I consider the motion as clockwise so you get a minus sign here this by comparing these two I get to omega is equal to minus gamma H naught that would imply that I have a clockwise motion here for the nuclear magnet. The same thing now can be represented in another fashion that is you as I said for a spin half this is shown more explicitly here we have two orientations in space as indicated m is equal to half here m is equal to minus half these states are represented conventionally as alpha and the beta states for the two spin system. The interaction energy is given by this kind of an equation the interaction energy is equal to mu dot H naught both are vectors here this is a vector and this is a vector here and therefore if you take the dot product of this you get minus mu z H naught. Now nullity is the minus sign here now if alpha is the state which you are considering m is positive so therefore mu z is positive and then therefore the energy here becomes less compared to what it was in the absence of the field and that is indicated by the minus sign therefore this goes down in the energy compared to what it was in the absence of the field and this state which is minus half here therefore mu z will be negative because this will have a negative z component this will be negative therefore this interaction energy becomes positive therefore this one goes up in the energy the two states earlier the two orientations were degenerate in the earlier case in the absence of the field when you apply field these degeneracy gets lifted because of the interaction between the magnetic moment and the field and once this is lifted the two energy states will get separated like this and now therefore the spins will have to redistribute themselves between the two states the ensemble if you have a sample we consist of trillions of nuclear spins and they will be populated in these two energy states and that follows what is called the Boltzmann distribution. Typically one thinks that if you have a nuclear spins which have the spin of half often one uses what is called as the Fermi Dirac statistics but if there is a spin of one often one uses Bose Einstein statistics but here we will see that we can use Boltzmann statistics for both kinds of spins and this has this is the consequence of the fact that this energy difference is very small compared to the sample temperatures what we might have we will discuss that later. So for all the spins we can discuss the distribution of the spins in terms of Boltzmann statistics and this has important implications for the signal strength. So I think with this we will stop here we have covered the basic aspects of the properties of the nuclei we talked about spin angular momentum and nuclear magnetic moment both are intrinsic properties of the nucleus because that is how the nuclear structure tells us and these are quantized angular momentum is quantized nuclear magnetic moment can take also certain specific values it can have certain orientations in space in the absence of any external magnetic field the all the orientations are degenerate when you apply a field the degeneracy gets lifted and the different orientations have different energies the spins in a ensemble get distributed according to the Boltzmann statistics this will have important implications for the signals that will you observe and that will be discussed later. Thank you.